Predictive Control

for linear and hybrid systems

F. Borrelli, A. Bemporad, M. Morari

October 26, 2016

2
 i

to Maryan, Federica and Marina

and all our families
ii

Preface
Dynamic optimization has become a standard tool for decision making in a wide
range of areas. The search for the most fuel-efficient strategy to send a rocket into
orbit or the most economical way to start up a chemical production facility can
be expressed as dynamic optimization problems that are solved almost routinely
nowadays.
The basis for these dynamic optimization problems is a dynamic model, for
example,
x(k + 1) = g(x(k), u(k)), x(0) = x0
that describes the evolution of the state x(k) with time, starting from the initial
condition x(0), as it is affected by the manipulated input u(k). Here g(x, u) is
some nonlinear function. Throughout the book we are assuming that this discrete-
time description, i.e., the model of the underlying system, is available. The goal
of the dynamic optimization procedure is to find the vector of manipulated inputs
UN = [u(0)′ , ..., u(N − 1)′ ]′ such that the objective function is optimized over some
time horizon N , typically
N
X −1
minUN q(x(k), u(k)) + p(x(N ))
k=0

The terms q(x, u) and and p(x) are referred to as the stage cost and terminal cost,
respectively. Many practical problems can be put into this form and many algo-
rithms and software packages are available to determine the optimal solution vector
∗
UN , the optimizer. The various algorithms exploit the structure of the particular
problem, e.g., linearity and convexity, so that even large problems described by
complex models and involving many degrees of freedom can be solved efficiently
and reliably.
One difficulty with this idea is that, in practice, the sequence of u(0), u(1), ...,
which is obtained by this procedure cannot be simply applied. The model of the
system predicting its evolution is usually inaccurate and the system may be affected
by external disturbances that may cause its path to deviate significantly from the
one that is predicted. Therefore, it is common practice to measure the state after
some time period, say one time step, and to solve the dynamic optimization problem
again, starting from the measured state x(1) as the new initial condition. This
feedback of the measurement information to the optimization endows the whole
procedure with a robustness typical for closed-loop systems.
What we have described above is usually referred to as Model Predictive Control
(MPC), but other names like Open Loop Optimal Feedback and Reactive Schedul-
ing have been used as well. Over the last 25 years MPC has evolved to dominate
the process industry, where it has been employed for thousands of problems [241].
The popularity of MPC stems from the fact that the resulting operating strategy
respects all the system and problem details, including interactions and constraints,
something that would be very hard to accomplish in any other way.
Indeed, often MPC is used for the regulatory control of large multivariable
linear systems with constraints, where the objective function is not related to an
economical objective, but is simply chosen in a mathematically convenient way,
 iii

namely quadratic in the states and inputs, to yield a “good” closed-loop response.
Again, there is no other controller design method available today for such systems,
that provides constraint satisfaction and stability guarantees.
One limitation of MPC is that running the optimization algorithm on-line at
each time step requires substantial time and computational resources. Today, fast
computational platforms together with advances in the field of operation research
and optimal control have enlarged in a very significant way the scope of appli-
cability of MPC to fast-sampled applications. One approach is to use tailored
optimization routines which exploit both the structure of the MPC problem and
the architecture of the embedded computing platform to implement MPC in the
order of milliseconds.
The second approach is to have the result of the optimization pre-computed
and stored for each x in the form of a look-up table or as an algebraic function
u(k) = f (x(k)) which can be easily evaluated. In other words, we want to determine
the (generally nonlinear) feedback control law f (x) that generates the optimal
u(k) = f (x(k)) explicitly and not just implicitly as the result of an optimization
problem. It requires the solution of the Bellman equation and has been a long
standing problem in optimal control. A clean simple solution exists only in the case
of linear systems with a quadratic objective function, where the optimal controller
turns out to be a linear function of the state (Linear Quadratic Regulator, LQR).
For all other cases a solution of the Bellman equation was considered prohibitive
except for systems of low dimension (2 or 3), where a look-up table can be generated
by gridding the state space and solving the optimization problem off-line for each
grid point.
The major contribution of this book is to show how the nonlinear optimal feed-
back controller can be calculated efficiently for some important classes of systems,
namely linear systems with constraints and switched linear systems or, more gen-
erally, hybrid systems. Traditionally, the design of feedback controllers for linear
systems with constraints, for example anti-windup techniques, was ad hoc requiring
both much experience and trial and error. Though significant progress has been
achieved on anti-windup schemes over the last decade, these techniques deal with
input constraints only and cannot be extended easily.
The classes of constrained linear systems and linear hybrid systems treated in
this book cover many, if not most, practical problems. The new design techniques
hold the promise to lead to better performance and a dramatic reduction in the
required engineering effort.
The book is structured in five parts. Chapters marked with a star (∗) are
considered “advanced” material and can be skipped without compromising the
understanding of the fundamental concepts presented in the book.
• In the first part of the book (Part I) we recall the main concepts and results of
convex and discrete optimization. Our intent is to provide only the necessary
background for the understanding of the rest of the book. The material of
this part follows closely the presentation from the following books and lecture
notes: “Convex optimization” by Boyd and Vandenberghe [65], “Nonlinear
Programming Theory and Algorithms” by Bazaraa, Sherali and Shetty [27],
“LMIs in Control” by Scherer and Weiland [258] and “Lectures on Polytopes”
by Ziegler [296].
iv

Continuous problems as well as integer and mixed-integer problems are pre-

sented in Chapter 1. Chapter 1 also discusses the classical results of Lagrange
duality. In Chapter 2 Linear and Quadratic programs are presented together
with their properties and some fundamental results. Chapter 3 introduces
algorithms for the solution of unconstrained and constrained optimization
problems. We only discuss those that are important for the problems en-
countered in this book and explain the underlying concepts. Since polyhedra
are the fundamental geometric objects used in this book, Part I closes with
Chapter 4 where we introduce the main definitions and the algorithms, which
describe standard operations on polyhedra.

• The second part of the book (Part II) is a self-contained introduction to

multi-parametric programming. In our framework, parametric programming
is the main technique used to study and compute state feedback optimal
control laws. In fact, we formulate the finite time optimal control problems
as mathematical programs where the input sequence is the optimization vec-
tor. Depending on the dynamical model of the system, the nature of the
constraints, and the cost function used, a different mathematical program is
obtained. The current state of the dynamical system enters the cost function
and the constraints as a parameter that affects the solution of the mathemati-
cal program. We study the structure of the solution as this parameter changes
and we describe algorithms for solving multi-parametric linear, quadratic and
mixed integer programs. They constitute the basic tools for computing the
state feedback optimal control laws for these more complex systems in the
same way as algorithms for solving the Riccati equation are the main tools for
computing optimal controllers for linear systems. In Chapter 5 we introduce
the concept of multiparametric programming and we recall the main results
of nonlinear multiparametric programming. Then, in Chapter 6 we describe
three algorithms for solving multiparametric linear programs (mp-LP), multi-
parametric quadratic programs (mp-QP) and multiparametric mixed-integer
linear programs (mp-MILP).

• In the third part of the book (Part III) we introduce the general class of
optimal control problems studied in the book. Chapter 7 contains the basic
definitions and essential concepts. Chapter 8 presents standard results on
Linear Quadratic Optimal Control while in Chapter 9 unconstrained optimal
control problems for linear systems with cost functions based on 1 and ∞
norms are analyzed.

• In the fourth part of the book (Part IV) we focus on linear systems with
polyhedral constraints on inputs and states. We start with a self-contained
introduction to controllability, reachability and invariant set theory in Chap-
ter 10. The chapter focuses on computational algorithms for constrained
linear systems and constrained linear systems subject to additive and para-
metric uncertainty.
In Chapter 11 we study finite time and infinite time constrained optimal
control problems with cost functions based on 2, 1 and ∞ norms. We first
show how to transform them into LP or QP optimization problems for a fixed
 v

initial condition. Then we show that the solution to all these optimal control
problems can be expressed as a piecewise affine state feedback law. Moreover,
the optimal control law is continuous and the value function is convex and
continuous. The results form a natural extension of the theory of the Linear
Quadratic Regulator to constrained linear systems.
Chapter 12 presents the concept of Model Predictive Control. Classical feasi-
bility and stability issues are shown through simple examples and explained
by using invariant set methods. Finally we show how they can be addressed
with a proper choice of the terminal constraints and the cost function.
The result in Chapter 11 and Chapter 12 have important consequences for
the implementation of MPC laws. Precomputing off-line the explicit piece-
wise affine feedback policy reduces the on-line computation for the receding
horizon control law to a function evaluation, therefore avoiding the on-line
solution of a mathematical program. However, the number of polyhedral re-
gions of the explicit optimal control laws could grow exponentially with the
number of constraints in the optimal control problem. Chapter 13 discusses
approaches to define approximate explicit control laws of desired complexity
that provide certificates of recursive feasibility and stability.
Chapter 14 focuses on efficient on-line methods for the computation of RHC
control laws. If the state-feedback solution is available explicitly, we present
efficient on-line methods for the evaluation of explicit piecewise affine control
laws. In particular, we present algorithms to reduce its storage demands and
computational complexity. If the on-line solution of a quadratic or linear
program is preferred, we briefly discuss how to improve the efficiency of a
mathematical programming solver by exploiting the structure of the RHC
control problem.
Part IV closes with Chapter 15 where we address the robustness of the opti-
mal control laws. We discuss min-max control problems for uncertain linear
systems with polyhedral constraints on inputs and states and present an ap-
proach to compute their state feedback solutions. Robustness is achieved
against additive norm-bounded input disturbances and/or polyhedral para-
metric uncertainties in the state space matrices.
• In the fifth part of the book (Part V) we focus on linear hybrid systems. We
give an introduction to the different formalisms used to model hybrid systems
focusing on computation-oriented models (Chapter 16). In Chapter 17 we
study finite time optimal control problems with cost functions based on 2, 1
and ∞ norms. The optimal control law is shown to be, in general, piecewise
affine over non-convex and disconnected sets. Along with the analysis of the
solution properties we present algorithms that compute the optimal control
law for all the considered cases.

Berkeley Francesco Borrelli

Lucca Alberto Bemporad
Zurich Manfred Morari
July 2016
vi

Acknowledgements
• Large parts of the material presented in this book are extracted from the work
of the authors Francesco Borrelli, Alberto Bemporad and Manfred Morari.
• Several sections contain the results of the PhD theses of Miroslav Baric, Mato
Baotic, Fabio Torrisi, Domenico Mignone, Pascal Grieder, Eric Kerrigan,
Tobias Geyer and Frank J. Christophersen.

• We are extremely grateful to Martin Herceg for his meticulous work on the
Matlab R
examples and all the book figures.
• A special thanks goes to Michal Kvasnica, Stefan Richter, Valerio Turri,
Thomas Besselmann and Rick Meyer for their help on the construction of the
examples.
• Our gratitude also goes to the colleagues who have carefully read preliminary
versions of the book and gave us suggestions on how to improve them. They
include Dimitri Bertsekas, Miroslav Fikar, Paul Goulart, Per Olof Gutman,
Diethard Klatte, Bill Levine, David Mayne and all the students of our classes
on Model Predictive Control.
• The authors of the LaTex book style files and macros are Stephen Boyd
and Lieven Vandenberghe. We are most grateful to Stephen and Lieven for
sharing them with us.
• Over many years ABB provided generous financial support making possible
the fundamental research reported in this book.
• The authors of Chapter 3 are Dr. Alexander Domahidi and Dr. Stefan Richter
(domahidi@embotech.com, richters@alumni.ethz.ch).
• The author of Chapter 13 is Prof. Colin N. Jones (colin.jones@epfl.ch).
Contents

I Basics of Optimization 1
1 Main Concepts 3
1.1 Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Continuous Problems . . . . . . . . . . . . . . . . . . . . . . 4
Active, Inactive and Redundant Constraints . . . . . . . . . . 5
Problems in Standard Forms . . . . . . . . . . . . . . . . . . 5
Eliminating Equality Constraints . . . . . . . . . . . . . . . . 5
Problem Description . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.2 Integer and Mixed-Integer Problems . . . . . . . . . . . . . . 6
1.2 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Operations preserving convexity . . . . . . . . . . . . . . . . 7
Linear and quadratic convex functions . . . . . . . . . . . . . 8
Convex optimization problems . . . . . . . . . . . . . . . . . 8
1.3 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 Optimality Conditions for Unconstrained Problems . . . . . . 9
1.4 Lagrange Duality Theory . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.1 Strong Duality and Constraint Qualifications . . . . . . . . . 12
1.4.2 Certificate of Optimality . . . . . . . . . . . . . . . . . . . . . 13
1.5 Complementary Slackness . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 Karush-Kuhn-Tucker Conditions . . . . . . . . . . . . . . . . . . . . 14
1.6.1 Geometric Interpretation of KKT Conditions . . . . . . . . . 17

2 Linear and Quadratic Optimization 19

2.1 Polyhedra and Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Geometric Interpretation and Solution Properties . . . . . . . 20
2.2.2 Dual of LP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.3 KKT condition for LP . . . . . . . . . . . . . . . . . . . . . . 23
2.2.4 Active Constraints and Degeneracies . . . . . . . . . . . . . . 23
2.2.5 Convex Piecewise Affine Optimization . . . . . . . . . . . . . 26
2.3 Quadratic Programming . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.1 Geometric Interpretation and Solution Properties . . . . . . . 28
2.3.2 Dual of QP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.3 KKT conditions for QP . . . . . . . . . . . . . . . . . . . . . 29
2.3.4 Active Constraints and Degeneracies . . . . . . . . . . . . . . 30
viii Contents

2.3.5 Constrained Least-Squares Problems . . . . . . . . . . . . . . 30

2.4 Mixed-Integer Optimization . . . . . . . . . . . . . . . . . . . . . . . 30

3 Numerical Methods for Optimization 33

3.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Unconstrained Optimization . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.1 Gradient Methods . . . . . . . . . . . . . . . . . . . . . . . . 36
Classic Gradient Method . . . . . . . . . . . . . . . . . . . . 36
Fast Gradient Method . . . . . . . . . . . . . . . . . . . . . . 39
Strongly Convex Problems . . . . . . . . . . . . . . . . . . . . 39
Preconditioning . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Stopping Criteria . . . . . . . . . . . . . . . . . . . . . . . . . 42
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.2 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . 43
Preconditioning and Affine Invariance . . . . . . . . . . . . . 45
Newton’s Method with Linear Equality Constraints . . . . . 45
3.2.3 Line Search Methods . . . . . . . . . . . . . . . . . . . . . . . 46
3.3 Constrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3.1 Gradient Projection Methods . . . . . . . . . . . . . . . . . . 47
Solution in Dual Domain . . . . . . . . . . . . . . . . . . . . 49
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3.2 Interior Point Methods . . . . . . . . . . . . . . . . . . . . . . 52
Primal barrier methods . . . . . . . . . . . . . . . . . . . . . 52
Primal-dual methods . . . . . . . . . . . . . . . . . . . . . . . 57
Relation of Primal-Dual to Primal Barrier Methods . . . . . 62
3.3.3 Active Set Methods . . . . . . . . . . . . . . . . . . . . . . . 62
Active Set Method for LP (Simplex Method) . . . . . . . . . 63
Active Set Method for QP . . . . . . . . . . . . . . . . . . . . 67

4 Polyhedra and P-collections 71

4.1 General Set Definitions and Operations . . . . . . . . . . . . . . . . 71
4.2 Polyhedra and Representations . . . . . . . . . . . . . . . . . . . . . 73
4.3 Polytopal Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.1 Functions on Polytopal Complexes . . . . . . . . . . . . . . . 77
4.4 Basic Operations on Polytopes . . . . . . . . . . . . . . . . . . . . . 78
4.4.1 Minimal Representation . . . . . . . . . . . . . . . . . . . . . 78
4.4.2 Convex Hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.4.3 Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.4.4 Vertex Enumeration . . . . . . . . . . . . . . . . . . . . . . . 80
4.4.5 Chebychev Ball . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4.6 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.4.7 Set-Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.4.8 Pontryagin Difference . . . . . . . . . . . . . . . . . . . . . . 85
4.4.9 Minkowski Sum . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4.10 Polyhedra Union . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4.11 Affine Mappings and Polyhedra . . . . . . . . . . . . . . . . 88
4.5 Operations on P-collections . . . . . . . . . . . . . . . . . . . . . . . 88
Contents ix

4.5.1 Set-Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.5.2 Polytope Covering . . . . . . . . . . . . . . . . . . . . . . . . 91
4.5.3 Union of P-collections . . . . . . . . . . . . . . . . . . . . . . 92

II Multiparametric Programming 93
5 Multiparametric Nonlinear Programming 95
5.1 Introduction to Multiparametric Programs . . . . . . . . . . . . . . . 95
5.2 General Results for Multiparametric Nonlinear Programs . . . . . . 98

6 Multiparametric Programming: a Geometric Approach 107

6.1 Multiparametric Programs with Linear Constraints . . . . . . . . . . 107
6.1.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.1.2 Definition of Critical Region . . . . . . . . . . . . . . . . . . . 108
6.1.3 Reducing the Dimension of the Parameter Space . . . . . . . 109
6.2 Multiparametric Linear Programming . . . . . . . . . . . . . . . . . 110
6.2.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.2.2 Critical Regions, Value Function and Optimizer: Local Prop-
erties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.2.3 Propagation of the Set of Active Constraints . . . . . . . . . 115
6.2.4 Non-unique Optimizer∗ . . . . . . . . . . . . . . . . . . . . . 117
6.2.5 Value Function and Optimizer: Global Properties . . . . . . . 119
6.2.6 mp-LP Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 123
6.3 Multiparametric Quadratic Programming . . . . . . . . . . . . . . . 126
6.3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.3.2 Critical Regions, Value Function and Optimizer: Local Prop-
erties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.3.3 Propagation of the Set of Active Constraints . . . . . . . . . 131
6.3.4 Value Function and Optimizer: Global Properties . . . . . . . 132
6.3.5 mp-QP Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 135
6.4 Multiparametric Mixed-Integer Linear Programming . . . . . . . . . 137
6.4.1 Formulation and Properties . . . . . . . . . . . . . . . . . . . 137
6.4.2 Geometric Algorithm for mp-MILP . . . . . . . . . . . . . . . 138
Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
mp-LP subproblem . . . . . . . . . . . . . . . . . . . . . . . . 139
MILP subproblem . . . . . . . . . . . . . . . . . . . . . . . . 140
Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.4.3 Solution Properties . . . . . . . . . . . . . . . . . . . . . . . . 141
6.5 Multiparametric Mixed-Integer Quadratic Programming . . . . . . . 141
6.5.1 Formulation and Properties . . . . . . . . . . . . . . . . . . . 141
6.6 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

III Optimal Control 145

7 General Formulation and Discussion 147
7.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
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7.2 Solution via Batch Approach . . . . . . . . . . . . . . . . . . . . . . 149

7.3 Solution via Recursive Approach . . . . . . . . . . . . . . . . . . . . 150
7.4 Optimal Control Problem with Infinite Horizon . . . . . . . . . . . . 152
7.4.1 Value Function Iteration . . . . . . . . . . . . . . . . . . . . . 153
7.4.2 Receding Horizon Control . . . . . . . . . . . . . . . . . . . . 154
7.5 Lyapunov Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.5.1 General Stability Conditions . . . . . . . . . . . . . . . . . . 156
7.5.2 Quadratic Lyapunov Functions for Linear Systems . . . . . . 160
7.5.3 1/∞ Norm Lyapunov Functions for Linear Systems . . . . . . 161

8 Linear Quadratic Optimal Control 163

8.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
8.2 Solution via Batch Approach . . . . . . . . . . . . . . . . . . . . . . 164
8.3 Solution via Recursive Approach . . . . . . . . . . . . . . . . . . . . 165
8.4 Comparison of the Two Approaches . . . . . . . . . . . . . . . . . . 166
8.5 Infinite Horizon Problem . . . . . . . . . . . . . . . . . . . . . . . . . 168

9 Linear 1/∞ Norm Optimal Control 171

9.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
9.2 Solution via Batch Approach . . . . . . . . . . . . . . . . . . . . . . 172
9.3 Solution via Recursive Approach . . . . . . . . . . . . . . . . . . . . 175
9.4 Comparison Of The Two Approaches . . . . . . . . . . . . . . . . . . 177
9.5 Infinite Horizon Problem . . . . . . . . . . . . . . . . . . . . . . . . . 178

IV Constrained Optimal Control of Linear Systems 181

10 Controllability, Reachability and Invariance 183
10.1 Controllable and Reachable Sets . . . . . . . . . . . . . . . . . . . . 183
10.1.1 Computation of Controllable and Reachable Sets . . . . . . . 185
10.2 Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
10.3 Robust Controllable and Reachable Sets . . . . . . . . . . . . . . . . 196
10.3.1 Linear Systems with Additive Uncertainty and without In-
puts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
10.3.2 Linear Systems with Additive Uncertainty and Inputs . . . . 199
10.3.3 Linear Systems with Parametric Uncertainty . . . . . . . . . 202
10.4 Robust Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 205

11 Constrained Optimal Control 211

11.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
11.2 Feasible Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
11.3 2-Norm Case Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 218
11.3.1 Solution via QP . . . . . . . . . . . . . . . . . . . . . . . . . 218
11.3.2 State Feedback Solution via Batch Approach . . . . . . . . . 219
11.3.3 State Feedback Solution via Recursive Approach . . . . . . . 222
11.3.4 Infinite Horizon Problem . . . . . . . . . . . . . . . . . . . . 222
11.3.5 CLQR Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 224
11.3.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
Contents xi

11.4 1-Norm and ∞-Norm Case Solution . . . . . . . . . . . . . . . . . . 229

11.4.1 Solution via LP . . . . . . . . . . . . . . . . . . . . . . . . . . 230
11.4.2 State Feedback Solution via Batch Approach . . . . . . . . . 231
11.4.3 State Feedback Solution via Recursive Approach . . . . . . . 233
11.4.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
11.4.5 Infinite-Time Solution . . . . . . . . . . . . . . . . . . . . . . 237
11.5 State Feedback Solution, Minimum-Time Control . . . . . . . . . . . 238
11.6 Comparison of the Design Approaches and Controllers . . . . . . . . 239

12 Receding Horizon Control 243

12.1 RHC Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
12.2 RHC Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 244
12.3 RHC Main Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
12.3.1 Feasibility of RHC . . . . . . . . . . . . . . . . . . . . . . . . 251
12.3.2 Stability of RHC . . . . . . . . . . . . . . . . . . . . . . . . . 254
12.4 State Feedback Solution of RHC, 2-Norm Case . . . . . . . . . . . . 257
12.5 State Feedback Solution of RHC, 1, ∞-Norm Case . . . . . . . . . . 260
12.6 Tuning and Practical Use . . . . . . . . . . . . . . . . . . . . . . . . 262
12.7 Offset-Free Reference Tracking . . . . . . . . . . . . . . . . . . . . . 267
12.8 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

13 Approximate Receding Horizon Control 277

13.1 Stability of Approximate Receding Horizon Control . . . . . . . . . . 278
13.2 Barycentric Interpolation . . . . . . . . . . . . . . . . . . . . . . . . 281
13.2.1 Feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
13.2.2 Suboptimality . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
13.2.3 Constructive Algorithm . . . . . . . . . . . . . . . . . . . . . 285
13.3 Partitioning and Interpolation Methods . . . . . . . . . . . . . . . . 286
13.3.1 Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
13.3.2 Outer Polyhedral Approximation . . . . . . . . . . . . . . . . 287
Implicit Double Description Algorithm: Computing an Outer
Polyhedral Approximation of a Convex Set . . . . . 288
Approximate Control Law from Polyhedral Approximation . 290
Barycentric Functions for Polytopes . . . . . . . . . . . . . . 291
13.3.3 Second-Order Interpolants . . . . . . . . . . . . . . . . . . . . 291

14 On-line Control Computation 303

14.1 Storage and On-line Evaluation of the PWA Control Law . . . . . . 303
14.1.1 Efficient Implementation, 1, ∞-Norm Case . . . . . . . . . . . 305
14.1.2 Efficient Implementation, 2-Norm Case . . . . . . . . . . . . 306
Generating a PWA descriptor function from the value function311
Generating a PWA descriptor function from the optimal inputs312
14.1.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
CFTOC based on LP . . . . . . . . . . . . . . . . . . . . . . 313
CFTOC based on QP . . . . . . . . . . . . . . . . . . . . . . 313
14.1.4 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . 314
14.2 Gradient Projection Methods Applied to MPC . . . . . . . . . . . . 314
14.2.1 Input-Constrained MPC . . . . . . . . . . . . . . . . . . . . . 315
xii Contents

14.2.2 Input- and State-Constrained MPC . . . . . . . . . . . . . . 316

14.3 Interior Point Method Applied to MPC . . . . . . . . . . . . . . . . 317

15 Constrained Robust Optimal Control 319

15.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
15.1.1 State Feedback Solutions Summary . . . . . . . . . . . . . . . 325
15.2 Feasible Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
15.3 State Feedback Solution, Nominal Cost . . . . . . . . . . . . . . . . 332
15.4 State Feedback Solution, Worst-Case Cost, 1-Norm and ∞-Norm Case333
15.4.1 Batch Approach: Open-Loop Predictions . . . . . . . . . . . 334
15.4.2 Recursive Approach: Closed-Loop Predictions . . . . . . . . . 334
15.4.3 Solution to CROC-CL and CROC-OL via mp-MILP* . . . . 336
15.5 Parametrizations of the Control Policies . . . . . . . . . . . . . . . . 338
15.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
15.7 Robust Receding Horizon Control . . . . . . . . . . . . . . . . . . . . 346
15.8 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

V Constrained Optimal Control of Hybrid Systems 349

16 Models of Hybrid Systems 351
16.1 Models of Hybrid Systems . . . . . . . . . . . . . . . . . . . . . . . . 351
16.2 Piecewise Affine Systems . . . . . . . . . . . . . . . . . . . . . . . . . 352
16.2.1 Modeling Discontinuities . . . . . . . . . . . . . . . . . . . . . 354
16.2.2 Binary States, Inputs, and Outputs . . . . . . . . . . . . . . 355
16.3 Discrete Hybrid Automata . . . . . . . . . . . . . . . . . . . . . . . . 358
16.3.1 Switched Affine System (SAS) . . . . . . . . . . . . . . . . . 359
16.3.2 Event Generator (EG) . . . . . . . . . . . . . . . . . . . . . . 359
16.3.3 Boolean Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 360
16.3.4 Finite State Machine (FSM ) . . . . . . . . . . . . . . . . . . 361
16.3.5 Mode Selector . . . . . . . . . . . . . . . . . . . . . . . . . . 362
16.3.6 DHA Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . 362
16.4 Logic and Mixed-Integer Inequalities . . . . . . . . . . . . . . . . . . 363
16.4.1 Transformation of Boolean Relations . . . . . . . . . . . . . . 363
16.4.2 Translating DHA Components into Linear Mixed-Integer Re-
lations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
16.5 Mixed Logical Dynamical Systems . . . . . . . . . . . . . . . . . . . 365
16.6 Model Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
16.7 The HYSDEL Modeling Language . . . . . . . . . . . . . . . . . . . 367
16.8 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

17 Optimal Control of Hybrid Systems 377

17.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
17.2 Properties of the State Feedback Solution, 2-Norm Case . . . . . . . 379
17.3 Properties of the State Feedback Solution, 1, ∞-Norm Case . . . . . 385
17.4 Computation of the Optimal Control Input via Mixed Integer Pro-
gramming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
17.4.1 Mixed-Integer Optimization Methods . . . . . . . . . . . . . 390
Contents xiii

17.5 State Feedback Solution via Batch Approach . . . . . . . . . . . . . 392

17.6 State Feedback Solution via Recursive Approach . . . . . . . . . . . 392
17.6.1 Preliminaries and Basic Steps . . . . . . . . . . . . . . . . . . 392
17.6.2 Multiparametric Programming with Multiple Quadratic Func-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
17.6.3 Algorithmic Solution of the Bellman Equations . . . . . . . . 397
17.6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
17.7 Discontinuous PWA systems . . . . . . . . . . . . . . . . . . . . . . . 402
17.8 Receding Horizon Control . . . . . . . . . . . . . . . . . . . . . . . . 403
17.8.1 Stability and Feasibility Issues . . . . . . . . . . . . . . . . . 404
17.8.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

Bibliography 419
xiv Contents

Symbols and Acronyms

Logic Operators and Functions

A⇒B A implies B, i.e., if A is true then B must be true

A⇔B A implies B and B implies A, i.e., A is true if and only if (iff) B is true

Sets

R (R+ ) Set of (non-negative) real numbers

N (N+ ) Set of (non-negative) integers
Rn Set of real vectors with n elements
Rn×m Set of real matrices with n rows and m columns

Algebraic Operators and Matrices

A′ Transpose of matrix A
A−1 Inverse of matrix A
A† Generalized Inverse of A, A† = (A′ A)−1 A′
det(A) Determinant of matrix A
A ≻ ()0 A symmetric positive (semi)definite matrix, x′ Ax > (≥)0, ∀x 6= 0
A ≺ ()0 A symmetric negative (semi)definite matrix, x′ Ax < (≤)0, ∀x 6= 0.
Ai i-th row of matrix A
xi i-th element of vector x, or vector x at time-step i depending on the context
x ∈ Rn , x > 0 (x ≥ 0) True iff xi > 0 (xi ≥ 0) ∀ i = 1, . . . , n
x ∈ Rn , x < 0 (x ≤ 0) True iff xi < 0 (xi ≤ 0) ∀ i = 1, . . . , n
|x|, x ∈ R Absolute value of x
kxk Any vector norm of x pPn
kxk2 Euclidian norm of vector x ∈ Rn , kxk2 = i=1 |xP
i|
2
n
kxk1 Sum of absolute elements of vector x ∈ Rn , kxk1 = i=1 |xi |
kxk∞ Largest absolute value of the vector x ∈ Rn , kxk∞ = maxi∈{1,...,n}
P |xi |
kSk∞ Matrix ∞-norm of S ∈ Cm×n , i.e., kSk∞ = maxi∈{1,...,m} nj=1 |si,j |
P
kSk1 Matrix 1-norm of S ∈ Cm×n , i.e., kSk1 = maxj∈{1,...,n} m i=1 |si,j |
I Identity matrix
1 Vector of ones, 1 = [1 1 . . . 1]′
0 Vector of zeros, 0 = [0 0 . . . 0]′
Contents xv

Set Operators and Functions

∅ The empty set
: “Such that”
∂P The boundary of P
int(P) The interior of P, i.e., int(P) = P \ ∂P
|P| The cardinality of P, i.e., the number of elements in P
P ∩Q Set intersection P ∩ Q = {x : x ∈ P and x ∈ Q}
P ∪Q
S Set union P ∪ Q = {x : S x ∈ P or x ∈ Q}
r∈{1,...,R} Pr Union of R sets Pr , i.e., r∈{1,...,R} Pr = {x : x ∈ P0 or . . . or x ∈ PR }
Pc Complement of the set P, P c = {x : x ∈ / P}
P \Q Set difference P \ Q = {x : x ∈ P and x ∈ / Q}
P⊆Q The set P is a subset of Q, x ∈ P ⇒ x ∈ Q
P⊂Q The set P is a strict subset of Q, x ∈ P ⇒ x ∈ Q and ∃x ∈ (Q \ P)
P⊇Q The set P is a superset of Q
P⊃Q The set P is a strict superset of Q
P ⊖Q Pontryagin difference P ⊖ Q = {x : x + q ∈ P, ∀q ∈ Q}
P ⊕Q Minkowski sum P ⊕ Q = {x + q : x ∈ P, q ∈ Q}
f (x) With abuse of notation denotes the value of the function f at x or
the function f , f : x → f (x).
We use the notation f : Rn → Rs to mean that f is a Rs -valued function on
some subset of Rn , its domain, which we denote by dom f .
f (x) continuous f : Rn → R continuous for all x ∈ Rn , f (0) = 0 and f (x) > 0 ∀x ∈ Rn \ {0}
and positive definite
f (x) ≻ 0 f (x) continuous and positive definite
f (x) continuous f : Rn → R continuous for all x ∈ Rn , f (x) ≥ 0 ∀x ∈ Rn
and positive semi-definite
f (x)  0 f (x) continuous and positive semi-definite
xvi Contents

Acronyms
ARE Algebraic Riccati Equation
CLQR Constrained Linear Quadratic Regulator
CFTOC Constrained Finite Time Optimal Control
CITOC Constrained Infinite Time Optimal Control
DP Dynamic Program(ming)
LMI Linear Matrix Inequality
LP Linear Program(ming)
LQR Linear Quadratic Regulator
LTI Linear Time Invariant
MILP Mixed Integer Linear Program
MIQP Mixed Integer Quadratic Program
MPC Model Predictive Control
mp-LP multi-parametric Linear Program
mp-QP multi-parametric Quadratic Program
PWA Piecewise Affine
PPWA Piecewise Affine on Polyhedra
PWP Piecewise Polynomial
PWQ Piecewise Quadratic
QP Quadratic Program(ming)
RHC Receding Horizon Control
rhs right-hand side
SDP Semi Definite Program(ming)
 Part I

Basics of Optimization
 Chapter 1

Main Concepts

In this chapter we recall the main concepts and definitions of continuous and dis-
crete optimization. Our intent is to provide only the necessary background for
the understanding of the rest of the book. The notions of feasibility, optimality,
convexity and active constraints introduced in this chapter will be widely used in
this book.

1.1 Optimization Problems

An optimization problem is generally formulated as

inf z f (z)
(1.1)
subj. to z ∈ S ⊆ Z,

where the vector z collects the decision variables, Z is the optimization problem
domain, S ⊆ Z is the set of feasible or admissible decisions. The function f : Z → R
assigns to each decision z a cost f (z) ∈ R. We will often use the following shorter
form of problem (1.1)
inf f (z). (1.2)
z∈S⊆Z

Solving problem (1.2) means to compute the least possible cost f ∗

f ∗ = inf f (z).
z∈S

The number f ∗ is the optimal value of problem (1.2), i.e.

f (z) ≥ f (z ∗ ) = f ∗ ∀z ∈ S, with z ∗ ∈ S,

or the greatest lower bound of f (z) over the set S:

f (z) > f ∗ ∀z ∈ S and (∀ε > 0 ∃z ∈ S : f (z) ≤ f ∗ + ε).

4 1 Main Concepts

If f ∗ = −∞ we say that the problem is unbounded below. If the set S is empty

then the problem is said to be infeasible and we set f ∗ = +∞ by convention. If
S = Z the problem is said to be unconstrained.
In general, one is also interested in finding an optimal solution, that is in finding
a decision whose associated cost equals the optimal value, i.e., z ∗ ∈ S with f (z ∗ ) =
f ∗ . If such z ∗ exists, then we rewrite problem (1.2) as

f ∗ = min f (z) (1.3)

z∈S

and z ∗ is called an optimizer or global optimizer or optimal solution. Minimizer or

global minimizer are also used to refer to an optimizer of a minimization problem.
The set of all optimal solutions is denoted by

argminz∈S f (z) = {z ∈ S : f (z) = f ∗ } .

A problem of determining whether the set of feasible decisions is empty and, if not,
to find a point which is feasible, is called a feasibility problem.

1.1.1 Continuous Problems

In continuous optimization the problem domain Z is a subset of the finite-dimensional
Euclidian vector-space Rs and the subset of admissible vectors is defined through
a list of equality and inequality constraints:
inf z f (z)
subj. to gi (z) ≤ 0 for i = 1, . . . , m
(1.4)
hj (z) = 0 for j = 1, . . . , p
z ∈ Z,
where f, g1 , . . . , gm , h1 , . . . , hp are real-valued functions defined over Rs , i.e., f :
Rs → R, gi : Rs → R, hi : Rs → R. The domain Z is the intersection of the
domains of the cost and constraint functions:

Z = {z ∈ Rs : z ∈ dom f, z ∈ dom gi , i = 1, . . . , m, z ∈ dom hj , j = 1, . . . , p}.

(1.5)
In the sequel we will consider the constraint z ∈ Z implicit in the optimization
problem and often omit it. Problem (1.4) is unconstrained if m = p = 0.
The inequalities gi (z) ≤ 0 are called inequality constraints and the equations
hi (z) = 0 are called equality constraints. A point z̄ ∈ Rs is feasible for problem (1.4)
if: (i) it belongs to Z, (ii) it satisfies all inequality and equality constraints, i.e.,
gi (z̄) ≤ 0, i = 1, . . . , m, hj (z̄) = 0, i = j, . . . , p. The set of feasible vectors is

S = {z ∈ Rs : z ∈ Z, gi (z) ≤ 0, i = 1, . . . , m, hj (z) = 0, j = 1, . . . , p}. (1.6)

Problem (1.4) is a continuous finite-dimensional optimization problem (since

Z is a finite-dimensional Euclidian vector space). We will also refer to (1.4) as
a nonlinear mathematical program or simply nonlinear program. Let f ∗ be the
optimal value of problem (1.4). An optimizer, if it exists, is a feasible vector z ∗
with f (z ∗ ) = f ∗ .
1.1 Optimization Problems 5

A feasible point z̄ is locally optimal for problem (1.4) if there exists an R > 0
such that
f (z̄) = inf z f (z)
subj. to gi (z) ≤ 0 for i = 1, . . . , m
hi (z) = 0 for i = 1, . . . , p (1.7)
kz − z̄k ≤ R
z ∈ Z.
Roughly speaking, this means that z̄ is the minimizer of f (z) in a feasible neigh-
borhood of z̄ defined by kz − z̄k ≤ R. The point z̄ is called a local optimizer or
local minimizer.

Active, Inactive and Redundant Constraints

Consider a feasible point z̄. We say that the i-th inequality constraint gi (z) ≤ 0 is
active at z̄ if gi (z̄) = 0. If gi (z̄) < 0 we say that the constraint gi (z) ≤ 0 is inactive
at z̄. Equality constraints are always active for all feasible points.
We say that a constraint is redundant if removing it from the list of constraints
does not change the feasible set S. This implies that removing a redundant con-
straint from problem (1.4) does not change its solution.

Problems in Standard Forms

Optimization problems can be cast in several forms. In this book we use the
form (1.4) where we adopt the convention to minimize the cost function and to
have the right-hand side of the inequality and equality constraints equal to zero.
Any problem in a different form (e.g., a maximization problem or a problem with
“box constraints”) can be transformed and arranged into this form. The interested
reader is referred to Chapter 4 of [65] for a detailed discussion on transformations
of optimization problems into different standard forms.

Eliminating Equality Constraints

Often in this book we will restrict our attention to problems without equality
constraints, i.e., p = 0

inf z f (z)
(1.8)
subj. to gi (z) ≤ 0 for i = 1, . . . , m.

The simplest way to remove equality constraints is to replace them with two in-
equalities for each equality, i.e., hi (z) = 0 is replaced by hi (z) ≤ 0 and −hi (z) ≤ 0.
Such a method, however, can lead to poor numerical conditioning and may ruin
the efficiency and accuracy of a numerical solver.
If one can explicitly parameterize the solution of the equality constraint hi (z) =
0, then the equality constraint can be eliminated from the problem. This process
can be described in a simple way for linear equality constraints. Assume the equal-
ity constraints to be linear, Az − b = 0, with A ∈ Rp×s . If Az = b is inconsistent
then the problem is infeasible. The general solution of the equation Az = b can be
expressed as z = F x + z0 where F is a matrix of full rank whose spanned space
coincides with the null space of the A matrix, i.e., R(F ) = N (A), F ∈ Rs×k ,
6 1 Main Concepts

where k is the dimension of the null space of A. The variable x ∈ Rk is the new
optimization variable and the original problem becomes

inf x f (F x + z0 )
(1.9)
subj. to gi (F x + z0 ) ≤ 0 for i = 1, . . . , m.

We want to point out that in some cases the elimination of equality constraints
can make the problem harder to analyze and understand and can make a solver
less efficient. In large problems it can destroy useful structural properties of the
problem such as sparsity. Some advanced numerical solvers perform elimination
automatically.

Problem Description
The functions f, gi and hi can be available in analytical form or can be described
through an oracle model (also called “black box” or “subroutine” model). In an
oracle model f, gi and hi are not known explicitly but can be evaluated by querying
the oracle. Often the oracle consists of subroutines which, called with the argument
z, return f (z), gi (z) and hi (z) and their gradients ∇f (z), ∇gi (z), ∇hi (z). In
the rest of the book we assume that analytical expressions of the cost and the
constraints of the optimization problem are available.

1.1.2 Integer and Mixed-Integer Problems

If the decision set Z in the optimization problem (1.2) is finite, then the optimiza-
tion problem is called combinatorial or discrete. If Z ⊆ {0, 1}s , then the problem
is said to be integer.
If Z is a subset of the Cartesian product of an integer set and a real Euclidian
space, i.e., Z ⊆ {[zc , zb ] : zc ∈ Rsc , zb ∈ {0, 1}sb }, then the problem is said to be
mixed-integer. The standard formulation of a mixed-integer nonlinear program is

inf [zc ,zb ] f (zc , zb )

subj. to gi (zc , zb ) ≤ 0 for i = 1, . . . , m
(1.10)
hj (zc , zb ) = 0 for j = 1, . . . , p
zc ∈ Rsc , zb ∈ {0, 1}sb

where f, g1 , . . . , gm , h1 , . . . , hp are real-valued functions defined over Z.

For combinatorial, integer and mixed-integer optimization problems all defini-
tions introduced in the previous section apply.

1.2 Convexity
A set S ∈ Rs is convex if

λz1 + (1 − λ)z2 ∈ S for all z1 ∈ S, z2 ∈ S and λ ∈ [0, 1].

1.2 Convexity 7

A function f : S → R is convex if S is convex and

f (λz1 + (1 − λ)z2 ) ≤ λf (z1 ) + (1 − λ)f (z2 )

for all z1 ∈ S, z2 ∈ S and λ ∈ [0, 1].

A function f : S → R is strictly convex if S is convex and

f (λz1 + (1 − λ)z2 ) < λf (z1 ) + (1 − λ)f (z2 )

for all z1 ∈ S, z2 ∈ S and λ ∈ (0, 1).

A twice differentiable function f : S → R is strongly convex if the Hessian

∇2 f (z) ≻ 0 for all z ∈ S.

A function f : S → R is concave if S is convex and −f is convex.

Operations preserving convexity

Various operations preserve convexity of functions and sets. A detailed list can be
found in Chapter 3.2 of [65]. A few operations used in this book are mentioned
below.

1. The intersection of an arbitrary number of convex sets is a convex set:

if S1 , S2 , . . . , Sk are convex, then S1 ∩ S2 ∩ . . . ∩ Sk is convex.

This property extends to the intersection of an infinite number of sets:

\
if Sn is convex ∀n ∈ N+ then Sn is convex.
n∈N+

The empty set is convex because it satisfies the definition of convexity.

2. The sub-level sets of a convex function f on S are convex:

if f (z) is convex then Sα = {z ∈ S : f (z) ≤ α} is convex ∀α ∈ R.

PN
3. If f1 , . . . , fN are convex functions, then i=1 αi fi is a convex function for all
αi ≥ 0, i = 1, . . . , N .

4. The composition of a convex function f (z) with an affine map z = Ax + b

generates a convex function f (Ax + b) of x:

if f (z) is convex then f (Ax + b) is convex on {x : Ax + b ∈ dom(f )}

5. Suppose f (x) = h(g(x)) = h(g1 (x), . . . , gk (x)) with h : Rk → R, gi : Rs → R.

Then,

(a) f is convex if h is convex, h is nondecreasing in each argument, and gi

are convex,
8 1 Main Concepts

(b) f is convex if h is convex, h is nonincreasing in each argument, and gi

are concave,
(c) f is concave if h is concave, h is nondecreasing in each argument, and
gi are concave.

6. The pointwise maximum of a set of convex functions is a convex function:

f1 (z), . . . , fk (z) convex functions ⇒ f (z) = max{f1 (z), . . . , fk (z)} is a convex function.

This property holds also when the set is infinite.

Linear and quadratic convex functions

1. A linear function f (z) = c′ z + r is both convex and concave.

2. A quadratic function f (z) = z ′ Hz + 2q ′ z + r is convex if and only if H  0.

3. A quadratic function f (z) = z ′ Hz + 2q ′ z + r is strictly convex if and only if

H ≻ 0. A strictly convex quadratic function is also strongly convex.

Convex optimization problems

The standard optimization problem (1.4) is said to be convex if the cost function f
is convex on Z and S is a convex set. A fundamental property of convex optimiza-
tion problems is that local optimizers are also global optimizers. This is proven
next.

Theorem 1.1 Consider a convex optimization problem and let z̄ be a local opti-
mizer. Then, z̄ is a global optimizer.

Proof: By hypothesis z̄ is feasible and there exists R such that

f (z̄) = min{f (z) : gi (z) ≤ 0 i = 1, . . . , m, hj (z) = 0, j = 1, . . . , p kz − z̄k ≤ R}.

(1.11)
Now suppose that z̄ is not globally optimal. Then, there exist a feasible y such
that f (y) < f (z̄), which implies that ky − z̄k > R. Now consider the point z given
by
R
z = (1 − θ)z̄ + θy, θ = .
2ky − z̄k
Then kz − z̄k = R/2 < R and by convexity of the feasible set z is feasible. By
convexity of the cost function f

f (z) ≤ (1 − θ)f (z̄) + θf (y) < f (z̄),

which contradicts (1.11). 

Theorem 1.1 does not make any statement about the existence of a solution to
problem (1.4). It merely states that all local minima of problem (1.4) are also global
minima. For this reason, convexity plays a central role in the solution of continuous
optimization problems. It suffices to compute a local minimum to problem (1.4)
 1.3 Optimality Conditions 9

to determine its global minimum. Convexity also plays a major role in most non-
convex optimization problems which are solved by iterating between the solutions
of convex sub-problems.
It is difficult to determine whether the feasible set S of the optimization prob-
lem (1.4) is convex or not except in special cases. For example, if the functions
g1 (z), . . . , gm (z) are convex and all the hi (z) (if any) are affine in z, then the feasible
set S in (1.6) is an intersection of convex sets and therefore convex. Moreover there
are non-convex problems which can be transformed into convex problems through
a change of variables and manipulations of cost and constraints. The discussion of
this topic goes beyond the scope of this overview on optimization. The interested
reader is referred to [65].

Remark 1.1 With the exception of trivial cases, integer and mixed-integer optimiza-
tion problems are always non-convex problems because {0, 1} is not a convex set.

1.3 Optimality Conditions

In general, an analytical solution to problem (1.4), restated below, does not exist.

inf z f (z)
subj. to gi (z) ≤ 0 for i = 1, . . . , m
(1.12)
hj (z) = 0 for j = 1, . . . , p
z ∈ Z.

Solutions are usually computed by iterative algorithms which start from an initial
guess z0 and at step k generate a point zk such that the sequence {f (zk )}k=0,1,2,...
converges to f ∗ as k increases. These algorithms iteratively use and/or solve condi-
tions for optimality, i.e., analytical conditions that a point z must satisfy in order
to be an optimizer. For instance, for convex, unconstrained optimization prob-
lems with a smooth cost function the most commonly used optimality criterion
requires the gradient to vanish at the optimizer, i.e., z is an optimizer if and only
if ∇f (z) = 0. In this chapter we summarize necessary and sufficient optimality
conditions for unconstrained and constrained optimization problems.

1.3.1 Optimality Conditions for Unconstrained Problems

The proofs of the theorems presented next can be found in Chapter 4 and Section
8.6.1 of [27].
Necessary conditions

Theorem 1.2 Suppose that f : Rs → R is differentiable at z̄. If there exists a

vector d such that ∇f (z̄)′ d < 0, then there exists a δ > 0 such that f (z̄+λd) <
f (z̄) for all λ ∈ (0, δ).
10 1 Main Concepts

The vector d in the theorem above is called a descent direction. At a given

point z̄ a descent direction d satisfies the condition ∇f (z̄)′ d < 0. Theorem 1.2
states that if a descent direction exists at a point z̄, then it is possible to
move from z̄ towards a new point z̃ whose associated cost f (z̃) is lower than
f (z̄). The direction of steepest descent ds at a given point z̄ is defined as the
normalized direction where ∇f (z̄)′ ds < 0 is minimized. The direction ds of
∇f (z̄)
steepest descent is ds = − k∇f (z̄)k .
Two corollaries of Theorem 1.2 are stated next.

Corollary 1.1 Suppose that f : Rs → R is differentiable at z̄. If z̄ is a local

minimizer, then ∇f (z̄) = 0.

Corollary 1.2 Suppose that f : Rs → R is twice differentiable at z̄. If z̄

is a local minimizer, then ∇f (z̄) = 0 and the Hessian ∇2 f (z̄) is positive
semidefinite.

Sufficient condition
Theorem 1.3 Suppose that f : Rs → R is twice differentiable at z̄. If
∇f (z̄) = 0 and the Hessian of f (z) at z̄ is positive definite, then z̄ is a
local minimizer.
Necessary and sufficient condition
Theorem 1.4 Suppose that f : Rs → R is differentiable at z̄. If f is convex,
then z̄ is a global minimizer if and only if ∇f (z̄) = 0.
When the optimization is constrained and the cost function is not sufficiently
smooth, the conditions for optimality become more complicated. The intent of this
chapter is to give an overview of some important optimality criteria for constrained
nonlinear optimization. The optimality conditions derived here will be the main
building blocks for the theory developed later in this book.

1.4 Lagrange Duality Theory

Consider the nonlinear program (1.12). Let f ∗ be the optimal value. Denote by Z
the domain of cost and constraints (1.5). Any feasible point z̄ provides an upper
bound to the optimal value f (z̄) ≥ f ∗ . Next we will show how to generate a lower
bound on f ∗ .
Starting from the standard nonlinear program (1.12) we construct another prob-
lem with different variables and constraints. The original problem (1.12) will be
called the primal problem while the new one will be called the dual problem. First,
we augment the objective function with a weighted sum of the constraints. In this
way the Lagrange dual function (or Lagrangian) L is obtained
L(z, u, v) = f (z) + u1 g1 (z) + · · · + um gm (z) +
+v1 h1 (z) + · · · + vp hp (z), (1.13)
1.4 Lagrange Duality Theory 11

where the scalars u1 , . . . , um , v1 , . . . , vp are real variables called dual variables or

Lagrange multipliers. We can write equation (1.13) in the compact form

L(z, u, v) = f (z) + u′ g(z) + v ′ h(z), (1.14)

where u = [u1 , . . . , um ]′ , v = [v1 , . . . , vp ]′ and L : Rs × Rm × Rp → R. The

components ui and vi are called dual variables. Note that the i-th dual variable
ui is associated with the i-th inequality constraint of problem (1.12), the i-th dual
variable vi is associated with the i-th equality constraint of problem (1.12).
Let z be a feasible point: for arbitrary vectors u ≥ 0 and v we trivially obtain
a lower bound on f (z)
L(z, u, v) ≤ f (z). (1.15)
We minimize both sides of equation (1.15)

inf L(z, u, v) ≤ inf f (z) (1.16)

z∈Z, g(z)≤0, h(z)=0 z∈Z, g(z)≤0, h(z)=0

in order to reconstruct the original problem on the right-hand side of the expression.
Since for arbitrary u ≥ 0 and v

inf L(z, u, v) ≤ inf L(z, u, v), (1.17)

z∈Z z∈Z, g(z)≤0, h(z)=0

we obtain
inf L(z, u, v) ≤ inf f (z). (1.18)
z∈Z z∈Z, g(z)≤0, h(z)=0

Equation (1.18) implies that for arbitrary u ≥ 0 and v the solution to

inf L(z, u, v) (1.19)

z∈Z

provides us with a lower bound to the original problem. The “best” lower bound
is obtained by maximizing problem (1.19) over the dual variables

sup inf L(z, u, v) ≤ inf f (z).

(u,v), u≥0 z∈Z z∈Z, g(z)≤0, h(z)=0

Define the dual cost d(u, v) as follows

d(u, v) = inf L(z, u, v) ∈ [−∞, +∞]. (1.20)

z∈Z

Then the Lagrange dual problem is defined as

sup d(u, v), (1.21)

(u,v), u≥0

and its optimal solution, if it exists, is denoted by (u∗ , v ∗ ). The dual cost d(u, v)
is the optimal value of an unconstrained optimization problem. Problem (1.20)
is called the Lagrange dual subproblem. Only points (u, v) with d(u, v) > −∞
are interesting for the Lagrange dual problem. A point (u, v) will be called dual
feasible if u ≥ 0 and d(u, v) > −∞. d(u, v) is always a concave function since
it is the pointwise infimum of a family of affine functions of (u, v). This implies
12 1 Main Concepts

that the dual problem is a convex optimization problem (max of a concave function
over a convex set) even if the original problem is not convex. Therefore, it is easier
in principle to solve the dual problem than the primal (which is in general non-
convex). However, in general, the solution to the dual problem is only a lower
bound of the primal problem:

sup d(u, v) ≤ inf f (z).

(u,v), u≥0 z∈Z, g(z)≤0, h(z)=0

Such a property is called weak duality. In a simpler form, let f ∗ and d∗ be the
primal and dual optimal value, respectively,

f∗ = inf f (z), (1.22a)

z∈Z, g(z)≤0, h(z)=0

d∗ = sup d(u, v), (1.22b)

(u,v), u≥0

then, we always have

f ∗ ≥ d∗ (1.23)
∗ ∗
and the difference f − d is called the optimal duality gap. The weak duality
inequality (1.23) holds also when d∗ and f ∗ are infinite. For example, if the primal
problem is unbounded below, so that f ∗ = −∞, we must have d∗ = −∞, i.e., the
Lagrange dual problem is infeasible. Conversely, if the dual problem is unbounded
above, so that d∗ = +∞, we must have f ∗ = +∞, i.e., the primal problem is
infeasible.

Remark 1.2 We have shown that the dual problem is a convex optimization problem
even if the original problem is non-convex. As stated in this section, for non-convex
optimization problems it is “easier” to solve the dual problem than the primal prob-
lem. However the evaluation of d(ū, v̄) at a point (ū, v̄) requires the solution of the
non-convex unconstrained optimization problem (1.20), which, in general, is “not
easy”.

1.4.1 Strong Duality and Constraint Qualifications

If d∗ = f ∗ , then the duality gap is zero and we say that strong duality holds:

sup d(u, v) = inf f (z). (1.24)

(u,v), u≥0 z∈Z, g(z)≤0, h(z)=0

This means that the best lower bound obtained by solving the dual problem coin-
cides with the optimal cost of the primal problem. In general, strong duality does
not hold, even for convex primal problems. Constraint qualifications are condi-
tions on the constraint functions which imply strong duality for convex problems.
A detailed discussion on constraint qualifications can be found in Chapter 5 of [27].
A well known simple constraint qualification is “Slater’s condition”:

Definition 1.1 (Slater’s condition) Consider problem (1.12). There exists ẑ ∈

Rs which belongs to the relative interior of the problem domain Z, which is feasible
 1.5 Complementary Slackness 13

(g(ẑ) ≤ 0, h(ẑ) = 0) and for which gj (ẑ) < 0 for all j for which gj is not an affine
function.

Remark 1.3 Note that Slater’s condition reduces to feasibility when all inequality
constraints are linear and Z = Rn .

Theorem 1.5 (Slater’s theorem) Consider the primal problem (1.22a) and its
dual problem (1.22b). If the primal problem is convex, Slater’s condition holds and
f ∗ is bounded then d∗ = f ∗ .

1.4.2 Certificate of Optimality

Consider the (primal) optimization problem (1.12) and its dual (1.21). Any feasible
point z gives us information about an upper bound on the cost, i.e., f ∗ ≤ f (z).
If we can find a dual feasible point (u, v) then we can establish a lower bound
on the optimal value of the primal problem: d(u, v) ≤ f ∗ . In summary, without
knowing the exact value of f ∗ we can give a bound on how suboptimal a given
feasible point is. In fact, if z is primal feasible and (u, v) is dual feasible then
d(u, v) ≤ f ∗ ≤ f (z). Therefore z is ε-suboptimal, with ε equal to the primal-dual
gap, i.e., ε = f (z) − d(u, v).
The optimal value of the primal (and dual) problems will lie in the same interval
f ∗ ∈ [d(u, v), f (z)] and d∗ ∈ [d(u, v), f (z)].
For this reason (u, v) is also called a certificate that proves the (sub)optimality
of z. Optimization algorithms make extensive use of such criteria. Primal-dual
algorithms iteratively solve primal and dual problems and generate a sequence of
primal and dual feasible points zk , (uk , vk ), k ≥ 0 until a certain ε is reached. The
condition
f (zk ) − d(uk , vk ) < ε,
for terminating the algorithm guarantees that when the algorithm terminates, zk
is ε-suboptimal. If strong duality holds the condition can be met for arbitrarily
small tolerances ε.

1.5 Complementary Slackness

Consider the (primal) optimization problem (1.12) and its dual (1.21). Assume that
strong duality holds. Suppose that z ∗ and (u∗ , v ∗ ) are primal and dual feasible
with zero duality gap (hence, they are primal and dual optimal):
f (z ∗ ) = d(u∗ , v ∗ ).
By definition of the dual problem, we have


f (z ∗ ) = inf f (z) + u∗ ′ g(z) + v ∗ ′ h(z) .
z∈Z
14 1 Main Concepts

Therefore

f (z ∗ ) ≤ f (z ∗ ) + u∗ ′ g(z ∗ ) + v ∗ ′ h(z ∗ ), (1.25)

and since h(z ∗ ) = 0, u∗ ≥ 0 and g(z ∗ ) ≤ 0 we have

f (z ∗ ) ≤ f (z ∗ ) + u∗ ′ g(z ∗ ) ≤ f (z ∗ ). (1.26)
Pm
From the last equation we can conclude that u∗ ′ g(z ∗ ) = i=1 u∗i gi (z ∗ ) = 0 and
since u∗i ≥ 0 and gi (z ∗ ) ≤ 0, we have

u∗i gi (z ∗ ) = 0, i = 1, . . . , m. (1.27)

Conditions (1.27) are called complementary slackness conditions. Complemen-

tary slackness conditions can be interpreted as follows. If the i-th inequality con-
straint of the primal problem is inactive at the optimum (gi (z ∗ ) < 0), then the
i-th dual optimizer has to be zero (u∗i = 0). Vice versa, if the i-th dual optimizer
is different from zero (u∗i > 0), then the i-th constraint is active at the optimum
(gi (z ∗ ) = 0).
Relation (1.27) implies that the inequality in (1.25) holds as equality
 
X X X X
f (z ∗ ) + u∗i gi (z ∗ ) + vj∗ hj (z ∗ ) = min f (z) + u∗i gi (z) + vj∗ hj (z) .
z∈Z
i j i j
(1.28)
Therefore, complementary slackness conditions implies that z ∗ is a minimizer of
L(z, u∗ , v ∗ ).

1.6 Karush-Kuhn-Tucker Conditions

Consider the (primal) optimization problem (1.12) and its dual (1.21). Assume
that strong duality holds. Assume that the cost functions and constraint func-
tions f , gi , hi are differentiable. Let z ∗ and (u∗ , v ∗ ) be primal and dual optimal
points, respectively. Complementary slackness conditions implies that z ∗ minimizes
L(z, u∗ , v ∗ ) under no constraints (equation (1.28)). Since f , gi , hi are differentiable,
the gradient of L(z, u∗, v ∗ ) must be zero at z ∗
X X
∇f (z ∗ ) + u∗i ∇gi (z ∗ ) + vj∗ ∇hj (z ∗ ) = 0.
i j

In summary, the primal and dual optimal pair z ∗ , (u∗ , v ∗ ) of an optimization

problem with differentiable cost and constraints and zero duality gap, have to
1.6 Karush-Kuhn-Tucker Conditions 15

satisfy the following conditions:

m
X p
X
∇f (z ∗ ) + u∗i ∇gi (z ∗ ) + vj∗ ∇hi (z ∗ ) = 0, (1.29a)
i=1 j=1
u∗i gi (z ∗ ) = 0, i = 1, . . . , m (1.29b)
u∗i ≥ 0, i = 1, . . . , m (1.29c)
gi (z ∗ ) ≤ 0, i = 1, . . . , m (1.29d)
hj (z ∗ ) = 0, j = 1, . . . , p (1.29e)

where equations (1.29d)-(1.29e) are the primal feasibility conditions, equation (1.29c)
is the dual feasibility condition and equation (1.29b) are the complementary slack-
ness conditions.
Conditions (1.29a)-(1.29e) are called the Karush-Kuhn-Tucker (KKT) condi-
tions. We have shown that the KKT conditions are necessary conditions for any
primal-dual optimal pair if strong duality holds and the cost and constraints are
differentiable, i.e., any primal and dual optimal points z ∗ , (u∗ , v ∗ ) must satisfy
the KKT conditions (1.29). If the primal problem is also convex then the KKT
conditions are sufficient, i.e., a primal dual pair z ∗ , (u∗ , v ∗ ) which satisfies condi-
tions (1.29a)-(1.29e) is a primal dual optimal pair with zero duality gap.
There are several theorems which characterize primal and dual optimal points
z ∗ and (u∗ , v ∗ ) by using KKT conditions. They mainly differ on the type of
constraint qualification chosen for characterizing strong duality. Next we report
just two examples.
If a convex optimization problem with differentiable objective and constraint
functions satisfies Slater’s condition, then the KKT conditions provide necessary
and sufficient conditions for optimality.

Theorem 1.6 [27, p. 244] Consider problem (1.12) and let Z be a nonempty set
of Rs . Suppose that problem (1.12) is convex and that cost and constraints f , gi and
hi are differentiable at a feasible z ∗ . If problem (1.12) satisfies Slater’s condition
then z ∗ is optimal if and only if there are (u∗ , v ∗ ) that, together with z ∗ , satisfy
the KKT conditions (1.29).
If a convex optimization problem with differentiable objective and constraint func-
tions has a linearly independent set of active constraints gradients, then the KKT
conditions provide necessary and sufficient conditions for optimality.

Theorem 1.7 (Section 4.3.7 in [27]) Consider problem (1.12) and let Z be a
nonempty open set of Rs . Let z ∗ be a feasible solution and A = {i : gi (z ∗ ) =
0} be the set of active constraints at z ∗ . Suppose cost and constraints f , gi are
differentiable at z ∗ for all i and that hj are continuously differentiable at z ∗ for
all j. Further, suppose that ∇gi (z ∗ ) for i ∈ A and ∇hj (z ∗ ) for j = 1, . . . , p, are
linearly independent. If z ∗ , (u∗ , v ∗ ) are primal and dual optimal points, then they
satisfy the KKT conditions (1.29). In addition, if problem (1.12) is convex, then
z ∗ is optimal if and only if there are (u∗ , v ∗ ) that, together with z ∗ , satisfy the
KKT conditions (1.29).
16 1 Main Concepts

z2

(1, 1)

∇g2 (1, 0)

∇f (1, 0)

z1

∇g1 (1, 0)

(1, −1)

Figure 1.1 Example 1.1. Constraints, feasible set and gradients. The fea-
sible set is the point (1, 0) that satisfies neither Slater’s condition nor the
constraint qualification condition in Theorem 1.7.

The KKT conditions play an important role in optimization. In a few special cases
it is possible to solve the KKT conditions (and therefore, the optimization problem)
analytically. Many algorithms for convex optimization are conceived as, or can be
interpreted as, methods for solving the KKT conditions as Boyd and Vandenberghe
observe in [65].
The following example [27] shows a convex problem where the KKT conditions
are not fulfilled at the optimum. In particular, both the constraint qualifications
of Theorem 1.7 and Slater’s condition in Theorem 1.6 are violated.

Example 1.1 [27, p. 196] Consider the convex optimization problem

min z1
(1.30)
subj. to (z1 − 1)2 + (z2 − 1)2 ≤ 1
(z1 − 1)2 + (z2 + 1)2 ≤ 1

From the graphical interpretation in Figure 1.1 it is immediate that the feasible set is a
single point z̄ = [1, 0]′ . The optimization problem does not satisfy Slater’s conditions
and moreover z̄ does not satisfy the constraint qualifications in Theorem 1.7. At
the optimum z̄ equation (1.29a) cannot be satisfied for any pair of nonnegative real
numbers u1 and u2 .
 1.6 Karush-Kuhn-Tucker Conditions 17

∇g1 (z1 ) −∇f (z1 )

∇g2 (z1 )
z1

g2
0

(z
)≤

≤ )
g1 (z

0
−∇f (z2 )
g(z) ≤ 0 ∇g2 (z2 )

z2 ∇g3 (z2 )
0
) ≤
(z
g3
Figure 1.2 Geometric interpretation of KKT conditions [27].

1.6.1 Geometric Interpretation of KKT Conditions

A geometric interpretation of the KKT conditions is depicted in Figure 1.2 for an
optimization problem in two dimensions with inequality constraints and no equality
constraints.
Equation (1.29a) and equation (1.29c) can be rewritten as
X
− ∇f (z) = ui ∇gi (z), ui ≥ 0, (1.31)
i∈A

where A = {1, 2} at z1 and A = {2, 3} at z2 . This means that the negative gradient
of the cost at the optimum −∇f (z ∗ ) (which represents the direction of steepest
descent) has to belong to the cone spanned by the gradients of the active constraints
∇gi , (since inactive constraints have the corresponding Lagrange multipliers equal
to zero). In Figure 1.2, condition (1.31) is not satisfied at z2 . In fact, one can
move within the set of feasible points g(z) ≤ 0 and decrease f , which implies that
z2 is not optimal. At point z1 , on the other hand, the cost f can only decrease
if some constraint is violated. Any movement in a feasible direction increases the
cost. Conditions (1.31) are fulfilled and hence z1 is optimal.
18 1 Main Concepts
 Chapter 2

Linear and Quadratic

Optimization

This chapter focuses on two widely known and used subclasses of convex optimiza-
tion problems: linear and quadratic programs. They are popular because many
important practical problems can be formulated as linear or quadratic programs
and because they can be solved efficiently. Also, they are the basic building blocks
of many other optimization algorithms. This chapter presents their formulation
together with their main properties and some fundamental results.

2.1 Polyhedra and Polytopes

We first introduce a few concepts needed for the geometric interpretation of linear
an quadratic optimization. They will be discussed in more detail in Section 4.2.
A polyhedron P in Rn denotes an intersection of a finite set of closed halfspaces
in Rn :
P = {x ∈ Rn : Ax ≤ b}, (2.1)

where Ax ≤ b is the usual shorthand form for a system of inequalities, namely

a′i x ≤ bi , i = 1, . . . , m, where a′1 , . . . , a′m are the rows of A, and b1 , . . . , bm are
the components of b. A polytope is a bounded polyhedron. In Figure 2.1 a two-
dimensional polytope is plotted.
A linear inequality c′ z ≤ c0 is said to be valid for P if it is satisfied for all points
z ∈ P. A face of P is any nonempty set of the form

F = P ∩ {z ∈ Rs : c′ z = c0 }, (2.2)

where c′ z ≤ c0 is a valid inequality for P. All faces of P satisfying F ⊂ P are called

proper faces and have dimension less than dim(P). The faces of dimension 0,1,
dim(P)-2 and dim(P)-1 are called vertices, edges, ridges, and facets, respectively.
20 2 Linear and Quadratic Optimization

a3′
x≤

4
b3

≤b
a4′ x

b2
≤
a1′ x ≤

2x
a′
b1

Figure 2.1 Polytope. A polytope is a bounded polyhedron defined by the in-

tersection of closed halfspaces. The planes (here lines) defining the boundary
of the halfspaces are a′i x − bi = 0.

2.2 Linear Programming

When the cost and the constraints of the continuous optimization problem (1.4)
are affine, then the problem is called a linear program (LP). The most general form
of a linear program is
inf z c′ z
(2.3)
subj. to Gz ≤ w,
where G ∈ Rm×s , w ∈ Rm . Linear programs are convex optimization problems.
Two other common forms of linear programs include both equality and inequal-
ity constraints:
inf z c′ z
subj. to Gz ≤ w (2.4)
Az = b,
where A ∈ Rp×s , b ∈ Rp , or only equality constraints and positive variables:

inf z c′ z
subj. to Az = b (2.5)
z ≥ 0.

By standard simple manipulations [65, p. 146] it is always possible to convert one

of the three forms (2.3), (2.4) and (2.5) into the others.

2.2.1 Geometric Interpretation and Solution Properties

Let P be the feasible set (1.6) of problem (2.3). As Z = Rs , this implies that P
is a polyhedron defined by the inequality constraints in (2.3). If P is empty, then
2.2 Linear Programming 21

P
P

z∗

k1 k1
k2 k2
k3 k3
k4 k4

(a) Case 1. Solution un- (b) Case 2. Unique optimizer z ∗ .

bounded.

Z∗

k4
k3
k2
k1
(c) Case 3. Multiple optimizers Z ∗ .

Figure 2.2 Linear Program. Geometric Interpretation, level curve parame-

ters ki < ki−1 .

the problem is infeasible. We will assume for the following discussion that P is
not empty. Denote by f ∗ the optimal value and by Z ∗ the set of optimizers of
problem (2.3)
Z ∗ = argminz∈P c′ z.
Three cases can occur.

Case 1. The LP solution is unbounded, i.e., f ∗ = −∞.

Case 2. The LP solution is bounded, i.e., f ∗ > −∞ and the optimizer is unique.
z ∗ = Z ∗ is a singleton.

Case 3. The LP solution is bounded and there are multiple optima. Z ∗ is an

subset of Rs which can be bounded or unbounded.

The two-dimensional geometric interpretation of the three cases discussed above

is depicted in Figure 2.2.
22 2 Linear and Quadratic Optimization

The level curves of the cost function c′ z are represented by the parallel lines.
All points z belonging both to the line c′ z = ki and to the polyhedron P are feasible
points with an associated cost ki , with ki < ki−1 . Solving (2.3) amounts to finding
a feasible z which belongs to the level curve with the smallest cost ki . Since the
gradient of the cost is c, the direction of steepest descent is −c/kck.
Case 1 is depicted in Figure 2.2(a). The feasible set P is unbounded. One can
move in the direction of steepest descent −c and be always feasible, thus decreasing
the cost to −∞. Case 2 is depicted in Figure 2.2(b). The optimizer is unique and
it coincides with one of the vertices of the feasible polyhedron. Case 3 is depicted
in Figure 2.2(c). The whole facet of the feasible polyhedron P is optimal, i.e., the
cost for any point z belonging to the facet equals the optimal value f ∗ . In general,
the optimal facet will be a facet of the polyhedron P parallel to the hyperplane
c′ z = 0.
From the analysis above we can conclude that the optimizers of any bounded
LP always lie on the boundary of the feasible polyhedron P.

2.2.2 Dual of LP

Consider the LP (2.3)

inf z c′ z
(2.6)
subj. to Gz ≤ w,

with z ∈ Rs and G ∈ Rm×s .

The Lagrange function as defined in (1.14) is

L(z, u) = c′ z + u′ (Gz − w).

The dual cost is


−u′ w if − G′ u = c
d(u) = inf L(z, u) = inf (c′ + u′ G)z − u′ w =
z z −∞ if − G′ u 6= c.

Since we are interested only in cases where d is finite, from the relation above we
conclude that the dual problem is

supu −u′ w
subj. to −G′ u = c (2.7)
u ≥ 0,

which can be rewritten as

inf u w′ u
subj. to G′ u = −c (2.8)
u ≥ 0.

Note that for LPs feasibility implies strong duality (Remark 1.3).
 2.2 Linear Programming 23

2.2.3 KKT condition for LP

The KKT conditions (1.29a)-(1.29e) for the LP (2.3) become

G′ u + c = 0, (2.9a)
ui (Gi z − wi ) = 0, i = 1, . . . , m (2.9b)
u ≥ 0, (2.9c)
Gz − w ≤ 0. (2.9d)

They are: stationarity condition (2.9a), complementary slackness conditions (2.9b),

dual feasibility (2.9c) and primal feasibility (2.9d). Often dual feasibility in linear
programs refers to both (2.9a) and (2.9c).

2.2.4 Active Constraints and Degeneracies

Consider the LP (2.3). Let I = {1, . . . , m} be the set of constraint indices. For any
A ⊆ I, let GA and wA be the submatrices of G and w, respectively, comprising
the rows indexed by A and denote with Gj and wj the j-th row of G and w,
respectively. Let z be a feasible point and consider the set of active and inactive
constraints at z:
A(z) = {i ∈ I : Gi z = wi }
(2.10)
N A(z) = {i ∈ I : Gi z < wi }.
From (2.10) we have
GA(z∗ ) z ∗ = wA(z∗ )
(2.11)
GN A(z∗ ) z ∗ < wN A(z∗ ) .

Definition 2.1 (Linear Independence Constraint Qualification (LICQ)) We

say that LICQ holds at z ∗ if the matrix GA(z∗ ) has full row rank.

Lemma 2.1 Assume that the feasible set P of problem (2.3) is bounded. If the
LICQ is violated at z1∗ ∈ Z ∗ then there exists z2∗ ∈ Z ∗ such that |A(z2∗ )| > s.

Consider z1∗ ∈ Z ∗ on an optimal facet. Lemma 2.1 states the simple fact that if
LICQ is violated at z1∗ , then there is a vertex z2∗ ∈ Z ∗ on the same facet where
|A(z2∗ )| > s. Thus, violation of LICQ is equivalent to having more than s constraints
active at an optimal vertex.

Definition 2.2 The LP (2.3) is said to be primal degenerate if there exists a z ∗ ∈

Z ∗ such that the LICQ does not hold at z ∗ .

Figure 2.3 depicts a case of primal degeneracy with four constraints active at the
optimal vertex, i.e., more than the minimum number two.
24 2 Linear and Quadratic Optimization

3
5

P 2

z∗

1
4

Figure 2.3 Primal Degeneracy in a Linear Program.

Definition 2.3 The LP (2.3) is said to be dual degenerate if its dual problem is
primal degenerate.

The LICQ condition is invoked so that the optimization problem is well behaved
in a way we will explain next. Let us look at the equality constraints of the
dual problem (2.8) at the optimum G′ u∗ = −c or equivalently G′A u∗A = −c since
u∗N A = 0. If LICQ is satisfied then the equality constraints will allow only a unique
optimizer u∗ . If LICQ is not satisfied, then the dual may have multiple optimizers.
Thus we have the following Lemma.

Lemma 2.2 If the primal problem (2.3) is not degenerate then the dual problem
has a unique optimizer. If the dual problem (2.8) is not degenerate, then the primal
problem has a unique optimizer.

Multiple dual optimizers imply primal degeneracy and multiple primal optimizers
imply dual degeneracy but the reverse is not true as we will illustrate next. In
other words LICQ is only a sufficient condition for uniqueness of the dual.

Example 2.1 Primal and dual degeneracies

Consider the following pair of primal and dual LPs

Primal
inf [−1 − 1]x
   
0 1 1 (2.12)
 1 1   1 
 x ≤  
 −1 0   0 
0 −1 0
 2.2 Linear Programming 25

Dual
inf [1 1 0 0]u
   
0 1 −1 0 1 (2.13)
u=
1 1 0 −1 1
u≥0
Substituting for u3 and u4 from the equality constraints in (2.13) we can rewrite the
dual as
Dual  
u1
inf [1 1]
  u2  
1 0   0 (2.14)
 0 1  u1  1 
   
 1 1  u2 ≥  1 
0 1 0

The situation is portrayed in Figure 2.4.

x2 u2

2
primal solution dual solution

max

min

1 x1 u1
(a) Primal LP. (b) Dual LP.

Figure 2.4 Example 2.1. LP with Primal and Dual Degeneracy. The vectors
max and min point in the direction for which the objective improves. The
feasible sets are shaded.

Consider the two solutions for the primal LP denoted with 1 and 2 in Figure 2.4(a)
and referred to as “basic” solutions. Basic solution 1 is primal non-degenerate, since
it is defined by exactly as many active constraints as there are variables. Basic solu-
tion 2 is primal degenerate, since it is defined by three active constraints, i.e., more
than two. Any convex combination of optimal solutions 1 and 2 is also optimal. This
continuum of optimal solutions in the primal problem corresponds to a degenerate
solution in the dual space, that is, the dual problem is primal-degenerate. Hence
the primal problem is dual-degenerate. In conclusion, Figures 2.4(a) and 2.4(b) show
an example of a primal problem with multiple optima and the corresponding dual
26 2 Linear and Quadratic Optimization

problem being primal degenerate.

Next we want to show that the statement “if the dual problem is primal degenerate
then the primal problem has multiple optima” is, in general, not true. Switch dual
and primal problems, i.e., call the “dual problem” primal problem and the “primal
problem” dual problem (this can be done since the dual of the dual problem is the
primal problem). Then, we have a dual problem which is primal degenerate in solution
2 while the primal problem does not present multiple optimizers.

2.2.5 Convex Piecewise Affine Optimization

Consider a continuous and convex piecewise affine function f : R ⊆ Rs → R:

f (z) = c′i z + di if z ∈ Ri , i = 1, . . . , p, (2.15)

Sp
where {Ri }pi=1 are polyhedral sets with disjoint interiors, R = i=1 Ri is a poly-
hedron and ci ∈ Rs , di ∈ R. Then, f (z) can be rewritten as [257]

f (z) = max {c′i z + di }, z ∈ R, (2.16)

i=1,...,k

or [65]:
f (z) = minε ε
c′i z + di ≤ ε, i = 1, . . . , k (2.17)
z ∈ R.
See Figure 2.5 for an illustration of the idea.
Consider the following optimization problem

f ∗ = minz f (z)
subj. to Gz ≤ w (2.18)
z ∈ R,

where the cost function has the form (2.15). Substituting (2.17) this becomes

f∗ = minz,ε ε
subj. to Gz ≤ w
(2.19)
c′i z + di ≤ ε, i = 1, . . . , k
z ∈ R.

The previous result can be extended to the sum of continuous and convex
piecewise affine functions. Let f : R ⊆ Rs → R be defined as:
r
X
f (z) = f j (z), (2.20)
j=1

with
′
f j (z) = max {cji z + dji }, z ∈ R. (2.21)
i=1,...,kj
 2.3 Quadratic Programming 27

c′4 z + d4

c′1 z + d1 c′3 z + d3

f (z)
c′2 z + d2

R1 R2 R3 R4

Figure 2.5 Convex PWA function described as the max of affine functions.

Then the following optimization problem

f ∗ = minz f (z)
subj. to Gz ≤ w (2.22)
z ∈ R,

where the cost function has the form (2.20), can be solved by the following linear
program:
f ∗ = minz,ε1 ,...,εr ε1 + · · · + εr
subj. to Gz ≤ w
′
c1i z + d1i ≤ ε1 , i = 1, . . . , k 1
′
c2i z + d2i ≤ ε2 , i = 1, . . . , k 2 (2.23)
..
.
cri ′ z + dri ≤ εr , i = 1, . . . , k r
z ∈ R.

Remark 2.1 Note that the results of this section can be immediately applied to the
minimization of one or infinity norms. For any y ∈ R, |y| = max {y, −y}. Therefore
for any Q ∈ Rk×s and p ∈ Rk :

kQz − pk∞ = max{Q1 ′ z + p1 , −Q1 ′ z − p1 , . . . , Qk ′ z + pk , −Qk ′ z − pk },

and
k
X k
X
kQz − pk1 = |Qi ′ z + pi | = max{Qi ′ z + pi , −Qi ′ z − pi }.
i=1 i=1

2.3 Quadratic Programming

The continuous optimization problem (1.4) is called a quadratic program (QP) if the
constraint functions are affine and the cost function is a convex quadratic function.
28 2 Linear and Quadratic Optimization

z∗
z∗

P P

(a) Case 1. Optimizer z ∗ in interior of P. (b) Case 2. Optimizer z ∗ on boundary of P.

Figure 2.6 Geometric interpretation of the Quadratic Program solution.

In this book we will use the form:

1 ′ ′
minz 2 z Hz + q z + r (2.24)
subj. to Gz ≤ w,

where z ∈ Rs , H = H ′ ≻ 0 ∈ Rs×s , q ∈ Rs , G ∈ Rm×s . In (2.24) the constant

term can be omitted if one is only interested in the optimizer.
Other QP forms often include equality and inequality constraints:
1 ′
minz 2 z Hz+ q′ z + r
subj. to Gz ≤ w (2.25)
Az = b.

2.3.1 Geometric Interpretation and Solution Properties

Let P be the feasible set (1.6) of problem (2.24). As Z = Rs , this implies that P is
a polyhedron defined by the inequality constraints in (2.24). The two dimensional
geometric interpretation is depicted in Figure 2.6. The level curves of the cost
function 12 z ′ Hz+q ′ z+r are represented by the ellipsoids. All the points z belonging
both to the ellipsoid 12 z ′ Hz + q ′ z + r = ki and to the polyhedron P are feasible
points with an associated cost ki . The smaller the ellipsoid, the smaller is its cost
ki . Solving (2.24) amounts to finding a feasible z which belongs to the level curve
with the smallest cost ki . Since H is strictly positive definite, the QP (2.24) cannot
have multiple optima nor unbounded solutions. If P is not empty the optimizer is
unique. Two cases can occur if P is not empty:

Case 1. The optimizer lies strictly inside the feasible polyhedron (Figure 2.6(a)).

Case 2. The optimizer lies on the boundary of the feasible polyhedron (Fig-
ure 2.6(b)).
 2.3 Quadratic Programming 29

In case 1 the QP (2.24) is unconstrained and we can find the minimizer by

setting the gradient equal to zero
Hz ∗ + q = 0. (2.26)
Since H ≻ 0 we obtain z ∗ = −H −1 q.

2.3.2 Dual of QP
Consider the QP (2.24)
1 ′
minz 2 z Hz+ q′ z
subj. to Gz ≤ w.
The Lagrange function as defined in (1.14) is
1 ′
L(z, u) = z Hz + q ′ z + u′ (Gz − w).
2
The dual cost is
1 ′
d(u) = min z Hz + q ′ z + u′ (Gz − w) (2.27)
z 2
and the dual problem is
1
max min z ′ Hz + q ′ z + u′ (Gz − w). (2.28)
u≥0 z 2

For a given u the Lagrange function 12 z ′ Hz + q ′ z + u′ (Gz − w) is convex. Therefore

it is necessary and sufficient for optimality that the gradient is zero
Hz + q + G′ u = 0.
From the equation above we can derive z = −H −1 (q + G′ u) and, substituting this
in equation (2.27), we obtain:
1 1
d(u) = − u′ (GH −1 G′ )u − u′ (w + GH −1 q) − q ′ H −1 q. (2.29)
2 2
By using (2.29) the dual problem (2.28) can be rewritten as:
1 ′
minu 2 u (GH
−1 ′
G )u + u′ (w + GH −1 q) + 12 q ′ H −1 q
(2.30)
subj. to u ≥ 0.
Note that for convex QPs feasibility implies strong duality (Remark 1.3).

2.3.3 KKT conditions for QP

Consider the QP (2.24). Then, ∇f (z) = Hz + q, gi (z) = Gi z − wi (where Gi is the
i-th row of G), ∇gi (z) = G′i . The KKT conditions become
Hz + q + G′ u = 0 (2.31a)
ui (Gi z − wi ) = 0, i = 1, . . . , m (2.31b)
u ≥ 0 (2.31c)
Gz − w ≤ 0. (2.31d)
30 2 Linear and Quadratic Optimization

2.3.4 Active Constraints and Degeneracies

Consider the definition of active set A(z) in (2.10). Note that A(z ∗ ) may be empty
in the case of a QP. We define primal and dual degeneracy as in the LP case.

Definition 2.4 The QP (2.24) is said to be primal degenerate if there exists a

z ∗ ∈ Z ∗ such that the LICQ does not hold at z ∗ .
Note that if the QP (2.24) is not primal degenerate, then the dual QP (2.30) has
a unique solution since u∗N A = 0 and (GA H −1 G′A ) is invertible.

Definition 2.5 The QP (2.24) is said to be dual degenerate if its dual problem is
primal degenerate.
We note from (2.30) that all the constraints are independent. Therefore LICQ
always holds for dual QPs and dual degeneracy can never occur for QPs with
H ≻ 0.

2.3.5 Constrained Least-Squares Problems

The problem of minimizing the convex quadratic function

kAz − bk22 = z ′ A′ Az − 2b′ Az + b′ b (2.32)

is an (unconstrained) QP. It arises in many fields and has many names, e.g., linear
regression or least-squares approximation. From (2.26) we find the minimizer

z ∗ = (A′ A)−1 A′ b = A† b,

where A† is the generalized inverse of A. When linear inequality constraints are

added, the problem is called constrained linear regression or constrained least-
squares, and there is no longer a simple analytical solution. As an example we can
consider regression with lower and upper bounds on the variables, i.e.,

minz kAz − bk22

(2.33)
subj. to li ≤ zi ≤ ui , i = 1, . . . , n,

which is a QP. In Chapter 6.3.1 we will show how to compute an analytical solu-
tion to the constrained least-squares problem. In particular we will show how to
compute the solution z ∗ as a function of b, ui and li .

2.4 Mixed-Integer Optimization

As discussed in Section 1.1.2, if the decision set Z in the optimization problem (1.2)
is the Cartesian product of a binary set and a real Euclidian space, i.e., Z ⊆
{[zc , zb ] : zc ∈ Rsc , zb ∈ {0, 1}sb }, then the optimization problem is said to be
2.4 Mixed-Integer Optimization 31

mixed-integer. In this section Mixed Integer Linear Programming (MILP) and

Mixed Integer Quadratic Programming (MIQP) are introduced.
When the cost of the optimization problem (1.10) is quadratic and the con-
straints are affine, then the problem is called a mixed integer quadratic program
(MIQP). The most general form of an MIQP is
1 ′ ′
inf [zc ,zb ] 2 z Hz + q z +r
subj. to Gc zc + Gb zb ≤w
Ac zc + Ab zb = b (2.34)
zc ∈ Rsc , zb ∈ {0, 1}sb
z = [zc , zb ],

where H  0 ∈ Rs×s , Gc ∈ Rm×sc , Gb ∈ Rm×sb , w ∈ Rm , Ac ∈ Rp×sc , Ab ∈

Rp×sb , b ∈ Rp and s = sc + sd . Mixed integer quadratic programs are nonconvex
optimization problems, in general. When H = 0 the problem is called a mixed
integer linear program (MILP). Often the term r is omitted from the cost since
it does not affect the optimizer, but r has to be considered when computing the
optimal value. In this book we will often use the form of MIQP with inequality
constraints only
inf [zc ,zb ] 21 z ′ Hz + q ′ z + r
subj. to Gc zc + Gb zb ≤ w
(2.35)
zc ∈ Rsc , zb ∈ {0, 1}sb
z = [zc , zb ].
The general form can always be translated into the form with inequality constraints
only by standard simple manipulations.
For a fixed integer value z̄b of zb , the MIQP (2.34) becomes a quadratic program:
1 ′ ′
inf zc 2 zc Hc zc + qc (zb ) zc + k(zb )
subj. to Gc zc ≤ w − Gb z̄b
(2.36)
Ac zc = beq − Ab z̄b
zc ∈ Rsc .

Therefore the most obvious way to interpret and solve an MIQP is to enumerate
all the 2sb integer values of the variable zb and solve the corresponding QPs. By
comparing the 2sb optimal costs one can derive the optimizer and the optimal cost
of the MIQP (2.34). Although this approach is not used in practice, it gives a
simple way for proving what is stated next. Let Pz̄b be the feasible set (1.6) of
problem (2.35) for a fixed zb = z̄b . The cost is a quadratic function defined over
Rsc and Pz̄b is a polyhedron defined by the inequality constraints

Gc zc ≤ w − Gb z̄b . (2.37)

Denote by f ∗ the optimal value and by Z ∗ the set of optimizers of problem (2.34)
If Pz̄b is empty for all z̄b , then the problem (2.35) is infeasible. Five cases can occur
if Pz̄b is not empty for at last one z̄b ∈ {0, 1}sb :

Case 1. The MIQP solution is unbounded, i.e., f ∗ = −∞. This cannot happen if
Hc ≻ 0.
32 2 Linear and Quadratic Optimization

Case 2. The MIQP solution is bounded, i.e., f ∗ > −∞ and the optimizer is
unique. Z ∗ is a singleton.
Case 3. The MIQP solution is bounded and there are infinitely many optimizers
corresponding to the same integer value. Z ∗ is the Cartesian product of an
infinite dimensional subset of Rs and an integer number zb∗ . This cannot
happen if Hc ≻ 0.
Case 4. The MIQP solution is bounded and there are finitely many optimizers
corresponding to different integer values. Z ∗ is a finite set of optimizers
∗ ∗ ∗ ∗
{(z1,c , z1,b ), . . . , (zN,c , zN,b )}.
Case 5. The union of Case 3 and Case 4.
 Chapter 3

Numerical Methods for

Optimization

Contributed by Dr. Alexander Domahidi and Dr. Stefan Richter

ETH Zurich
alex.domahidi@gmail.com, stefan.richter@alumni.ethz.ch

There is a great variety of algorithms for the solution of unconstrained and

constrained optimization problems. In this chapter we introduce those algorithms,
which are important for the problems encountered in this book and explain the
underlying concepts. In particular, we present numerical methods for finding a
minimizer z ∗ to the optimization problem

minz f (z)
(3.1)
subj. to z ∈ S,

where both the objective function f : Rs → R and the feasible set S are convex,
hence (3.1) is a convex program. We will assume that if the problem is feasible, an
optimizer z ∗ ∈ S exists, i.e., f ∗ = f (z ∗ ).

3.1 Convergence
Many model predictive control problems for linear systems have the form of prob-
lem (3.1) with a linear or convex quadratic objective function and a polyhedral
feasible set S defined by linear equalities and inequalities. In all but the simplest
cases, analytically finding a minimizer z ∗ is impossible. Hence, one has to resort
to numerical methods, which compute an approximate minimizer that is ‘good
enough’ (we give a precise definition of this term below). Any such method pro-
ceeds in an iterative manner, i.e., starting from an initial guess z 0 , it computes a
34 3 Numerical Methods for Optimization

sequence {z k }k=k
k=1
max
, where


z k+1 = Ψ z k , f, S ,

with Ψ being an update rule depending on the method. For a method to be useful,
it should terminate after kmax iterations and return an approximate minimizer
z kmax that satisfies


|f z kmax − f (z ∗ ) | ≤ ǫ and dist(z kmax , S) ≤ δ,

where
dist(z, S) = min ky − zk
y∈S

is the shortest distance between a point and a set in Rs measured by the norm k · k.
The parameters ǫ, δ > 0 define the required accuracy of the approximate minimizer.
There exist mainly two classes of optimization methods for solving (3.1): first-
order methods that make use of first-order information of the objective function
(subgradients or gradients) and second-order methods using in addition second-
order information (Hessians). Note that either method may also use zero-order
information, i.e., the objective value itself, for instance, to compute step sizes.
In the following list, key aspects that are important for any optimization method
are summarized. Some of these aspects are discussed in detail for the corresponding
optimization methods later in this chapter.

• Convergence: Is kmax finite for all accuracies ǫ, δ > 0?

• Convergence rate: How do errors |f z k − f (z ∗ ) | and dist(z k , S) depend on
the iteration counter k?

• Feasibility: Does the method produce feasible iterates, i.e., dist(z k , S) = 0

for all iterations k?

• Numerical robustness: Do the properties above change in presence of finite

precision arithmetics when using fixed or floating point number representa-
tions?

• Warm-starting: Can the method take advantage of z 0 being close to a mini-

mizer z ∗ ?

• Preconditioning: Is it possible to transform (3.1) into an equivalent problem

that can be solved in fewer iterations?

• Computational complexity: How many arithmetic operations are needed for

computing the approximate solution z kmax ?

The speed or rate of convergence relates to the second point in the list above
and is an important distinctive feature between various methods. We present a
definition of convergence rates next.
 3.2 Unconstrained Optimization 35

Definition 3.1 (Convergence Rates (cf. [223, Section 2.2])) Let {ek } be a
sequence of positive reals that converges to 0. The convergence rate of this sequence
is called linear if there exists a constant q ∈ (0, 1) such that

ek+1
lim sup ≤ q.
k→∞ ek
If q = 1, the convergence rate is said to be sublinear, whereas if q = 0, the sequence
has a superlinear convergence rate. A superlinearly converging sequence converges
with order p if
ek+1
lim sup p ≤ C,
k→∞ ek
where C is a positive constant, not necessarily less than 1. In particular, superlinear
convergence with order p = 2 is called quadratic convergence.
For an optimization method, the sequence {ek } in the definition above can be
any sequence of errors associated with the particular method, e.g. ek can be defined
as the absolute error in the objective value


ek = |f z k − f (z ∗ ) |

or any other nonnegative error measure that vanishes only at a minimizer z ∗ .

In the previous classification of convergence rates, sublinear convergence is even-
tually slower than linear convergence which itself is eventually slower than super-
linear convergence. Within superlinear convergence rates, the ones with higher
order converge faster eventually. However, for a fixed number of iterations, the ac-
tual ordering of convergence rates also depends on constants q and C, for instance,
a linearly converging sequence with a small value of q might have a smaller error
after 10 steps than a quadratically converging sequence.
In the remainder of this chapter we first introduce methods that solve the
unconstrained problem, i.e., S = Rs , in Section 3.2. They are the prerequisite
for understanding constrained optimization methods, i.e., S ⊂ Rs , which are then
presented in Section 3.3.

3.2 Unconstrained Optimization

In unconstrained smooth optimization we are interested in solving the problem

min f (z), (3.2)

z

where f : Rs → R is convex and continuously differentiable.

For the purpose of this section, we focus on so-called descent methods for un-
constrained optimization. These methods obtain the next iterate z k+1 from the
current iterate z k by taking a step of size hk > 0 along a certain direction dk :

z k+1 = z k + hk dk (3.3)
36 3 Numerical Methods for Optimization

Particular methods differ in the way dk and hk are chosen. In general, dk has to
be a descent direction, i.e., for all z k 6= z ∗ it must satisfy
∇f (z k )′ dk < 0. (3.4)
The step sizes hk have to be chosen carefully in order to ensure a sufficiently large
decrease in the function value, which is crucial for convergence of descent methods.
Since solving for the optimal step size,
hk,∗ ∈ arg min f (z k + hdk ),
h≥0

is generally too expensive, inexact line search methods are used instead. They
compute hk cheaply and achieve a reasonably good decrement in the function
value ensuring convergence of the algorithm. We discuss line search methods in
Section 3.2.3.
In the following, we present two of the most important instances of descent
methods: the classic gradient method in Section 3.2.1 and Newton’s method in
Section 3.2.2. We include Nesterov’s fast gradient method in the discussion along
with gradient methods, although it is not a descent method. We shall see that
gradient methods, under certain assumptions on the gradient of f , allow for a
constant step size, which results in an extremely simple and efficient algorithm in
practice. Newton’s method, on the contrary, requires a line search to ensure global
convergence, but it has a better (local) convergence rate than gradient methods.

3.2.1 Gradient Methods

Classic Gradient Method

One property of the gradient of f evaluated at z k , ∇f (z k ), is that it points into
the direction of steepest local ascent. The main idea of gradient methods for
minimization is therefore to use the anti-gradient direction as a descent direction,
dk = −∇f (z k ),
trivially satisfying (3.4). Before turning to the question of appropriate step sizes hk ,
we introduce an additional assumption on the gradient of f . For the remainder of
this section, we assume that f has a Lipschitz continuous gradient, i.e., that f is
so-called L-smooth. All of the upcoming results regarding Lipschitz continuity of
the gradient are taken from [220, Section 1.2.2].

Definition 3.2 (L-Smoothness of a Function) Let f : Rs → R be once conti-

nuously differentiable on Rs . The gradient ∇f is Lipschitz continuous on Rs with
Lipschitz constant L > 0 if for all pairs (z, y) ∈ Rs × Rs
k∇f (z) − ∇f (y)k ≤ L kz − yk .
If function f is twice continuously differentiable, L-smoothness can be charac-
terized using the largest eigenvalue of the Hessian of f . This alternative charac-
terization can be useful in order to compute the actual value of the Lipschitz
constant.
3.2 Unconstrained Optimization 37

f (z)
f (y) + ∇f (y)′ (z − y) + L
2
kz − yk2
2

1 f (y)

f (y) + ∇f (y)′ (z − y) + µ2 kz − yk2

0
-1 -0.5 0 0.25 0.5 1
z

Figure 3.1 Upper and lower bounds on functions: The quadratic upper
bound stems from L-smoothness of f (Definition 3.2), while the quadratic
lower bound is obtained from strong convexity of f (Theorem 3.1). In this
example, f (z) = z 2 + 12 z + 1 and we have chosen L = 3 and µ = 1 for
illustration.

Lemma 3.1 (L-Smoothness: Second-Order Characterization) Let function

f : Rs → R be twice continuously differentiable on Rs . Then the gradient ∇f is
Lipschitz continuous on Rs with Lipschitz constant L > 0 if and only if for all
z ∈ Rs
2
∇ f (z) ≤ L.

The most important consequence of Lipschitz continuity of the gradient is that

f can be globally upper-bounded by a quadratic. This is expressed in the next
lemma and illustrated in Figure 3.1.

Algorithm 3.1 Gradient method for smooth convex optimization

Input: Initial iterate z 0 ∈ Rs , Lipschitz constant L of ∇f
Output: Point close to z ∗
Repeat
z k+1 ← z k − L1 ∇f (z k )
Until stopping criterion is satisfied

Lemma 3.2 (Descent Lemma) Let f be L-smooth on Rs . Then for any pair
(z, y) ∈ Rs × Rs we have

L 2
f (z) ≤ f (y) + ∇f (y)′ (z − y) + kz − yk . (3.5)
2
38 3 Numerical Methods for Optimization

We now return to the choice of step sizes hk in (3.3). Notice that interpreting the
next iterate z k+1 as the result of minimizing a quadratic at the previous iterate z k ,

z k+1 = arg min f¯(z, z k ), (3.6)

z

where we define
1
f¯(z, z k ) = f (z k ) + ∇f (z k )′ (z − z k ) + k kz − z k k2 , (3.7)
2h
yields after some standard calculus the classic gradient update rule

z k+1 = z k − hk ∇f (z k ).

The quadratic function f¯(z, z k ) in (3.7) is parametrized by the step length hk .

Choosing it as
1
hk = ,
L
the quadratic f¯(z, z k ) becomes an upper bound on f (Lemma 3.2). Convergence
of the gradient method can then be shown using the relation
(1) (2)
f (z k+1 ) ≤ f¯(z k+1 , z k ) ≤ f¯(z k , z k ) = f (z k ), (3.8)

where the first and second inequality are due to (3.5) and (3.6) respectively. In
fact, the second inequality is strict for z k 6= z ∗ .
The gradient method with constant step size summarized in Algorithm 3.1 can
be shown to converge sublinearly with the number of iterations growing propor-
tionally to L/ǫ · kz 0 − z ∗ k2 [220, Corollary 2.1.2]. The required level of suboptimal-


ity ǫ > 0 is hereby defined in terms of the error in the objective value f z k − f ∗ .
Note that in case of a constant step size, the gradient method is an extremely sim-
ple, division-free algorithm (given that the gradient ∇f can be computed without
divisions).

Remark 3.1 If the Lipschitz constant L is unknown, a simple strategy is to start

with an initial guess L̃ > 0, solve for z k+1 according to (3.6) with hk = 1/L̃ and then
check whether the first inequality in (3.8) holds true. If so, the quadratic objective
in (3.6) was a (local) upper bound of f and minimization consequently leads to a
decrease in the function value, i.e., f (z k+1 ) < f (z k ). Otherwise, the current guess
for the Lipschitz constant needs to be increased, e.g. L̃ = 2L̃, and the procedure
repeated. Termination is guaranteed because f was assumed L-smooth.

Algorithm 3.2 Fast gradient method for smooth convex optimization

√
Input: Initial iterates z 0 ∈ Rs , y 0 = z 0 ; α0 = 12 ( 5 − 1), Lipschitz constant L of ∇f
Output: Point close to z ∗
Repeat
z k+1 ← y k − L1 ∇f (y k )
k
p 
αk+1 ← α2 αk 2 + 4 − αk
3.2 Unconstrained Optimization 39

αk (1−αk )
βk ← αk 2 +αk+1
y k+1 ← z k+1
+ β k (z k+1 − z k )
Until stopping criterion is satisfied

Fast Gradient Method

An accelerated version of the gradient method, Nesterov’s fast gradient method
[220, Section 2.2], is given in Algorithm 3.2. The basic idea behind this method is
to take the gradient at an affine combination of the previous two iterates, i.e., the
gradient is evaluated at

y k = z k + β k−1 (z k − z k−1 )

for some carefully chosen β k−1 ∈ (0, 1). This two-step method can be shown
to
p converge sublinearly with the number of iterations growing proportionally to
L/ǫ · kz 0 − z ∗ k [220, Theorem 2.2.2]. This is the best convergence rate for
L-smooth problems using only first-order information (see [220, Section 2.1] for
details). The mathematical reasoning behind the improved convergence rate and
optimality of the fast gradient method amongst all first-order methods is quite
involved and beyond the scope of this book.

Example 3.1 Gradient method vs. fast gradient method for quadratic functions
Consider the optimization problem given by
1
min z ′ Hz + q ′ z,
2
where the Hessian and the linear cost vector are given by
   
0.2 0.2 −0.1
H= and q = .
0.2 2 0.1

For quadratic functions, a tight Lipschitz constant L of the gradient can be deter-
mined by computing the largest eigenvalue of the Hessian (Lemma 3.1), i.e.

L = λmax (H), (3.9)

here L = 2.022. The upper left plot in Figure 3.2 shows the first 10 iterations of
the gradient method with constant step size 1/L (Algorithm 3.1). The upper right
and lower left plot show the first 10 iterations using larger (but constant) step sizes,
which correspond to underestimating the tight Lipschitz constant L. It can be shown
that if L is underestimated by a factor of 2 or more, convergence is lost (cf. lower left
plot). The lower right plot shows the iterates of the fast gradient method using the
tight Lipschitz constant (Algorithm 3.2). Contrary to the gradient method, the fast
gradient method generates a non-monotonically decreasing error sequence.

Strongly Convex Problems

In the case where the objective function f in (3.2) is also strongly convex, linear
convergence rates can be established for both the gradient method and the fast
40 3 Numerical Methods for Optimization

3 3
2 0 2 0
1 123 1 2
z2
0 0 1 3

z2
-1 -1
-2 -2
-3 -3
-5 0 5 -5 0 5
z1 z1
1 4
(a) GM. Step size L
. (b) GM. Step size 3L
.

3 3
2 0 2 2 0
1 1 1 3 2
0 0
z2

z2
-1 1 3 -1
-2 -2
-3 -3
-5 0 5 -5 0 5
z1 z1
2 1
(c) GM. Step size L
. (d) FGM. Step size L
.

Figure 3.2 Example 3.1. First 10 iterations of the gradient method (GM)
with constant step size 1/L using the tight, a slightly underestimated and a
strongly underestimated Lipschitz constant L. Also shown are the iterates
of the fast gradient method (FGM) using the tight Lipschitz constant. The
contour lines indicate the function value of each iterate. Note that the
gradients are orthogonal to the contour lines.

gradient method. The following theorems from [151, Section B] characterize strong
convexity whenever the function is once or twice continuously differentiable. For
an illustration of this property see Figure 3.1.

Theorem 3.1 (Strong Convexity: First-Order Characterization) Let func-

tion f : Rs → R be once continuously differentiable on Rs . Then f is strongly con-
vex on Rs with convexity parameter µ > 0 if and only if for all pairs (z, y) ∈ Rs ×Rs
µ 2
f (z) ≥ f (y) + ∇f (y)′ (z − y) + kz − yk .
2

Theorem 3.2 (Strong Convexity: Second-Order Characterization) Let f :

Rs → R be twice continuously differentiable on Rs . Then f is strongly convex on
Rs with parameter µ > 0 if and only if for all z ∈ Rs

∇2 f (z)  µIs .

Algorithm 3.3 Fast gradient method for smooth strongly convex optimization
3.2 Unconstrained Optimization 41

p
Input: Initial iterates z 0 ∈ Rs , y 0 = z 0 ; 0 < µ/L ≤ α0 < 1, Lipschitz constant L
of ∇f , strong convexity parameter µ of f
Output: Point close to z ∗
Repeat
z k+1 ← y k − 1
L
∇f (y k )
k+1 2 2 µαk+1
Compute α ∈ (0, 1): αk+1 = (1 − αk+1 )αk + L
k k
α (1−α )
βk ← αk 2 +αk+1
y k+1 ← z k+1
+ β k (z k+1 − z k )
Until stopping criterion is satisfied

From Theorem 3.1 it follows that a differentiable strongly convex function can
be lower-bounded by a quadratic. Relating this to the quadratic upper bound in
Lemma 3.2, the inequality
L
κf = ≥1
µ
follows intuitively. The constant κf is the so-called condition number of the L-
smooth and strongly convex function f .
With the additional assumption of strong convexity, it can be shown that the
gradient method in Algorithm 3.1 converges linearly with the number of iterations
growing proportionally to κf · ln(1/ǫ) [220, Theorem 2.1.15] with ǫ > 0 being the
required level of suboptimality in terms of the error in the objective. In order
for the fast gradient method in Algorithm 3.2 to achieve linear convergence rate,
it either needs to be modified or restarted after a number of steps that depends
on the condition number κf . In both cases it can be shown that the number
√
of iterations grows proportionally to κf · ln(1/ǫ) [220, Theorem 2.2.2]. This is
the best convergence rate for L-smooth and strongly convex problems using only
first-order information (see [220, Section 2.1] for details). A modified fast gradient
method that takes strong convexity into account explicitly and thus does not need
restarting is given in Algorithm 3.3.

Example 3.2 Strong convexity parameter of a quadratic function

Let f : Rs → R be a quadratic function defined as
1 ′
f (z) = z Hz + q ′ z
2
with H ≻ 0. Then, by Theorem 3.2, the (tight) parameter for strong convexity µ is
the smallest eigenvalue of the Hessian H, i.e.

µ = λmin (H).

For the Hessian given in Example 3.1 we therefore have µ = 0.178, and the condition
number is κf = 11.36. From the convergence results for the gradient method and the
fast gradient method for the strongly convex case it follows that condition numbers
close to 1 lead to fast convergence. So, for this example, fast convergence can be
guaranteed for both methods.
42 3 Numerical Methods for Optimization

Preconditioning
In the previous section we have seen that for gradient methods the number of ite-
rations to find an approximate minimizer depends on the condition number κf .
In practice, a high condition number usually indicates that many iterations of Al-
gorithm 3.1 and Algorithm 3.3 are needed to achieve the specified accuracy. To
improve the conditioning of the problem and thus to lower the amount of computa-
tion needed to obtain an approximate minimizer, one can transform problem (3.2)
into an equivalent problem by a variable transformation

y = Pz

with P invertible. The aim is to choose the preconditioner P such that the new
objective function h(y) = f (P −1 y) has a smaller condition number, i.e., κh < κf .
Intuitively, the best choice for P is one that achieves a circular shape of the level
sets of the objective function or at least comes close to it. Mathematically speak-
ing, the preconditioner matrix P should be such that κh ≈ 1. Finding optimal
preconditioners for which κh = 1 is impossible for all but the simplest cases. We
give one such example in the following.

Example 3.3 Optimal preconditioner for a quadratic function

Let f (z) = 12 z ′ Hz + q ′ z with Hessian H ≻ 0. It is easily verified that P = H 1/2 is
1
the optimal preconditioner for which κh = 1, since h(y) = 12 y ′ y + q ′ H − 2 y. However,
computing H 1/2 and its inverse is more expensive than minimizing f , which only
requires the inverse H −1 . So nothing has been gained from preconditioning in this
example.

Stopping Criteria
Stopping criteria test whether the current iterate z k is optimal or, more practically,
has attained an acceptable level of suboptimality. In optimization, the most natural
stopping criterion is

f (z k ) − f ∗ ≤ ǫ, (3.10)

where ǫ > 0 is an accuracy measure specified by the user. However, since the
optimal value f ∗ is rarely known, we look for a computable lower bound f k,∗
(≤ f ∗ ), so that we arrive at the evaluable sufficient condition for (3.10)

f (z k ) − f k,∗ ≤ ǫ.

The latter stopping criterion works for any ǫ > 0 if the lower bounds f k,∗ converge
to the optimal value f ∗ for an increasing iteration counter k.
For differentiable convex functions such lower bounds can be constructed right
from the definition of convexity, i.e.

f ∗ ≥ f (z k ) + ∇f (z k )′ (z ∗ − z k )
≥ f (z k ) − k∇f (z k )kkz ∗ − z k k,
 3.2 Unconstrained Optimization 43

where the last inequality is due to Cauchy-Schwarz. Observe that for any conver-
ging method, this lower bound converges to f ∗ from below since the gradient at
the minimizer z ∗ vanishes in unconstrained optimization.
For many problems, the distance between an iterate and the minimizer can be
upper-bounded, i.e., kz ∗ − z k k ≤ R for all iterations k. Consequently, we end up
with the evaluable stopping criterion

k∇f (z k )k R ≤ ǫ,

which is sufficient for (3.10).

More sophisticated stopping criteria can be derived if properties such as strong
convexity of function f are exploited. For more details, we refer the interested
reader to [250, Section 6.3].

Summary
In this section, we have discussed the concept of descent methods and presented
one of the most important instances, the classic gradient method for smooth con-
vex optimization. Although not a descent method in its basic form, Nesterov’s fast
gradient method was included due to its simplicity and improved speed of conver-
gence. Both methods are particularly useful in the strongly convex case, where
they converge linearly, with the fast gradient method attaining the best theoretical
convergence rate possible for first-order methods. Despite a slower theoretical (and
most of the time also practical) convergence, the classic gradient method is well-
suited for processors with fixed-point arithmetic, for instance field-programmable
gate arrays (FPGAs), because of its numerical robustness against inaccurate com-
putations. For the fast gradient method, recent research, e.g. [97], indicates that
care has to be taken if inexact gradients are used. When using gradient methods
in practice, preconditioning should be applied to decrease the condition number of
the problem in order to decrease the number of iterations needed. The latter can
also be achieved by warm-starting, i.e., starting the methods from a point z 0 close
to a minimizer z ∗ . This follows immediately from the dependence of the number of
iterations on the distance kz 0 − z ∗ k. More information and links to the literature
can be found in [250].

3.2.2 Newton’s Method

The (fast) gradient method uses only first-order information of the function f . A
much more quickly converging algorithm, at least locally, can be obtained when
using second-order information, i.e., the Hessian ∇2 f . In the following, we as-
sume f to be µ-strongly convex; hence the Hessian ∇2 f is positive definite on Rs .
Furthermore we assume that the Hessian ∇2 f is Lipschitz continuous on Rs with
Lipschitz constant M, i.e.

k∇2 f (z) − ∇2 f (y)k ≤ M kz − yk ∀z, y ∈ Rs . (3.11)

Intuitively, the constant M measures how well f can be approximated by a quadratic

function (M = 0 for a quadratic function f ).
44 3 Numerical Methods for Optimization

Algorithm 3.4 Newton’s method for (3.2) (strongly convex case)

Input: Initial iterate z 0 ∈ Rs
Output: Point close to z ∗
Repeat
 −1
Newton direction: dk ← − ∇2 f (z k ) ∇f (z k )
Line search: find h > 0 such that f (z + hk dk ) < f (z k )
k k

z k+1 ← z k + hk dk
Until stopping criterion is satisfied

The main idea of Newton’s method is to approximate the function f by a local

quadratic model θ(d) based on a second-order Taylor expansion,
1
f (z k + d) ≈ f (z k ) + ∇f (z k )′ d + d′ ∇2 f (z k )d = θ(d), (3.12)
2
where ∇2 f (z k ) is the Hessian of f evaluated at the current iterate z k and d is a
search direction. The latter is chosen to minimize the quadratic model θ(d). From
the optimality condition ∇d θ(d) = 0, the so-called Newton direction
 −1
dN (z k ) = − ∇2 f (z k ) ∇f (z k ) (3.13)

is obtained. The Newton direction is a descent direction, as it satisfies (3.4):

∇f (z k )′ dN (z k ) = −∇f (z k )′ [∇2 f (z k )]−1 ∇f (z k ) < 0 (3.14)

 −1
for all z k 6= z ∗ since, by strong convexity, the inverse of the Hessian ∇2 f (z)
exists for all z ∈ Rs and is positive definite. If the Lipschitz constant of the Hessian,
M , is small, this method works very well, since the quadratic model remains a good
approximation even if a full Newton step is taken. However, the quadratic model
θ(d) is in general neither an upper nor a lower bound on f , and consequently a line
search (Section 3.2.3) must be used to ensure a sufficient decrease in the function
value, which results in global convergence of the algorithm. Newton’s method is
summarized in Algorithm 3.4. It can be shown to converge globally in two phases
(for technical details see [65, Section 9.5] and [220, Section 1.2.4]):
1. Damped Newton phase: When the iterates are “far away” from the optimal
point, the function value is decreased sublinearly, i.e., it can be shown that
there exist constants η, γ (with 0 < η < µ2 /M and γ > 0) such that f (z k+1 )−
f (z k ) ≤ −γ for all z k for which k∇f (z k )k2 ≥ η. The term damped is used in
this context since during this phase, the line search often returns step sizes
hk < 1, i.e., no full Newton steps are taken.
2. Quadratic convergence phase: If an iterate is “sufficiently close” to the opti-
mum, i.e., k∇f (z k )k2 < η, a full Newton step is taken, i.e., the line search
returns hk = 1. The set {z | k∇f (z)k2 < η} is called the quadratic con-
vergence zone, and once an iterate z k is in this zone, all subsequent iterates
remain in it. The function value f (z k ) converges quadratically (i.e., with the
iteration number growing proportionally to kmax ∼ O (ln ln 1/ǫ)) to f ∗ .
3.2 Unconstrained Optimization 45

The quadratic convergence property makes Newton’s method one of the most
powerful algorithms for unconstrained optimization of twice continuously differen-
tiable functions. (The method is also widely employed for solving nonlinear equa-
tions.) As a result, it is used in interior point methods for constrained optimization
to solve the (unconstrained or equality constrained) subproblems in barrier meth-
ods or to generate primal-dual search directions. We discuss these in detail in
Section 3.3.2.
A disadvantage of Newton’s method is the need to form the Hessian ∇2 f (z k ),
which can be numerically ill-conditioned, and to solve the linear system
∇2 f (z k )dN = −∇f (z k ) (3.15)
for the Newton direction dN , which costs O(s3 ) floating point operations in general.
Each iteration is therefore significantly more expensive than one iteration of any
of the gradient methods.

Preconditioning and Affine Invariance

Newton’s method is affine invariant, i.e., a linear (or affine) transformation of
variables as discussed in Section 3.2.1 will lead to the same iterates subject to
transformation, i.e.
z k = P y k ∀k = 0, . . . , kmax . (3.16)
Therefore, the number of steps to reach a desired accuracy of the solution does not
change. Consequently, preconditioning has no effect when using Newton’s method.

Newton’s Method with Linear Equality Constraints

Equality constraints are naturally handled in Newton’s method as follows. Assume
we want to solve the problem
minimize f (z)
(3.17)
subject to Az = b

with A ∈ Rp×s (assuming full rank p) and b ∈ Rp . As discussed above, the Newton
direction is given by the minimization of the quadratic model θ(d) at the current
feasible iterate z k (i.e., Az k = b) with an additional equality constraint:
1 ′ 2
dN = arg min d ∇ f (z k )d + ∇f (z k )′ d + f (z k )
d 2 (3.18)
subj. to Ad = 0
Since d is restricted to lie in the nullspace of A, we ensure that any iterate gener-
ated by Newton’s method will satisfy Az k+1 = A(z k + hk dN (z k )) = b. From the
optimality conditions of (3.18), it follows that dN is obtained from the solution to
the linear system of dimension s + p:
 2    
∇ f (z) A′ d −∇f (z)
= , (3.19)
A 0 y 0
where y ∈ Rp are the Lagrange multipliers associated with the equality constraint
Ad = 0.
46 3 Numerical Methods for Optimization

3.2.3 Line Search Methods

A line search determines the step length hk that is taken from the current iterate
z k ∈ Rs along a search direction ∆z k ∈ Rs to the next iterate,

z k+1 = z k + hk ∆z k . (3.20)

In the most general setting, a merit function ϕ : Rs → R is employed to measure

the quality of the step taken to determine a value of the step size hk that ensures
convergence of the optimization algorithm. Merit functions can, for example, be
the objective function f , or the duality measure µ (3.44) in primal-dual interior
point methods.
The general line search problem can be written as a one dimensional optimiza-
tion problem over the step size h,

h∗ ∈ arg min ϕ(z k + h∆z k ) , (3.21)

h≥0

minimizing the merit function along the ray {z k + h∆z k } for h ≥ 0. It is often too
expensive to solve this problem exactly, and therefore usually inexact line search
methods are employed in practice that return a step size hk that approximates
the optimal step size h∗ from (3.21) sufficiently well. In the following, we give a
brief overview on practical inexact line search methods; see [224] for a thorough
treatment of the subject.

Wolfe Conditions
In order to ensure convergence of descent methods for unconstrained optimization,
for example Newton’s method presented in Section 3.2, certain conditions on the
selected step size hk must be fulfilled. Popular conditions for inexact line searches
are the so-called Wolfe conditions [223]

f (z k + hk ∆z k ) ≤ f (z k ) + c1 hk ∇f (z k )′ ∆z k (3.22a)
k k k ′ k k ′ k
∇f (z + h ∆z ) ∆z ≥ c2 ∇f (z ) ∆z (3.22b)

with 0 < c1 < c2 < 1. The first condition (3.22a) is called Armijo’s condition [9],
and ensures that a sufficient decrease in the objective is achieved. The second
condition, the curvature condition (3.22b), ensures that the step length is not too
small [223].
One of the most popular inexact line search methods that fulfills the Wolfe
conditions is the backtracking line search as discussed in the following.

Backtracking Line Search for Newton’s Method

The backtracking line search for Newton’s method is given in Algorithm 3.5. It
starts with a unit step size, which is iteratively reduced by a factor t ∈ (0, 1) until
Armijo’s condition (3.22a) is satisfied. It can be shown that it is not necessary to
check the curvature condition because it is implied by the backtracking rule [223,
Algorithm 3.1].
 3.3 Constrained Optimization 47

Algorithm 3.5 Backtracking Line Search for Newton’s Method (Algorithm 3.4) [223,
Algorithm 3.1]
Input: Current iterate z k , descent direction ∆z k , constants c ∈ (0, 1) and t ∈ (0, 1)
Output: Step size hk ensuring Wolfe conditions (3.22)
hk ← 1
Repeat
hk ← thk
Until Armijo’s condition (3.22a) holds, i.e., f (z k +hk ∆z k ) ≤ f (z k )+chk ∇f (z k )′ ∆z k

Backtracking Line Search for Constrained Optimization

In order to obtain a line search method that ensures in addition feasibility of the
next iterate z k+1 , the backtracking procedure is preceeded by finding 0 < h̄ ≤ 1
such that z k + h̄∆z k is feasible [65, pp. 465]. The backtracking then starts with
hk = h̄ in Algorithm 3.5.

3.3 Constrained Optimization

In this section we present three methods to solve the constrained optimization
problem (1.1): gradient projection, interior point and active set methods.

3.3.1 Gradient Projection Methods

Gradient methods presented in Section 3.2.1 for unconstrained L-smooth convex
optimization (or strongly convex optimization with parameter µ) have a natural
extension to the constrained problem
minz f (z)
(3.23)
subj. to z∈S
where S is a convex subset of Rs . The treatment of constraints within first-order
methods is best explained in terms of the classic gradient method in Algorithm 3.1;
the accelerated versions (Algorithm 3.2 and Algorithm 3.3) can be adapted accord-
ing to the same principle.
As in the unconstrained case, we begin by rewriting the gradient update rule
as a minimization of a quadratic upper bound of f defined in (3.7) with hk = 1/L,
but restrict the minimization to the set S:
L
z k+1 = arg min f (z k ) + ∇f (z k )′ (z − z k ) + kz − z k k2 . (3.24)
z∈S 2
The solution to this constrained minimization problem can be recast as
1
z k+1 = πS (z k − ∇f (z k )), (3.25)
L
48 3 Numerical Methods for Optimization

where πS denotes the projection operator for set S defined as

1
πS (z) = arg min ky − zk2 . (3.26)
y∈S 2

Algorithm 3.6 Gradient method for constrained smooth convex optimization

Input: Initial iterate z 0 ∈ S, Lipschitz constant L of ∇f
Output: Point close to z ∗
Repeat


z k+1 ← πS z k − L1 ∇f (z k )
Until stopping criterion is satisfied

Algorithm 3.7 Fast gradient method for constrained smooth strongly convex opti-
mization p
Input: Initial iterates z 0 ∈ S, y 0 = z 0 ; 0 < µ/L ≤ α0 < 1, Lipschitz constant L
of ∇f , strong convexity parameter µ of f
Output: Point close to z ∗
Repeat


z k+1 ← πS y k − 1
L
∇f (y k )
2 2 µαk+1
Compute αk+1 ∈ (0, 1): αk+1 = (1 − αk+1 )αk + L
k k
α (1−α )
βk ← αk 2 +αk+1
y k+1 ← z k+1
+ β k (z k+1 − z k )
Until stopping criterion is satisfied

The expressions (3.24) and (3.25) for z k+1 can be shown to be equivalent by
deriving the optimality conditions and noting that they coincide. Thus, for con-
strained optimization, Algorithms 3.1, 3.2 and 3.3 remain unchanged, except that
the right hand side of the assignment in the first line of these algorithms is pro-
jected onto set S. The resulting algorithmic schemes are stated in Algorithm 3.6
(gradient method) and Algorithm 3.7 (fast gradient method for the strongly convex
case) and illustrated in Figure 3.3. It can be shown that for both Algorithm 3.6
and Algorithm 3.7 the same convergence results hold as in the unconstrained case
([220, Section 2.2.4]).
Whenever the projection operator for set S can be evaluated efficiently, as for
the convex sets listed in Table 3.1, gradient projection methods have been shown
to work very well in practice. Such sets, which are also referred to as ‘simple’
sets in the literature, can be translated, uniformly scaled and/or rotated without
complicating the projection as stated in the next lemma.

Lemma 3.3 (Projection on Translated, Scaled and Rotated Convex Set)

Let S ⊆ Rs be a closed convex set and πS its associated projection operator. Pro-
jection of point z̄ ∈ Rs on the set

Sb = {z ∈ Rs | z = γW y + c, y ∈ S}
3.3 Constrained Optimization 49

1
zk − k
L ∇f (z ) 1
yk − k
L ∇f (y )

yk
z k+1 z k+1

zk zk

S z k−1 S

(a) Gradient method. (b) Fast gradient method.

Figure 3.3 Classic gradient method (left) and fast gradient method (right)
for the constrained case. After computing the standard update rule from
the unconstrained case, a projection onto the feasible set is applied.

that is obtained from set S by translation with offset c ∈ Rs , positive scaling by

γ > 0 and a linear transformation with an orthonormal matrix W ∈ Rs×s , can be
accomplished by


πSb (z̄) = γW πS γ −1 W ′ (z̄ − c) + c .

If the feasible set S is polyhedral and does not have a computationally efficient
projection operator, it can alternatively be pre-computed by means of multipara-
metric programming (see Chapter 6) with the point to project as the parameter.
Another possibility is to solve the original problem in the dual domain instead as
discussed next.

Solution in Dual Domain

If set S is such that the projection (3.26) cannot be carried out efficiently, one can
still use first-order methods to solve the constrained problem (3.23) by adopting a
dual approach. In the following, we assume that set S can be written as

S = {z ∈ Rs | Az = b, z ∈ K} , (3.27)

where K is a convex subset of Rs that allows for an efficient projection and the
pair (A, b) is from Rp×s × Rp . Many practically relevant sets can be represented in
terms of (3.27), i.e., as the intersection of a ‘simple’ convex set and an affine set,
e.g. polyhedra. We can now rewrite the original problem (3.23) as

min f (z)
z
subj. to Az = b
z ∈ K.
50 3 Numerical Methods for Optimization

Table 3.1 Euclidean norm projection operators of selected convex sets in Rs .

Set Definition Projection Operator

affine set (
S = {z ∈ Rs | Az = b} z + A′ (AA′ )−1 (b − Az) if rank A = p,
πS (z) = ′†

z + A′ A A† b − z otherwise
with (A, b) ∈ Rp×s × Rp

(
nonnegative orthant zi if zi ≥ 0,
(πS (z))i = i = 1, . . . , s
S = {z ∈ Rs | z ≥ 0} 0 otherwise


rectangle 
li if zi < li ,
S = {z ∈ Rs | l ≤ z ≤ u} (πS (z))i = zi if li ≤ zi ≤ ui , i = 1, . . . , s


with (l, u) ∈ Rs × Rs ui if zi > ui
(
2-norm ball z
r kzk if kzk > r,
s πS (z) =
S = {z ∈ R | kzk ≤ r} , r ≥ 0 z otherwise

Using a technique called partial Lagrange relaxation [52, Section 4.2.2] we eliminate
the complicating equality constraints and define the (concave) dual function as
d(v) = min f (z) + v ′ (Az − b), (3.28)
z∈K

where v ∈ Rp . The concave dual problem is then given by

d∗ = max d(v), (3.29)
v

which we also call the outer problem.

In order to apply any of the gradient methods for unconstrained optimization
in Section 3.2 for solving the outer problem, the dual function d(v) needs to fulfill
two requirements: (i) differentiability, i.e., the gradient ∇d(v) must exist for all
v, and (ii) Lipschitz continuity of the gradient. Next, we will show that if f is
strongly convex, both requirements are indeed fulfilled.
In the following it is assumed that the objective function f is strongly convex.
Strong convexity ensures uniqueness of the minimizer z ∗ (v) in the definition of the
dual function in (3.28). It is this uniqueness of z ∗ (v) that leads to a differentiable
dual function as shown in [52, Danskin’s Theorem, Proposition B.25]. Specifically,
it can be proved that
∇d(v) = Az ∗ (v) − b,
i.e., every evaluation of the dual gradient requires the solution of the so-called inner
problem (3.28).
The Lipschitz constant Ld of the dual gradient can be upper bounded by
2
kAk
Ld ≤
µ
3.3 Constrained Optimization 51

as stated in [221, Theorem 1]. This upper bound is sufficient for the step size
computation for any gradient method (note that the step size is given by 1/Ld ).
However, the less conservative the Lipschitz constant is, the larger the step size
and the smaller the number of iterations. In [251, Theorem 7] it is shown that for
the case of quadratic objective functions f (z) = 12 z ′ Hz + q ′ z, H ≻ 0, and under
mild assumptions, the tight Lipschitz constant of the dual gradient is
1
L∗d = kAH − 2 k2 .

So, strong convexity of the objective function f is key for using the gradient
methods in Section 3.2.1 for the solution of the dual problem (3.29). If f is not
strongly convex, there are different approaches available in the literature, e.g. a
smoothing approach for the dual function [221].
Note that the (unique) primal minimizer z ∗ can be recovered from the maxi-
mizer of the dual problem v ∗ , assuming it exists, by

z ∗ = z ∗ (v ∗ ).

It is also important to notice that, in general, the dual function d(v) lacks strong
concavity, so only sublinear convergence rates (instead of linear rates for the primal
problem) can be guaranteed for the solution of the dual problem. This and the
fact that each evaluation of the dual gradient requires solving the inner problem
can make the dual approach, as presented in this section, slow in practice. More
advanced first-order methods can significantly speed up convergence of the dual
approach, e.g. the alternating direction method of multipliers (ADMM). For more
details on ADMM, the interested reader is referred to the survey paper [64].

Summary
This section has introduced the generalization of gradient methods to the con-
strained case, the so-called gradient projection methods. As the name suggests, the
generalization comes from having to compute the projection of a gradient step onto
the feasible set. Apart from this additional operation, these methods are identical
to the ones introduced for the unconstrained case.
There exist many important convex sets for which the projection operator can
be evaluated in an efficient way, some of which are stated in Table 3.1. More sets
can be found, e.g. in [250, Section 5.4], which also contains further literature links.
One important consequence of having a set constraint is that a full preconditioner
matrix P (Section 3.2.1 and Example 3.3) is not admissible anymore since in the
new basis the set can lose its favorable projection properties. For instance, consider
a box constraint with its projection operator given in Table 3.1. Only for a diagonal
preconditioner the set remains a box in the new basis and thus easy to project.
For quadratic objective functions, finding the best preconditioner under set-related
structure constraints turns out to be a convex semi-definite program and hence can
be solved efficiently [250, Section 8.4].
Finally, we point the reader to the recent proximity gradient methods, which
can be understood as a natural generalization of gradient projection methods. An
introduction to these methods and literature links can be found in [250, Chapter
5] and [228].
52 3 Numerical Methods for Optimization

3.3.2 Interior Point Methods

Today interior point methods are among the most widely used numerical methods
for solving convex optimization problems. This is the result of 25 years of intensive
research that has been initiated by Karmarkar’s seminal paper in 1984 on solving
LPs [166]. In 1994, Nesterov and Nemirovskii generalized interior point methods
to nonlinear convex problems, including second-order cone and semi-definite pro-
gramming [222]. Issues like numerical ill-conditioning, efficient and stable solution
of linear systems etc. are now well understood, and numerous free and commer-
cial codes are available that can solve linear and quadratic programs with high
reliability.
We first discuss barrier methods, which were the first polynomial-time algo-
rithms for linear programming of practical relevance. Then we present modern
primal-dual method. This powerful class of interior point methods forms the basis
of almost all implementations today.
Throughout this section, we consider the primal P-IPM problem

minz f (z)
subj. to Az = b (3.30)
g(z) ≤ 0

where z ∈ Rs are the primal variables, the functions f : Rs → R and g : Rs → Rm

are convex and twice continuously differentiable (for g, this holds component-wise
for each function gi , i = 1, . . . , m). We assume that there exists a strictly feasible
point z 0 with respect to g(z) ≤ 0, i.e., the set

S = {z ∈ Rs | gi (z) ≤ 0, i = 1, . . . , m}, (3.31)

has a non-empty interior. We furthermore assume that a minimizer z ∗ exists and

that it is attained.

Primal barrier methods

The main idea of the barrier method is to convert the constrained optimization P-
IPM problem (3.30) into an unconstrained problem (with respect to inequalities)
by means of a barrier function Φg : Rs → R. We denote by P (µ) the problem

z ∗ (µ) ∈ arg min f (z) + µΦg (z)

(3.32)
subj. to Az = b

where µ > 0 is called barrier parameter. The purpose of the barrier function Φg is
to “trap” an optimal solution of problem P (µ) (3.32), which we denote by z ∗ (µ), in
the set S. Thus, Φg (z) must take on the value +∞ whenever gi (z) > 0 for some i,
and a finite value otherwise. As a result, z ∗ (µ) is feasible with respect to S, but it
differs from z ∗ since we have perturbed the objective function by the barrier term.
With the above idea in mind, we are now ready to outline the (primal) barrier
method summarized in Algorithm 3.8. Starting from (a possibly large) µ0 , a solu-
tion z ∗ (µ0 ) to problem P (µ0 ) is computed, for example by any of the methods for
unconstrained optimization presented in Section 3.2. For this reason, Φg should be
3.3 Constrained Optimization 53

twice continuously differentiable, allowing one to apply Newton’s method. After

z ∗ (µ0 ) has been computed, the barrier parameter is decreased by a constant fac-
tor, µ1 = µ0 /κ (with κ > 1), and z ∗ (µ1 ) is computed. This procedure is repeated
until µ has been sufficiently decreased. It can be shown under mild conditions
that z ∗ (µ) → z ∗ from the interior of S as µ → 0. The points z ∗ (µ) form the
so-called central path, which is a continuously differentiable curve in S that ends in
the solution set (see Example 3.4). Solving a subproblem P (µ) is therefore called
centering.
If the decrease factor κ is not too large, z ∗ (µk ) will be a good starting point
for solving P (µk+1 ), and therefore Newton’s method can be applied to P (µk+1 )
to exploit the local quadratic convergence rate. Note that the equality constraints
in (3.30) have been preserved in problem P (µ) (3.32), since they do not cause
substantial difficulties for unconstrained methods as shown in Section 3.2. In par-
ticular, we refer the reader to Section 3.2.2 for the equality constrained Newton
method.

Logarithmic barrier
So far, we have not specified which function to use as a barrier. For example, the
indicator function (
0 if g(z) ≤ 0
Ig (z) = , (3.33)
+∞ otherwise
trivially achieves the purpose of a barrier, but it is not useful since methods for
smooth convex optimization as discussed in Section 3.2 require the barrier function
to be convex and continuously differentiable. A twice continuously differentiable
convex barrier function that approximates Ig well is the logarithmic barrier function
m
X
Φg (z) = − ln (−gi (z)) , (3.34)
i=1

with domain {z ∈ Rs | gi (z) < 0 ∀ i = 1, . . . , m} (the logarithmic barrier confines

z to the interior of the set S), gradient
m
X 1
∇Φg (z) = ∇gi (z) (3.35)
i=1
−gi (z)

and continuous Hessian

m
X 1 1
∇2 Φg (z) = ∇gi (z)∇gi (z)′ + ∇2 gi (z), (3.36)
i=1
gi (z)2 −gi (z)

which makes it possible to use Newton’s method for solving the barrier subprob-
lems. The logarithmic barrier function is used in almost all implementations of
barrier methods. In fact, the existence of polynomial-time algorithms for convex
optimization is closely related to the existence of barrier functions for the under-
lying feasible sets [222].

Algorithm 3.8 Barrier interior point method for (3.30)

54 3 Numerical Methods for Optimization

Input: Strictly feasible initial iterate z 0 w.r.t. g(z) ≤ 0, µ0 , κ > 1, tolerance ǫ > 0
Output: Point close to z ∗
Repeat
Compute z ∗ (µk ) by minimizing f (z) + µk Φg (z) subject to Az = b start-
ing from the previous solution z k−1 (usually by Newton’s method, Algo-
rithm 3.4). This is called ‘centering step’.
Update: z k ← z ∗ (µk )
Stopping criterion: Stop if mµk < ǫ
Decrease barrier parameter: µk+1 ← µk /κ
Until stopping criterion is satisfied

Example 3.4 Central Path of a QP

Consider the QP
1 ′
min 2
z [ 20 01 ] z + [ 4 1
2 ]z
" −1 2.0588 # " 1.0890 # (3.37)
−1 −1.7527 1.0623
subj. to −3.9669 1 z ≤ 2.0187
1.1997 −1.3622 1
1.1673 1.3111 1

The solutions of P (µ) (3.32) for this Quadratic Program for different values of µ are
depicted in Fig. 3.4. For large µ, the level sets are almost identical to the level sets of
Φg (z), i.e., z ∗ (5) is very close to the analytic center z a = arg min Φg (z) of the feasible
set. As µ decreases, z ∗ (µ) approaches z ∗ , by following the central path.

Example 3.5 Barrier method for QPs

In the following, we derive the barrier method for quadratic programs of the form
1 ′
minimize 2
+ q′ z
z Hz
subject to Az = b, (3.38)
Gz ≤ w

with H ≻ 0, A ∈ Rp×s , b ∈ Rp , G ∈ Rm×s and w ∈ Rm from the general method

described above. The logarithmic barrier function for linear inequalities takes the
form
P Pm
Φg (z) =− m i=1 ln(wi − Gi z), ∇Φg (z) =
1 ′
i=1 wi −Gi z Gi ,
2
P m 1 ′ (3.39)
∇ Φg (z) = i=1 (w −G z)2 Gi Gi
i i

where Gi denotes the i-th row of G and wi the i-th element of w. Hence problem
P (µ) (3.32) takes the form

Xm
1 ′
min z Hz + q ′ z − µ ln(wi − Gi z)
z 2 i=1 (3.40)
subj. to Az = b,

which we solve by Newton’s method. Using the formula for the Newton direction dN
with equality constraints (3.19), we arrive at the following linear system of size s + p,
3.3 Constrained Optimization 55
z2

z2
•
•

z1 z1
(a) Level curves of f (z) + (b) Level curves of f (z) +
µΦg (z) for µ = 5 µΦg (z) for µ = 0.25
z2
z2

•
•
• •

z1 z1
(c) Level curves of f (z) + (d) Central path starting at an-
µΦg (z) for µ = 0.0125 alytic center and moving to-
ward solution of (3.37) as µ de-
creases

Figure 3.4 Example 3.4. Illustration of interior point method for example
problem (3.37). Black dots are the solutions z ∗ (µ).
56 3 Numerical Methods for Optimization

which has to be solved in each Newton step at the current iterate z:

 P    P ′
H +µ m 1 ′
i=1 (wi −Gi z)2 Gi Gi A ′ dN Hz + q + µ m 1
i=1 wi −Gi z Gi
=− .
A 0 y 0
(3.41)
The resulting barrier method for solving (3.38) is summarized in Algorithm 3.9. The
inner loop resembles the Newton iterations of the algorithm, computing z ∗ (µ). Note
that a primal feasible line search must be employed to ensure that the iterates remain
in the interior of S, see Section 3.2.3 for details.

Algorithm 3.9 Barrier interior point method for QPs (3.38)

Input: Strictly feasible initial iterate z 0 w.r.t. Az = b, Gz ≤ w, µ0 , κ > 1, tolerance
ǫ>0
Output: Point close to z ∗
Repeat
Initialize Newton’s method with z k,0 ← z k−1 . Set l = 0.
Loop
Newton direction: solve (3.41) for z = z k,l to obtain dk,l
N (z
k,l
).
k,l
Feasible line search to obtain step size h (see Section 3.2.3).
Update: z k,l+1 ← z k,l + hk,l dk,l
N
End inner loop: break if l = lmax , otherwise set l = l + 1 and continue.
Endloop
Update: z k ← z k,lmax
Stopping criterion: Stop if mµk < ǫ
Decrease barrier parameter: µk+1 ← µk /κ
Until stopping criterion

We conclude this section with a few remarks on the barrier method. The main
computational burden is the computation of the Newton directions by solving linear
systems of type (3.41). This is usually done using direct methods (matrix factor-
izations), but iterative linear solvers can be used as well. In most applications the
matrices involved have a special sparsity pattern, which can be exploited to signif-
icantly speed up the computation of the Newton direction. We discuss structure
exploitation for solving (3.41) arising from MPC problems in Section 14.3.
If the barrier parameter µ is decreased too quickly, many inner iterations are
needed, but only a few outer iterations. The situation is reversed if µ is decreased
only slowly, which leads to a quick convergence of the inner problems (usually in
one or two Newton steps), but more outer iterations are needed. This tradeoff is
not much of an issue in practice. Usually, the total number of Newton steps is in
the range of 20 − 40, essentially independent of the conditioning of the problem.
The barrier method can be shown to converge linearly; more details can be found
in [65, Section 11.3].
Finally, we would like to point out that the barrier method is a feasible method,
which means that it starts with a feasible point and all iterates remain feasible.
This complicates the implementation in practice, as a line search must be employed
that maintains feasibility, and a strictly feasible initial iterate z 0 has to be supplied
3.3 Constrained Optimization 57

by the user or computed first by what is called a Phase I method. See [65] for
more details.
Computational experience has shown that modern primal-dual methods are sig-
nificantly more effective with only little additional computational cost when com-
pared to the barrier method. Therefore, primal-dual methods are now predominant
in commercial codes. They also allow infeasible iterates, that is, the equality and
inequality constraints on the primal variables are satisfied only at convergence,
which alleviates the necessity for finding a feasible initial point. We discuss this
class of methods next.

Primal-dual methods

The general idea of primal-dual interior point methods is to solve the Karush-
Kuhn-Tucker (KKT) conditions by a modified version of Newton’s method. The
nonlinear KKT equations represent necessary and sufficient conditions for optimal-
ity for convex problems under the assumptions discussed in Section 1.6. For the
P-IPM problem (3.30), the KKT conditions are

∇f (z) + A′ v + G(z)′ u = 0 (3.42a)

Az − b = 0 (3.42b)
g(z) + s = 0 (3.42c)
si ui = 0, i = 1, . . . , m (3.42d)
(s, u) ≥ 0 (3.42e)

where the vector s ∈ Rm + denotes slack variables for the inequality constraints
g(z) ≤ 0, and u ∈ Rm + is the vector of the associated Lagrange multipliers. The
vector v ∈ Rp denotes the Lagrange multipliers for the equality constraints Az = b,
and the matrix G(z) ∈ Rm×s is the Jacobian of g evaluated at z. The variables z
and s are from the primal, the variables v and u from the dual space.
If a primal-dual pair of variables (z ∗ , v ∗ , u∗ , s∗ ) is found that satisfies these
conditions, the corresponding primal variables (z ∗ , s∗ ) are optimal for (3.30) (and
the dual variables (v ∗ , u∗ ) are optimal for the dual of (3.30)). Since Newton’s
method is a powerful tool for solving nonlinear equations, the idea of applying it to
the system of nonlinear KKT equations is apparent. However, certain modifications
to a pure Newton method are necessary to obtain a useful method for constrained
optimization. The name primal-dual indicates that the algorithm operates in both
the primal and dual space.
There are many variants of primal-dual interior point methods. For a thorough
treatment of the theory we refer the reader to the book by Wright [290], which
gives a comprehensive overview of the different approaches. In this section, after
introducing the general framework of primal-dual methods, we restrict ourselves
to the presentation of a variant of Mehrotra’s predictor-corrector method [206],
because it forms the basis of most existing implementations of primal-dual interior
point methods for convex optimization and has proven particularly effective in
practice. The method we present here allows infeasible starting points.
58 3 Numerical Methods for Optimization

Central Path
We start by outlining the basic idea. Most methods track the central path, which we
have encountered already in barrier methods, to the solution set. These methods
are called path-following methods. In the primal-dual space, the central path
is defined as the set of points (z, v, u, s), for which the following relaxed KKT
conditions hold:

∇f (z) + A′ v + G(z)′ u = 0 (3.43a)

Az − b = 0 (3.43b)
g(z) + s = 0 (3.43c)
si ui = τ, i = 1, . . . , m, τ > 0 (3.43d)
(s, u) > 0 (3.43e)

In (3.43) as compared to (3.42), the complementarity condition (3.42d) is relaxed

by a scalar τ > 0, the path parameter, and slacks s and multipliers u are required
to be positive instead of nonnegative, i.e., s and u lie in the interior of the positive
orthant. The main idea of primal-dual methods is to solve (3.43) for successively
decreasing values of τ , and thereby to generate iterates (z k , v k , uk , sk ) that ap-
proach (z ∗ , v ∗ , u∗ , s∗ ) as τ → 0. The similarity with the barrier method in this
regard is evident. However, the Newton step is modified in primal-dual interior
point methods, as we shall see below.

Measure of Progress
Similar to primal barrier methods, the subproblem (3.43) is usually solved only
approximately, here by applying only one Newton step. As a result, the generated
iterates are not exactly on the central path, i.e., si ui 6= τ for some i. Hence a more
useful measure than the path parameter τ for the progress of the algorithm is the
average value of the complementarity condition (3.43d),

µ = (s′ u)/m (3.44)

– evidently, µ → 0 implies τ → 0. In interior point nomenclature, µ is called the

duality measure, as it corresponds to the duality gap scaled by 1/m if all other
equalities in (3.43) are satisfied. To be able to reduce µ substantially (at least by a
constant factor) at each iteration of the method, the pure Newton step is enhanced
with a centering component, as discussed next.

Newton Directions and Centering

A search direction is obtained by linearizing (3.43) at the current iterate (z, v, u, s)
and solving
    
H(z, u) A′ G(z)′ 0 ∆z ∇f (z) + A′ v + G(z)′ u
 A 0 0 0    Az − b 
   ∆v  = −   (3.45)
 G(z) 0 0 I   ∆u   g(z) + s 
0 0 S Z ∆s Su − ν
3.3 Constrained Optimization 59

u1

current iterate centering direction

(z, v, u, s)

Newton direction
(predictor)
∆(z, v, u, s)(0) central path C
resulting
search direction
∆(z, v, u, s)(σµ) optimal point
s1

Figure 3.5 Affine-scaling (pure Newton) and centering search directions of

classic primal-dual methods. The Newton direction makes only progress
towards the solution when it is computed from a point close to the central
path, hence centering must be added to the pure Newton step to ensure that
the iterates stay sufficiently close to the central path.

where S = diag(s1 , . . . , sm ) and Z = diag(u1 , . . . , um ), the (1,1) block in the coef-

ficient matrix is
Xm
H(z, u) = ∇2 f (z) + ui ∇2 gi (z) (3.46)
i=1

and the vector ν ∈ Rm allows a modification of the right-hand side, thereby gen-
erating different search directions that depend on the particular choice of ν. For
brevity, we denote the solution to (3.45) by ∆[z, v, u, s](ν).
The vector ν can take a value between the following two extremes:

• If ν = 0, the right hand side in (3.45) corresponds to the residual of the KKT
conditions (3.42), i.e., ∆[z, v, u, s](0) is the pure Newton direction that aims
at satisfying the KKT conditions (3.42) in one step based on a linearization.
This direction is called affine-scaling direction. However, pure Newton steps
usually decrease some products si ui prematurely towards zero, i.e., the gen-
erated iterate is close to the boundary of the feasible set (if si ui ≈ 0, either
si or ui must be small). This is disadvantageous since from such points only
very small steps can be taken, and convergence is prohibitively slow. Hence
the iterates must be centered by bringing them closer to the central path.

• If ν = µ1, the complementarity condition (3.43d) is modified such that the

resulting search direction aims at making the individual products si ui equal
to their average value µ. This is called centering. A centering step does not
decrease the average value, so no progress towards the solution is made.

A tradeoff between the two goals of progressing towards the solution and centering
is achieved by computing search directions ∆[z, v, u, s](ν) by solving (3.45) for

ν = σµ1 (3.47)
60 3 Numerical Methods for Optimization

where σ ∈ (0, 1) is called the centering parameter. This is the key difference
from pure Newton steps, and the main ingredient (together with tracking the
central path) in making Newton’s method work efficiently for solving the non-
linear KKT system (3.42). Figure 3.5 depicts this combination of search directions
schematically, and a principle framework for primal-dual methods is given in Al-
gorithm (3.10). We will instantiate it with Mehrotra’s method below.
In the discussion above, we have excluded almost all theoretical aspects con-
cerned with the convergence of primal-dual methods. For example, the terms
“sufficiently close to the central path” and “sufficient reduction in µ” have precise
mathematical counterparts that allow one to analyze the convergence properties.
The book by Wright [290] is a good starting point for the interested reader.

Algorithm 3.10 Primal-dual interior point methods for the P-IPM problem (3.30)
Input: Initial iterates z 0 , v 0 , u0 > 0, s0 > 0, Centering parameter σ ∈ (0, 1)
Output: Point close to z ∗
Repeat
′
Duality measure: µk ← sk uk /m
Search direction: compute ∆[z k , v k , uk , sk ](σµk 1) by solving (3.45) at the
current iterate
Choose step length hk such that sk+1 > 0 and uk+1 > 0
Update: (z k+1 , v k+1 , uk+1 , sk+1 ) ← (z k , v k , uk , sk )+hk ∆[z k , v k , uk , sk ](σ k µk 1)
Until stopping criterion

Predictor-Corrector Methods∗
Modern methods use the full Newton step ∆[z, v, u, s](0) merely as a predictor to
estimate the error made by using a linear model of the central path. If a full step
along the affine-scaling direction were taken, the complementarity condition would
evaluate to
(S + ∆S aff )(u + ∆uaff ) = Su + ∆S aff u + S∆uaff +∆S aff ∆uaff (3.48)
| {z }
=0

where capital letters denote diagonal matrices constructed from the correspond-
ing vectors. The first three terms sum up to zero by the last equality of (3.45).
Hence the full Newton step produces an error of ∆S aff ∆uaff in the complemen-
tarity condition. In order to compensate for this error, a corrector direction
∆[z, v, u, s](−∆S aff ∆uaff ) can be added to the Newton direction. This compen-
sation is not perfect as the predictor direction is only an approximation; neverthe-
less, predictor-corrector methods work usually better than single-direction meth-
ods. The principle of predictor-corrector methods is depicted in Fig. 3.6.
An important contribution of Mehrotra [206] was to define an adaptive rule for
choosing the centering parameter σ based on the information from the predictor
step. His rule
σ = (µaff /µ)3 (3.49)
increases the centering if the progress would be poor, that is, if the predicted value
of the duality measure, µaff , is not significantly smaller than µ. If good progress
3.3 Constrained Optimization 61

u1

current iterate resulting search

(z, y, u, s) direction (Mehrotra)
∆(z, y, u, s)(σµ1 − ∆S aff ∆uaff )
Newton direction
(predictor)
∆(z, y, u, s)(0)
central path C
predicted point centering
(uaff , saff ) + corrector
direction optimal point
s1

Figure 3.6 Search direction generation in predictor-corrector methods. The

affine search direction can be used to correct for linearization errors, which
results in better performance in practice due to closer tracking of the central
path.

can be made, then σ ≪ 1, and the direction along which the iterate is moved is
closer to the pure Newton direction. The corrector and centering direction can be
calculated in one step because ∆[z, v, u, s](ν) is linear in ν; the final search direction
is therefore given by ∆[z, v, u, s](σµ1 − ∆S aff ∆uaff ).
A variant of Mehrotra’s predictor-corrector method that works well in practice
is given in Algorithm 3.11. The parameter γ is a safeguard against numerical errors,
keeping the iterates away from the boundary. Because of the predictor-corrector
scheme, two linear systems have to be solved in each iteration. Since the coefficient
matrix is the same for both solves, direct methods are preferred to solve the linear
systems in Steps 3 and 7 of the algorithm, as the computational bottleneck of the
method is the factorization of the coefficient matrix. Once it has been factored in
Step 3, the computational cost of the additional solve in Step 7 is negligible, and
is more than compensated by the savings in the number of iterations.

Algorithm 3.11 Mehrotra primal-dual interior point method for (3.30)

Input: Initial iterates z 0 , v 0 , u0 > 0, s0 > 0, γ < 1 (typically γ = 0.99)
Output: Point close to z ∗
Repeat
′
Duality measure: µk ← sk uk /m
Newton direction (predictor): compute ∆[z k,aff , v k,aff , uk,aff , sk,aff ](0) by
solving (3.45) at the current iterate
Line search: hk,aff ← max{h ∈ [0, 1] | sk + h∆sk,aff ≥ 0, uk + h∆uk,aff ≥ 0}
Predicted (affine) duality measure: µk,aff ← (sk +hk,aff ∆sk,aff )′ (uk +hk,aff ∆uk,aff )/m
Centering parameter: σ k ← (µk,aff /µk )3
Final search direction: compute ∆[z k , v k , uk , sk ](σ k µk 1 − ∆S k,aff ∆uk,aff )
by solving (3.45) at current iterate
Line search: hk ← max{h ∈ [0, 1] | sk + h∆sk ≥ 0, uk + h∆uk ≥ 0}
Update: (z k+1 , v k+1 , uk+1 , sk+1 ) ← (z k , v k , uk , sk )+γhk ∆[z k , v k , uk , sk ](σ k µk 1−
∆S k,aff ∆uk,aff )
62 3 Numerical Methods for Optimization

Until stopping criterion is satisfied

Relation of Primal-Dual to Primal Barrier Methods

An insightful relation between primal barrier and primal-dual methods can be
established as follows. Writing the optimality conditions for problem P (µ) (3.32)
gives

∇f (z) + µ∇Φg (z) + A′ v = 0 (3.50a)

Az = b (3.50b)
g(z) ≤ 0 (3.50c)

where g(z) ≤ 0 has been added for the barrier function Φg to be well defined. If
one now defines
u = µ diag(−g(z))−1 1 , (3.51)
the conditions (3.50) become
m
X
∇f (z) + ui ∇gi (z) + A′ v = 0 (3.52a)
i=1
Az = b (3.52b)
g(z) + s = 0 (3.52c)
Su = µ1 (3.52d)
(s, u) ≥ 0 (3.52e)

where we have introduced slack variables s and used the expression for the gradient
of the barrier function Φg and the relations s = −g(z), S = diag(s). Using
m
X
ui ∇gi (z) = G(z)′ u , (3.53)
i=1

where G(z) is the Jacobian of g evaluated at the point z, yields the same relaxed
optimality conditions as (3.43).
Hence if subproblems P (µ) (3.32) and (3.43) were solved exactly assuming (3.51),
the primal iterates of the primal-barrier method and the primal-dual method would
coincide (provided that the path parameters coincide, i.e., τ = µ). However, primal-
dual methods are in practice more efficient than primal barrier methods, since they
generate search directions using information from the dual space, and therefore the
iterates generated by the two algorithms do not coincide in general.

3.3.3 Active Set Methods

In this section we will discuss methods for solving the problem

minz f (z)
subj. to z ∈ S,
3.3 Constrained Optimization 63

where the feasible set S ⊂ Rs is a polyhedron, i.e., a set defined by linear equali-
ties and inequalities, and the objective f is a linear function (linear programming
(LP)) or a convex quadratic function (quadratic programming (QP)). As the name
indicates, active set methods aim to identify the set of active constraints at the
solution. Once this set is known, a solution to the problem can be easily compted.
Since the number of potentially visited active sets depends combinatorially on the
number of decision variables and constraints, these methods have a worst case com-
plexity that is exponential in the problem size (as opposed to first-order and interior
point methods presented in Sections 3.3.1 and 3.3.2, which both have polynomial
complexity). However, active set methods work quite well in practice showing the
worst case number of iterations only on pathological problem instances. Also, their
underlying concepts are important ingredients in multi-parametric programming
discussed in Chapter 6.

Active Set Method for LP (Simplex Method)

In the following we describe an active set method for the LP

minz c′ z
(3.54)
subj. to Gz ≤ w

where for the sake of simplicity we assume that this LP has a solution, i.e., it is
neither unbounded below nor infeasible. Although we consider the case where the
feasible set is described by linear inequalities only, the same method can be used
to solve problems of type
maxu −w′ u
subj. to G′ u = −c (3.55)
u ≥ 0,

which is another LP with linear equality and inequality constraints. In particular,

(3.55) is the dual problem associated with (3.54) and plays a central role in the
active set method for the solution of the primal problem. In fact, the dual will be
solved simultaneously with the primal problem as will be shown below.
Central to active set methods for LP is the observation that a solution is always
attained at a vertex of the polyhedral feasible set. This fact is often referred to as
the fundamental theorem of linear programming and gives reason to the methods’
alternative naming simplex methods. A simple strategy would be to enumerate all
vertices of the polyhedron and declare the vertex with the smallest cost as solution.
However, one can do better by only visiting those vertices that improve the cost
over the previous ones. This is the main idea behind active set methods for LP.
Interestingly, it was observed that the simplex method finds a solution in about 2m
to 3m iterations for most practical problems where m ≥ s denotes the number of
inequalities in (3.54). Nevertheless, pathological polyhedral sets exist which require
the method to visit all vertices for certain cost vectors, e.g. the Klee-Minty cube
which is a polyhedral set in Rs with 2s vertices [26].
Before describing the algorithmic scheme of the simplex method for the solution
64 3 Numerical Methods for Optimization

of (3.54), we state the associated necessary and sufficient KKT conditions

G′ u + c = 0, (3.56a)
ui (Gi z − wi ) = 0, i = 1, . . . , m, (3.56b)
u ≥ 0, (3.56c)
Gz − w ≤ 0. (3.56d)

These conditions are composed of primal feasibility (3.56d), dual feasibility (3.56a),
(3.56c) and complementarity conditions (3.56b). In the simplex method, primal
feasibility and the complementarity conditions are maintained throughout the it-
erations whereas dual feasibility is sacrificed and will only be satisfied whenever a
solution is found. Consequently, the method not only returns a primal minimizer z ∗
but also a certificate for optimality u∗ which at the same time is a solution to the
dual problem (3.55).

Initialization
Assume that a vertex z of the feasible set is given. At every vertex, at least s
inequalities are active. For the initial active set A we select exactly s indices
of active inequalities such that the associated submatrix GA is invertible.
The remaining ones are put into the set N A, i.e., A ∪ N A = {1, 2, . . . , m}
and A ∩ N A = ∅. Hence, the vertex z is characterized by

GA z = wA , (3.57a)
GN A z ≤ wN A , (3.57b)

with wA and wN A being defined accordingly. Similarly, the simplex method

initializes a dual multiplier u such that

G′A uA = −c,
uN A = 0.

Thus, the vertex z and the dual multiplier u fulfill the primal feasibility and
complementarity conditions of the KKT conditions (3.56) but not necessarily
dual feasibility as uA = −G′−1
A c is nonnegative only when a solution is found.

Main Loop
If uA ≥ 0 (Optimal solution found)
The pair (z, u) satisfies the KKT conditions (3.56) and thus is an optimal
primal/dual pair.
Else (Pivoting)
A leaving index l ∈ A is selected such that the related dual variable is
negative, i.e., ul < 0. It can be shown that in this case an entering index
e ∈ N A can be found such that after interchanging these indices, i.e.

A = A \ {l} ∪ {e},
NA = N A \ {e} ∪ {l},
3.3 Constrained Optimization 65

the next vertex has a cost that is never worse than the cost at the
previous vertex. In fact, if the previous vertex z is non-degenerate,
i.e., all inequalities in (3.57b) are strict, then the cost improves strictly.
The systematic interchange of indices between sets A and N A is called
pivoting. Pivoting maintains primal feasibility and the complementarity
conditions.

The simplex method described above terminates in a finite number of iterations

if all visited vertices are non-degenerate. This follows from strict improvement of
the cost in every pivoting step, the finite number of vertices and our assumption
that there exists a solution. In the remaining part of this section, we shortly
summarize important aspects for every practical implementation of the simplex
method that were not discussed before in detail and at the end provide links to the
literature.

• Initialization and detection of infeasibility For initialization, the simplex

method requires a vertex and an associated set of linearly independent con-
straint vectors (we call such an active set admissible from here on). In general,
determining both is as hard as solving the original LP. For this reason, sim-
plex implementations often resort to a two-phase approach: In Phase 1, an
auxiliary LP is solved for which an initial vertex and admissible active set
are easy to spot and which has an optimal solution and associated active set
that can be used to initialize an LP in Phase 2. From the solution of the
latter, an optimizer for the original LP can then be reconstructed.
For the LP in (3.54), a possible auxiliary Phase 1 LP is

miny,z+ ,z− y
subj. to Gz+ − Gz− − 1y ≤ w, (3.60)
z+ , z− , y ≥ 0.

It can be easily verified that the vector (z+ , z− , y) with z+ = z− = 0 and

y = min{y ≥ 0 | y ≥ −wi , i = 1, . . . , m} is a vertex of the feasible set in (3.60).
In order to find an admissible active set, we distinguish two cases: If w ≥ 0,
then y = 0 and the admissible active set is composed of all indices referring
to the non-negativity inequalities in (3.60). Otherwise, there exists at least
one index i such that wi < 0 which implies y > 0. In this case, the admissible
active set is determined by all indices referring to non-negativity inequalities
for variables z+ and z− and a single index referring to one of the active
constraints in Gz+ − Gz− − 1y ≤ w at the initial vertex. This active set can
be shown to lead to a constraint submatrix that is always invertible, so, the
auxiliary LP can be solved by the simplex method next.
Let us denote the optimal solution to (3.60) as (y ∗ , z+
∗ ∗
, z− ). If y ∗ is positive,
∗
then the original LP (3.54) is infeasible. Only if y = 0, a feasible solution
∗ ∗
exists and is given by z+ − z− . Since the solution to the Phase 1 problem is
a vertex and comes with an admissible active set, it can be used right away
66 3 Numerical Methods for Optimization

for the initialization of the simplex method for solving the Phase 2 problem

miny,z+ ,z− c′ z + − c′ z −
subj. to Gz+ − Gz− − 1y ≤ w,
(3.61)
z+ , z− , y ≥ 0
y ≤ 0,
∗∗ ∗∗
which is equivalent to the original LP (3.54): Every solution (0, z+ , z− )
∗∗ ∗∗
to (3.61) implies a solution z+ − z− to the original problem and vice versa.

• Detection of unboundedness An LP is unbounded whenever there exists a

feasible sequence {z k }, Gz k ≤ w, such that c′ z k → −∞. The simplex method
can detect unboundedness quite easily by identifying a descent direction along
which the objective decreases without bound.

• Convergence in case of degeneracy A vertex is degenerate whenever more

than s inequalities are active at this point. Degeneracy does not prevent
active set updates but does prevent improving the cost from iterate to iterate
by revisiting the same vertex. In the worst case, this can lead to cycling, i.e.,
the simplex method visits the same vertex infinitely often and thus fails to
find an optimal solution. The main methodology to overcome cycling is to
slightly perturb the problem data such that all visited vertices are guaranteed
to be non-degenerate and to recover the original (non-perturbed) solution in
a final step.

• Pivoting rules In the simplex method presented above, a leaving index is

selected from the active set such that the associated dual variable is negative.
In fact, there can be more than one such index and different selection rules
lead to different next vertices and thus progress in the cost. Many selection
or pivoting rules for the leaving index exist, e.g. choosing the index that
corresponds to the most negative dual variable as proposed in the original
simplex method by Dantzig [92], and consequently lead to a whole family of
different simplex methods.

• Linear algebra Efficiency of the simplex method depends not only on the
number of visited vertices, which is unknown in advance and can be expo-
nential in the problem size in the worst case, but to a great extent on efficient
linear algebra routines for the computation of new vertices when doing pivo-
ting steps. Since in the simplex method only a single index is removed and
added to the active set A in every iteration, specific linear algebra routines
can be used to update the factorization of the constraint submatrix GA .

There is a great amount of literature available on linear programming and the

simplex method. One of the earliest books on this topic is by G.B. Dantzig [92]
who developed the simplex method in 1947 and is generally known as the father of
linear programming. Other references in this field are [217], [280] and [223, Chapter
13]. The material for the Phase 1 /Phase 2 part of this section is based on the latter
two references. Finally, it should be noted that the simplex method consistently
ranks among the ten most important algorithms of the 20th century.
3.3 Constrained Optimization 67

Active Set Method for QP

In this section we consider an active set method for the solution of the convex QP
1 ′
minz 2 z Hz+ q′ z
(3.62)
subj. to Gz ≤ w.

We restrict the discussion to the case when the Hessian H is positive definite and
the feasible set determined by the linear inequality constraints is non-empty. These
assumptions imply that both a unique solution z ∗ to (3.62) and unique solutions
to all the subproblems encountered in the active set method exist (this will become
clear later in this section). Another consequence of positive definiteness of the
Hessian is that the presented active set method can be shown to never cycle, i.e.,
it can be guaranteed that the solution is found in a finite number of iterations
without any safeguard against cycling, in contrast to the LP case. Note that we do
not consider explicit linear equality constraints in order to facilitate a streamlined
presentation of the main concepts.
Active set methods for QP share many features of active set methods for LP.
They too identify the set of active inequality constraints at the optimal solution
and return a certificate of optimality u∗ which is a solution to the dual problem

maxu − 21 (q + G′ u)′ H −1 (q + G′ u) − w′ u
subj. to u ≥ 0.

The dual problem is also a QP. If, however, the constraint matrix G has more rows
than columns (this is the usual case) then the Hessian of the concave objective is
negative semi-definite and thus the dual solution might not be unique.
The principal feature that distinguishes the QP from the LP is that the search
for a solution cannot be restricted to the vertices of the polyhedral feasible set.
The solution can also be attained on an edge, a face or even in the interior. These
possibilities, in the non-degenerate case, correspond to differently sized active sets,
i.e., the size is s (solution on a vertex), s − 1 (solution on an edge), s − j where
j ∈ {2, 3, . . . , s − 1} (solution on a face) or the active set might even be empty
(solution in the interior). So, there is a combinatorial number of potential active
sets A, each corresponding to a QP with equality constraints
1 ′ ′
minz 2 z Hz + q z (3.63)
subj. to GA z = wA ,

where GA is the submatrix of constraint matrix G according to the active set A.

The r.h.s. vector wA is defined accordingly.
The main idea behind active set methods for QP is to select active sets in a
way such that the corresponding iterates stay primal feasible and lead to a cost
that decreases monotonically. By doing so, one usually circumvents the brute force
approach of checking all possible active sets although there exists no active set
method that excludes this possibility by mathematical analysis, i.e., no active set
method for QP with polynomial complexity is known.
A non-constant active set size has two important consequences: First of all,
there is no interchange of indices between set A and the complementary set N A
as in the LP case, i.e., in every iteration of the active set method for QP an index
68 3 Numerical Methods for Optimization

corresponding to a row of constraint matrix G is either added or removed from the

current active set or no action is taken at all. Second, a solution to the equality-
constrained QP (3.63) associated with every active set is not necessarily a feasible
iterate with respect to the original problem (3.62). So, additional measures must
be taken in order to maintain primal feasibility of the iterates.
For the understanding of the active set method presented next, it is instructive
to consider the necessary and sufficient KKT conditions for problem (3.62)

Hz + q + G′ u = 0, (3.64a)
ui (Gi z − wi ) = 0, i = 1, . . . , m, (3.64b)
u ≥ 0, (3.64c)
Gz − w ≤ 0. (3.64d)

The active set method maintains primal feasibility (3.64d) as well as stationa-
rity (3.64a) and the complementarity conditions (3.64b). The dual variable u is
feasible (3.64c) only when the active set at the optimal solution is identified. Note,
however, that no dual iterates are maintained, in contrast to the LP case, since they
are not required in every iteration. Only at times when the termination criterion
needs to be verified or an index removed from the current active set, dual variables
are computed.

Initialization
Assume that a primal feasible point z is given. Define the initial active set A
as a non-empty subset of all indices of active inequalities at z such that the
associated submatrix GA has full row rank. In fact, it suffices to choose only
a single index corresponding to any active inequality. The remaining indices
are put into set N A, i.e., N A = {1, 2, . . . , m}\A.
Main Loop
Minimize the quadratic cost over the affine set determined by the active set A,
i.e., solve (3.63). For the sake of convenience, we substitute the decision
variable in (3.63) by z + δ, i.e., the sum of the current iterate and an offset,
and obtain the optimal offset δ ∗ from
1 ′
minδ 2 δ Hδ + (q + Hz)′ δ
(3.65)
subj. to GA δ = 0.

If δ ∗ = 0
Compute the Lagrange multipliers uA for the equality constraints in (3.65).
If uA ≥ 0 (Optimal solution found)
The pair (z, u), where u is partitioned into uA and uN A = 0, satis-
fies the KKT conditions (3.64) and thus is an optimal primal/dual
pair. This follows from the fact that the pair (z, uA ) is an opti-
mal primal/dual solution for (3.65) and so satisfies the stationarity
condition
Hz + q + G′A uA = 0.
By definition of uN A as the zero vector, this implies that the sta-
tionarity condition for the original QP (3.64a) is satisfied as well.
3.3 Constrained Optimization 69

Else (Remove index from active set)

An index l ∈ A is removed from the active set such that the related
dual variable is negative, i.e., ul < 0, and the next iterate is defined
as the previous one, i.e.

A = A \ {l},
NA = N A ∪ {l},
z = z.

It can be shown that by choosing the leaving index this way, an

offset δ ∗ is obtained in the next iteration from (3.65) that is a descent
direction with respect to the quadratic cost. Also, δ ∗ turns out
to be a feasible direction with respect to the removed inequality
constraint, i.e., the leaving constraint will not be added again in
the next iteration.
Else (δ ∗ 6= 0)
The minimizer z + δ ∗ of the quadratic cost over the affine set might be
infeasible with respect to the original problem (3.62). So, we need to
find the largest step size α∗ ∈ [0, 1] such that z + α∗ δ ∗ remains feasible.
Note that this step size can be computed explicitly as
n wi − G′i z o
α∗ = min 1, min′ . (3.67)
i∈N A: Gi δ ∗ >0 G′i δ ∗

So, a new primal iterate is obtained from

z = z + α∗ δ ∗ .

If α∗ = 1 (No active set change)

A = A
NA = NA

Else (Add blocking index)

Assume that the inequality corresponding to index e ∈ N A prevents
w −G′ z
us from taking a full step, i.e., α∗ = eG′ δ∗e < 1. Then we update
e
the active set by adding this index, i.e.

A = A ∪ {e},
NA = N A \ {e}.

In the following, we shortly describe how to initialize the active set method and
how to obtain a solution to the equality-constrained QP (3.65). Note that this
part, same as the described active set method, is based on [223, Chapter 16].
• Initialization and detection of infeasibility Initialization of the active set
method for QP is less restrictive than for LP since only a feasible point of
the polyhedral feasible set, as opposed to a vertex, is required. Such a point
70 3 Numerical Methods for Optimization

can be obtained either from practical insight or from solving the Phase 1 LP
in (3.60). An alternative approach is to solve
1 ′
minz,ǫ 2 z Hz + q ′ z + ρ1 ǫ ′ ǫ + ρ2 1 ′ ǫ
subj. to Gz ≤ w + ǫ
ǫ ≥ 0,

for which a feasible point can be spotted easily and which can be shown to
have the same solution as (3.62) if ρ1 ≥ 0 and ρ2 > ku∗ k∞ , where u∗ is a
Lagrange multiplier for the inequality constraint in (3.62). This follows from
the theory of exact penalty functions (Section 12.6, page 265).
Since the Lagrange multiplier is usually unknown, one starts with a guess for
the weight ρ2 and solves the problem. The value was chosen big enough if
at the solution ǫ∗ = 0, otherwise, the weight needs to be increased and the
problem solved again. If no big enough weight can be found such that ǫ∗ = 0,
the problem is infeasible.

• Linear algebra Key to good performance of an active set method is an

efficient solution of the equality-constrained QP in (3.65). Notice that a
solution to (3.65) satisfies the KKT conditions
  ∗   
H G′A δ −(q + Hz)
= . (3.71)
GA 0 uA 0

It turns out that the so-called KKT matrix in (3.71) is invertible if the Hes-
sian H is positive definite and the submatrix GA has full row rank. Both
conditions are fulfilled in our case. In particular, whenever the method is ini-
tialized with an active set that leads to full row rank of GA , this property can
be shown to be maintained throughout all iterations without requiring extra
checks. In this case, a good way for solving (3.71) is the Schur complement
method. Since H can be inverted, we can express the primal solution as


δ ∗ = −H −1 G′A uA + q + Hz ,

which leads to the equation

GA H −1 G′A uA = −GA H −1 (q + Hz)

for the Lagrange multiplier uA . Since GA H −1 G′A is positive definite, we can

solve for uA using a Cholesky factorization and compute δ ∗ afterwards. Note
that the factorization needs to be updated every time the active set changes.
Since at most one index is added or removed per iteration, efficient update
schemes can be used that circumvent a full factorization.
 Chapter 4

Polyhedra and P-collections

Polyhedra are the fundamental geometric objects used in this book. There is a
vast body of literature related to polyhedra because they are important for a wide
range of applications. In this chapter we introduce the main definitions and the
algorithms which describe some operations on polyhedra. Our intent is to provide
only the necessary elements for the readers interested in reproducing the control
algorithms reported in this book. Most definitions given here are standard. For
additional details the reader is referred to [296, 135, 112].

4.1 General Set Definitions and Operations

An n-dimensional ball B(xc , ρ) is the set B(xc , ρ) = {x ∈ Rn : kx − xc k2 ≤ ρ}.
The vector xc is the center of the ball and ρ is the radius.
Affine sets are sets described by the solutions of a system of linear equations:

F = {x ∈ Rn : Ax = b, with A ∈ Rm×n , b ∈ Rm }. (4.1)

If F is an affine set and x̄ ∈ F , then the translated set V = {x − x̄ : x ∈ F } is a

subspace.
The affine combination of a finite set of P points x1 , . . . , xk belonging to Rn is
k
defined as the point λ x + . . . + λ x where i=1 λi = 1.
1 1 k k
n
The affine hull of K ⊆ R is the set of all affine combinations of points in K
and it is denoted as aff(K):
k
X
aff(K) = {λ1 x1 + . . . + λk xk : xi ∈ K, i = 1, . . . , k, λi = 1}. (4.2)
i=1

The affine hull of K is the smallest affine set that contains K in the following sense:
if S is any affine set with K ⊆ S, then aff(K)⊆ S.
The dimension of an affine set, affine combination or affine hull is the dimen-
sion of the largest ball of radius ρ > 0 included in the set.
72 4 Polyhedra and P-collections

Example 4.1 The set

F = {x ∈ R2 : x1 + x2 = 1}
is an affine set in R2 of dimension one. The points x1 = [0, 1]′ and x2 = [1, 0]′ belong
to the set F. The point x̄ = −0.2x1 + 1.2x2 = [1.2, −0.2]′ is an affine combination of
points x1 and x2 . The affine hull of x1 and x2 , aff({x1 , x2 }), is the set F.

Convex sets have been defined in Section 1.2.

The convex combination of a finite set ofPpoints x1 , . . . , xk belonging to Rn is
k
defined as the point λ1 x1 + . . . + λk xk where i i
i=1 λ = 1 and λ ≥ 0, i = 1, . . . , k.
n
The convex hull of a set K ⊆ R is the set of all convex combinations of points
in K and it is denoted as conv(K):
k
X
conv(K) = {λ1 x1 + . . . + λk xk : xi ∈ K, λi ≥ 0, i = 1, . . . , k, λi = 1}. (4.3)
i=1

The convex hull of K is the smallest convex set that contains K in the following
sense: if S is any convex set with K ⊆ S, then conv(K)⊆ S.

Example 4.2 Consider three points x1 = [1, 1]′ , x2 = [1, 0]′ , x3 = [0, 1]′ in R2 . The
point x̄ = λ1 x1 +λ2 x2 +λ3 x3 with λ1 = 0.2, λ2 = 0.2, λ3 = 0.6 is x̄ = [0.4, 0.8]′ and it
is a convex combination of the points {x1 , x2 , x3 }. The convex hull of {x1 , x2 , x3 }
is the triangle plotted in Figure 4.1. Note that any set in R2 strictly contained in the
triangle and containing {x1 , x2 , x3 } is non-convex. This illustrates that the convex
hull is the smallest set that contains these three points.

2
1.5
1
x̄
x2

0.5
0
-0.5 0 0.5 1.5
-0.5 1 2
x1
Figure 4.1 Example 4.2. Illustration of the convex hull of three points x1 =
[1, 1]′ , x1 = [1, 0]′ , x3 = [0, 1]′ .

A cone spanned by a finite set of points K = {x1 , . . . , xk } is defined as

k
X
cone(K) = { λi xi , λi ≥ 0, i = 1, . . . , k}. (4.4)
i=1
 4.2 Polyhedra and Representations 73

We define cone(K) = {0} if K is the empty set.

Example 4.3 Consider three points x1 = [1, 1, 1]′ , x2 = [1, 2, 1]′ , x3 = [1, 1, 2]′ in R3 .
The cone spanned by {x1 , x2 , x3 } is an unbounded set. It is depicted in Figure 4.2.

4
2
x3

0
4
2
0 2 4
x2 0
x1
Figure 4.2 Example 4.3. Illustration of a cone spanned by the three points
x1 = [1, 1, 1]′ , x2 = [1, 2, 1]′ , x3 = [1, 1, 2]′ .

The Minkowski sum of two sets P, Q ⊆ Rn is defined as

P ⊕ Q = {x + y : x ∈ P, y ∈ Q}. (4.5)
By definition, any point in P ⊕ Q can be written as the sum of two points, one in
P and one in Q. For instance, the Minkowski sum of two balls (P and Q) centered
at the origin and with radius 1, is a ball (P ⊕ Q) centered at the origin and with
radius 2.

4.2 Polyhedra and Representations

Some of the concepts here have been briefly touched in Section 2.1. In the following
we give two definitions of a polyhedron. They are mathematically equivalent but
they lead to two different polyhedron representations. The proof of equivalence is
not trivial and can be found in [296].
An H-polyhedron P in Rn denotes an intersection of a finite set of closed half-
spaces in Rn :
P = {x ∈ Rn : Ax ≤ b}, (4.6)
where Ax ≤ b is the usual shorthand form for a system of inequalities, namely
a′i x ≤ bi , i = 1, . . . , m, where a1 , . . . , am are the rows of A, and b1 , . . . , bm are the
components of b. In Figure 4.3 a two-dimensional H-polyhedron is plotted.
A V-polyhedron P in Rn denotes the Minkowski sum (defined in (4.5)) of the
convex hull of a finite set of points {V1 , . . . , Vk } of Rn and the cone generated by
a finite set of vectors {y1 , . . . , yk′ } of Rn :
P = conv(V ) ⊕ cone(Y ), (4.7)
74 4 Polyhedra and P-collections

a3′
x≤
b3

b2
≤
2x
a1′ x ≤

a′
b1

Figure 4.3 H-polyhedron. The planes (here lines) defining the boundary of
the halfspaces are a′i x − bi = 0.

a′
3x
≤
4

b3
≤b
a4′ x

b2
≤

a1′ x ≤
2x
a′

b1

Figure 4.4 H-polytope. The planes (here lines) defining the halfspaces are
a′i x − bi = 0.

′
for some V = [V1 , . . . , Vk ] ∈ Rn×k , Y = [y1 , . . . , yk′ ] ∈ Rn×k . The main theorem
for polyhedra states that any H-polyhedron can be converted into a V-polyhedron
and vice-versa [296, p.30].
An H-polytope is a bounded H-polyhedron (in the sense that it does not contain
any ray {x + ty : t ≥ 0}). In Figure 4.4 a two-dimensional H-polytope is plotted.
A V-polytope is a bounded V-polyhedron

P = conv(V ) = {V λ | λ ≥ 0, 1′ λ = 1 } . (4.8)

In Section 4.4.4 we show how to convert any H-polytope into a V-polytope and
vice versa.
4.2 Polyhedra and Representations 75

The dimension of a polytope (polyhedron) P is the dimension of its affine hull

and is denoted by dim(P). We say that a polytope P ⊂ Rn , P = {x ∈ Rn :
P x x ≤ P c }, is full-dimensional if dim(P) = n or, equivalently, if it is possible to
fit a nonempty n-dimensional ball in P,

∃x ∈ Rn , ǫ > 0 : B(x, ǫ) ⊂ P, (4.9)

or, equivalently,

∃x ∈ Rn , ǫ > 0 : kδk2 ≤ ǫ ⇒ P x (x + δ) ≤ P c . (4.10)

Otherwise, we say that polytope P is lower-dimensional. A polytope is referred to

as empty if
∄x ∈ Rn : P x x ≤ P c . (4.11)
Furthermore, if kPix k2 = 1, where Pix denotes the i-th row of a matrix P x , we say
that the polytope P is normalized.
Let P be a polyhedron. A linear inequality c′ z ≤ c0 is said to be valid for P if
it is satisfied for all points z ∈ P. A face of P is any nonempty set of the form

F = P ∩ {z ∈ Rs : c′ z = c0 }, (4.12)

where c′ z ≤ c0 is a valid inequality for P. The dimension of a face is the dimension

of its affine hull. For the valid inequality 0z ≤ 0 we get that P is a face of P. All
faces of P satisfying F ⊂ P are called proper faces and have dimension less than
dim(P). The faces of dimension 0,1, dim(P)-2 and dim(P)-1 are called vertices,
edges, ridges, and facets, respectively. The set V of all the vertices of a polytope
P will be denoted as
V = extreme(P).
The next theorem summarizes basic facts about faces.

Theorem 4.1 Let P be a polytope, V the set of all its vertices and F a face.
1. P is the convex hull of its vertices: P=conv(V ).
2. F is a polytope.
3. Every intersection of faces of P is a face of P.
4. The faces of F are exactly the faces of P that are contained in F .
5. The vertices of F are exactly the vertices of P that are contained in F .
A d-simplex is a polytope of Rd with d + 1 vertices.
In this book we will work mostly with H-polyhedra and H-polytopes. This
choice has important consequences for algorithm implementations and their com-
plexity. As a simple example, a unit cube in Rd can be described through 2d
equalities as an H-polytope, but requires 2d points in order to be described as
a V-polytope. We want to mention here that many efficient algorithms that
work on polyhedra require both H and V representations. In Figure 4.5 the H-
representation and the V-representation of the same polytope in two dimensions
are plotted.
76 4 Polyhedra and P-collections

x2

x2
x1 x1
(a) Example of V-representation. (b) Example of H-representation.
The vertices V1P , . . . , V7P are de- The hyperplanes Pix x = Pic , i =
picted as dots. 1, . . . , 7 are depicted as lines.

Figure 4.5 Illustration of a polytope in H- and V- representation.

Consider Figure 4.5(b) and notice that no inequality can be removed without
changing the polyhedron. Inequalities which can be removed without changing the
polyhedron described by the original set are called redundant. The representation
of an H-polyhedron is minimal if it does not contain redundant inequalities. De-
tecting whether an inequality is redundant for an H-polyhedron requires solving a
linear program, as described in Section 4.4.1.

By definition a polyhedron is a closed set. In this book we will also work with
sets which are not closed but whose closure is a polyhedron. For this reason the
two following non-standard definitions will be useful.

Definition 4.1 (Open Polyhedron (Polytope)) A set C ⊆ Rn is called an

open polyhedron (polytope) if it is open and its closure is a polyhedron (polytope).

Definition 4.2 (Neither Open nor Closed Polyhedron (Polytope)) A set C ⊆

Rn is called neither an open nor a closed polyhedron (polytope) if it is neither open
nor closed and its closure is a polyhedron (polytope).

4.3 Polytopal Complexes

According to our definition every polytope represents a convex, compact (i.e.,
bounded and closed) set. In this book we will also encounter sets that are dis-
joint or non-convex but can be represented by the union of a finite number of
polytopes. Therefore, it is useful to define the following mathematical concepts.
 4.3 Polytopal Complexes 77

Definition 4.3 (P-collection) A set C ⊆ Rn is called a P-collection if it is a

collection of a finite number of n-dimensional polyhedra, i.e.

C = {Ci }N
i=1 ,
C
(4.13)

where Ci = {x ∈ Rn : Cix x ≤ Cic }, dim(Ci ) = n, i = 1, . . . , NC , with NC < ∞.

Definition 4.4 (Underlying Set) The underlying set of a P-collection C = {Ci }N C

i=1
is the
[ N
[ C

C= P= Ci . (4.14)
P∈C i=1

Example 4.4 A collection C = {[−2, −1], [0, 2], [2, 4]} is a P-collection in R1 with the
underlying set C = [−2, −1] ∪ [0, 4]. As another example, C = {[−2, 0], [−1, 1], [0, 2]}
is a P-collection in R1 with underlying set C = [−2, 2]. Clearly, polytopes that define
a P-collection can overlap, the underlying set can be disconnected and non-convex.

Usually it is clear from the context if we are talking about the P-collection or
referring to the underlying set of a P-collection. Therefore for simplicity, we use
the same notation for both.

Definition 4.5 (Strict Polyhedral Partition) A collection of sets {Ci }N i=1 is a

C
SNC
strict partition of a set C if (i) i=1 Ci = C and (ii) Ci ∩ Cj = ∅, ∀i 6= j. Moreover
{Ci }N NC
i=1 is a strict polyhedral partition of a polyhedral set C if {Ci }i=1 is a strict
C

¯ ¯
partition of C and Ci is a polyhedron for all i, where Ci denotes the closure of the
set Ci .

Definition 4.6 (Polyhedral Partition) A collection of sets {Ci }N i=1 is a parti-

C
SNC
tion of a set C if (i) i=1 Ci = C and (ii) (Ci \∂Ci ) ∩ (Cj \∂Cj ) = ∅, ∀i 6= j. Moreover
{Ci }N NC
i=1 is a polyhedral partition of a polyhedral set C if {Ci }i=1 is a partition of C
C

and Ci is a polyhedron for all i. The set ∂Cj is the boundary of the set Cj .
Note that in a strict polyhedral partition of a polyhedron some of the sets Ci
must be open or neither open nor closed. In a partition all the sets may be closed
and points on the closure of a particular set may also belong to one or several other
sets. Also, note that a polyhedral partition is a special class of a P-collection.

4.3.1 Functions on Polytopal Complexes

Definition 4.7 A function h(θ) : Θ → R, where Θ ⊆ Rs , is piecewise affine

(PWA) if there exists a strict partition R1 ,. . . ,RN of Θ and h(θ) = H i θ + k i ,
∀θ ∈ Ri , i = 1, . . . , N .

Definition 4.8 A function h(θ) : Θ → R, where Θ ⊆ Rs , is piecewise affine on

polyhedra (PPWA) if there exists a strict polyhedral partition R1 ,. . . ,RN of Θ and
h(θ) = H i θ + k i , ∀θ ∈ Ri , i = 1, . . . , N .
78 4 Polyhedra and P-collections

Definition 4.9 A function h(θ) : Θ → R, where Θ ⊆ Rs , is piecewise quadratic

′
(PWQ) if there exists a strict partition R1 ,. . . ,RN of Θ and h(θ) = θ′ H i θ+k i θ+li ,
∀θ ∈ Ri , i = 1, . . . , N .

Definition 4.10 A function h(θ) : Θ → R, where Θ ⊆ Rs , is piecewise quadratic

on polyhedra (PPWQ) if there exists a strict polyhedral partition R1 ,. . . ,RN of Θ
and h(θ) = θ′ H i θ + k i θ + li , ∀θ ∈ Ri , i = 1, . . . , N .

As long as the function h(θ) we are defining is continuous it is not important

if the partition constituting the domain of the function is strict or not. If the
function is discontinuous at points on the closure of a set, then this function can
only be defined if the partition is strict. Otherwise we may obtain several conflicting
definitions of function values (or set-valued functions). Therefore for the statement
of theoretical results involving discontinuous functions we will always assume that
the partition is strict. For notational convenience, however, when working with
continuous functions we will make use of partitions rather than strict partitions.

4.4 Basic Operations on Polytopes

We will now define some basic operations and functions on polytopes. Note that
although we focus on polytopes and polytopic objects most of the operations de-
scribed here are directly (or with minor modifications) applicable to polyhedral ob-
jects. Additional details on polytope computation can be found in [296, 135, 112].
All operations and functions described in this chapter are contained in the MPT
toolbox [149].

4.4.1 Minimal Representation

We say that a polytope P ⊂ Rn , P = {x ∈ Rn : P x x ≤ P c } is in a minimal

representation if the removal of any row in P x x ≤ P c would change it (i.e., if
there are no redundant constraints). The computation of the minimal representa-
tion (henceforth referred to as polytope reduction) of polytopes is discussed in [112].
The redundancy of each constraint is checked, which generally requires the solution
of one LP for each half-space defining the non-minimal representation of P. We
summarize this simple implementation of the polytope reduction in Algorithm 4.1.
An improved algorithm for polytope reduction is discussed in [269] where the au-
thors combine the procedure outlined in Algorithm 4.1 with heuristic methods,
such as bounding-boxes and ray-shooting, to discard redundant constraints more
efficiently.
It is straightforward to see that a normalized, full-dimensional polytope P has
a unique minimal representation. Note that ‘unique’ here means that for P = {x ∈
Rn : P x x ≤ P c } the matrix [P x P c ] consists of the unique set of row vectors,
the rows’ order is irrelevant. This fact is very useful in practice. Normalized, full-
dimensional polytopes in a minimal representation allow us to avoid any ambiguity
 4.4 Basic Operations on Polytopes 79

when comparing them and very often speed-up other polytope manipulations.

Algorithm 4.1 Polytope in minimal representation

Input P = {x : P x x ≤ P c }, with P x ∈ RnP ×n , P c ∈ RnP
Output Q = {x : Qx x ≤ Qc } = minrep(P)
I ← {1, . . . , nP }
For i = 1 to nP
I ← I \ {i}
f ∗ ← max Pix x, subj. to x
P(I) c
x ≤ P(I) , Pix x ≤ Pic + 1
x
If f ∗ > Pic Then I ← I ∪ {i}
End
Qx = PIx , Qc = PIc

4.4.2 Convex Hull

The convex hull of a set of points V = {V i }N i n

i=1 , with V ∈ R , is a polytope defined
V

as
NV
X NV
X
n i i i
conv(V ) = {x ∈ R :x= α V , α ≥ 0, αi = 1}. (4.15)
i=1 i=1

The convex hull of a union of polytopes Ri ⊂ Rn , i = 1, . . . , NR , is a polytope

N
! NR NR
[R X X
conv Ri = {x ∈ Rn : x = αi xi , xi ∈ Ri , αi ≥ 0, αi = 1}. (4.16)
i=1 i=1 i=1

An illustration of the convex hull operation is given in Figure 4.6. Construction

of the convex hull of a set of polytopes is an expensive operation which is expo-
nential in the number of facets of the original polytopes. An efficient software
implementation is available in [111] and used in the MPT toolbox [149].

4.4.3 Envelope
The envelope of two H-polyhedra P = {x ∈ Rn : P x x ≤ P c } and Q = {x ∈ Rn :
Qx x ≤ Qc } is an H-polyhedron

env(P, Q) = {x ∈ Rn : P¯x x ≤ P¯c , Q¯x x ≤ Q̄c }, (4.17)

where P¯x x ≤ P¯c is the subsystem of P x x ≤ P c obtained by removing all the

inequalities not valid for the polyhedron Q, and Q¯x x ≤ Q̄c is defined in a similar
way with respect to Qx x ≤ Qc and P [38]. In a similar fashion, the definition can
be extended to the case of the envelope of a P-collection. An illustration of the
envelope operation is depicted in Figure 4.7.
80 4 Polyhedra and P-collections

The computation of the envelope is relatively cheap since it only requires the
solution of one LP for each facet of P and Q. In particular, if a facet of Q (and P)
is found to be “non redundant” for the polytope P (for the polytope Q ) then it is
not part of the envelope. A version of Algorithm 4.1 can be used. Note that the
envelope of two (or more) polytopes is not necessarily a bounded set (e.g., when
P ∪ Q is shaped like a star).

R1 R2 conv(R)
x2

x2
x1 x1
S
(a) R = i∈{1,2} Ri . (b) Convex hull of R.

Figure 4.6 Illustration of the convex hull operation.

R1 R2 env(R)
x2

x2

x1 x1
S
(a) R = i∈{1,2} Ri . (b) Envelope env(R).

Figure 4.7 Illustration of the envelope operation.

4.4.4 Vertex Enumeration

The operation of extracting the vertices V = {V i }N i=1 of a polytope P given in H-
V

representation is referred to as vertex enumeration and denoted as V = extreme(P).

The necessary computational effort is exponential in the number of facets. The
work in [12] classifies algorithms for vertex enumeration in two groups: graph
traversal and incremental. Graph traversal algorithms construct the graph of ver-
tices and edges of the polyhedron starting from a vertex and finding the other
vertices by traversing the graph. Going form one vertex to another is equivalent
 4.4 Basic Operations on Polytopes 81

to move from one base to another base through pivoting in the simplex algorithm.
The reverse search [11] approach belongs to this class. Incremental algorithms start
from a subset of the polyhedron halfspaces and its associated vertices and itera-
tively intersect the remaining halfspaces to compute a new set of vertices. The
double description method [114] belongs to this class. An efficient implementa-
tion of the double description method is available in [111] and is used in the MPT
toolbox [149].
Converting a V-polytope into an H-polytope corresponds to the convex hull
computation. It can also be done via vertex enumeration. This procedure is based
upon the polar dual [296, p. 61].
Alternatively, using the polar dual, an H-polytope can be converted into a V-
polytope by a convex hull computation. Because of this “symmetry” we say that
vertex enumeration and convex hull computation are dual to each other.

4.4.5 Chebychev Ball

The Chebychev Ball of a polytope P = {x ∈ Rn : P x x ≤ P c }, with P x ∈ RnP ×n ,

P c ∈ RnP , corresponds to the largest radius ball B(xc , R) with center xc , such that
B(xc , R) ⊂ P. The center and radius of the Chebychev ball can be easily found by
solving the following linear optimization problem

max R (4.18a)
xc ,R

subj. to Pix xc + RkPix k2 ≤ Pic , i = 1, . . . , nP , (4.18b)

where Pix denotes the i-th row of P x . This can be proven as follows. Any point x
of the ball can be written as x = xc + v where v is a vector of length less or equal to
R. Therefore the center and radius of the Chebychev ball can be found by solving
the following optimization problem

max R (4.19a)
xc ,R

subj. to Pix (xc + v) ≤ Pic , ∀ v such that kvk2 ≤ R, i = 1, . . . , nP . (4.19b)

Consider the i-th constraint

Pix (xc + v) ≤ Pic , ∀ v such that kvk2 ≤ R.

This can be written as

Pix xc ≤ Pic − Pix v, ∀ v such that kvk2 ≤ R. (4.20)

Constraint (4.20) is satisfied ∀ v such that kvk2 ≤ R if and only if it is satisfied

P x′
at v = kP ix k2 R. Therefore we can rewrite the optimization problem (4.19) as the
i
linear program (4.18).
If the radius obtained by solving (4.18) is R = 0, then the polytope is lower-
dimensional. If R < 0, then the polytope is empty. Therefore, an answer to the
question “is polytope P full-dimensional/empty?” is obtained at the cost of only
82 4 Polyhedra and P-collections

one linear program. Furthermore, for a full-dimensional polytope we also get a

point xc that is in the strict interior of P. However, the center of a Chebyshev Ball
xc in (4.18) is not necessarily a unique point (e.g., when P is a rectangle). There
are other types of unique interior points one could compute for a full-dimensional
polytope, e.g., center of the largest volume ellipsoid, analytic center, etc., but those
computations involve the solution of Semi-Definite Programs (SDPs) and therefore
may be more expensive than the Chebyshev Ball computation [65]. An illustration
of the Chebyshev Ball is given in Figure 4.8.

x2

x1
Figure 4.8 Illustration of the Chebychev ball contained in a polytope P.

4.4.6 Projection
Given a polytope P = {[x′ y ′ ]′ ∈ Rn+m : P x x+ P y y ≤ P c } ⊂ Rn+m the projection
onto the x-space Rn is defined as
projx (P) = {x ∈ Rn : ∃y ∈ Rm : P x x + P y y ≤ P c }. (4.21)
An illustration of a projection operation is given in Figure 4.9. Current projection
x3

x2 x1
Figure 4.9 Illustration of the projection of a 3-dimensional polytope P onto
the plane x3 = 0.

methods that can operate in general dimensions can be grouped into four classes:
 4.4 Basic Operations on Polytopes 83

Fourier elimination [82, 169], block elimination [15], vertex based approaches [113]
and wrapping-based techniques [161]. For a good introduction to projection, we
refer the reader to [161] and the references therein.

4.4.7 Set-Difference
The set-difference of two polytopes Y and R0

R = Y \ R0 = {x ∈ Rn : x ∈ Y, x ∈
/ R0 }, (4.22)

in general, can beSma nonconvex and disconnected

Sm S set T
and can be described as a P-
collection
Sm R = i=1 R i , where Y = i=1 R i (R 0 Y). The P-collection R =
i=1 Ri can be computed by consecutively inverting the half-spaces defining R0
as described in the following Theorem 4.2.
Note that here we use the term P-collection in the dual context of both P-
collection and its underlying set (Definitions 4.3 and 4.4). The precise statement
would say that R = Y \ R0 , where R is the underlying set of the P-collection R =
{Ri }mi=1 . However, whenever it is clear from context, we will use the former, more
compact form.

Theorem 4.2 [44] Let Y ⊆TRn be a polyhedron, R0 = {x ∈ Rn : Ax ≤ b}, and

R̄0 = {x ∈ Y : Ax ≤ b} = R0 Y, where b ∈ Rm , R0 6= ∅ and Ax ≤ b is a minimal
representation of R0 . Also let
 
Ai x > bi
Ri = x∈Y : , i = 1, . . . , m, j = 1, . . . , m − 1.
Aj x ≤ bj , ∀j < i
Sm
Let R = i=1 Ri . Then, R is a P-collection and {R̄0 , R1 , . . . , Rm } is a strict
polyhedral partition of Y.
Proof: (i) We want to prove that given an x ∈ Y, then either x belongs to R̄0
or to Ri for some i but not both. If x ∈ R̄0 , we are done. Otherwise, there exists
an index i such that Ai x > bi . Let i∗ = min{i : Ai x > bi }. Then x ∈ Ri∗ , as
i≤m
∗ ∗
Ai x > bi and Aj x ≤ bj , ∀j < i∗ , by definition of i∗ .
(ii) Let x ∈ R̄0 . Then there does not exist any i such that Ai x > bi , which
implies that x 6∈ Ri , ∀i ≤ m. Let x ∈ Ri and take i > j. Because x ∈ Ri , by
definition of Ri (i > j) Aj x ≤ bj , which implies that x 6∈ Rj . 

As an illustration for the procedure proposed in Theorem 4.2 consider the two-
dimensional case depicted in Figure 4.10(a). Here Y is defined by the inequalities
{x− + − +
1 ≤ x1 ≤ x1 , x2 ≤ x2 ≤ x2 }, and R0 by the inequalities {g1 ≤ 0, . . . , g5 ≤ 0}
where g1 , . . ., g5 are linear in x. The procedure consists of considering one by one
the inequalities which define R0 . Considering, for example, the inequality g1 ≤ 0,
the first set of the rest of the region Y\R0 is given by R1 = {g1 ≥ 0, x1 ≥ x− −
1 , x2 ≤
+
x2 ≤ x2 }, which is obtained by reversing the sign of the inequality g1 ≤ 0 and
removing redundant constraints in Y (see Figure 4.10(b)). Thus, by considering
84 4 Polyhedra and P-collections

the rest of
S5the inequalities we get the partition of the rest of the parameter space
Y\R0 = i=1 Ri , as reported in Figure 4.10(d).

x2 x2
x+ Y x+ Y
2 2

g1 ≥ 0 g1 ≤ 0

R0 R0
R1

x−
2 x−
2
x1 x1
x−
1 x+
1 x−
1 x+
1

(a) Set of parameters Y and initial set R0 . (b) Partition of Y\R0 - Step 1.

x2 x2
x+ Y x+ Y
2 2
R2 R2
g1 ≤ 0
g2 ≥ 0
R0 R0
R1 R1 R3
R5
R4

x−
2 x−
2
x1 x1
x−
1 x+
1 x−
1 x+
1

(c) Partition of Y\R0 - Step 2. (d) Final partition of Y\R0 .

Figure 4.10 Two dimensional example illustrating Theorem 4.2. Partition

of the rest of the space Y\R0 .

Remark 4.1 The set difference of two intersecting polytopes P and Q (or any closed
sets) is not a closed set. This means that some borders of polytopes Ri from a P-
collection R = P \ Q are open, while other borders are closed. Even though it is
possible to keep track of the origin of particular borders of Ri , thus specifying if they
are open or closed, we are not doing so in the algorithms described in this book nor
in MPT [149]. In computations, we will henceforth only consider the closure of the
sets Ri .

The set difference between two P-collections P and Q can be computed as

 4.4 Basic Operations on Polytopes 85

described in [22, 134, 243].

4.4.8 Pontryagin Difference

The Pontryagin difference (also known as Minkowski difference) of two polytopes

P and Q is a polytope

P ⊖ Q = {x ∈ Rn : x + q ∈ P, ∀q ∈ Q}. (4.23)

The Pontryagin difference can be efficiently computed for polytopes by solving a

sequence of LPs as follows. Define the P and Q as

P = {y ∈ Rn : P y y ≤ P b }, Q = {z ∈ Rn : Qz z ≤ Qb }, (4.24)

then

W =P ⊖Q (4.25a)
n y b y
= {x ∈ R : P x ≤ P − H(P , Q)}, (4.25b)

where the i-th element of H(P y , Q) is

Hi (P y , Q) = max Piy x, (4.26)

x∈Q

and Piy is the i-th row of the matrix P y . Note that for special cases (e.g., when Q
is a hypercube), more efficient computational methods exist [175]. An illustration
of the Pontryagin difference is given in Figure 4.11a.

8 P ⊕Q
6 P 6 P
4 P ⊖Q
2 2
Q Q
x2

x2

-2 -2
-4
-6 -6
-8
-6 -4 -2 2 4 6 -8 -6 -2 2 6 8
x1 x1
(a) Illustration of the Pontrya- (b) Illustration of the
gin difference P ⊖ Q. Minkowski sum P ⊕ Q.

Figure 4.11 Illustration of the Pontryagin difference and Minkowski sum

operations.
86 4 Polyhedra and P-collections

4.4.9 Minkowski Sum

The Minkowski sum of two polytopes P and Q is a polytope

P ⊕ Q = {y + z ∈ Rn : y ∈ P, z ∈ Q}. (4.27)
The Minkowski sum is a computationally expensive operation which requires either
vertex enumeration and convex hull computation in n-dimensions or a projection
from 2n down to n dimensions. The implementation of the Minkowski sum via
projection is described below. If the polytopes are defined by
P = {y ∈ Rn : P y y ≤ P c }, Q = {z ∈ Rn : Qz z ≤ Qc },
then it holds that
W = P ⊕Q
n o
= x ∈ Rn : x = y + z, P y y ≤ P c , Qz z ≤ Qc , y, z ∈ Rn
n o
= x ∈ Rn : ∃y ∈ Rn , subj. to P y y ≤ P c , Qz (x − y) ≤ Qc
n      c o
n n 0 Py x P
= x ∈ R : ∃y ∈ R , subj. to ≤
Qz −Qz y Qc
!
n x 
0 Py
    c o
x P
n+n
= projx ∈R : ≤ .
y Qz −Qz y Qc

Both the projection and vertex enumeration based methods are implemented in
the MPT toolbox [149]. An illustration of the Minkowski sum is given in Figure
4.11b.

Remark 4.2 The Minkowski sum is not the complement of the Pontryagin difference.
For two polytopes P and Q, it holds that (P ⊖ Q) ⊕ Q ⊆ P. This is illustrated in
Figure 4.12.

4.4.10 Polyhedra Union

Consider the following basic problem in polyhedral computation: given two poly-
hedra P ⊂ Rn and Q ⊂ Rn , decide whether their union is convex, and, if so,
compute it. There are three classes of algorithms for the given problem depending
on their representation: (1) P and Q are given in H-representation, (2) P and Q
are given in V-representation, (3) both H- and V-representations are available for
P and Q. Next we present an algorithm for case (1). Case (2), case (3) and the
computational complexity of all three cases are discussed in [38].
Recall the definition of envelope in Section 4.4.3. By definition, it is easy to see
that env(P, Q) is convex and that
P ∪ Q ⊆ env(P, Q). (4.28)
We have the following theorem.
 4.4 Basic Operations on Polytopes 87

3 3 3
P P (P ⊖ Q) ⊕ Q
2 2
1
Q P ⊖Q P ⊖Q
x2

x2

x2
-1
-2 -2
-3 -3 -3
-3 -1 1 3 -3 -2 2 3 -3 -2 2 3
x1 x1 x1
(a) Two polytopes P and Q. (b) Polytope P and Pontryagin (c) Polytope P ⊖ Q and the set
difference P ⊖ Q. (P ⊖ Q) ⊕ Q.

Figure 4.12 Illustration that the Minkowski sum is not the complement of
the Pontryagin difference. (P ⊖ Q) ⊕ Q ⊆ P.

Theorem 4.3 P ∪ Q is convex ⇔ P ∪ Q = env(P, Q).

Algorithm 4.2 checks for points x∗ ∈ env(P, Q) outside P ∪ Q.

Algorithm 4.2 Algorithm for recognizing the convexity of P ∪ Q

Input P, Q
Output Flag convex, if convex=true H-representation of P ∪ Q
convex ← true
Construct env(P, Q) by removing non-valid constraints (see Fig. 4.7)
Let Ãx ≤ α̃, B̃x ≤ β̃ be the set of removed constraints from P and Q
Let env(P, Q) = {x : Cx ≤ γ} the resulting envelope
Remove from env(P, Q) possible duplicates (Bj , βj ) = (σAi , σαi ), σ > 0
For each pair Ãi x ≤ α̃i , B̃j x ≤ β̃j Do
Determine ǫ∗ by solving the LP

ǫ∗ = max(x,ǫ) ǫ
subj. to Ãi x ≥ α̃i + ǫ
B̃j x ≥ β̃j + ǫ
Cx ≤ γ

If ǫ∗ > 0 Stop, Return convex=false

Return env(P, Q)

Note that if ǫ∗ = 0 for each i, j, then the union is convex and equals env(P, Q).
On the other hand, ǫ∗ > 0 indicates the existence of a point x ∈ env(P, Q) outside
P ∪ Q.
For recognizing convexity and computing the union of k polyhedra, the test
can be modified by checking each k-tuple of removed constraints. Let m̃1 , . . ., m̃k
88 4 Polyhedra and P-collections

be the number of removed constrains from the polyhedra

Qk P1 , . . . , Pk , respectively.
Then similarly to the for loop in Algorithm 4.2, i=1 m̃i linear programs need to
be solved in the general case.

4.4.11 Affine Mappings and Polyhedra

This section deals with the composition of affine mappings and polyhedra. Consider
a polyhedron P = {x ∈ Rn : P x x ≤ P c }, with P x ∈ RnP ×n and an affine mapping
f (z)
f : z ∈ Rm 7→ Az + b, A ∈ RmA ×m , b ∈ RmA . (4.29)
Let mA = n. We define the composition of P and f as the following polyhedron

P ◦ f = {z ∈ Rm : P x f (z) ≤ P c } = {z ∈ Rm : P x Az ≤ P c − P x b}. (4.30)

Let m = n. We define the composition of f and P as the following polyhedron

f ◦ P = {y ∈ RmA : ∃x ∈ P and y = Ax + b }. (4.31)

The polyhedron f ◦ P in (4.31) can be computed as follows. Let us write P in

V-representation
P = conv(V ), (4.32)
and let us map the set of vertices V = {V1 , . . . , Vk } through the transformation f .
Because the transformation is affine, the set f ◦ P is simply the convex hull of the
transformed vertices

f ◦ P = conv(F ), F = {AV1 + b, . . . , AVk + b}. (4.33)

The polyhedron f ◦ P in (4.31) can be computed immediately if mA = m = n and

A is invertible. In this case, from the definition in (4.31), x = A−1 y − A−1 b and
therefore
f ◦ P = {y ∈ RmA : P x A−1 y ≤ P c + P x A−1 b}. (4.34)
Vertex enumeration can be avoided even if A is not invertible and mA ≥ m = n by
using a QR decomposition of the matrix A.

Remark 4.3 Often in the literature the symbol “◦” is omitted for linear maps f = Az.
Therefore, AP refers to the operation A ◦ P and PA refers to the operation P ◦ A.

4.5 Operations on P-collections

This section covers some results and algorithms which are specific to operations
with P-collections. P-collections are unions of polytopes (see Definition 4.3) and
therefore the set of points contained in a P-collection can be represented in an
infinite number of ways, i.e., the P-collection representation is not unique. For
 4.5 Operations on P-collections 89

example, one can subdivide any polytope P into a number of smaller polytopes
whose union is a P-collection which covers P. Note that the complexity of all
subsequent computations depends strongly on the number of polytopes representing
a P-collection. The smaller the cardinality of a P-collection, the more efficient the
computations. The reader is referred to [243, 242] for proofs and comments on
computational efficiency.

4.5.1 Set-Difference

The first two results given here show how the set difference of a P-collection and a
P-collection (or polyhedron) may be computed.
S
Lemma 4.1 Let C = j∈{1,...,J} Cj be a P-collection, where all the Cj , j ∈ {1, . . . , J},
S
are nonempty polyhedra. If D is a nonempty polyhedron, then C\D = j∈{1,...,J} (Cj \
D) is a P-collection.
S S
Lemma 4.2 Let the sets C = j∈{1,...,J} Cj and D = y=1,...,Y Dy be P-collections,
where all the Cj , j ∈ {1, . . . , J}, and Dy , y ∈ {1, . . . , Y }, are nonempty polyhedra.
If E0 = C and Ey = Ey−1 \ Dy , y ∈ {1, . . . , Y } then C \ D = EY is a P-collection.

The condition C ⊆ D can be easily verified since C ⊆ D ⇔ C \ D = ∅. Similarly

C = D is also easily verified since

C = D ⇔ (C \ D = ∅ and D \ C = ∅).

Next, an algorithm for computing the Pontryagin difference of a P-collection and

a polytope is presented. If S and B are two subsets of Rn then S ⊖B = [S c ⊕ (−B)]c
where (·)c denotes the set complement. The following algorithm implements the
computation of the Pontryagin difference of a P-collection C = ∪j∈{1,...,J} Cj , where
Cj , j ∈ {1, . . . , J} are polytopes in Rn , and a polytope B ⊂ Rn .

Algorithm 4.3 Pontryagin Difference for P-collections, C ⊖ B

Input: P-collection C, polytope B
Output: P-collection G ← C ⊖ B
H ← env(C) (or H ← conv(C))
D ← H⊖B
E ←H\C
F ← E ⊕ (−B)
G ←D\F

Remark 4.4 Note that H in Algorithm 4.3 can be any convex set containing the
P-collection C. The computation of H is generally more efficient if the envelope
operation is used instead of convex hull.
90 4 Polyhedra and P-collections

S S
Remark 4.5 It is important to note that ( j∈{1,...,J } Cj ) ⊖ B =
6 j∈{1,...,J } (Cj ⊖ B),
where B and Cj are polyhedra; hence, the relatively high computational effort of
computing the Pontryagin difference of a P-collection and a polytope.

C2 C1

B
x2

x2
x1 x1
S
(a) j∈{1,...,J } Cj and B. (b) H = conv(C).

E
H
D
x2

x2

x1 x1
(c) D = H ⊖ B. (d) E = H \ C.

F F D
x2

x2

x1 x1
(e) F = E ⊕ (−B). (f) G = D \ F .

Figure 4.13 Illustration of Algorithm 4.3. Computing the set difference

G = C ⊖ B between the P-collection C = C1 ∪ C2 and the polytope B.
 4.5 Operations on P-collections 91

Theorem 4.4 (Computation of Pontryagin Difference, [242]) The output of

Algorithm 4.3 is G = C ⊖ B.
Proof: It holds by definition that

D = H ⊖ B = {x : x + w ∈ H, ∀w ∈ B},
E = H \ C = {x : x ∈ H and x ∈
/ C}.

By the definition of the Minkowski sum:

F = E ⊕ (−B) = {x : x = z + w, z ∈ E, w ∈ (−B)}
= {x : ∃w ∈ (−B), subj. to x − w ∈ E}.

By definition of the set difference:

D \ F = {x : x ∈ D and x ∈
/ F}
= {x ∈ D : ∄ w ∈ B s.t. x + w ∈ E}
= {x ∈ D : x + w ∈
/ E, ∀w ∈ B}.

From the definition of the set D:

D \ F = {x : x + w ∈ H and x + w ∈
/ E, ∀w ∈ B}

and from the definition of the set E and because C ⊆ H:

D \ F = {x : x + w ∈ H and (x + w ∈
/ H or x + w ∈ C) ∀w ∈ B}
= {x : x + w ∈ C, ∀w ∈ B}
= C ⊖ B.


Algorithm 4.3 is illustrated on a sample P-collection in Figures 4.13(a) to 4.13(f).

Remark 4.6 It should be noted that Algorithm 4.3 for computation of the Pontrya-
gin difference is conceptually similar to the algorithm proposed in [263, 172]. The
envelope operation H = env(C) employed in Algorithm 4.3 might reduce the number
of sets obtained when computing H \ C, which in turn results in fewer Minkowski set
additions. Since the computation of a Minkowski set addition is expensive, a runtime
improvement can be expected.

4.5.2 Polytope Covering

The problem of checking if some P polytope is covered with the union of other
polytopes, i.e., a P-collection Q = ∪i Qi is discussed in this section. We consider
two related problems:
92 4 Polyhedra and P-collections

polycover: Check if P ⊆ Q, and

regiondiff: Compute P-collection R = P \ Q.
Clearly, polycover is just a special case of regiondiff, where the resulting P-
collection R = ∅. Also, it is straightforward to extend the above problems to the
case where both P and Q are both P-collections.
One idea of solving the polycover problem is inspired by the following observa-
tion
P ⊆ Q ⇔ P = ∪i (P ∩ Qi ).
Therefore, we could create Ri = P ∩ Qi , for i = 1, . . . , NQ and then compute the
union of the collection of polytopes {Ri } by using the polyunion algorithm for
computing the convex union of H-polyhedra reported discussed in Section 4.4.10.
If polyunion succeeds (i.e., the union is a convex set) and the resulting polytope is
equal to P then P is covered by Q, otherwise it is not. However, this approach is
computationally very expensive. More details can be found in [123].

4.5.3 Union of P-collections

Consider a P-collection P = {Pi }pi=1 . We study the problem of finding a minimal
representation of P by merging one or more polyhedra belonging to the P-collection.
Clearly one could use the polyunion algorithm presented in Section 4.4.10 for all
possible subsets of the P-collection and solve the problem by comparing all solu-
tions. However this approach is not computationally efficient.
Our interest in this problem will be clear later in this book (Section 11.1) when
computing the PWA state feedback control law to optimal control problems. Once
the PWA state feedback control law has been derived, the memory requirement and
the on-line computation time are linear in the number of polyhedra of the feedback
law when using standard brute force search. Therefore, we will be interested in the
problem of finding a minimal representation of piecewise affine (PWA) systems,
or more specifically, for a given PWA system, we solve the problem of deriving a
PWA system, that is both equivalent to the former and minimal in the number of
regions. This is done by associating a different color to a different feedback law,
and then collecting the polyhedra with the same color. Then, for a given color, we
try to merge the corresponding P-collection. If the number of polyhedra with the
same affine dynamic is large, the number of possible polyhedral combinations for
merging explodes. As most of these unions are not convex or even not connected
and thus cannot be merged, trying all combinations using standard techniques
based on linear programming (LP) is prohibitive. Furthermore, our objective here
is not only to reduce the number of polyhedra but rather to find the minimal and
thus optimal number of disjoint polyhedra. This problem is known to be N P-hard.
In [123] details on the solutions of these problems can be found.
 Part II

Multiparametric Programming
 Chapter 5

Multiparametric Nonlinear
Programming

The operations research community has addressed parameter variations in mathe-

matical programs at two levels: sensitivity analysis, which characterizes the change
of the solution with respect to small perturbations of the parameters, and para-
metric programming, where the characterization of the solution for a full range of
parameter values is studied. In this chapter we introduce the concept of multi-
parametric programming and recall the main results of nonlinear multiparametric
programming. The main goal is to make the reader aware of the complexities of
general multiparametric nonlinear programming. Later in this book we will use
multiparametric programming to characterize and compute the state feedback so-
lution of optimal control problems. There we will only make use of Corollary 5.1 for
multiparametric linear programs and Corollary 5.2 for multiparametric quadratic
programs, which show that these specific programs are “well behaved”.

5.1 Introduction to Multiparametric Programs

Consider the mathematical program

J ∗ (x) = inf J(z, x)

z
subj. to g(z, x) ≤ 0

where z is the optimization vector and x is a vector of parameters. We are interested

in studying the behavior of the value function J ∗ (x) and the optimizer z ∗ (x) as
we vary the parameter x. Mathematical programs where x is a scalar are referred
to as parametric programs, while programs where x is a vector are referred to as
multiparametric programs.
There are several reasons to look for efficient solvers of multiparametric pro-
grams. Typically, mathematical programs are affected by uncertainties due to
factors that are either unknown or that will be decided later. Parametric program-
96 5 Multiparametric Nonlinear Programming

ming systematically subdivides the space of parameters into characteristic regions,

which depict the feasibility and corresponding performance as a function of the
uncertain parameters, and hence provide the decision maker with a complete map
of various outcomes.
Our interest in multiparametric programming arises from the field of system
theory and optimal control. For example, for discrete-time dynamical systems,
finite time constrained optimal control problems can be formulated as mathematical
programs where the cost function and the constraints are functions of the initial
state of the dynamical system. In particular, Zadeh and Whalen [293] appear to
have been the first ones to express the optimal control problem for constrained
discrete-time linear systems as a linear program. We can interpret the initial state
as a parameter. By using multiparametric programming we can characterize and
compute the solution of the optimal control problem explicitly as a function of the
initial state.
We are further motivated by the model predictive control (MPC) technique.
MPC is very popular in the process industry for the automatic regulation of process
units under operating constraints, and has attracted a considerable research effort
in the last two decades. MPC requires an optimal control problem to be solved
on-line in order to compute the next command action. This mathematical program
depends on the current sensor measurements. The computation effort can be moved
off-line by solving multiparametric programs, where the control inputs are the
optimization variables and the measurements are the parameters. The solution of
the parametric program problem is a control law describing the control inputs as
function of the measurements. Model Predictive Control and its multiparametric
solution are discussed in Chapter 12.
In the following we will present several examples that illustrate the parametric
programming problem and hint at some of the issues that need to be addressed by
the solvers.

Example 5.1 Consider the parametric quadratic program

J ∗ (x) = min J(z, x) = 21 z 2 + 2xz + 2x2

z
subj. to z ≤ 1 + x,

where x ∈ R. Our goals are:

1. to find z ∗ (x) = arg minz J(z, x),

2. to find all x for which the problem has a solution, and
3. to compute the value function J ∗ (x).

The Lagrangian is

1 2
L(z, x, u) = z + 2xz + 2x2 + u(z − x − 1)
2

and the KKT conditions are (see Section 2.3.3 for KKT conditions for quadratic
5.1 Introduction to Multiparametric Programs 97

programs)

z + 2x + u =0 (5.1a)
u(z − x − 1) =0 (5.1b)
u ≥0 (5.1c)
z−x−1 ≤ 0. (5.1d)

Consider (5.1) and the two strictly complementary cases:

 ∗
z + 2x + u = 0  z =x+1
A. z − x − 1 = 0 ⇒ J ∗ = 92 x2 + 3x + 1
2

u≥0 x ≤ − 13
 ∗ (5.2)
z + 2x + u = 0  z = −2x
B. z − x − 1 < 0 ⇒ J∗ = 0

u=0 x > − 13

This solution is depicted in Figure 5.1.

2
J ∗ (x)
1

0
z ∗ (x)
-1

-2
-2 -1 0 1 2
x
Figure 5.1 Example 5.1. Optimizer z ∗ (x) and value function J ∗ (x) as a
function of the parameter x.

The above simple procedure, which required nothing but the solution of the KKT
conditions, yielded the optimizer z ∗ (x) and the value function J ∗ (x) for all values
of the parameter x. The set of admissible parameter values was divided into two
critical regions, defined by x ≤ − 13 and x > − 31 . In the region x ≤ − 31 the
inequality constraint is active and the Lagrange multiplier is greater or equal than
zero, in the other region x > − 13 the inequality constraint is not active and the
Lagrange multiplier is equal to zero.
In general, when there are more than one inequality constraints a critical region
is defined by the set of inequalities that are active in the region. Throughout a
critical region the conditions for optimality derived from the KKT conditions do
not change. For our example, in each critical region the optimizer z ∗ (x) is affine and
the value function J ∗ (x) is quadratic. Thus, considering all x, z ∗ (x) is piecewise
affine and J ∗ (x) is piecewise quadratic. Both z ∗ (x) and J ∗ (x) are continuous, but
z ∗ (x) is not continuously differentiable.
98 5 Multiparametric Nonlinear Programming

In much of this book we will be interested in two questions: how to find the value
function J ∗ (x) and the optimizer z ∗ (x) and what are their structural properties,
e.g., continuity, differentiability and convexity. Such questions have been addressed
for general nonlinear multiparametric programming by several authors in the past
(see [18] and references therein), by making use of quite involved mathematical
theory based on the continuity of point-to-set maps. The concept of point-to-set
maps will not be much used in this book. However, it represents a key element for a
rigorous mathematical description of the properties of a nonlinear multiparametric
program and hence a few key theoretical results for nonlinear multiparametric
programs based on the point-to-set map formalism will be discussed in this chapter.

5.2 General Results for Multiparametric Nonlinear Programs

Consider the nonlinear mathematical program dependent on a parameter x ap-
pearing in the cost function and in the constraints

J ∗ (x) = inf J(z, x)

z (5.3)
subj. to g(z, x) ≤ 0,

where z ∈ Z ⊆ Rs is the optimization vector, x ∈ X ⊆ Rn is the parameter vector,

J : Rs × Rn → R is the cost function and g : Rs × Rn → Rng are the constraints.
We denote by gi (z, x) the i-th component of the vector-valued function g(z, x).
A small perturbation of the parameter x in the mathematical program (5.3) can
cause a variety of results. Depending on the properties of the functions J and g the
solution z ∗ (x) may vary smoothly or change abruptly as a function of x. Denote
by 2Z the set of subsets of Z. We denote by R the point-to-set map which assigns
to a parameter x ∈ X the (possibly empty) set R(x) of feasible variables z ∈ Z,
R : X 7→ 2Z
R(x) = {z ∈ Z : g(z, x) ≤ 0}, (5.4)
by K∗ the set of feasible parameters

K∗ = {x ∈ X : R(x) 6= ∅}, (5.5)

by J ∗ (x) the real-valued function that expresses the dependence of the minimum
value of the objective function over K∗ on x

J ∗ (x) = inf {J(z, x) : z ∈ R(x)}, (5.6)

z

and by Z ∗ (x) the point-to-set map which assigns the (possibly empty) set of opti-
mizers z ∗ ∈ 2Z to a parameter x ∈ X

Z ∗ (x) = {z ∈ R(x) : J(z, x) = J ∗ (x)}. (5.7)

J ∗ (x) will be referred to as optimal value function or simply value function, Z ∗ (x)
will be referred to as the optimal set. If Z ∗ (x) is a singleton for all x, then z ∗ (x) =
Z ∗ (x) will be called optimizer function. We remark that R and Z ∗ are set-valued
5.2 General Results for Multiparametric Nonlinear Programs 99

functions. As discussed in the notation section, with abuse of notation J ∗ (x) and
Z ∗ (x) will denote both the functions and the value of the functions at the point x.
The context will make clear which notation is being used.
The book by Bank and coauthors [18] and Chapter 2 of [107] describe conditions
under which the solution of the nonlinear multiparametric program (5.3) is locally
well behaved and establish properties of the optimal value function and of the
optimal set. The description of such conditions requires the definition of continuity
of point-to-set maps. Before introducing this concept we will show through two
simple examples that continuity of the constraints gi (z, x) with respect to z and
x is not enough to imply any “regularity” of the value function and the optimizer
function.

Example 5.2 [18, p. 12]

Consider the following problem:

J ∗ (x) = minz x2 z 2 − 2x(1 − x)z

subj. to z≥0 (5.8)
0 ≤ x ≤ 1.

Cost and constraints are continuous and continuously differentiable. For 0 < x ≤ 1
the optimizer function is z ∗ = (1 − x)/x and the value function is J ∗ (x) = −(1 − x)2 .
For x = 0, the value function is J ∗ (x) = 0 while the optimal set is Z ∗ = {z ∈ R : z ≥
0}. Thus, the value function is discontinuous at 0 and the optimal set is single-valued
for all 0 < x ≤ 1 and set-valued for x = 0.

Example 5.3 Consider the following problem:

J ∗ (x) = inf z
z
subj. to zx ≥ 0 (5.9)
−10 ≤ z ≤ 10
−10 ≤ x ≤ 10,

where z ∈ R and x ∈ R. For each fixed x the set of feasible z is a segment. The
point-to-set map R(x) is plotted in Figure 5.2(a). The function g1 : (z, x) 7→ zx is
continuous. Nevertheless, the value function J ∗ (x) = z ∗ (x) has a discontinuity at the
origin as can be seen in Figure 5.2(b).

Example 5.4 Consider the following problem:

J ∗ (x) = inf −z1

z1 ,z2
subj. to g1 (z1 , z2 ) + x ≤ 0 (5.10)
g2 (z1 , z2 ) + x ≤ 0,

where examples of the functions g1 (z1 , z2 ) and g2 (z1 , z2 ) are plotted in Figures 5.3(a)–
5.3(c). Figures 5.3(a)–5.3(c) also depict the point-to-set map R(x) = {[z1 , z2 ] ∈
R2 |g1 (z1 , z2 ) + x ≤ 0, g2 (z1 , z2 ) + x ≤ 0} for three fixed x. Starting from x = x̄1 ,
as x increases, the domain of feasibility in the space z1 , z2 shrinks; at the beginning
it is connected (Figure 5.3(a)), then it becomes disconnected (Figure 5.3(b)) and
eventually connected again (Figure 5.3(c)). No matter how smooth one chooses the
100 5 Multiparametric Nonlinear Programming

10 10

5 5

J ∗ (x)
R(x)
0 0

-5 -5

-10 -10
-10 -5 0 5 10 -10 -5 0 5 10
x x
(a) Point-to-set map R(x). (b) Value function J ∗ (x).

Figure 5.2 Example 5.3. Point-to-set map and value function.

functions g1 and g2 , the value function J ∗ (x) = −z1∗ (x) will have a discontinuity at
x = x̄3 .

Examples 5.2, 5.3, 5.4 show the case of simple and smooth constraints which lead
to a discontinuous behavior of value function and in Examples 5.3, 5.4 we also
observe discontinuity of the optimizer function. The main causes are:
• in Example 5.2 the feasible vector space Z is unbounded (z ≥ 0),
• in Examples 5.3 and 5.4 the feasible point-to-set map R(x) (defined in (5.4))
is discontinuous, as defined precisely below.
In the next sections we discuss both cases in detail.

Continuity of Point-to-Set Maps

Consider a point-to-set map R : x ∈ X 7→ R(x) ∈ 2Z . We give the following
definitions of open and closed maps according to Hogan [152]:

Definition 5.1 The point-to-set map R(x) is open at a point x̄ ∈ K∗ if for all
sequences {xk } ⊂ K∗ with xk → x̄ and for all z̄ ∈ R(x̄) there exists an integer m
and a sequence {z k } ∈ Z such that z k ∈ R(xk ) for k ≥ m and z k → z̄.

Definition 5.2 The point-to-set map R(x) is closed at a point x̄ ∈ K∗ if for each
pair of sequences {xk } ⊂ K∗ , and z k ∈ R(xk ) with the properties

xk → x̄, z k → z̄.

it follows that z̄ ∈ R(x̄).

We define the continuity of a point-to-set map according to Hogan [152] as follows:
5.2 General Results for Multiparametric Nonlinear Programs 101

z2 z2

g1 (z1 , z2 ) = x̄1
g1 (z1 , z2 ) = x̄2

z1 z1

g2 (z1 , z2 ) = x̄2
g2 (z1 , z2 ) = x̄1

(a) Set R(x̄1 ) shaded in gray. (b) Set R(x̄2 ) shaded in gray.

z2 J ∗ (x)

g1 (z1 , z2 ) = x̄3

z1 x̄1 x̄3 x

g2 (z1 , z2 ) = x̄3

(c) Set R(x̄3 ) shaded in gray. (d) Value function J ∗ (x).

Figure 5.3 Example 5.4. Problem (5.10). (a)-(c) Projections of the point-to-
set map R(x) for three values of the parameter x: x̄1 < x̄2 < x̄3 ; (d) Value
function J ∗ (x).

Definition 5.3 The point-to-set map R(x) is continuous at a point x̄ in K∗ if it

is both open and closed at x̄. R(x) is continuous in K∗ if R(x) is continuous at
every point x in K∗ .
The definitions above are illustrated through two examples.

Example 5.5 Consider

R(x) = {z ∈ R : z ∈ [0, 1] if x < 1, z ∈ [0, 0.5] if x ≥ 1.}
The point-to-set map R(x) is plotted in Figure 5.4. It is easy to see that R(x) is
not closed but open. In fact, if one considers a sequence {xk } that converges to
102 5 Multiparametric Nonlinear Programming

x̄ = 1 from the left and extracts the sequence {z k } plotted in Figure 5.4 converging
to z̄ = 0.75, then z̄ ∈
/ R(x̄) since R(1) = [0, 0.5].
z
1
{zk }

0.5

{xk }
1 x

Figure 5.4 Example 5.5. Open and not closed point-to-set map R(x).

Example 5.6 Consider

R(x) = {z ∈ R : z ∈ [0, 1] if x ≤ 1, z ∈ [0, 0.5] if x > 1}

The point-to-set map R(x) is plotted in Figure 5.5. It is easy to verify that R(x) is
closed but not open. Choose z̄ = 0.75 ∈ R(x̄). Then, for any sequence {xk } that
converges to x̄ = 1 from the right, one is not able to construct a sequence {z k } ∈ Z
such that z k ∈ R(xk ) and z k → z̄. In fact, such sequence z k will always be bounded
between 0 and 0.5.
z
1

0.5
{zk }
{xk }
1 x

Figure 5.5 Example 5.6. Closed and not open point-to-set map R(x).

Remark 5.1 We remark that “upper semicontinuous” and “lower semicontinuous”

definitions of point-to-set map are sometimes preferred to open and closed defini-
tions [47, p. 109]. In [18, p. 25] nine different definitions for the continuity of
point-to-set maps are introduced and compared. We will not give any details on this
subject and refer the interested reader to [18, p. 25].

The examples above are only illustrative. In general, it is difficult to test if a

set is closed or open by applying the definitions. Several authors have proposed
sufficient conditions on gi which imply the continuity of R(x). In the following we
introduce a theorem which summarizes the main results of [254, 93, 152, 47, 18].
5.2 General Results for Multiparametric Nonlinear Programs 103

Theorem 5.1 If Z is convex, if each component gi (z, x) of g(z, x) is continuous

on Z × x̄ and convex in z for each x̄ ∈ X and if there exists a z̄ such that g(z̄, x̄) < 0,
then R(x) is continuous at x̄.
The proof is given in [152, Theorems 10 and 12]. An equivalent proof can be also
derived from [18, Theorem 3.1.1 and Theorem 3.1.6]. 

Remark 5.2 Note that convexity in z for each x is not enough to imply the continuity
of R(x) everywhere in K∗ . In [18, Example 3.3.1 on p. 53] an example illustrating
this is presented. We remark that in Example 5.3 the origin does not satisfy the last
hypothesis of Theorem 5.1.

Remark 5.3 If the assumptions of Theorem 5.1 hold at each x̄ ∈ X then one can
extract a continuous single-valued function (often called a “continuous selection”) r :
X 7→ R such that r(x) ∈ R(x), ∀x ∈ X , provided that Z is finite-dimensional. Note
that convexity of R(x) is a critical assumption [18, Corollary 2.3.1]. The following
example shows a point-to-set map R(x) not convex for a fixed x which is continuous
but has no continuous selection [18, p. 29]. Let Λ be the unit disk in R2 , define R(x)
as
1
x ∈ Λ 7→ R(x) = {z ∈ Λ : kz − xk2 ≥ }. (5.11)
2
It can be shown that the point-to-set map R(x) in (5.11) is continuous according to
Definition 5.3. In fact, for a fixed x̄ ∈ Λ the set R(x̄) is the set of points in the unit
disk outside the disk centered in x̄ and of radius 0.5 (next called the half disk ); small
perturbations of x̄ yield small translations of the half disk inside the unit disk for
all x̄ ∈ Λ. However R(x) has no continuous selection. Assume that there exists a
continuous selection r : x ∈ Λ 7→ r(x) ∈ Λ. Then, there exists a point x∗ such that
x∗ = r(x∗ ). Since r(x) ∈ R(x), ∀x ∈ Λ, there exists a point x∗ such that x∗ ∈ R(x∗ ).
This is not possible since for all x∗ ∈ Λ, x∗ ∈
/ R(x∗ ) (recall that R(x∗ ) is set of points
in the unit disk outside the disk centered in x∗ and of radius 0.5).

Remark 5.4 Let Λ be the unit disk in R, define R(x) as

1
x ∈ Λ 7→ R(x) = {z ∈ Λ : |z − x| ≥ }. (5.12)
2
R(x) is closed and not open and it has no continuous selection.

Remark 5.5 Based on [18, Theorem 3.2.1-(I) and Theorem 3.3.3], the hypotheses of
Theorem 5.1 can be relaxed for affine gi (z, x). In fact, affine functions are weakly
analytic functions according to [18, p. 47]. Therefore, we can state that if Z is convex,
if each component gi (z, x) of g(z, x) is an affine function, then R(x) is continuous at
x̄ for all x̄ ∈ K∗ .

Properties of the Value Function and Optimal Set

Consider the following definition

104 5 Multiparametric Nonlinear Programming

Definition 5.4 A point-to-set map R(x) is said to be uniformly compactS near x̄

if there exist a neighborhood N of x̄ such that the closure of the set x∈N R(x) is
compact.
Now we are ready to state the two main theorems on the continuity of the value
function and of the optimizer function.

Theorem 5.2 [152, Theorem 7] Consider problem (5.3)–(5.4). If R(x) is a con-

tinuous point-to-set map at x̄ and uniformly compact near x̄ and if J is continuous
on x̄ × R(x̄), then J ∗ is continuous at x̄.

Theorem 5.3 [152, Corollary 8.1] Consider problem (5.3)–(5.4). If R(x) is a

continuous point-to-set map at x̄, J is continuous on x̄ × R(x̄), Z ∗ is nonempty
and uniformly compact near x̄, and Z ∗ (x̄) is single valued, then Z ∗ is continuous
at x̄.

Remark 5.6 Equivalent results of Theorems 5.2 and 5.3 can be found in [47, p. 116]
and [18, Chapter 4.2].

Example 5.7 Example 5.2 revisited

Consider Example 5.2. The feasible map R(x) is unbounded and therefore it does not
satisfy the assumptions of Theorem 5.2 (since it is not uniformly compact). Modify
Example 5.2 as follows:

J ∗ (x) = minz x2 z 2 − 2x(1 − x)z

subj. to 0≤z≤M (5.13)
0≤x≤1

with M ≥ 0. The solution can be computed immediately. For 1/(1 + M ) < x ≤ 1

the optimizer function is z ∗ = (1 − x)/x and the value function is J ∗ (x) = −(1 − x)2 .
For 0 < x ≤ 1/(1 + M ), the value function is J ∗ (x) = x2 M 2 − 2M x(1 − x) and the
optimizer function is z ∗ = M . For x = 0, the value function is J ∗ (x) = 0 while the
optimal set is Z ∗ = {z ∈ R : 0 ≤ z ≤ M }.
No matter how large we choose M , the value function and the optimal set are con-
tinuous for all x ∈ [0, 1].

Example 5.8 Example 5.3 revisited

Consider Example 5.3. The feasible map R(x) is not continuous at x = 0 and therefore
it does not satisfy the assumptions of Theorem 5.2. Modify Example 5.3 as follows:

J ∗ (x) = inf z
z
subj. to zx ≥ −ε (5.14)
−10 ≤ z ≤ 10
−10 ≤ x ≤ 10,

where ε > 0. The value function and the optimal set are depicted in Figure 5.6 for
ε = 1. No matter how small we choose ε, the value function and the optimal set are
continuous for all x ∈ [−10, 10]
5.2 General Results for Multiparametric Nonlinear Programs 105

10 10

5 5

J ∗ (x)
R(x)

0 0

-5 -5

-10 -10
-10 -5 0 5 10 -10 -5 0 5 10
x x
(a) Point-to-set map R(x). (b) Value function J ∗ (x).

Figure 5.6 Example 5.8. Point-to-set map and value function.

The following corollaries consider special classes of parametric problems.

Corollary 5.1 (mp-LP) Consider the special case of the multiparametric pro-
gram (5.3). where the objective and the constraints are linear
J ∗ (x) = min c′ z
z (5.15)
subj. to Gz ≤ w + Sx,
and assume that there exists an x̄ and z ∗ (x̄) with a bounded cost J ∗ (x̄). Then, K∗
is a nonempty polyhedron, J ∗ (x) is a continuous and convex function on K∗ and
the optimal set Z ∗ (x) is a continuous point-to-set map on K∗ .
Proof: See Theorem 5.5.1 in [18] and the bottom of page 138 in [18]. 

Corollary 5.2 (mp-QP) Consider the special case of the multiparametric pro-
gram (5.3). where the objective is quadratic and the constraints are linear
1 ′
J ∗ (x) = min 2 z Hz + z ′F
z (5.16)
subj. to Gz ≤ w + Sx,
and assume that H ≻ 0 and that there exists (z̄, x̄) such that Gz̄ ≤ w + S x̄. Then,
K∗ is a nonempty polyhedron, J ∗ (x) is a continuous and convex function on K∗
and the optimizer function z ∗ (x) is continuous in K∗ .
Proof: See Theorem 5.5.1 in [18] and the bottom of page 138 in [18]. 

Remark 5.7 We remark that Corollary 5.1 requires the existence of optimizer z ∗ (x̄)
with a bounded cost. This is implicitly guaranteed in the mp-QP case since in
Corollary 5.2 the matrix H is assumed to be strictly positive definite. Moreover, the
existence of an optimizer z ∗ (x̄) with a bounded cost guarantees that J ∗ (x) is bounded
for all x in K∗ . This has been proven in [115, p. 178, Theorem 1] for the mp-LP case
and it is immediate to prove for the mp-QP case.
106 5 Multiparametric Nonlinear Programming

Remark 5.8 Both Corollary 5.1 (mp-LP) and Corollary 5.2 (mp-QP) could be formu-
lated stronger: J ∗ and Z ∗ are even Lipschitz-continuous. J ∗ is also piecewise affine
(mp-LP) or piecewise quadratic (mp-QP), and for the mp-QP z ∗ (x) is piecewise
affine. For the linear case, Lipschitz continuity is known from Walkup-Wets [283] as
a consequence of Hoffman’s theorem. For the quadratic case, Lipschitz continuity fol-
lows from Robinson [253], as e.g., shown by Klatte and Thiere [178]. The “piecewise”
properties are consequences of local stability analysis of parametric optimization,
e.g. [107, 18, 200] and are the main focus of the next chapter.
 Chapter 6

Multiparametric Programming:
a Geometric Approach

In this chapter we will concentrate on multiparametric linear programs (mp-LP),

multiparametric quadratic programs (mp-QP) and multiparametric mixed-integer
linear programs (mp-MILP).
The main idea of the multiparametric algorithms presented in this chapter is
to construct a critical region in a neighborhood of a given parameter by using nec-
essary and sufficient conditions for optimality, and then to recursively explore the
parameter space outside such a region. For this reason the methods are classified as
“geometric”. All the algorithms are easy to implement once standard solvers are
available: linear programming, quadratic programming and mixed-integer linear
programming for solving mp-LP, mp-QP and mp-MILP, respectively. A literature
review is presented in Section 6.6.

6.1 Multiparametric Programs with Linear Constraints

6.1.1 Formulation

Consider the multiparametric program

J ∗ (x) = min J(z, x)
z (6.1)
subj. to Gz ≤ w + Sx,
where z ∈ Rs are the optimization variables, x ∈ Rn is the vector of parameters,
J(z, x) : Rs+n → R is the objective function and G ∈ Rm×s , w ∈ Rm , and
S ∈ Rm×n . Given a closed and bounded polyhedral set K ⊂ Rn of parameters,
K = {x ∈ Rn : T x ≤ N }, (6.2)
we denote by K∗ ⊆ K the region of parameters x ∈ K such that (6.1) is feasible:
K∗ = {x ∈ K : ∃z satisfying Gz ≤ w + Sx}. (6.3)
108 6 Multiparametric Programming: a Geometric Approach

In this book we assume that

1. the constraint x ∈ K is included in the constraints Gz ≤ w + Sx.
2. the polytope K is full-dimensional. Otherwise we can reformulate the problem
with a smaller set of parameters such that K becomes full-dimensional.
3. S has full column rank. Otherwise we can again reformulate the problem in
a smaller set of parameters.

Theorem 6.1 Consider the multiparametric problem (6.1). If the domain of J(z, x)
is Rs+n then K∗ is a polytope.
Proof: K∗ is the projection of the set Gz − Sx ≤ w on the x space intersected
with the polytope K. 
For any given x̄ ∈ K∗ , J ∗ (x̄) denotes the minimum value of the objective func-
tion in problem (6.1) for x = x̄. The function J ∗ : K∗ → R expresses the depen-
dence of the minimum value of the objective function on x, J ∗ (x) is called the value
s s
function. The set-valued function Z ∗ : K∗ → 2R , where 2R is the set of subsets of
Rs , describes for any fixed x ∈ K∗ the set Z ∗ (x) of optimizers z ∗ (x) yielding J ∗ (x).
We aim to determine the feasible set K∗ ⊆ K of parameters, the expression of
the value function J ∗ (x) and the expression of one of the optimizers z ∗ (x) ∈ Z ∗ (x).

6.1.2 Definition of Critical Region

Consider the multiparametric program (6.1). Let I = {1, . . . , m} be the set of
constraint indices. For any A ⊆ I, let GA and SA be the submatrices of G and S,
respectively, comprising the rows indexed by A and denote with Gj , Sj and wj the
j-th row of G, S and w, respectively. We define CRA as the set of parameters x
for which the same set A of constraints is active at the optimum. More formally
we have the following definitions.

Definition 6.1 The optimal partition of I at x is the partition (A(x), N A(x))

where
A(x) = {j ∈ I : Gj z ∗ (x) − Sj x = wj for all z ∗ (x) ∈ Z ∗ (x)}
N A(x) = {j ∈ I : exists z ∗ (x) ∈ Z ∗ (x) satisfying Gj z ∗ (x) − Sj x < wj }.

It is clear that A(x) and N A(x) are disjoint and their union is I.

Definition 6.2 Consider a set A ⊆ I. The critical region associated with the set
of active constraints A is defined as
CRA = {x ∈ K∗ : A(x) = A}. (6.4)

The set CRA is the set of all parameters x such that the constraints indexed by A
are active at the optimum of problem (6.1). Our first objective is to work with full-
dimensional critical regions. For this reason, we discuss next how the dimension of
the parameter space can be reduced in case it is not full-dimensional.
 6.1 Multiparametric Programs with Linear Constraints 109

6.1.3 Reducing the Dimension of the Parameter Space

It may happen that the set of inequality constraints in (6.1) contains some “hidden”
or “implicit” equality constraints as the following example shows.

Example 6.1
minz J(z,
 x)
 z1 + z2
 ≤ 9 − x1 − x2



 z1 − z2 ≤ 1 − x1 − x2


 z1 + z2 ≤ 7 + x1 + x2
(6.5)
subj. to z1 − z2 ≤ −1 + x1 + x2



 −z1 ≤ −4



 −z 2 ≤ −4

z1 ≤ 20 − x2 ,
where K = {[x1 , x2 ]′ ∈ R2 : −100 ≤ x1 ≤ 100, −100 ≤ x2 ≤ 100}. The reader can
check that all feasible values of x1 , x2 , z1 , z2 satisfy

z1 + z2 = 9 − x 1 − x 2
z1 − z2 = 1 − x 1 − x 2
z1 + z2 = 7 + x 1 + x 2
z1 − z2 = −1 + x1 + x2 (6.6)
z1 = 4
z2 = 4
z1 ≤ 20 − x2 ,

where we have identified many of the inequalities to be hidden equalities. This can
be simplified to
x1 + x2 = 1
(6.7)
x2 ≤ 16.
Thus  
 x1 + x2 = 1 
K∗ = [x1 , x2 ]′ ∈ R2 : −100 ≤ x1 ≤ 100 . (6.8)
 
−100 ≤ x2 ≤ 16
The example shows that the polytope K∗ is contained in a lower dimensional subspace
of K, namely a line segment in R2 .

Our goal is to identify the hidden equality constraints (as in (6.6)) and use
them to reduce the dimension of the parameter space (as in (6.7)) for which the
multiparametric program needs to be solved.

Definition 6.3 A hidden equality of the polyhedron C = {ξ ∈ Rs : Bξ ≤ v} is an

inequality Bi ξ ≤ vi such that Bi ξ¯ = vi ∀ξ¯ ∈ C.

To find hidden equalities we need to solve

vi∗ = min Bi ξ
subj. to Bξ ≤ v,

for all constraints i = 1, . . . , m. If vi∗ = vi , then Bi ξ = vi is a hidden equality.

110 6 Multiparametric Programming: a Geometric Approach

We can apply this procedure with ξ = [ xz ] to identify the hidden equalities in

the set of inequalities in (6.1)
Gz ≤ w + Sx,
to obtain
Gnh z ≤ wnh + Snh x (6.9a)
Gh z = wh + Sh x, (6.9b)
where we have partitioned G, w and S to reflect the hidden equalities. In order to
find the equality constraints involving only the parameter x that allow us to reduce
the dimension of x, we need to project the equalities (6.9b) onto the parameter
space. Let the singular value decomposition of Gh be
 ′ 
V1
Gh = [U1 U2 ] Σ ,
V2′
where the columns of U2 are the singular vectors associated with the zero singular
values. Then the matrix U2′ defines the projection of (6.9b) onto the parameter
space, i.e.
U2′ Gh z = 0 = U2′ wh + U2′ Sh x. (6.10)
We can use this set of equalities to replace the parameter x ∈ Rn with a set of
n′ = n − rank(U2′ Sh ) new parameters in (6.9a) which simplifies the parametric
program (6.1).
In the rest of this book we always assume that the multiparametric program has
been preprocessed using the ideas of this section so that the number of independent
parameters is reduced as much as possible and K∗ is full-dimensional.

6.2 Multiparametric Linear Programming

6.2.1 Formulation

Consider the special case of the multiparametric program (6.1) where the objective
is linear
J ∗ (x) = min c′ z
z (6.11)
subj. to Gz ≤ w + Sx.
All the variables were defined in Section 6.1.1. Our goal is to find the value function
J ∗ (x) and an optimizer function z ∗ (x) for x ∈ K∗ . Note that K∗ can be determined
as discussed in Theorem 6.1. As suggested through Example 5.1 our search for
these functions proceeds by partitioning the set of feasible parameters into critical
regions. This is shown through a simple example next.

Example 6.2 Consider the parametric linear program

J ∗ (x) = min z+1
z
subj. to z ≥1+x
z ≥ 0,
6.2 Multiparametric Linear Programming 111

where z ∈ R and x ∈ R. Our goals are:

1. to find z ∗ (x) = arg minz, z≥0, z≥1+x z + 1,

2. to find all x for which the problem has a solution, and
3. to compute the value function J ∗ (x).

The Lagrangian is

L(z, x, u1 , u2 ) = z + u1 (−z + x + 1) + u2 (−z)

and the KKT conditions are (see Section 2.2.3 for KKT conditions for linear pro-
grams)

−u1 − u2 = −1 (6.12a)
u1 (−z + x + 1) =0 (6.12b)
u2 (−z) =0 (6.12c)
u1 ≥0 (6.12d)
u2 ≥0 (6.12e)
−z + x + 1 ≤0 (6.12f)
−z ≤ 0. (6.12g)

Consider (6.12) and the three complementary cases:

u1 + u2 = 1  ∗

 z =1+x
−z + x + 1 = 0  ∗
u1 = 1, u∗2 = 0
A. −z < 0 ⇒

 J∗ = 2 + x
u1 > 0 
x > −1
u2 = 0

u1 + u2 = 1  ∗

 z =0
−z + x + 1 < 0  ∗
u1 = 0, u∗2 = 1
B. −z = 0 ⇒ (6.13)

 J∗ = 1
u1 = 0 
x < −1
u2 > 0

u1 + u2 = 1  ∗

 z =0
−z + x + 1 = 0  ∗
u1 ≥ 0, u2 ≥ 0, u∗1 + u∗2 = 1
C. −z = 0 ⇒
 J∗ = 1

u1 ≥ 0 
x = −1
u2 ≥ 0

This solution is depicted in Figure 6.1.

The above simple procedure, which required nothing but the solution of the KKT
conditions, yielded the optimizer z ∗ (x) and the value function J ∗ (x) for all values
of the parameter x. The set of admissible parameters values was divided into three
critical regions, defined by x < −1, x > −1 and x = −1. In the region x > −1
the first inequality constraint is active (z = 1 + x) and the Lagrange multiplier u1
is greater than zero, in the second region x < −1 the second inequality constraint
is active (z = 0) and the Lagrange multiplier u2 is greater than zero. In the third
region x = −1 both constraints are active and the Lagrange multipliers belong to
the set u∗1 ≥ 0, u2 ≥ 0, u∗1 + u∗2 = 1. Throughout a critical region the conditions
112 6 Multiparametric Programming: a Geometric Approach

J ∗ (x)
1
0
z ∗ (x)

-1
x
Figure 6.1 Example 6.2. Optimizer z ∗ (x) and value function J ∗ (x) as a
function of the parameter x.

for optimality derived from the KKT conditions do not change. For our example, in
each critical region the optimizer z ∗ (x) is affine and the value function J ∗ (x) is also
affine.

6.2.2 Critical Regions, Value Function and Optimizer: Local Properties

Consider Definition 6.2 of a critical region. In this section we show that critical re-
gions of mp-LP are polyhedra. We use primal feasibility to derive the H-polyhedral
representation of the critical regions, the complementary slackness conditions to
compute an optimizer z ∗ (x), and the dual problem of (6.11) to derive the optimal
value function J ∗ (x) inside each critical region.
Consider the dual problem of (6.11):
min (w + Sx)′ u
u
subj. to G′ u = −c (6.14)
u ≥ 0.
The dual feasibility, complementary slackness and primal feasibility conditions for
problems (6.11), (6.14) are
G′ u + c = 0, u ≥ 0 (6.15a)
ui (Gi z − wi − Si x) = 0, i = 1, . . . , m (6.15b)
Gz − w − Sx ≤ 0. (6.15c)
Let us assume that we have determined the optimal partition

(A, N A) = (A(x∗ ), N A(x∗ ))

for some x∗ ∈ K∗ . The primal feasibility condition (6.15c) can be rewritten as
GA z ∗ − SA x = wA (6.16a)
GN A z ∗ − SN A x < wN A . (6.16b)
6.2 Multiparametric Linear Programming 113

Here (6.16a) assigns to a parameter x an optimizer z ∗ (x) (in general, a set of

optimizers z ∗ (x)) and (6.16b) defines the critical region CRA in the parameter
space K∗ for which this assignment is valid. In the simplest case assume that GA
is square and of full rank. From (6.16a) we find

z ∗ = G−1
A (SA x + wA ), (6.17)

and substituting in (6.16b) we obtain the set of inequalities that defines the critical
region CRA of parameters x for which the optimizer is (6.17):

CRA = x : GN A G−1 A (SA x + wA ) − SN A x < wN A . (6.18)

The following derivation deals with the general case when the optimizer z ∗ (x)
is not necessarily unique and GA is not invertible. We know from (6.16a) that for
x ∈ CRA
GA z ∗ − SA x = wA .
For another x̄ ∈ CRA we require

GA z̄ − SA x̄ = wA ,

or for the perturbation x̃ = x − x̄

GA z̃ − SA x̃ = 0.

This equation will have a solution z̃ for arbitrary values of x̃, i.e., a full dimensional
CRA if
rank[GA ] = rank[GA SA ].
Otherwise the dimension over which x̃ can vary will be restricted to

n − (rank[GA SA ] − rank[GA ]).

Next we will show how to construct these restrictions on the variation of x.

Let l = rank GA and consider the QR decomposition of GA
 
U1 U2
GA = Q ,
0|A|−l×l 0|A|−l×|A|−l

where Q ∈ R|A|×|A| is a unitary

 matrix, U1 ∈ Rl×l is a full-rank square matrix
P q
and U2 ∈ Rl×|A|−l . Let = −Q−1 SA and = Q−1 wA . From (6.16a) we
D r
obtain  
  z∗  
U1 U2 P  1∗  q
z2 = . (6.19)
0|A|−l×l 0|A|−l×|A|−l D r
x
We partition (6.16b) accordingly
 
  z1∗
E F − SN A x < wN A . (6.20)
z2∗
114 6 Multiparametric Programming: a Geometric Approach

Calculating z1∗ from (6.19) we obtain

z1∗ = U1−1 (−U2 z2∗ − P x + q), (6.21)

which substituted in (6.20) gives:

(F − EU1−1 U2 )z2∗ + (SN A − EU1−1 P )x < wN A − EU1−1 q. (6.22)

The critical region CRA can be equivalently written as:

   
z2
CRA = x ∈ Px : ∃z2 such that ∈ Pz2 x , (6.23)
x

where:

Px = {x : Dx = r} , (6.24)
  
z2
Pz2 x = : (F − EU1−1 U2 )z2 + (SN A − EU1−1 P )x < wN A − EU1−1 q .
x
(6.25)

In other words:
CRA = Px ∩ projx (Pz2 x ). (6.26)
∗
In summary, if the optimizer z (x) is not unique then it can be expressed as
z1∗ (x) (6.21), where the set of optimizers is characterized by z2∗ (x), which is allowed
to vary within the set Pz2 x (6.25). We have also constructed the critical region
CRA (6.26), where z1∗ (x) remains optimal. If GA is not invertible and the matrix
D is not empty then the critical region is not full dimensional.
We can now state some fundamental properties of the critical regions, value
function and optimizer inside a critical region.

Theorem 6.2 Let (A, N A) = (A(x∗ ), N A(x∗ )) for some x∗ ∈ K∗

 
i) CRA is an open polyhedron of dimension d where d = n − rank GA SA +
rank GA . If d = 0 then CRA = {x∗ }.
ii) If rank GA = s (recall that z ∈ Rs ) then the optimizer z ∗ (x) is unique and
given by an affine function of the state inside CRA , i.e., z ∗ (x) = Fi x + gi for
all x ∈ CRA .
iii) If the optimizer is not unique in CRA then Z ∗ (x) is an open polyhedron for
all x ∈ CRA .
iv) J ∗ (x) is an affine function of the state, i.e., J ∗ (x) = c′i x+di for all x ∈ CRA .
Proof:
i) Polytope Pz2 x (6.25) is open and nonempty, therefore it is full-dimensional
in the (z2 , x) space and dim projx (Pz2 x ) = n. Also,
 
dim Px = n − rank D = n − (rank GA SA − rank GA ).
 6.2 Multiparametric Linear Programming 115

Since the intersection of Px and projx (Pz2 x ) is nonempty (it contains at least
the point x∗ ) we can conclude that
 
dim CRA = n − rank GA SA + rank GA .

Since we assumed that the set K in (6.2) is bounded, CRA is bounded. This
implies that CRA is an open polytope since it is the intersection of an open
polytope and the subspace Dx = r. In general, if we allow K in (6.2) to be
unbounded, then the critical region CRA can be unbounded.

ii) Consider (6.21) and recall that l = rank GA . If l = s, then the primal
optimizer is unique, U2 is an empty matrix and

z ∗ (x) = z1∗ (x) = U1−1 (−P x + q). (6.27)

iii) If the primal optimizer is not unique in CRA then Z ∗ (x) in CRA is the
following point-to-set map: Z ∗ (x) = {[z1 , z2 ] : z2 ∈ Pz2 x , U1 z1 +U2 z2 +P x =
q)}. Z ∗ (x) is an open polyhedron since Pz2 x is open.

iv) Consider the dual problem (6.14) and one of its optimizer u∗0 for x = x∗ . By
definition of a critical region u∗0 remains optimal for all x ∈ CRA . Therefore
the value function in CRA is

J ∗ (x) = (w + Sx)′ u∗0 , (6.28)

which is an affine function of x on CRA .

Remark 6.1 If the optimizer is unique then the computation of CRA in (6.26) does
not require the projection of the set Pz2 x in (6.25). In fact, U2 and F are empty
matrices and

CRA = Pz2 x = x : Dx = r, (SNA − EU −1 P )x < wNA − EU −1 q . (6.29)

If D is also empty, then the critical region is full dimensional.

6.2.3 Propagation of the Set of Active Constraints

The objective of this section is to briefly describe the propagation of the set of active
constraints when moving from one full-dimensional critical region to a neighboring
full-dimensional critical region. We will use a simple example in order to illustrate
the main points.
116 6 Multiparametric Programming: a Geometric Approach

Example 6.3 Consider the mp-LP problem

minz1 ,z2 ,z3 ,z4 z1 + z2



 −z1 ≤ x1 + x2

 −z
 1 − z3 ≤ x2



 −z1 ≤ −x1 − x2



 −z1 + z3 ≤ −x2



 −z2 − z3 ≤ x1 + 2x2


−z2 − z3 − z4 ≤ x2 (6.30)
subj. to

 −z2 + z3 ≤ −x1 − 2x2



 −z2 + z3 + z4 ≤ −x2



 z3 ≤ 1



 −z 3 ≤ 1



 z4 ≤ 1

−z4 ≤ 1,

where K is given by
−2.5 ≤ x1 ≤ 2.5
(6.31)
−2.5 ≤ x2 ≤ 2.5.
A solution to the mp-LP problem is shown in Figure 6.2 and the constraints which
are active in each associated critical region are reported in Table 6.1. Clearly, as
z ∈ R4 , CR1 = CR{1,4,5,6,7,8} and CR2 = CR{2,3,5,6,7,8} are primal degenerate
full-dimensional critical regions.

Critical Region Value function

CR1=CR{1,4,5,6,7,8} −x1 − x2
CR2=CR{2,3,5,6,7,8} x1 + x2
CR3=CR{1,5,6,9} −2x1 − 3x2 − 1
CR4=CR{4,6,7,10} x1 + 3x2 − 2
CR5=CR{1,5,8,12} −1.5x1 − 1.5x2 + 0.5
CR6=CR{2,5,6,9} −x1 − 3x2 − 2
CR7=CR{3,6,7,10} 2x1 + 3x2 − 1
CR8=CR{3,6,7,11} 1.5x1 + 1.5x2 − 0.5
CR9=CR{1,5,9,12} −2x1 − 3x2 − 1
CR10=CR{4,7,10,12} x1 + 3x2 − 2
CR11=CR{2,5,9,12} −x1 − 3x2 − 2
CR12=CR{3,7,10,12} 2x1 + 3x2 − 1

Table 6.1 Example 6.3. Critical regions and corresponding value function.

By observing Figure 6.2 and Table 6.1 we notice the following. Under no primal
and dual degeneracy, (i) full critical regions are described by a set of active con-
straints of dimension n, (ii) two neighboring full-dimensional critical regions CRAi
and CRAj have Ai and Aj differing only in one constraint, (iii) CRAi and CRAj
will share a facet which is a primal degenerate critical region CRAp of dimension
n − 1 with Ap = Ai ∪ Aj . In Example 6.3, CR7 ad CR12 are two full-dimensional
and neighboring critical regions and the corresponding active set differs only in one
 6.2 Multiparametric Linear Programming 117

2.5
10
12
4
5
1 7
x2

2 8

9 3

6
11
-2.5
-2.5 2.5
x1

Figure 6.2 Example 6.3. Polyhedral partition of the parameter space.

constraint (constraint 6 in CR7 and constraint 12 in CR12). They share a facet

which is a one dimensional critical region (an open line), CR13=CR{3,6,7,10,12} .
(The solution depicted in Figure 6.2 and detailed in Table 6.1 contains only full-
dimensional critical regions.)
If primal and/or dual degeneracy occur, then the situation becomes more com-
plex. In particular, in the case of primal degeneracy, it might happen that full-
dimensional critical regions are described by more than s active constraints (CR1
or CR2 in Example 6.3). In case of dual degeneracy, it might happen that full-
dimensional critical regions are described by fewer than s active constraints. Details
on this case are provided in the next section.

6.2.4 Non-unique Optimizer∗

If rank[GA ] < s then Z ∗ (x) is not a singleton and the projection of the set Pz2 x
in (6.25) is required in order to compute the critical region CRA(x∗ ) , which is “ex-
pensive” (see Section 6.1.1 for a discussion of polyhedra projection). It is preferable
to move on the optimal facet to a vertex and to construct the critical region starting
with this optimizer. This is explained next.
If one needs to determine one possible optimizer z ∗ (·) in the dual degenerate
region CRA(x∗ ) the following simple method can be used. Choose a particular
b ∗ ) ⊃ A(x∗ )
optimizer which lies on a vertex of the feasible set, i.e., determine set A(x
of active constraints for which rank(GA(x d b ∗ of
b ∗ ) ) = s, and compute a subset CRA(x )
the dual degenerate critical region (namely, the subset of parameters x such that
118 6 Multiparametric Programming: a Geometric Approach

x2

CR{2}

CR{1,5}
CR{1,3}

x1

Figure 6.3 Example 6.4. Polyhedral partition of the parameter space corre-
sponding to the solution.

b ∗ ) are active at the optimizer, which is not a critical region in

the constraints A(x
the sense of Definition 6.2). Within CRd b ∗ , the piecewise linear expression of an
A(x )
d b ∗ is defined by (6.29).
optimizers z ∗ (x) is available from (6.27) and CR A(x )

The algorithm proceeds by exploring the space surrounding CR d b ∗ until

A(x )
CRA(x∗ ) is covered. The arbitrariness in choosing an optimizer leads to differ-
ent ways of partitioning CRA(x∗ ) , where the partitions, in general, may overlap.
Nevertheless, in each region a unique optimizer is defined. The storing of over-
lapping regions can be avoided by intersecting each new region (inside the dual
degenerate region) with the current partition computed so far. This procedure is
illustrated in the following example.

Example 6.4 Consider the following mp-LP reported in [115, p. 152]

min −2z1 − z2


 z1 + 3z2 ≤ 9 − 2x1 + x2


 2z1 + z2 ≤ 8 + x1 − 2x2
(6.32)
subj. to z1 ≤ 4 + x1 + x2



 −z1 ≤ 0

−z2 ≤ 0,

where K is given by:

−10 ≤ x1 ≤ 10
(6.33)
−10 ≤ x2 ≤ 10.

The solution is represented in Figure 6.3 and the critical regions are listed in Table 6.2.

The critical region CR{2} is related to a dual degenerate solution with non-
unique optimizers. The analytical expression of CR{2} is obtained by projecting
 6.2 Multiparametric Linear Programming 119

Region Optimizer Value Function

CR{2} not single valued −x1 + 2x2 − 8
CR{1,5} z1∗ = −2x1 + x2 + 9, z2∗ = 0 4x1 − 2x2 − 18
CR{1,3} z1∗ = x1 + x2 + 4, z2∗ = −x1 + 5/3 −x1 − 2x2 − 29/3

Table 6.2 Example 6.4. Critical regions and corresponding optimal value.

the H-polyhedron
z1 + 3z2 + 2x1 − x2 < 9
2z1 + z2 − x1 + 2x2 = 8
z1 − x1 − x2 < 4 (6.34)
−z1 < 0
−z2 < 0
on the parameter space to obtain:
 

 2.5x1 − 2x2 ≤ 5 

 
−0.5x1 + x2 ≤ 4
CR{2} = [x1 , x2 ] : , (6.35)

 −12x2 ≤ 5 

 
−x1 − x2 ≤ 4

which is effectively the result of (6.26). For all x ∈ CR{2} , only one constraint is
active at the optimum, which makes the optimizer not unique.
Figures 6.4, 6.5 and 6.6 show two possible ways of covering CR{2} without using
projection. The generation of overlapping regions is avoided by intersecting each
new region with the current partition computed so far, as shown in Figure 6.7 where
g {2,4} and CR
CR g {2,1} represent the intersected critical regions. In Figures 6.4, 6.5
the regions are overlapping, and in Figure 6.7 artificial cuts are introduced at
the boundaries inside the degenerate critical region CR{2} . No artificial cuts are
introduced in Figure 6.6 because the CR g {2,3} and CRg {2,5} happen to be non-
overlapping.

6.2.5 Value Function and Optimizer: Global Properties

In this section we discuss global properties of the value function J ∗ (x), optimizer
z ∗ (x), and of the set K∗ .

Theorem 6.3 Assume that for a fixed x0 ∈ K there exists a finite optimal solution
z ∗ (x0 ) of (6.11). Then, for all x ∈ K, (6.11) has either a finite optimum or no
feasible solution.
Proof: Consider the mp-LP (6.11) and assume by contradiction that there
exist x0 ∈ K and x̄ ∈ K with a finite optimal solution z ∗ (x0 ) and an unbounded
solution z ∗ (x̄). Then the dual problem (6.14) for x = x̄ is infeasible. This implies
that the dual problem will be infeasible for all real vectors x since x enters only in
the cost function. This contradicts the hypothesis since the dual problem (6.14)
for x = x0 has a finite optimal solution. 
120 6 Multiparametric Programming: a Geometric Approach

x2
d {2,5}
CR

x̄1

z2 c′ x = const
x1

1
4 2
5
z1
d {2,5} ⊂ CR{2} , and below
(a) First region CR
the feasible set in the z-space corresponding
d {2,5} .
to x̄1 ∈ CR

x2 d {2,4}
CR

x̄2

z2
c′ x = const
x1
2

4 3
5
z1
d {2,4} ⊂ CR{2} , and be-
(b) Second region CR
low the feasible set in the z-space correspond-
d {2,4} .
ing to x̄2 ∈ CR

Figure 6.4 Example 6.4. A possible sub-partitioning of the degenerate region

CR2 where the regions CR d {2,5} (Fig. 6.4(a)), CRd {2,4} (Fig. 6.4(b)) and
d {2,1} (Fig. 6.5(a)) are overlapping. Note that below each picture the
CR
feasible set and the level set of the value function in the z-space are depicted
for a particular choice of the parameter x indicated by a point marked with
×.

Theorem 6.4 The set of all parameters x such that the LP (6.11) has a finite
optimal solution z ∗ (x) equals K∗ .
6.2 Multiparametric Linear Programming 121

x2

d {2,1}
CR
x̄3
z2
1 c′ x = const
x1
4 2

3
5
z1
d {2,1} ⊂ CR{2} , and below
(a) Third region CR
the feasible set in the z-space corresponding
d {2,1} .
to x̄3 ∈ CR

x2 d {2,5}
CR
d {2,4}
CR

d {2,1}
CR

x1
(b) Final partition of CR{2} . Note that the
d {2,5} is hidden by region CR
region CR d {2,4}
d {2,1} .
and region CR

Figure 6.5 Example 6.4. A possible sub-partitioning of the degenerate region

CR2 where the regions CR d {2,5} (Fig. 6.4(a)), CRd {2,4} (Fig. 6.4(b)) and
d
CR{2,1} (Fig. 6.5(a)) are overlapping. Note that below each picture the
feasible set and the level set of the value function in the z-space are depicted
for a particular choice of the parameter x indicated by a point marked with
×.

Proof: It follows directly from from Theorem 6.1 and Theorem 6.3. 
Note that from Definition 6.3 K∗ is the set of feasible parameters. However the
LP (6.11) might be unbounded for some x ∈ K∗ . Theorem 6.4 excludes this case.
The following Theorem 6.5 summarizes the properties enjoyed by the multi-
parametric solution.
122 6 Multiparametric Programming: a Geometric Approach

x2 d {2,5}
CR

x1
d {2,5} ⊂ CR{2} .
(a) First region CR

x2

d {2,3}
CR

x1
d {2,3} ⊂ CR{2} .
(b) Second region CR

d {2,5} and
Figure 6.6 Example 6.4. A possible solution where the regions CR
d
CR{2,3} are non-overlapping

Theorem 6.5 The function J ∗ (·) is convex and piecewise affine over K∗ . If the
optimizer z ∗ (x) is unique for all x ∈ K∗ , then the optimizer function z ∗ : K∗ → Rs
is continuous and piecewise affine. Otherwise it is always possible to define a
continuous and piecewise affine optimizer function z ∗ such that z ∗ (x) ∈ Z ∗ (x) for
all x ∈ K∗ .

Remark 6.2 In Theorem 6.5, the piecewise affine property of optimizer and value
function follows immediately from Theorem 6.2 and from the enumeration of all
possible combinations of active constraint sets. Convexity of J ∗ (·) and continuity of
Z ∗ (x) follows from standard results on multiparametric programs (see Corollary 5.1).
In the presence of multiple optimizers, the proof of existence of a continuous and
piecewise affine optimizer function z ∗ such that z ∗ (x) ∈ Z ∗ (x) for all z ∈ K∗ is more
 6.2 Multiparametric Linear Programming 123

x2 d {2,5}
CR

g {2,4}
CR
g {2,1}
CR

x1

Figure 6.7 Example 6.4. A possible solution where CR g {2,4} is obtained

d d g {2,1} by
by intersecting CR{2,4} with the complement of CR{2,5} , and CR
d d d
intersecting CR{2,1} with the complement of CR{2,5} and CR{2,4}

involved and we refer the reader to [115, p. 180].

6.2.6 mp-LP Algorithm

The goal of an mp-LP algorithm is to determine the partition of K∗ into full

dimensional critical regions CRAi , and to find the expression of the functions J ∗ (·)
and z ∗ (·) for each critical region. An mp-LP algorithm has two components: the
“active set generator” and the “KKT solver”. The active set generator computes
the set of active constraints Ai . The KKT solver computes CRAi and the expression
of J ∗ (·) and z ∗ (·) in CRAi as explained in Theorem 6.2.
The active set generator is the critical part. In principle, one could simply
generate all the possible combinations of active sets. However, in many problems
only a few active constraints sets generate full-dimensional critical regions inside
the region of interest K. Therefore, the goal is to design an active set generator
algorithm which computes only the active sets Ai with associated full-dimensional
critical regions covering only K∗ . This avoids the combinatorial explosion of a
complete enumeration. Also, we will use the technique described in Section 6.2.4
in order to avoid expressions for non-unique optimizers (6.21) with U2 non-empty.
Next we will describe one possible implementation of an mp-LP algorithm.
In order to start solving the mp-LP problem, we need an initial vector x0
inside the polyhedral set K∗ of feasible parameters. A possible choice for x0 is the
Chebychev center (see Section 4.4.5) of K∗ , i.e., x0 solving the following LP:

maxx,z,ǫ ǫ
subj. to Ti x + ǫkTi k2 ≤ Ni , i = 1, . . . , nT (6.36)
Gz − Sx ≤ w,

where nT is the number of rows Ti of the matrix T defining the set K in (6.2).
124 6 Multiparametric Programming: a Geometric Approach

If ǫ ≤ 0, then the LP problem (6.11) is infeasible for all x in the interior of K.

Otherwise, we solve the primal and dual problems (6.11), (6.14) for x = x0 . Let
z0∗ and u∗0 be the optimizers of the primal and the dual problem, respectively. The
value z0∗ defines the following optimal partition

A(x0 ) = {j ∈ I : Gj z0∗ − Sj x0 − wj = 0}
(6.37)
NA(x0 ) = {j ∈ I : Gj z0∗ − Sj x0 − wj < 0}

and consequently the critical region CRA(x0 ) . Once the critical region CRA(x0 )
has been defined, the rest of the space Rrest = K\CRA(x0 ) has to be explored and
new critical regions generated. An approach for generating a polyhedral parti-
tion {R1 , . . . , Rnr est } of the rest of the space Rrest is described in Theorem 4.2 in
Section 4.4.7. The procedure proposed in Theorem 4.2 for partitioning the set of
parameters allows one to recursively explore the parameter space. Such an iterative
procedure terminates after a finite time, as the number of possible combinations of
active constraints decreases with each iteration. The following two issues need to
be considered:

1. The partitioning in Theorem 4.2 defines new polyhedral regions Rk to be

explored that are not related to the critical regions which still need to be
determined. This may split some of the critical regions, due to the artificial
cuts induced by Theorem 4.2. Post-processing can be used to join cut critical
regions [44]. As an example, in Figure 6.8 the critical region CR{3,7} is
discovered twice, one part during the exploration of R1 and the second part
during the exploration of R2 .
Although algorithms exist for convexity recognition and computation of the
union of polyhedra, the post-processing operation is computationally expen-
sive. Therefore, it is more efficient not to intersect the critical region obtained
by (6.29) with halfspaces generated by Theorem 4.2, which is only used to
drive the exploration of the parameter space. Then, no post processing is
needed to join subpartitioned critical regions.
On the other hand, some critical regions may appear more than once. Dupli-
cates can be uniquely identified by the set of active constraints A(x) and can
be easily eliminated. To this aim, in the implementation of the algorithm we
keep a list of all the critical regions which have already been generated in or-
der to avoid duplicates. In Figure 6.8 the critical region CR{3,7} is discovered
twice but stored only once.

2. If a region is generated which is not full-dimensional we want to avoid further

recursion of the algorithm not producing any full-dimensional critical region,
and therefore lengthening the number of steps required to determine the
solution to the mp-LP. We perturb the parameter x0 by a random vector
with length smaller than the Chebychev radius of the polyhedral region Rk
where we are looking for a new critical region. This will ensure that the
perturbed vector is still contained in Rk .

Based on the above discussion, the mp-LP solver can be summarized in the
following recursive Algorithm 6.1. Note that the algorithm generates a partition
6.2 Multiparametric Linear Programming 125

x2 CR{3,7}
x+
2
R2

CR{6,7}
R1

x−
2

x−
1 x+
1 x1

Figure 6.8 Example of a critical region explored twice

of the state space which is not strict. The algorithm could be modified to store
the critical regions as defined in (6.4) which are open sets, instead of storing their
closure. In this case the algorithm would have to explore and store all the crit-
ical regions that are not full-dimensional in order to generate a strict polyhedral
partition of the set of feasible parameters. From a practical point of view such a
procedure is not necessary since the value function and the optimizer are continuous
functions of x.

Algorithm 6.1 mp-LP Algorithm (non-degenerate case)

Input: Matrices c, G, w, S of the mp-LP (6.11) and set K in (6.2)
Output: Multiparametric solution to the mp-LP (6.11)
Execute partition(K)
End
Function partition(Y )
Let x0 ∈ Y and ǫ be the solution to the LP (6.36)
If ǫ ≤ 0 Then exit (no full-dimensional CR is in Y )
Solve the LP (6.11), (6.14) for x = x0
Let A(x0 ) be the set of active constraints as in (6.37)
Determine z ∗ (x) from (6.17) and CRA(x0 ) from (6.18)
Partition the rest of the region as in Theorem 4.2
For each new sub-region Ri , Do partition(Ri )
End function

Remark 6.3 In the degenerate case z ∗ (x) and CRA(x0 ) are given by (6.21) and (6.26),
respectively. As remarked in Section 6.2.2, if rank(D) > 0 the region CRA(x0 ) is not
full-dimensional and therefore not of interest. To use Algorithm 6.1 after computing
U, P, D if D 6= 0 one should compute a random vector ǫ ∈ Rn smaller than the
Chebychev radius of Y and such that the LP (6.11) is feasible for x0 + ǫ and then
repeat step where A(x0 ) is computed with x0 ← x0 + ǫ.
126 6 Multiparametric Programming: a Geometric Approach

Remark 6.4 The algorithm determines the partition of K recursively. After the first
critical region is found, the rest of the region in K is partitioned into polyhedral sets
{Ri } as in Theorem 4.2. By using the same method, each set Ri is further partitioned,
and so on.

6.3 Multiparametric Quadratic Programming

6.3.1 Formulation

In this section we investigate multiparametric quadratic programs (mp-QP), a spe-

cial case of the multiparametric program (6.1) where the objective is a quadratic
function
J ∗ (x) = min J(z, x) = 12 z ′ Hz
z (6.38)
subj. to Gz ≤ w + Sx.
All the variables were defined in Section 6.1.1. We assume H ≻ 0. Our goal is to
find the value function J ∗ (x) and the optimizer function z ∗ (x) in K∗ . Note that
K∗ can be determined by projection as discussed in Theorem 6.1. As suggested
through Example 5.1 our search for these functions proceeds by partitioning the set
of feasible parameters into critical regions. Note that the more general problem with
J(z, x) = 12 z ′ Hz + x′ F z can always be transformed into an mp-QP of form (6.38)
by using the variable substitution z̃ = z + H −1 F ′ x.
As in the previous sections, we denote with the subscript j the j-th row of a
matrix or j-th element of a vector. Also, J = {1, . . . , m} is the set of constraint
indices and for any A ⊆ J, GA , wA and SA are the submatrices of G, w and S,
respectively, consisting of the rows indexed by A. Without loss of generality we
will assume that K∗ is full-dimensional (if it is not, then the procedure described in
Section 6.1.3 can be used to obtain a full-dimensional K∗ in a reduced parameter
space).

Example 6.5 Consider the parametric quadratic program

J ∗ (x) = min J(z, x) = 12 z 2

z
subj. to z ≤ 1 + 3x,

where z ∈ R and x ∈ R. Our goals are:

1. to find z ∗ (x) = arg minz, z≤1+3x J(z, x),

2. to find all x for which the problem has a solution, and
3. to compute the value function J ∗ (x).

The Lagrangian function is

1 2
L(z, x, u) = z + u(z − 3x − 1)
2
6.3 Multiparametric Quadratic Programming 127

and the KKT conditions are (see Section 2.3.3 for KKT conditions for quadratic
programs)

z+u =0 (6.39a)
u(z − 3x − 1) =0 (6.39b)
u ≥0 (6.39c)
z − 3x − 1 ≤ 0. (6.39d)

Consider (6.39) and the two strictly complementary cases:

 ∗

 z = 3x + 1
z+u=0  ∗
u = −3x − 1
A. z − 3x − 1 = 0 ⇒ ∗

 J = 29 x2 + 3x + 1
u≥0  2
x ≤ − 13
 ∗ (6.40)

 z =0
z+u=0  ∗
u =0
B. z − 3x − 1 < 0 ⇒
 J∗ = 0

u=0 
x > − 31 .

This solution is depicted in Figure 6.9. The above simple procedure, which required
nothing but the solution of the KKT conditions, yielded the optimizer z ∗ (x) and
the value function J ∗ (x) for all values of the parameter x. The set of admissible
parameters values was divided into two critical regions, defined by x ≤ − 13 and
x > − 31 . In the region x ≤ − 31 the inequality constraint is active and the Lagrange
multiplier is greater or equal than zero, in the other region x > − 31 the inequality
constraint is not active and the Lagrange multiplier is equal to zero. Note that for
x = − 31 the inequality constraint z − 3x − 1 ≤ 0 is active at z ∗ and the Lagrange
multiplier is equal to zero.

Throughout a critical region the conditions for optimality derived from the KKT
conditions do not change. In each critical region the optimizer z ∗ (x) is affine and the
value function J ∗ (x) is quadratic. Both z ∗ (x) and J ∗ (x) are continuous.

2
J ∗ (x)
1

0
z ∗ (x)
-1

-2
-2 -1 0 1 2
x
Figure 6.9 Example 6.5. Optimizer z ∗ (x) and value function J ∗ (x) as a
function of the parameter x.
128 6 Multiparametric Programming: a Geometric Approach

6.3.2 Critical Regions, Value Function and Optimizer: Local Properties

Consider the definition of a critical region given in Section 6.1.2. We show next that
critical regions of mp-QP are polyhedra. We use the KKT conditions (Section 1.6)
to derive the H-polyhedral representation of the critical regions and to compute the
optimizer function z ∗ (x) and the value function J ∗ (x) inside each critical region.
The following theorem introduces fundamental properties of critical regions, the
value function and the optimizer inside a critical region.

Theorem 6.6 Let (A, N A) = (A(x̄), N A(x̄)) for some x̄ ∈ K∗ , Then

i) the closure of CRA is a polyhedron.

ii) z ∗ (x) is an affine function of the state inside CRA , i.e., z ∗ (x) = Fi x + gi for
all x ∈ CRA .

iii) J ∗ (x) is a quadratic function of the state inside CRA , i.e., J ∗ (x) = x′ Mi x +
c′i x + di for all x ∈ CRA .

Proof: The first-order Karush-Kuhn-Tucker (KKT) optimality conditions (see

Section 2.3.3) for the mp-QP are given by

Hz ∗ + G′ u∗ = 0, u ∈ Rm (6.41a)
u∗i (Gi z ∗ − wi − Si x) = 0, i = 1, . . . , m (6.41b)
u∗ ≥ 0 (6.41c)
Gz ∗ − w − Sx ≤ 0. (6.41d)

Let us assume that we have determined the optimal partition

(A, N A) = (A(x̄), N A(x̄))

for some x̄ ∈ K∗ . The primal feasibility condition (6.41d) can be rewritten as

GA z ∗ − SA x = wA (6.42a)
∗
GN A z − SN A x < wN A . (6.42b)

We solve (6.41a) for z ∗

z ∗ = −H −1 G′ u∗ (6.43)
and substitute the result into (6.41b) to obtain the complementary slackness con-
ditions
u∗i (−Gi H −1 G′ u∗ − wi − Si x) = 0, i = 1, . . . , m. (6.44)
Let u∗N A and u∗A denote the Lagrange multipliers corresponding to inactive and
active constraints, respectively. For inactive constraints u∗N A = 0. For active
constraints:
(−GA H −1 GA ′ )u∗A − wA − SA x = 0. (6.45)
6.3 Multiparametric Quadratic Programming 129

If the set of active constraint A is empty, then u∗ = u∗N A = 0 and therefore z ∗ = 0

which implies that the critical region CRA is

CRA = {x : Sx + w > 0}. (6.46)

Otherwise we distinguish two cases.

Case 1: LICQ holds, i.e., the rows of GA are linearly independent. This implies
that (GA H −1 GA ′ ) is a square full rank matrix and therefore

u∗A = −(GA H −1 GA ′ )−1 (wA + SA x), (6.47)

where GA , wA , SA correspond to the set of active constraints A. Thus u∗ is an

affine function of x. We can substitute u∗A from (6.47) into (6.43) to obtain

z ∗ = H −1 GA ′ (GA H −1 GA ′ )−1 (wA + SA x) (6.48)

and note that z ∗ is also an affine function of x. J ∗ (x) = 12 z ∗ (x)′ Hz ∗ (x) and
therefore it is a quadratic function of x. The critical region CRA is computed by
substituting z ∗ from (6.48) in the primal feasibility conditions (6.42b)

Pp = {x : GN A H −1 GA ′ (GA H −1 GA ′ )−1 (wA + SA x) < wN A + SN A x}, (6.49)

and the Lagrange multipliers from (6.47) in the dual feasibility conditions (6.41c)

Pd = {x : −(GA H −1 GA ′ )−1 (wA + SA x) ≥ 0}. (6.50)

In conclusion, the critical region CRA is the intersection of Pp and Pd :

CRA = {x : x ∈ Pp , x ∈ Pd }.ehre (6.51)

Obviously, the closure of CRA is a polyhedron in the x-space.

The polyhedron Pp is open and non empty (it contains at least the point x̄).
Therefore it is full-dimensional in the x space. This implies that dim CRA =
dim Pd .

Case 2: LICQ does not hold, the rows of GA are not linearly independent. For
instance, this happens when more than s constraints are active at the optimizer
z ∗ (x̄) ∈ Rs , i.e., in a case of primal degeneracy. In this case the vector of Lagrange
multipliers u∗ might not be uniquely defined, as the dual problem of (6.38) is not
strictly convex. Note that dual degeneracy and nonuniqueness of z ∗ (x̄) cannot
occur, as H ≻ 0.
Using the same arguments as in Section 6.2.2, equation (6.45) will allow a
full-dimensional critical region only if rank[(GA H −1 GA ′ ) SA ] = rank[GA H −1 GA ′ ].
Indeed the dimension of the critical region will be

n − (rank[(GA H −1 GA ′ ) SA ] − rank[GA H −1 GA ′ ]).

We will now show how to compute the critical region.

Let l = rank GA and consider the QR decomposition of −GA H −1 G′A
 
U1 U2
−GA H −1 G′A = Q ,
0|A|−l×l 0|A|−l×|A|−l
130 6 Multiparametric Programming: a Geometric Approach

where Q ∈ R|A|×|A| is a unitary

 matrix, U1 ∈ Rl×l is a full-rank square matrix
P q
and U2 ∈ Rl×|A|−l . Let = −Q−1 SA and = Q−1 wA . From (6.45) we
D r
obtain  
  u∗A,1  
U1 U2 P  ∗  q
uA,2 = . (6.52)
0|A|−l×l 0|A|−l×|A|−l D r
x
We compute u∗A,1 from (6.52)

u∗A,1 = U1−1 (−U2 u∗A,2 − P x + q). (6.53)

 ′
Finally, we can substitute u∗A = u∗A,1 ′ u∗A,2 ′ from (6.53) into (6.43) to obtain
 
∗ −1 ′ U1−1 (U2 u∗A,2 + P x − q)
z =H GA . (6.54)
u∗A,2

The optimizer z ∗ is unique and therefore independent of the choice of u∗A,2 ≥ 0.

Setting u∗A,2 = 0 we obtain

z ∗ = H −1 GA,1 ′ U1−1 (P x − q), (6.55)

 ′
where we have partitioned GA as GA = G′A,1 G′A,2 and GA,1 ∈ Rl×s .
Note that z ∗ is an affine function of x. J ∗ (x) = 12 z ∗ (x)′ Hz ∗ (x) and therefore
is a quadratic function of x.
The critical region CRA is computed by substituting z ∗ from (6.55) in the
primal feasibility conditions (6.42b)

Pp = {x : GN A (H −1 GA,1 ′ U1−1 (P x − q)) < wN A + SN A x}, (6.56)

and the Lagrange multipliers from (6.53) in the dual feasibility conditions (6.41c)

Pu∗A,2 ,x = {[ u∗A,2 , x] : U1−1 (−U2 u∗A,2 − P x + q) ≥ 0, u∗A,2 ≥ 0}. (6.57)

The critical region CRA is the intersection of the sets Dx = r, Pp and projx (Pu∗A,2 ,x ):

CRA = {x : Dx = r, x ∈ Pp , x ∈ projx (Pu∗A,2 ,x )}. (6.58)

The closure of CRA is a polyhedron in the x-space. 

Remark 6.5 In general, the critical region polyhedron (6.49)-(6.51) is open on facets
arising from primal feasibility and closed on facets arising from dual feasibility.

Remark 6.6 If D in (6.52) is nonzero, then from (6.58), CRA is a lower dimensional
region, which, in general, corresponds to a common boundary between two or more
full-dimensional regions.
 6.3 Multiparametric Quadratic Programming 131

6.3.3 Propagation of the Set of Active Constraints

The objective of this section is to briefly describe the propagation of the set of active
constraints when moving from one full-dimensional critical region to a neighboring
full-dimensional critical region. We will use a simple example in order to illustrate
the main points.

Example 6.6 Consider the mp-QP problem

J ∗ (x) = min 1 ′
2
z Hz + x′ F z
z (6.59)
subj. to Gz ≤ w + Sx,

with 1 
1
H = [ 10 01 ] , F = 1 −1 (6.60)
and  1 0   0 0
 1
−1 0 0 0 1
G=  0
0 1 
−1 , S=  00 0
0 , w =  11  , (6.61)
1 −1 1 1 0
−1 1 −2 1 0

where K is given by
−1 ≤ x1 ≤ 1
(6.62)
−1 ≤ x2 ≤ 1.
A solution to the mp-QP problem is shown in Figure 6.10 and the constraints which
are active in each associated full-dimensional critical region are reported in Table 6.3.

Critical Region Active Constraints

CR1 {5}
CR2 {3,5}
CR3 {}
CR4 {3}
CR5 {6}
CR6 {3,6}
CR7 {4,6}

Table 6.3 Example 6.6. Critical regions and corresponding set of active
constraints.

Since A3 is empty in CR3 we can conclude from (6.46) that the facets of CR3
are facets of primal feasibility and therefore do not belong to CR3. In general,
as discussed in Remark 6.5 critical regions are open on facets arising from primal
feasibility and closed on facets arising from dual feasibility. Next we focus on the
closure of the critical regions.
By observing Figure 6.10 and Table 6.3 we notice that as we move away from
region CR3 (corresponding to no active constraints), the number of active con-
straints increases. In particular, for any two neighboring full-dimensional critical
regions CRAi and CRAj we have Ai ⊂ Aj and |Ai | = |Aj | − 1 or Aj ⊂ Ai and
|Ai | = |Aj | + 1. This means that as one moves from one full-dimensional region
132 6 Multiparametric Programming: a Geometric Approach

1 6

4
5 7

3
x2

1
2

-1

-1 1
x1

Figure 6.10 Example 6.6. Polyhedral partition of the parameter space.

to a neighboring full-dimensional region, one constraint is either added to the list

of active constraints or removed from it. This happens for instance when moving
from CR3 to CR4, CR5, CR1 or from CR1 to CR2, or from CR5 to CR7.
The situation is more complex if LICQ does not hold everywhere in K∗ . In
particular, there might exist two neighboring full-dimensional critical regions CRAi
and CRAj where Ai and Aj do not share any constraint and with |Ai | = |Aj |. Also,
CRAi might have multiple neighboring region on the same facet. In other words,
it can happen that the intersection of the closures of two adjacent full-dimensional
critical regions is a not a facet of both regions but only a subset of it. We refer the
reader to [268] for more details on such a degenerate condition.

6.3.4 Value Function and Optimizer: Global Properties

The convexity of the value function J ∗ (x) and the continuity of the solution z ∗ (x)
follow from the general results on multiparametric programming (Corollary 5.2).
In the following we present an alternate simple proof.

Theorem 6.7 Consider the multiparametric quadratic program (6.38) and let H ≻
0. Then the optimizer z ∗ (x) : K∗ → Rs is continuous and piecewise affine on
polyhedra, in particular it is affine in each critical region, and the optimal solution
J ∗ (x) : K∗ → R is continuous, convex and piecewise quadratic on polyhedra.

Proof: We first prove convexity of J ∗ (x). Take generic x1 , x2 ∈ K∗ , and let

J (x1 ), J ∗ (x2 ) and z1 , z2 the corresponding optimal values and minimizers. Let
∗
6.3 Multiparametric Quadratic Programming 133

xα = αx1 + (1 − α)x2 and zα = αz1 + (1 − α)z2 . By optimality of J ∗ (xα ),

1 ′
J ∗ (xα ) ≤ z Hzα .
2 α
Hence
1 1 1
J ∗ (xα ) − [αz1′ Hz1 + (1 − α)z2′ Hz2 ] ≤ zα′ Hzα − [αz1′ Hz1 + (1 − α)z2′ Hz2 ].
2 2 2
We can upperbound the latter term as follows:
1 ′
2 zα Hzα − 12 [αz1′ Hz1 + (1 − α)z2′ Hz2 ] = 12 [α2 z1′ Hz1 + (1 − α)2 z2′ Hz2 +
+2α(1 − α)z2′ Hz1 − αz1′ Hz1 − (1 − α)z2′ Hz2 ] =
= − 21 α(1 − α)(z1 − z2 )′ H(z1 − z2 ) ≤ 0.
In conclusion

J ∗ (αx1 + (1 − α)x2 ) ≤ αJ ∗ (x1 ) + (1 − α)J ∗ (x2 ), ∀x1 , x2 ∈ K, ∀α ∈ [0, 1],

which proves the convexity of J ∗ (x) on K∗ .

Within the closed polyhedral regions CRi in K∗ the solution z ∗ (x) is affine (6.48)
by Theorem 6.6. The boundary between two regions belongs to both closed regions.
Since H ≻ 0, the optimum is unique and hence the solution must be continuous
across the boundary. Therefore z ∗ (x) : K∗ → Rs is continuous and piecewise affine
on polyhedra. The fact that J ∗ (x) is continuous and piecewise quadratic follows
trivially. 
We can easily derive a result similar to Theorem 6.7 for the dual function d∗ (x)
(see (6.38) and (2.28))
 
∗ 1 ′ −1 ′ ′
d (x) = max d(u) = − u (GH G )u − u (w + Sx) . (6.63)
u≥0 2

Theorem 6.8 Consider the multiparametric quadratic program (6.38) with H ≻ 0

and assume that LICQ holds ∀ x ∈ K∗ . Then the optimizer u∗ (x) : K∗ → Rm
of the dual (6.63) is continuous and piecewise affine on polyhedra, in particular
it is affine in each critical region, and the optimal solution d∗ (x) : K∗ → R is
continuous, convex and piecewise quadratic on polyhedra.
Proof: The proof follows along the lines of the proof of Theorem 6.7 once we
know that the optimizer u∗ (x) is unique because of the LICQ assumptions. 

Theorem 6.9 Assume that the mp-QP problem (6.38) is not degenerate, then the
value function J ∗ (x) in (6.38) is continuously differentiable (C (1) ).
Proof: The dual of (6.38) is
1
d∗ (x) = max − u′ (GH −1 G′ )u − u′ (w + Sx). (6.64)
u≥0 2
By strong duality we have
1
J ∗ (x) = d∗ (x) = − u∗ (x)′ (GH −1 G′ )u∗ (x) − u∗ (x)′ (w + Sx). (6.65)
2
134 6 Multiparametric Programming: a Geometric Approach

For the moment let us restrict our attention to the interior of a critical region.
From (6.47) we know that u∗ (x) is continuously differentiable. Therefore
 ∗ ′  ∗ ′
∂u −1 ′ ∗ ∂u
∇x J(x) = − (x) GH G u (x) − (x) (w + Sx) − S ′ u∗ (x) =
∂x ∂x
 ∗ ′
∂u
= (x) (−GH −1 G′ u∗ (x) − w − Sx) − S ′ u∗ (x).
∂x
(6.66)
Using (6.43)
z ∗ = −H −1 G′ u∗ , (6.67)
we get
 ′
∂u∗
∇x J(x) = −S ′ u∗ (x) + (x) (Gz ∗ − w − Sx) =
∂x ′
P ∂u∗i
= ′ ∗
−S u (x) + (x) (Gi z ∗ − wi − Si x) +
i∈A ∂x | {z }
 ∗ ′ 0 (6.68)
P ∂ui
+ (x) (Gi z ∗ − wi − Si x) =
i∈N A ∂x
| {z }
0
= −S ′ u∗ (x).

Continuous differentiability of J ∗ in the interior of a critical region follows from (6.47).

Continuous differentiability of J ∗ at the boundary follows from the fact that u∗ (x)
is continuous (Theorem 6.8) for all x in the interior of K∗ [20]. 
Note that in case of degeneracy the value function J ∗ (x) in (6.38) may not be C (1) .
The following counterexample was given in [48].

Example 6.7
Consider the mp-QP (6.38) with
 
3 3 −1
H= 3 11 23 
 −1 23 75     
1 1 1 1 0
 1 3 5   0   1 
      (6.69)
 −1 −1 −1   −1   0 
     
G= −1 −3 −5

 w=
 0 
 S=
 −1 

 −1 0 0   0   0 
     
 0 −1 0   0   0 
0 0 −1 0 0

and K = {x ∈ R|1 ≤ x ≤ 5}.

The problem was solved by using Algorithm 6.2 described next. The solution com-
prises 5 critical regions. The critical regions and the expression of the value function
are reported in Table 6.4. The reader can verify that the value function is not con-
tinuously differentiable at x = 3. Indeed, at x = 3 the LICQ condition does not hold
and therefore, the hypothesis of Theorem 6.9 is not fulfilled.
 6.3 Multiparametric Quadratic Programming 135

Region Optimal value

CR{1,2,3,4,6} = {x : 1 ≤ x < 1.5} 2.5x2 − 6x + 5
CR{1,2,3,4} = {x : 1.5 ≤ x ≤ 2} 0.5x2 + 0.5
CR{1,2,3,4,7} = {x : 2 < x < 3} x2 − 2x + 2.5
CR{1,2,3,4,5,7} = {x : x = 3} x2 − 2x + 2.5
CR{1,2,3,4,5} = {x : 3 < x ≤ 5} 5x2 − 24x + 32.5

Table 6.4 Critical regions and value function corresponding to the solution
of Example 6.7

6.3.5 mp-QP Algorithm

The goal of an mp-QP algorithm is to determine the partition of K∗ into critical
regions CRi , and find the expression of the functions J ∗ (·) and z ∗ (·) for each critical
region. An mp-QP algorithm has two components: the “active set generator” and
the “KKT solver”. The active set generator computes the set of active constraints
Ai . The KKT solver computes CRAi and the expression of J ∗ (·) and z ∗ (·) in
CRAi as explained in Theorem 6.6. The active set generator is the critical part.
In principle, one could simply generate all the possible combinations of active
sets. However, in many problems only a few active constraints sets generate full-
dimensional critical regions inside the region of interest K. Therefore, the goal is
to design an active set generator algorithm which computes only the active sets Ai
with the associated full-dimensional critical regions covering only K∗ .
Next an implementation of an mp-QP algorithm is described. See Section 6.6
for a literature review on alternative approaches to the solution of an mp-QP.
In order to start solving the mp-QP problem, we need an initial vector x0
inside the polyhedral set K∗ of feasible parameters. A possible choice for x0 is the
Chebychev center (see Section 4.4.5) of K∗ obtained as a solution of

maxx,z̄,ǫ ǫ
subj. to Ti x + ǫkTi k2 ≤ Ni , i = 1, . . . , nT (6.70)
Gz̄ − Sx ≤ w

where nT is the number of rows Ti of the matrix T defining the set K in (6.2).
If ǫ ≤ 0, then the QP problem (6.38) is infeasible for all x in the interior of K.
Otherwise, we set x = x0 and solve the QP problem (6.38), in order to obtain the
corresponding optimal solution z0∗ . Such a solution is unique, because H ≻ 0. The
value of z0∗ defines the following optimal partition

A(x0 ) = {j ∈ J : Gj z0∗ − Sj x0 − wj = 0}
(6.71)
NA(x0 ) = {j ∈ J : Gj z0∗ − Sj x0 − wj < 0}

and consequently the critical region CRA(x0 ) . Once the critical region CRA(x0 )
has been defined, the rest of the space Rrest = K\CRA(x0 ) has to be explored and
new critical regions generated. An approach for generating a polyhedral partition
{R1 , . . . , Rnrest } of the rest of the space Rrest is described in Theorem 4.2. Theo-
rem 4.2 provides a way of partitioning the non-convex set K \ CR0 into polyhedral
136 6 Multiparametric Programming: a Geometric Approach

subsets Ri . For each Ri , a new vector xi is determined by solving the LP (6.70),

and, correspondingly, an optimum zi∗ , a set of active constraints Ai , and a critical
region CRi . The procedure proposed in Theorem 4.2 for partitioning the set of
parameters allows one to recursively explore the parameter space. Such an iterative
procedure terminates after a finite time, as the number of possible combinations
of active constraints decreases with each iteration. Two main elements need to be
considered:

1. As for the mp-LP algorithm, the partitioning in Theorem 4.2 defines new
polyhedral regions Rk to be explored that are not related to the critical
regions which still need to be determined. This may split some of the critical
regions, due to the artificial cuts induced by Theorem 4.2. Post-processing
can be used to join cut critical regions [44]. As an example, in Figure 6.8 the
critical region CR{3,7} is discovered twice, one part during the exploration of
R1 and the second part during the exploration of R2 .
Although algorithms exist for convexity recognition and computation of the
union of polyhedra, the post-processing operation is computationally expen-
sive. Therefore, it is more efficient not to intersect the critical region obtained
by (6.29) with halfspaces generated by Theorem 4.2, which is only used to
drive the exploration of the parameter space. Then, no post processing is
needed to join subpartitioned critical regions. On the other hand, some criti-
cal regions may appear more than once. Duplicates can be uniquely identified
by the set of active constraints A(x) and can be easily eliminated. To this
aim, in the implementation of the algorithm we keep a list of all the critical
regions which have already been generated in order to avoid duplicates. In
Figure 6.8 the critical region CR{3,7} is discovered twice but stored only once.

2. If Case 2 occurs in Section 6.3.2 and D is nonzero, CRA is a lower dimen-

sional critical region (see Remark 6.6). Therefore we do not need to explore
the actual combination GA , SA , wA . On the other hand, if D = 0 the
KKT conditions do not lead directly to (6.49)–(6.50). In this case, a full-
dimensional critical region can be obtained from (6.58) by projecting the set
Px,u∗A,2 in (6.57).

Based on the discussion and results above, the main steps of the mp-QP solver
are outlined in the following algorithm.

Algorithm 6.2 mp-QP Algorithm

Input: Matrices H, G, w, S of problem (6.38) and set K in (6.2)
Output: Multiparametric solution to problem (6.38)
Execute partition(K)
end

Function partition(Y )
Let x0 ∈ Y and ǫ be the solution to the LP (6.70);
If ǫ ≤ 0 Then exit (no full-dimensional CR is in Y )
Solve the QP (6.38) for x = x0 to obtain (z0∗ , u∗0 )
 6.4 Multiparametric Mixed-Integer Linear Programming 137

Determine the set of active constraints A when z = z0∗ , x = x0 , and build

GA , wA , SA
If GA has full row rank Then
Determine u∗A (x), z ∗ (x) from (6.47) and (6.48)
Characterize the CR from (6.49) and (6.50)
Else
Determine z ∗ (x) from (6.55)
Characterize the CR from (6.58)
End
Partition the rest of the region as in Theorem 4.2
For each new sub-region Ri , Do partition(Ri )
End function

Remark 6.7 If rank(D) > 0 in Algorithm 6.2, the region CRA(x0 ) is not full-
dimensional. To avoid further recursion in the algorithm which does not produce
any full-dimensional critical region, one should compute a random vector ǫ ∈ Rn
smaller than the Chebychev radius of Y and such that the QP (6.38) is feasible for
x0 + ǫ and then repeat step where A(x0 ) is computed with x0 ← x0 + ǫ.

Remark 6.8 The algorithm solves the mp-QP problem by partitioning the given
parameter set K into Nr closed polyhedral regions. Note that the algorithm generates
a partition of the state space which is not strict. The algorithm could be modified
to store the critical regions as defined in Section 6.1.2. (which are neither closed nor
open as proven in Theorem 6.6) instead of storing their closure. This can be done by
keeping track of which facet belongs to a certain critical region and which not. From
a practical point of view, such a procedure is not necessary since the value function
and the optimizer are continuous functions of x.

Remark 6.9 The proposed algorithm does not apply to the case when H  0 and when
the optimizer may not be unique. In this case one may resort to “regularization”,
i.e., adding an appropriate “small” quadratic term, or follow the ideas of [231].

6.4 Multiparametric Mixed-Integer Linear Programming

6.4.1 Formulation and Properties

Consider the mp-LP

J ∗ (x) = min {J(z, x) = c′ z}
z (6.72)
subj. to Gz ≤ w + Sx,
where z ∈ Rs are the optimization variables, x ∈ Rn is the vector of parameters,
G ∈ Rm×s , w ∈ Rm , and S ∈ Rm×n . When we restrict some of the optimization
138 6 Multiparametric Programming: a Geometric Approach

variables to be 0 or 1, z = {zc , zd }, zc ∈ Rsc , zd ∈ {0, 1}sd and s = sc + sd , we

refer to (6.72) as a (right-hand side) multiparametric mixed-integer linear program
(mp-MILP).

6.4.2 Geometric Algorithm for mp-MILP

Consider the mp-MILP (6.72). Given a closed and bounded polyhedral set K ⊂ Rn
of parameters,
K = {x ∈ Rn : T x ≤ N }, (6.73)
we denote by K∗ ⊆ K the region of parameters x ∈ K such that the MILP (6.72)
is feasible and the optimum J ∗ (x) is finite. For any given x̄ ∈ K∗ , J ∗ (x̄) denotes
the minimum value of the objective function in problem (6.72) for x = x̄. The
function J ∗ : K∗ → R will denote the function which expresses the dependence on
x of the minimum value of the objective function over K∗ , J ∗ will be called the
sc s
value function. The set-valued function Z ∗ : K∗ → 2R × 2{0,1} d will describe for
∗ ∗ ∗
any fixed x ∈ K the set of optimizers z (x) related to J (x).
We aim to determine the region K∗ ⊆ K of feasible parameters x and to find the
expression of the value function J ∗ (x) and the expression of an optimizer function
z ∗ (x) ∈ Z ∗ (x).
Two main approaches have been proposed for solving mp-MILP problems. In
[1], the authors develop an algorithm based on branch and bound (B&B) methods.
At each node of the B&B tree an mp-LP is solved. The solution at the root
node where all the binary variables are relaxed to the interval [0,1] represents a
valid lower bound, while the solution at a node where all the integer variables
have been fixed represents a valid upper bound. As in standard B&B methods,
the complete enumeration of combinations of 0-1 integer variables is avoided by
comparing the multiparametric solutions, and by fathoming the nodes where there
is no improvement of the value function.
In [101] an alternative algorithm was proposed, which will be detailed in this
section. Problem (6.72) is alternatively decomposed into an mp-LP and an MILP
subproblem. In one step the values of the binary variable are fixed for a region
and an mp-LP is solved. Its solution provides a parametric upper bound to the
value function J ∗ (x) in the region. In the other step, the parameters x are treated
as additional free variables and an MILP is solved. In this way a parameter x is
found and an associated new integer vector which improves the value function at
this point.
The algorithm is composed of an initialization step, and a recursion between
the solution of an mp-LP subproblem and an MILP subproblem.

Initialization
Solve the following MILP problem

min c′ z
{z,x}
subj. to Gz − Sx ≤ w (6.74)
x ∈ K,
6.4 Multiparametric Mixed-Integer Linear Programming 139

where x is treated as an independent variable. If the MILP (6.74) is infeasible then

the mp-MILP (6.72) admits no solution, i.e., K∗ = ∅; otherwise its solution z ∗ , x∗
provides an integer variable z ∗ that is feasible at point x∗ .
At step j = 0, set: N0 = 1, CR1 = K, Z1 = ∅, J¯1 = +∞, Nbi = 0, z̄d11 = z̄d .

mp-LP subproblem
For each CRi we solve the following mp-LP problem

J˜i (x) = min c′ z

z
subj. to Gz ≤ w + Sx (6.75)
zd = z̄dNi bi +1
x ∈ CRi .

By Theorem 6.5, the solution of mp-LP (6.75) provides a partition of CRi into
polyhedral regions Rik , k = 1, . . . , NRi and a PWA value function

˜ k (x) = ck ′ x + pk ) if x ∈ Rk , k = 1, . . . , NRi
J˜i (x) = (JR (6.76)
i i i i

where JR ˜ ji (x) = +∞ in Rj if the integer variable z̄di is not feasible in Rj and a

i i
PWA continuous optimizer z ∗ (x) (z ∗ (x) is not defined in Rij if JR˜ j (x) = +∞).
i
The function J˜i (x) will be an upper bound of J ∗ (x) for all x ∈ CRi . Such
a bound J˜i (x) on the value function has to be compared with the current bound
J¯i (x) in CRi in order to obtain the lowest of the two parametric value functions
and to update the bound.
While updating J¯i (x) three cases are possible:
˜ ki (x) ∀x ∈ Rk if (JR
1. J¯i (x) = JR ˜ ki (x) ≤ J¯i (x) ∀ x ∈ Rk ).
i i

˜ k (x) ≥ J¯i (x) ∀ x ∈ Rk ).

2. J¯i (x) = J¯i (x) ∀x ∈ Rik (if JR i i
( k
J¯i (x) ∀x ∈ (Rik )1 = {x ∈ Rik : JR ˜ (x) ≥ J¯i (x)}
3. J¯i (x) = k
i
k
JRi (x) ∀x ∈ (Ri )2 = {x ∈ Ri : JRi (x) ≤ J¯i (x)}.
˜ k k ˜

The three cases above can be distinguished by using a simple linear program. We
add the constraint JR ˜ ki (x) ≤ J¯i (x) to the constraints defining Rk and tests its
i
redundancy (Section 4.4.1). If it is redundant we have case 1, if it is infeasible
we have case 2. Otherwise JR ˜ ki (x) = J¯i (x) defines the facet separating (Rk )1 and
i
(Rik )2 . In the third case, the region Rik is partitioned into two regions (Rik )1 and
(Rik )2 which are convex polyhedra since JR ˜ ki (x) and J¯i (x) are affine functions of
x.
After the mp-LP (6.75) has been solved for all i = 1, . . . , Nj (the subindex
j denotes that we are at step j of the recursion) and the value function has been
updated, each initial region CRi has been subdivided into at most 2NRi polyhedral
regions Rik and possibly (Rik )1 and (Rik )2 with a corresponding updated parametric
bound on the value function J¯i (x). For each Rik , (RSik )1 and (Rik )2 we define the
set of integer variables already explored as Zi = Zi z̄dNi bi +1 , N bi = N bi + 1. In
the sequel the polyhedra of the new partition will be referred to as CRi .
140 6 Multiparametric Programming: a Geometric Approach

MILP subproblem
At step j for each critical region CRi (note that these CRi are the output of the
previous phase) we solve the following MILP problem

min c′ z (6.77)
{z,x}

subj. to Gz − Sx ≤ w (6.78)
c′ z ≤ J¯i (x) (6.79)
zd 6= z̄dki , k = 1, . . . , N bi (6.80)
x ∈ CRi , (6.81)

where constraints (6.80) prohibit integer solutions that have been already analyzed
in CRi from appearing again and constraint (6.79) excludes integer solutions with
higher values than the current upper bound. If problem (6.81) is infeasible then
the region CRi is excluded from further recursion and the current upper bound
represents the final solution. If problem (6.81) is feasible, then the discrete optimal
component zd∗i is stored and represents a feasible integer variable that is optimal
at least in one point of CRi .

Recursion
At step j we have stored

1. A list of Nj polyhedral regions CRi and for each of them an associated

parametric affine upper bound J¯i (x) (J¯i (x) = +∞ if no integer solution has
been found yet in CRi ).

2. For each CRi a set of integer variables Zi = z̄d0i , . . . , z̄dNi bi , that have already
been explored in the region CRi .

3. For each CRi an integer feasible variable z̄dNi bi +1 ∈

/ Zi such that there exists zc
¯
and x̂ ∈ CRi for which Gz ≤ w + S x̂ and c z < Ji (x̂) where z = {zc , z̄dNi bi +1 }.
′

That is, z̄dNi bi +1 is an integer variable that improves the current bound for at
least one point of the current polyhedron.

For all the regions CRi not excluded from the MILP’s subproblem (6.77)-(6.81)
the algorithm continues to iterate between the mp-LP (6.75) with z̄dNi bi +1 = zd∗i
and the MILP (6.77)-(6.81). The algorithm terminates when all the MILPs (6.77)-
(6.81) are infeasible.
Note that the algorithm generates a partition of the state space. Some param-
eter x could belong to the boundary of several regions. Differently from the LP
and QP case, the value function may be discontinuous and therefore such a case
has to be treated carefully. If a point x belongs to different critical regions, the
expressions of the value function associated with such regions have to be compared
in order to assign to x the right optimizer. Such a procedure can be avoided by
keeping track of which facet belongs to a certain critical region and which not.
Moreover, if the value functions associated with the regions containing the same
parameter x coincide this may imply the presence of multiple optimizers.
 6.5 Multiparametric Mixed-Integer Quadratic Programming 141

6.4.3 Solution Properties

The following properties of J ∗ (x) and Z ∗ (x) follow easily from the algorithm de-
scribed above.

Theorem 6.10 Consider the mp-MILP (6.72). The set K∗ is the union of a finite
number of (possibly open) polyhedra and the value function J ∗ is piecewise affine
on polyhedra. If the optimizer z ∗ (x) is unique for all x ∈ K∗ , then the optimizer
functions zc∗ : K∗ → Rsc and zd∗ : K∗ → {0, 1}sd are piecewise affine and piecewise
constant, respectively, on polyhedra. Otherwise, it is always possible to define a
piecewise affine optimizer function z ∗ (x) ∈ Z ∗ (x) for all x ∈ K∗ .

Note that, differently from the mp-LP case, the set K∗ can be non-convex and
even disconnected.

6.5 Multiparametric Mixed-Integer Quadratic Programming

6.5.1 Formulation and Properties

Consider the mp-QP

J ∗ (x) = min J(z, x) = z ′ H1 z + c′1 z

z (6.82)
subj. to Gz ≤ w + Sx,

When we restrict some of the optimization variables to be 0 or 1, z = [zc′ , zd′ ]′ ,

where zc ∈ Rsc , zd ∈ {0, 1}sd , we refer to (6.82) as a multiparametric mixed-integer
quadratic program (mp-MIQP). Given a closed and bounded polyhedral set K ⊂ Rn
of parameters,
K = {x ∈ Rn : T x ≤ N }, (6.83)

we denote by K∗ ⊆ K the region of parameters x ∈ K such that the MIQP (6.82) is

feasible and the optimum J ∗ (x) is finite. For any given x̄ ∈ K∗ , J ∗ (x̄) denotes the
minimum value of the objective function in problem (6.82) for x = x̄. The value
function J ∗ : K∗ → R denotes the function which expresses the dependence on x
of the minimum value of the objective function over K∗ . The set-valued function
sc s
Z ∗ : K∗ → 2R × 2{0,1} d describes for any fixed x ∈ K∗ the set of optimizers z ∗ (x)
∗
related to J (x).
We aim at determining the region K∗ ⊆ K of feasible parameters x and at
finding the expression of the value function J ∗ (x) and the expression of an optimizer
function z ∗ (x) ∈ Z ∗ (x).
We show with a simple example that the geometric approach discussed in this
chapter cannot be used for solving mp-MIQPs.
142 6 Multiparametric Programming: a Geometric Approach

Example 6.8 Suppose z1 , z2 , x1 , x2 ∈ R and δ ∈ {0, 1}, then the following mp-MIQP

J ∗ (x1 , x2 ) = minz1 ,z2 ,δ z12 + z22 − 25δ + 100

     
1 0 10 1 0 10
 −1 0 10   −1 0   10 
     
 0 1 10   0 1   10 
     
 0 −1 10   0 −1   10 
     
 1 0 −10    0 0   0 
  z1     
 −1 0 −10   0 0  x1  0 
subj. to    z2  ≤    
 0 1 −10   0 0  x2 +  0 
  δ    
 0 −1 −10   0 0   0 
     
 0 0 0   1 0   10 
     
 0 0 0   −1 0   10 
     
 0 0 0   0 1   10 
0 0 0 0 −1 10
(6.84)
can be simply solved by noting that for δ = 1 z1 = x1 and z2 = x2 while for δ = 0
z1 = z2 = 0. By comparing the value functions associated with δ = 0 and δ = 1 we
obtain two critical regions
CR1 = {x1 , x2 ∈ R : x21 + x22 ≤ 25}
(6.85)
CR2 = {x1 , x2 ∈ R : − 10 ≤ x1 ≤ 10, − 10 ≤ x2 ≤ 10, x21 + x22 > 25},

x1 if [x1 , x2 ] ∈ CR1
z1∗ (x1 , x2 ) =
 0 if [x1 , x2 ] ∈ CR2
(6.86)
∗ x2 if [x1 , x2 ] ∈ CR1
z2 (x1 , x2 ) = ,
0 if [x1 , x2 ] ∈ CR2
and the parametric value function
 2
x1 + x22 + 75 if [x1 , x2 ] ∈ CR1
J ∗ (x1 , x2 ) = (6.87)
100 if [x1 , x2 ] ∈ CR2 .
The two critical regions and the value function are depicted in Figure 6.11.

Example 6.8 demonstrates that, in general, the critical regions of an mp-MIQP

cannot be decomposed into convex polyhedra. Therefore the method of partition-
ing the rest of the space presented in Theorem 4.2 cannot be applied here. In
Chapter 17 we will present an algorithm that efficiently solves specific mp-MIQPs
that stem from the optimal control of discrete-time hybrid systems.

6.6 Literature Review

Many of the theoretical results on parametric programming can be found in[18,
107, 47, 116].

The first method for solving parametric linear programs was proposed by Gass
and Saaty [120], and since then extensive research has been devoted to sensitivity
6.6 Literature Review 143

J ∗ (x)
CR1
x2

CR2

x1 x2 x1
(a) Critical Regions. (b) Value function.

Figure 6.11 Example 6.8. Solution to the mp-MIQP (6.84).

and (multi)-parametric linear analysis, as attested by the hundreds of references

in [115] (see also [116] for recent advances in the field). One of the first methods
for solving multiparametric linear programs (mp-LPs) was formulated by Gal and
Nedoma [117]. The method constructs the critical regions iteratively, by visiting
the graph of bases associated with the LP tableau of the original problem. Many
of the results on mp-LP presented in this book can be found in [115, p. 178-180].
Note that in [117, 115] a critical region is defined as a subset of the param-
eter space on which a certain basis of the linear program is optimal. The al-
gorithm proposed in [117] for solving multiparametric linear programs generates
non-overlapping critical regions by generating and exploring the graph of bases. In
the graph of bases the nodes represent optimal bases of the given multiparametric
problem and two nodes are connected by an edge if it is possible to pass from one
basis to another by one pivot step (in this case the bases are called neighbors).
In this book we use the definition (6.4) of critical regions which is not associated
with the bases but with the set of active constraints and it is directly related to
the definition given in [2, 207, 116].

The solution to multiparametric quadratic programs has been studied in detail

in [18, Chapter 5]. In [44] Bemporad and coauthors presented a simple method
for solving mp-QPs. The method constructs a critical region in a neighborhood
of a given parameter, by using the Karush-Kuhn-Tucker conditions for optimality,
and then recursively explores the parameter space outside such a region. Other
algorithms for solving mp-QPs have been proposed by Seron, DeDoná and Good-
win in [262, 99] in parallel with the study of Bemporad and coauthors in [44], by
Tondel, Johansen and Bemporad in [274] and by Baotic in [19]. All these algo-
rithms are based on an iterative procedure that builds up the parametric solution
by generating new polyhedral regions of the parameter space at each step. The
methods differ in the way they explore the parameter space, that is, the way they
identify active constraints corresponding to the critical regions neighboring to a
given critical region, i.e., in the “active set generator” component.
In [262, 99] the authors construct the unconstrained critical region and then
144 6 Multiparametric Programming: a Geometric Approach

generate neighboring critical regions by enumerating all possible combinations of

active constraints.
In [274] the authors explore the parameter space outside a given region CRi
by examining its set of active constraints Ai . The critical regions neighboring to
CRi are constructed by elementary operations on the active constraints set Ai
that can be seen as an equivalent “pivot” for the quadratic program. For this
reason the method can be considered as an extension of the method of Gal [115]
to multiparametric quadratic programming.
In [19] the author uses a direct exploration of the parameter space as in [44]
but he avoids the partition of the state space described in Theorem 4.2. Given a
polyhedral critical region CRi , the procedure goes through all its facets and gen-
erates the Chebychev center of each facet. For each facet Fi a new parameter xiε
is generated, by moving from the center of the facet in the direction of the normal
to the facet by a small step. If such parameter xiε is infeasible or is contained in
a critical region already stored, then the exploration in the direction of Fi stops.
Otherwise, the set of active constraints corresponding to the critical region sharing
the facet Fi with the region CRi is found by solving a QP for the new parameter
xiε .

In [1, 101] two approaches were proposed for solving mp-MILP problems. In
both methods the authors use an mp-LP algorithm and a branch and bound strat-
egy that avoids the complete enumeration of combinations of 0-1 integer variables
by comparing the available bounds on the multiparametric solutions.
 Part III

Optimal Control
 Chapter 7

General Formulation and

Discussion

In this chapter we introduce the optimal control problem we will be studying in

a very general form. We want to communicate the basic definitions and essential
concepts. We will sacrifice mathematical precision for the sake of simplicity. In later
chapters we will study specific versions of this problem for specific cost functions
and system classes in greater detail.

7.1 Problem Formulation

We consider the nonlinear time-invariant system

x(t + 1) = g(x(t), u(t)), (7.1)

subject to the constraints

h(x(t), u(t)) ≤ 0 (7.2)
n m
at all time instants t ≥ 0. In (7.1)–(7.2), x(t) ∈ R and u(t) ∈ R are the state and
input vector, respectively. Inequality (7.2) with h : Rn × Rm → Rnc expresses the
nc constraints imposed on the input and the states. These may be simple upper
and lower bounds or more complicated expressions. We assume that the origin is
an equilibrium point (g(0, 0) = 0) in the interior of the feasible set, i.e., h(0, 0) < 0.
We assumed the system to be specified in discrete time. One reason is that we
are looking for solutions to engineering problems. In practice, the controller will
almost always be implemented through a digital computer by sampling the variables
of the system and transmitting the control action to the system at discrete time
points. Another reason is that for the solution of the optimal control problems for
discrete-time systems we will be able to make ready use of powerful mathematical
programming software.
We want to caution the reader, however, that in many instances the discrete
time model is an approximation of the continuous time model. It is generally
148 7 General Formulation and Discussion

difficult to derive “good” discrete time models from nonlinear continuous time
models, and especially so when the nonlinear system has discontinuities as would
be the case for switched systems. We also note that continuous time switched
systems can exhibit behavioral characteristics not found in discrete-time systems,
for example, an ever increasing number of switches in an ever decreasing time
interval (Zeno behavior [127]).
We define the following performance objective or cost function from time instant
0 to time instant N
N
X −1
J0→N (x0 , U0→N ) = p(xN ) + q(xk , uk ), (7.3)
k=0

where N is the time horizon and xk denotes the state vector at time k obtained by
starting from the measured state x0 = x(0) and applying to the system model
xk+1 = g(xk , uk ), (7.4)
the input sequence u0 , . . . , uk−1 . From this sequence we define the vector of future
inputs U0→N = [u′0 , . . . , u′N −1 ]′ ∈ Rs , s = mN . The terms q(xk , uk ) and p(xN )
are referred to as stage cost and terminal cost, respectively, and are assumed to be
positive definite (q ≻ 0, p ≻ 0):
p(x, u) > 0 ∀x 6= 0, u 6= 0, p(0, 0) = 0
q(x, u) > 0 ∀x =
6 0, u =6 0, q(0, 0) = 0.
The form of the cost function (7.3) is very general. If a practical control objective
can be expressed as a scalar function then this function usually takes the indicated
form. Specifically, we consider the following constrained finite time optimal control
(CFTOC) problem.
∗
J0→N (x0 ) = minU0→N J0→N (x0 , U0→N )
subj. to xk+1 = g(xk , uk ), k = 0, . . . , N − 1
h(xk , uk ) ≤ 0, k = 0, . . . , N − 1 (7.5)
xN ∈ Xf
x0 = x(0).
Here Xf ⊆ Rn is a terminal region that we want the system states to reach at the
end of the horizon. The terminal region could be the origin, for example. We define
X0→N ⊆ Rn to be the set of initial conditions x(0) for which there exists an input
vector U0→N so that the inputs u0 , . . . , uN −1 and the states x0 , . . . , xN satisfy the
model xk+1 = g(xk , uk ) and the constraints h(xk , uk ) ≤ 0 and that the state xN
lies in the terminal set Xf .
We can determine this set of feasible initial conditions in a recursive manner.
Let us denote with Xj→N the set of states xj at time j which can be steered into
Xf at time N , i.e., for which the model xk+1 = g(xk , uk ) and the constraints
h(xk , uk ) ≤ 0 are feasible for k = j, . . . , N − 1 and xN ∈ Xf . This set can be
defined recursively by
Xj→N = {x ∈ Rn : ∃u such that (h(x, u) ≤ 0, and g(x, u) ∈ Xj+1→N )},
j = 0, . . . , N − 1 (7.6)
XN →N = Xf . (7.7)
 7.2 Solution via Batch Approach 149

The set X0→N is the final result of these iterations starting with Xf .
∗
The optimal cost J0→N (x0 ) is also called value function. In general, the problem
(7.3)–(7.5) may not have a minimum. We will assume that there exists a minimum.
This is the case, for example, when the set of feasible input vectors U0→N (defined
by h and Xf ) is compact and when the functions g, p and q are continuous. Also,
∗ ∗
there might be several input vectors U0→N which yield the minimum (J0→N (x0 ) =
∗ ∗
J0→N (x0 , U0→N )). In this case we will define one of them as the minimizer U0→N .
Note that throughout the book we will distinguish between the current state
x(k) of system (7.1) at time k and the variable xk in the optimization problem (7.5),
that is the predicted state of system (7.1) at time k obtained by starting from the
state x0 and applying to system (7.4) the input sequence u0 , . . . , uk−1 . Analogously,
u(k) is the input applied to system (7.1) at time k while uk is the k-th optimization
variable of the optimization problem (7.5). Clearly, x(k) = xk for any k if u(k) = uk
for all k (under the assumption that our model is perfect).
In the rest of this chapter we will be interested in the following questions related
to the general optimal control problem (7.3)–(7.5).

• Solution. We will show that the problem can be expressed and solved either
as one general nonlinear programming problem, or in a recursive manner by
invoking Bellman’s Principle of Optimality.

• Infinite horizon. We will investigate if a solution exists as N → ∞, the prop-

erties of this solution, and how it can be obtained or at least approximated
by using a receding horizon technique.

7.2 Solution via Batch Approach

If we write the equality constraints appearing in (7.5) explicitly

x1 = g(x(0), u0 )
x2 = g(x1 , u1 )
.. (7.8)
.
xN = g(xN −1 , uN −1 ),

then the optimal control problem (7.3)–(7.5), rewritten below

∗
PN −1
J0→N (x0 ) = minU0→N p(xN ) + k=0 q(xk , uk )
subj. to x1 = g(x0 , u0 )
x2 = g(x1 , u1 )
..
. (7.9)
xN = g(xN −1 , uN −1 )
h(xk , uk ) ≤ 0, k = 0, . . . , N − 1
xN ∈ Xf
x0 = x(0)
150 7 General Formulation and Discussion

is recognized more easily as a general nonlinear programming problem with vari-

ables u0 , . . . , uN −1 and x1 , . . . , xN .
As an alternative we may try to eliminate the state variables and equality
constraints (7.8) by successive substitution so that we are left with u0 , . . . , uN −1
as the only decision variables. For example, we can express x2 as a function of
x(0), u0 and u1 only, by eliminating the intermediate state x1

x2 = g(x1 , u1 )
(7.10)
x2 = g(g(x(0), u0 ), u1 ).

Except when the state equations are linear this successive substitution may become
complex. Even when they are linear it may be bad from a numerical point of view.
Either with or without successive substitution the solution of the nonlinear pro-
′ ′
∗
gramming problem is a sequence of present and future inputs U0→N = [u∗0 , . . . , u∗N −1 ]′
determined for the particular initial state x(0).

7.3 Solution via Recursive Approach

The recursive approach, Bellman’s dynamic programming technique, rests on a sim-
ple idea, the principle of optimality. It states that for a trajectory x0 , x∗1 , . . . , x∗N to
be optimal, the trajectory starting from any intermediate point x∗j , i.e., x∗j , x∗j+1 , . . . , x∗N ,
0 ≤ j ≤ N − 1, must be optimal.
Consider the following example to provide an intuitive justification [53]. Sup-
pose that the fastest route from Los Angeles to Boston passes through Chicago.
Then the principle of optimality formalizes the obvious fact that the Chicago to
Boston portion of the route is also the fastest route for a trip that starts from
Chicago and ends in Boston.
We can utilize the principle of optimality for the optimal control problem we
are investigating. We define the cost over the reduced horizon from j to N

N
X −1
Jj→N (xj , uj , uj+1 , . . . , uN −1 ) = p(xN ) + q(xk , uk ), (7.11)
k=j

∗
also called the cost-to-go. Then the optimal cost-to-go Jj→N is

∗
Jj→N (xj ) = minuj ,uj+1 ,...,uN −1 Jj→N (xj , uj , uj+1 , . . . , uN −1 )
subj. to xk+1 = g(xk , uk ), k = j, . . . , N − 1
(7.12)
h(xk , uk ) ≤ 0, k = j, . . . , N − 1
xN ∈ Xf .

∗
Note that the optimal cost-to-go Jj→N (xj ) depends only on the initial state xj .
∗
The principle of optimality implies that the optimal cost-to-go Jj−1→N from
time j − 1 to the final time N can be found by minimizing the sum of the stage
7.3 Solution via Recursive Approach 151

∗
cost q(xj−1 , uj−1 ) and the optimal cost-to-go Jj→N (xj ) from time j onwards:

∗ ∗
Jj−1→N (xj−1 ) = min q(xj−1 , uj−1 ) + Jj→N (xj )
uj−1
subj. to xj = g(xj−1 , uj−1 ) (7.13)
h(xj−1 , uj−1 ) ≤ 0
xj ∈ Xj→N .

Here the only decision variable left for the optimization is uj−1 , the input at time
j − 1. All the other inputs u∗j , . . . , u∗N −1 have already been selected optimally to
∗
yield the optimal cost-to-go Jj→N (xj ). We can rewrite (7.13) as

∗ ∗
Jj−1→N (xj−1 ) = min q(xj−1 , uj−1 ) + Jj→N (g(xj−1 , uj−1 ))
uj−1
subj. to h(xj−1 , uj−1 ) ≤ 0 (7.14)
g(xj−1 , uj−1 ) ∈ Xj→N ,

making the dependence of xj on the initial state xj−1 explicit.

The optimization problem (7.14) suggests the following recursive algorithm
backwards in time to determine the optimal control law. We start with the terminal
cost and constraint
∗
JN →N (xN ) = p(xN ) (7.15)
XN →N = Xf , (7.16)

and then proceed backwards

∗ ∗
JN −1→N (xN −1 ) = min q(xN −1 , uN −1 ) + JN →N (g(xN −1 , uN −1 ))
uN −1
subj. to h(xN −1 , uN −1 ) ≤ 0,
g(xN −1 , uN −1 ) ∈ XN →N
..
.
∗ ∗
J0→N (x0 ) = min q(x0 , u0 ) + J1→N (g(x0 , u0 ))
u0
subj. to h(x0 , u0 ) ≤ 0,
g(x0 , u0 ) ∈ X1→N
x0 = x(0).
(7.17)
This algorithm, popularized by Bellman, is referred to as dynamic programming.
The dynamic programming problem is appealing because it can be stated compactly
and because at each step the optimization takes place over one element uj of the
optimization vector only. This optimization is rather complex, however. It is not a
standard nonlinear programming problem, since we have to construct the optimal
∗
cost-to-go Jj→N (xj ), a function defined over the subset Xj→N of the state space.
In a few special cases we know the type of function and we can find it efficiently.
For example, in the next chapter we will cover the case when the system is linear
and the cost is quadratic. Then the optimal cost-to-go is also quadratic and can be
constructed rather easily. Later in the book we will show that, when constraints
are added to this problem, the optimal cost-to-go becomes piecewise quadratic and
efficient algorithms for its construction are also available.
152 7 General Formulation and Discussion

In general, however, we may have to resort to a “brute force” approach to

∗
construct the cost-to-go function Jj−1→N and to solve the dynamic program. Let us
∗
assume that at time j−1 the cost-to-go Jj→N is known and discuss how to construct
∗ ∗
an approximation of Jj−1→N . With Jj→N known, for a fixed xj−1 the optimization
problem (7.14) becomes a standard nonlinear programming problem. Thus, we
can define a grid in the set Xj−1→N of the state space and compute the optimal
cost-to-go function on each grid point. We can then define an approximate value
function J˜j−1→N
∗
(xj−1 ) at intermediate points via interpolation. The complexity
of constructing the cost-to-go function in this manner increases rapidly with the
dimension of the state space (“curse of dimensionality”).
The extra benefit of solving the optimal control problem via dynamic pro-
∗
gramming is that we do not only obtain the vector of optimal inputs U0→N for a
particular initial state x(0) as with the batch approach. At each time j the optimal
cost-to-go function defines implicitly a nonlinear feedback control law.

u∗j (xj ) = arg min ∗

q(xj , uj ) + Jj+1→N (g(xj , uj ))
uj
subj. to h(xj , uj ) ≤ 0, (7.18)
g(xj , uj ) ∈ Xj+1→N .

For a fixed xj this nonlinear programming problem can be solved quite easily in
order to find u∗j (xj ). Because the optimal cost-to-go function Jj→N
∗
(xj ) changes
with time j, the nonlinear feedback control law is time-varying.

7.4 Optimal Control Problem with Infinite Horizon

We are interested in the optimal control problem (7.3)–(7.5) as the horizon N
approaches infinity.
∞
X
∗
J0→∞ (x0 ) = minu0 ,u1 ,... q(xk , uk )
k=0
subj. to xk+1 = g(xk , uk ), k = 0, . . . , ∞ (7.19)
h(xk , uk ) ≤ 0, k = 0, . . . , ∞
x0 = x(0).

We define the set of initial conditions for which this problem has a solution.

X0→∞ = {x(0) ∈ Rn : Problem (7.19) is feasible and J0→∞ ∗

(x(0)) < +∞}.
(7.20)
∗
For the value function J0→∞ (x0 ) to be finite it must hold that

lim q(xk , uk ) = 0,
k→∞

and because q(xk , uk ) > 0 for all (xk , uk ) 6= 0

lim xk = 0
k→∞
 7.4 Optimal Control Problem with Infinite Horizon 153

and
lim uk = 0.
k→∞

Thus the sequence of control actions generated by the solution of the infinite horizon
problem drives the system to the origin. For this solution to exists the system must
be, loosely speaking, stabilizable.
Using the recursive dynamic programming approach we can seek the solution
of the infinite horizon optimal control problem by increasing N until we observe
convergence. If the dynamic programming algorithm converges as N → ∞ then
(7.14) becomes the Bellman equation

J ∗ (x) = minu q(x, u) + J ∗ (g(x, u))

subj. to h(x, u) ≤ 0 (7.21)
g(x, u) ∈ X0→∞

This procedure of simply increasing N may not be well behaved numerically

and it may also be difficult to define a convergence criterion that is meaningful for
the control problem. We will describe a method, called Value Function Iteration,
in the next section.
An alternative is receding horizon control which can yield a time invariant con-
troller guaranteeing convergence to the origin without requiring N → ∞. We will
describe this important idea later in this chapter.

7.4.1 Value Function Iteration

Once the value function J ∗ (x) is known, the nonlinear feedback control law u∗ (x)
is defined implicitly by (7.21)

u∗ (x) = arg minu q(x, u) + J ∗ (g(x, u))

subj. to h(x, u) ≤ 0 (7.22)
g(x, u) ∈ X0→∞ .

It is time invariant and guarantees convergence to the origin for all states in X0→∞ .
For a given x ∈ X0→∞ , u∗ (x) can be found from (7.21) by solving a standard
nonlinear programming problem.
In order to find the value function J ∗ (x) we need to solve (7.21). We can start
with some initial guess J˜0∗ (x) for the value function and an initial guess X˜0 for the
region in the state space where we expect the infinite horizon problem to converge
and iterate. Then at iteration i + 1 solve

J˜i+1
∗
(x) = minu q(x, u) + J˜i∗ (g(x, u))
subj. to h(x, u) ≤ 0 (7.23)
g(x, u) ∈ X̃i

X̃i+1 = {x ∈ Rn : ∃u (h(x, u) ≤ 0, and g(x, u) ∈ X̃i )}. (7.24)

Again, here i is the iteration index and does not denote time. This iterative pro-
cedure is called value function iteration. It can be executed as follows. Let us
154 7 General Formulation and Discussion

assume that at iteration step i we gridded the set X̃i and that J˜i∗ (x) is known
at each grind point from the previous iteration. We can approximate J˜i∗ (x) at
intermediate points via interpolation. For a fixed point x̄ the optimization prob-
lem (7.23) is a nonlinear programming problem yielding J˜i∗ (x̄). In this manner the
approximate value function J˜i∗ (x) can be constructed at all grid points and we can
proceed to the next iteration step i + 1.

7.4.2 Receding Horizon Control

Receding Horizon Control will be covered in detail in Chapter 12. Here we illustrate
the main idea and discuss the fundamental properties.
Assume that at time t = 0 we determine the control action u0 by solving
∗
the finite horizon optimal control problem (7.3)-(7.5). If J0→N (x0 ) converges to
∗
J0→∞ (x0 ) as N → ∞ then the effect of increasing N on the value of u0 should
diminish as N → ∞. Thus, intuitively, instead of making the horizon infinite we
can get a similar behavior when we use a long, but finite horizon N, and repeat
this optimization at each time step, in effect moving the horizon forward (mov-
ing horizon or receding horizon control). We can use the batch or the dynamic
programming approach.
Batch approach. We solve an optimal control problem with horizon N yielding a
sequence of optimal inputs u∗0 , . . . , u∗N −1 , but we would implement only the first one
of these inputs u∗0 . At the next time step we would measure the current state and
then again solve the N -step problem with the current state as new initial condition
x0 . If the horizon N is long enough then we expect that this approximation of the
infinite horizon problem should not matter and the implemented sequence should
drive the states to the origin.
Dynamic programming approach. We always implement the control u0 obtained
from the optimization problem
∗ ∗
J0→N (x0 ) = min q(x0 , u0 ) + J1→N (g(x0 , u0 ))
u0
subj. to h(x0 , u0 ) ≤ 0, (7.25)
g(x0 , u0 ) ∈ X1→N ,
x0 = x(0)
∗
where J1→N (g(x0 , u0 )) is the optimal cost-to-go from the state x1 = g(x0 , u0 ) at
time 1 to the end of the horizon N .
If the dynamic programming iterations converge as N → ∞, then for a long,
but finite horizon N we expect that this receding horizon approximation of the
infinite horizon problem should not matter and the resulting controller will drive
the system asymptotically to the origin.
In both the batch and the recursive approach, however, it is not obvious how
long N must be for the receding horizon controller to inherit these desirable con-
vergence characteristics. Indeed, for computational simplicity we would like to
keep N small. We will argue next that the proposed control scheme guarantees
convergence just like the infinite horizon variety if we impose a specific terminal
constraint, for example, if we require the terminal region to be the origin Xf = 0.
7.4 Optimal Control Problem with Infinite Horizon 155

From the principle of optimality we know that

∗ ∗
J0→N (x0 ) = min q(x0 , u0 ) + J1→N (x1 ). (7.26)
u0

Assume that we are at x(0) at time 0 and implement the optimal u∗0 that takes us
to the next state x1 = g(x(0), u∗0 ). At this state at time 1 we postulate to use over
the next N steps the sequence of optimal moves determined at the previous step
followed by zero: u∗1 , . . . , u∗N −1 , 0. This sequence is not optimal but the associated
cost over the shifted horizon from 1 to N + 1 can be easily determined. It consists
∗
of three parts: 1) the optimal cost J0→N (x0 ) from time 0 to N computed at time 0,
minus 2) the stage cost q(x0 , u0 ) at time 0 plus 3) the cost at time N + 1. But this
last cost is zero because we imposed the terminal constraint xN = 0 and assumed
uN = 0. Thus the cost over the shifted horizon for the assumed sequence of control
moves is
∗
J0→N (x0 ) − q(x0 , u0 ).
Because this postulated sequence of inputs is not optimal at time 1
∗ ∗
J1→N +1 (x1 ) ≤ J0→N (x0 ) − q(x0 , u0 ).

∗ ∗
Because the system and the objective are time invariant J1→N +1 (x1 ) = J0→N (x1 )
so that
∗ ∗
J0→N (x1 ) ≤ J0→N (x0 ) − q(x0 , u0 ).
∗ ∗
As q ≻ 0 for all (x, u) 6= (0, 0), the sequence of optimal costs J0→N (x0 ), J0→N (x1 ), . . .
∗
is strictly decreasing for all (x, u) 6= (0, 0). Because the cost J0→N ≥ 0 the sequence
∗ ∗
J0→N (x0 ), J0→N (x1 ), . . . (and thus the sequence x0 , x1 ,. . .) is converging. Thus we
have established the following important theorem.

Theorem 7.1 (Convergence of Receding Horizon Control) At time step j

consider the cost function
j+N
X
Jj→j+N (xj , uj , uj+1 , . . . , uj+N −1 ) = q(xk , uk ), q ≻ 0 (7.27)
k=j

and the CFTOC problem

∗
Jj→j+N (xj ) = Jj→j+N (xj , uj , uj+1 , . . . , uj+N −1 )
minuj ,uj+1 ,...,uj+N −1
subj. to xk+1 = g(xk , uk )
h(xk , uk ) ≤ 0, k = j, . . . , j + N − 1
xN = 0
(7.28)
Assume that only the optimal u∗j is implemented. At the next time step j + 1 the
CFTOC problem is solved again starting from the resulting state xj+1 = g(xj , u∗j ).
Assume that the CFTOC problem (7.28) has a solution for every state xj , xj+1 , . . .
resulting from the control policy. Then the system will converge to the origin as
j → ∞.
156 7 General Formulation and Discussion

Thus we have established that a receding horizon controller with terminal con-
straint xN = 0 has the same desirable convergence characteristics as the infinite
horizon controller. At first sight the theorem appears very general and power-
ful. It is based on the implicit assumption, however, that at every time step the
CFTOC problem has a solution. Infeasibility would occur, for example, if the un-
derlying system is not stabilizable. It could also happen that the constraints on
the inputs which restrict the control action prevent the system from reaching the
terminal state in N steps. In Chapter 12 we will present special formulations of
problem (7.28) such that feasibility at the initial time guarantees feasibility for
all future times. Furthermore in addition to asymptotic convergence to the ori-
gin we will establish stability for the closed-loop system with the receding horizon
controller.

Remark 7.1 For the sake of simplicity in the rest of the book we will use the following
shorter notation
Jj∗ (xj ) = Jj→N
∗
(xj ), j = 0, . . . , N
∗ ∗
J∞ (x0 ) = J0→∞ (x0 )
Xj = Xj→N , j = 0, . . . , N (7.29)
X∞ = X0→∞
U0 = U0→N
and use the original notation only if needed.

7.5 Lyapunov Stability

While asymptotic convergence limk→∞ xk = 0 is a desirable property, it is gen-
erally not sufficient in practice. We would also like a system to stay in a small
neighborhood of the origin when it is disturbed slightly. Formally this is expressed
as Lyapunov stability.

7.5.1 General Stability Conditions

Consider the autonomous system

xk+1 = g(xk ) (7.30)

with g(0) = 0.

Definition 7.1 (Lyapunov Stability) The equilibrium point x = 0 of system (7.30)

is
- stable (in the sense of Lyapunov) if, for each ε > 0, there is δ > 0 such that

kx0 k < δ ⇒ kxk k < ε, ∀k ≥ 0 (7.31)

- unstable if not stable

7.5 Lyapunov Stability 157

- asymptotically stable in Ω ⊆ Rn if it is stable and

lim xk = 0, ∀x0 ∈ Ω (7.32)

k→∞

- globally asymptotically stable if it is asymptotically stable and Ω = Rn

- exponentially stable if it is stable and there exist constants α > 0 and γ ∈

(0, 1) such that

kx0 k < δ ⇒ kxk k ≤ αkx0 kγ k , ∀k ≥ 0. (7.33)

The ε-δ requirement for stability (7.31) takes a challenge-answer form. To

demonstrate that the origin is stable, for any value of ε that a challenger may
chose (however small), we must produce a value of δ such that a trajectory starting
in a δ neighborhood of the origin will never leave the ε neighborhood of the origin.

Remark 7.2 If in place of system (7.30), we consider the time-varying system xk+1 =
g(xk , k), then δ in Definition 7.1 is a function of ε and k, i.e., δ = δ(ε, k) > 0. In this
case, we introduce the concept of “uniform stability”. The equilibrium point x = 0
is uniformly stable if, for each ε > 0, there is δ = δ(ε) > 0 (independent from k) such
that
kx0 k < δ ⇒ kxk k < ε, ∀k ≥ 0. (7.34)

The following example shows that Lyapunov stability and convergence are, in
general, different properties.

Example 7.1 Consider the following system with one state x ∈ R:

xk+1 = xk (xk − 1 − |xk − 1|). (7.35)

The state x = 0 is an equilibrium for the system. For any state x ∈ [−1, 1] we have
(x − 1) ≤ 0 and the system dynamics (7.35) become

xk+1 = 2x2k − 2xk . (7.36)

System (7.36) generates oscillating and diverging trajectories for any x0 ∈ (−1, 1)\{0}.
Any such trajectory will enter in finite time T the region with x ≥ 1. In this region
the system dynamics (7.35) become

xk+1 = 0, ∀ k ≥ T. (7.37)

Therefore the origin is not Lyapunov stable, however the system converges to the
origin for all x0 ∈ (−∞, +∞).

Usually to show Lyapunov stability of the origin for a particular system one
constructs a so called Lyapunov function, i.e., a function satisfying the conditions
of the following theorem.
158 7 General Formulation and Discussion

Theorem 7.2 Consider the equilibrium point x = 0 of system (7.30). Let Ω ⊂ Rn

be a closed and bounded set containing the origin. Assume there exists a function
V : Rn → R continuous at the origin, finite for every x ∈ Ω, and such that
V (0) = 0 and V (x) > 0, ∀x ∈ Ω \ {0} (7.38a)
V (xk+1 ) − V (xk ) ≤ −α(xk ) ∀xk ∈ Ω \ {0} (7.38b)
where α : Rn → R is a continuous positive definite function. Then x = 0 is
asymptotically stable in Ω.

Definition 7.2 A function V (x) satisfying conditions (7.38a)-(7.38b) is called a

Lyapunov Function.
The main idea of Theorem 7.2 can be explained as follows. We aim to find a scalar
function V (x) that captures qualitative characteristics of the system response, and,
in particular, its stability. We can think of V as an energy function that is zero
at the origin and positive elsewhere (condition (7.38a)). Condition (7.38b) of The-
orem 7.2 requires that for any state xk ∈ Ω, xk 6= 0 the energy decreases as the
system evolves to xk+1 .
A proof of Theorem 7.2 for continuous dynamical systems can be found in [184].
The continuity assumptions on the dynamical system g is not used in the proof
in [247, p. 609]. Assumption in equation (7.38b), however, together with the
continuity of V (·) at the origin, implies that g(·) must be continuous at the origin.
Theorem 7.2 states that if we find an energy function which satisfies the two
conditions (7.38a)-(7.38b), then the system states starting from any initial state
x0 ∈ Ω will eventually settle to the origin.
Note that Theorem 7.2 is only sufficient. If condition (7.38b) is not satisfied
for a particular choice of V nothing can be said about stability of the origin.
Condition (7.38b) of Theorem 7.2 can be relaxed to allow α to be a continuous
positive semi-definite (psd) function:
V (xk+1 ) − V (xk ) ≤ −α(xk ), ∀xk 6= 0, α continuous and psd. (7.39)
Condition (7.39) along with condition (7.38a) are sufficient to guarantee stability
of the origin as long as the set {xk : V (g(xk )) − V (xk ) = 0} contains no trajectory
of the system xk+1 = g(xk ) except for xk = 0 for all k ≥ 0. This relaxation
of Theorem 7.2 is the so called Barbashin-Krasovski-LaSalle principle [183]. It
basically means that V (xk ) may stay constant and non zero at one or more time
instants as long as it does not do so at an equilibrium point or periodic orbit of
the system.
A similar result as Theorem 7.2 can be derived for global asymptotic stability,
i.e., Ω = Rn .

Theorem 7.3 Consider the equilibrium point x = 0 of system (7.30). Assume

there exists a function V : Rn → R continuous at the origin, finite for every
x ∈ Rn , and such that
kxk → ∞ ⇒ V (x) → ∞ (7.40a)
V (0) = 0 and V (x) > 0, ∀x 6= 0 (7.40b)
V (xk+1 ) − V (xk ) ≤ −α(xk ) ∀xk 6= 0 (7.40c)
7.5 Lyapunov Stability 159

where α : Rn → R is a continuous positive definite function. Then x = 0 is globally

asymptotically stable.

Definition 7.3 A function V (x) satisfying condition (7.40a) is said to be radially

unbounded.

Definition 7.4 A radially unbounded Lyapunov function is called a Global Lya-

punov function.

Note that it was not enough just to restate Theorem 7.2 with Ω = Rn but we
also have to require V (x) to be radially unbounded to guarantee global asymptotic
stability. To motivate this condition consider the candidate Lyapunov function for
a system in R2 [176]
x21
V (x) = + x22 , (7.41)
1 + x21

1
V (x)

x2

-1-4 0
-2 2 4
x2 x1 x1
(a) Lyapunov function V (x). (b) Level curves of V (x).

Figure 7.1 Lyapunov function (7.41).

which is depicted in Fig. 7.1, where x1 and x2 denote the first and second compo-
nents of the state vector x, respectively. V (x) in (7.41) is not radially unbounded
as for x2 = 0
lim V (x) = 1.
x1 →∞

For this Lyapunov function even if condition (7.40c) is satisfied, the state x may
escape to infinity. Condition (7.40c) of Theorem 7.3 guarantees that the level sets
Ωc of V (x) (Ωc = {x ∈ Rn : V (x) ≤ c}) are closed.
The construction of suitable Lyapunov functions is a challenge except for linear
systems. First of all one can quite easily show that for linear systems Lyapunov
stability agrees with the notion of stability based on eigenvalue location.

Theorem 7.4 A linear system xk+1 = Axk is globally asymptotically stable in the
sense of Lyapunov if and only if all its eigenvalues are strictly inside the unit circle.

We also note that stability is always “global” for linear systems.

160 7 General Formulation and Discussion

7.5.2 Quadratic Lyapunov Functions for Linear Systems

A simple effective Lyapunov function for linear systems is

V (x) = x′ P x, P ≻ 0 (7.42)

which satisfies conditions (7.40a)-(7.40b) of Theorem 7.3. In order to test condi-

tion (7.40c) we compute

V (xk+1 )−V (xk ) = x′k+1 P xk+1 −x′k P xk = x′k A′ P Axk −x′k P xk = x′k (A′ P A−P )xk .
(7.43)
Therefore condition (7.40c) is satisfied if P ≻ 0 can be found such that

A′ P A − P = −Q, Q ≻ 0. (7.44)

Equation (7.44) is referred to as discrete-time Lyapunov equation. The following

Theorem [75, p. 211] shows that P ≻ 0 satisfying (7.44) exists if and only if the
linear system is asymptotically stable.

Theorem 7.5 Consider the linear system xk+1 = Axk . Equation (7.44) has a
unique solution P ≻ 0 for any Q ≻ 0 if and only if A has all eigenvalues strictly
inside the unit circle.
Thus, a quadratic form x′ P x is always a suitable Lyapunov function for linear
systems and an appropriate P can be found by solving (7.44) for a chosen Q ≻
0 iff the system’s eigenvalues lie inside the unit circle. For nonlinear systems,
determining a suitable form for V (x) is generally difficult.
For a stable linear system xk+1 = Axk , P turns out to be the infinite time cost
matrix ∞
X
J∞ (x0 ) = x′k Qxk = x′0 P x0 (7.45)
k=0

as we can easily show. From

J∞ (x1 ) − J∞ (x0 ) = x′1 P x1 − x′0 P x0 = x′0 A′ P Ax0 − x′0 P x0 = −x′0 Qx0 (7.46)

we recognize that P is the solution of the Lyapunov equation (7.44). In other

words the infinite time cost (7.45) is a Lyapunov function for the linear system
xk+1 = Axk .
The conditions of Theorem 7.5 can be relaxed as follows.

Theorem 7.6 Consider the linear system xk+1 = Axk . Equation (7.44) has a
unique solution P ≻ 0 for any Q = C ′ C  0 if and only if A has all eigenvalues
inside the unit circle and (C, A) is observable.
We can prove Theorem 7.6 in the same way as Callier and Desoer [75, p. 211]
proved Theorem 7.5. In Theorem 7.6 we do not require that the Lyapunov function
decreases at every time step, i.e., we allow Q to be positive semidefinite. To
understand this, let us assume that for a particular system state x̄, V does not
decrease, i.e., x̄′ Qx̄ = (C x̄)′ (C x̄) = 0. Then at the next time steps we have the rate
 7.5 Lyapunov Stability 161

of decrease (CAx̄)′ (CAx̄), (CA2 x̄)′ (CA2 x̄), . . .. If the system (C, A) is observable
then for all x̄ 6= 0
 
x̄′ C (CA)′ (CA2 )′ · · · (CAn−1 )′ 6= 0, (7.47)
which implies that after at most (n − 1) steps the rate of decrease will become
nonzero. This is a special case of the Barbashin-Krasovski-LaSalle principle. Note
that for C square and nonsingular Theorem 7.6 reduces to Theorem 7.5.
Similarly we can analyze the controlled system xk+1 = Axk + Buk with uk =
F xk and the infinite time cost
∞
X
J∞ (x0 ) = x′k Qxk + u′k Ruk (7.48)
k=0

with Q = C ′ C and R = D′ D with det(D) 6= 0. We can rewrite the cost as

∞
X ∞
X  ′
′ ′ ′ C  
J∞ (x0 ) = xk (Q + F RF )xk = xk C DF xk (7.49)
DF
k=0 k=0

for the controlled system xk+1 = (A + BF )xk . The infinite time cost matrix P can
now be found from the Lyapunov equation
 ′
′ C  
(A + BF ) P (A + BF ) − P = C DF . (7.50)
DF
According to Theorem
 7.6 the solution
  P is unique and positive definite iff (A+BF )
C
is stable and , (A + BF ) is observable. This follows directly from the
DF
observability of (C, A). If (C, A) is observable, then so is (C, A + BF ) because
feedback does not affect observability. Observability is also not affected by adding
the observed outputs DF x.
From (7.44) it follows that for stable systems and for a chosen Q ≻ 0 one can
always find P ≻ 0 solving
A′ P A − P + Q  0. (7.51)
This Lyapunov inequality shows that for a stable system we can always find a P
such that V (x) = x′ P x decreases at a desired “rate” indicated by Q. We will need
this result later to prove stability of receding horizon control schemes.

7.5.3 1/∞ Norm Lyapunov Functions for Linear Systems

For p = {1, ∞} the function

V (x) = kP xkp
l×n
with P ∈ R of full column rank satisfies the requirements (7.40a), (7.40b)
of a Lyapunov function. It can be shown that a matrix P can be found such
that condition (7.40c) is satisfied for the system xk+1 = Axk if and only if the
eigenvalues of A are inside the unit circle. The number of rows l necessary in P
depends on the system. The techniques to construct P are based on the following
theorem [177, 237].
162 7 General Formulation and Discussion

Theorem 7.7 Let P ∈ Rl×n with rank(P ) = n and p ∈ {1, ∞}. The function

V (x) = kP xkp (7.52)

is a Lyapunov function for the discrete-time system

xk+1 = Axk , k ≥ 0, (7.53)

if and only if there exists a matrix H ∈ Rl×l , such that

P A = HP, (7.54a)
kHkp < 1. (7.54b)

An effective method to find both H and P was proposed by Christophersen in [88].

To prove the stability of receding horizon control, later in this book, we will
need to find a P̃ such that

kP̃ Axk∞ − kP̃ xk∞ + kQxk∞ ≤ 0, ∀x ∈ Rn . (7.55)

Note that the inequality (7.55) is equivalent to the Lyapunov inequality (7.51)
when the squared two-norm is replaced by the 1− or ∞−norm. Once we have
constructed a P and H to fulfill the conditions of Theorem 7.7 we can easily find
P̃ to satisfy (7.55) according to the following lemma.

Lemma 7.1 Let P and H be matrices satisfying conditions (7.54), with P full
column rank. Let σ = 1 − kHk∞ , ρ = kQP # k∞ , where P # = (P ′ P )−1 P ′ is the
left pseudoinverse of P . Then, the square matrix
ρ
P̃ = P (7.56)
σ
satisfies condition (7.55).
Proof: Since P̃ satisfies P̃ A = H P̃ , we obtain −kP̃ xk∞ +kP̃ Axk∞ +kQxk∞ =
−kP̃ xk∞ + kH P̃ xk∞ + kQxk∞ ≤ (kHk∞ − 1)kP̃ xk∞ + kQxk∞ ≤ (kHk∞ −
1)kP̃ xk∞ + kQP #k∞ kP xk∞ = 0. Therefore, (7.55) is satisfied. 
 Chapter 8

Linear Quadratic Optimal

Control

In this chapter we study the finite time and infinite time optimal control problem
for unconstrained linear systems with quadratic objective functions. We derive the
structure of the optimal control law by using two approaches: the batch approach
and the dynamic programming approach. For problems with quadratic objective
functions we obtain the well known Algebraic Riccati Equations.

8.1 Problem Formulation

We consider a special case of the problem stated in the last chapter, where the
system is linear and time-invariant

x(t + 1) = Ax(t) + Bu(t). (8.1)

Again, x(t) ∈ Rn and u(t) ∈ Rm are the state and input vectors respectively.
We define the following quadratic cost function over a finite horizon of N steps
N
X −1
J0 (x0 , U0 ) = x′N P xN + x′k Qxk + u′k Ruk , (8.2)
k=0

where xk denotes the state vector at time k obtained by starting from the state
x0 = x(0) and applying to the system model

xk+1 = Axk + Buk (8.3)

the input sequence u0 , . . . , uk−1 . Consider the finite time optimal control problem

J0∗ (x(0)) = minU0 J0 (x(0), U0 )

subj. to xk+1 = Axk + Buk , k = 0, 1, . . . , N − 1 (8.4)
x0 = x(0).
164 8 Linear Quadratic Optimal Control

In (8.4) U0 = [u′0 , . . . , u′N −1 ]′ ∈ Rs , s = mN is the decision vector containing

all future inputs. We will assume that the state penalty is positive semi-definite
Q = Q′  0, P = P ′  0 and the input penalty is positive definite R = R′ ≻ 0.
As introduced in the previous chapter we will present two alternate approaches
to solve problem (8.4), the batch approach and the recursive approach using dy-
namic programming.

8.2 Solution via Batch Approach

First we write the equality constraints (8.4) explicitly to express all future states
x1 , x2 , . . . as a function of the future inputs u0 , u1 , . . . and then we eliminate all
intermediate states by successive substitution to obtain
     
x(0) I 0 ... ... 0  
     u0
 x1   A   B 0 ... 0   .. 
 ..   ..   .. . .  . 

 . =
  .  x(0) + 
  AB . . . .. 

 .
 (8.5)
      ..
.. .. .. .. . .  . 
 .   .   . . . . .. 
uN −1
xN AN AN −1 B ... ... B
| {z } | {z } | {z }
X Sx Su

Here all future states are explicit functions of the present state x(0) and the future
inputs u0 , u1 , u2 , . . . only. By defining the appropriate quantities we can rewrite
this expression compactly as

X = S x x(0) + S u U0 . (8.6)

Using the same notation the objective function can be rewritten as

J(x(0), U0 ) = X ′ Q̄X + U0 ′ R̄U0 , (8.7)

where Q̄ = blockdiag{Q, . . . , Q, P }, Q̄  0, and R̄ = blockdiag{R, . . . , R}, R̄ ≻ 0.

Substituting (8.6) into the objective function (8.7) yields

J0 (x(0), U0 ) = (S x x(0) + S u U0 )′ Q̄ (S x x(0) + S u U0 ) + U0 ′ R̄U0

= U0 ′ (S u′ Q̄S u + R̄) U0 + 2x′ (0) (S x′ Q̄S u ) U0 + x′ (0) (S x′ Q̄S x ) x(0)
| {z } | {z } | {z }
H F Y
= U0 ′ HU0 + 2x′ (0)F U0 + x′ (0)Y x(0).
(8.8)
Because R̄ ≻ 0, also H ≻ 0. Thus J0 (x(0), U0 ) is a positive definite quadratic
function of U0 . Therefore, its minimum can be found by computing its gradient
and setting it to zero. This yields the optimal vector of future inputs

U0∗ (x(0)) = −H −1 F ′ x(0)

−1 u ′ (8.9)
= − S u ′ Q̄S u + R̄ S Q̄S x x(0).
 8.3 Solution via Recursive Approach 165

With this choice of U0 the optimal cost is

J0∗ (x(0)) = −x(0)h′ F H −1 F ′ x(0) + x(0)′ Y x(0) i

−1 u ′
= x(0)′ S x ′ Q̄S x − S x ′ Q̄S u S u ′ Q̄S u + R̄ S Q̄S x x(0).
(8.10)
Note that the optimal vector of future inputs U0∗ (x(0)) is a linear function (8.9) of
the initial state x(0) and the optimal cost J0∗ (x(0)) is a quadratic function (8.10)
of the initial state x(0).

8.3 Solution via Recursive Approach

Alternatively, we can use dynamic programming to solve the same problem in a
recursive manner. We define the optimal cost Jj∗ (xj ) for the N − j step problem
starting from state xj by
N
X −1
∆
Jj∗ (xj ) = min x′N P xN + x′k Qxk + u′k Ruk . (8.11)
uj ,...,uN −1
k=j

According to the principle of optimality the optimal one step cost-to-go can be
obtained from
∗
JN −1 (xN −1 ) = min x′N PN xN + x′N −1 QxN −1 + u′N −1 RuN −1 (8.12)
uN −1

xN = AxN −1 + BuN −1
subj. to (8.13)
PN = P.

Substituting (8.13) into the objective function (8.12),

∗
 ′ ′
JN −1 (xN −1 ) = minuN −1 xN −1 (A PN A + Q)xN −1
+ 2x′N −1 A′ PN BuN −1 (8.14)
+ u′N −1 (B ′ PN B + R)uN −1 .

We note that the cost-to-go JN −1 (xN −1 ) is a positive definite quadratic function

of the decision variable uN −1 . We find the optimum by setting the gradient to zero
and obtain the optimal input

u∗N −1 = −(B ′ PN B + R)−1 B ′ PN A xN −1 (8.15)

| {z }
FN −1

and the one-step optimal cost-to-go

∗ ′
JN −1 (xN −1 ) = xN −1 PN −1 xN −1 , (8.16)

where we have defined

PN −1 = A′ PN A + Q − A′ PN B(B ′ PN B + R)−1 B ′ PN A. (8.17)

166 8 Linear Quadratic Optimal Control

At the next stage, consider the two-step problem from time N − 2 forward:
∗ ′ ′ ′
JN −2 (xN −2 ) = min xN −1 PN −1 xN −1 + xN −2 QxN −2 + uN −2 RuN −2 (8.18)
uN −2

xN −1 = AxN −2 + BuN −2 . (8.19)

We recognize that (8.18), (8.19) has the same form as (8.12), (8.13). Therefore we
can state the optimal solution directly.

u∗N −2 = −(B ′ PN −1 B + R)−1 B ′ PN −1 A xN −2 . (8.20)

| {z }
FN −2

The optimal two-step cost-to-go is

∗ ′
JN −2 (xN −2 ) = xN −2 PN −2 xN −2 , (8.21)
where we defined
PN −2 = A′ PN −1 A + Q − A′ PN −1 B(B ′ PN −1 B + R)−1 B ′ PN −1 A. (8.22)
Continuing in this manner, at some arbitrary time k the optimal control action is
u∗ (k) = −(B ′ Pk+1 B + R)−1 B ′ Pk+1 Ax(k),
(8.23)
= Fk x(k), for k = 0, . . . , N − 1,
where
Pk = A′ Pk+1 A + Q − A′ Pk+1 B(B ′ Pk+1 B + R)−1 B ′ Pk+1 A (8.24)
and the optimal cost-to-go starting from the measured state x(k) is
Jk∗ (x(k)) = x′ (k)Pk x(k). (8.25)
Equation (8.24) (called Discrete Time Riccati Equation or Riccati Difference Equa-
tion - RDE) is initialized with PN = P and is solved backwards, i.e., starting with
PN and solving for PN −1 , etc. Note from (8.23) that the optimal control action
u∗ (k) is obtained in the form of a feedback law as a linear function of the measured
state x(k) at time k. The optimal cost-to-go (8.25) is found to be a quadratic
function of the state at time k.

Remark 8.1 According to Section 7.4.2, the receding horizon control policy is ob-
tained by solving problem (8.4) at each time step t with x0 = x(t). Consider the
state feedback solution u∗ (k) in (8.23) to problem (8.4). Then, the receding horizon
control policy is:
u∗ (t) = F0 x(t), t ≥ 0 (8.26)

8.4 Comparison of the Two Approaches

We will compare the batch and the recursive dynamic programming approach in
terms of the results and the methods used to obtain the results.
8.4 Comparison of the Two Approaches 167

Most importantly we observe that the results obtained by the two methods are
fundamentally different. The batch approach yields a formula for the sequence of
inputs as a function of the initial state.

−1 u′
U0∗ = − S u′ Q̄S u + R̄ S Q̄S x x(0). (8.27)

The recursive dynamic programming approach yields a feedback policy, i.e., a se-
quence of feedback laws expressing at each time step the control action as a function
of the state at that time.

u∗ (k) = Fk x(k), for k = 0, . . . , N − 1. (8.28)

As this expression implies, we determine u(k) at each time k as a function of the

current state x(k) rather than use a u(k) precomputed at k = 0 as in the batch
method. If the state evolves exactly according to the linear model (8.3) then the
sequence of control actions u(k) obtained from the two approaches is identical. In
practice, the result of applying the sequence (8.27) in an open-loop fashion may
be rather different from applying the time-varying feedback law (8.28) because the
model (8.1) for predicting the system states may be inaccurate and the system may
be subject to disturbances not included in the model. We expect the application
of the feedback law to be more robust because at each time step the observed state
x(k) is used to determine the control action rather than the state xk predicted at
time t = 0.
We note that we can get the same feedback effect with the batch approach
if we recalculate the optimal open-loop sequence at each time step j with the
current measurement as initial condition. In this case we need to solve the following
optimization problem
PN −1
Jj∗ (x(j)) = minuj ,··· ,uN −1 x′N P xN + k=j x′k Qxk + u′k Ruk
(8.29)
subj. to xj = x(j),

where we note that the horizon length is shrinking at each time step.
As seen from (8.27) the solution to (8.29) relates the sequence of inputs u∗j , u∗j+1 , . . .
to the state x(j) through a linear expression. The first part of this expression yields
again the optimal feedback law (8.28) at time j, u∗ (j) = Fj x(j).
Here the dynamic programming approach is clearly a more efficient way to gen-
erate the feedback policy because it only uses a simple matrix recursion (8.24).
Repeated application of the batch approach, on the other hand, requires the re-
peated inversion of a potentially large matrix in (8.27). For such inversion, however,
one can take advantage of the fact that only a small part of the matrix H changes
at every time step.
What makes dynamic programming so effective here is that in this special case,
where the system is linear and the objective is quadratic, the optimal cost-to-go,
the value function Jj∗ (x(j)) has a very simple form: it is quadratic. If we make the
problem only slightly more complicated, e.g., if we add constraints on the inputs or
states, the value function can still be constructed, but it is much more complex. In
general, the value function can only be approximated as discussed in the previous
chapter. Then a repeated application of the batch policy, where we resolve the
optimization problem at each time step is an attractive alternative.
168 8 Linear Quadratic Optimal Control

8.5 Infinite Horizon Problem

For continuous processes operating over a long time period it would be interesting
to solve the following infinite horizon problem.
∞
X
∗
J∞ (x(0)) = min x′k Qxk + u′k Ruk . (8.30)
u0 ,u1 ,...
k=0

Since the prediction must be carried out to infinity, application of the batch method
becomes impossible. On the other hand, derivation of the optimal feedback law via
dynamic programming remains viable. We can initialize the RDE (8.24)

Pk = A′ Pk+1 A + Q − A′ Pk+1 B(B ′ Pk+1 B + R)−1 B ′ Pk+1 A (8.31)

with the terminal cost matrix P0 = Q and solve it backwards for k → −∞. Let
us assume for the moment that the iterations converge to a solution P∞ . Such P∞
would then satisfy the Algebraic Riccati Equation (ARE)

P∞ = A′ P∞ A + Q − A′ P∞ B(B ′ P∞ B + R)−1 B ′ P∞ A. (8.32)

Then the optimal feedback control law is

u∗ (k) = −(B ′ P∞ B + R)−1 B ′ P∞ A x(k), k = 0, · · · , ∞ (8.33)

| {z }
F∞

and the optimal infinite horizon cost is

∗
J∞ (x(0)) = x(0)′ P∞ x(0). (8.34)

Controller (8.33) is referred to as the asymptotic form of the Linear Quadratic

Regulator (LQR) or the ∞-horizon LQR.
Convergence of the RDE has been studied extensively. A nice summary of
the various results can be found in Appendix E of the book by Goodwin and
Sin [129]. Intuitively we expect that the system (8.3) must be controllable so that
all states can be affected by the control and that the cost function should capture
the behavior of all the states, e.g., that Q ≻ 0. These conditions are indeed
sufficient for the RDE to converge and to yield a stabilizing feedback control law.
Less restrictive conditions are possible as stated in the following theorem.

Theorem 8.1 [190, Theorem 2.4-2] If (A, B) is a stabilizable pair and (Q1/2 , A) is
an observable pair, then the Riccati difference equation (8.31) with P0  0 converges
to the unique positive definite solution P∞ of the ARE (8.32) and all the eigenvalues
of (A + BF∞ ) lie strictly inside the unit circle.
∗
The first condition is clearly necessary for J∞ (and P∞ ) to be finite. To un-
derstand the second condition, we write the state dependent term in the objective
function as x′ Qx = (x′ Q1/2 )(Q1/2 x). Thus not the state but the “output” (Q1/2 x)
is penalized in the objective. Therefore the second condition ((Q1/2 , A) observable)
requires that this output captures all system modes. In this manner convergence
of the output (Q1/2 x) implies convergence of the state to zero.
8.5 Infinite Horizon Problem 169

From Section 7.5.2 we know that the optimal infinite horizon cost (8.34) is a
Lyapunov function for the system xk+1 = Axk + Buk with uk = F∞ xk and satisfies

(A + BF∞ )′ P∞ (A + BF∞ ) − P∞ = Q + F∞
′
RF∞ .

The reader can verify that substituting F∞ from (8.33) we recover the Riccati
equation (8.32).
170 8 Linear Quadratic Optimal Control
 Chapter 9

Linear 1/∞ Norm Optimal

Control

In this chapter we study the finite time and infinite time optimal control problem
for unconstrained linear systems with convex piecewise linear objective functions.
We derive the structure of the optimal control law by using two approaches: the
Batch approach and the Dynamic Programming approach.

9.1 Problem Formulation

We consider a special case of the problem stated in Chapter 7, where the system
is linear and time-invariant
x(t + 1) = Ax(t) + Bu(t). (9.1)
Again, x(t) ∈ Rn and u(t) ∈ Rm are the state and input vector respectively.
We define the following piecewise linear cost function over a finite horizon of N
steps
N
X −1
J0 (x0 , U0 ) = kP xN kp + kQxk kp + kRuk kp (9.2)
k=0
with p = 1 or p = ∞ and where xk denotes the state vector at time k obtained by
starting from the state x0 = x(0) and applying to the system model
xk+1 = Axk + Buk (9.3)
the input sequence u0 , . . . , uk−1 . The weighting matrices in (9.2) could have an
arbitrary number of rows. For simplicity of notation we will assume Q ∈ Rn×n ,
R ∈ Rm×m and P ∈ Rr×n . Consider the finite time optimal control problem
J0∗ (x(0)) = minU0 J0 (x(0), U0 )
subj. to xk+1 = Axk + Buk , k = 0, 1, . . . , N − 1 (9.4)
x0 = x(0).
172 9 Linear 1/∞ Norm Optimal Control

In (9.4) U0 = [u′0 , . . . , u′N −1 ]′ ∈ Rs , s = mN is the decision vector containing all

future inputs.
We will present two different approaches to solve problem (9.2)-(9.4), the batch
approach and the recursive approach using dynamic programming. Unlike in the
2-norm case presented in the previous chapter, there does not exist a simple closed-
form solution of problem (9.2)-(9.4). In this chapter we will show how to use
multiparametric linear programming to compute the solution to problem (9.2)-
(9.4). We will concentrate on the use of the ∞-norm, the results can be extended
easily to cost functions based on the 1-norm or mixed 1/∞ norms.

9.2 Solution via Batch Approach

First we write the equality constraints (9.4) explicitly to express all future states
x1 , x2 , . . . as a function of the future inputs u1 , u2 , . . . and then we eliminate all
intermediate states by using

Pk−1
xk = Ak x0 + j=0 Aj Buk−1−j (9.5)

so that all future states are explicit functions of the present state x(0) and the
future inputs u0 , u1 , u2 , . . . only.
The optimal control problem (9.4) with p = ∞ can be rewritten as a linear
program by using the following standard approach (see e.g. [79]). The sum of
components of any vector {εx0 , . . . , εxN , εu0 , . . . , εuN −1 } that satisfies

−1n εxk ≤ Qxk , k = 0, 1, . . . , N − 1

−1n εxk ≤ −Qxk , k = 0, 1, . . . , N − 1
−1r εxN ≤ P xN ,
(9.6)
−1r εxN ≤ −P xN ,
−1m εuk ≤ Ruk , k = 0, 1, . . . , N − 1
−1m εuk ≤ −Ruk , k = 0, 1, . . . , N − 1

. . 1}]′ , and the inequali-

forms an upper bound on J0 (x(0), U0 ), where 1k = [1| .{z
k
ties (9.6) hold componentwise. It is easy to prove that the vector
z0 = {εx0 , . . . , εxN , εu0 , . . . , εuN −1 , u′0 , . . . , u′N −1 } ∈ Rs , s = (m+ 2)N + 1, that satisfies
equations (9.6) and simultaneously minimizes J(z0 ) = εx0 + . . . + εxN + εu0 + . . . +
εuN −1 also solves the original problem (9.4), i.e., the same optimum J0∗ (x(0)) is
achieved [293, 79]. Therefore, problem (9.4) can be reformulated as the following
9.2 Solution via Batch Approach 173

LP problem

min εx0 + . . . + εxN + εu0 + . . . + εuN −1 (9.7a)

z0
 
k−1
X
subj. to −1n εxk ≤ ±Q Ak x0 + Aj Buk−1−j  (9.7b)
j=0
 
N
X −1
−1r εxN ≤ ±P AN x0 + Aj BuN −1−j  (9.7c)
j=0

−1m εuk ≤ ±Ruk (9.7d)

k = 0, . . . , N − 1
x0 = x(0), (9.7e)

where constraints (9.7b)–(9.7d) are componentwise, and ± means that the con-
straint appears once with each sign, as in (9.6).

Remark 9.1 The cost function (9.2) with p = ∞ can be interpreted as a special case
of a cost function with 1-norm over time and ∞-norm over space. For instance, the
dual choice (∞-norm over time and 1-norm over space) leads to the following cost
function

J0 (x(0), U0 ) = max {kQxk k1 + kRuk k1 }. (9.8)

k=0,...,N

We remark that any combination of 1- and ∞-norms leads to a linear program. In

general, ∞-norm over time could result in a poor closed-loop performance (only the
largest state deviation and the largest input would be penalized over the prediction
horizon), while 1-norm over space leads to an LP with a larger number of variables.

The results of this chapter hold for any combination of 1- and ∞-norms over time
and space. Clearly the LP formulation will differ from the one in (9.7). For instance,
the 1−norm in space requires the introduction of nN slack variables for the terms
kQxk k1 , εk,i ≥ ±Qi xk k = 0, 2, . . . , N − 1, i = 1, 2, . . . , n, plus r slack variables for
the terminal penalty kP xN k1 , εN,i ≥ ±Pi xN i = 1, 2, . . . , r, plus mN slack variables
for the input terms kRuk k1 , εuk,i ≥ ±Ri uk k = 0, 1, . . . , N − 1, i = 1, 2, . . . , m. Here
we have used the notation Mi to denote the i-th row of matrix M .

Problem (9.7) can be rewritten in the more compact form

min c′0 z0
z0
(9.9)
subj. to Gε z0 ≤ wε + Sε x(0),
174 9 Linear 1/∞ Norm Optimal Control

where c0 ∈ Rs and Gε ∈ Rq×s , Sε ∈ Rq×n and wε ∈ Rq are

N+1 N mN
z }| { z }| { z }| {
c0 = [1 . . . 1 1 . . . 1 0 . . .
 0]′ 
N+1 N mN
z }| { z }| { z }| {
 
 −1n 0 ... 0 0 ... 0 0 0 ... 0 
 
 −1n 0 ... 0 0 ... 0 0 0 ... 0 
 
 0 −1n ... 0 0 ... 0 QB 0 ... 0 
 
 0 −1n ... 0 0 ... 0 −QB 0 ... 0 
 
 ... ... ... ... ... ... ... ... ... ... ... 
 
 0 . . . −1n 0 0 ... 0 QAN−2 B QAN−3 B ... 0 
 
Gǫ = 
 0 . . . −1n 0 0 ... 0 −QAN−2 B −QAN−3 B ... 0 

 0 ... 0 −1r 0 ... 0 P AN−1 B P AN−2 B ... PB 
 
 0 . . . 0 −1r 0 ... 0 −P AN−1 B −P AN−2 B ... −P B 
 
 0 0 ... 0 −1m ... 0 R 0 ... 0 
 
 0 0 ... 0 −1m ... 0 −R 0 ... 0 
 
 ... . . . . .. ... ... ... ... ... ... ... ... 
 
 0 0 ... 0 0 ... −1m 0 0 ... R 
0 0 ... 0 0 ... −1m 0 0 ... −R
2nN+2r 2mN
z }| { z }| {
wǫ = [0 . . . 0 0 . . . 0]′
2nN 2r 2m
z }| { z }| { z }
Sǫ = [ −Q′ Q ′
(−QA) ′
(QA) ′
(−QA2 )′ ... (−QA N−1 ′
) (QA N−1 ′ ′
)) (−P AN )′ (P AN )′ 0′m .
(9.10)
Note that in (9.10) we include the zero vector wǫ to make the notation consistent
with the one used in Section 6.2.
By treating x(0) as a vector of parameters, the problem (9.9) becomes a multi-
parametric linear program (mp-LP) that can be solved as described in Section 6.2.
Once the multiparametric problem (9.7) has been solved, the explicit solution
z0∗ (x(0)) of (9.9) is available as a piecewise affine function of x(0), and the op-
timal control law U0∗ is also available explicitly, as the optimal input U0∗ consists
simply of the last part of z0∗ (x(0))

U0∗ (x(0)) = [0 . . . 0 Im Im . . . Im ]z0∗ (x(0)). (9.11)

Theorem 6.5 states that there exists a continuous and PPWA solution z0∗ (x)
of the mp-LP problem (9.9). Clearly the same properties are inherited by the
controller. The following Corollaries of Theorem 6.5 summarize the analytical
properties of the optimal control law and the value function.

Corollary 9.1 There exists a control law U0∗ = f¯0 (x(0)), f¯0 : Rn → Rm , obtained
as a solution of the optimal control problem (9.2)-(9.4) with p = 1 or p = ∞, which
is continuous and PPWA

f¯0 (x) = F̄0i x if x ∈ CR0i , i = 1, . . . , N0r , (9.12)

where the polyhedral sets CR0i = {H0i x ≤ 0}, i = 1, . . . , N0r , are a partition of Rn .
Note that in Corollary 9.1 the control law is linear (not affine) and the critical
regions have a conic shape (CR0i = {H0i x ≤ 0}). This can be proven immediately
 9.3 Solution via Recursive Approach 175

from the results in Section 6.2 by observing that the constant term wǫ at the
right-hand side on the mp-LP problem (9.9) is zero.

Corollary 9.2 The value function J ∗ (x) obtained as a solution of the optimal
control problem (9.2)-(9.4) is convex and PPWA.

Remark 9.2 Note that if the optimizer of problem (9.4) is unique for all x(0) ∈ Rn ,
then Corollary 9.1 reads: “ The control law U ∗ (0) = f¯0 (x(0)), f¯0 : Rn → Rm ,
obtained as a solution of the optimal control problem (9.2)-(9.4) with p = 1 or
p = ∞, is continuous and PPWA,. . .”. From the results of Section 6.2 we know that
in case of multiple optimizers for some x(0) ∈ Rn , a control law of the form (9.12)
can always be computed.

9.3 Solution via Recursive Approach

Alternatively we can use dynamic programming to solve the same problem in a
recursive manner. We define the optimal cost Jj∗ (xj ) for the N − j step problem
starting from state xj by
N
X −1
Jj∗ (xj ) = min kP xN k∞ + kQxk k∞ + kRuk k∞ .
uj ,··· ,uN −1
k=j

According to the principle of optimality the optimal one step cost-to-go can be
obtained from
∗
JN −1 (xN −1 ) = min kPN xN k∞ + kQxN −1 k∞ + kRuN −1 k∞
uN −1
(9.13)

xN = AxN −1 + BuN −1
(9.14)
PN = P.
Substituting (9.14) into the objective function (9.13), we have
∗
JN −1 (xN −1 ) = min kPN (AxN −1 +BuN −1 )k∞ +kQxN −1 k∞ +kRuN −1 k∞ . (9.15)
uN −1

We find the optimum by solving the mp-LP

min εxN −1 + εxN + εuN −1 (9.16a)

εx x u
N −1 ,εN ,εN −1 ,uN −1

subj. to −1n εxN −1 ≤ ±QxN −1 (9.16b)

−1rN εxN ≤ ±PN [AxN −1 + BuN −1 ] (9.16c)
−1m εuN −1 ≤ ±RuN −1, (9.16d)

∗
where rN is the number of rows of the matrix PN . By Theorem 6.5, JN −1 is a
convex and piecewise affine function of xN −1 , the corresponding optimizer u∗N −1
176 9 Linear 1/∞ Norm Optimal Control

is piecewise affine and continuous, and the feasible set XN −1 is Rn . We use the
equivalence of representation between convex and PPWA functions and infinity
norm (see Section 2.2.5) to write the one-step optimal cost-to-go as

∗
JN −1 (xN −1 ) = kPN −1 xN −1 k∞ (9.17)

with PN −1 defined appropriately. At the next stage, consider the two-step problem
from time N − 2 forward:

∗
JN −2 (xN −2 ) = min kPN −1 xN −1 k∞ + kQxN −2 k∞ + kRuN −2 k∞ (9.18)
uN −2

xN −1 = AxN −2 + BuN −2 . (9.19)

We recognize that (9.18), (9.19) has the same form as (9.13), (9.14). Therefore
we can compute the optimal solution again by solving the mp-LP

min εxN −2 + εxN −1 + εuN −2 (9.20a)

εx x u
N −2 ,εN −1 ,εN −2 ,uN −2

subj. to −1n εxN −2 ≤ ±QxN −2 (9.20b)

x
−1rN −1 εN −1 ≤ ±PN −1 [AxN −2 + BuN −2 ] (9.20c)
−1m εuN −2 ≤ ±RuN −2 , (9.20d)

where rN −1 is the number of rows of the matrix PN −1 . The optimal two-step

cost-to-go is
∗
JN −2 (xN −2 ) = kPN −2 xN −2 k∞ , (9.21)

Continuing in this manner, at some arbitrary time k the optimal control action is

u∗ (k) = fk (x(k)), (9.22)

where fk (x) is continuous and PPWA

fk (x) = Fki x if Hki x ≤ 0, i = 1, . . . , Nkr , (9.23)

where the polyhedral sets {Hki x ≤ 0}, i = 1, . . . , Nkr , are a partition of Rn . The
optimal cost-to-go starting from the measured state x(k) is

Jk∗ (x(k)) = kPk x(k)k∞ . (9.24)

Here we have introduced the notation Pk to express the optimal cost-to-go Jk∗ (x(k)) =
kPk x(k)k∞ from time k to the end of the horizon N . We also remark that the rows
of Pk correspond to the different affine functions constituting Jk∗ and thus their
number varies with the time index k. Clearly, we do not have a closed form as
for the 2-norm with the Riccati Difference Equation (8.24) linking cost and control
law at time k given their value at time k − 1.
 9.4 Comparison Of The Two Approaches 177

9.4 Comparison Of The Two Approaches

We will compare the batch and the recursive dynamic programming approach in
terms of the results and the methods used to obtain the results. Most importantly
we observe that the results obtained by the two methods are fundamentally differ-
ent. The batch approach yields a formula for the sequence of inputs as a function
of the initial state.

U0∗ = F̄0i x(0) if H̄0i x(0) ≤ 0, i = 1, . . . , N̄0r . (9.25)

The recursive dynamic programming approach yields a feedback policy, i.e., a se-
quence of feedback laws expressing at each time step the control action as a function
of the state at that time.

u∗ (k) = Fki x(k) if Hki x(k) ≤ 0, i = 1, . . . , Nkr for k = 0, . . . , N − 1. (9.26)

As this expression implies, we determine u(k) at each time k as a function of the

current state x(k) rather than use a u(k) precomputed at k = 0 as in the batch
method. If the state evolves exactly according to the linear model (9.3) then the
sequence of control actions u(k) obtained from the two approaches is identical. In
practice, the result of applying the sequence (9.25) in an open-loop fashion may
be rather different from applying the time-varying feedback law (9.26) because the
model (9.3) for predicting the system states may be inaccurate and the system may
be subject to disturbances not included in the model. We expect the application
of the feedback law to be more robust because at each time step the observed state
x(k) is used to determine the control action rather than the state xk predicted at
time t = 0.
We note that we can get the same feedback effect with the batch approach
if we recalculate the optimal open-loop sequence at each time step j with the
current measurement as initial condition. In this case we need to solve the following
optimization problem

PN −1
Jj∗ (x(j)) = minuj ,··· ,uN −1 kP xN k∞ + k=j kQxk k∞ + kRuk k∞
(9.27)
subj. to xj = x(j),

where we note that the horizon length is shrinking at each time step.
As seen from (9.25) the solution to (9.27) relates the sequence of inputs u∗j , u∗j+1 , . . .
to the state x(j) through a linear expression. The first part of this expression yields
again the optimal feedback law (9.26) at time j, u∗ (j) = fj (x(j)).
Here the dynamic programming approach is clearly a more efficient way to
generate the feedback policy because it requires the solution of a small mp-LP
problem (9.7) for each time step. Repeated application of the batch approach, on
the other hand, requires repeatedly the solution of a larger mp-LP for each time
step.
178 9 Linear 1/∞ Norm Optimal Control

9.5 Infinite Horizon Problem

For continuous processes operating over a long time period it would be interesting
to solve the following infinite horizon problem.

∞
X
∗
J∞ (x(0)) = min kQxk k∞ + kRuk k∞ . (9.28)
u(0),u(1),...
k=0

Since the prediction must be carried out to infinity, application of the batch method
becomes impossible. On the other hand, derivation of the optimal feedback law
via dynamic programming remains viable. We can use the dynamic programming
formulation

kPj xj k∞ = min kPj+1 xj+1 k∞ + kQxj k∞ + kRuj k∞ (9.29)

uj

xj+1 = Axj + Buj (9.30)

with the terminal cost matrix P0 = Q and solve it backwards for k → −∞. Let
us assume for the moment that the iterations converge to a solution P∞ in a finite
number of iterations. Then the optimal feedback control law is time-invariant and
piecewise linear

u∗ (k) = F i x(k) if H i x ≤ 0, i = 1, . . . , N r (9.31)

and the optimal infinite horizon cost is

∗
J∞ (x(0)) = kP∞ x(0)k∞ . (9.32)

∗
In general, the infinite time optimal cost J∞ (x(0)) and the optimal feedback con-
trol law are not necessarily piecewise linear (with a finite number of regions). Con-
vergence of the recursive scheme (9.29) has been studied in detail in [87]. If this
recursive scheme converges and Q and R are of full column rank, then the resulting
control law (9.31) stabilizes the system (see Section 7.4).

Example 9.1 Consider the double integrator system

    
1 1 0
x(t + 1) = x(t) + u(t) (9.33)
0 1 1

The aim is to compute the infinite horizon

 optimal controller that solves the opti-
1 0
mization problem (9.28) with Q = and R = 20.
0 1

The dynamic programming iteration (9.29) converges after 18 iterations to the fol-
9.5 Infinite Horizon Problem 179

lowing optimal solution:

  −0.10 −1.00 

 [ 9.44 29.44 ] x if −0.71 −0.71 x ≤ [ 00 ] (Region #1)



  

 [ 9.00 25.00 ] x if 0.10 1.00
x ≤ [ 00 ] (Region #2)

 −0.11 −0.99





 [ −1.00 19.00 ] x if [ −0.45 −0.89
≤ [ 00 ]

 0.71 0.71 ] x (Region #3)



  

 0.11 0.99 0

 [ 8.00 16.00 ] x if −0.12 −0.99 x≤[ 0 ] (Region #4)







 [ −2.00 17.00 ] x if [ −0.32
0.45
−0.95
0.89 ] x ≤ [ 00 ] (Region #5)



  

 0.12 0.99

 [ 7.00 8.00 ] x if −0.14 −0.99 x ≤ [ 00 ] (Region #6)



  

 0.32 0.95

 [ −3.00 14.00 ] x if −0.24 −0.97 x ≤ [ 00 ] (Region #7)





  0.14 0.99


 [ 6.00 1.00 ] x if x ≤ [ 00 ] (Region #8)


−0.16 −0.99



  

 [ −4.00 10.00 ] x if 0.24 0.97
x ≤ [ 00 ] (Region #9)

 −0.20 −0.98



  

 [ 5.00 −5.00 ] x if 0.16 0.99
x ≤ [ 00 ] (Region #10)

 −0.20 −0.98

u=  0.20 0.98


 [ −5.00 5.00 ] x if −0.16 −0.99 x ≤ [ 00 ] (Region #11)



  

 0.20 0.98

 [ 4.00 −10.00 ] x if −0.24 −0.97 x ≤ [ 00 ] (Region #12)



  

 0.16 0.99

 [ −6.00 −1.00 ] x if −0.14 −0.99 x ≤ [ 00 ] (Region #13)



  

 0.24 0.97

 [ 3.00 −14.00 ] x if −0.32 −0.95 x ≤ [ 00 ] (Region #14)





  

 [ −7.00 −8.00 ] x if 0.14 0.99
x ≤ [ 00 ] (Region #15)


−0.12 −0.99



  

 [ 2.00 −17.00 ] x if 0.32 0.95
x ≤ [ 00 ] (Region #16)

 −0.45 −0.89



  

 [ −8.00 −16.00 ] x if 0.12 0.99
x ≤ [ 00 ] (Region #17)

 −0.11 −0.99



  

 [ 1.00 −19.00 ] x if 0.45 0.89
x ≤ [ 00 ] (Region #18)

 −0.71 −0.71



  

 [ −9.00 −25.00 ] x if 0.11 0.99
x ≤ [ 00 ] (Region #19)

 −0.10 −1.00





 [ −9.44 −29.44 ] x if [ 0.10 1.00
≤ [ 00 ]

 0.71 0.71 ] x (Region #20)
180 9 Linear 1/∞ Norm Optimal Control

with P∞ equal to
 
9.44 29.44
 9.00 25.00 
 
−1.00 19.00 
 
 8.00 16.00 
 
−2.00 17.00 
 
 7.00 8.00 
 
−3.00 14.00 
 
 6.00 1.00 
 
−4.00 10.00 
 
 5.00 −5.00 
P∞ =
−5.00
 (9.34)
 5.00 

 4.00 −10.00
 
−6.00 −1.00 
 
 3.00 −14.00
 
−7.00 −8.00 
 
 2.00 −17.00
 
−8.00 −16.00
 
 1.00 −19.00
 
−9.00 −25.00
−9.44 −29.44
Note that P∞ in (9.34) has 20 rows corresponding to the 20 linear terms (or pieces)
∗
of the piecewise linear value function J∞ (x) = kP∞ xk∞ for x ∈ R2 . For instance,
∗ ∗
J∞ (x) in region 1 is J∞ (x) = 9.44x1 + 29.44x2 , where x1 and x2 denote the first
and second component of the state vector x, respectively. Note that each linear term
∗
appears twice, with positive and negative sign. Therefore J∞ (x) can be written in
minimal form as the infinity norm of a matrix kP̃∞ xk∞ with P̃∞ being a matrix with
∗
ten rows. The value function J∞ (x) = kP∞ xk∞ is plotted in Figure 9.1.
J ∗ (x)

x2 x1
Figure 9.1 Example 9.1. ∞-norm objective function, ∞-horizon controller.
Piecewise linear optimal cost (value function) and corresponding polyhedral
partition.
 Part IV

Constrained Optimal Control of

Linear Systems
 Chapter 10

Controllability, Reachability
and Invariance

This chapter is a self-contained introduction to controllability, reachability and

invariant set theory. N -steps reachable sets are defined for autonomous systems.
They represent the set of states which a system can evolve to after N steps. N -
steps controllable sets are defined for systems with inputs. They represent the set
of states which a system can be steered to after N steps.
Invariant sets are the infinite time versions of N -steps controllable and reachable
sets. Invariant sets are computed for autonomous systems. These types of sets
are useful to answer questions such as: “For a given feedback controller u = k(x),
find the set of states whose trajectory will never violate the system constraints”.
Control invariant sets are defined for systems subject to external inputs. These
types of sets are useful to answer questions such as: “Find the set of states for which
there exists a controller such that the system constraints are never violated”.
This chapter focuses on computational tools for constrained linear systems and
constrained linear systems subject to additive and parametric uncertainty. A thor-
ough presentation of the basic notions and algorithms presented in this chapter can
be found in the book by Blanchini and Miani [59].

10.1 Controllable and Reachable Sets

In this section we deal with two types of systems, namely, autonomous systems:
x(t + 1) = ga (x(t)), (10.1)
and systems subject to external inputs:
x(t + 1) = g(x(t), u(t)). (10.2)
Both systems are subject to state and input constraints
x(t) ∈ X , u(t) ∈ U, ∀ t ≥ 0. (10.3)
184 10 Controllability, Reachability and Invariance

The sets X and U are polyhedra.

Definition 10.1 For the autonomous system (10.1) we denote the precursor set
to the set S as

http : //www.mpc.berkeley.edu/mpc − course − material/bookexamples/P reReachAut.m Pre(S) = {x

(10.4)

Pre(S) is the set of states which evolve into the target set S in one time step.

Definition 10.2 For the system (10.2) we denote the precursor set to the set S
as
Pre(S) = {x ∈ Rn : ∃u ∈ U s.t. g(x, u) ∈ S}. (10.5)

For a system with inputs, Pre(S) is the set of states which can be driven into the
target set S in one time step while satisfying input and state constraints.

Definition 10.3 For the autonomous system (10.1) we denote the successor set
from the set S as

Suc(S) = {x ∈ Rn : ∃ x(0) ∈ S s.t. x = ga (x(0))}.

Definition 10.4 For the system (10.2) with inputs we will denote the successor
set from the set S as

Suc(S) = {x ∈ Rn : ∃ x(0) ∈ S, ∃ u(0) ∈ U s.t. x = g(x(0), u(0))}.

Therefore, all the states contained in S are mapped into the set Suc(S) under the
map ga or under the map g for some input u ∈ U.

Remark 10.1 The sets Pre(S) and Suc(S) are also denoted as ‘one-step backward-
reachable set’ and ‘one-step forward-reachable set’, respectively, in the literature.

N -step controllable and reachable sets are defined by iterating Pre(·) and Suc(·)
computations, respectively.

Definition 10.5 (N -Step Controllable Set KN (S)) For a given target set S ⊆
X , the N -step controllable set KN (S) of the system (10.1) or (10.2) subject to the
constraints (10.3) is defined recursively as:

Kj (S) = Pre(Kj−1 (S)) ∩ X , K0 (S) = S, j ∈ {1, . . . , N } (10.6)

From Definition 10.5, all states x0 of the system (10.1) belonging to the N -Step
Controllable Set KN (S) will evolve to the target set S in N steps, while satisfying
state constraints.
Also, all states x0 of the system (10.2) belonging to the N -Step Controllable
Set KN (S) can be driven, by a suitable control sequence, to the target set S in N
steps, while satisfying input and state constraints.
 10.1 Controllable and Reachable Sets 185

Definition 10.6 (N -Step Reachable Set RN (X0 )) For a given initial set X0 ⊆
X , the N -step reachable set RN (X0 ) of the system (10.1) or (10.2) subject to the
constraints (10.3) is defined as:

Ri+1 (X0 ) = Suc(Ri (X0 )) ∩ X , R0 (X0 ) = X0 , i = 0, . . . , N − 1 (10.7)

From Definition 10.6, all states x0 belonging to X0 will evolve to the N -step reach-
able set RN (X0 ) in N steps.

10.1.1 Computation of Controllable and Reachable Sets

Next we will show through simple examples the main steps involved in the com-
putation of controllable and reachable sets for constrained linear systems. Later in
this section we will provide compact formulas based on polyhedral operations.

Example 10.1 Consider the second order autonomous stable system

 
0.5 0
x(t + 1) = Ax(t) = x(t) (10.8)
1 −0.5

subject to the state constraints

    
−10 10
x(t) ∈ X = x : ≤x≤ , ∀t ≥ 0. (10.9)
−10 10

The set Pre(X ) can be obtained as follows: Since the set X is a polytope, it can be
represented as a H-polytope (Section 4.2)

X = {x : Hx ≤ h}, (10.10)

where    
1 0 10
0 1 10
H=
−1
 and h =   .
0 10
0 −1 10

By using this H-presentation and the system equation (10.8), the set Pre(X ) can be
derived:

Pre(X ) = {x : Hga (x) ≤ h} (10.11)

= {x : HAx ≤ h} . (10.12)

The set (10.12) may contain redundant inequalities which can be removed by using
Algorithm 4.1 in Section 4.4.1 to obtain its minimal representation. Note that by
using the notation in Section 4.4.11, the set Pre(X ) in (10.12) is simply X ◦ A.
The set Pre(X ) is
    

 1 0 20 

 1 10
 −0.5x ≤   .
Pre(X ) = x : −1

 0  20

 
−1 −0.5 10
186 10 Controllability, Reachability and Invariance

The one-step controllable set to X , K1 (X ) = Pre(X ) ∩ X is

    
1 
 0 10 

0 
 10
 
 1   


−1  10
0 
Pre(X ) ∩ X = x : 
0
x ≤  

  −1 
10
 

 1 10


 −0.5 

 
−1 0.5 10

and it is depicted in Fig. 10.1.

10

x2 5
0 Pre(X ) ∩ X
-5
X
-10
-10 -5 0 5 10
x1

Figure 10.1 Example 10.1. One-step controllable set Pre(X ) ∩ X for system
(10.8) under constraints (10.9).

The set Suc(X ) is obtained by applying the map A to the set X . Let us write X in
V-representation (see Section 4.1)

X = conv(V ), (10.13)

and let us map the set of vertices V through the transformation A. Because the
transformation is linear, the successor set is simply the convex hull of the transformed
vertices
Suc(X ) = A ◦ X = conv(AV ). (10.14)
We refer the reader to Section 4.4.11 for a detailed discussion on linear transforma-
tions of polyhedra.
The set Suc(X ) in H-representation is
    

 1 0 5 
 −1  5 
 0 x ≤  
Suc(X ) = x : 1

 −0.5 2.5

 
−1 0.5 2.5

and is depicted in Fig. 10.2.

Example 10.2 Consider the second order unstable system

   
1.5 0 1
x(t + 1) = Ax + Bu = x(t) + u(t) (10.15)
1 −1.5 0
10.1 Controllable and Reachable Sets 187

10
5

)
x2
0

c(X
-5

Su
X
-10

-10 -5 0 5 10
x1

Figure 10.2 Example 10.1. Successor set for system (10.8).

subject to the input and state constraints

u(t) ∈ U = {u : − 5 ≤ u ≤ 5} , ∀t ≥ 0 (10.16a)
    
−10 10
x(t) ∈ X = x : ≤x≤ , ∀t ≥ 0. (10.16b)
−10 10

For the non-autonomous system (10.15), the set Pre(X ) can be computed using the
H-representation of X and U,

X = {x : Hx ≤ h}, U = {u : Hu u ≤ hu }, (10.17)

to obtain

Pre(X ) = x ∈ R2 : ∃u ∈ U s.t. g(x, u) ∈ X , (10.18)
      
HA HB x h
= x ∈ R2 : ∃u ∈ R s.t. ≤ . (10.19)
0 Hu u hu

The half-spaces in (10.19) define a polytope in the state-input space, and a projection
operation (see Section 4.4.6) is used to derive the half-spaces which define Pre(X ) in
the state space. The one-step controllable set Pre(X ) ∩ X
   
1 0 10
0 1  10
   
−1 0   
  x ≤ 10
0 −1  10
   
1 −1.5  10
−1 1.5 10
is depicted in Fig. 10.3.
Note that by using the definition of the Minkowski sum given in Section 4.4.9 and
the affine operation on polyhedra in Section 4.4.11 we can write the operations in
(10.19) compactly as follows:

Pre(X ) = {x : ∃u ∈ U s.t. Ax + Bu ∈ X }
{x : y = Ax + Bu, y ∈ X , u ∈ U}
{x : Ax = y + (−Bu), y ∈ X , u ∈ U}
(10.20)
{x : Ax ∈ C, C = X ⊕ (−B) ◦ U}
{x : x ∈ C ◦ A, C = X ⊕ (−B) ◦ U}
{x : x ∈ (X ⊕ (−B) ◦ U) ◦ A} .
188 10 Controllability, Reachability and Invariance

10
5
0

x2
Pre(X ) ∩ X
-5
X
-10
-10 -5 0 5 10
x1

Figure 10.3 Example 10.2. One-step controllable set Pre(X ) ∩ X for system
(10.15) under constraints (10.16).

The set Suc(X ) = {Ax + Bu ∈ R2 : x ∈ X , u ∈ U} is obtained by applying the map

A to the set X and then considering the effect of the input u ∈ U. As shown before,
A ◦ X = conv(AV ) (10.21)
and therefore
Suc(X ) = {y + Bu : y ∈ A ◦ X , u ∈ U}.
We can use the definition of the Minkowski sum given in Section 4.4.9 and rewrite
the set Suc(X ) as
Suc(X ) = (A ◦ X ) ⊕ (B ◦ U).
We can compute the Minkowski sum via projection or vertex enumeration as ex-
plained in Section 4.4.9 and obtain the set Suc(X ) in H-representation
    

 1 0 20  

 −1  20 

  0  



 0  25 
1 
Suc(X ) = x :  0
x ≤   ,

  −1 
 25 
 

 1 27.5 

 −1.5 

 
−1 1.5 27.5
which is depicted in Fig. 10.4.

In summary, the sets Pre(X ) and Suc(X ) are the results of linear operations
on the polyhedra X and U and therefore are polyhedra. By using the definition of
the Minkowski sum given in Section 4.4.9 and of affine operation on polyhedra in
Section 4.4.11 we can compactly summarize the Pre and Suc operations on linear
systems in Table 10.1.
The N -step controllable set KN (S) and the N -step reachable set RN (X0 ) can
be computed by using their recursive formulas (10.6), (10.7) and computing the
Pre and Suc operations as in Table 10.1.

Example 10.3 Consider the second order unstable system (10.15) subject to the input
and state constraints (10.16). Consider the target set
    
−1 1
S= x : ≤x≤ .
−1 1
10.1 Controllable and Reachable Sets 189

30
20
10
Suc(X )

x2
0
-10
-20
-30
-30 -20 -10 0 10 20 30
x1

Figure 10.4 Example 10.2. Successor set for system (10.15) under constraints
(10.16).

x(t + 1) = Ax(t) x(t + 1) = Ax(t) + Bu(t)

Pre(X ) X ◦A (X ⊕ (−B ◦ U)) ◦ A
Suc(X ) A◦X (A ◦ X ) ⊕ (B ◦ U)

Table 10.1 Pre and Suc operations for linear systems subject to polyhedral
state and input constraints x(t) ∈ X , u(t) ∈ U

The N -step controllable set KN (S) of the system (10.15) subject to the constraints (10.16)
can be computed by using the recursive formula (10.6)
Kj (S) = Pre(Kj−1 (S)) ∩ X , K0 (S) = S, j = 1, . . . , N
and the steps described in Example 10.2 to compute the Pre(·) set.
The sets Kj (S) for j = 1, 2, 3, 4 are depicted in Fig. 10.5.

K4 (S) K3 (S) K2 (S) K1 (S) S

Figure 10.5 Example 10.3. Controllable sets Kj (S) for system (10.15) under
constraints (10.16) for j = 1, 2, 3, 4. Note that the sets are shifted along the
x-axis for a clearer visualization.

Example 10.4 Consider the second order unstable system (10.15) subject to the input
and state constraints (10.16). Consider the initial set
    
−1 1
X0 = x : ≤x≤ .
−1 1
190 10 Controllability, Reachability and Invariance

The N -step reachable set RN (X0 ) of the system (10.15) subject to the constraints (10.16)
can be computed by using the recursively formula (10.7)

Rj+1 (X0 ) = Suc(Rj (X0 )) ∩ X , R0 (X0 ) = X0 , j = 0, . . . , N − 1

and the steps described in Example 10.2 to compute the Suc(·) set.
The sets Rj (X0 )) for j = 1, 2, 3, 4 are depicted in Fig. 10.6. The sets are shifted along
the x-axis for a clearer visualization.

X0 R1 (X0 ) R2 (X0 ) R3 (X0 ) R4 (X0 )

Figure 10.6 Example 10.3. Reachable sets Rj (X0 )) for system (10.15) under
constraints (10.16) for j = 1, 2, 3, 4.

10.2 Invariant Sets

Consider the constrained autonomous system (10.1) and the constrained system
subject to external inputs (10.2) defined in Section 10.1.
Two different types of sets are considered in this section: invariant sets and
control invariant sets. We will first discuss invariant sets.

Positive Invariant Sets

Invariant sets are used for characterizing the behavior of autonomous systems.
These types of sets are useful to answer questions such as: “For a given feedback
controller u = f (x), find the set of initial states whose trajectory will never violate
the system constraints”. The following definitions, derived from [172, 58, 54, 49,
179, 137, 139, 140], introduce the different types of invariant sets.

Definition 10.7 (Positive Invariant Set) A set O ⊆ X is said to be a positive

invariant set for the autonomous system (10.1) subject to the constraints in (10.3),
if
x(0) ∈ O ⇒ x(t) ∈ O, ∀t ∈ N+ .
10.2 Invariant Sets 191

Definition 10.8 (Maximal Positive Invariant Set O∞ ) The set O∞ ⊆ X is

the maximal invariant set of the autonomous system (10.1) subject to the con-
straints in (10.3) if O∞ is invariant and O∞ contains all the invariant sets con-
tained in X .

Remark 10.2 The maximal invariant sets defined here are often referred to as ‘maxi-
mal admissible sets’ or ‘maximal output admissible sets’ in the literature (e.g., [124]),
depending on whether the system state or output is constrained.

Remark 10.3 Note that, in general, the nonlinear system (10.1) may have multi-
ple equilibrium points, and thus O∞ might be the union of disconnected sets each
containing an equilibrium point.

Theorem 10.1 (Geometric condition for invariance [100]) A set O ⊆ X is

a positive invariant set for the autonomous system (10.1) subject to the constraints
in (10.3), if and only if
O ⊆ Pre(O). (10.22)

Proof: We prove both the necessary and sufficient parts by contradiction. (⇐:)
If O * Pre(O) then ∃x̄ ∈ O such that x̄ ∈ / Pre(O). From the definition of Pre(O),
ga (x̄) ∈
/ O and thus O is not positive invariant. (⇒:) If O is not a positive invariant
set then ∃x̄ ∈ O such that ga (x̄) ∈/ O. This implies that x̄ ∈ O and x̄ ∈ / Pre(O)
and thus O * Pre(O) 
It is immediate to prove that condition (10.22) of Theorem 10.1 is equivalent
to the following condition
Pre(O) ∩ O = O. (10.23)
Based on condition (10.23), the following algorithm provides a procedure for com-
puting the maximal positive invariant subset O∞ for system (10.1),(10.3) [10, 49,
172, 124].

Algorithm 10.1 Computation of O∞

Input: ga , X
Output: O∞
Ω0 ← X , k ← −1
Repeat
k ←k+1
Ωk+1 ← Pre(Ωk ) ∩ Ωk
Until Ωk+1 = Ωk
O∞ ← Ωk

Algorithm 10.1 generates the set sequence {Ωk } satisfying Ωk+1 ⊆ Ωk , ∀k ∈ N and
it terminates when Ωk+1 = Ωk . If it terminates, then Ωk is the maximal positive
invariant set O∞ for the system (10.1)-(10.3). If Ωk = ∅ for some integer k then
the simple conclusion is that O∞ = ∅.
192 10 Controllability, Reachability and Invariance

In general, Algorithm 10.1 may never terminate. If the algorithm does not
terminate in a finite number of iterations, it can be proven that [179]

O∞ = lim Ωk .
k→+∞

Conditions for finite time termination of Algorithm 10.1 can be found in [124].
A simple sufficient condition for finite time termination of Algorithm 10.1 requires
the system ga (x) to be linear and stable, and the constraint set X to be bounded
and to contain the origin.

Example 10.5 Consider the second order stable system in Example 10.1. The maxi-
mal positive invariant set of system (10.8) subject to constraints (10.9)
   
1 0 10
 0 1   10 
   
 −1 0   
  x ≤  10 
 0 −1    
  10 
 1 −0.5   10 
−1 0.5 10

is depicted in Fig. 10.7.

10

5
x2

0 O∞

-5
X
-10
-10 -5 0 5 10
x1
Figure 10.7 Example 10.5. Maximal Positive Invariant Set of system (10.8)
under constraints (10.9).

Note from the previous discussion of the example and from Figure 10.1 that
here the maximal positive invariant set O∞ is obtained after a single step of Algo-
rithm 10.1, i.e.
O∞ = Ω1 = Pre(X ) ∩ X .

Control Invariant Sets

Control invariant sets are defined for systems subject to external inputs. These
types of sets are useful to answer questions such as: “Find the set of initial states for
which there exists a controller such that the system constraints are never violated”.
10.2 Invariant Sets 193

The following definitions, adopted from [172, 58, 54, 49, 179], introduce the different
types of control invariant sets.

Definition 10.9 (Control Invariant Set) A set C ⊆ X is said to be a control

invariant set for the system (10.2) subject to the constraints in (10.3), if

x(t) ∈ C ⇒ ∃u(t) ∈ U such that g(x(t), u(t)) ∈ C, ∀t ∈ N+ .

Definition 10.10 (Maximal Control Invariant Set C∞ ) The set C∞ ⊆ X is

said to be the maximal control invariant set for the system (10.2) subject to the
constraints in (10.3), if it is control invariant and contains all control invariant
sets contained in X .

Remark 10.4 The geometric conditions for invariance (10.22), (10.23) hold for control
invariant sets.

The following algorithm provides a procedure for computing the maximal con-
trol invariant set C∞ for system (10.2),(10.3) [10, 49, 172, 124].

Algorithm 10.2 Computation of C∞

Input: g, X , U
Output: C∞
Ω0 ← X , k ← −1
Repeat
k ←k+1
Ωk+1 ← Pre(Ωk ) ∩ Ωk
Until Ωk+1 = Ωk
C∞ ← Ωk+1

Algorithm 10.2 generates the set sequence {Ωk } satisfying Ωk+1 ⊆ Ωk , ∀k ∈ N. Al-
gorithm 10.2 terminates when Ωk+1 = Ωk . If it terminates, then Ωk is the maximal
control invariant set C∞ for the system (10.2)-(10.3). In general, Algorithm 10.2
may never terminate [10, 49, 172, 164]. If the algorithm does not terminate in a fi-
nite number of iterations, in general, convergence to the maximal control invariant
set is not guaranteed
C∞ 6= lim Ωk . (10.24)
k→+∞

The work in [50] reports examples of nonlinear systems where (10.24) can be ob-
served. A sufficient condition for the convergence of Ωk to C∞ as k → +∞ requires
the polyhedral sets X and U to be bounded and the system g(x, u) to be continu-
ous [50].

Example 10.6 Consider the second order unstable system in Example 10.2. Algo-
rithm 10.2 is used to compute the maximal control invariant set of system (10.15)
194 10 Controllability, Reachability and Invariance

subject to constraints (10.16). Algorithm 10.2 terminates after 45 iterations and the
maximal control invariant set C∞ is:
   
0 1 4
 0 −1   4 
   
 0.55 −0.83   
  x ≤  2.22  .
 −0.55 0.83   2.22 
   
 1 0   10 
−1 0 10

The results of the iterations and C∞ are depicted in Fig. 10.8.

10

5
x2

0 C∞

-5
X
-10
-10 -5 0 5 10
x1
Figure 10.8 Example 10.6. Maximal Control Invariant Set of system (10.15)
subject to constraints (10.16).

Definition 10.11 (Finitely determined set) Consider Algorithm 10.1 (Algo-

rithm 10.2). The set O∞ (C∞ ) is finitely determined if and only if ∃ i ∈ N such
that Ωi+1 = Ωi . The smallest element i ∈ N such that Ωi+1 = Ωi is called the
determinedness index.

Remark 10.5 From the results in Section 10.1.1, for linear system with linear con-
straints the sets O∞ and C∞ are polyhedra if they are finitely determined.

For all states contained in the maximal control invariant set C∞ there exists
a control law such that the system constraints are never violated. This does not
imply that there exists a control law which can drive the state into a user-specified
target set. This issue is addressed in the following by introducing the concepts of
maximal controllable sets and stabilizable sets.

Definition 10.12 (Maximal Controllable Set K∞ (O)) For a given target set
O ⊆ X , the maximal controllable set K∞ (O) for system (10.2) subject to the con-
straints in (10.3) is the union of all N -step controllable sets KN (O) contained in
X (N ∈ N).

We will often deal with controllable sets KN (O) where the target O is a control
invariant set. They are special sets, since in addition to guaranteeing that from
10.2 Invariant Sets 195

KN (O) we reach O in N steps, one can ensure that once it has reached O, the
system can stay there at all future time instants.

Definition 10.13 (N -step (Maximal) Stabilizable Set) For a given control

invariant set O ⊆ X , the N -step (maximal) stabilizable set of the system (10.2)
subject to the constraints (10.3) is the N -step (maximal) controllable set KN (O)
(K∞ (O)).
The set K∞ (O) contains all states which can be steered into the control invariant
set O and hence K∞ (O) ⊆ C∞ . The set K∞ (O) ⊆ C∞ can be computed as
follows [58, 50]:

Algorithm 10.3 Computation of K∞ (O)

Input: g, X , U
Output: K∞ (O)
K0 ← O, where O is a control invariant set
c ← −1
Repeat
c←c+1
T
Kc+1 ← Pre(Kc ) X
Until Kc+1 = Kc
K∞ (O) ← Kc

Since O is control invariant, it holds ∀c ∈ N that Kc (O) is control invariant and

Kc ⊆ Kc+1 . Note that Algorithm 10.3 is not guaranteed to terminate in finite time.

Remark 10.6 In general, the maximal stabilizable set K∞ (O) is not equal to the
maximal control invariant set C∞ , even for linear systems. K∞ (O) ⊆ C∞ for all
control invariant sets O. The set C∞ \ K∞ (O) includes all initial states from which
it is not possible to steer the system to the stabilizable region K∞ (O) and hence O.

Example 10.7 Consider the simple constrained one-dimensional system

x(t + 1) = 2x(t) + u(t) (10.25a)
|x(t)| ≤ 1, and |u(t)| ≤ 1 (10.25b)
and the state feedback control law
  
 1 if x(t) ∈ −1, − 21
u(t) = −2x(t) if x(t) ∈ − 21 , 12 (10.26)

−1 if x(t) ∈ 12 , 1 .
The closed-loop system has three equilibria at −1, 0, and 1 and system (10.25) is
always feasible for all initial states in [−1, 1] and therefore C∞ = [−1, 1]. The
equilibrium point 0 is, however, asymptotically stable only for the open set (−1, 1).
In fact, u(t) and any other feasible control law cannot stabilize the system from x = 1
and from x = −1 and therefore when O = ∅ then K∞ (O) = (−1, 1) ⊂ C∞ . We note
in this example that the maximal stabilizable set is open. One can easily argue that,
in general, if the maximal stabilizable set is closed then it is equal to the maximal
control invariant set.
196 10 Controllability, Reachability and Invariance

10.3 Robust Controllable and Reachable Sets

In this section we deal with two types of systems, namely, autonomous systems

x(k + 1) = ga (x(k), w(k)), (10.27)

and systems subject to external controllable inputs

x(k + 1) = g(x(k), u(k), w(k)). (10.28)

Both systems are subject to the disturbance w(k) and to the constraints

x(k) ∈ X , u(k) ∈ U, w(k) ∈ W ∀ k ≥ 0. (10.29)

The sets X and U are and W are polyhedra.

Definition 10.14 For the autonomous system (10.27) we will denote the robust
precursor set to the set S as

Pre(S, W) = {x ∈ Rn : ga (x, w) ∈ S, ∀w ∈ W}. (10.30)

Pre(S, W) defines the set of states of system (10.27) which evolve into the target
set S in one time step for all possible disturbance w ∈ W.

Definition 10.15 For the system (10.28) we will denote the robust precursor set
to the set S as

Pre(S, W) = {x ∈ Rn : ∃u ∈ U s.t. g(x, u, w) ⊆ S, ∀w ∈ W}. (10.31)

For a system with inputs, Pre(S, W) is the set of states which can be robustly
driven into the target set S in one time step for all admissible disturbances.

Definition 10.16 For the autonomous system (10.27) we will denote the robust
successor set from the set S as

Suc(S, W) = {x ∈ Rn : ∃ x(0) ∈ S, ∃ w ∈ W such that x = ga (x(0), w)}.

Definition 10.17 For the system (10.28) with inputs we will denote the robust
successor set from the set S as

Suc(S, W) = {x ∈ Rn : ∃ x(0) ∈ S, ∃ u ∈ U, ∃ w ∈ W, such that x = g(x(0), u, w)}.

Thus, all the states contained in S are mapped into the set Suc(S, W) under the
map ga for all disturbances w ∈ W, and under the map g for all inputs u ∈ U and
for all disturbances w ∈ W.

Remark 10.7 The sets Pre(S, W) and Suc(S, W) are also denoted as ‘one-step robust
backward-reachable set’ and ‘one-step robust forward-reachable set’, respectively, in
the literature.
 10.3 Robust Controllable and Reachable Sets 197

N -step robust controllable and robust reachable sets are defined by iterating
Pre(·, ·) and Suc(·, ·) computations, respectively.

Definition 10.18 (N -Step Robust Controllable Set KN (S, W)) For a given
target set S ⊆ X , the N -step robust controllable set KN (S, W) of the system (10.27)
or (10.28) subject to the constraints (10.29) is defined recursively as:

Kj (S, W) = Pre(Kj−1 (S, W), W) ∩ X , K0 (S, W) = S, j ∈ {1, . . . , N }. (10.32)

From Definition 10.18, all states x0 belonging to the N -Step Robust Controllable
Set KN (S, W) can be robustly driven, through a time-varying control law, to the
target set S in N steps, while satisfying input and state constraints for all possible
disturbances.
N -step robust reachable sets are defined analogously to N -step robust control-
lable set.

Definition 10.19 (N -Step Robust Reachable Set RN (X0 , W)) For a given ini-
tial set X0 ⊆ X , the N -step robust reachable set RN (X0 , W) of the system (10.27)
or (10.28) subject to the constraints (10.29) is defined recursively as:

Ri+1 (X0 , W) = Suc(Ri (X0 , W), W) ∩ X , R0 (X0 , W) = X0 , i = 0, . . . , N − 1.

(10.33)
From Definition 10.19, all states x0 belonging to X0 will evolve to the N -step robust
reachable set RN (X0 , W) in N steps.
Next we will show through simple examples the main steps involved in the
computation of robust controllable and robust reachable sets for certain classes of
uncertain constrained linear systems.

10.3.1 Linear Systems with Additive Uncertainty and without Inputs

Example 10.8 Consider the second order autonomous system

 
0.5 0
x(t + 1) = Ax(t) + w(t) = x(t) + w(t) (10.34)
1 −0.5

subject to the state constraints

    
−10 10
x(t) ∈ X = x : ≤x≤ , ∀t ≥ 0, (10.35)
−10 10

and where the additive disturbance belongs to the set

    
−1 1
w(t) ∈ W = w : ≤w≤ , ∀t ≥ 0. (10.36)
−1 1

The set Pre(X , W) can be obtained as described next. Since the set X is a polytope,
it can be represented as an H-polytope (Section 4.2)

X = {x : Hx ≤ h}, (10.37)
198 10 Controllability, Reachability and Invariance

where    
1 0 10
0 1 10
H=
−1
 and h =   .
0 10
0 −1 10

By using this H-presentation and the system equation (10.34), the set Pre(X , W) can
be rewritten as

Pre(X , W) = {x : Hga (x, w) ≤ h, ∀w ∈ W} (10.38a)

= {x : HAx ≤ h − Hw, ∀w ∈ W} . (10.38b)

The set (10.38) can be represented as a the following polyhedron

Pre(X , W) = {x ∈ Rn : HAx ≤ h̃} (10.39)

with
h̃i = min (hi − Hi w). (10.40)
w∈W

In general, a linear program is required to solve problems

  (10.40). In this example Hi
9
9
and W have simple expressions and we get h̃ =  
9. The set (10.39) might contain
9
redundant inequalities which can be removed by using Algorithm 4.1 in Section 4.4.1
to obtain its minimal representation.
The set Pre(X , W) is
    

 1 0 18 
 1  
 −0.5
x ≤  9  .
Pre(X , W) = x : 

 −1 0  18

 
−1 0.5 9

The one-step robust controllable set K1 (X , W) = Pre(X , W) ∩ X is

    

 1 0 10 

 0  10 

  1    
 −1  10
0
Pre(X , W) ∩ X = x :  0
x ≤  

  −1 
10
 

 1   9 

 −0.5 

 
−1 0.5 9

and is depicted in Fig. 10.9.

Note that by using the definition of the Pontryagin difference given in Section 4.4.8
and affine operations on polyhedra in Section 4.4.11 we can compactly summarize
the operations in (10.38) and write the set Pre in (10.30) as

Pre(X , W) = {x ∈ Rn : Ax + w ∈ X , ∀w ∈ W} = {x ∈ Rn : Ax ∈ X ⊖ W} =
= (X ⊖ W) ◦ A.

The set

Suc(X , W) = {y : ∃x ∈ X , ∃w ∈ W such that y = Ax + w} (10.41)

 10.3 Robust Controllable and Reachable Sets 199

10
5

x2
0 Pre(X , W) ∩ X
-5
X
-10
-10 -5 0 5 10
x1

Figure 10.9 Example 10.8. One-step robust controllable set Pre(X , W) ∩ X

for system (10.34) under constraints (10.35)-(10.36).

is obtained by applying the map A to the set X and then considering the effect of
the disturbance w ∈ W. Let us write X in V-representation (see Section 4.1)
X = conv(V ), (10.42)
and let us map the set of vertices V through the transformation A. Because the
transformation is linear, the composition of the map A with the set X , denoted as
A ◦ X , is simply the convex hull of the transformed vertices
A ◦ X = conv(AV ). (10.43)
We refer the reader to Section 4.4.11 for a detailed discussion on linear transforma-
tions of polyhedra. Rewrite (10.41) as
Suc(X , W) = {y ∈ Rn : ∃ z ∈ A ◦ X , ∃w ∈ W such that y = z + w}.
We can use the definition of the Minkowski sum given in Section 4.4.9 and rewrite
the Suc set as
Suc(X , W) = (A ◦ X ) ⊕ W.
We can compute the Minkowski sum via projection or vertex enumeration as ex-
plained in Section 4.4.9. The set Suc(X , W) in H-representation is
    

 1 −0.5 4  

 0 16

  −1   


 −1   6 
 0   
Suc(X , W) = x :  x≤  ,

 −1 0.5 
  4 

   16 

 0 1 

 
1 0 6
and is depicted in Fig. 10.10.

10.3.2 Linear Systems with Additive Uncertainty and Inputs

Example 10.9 Consider the second order unstable system

    
1.5 0 1
x(t + 1) = Ax + Bu = x(t) + u(t) + w(t) (10.44)
1 −1.5 0
200 10 Controllability, Reachability and Invariance

10

)
,W
x2
0

c(X
Su
X
-10

-10 0 10
x1

Figure 10.10 Example 10.8. Robust successor set for system (10.34) under
constraints (10.36).

subject to the input and state constraints

u(t) ∈ U = {u : − 5 ≤ u ≤ 5} , ∀t ≥ 0 (10.45a)
    
−10 10
x(t) ∈ X = x : ≤x≤ , ∀t ≥ 0, (10.45b)
−10 10

where
w(t) ∈ W = {w : − 1 ≤ w ≤ 1} , ∀t ≥ 0. (10.46)

For the non-autonomous system (10.44), the set Pre(X , W) can be computed using
the H-presentation of X and U,

X = {x : Hx ≤ h}, U = {u : Hu u ≤ hu }, (10.47)

to obtain

Pre(X , W) = x ∈ R2 : ∃u ∈ U s.t. Ax + Bu + w ∈ X , ∀ w ∈ W (10.48a)
      
HA HB x h − Hw
= x ∈ R2 : ∃u ∈ R s.t. ≤ , ∀w∈W .
0 Hu u hu
(10.48b)

As in Example 10.8, the set Pre(X , W) can be compactly written as

      
2 HA HB x h̃
Pre(X , W) = x ∈ R : ∃u ∈ R s.t. ≤ , (10.49)
0 Hu u hu
where
h̃i = min (hi − Hi w). (10.50)
w∈W

In general, a linear program is required to solve problems

  (10.50). In this example
9
9
Hi and W have simple expressions and we get h̃ =  
9.
9
The halfspaces in (10.49) define a polytope in the state-input space, and a projection
operation (see Section 4.4.6) is used to derive the halfspaces which define Pre(X , W)
10.3 Robust Controllable and Reachable Sets 201

in the state space. The set Pre(X , W) ∩ X is depicted in Fig. 10.11 and reported
below:    
1 0 9.3
−1 0  9.3
   
1 −1.5  
 x ≤  9 .
−1 1.5  9
   
0 1   10 
0 −1 10

10
5
x2

0 Pre(X , W) ∩ X
-5
X
-10
-10 -5 0 5 10
x1

Figure 10.11 Example 10.9. One-step robust controllable set Pre(X , W) ∩ X

for system (10.44) under constraints (10.45)-(10.46).

Note that by using the definition of a Minkowski sum given in Section 4.4.9
and the affine operation on polyhedra in Section 4.4.11 we can compactly write the
operations in (10.48) as follows:

Pre(X , W) = {x : ∃u ∈ U s.t. Ax + Bu + w ∈ X , ∀ w ∈ W}
= {x : ∃y ∈ X , ∃u ∈ U s.t. y = Ax + Bu + w, ∀ w ∈ W}
= {x : ∃y ∈ X , ∃u ∈ U s.t. Ax = y + (−Bu) − w, ∀ w ∈ W}
= {x : Ax ∈ C and C = X ⊕ (−B) ◦ U ⊖ W}
= {x : x ∈ C ◦ A, C = X ⊕ (−B) ◦ U ⊖ W}
= {x : x ∈ ((X ⊖ W) ⊕ (−B ◦ U)) ◦ A} .
(10.51)

Remark 10.8 Note that in (10.51) we have used the fact that if a set S is described
as S = {v : ∃z ∈ Z, s.t. v = z − w, ∀ w ∈ W}, then S = {v : ∃z ∈ Z, s.t. z =
v + w, ∀ w ∈ W} or S = {v : v + w ∈ Z, ∀ w ∈ W} = Z ⊖ W. Also, to derive
the last equation of (10.51) we have used the associative property of the Pontryagin
difference.

The set Suc(X , W) = {y : ∃x ∈ X , ∃u ∈ U, ∃w ∈ W s.t. y = Ax + Bu + w}

is obtained by applying the map A to the set X and then considering the effect
of the input u ∈ U and of the disturbance w ∈ W. We can use the definition of
Minkowski sum given in Section 4.4.9 and rewrite Suc(X , W) as

Suc(X , W) = (A ◦ X ) ⊕ (B ◦ U) ⊕ W.
202 10 Controllability, Reachability and Invariance

30
20
10

x2
0 Suc(X , W)
-10
-20
-30
-30 -20 -10 0 10 20 30
x1

Figure 10.12 Example 10.9. Robust successor set for system (10.44) under
constraints (10.45)-(10.46).

The set Suc(X , W) is depicted in Fig. 10.12.

In summary, for linear systems with additive disturbances the sets Pre(X , W)
and Suc(X , W) are the results of linear operations on the polytopes X , U and
W and therefore are polytopes. By using the definition of Minkowski sum given
in Section 4.4.9, Pontryagin difference given in Section 4.4.8 and affine operation
on polyhedra in Section 4.4.11 we can compactly summarize the operations in
Table 10.2. Note that the summary in Table 10.2 applies also to the class of
˜ where d˜ ∈ W̃. This can be transformed
systems x(k + 1) = Ax(t) + Bu(t) + E d(t)
into x(k + 1) = Ax(t) + Bu(t) + w(t) where w ∈ W = E ◦ W̃.

x(t + 1) = Ax(t) + w(t) x(t + 1) = Ax(t) + Bu(t) + w(t)

Pre(X , W) (X ⊖ W) ◦ A ((X ⊖ W) ⊕ (−B ◦ U)) ◦ A
Suc(X , W) (A ◦ X ) ⊕ W (A ◦ X ) ⊕ (B ◦ U) ⊕ W

Table 10.2 Pre and Suc operations for uncertain linear systems subject to
polyhedral input and state constraints x(t) ∈ X , u(t) ∈ U with additive
polyhedral disturbances w(t) ∈ W

The N -step robust controllable set KN (S, W) and the N -step robust reachable
set RN (X0 , W) can be computed by using their recursive formulas (10.32), (10.33)
and computing the Pre and Suc operations as described in Table 10.2.

10.3.3 Linear Systems with Parametric Uncertainty

The next Lemma 10.1 will help us computing Pre and Suc sets for linear systems
with parametric uncertainty.

Lemma 10.1 Let g : Rnz × Rn × Rnw → Rng be a function of (z, x, w) convex in w

for each (z, x). Assume that the variable w belongs to the polytope W with vertices
{w̄i }ni=1
W
. Then, the constraint
g(z, x, w) ≤ 0 ∀w ∈ W (10.52)
10.3 Robust Controllable and Reachable Sets 203

is satisfied if and only if

g(z, x, w̄i ) ≤ 0, i = 1, . . . , nW . (10.53)

Proof: Easily follows from the fact that the maximum of a convex function
over a compact convex set is attained at an extreme point of the set. 
Lemma 10.2 shows how to reduce the number of constraints in (10.53) for a
specific class of constraint functions.

Lemma 10.2 Assume g(z, x, w) = g 1 (z, x) + g 2 (w). Then the constraint (10.52)
 ′
can be replaced by g 1 (z, x) ≤ −ḡ, where ḡ = ḡ1 , . . . , ḡng is a vector whose i-th
component is
ḡi = max gi2 (w), (10.54)
w∈W

and gi2 (w) denotes the i-th component of g 2 (w).

Example 10.10 Consider the second order autonomous system

 
0.5 + wp (t) 0
x(t + 1) = A(wp (t))x(t) + wa (t) = x(t) + wa (t)
1 −0.5
(10.55)
subject to the constraints
    
−10 10
x(t) ∈ X = x : ≤x≤ , ∀t ≥ 0
 −10
  10  
a a a −1 a 1 (10.56)
w (t) ∈ W = w : ≤w ≤ , ∀t ≥ 0
−1 1
wp (t) ∈ W p = {wp : 0 ≤ wp ≤ 0.5} , ∀t ≥ 0.

Let w = [wa ; wp ] and W = W a × W p . The set Pre(X , W) can be obtained as follows.

The set X is a polytope and it can be represented as an H-polytope (Section 4.2)

X = {x : Hx ≤ h}, (10.57)

where    
1 0 10
0 1  

H=  and h = 10 .
−1 0 10
0 −1 10

By using this H-presentation and the system equation (10.55), the set Pre(X , W) can
be rewritten as

Pre(X , W) = x : Hga (x, w) ≤ h, ∀w ∈ W (10.58)
= {x : HA(wp )x ≤ h − Hwa , ∀wa ∈ W a , wp ∈ W p } .
(10.59)

By using Lemmas 10.1 and 10.2, the set (10.59) can be rewritten as a the polytope
   
HA(0) h̃
x ∈ Pre(X , W) = {x ∈ Rn : x≤ } (10.60)
HA(0.5) h̃
204 10 Controllability, Reachability and Invariance

with
h̃i = amin a (hi − Hi wa ), i = 1, . . . , 4. (10.61)
w ∈W

The set Pre(X , W) ∩ X is depicted in Fig. 10.13 and reported below:

   
1 0 9
 0.89 −0.44  
 8.049
 1 0   9 
 x≤ 
−0.89 0.44  8.049 .
   
 0 1   10 
0 −1 10

10
5
x2

0 Pre(X , W) ∩ X
-5
X
-10
-10 -5 0 5 10
x1

Figure 10.13 Example 10.10. One-step robust controllable set Pre(X , W)∩X
for system (10.55) under constraints (10.56).

Example 10.11 Consider the second order autonomous system

 
0.5 + wp (t) 0
x(t + 1) = A(wp (t))x(t) = x(t) (10.62)
1 −0.5

subject to the constraints

    
−10 10
x(t) ∈ X = x : ≤x≤ , ∀t ≥ 0
−10 10 (10.63)
p p p p
w (t) ∈ W = {w : 0 ≤ w ≤ 0.5} , ∀t ≥ 0.

Let w = [wp ] and W = W p . The set Suc(X , W) can be written as infinite union of
reachable sets
[
Suc(X , W) = Suc(X , w̄) (10.64)
w̄∈W

where Suc(X , w̄) is computed as described in Example 10.1 for the system A(w̄). In
general, the union in equation (10.64) generates a non-convex set, as can be seen
in Fig. 10.14. Non convexity of successor sets for parametric linear systems is also
discussed in [59][Sec. 6.1.2].
 10.4 Robust Invariant Sets 205

10

)
,W
x2
0

c(X
Su
-10

-10 0 10
x1

Figure 10.14 Example 10.11. Robust successor set Suc(X , W) for system
(10.62) under constraints (10.63).

10.4 Robust Invariant Sets

Two different types of sets are considered in this chapter: robust invariant sets and
robust control invariant sets. We will first discuss robust invariant sets.

Robust Positive Invariant Sets

Robust invariant sets are computed for autonomous systems. These types of sets
are useful to answer questions such as: “For a given feedback controller u = f (x),
find the set of states whose trajectory will never violate the system constraints for
all possible disturbances”. The following definitions introduce the different types
of robust invariant sets.

Definition 10.20 (Robust Positive Invariant Set) A set O ⊆ X is said to be

a robust positive invariant set for the autonomous system (10.27) subject to the
constraints (10.29), if

x(0) ∈ O ⇒ x(t) ∈ O, ∀w(t) ∈ W, t ∈ N+ .

Definition 10.21 (Maximal Robust Positive Invariant Set O∞ ) The set O∞ ⊆

X is the maximal robust invariant set of the autonomous system (10.27) subject
to the constraints (10.29) if O∞ is a robust invariant set and O∞ contains all the
robust positive invariant sets contained in X .

Theorem 10.2 (Geometric condition for invariance) A set O ⊆ X is a ro-

bust positive invariant set for the autonomous system (10.27) subject to the con-
straints (10.29), if and only if

O ⊆ Pre(O, W). (10.65)

The proof of Theorem 10.2 follows the same lines of the proof of Theorem 10.1. 
206 10 Controllability, Reachability and Invariance

It is immediate to prove that condition (10.65) of Theorem 10.2 is equivalent

to the following condition
Pre(O, W) ∩ O = O. (10.66)
Based on condition (10.66), the following algorithm provides a procedure for com-
puting the maximal robust positive invariant subset O∞ for system (10.27)-(10.29)
(for reference to proofs and literature see Section 10.1).

Algorithm 10.4 Computation of O∞

Input: ga , X , W
Output: O∞
Ω0 ← X , k ← −1
Repeat
k ←k+1
Ωk+1 ← Pre(Ωk , W) ∩ Ωk
Until Ωk+1 = Ωk
O∞ ← Ωk

Algorithm 10.4 generates the set sequence {Ωk } satisfying Ωk+1 ⊆ Ωk , ∀k ∈ N and
it terminates when Ωk+1 = Ωk . If it terminates, then Ωk is the maximal robust
positive invariant set O∞ for system (10.27)-(10.29). If Ωk = ∅ for some integer k
then the simple conclusion is that O∞ = ∅.
In general, Algorithm 10.4 may never terminate. If the algorithm does not
terminate in a finite number of iterations, it can be proven that [179]
O∞ = lim Ωk .
k→+∞

Conditions for finite time termination of Algorithm 10.4 can be found in [124]. A
simple sufficient condition for finite time termination of Algorithm 10.1 requires the
system ga (x, w) to be linear and stable, and the constraint set X and disturbance
set W to be bounded and to contain the origin.

Example 10.12 Consider the second order stable system in Example 10.8
 
0.5 0
x(t + 1) = Ax(t) + w(t) = x(t) + w(t) (10.67)
1 −0.5
subject to the constraints
    
−10 10
x(t) ∈ X = x : ≤x≤ , ∀t ≥ 0
 −10   10 
(10.68)
−1 1
w(t) ∈ W = w : ≤w≤ , ∀t ≥ 0.
−1 1
The maximal robust positive invariant set of system (10.67) subject to constraints (10.68)
is depicted in Fig. 10.15 and reported below:
   
0.89 −0.44 8.04
 −0.89 0.44   8.04 
   
 −1 0   
  x ≤  10  .
 0 −1   10 
   
 1 0   10 
0 1 10
10.4 Robust Invariant Sets 207

10
5

x2
0 O∞
-5
X
-10
-10 -5 0 5 10
x1

Figure 10.15 Example 10.12. Maximal Robust Positive Invariant Set of

system (10.67) subject to constraints (10.68).

Robust Control Invariant Sets

Robust control invariant sets are defined for systems subject to controllable inputs.
These types of sets are useful to answer questions such as: “Find the set of states for
which there exists a controller such that the system constraints are never violated
for all possible disturbances”. The following definitions introduce the different
types of robust control invariant sets.

Definition 10.22 (Robust Control Invariant Set) A set C ⊆ X is said to be a

robust control invariant set for the system (10.28) subject to the constraints (10.29),
if

x(t) ∈ C ⇒ ∃u(t) ∈ U such that g(x(t), u(t), w(t)) ∈ C, ∀ w(t) ∈ W, ∀ t ∈ N+ .

Definition 10.23 (Maximal Robust Control Invariant Set C∞ ) The set C∞ ⊆

X is said to be the maximal robust control invariant set for the system (10.28) sub-
ject to the constraints (10.29), if it is robust control invariant and contains all
robust control invariant sets contained in X .

Remark 10.9 The geometric conditions for invariance (10.65)-(10.66) hold for control
invariant sets.

The following algorithm provides a procedure for computing the maximal robust
control invariant set C∞ for system (10.28)-(10.29).

Algorithm 10.5 Computation of C∞

Input: g, X , U, W
Output: C∞
208 10 Controllability, Reachability and Invariance

Ω0 ← X , k ← −1
Repeat
k ←k+1
Ωk+1 ← Pre(Ωk , W) ∩ Ωk
Until Ωk+1 = Ωk
C∞ ← Ωk+1

Algorithm 10.5 generates the set sequence {Ωk } satisfying Ωk+1 ⊆ Ωk , ∀k ∈ N.

Algorithm 10.5 terminates when Ωk+1 = Ωk . If it terminates, then Ωk is the
maximal robust control invariant set C∞ for the system (10.28)-(10.29).
In general, Algorithm 10.5 may never terminate [10, 49, 172, 164]. If the algo-
rithm does not terminate in a finite number of iterations, in general, convergence
to the maximal robust control invariant set is not guaranteed
C∞ 6= lim Ωk . (10.69)
k→+∞

The work in [50] reports examples of nonlinear systems where (10.69) can be ob-
served. A sufficient condition for the convergence of Ωk to C∞ as k → +∞ requires
the polyhedral sets X , U and W to be bounded and the system g(x, u, w) to be
continuous [50].

Example 10.13 Consider the second order unstable system in Example 10.9. The
maximal robust control invariant set of system (10.44) subject to constraints (10.45)-
(10.46) is an empty set. If the uncertain set (10.46) is replaced with
w(t) ∈ W = {w : − 0.1 ≤ w ≤ 0.1} , ∀t ≥ 0
the maximal robust control invariant set is
   
0 1 3.72
 0 −1   3.72 
 x ≤  
 0.55 −0.83   2.0 
−0.55 0.83 2.0
which is depicted in Fig. 10.16.

Definition 10.24 (Finitely determined set) Consider Algorithm 10.4 (10.5).

The set O∞ (C∞ ) is finitely determined if and only if ∃ i ∈ N such that Ωi+1 = Ωi .
The smallest element i ∈ N such that Ωi+1 = Ωi is called the determinedness index.
For all states contained in the maximal robust control invariant set C∞ there
exists a control law, such that the system constraints are never violated for all
feasible disturbances. This does not imply that there exists a control law which
can drive the state into a user-specified target set. This issue is addressed in the
following by introducing the concept of robust controllable and stabilizable sets.

Definition 10.25 (Maximal Robust Controllable Set K∞ (O, W)) For a given
target set O ⊆ X , the maximal robust controllable set K∞ (O, W) for the system
(10.28) subject to the constraints (10.29) is the union of all N -step robust control-
lable sets contained in X for N ∈ N.
10.4 Robust Invariant Sets 209

10
5

x2
0 C∞
-5
X
-10
-10 -5 0 5 10
x1

Figure 10.16 Example 10.13. Maximal Robust Control Invariant Set of

system (10.44) subject to constraints (10.45).

Robust controllable sets KN (O, W) where the target O is a robust control invariant
set are special sets, since in addition to guaranteeing that from KN (O, W) we
robustly reach O in N steps, one can ensure that once reached O, the system can
stay there at all future time instants and for all possible disturbance realizations.

Definition 10.26 (N -step (Maximal) Robust Stabilizable Set) For a given

robust control invariant set O ⊆ X , the N -step (maximal) robust stabilizable set
of the system (10.28) subject to the constraints (10.29) is the N -step (maximal)
robust controllable set KN (O, W) (K∞ (O, W)).

The set K∞ (O, W) contains all states which can be robustly steered into the
robust control invariant set O and hence K∞ (O, W) ⊆ C∞ . The set K∞ (O, W) ⊆
C∞ can be computed as follows:

Algorithm 10.6 Computation of K∞ (O, W)

Input: ga , X , W
Output: K∞ (O, W)
K0 ← O, where O is a robust control invariant set
c ← −1
Repeat
c←c+1
T
Kc+1 ← Pre(Kc , W) X
Until Kc+1 = Kc
K∞ (O, W) = Kc

Since O is robust control invariant, it holds ∀c ∈ N that Kc is robust control

invariant and Kc ⊆ Kc+1 . Note that Algorithm 10.6 is not guaranteed to terminate
in finite time.
210 10 Controllability, Reachability and Invariance
 Chapter 11

Constrained Optimal Control

In this chapter we study the finite time and infinite time optimal control problem for
linear systems with linear constraints on inputs and state variables. We establish
the structure of the optimal control law and derive algorithms for its computation.
For finite time problems with linear and quadratic objective functions we show that
the time varying feedback law is piecewise affine and continuous. The value function
is a convex piecewise linear for linear objective functions and convex piecewise
quadratic for quadratic objective functions.
We describe how the optimal control action for a given initial state can be
computed by means of linear or quadratic programming. We also describe how
the optimal control law can be computed by means of multiparametric linear or
quadratic programming. Finally we show how to compute the infinite time optimal
controller for linear and quadratic objective functions and prove that, when it
exists, the infinite time controller inherits all the structural properties of the finite
time optimal controller.

11.1 Problem Formulation

Consider the linear time-invariant system

x(t + 1) = Ax(t) + Bu(t), (11.1)

where x(t) ∈ Rn , u(t) ∈ Rm are the state and input vectors, respectively, subject
to the constraints
x(t) ∈ X , u(t) ∈ U, ∀t ≥ 0. (11.2)
The sets X ⊆ Rn and U ⊆ Rm are polyhedra.

Remark 11.1 The results of this chapter also hold for more general forms of linear
constraints such as mixed input and state constraints

[x(t)′ , u(t)′ ] ∈ Px,u , (11.3)

212 11 Constrained Optimal Control

where Px,u is a polyhedron in Rn+m of mixed input and state constraints over a finite
time, or, even more general, constraints of the type:
[x(0)′ , . . . , x(N − 1)′ , u(0)′ , . . . , u(N − 1)′ ] ∈ Px,u,N , (11.4)
where Px,u,N is a polyhedron in RN(n+m) . Note that constraints of the type (11.4)
can arise, for example, from constraints on the input rate ∆u(t) = u(t) − u(t − 1). In
this chapter, for the sake of simplicity, we will use the less general form (11.2).

Define the cost function

N
X −1
J0 (x(0), U0 ) = p(xN ) + q(xk , uk ), (11.5)
k=0

where xk denotes the state vector at time k obtained by starting from the state
x0 = x(0) and applying to the system model
xk+1 = Axk + Buk (11.6)
the input sequence u0 , . . . , uk−1 .
If the 1-norm or ∞-norm is used in the cost function (11.5), then we set p(xN ) =
kP xN kp and q(xk , uk ) = kQxk kp + kRuk kp with p = 1 or p = ∞ and P , Q, R full
column rank matrices. Cost (11.5) is rewritten as
N
X −1
J0 (x(0), U0 ) = kP xN kp + kQxk kp + kRuk kp . (11.7)
k=0

If the squared Euclidian norm is used in the cost function (11.5), then we set
p(xN ) = x′N P xN and q(xk , uk ) = x′k Qxk + u′k Ruk with P  0, Q  0 and R ≻ 0.
Cost (11.5) is rewritten as
N
X −1
J0 (x(0), U0 ) = x′N P xN + x′k Qxk + u′k Ruk . (11.8)
k=0

Consider the constrained finite time optimal control problem (CFTOC)

J0∗ (x(0)) = minU0 J0 (x(0), U0 )
subj. to xk+1 = Axk + Buk , k = 0, . . . , N − 1
xk ∈ X , uk ∈ U, k = 0, . . . , N − 1 (11.9)
xN ∈ Xf
x0 = x(0),
where N is the time horizon and Xf ⊆ Rn is a terminal polyhedral region. In (11.5)–
(11.9) U0 = [u′0 , . . . , u′N −1 ]′ ∈ Rs , s = mN is the optimization vector. We denote
with X0 ⊆ X the set of initial states x(0) for which the optimal control prob-
lem (11.5)–(11.9) is feasible, i.e.,
X0 = {x0 ∈ Rn : ∃(u0 , . . . , uN −1 ) such that xk ∈ X , uk ∈ U, k = 0, . . . , N − 1,
xN ∈ Xf where xk+1 = Axk + Buk , k = 0, . . . , N − 1}.
(11.10)
11.1 Problem Formulation 213

Remark 11.2 Note that we distinguish between the current state x(k) of system (11.1)
at time k and the variable xk in the optimization problem (11.9), that is the predicted
state of system (11.1) at time k obtained by starting from the state x0 = x(0) and
applying to system (11.6) the input sequence u0 , . . . , uk−1 . Analogously, u(k) is the
input applied to system (11.1) at time k while uk is the k-th optimization variable of
the optimization problem (11.9).

If we use cost (11.8) with the squared Euclidian norm and set

{(x, u) ∈ Rn+m : x ∈ X , u ∈ U} = Rn+m , Xf = Rn , (11.11)

problem (11.9) becomes the standard unconstrained finite time optimal control
problem (Chapter 8) whose solution (under standard assumptions on A, B, P , Q
and R) can be expressed through the time varying state feedback control law (8.28)

u∗ (k) = Fk x(k) k = 0, . . . , N − 1. (11.12)

From (8.25) the optimal cost is given by

J0∗ (x(0)) = x(0)′ P0 x(0). (11.13)

If we let N → ∞ as discussed in Section 8.5, then problem (11.8), (11.9), (11.11)

becomes the standard infinite horizon linear quadratic regulator (LQR) problem
whose solution (under standard assumptions on A, B, P , Q and R) can be expressed
as the state feedback control law (see (8.33))

u∗ (k) = F∞ x(k), k = 0, 1, . . . (11.14)

In the following chapters we will show that the solution to problem (11.9) can
again be expressed in feedback form where now u∗ (k) is a continuous piecewise
affine function on polyhedra of the state x(k), i.e., u∗ (k) = fk (x(k)) where

fk (x) = Fkj x + gkj if Hkj x ≤ Kkj , j = 1, . . . , Nkr . (11.15)

Matrices Hkj and Kkj in equation (11.15) describe the j-th polyhedron CRkj = {x ∈
Rn : Hkj x ≤ Kkj } inside which the feedback optimal control law u∗ (k) at time
k has the affine form Fkj x + gkj . The set of polyhedra CRkj , j = 1, . . . , Nkr is a
polyhedral partition of the set of feasible states Xk of problem (11.9) at time k. The
sets Xk are discussed in detail in the next section. Since the functions fk (x(k)) are
continuous, the use of polyhedral partitions rather than strict polyhedral partitions
(Definition 4.5) will not cause any problem, indeed it will simplify the exposition.
In the rest of this chapter we will characterize the structure of the value function
and describe how the optimal control law can be efficiently computed by means of
multiparametric linear and quadratic programming. We will distinguish the cases
1- or ∞-norm and squared 2-norm.
214 11 Constrained Optimal Control

11.2 Feasible Solutions

We denote with Xi the set of states xi at time i for which (11.9) is feasible, for
i = 0, . . . , N . The sets Xi for i = 0, . . . , N play an important role in the solution
of (11.9). They are independent of the cost function (as long as it guarantees the
existence of a minimum) and of the algorithm used to compute the solution to
problem (11.9). There are two ways to rigourously define and compute the sets Xi :
the batch approach and the recursive approach. In the batch approach

Xi = {xi ∈ X : ∃(ui , . . . , uN −1 ) such that xk ∈ X , uk ∈ U, k = i, . . . , N − 1,

xN ∈ Xf where xk+1 = Axk + Buk , k = i, . . . , N − 1}.
(11.16)
The definition of Xi in (11.16) requires that for any initial state xi ∈ Xi there exists
a feasible sequence of inputs Ui = [u′i , . . . , u′N −1 ] which keeps the state evolution
in the feasible set X at future time instants k = i + 1, . . . , N − 1 and forces xN
into Xf at time N . Clearly XN = Xf . Next we show how to compute Xi for
i = 0, . . . , N − 1. Let the state and input constraint sets X , Xf and U be the H-
polyhedra Ax x ≤ bx , Af xN ≤ bf , Au u ≤ bu , respectively. Define the polyhedron
Pi for i = 0, . . . , N − 1 as follows

Pi = {(Ui , xi ) ∈ Rm(N −i)+n : Gi Ui − Ei xi ≤ wi }, (11.17)

where Gi , Ei and wi are defined as follows

     
Au 0 ... 0 0 bu
 0 A u ... 0   0   bu 
     
 .. .. .. ..   ..   .. 
 . . . .   .   . 
     
 0 0 . . . A   0   bu 
 u     
Gi = 
 0 0 ... 0  , Ei = 
 −Ax

 , wi =  bx
 
.

 Ax B 0 ... 0  −Ax A 
  bx 
     
 Ax AB A B ... 0  −Ax A2 
  bx 
 x     
 . . .. ..   ..   . 
 .. .. . .   .   .. 
Af AN −i−1 B Af AN −i−2 B . . . Af B −Af A N −i
bf
(11.18)
The set Xi is a polyhedron as it is the projection of the polyhedron Pi in (11.17)-
(11.18) on the xi space.
In the recursive approach,

Xi = {x ∈ X : ∃u ∈ U such that Ax + Bu ∈ Xi+1 },

i = 0, . . . , N − 1
XN = Xf . (11.19)

The definition of Xi in (11.19) is recursive and requires that for any feasible initial
state xi ∈ Xi there exists a feasible input ui which keeps the next state Axi + Bui
in the feasible set Xi+1 . It can be compactly written as

Xi = Pre(Xi+1 ) ∩ X . (11.20)
11.2 Feasible Solutions 215

Initializing XN to Xf and solving (11.19) backward in time yields the same sets Xi
as the batch approach. This recursive formulation, however, leads to an alternative
approach for computing the sets Xi . Let Xi be the H-polyhedra AXi x ≤ bXi . Then
the set Xi−1 is the projection of the following polyhedron
     
Au 0 bu
 0  ui +  Ax  xi ≤  bx  (11.21)
AXi B AXi A bXi
on the xi space.
Consider problem (11.9). The set X0 is the set of all initial states x0 for
which (11.9) is feasible. The sets Xi with i = 1, . . . , N − 1 are hidden. A
given Ū0 = [ū0 , . . . , ūN −1 ] is feasible for problem (11.9) if and only if at all time
instants i, the state xi obtained by applying ū0 , . . . , ūi−1 to the system model
xk+1 = Axk + Buk with initial state x0 ∈ X0 belongs to Xi . Also, Xi is the set of
feasible initial states for problem
PN −1
Ji∗ (x(0)) = minUi p(xN ) + k=i q(xk , uk )
subj. to xk+1 = Axk + Buk , k = i, . . . , N − 1
(11.22)
xk ∈ X , uk ∈ U, k = i, . . . , N − 1
xN ∈ Xf .
Next, we provide more insights into the set Xi by using the invariant set theory
of Section 10.2. We consider two cases: (1) Xf = X which corresponds to effectively
“removing” the terminal constraint set and (2) Xf chosen to be a control invariant
set.

Theorem 11.1 [172, Theorem 5.3]. Let the terminal constraint set Xf be equal
to X . Then,
1. The feasible set Xi , i = 0, . . . , N − 1 is equal to the (N − i)-step controllable
set:
Xi = KN −i (X ).
2. The feasible set Xi , i = 0, . . . , N − 1 contains the maximal control invariant
set:
C∞ ⊆ Xi .
3. The feasible set Xi is control invariant if and only if the maximal control
invariant set is finitely determined and N − i is equal to or greater than its
determinedness index N̄ , i.e.
Xi ⊆ P re(Xi ) ⇔ C∞ = KN −i (X ) for all i ≤ N − N̄ .

4. Xi ⊆ Xj if i < j for i = 0, . . . , N − 1. The size of the feasible set Xi stops

decreasing (with decreasing i) if and only if the maximal control invariant set
is finitely determined and N − i is larger than its determinedness index, i.e.
Xi ⊂ Xj if N − N̄ < i < j < N.
Furthermore,
Xi = C∞ if i ≤ N − N̄ .
216 11 Constrained Optimal Control

Theorem 11.2 [172, Theorem 5.4]. Let the terminal constraint set Xf be a control
invariant subset of X . Then,
1. The feasible set Xi , i = 0, . . . , N − 1 is equal to the (N − i)-step stabilizable
set:
Xi = KN −i (Xf ).

2. The feasible set Xi , i = 0, . . . , N − 1 is control invariant and contained within

the maximal control invariant set:

Xi ⊆ C∞ .

3. Xi ⊇ Xj if i < j, i = 0, . . . , N − 1. The size of the feasible Xi set stops

increasing (with decreasing i) if and only if the maximal stabilizable set is
finitely determined and N − i is larger than its determinedness index, i.e.

Xi ⊃ Xj if N − N̄ < i < j < N.

Furthermore,
Xi = K∞ (Xf ) if i ≤ N − N̄ .

Remark 11.3 Theorems 11.1 and 11.2 help us understand how the feasible sets Xi
propagate backward in time as a function of the terminal set Xf . In particular,
when Xf = X the set Xi shrinks as i becomes smaller and stops shrinking when it
becomes the maximal control invariant set. Also, depending on i, either it is not a
control invariant set or it is the maximal control invariant set. We have the opposite
if a control invariant set is chosen as terminal constraint Xf . The set Xi grows as
i becomes smaller and stops growing when it becomes the maximal stabilizable set.
Both cases are shown in the Example 11.1 below.

Remark 11.4 In this section we investigated the behavior of Xi as i varies for a

fixed horizon N . Equivalently, we could study the behavior of X0 as the horizon N
varies. Specifically, the sets X0→N1 and X0→N2 with N2 > N1 are equal to the sets
XN2 −N1 →N and X0→N , respectively, with N = N2 .

Example 11.1 Consider the double integrator

    
 1 1 0
x(t + 1) = x(t) + u(t)
  0 1  1 (11.23)
y(t) = 1 0 x(t)

subject to the input constraints

− 1 ≤ u(k) ≤ 1 for all k ≥ 0 (11.24)

and the state constraints

   
−5 5
≤ x(k) ≤ for all k ≥ 0. (11.25)
−5 5

We compute the feasible sets Xi and plot them in Figure 11.1 in two cases.
 11.2 Feasible Solutions 217

Case 1. Xf is the control invariant set

   
−0.32132 −0.94697 0.3806
 0.32132 0.94697   
  x ≤  0.3806  . (11.26)
 1 0   2.5 
−1 0 2.5

After six iterations the sets Xi converge to the following K∞ (Xf )

   
−0.44721 −0.89443 2.6833
 −0.24254 −0.97014   2.6679 
   
 −0.31623 −0.94868   2.5298 
   
 0.24254 0.97014   2.6679 
   
 0.31623 0.94868   2.5298 
 x ≤  . (11.27)
 0.44721 0.89443   2.6833 
   
 1 0   5 
   
 −1 0   5 
   
 0.70711 0.70711   3.5355 
−0.70711 −0.70711 3.5355

Note that in this case C∞ = K∞ (Xf ) and the determinedness index is six.

Case 2. Xf = X . After six iterations the sets Xi converge to K∞ (Xf ) in (11.27).

6 6
4 4
2 2
x2

x2

0 0
-2 -2
-4 -4
-6-6 -2 2 6 -6-6 -2 2 6
-4 0 4 -4 0 4
x1 x1
(a) Case 1: Xf = control invariant (b) Case 2: Xf = X .
set in (11.26).

Figure 11.1 Example 11.1. Propagation of the feasible sets Xi to arrive at

C∞ (shaded dark).
218 11 Constrained Optimal Control

11.3 2-Norm Case Solution

Consider problem (11.9) with J0 (·) defined by (11.8). In this chapter we always
assume that Q = Q′  0, R = R′ ≻ 0, P = P ′  0.

N
X −1
J0∗ (x(0)) = minU0 J0 (x(0), U0 ) = x′N P xN + x′k Qxk + u′k Ruk
k=0
subj. to xk+1 = Axk + Buk , k = 0, . . . , N − 1 (11.28)
xk ∈ X , uk ∈ U, k = 0, . . . , N − 1
xN ∈ Xf
x0 = x(0).

11.3.1 Solution via QP

As shown in Section 8.2, problem (11.28) can be rewritten as

J0∗ (x(0)) = min J0 (x(0), U0 ) = U0′ HU0 + 2x′ (0)F U0 + x′ (0)Y x(0)
U0  
= min J0 (x(0), U0 ) = [U0′ x′ (0)] H F ′ [U ′ x(0)′ ]′
F Y 0
U0

subj. to G0 U0 ≤ w0 + E0 x(0),
(11.29)
with G0 , w0 and E0 defined in (11.18) for i = 0 and  H, F , Y defined in (8.8). As
J0 (x(0), U0 ) ≥ 0 by definition it follows that H F ′  0.
F Y
For a given vector x(0) the optimal input sequence U0∗ solving problem (11.29)
can be computed by using a Quadratic Program (QP) solver (see Section 2.3 for
QP definition and properties and Chapter 3 for fast numerical methods for solving
QPs).
To obtain the problem (11.29) we have eliminated the state variables and equal-
ity constraints xk+1 = Axk + Buk by successive substitution so that we are left
with u0 , . . . , uN −1 as the only decision variables and x(0) as a parameter vector.
In general, it might be more efficient to solve a QP problem with equality and
inequality constraints so that sparsity can be exploited. To this aim we can define
the variable z̃ as
 ′
z̃ = x′1 . . . x′N u′0 . . . u′N −1

and rewrite problem (11.28) as

h i
J0∗ (x(0)) = min [z̃ ′ x(0)′ ] H̄ 0
0 Q [z̃ ′ x(0)′ ]′
z̃
subj. to G0,eq z̃ = E0,eq x(0) (11.30)
G0,in z̃ ≤ w0,in + E0,in x(0)

For a given vector x(0) the optimal input sequence U0∗ solving problem (11.30) can
be computed by using a Quadratic Program (QP) solver.
To obtain problem (11.30) we have rewritten the equalities from system dy-
 11.3 2-Norm Case Solution 219

namics xk+1 = Axk + Buk as G0,eq z̃ = E0,eq x(0) where

 
I −B  
 −A  A
 I −B   0 
 −A I −B   
G0,eq =   , E0,eq =  .. ,
 . .. . .. ..   . 
 . 
0
−A I −B

and rewritten state and input constraints as G0,in z̃ ≤ w0,in + E0,in x(0) where
 
0 0  
 Ax 0  bx
 0   bx 
 Ax 0   
   .. 
 . .. . ..   . 
   
   bx 
 Ax 0   
   bf 

G0,in =  Af 
0  , w0,in =  ,
 bu 
 0 Au   
   bu 
 0 Au   
   . 
 .. ..   .. 
 . .   
   bu 
 0 Au 
bu
0 Au
 ′
E0,in = −A′x 0 ··· 0 ,
and constructed the cost matrix H̄ as
 
Q
 .. 
 . 
 
 Q 
 
H̄ = 
 P .

 R 
 
 .. 
 . 
R

11.3.2 State Feedback Solution via Batch Approach

As shown in Section 8.2, problem (11.28) can be rewritten as
 
J0∗ (x(0)) = min J0 (x(0), U0 ) = [U0′ x′ (0)] H F ′ [U ′ x(0)′ ]′
F Y 0
U0
(11.31)
subj. to G0 U0 ≤ w0 + E0 x(0),

with G0 , w0 and E0 defined in (11.18) for i = 0 and H, F , Y defined in (8.8). As

J0 (x(0), U0 ) ≥ 0 by definition it follows that H F ′  0. Note that the optimizer
F Y
∗
U0 is independent of the term involving Y in (11.31).
220 11 Constrained Optimal Control

We view x(0) as a vector of parameters and our goal is to solve (11.31) for all
values of x(0) ∈ X0 and to make this dependence explicit. The computation of
the set X0 of initial states for which problem (11.31) is feasible was discussed in
Section 11.2.
Before proceeding further, it is convenient to define
z = U0 + H −1 F ′ x(0), (11.32)
z ∈ Rs , remove x(0)′ Y x(0) and to transform (11.31) to obtain the equivalent
problem
Jˆ∗ (x(0)) = min z ′ Hz
z (11.33)
subj. to G0 z ≤ w0 + S0 x(0),
where S0 = E0 + G0 H −1 F ′ , and Jˆ∗ (x(0)) = J0∗ (x(0))− x(0)′ (Y − F H −1 F ′ )x(0). In
the transformed problem the parameter vector x(0) appears only on the right-hand
side of the constraints.
Problem (11.33) is a multiparametric quadratic program that can be solved by
using the algorithm described in Section 6.3.1. Once the multiparametric prob-
lem (11.33) has been solved, the solution U0∗ = U0∗ (x(0)) of CFTOC (11.28) and
therefore u∗ (0) = u∗ (x(0)) is available explicitly as a function of the initial state
x(0) for all x(0) ∈ X0 .
Theorem 6.7 states that the solution z ∗ (x(0)) of the mp-QP problem (11.33) is
a continuous and piecewise affine function on polyhedra of x(0). Clearly the same
properties are inherited by the controller. The following corollaries of Theorem 6.7
establish the analytical properties of the optimal control law and of the value
function.

Corollary 11.1 The control law u∗ (0) = f0 (x(0)), f0 : Rn → Rm , obtained as a

solution of the CFTOC (11.28) is continuous and piecewise affine on polyhedra
f0 (x) = F0j x + g0j if x ∈ CR0j , j = 1, . . . , N0r , (11.34)
where the polyhedral sets CR0j = {x ∈ Rn : H0j x ≤ K0j }, j = 1, . . . , N0r are a
partition of the feasible polyhedron X0 .
Proof: From (11.32) U0∗ (x(0)) = z ∗ (x(0)) − H −1 F ′ x(0). From Theorem 6.7
we know that z ∗ (x(0)), solution of (11.33), is PPWA and continuous. As U0∗ (x(0))
is a linear combination of a linear function and a PPWA function, it is PPWA. As
U0∗ (x(0)) is a linear combination of two continuous functions it is continuous. In
particular, these properties hold for the first component u∗ (0) of U0∗ . 

Remark 11.5 Note that, as discussed in Remark 6.8, the critical regions defined
in (6.4) are in general sets that are neither closed nor open. In Corollary 11.1 the
polyhedron CR0i describes the closure of a critical region. The function f0 (x) is
continuous and therefore it is simpler to use a polyhedral partition rather than a
strict polyhedral partition.

Corollary 11.2 The value function J0∗ (x(0)) obtained as solution of the CFTOC (11.28)
is convex and piecewise quadratic on polyhedra. Moreover, if the mp-QP prob-
lem (11.33) is not degenerate, then the value function J0∗ (x(0)) is C (1) .
11.3 2-Norm Case Solution 221

 
Proof: By Theorem 6.7 Jˆ∗ (x(0)) is a convex function of x(0). As H F ′  0,
F Y
its Schur complement1 Y − F H −1 F ′  0, and therefore J0∗ (x(0)) = Jˆ∗ (x(0)) +
x(0)′ (Y − F H −1 F ′ )x(0) is a convex function, because it is the sum of convex
functions. If the mp-QP problem (11.33) is not degenerate, then Theorem 6.9
implies that Jˆ∗ (x(0)) is a C (1) function of x(0) and therefore J0∗ (x(0)) is a C (1)
function of x(0). The results of Corollary 11.1 imply that J0∗ (x(0)) is piecewise
quadratic. 

Remark 11.6 The relation between the design parameters of the optimal control
problem (11.28) and the degeneracy of the mp-QP problem (11.33) is complex, in
general.

The solution of the multiparametric problem (11.33) provides the state feed-
back solution u∗ (k) = fk (x(k)) of CFTOC (11.28) for k = 0 and it also provides
the open-loop optimal control u∗ (k) as function of the initial state, i.e., u∗ (k) =
u∗ (k, x(0)). The state feedback PPWA optimal controllers u∗ (k) = fk (x(k)) with
fk : Xk 7→ U for k = 1, . . . , N are computed in the following way. Consider the
same CFTOC (11.28) over the shortened time-horizon [i, N ]

N
X −1
minUi x′N P xN + x′k Qxk + u′k Ruk
k=i
subj. to xk+1 = Axk + Buk , k = i, . . . , N − 1 (11.35)
xk ∈ X , uk ∈ U, k = i, . . . , N − 1
xN ∈ Xf
xi = x(i),

where Ui = [u′i , . . . , u′N −1 ]. As defined in (11.16) and discussed in Section 11.2,

Xi ⊆ Rn is the set of initial states x(i) for which the optimal control problem (11.35)
is feasible. We denote by Ui∗ the optimizer of the optimal control problem (11.35).
Problem (11.35) can be translated into the mp-QP

min Ui ′ Hi Ui + 2x′ (i)Fi Ui + x′ (i)Yi x(i)

(11.36)
subj. to Gi Ui ≤ wi + Ei x(i),

where Hi = Hi′ ≻ 0, Fi , Yi are appropriately defined for each i and Gi , wi , Ei are

defined in (11.18). The first component of the multiparametric solution of (11.36)
has the form
u∗i (x(i)) = fi (x(i)), ∀x(i) ∈ Xi , (11.37)
where the control law fi : Rn → Rm , is continuous and PPWA

fi (x) = Fij x + gij if x ∈ CRij , j = 1, . . . , Nir , (11.38)

B′
 
1 Let A
X = and A ≻ 0. Then X  0 if and only if the Schur complement S =
B C
C − BA−1 B ′  0.
222 11 Constrained Optimal Control

and where the polyhedral sets CRij = {x ∈ Rn : Hij x ≤ Kij }, j = 1, . . . , Nir

are a partition of the feasible polyhedron Xi . Therefore the feedback solution
u∗ (k) = fk (x(k)), k = 0, . . . , N − 1 of the CFTOC (11.28) is obtained by solving
N mp-QP problems of decreasing size. The following corollary summarizes the
final result.

Corollary 11.3 The state feedback control law u∗ (k) = fk (x(k)), fk : Xk ⊆ Rn →

U ⊆ Rm , obtained as a solution of the CFTOC (11.28) and k = 0, . . . , N − 1 is
time-varying, continuous and piecewise affine on polyhedra
fk (x) = Fkj x + gkj if x ∈ CRkj , j = 1, . . . , Nkr , (11.39)
where the polyhedral sets CRkj = {x ∈ Rn : Hkj x ≤ Kkj }, j = 1, . . . , Nkr are a
partition of the feasible polyhedron Xk .

11.3.3 State Feedback Solution via Recursive Approach

Consider the dynamic programming formulation of the CFTOC (11.28)

Jj∗ (xj ) = min x′j Qxj + u′j Ruj + Jj+1
∗
(Axj + Buj )
uj
subj. to xj ∈ X , uj ∈ U, (11.40)
Axj + Buj ∈ Xj+1
for j = 0, . . . , N − 1, with boundary conditions
∗
JN (xN ) = x′N P xN (11.41)
XN = Xf , (11.42)
where Xj denotes the set of states x for which the CFTOC (11.28) is feasible at time
∗
j (as defined in (11.16)). Note that according to Corollary 11.2, Jj+1 (Axj +Buj ) is
piecewise quadratic for j < N − 1. Therefore (11.40) is not simply an mp-QP and,
contrary to the unconstrained case (Section 8.2), the computational advantage of
the iterative over the batch approach is not obvious. Nevertheless an algorithm
was developed and can be found in Section 17.6.

11.3.4 Infinite Horizon Problem

Assume Q ≻ 0, R ≻ 0 and that the constraint sets X and U contain the origin
in their interior2 Consider the following infinite-horizon linear quadratic regulation
problem with constraints (CLQR)
∞
X
∗
J∞ (x(0)) = minu0 ,u1 ,... x′k Qxk + u′k Ruk
k=0
subj. to xk+1 = Axk + Buk , k = 0, . . . , ∞ (11.43)
xk ∈ X , uk ∈ U, k = 0, . . . , ∞
x0 = x(0)
2 As in the unconstrained case, the assumption Q ≻ 0 can be relaxed by requiring that (Q1/2 , A)

is observable (Section 8.5).

11.3 2-Norm Case Solution 223

and the set (see Remark 7.1 for notation)

X∞ = {x(0) ∈ Rn : Problem (11.43) is feasible and J∞

∗
(x(0)) < +∞}. (11.44)

Because Q ≻ 0, R ≻ 0 any optimizer u∗k of problem (11.43) must converge to the

origin (u∗k → 0) and so must the state trajectory resulting from the application
of u∗k (x∗k → 0). Thus the origin x = 0, u = 0 must lie in the interior of the
constraint set (X , U) (if the origin were not contained in the constraint set then
∗
J∞ (x(0)) would be infinite). For this reason, the set X∞ in (11.44) is the maximal
stabilizable set K∞ (O) of system (11.1) subject to the constraints (11.2) with O
being the origin (Definition 10.13).
If the initial state x0 = x(0) is sufficiently close to the origin, then the con-
straints will never become active and the solution of problem (11.43) will yield the
same control input as the unconstrained LQR (8.33). More formally we can define
a corresponding invariant set around the origin.
LQR
Definition 11.1 (Maximal LQR Invariant Set O∞ ) Consider the system x(k+
n
1) = Ax(k) + Bu(k). O∞ ⊆ R denotes the maximal positively invariant set for
LQR

the autonomous constrained linear system:

x(k + 1) = (A + BF∞ )x(k), x(k) ∈ X , u(k) ∈ U, ∀ k ≥ 0,

where u(k) = F∞ x(k) is the unconstrained LQR control law (8.33) obtained from
the solution of the ARE (8.32).
Therefore, from the previous discussion, there is some finite time N̄ (x0 ), de-
LQR
pending on the initial state x0 , at which the state enters O∞ and after which the
∗ ∗
system evolves in an unconstrained manner (xk ∈ X , uk ∈ U, ∀k > N̄ ). This con-
sideration allows us to split problem (11.43) into two parts by using the dynamic
programming principle, one up to time k = N̄ where the constraints may be active
and one for longer times k > N̄ where there are no constraints.

N̄
X −1
∗
J∞ (x(0)) = minu0 ,u1 ,... x′k Qxk + u′k Ruk + JN̄
∗
→∞ (xN̄ )
k=0
subj. to xk ∈ X , uk ∈ U, k = 0, . . . , N̄ − 1 (11.45)
xk+1 = Axk + Buk , k ≥ 0
x0 = x(0),

where
∞
X
∗
JN̄→∞ (xN̄ ) = minuN̄ ,uN̄ +1 ,... x′k Qxk + u′k Ruk
k=N̄
subj. to xk+1 = Axk + Buk , k ≥ N̄ (11.46)

= x′N̄ P∞ xN̄ .

This key insight due to Sznaier and Damborg [272] is formulated precisely in the
following.
224 11 Constrained Optimal Control

Theorem 11.3 (Equality of Finite and Infinite Optimal Control, [260]) For
any given initial state x(0), the solution to (11.45, 11.46) is equal to the infinite
time solution of (11.43), if the terminal state xN̄ of (11.45) lies in the positive
LQR
invariant set O∞ and no terminal set constraint is applied in (11.45), i.e., the
LQR
state ‘voluntarily’ enters the set O∞ after N̄ steps.

Theorem 11.3 suggests that we can obtain the infinite horizon constrained linear
quadratic regulator CLQR by solving the finite horizon problem for a horizon of N̄
with a terminal weight of P = P∞ and no terminal constraint. The critical question
of how to determine N̄ (x0 ) or at least an upper bound was studied by several
researchers. Chmielewski and Manousiouthakis [85] presented an approach that
provides a conservative estimate Nest of the finite horizon N̄ (x0 ) for all x0 belonging
to a compact set of initial conditions S ⊆ X∞ = K∞ (0) (Nest ≥ N̄S (x0 ), ∀x0 ∈ S).
They solve a single, finite dimensional, convex program to obtain Nest . Their
estimate can be used to compute the PWA solution of (11.45) for a particular set
S.
Alternatively, the quadratic program with horizon Nest can be solved to deter-
mine u∗0 , u∗1 , . . . , u∗Nest for a particular x(0) ∈ S. For a given initial state x(0),
rather then a set S, Scokaert and Rawlings [260] presented an algorithm that at-
tempts to identify N̄ (x(0)) iteratively. In summary we can state the following
Theorem.

Theorem 11.4 (Explicit solution of CLQR) Assume that (A, B) is a stabi-

lizable pair and (Q1/2 , A) is an observable pair, R ≻ 0. The state feedback so-
lution to the CLQR problem (11.43) in a compact set of the initial conditions
S ⊆ X∞ = K∞ (0) is time-invariant, continuous and piecewise affine on polyhedra

u∗ (k) = f∞ (x(k)), f∞ (x) = F j x + g j if j

x ∈ CR∞ r
, j = 1, . . . , N∞ , (11.47)
j
where the polyhedral sets CR∞ = {x ∈ Rn : H j x ≤ K j }, j = 1, . . . , N∞
r
are a
finite partition of the feasible compact polyhedron S ⊆ X∞ .

As argued previously, the complexity of the solution manifested by the number

of polyhedral regions depends on the chosen horizon. As the various discussed
techniques yield an Nest that may be too large by orders of magnitude this is not
a viable proposition. An efficient algorithm for computing the PPWA solution to
the CLQR problem is presented next.

11.3.5 CLQR Algorithm

In this section we will sketch an efficient algorithm to compute the PWA solution
to the CLQR problem in (11.43) for a given set S of initial conditions. Details
are available in [132, 133]. As a side product, the algorithm also computes N̄S ,
the shortest horizon N̄ for which the problem (11.45), (11.46) is equivalent to the
infinite horizon problem (11.43).
The idea is as follows. For the CFTOC problem (11.28) with a horizon N with
no terminal constraint (Xf = Rn ) and terminal cost P = P∞ , where P∞ is the
11.3 2-Norm Case Solution 225

solution to the ARE (8.32), we solve an mp-QP and obtain the PWA control law.
From Theorem 11.3 we can conclude that for all states which enter the invariant set
LQR
O∞ introduced in Definition 11.1 with the computed control law in N steps, the
infinite-horizon problem has been solved. For these states, which we can identify
via a reachability analysis, the computed feedback law is infinite-horizon optimal.
In more detail, we start the procedure by computing the Maximal LQR Invari-
LQR LQR
ant Set O∞ introduced in Definition 11.1, the polyhedron P0 = O∞ = {x ∈
n LQR
R : H0 x ≤ K0 }. Figure 11.2(a) depicts O∞ . Then, the algorithm finds a point

3 3
2 2
Step ε over facet P1
1 1
x2

x2
0 LQR
O∞ 0 LQR
O∞
-1 -1
-2 -2
-3-3 -1 1 3 -3-3 -1 1 3
-2 0 2 -2 0 2
x1 x1
(a) Compute positive invariant (b) Solve QP for new point with
LQR horizon N = 1 to create the first
region O∞ and step over facet
with step-size ǫ. constrained region P1 .

3
2 Reachability subset IT P 11
1
x2

0 LQR
O∞
-1
-2
-3
-3 -2 -1 0 1 2 3
x1
(c) Compute reachability subset
of P1 to obtain IT P 11 .

Figure 11.2 CLQR Algorithm. Region Exploration.

LQR
x̄ by stepping over a facet of O∞ with a small step ǫ, as described in [19]. If
(11.28) is feasible for horizon N = 1 (terminal set constraint Xf = Rn , terminal
cost P = P∞ and x(0) = x̄), the active constraints will define the neighboring
polyhedron P1 = {x ∈ Rn : H1 x ≤ K1 } (x̄ ∈ P1 , see Figure 11.2(b)) [44]. By
Theorem 11.3, the finite time optimal solution computed above equals the infinite
LQR
time optimal solution if x1 ∈ O∞ . Therefore we extract from P1 the set of points
LQR
that will enter O∞ in N = 1 time-steps, provided that the optimal control law as-
226 11 Constrained Optimal Control

sociated with P1 (i.e., U1∗ = F1 x(0) + g1 ) is applied. The Infinite Time Polyhedron
(IT P 11 ) is therefore defined by the intersection of the following two polyhedra:
LQR
x1 ∈ O∞ , x1 = Ax0 + BU1∗ , (11.48a)
x0 ∈ P1 . (11.48b)

Equation (11.48a) is the reachability constraint and (11.48b) defines the set of
states for which the computed feedback law is feasible and optimal over N = 1
steps (see [44] for details). The intersection is the set of points for which the
control law is infinite time optimal.
A general step r of the algorithm involves stepping over a facet to a new point x̄
∗
and determining the polyhedron Pr and the associated control law (UN = Fr x(0)+
gr ) from (11.28) with horizon N. Then we extract from Pr the set of points that
LQR
will enter O∞ in N time-steps, provided that the optimal control law associated
with Pr is applied. The Infinite Time Polyhedron (IT P N r ) is therefore defined by
the intersection of the following two polyhedra:
LQR
xN ∈ O∞ (11.49a)
x0 ∈ Pr . (11.49b)

This intersection is the set of points for which the control law is infinite time
optimal. Note that, as for x1 in the one-step case, xN in (11.49a) can be described
∗
as a linear function of x0 by substituting the feedback sequence UN = Fr x0 + gr
into the LTI system dynamics (11.1).
We continue exploring the facets increasing N when necessary. The algorithm
terminates when we have covered S or when we can no longer find a new feasible
polyhedron Pr . The following theorem shows that the algorithm also provides the
horizon N̄S for compact sets. Exact knowledge of N̄S can serve to improve the
performance of a wide array of algorithms presented in the literature.

Theorem 11.5 (Exact Computation of N̄S , [132, 133]) If we explore any given
compact set S with the proposed algorithm, the largest resulting horizon is equal to
N̄S , i.e.,
N̄S = max N.
IT P N
r r=0,...,R

Often the proposed algorithm is more efficient than standard multi-parametric

solvers, even if finite horizon optimal controllers are sought. The initial polyhedral
representation Pr contains redundant constraints which need to be removed in order
to obtain a minimal representation of the controller region. The intersection with
the reachability constraint, as proposed here, can simplify this constraint removal.

11.3.6 Examples

Example 11.2 Consider the double integrator (11.23). We want to compute the
state feedback optimal controller that solves problem (11.28) with N = 6, Q = [ 10 01 ],
11.3 2-Norm Case Solution 227

R = 0.1, P is equal to the solution of the Riccati equation (8.32), Xf = R2 . The

input constraints are
− 1 ≤ u(k) ≤ 1, k = 0, . . . , 5 (11.50)
and the state constraints
   
−10 10
≤ x(k) ≤ , k = 0, . . . , 5. (11.51)
−10 10

This task is addressed as shown in Section (11.3.2). The feedback optimal solution
u∗ (0), . . . , u∗ (5) is computed by solving six mp-QP problems and the corresponding
polyhedral partitions of the state space are depicted in Fig. 11.3. Only the last two
optimal control moves are reported below:
  −0.35 −0.94
  0.61


 [ −0.58 −1.55 ] x(5) if 0.35 0.94
x(5) ≤ 0.61
(Region #1)

 1.00 0.00 10.00

 −1.00 0.00 10.00



 "

 0.35 0.94 # " −0.61 #

 0.00 −1.00 10.00
 1.00 if −0.71 −0.71 x(5) ≤ 7.07 (Region #2)
u∗ (5) = 1.00
−1.00
0.00
0.00
10.00
10.00



 " −0.35 −0.94 # " −0.61 #



 1.00 0.00 10.00

 − 1.00 if 0.00 1.00 x(5) ≤ (Region #3)


10.00


−1.00 0.00 10.00
 0.71 0.71 7.07

  −0.35 −0.94
  0.61 

 [ −0.58 −1.55 ] x(4) if 0.35 0.94
x(4) ≤ 0.61
(Region #1)

 −0.77 −0.64 2.43

 0.77 0.64 2.43



 

 0.29 0.96   −0.98 

 0.00 −1.00 10.00

  −0.71 −0.71 

 1.00 if −0.45 −0.89 x(4) ≤ 7.07
4.92
 (Region #2)

 1.00 0.00 10.00

 −1.00 0.00 10.00





  0.29 0.96   −0.37 

 0.35 0.94


−0.61

  −0.71
0.00 −1.00   10.00 

 1.00 if  −0.45 −0.71  x(4) ≤  7.07  (Region #3)

 −0.89 4.92

 1.00 0.00 10.00

 −1.00 0.00 10.00



  −0.29 −0.96   −0.98 

 1.00 0.00 10.00
∗
u (4) = − 1.00 if  −1.00
0.00 1.00  x(4) ≤  10.00  (Region #4)


0.00 10.00

 0.71 0.71 7.07

 0.45 0.89 4.92



  −0.29 −0.96   −0.37 




−0.35 −0.94 −0.61

  1.00 0.00   10.00 

 − 1.00 if  0.00 1.00  x(4) ≤  10.00  (Region #5)

 −1.00 0.00 10.00

 0.71 0.71 7.07

 0.45 0.89 4.92



  −0.29   

 −0.96 0.98

 [ −0.44 −1.43 ] x(4) − 0.46 if 0.29 0.96
x(4) ≤ 0.37
(Region #6)

 −0.77 −0.64 −2.43

 1.00 0.00 10.00



  −0.29   

 −0.96 0.37

 [ −0.44 −1.43 ] x(4) + 0.46 if 0.29 0.96
x(4) ≤ 0.98
(Region #7)

 0.77 0.64 −2.43

 −1.00 0.00 10.00
228 11 Constrained Optimal Control

10 10

5 5
x2 (0)

x2 (1)
0 0

-5 -5

-10 -10
-10 -5 0 5 10 -10 -5 0 5 10
x1 (0) x1 (1)
(a) Partition of the state space (b) Partition of the state space
for the affine control law u∗ (0) for the affine control law u∗ (1)
(N0r = 13). (N1r = 13).

10 10

5 5
x2 (2)

0 x2 (3) 0

-5 -5

-10
-10 0 10 -10
-10 0 10
-5 5 -5 5
x1 (2) x1 (3)
(c) Partition of the state space (d) Partition of the state space
for the affine control law u∗ (2) for the affine control law u∗ (3)
(N2r = 13). (N3r = 11).

10 10

5 5
x2 (4)

x2 (5)

0 0

-5 -5

-10
-10 0 10 -10
-10 0 10
-5 5 -5 5
x1 (4) x1 (5)
(e) Partition of the state space (f) Partition of the state space
for the affine control law u∗ (4) for the affine control law u∗ (5)
(N4r = 7). (N5r = 3).

Figure 11.3 Example 11.2. Double Integrator, 2-norm objective function,

horizon N = 6. Partition of the state space for the time-varying optimal
control law. Polyhedra with the same control law were merged.
 11.4 1-Norm and ∞-Norm Case Solution 229

Note that by increasing the horizon N , the control law changes only far away from the
origin, the larger N the more in the periphery. This must be expected from the results
of Section 11.3.5. The control law does not change anymore with increasing N in the
set where the CFTOC law becomes equal to the constrained infinite-horizon linear
quadratic regulator (CLQR) problem. This set gets larger as N increases [85, 260].

Example 11.3 The infinite time CLQR (11.43) was determined for Example 11.2
by using the approach presented in Section 11.3.4. The resulting N̄S is 12. The
state space is divided into 117 polyhedral regions and is depicted in Fig. 11.4(a). In
Fig. 11.4(b) the same control law is represented where polyhedra with the same affine
control law were merged.

10 10

5 5
x2

x2

0 0

-5 -5

-10 -5 5 -10 -5 5
-10 0 10 -10 0 10
x1 x1
(a) Partition before merging (b) Partition after merging
r r
(N∞ = 117). (N∞ = 13).

Figure 11.4 Example 11.3. Double Integrator, 2-norm objective function,

horizon N = ∞. Partition of the state space for the time invariant optimal
control law.

11.4 1-Norm and ∞-Norm Case Solution

Next we consider problem (11.9) with J0 (·) defined by (11.7) with p = 1 or p = ∞.

In the following section we will concentrate on the ∞-norm, the results can be
extended easily to cost functions with 1-norm or mixed 1/∞ norms.
230 11 Constrained Optimal Control

N
X −1
J0∗ (x(0)) = minU0 J0 (x(0), U0 ) = kP xN kp + kQxk kp + kRuk kp
k=0
subj. to xk+1 = Axk + Buk , k = 0, . . . , N − 1
xk ∈ X , uk ∈ U, k = 0, . . . , N − 1
xN ∈ Xf
x0 = x(0).
(11.52)

11.4.1 Solution via LP

The optimal control problem (11.52) with p = ∞ can be rewritten as a linear
program by using the approach presented in Section 9.2. Therefore, problem (11.52)
can be reformulated as the following LP problem

min εx0 + . . . + εxN + εu0 + . . . + εuN −1 (11.53a)

z0
 
k−1
X
subj. to −1n εxk ≤ ±Q Ak x0 + Aj Buk−1−j  , (11.53b)
j=0
 
N
X −1
−1r εxN ≤ ±P AN x0 + Aj BuN −1−j  , (11.53c)
j=0

−1m εuk ≤ ±Ruk , (11.53d)

k−1
X
Ak x0 + Aj Buk−1−j ∈ X , uk ∈ U, (11.53e)
j=0
N
X −1
AN x0 + Aj BuN −1−j ∈ Xf , (11.53f)
j=0
k = 0, . . . , N − 1
x0 = x(0), (11.53g)

where constraints (11.53b)–(11.53f) are componentwise, and ± means that the

constraint appears once with each sign.
Problem (11.53) can be rewritten in the more compact form

min c′0 z0
z0 (11.54)
subj. to Ḡ0 z0 ≤ w̄0 + S̄0 x(0),

where z0 = {εx0 , . . . , εxN , εu0 , . . . , εuN −1 , u′0 , . . . , u′N −1 } ∈ Rs , s = (m + 1)N + N + 1

and  x     
Gε Guε Gcε Sε wε
Ḡ0 = , S̄0 = , w̄0 = , (11.55)
0 0 G0 E0 w0
 11.4 1-Norm and ∞-Norm Case Solution 231

where [Gxε , Guε , Gcε ] is the block partition of the matrix Gε into the three parts
corresponding to the variables εxi , εui and ui , respectively. The vector c0 and the
submatrices Gǫ , wǫ , Sǫ associated with the constraints (11.53b)-(11.53d) are defined
in (9.10). The matrices G0 , w0 and E0 are defined in (11.18) for i = 0.
For a given vector x(0) the optimal input sequence U0∗ solving problem (11.54)
can be computed by using a Linear Program (LP) solver (see Section 2.2 for LP
definition and properties and Chapter 3 for fast numerical methods for solving
LPs).
To obtain the problem (11.54) we have eliminated the state variables and equal-
ity constraints xk+1 = Axk + Buk by successive substitution so that we are left
with u0 , . . . , uN −1 and the slack variables ǫ as the only decision variables, and x(0)
as a parameter vector. As in the 2-norm case, it might be more efficient to solve
an LP problem with equality and inequality constraints so that sparsity can be
exploited. We omit the details and refer the reader to the construction of the QP
problem without substitution in section 11.3.1.

11.4.2 State Feedback Solution via Batch Approach

As shown in the previous section, problem (11.52) can be rewritten in the compact
form
min c′0 z0
z0 (11.56)
subj. to Ḡ0 z0 ≤ w̄0 + S̄0 x(0),
where z0 = {εx0 , . . . , εxN , εu0 , . . . , εuN −1 , u′0 , . . . , u′N −1 } ∈ Rs , s = (m + 1)N + N + 1.
As in the 2-norm case, by treating x(0) as a vector of parameters, problem (11.56)
becomes a multiparametric linear program (mp-LP) that can be solved as described
in Section 6.2. Once the multiparametric problem (11.56) has been solved, the
explicit solution z0∗ (x(0)) of (11.56) is available as a piecewise affine function of
x(0), and the optimal control law u∗ (0) is also available explicitly, as the optimal
input u∗ (0) consists simply of m components of z0∗ (x(0))

u∗ (0) = [0 . . . 0 Im 0 . . . 0]z0∗ (x(0)). (11.57)

Theorem 6.5 states that there always exists a continuous PPWA solution z0∗ (x)
of the mp-LP problem (11.56). Clearly the same properties are inherited by the
controller. The following Corollaries of Theorem 6.5 summarize the analytical
properties of the optimal control law and of the value function.

Corollary 11.4 There exists a control law u∗ (0) = f0 (x(0)), f0 : Rn → Rm ,

obtained as a solution of the CFTOC (11.52) with p = 1 or p = ∞, which is
continuous and PPWA

f0 (x) = F0j x + g0j if x ∈ CR0j , j = 1, . . . , N0r , (11.58)

where the polyhedral sets CR0j = {H0j x ≤ k0j }, j = 1, . . . , N0r , are a partition of the
feasible set X0 .
232 11 Constrained Optimal Control

Corollary 11.5 The value function J ∗ (x) obtained as a solution of the CFTOC (11.52)
is convex and PPWA.

Remark 11.7 Note that if the optimizer of problem (11.52) is unique for all x(0) ∈ X0 ,
then Corollary 11.4 reads: “The control law u∗ (0) = f0 (x(0)), f0 : Rn → Rm ,
obtained as a solution of the CFTOC (11.52) with p = 1 or p = ∞, is continuous
and PPWA,. . .”. From the results of Section 6.2 we know that in case of multiple
optimizers for some x(0) ∈ X0 , a continuous control law of the form (11.58) can
always be computed.

The multiparametric solution of (11.56) provides the open-loop optimal se-

quence u∗ (0), . . . , u∗ (N − 1) as an affine function of the initial state x(0). The
state feedback PPWA optimal controllers u∗ (k) = fk (x(k)) with fk : Xk 7→ U for
k = 1, . . . , N are computed in the following way. Consider the same CFTOC (11.52)
over the shortened time horizon [i, N ]
N
X −1
minUi kP xN kp + kQxk kp + kRuk kp
k=i
subj. to xk+1 = Axk + Buk , k = i, . . . , N − 1 (11.59)
xk ∈ X , uk ∈ U, k = i, . . . , N − 1
xN ∈ Xf
xi = x(i),

where Ui = [u′i , . . . , u′N −1 ] and p = 1 or p = ∞. As defined in (11.16) and discussed

in Section 11.2, Xi ⊆ Rn is the set of initial states x(i) for which the optimal control
problem (11.59) is feasible. We denote by Ui∗ one of the optimizers of the optimal
control problem (11.59).
Problem (11.59) can be translated into the mp-LP

min c′i zi
zi (11.60)
subj. to Ḡi zi ≤ w̄i + S̄i x(i),

where zi = {εxi , . . . , εxN , εui , . . . , εuN −1 , u′i , . . . , u′N −1 } and ci , Ḡi , S̄i , w̄i , are appro-
priately defined for each i. The component u∗i of the multiparametric solution
of (11.60) has the form

u∗i (x(i)) = fi (x(i)), ∀x(i) ∈ Xi , (11.61)

where the control law fi : Rn → Rm , is continuous and PPWA

fi (x) = Fij x + gij if x ∈ CRij , j = 1, . . . , Nir (11.62)

and where the polyhedral sets CRij = {x ∈ Rn : Hij x ≤ Kij }, j = 1, . . . , Nir

are a partition of the feasible polyhedron Xi . Therefore the feedback solution
u∗ (k) = fk (x(k)), k = 0, . . . , N − 1 of the CFTOC (11.52) with p = 1 or p = ∞ is
obtained by solving N mp-LP problems of decreasing size. The following corollary
summarizes the final result.
 11.4 1-Norm and ∞-Norm Case Solution 233

Corollary 11.6 There exists a state feedback control law u∗ (k) = fk (x(k)), fk :
Xk ⊆ Rn → U ⊆ Rm , solution of the CFTOC (11.52) for p = 1 or p = ∞
and k = 0, . . . , N − 1 which is time-varying, continuous and piecewise affine on
polyhedra
fk (x) = Fkj x + gkj if x ∈ CRkj , j = 1, . . . , Nkr , (11.63)

where the polyhedral sets CRkj = {x ∈ Rn : Hkj x ≤ Kkj }, j = 1, . . . , Nkr are a

partition of the feasible polyhedron Xk .

11.4.3 State Feedback Solution via Recursive Approach

Consider the dynamic programming formulation of (11.52) with J0 (·) defined by (11.7)
with p = 1 or p = ∞

Jj∗ (xj ) = minuj ∗

kQxj kp + kRuj kp + Jj+1 (Axj + Buj )
subj. to xj ∈ X , uj ∈ U (11.64)
Axj + Buj ∈ Xj+1 ,

for j = 0, . . . , N − 1, with boundary conditions

∗
JN (xN ) = kP xN kp (11.65)
XN = Xf . (11.66)

Unlike for the 2-norm case the dynamic program (11.64)-(11.66) can be solved as
explained in the next theorem.

Theorem 11.6 The state feedback piecewise affine solution (11.63) of the CFTOC (11.52)
for p = 1 or p = ∞ is obtained by solving the optimization problem (11.64)-(11.66)
via N mp-LPs.

Proof: Consider the first step j = N − 1 of dynamic programming (11.64)-

(11.66)
∗ ∗
JN −1 (xN −1 ) = minuN −1 kQxN −1 kp + kRuN −1 kp + JN (AxN −1 + BuN −1 )
subj. to xN −1 ∈ X , uN −1 ∈ U
AxN −1 + BuN −1 ∈ Xf .
(11.67)
∗ ∗
JN −1 (xN −1 ), u N −1 (xN −1 ) and XN −1 are computable via the mp-LP:

∗
JN −1 (xN −1 ) = minµ,uN −1 µ
subj. to µ ≥ kQxN −1 kp + kRuN −1kp + kP (AxN −1 + BuN −1 )kp
xN −1 ∈ X , uN −1 ∈ U
AxN −1 + BuN −1 ∈ Xf .
(11.68)
The constraint µ ≥ kQxN −1 kp + kRuN −1 kp + kP (AxN −1 + BuN −1 )kp in (11.68) is
converted into a set of linear constraints as discussed in Remark 2.1 of Section 2.2.5.
234 11 Constrained Optimal Control

For instance, if p = ∞ we follow the approach of Section 9.3 and rewrite the
constraint as
µ ≥ εxN −1 + εuN −1 + εxN
x
−1n εN −1 ≤ ±QxN −1
(11.69)
−1m εuN −1 ≤ ±RuN −1
x
−1rN εN ≤ ±PN [AxN −1 + BuN −1 ]
∗
By Theorem 6.5, JN −1 is a convex and piecewise affine function of xN −1 , the
corresponding optimizer u∗N −1 is piecewise affine and continuous, and the feasible
∗
set XN −1 is a polyhedron. Without any loss of generality we assume JN −1 to be
∗ ′
described as follows: JN −1 (xN −1 ) = maxi=1,...,nN −1 {ci xN −1 +di } (see Section 2.2.5
for convex PPWA functions representation) where nN −1 is the number of affine
∗
components comprising the value function JN −1 . At step j = N − 2 of dynamic
programming (11.64)-(11.66) we have

∗ ∗
JN −2 (xN −2 ) = minuN −2 kQxN −2 kp + kRuN −2 kp + JN −1 (AxN −2 + BuN −2 )
subj. to xN −2 ∈ X , uN −2 ∈ U
AxN −2 + BuN −2 ∈ XN −1 .
(11.70)
∗
Since JN −1 (x) is a convex and piecewise affine function of x, the problem (11.70)
can be recast as the following mp-LP (see Section 2.2.5 for details)

∗
JN −2 (xN −2 ) = minµ,uN −2 µ
subj. to µ ≥ kQxN −2 kp + kRuN −2 kp + ci (AxN −2 + BuN −2 ) + di
i = 1, . . . , nN −1
xN −2 ∈ X , uN −2 ∈ U
AxN −2 + BuN −2 ∈ XN −1 .
(11.71)
∗ ∗
JN −2 (xN −2 ), u N −2 (x N −2 ) and X N −2 are computed by solving the mp-LP (11.71).
∗
By Theorem 6.5, JN −2 is a convex and piecewise affine function of xN −2 , the
corresponding optimizer u∗N −2 is piecewise affine and continuous, and the feasible
set XN −2 is a convex polyhedron.
The convexity and piecewise linearity of Jj∗ and the polyhedra representation
of Xj still hold for j = N − 3, . . . , 0 and the procedure can be iterated backwards
in time, proving the theorem. 

Consider the state feedback piecewise affine solution (11.63) of the CFTOC (11.52)
for p = 1 or p = ∞ and assume we are interested only in the optimal controller at
time 0. In this case, by using duality arguments we can solve the equations (11.64)-
(11.66) by using vertex enumerations and one mp-LP. This is proven in the next
theorem.

Theorem 11.7 The state feedback piecewise affine solution (11.63) at time k = 0
of the CFTOC (11.52) for p = 1 or p = ∞ is obtained by solving the optimization
problem (11.64)-(11.66) via one mp-LP.
11.4 1-Norm and ∞-Norm Case Solution 235

Proof: Consider the first step j = N − 1 of dynamic programming (11.64)-

(11.66)
∗ ∗
JN −1 (xN −1 ) = minuN −1 kQxN −1 kp + kRuN −1 kp + JN (AxN −1 + BuN −1 )
subj. to xN −1 ∈ X , uN −1 ∈ U
AxN −1 + BuN −1 ∈ Xf
(11.72)
and the corresponding mp-LP:
∗
JN −1 (xN −1 ) = minµ,uN −1 µ
subj. to µ ≥ kQxN −1 kp + kRuN −1kp + kP (AxN −1 + BuN −1 )kp
xN −1 ∈ X , uN −1 ∈ U
AxN −1 + BuN −1 ∈ Xf .
(11.73)
∗
By Theorem 6.5, JN −1 is a convex and piecewise affine function of xN −1 , and the
∗
feasible set XN −1 is a polyhedron. JN −1 and X N −1 are computed without explicitly
solving the mp-LP (11.73). Rewrite problem (11.73) in the more compact form

min c′N −1 zN −1
zN −1 (11.74)
subj. to ḠN −1 zN −1 ≤ w̄N −1 + S̄N −1 xN −1 ,

where zN −1 collects the optimization variables µ, uN −1 and the auxiliary variables

need to transform the constraint µ ≥ kQxN −1 kp + kRuN −1 kp + kP (AxN −1 +
BuN −1 )kp into a set of linear constraints. Consider the Linear Program dual
of (11.74)
maxv −(w̄N −1 + S̄N −1 xN −1 )′ v
subj. to Ḡ′N −1 v = −cN −1 (11.75)
v ≥ 0.

Consider the dual feasibility polyheron Pd = {v ≥ 0 : Ḡ′N −1 v = −cN −1 }. Let

{V1 , . . . , Vk } be the vertices of Pd and {y1 , . . . , ye } be the rays of Pd . Since we have
a zero duality gap, we have that
∗ ′
JN −1 (xN −1 ) = max {−(w̄N −1 + S̄N −1 xN −1 ) Vi }
i=1,...,k

i.e.,
∗ ′ ′
JN −1 (xN −1 ) = max {−(Vi S̄N −1 )xN −1 − w̄N −1 Vi }.
i=1,...,k

Recall that if the dual of a mp-LP is unbounded, then the primal is infeasible
(Theorem 6.3). For this reason the feasible set XN −1 is obtained by requiring that
the cost of (11.75) does not increase in the direction of the rays:

XN −1 = {xN −1 : − (w̄N −1 + S̄N −1 xN −1 )′ yi ≤ 0, ∀ i = 1, . . . , e}

∗
with JN −1 (xN −1 ) and XN −1 available, we can iterate the procedure backwards in
time for steps N − 2, N − 3, . . . , 1. At step 0 one mp-LP will be required in order
to compute u∗0 (x(0)). 
236 11 Constrained Optimal Control

11.4.4 Example

Example 11.4 Consider the double integrator system (11.23). We want to compute
the
 state feedback  controller that solves (11.52) with p = ∞, N = 6, P =
 optimal
1 0 1 0
,Q= , R = 0.8, subject to the input constraints
0 1 0 1

U = {u ∈ R : − 1 ≤ u ≤ 1} (11.76)

and the state constraints

   
−10 10
X = {x ∈ R2 : ≤x≤ } (11.77)
−10 10

and Xf = X . The optimal feedback solution u∗ (0), . . . , u∗ (5) was computed by solving
six mp-LP problems and the corresponding polyhedral partitions of the state space
are depicted in Fig. 11.5, where polyhedra with the same control law were merged.
Only the last optimal control move is reported below:
  0.45 0.89
  0.00


 0 if 1.00 0.00
x(5) ≤ 0.00
(Region #1)

 −0.71 −0.71 7.07

 −1.00 −0.00 10.00



  −0.45   

 −0.89 0.00

 −1.00 0.00 0.00

 0 if 0.71 0.71 x(5) ≤ 7.07 (Region #2)

 1.00 −0.00 10.00



  −0.45   

 −0.89 0.00

 0.45 0.89 0.45

 [ −1.00 −2.00 ] x(5) if 0.71 0.71 x(5) ≤ 0.00 (Region #3)

 −1.00 −0.00


10.00



  −0.45 −0.89
  


0.45

 [ −1.00 −2.00 ] x(5) if 0.45 0.89
x(5) ≤ 0.00
(Region #4)


−0.71 −0.71 0.00

 1.00 −0.00 10.00

u∗ (5) =  −0.71 −0.71

  0.00


 −1.00 0.00 1.00

 [ 1.00 0.00 ] x(5) if 1.00 0.00 x(5) ≤ 0.00 (Region #5)

 −0.00 1.00 10.00



    

 0.71 0.71 0.00

 −1.00 0.00

 [ 1.00 0.00 ] x(5) if x(5) ≤ 0.00
(Region #6)


1.00 0.00 1.00


−0.00 −1.00 10.00



  0.45 0.89
  −0.45 



 1.00 if −1.00 0.00
x(5) ≤ −1.00
(Region #7)

 −0.00 −1.00 10.00

 1.00 −0.00 10.00



  −0.45   −0.45 

 −0.89

 − 1.00 if 1.00 0.00
x(5) ≤ −1.00
(Region #8)

 −1.00 −0.00 10.00

 −0.00 1.00 10.00

(11.78)
Note that the controller (11.78) is piecewise linear around the origin. In fact, the
origin belongs to multiple regions (1 to 6). Note that the number Nir of regions is not
always increasing with decreasing i (N5r = 8, N4r = 12, N3r = 12, N2r = 26, N1r = 28,
N0r = 26). This is due to the merging procedure, before merging we have N5r = 12,
N4r = 22, N3r = 40, N2r = 72, N1r = 108, N0r = 152.
 11.4 1-Norm and ∞-Norm Case Solution 237

11.4.5 Infinite-Time Solution

Assume that Q and R have full column rank and that the constraint sets X and U
contain the origin in their interior. Consider the following infinite-horizon problem
with constraints
∞
X
∗
J∞ (x(0)) = minu0 ,u1 ,... kQxk kp + kRuk kp
k=0
subj. to xk+1 = Axk + Buk , k = 0, . . . , ∞ (11.79)
xk ∈ X , uk ∈ U, k = 0, . . . , ∞
x0 = x(0)

and the set

X∞ = {x(0) ∈ Rn : Problem (11.79) is feasible and J∞

∗
(x(0)) < +∞}. (11.80)

Because Q and R have full column rank, any optimizer u∗k of problem (11.43) must
converge to the origin (u∗k → 0) and so must the state trajectory resulting from the
application of u∗k (x∗k → 0). Thus the origin x = 0, u = 0 must lie in the interior
of the constraint set (X , U). (If the origin were not contained in the constraint
∗
set then J∞ (x(0)) would be infinite.) Furthermore, if the initial state x0 = x(0)
is sufficiently close to the origin, then the state and input constraints will never
become active and the solution of problem (11.43) will yield the unconstrained
optimal controller (9.31).
The discussion for the solution of the infinite horizon constrained linear quadratic
regulator (Section 11.3.4) by means of the batch approach can be repeated here
with one precaution. Since the unconstrained optimal controller (if it exists) is
PPWA the computation of the Maximal Invariant Set for the autonomous con-
strained piecewise linear system is more involved and requires algorithms which
will be presented later in Chapter 17.
Differently from the 2-norm case, here the use of dynamic programming for
computing the infinite horizon solution is a viable alternative to the batch approach.
Convergence conditions for the dynamic programming strategy and convergence
guarantees for the resulting possibly discontinuous closed-loop system are given
in [87]. A computationally efficient algorithm to obtain the infinite time optimal
solution, based on a dynamic programming exploration strategy with a multi-
parametric linear programming solver and basic polyhedral manipulations, is also
presented in [87].

Example 11.5 We consider the double integrator system (11.23) from Example 11.4
with N = ∞.

The partition of the state space for the time invariant optimal control law is shown in
Fig. 11.6(a) and consists of 202 polyhedral regions. In Fig. 11.6(b) the same control
law is represented where polyhedra with the same affine control law were merged.
238 11 Constrained Optimal Control

11.5 State Feedback Solution, Minimum-Time Control

In this section we consider the solution of minimum-time optimal control problems
J0∗ (x(0)) = min N
U0 ,N
subj. to xk+1 = Axk + Buk , k = 0, . . . , N − 1
xk ∈ X , uk ∈ U, k = 0, . . . , N − 1 (11.81)
xN ∈ Xf
x0 = x(0),
where Xf ⊂ Rn is a terminal target set to be reached in minimum time.
We can find the controller that brings the states into Xf in one time step by
solving the following multiparametric program
min c(x0 , u0 )
u0
subj. to x1 = Ax0 + Bu0 (11.82)
x0 ∈ X , u0 ∈ U
x1 ∈ Xf ,

where c(x0 , u0 ) is any convex quadratic function. Let us assume that the solution of
1
the multiparametric program generates R1 regions {Pr1 }R r=1 with the affine control
1 1
law u0 = Fr x + gr in each region r. By construction we have

X0 = K1 (Xf ).

Continuing to set up simple multiparametric programs bring the states into Xf in

2, 3, . . . steps, we have for step j
min c(x0 , u0 )
u0
subj. to x1 = Ax0 + Bu0 (11.83)
x0 ∈ X , u0 ∈ U
x1 ∈ Kj−1 (Xf ),
j
which yields Rj regions {Prj }R j j
r=1 with the affine control law u0 = Fr x + gr in each
region r. By construction we have

X0 = Kj (Xf ).

Thus to obtain K1 (Xf ), . . . , KN (Xf ) we need to solve N multiparametric programs

with a prediction horizon of 1. Since the overall complexity of a multiparametric
program is exponential in N , this scheme can be exploited to yield controllers
of lower complexity than the optimal control schemes introduced in the previous
sections.
Since N multiparametric programs have been solved, the controller regions over-
lap in general. In order to achieve minimum time behavior, the feedback law asso-
ciated with the region computed for the smallest number of steps c, is selected for
any given state x.

Algorithm 11.1 On-line computation of minimum-time control input

 11.6 Comparison of the Design Approaches and Controllers 239

Input: State measurement x, N controller partitions solution to (11.83) for j =

1, . . . , N
Output: Minimum time control action u(x)
Find controller partition cmin = minc∈{0,...,N} c, s.t. x ∈ Kc (Xf )
Find controller region r, such that x ∈ Prcmin and compute u = Frcmin x + grcmin
Return u

Note that the region identification for this type of controller partition is much
more efficient than simply checking all the regions. The two steps of “finding a
controller partition” and “finding a controller region” in Algorithm 11.1 correspond
to two levels of a search tree, where the search is first performed over the feasible
c
sets Kc (Xf ) and then over the controller partition {Prc }R r=1 . Furthermore, one
i
may discard all regions
i
S Pr which are completely covered by previously computed
controllers (i.e., Pr ⊆ j∈{1,...,i−1} Kj (Xf )) since they are not time optimal.

Example 11.6 Consider again the double integrator from Example 11.2. The Minimum-
Time Controller is computed that steers the system to the Maximal LQR Invariant
LQR
Set O∞ in the minimum number of time steps N . The Algorithm terminated af-
LQR
ter 11 iterations, covering the Maximal Controllable Set K∞ (O∞ ). The resulting
controller is defined over 33 regions. The regions are depicted in Fig. 11.7(a). The
LQR
Maximal LQR Invariant Set O∞ is the central shaded region.
The control law on this partition is depicted in Fig. 11.7(b). Note that, in general,
the minimum-time control law is not continuous as can be seen in Fig. 11.7(b).

11.6 Comparison of the Design Approaches and Controllers

For the design, the storage and the on-line execution time of the control law we
are interested in the controller complexity. All the controllers we discussed in this
chapter are PPWA. Their storage and execution times are closely related to the
number of polyhedral regions, which is related to the number of constraints in the
mp-LP or mp-QP.
When the control objective is expressed in terms of the 1- or ∞- norm, it
is translated into a set of constraints in the mp-LP. When the control objective
is expressed in terms of the 2-norm such a translation is not necessary. Thus
controllers minimizing the 1- or ∞- norm do generally involve more regions than
those minimizing the 2-norm and therefore tend to be more complex.
In mp-LPs degeneracies are common and have to be taken care of in the re-
spective algorithms. Strictly convex mp-QPs for which the Linear Independent
Constraint Qualification holds are comparatively well behaved with unique solu-
tions and full-dimensional polyhedral regions. Only for the 1-norm and ∞-norm
objective, however, we can use for the controller design an efficient Dynamic Pro-
gramming scheme involving a sequence of mp-LPs.
240 11 Constrained Optimal Control

As we let the horizon N go to infinity, the number of regions stays finite for the
2-norm objective, for the 1- and ∞-norm nothing can be said in general. In the
examples we have usually observed a finite number, but cases can be constructed,
where the number can be proven to be infinite.
As we will argue in the following chapter, infinite horizon controllers based on
the 2-norm render the closed-loop system exponentially stable, while controllers
based on the 1- or ∞-norm render the closed-loop system stable, only.
Among the controllers proposed in this chapter, the minimum-time controller
is usually the least complex involving the smallest number of regions. Minimum
time controllers are often considered to be too aggressive for practical use. The
controller here is different, however. It is minimum time only until it reaches the
terminal set Xf . Inside the terminal set a different unconstrained control law can
be used. Overall the operation of the minimum-time controller is observed to be
very similar to that of the other infinite time controllers in this chapter.
11.6 Comparison of the Design Approaches and Controllers 241

10 10

5 5
x2 (0)

x2 (1)
0 0

-5 -5

-10 -10
-10 -5 0 5 10 -10 -5 0 5 10
x1 (0) x1 (1)
(a) Partition of the state space for (b) Partition of the state space
the affine control law u∗ (0)(N0r = for the affine control law u∗ (1)
26). (N1r = 28).

10 10

5 5
x2 (2)

x2 (3)

0 0

-5 -5

-10
-10 0 10 -10
-10 0 10
-5 5 -5 5
x1 (2) x1 (3)
(c) Partition of the state space (d) Partition of the state space
for the affine control law u∗ (2) for the affine control law u∗ (3)
(N2r = 26). (N3r = 12)

10 10

5 5
x2 (4)

x2 (5)

0 0

-5 -5

-10
-10 0 10 -10
-10 0 10
-5 5 -5 5
x1 (4) x1 (5)
(e) Partition of the state space (f) Partition of the state space
for the affine control law u∗ (4) for the affine control law u∗ (5)
(N4r = 12). (N5r = 8).

Figure 11.5 Example 11.4. Double Integrator, ∞-norm objective function,

horizon N = 6. Partition of the state space for the time-varying optimal
control law. Polyhedra with the same control law were merged.
242 11 Constrained Optimal Control

10 10

5 5
x2

x2
0 0

-5 -5

-10 -5 5 -10 -5 5
-10 0 10 -10 0 10
x1 x1
(a) Partition before merging (b) Partition after merging
r r
(N∞ = 202). (N∞ = 26).

Figure 11.6 Example 11.5. Double Integrator, ∞-norm objective function,

horizon N = ∞. Partition of the state space for the time invariant optimal
control law.

10

5
1
x2

0
0
u

-5 -5
-1 0
10 5
-10 0 -5 -10 5
-10 -5 0 5 10 x2
x1 x1
(a) Minimum-Time State Space (b) Minimum-Time Control Law
Partition (N r =33). (discontinuous).

Figure 11.7 Example 11.6. Double Integrator, minimum-time objective func-

tion. Partition of the state space and control law. Maximal LQR invariant
LQR
set O∞ is the central shaded region in (a).
 Chapter 12

Receding Horizon Control

In this chapter we review the basics of Receding Horizon Control (RHC). In the first
part we discuss the stability and the feasibility of RHC and we provide guidelines
for choosing the terminal weight so that closed-loop stability is achieved.
The second part of the chapter focuses on the RHC implementation. Since
RHC requires at each sampling time to solve an open-loop constrained finite time
optimal control problem as a function of the current state, the results of the previous
chapters imply two possible approaches for RHC implementation.
In the first approach a mathematical program is solved at each time step for
the current initial state. In the second approach the explicit piecewise affine feed-
back policy (that provides the optimal control for all states) is precomputed off-line.
This reduces the on-line computation of the RHC law to a function evaluation, thus
avoiding the on-line solution of a quadratic or linear program. This technique is
attractive for a wide range of practical problems where the computational complex-
ity of on-line optimization is prohibitive. It also provides insight into the structure
underlying optimization-based controllers, describing the behaviour of the RHC
controller in different regions of the state space. Moreover, for applications where
safety is crucial, the correctness of a piecewise affine control law is easier to verify
than that of a mathematical program solver.

12.1 RHC Idea

In the previous chapter we discussed the solution of constrained finite time and
infinite time optimal control problems for linear systems. An infinite horizon sub-
optimal controller can be designed by repeatedly solving finite time optimal control
problems in a receding horizon fashion as described next. At each sampling time,
starting at the current state, an open-loop optimal control problem is solved over
a finite horizon (top diagram in Figure 12.1). The computed optimal manipulated
input signal is applied to the process only during the following sampling interval
[t, t + 1]. At the next time step t + 1 a new optimal control problem based on
new measurements of the state is solved over a shifted horizon (bottom diagram
244 12 Receding Horizon Control

past future

reference predicted outputs

u(t) manipulated inputs

t t+1 t+N

predicted outputs

u(t + 1) manipulated inputs

t + 1t + 2 t+1+N

Figure 12.1 Receding Horizon Idea.

in Figure 12.1). The resulting controller is referred to as a Receding Horizon Con-

troller (RHC). A receding horizon controller where the finite time optimal control
law is computed by solving an optimization problem on-line is usually referred to
as Model Predictive Control (MPC).

12.2 RHC Implementation

Consider the problem of regulating to the origin the discrete-time linear time-
invariant system

x(t + 1) = Ax(t) + Bu(t), (12.1)

where x(t) ∈ Rn , u(t) ∈ Rm are the state and input vectors, respectively, subject
to the constraints

x(t) ∈ X , u(t) ∈ U, ∀t ≥ 0, (12.2)

where the sets X ⊆ Rn and U ⊆ Rm are polyhedra. Receding Horizon Control

(RHC) approaches such a constrained regulation problem in the following way.
Assume that a full measurement or estimate of the state x(t) is available at the
12.2 RHC Implementation 245

current time t. Then the finite time optimal control problem

N
X −1
Jt∗ (x(t)) = minUt→t+N |t Jt (x(t), Ut→t+N |t ) = p(xt+N |t ) + q(xt+k|t , ut+k|t )
k=0
subj. to xt+k+1|t = Axt+k|t + But+k|t , k = 0, . . . , N − 1
xt+k|t ∈ X , ut+k|t ∈ U, k = 0, . . . , N − 1
xt+N |t ∈ Xf
xt|t = x(t)
(12.3)
is solved at time t, where Ut→t+N |t = {ut|t , . . . , ut+N −1|t } and where xt+k|t denotes
the state vector at time t + k predicted at time t obtained by starting from the
current state xt|t = x(t) and applying to the system model
xt+k+1|t = Axt+k|t + But+k|t
the input sequence ut|t , . . . , ut+k−1|t . Often the symbol xt+k|t is read as “the state
x at time t + k predicted at time t”. Similarly, ut+k|t is read as “the input u at
time t + k computed at time t”. For instance, x3|1 represents the predicted state
at time 3 when the prediction is done at time t = 1 starting from the current state
x(1). It is different, in general, from x3|2 which is the predicted state at time 3
when the prediction is done at time t = 2 starting from the current state x(2).
∗ ∗ ∗
Let Ut→t+N |t = {ut|t , . . . , ut+N −1|t } be the optimal solution of (12.3) at time t
∗ ∗
and Jt (x(t)) the corresponding value function. Then, the first element of Ut→t+N |t
is applied to system (12.1)
u(t) = u∗t|t (x(t)). (12.4)
The optimization problem (12.3) is repeated at time t + 1, based on the new state
xt+1|t+1 = x(t + 1), yielding a moving or receding horizon control strategy.
Let ft : Rn → Rm denote the receding horizon control law that associates
the optimal input u∗t|t to the current state x(t), ft (x(t)) = u∗t|t (x(t)). Then the
closed-loop system obtained by controlling (12.1) with the RHC (12.3)-(12.4) is
x(k + 1) = Ax(k) + Bfk (x(k)) = fcl (x(k), k), k ≥ 0. (12.5)
Note that the notation used in this chapter is slightly different from the one
used in Chapter 11. Because of the receding horizon strategy, there is the need
to distinguish between the input u∗ (t + k) applied to the plant at time t + k, and
optimizer u∗t+k|t of the problem (12.3) at time t + k obtained by solving (12.3) at
time t with xt|t = x(t).
Consider problem (12.3). As the system, the constraints and the cost function
are time-invariant, the solution to problem (12.3) is a time-invariant function of
the initial state x(t). Therefore, in order to simplify the notation, we can set t = 0
in (12.3) and remove the term “|0” since it is now redundant and rewrite (12.3) as
N
X −1
J0∗ (x(t)) = minU0 J0 (x(t), U0 ) = p(xN ) + q(xk , uk )
k=0
subj. to xk+1 = Axk + Buk , k = 0, . . . , N − 1 (12.6)
xk ∈ X , uk ∈ U, k = 0, . . . , N − 1
xN ∈ Xf
x0 = x(t),
246 12 Receding Horizon Control

where U0 = {u0 , . . . , uN −1 } and the notation in Remark 7.1 applies. Similarly

as in previous chapters, we will focus on two classes of cost functions. If the 1-
norm or ∞-norm is used in (12.6), then we set p(xN ) = kP xN kp and q(xk , uk ) =
kQxk kp + kRuk kp with p = 1 or p = ∞ and P , Q, R full column rank matrices.
The cost function is rewritten as
N
X −1
J0 (x(0), U0 ) = kP xN kp + kQxk kp + kRuk kp . (12.7)
k=0

If the squared Euclidian norm is used in (12.6), then we set p(xN ) = x′N P xN and
q(xk , uk ) = x′k Qxk + u′k Ruk with P  0, Q  0 and R ≻ 0. The cost function is
rewritten as
N
X −1
J0 (x(0), U0 ) = x′N P xN + x′k Qxk + u′k Ruk . (12.8)
k=0

The control law (12.4)

u(t) = f0 (x(t)) = u∗0 (x(t)) (12.9)

and closed-loop system (12.5)

x(k + 1) = Ax(k) + Bf0 (x(k)) = fcl (x(k)), k ≥ 0 (12.10)

are time-invariant as well.

Note that the notation in (12.6) does not allow us to distinguish at which time
step a certain state prediction or optimizer is computed and is valid for time-
invariant problems only. Nevertheless, we will prefer the RHC notation in (12.6)
to the one in (12.3) in order to simplify the exposition.
Compare problem (12.6) and the CFTOC (11.9). The only difference is that
problem (12.6) is solved for x0 = x(t), t ≥ 0 rather than for x0 = x(0). For this
reason we can make use of all the results of the previous chapter. In particular, X0
denotes the set of feasible states x(t) for problem (12.6) as defined and studied in
Section 11.2. Recall from Section 11.2 that X0 is a polyhedron.
From the above explanations it is clear that a fixed prediction horizon is shifted
or recedes over time, hence its name, receding horizon control. The procedure of
this on-line optimal control technique is summarized in the following algorithm.

Algorithm 12.1 On-line receding horizon control

Input: State x(t) at time instant t
Output: Receding horizon control input u(x(t))
Obtain U0∗ (x(t)) by solving the optimization problem (12.6)
If ‘problem infeasible’ Then stop
Return the first element u∗0 of U0∗
12.2 RHC Implementation 247

Example 12.1 Consider the double integrator system (11.23) rewritten below:
   
1 1 0
x(t + 1) = x(t) + u(t) (12.11)
0 1 1

The aim is to compute the receding horizon controller that solves the optimization
′ ′ ′
problem
  (12.6) with p(xN ) = xN P xN , q(xk , uk ) = xk Qxk + uk Ruk N = 3, P = Q =
1 0
, R = 10, Xf = R2 subject to the input constraints
0 1

− 0.5 ≤ u(k) ≤ 0.5, k = 0, . . . , 3 (12.12)

and the state constraints

   
−5 5
≤ x(k) ≤ , k = 0, . . . , 3. (12.13)
−5 5

The QP problem associated with the RHC has the form (11.31) with
h 13.50 −10.00 −0.50 i  
H = −10.00 22.00 −10.00 , F = −10.50 10.00 −0.50
−20.50 10.00 9.50 , Y = [ 14.50 23.50
23.50 54.50 ] (12.14)
−0.50 −10.00 31.50

and
 0.50 −1.00 0.50   0.50 0.50   0.50 
−0.50 1.00 −0.50 −0.50 −0.50 0.50
 −0.50
−0.50 0.00 0.50   −0.50
0.50 0.50 
 5.00 
 0.00 −0.50   −0.50 
 5.00 
 0.50 0.00 −0.50   −0.50 −0.50 
 5.00 
 0.50 0.00 0.50   0.50 0.50   5.00 
 −1.00 0.00 0.00   0.00 0.00   5.00 
5.00
 0.00 −1.00 0.00   0.00 0.00   5.00 
 1.00 0.00 0.00   0.00 0.00   5.00 
 0.00 1.00 0.00   0.00 0.00   
 0.00 −1.00  ,  0.00 0.00 
G0 =  0.00 0.00
1.00 
E0 =  0.00 0.00  , w0 =  0.50 
 0.50  (12.15)
 0.00 0.00   1.00 1.00   5.00 
 −0.50 0.00 0.00   −0.50 −0.50   5.00 
 0.00 0.00 0.50   −1.00 −1.00   5.00 
 0.50 0.00 0.00   0.50   5.00 
 0.00 −0.50   −0.50 0.50 
 0.50 
 −0.50 0.00 0.50   0.50 −1.50 
 0.50 
 0.50 0.00 −0.50   1.50 
 5.00 
 0.00 0.00 0.00   0.00
1.00 0.00 
1.00 5.00
0.00 0.00 0.00 5.00
0.00 0.00 0.00 −1.00 0.00
0.00 0.00 0.00 0.00 −1.00 5.00

The RHC (12.6)-(12.9) algorithm becomes Algorithm 12.2. We refer to the solution
U0∗ of the QP (11.31) as [U0∗ , Flag] = QP(H, 2F ′ x(0), G0 , w0 + E0 x(0)) where “Flag”
indicates if the QP was found to be feasible or not.

Algorithm 12.2 QP-based on-line receding horizon control

Input: State x(t) at time instant t
Output: Receding horizon control input u(x(t))
Compute F̃ = 2F ′ x(t) and w̃0 = w0 + E0 x(t)
Obtain U0∗ (x(t)) by solving the optimization problem [U0∗ , Flag] = QP(H, F̃ , G0 , w̃0 )
If Flag=infeasible Then stop
Return the first element u∗0 of U0∗

Fig. 12.2 shows two closed-loop trajectories starting at state x(0) = [−4.5, 2] and
x(0) = [−4.5, 3]. The trajectory starting from x(0) = [−4.5, 2] converges to the
origin and satisfies input and state constraints. The trajectory starting from x(0) =
248 12 Receding Horizon Control

x2
1

0
-5 -4 -3 -2 -1 0 1 2 3 4 5
x1
Figure 12.2 Example 12.1. Closed-loop trajectories realized (solid) and pre-
dicted (dashed) for two initial states x(0)=[-4.5,2] (boxes) and x(0)=[-4.5,3]
(circles).

[−4.5, 3] stops at x(2) = [1, 2] because of infeasibility. At each time step, the open-
loop predictions are depicted with dashed lines. This shows that the closed-loop
trajectories are different from the open-loop predicted trajectories because of the
receding horizon nature of the controller.
In Fig. 12.3(a) the feasible state space was gridded and each point of the grid was
marked with a square if the RHC law (12.6)-(12.9) generates feasible closed-loop tra-
jectories and with a circle if it does not. The set of all initial conditions generating
feasible closed-loop trajectories is the maximal positive invariant set O∞ of the au-
tonomous system (12.10). We remark that this set is different from the set X0 of
feasible initial conditions for the QP problem (11.31) with matrices (12.15). Both
sets O∞ and X0 are depicted in figure 12.3(b). The computation of f0 is discussed
later in this chapter. Because of the nonlinear nature of f0 , the computation of O∞
for the system (12.10) is not an easy task. Therefore we will show how to choose a
terminal invariant set Xf such that O∞ = X0 is guaranteed automatically.
Note that a feasible closed-loop trajectory does not necessarily converge to the origin.
Feasibility, convergence and stability of RHC are discussed in detail in the next
sections. Before that we want to illustrate these issues through another example.

Example 12.2 Consider the unstable system

   
2 1 1
x(t + 1) = x(t) + u(t) (12.16)
0 0.5 0

with the input constraints

− 1 ≤ u(k) ≤ 1, k = 0, . . . , N − 1 (12.17)

and the state constraints

   
−10 10
≤ x(k) ≤ , k = 0, . . . , N − 1. (12.18)
−10 10

In the following, we study the receding horizon control problem (12.6) with p(xN ) =
x′N P xN , q(xk , uk ) = x′k Qxk + u′k Ruk for different horizons N and weights R. We
replacemen

12.2 RHC Implementation 249

5 5
4 4
3 3
2 2
1 1

x2
x2

0 0
-1 -1
-2 -2
-3 -3
-4 -4
-5 -5
-5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5
x1 x1
(a) Boxes (Circles) are initial (b) Maximal positive invariant set
points leading (not leading) to O∞ (gray) and set of initial fea-
feasible closed-loop trajectories. sible states X0 (white and gray).

Figure 12.3 Example 12.1. Double integrator with RHC.

set Q = I and omit both the terminal set constraint and the terminal weight, i.e.,
Xf = R2 , P = 0.
Fig. 12.4 shows closed-loop trajectories for receding horizon control loops that were
obtained with the following parameter settings

• Setting 1: N = 2, R = 10
• Setting 2: N = 3, R = 2
• Setting 3: N = 4, R = 1

For Setting 1 (Fig. 12.4(a)) there is evidently no initial state that can be steered to
the origin. Indeed, it turns out, that all non-zero initial states x(0) ∈ R2 diverge
from the origin and eventually become infeasible. In contrast, Setting 2 leads to a
receding horizon controller, that manages to get some of the initial states converge
to the origin, as seen in Fig. 12.4(b). Finally, Fig. 12.4(c) shows that Setting 3 can
expand the set of those initial states that can be brought to the origin.

Note the behavior of particular initial states:

1. Closed-loop trajectories starting at state x(0) = [−4, 7] behave differently de-

pending on the chosen setting. Both Setting 1 and Setting 2 cannot bring this
state to the origin, but the controller with Setting 3 succeeds.
2. There are initial states, e.g. x(0) = [−4, 8.5], that always lead to infeasible
trajectories independent of the chosen settings. It turns out, that no setting
can be found that brings those states to the origin.

These results illustrate that the choice of parameters for receding horizon control
influences the behavior of the resulting closed-loop trajectories in a complex manner.
A better understanding of the effect of parameter changes can be gained from an
inspection of maximal positive invariant sets O∞ for the different settings, and the
maximal control invariant set C∞ as depicted in Fig. 12.5.
The maximal positive invariant set stemming from Setting 1 only contains the origin
(O∞ = {0}) which explains why all non-zero initial states diverge from the origin.
250 12 Receding Horizon Control

10 10

5 5
x2

x2
0 0

-5 -5

-10 -10
-8 -4 0 4 8 -8 -4 0 4 8
x1 x1
(a) Setting 1 : N = 2, R = 10. (b) Setting 2 : N = 3, R = 2.

10

5
x2

-5

-10
-8 -4 0 4 8
x1
(c) Setting 3 : N = 4, R = 1.

Figure 12.4 Example 12.2. Closed-loop trajectories for different settings of

horizon N and weight R. Boxes (Circles) are initial points leading (not
leading) to feasible closed-loop trajectories.

For Setting 2 the maximal positive invariant set has grown considerably, but does
not contain the initial state x(0) = [−4, 7], thus leading to infeasibility eventually.
Setting 3 leads to a maximal positive invariant set that contains this state and thus
keeps the closed-loop trajectory inside this set for all future time steps.
From Fig. 12.5 we also see that a trajectory starting at x(0) = [−4, 8.5] cannot be
kept inside any bounded set by any setting (indeed, by any controller) since it is
outside the maximal control invariant set C∞ .

12.3 RHC Main Issues

If we solve the receding horizon problem for the special case of an infinite hori-
zon (setting N = ∞ in (12.6) as we did for LQR in Section 8.5 and CLQR in
Section 11.3.4) then it is almost immediate that the closed-loop system with this
controller has some nice properties. Most importantly, the differences between
 12.3 RHC Main Issues 251

10
x(0) = [−4, 8.5]
x(0) = [−4, 7]
5

x2
0

-5

-10
-4 0 4
x1
Figure 12.5 Example 12.2. Maximal positive invariant sets O∞ for different
parameter settings: Setting 1 (origin), Setting 2 (dark-gray) and Setting 3
(gray and dark-gray). Also depicted is the maximal control invariant set C∞
(white and gray and dark-gray).

the open-loop predicted and the actual closed-loop trajectories observed in Exam-
ple 12.1 disappear. As a consequence, if the optimization problem is feasible, then
the closed-loop trajectories will be feasible for all times. If the optimization prob-
lem has a finite solution, then in closed-loop the states and inputs will converge to
the origin asymptotically.
In RHC, when we solve the optimization problem over a finite horizon repeat-
edly at each time step, we hope that the controller resulting from this “short-
sighted” strategy will lead to a closed-loop behavior that mimics that of the infi-
nite horizon controller. The examples in the last section indicated that at least two
problems may occur. First of all, the controller may lead us into a situation where
after a few steps the finite horizon optimal control problem that we need to solve
at each time step is infeasible, i.e., that there does not exist a sequence of control
inputs for which the constraints are obeyed. Second, even if the feasibility prob-
lem does not occur, the generated control inputs may not lead to trajectories that
converge to the origin, i.e., that the closed-loop system is asymptotically stable.
In general, stability and feasibility are not ensured by the RHC law (12.6)-
(12.9). In principle, we could analyze the RHC law for feasibility, stability and
convergence but this is difficult as the examples in the last section illustrated.
Therefore, conditions will be derived on how the terminal weight P and the terminal
constraint set Xf should be chosen such that closed-loop stability and feasibility
are ensured.

12.3.1 Feasibility of RHC

The examples in the last section illustrate that feasibility at the initial time x(0) ∈
X0 does not necessarily imply feasibility for all future times. It is desirable to
design a RHC such that feasibility for all future times is guaranteed, a property we
refer to as persistent feasibility.
We would like to gain some insight when persistent feasibility occurs and how it
is affected by the formulation of the control problem and the choice of the controller
252 12 Receding Horizon Control

parameters. Let us first recall the various sets introduced in Section 10.2 and
Section 11.2 and how they are influenced.
C∞ : The maximal control invariant set C∞ is only affected by the sets X and U,
the constraints on states and inputs. It is the largest set over which we can
expect any controller to work.
X0 : A control input U0 can only be found, i.e., the control problem is feasible,
if x(0) ∈ X0 . The set X0 depends on X and U, on the controller horizon N
and on the controller terminal set Xf . It does not depend on the objective
function and it has generally no relation with C∞ (it can be larger, smaller,
etc.).
O∞ : The maximal positive invariant set for the closed-loop system depends on
the controller and as such on all parameters affecting the controller, i.e., X , U,
N , Xf and the objective function with its parameters P , Q and R. Clearly
O∞ ⊆ X0 because if it were not there would be points in O∞ for which
the control problem is not feasible. Because of invariance, the closed-loop is
persistently feasible for all states x(0) ∈ O∞ . Clearly, O∞ ⊆ C∞ .
We can now state necessary and sufficient conditions guaranteeing persistent fea-
sibility by means of invariant set theory.

Lemma 12.1 Let O∞ be the maximal positive invariant set for the closed-loop
system x(k + 1) = fcl (x(k)) in (12.10) with constraints (12.2). The RHC problem
is persistently feasible if and only if X0 = O∞ .
Proof: For the RHC problem to be persistently feasible X0 must be positive
invariant for the closed-loop system. We argued above that O∞ ⊆ X0 . As the
positive invariant set X0 cannot be larger than the maximal positive invariant set
O∞ , it follows that X0 = O∞ . 
As X0 does not depend on the controller parameters P , Q and R but O∞ does,
the requirement X0 = O∞ for persistent feasibility shows that, in general, only
some P , Q and R are allowed. The parameters P , Q and R affect the performance.
The complex effect they have on persistent feasibility makes their choice extremely
difficult for the design engineer. In the following we will remedy this undesirable
situation. We will make use of the following important sufficient condition for
persistent feasibility.

Lemma 12.2 Consider the RHC law (12.6)-(12.9) with N ≥ 1. If X1 is a control

invariant set for system (12.1)-(12.2) then the RHC is persistently feasible. Also,
O∞ = X0 is independent of P , Q and R.
Proof: If X1 is control invariant then, by definition, X1 ⊆ Pre(X1 ). Also recall
that Pre(X1 ) = X0 from the properties of the feasible sets in equation (11.20)
(note that Pre(X1 ) ∩ X = Pre(X1 ) from control invariance). Pick some x ∈ X0
and some feasible control u for that x and define x+ = Ax + Bu ∈ X1 . Then
x+ ∈ X1 ⊆ Pre(X1 ) = X0 . As u was arbitrary (as long as it is feasible) x+ ∈ X0
for all feasible u. As X0 is positive invariant, X0 = O∞ from Lemma 12.1. As X0
is positive invariant for all feasible u, O∞ does not depend on P , Q and R. 
12.3 RHC Main Issues 253

Note that in the proof of Lemma 12.2, persistent feasibility does not depend on the
input u as long as it is feasible. For this reason, sometimes in the literature this
property is referred to “persistently feasible for all feasible u”.
We can use Lemma 12.2 in the following manner. For N = 1, X1 = Xf . If we
choose the terminal set to be control invariant then X0 = O∞ and RHC will be
persistently feasible independent of chosen control objectives and parameters. Thus
the designer can choose the parameters to affect performance without affecting
persistent feasibility. A control horizon of N = 1 is often too restrictive, but we
can easily extend Lemma 12.2.

Theorem 12.1 Consider the RHC law (12.6)-(12.9) with N ≥ 1. If Xf is a

control invariant set for system (12.1)-(12.2) then the RHC is persistently feasible.

Proof: If Xf is control invariant, then XN −1 , XN −2 , . . . , X1 are control invari-

ant and Lemma 12.2 establishes persistent feasibility for all feasible u. 

Corollary 12.1 Consider the RHC law (12.6)-(12.9) with N ≥ 1. If there exists
i ∈ [1, N ] such that Xi is a control invariant set for system (12.1)-(12.2), then the
RHC is persistently feasible for all cost functions.

Proof: Follows directly from the proof of Theorem 12.1. 

Recall that Theorem 11.2 together with Remark 11.4 define the properties of
the set X0 as N varies. Therefore, Theorem 12.1 and Corollary 12.1 provide also
guidelines on the choice of the horizon N for guaranteeing persistent feasibility
for all feasible u. For instance, if the RHC problem (12.6) for N = N̄ yields a
control invariant set X0 , then from Theorem 11.2 the RHC law (12.6)-(12.9) with
N = N̄ + 1 is persistently feasible for all feasible u. Moreover, from Corollary 12.1
the RHC law (12.6)-(12.9) with N ≥ N̄ + 1 is persistently feasible for all feasible
u.

Corollary 12.2 Consider the RHC problem (12.6)-(12.9). If N is greater than

the determinedness index N̄ of K∞ (Xf ) for system (12.1)-(12.2), then the RHC is
persistently feasible.

Proof: The feasible set Xi for i = 1, . . . , N − 1 is equal to the (N − i)-step con-

trollable set Xi = KN −i (Xf ). If the maximal controllable set is finitely determined
then Xi = K∞ (Xf ) for i ≤ N − N̄ . Note that K∞ (Xf ) is control invariant. Then
persistent feasibility for all feasible u follows from Corollary 12.1. 
Persistent feasibility does not guarantee that the closed-loop trajectories con-
verge towards the desired equilibrium point. From Theorem 12.1 it is clear that
one can only guarantee that x(k) ∈ X1 for all k > 0 if x(0) ∈ X0 .
One of the most popular approaches to guarantee persistent feasibility and
stability of the RHC law (12.6)-(12.9) makes use of a control invariant terminal set
Xf and a terminal cost P which drives the closed-loop optimal trajectories towards
the origin. A detailed discussion follows in the next section.
254 12 Receding Horizon Control

12.3.2 Stability of RHC

In this section we will derive the main stability result for RHC. Our objective is
to find a Lyapunov function for the closed-loop system. We will show next that if
the terminal cost and constraint are appropriately chosen, then the value function
J0∗ (·) is a Lyapunov function.

Theorem 12.2 Consider system (12.1)-(12.2), the RHC law (12.6)-(12.9) and
the closed-loop system (12.10). Assume that

(A0) The stage cost q(x, u) and terminal cost p(x) are continuous and positive
definite functions.

(A1) The sets X , Xf and U contain the origin in their interior and are closed.

(A2) Xf is control invariant, Xf ⊆ X .

(A3) min (−p(x) + q(x, v) + p(Ax + Bv)) ≤ 0, ∀x ∈ Xf .

v∈U , Ax+Bv∈Xf

Then, the origin of the closed-loop system (12.10) is asymptotically stable with
domain of attraction X0 .

Proof: From hypothesis (A2), Theorem 12.1 and Lemma 12.1, we conclude
that X0 = O∞ is a positive invariant set for the closed-loop system (12.10) for
any choice of the cost function. Thus persistent feasibility for any feasible input is
guaranteed in X0 .
Next we prove convergence and stability. We establish that the function J0∗ (·)
in (12.6) is a Lyapunov function for the closed-loop system. Because the cost J0 ,
the system and the constraints are time-invariant we can study the properties of
J0∗ between step k = 0 and step k + 1 = 1.
Consider problem (12.6) at time t = 0. Let x(0) ∈ X0 and let U0∗ = {u∗0 , . . . , u∗N −1 }
be the optimizer of problem (12.6) and x0 = {x(0), x1 , . . . , xN } be the corre-
sponding optimal state trajectory. After the implementation of u∗0 we obtain
x(1) = x1 = Ax(0) + Bu∗0 . Consider now problem (12.6) for t = 1. We will con-
struct an upper bound on J0∗ (x(1)). Consider the sequence Ũ1 = {u∗1 , . . . , u∗N −1 , v}
and the corresponding state trajectory resulting from the initial state x(1), x̃1 =
{x1 , . . . , xN , AxN + Bv}. Because xN ∈ Xf and (A2) there exists a feasible v such
that xN +1 = AxN + Bv ∈ Xf and with this v the sequence Ũ1 = {u∗1 , . . . , u∗N −1 , v}
is feasible. Because Ũ1 is not optimal J0 (x(1), Ũ1 ) is an upper bound on J0∗ (x(1)).
Since the trajectories generated by U0∗ and Ũ1 overlap, except for the first and
last sampling intervals, it is immediate to show that

J0∗ (x(1)) ≤ J0 (x(1), Ũ1 ) = J0∗ (x(0)) − q(x0 , u∗0 ) − p(xN )+

(12.19)
(q(xN , v) + p(AxN + Bv)).

Let x = x0 = x(0) and u = u∗0 . Under assumption (A3) equation (12.19) becomes

J0∗ (Ax + Bu) − J0∗ (x) ≤ −q(x, u), ∀x ∈ X0 . (12.20)

12.3 RHC Main Issues 255

Equation (12.20) and the hypothesis (A0) on the stage cost q(·) ensure that J0∗ (x)
strictly decreases along the state trajectories of the closed-loop system (12.10)
for any x ∈ X0 , x 6= 0. In addition to the fact that J0∗ (x) decreases, J0∗ (x) is
lower-bounded by zero and since the state trajectories generated by the closed-loop
system (12.10) starting from any x(0) ∈ X0 lie in X0 for all k ≥ 0, equation (12.20)
is sufficient to ensure that the state of the closed-loop system converges to zero as
k → 0 if the initial state lies in X0 . We have proven (i).
In order to prove stability via Theorem 7.2 we have to establish that J0∗ (x)
is a Lyapunov function. Positivity holds by the hypothesis (A0), decrease follows
from (12.20). For continuity at the origin we will show that J0∗ (x) ≤ p(x), ∀x ∈ Xf
and as p(x) is continuous at the origin (by hypothesis (A0)) J0∗ (x) must be con-
tinuous as well. From assumption (A2), Xf is control invariant and thus for any
x ∈ Xf there exists a feasible input sequence {u0 , . . . , uN −1 } for problem (12.6)
starting from the initial state x0 = x whose corresponding state trajectory is
{x0 , x1 , . . . , xN } stays in Xf , i.e., xi ∈ Xf ∀ i = 0, . . . , N . Among all the afore-
mentioned input sequences {u0 , . . . , uN −1 } we focus on the one where ui satisfies
assumption (A3) for all i = 0, . . . , N − 1. Such a sequence provides an upper bound
on the function J0∗ :
N −1
!
X
∗
J0 (x0 ) ≤ q(xi , ui ) + p(xN ), xi ∈ Xf , i = 0, . . . , N, (12.21)
i=0

which can be rewritten as

P 
N −1
J0∗ (x0 ) ≤ i=0 q(xi , u i ) + p(xN )
P 
N −1
= p(x0 ) + i=0 q(xi , ui ) + p(xi+1 ) − p(xi ) xi ∈ Xf , i = 0, . . . , N,
(12.22)
which from assumption (A3) yields
J0∗ (x) ≤ p(x), ∀x ∈ Xf . (12.23)
In conclusion, there exist a finite time in which any x ∈ X0 is steered to a level
set of J0∗ (x) contained in Xf after which convergence to and stability of the origin
follows. 

Remark 12.1 The assumption on the positive definiteness of the stage cost q(·) in
Theorem 12.2 can be relaxed as in standard optimal control. For instance for the
1
2-norm based cost function (12.8) one can allow Q  0 with (Q 2 , A) observable.

Remark 12.2 The procedure outlined in Theorem 12.2 is, in general, conservative
because it requires the introduction of an artificial terminal set Xf to guarantee
persistent feasibility and a terminal cost to guarantee stability. Requiring xN ∈ Xf
usually decreases the size of the region of attraction X0 = O∞ . Also the performance
may be negatively affected.

Remark 12.3 A function p(x) satisfying assumption (A3) of Theorem 12.2 is often
called control Lyapunov function.
256 12 Receding Horizon Control

The hypothesis (A2) of Theorem 12.2 is required for guaranteeing persistent

feasibility as discussed in Section 12.3.1. In some part of the literature the con-
straint Xf is not used. However, in this literature the terminal region constraint
Xf is implicit. In fact, it is typically required that the horizon N is sufficiently
large to ensure feasibility of the RHC (12.6)–(12.9) at all time instants t. Tech-
nically this means that N has to be greater than the determinedness index N̄ of
system (12.1)-(12.2) which by Corollary 12.2 guarantees persistent feasibility for
all inputs. We refer the reader to Section 12.3.1 for more details on feasibility.
Next we will show a few simple choices for P and Xf satisfying the hypothesis
(A2) and (A3) of Theorem 12.2.

Stability, 2-Norm case

Consider system (12.1)-(12.2), the RHC law (12.6)-(12.9), the cost function (12.8)
and the closed-loop system (12.10). A simple choice for Xf is the maximal positive
invariant set (see Section 10.1) for the closed-loop system x(k+1) = (A+BF∞ )x(k)
where F∞ is the associated unconstrained infinite time optimal controller (8.33).
With this choice the assumption (A3) in Theorem 12.2 becomes

x′ (A′ (P − P B(B ′ P B + R)−1 BP )A + Q − P )x ≤ 0, ∀x ∈ Xf , (12.24)

which is satisfied as an equality if P is chosen as the solution P∞ of the algebraic

Riccati equation (8.32) for system (12.1).
In general, instead of F∞ we can choose any controller F which stabilizes A +
BF . With v = F x the assumption (A3) in Theorem 12.2 becomes

− P + (Q + F ′ RF ) + (A + BF )′ P (A + BF ) ≤ 0. (12.25)

It is satisfied as an equality if we choose P as a solution of the corresponding

Lyapunov equation.
We learned in Section 7.5.2 that P satisfying (12.24) or (12.25) as equalities
expresses the infinite horizon cost
∞
X
∗
J∞ (x0 ) = x′0 P x0 = x′k Qxk + u′k Ruk . (12.26)
k=0

In summary, we conclude that the closed-loop system (12.10) is stable if there

exist a controller F which stabilizes the unconstrained system inside the controlled
invariant terminal region Xf and if the infinite horizon cost x′ P x incurred with
this controller is used in the cost function (12.8). Thus the two terms in the
objective (12.8) reflect the infinite horizon cost, one the initial finite horizon cost
when the controller is constrained and the second one the cost for the infinite
tail of the trajectory incurred after the system enters Xf and the controller is
unconstrained.
If the open loop system (12.1) is asymptotically stable, then we may even select
F = 0. Note that depending on the choice of the controller the controlled invariant
terminal region Xf changes.
 12.4 State Feedback Solution of RHC, 2-Norm Case 257

For any of the discussed choices for F and Xf stability implies exponential
stability. The argument is simple. As the system is closed-loop stable it enters
the terminal region in finite time. If Xf is chosen as suggested, the closed-loop
system is unconstrained after entering Xf . For an unconstrained linear system the
convergence to the origin is exponential.

Stability, 1, ∞-Norm case

Consider system (12.1)-(12.2), the RHC law (12.6)-(12.9), the cost function (12.7)
and the closed-loop system (12.10). Let p = 1 or p = ∞. If system (12.1) is
asymptotically stable, then Xf can be chosen as the positively invariant set of the
autonomous system x(k + 1) = Ax(k) subject to the state constraints x ∈ X .
Therefore in Xf the input 0 is feasible and the assumption (A3) in Theorem 12.2
becomes
− kP xkp + kP Axkp + kQxkp ≤ 0, ∀x ∈ Xf , (12.27)
which is the corresponding Lyapunov inequality for the 1, ∞-norm case (7.55)
whose solution has been discussed in Section 7.5.3.
In general, if the unconstrained optimal controller (9.31) exists it is PPWA. In
this case the computation of the maximal invariant set Xf for the closed-loop PWA
system
x(k + 1) = (A + F i )x(k) if H i x ≤ 0, i = 1, . . . , N r (12.28)
is more involved. However if such Xf can be computed it can be used as terminal
constraint in Theorem 12.2. With this choice the assumption (A3) in Theorem 12.2
is satisfied by the infinite time unconstrained optimal cost matrix P∞ in (9.32).

12.4 State Feedback Solution of RHC, 2-Norm Case

The state feedback receding horizon controller (12.9) with cost (12.8) for sys-
tem (12.1) is
u(t) = f0∗ (x(t)), (12.29)
where f0∗ (x0 ) : Rn → Rm is the piecewise affine solution to the CFTOC (12.6) and
is obtained as explained in Section 11.3.
We remark that the implicit form (12.6) and the explicit form (12.29) describe
the same function, and therefore the stability, feasibility, and performance proper-
ties mentioned in the previous sections are automatically inherited by the piecewise
affine control law (12.29). Clearly, the explicit form (12.29) has the advantage of
being easier to implement, and provides insight into the type of controller action
in different regions CRi of the state space.

Example 12.3 Consider the double integrator system (12.11) subject to the input
constraints
− 1 ≤ u(k) ≤ 1 (12.30)
258 12 Receding Horizon Control

and the state constraints

− 10 ≤ x(k) ≤ 10. (12.31)

We want to regulate the system to the origin by using the RHC problem (12.6)–(12.9)
with cost (12.8), Q = [ 10 01 ], R = 0.01, and P = P∞ where P∞ solves the algebraic
Riccati equation (8.32). We consider three cases:

Case 1. N = 2, Xf = 0,
Case 2. N = 2, Xf is the positively invariant set of the closed-loop system x(k + 1) =
(A + BF∞ ) where F∞ is the infinite time unconstrained optimal controller (8.33).
Case 3. No terminal state constraints: Xf = R2 and N = 6 = determinedness in-
dex+1.

From the results presented in this chapter, all three cases guarantee persistent fea-
sibility for all cost functions and asymptotic stability of the origin with region of
attraction X0 (with X0 different for each case). Next we will detail the matrices of
the quadratic program for the on-line solution as well as the explicit solution for the
three cases.

Case 1: Xf = 0. The mp-QP problem associated with the RHC has the form (11.31)
with
h i
 19.08 8.55
 −10.57 −5.29  10.31 9.33

H= 8.55 5.31 , F = −10.59 −5.29 , Y = 9.33 10.37 (12.32)

and

 0.00 −1.00 
 0.00 0.00 
0.00 0.00
 1.00 
0.00 1.00 1.00
0.00 0.00   1.00 1.00 
 −1.00 0.00   −1.00 −1.00   10.00
 10.00 


 0.00 0.00   −1.00 −1.00 
 1.00 0.00 
 1.00 1.00   10.00 
 −1.00 0.00 
 0.00 0.00   10.00 
 −1.00  −1.00 −1.00 
  10.00 
−1.00    10.00 
 1.00 0.00 
  0.00 0.00   10.00 

 1.00 1.00   1.00 1.00   10.00 
 0.00 0.00   1.00 1.00   5.00 
 −1.00 0.00 
 −1.00 −1.00   5.00 
 0.00  −1.00 −1.00   5.00 
0.00   1.00 1.00  5.00 
 1.00   
G0 = 0.00  , E0 = w0 =  (12.33)
 −1.00 −2.00  ,

 −1.00 0.00   1.00 
 1.00 0.00 
  1.00 2.00   1.00 

 0.00 0.00   1.00 0.00   10.00 
 0.00 0.00   0.00 1.00   10.00 
 0.00 0.00 
 −1.00 0.00   10.00 
 0.00 0.00 
 0.00 −1.00 
  10.00 
 0.00   5.00 
0.00   1.00 0.00   5.00 
 0.00 0.00   0.00 1.00   5.00 
 0.00 0.00   −1.00 0.00   
 0.00 0.00 
  0.00 −1.00   5.00 
 1.00 0.00   0.00 0.00 
  0.00 
 1.00 1.00  1.00 0.00
1.00 0.00
−1.00 0.00 0.00 0.00
−1.00 −1.00 0.00
−1.00 −1.00

The corresponding polyhedral partition of the state space is depicted in Fig. 12.6(a).
12.4 State Feedback Solution of RHC, 2-Norm Case 259

x2

x2
x1 x1
(a) Polyhedral partition of X0 , (b) Polyhedral partition of X0 ,
Case 1. N = 2, Xf = 0, N0r = 7. LQR
Case 2. N = 2, Xf = O∞ ,
r
N0 = 9.
x2

x1
(c) Polyhedral partition of X0 ,
Case 3. N = 6, Xf = R2 , N0r =
13.

Figure 12.6 Example 12.3. Double integrator. RHC with 2-norm. Region
of attraction X0 for different horizons N and terminal regions Xf .

The RHC law is:

  0.70 0.71
  0.00 

 −0.70 −0.71 0.00
 [ −0.61
 −1.61 ] x if −0.70 −0.71 x≤ 0.00 (Region #1)

 0.70 0.71 0.00



  −0.71

 −0.71   0.00


 −0.70 −0.71 −0.00

  −0.45 −0.89   0.45 

 [ −1.00 −2.00 ] x if x ≤ (Region #2)


0.45 0.89 0.45


0.71 0.71 0.71

 −0.70 −0.71 −0.00



 h i h i

 0.45 0.89
x≤
0.45
 [ −1.00
 −2.00 ] x if −0.70 −0.71 −0.00 (Region #3)

 0.71 0.71 −0.00



    

 −0.72 0.39
0.70
0.92
0.71
0.54
0.00
[ −1.72 ] x if −0.70 −0.71 x≤ 0.00 (Region #4)
u= 0.70 0.71 −0.00



    

 0.45 0.89 0.45

 −0.71 −0.71 0.71

  −0.45
0.70 0.71   −0.00 

 [ −1.00 −2.00 ] x if −0.89 x ≤ 0.45 (Region #5)

 0.00


0.71 0.71

 0.70 0.71 −0.00



 h −0.45 −0.89
i h 0.45
i

 x≤
 [
 −1.00 −2.00 ] x if −0.71 −0.71 −0.00 (Region #6)

 0.70 0.71 −0.00



  −0.39 −0.92   


0.54

 [ −0.72 −1.72 ] x if 0.70 0.71
x≤ 0.00
(Region #7)

 −0.70 −0.71 0.00

 −0.70 −0.71 −0.00
260 12 Receding Horizon Control

The union of the regions depicted in Fig. 12.6(a) is X0 . From Theorem 12.2, X0 is
also the domain of attraction of the RHC law.
Case 2: Xf positively invariant set. The set Xf is
 −0.35617 −0.93442   0.58043 
Xf = {x ∈ R2 : 0.35617 0.93442
0.71286 0.70131 x ≤ 0.58043
1.9049 }. (12.34)
−0.71286 −0.70131 1.9049

The corresponding polyhedral partition of the state space is depicted in Fig. 12.6(b).
The union of the regions depicted in Fig. 12.6(b) is X0 . Note that from Theorem 12.2
the set X0 is also the domain of attraction of the RHC law.
Case 3: Xf = Rn , N = 6. The corresponding polyhedral partition of the state space
is depicted in Fig. 12.6(c).
Comparing the feasibility regions X0 in Figure 12.6 we notice that in Case 2 we obtain
a larger region than in Case 1 and that in Case 3 we obtain a feasibility region larger
than Case 1 and Case 2. This can be easily explained from the theory presented in
this and the previous chapter. In particular we have seen that if a control invariant
set is chosen as terminal constraint Xf , the size of the feasibility region increases
with the number of control moves (increase from Case 2 to Case 3) (Remark 11.3).
Actually in Case 3, X0 = K∞ (Xf ) with Xf = R2 , the maximal controllable set. Also,
the size of the feasibility region increases with the size of the target set (increase from
Case 1 to Case 2).

12.5 State Feedback Solution of RHC, 1, ∞-Norm Case

The state feedback receding horizon controller (12.6)–(12.9) with cost (12.7) for
system (12.1) is
u(t) = f0∗ (x(t)) (12.35)
where f0∗ (x0 ) : Rn → Rm is the piecewise affine solution to the CFTOC (12.6)
and is computed as explained in Section 11.4. As in the 2-norm case the explicit
form (12.35) has the advantage of being easier to implement, and provides insight
into the type of control action in different regions CRi of the state space.

Example 12.4 Consider the double integrator system (12.11) subject to the input
constraints
− 1 ≤ u(k) ≤ 1 (12.36)
and the state constraints
− 5 ≤ x(k) ≤ 5. (12.37)
We want to regulate the system to the origin by using the RHC controller (12.6)–
(12.9) with cost (12.7), p = ∞, Q = [ 10 01 ], R = 20. We consider two cases:
Case 1. Xf = Rn , N = 6 (determinedness index+1) and P = Q
Case 2. Xf = Rn , N = 6 and P = P∞ given in (9.34) measuring the infinite time
unconstrained optimal cost in (9.32).
From Corollary 12.2 in both cases persistent feasibility is guaranteed for all cost func-
tions and X0 = C∞ . However, in Case 1 the terminal cost P does not satisfy (12.27)
which is assumption (A3) in Theorem 12.2 and therefore the convergence to and the
12.5 State Feedback Solution of RHC, 1, ∞-Norm Case 261

stability of the origin cannot be guaranteed. In order to satisfy assumption (A3) in

Theorem 12.2, in Case 2 we select the terminal cost to be equal to the infinite time
unconstrained optimal cost computed in Example 9.1.

Next we will detail the explicit solutions for the two cases.
Case 1. The LP problem associated with the RHC has the form (11.56) with Ḡ0 ∈
R124×18 , S̄0 ∈ R124×2 and c′ = [06 112 ]. The corresponding polyhedral partition of
the state space is depicted in Fig. 12.7(a). The RHC law is:
x2

x2

x1 x1
(a) Polyhedral partition of X0 , (b) Polyhedral partition of X0 ,
Case 1. Case 2.
x2
x2

x1 x1
(c) Closed-loop Trajectories, Case (d) Closed-loop Trajectories,
1. Convergence to origin to guar- Case 2.
anteed.

Figure 12.7 Example 12.4. Double integrator. RHC with ∞-norm cost
function, behaviour for different terminal weights.
262 12 Receding Horizon Control

  0.16 0.99
  0.82 

 0 if −0.16 −0.99
x≤ 0.82
(Region #1)

 −1.00 0.00 5.00

 1.00 0.00 5.00



    

 1.00 0.00 5.00

 −0.16 −0.99 −0.82

 [ −0.29 −1.71 ] x + 1.43 if −1.00 0.00 x≤ 5.00 (Region #2)

 0.16 0.99 1.40



  −0.16

 −0.99   −1.40 

 1.00 0.00 5.00

  −1.00
0.71 0.71   4.24 

  0.00   5.00 

 − 1.00 if  0.20 0.98  x ≤ 2.94  (Region #3)

  0.16 0.99   

 0.24 0.97
3.04
2.91
0.45 0.89 3.35
u= 0.32 0.95 3.00



  −1.00   

 0.00 5.00

 [ −0.29 −1.71 ] x − if 0.16 0.99
x≤ −0.82
(Region #4)

 1.43 1.00 0.00 5.00

 −0.16 −0.99 1.40



  −0.32

 −0.95   3.00


 −0.24 −0.97 2.91

  −0.20 −0.98   2.94 

  −0.16 −0.99   3.04 

 if  −1.00 0.00  x ≤  5.00  (Region #5)


1.00
 0.16 0.99   −1.40 



 −0.71 −0.71 4.24

 −0.45 −0.89 3.35

 1.00 0.00 5.00

The union of the regions depicted in Fig. 12.7(a) is X0 and is shown in white in
Fig. 12.7(c). Since N is equal to the determinedness index plus one, X0 is a positive
invariant set for the closed-loop system and thus persistent feasibility is guaranteed
for all x(0) ∈ X0 . However, it can be noticed from Fig. 12.7(c) that convergence to
the origin is not guaranteed. Starting from the initial conditions [-4,2], [-2, 2], [0,
0.5], [4,-2], [-1,-1] and [2,-0.5], the closed-loop system converges to either [−5, 0] or
[5, 0].
Case 2. The LP problem associated with the RHC has the form (11.56) with Ḡ0 ∈
R174×18 , S̄0 ∈ R174×2 and c′ = [06 112 ]. The RHC law is defined over 21 regions and
the corresponding polyhedral partition of the state space is depicted in Fig. 12.7(b).
The union of the regions depicted in Fig. 12.7(b) is X0 and is shown in white in
Fig. 12.7(d). Since N is equal to the determinedness index plus one, X0 is a positive
invariant set for the closed-loop system and thus persistent feasibility is guaranteed
for all x(0) ∈ X0 . Convergence to the origin is also guaranteed by the choice of P as
shown by the closed-loop trajectories in Fig. 12.7(d) starting from the same initial
conditions as in Case 1.

12.6 Tuning and Practical Use

Receding Horizon Control in the different variants and with the many parameter
options as introduced in this chapter is a very powerful control technique to address
complex control problems in practice. At present there is no other technique to
design controllers for general large linear multivariable systems with input and
output constraints with a stability guarantee. The fact that there is no effective
12.6 Tuning and Practical Use 263

tool to analyze the stability of large constrained multivariable systems makes this
stability guarantee by design so important. The application of RHC in practice,
however, requires the designer to make many choices. In this section we will offer
some guidance.

Objective Function

The squared 2-norm is employed more often as an indicator of control quality in

the objective function than the 1- or ∞-norm. The former has many advantages.
The 1- or ∞-norm formulation leads to an LP (11.56) from which the opti-
mal control action is determined. The solution lies always at the intersection of
constraints and changes discontinuously as the tuning parameters are varied. This
makes the formulation of the control problem (what constraints need to be added
for good performance?) and the choice of the weights in the objective function
difficult.
The optimizer may be non-unique in the case of dual degeneracy. This may
cause the control action to vary randomly as the solver picks up different optimizers
in succeeding time steps.
The construction of the control invariant target set Xf and of a suitable ter-
minal cost p(xN ) needed in Theorem 12.2 is more difficult for the 1- or ∞-norm
formulation because the unconstrained controller is PPWA (Section 12.3.2).
If the optimal control problem is stated in an ad hoc manner by trial and
error rather than on the basis of Theorem 12.2 undesirable response characteristics
may result. The controller may cease to take any action and the system may
get stuck. This happens if any control action leads to a short-term increase in
the objective function, which can occur, for example, for a system with inverse
response characteristics.
Finally, the 1- or ∞-norm formulation involves many more constraints than the
2-norm formulation. In general, this will lead to a larger number of regions of the
explicit control law. These regions do not need to be stored, however, but only
the value function and the controller, and the search for the region containing the
present state can be executed very efficiently (Chapter 14).
Despite the discussed deficiencies it may be advantageous to use the 1- or ∞-
norm in cases when the control objective is not just an indirect tool to achieve a
control specification but reflects the economics of the process. For example, the
costs of electrical power depend sometimes on peak power that can be captured
with the ∞-norm. However, even in these cases, quadratic terms are often added
to the objective for “regularization”.

Design via Theorem 12.2

First, we need to choose the horizon length N and the control invariant target
set Xf . Then we can vary the parameters Q and R freely to affect the control
performance in the same spirit as we do for designing an LQR. Stability is assured
as long as we adjust P according to the outlined procedures when changing Q and
264 12 Receding Horizon Control

R.
The longer the horizon N , the larger the maximal controllable set KN (Xf ) over
which the closed-loop system is guaranteed to be able to operate (this is true as
long as the horizon is smaller than the determinedness index of K∞ (Xf )). On the
other hand, with the control horizon increases the on-line computational effort or
the complexity of the explicit controller determined off-line.
We need to keep in mind that the terminal set Xf is introduced artificially
for the sole purpose of leading to a sufficient condition for persistent feasibility.
We want it to be large so that it does not compromise closed-loop performance.
The larger Xf , the larger KN (Xf ). Though it is simplest to choose Xf = 0, it is
undesirable unless N is chosen large. Ideally Xf should be the maximal control
invariant set achieved with the unconstrained controller.
Specifically, we first design the unconstrained optimal controller as suggested
at the end of Section 12.3.2. From this construction we also obtain the terminal
cost satisfying condition (A3) of Theorem 12.2 to use in the RHC design. Then
we determine the maximal positive invariant set for the closed-loop system with
the unconstrained controller and use this set as Xf . This set is usually difficult
to compute for systems of large dimension with the algorithms we introduced in
Section (10.2).
Note that for stable systems without state constraints K∞ (Xf ) = Rn always,
i.e., the choice of Xf is less critical. For unstable systems K∞ (Xf ) is the region
over which the system can operate stably in the presence of input constraints and
is of eminent practical importance.

State/Output Constraints

State constraints arise from practical restrictions on the allowed operating range
of the system. Thus, contrary to input constraints, they are rarely “hard”. They
can lead to complications in the controller implementation, however. As it can
never be excluded that the state of the real system moves outside the constraint
range chosen for the controller design, special provisions must be made (patches in
the algorithm) to move the state back into the range. This is difficult and these
types of patches are exactly what one wanted to avoid by choosing MPC in the
first place.
Thus, typically, state constraints are “softened” in the MPC formulation. For
example,
x ≤ xmax

is approximated by
x ≤ xmax + ǫ, ǫ ≥ 0,

and a term l(ǫ) is added to the objective function to penalize violations of the
constraint. This formulation may, however, lead to a violation of the constraint
even when a feasible input exists that avoids it but is not optimal for the new
objective function with the penalty term added. Let us analyze how to choose l(·)
such that this does not occur by using the exact penalty method [173].
12.6 Tuning and Practical Use 265

As an example take

J∗ = minz f (z)
(12.38)
subj. to g(z) ≤ 0,

where g(z) is assumed to be scalar for simplicity and f (z) to be strictly convex.
Let us soften the constraints and add the penalty as suggested

p(ǫ) = minz f (z) + l(ǫ)

(12.39)
subj. to g(z) ≤ ǫ,

where ǫ ≥ 0. We want to choose the penalty l(ǫ) so that by minimizing the

augmented objective we recover the solution to the original problem if it exists:

p(0) = J∗
and
arg minǫ≥0 p(ǫ) = 0

which are equivalent to

p(0) = J ∗ (12.40)
and
p(ǫ) > p(0), ∀ǫ > 0. (12.41)

For (12.40) we require l(0) = 0. To construct l(ǫ) to satisfy (12.41) we assume that
strong duality holds and u∗ exists so that

J ∗ = min(f (z) + u∗ g(z)), (12.42)

z

where u∗ is an optimal dual variable. As the optimizer z ∗ of (12.38) satisfies

g(z ∗ ) ≤ 0 we can add a redundant constraint without affecting the solution

J ∗ = minz (f (z) + u∗ g(z))

∀ǫ ≥ 0. (12.43)
subj. to g(z) ≤ ǫ

Then we can state the bounds

J∗ ≤ minz (f (z) + u∗ ǫ) p(ǫ) = minz(f (z) + uǫ)
< ∀ǫ > 0, u > u∗ .
subj. to g(z) ≤ ǫ g(z) ≤ ǫ
subj. to
(12.44)
Thus l(ǫ) = uǫ with u > u∗ ≥ 0 is a possible penalty term satisfying the require-
ments (12.40)-(12.41).
Because of smoothness

l(ǫ) = uǫ + vǫ2 , u > u∗ , v > 0 (12.45)

is preferable. On the other hand, note that l(ǫ) = vǫ2 does not satisfy an inequality
of the form (12.44). Therefore it should not be used as it can lead to optimizers
z ∗ in (12.39) which violate g(z) ≤ 0 even if a feasible optimizer to the original
problem exists.
266 12 Receding Horizon Control

These ideas can be extended to multiple constraints gj (z) ≤ 0, j = 1, . . . , r via

the penalty term
r
X r
X
l(ǫ) = u ǫj + v ǫ2j , (12.46)
j=1 j=1

where

u> max u∗j , v ≥ 0. (12.47)

j∈{1,...,r}

Formulations also exist where the necessary constraint violations are following
a prescribed order so that less important constraints are violated first [174].

Time-varying references, constraints, disturbances and system pa-

rameters

The standard RHC formulation (12.6) can be easily extended to include these
features. Known disturbances are simply included in the prediction model. If the
system states are not to return to the origin, but some output y is to follow some
trajectory r, then appropriate penalties of the error e = y − r are included in the
control objective. How to do this and to achieve offset-free tracking is described in
Section 12.7. Theorem 12.2 can be used to design the controller in these cases.
If the constraints are time-varying then X and U become time-varying. For
example, the constraints may be shaped like a funnel tightening towards the end
of the horizon.
If the underlying system is nonlinear, one often uses a locally linearized model
for the prediction and update it at each time step. Note that the results of The-
orem 12.2 do not apply when the system model and/or the constraints are time-
varying.
In all these cases, the optimization problem to be solved on line, (11.31) or (11.56),
does not change in structure but some of the defining matrices will now change at
each time step, which will increase the necessary on-line computational effort some-
what.
If the controller is to be computed explicitly off-line, then all the varying pa-
rameters (disturbances, constraints, references), which we will denote by θ, be-
come parameters in the multi-parametric optimization problem and the resulting
controller becomes an explicit function of them: u(t) = F (x(t), θ). As we have
learned, the complexity of the solution of mp-QP and mp-LP problems depends
primarily on the number of constraints. If the number of parameters affects the
number of constraints then this may only be possible for a relatively small number
of parameters. Thus, the possibilities to take into account time-varying control
problem features in explicit MPC are rather limited. Time-varying models cannot
be handled at all.
 12.7 Offset-Free Reference Tracking 267

Multiple Horizons and Move-Blocking

Unfortunately, as we started to discuss in the previous paragraphs, in challenging

applications, the design procedure implied by Theorem 12.2 may not be strictly
applicable. Then the closed-loop behavior with the RHC has to be analyzed by
other means, in the worst case through extensive simulation studies. In principle,
this offers us also more freedom in the problem formulation and the choice of the
parameters, for example, in order to reduce the computational complexity. The
basic RHC formulation (12.6) may be modified as follows:
Ny −1
X
min p(xNy ) + q(xk , uk )
U0
k=0
subj. to xk+1 = Axk + Buk , k = 0, . . . , Ny − 1 (12.48)
xk ∈ X , k = 0, . . . , Nc
uk ∈ U, k = 0, . . . , Nu
uk = Kxk , Nu < k < Ny

where K is some feedback gain, Ny , Nu , Nc are the prediction, input, and state
constraint horizons, respectively, with Nu ≤ Ny and Nc ≤ Ny . This formulation
reduces the number of constraints and as a consequence makes the long horizon
prediction used in the optimization less accurate as it is not forced to obey all the
constraints. As this approximation affects only the states far in the future, it is
hoped that it will not influence significantly the present control action.
Generalized Predictive Control (GPC) in its most common formulation [89] has
multiple horizons as an inherent feature but does not include constraints. Experi-
ence and theoretical analysis [56] have shown that it is very difficult to choose all
these horizons that affect not only performance but even stability in a non-intuitive
fashion. Thus, for problems where constraints are not important and the adaptive
features of GPC are not needed, it is much preferable to resort to the well estab-
lished LQR and LQG controllers for which a wealth of stability, performance and
robustness results have been established.
Another more effective way to reduce the computational effort is move-blocking
where the manipulated variables are assumed to be fixed over time intervals in
the future thus reducing the degrees of freedom in the optimization problem. By
choosing the blocking strategies carefully, the available RHC stability results remain
applicable [72].

12.7 Offset-Free Reference Tracking

This section describes how the RHC problem has to be formulated to track con-
stant references without offset under model mismatch. We distinguish between the
number p of measured outputs, the number r of outputs which one desires to track
(called “tracked outputs”), and the number nd of disturbances. First we summa-
rize the conditions that need to be satisfied to obtain offset-free RHC by using the
arguments of the internal model principle. Then we provide a simple proof of zero
268 12 Receding Horizon Control

steady-state offset when r ≤ p = nd . Extensive treatment of reference tracking

for RHC can be found in in [13, 218, 225, 226, 198]. Consider the discrete-time
time-invariant system

 xm (t + 1) = f (xm (t), u(t))
ym (t) = g(xm (t)) (12.49)

z(t) = Hym (t).
In (12.49), xm (t) ∈ Rn , u(t) ∈ Rm and ym (t) ∈ Rp are the state, input, measured
output vector, respectively. The controlled variables z(t) ∈ Rr are a linear combi-
nation of the measured variables for which offset-free behaviour is sought. Without
any loss of generality we assume H to have full row rank.
The objective is to design an RHC based on the linear system model (12.1)
of (12.49) in order to have z(t) track r(t), where r(t) ∈ Rr is the reference signal,
which we assume to converge to a constant, i.e., r(t) → r∞ as t → ∞. We require
zero steady-state tracking error, i.e., (z(t) − r(t)) → 0 for t → ∞.

The Observer Design

The plant model (12.1) is augmented with a disturbance model in order to capture
the mismatch between (12.49) and (12.1) in steady state. Several disturbance
models have been presented in the literature [13, 205, 193, 226, 225, 294]. Here we
follow [226] and use the form:

 x(t + 1) = Ax(t) + Bu(t) + Bd d(t)
d(t + 1) = d(t) (12.50)

y(t) = Cx(t) + Cd d(t)
with d(t) ∈ Rnd . With abuse of notation we have used the same symbols for state
and outputs of system (12.1) and system (12.50). Later we will focus on specific
versions of the model (12.50).
The observer estimates both states and disturbances based on this augmented
model. Conditions for the observability of (12.50) are given in the following theo-
rem.

Theorem 12.3 [215, 216, 226, 13] The augmented system (12.50) is observable
if and only if (C, A) is observable and
 
A − I Bd
(12.51)
C Cd
has full column rank.
Proof: From the Hautus observability condition system (12.50) is observable
iff  ′ 
A − λI 0 C′
has full row rank ∀λ. (12.52)
Bd′ I − λI Cd′
Again from the Hautus condition, the first set of rows is linearly independent iff
(C, A) is observable. The second set of rows is linearly independent from the first
12.7 Offset-Free Reference Tracking 269

n rows except possibly for λ = 1. Thus, for the augmented system the Hautus
condition needs to be checked for λ = 1 only, where it becomes (12.51). 

Remark 12.4 Note that for condition (12.51) to be satisfied the number of distur-
bances in d needs to be smaller or equal to the number of available measurements in y,
nd 6 p. Condition (12.51) can be nicely interpreted. It requires that the model of the
disturbance effect on the output d → y must not have a zero at (1, 0). Alternatively
we can look at the steady state of system (12.50)
    
A − I Bd x∞ 0
= , (12.53)
C Cd d∞ y∞

where we have denoted the steady state values with a subscript ∞ and have omit-
ted the forcing term u for simplicity. We note that from the observability condi-
tion (12.51) for system (12.50) equation (12.53) is required to have a unique solution,
which means, that we must be able to deduce a unique value for the disturbance d∞
from a measurement of y∞ in steady state.

The following corollary follows directly from Theorem 12.3.

Corollary 12.3 The augmented system (12.50) with nd = p and Cd = I is ob-

servable if and only if (C, A) is observable and
 
A − I Bd
det = det(A − I − Bd C) 6= 0. (12.54)
C I

Remark 12.5 We note here how the observability requirement restricts the choice of
the disturbance model. If the plant has no integrators, then det (A − I) 6= 0 and we
can choose Bd = 0. If the plant has integrators then Bd has to be chosen specifically
to make det (A − I − Bd C) 6= 0.

The state and disturbance estimator is designed based on the augmented model
as follows:
        
x̂(t + 1) A Bd x̂(t) B Lx ˆ
ˆ + 1) = 0 I
d(t ˆ
d(t)
+
0
u(t)+
Ld
(−ym (t)+C x̂(t)+Cd d(t)),
(12.55)
where Lx and Ld are chosen so that the estimator is stable. We remark that the
results below are independent of the choice of the method for computing Lx and
Ld . We then have the following property.

Lemma 12.3 Suppose the observer (12.55) is stable. Then, rank(Ld ) = nd .

Proof: From (12.55) it follows

        
x̂(t + 1) A + Lx C Bd + Lx Cd x̂(t) B Lx
ˆ + 1) = ˆ + u(t) − ym (t).
d(t Ld C I + Ld Cd d(t) 0 Ld
(12.56)
270 12 Receding Horizon Control

By stability, the observer has no poles at (1, 0) and therefore

 
A − I + Lx C Bd + Lx Cd
det 6= 0. (12.57)
Ld C Ld Cd
For (12.57) to hold, the last nd rows of the matrix have to be of full row rank.
A necessary condition is that Ld has full row rank. 
In the rest of this section we will focus on the case nd = p.

Lemma 12.4 Suppose the observer (12.55) is stable. Choose nd = p. The steady
state of the observer (12.55) satisfies:
    
A−I B x̂∞ −Bd dˆ∞
= , (12.58)
C 0 u∞ ym,∞ − Cd dˆ∞
where ym,∞ and u∞ are the steady state measured output and input of the sys-
tem (12.49), x̂∞ and dˆ∞ are state and disturbance estimates from the observer (12.55)
at steady state, respectively.
Proof: From (12.55) we note that the disturbance estimate dˆ converges only if
Ld (−ym,∞ + C x̂∞ + Cd dˆ∞ ) = 0. As Ld is square by assumption and nonsingular
by Lemma 12.3 this implies that at steady state, the observer estimates (12.55)
satisfy
− ym,∞ + C x̂∞ + Cd dˆ∞ = 0. (12.59)
Equation (12.58) follows directly from (12.59) and (12.55). 

The MPC design

Denote by z∞ = Hym,∞ and r∞ the tracked measured outputs and their references
at steady state, respectively. For offset-free tracking at steady state we want z∞ =
r∞ . The observer condition (12.58) suggests that at steady state the MPC should
satisfy     
A−I B x∞ −Bd dˆ∞
= , (12.60)
HC 0 u∞ r∞ − HCd dˆ∞
where x∞ is the MPC  state at steady
 state. For x∞ and u∞ to exist for any dˆ∞
A−I B
and r∞ the matrix must be of full row rank which implies m ≥ r.
HC 0
The MPC is designed as follows
N
X −1
minU0 (xN − x̄t )′ P (xN − x̄t ) + (xk − x̄t )′ Q(xk − x̄t ) + (uk − ūt )′ R(uk − ūt )
k=0
subj. to xk+1 = Axk + Buk + Bd dk , k = 0, . . . , N
xk ∈ X , uk ∈ U, k = 0, . . . , N − 1
xN ∈ Xf
dk+1 = dk , k = 0, . . . , N − 1
x0 = x̂(t)
ˆ
d0 = d(t),
(12.61)
12.7 Offset-Free Reference Tracking 271

with the targets ūt and x̄t given by

    
A−I B x̄t ˆ
−Bd d(t)
= ˆ (12.62)
HC 0 ūt r(t) − HCd d(t)

and where Q  0, R ≻ 0, and P ≻ 0.

Let U ∗ (t) = {u∗0 , . . . , u∗N −1 } be the optimal solution of (12.61)-(12.62) at time
t. Then, the first sample of U ∗ (t) is applied to system (12.49)

u(t) = u∗0 . (12.63)

ˆ r(t)) = u∗ (x̂(t), d(t),

Denote by c0 (x̂(t), d(t), ˆ r(t)) the control law when the esti-
0
ˆ
mated state and disturbance are x̂(t) and d(t), respectively. Then the closed-loop
system obtained by controlling (12.49) with the MPC (12.61)-(12.62)-(12.63) and
the observer (12.55) is:

ˆ r(t)))
x(t + 1) = f (x(t), c0 (x̂(t), d(t),
ˆ + Bc0 (x̂(t), d(t),
x̂(t + 1) = (A + Lx C)x̂(t) + (Bd + Lx Cd )d(t) ˆ r(t)) − Lx ym (t)
ˆ + 1) = Ld C x̂(t) + (I + Ld Cd )d(t)
d(t ˆ − Ld ym (t).
(12.64)
Often in practice, one desires to track all measured outputs with zero offset.
Choosing nd = p = r is thus a natural choice. Such a zero-offset property continues
to hold if only a subset of the measured outputs are to be tracked, i.e., nd = p > r.
Next we provide a very simple proof for offset-free control when nd = p.

Theorem 12.4 Consider the case nd = p. Assume that for r(t) → r∞ as t → ∞,

the MPC problem (12.61)-(12.62) is feasible for all t ∈ N+ , unconstrained for t ≥ j
with j ∈ N+ and the closed-loop system (12.64) converges to x̂∞ , dˆ∞ , ym,∞ , i.e.,
ˆ → dˆ∞ , ym (t) → ym,∞ as t → ∞. Then z(t) = Hym (t) → r∞ as
x̂(t) → x̂∞ , d(t)
t → ∞.

Proof: Consider the MPC problem (12.61)-(12.62). At steady state u(t) →

u∞ = c0 (x̂∞ , dˆ∞ , r∞ ), x̄t → x̄∞ and ūt → ū∞ . Note that the steady state con-
troller input u∞ (computed and implemented) might be different from the steady
state target input ū∞ .
The asymptotic values x̂∞ , x̄∞ , u∞ and ū∞ satisfy the observer conditions
(12.58)
    
A−I B x̂∞ −Bd dˆ∞
= (12.65)
C 0 u∞ ym,∞ − Cd dˆ∞
and the controller requirement (12.62)
    
A−I B x̄∞ −Bd dˆ∞
= . (12.66)
HC 0 ū∞ r∞ − HCd dˆ∞

Define δx = x̂∞ − x̄∞ , δu = u∞ − ū∞ and the offset ε = z∞ − r∞ . Notice that the
steady state target values x̄∞ and ū∞ are both functions of r∞ and dˆ∞ as given
272 12 Receding Horizon Control

by (12.66). Left multiplying the second row of (12.65) by H and subtracting (12.66)
from the result, we obtain

(A − I)δx + Bδu = 0
(12.67)
HCδx = ε.

Next we prove that δx =0 and thus ε = 0.

Consider the MPC problem (12.61)-(12.62) and the following change of variables
δxk = xk − x̄t , δuk = uk − ūt . Notice that Hyk − r(t) = HCxk + HCd dk − r(t) =
HCδxk + HC x̄t + HCd dk − r(t) = HCδxk from condition (12.62) with d(t) ˆ = dk .
Similarly, one can show that δxk+1 = Aδxk + Bδuk . Then, around the origin where
all constraints are inactive the MPC problem (12.61) becomes:
N
X −1
minδu0 ,...,δuN −1 δx′N P δxN + δx′k Qδxk + δu′k Rδuk
k=0
subj. to δxk+1 = Aδxk + Bδuk , 0≤k≤N (12.68)
δx0 = δx(t),
δx(t) = x̂(t) − x̄t .

Denote by KMP C the unconstrained MPC controller (12.68), i.e., δu∗0 = KMP C δx(t).
At steady state δu∗0 → u∞ − ū∞ = δu and δx(t) → x̂∞ − x̄∞ = δx. Therefore, at
steady state, δu = KMP C δx. From (12.67)

(A − I + BKMP C )δx = 0. (12.69)

By assumption the unconstrained system with the MPC controller converges. Thus
KMP C is a stabilizing control law, which implies that (A − I + BKMP C ) is non-
singular and hence δx = 0. 

Remark 12.6 Theorem 12.4 was proven in [226] by using a different approach.

Remark 12.7 Theorem 12.4 can be extended to prove local Lyapunov stability of
the closed-loop system (12.64) under standard regularity assumptions on the state
update function f in (12.64) [204].

Remark 12.8 The proof of Theorem 12.4 assumes only that the models used for the
control design (12.1) and the observer design (12.50) are identical in steady state in
the sense that they give rise to the same relation z = z(u, d, r). It does not make
any assumptions about the behavior of the real plant (12.49), i.e., the model-plant
mismatch, with the exception that the closed-loop system (12.64) must converge to
a fixed point. The models used in the controller and the observer could even be
different as long as they satisfy the same steady state relation.

Remark 12.9 If condition (12.62) does not specify x̄t and ūt uniquely, it is customary
to determine x̄t and ūt through an optimization problem, for example, minimizing
the magnitude of ūt subject to the constraint (12.62) [226].
12.7 Offset-Free Reference Tracking 273

Remark 12.10 Note that in order to achieve no offset we augmented the model of the
plant with as many disturbances (and integrators) as we have measurements (nd = p)
(cf. equation (12.56)). Our design procedure requires the addition of p integrators
even if we wish to control only a subset of r < p measured variables. This is actually
not necessary as we suspect from basic system theory. The design procedure for the
case nd = r < p is, however, more involved [198].

If the squared 2-norm in the objective function of (12.61) is replaced with a

P −1
1- or ∞- norm (kP (xN − x̄t )kp + N k=0 kQ(xk − x̄t )kp + kR(uk − ūt )kp , where
p = 1 or p = ∞), then our results continue to hold. In particular, Theorem 12.4
continues to hold. The unconstrained MPC controlled KMP C in (12.68) will be
piecewise linear around the origin [44]. In particular, around the origin, δu∗ (t) =
δu∗0 = KMP C (δx(t)) is a continuous piecewise linear function of the state variation
δx:
KMP C (δx) = F j δx if H j δx ≤ K j , j = 1, . . . , N r , (12.70)
where H j and K j in equation (12.70) are the matrices describing the j-th polyhe-
dron CRj = {δx ∈ Rn : H j δx ≤ K j } inside which the feedback optimal control
law δu∗ (t) has the linear form F j δx(k). The polyhedra CRj , j = 1, . . . , N r are a
partition of the set of feasible states of problem (12.61) and they all contain the
origin.

Explicit Controller
Examining (12.61), (12.62) we note that the control law depends on x̂(t), d(t) ˆ and
r(t). Thus in order to achieve offset free tracking of r outputs out of p measurements
we had to add the p + r “parameters” d(t) ˆ and r(t) to the usual parameters x̂(t).
There are more involved RHC design techniques to obtain offset-free control
for models with nd < p and in particular, with minimum order disturbance models
nd = r. The total size of the parameter vector can thus be reduced to n + 2r.
This is significant only if a small subset of the plant outputs are to be controlled.
A greater reduction of parameters can be achieved by the following method. By
Corollary 12.3, we are allowed to choose Bd = 0 in the disturbance model if the
plant has no integrators. Recall the target conditions 12.62 with Bd = 0
    
A−I B x̄t 0
= ˆ . (12.71)
HC 0 ūt r(t) − HCd d(t)

Clearly, any solution to (12.71) can be parameterized by r(t) − HCd d(t). ˆ The
ˆ
explicit control law is written as u(t) = c0 (x̂(t), r(t) − HCd d(t)), with only n + r
parameters. Since the observer is unconstrained, complexity is much less of an
issue. Hence, a full disturbance model with nd = p can be chosen to yield offset-
free control.

Remark 12.11 The choice of Bd = 0 might be limiting in practice. In [14], the

authors have shown that for a wide range of systems, if Bd = 0 and a Kalman filter is
chosen as observer, then the closed-loop system might suffer a dramatic performance
deterioration.
274 12 Receding Horizon Control

Delta Input (δu) Formulation.

In the δu formulation, the MPC scheme uses the following linear time-invariant
system model of (12.49):

 x(t + 1) = Ax(t) + Bu(t)
u(t) = u(t − 1) + δu(t) (12.72)

y(t) = Cx(t).
System (12.72) is controllable if (A, B) is controllable. The δu formulation often
arises naturally in practice when the actuator is subject to uncertainty, e.g., the
exact gain is unknown or is subject to drift. In these cases, it can be advantageous
to consider changes in the control value as input to the plant. The absolute control
value is estimated by the observer, which is expressed as follows
        
x̂(t + 1) A B x̂(t) B Lx
= + δu(t) + (−ym (t) + C x̂(t)).
û(t + 1) 0 I û(t) I Lu
(12.73)
The MPC problem is readily modified
minδu0 ,...,δuN −1 kyk − rk k2Q + kδuk k2R
subj. to xk+1 = Axk + Buk , k ≥ 0
yk = Cxk k ≥ 0
xk ∈ X , uk ∈ U, k = 0, . . . , N − 1
(12.74)
xN ∈ Xf
uk = uk−1 + δuk , k ≥ 0
u−1 = û(t)
x0 = x̂(t).
The control input applied to the system is
u(t) = δu∗0 + u(t − 1). (12.75)
The input estimate û(t) is not necessarily equal to the actual input u(t). This
scheme inherently achieves offset-free control, there is no need to add a disturbance
model. To see this, we first note that δu∗0 = 0 in steady-state. Hence our analysis
applies as the δu formulation is equivalent to a disturbance model in steady-state.
This is due to the fact that any plant/model mismatch is lumped into û(t). Indeed
this approach is equivalent to an input disturbance model (Bd = B, Cd = 0). If
in (12.74) the measured u(t) were substituted for its estimate, i.e., u−1 = u(t − 1),
then the algorithm would show offset.
In this formulation the computation of a target input ūt and state x̄t is not
required. A disadvantage of the formulation is that it is not applicable when there
is an excess of manipulated variables u compared to measured variables y, since
detectability of the augmented system (12.72) is then lost.

Minimum-Time Controller
In minimum-time control, the cost function minimizes the predicted number of
steps necessary to reach a target region, usually the invariant set associated with
 12.8 Literature Review 275

the unconstrained LQR controller [167]. This scheme can reduce the on-line com-
putation time significantly, especially for explicit controllers (Section 11.5). While
minimum-time MPC is computed and implemented differently from standard MPC
controllers, there is no difference between the two control schemes at steady-state.
In particular, one can choose the target region to be the unconstrained region
of (12.61)-(12.62). When the state and disturbance estimates and reference are
within this region, the control law is switched to (12.61)-(12.62). The analysis and
methods presented in this section therefore apply directly.

12.8 Literature Review

Although the basic idea of receding horizon control can be found in the theoretical
work of Propoi [239] in 1963 it did not gain much attention until the mid-1970s,
when Richalet and coauthors [248, 249] introduced their “Model Predictive Heuris-
tic Control (MPHC)”). Independently, in 1969 Charles Cutler proposed the con-
cept to his PhD advisor Dr. Huang at the University of Houston. In 1973 Cutler
implemented it successfully in the Shell Refinery in New Orleans [90].
Several years later Cutler and Ramaker [91] described this predictive control
algorithm named Dynamic Matrix Control (DMC) in the literature. It has been
hugely successful in the petro-chemical industry. A vast variety of different method-
ologies with different names followed, such as Quadratic Dynamic Matrix Con-
trol (QDMC), Adaptive Predictive Control (APC), Generalized Predictive Control
(GPC), Sequential Open Loop Optimization (SOLO), and others.
While the mentioned algorithms are seemingly different, they all share the same
structural features: a model of the plant, the receding horizon idea, and an opti-
mization procedure to obtain the control action by optimizing the system’s pre-
dicted evolution.
Some of the first industrial MPC algorithms like IDCOM [249] and DMC [91]
were developed for constrained MPC with quadratic performance indices. However,
in those algorithms input and output constraints were treated in an indirect ad-hoc
fashion. Only later, algorithms like QDMC [118] overcame this limitation by em-
ploying quadratic programming to solve constrained MPC problems with quadratic
performance indices. During the same period, the use of linear programming was
studied by Gutman and coauthors [137, 138, 140].
An extensive theoretical effort was devoted to analyze receding horizon control
schemes, provide conditions for guaranteeing feasibility and closed-loop stability,
and highlight the relations between MPC and the linear quadratic regulator ([204,
203]). Theorem 12.2 in this book is the main result on feasibility and stability of
MPC and was adopted from these publications.
The idea behind Theorem 12.2 dates back to Keerthi and Gilbert [168], the first
researchers to propose specific choices for the terminal cost P and the terminal
constraint Xf , namely Xf = 0 and P = 0. Under these assumptions Keerthi
and Gilbert prove the stability for general nonlinear performance functions and
nonlinear models. Their work has been followed by many other stability conditions
for RHC including those in [168, 34, 35, 154, 84]. If properly analyzed all these
276 12 Receding Horizon Control

results are based on the same concepts as Theorem 12.2.

In the past fifty years, the richness of theoretical and computational issues sur-
rounding MPC has generated a large number of research studies that appeared in
conference proceedings, archival journals and research monographs. The preced-
ing paragraphs are not intended to summarize this vast literature. We refer the
interested reader to some books on Model Predictive Control [76, 197, 247] that
appeared in the last decade.

For complex constrained multivariable control problems, model predictive con-

trol has long been the accepted standard in the process industries [240, 241]. The
results reported in this book have greatly reduced the on-line computational effort
and have opened up this methodology to other application areas where hardware
speed and costs are dominant - unlike in process control.
Chapter 13

Approximate Receding Horizon

Control

Contributed by Prof. Colin N. Jones

Ecole Polytechnique Fédérale de Lausanne
colin.jones@epfl.ch

This chapter explains the construction of approximate explicit control laws of

desired complexity that provide certificates of recursive feasibility and stability.
The methods introduced all have the following properties:

1. Operations are only on the problem data, and the methods do not require
the computation of the explicit control law.

2. Any convex parametric problem can be approximated, not just those for
which the explicit solution can be computed.

3. The desired level of suboptimality or the desired complexity (in terms of

online storage and/or FLOPS) can be specified.

We begin by assuming that a controller with the desired properties has already
been defined via one of the predictive control methods introduced in Chapters 12,
but that solving the requisite optimization problem online, or evaluating the ex-
plicit solution offline, is not possible due to computational limitations. The chapter
begins by listing the specific assumptions on this optimal control law, which is to
be approximated. Then, the key theorem is introduced, which provides sufficient
conditions under which an approximate control law will be stabilizing and persis-
tently feasible. Section 13.2 describes an abstract algorithm that will provide these
properties, and Section 13.3 reviews a number of specific implementations of this
algorithm that have been proposed in the literature.
278 13 Approximate Receding Horizon Control

13.1 Stability of Approximate Receding Horizon Control

Recall problem (12.6)
N
X −1
J0∗ (x(t)) = minU0 J0 (x(t), U0 ) = p(xN ) + q(xk , uk )
k=0
subj. to xk+1 = Axk + Buk , k = 0, . . . , N − 1 (13.1)
xk ∈ X , uk ∈ U, k = 0, . . . , N − 1
xN ∈ Xf
x0 = x(t).

The cost function is either

N
X −1
J0 (x(0), U0 ) = kP xN kp + kQxk kp + kRuk kp (13.2)
k=0

with p = 1 or p = ∞, or
N
X −1
J0 (x(0), U0 ) = x′N P xN + x′k Qxk + u′k Ruk . (13.3)
k=0

Denote a suboptimal feasible solution of (13.1) by Ũ0 = {ũ0 , . . . , ũN −1 }. The

corresponding suboptimal MPC law is

ũ(t) = f˜0 (x(t)) = ũ0 (x(t)) (13.4)

and the closed-loop system

x(k + 1) = Ax(k) + B f˜0 (x(k)) = f˜cl (x(k)), k ≥ 0. (13.5)

In the following we will prove that the stability of receding horizon control is
preserved when the suboptimal control law ũ0 (x(t)) is used as long as the level of
suboptimality is less than the stage cost, i.e.

J0 (x, Ũ0 (x)) < J0∗ (x) + q(x, 0).

This is stated more precisely in the following Theorem 13.1.

Theorem 13.1 Assume

(A0) The stage cost q(x, u) and terminal cost p(x) are continuous and positive
definite functions.

(A1) The sets X , Xf and U contain the origin in their interior and are closed.

(A2) Xf is control invariant, Xf ⊆ X .

(A3) min (−p(x) + q(x, v) + p(Ax + Bv)) ≤ 0, ∀x ∈ Xf .

v∈U , Ax+Bv∈Xf
13.1 Stability of Approximate Receding Horizon Control 279

(A4) The suboptimal Ũ0 (x) satisfies J0 (x, Ũ0 (x)) ≤ J0∗ (x) + γ(x), ∀x ∈ X0 ,
γ  0.

(A5) γ(x) − q(x, 0) ≺ 0, ∀x ∈ X0 , x 6= 0.

Then, the origin of the closed-loop system (13.5) is asymptotically stable with do-
main of attraction X0 .

Proof: The proof follows the proof of Theorem 12.2. We focus only on the
arguments for convergence, which are different for the suboptimal MPC controller.
We establish that the function J0∗ (·) in (13.1) strictly decreases for the closed-loop
system. Because the cost J0 , the system and the constraints are time-invariant we
can study the properties of J0∗ between step k = 0 and step k + 1 = 1.
Consider problem (13.1) at time t = 0. Let x(0) ∈ X0 and let Ũ0 = {ũ0 , . . . , ũN −1 }
be a feasible suboptimal solution to problem (13.1) and x0 = {x(0), x1 , . . . , xN } be
the corresponding state trajectory. We denote by J˜0 (x(0)) the cost at x(0) when
the feasible Ũ0 is applied, J˜0 (x(0)) = J0 (x(0), Ũ0 ).
After the implementation of ũ0 we obtain x(1) = x1 = Ax(0) + B ũ0 . Consider
now problem (13.1) for t = 1. We will construct an upper bound on J0∗ (x(1)).
Consider the sequence Ũ1 = {ũ1 , . . . , ũN −1 , v} and the corresponding state trajec-
tory resulting from the initial state x(1), x̃1 = {x1 , . . . , xN , AxN + Bv}. Because
xN ∈ Xf and (A2) there exists a feasible v such that xN +1 = AxN + Bv ∈ Xf
and with this v the sequence Ũ1 = {ũ1 , . . . , ũN −1 , v} is feasible. Because Ũ1 is not
optimal J0 (x(1), Ũ1 ) is an upper bound on J0∗ (x(1)).
Since the trajectories generated by Ũ0 and Ũ1 overlap, except for the first and
last sampling intervals, it is immediate to show that

J0∗ (x(1)) ≤ J0 (x(1), Ũ1 ) = J˜0 (x(0)) − q(x0 , ũ0 ) − p(xN )+

(13.6)
(q(xN , v) + p(AxN + Bv)).

Let x = x0 = x(0) and u = ũ0 . Under assumption (A3) equation (13.6) becomes

J0∗ (Ax + Bu) ≤ J˜0 (x) − q(x, u), ∀x ∈ X0 . (13.7)

Under assumption (A4) equation (13.7) becomes

J0∗ (Ax + Bu) − J0∗ (x) ≤ γ(x) − q(x, u) ≤ γ(x) − q(x, 0), ∀x ∈ X0 . (13.8)

Equation (13.8), the hypothesis (A0) on the matrices Q and R and the hypothesis
(A5) ensure that J0∗ (x) strictly decreases along the state trajectories of the closed-
loop system (13.5) for any x ∈ X0 , x 6= 0. 

Theorem 13.1 has introduced sufficient conditions for stability for the approxi-
mate solution Ũ0 (x) based on its level of suboptimality specified by the assumptions
(A4) and (A5). In the following section we describe how these conditions can be
met by sampling the optimal control law for different values of x and then using a
technique called barycentric interpolation.
280 13 Approximate Receding Horizon Control

To simplify the discussion we will adopt the notation for the general multipara-
metric programming problem from Chapter 5. We consider the multiparametric
program
J ∗ (x) = inf J(z, x)
z (13.9)
subj. to (z, x) ∈ C,
where
C = {z, x : g(z, x) ≤ 0}, (13.10)
z ∈ Z ⊆ Rs is the optimization vector, x ∈ X ⊆ Rn is the parameter vector,
J : Rs × Rn → R is the cost function and g : Rs × Rn → Rng are the constraints.
We assume throughout this chapter that the set C is convex, and that the func-
tion J(z, x) is simultaneously convex in z and x. Note that this implies that the
optimization problem (13.9) is convex for each fixed value of the parameter x. For
a particular value of the parameter x, we denote the set of feasible points as

R(x) = {z ∈ Z : g(z, x) ≤ 0}, (13.11)

and define the feasible set as the set of parameters for which (13.9) has a solution

K∗ = {x ∈ X : R(x) 6= ∅}, (13.12)

We denote by J ∗ (x) the real-valued function that expresses the dependence of the
minimum value of the objective function over K∗ on x

J ∗ (x) = inf {J(z, x) : z ∈ R(x)}, (13.13)

z

and by Z ∗ (x) the point-to-set map which assigns the (possibly empty) set of opti-
mizers z ∗ ∈ 2Z to a parameter x ∈ X

Z ∗ (x) = {z ∈ R(x) : J(z, x) = J ∗ (x)}. (13.14)

We make the following assumptions:

Convexity The function J ∗ (x) is continuous and convex.

Uniqueness The set of optimizers Z ∗ (x) is a singleton for each value of the pa-
rameter. The function z ∗ (x) = Z ∗ (x) will be called optimizer function.

Continuity The function z ∗ (x) is continuous.

Note that the solutions J ∗ (x), Z ∗ (x) and z ∗ (x) of problem (13.1) satisfy these
conditions. We emphasize, however, that the approximate methods introduced in
the following are applicable also to more general MPC problems as long as these
conditions of convexity, uniqueness and continuity are met.
For the approximation methods introduced below convexity of the function
γ(x)  0 (Theorem 13.1, assumptions (A4)-(A5)) bounding the level of subopti-
mality is necessary. It is not required, however, for the closed-loop stability with
the approximate control law.
 13.2 Barycentric Interpolation 281

13.2 Barycentric Interpolation

Schemes have been proposed in the literature that compute an approximate solution
to various problems of the form (13.9) based on interpolating the optimizer z ∗ (vi )
at a number of samples of the parameter V = {vi } ⊂ K∗ . The goal of interpolated
control is to define a function z̃(x) over the convex hull of the sampled points
conv(V ), which has the required properties laid out in Theorem 13.1.
Section 13.2.1 will introduce the notion of barycentric interpolation, and will
demonstrate that such an approach will lead to an approximate function z̃(x),
which is feasible
z̃(x) ∈ R(x), ∀x ∈ conv(V ). (13.15)
Then Section 13.2.2 introduces a convex optimization problem, whose optimal
value, if positive, verifies a specified level of suboptimality
J(z̃(x), x) ≤ J ∗ (x) + γ(x), ∀x ∈ conv(V ). (13.16)
By Theorem 13.1 the stability of the resulting suboptimal interpolated control law
follows.

13.2.1 Feasibility

The class of interpolating functions z̃(x) satisfying (13.15) is called the class of
barycentric functions.

Definition 13.1 (Barycentric function) Let V = conv(v1 , . . . , vn ) ⊂ Rn be a

polytope. The set of functions wv (x), v ∈ extreme(V ) is called barycentric if three
conditions hold for all x ∈ V
wv (x) ≥ 0 positivity (13.17a)
X
wv (x) = 1 partition of unity (13.17b)
v∈extreme(V )
X
vwv (x) = x linear precision (13.17c)
v∈extreme(V )

Many methods have been proposed in the literature to generate barycentric

functions, several of which are detailed in Section 13.3, when we introduce con-
structive methods for generating approximate controllers. Figure 13.1 illustrates
several standard methods of barycentric interpolation.
Given a set of samples V , and a set of barycentric interpolating functions
{wv (x)}, we can now define an interpolated approximation z̃(x):
X
z̃(x) = z ∗ (v)wv (x) (13.18)
v∈extreme(V )

The key statement that can be made about functions defined via barycentric
interpolation is that the function values lie within the convex hull of the function
values at the sample points. This property is illustrated in Figure 13.2, and formally
defined in the following Lemma.
282 13 Approximate Receding Horizon Control

(a) Linear interpolation. (b) Wavelet interpolation.

(c) Barycentric interpolation de-

fined in Lemma 13.5.

Figure 13.1 Examples of different barycentric interpolations.

Lemma 13.1 (Convex Hull Property of Barycentric Interpolation) If V =

{v0 , . . . , vm } ⊂ K∗ , z ∗ (vi ) is the optimizer of (13.9) for the parameter vi , and
{wv (x)} is a set of barycentric interpolating functions, then

X
z̃(x) = z ∗ (v)wv (x) ∈ R(x) for all x ∈ conv(V ).
v∈V

Proof: The statement holds if for each x ∈ conv(V ) there exists a set of positive
multipliers λ0 , . . . , λm such that

  m
X  
x vi
= λi
z̃(x) z ∗ (vi )
i=0

Pm
and i=0 λi = 1. The properties of barycentric functions (13.17) clearly satisfy
this requirement. 
 13.2 Barycentric Interpolation 283

Figure 13.2 Illustration of the Convex-Hull Property of Barycentric Inter-

polation. The first image shows a number of samples of a function, the
second the convex hull of these sample points and the third, a barycentric
interpolation that lies within this convex hull.

13.2.2 Suboptimality

Next we investigate the level of suboptimality obtained for a barycentric inter-

polation. First we demonstrate that the value of the function J (z, x) for the
approximate solution z̃(x) is upper-bounded by the barycentric interpolation of its
optimal value taken at the sample points. Then we introduce a convex optimization
problem, whose solution upper-bounds the approximation error.

Lemma 13.2 If z̃(x) is a barycentric interpolation as defined in (13.18), then the

following condition holds:

X
J ∗ (x) ≤ J (z̃(x), x) ≤ wv (x)J ∗ (v) for all x ∈ conv(V ).
v∈V

Proof: The lower bound follows from the fact that z̃(x) is feasible for the
optimization problem for all x ∈ V (Lemma 13.1).
284 13 Approximate Receding Horizon Control

The upper bound follows from convexity of the function J (z, x). Consider a
fixed value of the parameter x ∈ conv(V ):
!
X X
J (z̃(x), x) = J wv (x)z ∗ (v), wv (x)v
v∈V v∈V
X
∗
≤ wv (x)J (z (v) , v)
v∈V
X
= wv (x)J ∗ (v).
v∈V

The inequality in the above sequence follows because for a fixed value of the pa-
rameter x, the barycentric interpolating functions {wv } are all positive, and sum
to one. 
The previous lemma demonstrates that due to the convexity of the value func-
tion J (z, x), the interpolation of the optimal value function at the sample points
provides an upper bound of the value function evaluated at the interpolated approx-
imation z̃(x) (Fig. 13.3). We can use this fact to formulate a convex optimization
problem, whose solution will provide a certificate for (13.16).

Theorem 13.2 If δV ≥ 0, then J (z̃(x), x) ≤ J ∗ (x) + γ(x) for all x ∈ conv(V ),

where
X
δV = min J (z, x) + γ(x) − λv J ∗ (v) (13.19)
x,z,λv
v∈V
X X
s.t. x = λv v, λv ≥ 0, λv = 1
v∈V v∈V
(x, z) ∈ C

Proof: We see that the suboptimality bound is met within the convex hull of
V if and only if the following value is non-negative:

min J ∗ (x) + γ(x) − J (z̃(x), x) (13.20)

λv ,x∈conv(V )

Lemma 13.2 provides a lower-bound

X
(13.20) ≥ min J ∗ (x) + γ(x) − wv (x)J ∗ (v)
x∈conv(V )
v∈V

and we obtain a further relaxation / lower bound by optimizing over all possible
barycentric functions
X
(13.20) ≥ min J ∗ (x) + γ(x) − λv J ∗ (v)
x∈conv(V )
v∈V
X X
s.t. x= λv v, λv ≥ 0, λv = 1.
v∈V v∈V

Recalling that J ∗ (x) = minz J (z, x) s.t. (x, z) ∈ C provides the desired result. 
 13.2 Barycentric Interpolation 285

J ⋆ (v1 )
P
wv (x)J ⋆ (v)

J(z̃(x), x) J ⋆ (v2 )
J ⋆ (x)
v1 v2

Figure 13.3 Illustration of the relationship between the optimal value func-
tion J ∗ (x), the value function evaluated for the sub-optimal function z̃(x)
and the convex hull of the optimal value at the sample points.

In this section we have demonstrated that if the approximate solution z̃(x) is

generated via barycentric interpolation, it will satisfy (13.15) and that the con-
dition for suboptimality (13.16) can be verified by solving a convex optimization
problem. In the next section, we will introduce several constructive techniques to
incrementally build approximate barycentric interpolating functions.

13.2.3 Constructive Algorithm

In this section, we introduce a simple high-level algorithm to incrementally con-

struct a barycentric function that satisfies conditions (13.15)- (13.16) and therefore
can be used in Theorem 13.1 to generate a stabilizing control law. The idea is
simple: begin with a set of sample points and solve the optimization problem in
Theorem 13.2 to test if the stability condition is met. If it is not, then sample an
additional set of points, sub-divide the resulting region into sub-regions and repeat
the procedure for each such generated region. In the following we formalize this
idea, and demonstrate that this procedure will terminate.

Algorithm 13.1 Approximate Parametric Programming with Stability Guarantee

Input: Parametric Optimization Problem (13.9). Initial set of samples V
Output: Sample points V ⊂ K∗ and set of subsets R = {Rj , j = 1, . . .}, Rj ⊆ V
such that {conv(Rj ), j = 1, . . .} forms a partition of conv(V ) and δRj ≥ 0 for all j
Let R ← V
Repeat
R∗ ← argminRj ∈R δRj (from problem (13.19))
If δR∗ < 0 Then
V ← V ∪ {xR∗ }, where xR∗ is the optimizer of (13.19) for region R∗
R ← partition(V, R)
Until δR∗ ≥ 0
Return R
286 13 Approximate Receding Horizon Control

Algorithm 13.1 outlines at a high level a method for constructing a polytopic

partition (Definition 4.6) of a set of initial samples, where each polytope in the
partition satisfies the suboptimality condition of Theorem 13.1, and therefore any
barycentrically interpolated control law defined over the partition will be stabiliz-
ing. One can see that in each iteration of the algorithm, the region R∗ with the
worst fit measured in terms of δR∗ is computed, and the parameter xR∗ at which
this maximum error occurs is added to the samples. The sample points V are then
updated to include this new point, and the set R is re-computed. This set R defines
a partition splitting conv(V ) into a set of polytopes {conv(Rj ), j = 1, . . .}. Once
this key function “partition” is defined, this algorithm can be directly implemented.
Several methods have been proposed in the literature that roughly follow the
scheme of Algorithm 13.1 [37, 159, 3, 156]. Many of these approaches differ in
the details by adding caching steps to minimize computation of δR , add multiple
points at a time, or take an incremental re-partitioning approach. All of these
procedures can significantly improve the computation speed of the algorithm, and
should therefore be incorporated into any practical approach.

13.3 Partitioning and Interpolation Methods

This section outlines three approaches to partitioning the set of samples and the
associated barycentric interpolation methods. While many proposals have been
put forward in the literature, we focus here on three techniques: inner polytopic
approximation (triangulation), outer polytopic approximation (convex hull) and
second-order interpolants (hierarchical grids). For each partitioning approach, we
present an appropriate barycentric interpolation method.

13.3.1 Triangulation

The simplest approach to generate a barycentric function is to first triangulate

the sampled points V , and then define a piece-wise affine control law over the
triangulation. The resulting interpolation of the optimal value function at the
sampled points will result in a convex piecewise affine inner approximation of the
optimal value function, and the approximate control law will be a piecewise affine
function.

Definition 13.2 (Triangulation) A triangulation of a finite set of points V ⊂

Rn is a finite collection TV = {R0 , . . . , RL } such that

• Ri = conv(Vi ) is an n−dimensional simplex for some Vi ⊂ V

• conv(V ) = ∪Ri and int Ri ∩ int Rj = ∅ for all i 6= j

• If i 6= j then there is a common (possibly empty) face F of the boundaries of

Si and Sj such that Si ∩ Sj = F .
 13.3 Partitioning and Interpolation Methods 287

There are various triangulations possible; for example, the recursive triangula-
tion developed in [232, 37] has the strong property of generating a simple hierarchy
that can significantly speed online evaluation of the resulting control law. The De-
launay triangulation [109], which has the nice property of minimizing the number
of skinny triangles, or those with small angles is a common choice for which incre-
mental update algorithms are well-studied and readily available (i.e., computation
of TV ∪{v} given TV ). A particularly suitable Delaunay triangulation can be defined
by using the optimal cost function as a weighting term, which causes the resulting
triangulation to closely match the optimal partition [159].
Given a discrete set of states V , we can now define an interpolated control law
z̃(x; TV ) : Rn 7→ Rm .

Definition 13.3 If V ⊂ X ⊂ Rn is a finite set and T = {conv(V1 ), . . . , conv(VL )}

is a triangulation of V , then the interpolated function z̃(x; T ) : conv(V ) 7→ Rm is
X
z̃(x; T ) = z ∗ (v)λv , if x ∈ conv(Vj ) (13.21)
v∈Vj

P P
where λv ≥ 0, λv = 1 and x = v∈Vj vλv .

For each simplical region conv(Vj ) in the triangulation, the set of multipliers λ
is unique and can be found by solving the system of linear equations
 −1  
v1 ··· vn+1 x
λ=
1 ··· 1 1

where vi ∈ Vj .
The function z̃(x; T ) is piecewise affine, and given by:
 ′
z ∗ (v1 )  −1  
 ..  v1 ··· vn+1 x
z̃(x; T ) =  .  if x ∈ Vj = conv({v1 , . . . , vn+1 }).
1 ··· 1 1
z ∗ (vn+1 )

The simplical nature of each region immediately demonstrates that the inter-
polation 13.21 is barycentric.

13.3.2 Outer Polyhedral Approximation

The previous section introduced an approach that generates an inner polytopic

approximation of the optimal value function, which results in a triangulation of the
state space. This section introduces an outer polytopic approximation method that
will result in a general polytopic partition of the space, which is often significantly
less conservative than a triangulation, i.e., for the same number of samples it
provides a tighter approximation of the value function.
288 13 Approximate Receding Horizon Control

Implicit Double Description Algorithm: Computing an Outer Polyhedral Ap-

proximation of a Convex Set
We begin by introducing an algorithm, called the implicit double description method [162],
which computes an outer polyhedral approximation in an incremental fashion that
is greedy-optimal in terms of the Hausdorff metric defined below.
The epigraph of the optimal value function plus the specified desired approxi-
mation error is defined as:

epi J ∗,γ = {(x, J) : J ≥ J ∗ (x) + γ(x)}.

The epigraph of a convex function is a convex set. Our goal is to compute a

polyhedron P which is an inner approximation of the epigraph of the optimal
value function and an outer approximation of the optimal value function plus the
specified desired approximation error (Figure 13.4):

P J ⋆ (x) + γ(x)

J ⋆ (x)

Figure 13.4 Polyhedron P constructed to be an outer approximation of

epi J ∗,γ and an inner approximation of epi J ∗ .

epi J ∗ ⊇ P ⊇ epi J ∗,γ .

Once such an approximation is in hand, we will then use it to compute a polytopic

partition of K∗ over which the approximate control law will be defined.
The quality of approximation is measured in terms of the Hausdorff distance,
which measures the distance between two sets X and Y as:

dH (X, Y ) = max supx∈X inf y∈Y kx − yk2 , supy∈Y inf x∈X kx − yk2 .

Our goal is to compute the Hausdorff distance between our polytopic approxi-
mation, and epi J ∗,γ by solving a convex optimization problem. We first recall the
definition of the projection of a point v onto a convex set S

projS v = arg minx∈S kv − xk22

The following lemma states that we can compute the projection of a point
onto the epigraph of the optimal value function by solving a convex optimization
problem.
13.3 Partitioning and Interpolation Methods 289

Lemma 13.3 The projection of (v, Jv ) onto the epigraph epi J ∗,γ is the optimizer
of the following problem, (x∗ , J ∗ ):
min kx − vk22 + kJ (z, x) + γ(x) − Jv k22
z,x

s.t. (x, z) ∈ C.
We can now use the fact that the maximizer of a convex function over a convex
set will always occur at an extreme point of the set to write the Hausdorff distance
as the maximum of the projection of the vertices of the polyhedron P onto the
epigraph, which can be done by testing a finite number of points, since a polyhedron
has a finite number of vertices.

Lemma 13.4 If P is a polyhedron, and P ⊇ epi J ∗,γ is a convex set, then the
Hausdorff distance is:
dH (P, epi J ∗,γ ) = max(v,Jv )∈extreme(P ) kx∗ (v) − vk22 + kJ ∗ (v) + γ(x) − Jv k22
where (x∗ (v), J ∗ (v)) is the projection of (v, Jv ) onto the epigraph.
Algorithm 13.2 below will generate a sequence of outer polyhedral approxima-
tions of the epigraph that are monotonically decreasing in terms of the Hausdorff
distance. In each iteration of the algorithm, the Hausdorff distance is computed,
and the extreme point v of P at which the worst-case error occurs is removed from
P by adding a seperating hyperplane between the point v and the epigraph. The
addition of a seperating hyperplane ensures that the updated polyhedron remains
an outer approximation, and since the worst-case point is removed from the set,
we have that the approximation error is monotonically decreasing in the Hausdorff
distance. The algorithm terminates when all of the vertices of P are contained in
the epigraph of J ∗ . Figure 13.5 illustrates the process of a single cycle of the im-
plicit double description algorithm. Figure 13.6 shows a progression of steps until
the termination criterion extreme(P ) ⊂ epi J ∗ (x) is satisfied.

Figure 13.5 Example of one iteration of the implicit double-description al-

gorithm. Square: Point vmax at which the worst-case error occurs. Circle:
Point z ∗ (vmax ); Projection of vmax onto the set. Line: Separating hyper-
plane.

Algorithm 13.2 Approximate Parametric Programming with Stability Guarantee

290 13 Approximate Receding Horizon Control

Input: Parametric Optimization Problem (13.9). Initial polyhedron P ⊃ epi J ∗,γ

Output: Polyhedron P such that epi J ∗,γ ⊆ P ⊆ epi J ∗
Repeat
ForEach v ∈ extreme(P )
z ∗ (v) = minz∈epi J ∗,γ kv − zk22
vmax ← argmaxv∈extreme(P ) kv − z ∗ (v)k
P ← P ∩ {z : [z ∗ (vmax ) − v]′ (z − z ∗ (vmax )) ≤ 0}
Until extreme(P ) ⊂ epi J ∗ (x)

Approximate Control Law from Polyhedral Approximation

Once an outer approximation P of the epigraph has been computed, then an ap-
proximate control law can be defined as follows. For each facet F of P define the
polytope RF as the projection of the facet F onto the state space

RF = {x : ∃J, (x, J) ∈ F }.

The set of all such polytopes, {RF }, forms a partition of the feasible set of the
controller and we define our approximate control law over this set:
X
z̃(x) = u∗ (v)wv,F (x), if v ∈ RF (13.22)
v∈extreme(RF )

where wv,F (x) is a barycentric set of functions for the polytope RF . A method to
construct such a function will be described in the next sub-section.
The following theorem demonstrates that this approximate z̃(x) satisfies con-
dition (13.16).

Theorem 13.3 Let z̃(x) be defined as in (13.22) for a polyhedral approximation

P that satisfies the condition extreme(P ) ⊂ epi J ∗ (x), then condition (13.16) is
satisfied.

Proof:

J ∗ (x) ≤ J(z̃(x), x) Feasibility of barycentric interpolation

from Lemma 13.1
X
≤ J ∗ (v)wv,F (x), if v ∈ RF Convexity of J ∗
v∈extreme(RF )
X
≤ Jv wv,F (x), if v ∈ RF P ⊆ epi J ∗
(v,Jv )∈extreme(F )

≤ min J (v, Jv ) are all on a hyperplane

(x,J)∈P

≤ J ∗ (x) + γ(x) P ⊇ epi J ∗,γ


 13.3 Partitioning and Interpolation Methods 291

Barycentric Functions for Polytopes

Our goal is now to define an easily computable barycentric function for each poly-
tope RF . If the polytope RF is a simplex, then the barycentric function is unique,
linear and trivially computed and so we focus on the non-simplical case. In [285]
a very elegant method of computing a barycentric function for arbitrary polytopes
was proposed that can be put to use here.

Lemma 13.5 (Barycentric coordinates for polytopes [285]) Let S = conv(V ) ⊂

Rd be a polytope and for each simple vertex v of S, let bv (x) be the function
αv
bv (x) =
kv − xk2

where αv is the area of the polytope {y : [V − 1x′ ]y ≤ 0, (v − x)′ y = 1}; i.e., the
P dual of S − {x} corresponding to the vertex v − x. The
area of the facet of the polar
function wv (x) = bv (x)/ v bv (x) is barycentric over the polytope S.

The areas of the facets of the polar duals αv can be pre-computed offline and
stored. If there are d + 1 facets incident
 with the vertex
v (i.e., v is simplical),
then the area of the polar facet is det a0 · · · ad+1 , where {a0 , . . . , ad+1 } are
the normals of the incident facets. If the vertex is not simplical, then the area can
be easily computed by perturbing the incident facets [285]. Such computation is
straightforward because both the vertices and halfspaces of each region are available
due to the double-description representation.

13.3.3 Second-Order Interpolants

This section summarizes the hierarchical grid-based approach proposed in [270].

The main idea is to partition the space into a grid, which is then extremely fast
to evaluate online. We first introduce an interpolation method based on second-
order interpolants and demonstrate that it is barycentric. We then discuss a sparse
hierarchical gridding structure, which allows for extremely sparse storage of the
data and rapid evaluation of the function.
Our goal is to define a set of basis functions, one for each sampled vertex of
the grid, whose sum is barycentric within each hypercube of the grid. We begin
by defining the one-dimensional scaling function (hat function), with support on
[−1, 1] as:
(
1 − |x| if x ∈ [−1, 1]
φ(x) =
0 otherwise.

For a particular grid point v, with grid width of w, we can extend the hat function
by scaling and shifting:
 
x−v
φv,w (x) = φ .
w
292 13 Approximate Receding Horizon Control

We extend the function to d-dimensions by simply taking the product:

d
Y
φv,w (x) = φvi ,w (xi ),
i=1

where vi and xi are the ith components of the vectors v and x respectively.
The resulting basis function φ0,1 (x) is illustrated in two dimensions in Fig-
ure 13.7.
We now define our interpolated approximation as:
X
z̃(x) = z ∗ (v)φv,w (x)
v∈G

where w is the width of the grid, and G is the set of grid vertices.

Lemma 13.6 [270] Let R be the hypercube defined by

w
R= (B∞ ⊕ 1) ⊕ v
2
where B∞ = {x : kxk∞ ≤ 1} is the unit infinity-norm ball. The set of functions
{φv,w (x)} is barycentric within the set R.

Lemma 13.6 states that each hypercube in the grid is in fact barycentric. In
order to apply Algorithm 13.1 we need a method of re-partitioning the current
sample points. We do this via a hierarchical approach, in which each hypercube
of the grid is tested for compliance with condition (13.16) and, if it fails, is sub-
divided into 2d more hypercubes. The result is a hierarchical grid, that is very easy
to evaluate online.
What has been discussed so far is termed a ‘nodal’ representation of the approxi-
mate function. It is also possible to define what is termed a ‘hierarchical multi-scale’
representation, which, at each level of the re-gridding procedure, approximates only
the residual, rather than the original function. This approach generally results in
an extremely sparse representation of the data, and significantly ameliorates the
downsides of sampling on a grid. See [270] for more details.

Example 13.1 We demonstrate and compare the three approximation algorithms

introduced in this Chapter on a simple example. The result of this comparison
should not be generalized to other problems, since the performance of any given
approximation method will depend strongly on the problem being considered.
Consider the following system:
   
1 1 1
x(k + 1) = x(k) + u(k)
0 1 0.5

with state and input constraints:

x ∈ X = {x | kxk∞ ≤ 5} u ∈ U = {u | kuk∞ ≤ 1}.

13.3 Partitioning and Interpolation Methods 293

We approximate the control law resulting for a standard quadratic-cost MPC problem
with a horizon of 10 and weighting matrices taken to be
 
0.1 0
Q= , R = 1.
0 2

The MPC problem formulation includes a terminal set and terminal weight, which
are computed as discussed in Chapter 12 as the maximum invariant set for the system
when controlled with the optimal LQR control law.
The optimal value function J ∗ for this problem can be seen in Figure 13.8, along with
the approximation target J ∗ (x) + γ(x) for γ(x) = x′ Qx.
Figures 13.9-13.11 shows the evolution of the approximation algorithm using a trian-
gulation approach, and Figure 13.12 shows the resulting approximate control law and
the approximation error. The approximation error seems quite large but this is all
what is needed when the only objective is to retain stability rather than performance.
Figures 13.13-13.14 shows the same evolution, but for the double description/outer
approximation approach. One can see that far fewer sample points are required to
reach a stability certificate.
The example has also been approximated using the second-order interpolant method
until the stability certification was achieved. Figure 13.15 shows the resulting nodal
partition of the state space. Note that since it would take an infinite number of
hypercubes to approximate a non-orthogonal polytope, a soft-constrained version of
the problem is taken in the domain outside the feasible set (although the stability
certificate is valid only within the feasible set). One can see the approximate control
law, the resulting error and the resulting approximated value function in Figures 13.16
and 13.17, respectively.
Table 13.1 shows the required online data storage and computation time for the
various methods introduced in this chapter for the studied example, as well as the
average and worst-case approximation errors (based on samples). One can see that
for this small example, the second-order interpolants are clearly the best solution.
However, this should not be taken as a clear indicator, or a recommendation for
the method beyond the others, since this would require a thorough computational
comparison of the various methods (or their variants in the literature) on a wide
range of practical problems.

Table 13.1 Example 13.1. Comparison of the approximation methods intro-

duced in Section 13.3. The indicated Online FLOPS for triangulation are
based on linear search.

Data Online FLOPS Approx error kũ(x) − u∗ (x)k

(numbers) (avg) (max)
Triangulation 120 1,227 3.3% 24.6%
Polyhedral approx 720 752 2.5% 22.6%
Second-order interpolants 187 26 1.1% 8.3%
294 13 Approximate Receding Horizon Control

J ⋆ (x) + γ(x)
P

J ⋆ (x)

(a) Iteration 1.

J ⋆ (x) + γ(x)
P

J ⋆ (x)

(b) Iteration 2.

P J ⋆ (x) + γ(x)

J ⋆ (x)

(c) Last iteration.

Figure 13.6 Several steps of the Double Description algorithm with the ter-
mination criterion extreme(P ) ⊂ epi J ∗ (x) satisfied in the last step.
13.3 Partitioning and Interpolation Methods 295

1
φ0,1 (x) 0.8
0.6
0.4
0.2
0
1
0 1
-1 0
x2 -1
x1
Figure 13.7 Plot of the second order interpolant in 2D centered at zero, with
a width of one.

40
30
20
10
0
5
0 5
0
-5 -5
x2 x1
Figure 13.8 Example 13.1. Optimal cost J ∗ (x) and approximation stability
bound J ∗ (x) + γ(x) above it.
296 13 Approximate Receding Horizon Control

15 15
10 10
5 5
0 0
-5 -5
-10 -10
5 5
0 5 0 5
0 0
-5 -5 -5 -5
x2 x1 x2 x1
5 5
4 4
3 3
2 2
1 1
x2

0 x2 0
-1 -1
-2 -2
-3 -3
-4 -4
-5-5 5 -5-5 5
0 0
x1 x1

30 30
20 20
10 10
0 0
5 5
0 5 0 5
0 0
-5 -5 -5 -5
x2 x1 x2 x1
(a) Sub-optimal solution with 15 (b) Sub-optimal solution with 25
vertices. Error: δV = −17.8. vertices. Error: δV = −4.51.

Figure 13.9 Example 13.1. Error function −δV (top). Points marked are
the vertices with worst-case fit. Partition (triangulation) of the sampled
points (middle). Upper-bound on approximate value function (convex hull
of sample points) (bottom).
13.3 Partitioning and Interpolation Methods 297

15 15
10 10
5 5
0 0
-5 -5
-10 -10
5 5
0 5 0 5
0 0
-5 -5 -5 -5
x2 x1 x2 x1
5 5
4 4
3 3
2 2
1 1
x2

x2

0 0
-1 -1
-2 -2
-3 -3
-4 -4
-5-5 5 -5-5 5
0 0
x1 x1

30 30
20 20
10 10
0 0
5 5
0 5 0 5
0 0
-5 -5 -5 -5
x2 x1 x2 x1
(a) Sub-optimal solution with 35 (b) Sub-optimal solution with 45
vertices. Error: δV = −1.26. vertices. Error: δV = −0.82

Figure 13.10 Example 13.1. Error function −δV (top). Points marked are
the vertices with worst-case fit. Partition (triangulation) of the sampled
points (middle). Upper-bound on approximate value function (convex hull
of sample points) (bottom).
298 13 Approximate Receding Horizon Control

15 15
10 10
5 5
0 0
-5 -5
-10 -10
5 5
0 5 0 5
0 0
-5 -5 -5 -5
x2 x1 x2 x1
5 5
4 4
3 3
2 2
1 1
x2

x2
0 0
-1 -1
-2 -2
-3 -3
-4 -4
-5 -5
-5 0 5 -5 0 5
x1 x1

30 30
20 20
10 10
0 0
5 5
0 5 0 5
0 0
-5 -5 -5 -5
x2 x1 x2 x1
(a) Sub-optimal solution with 55 (b) Sub-optimal solution with 61
vertices. Error: δV = −0.23. vertices. Error: δV = 0, system
stable.

Figure 13.11 Example 13.1. Error function −δV (top). Points marked are
the vertices with worst-case fit. Partition (triangulation) of the sampled
points (middle). Upper-bound on approximate value function (convex hull
of sample points) (bottom).
13.3 Partitioning and Interpolation Methods 299

1 1
ũ(x) − u∗ (x)

0.5 0.5
0 0
-0.5 -0.5
-1 -1
5 5
0 0 -5 0 0 -5
-5 5 -5 5
x1 x2 x1 x2
(a) Approximated control law (b) Approximation error ũ(x) −
ũ(x). u∗ (x).

Figure 13.12 Example 13.1. Approximate control law using triangulation.

300 13 Approximate Receding Horizon Control

15 15
10 10
5 5
0 0
-5 -5
-10 -10
5 5
0 5 0 5
0 0
-5 -5 -5 -5
x2 x1 x2 x1
5 5
4 4
3 3
2 2
1 1
x2

0 x2 0
-1 -1
-2 -2
-3 -3
-4 -4
-5-5 5 -5-5 5
0 0
x1 x1

40 40
30 30
20 20
10 10
0 0
5 5
0 5 0 5
0 0
-5 -5 -5 -5
x2 x1 x2 x1
(a) Sub-optimal solution with 5 (b) Sub-optimal solution with 13
facets. Error: δV = −18.35. facets. Error: δV = −5.62.

Figure 13.13 Example 13.1. Error function −δV (top). Points marked are
the vertices with worst-case fit. Partition (polyhedral approximation) of
the sampled points (middle). Upper-bound on approximate value function
(convex hull of sample points) (bottom).
13.3 Partitioning and Interpolation Methods 301

15 15
10 10
5 5
0 0
-5 -5
-10 -10
5 5
0 5 0 5
0 0
-5 -5 -5 -5
x2 x1 x2 x1
5 5
4 4
3 3
2 2
1 1
x2

x2

0 0
-1 -1
-2 -2
-3 -3
-4 -4
-5 0 -5 0
-5 5 -5 5
x1 x1

40 40
30 30
20 20
10 10
0 0
5 5
0 5 0 5
0 0
-5 -5 -5 -5
x2 x1 x2 x1
(a) Sub-optimal solution with 21 (b) Example 13.1. Sub-optimal
facets. Error: δV = −0.77. solution with 31 facets. Error:
δV = 0, system stable.

Figure 13.14 Example 13.1. Error function −δV (top). Points marked are
the vertices with worst-case fit. Partition (polyhedral approximation) of
the sampled points (middle). Upper-bound on approximate value function
(convex hull of sample points) (bottom).
302 13 Approximate Receding Horizon Control

5
4
3
2
1

x2
0
-1
-2
-3
-4
-5 0 5
-5
x1
Figure 13.15 Example 13.1. Partition generated by the second-order inter-
polant method.

1 1
0.5 0.5
0 0
-0.5 -0.5
-1 -1
5 5
0 0 -5 0 0 -5
-5 5 -5 5
x1 x2 x1 x2
(a) Approximated control law (b) Approximation error ũ(x) −
ũ(x). u∗ (x).

Figure 13.16 Example 13.1. Approximate control law using second-order

interpolants.

30
20
J(x)

10
˜

0
5
0 5
-5 0
x2 -5
x1
˜
Figure 13.17 Example 13.1. Approximate value function J(x) using second-
order interpolants.
 Chapter 14

On-line Control Computation

In previous chapters we have shown how to compute the solution to constrained

finite time optimal control problems. Two approaches can be used. The first ap-
proach consists of solving a mathematical program for a given initial state. The
second approach employs multiparametric programming to compute the solution
as an explicit piecewise affine function of the initial state. This implies that for
constrained linear systems, a Receding Horizon Control (RHC) policy requires ei-
ther the on-line solution of a quadratic or linear program, or the evaluation of a
piecewise affine on polyhedra function.
In this chapter we focus on efficient on-line methods for the computation of RHC
control laws. The main drawback of explicit optimal control laws is that the number
of polyhedral regions can grow exponentially with the number of constraints in the
optimal control problem. Section 14.1 presents efficient on-line methods for the
evaluation of explicit piecewise affine control laws.
If the on-line solution of a quadratic or linear program is preferred, Sections 14.2
and 14.3 briefly discuss how to improve the efficiency of a mathematical program-
ming solver by exploiting the structure of the RHC control problem.

14.1 Storage and On-line Evaluation of the PWA Control

Law
In Chapter 11 we have shown how to compute the solution to the constrained finite
time optimal control (CFTOC) problem as an explicit piecewise affine function of
the initial state. Such a function is computed off-line by using a multiparametric
program solver, which divides the state space into polyhedral regions, and for each
region determines the linear gain and offset which produces the optimal control
action.
This method reveals its effectiveness when applied to Receding Horizon Control
(RHC). Having a precomputed solution as an explicit piecewise affine on polyhedra
(PPWA) function of the state vector reduces the on-line computation of the RHC
304 14 On-line Control Computation

control law to a function evaluation, therefore avoiding the on-line solution of a

quadratic or linear program.
The main drawback of such an explicit optimal control law is that the number
of polyhedral regions can grow dramatically with the number of constraints in the
optimal control problem. In this chapter we focus on efficient on-line methods for
the evaluation of such a piecewise affine control law.
The simplest way to implement the piecewise affine feedback laws is to store the
polyhedral cells {H i x ≤ K i }, perform on-line a search through them to locate the
one which contains x(t) and then look up the corresponding feedback gain (F i , g i )
(note that this procedure can be easily parallelized). This chapter presents imple-
mentation techniques which avoid the storage and the evaluation of the polyhedra
and can significantly reduce the on-line storage demands and computational com-
plexity of RHC. They exploit the properties of the value function and the piecewise
affine optimal control law of the constrained finite time optimal control problem.
Let the explicit optimal control law be:

u∗ (x) = F i x + g i , ∀x ∈ Pi , i = 1, . . . , N r , (14.1)
n i i
o
where F i ∈ Rm×n , g i ∈ Rm , and Pi = x ∈ Rn : H i x ≤ K i , H i ∈ RNc ×n , K i ∈ RNc ,
i = 1, . . . , N r is a polyhedral partition of X . In the following Hji denotes the j-th
row of the matrix H i , Kji denotes the j-th element of the vector K i and Nci is the
number of constraints defining the i-th polyhedron Pi . The on-line implementation
of the control law (14.1) is simply executed according to the following steps:

Algorithm 14.1
Input: State x(t) at time instant t
Output: Receding horizon control input u(x(t))
Search for the j-th polyhedron that contains x(t), (H j x(t) ≤ K j )
u(t) ← F j x(t) + g j
Return u(t), the j-th control law evaluated at x(t)

In Algorithm 14.1, step (2) is critical and it is the only step whose efficiency can
be improved. A simple implementation of step (2) would require searching for the
polyhedral region that contains the state x(t) as in the following algorithm.

Algorithm 14.2
r
Input: State x(t) at time instant t and polyhedral partition {Pi }N
i=1 of the control
law (14.1)
Output: Index j of the polyhedron Pj in the control law (14.1) containing x(t)
i ← 1, notfound← 1
While i ≤ N r and notfound
j ← 0, stillfeasible← 1
While j ≤ Nci and stillfeasible=1
If Hji x(t) > Kji Then stillfeasible← 0
Else j ← j + 1
 14.1 Storage and On-line Evaluation of the PWA Control Law 305

End
If stillfeasible=1 Then notfound← 0 Else i ← i + 1
End
Return j

Algorithm 14.2 requires the storage of all polyhedra Pi , i.e., (n + 1)NC real
numbers (n numbers for each row of the matrix H i plus one number for the cor-
PN r
responding element in the matrix K i ), NC = i=1 Nci , and in the worst case (the
state is contained in the last region of the list) it will give a solution after nNC
multiplications, (n − 1)NC sums and NC comparisons.
In this section, by using the properties of the value function, we show how
Algorithm 14.2 can be replaced by more efficient algorithms that avoid storing the
polyhedral regions Pi , i = 1, . . . , N r , therefore reducing significantly the storage
demand and the computational complexity.
In the following we will distinguish between optimal control based on LP and
optimal control based on QP.

14.1.1 Efficient Implementation, 1, ∞-Norm Case

From Corollary 11.5, the value function J ∗ (x) corresponding to the solution of the
CFTOC problem (12.48) with 1, ∞-norm is convex and PWA:
′
J ∗ (x) = T i x + V i , ∀x ∈ Pi , i = 1, . . . , N r . (14.2)

By exploiting the convexity of the value function the storage of the polyhedral
regions Pi can be avoided. From the equivalence of the representations of PWA
convex functions (see Section 2.2.5), the function J ∗ (x) in equation (14.2) can be
represented alternatively as

n ′ o r
J ∗ (x) = max T i x + V i , i = 1, . . . , N r for x ∈ X = ∪N
i=1 Pi . (14.3)

Thus, the polyhedral region Pj containing x can be identified simply by search-

′ r
ing for the maximum number in the list {T i x + V i }N
i=1 :

′
n ′ o
x ∈ Pj ⇔ T j x + V j = max T i x + V i , i = 1, . . . , N r . (14.4)

Therefore, instead of searching for the polyhedron j that contains the point x
via Algorithm 14.2, we can just store the value function and identify region j by
searching for the maximum in the list of numbers composed of the single affine
′
function T i x + V i evaluated at x (see Figure 14.1):
306 14 On-line Control Computation

′
T4 x + V 4

′ ′
T1 x + V 1 T3 x + V 3

J ∗ (x)
′
T2 x + V 2

P1 P2 P3 P4

Figure 14.1 Example for Algorithm 14.3 in one dimension: For a given point
x ∈ P3 (x = 5) we have J ∗ (x) = max(T 1 x + V 1 , . . . , T 4 x + V 4 ).

Algorithm 14.3
′
Input: State x(t) at time instant t and value function 14.2, T i x + V i , i = 1, . . . , N r
Output: Index j of the polyhedron Pj containing x(t) in the control law (14.1)
′
Compute the list L = {ni = T i x + V i , i = 1, . . . , N r }
Find j such that nj = maxni ∈L ni
Return j

The search Algorithm 14.3 requires the storage of (n + 1)N r real numbers
and will give a solution after nN r multiplications, (n − 1)N r sums, and N r − 1
comparisons. In Table 14.1 we compare the complexity of Algorithm 14.3 against
Algorithm 14.2 in terms of storage demand and number of flops. Algorithm 14.3
will outperform Algorithm 14.2 since typically NC ≫ N r .

Table 14.1 Complexity comparison of Algorithm 14.2 and Algorithm 14.3

Algorithm 14.2 Algorithm 14.3
Storage demand (real numbers) (n + 1)NC (n + 1)N r
Number of flops (worst case) 2nNC 2nN r

14.1.2 Efficient Implementation, 2-Norm Case

Consider the state feedback solution (11.15) of the CFTOC problem (11.9) with
p = 2. Theorem 11.2 states that the value function J ∗ (x) is convex and piecewise
quadratic on polyhedra and the simple Algorithm 14.3 described in the previous
subsection cannot be used here. Instead, a different approach is described below.
It uses a surrogate of the value function to uniquely characterize the polyhedral
partition of the optimal control law.
14.1 Storage and On-line Evaluation of the PWA Control Law 307

We will first establish the following general result: given a general polyhedral
partition of the state space, we can locate where the state lies (i.e., in which poly-
hedron) by using a search procedure based on the information provided by an
“appropriate” PWA continuous function defined over the same polyhedral parti-
tion. We will refer to such “appropriate” PWA function as a PWA descriptor
function. First we outline the properties of the PWA descriptor function and then
we describe the search procedure itself.
r
Let {Pi }Ni=1 be the polyhedral partition obtained by solving the mp-QP (11.31)
and denote by Ci = {j : Pj is a neighbor of Pi , j = 1, . . . , N r , j 6= i} the list of
all neighboring polyhedra of Pi . The list Ci has Nci elements and we denote by
Ci (k) its k-th element.

Definition 14.1 (PWA descriptor function) A continuous real-valued PWA

function
′
f (x) = fi (x) = Ai x + B i , if x ∈ Pi (14.5)
is called a descriptor function if

Ai 6= Aj , ∀j ∈ Ci , i = 1, . . . , Nr . (14.6)

Theorem 14.1 Let f (x) be a PWA descriptor function on the polyhedral partition
{Pi }N
i=1 .
r

i
Let Oi (x) ∈ RNc be a vector associated with region Pi , and let the j-th element
i
of O (x) be defined as

+1 fi (x) > fCi (j) (x)
Oji (x) = (14.7)
−1 fi (x) < fCi (j) (x).

Then Oi (x) has the following properties:

(i) Oi (x) = S i = const, ∀x ∈ Pi , i = 1, . . . , Nr .
(ii) Oi (x) 6= S i , ∀x ∈
/ Pi , i = 1, . . . , Nr .
Proof: Let F = Pi ∩ PCi (j) be the common facet of Pi and PCi (j) . Define the
linear function
gji (x) = fi (x) − fCi (j) (x). (14.8)
From the continuity of f (x) it follows that gji (x) = 0, ∀x ∈ F . As Pi and PCi (j) are
disjoint convex polyhedra and Ai 6= ACi (j) it follows that gji (ξi ) > 0 (or gji (ξi ) < 0,
but not both) for any interior point ξi of Pi . Similarly for any interior point ξCi (j)
of PCi (j) we have gji (ξCi (j) ) < 0 (or gij (ξi ) > 0, but not both). Consequently,
gji (x) = 0 is the separating hyperplane between Pi and PCi (j) .
(i) Because gji (x) = 0 is a separating hyperplane, the function gji (x) does not
change its sign for all x ∈ Pi , i.e., Oji (x) = sij , ∀x ∈ Pi with sij = +1 or sij = −1.
The same reasoning can be applied to all neighbors of Pi to get the vector S i =
i
{sij } ∈ RNc .
(ii) ∀x ∈/ Pi , ∃j ∈ Ci such that Hji x > Kji . Since gji (x) = 0 is a separating
hyperplane Oji (x) = −sij . 
308 14 On-line Control Computation

f4 (x)

f (x)
f3 (x)
f2 (x)

f1 (x)

P1 P2 P3 P4

Figure 14.2 Example for Algorithm 14.4 in one dimension: For a given point
x ∈ P2 (x = 4) we have O2 (x) = [−1 1]′ = S2 , while O1 (x) = 1 6= S1 = −1,
O3 (x) = [−1 − 1]′ 6= S3 = [1 − 1]′ , O4 (x) = 1 6= S4 = −1.

Equivalently, Theorem 14.1 states that

x ∈ Pi ⇔ Oi (x) = S i , (14.9)

which means that the function Oi (x) and the vector S i uniquely characterize Pi .
Therefore, to check on-line if the polyhedral region i contains the state x it is
sufficient to compute the binary vector Oi (x) and compare it with S i . Vectors
S i are calculated off-line for all i = 1, . . . , N r , by comparing the values of fi (x)
and fCi (j) (x) for j = 1, . . . , Nci , for a point x belonging to Pi , for instance, the
Chebychev center of Pi .

Remark 14.1 Theorem 14.1 does not hold if one stores the closures of critical regions
Pi (see Remark 6.8). In this case one needs to modify Theorem 14.1 as follows

(i) Oi (x) = S i = const, ∀x ∈ int(Pi ), i = 1, . . . , Nr ,

S
(ii) Oi (x) 6= S i , ∀x ∈
/ (int(Pi ) ∂Pi ), i = 1, . . . , Nr .

The results in the remainder of this section can be easily extended to this case.

In Figure 14.2 a one dimensional example illustrates the procedure with N r = 4

regions. The list of neighboring regions Ci and the vector S i can be constructed
by simply looking at the figure: C1 = {2}, C2 = {1, 3}, C3 = {2, 4}, C4 = {3},
S 1 = −1, S 2 = [−1 1]′ , S 3 = [1 − 1]′ , S 4 = −1. The point x = 4 is in region 2 and
we have O2 (x) = [−1 1]′ = S 2 , while O3 (x) = [−1 − 1]′ 6= S 3 , O1 (x) = 1 6= S 1 ,
O4 (x) = 1 6= S 4 . The failure of a match Oi (x) = S i provides information on a
good search direction(s). The solution can be found by searching in the direction
where a constraint is violated, i.e., one should check the neighboring region Pj for
which Oji (x) 6= sij .
The overall procedure is composed of two parts:
14.1 Storage and On-line Evaluation of the PWA Control Law 309

1. (off-line) Construction of the PWA function f (x) in (14.5) satisfying (14.6)

and computation of the list of neighbors Ci and the vector S i ,

2. (on-line) Execution of the following algorithm

Algorithm 14.4
Input: State x(t) at time instant t, list of neighboring regions Ci and the vectors S i
Output: Index i of the polyhedron Pi containing x(t) in the control law (14.1)
i ← 1, I = {1, . . . , Nr }, notfound← 1
While notfound and I 6= ∅
I ←I \i
Compute Oi (x)
If Oi (x) = S i Then notfound← 0
Else i ← Ci (q), where Oqi (x) 6= siq
End
Return i

Algorithm 14.4 does not require the storage of the polyhedra Pi , but only
the storage of one linear function fi (x) per polyhedron, i.e., N r (n + 1) real num-
bers and the list of neighbors Ci which requires NC integers. In the worst case,
Algorithm 14.4 terminates after N r n multiplications, N r (n − 1) sums and NC
comparisons.
In Table 14.2 we compare the complexity of Algorithm 14.4 against the standard
Algorithm 14.2 in terms of storage demand and number of flops.

Remark 14.2 Note that the computation of Oi (x) in Algorithm 14.4 requires the
evaluation of Nci linear functions, but the overall computation never exceeds N r linear
function evaluations. Consequently, Algorithm 14.4 will outperform Algorithm 14.2,
since typically NC ≫ N r .

Table 14.2 Complexity comparison of Algorithm 14.2 and Algorithm 14.4

Algorithm 14.2 Algorithm 14.4
Storage demand (real numbers) (n + 1)NC (n + 1)N r
Number of flops (worst case) 2nNC (2n − 1)N r + NC

Now that we have shown how to locate the polyhedron in which the state lies
by using a PWA descriptor function, we need a procedure for the construction of
such a function.
The image of the descriptor function is the set of real numbers R. In the
following we will show how the descriptor function can be generated from a vector-
valued function m : Rn → Rs . This general result will be used in the following
subsections.
310 14 On-line Control Computation

Definition 14.2 (Vector valued PWA descriptor function) A continuous vector-

valued PWA function
m(x) = Āi x + B̄ i , if x ∈ Pi , (14.10)
is called a vector-valued PWA descriptor function if

Āi 6= Āj , ∀j ∈ Ci , i = 1, . . . , Nr , (14.11)

where Āi ∈ Rs×n , B̄ i ∈ Rs .

Theorem 14.2 Given a vector-valued PWA descriptor function m(x) defined over
a polyhedral partition {Pi }N
i=1 it is possible to construct a PWA descriptor function
r

f (x) over the same polyhedral partition.

Proof: Let Ni,j be the null-space of (Āi −Āj )′ . Since by the definition Āi −Āj 6=
0 it follows that Ni,j is not full-dimensional, i.e., Ni,j ⊆ Rs−1 . Consequently, it
is always possible to find a vector w ∈ Rs such that w(Āi − Āj ) 6= 0 holds for all
i = 1, . . . , Nr and ∀j ∈ Ci . Clearly, f (x) = w′ m(x) is then a valid PWA descriptor
function. 
As shown in the proof of Theorem 14.2, once we have a vector-valued PWA
descriptor function, practically any randomly chosen vector w ∈ Rs is likely to be
satisfactory for the construction of PWA descriptor function. But, from a numerical
point of view, we would like to obtain a w that is as far away as possible from the
null-spaces Ni,j . We show one algorithm for finding such a vector w.
For a given vector-valued PWA descriptor function we form a set of vectors
ak ∈ Rs , kak k = 1, k = 1, . . . , NC /2, by choosing and normalizing one (and only
one) nonzero column from each matrix (Āi − Āj ), ∀j ∈ Ci , i = 1, . . . , Nr . The
vector w ∈ Rs satisfying the set of equations w′ ak 6= 0, k = 1, . . . , NC /2, can then
be constructed by using the following algorithm. Note that the index k goes to
NC /2 since the term (Āj − Āi ) is the same as (Āi − Āj ) and thus there is no need
to consider it twice.

Algorithm 14.5
Input: Vectors ai ∈ Rs , i = 1, . . . , N
Output: The vector w ∈ Rs satisfying the set of equations w′ ai 6= 0, i = 1, . . . , N
w ← [1, . . . , 1]′ , R ← 1
While k ≤ NC /2
d ← w ′ ak
If 0 ≤ d ≤ R Then w ← w + 21 (R − d)ak , R ← 12 (R + d)
If −R ≤ d < 0 Then w ← w − 12 (R + d)ak , R ← 12 (R − d)
End
Return w

Algorithm 14.5 is based on a construction of a sequence of balls

B = {x : x = w + r, krk2 ≤ R}. As depicted in Figure 14.3, Algorithm 14.5 starts
with the initial ball of radius R = 1, centered at w = [1, . . . , 1]′ . Iteratively one
hyperplane a′k x = 0 at the time is introduced and the largest ball B ′ ⊆ B that
14.1 Storage and On-line Evaluation of the PWA Control Law 311

aTi x ≥ 0
B
R′ B′
R aTi x ≤ 0
d

R−d

Figure 14.3 Illustration for Algorithm 14.5 in two dimensions.

does not intersect this hyperplane is designed. The center w of the final ball is the
vector w we wanted to construct, while R gives some information about the degree
of non-orthogonality: |w′ ak | ≥ R, ∀k.
In the following subsections we will show that the gradient of the value function,
and the optimizer, are vector-valued PWA descriptor functions and therefore we
can use Algorithm 14.5 for the construction of the PWA descriptor function.

Generating a PWA descriptor function from the value function

Let J ∗ (x) be the convex and piecewise quadratic (CPWQ) value function obtained
as a solution of the CFTOC (11.9) problem for p = 2:

J ∗ (x) = qi (x) = x′ Qi x + Ti ′ x + Vi , if x ∈ Pi , i = 1, . . . , N r . (14.12)

In Section 6.3.4 we have proven that for non-degenerate problems the value
function J ∗ (x) is a C (1) function. We can obtain a vector-valued PWA descriptor
function by differentiating J ∗ (x). We first need to introduce the following theorem.

Theorem 14.3 ([21]) Assume that the CFTOC (11.9) problem leads to a non-
degenerate mp-QP (11.33). Consider the value function J ∗ (x) in (14.12) and let
CRi , CRj be the closure of two neighboring critical regions corresponding to the
set of active constraints Ai and Aj , respectively, then

Qi − Qj  0 or Qi − Qj  0 and Qi 6= Qj (14.13)

and
Qi − Qj  0 iff Ai ⊂ Aj . (14.14)

Theorem 14.4 Consider the value function J ∗ (x) in (14.12) and assume that
the CFTOC (11.9) problem leads to a non-degenerate mp-QP (11.33). Then the
gradient m(x) = ∇J ∗ (x), is a vector-valued PWA descriptor function.

Proof: From Theorem 6.9 we see that m(x) is a continuous vector-valued PWA
function, while from equation (14.12) we get

m(x) = ∇J ∗ (x) = 2Qi x + Ti . (14.15)

312 14 On-line Control Computation

Since from Theorem 14.3 we know that Qi 6= Qj for all neighboring polyhedra,
it follows that m(x) satisfies all conditions for a vector-valued PWA descriptor
function. 
Combining results of Theorem 14.4 and Theorem 14.2 it follows that by using
Algorithm 14.5 we can construct a PWA descriptor function from the gradient of
the value function J ∗ (x).

Generating a PWA descriptor function from the optimal inputs

Another way to construct a descriptor function f (x) emerges naturally if we look at
the properties of the optimizer U0∗ (x) corresponding to the state feedback solution
of the CFTOC problem (11.9). From Theorem 6.7 it follows that the optimizer
U0∗ (x) is continuous in x and piecewise affine on polyhedra:

U0∗ (x) = li (x) = F i x + g i , if x ∈ Pi , i = 1, . . . , N r , (14.16)

where F i ∈ Rs×n and g i ∈ Rs . We will assume that Pi are critical regions (as
defined in Section 6.1.2). Before going further we need the following lemma.

Lemma 14.1 Consider the state feedback solution (14.16) of the CFTOC prob-
lem (11.9) and assume that the CFTOC (11.9) leads to a non-degenerate mp-
QP (11.33). Let Pi , Pj be two neighboring polyhedra, then F i 6= F j .

Proof: The proof is a simple consequence of Theorem 14.3. As in Theorem 14.3,

without loss of generality we can assume that the set of active constraints Ai
associated with the critical region Pi is empty, i.e., Ai = ∅. Suppose that the
optimizer is the same for both polyhedra, i.e., [F i g i ] = [F j g j ]. Then, the cost
functions qi (x) and qj (x) are also equal. From the proof of Theorem 14.3 this
implies that Pi = Pj , which is a contradiction. Thus we have [F i g i ] 6= [F j g j ].
Note that F i = F j cannot happen since, from the continuity of U0∗ (x), this would
imply g i = g j . Consequently we have F i 6= F j . 
From Lemma 14.1 and Theorem 14.2 it follows that an appropriate PWA de-
scriptor function f (x) can be calculated from the gradient of the optimizer U ∗ (x)
by using Algorithm 14.5.

Remark 14.3 Note that even if we are implementing a receding horizon control strat-
egy, the construction of the PWA descriptor function is based on the full optimization
vector U ∗ (x) and the corresponding matrices F i and g i .

Remark 14.4 In some cases the use of the optimal control profile U ∗ (x) for the
construction of descriptor function f (x) can be extremely simple. If there is a row
r, r ≤ m (m is the dimension of u) for which (F i )r 6= (F j )r , ∀i = 1 . . . , N r , ∀j ∈ Ci ,
′
it is enough to set Ai = (F i )r and B i = (g i )r , where (F i )r and (g i )r denote the i-th
row of the matrices F i and g i , respectively. The row r has to be the same for all
i = 1 . . . , N r . In this way we avoid the storage of the descriptor function altogether
since it is equal to one component of the control law, which is stored anyway.
 14.1 Storage and On-line Evaluation of the PWA Control Law 313

14.1.3 Example

As an example, we compare the performance of Algorithm 14.2, 14.3 and 14.4 on

CFTOC problem for the discrete-time system
    

 4 −1.5 0.5 −0.25 0.5

  4   
 0 0 0  x(t) +  0  u(t)
x(t + 1) =   0   0 
2 0 0 (14.17)



  0 0 0.5 0  0

y(t) = 0.083 0.22 0.11 0.02 x(t)

resulting from the linear system

1
y= u (14.18)
s4
sampled at Ts = 1, subject to the input constraint

− 1 ≤ u(t) ≤ 1 (14.19)

and the output constraint

− 10 ≤ y(t) ≤ 10. (14.20)

CFTOC based on LP
To regulate (14.17), we design a receding horizon controller based on the optimiza-
tion problem (12.6), (12.7)) where p = ∞, N = 2, Q = diag{5, 10, 10, 10}, R = 0.8,
P = 0, Xf = R4 . The PPWA solution of the mp-LP problem comprises 136 regions.
In Table 14.3 we report the comparison between the complexity of Algorithm 14.2
and Algorithm 14.3 for this example.
The average on-line evaluation effort of the PPWA solution for a set of 1000
random points in the state space is 2259 flops (Algorithm 14.2), and 1088 flops
(Algorithm 14.3). We note that the solution using Matlab R
LP solver (function
linprog.m with interior point algorithm and LargeScale set to ’off’) takes 25459
flops on average.

Table 14.3 Complexity comparison of Algorithm 14.2 and Algorithm 14.3

for the example in Section 14.1.3

Algorithm 14.2 Algorithm 14.3

Storage demand (real numbers) 5690 680
Number of flops (worst case) 9104 1088
Number of flops (average for 1000 random points) 2259 1088

CFTOC based on QP
To regulate (14.17), we design a receding horizon controller based on the optimiza-
tion problem (12.6), (12.7)) where N = 7, Q = I, R = 0.01, P = 0, Xf = R4 .
The PPWA solution of the mp-QP problem comprises 213 regions. We obtained
314 14 On-line Control Computation

a descriptor function from the value function and for this example the choice of
w = [1 0 0 0]′ is satisfactory. In Table 14.4 we report the comparison between the
complexity of Algorithm 14.2 and Algorithm 14.4 for this example.
The average computation effort of the PPWA solution for a set of 1000 ran-
dom points in the state space is 2114 flops (Algorithm 14.2), and 175 flops (Algo-
rithm 14.4). The solution of the corresponding quadratic program with Matlab R

QP solver (function quadprog.m and LargeScale set to ’off’) takes 25221 flops on
average.

Table 14.4 Complexity comparison of Algorithm 14.2 and Algorithm 14.4

for the example in Section 14.1.3

Algorithm 14.2 Algorithm 14.4

Storage demand (real numbers) 9740 1065
Number of flops (worst case) 15584 3439
Number of flops (average for 1000 random points) 2114 175

14.1.4 Literature Review

The problem considered in this chapter has been approached by several other re-
searchers. For instance in [275] the authors propose to organize the controller
gains of the PWA control law on a balanced search tree. By doing so, the search
for the region containing the current state has on average a logarithmic computa-
tion complexity although it is less efficient in terms of memory requirements. At
the expense of the optimality a similar computational complexity can be achieved
with the approximative point location algorithm described in [160].
The comparison of the proposed algorithms with other efficient solvers in the
literature, e.g., [21, 55, 256, 105, 209, 212, 284, 61, 25] requires the simultaneous
analysis of several issues such as speed of computation, storage demand and real
time code verifiability.

14.2 Gradient Projection Methods Applied to MPC

In this section, we investigate the gradient projection methods from Section 3.3.1
for the iterative solution of MPC problems with a quadratic cost function given as
PN −1
J0∗ (x(0)) = min x′N P xN + k=0 x′k Qxk + u′k Ruk
subj. to xk+1 = Axk + Buk , k = 0, . . . , N − 1
xk ∈ X , uk ∈ U, k = 0, . . . , N − 1 (14.21)
xN ∈ Xf
x0 = x(0).
 14.2 Gradient Projection Methods Applied to MPC 315

Note that 1- and ∞-norm cost functions (see Section 11.1) cannot be handled
by the gradient (projection) methods discussed in this book since they require a
smooth objective function.
In Section 14.2.1 we first investigate a special case of the general setup in (14.21)
which is input-constrained MPC. In this case, we impose constraints on the control
inputs only, i.e., for the state sets we have X = Xf = Rn . If there are no state
constraints, the MPC problem can be formulated with the sequence of inputs U0 =
[u′0 , . . . , u′N −1 ]′ as the optimization vector (so-called condensing) where at the same
time the feasible set U N = U ×. . .×U remains ‘simple’ if the input set U was simple
(recall from Section 3.3.1 that a ‘simple’ set is a convex set with a projection
operator that can be evaluated efficiently). This allows one to solve the MPC
problem right in the primal domain and thus to achieve linear convergence with
the gradient projection methods of this book.
In the case of input and state constraints, the feasible set of problem (14.21)
does not allow for an easy-to-evaluate projection operator in general, neither in
condensed form nor with the state sequence as an additional optimization vector
(so-called sparse formulation). However, gradient methods can still be applied to
the sparse formulation of the MPC problem if a dual approach, as discussed in
Section 3.3.1, is applied. However, only sublinear convergence can be achieved
in the dual domain. Section 14.2.2 will summarize problem formulation, impor-
tant aspects regarding the computation of the dual gradient and optimal step size
selection.

14.2.1 Input-Constrained MPC

In order to apply the classic or the fast gradient projection method from Sec-
tion 3.3.1, we first eliminate the states from the MPC problem (14.21) and rewrite
it in condensed form (see (8.8)) as
J0∗ (x(0)) = x′ (0)Y x(0) + minU0 U0 ′ HU0 + 2x′ (0)F U0
subj. to U0 ∈ U N ,
which resembles the general problem setup for gradient methods given by (3.23).
Since the objective function is twice continuously differentiable, L-smoothness and
strong convexity follow from Lemma (3.1) and Theorem (3.2), respectively. In fact,
the Lipschitz constant L and the strong convexity parameter µ are given by
L = 2 λmax (H), µ = 2 λmin (H) ,
and are both independent of the initial state x(0). Thus, no computationally
expensive eigenvalue computations are necessary during runtime if the Hessian H
stays constant.
From the definition of H in (8.8) it follows that it is positive definite whenever
the penalty matrix R is positive definite. In this case, the constant µ is positive
and both the classic gradient method given by Algorithm 3.6 and the fast gradient
projection method in Algorithm 3.7 converge linearly.
The projection operator for the feasible set U N has to be computed in every
iteration of any gradient projection method. In the MPC context, this operation
316 14 On-line Control Computation

translates to
 
2

πU N Yi − HYi + F ′ x(0) , (14.22)
L
where Yi is the previous iterate in case of the classic gradient projection method
or the previous secondary iterate in case of the fast gradient projection method.
Since the feasible set in input-constrained MPC is the direct product of N sets,
evaluation of the projection operator in (14.22) can be separated into N evaluations


of the projection operator πU , i.e., if we let Uf = Yi − L2 HYi + F ′ x(0) , then
 ′
πU N (Uf ) = πU (uf,1 )′ , . . . , πU (uf,N )′ ,
 ′
with Uf = u′f,1 , . . . , u′f,N and uf,k ∈ Rm , k = 1, . . . , N .

14.2.2 Input- and State-Constrained MPC

In the input- and state-constrained case, gradient projection methods cannot be
applied in the primal domain, since the projection operator for the feasible set
 
uk ∈ U, xk ∈ X , xN ∈ Xf  xk+1 = Axk + Buk , x0 = x(0), k = 0 . . . N − 1
is nontrivial in general, even if the convex sets U, X and Xf allow efficient projec-
tions individually. The inherent complication comes from the intersection of these
sets with the affine set, which is determined by the state update equation.
As an alternative, the input- and state-constrained MPC problem can be solved
in the dual domain by first relaxing the state update equation and then solving for
the associated optimal dual multiplier (or Lagrange multiplier) vector using gradi-
ent methods. This approach was thoroughly discussed in Section 3.3.1 for a generic
convex objective function and a feasible set that can be written as the intersec-
tion of an affine set and a simple set (see (3.27)). If we rewrite the MPC problem
in (14.21) analogously to the generic setup of Section 3.3.1 with the optimization
 ′
vector z = x′0 , x′1 , . . . , x′N , u′0 , . . . , u′N −1 , we obtain
J0∗ (x(0)) = min z ′M z
subj. to Gz = Ex(0) (14.23)
z ∈ K,
where the problem data are defined as

M = blockdiag Q, · · · , Q, P, R, · · · , R ,
 
−In 0 ··· ··· 0 0 ··· ··· 0
 A −I n 0 · · · 0 B 0 ··· 0
 
 .. .. .. .. .. .. 
G = −  0 . . . . 0 B . .,
 . . .. .. .. 
 .. . . A −In 0 . . . 0
0 ··· 0 A −In 0 ··· 0 B
 ′
E = In , 0n , . . . , 0n ,
K = X × . . . × X × Xf × U × . . . U .
 14.3 Interior Point Method Applied to MPC 317

The dual problem is an unconstrained maximization problem with the dual

gradient given as Gz ∗ (ν) − Ex(0), where

z ∗ (ν) = arg minz∈K z ′ M z + ν ′ (Gz − Ex(0)) . (14.24)

As a prerequisite for the above statement, the minimizer z ∗ (ν) must be unique for
every dual multiplier vector ν (see Section 3.3.1). A sufficient condition for unique-
ness of the minimizer is a strongly convex objective function in (14.24). In view of
the definition of matrix M this is the case whenever the penalty matrices Q, P and
R are positive definite, which is not a restrictive assumption in an application.
The so-called inner problem (14.24) needs to be solved in every outer iteration
of the classic or the fast gradient method in order to determine the dual gradient.
This can be accomplished in one of the following ways.

• Exact analytic solution: This is possible if all penalty matrices are positive
diagonal matrices and all sets X , Xf and U are boxes. In practice, this is a
frequent setup.

• Exact multiparametric solution: By the block-diagonal structure of matrix M

and the structure of set K, the inner problem can be separated into 2N + 1
subproblems. If all sets X , Xf and U are polyhedral, each subproblem can
be pre-solved by means of multiparametric programming with the gradient
of the linear objective term as the parameter.

• Approximate iterative solution: This is the most general approach for the
solution of the inner problem as it only requires simple sets X , Xf and U (not
necessarily polyhedral). Each of the 2N + 1 subproblems can then be solved
approximately by a gradient projection method. Note that convergence issues
of the outer gradient method might arise if the accuracy of the approximate
inner solution is too low.

Finally, we note that under mild assumptions, the tight, i.e., smallest possible,
Lipschitz constant of the dual gradient is given by Ld = 12 kGM −1/2 k2 which, same
as in the input-constrained case, is independent of the initial state x(0). The
Lipschitz constant determines the step size of the outer gradient method and being
able to compute a tight value ensures the best possible (sublinear) convergence
(see Section 3.3.1).

14.3 Interior Point Method Applied to MPC

All presented interior-point methods can be applied to linear model predictive con-
trol problems with convex constraints. To obtain an efficient method with low
solve times, it is crucial to exploit the sparsity structure of the resulting quadratic
program (or any other form of problem P-IPM (3.30)) when computing the search
directions. Therefore, when using interior point solvers, the MPC problem is for-
mulated as a sparse QP (sometimes with a different ordering of states and inputs
318 14 On-line Control Computation

in the optimization vector). As a result, general sparse routines for solving lin-
ear systems with KKT-structure can be applied to the Newton systems (3.41)
and (3.45), or specific linear system solvers can be used that exploit the MPC-
specific problem structure. A good starting point for details on this topic are the
references [284, 246, 98, 201, 110, 171, 170].
 Chapter 15

Constrained Robust Optimal

Control

In the previous chapters we assumed perfect model knowledge for the plant to be
controlled. This chapter focuses on the case when the model is uncertain. We
focus on discrete-time uncertain linear systems with linear constraints on inputs
and states. Uncertainty is modeled as additive disturbance and/or parametric
uncertainty in the state space matrices. We first introduce the concepts of robust
control design with “open-loop prediction” and “closed-loop prediction”. Then
we discuss how to robustly enforce constraints satisfaction, and present several
objective functions one might be interested in minimizing. For each robust control
design we discuss the conservatism and show how to obtain the optimal control
action by using convex or non-convex programming.
For robust controllers which can be implemented by using linear and quadratic
programming, we show that the robust optimal control law is a continuous and
piecewise affine function of the state vector. The robust state feedback controller
can be computed by means of multiparametric linear or quadratic programming.
Thus, when the optimal control law is implemented in a moving horizon scheme,
the on-line computation of the resulting controller requires simply the evaluation
of a piecewise affine function.

15.1 Problem Formulation

Consider the linear uncertain system

x(t + 1) = A(wp (t))x(t) + B(wp (t))u(t) + Ewa (t) (15.1)

where x(t) ∈ Rn and u(t) ∈ Rm are the state and input vectors, respectively,
subject to the constraints

x(t) ∈ X , u(t) ∈ U, ∀t ≥ 0. (15.2)

320 15 Constrained Robust Optimal Control

The sets X ⊆ Rn and U ⊆ Rm are polytopes. Vectors wa (t) ∈ Rna and wp (t) ∈ Rnp
are unknown additive disturbances and parametric uncertainties, respectively. The
disturbance vector is

w(t) = [wa (t)′ , wp (t)′ ]′ ∈ W ⊂ Rnw (15.3)

with nw = na + np . We assume that bounds on wa (t) and wp (t) are known. In

particular W = W a × W p with wa (t) ∈ W a and wp (t) ∈ W p , where W a ⊂ Rna and
W p ⊂ Rnp are given polytopes. We also assume that A(·), B(·) are affine functions
of wp
np np
X X
A(wp ) = A0 + Ai wcp,i , B(wp ) = B 0 + B i wcp,i (15.4)
i=1 i=1

where Ai ∈ Rn×n and B i ∈ Rn×m are given matrices for i = 0 . . . , np and wcp,i is
p,n
the i-th component of the vector wp , i.e., wp = [wcp,1 , . . . , wc p ].
Robust finite time optimal control problems can be formulated by optimizing
over “Open-Loop” policies or “Closed-Loop” policies. We will discuss the two cases
next.

Open-Loop Predictions

Define the cost function as

h P −1 i
J0 (x(0), U0 ) = JW p(xN ) + N
k=0 q(xk , u k ) (15.5)

PN −1
where the operation JW evaluates the cost p(xN ) + k=0 q(xk , uk ) for the set of
uncertainties {w0 , . . . , wN −1 } ∈ W × · · · × W and the input sequence U0 . The
following options are typically considered.

• Nominal Cost.
In the simplest case, the objective function is evaluated for a single distur-
p p
bance profile w̄0a , . . . , w̄N
a
−1 , w̄0 , . . . , w̄N −1 :
h PN −1 i
JW = p(xN ) + k=0 q(xk , uk )


 xk+1 = A(wkp )xk + B(wkp )uk + Ewka (15.6)

x0 = x(0)
where

 w a
= w̄ka , wkp = w̄kp
 k
k = 0, . . . , N − 1.

• Expected Cost.
Another option is to consider the probability density function f (w) of the
disturbance w:
Z
Probability [w(t) ∈ W] = 1 = f (w)dw.
w∈W
15.1 Problem Formulation 321

Then the expected value of a function g(w) of the disturbance w is defined

as: Z
Ew [g(w)] = g(w)f (w)dw.
w∈W

In this case we consider the expected cost over the admissible disturbance
set.
h P −1 i
JW = Ew0 ,...,wN −1 p(xN ) + N k=0 q(xk , u k )


 xk+1 = A(wkp )xk + B(wkp )uk + Ewka (15.7)

x0 = x(0)
where

 w a
∈ W a , wkp ∈ W p
 k
k = 0, . . . , N − 1.

• Worst Case Cost.

Finally the worst case cost may be of interest
h PN −1 i
JW = maxw0 ,...,wN −1 p(xN ) + k=0 q(xk , uk )


 xk+1 = A(wkp )xk + B(wkp )uk + Ewka (15.8)

x0 = x(0)
where p
 ka ∈ W a , wk ∈ W p
 w

k = 0, . . . , N − 1.

If the 1-norm or ∞-norm is used in the cost function of problems (15.6), (15.7)
and (15.8), then we set
p(xN ) = kP xN kp , q(xk , uk ) = kQxk kp + kRuk kp (15.9)
with p = 1 or p = ∞. If the squared Euclidian norm is used in the cost function of
problems (15.6), (15.7) and (15.8), then we set
p(xN ) = x′N P xN , q(xk , uk ) = x′k Qxk + u′k Ruk . (15.10)
Note that in (15.6), (15.7) and (15.8) xk denotes the state vector at time k obtained
by starting from the state x0 = x(0) and applying to the system model
xk+1 = A(wkp )xk + B(wkp )uk + Ewka
the input sequence u0 , . . . , uk−1 and the disturbance sequences wa = {w0a , . . . , wN
a
−1 },
p p p
w = {w0 , . . . , wN −1 }.
Consider the optimal control problem
J0∗ (x(0)) = minU0 J0 (x(0), U0 )
 

 xk+1 = A(wkp )xk + B(wkp )uk + Ewka 


 

 xk ∈ X , uk ∈ U 
∀wka ∈ W a , ∀wkp ∈ W p
subj. to k = 0, . . . , N − 1

 
 ∀k = 0, . . . , N − 1.

 xN ∈ Xf 

 
x0 = x(0)
(15.11)
322 15 Constrained Robust Optimal Control

where Xf ⊆ Rn is a terminal polyhedral region. In (15.11) N is the time horizon

and U0 = [u′0 , . . . , u′N −1 ]′ ∈ Rs , s = mN the vector of the input sequence. We
denote by U0∗ = {u∗0 , . . . , u∗N −1 } the optimal solution to (15.11).
We denote with XiOL ⊆ X the set of states xi for which the robust optimal
control problem (15.8)-(15.11) is feasible, i.e.,

XiOL = {xi ∈ X : ∃(ui , . . . , uN −1 ) such that xk ∈ X , uk ∈ U, k = i, . . . , N − 1,

xN ∈ Xf ∀ wka ∈ W a , ∀wkp ∈ W p k = i, . . . , N − 1,
where xk+1 = A(wkp )xk + B(wkp )uk + Ewka }.
(15.12)
Problem (15.11) minimizes the performance index JW subject to the constraint
that the input sequence must be feasible for all possible disturbance realizations.
In other words, the nominal, expected or worst-case performance is minimized with
the requirement of constraint fulfillment for all possible realizations of wa , wp .

Remark 15.1 Note that we distinguish between the current state x(k) of system (15.1)
at time k and the variable xk in the optimization problem (15.11), that is the predicted
state of system (15.1) at time k obtained by starting from the state x0 = x(0) and
applying to system xk+1 = A(wkp )xk +B(wkp )uk +Ewka the input sequence u0 , . . . , uk−1
and the disturbance sequences w0p , . . . , wk−1
p
, w0a , . . . , wk−1
a
. Analogously, u(k) is the
input applied to system (15.1) at time k while uk is the k-th optimization variable of
the optimization problem (15.11).

The formulation (15.11) is based on an open-loop prediction and thus referred to

as Constrained Robust Optimal Control with open-loop predictions (CROC-OL).
The optimal control problem (15.11) can be viewed as a game played between two
players: the controller U and the disturbance W [24, p. 266-272]. Regardless of the
cost function in (15.11) the player U tries to counteract any feasible disturbance
realization with just one single sequence {u0 , . . . , uN −1 }. This prediction model
does not consider that at the next time step, the player can measure the state x(1)
and “adjust” his input u(1) based on the current measured state. By not consid-
ering this fact, the effect of the uncertainty may grow over the prediction horizon
and may easily lead to infeasibility of the min problem (15.11). Alternatively, in
the closed-loop prediction scheme presented next, the optimization scheme takes
into account that the disturbance and the controller play one move at a time.
The game is played differently depending on the cost function JW :

• In problem (15.6), (15.11), among all control actions which robustly satisfy
the constraints, the player U chooses the one which minimizes a cost for a
“guessed” sequence played by W .

• In problem (15.7), (15.11), among all control actions which robustly satisfy
the constraints, the player U chooses the one which minimizes an expected
cost over all the sequences, which can be played by W .

• The optimal control problem (15.8) and (15.11) is somehow more involved
and can be viewed as the solution of a zero-sum dynamic game. The player
U plays first. Given the initial state x(0), U chooses his action over the
15.1 Problem Formulation 323

whole horizon {u0 , . . . , uN −1 }, reveals his plan to the opponent W , who de-
cides on his actions next {w0a , w0p , . . . , wN
a p
−1 , wN −1 } by solving (15.8). By
solving (15.8), (15.11), the player U selects the action corresponding to the
smallest worst-case cost.

Closed-Loop Predictions
The constrained robust optimal control problem based on closed-loop predictions
(CROC-CL) is defined as follows:
J0∗ (x(0)) = minπ0 (·),...,πN −1 (·) J0 (x(0), U0 )
 

 xk+1 = A(wkp )xk + B(wkp )uk + Ewka 


 


 xk ∈ X , uk ∈ U 

 
uk = πk (xk ) ∀wka ∈ W a , ∀wkp ∈ W p
subj. to

 k = 0, . . . , N − 1 
 ∀k = 0, . . . , N − 1.

 


 x N ∈ Xf 

 
x0 = x(0)
(15.13)
In (15.13) we allow the same cost functions as in (15.6), (15.7) and (15.8).
In problem (15.13) we look for a set of time-varying feedback policies π0 (·),. . .,
πN −1 (·) which minimizes the performance index JW and generates input sequences
π0 (x0 ),. . ., πN −1 (xN −1 ) satisfying state and input constraints for all possible dis-
turbance realizations.
We can formulate the search for the optimal policies π0 (·), . . . , πN −1 (·) as a
dynamic program.

Jj∗ (xj ) = min Jj (xj , uj )

uj
 
xj ∈ X , uj ∈ U
subj. to p p ∀wja ∈ W a , wjp ∈ W p
A(wj )xj + B(wj )uj + Ewja ∈ Xj+1
(15.14)

 
∗
Jj (xj , uj ) = JW q(xj , uj ) + Jj+1 (A(wjp )xj + B(wjp )uj + Ewja ) , (15.15)
for j = 0, . . . , N − 1 and with boundary conditions
∗
JN (xN ) = p(xN ) (15.16a)
XN = Xf , (15.16b)

where Xj denotes the set of states x for which (15.14)–(15.16) is feasible

Xj = {x ∈ X : ∃u ∈ U s.t. A(wp )x + B(wp )u + Ewa ∈ Xj+1 ∀wa ∈ W a , wp ∈ W p },
(15.17)
and the cost operator JW belongs to one of the following classes:
• Nominal Cost.
 
∗
JW = q(xj , uj ) + Jj+1 (A(w̄jp )xj + B(w̄jp )uj + E w̄ja ) . (15.18)
324 15 Constrained Robust Optimal Control

• Expected Cost.
 
∗ (A(wp )x + B(wp )u + Ewa ) .
JW = Ewja ∈W a ,wjp ∈W p q(xj , uj ) + Jj+1 j j j j j
(15.19)

• Worst Case Cost.

 
∗ (A(wp )x + B(wp )u + Ewa ) .
JW = maxwja ∈W a ,wjp ∈W p q(xj , uj ) + Jj+1 j j j j j
(15.20)

If the 1-norm or ∞-norm is used in the cost function of problems (15.18), (15.19)
and (15.20), then we set

p(xN ) = kP xN kp , q(xk , uk ) = kQxk kp + kRuk kp (15.21)

with p = 1 or p = ∞. If the squared Euclidian norm is used then we set

p(xN ) = x′N P xN , q(xk , uk ) = x′k Qxk + u′k Ruk . (15.22)

The reason for including constraints (15.14) in the minimization problem and
not in the inner problem (15.15) is that in (15.15) wja and wjp are free to act
regardless of the state constraints. On the other hand in (15.14), the input uj has
the duty to keep the state within the constraints (15.14) for all possible disturbance
realization. By solving problem (15.14) at step j of the dynamic program, we obtain
the function uj (xj ), i.e., the policy πj (·).
Again, the optimal control problem (15.14)–(15.15) can be viewed as a game
between two players: the controller U and the disturbance W . This time the player
U predicts that the game is going to be played as follows. At the generic time j
player U observes xj and responds with uj = πj (xj ). Player W observes (xj , uj )
and responds with wja and wjp . Therefore, regardless of the cost function in (15.15)
the player U counteracts any feasible disturbance realization wj with a feedback
controller πj (xj ). This prediction model takes into account that the disturbance
and the controller play one move at a time.
The game is played differently depending on the cost function JW .

• In problem (15.14) and (15.18), among all the feedback policies uj = πj (xj )
which robustly satisfy the constraints, the player U chooses the one which
minimizes a cost for a “guessed” action w̄j played by W .

• In problem (15.14) and (15.19), among all the feedback policies uj = πj (xj )
which robustly satisfy the constraints, the player U chooses the one which
minimizes the expected cost at time j.

• The optimal control problem (15.14)–(15.20) is somehow more involved and

can be viewed as the solution of a zero-sum dynamic game. The player U
plays first. At the generic time j player U observes xj and responds with
uj = πj (xj ). Player W observes (xj , uj ) and responds with the wja and wjp
which lead to the worst case cost. The player W will always play the worst
case action only if it has knowledge of both xj and uj = πj (xj ). In fact,
wja and wjp in (15.20) are a function of xj and uj . If U does not reveal his
 15.1 Problem Formulation 325

action to player W , then we can only claim that the player W might play
the worst case action. Problem (15.14)–(15.20) is meaningful in both cases.
Robust constraint satisfaction and worst case minimization will always be
guaranteed.

Example 15.1 Consider the system

xk+1 = xk + uk + wk (15.23)

where x, u and w are state, input and disturbance, respectively. Let uk ∈ {−1, 0, 1}
and wk ∈ {−1, 0, 1} be the feasible input and disturbance. Here {−1, 0, 1} denotes
the set with three elements: -1, 0 and 1. Let x(0) = 0 be the initial state. The
objective for player U is to play two moves in order to keep the state x2 at time 2
in the set [−1, 1]. If U is able to do so for any possible disturbance, then he will win
the game.
The open-loop formulation (15.11) is infeasible. In fact, in open-loop U can choose
from nine possible sequences: (0,0), (1,1), (-1,-1), (-1,1) (1,-1), (-1,0), (1,0), (0,1)
and (0,-1). For any of those sequences there will always exist a disturbance sequence
w0 , w1 which will bring x2 outside the feasible set [-1,1].
The closed-loop formulation (15.14)–(15.20) is feasible and has a simple feasible so-
lution: uk = −xk . In this case system (15.23) becomes xk+1 = wk and x2 = w1 lies
in the feasible set [−1, 1] for all admissible disturbances w1 . The optimal feedback
policy might be different from uk = −xk and depends on the choice of the objective
function.

15.1.1 State Feedback Solutions Summary

In the following sections we will describe how to compute the solution to CROC-OL
and CROC-CL problems. In particular we will show that the solution to CROC-
OL and CROC-CL problem can be expressed in feedback form where u∗ (k) is a
continuous piecewise affine function on polyhedra of the state x(k), i.e., u∗ (k) =
fk (x(k)) where

fk (x) = Fki x + gki if Hki x ≤ Kki , i = 1, . . . , Nkr . (15.24)

Hki and Kki in equation (15.24) are the matrices describing the i-th polyhedron
CRki = {x ∈ Rn : Hki x ≤ Kki } inside which the feedback optimal control law
u∗ (k) at time k has the affine form Fki x + gki .
The optimal solution is continuous and has the form (15.24) in the following
cases.

• Nominal or worst-case performance index based on 1- and ∞-norm,

– CROC-OL with no uncertainty in the dynamics matrix A (i.e., A(·) = A)

– CROC-CL

• Nominal performance index based on 2-norm,

326 15 Constrained Robust Optimal Control

– CROC-OL with no uncertainty in the dynamics matrix A (i.e., A(·) = A)

If CROC-OL problems are considered, the set of polyhedra CRki , i = 1, . . . , Nkr
is a polyhedral partition of the set of feasible states XkOL (15.12) at time k.
If CROC-CL problems are considered, the set of polyhedra CRki , i = 1, . . . , Nkr
is a polyhedral partition of the set of feasible states Xk of problem (15.17) at time
k.
The difference between the feasible sets XkOL and Xk associated with open-loop
prediction and closed-loop prediction, respectively, are discussed in the next section.

Remark 15.2 CROC-OL and CROC-CL problems with expected cost performance
index are not treated in this book. In general, the calculation of expected costs
over polyhedral domains requires approximations via sampling or bounding even for
disturbances characterized by simple probability distribution functions (e.g., uniform
or Gaussian). Once the cost has been approximated, one can easily modify the
approaches described in this chapter to solve CROC-OL and CROC-CL problems
with expected cost performance index.

15.2 Feasible Solutions

As in the nominal case (Section 11.2), there are two ways to define and compute
the robust feasible sets: the batch approach and the recursive approach. While for
systems without disturbances, both approaches yield the same result, in the robust
case the batch approach provides the feasible set XiOL of CROC with open-loop
predictions and the recursive approach provides the feasible set Xi of CROC with
closed-loop predictions. From the discussion in the previous sections, clearly we
have XiOL ⊆ Xi . We will detail the batch approach and the recursive approach next
and at the end of the section we will show how to modify the batch approach in
order to compute Xi .

Batch Approach: Open-Loop Prediction

Consider the set XiOL (15.12) of feasible states xi at time i for which (15.11) is
feasible, for i = 0, . . . , N , rewritten below
XiOL = {xi ∈ X : ∃(ui , . . . , uN −1 ) such that xk ∈ X , uk ∈ U, k = i, . . . , N − 1,
xN ∈ Xf ∀ wka ∈ W a , ∀ wkp ∈ W p k = i, . . . , N − 1,
where xk+1 = A(wkp )xk + B(wkp )uk + Ewka }.
Thus for any initial state xi ∈ XiOL there exists a feasible sequence of inputs
Ui = [u′i , . . . , u′N −1 ] which keeps the state evolution in the feasible set X at future
time instants k = i + 1, . . . , N − 1 and forces xN into Xf at time N for all feasible
disturbance sequences wka ∈ W a , wkp ∈ W p , k = i, . . . , N − 1. Clearly XNOL = Xf .
Next we show how to compute XiOL for i = 0, . . . , N −1. Let the state and input
constraint sets X , Xf and U be the H-polyhedra Ax x ≤ bx , Af x ≤ bf , Au u ≤ bu ,
15.2 Feasible Solutions 327

respectively. Assume that the disturbance sets are represented in terms of their
vertices: W a = conv{wa,1 , . . . , wa,nW a } and W p = conv{wp,1 , . . . , wp,nW p }. Define
Ui = [u′i , . . . , u′N −1 ] and the polyhedron Pi of robustly feasible states and input
sequences at time i,

Pi = {(Ui , xi ) ∈ Rm(N −i)+n : Gi Ui − Ei xi ≤ Wi }. (15.25)

In (15.25) Gi , Ei and Wi are obtained by collecting all the following inequalities:

• Input Constraints
Au uk ≤ bu , k = i, . . . , N − 1.

• State Constraints

Ax xk ≤ bx , k = i, . . . , N − 1 for all wla ∈ W a , wlp ∈ W p , l = i, . . . , k − 1.

(15.26)
• Terminal State Constraints

Af xN ≤ bf , for all wla ∈ W a , wlp ∈ W p , l = i, . . . , N − 1. (15.27)

Constraints (15.26)-(15.27) are enforced for all feasible disturbance sequences.

In order to do so, we rewrite constraints (15.26)-(15.27) at time k as a function of
xi and the input sequence Ui
 Pk−1 l−1 
k−1
Ax Πj=i A(wjp )xi + l=i Πj=i A(wjp )(B(wk−1−l
p a
)uk−1−l + Ewk−1−l ) ≤ bx .

(15.28)
In general, the constraint (15.28) is non-convex in the disturbance sequences
wia , . . . , wk−1
a
, wip , . . . , wk−1
p
because of the product Πj=ik−1
A(wjp ).
Assume that there is no uncertainty in the dynamics matrix A, i.e., A(·) = A.
In this case, since B(·) is an affine function of wp (cf. equation (15.4)) and since the
composition of a convex constraint with an affine map generates a convex constraint
(Section 1.2), we can use Lemma 10.1 to rewrite constraints (15.26) at time k as
 Pk−1 l−i−1 
p
Ax Ak−i−1 xi + l=i A (B(wk−1−l )uk−1−l + Ewk−1−l a
) ≤ bx ,
for all wja ∈ {wa,i }ni=1 Wa
, wjp ∈ {wp,i }ni=1
Wp
, ∀ j = i, . . . , k − 1, (15.29)
k = i, . . . , N − 1,

and imposing the constraints at all the vertices of the sets W a × W a × . . . × W a

| {z }
i,...,N −1
and W p × W p × . . . × W p . Note that the constraints (15.29) are now linear in xi
| {z }
i,...,N −1
and Ui . The same procedure can be repeated for constraint (15.27).

Remark 15.3 When only additive disturbances are present (i.e., np = 0), vertex
enumeration is not required and a set of linear programs can be used to transform
the constraints (15.26) into a smaller number of constraints than (15.29) as explained
in Lemma 10.2.
328 15 Constrained Robust Optimal Control

Once the matrices Gi , Ei and Wi have been computed, the set XiOL is a poly-
hedron and can be computed by projecting the polyhedron Pi in (15.25) on the xi
space.

Recursive Approach: Closed-Loop Prediction

In the recursive approach we have

Xi = {x ∈ X : ∃u ∈ U such that A(wip )x + B(wip )u + Ewia ∈ Xi+1 ,

∀wia ∈ W a , ∀wip ∈ W p } , i = 0, . . . , N − 1
XN = Xf . (15.30)

The definition of Xi in (15.30) is recursive and it requires that for any feasible
initial state xi ∈ Xi there exists a feasible input ui which keeps the next state
A(wip )x + B(wip )u + Ewia in the feasible set Xi+1 for all feasible disturbances wia ∈
W a , wip ∈ W p .
Initializing XN to Xf and solving (15.30) backward in time yields the feasible set
X0 for the CROC-CL (15.14)–(15.16) which, as shown in Example 15.1, is different
from X0OL .
Let Xi be the H-polyhedron AXi x ≤ bXi . Then the set Xi−1 is the projection
of the following polyhedron
     
Au 0 bu
 0  ui−1 +  Ax  xi−1 ≤  bx 
p p a
AXi B(wi ) AXi A(wi ) bXi − Ewi
for all wia ∈ {wa,i }ni=1
Wa
,
for all wip ∈ {wp,i }ni=1
Wp

(15.31)

on the xi−1 space. (Note that we have used Lemma 10.1 and imposed state con-
straints at all the vertices of the sets W a × W p .)

Remark 15.4 When only additive disturbances are present (i.e., np = 0), vertex
enumeration is not required and a set of linear programs can be used to transform the
constraints in (15.30) into a smaller number of constraints than (15.31) as explained
in Lemma 10.2.

The backward evolution in time of the feasible sets Xi enjoys the properties
described by Theorems 11.1 and 11.2 for the nominal case. In particular if a
robust controlled invariant set is chosen as terminal constraint Xf , then set Xi
grows as i becomes smaller and stops growing when it becomes the maximal robust
stabilizable set.

Theorem 15.1 Let the terminal constraint set Xf be a robust control invariant
subset of X . Then,
15.2 Feasible Solutions 329

1. The feasible set Xi , i = 0, . . . , N − 1 is equal to the (N − i)-step robust

stabilizable set:
Xi = KN −i (Xf , W).
2. The feasible set Xi , i = 0, . . . , N − 1 is robust control invariant and contained
within the maximal robust control invariant set:
Xi ⊆ C∞ .

3. Xi ⊇ Xj if i < j, i = 0, . . . , N − 1. The size of the feasible Xi set stops

increasing (with decreasing i) if and only if the maximal robust stabilizable
set is finitely determined and N − i is larger than its determinedness index
N̄ , i.e.
Xi ⊃ Xj if N − N̄ < i < j < N.
Furthermore,
Xi = K∞ (Xf , W) if i ≤ N − N̄ .

Batch Approach: Closed-Loop Prediction∗

The batch approach can be modified in order to obtain Xi instead of XiOL . The
main idea is to augment the number of inputs by allowing one input sequence for
each vertex l of the set containing all disturbance sequences over the horizon [0,N-
1]. We will explain the approach under the assumption that there is no parametric
uncertainty (w = wa ). From causality, the input uj is a function of x0 and the
sequence w0 , . . . , wj−1 . Therefore, u0 will be unique, u1 a function of the vertices
of the disturbance set W a , u2 a function of the vertices of the disturbance set
W a × W a , and uN −1 a function of |W a × W a{z × . . . × W a}. The generic l-th input
N −1
sequence can be written as
j
Ul = [u0 , ũj11 , ũj22 , . . . ũNN−1
−1
]
where the lower index denotes time, and the upper index is used to associate a differ-
ent input for every disturbance realization. For instance, at time 0 the extreme re-
alizations of additive uncertainty are nW a . At time 1 there is one input for each re-
alization and thus j1 ∈ {1, . . . , nW a }. At time 2 the number of extreme realizations
of the disturbance sequence w0 , w1 is n2W a and therefore j2 ∈ {1, . . . , n2W a }. By re-
peating this argument we have jk ∈ {1, . . . , nkW a }, k = 1, . . . , N −1. In total we will

have nN −1 2
W a different control sequences Ul and m 1 + nW a + nW a + · · · + nW a
N −1

control variables to optimize over.

Figure 15.1 depicts an example where the disturbance set has two vertices w1
and w2 (nW a = 2) and the horizon is N = 4. The continuous line represents the
occurrence of w1 , while the dashed line represents the occurrence of w2 . Over the
horizon there are eight possible scenarios associated to eight extreme disturbance
realizations. The state x̃j3 denotes the predicted state at time 3 associated to the
j-th disturbance scenario.
Since the approach is computationally demanding, we prefer to present the main
idea through a very simple example rather than including all the tedious details.
330 15 Constrained Robust Optimal Control

(x̃12 , ũ12 ) (x̃13 , ũ13 )

(x̃11 , ũ11 ) (x̃23 , ũ23 )
(x̃22 , ũ22 ) (x̃33 , ũ33 )
(x0 , u0 ) (x̃43 , ũ43 )
(x̃32 , ũ32 ) (x̃53 , ũ53 )
(x̃63 , ũ63 )
(x̃21 , ũ21 ) (x̃73 , ũ73 )
(x̃42 , ũ42 ) (x̃83 , ũ83 )

Figure 15.1 Batch Approach with Closed-Loop Prediction. State propaga-

tion and associated input variables. The disturbance set has two vertices w1
and w2 (nW a = 2) and the horizon is N = 4. The continuous line represents
the occurrence of w1 , while the dashed line represents the occurrence of w2 .
Over the horizon there are eight possible scenarios associated to eight ex-
treme disturbance realizations. The state x̃j3 denotes the predicted state at
time 3 associated to the j-th disturbance scenario.

Example 15.2 Consider the system

xk+1 = xk + uk + wk (15.32)

where x, u and w are state, input and disturbance, respectively. Let uk ∈ [−1, 1]
and wk ∈ [−1, 1] be the feasible input and disturbance. The objective for player U
is to play two moves in order to keep the state at time three x3 in the set Xf = [−1, 1].

• Batch approach
We rewrite the terminal constraint as

x3 = x0 + u0 + u1 + u2 + w0 + w1 + w2 ∈ [−1, 1]
(15.33)
for all w0 ∈ [−1, 1], w1 ∈ [−1, 1], w2 ∈ [−1, 1]

which by Lemma 10.1 becomes

−1 ≤ x0 + u0 + u1 + u2 + 3 ≤ 1
−1 ≤ x0 + u0 + u1 + u2 + 1 ≤ 1
(15.34)
−1 ≤ x0 + u0 + u1 + u2 − 1 ≤ 1
−1 ≤ x0 + u0 + u1 + u2 − 3 ≤ 1

which by removing redundant constraints becomes the constraint

2 ≤ x0 + u0 + u1 + u2 ≤ −2 (15.35)

which is infeasible and thus X0 = ∅. For the sake of clarity we continue with
the procedure as if we did not notice that constraint (15.35) was infeasible.
The set X0OL is the projection on the x0 space of the polyhedron P0 of combined
15.2 Feasible Solutions 331

feasible input and state constraints

     
1 0 0 0 1
 −1 0 0   0   1 
     
 0 1 0    0   1 
  u0    
 0 −1 0   0   1 
P0 = {(u0 , u1 , u2 , x0 ) ∈ R : 
4
 0
  u1 +
 
 x0 ≤ 
 
}.

 0 1  u2  0   1 
 0 0 −1   0   1 
     
 1 1 1   1   −2 
−1 −1 −1 −1 −2
The projection will provide an empty set since the last two constraints corre-
spond to the infeasible terminal state constraint (15.35).

• Recursive approach
For the recursive approach we have X3 = Xf = [−1, 1]. We rewrite the terminal
constraint as
x3 = x2 + u2 + w2 ∈ [−1, 1] for all w2 ∈ [−1, 1] (15.36)
which by Lemma 10.1 becomes
−1 ≤ x2 + u2 + 1 ≤ 1
(15.37)
−1 ≤ x2 + u2 − 1 ≤ 1
which by removing redundant constraints becomes
0 ≤ x2 + u2 ≤ 0.
The set X2 is the projection on the x2 space of the polyhedron
   
1 0   1
 −1 0   
  u2 ≤  1  (15.38)
 1 1  x2  0 
−1 −1 0
which yields X2 = [−1, 1]. Since X2 = X3 one can conclude that X2 is the
maximal controllable robust invariant set and X0 = X1 = X2 = [−1, 1].

• Batch approach with closed-loop predictions∗

The horizon N is three and therefore we have 2(N−1) = 4 different control
sequences: {u0 , ũ11 , ũ12 }, {u0 , ũ11 , ũ22 }, {u0 , ũ21 , ũ32 }, and {u0 , ũ21 , ũ42 }, cor-
responding to the extreme disturbance realizations {w0 = −1, w1 = −1, w2 },
{w0 = −1, w1 = 1, w2 }, {w0 = 1, w1 = −1, w2 } and {w0 = 1, w1 = 1, w2 },
respectively.
The terminal constraint is thus rewritten as
−1 ≤ x0 + u0 + ũ11 + ũ12 − 1 − 1 + w2 ≤ 1, ∀ w2 ∈ [−1, 1]
−1 ≤ x0 + u0 + ũ11 + ũ22 − 1 + 1 + w2 ≤ 1, ∀ w2 ∈ [−1, 1]
−1 ≤ x0 + u0 + ũ21 + ũ32 + 1 − 1 + w2 ≤ 1, ∀ w2 ∈ [−1, 1]
−1 ≤ x0 + u0 + ũ21 + ũ42 + 1 + 1 + w2 ≤ 1, ∀ w2 ∈ [−1, 1]
−1 ≤ u0 ≤ 1
−1 ≤ ũ11 ≤ 1 (15.39)
−1 ≤ ũ21 ≤ 1
−1 ≤ ũ12 ≤ 1
−1 ≤ ũ22 ≤ 1
−1 ≤ ũ32 ≤ 1
−1 ≤ ũ42 ≤ 1
332 15 Constrained Robust Optimal Control

which becomes by using Lemma 10.1

−1 ≤ x0 + u0 + ũ11 + ũ12 − 1 − 1 + 1 ≤ 1
−1 ≤ x0 + u0 + ũ11 + ũ12 − 1 − 1 − 1 ≤ 1
−1 ≤ x0 + u0 + ũ11 + ũ22 − 1 + 1 + 1 ≤ 1
−1 ≤ x0 + u0 + ũ11 + ũ22 − 1 + 1 − 1 ≤ 1
−1 ≤ x0 + u0 + ũ21 + ũ32 + 1 − 1 + 1 ≤ 1
−1 ≤ x0 + u0 + ũ21 + ũ32 + 1 − 1 − 1 ≤ 1
−1 ≤ x0 + u0 + ũ21 + ũ42 + 1 + 1 + 1 ≤ 1
−1 ≤ x0 + u0 + ũ21 + ũ42 + 1 + 1 − 1 ≤ 1 (15.40)
−1 ≤ u0 ≤ 1
−1 ≤ ũ11 ≤ 1
−1 ≤ ũ21 ≤ 1
−1 ≤ ũ12 ≤ 1
−1 ≤ ũ22 ≤ 1
−1 ≤ ũ32 ≤ 1
−1 ≤ ũ42 ≤ 1.

The set X0 can be obtained by projecting the polyhedron (15.40) in the

(x0 , u0 , ũ11 , ũ21 , ũ12 , ũ22 , ũ32 , ũ42 )-space on the x0 space. This yields X0 = [−1, 1],
the same as in the recursive approach.

15.3 State Feedback Solution, Nominal Cost

Theorem 15.2 Consider the CROC-OL (15.11) with cost (15.5), (15.6) and (15.9)
or (15.10). Assume that the parametric uncertainties are in the B matrix only
(A(wp ) = A). Then, there exists a solution u∗ (0) = f0OL (x(0)), f0 : Rn → Rm ,
which is continuous and PPWA

f0OL (x) = F0i x + g0i if x ∈ CR0i , i = 1, . . . , N0r (15.41)

where the polyhedral sets CR0i = {H0i x ≤ k0i }, i = 1, . . . , N0r , are a partition of the
feasible set X0OL . Moreover f0 can be found by solving an mp-LP for cost (15.9)
and an mp-QP for cost (15.10). The same result holds for u∗ (k) = fkOL (x(0)), k =
1, . . . , N − 1.

Proof: The proof can be easily derived from the results in Chapter 11. The
only difference is that the constraints have to be first “robustified” as explained in
Section 15.2. 

Theorem 15.3 There exists a state feedback control law u∗ (k) = fk (x(k)), fk :
Xk ⊆ Rn → U ⊆ Rm , solution of the CROC-CL (15.14)–(15.16) with cost (15.18),
(15.21) and k = 0, . . . , N −1 which is time-varying, continuous and piecewise affine
on polyhedra

fk (x) = Fki x + gki if x ∈ CRki , i = 1, . . . , Nkr (15.42)

 15.4 State Feedback Solution, Worst-Case Cost, 1-Norm and ∞-Norm Case 333

where the polyhedral sets CRki = {x ∈ Rn : Hki x ≤ Kki }, i = 1, . . . , Nkr are a

partition of the feasible polyhedron Xk . Moreover fi , i = 0, . . . , N − 1 can be found
by solving N mp-LPs.
Proof: The proof can be easily derived from the results in Chapter 11. The
only difference is that the constraints have to be first robustified as explained in
Section 15.2. 

15.4 State Feedback Solution, Worst-Case Cost, 1-Norm

and ∞-Norm Case
In this chapter we show how to find a state feedback solution to CROC problems
when a worst case performance index is used. The following results will be used in
the next sections.

Lemma 15.1 Let f : Rs ×Rn ×Rnw → R and g : Rs ×Rn ×Rnw → Rng be functions
of (z, x, w) convex in w for each (z, x). Assume that the variable w belongs to the
polyhedron W with vertices {w̄i }N
i=1 . Then the min-max multiparametric problem
W

J ∗ (x) = minz maxw∈W f (z, x, w)

(15.43)
subj. to g(z, x, w) ≤ 0 ∀w ∈ W
is equivalent to the multiparametric optimization problem
J ∗ (x) = minµ,z µ
subj. to µ ≥ f (z, x, w̄i ), i = 1, . . . , NW (15.44)
g(z, x, w̄i ) ≤ 0, i = 1, . . . , NW .

Proof: Easily follows from the fact that the maximum of a convex function
over a convex set is attained at an extreme point of the set, cf. also [259]. 

Corollary 15.1 If f is convex and piecewise affine in (z, x), i.e., f (z, x, w) =
maxi=1,...,nf {Li (w)z + Hi (w)x + Ki (w)} and g is linear in (z, x) for all w ∈ W,
g(z, x, w) = Lg (w)z + Hg (w)x + Kg (w) (with Lg (·), Hg (·), Kg (·), Li (·), Hi (·),
Ki (·), i = 1, . . . , nf , convex functions), then the min-max multiparametric prob-
lem (15.43) is equivalent to the mp-LP problem
J ∗ (x) = minµ,z µ
subj. to µ ≥ Lj (w̄i )z + Hj (w̄i )x + Kj (w̄i ), i = 1, . . . , NW , j = 1, . . . , nf
Lg (w̄i )z + Hg (w̄i )x ≤ −Kg (w̄i ), i = 1, . . . , NW .
(15.45)

Remark 15.5 As discussed in Lemma 10.2, in the case g(z, x, w) = g1 (z, x) + g2 (w),
the second constraint in (15.44) can be replaced by g1 (z, x) ≤ −ḡ, where ḡ =
′
ḡ1 , . . . , ḡng is a vector whose i-th component is

ḡi = max g2i (w), (15.46)

w∈W
334 15 Constrained Robust Optimal Control

and g2i (w) denotes the i-th component of g2 (w). Similarly, if f (z, x, w) = f1 (z, x) +
f2 (w), the first constraint in (15.44) can be replaced by µ ≥ f1 (z, x) + f¯, where

f¯i = max f2i (w). (15.47)

w∈W

Clearly, this has the advantage of reducing the number of constraints in the multi-
parametric program from NW ng to ng for the second constraint in (15.44) and from
NW nf to nf for the first constraint in (15.44).

In the following subsections we show how to solve CROC problems in state

feedback form by using multiparametric linear programming.

15.4.1 Batch Approach: Open-Loop Predictions

Theorem 15.4 Consider the CROC-OL (15.5), (15.8), (15.9), (15.11). Assume
that the parametric uncertainties are in the B matrix only (A(wp ) = A). Then,
there exists a solution u∗ (0) = f0OL (x(0)), f0OL : Rn → Rm , which is continuous
and PPWA
f0OL (x) = F0i x + g0i if x ∈ CR0i , i = 1, . . . , N0r (15.48)
where the polyhedral sets CR0i = {H0i x ≤ k0i }, i = 1, . . . , N0r , are a partition of the
feasible set X0OL . Moreover f0 can be found by solving an mp-LP. The same result
holds for u∗ (k) = fkOL (x(0)), k = 1, . . . , N − 1.
Pk−1 p
Proof: Since xk = Ak x0 + k=0 Ai [B(wk−1−i )uk−1−i + Ewk−1−ia
] is a linear
p p
function of the disturbances wa = {w0a , . . . , wN
a
−1 }, and w p
= {w 0 , . . . , w N −1 } for a
fixed input sequence and x0 , the cost function in the maximization problem (15.8) is
convex and piecewise affine with respect to the optimization vectors wa and wp and
the parameters U0 , x0 . The constraints in (15.11) are linear in U0 and x0 , for any
wa and wp . Therefore, by Corollary 15.1, problem (15.8)–(15.11) can be solved by
specifying the constraints (15.11) only at the vertices of the set W a ×W a ×. . .×W a
and W p × W p × . . . × W p and solving an mp-LP. The theorem follows from the
mp-LP properties described in Theorem 6.5. 

Remark 15.6 Note that Theorem 15.4 does not hold when parametric uncertainties
are present also in the A matrix. In this case the predicted state xk is a nonlinear
function of the vector wp .

Remark 15.7 In case of CROC-OL with additive disturbances only (wp (t) = 0) the
number of constraints in (15.11) can be reduced as explained in Remark 15.5.

15.4.2 Recursive Approach: Closed-Loop Predictions

Theorem 15.5 There exists a state feedback control law u∗ (k) = fk (x(k)), fk :
Xk ⊆ Rn → U ⊆ Rm , solution of the CROC-CL (15.14)–(15.17) with cost (15.20), (15.21)
15.4 State Feedback Solution, Worst-Case Cost, 1-Norm and ∞-Norm Case 335

and k = 0, . . . , N − 1 which is time-varying, continuous and piecewise affine on

polyhedra
fk (x) = Fki x + gki if x ∈ CRki , i = 1, . . . , Nkr (15.49)

where the polyhedral sets CRki = {x ∈ Rn : Hki x ≤ Kki }, i = 1, . . . , Nkr are a

partition of the feasible polyhedron Xk . Moreover fi , i = 0, . . . , N − 1 can be found
by solving N mp-LPs.

Proof: Consider the first step j = N − 1 of dynamic programming applied to

the CROC-CL problem (15.14)–(15.20) with cost (15.9)
∗
JN −1 (xN −1 ) = umin JN −1 (xN −1 , uN −1 ) (15.50)
N −1

 xN −1 ∈ X , uN −1 ∈ U
p p a
subj. to A(wN −1 )xN −1 + B(wN −1 )uN −1 + EwN −1 ∈ Xf
 a a p p
∀wN −1 ∈ W , ∀wN −1 ∈ W
(15.51)

 
 kQxN −1 kp + kRuN −1 kp + 
p
JN −1 (xN −1 , uN −1 ) = a maxp +kP (A(wN −1 )xN −1 + .
wN −1 ∈W a , wN −1 ∈W p  p a 
+B(wN −1 )u N −1 + Ew N −1 )k p
(15.52)

The cost function in the maximization problem (15.52) is piecewise affine and
a p
convex with respect to the optimization vector wN −1 , wN −1 and the parameters
uN −1 , xN −1 . Moreover, the constraints in the minimization problem (15.51) are
a p
linear in (uN −1 , xN −1 ) for all vectors wN −1 , wN −1 . Therefore, by Corollary 15.1,
∗ ∗
JN −1 (xN −1 ), uN −1 (xN −1 ) and XN −1 are computable via the mp-LP:
∗
JN −1 (xN −1 ) = min µ
µ,uN −1

subj. to µ ≥ kQxN −1 kp + kRuN −1 kp +

+ kP (A(w̄hp )xN −1 + B(w̄hp )uN −1 + E w̄ia )kp
(15.53a)
xN −1 ∈ X , uN −1 ∈ U (15.53b)
A(w̄hp )xN −1 + B(w̄hp )uN −1 + E w̄ia ∈ XN (15.53c)
∀i = 1, . . . , nW a , ∀h = 1, . . . , nW p .

where {w̄ia }ni=1 and {w̄hp }nh=1 are the vertices of the disturbance sets W a and
Wa Wp

p ∗
W , respectively. By Theorem 6.5, JN −1 is a convex and piecewise affine function
of xN −1 , the corresponding optimizer u∗N −1 is piecewise affine and continuous,
and the feasible set XN −1 is a convex polyhedron. Therefore, the convexity and
linearity arguments still hold for j = N −2, . . . , 0 and the procedure can be iterated
backwards in time j, proving the theorem. The theorem follows from the mp-LP
properties described in Theorem 6.5. 
336 15 Constrained Robust Optimal Control

Remark 15.8 Let na and nb be the number of inequalities in (15.53a) and (15.53c),
respectively, for any i and h. In case of additive disturbances only (nW p = 0) the
total number of constraints in (15.53a) and (15.53c) for all i and h can be reduced
from (na + nb )nW a nW p to na + nb as shown in Remark 15.5.

Remark 15.9 The closed-loop solution u∗ (k) = fk (x(k)) can be also obtained by using
the modified batch approach with closed-loop prediction as discussed in Section 15.2.
The idea there is to augment the number of free inputs by allowing one sequence
ũi0 , . . . ũiN−1 for each vertex i of the disturbance set W a × W a × . . . × W a . The large
| {z }
N−1
number of extreme points of such a set and the resulting large number of inputs and
constraints make this approach computationally hard.

15.4.3 Solution to CROC-CL and CROC-OL via mp-MILP*

Theorems 15.2-15.5 propose different ways of finding the PPWA solution to con-
strained robust optimal control by using multiparametric programs. The solution
approach presented in this section is more general than the one of Theorems 15.2-
15.5 as it does not exploit convexity, so that it may also be used in other contexts,
for instance for CROC-CL of hybrid systems, or for CROC of uncertain systems
of the type x(t + 1) = Ax(t) + Bu(t) + Ewa (t) where A and B belong to a finite
discrete set of known matrices.
Consider the multiparametric mixed-integer linear program (mp-MILP)
J ∗ (x) = min {J(z, x) = c′ z}
z (15.54)
subj. to Gz ≤ w + Sx.
where z = [zc , zd ], zc ∈ Rnc , zd ∈ {0, 1}nd , s = nc + nd , z ∈ Rs is the optimization
vector and x ∈ Rn is the vector of parameters.
For a given polyhedral set K ⊆ Rn of parameters, solving (15.54) amounts to
determining the set K∗ ⊆ K of parameters for which (15.54) is feasible, the value
function J : K∗ → R, and the optimizer function1 z ∗ : K∗ → Rs .
The properties of J ∗ (·) and z ∗ (·) have been analyzed in Section 6.4.1 and sum-
marized in Theorem 6.10. Below we state some properties based on these theorems.

Lemma 15.2 Let J : Rs × Rn → R be a continuous piecewise affine (possibly

nonconvex) function of (z, x),
J(z, x) = Li z + Hi x + Ki for [ xz ] ∈ Ri , (15.55)
SnJ
where {Ri }ni=1
are polyhedral sets with disjoint interiors, R = i=1 Ri is a (possibly
J

non-convex) polyhedral set and Li , Hi and Ki are matrices of suitable dimensions.

Then the multiparametric optimization problem
J ∗ (x) = minz J(z, x)
(15.56)
subj. to Cz ≤ c + Sx
1 In case of multiple solutions, we define z ∗ (x) as one of the optimizers.
15.4 State Feedback Solution, Worst-Case Cost, 1-Norm and ∞-Norm Case 337

is an mp-MILP.
Proof: By following the approach of [42] to transform piecewise affine func-
tions into a set of mixed-integer linear inequalities, introduce the auxiliary binary
optimization variables δi ∈ {0, 1}, defined as
 
[δi = 1] ↔ [ xz ] ∈ Ri , (15.57)
PnJ
where δi , i = 1, . . . , nJ , satisfy the exclusive-or condition i=1 δi = 1, and set
nJ
X
J(z, x) = qi (15.58)
i=1
qi = [Li z + Hi x + Ki ]δi (15.59)

where qi are auxiliary continuous optimization vectors. By transforming (15.57)–

(15.59) into mixed-integer linear inequalities [42], it is easy to rewrite (15.56) as a
multiparametric MILP. 

Next we present two theorems which describe how to use mp-MILP to solve
CROC-OL and CROC-CL problems.

Theorem 15.6 By solving two mp-MILPs, the solution U0∗ (x0 ) to the CROC-
OL (15.8), (15.9), (15.11) with additive disturbances only (np = 0) can be computed
in explicit piecewise affine form (15.48).
Proof: The objective function in the maximization problem (15.8) is convex
and piecewise affine with respect to the optimization vector wa = {w0a , . . . , wN
a
−1 }
and the parameters U = {u0 , . . . , uN −1 }, x0 . By Lemma 15.2, it can be solved via
mp-MILP. By Theorem 6.10, the resulting value function J is a piecewise affine
function of U and x0 and the constraints in (15.11) are a linear function of the
disturbance wa for any given U and x0 . Then, by Lemma 15.2 and Corollary 15.1
the minimization problem is again solvable via mp-MILP, and the optimizer U ∗ =
{u∗0 ,. . . ,u∗N −1 } is a piecewise affine function of x0 . 

Theorem 15.7 By solving 2N mp-MILPs, the solution of the CROC-CL (15.14)–

(15.17) problem with cost (15.20), (15.21) and additive disturbances only (np = 0)
can be obtained in state feedback piecewise affine form (15.49)
Proof: Consider the first step j = N − 1 of the dynamic programming solution
to CROC-CL. The cost-to-go at step N − 1 is p(xN ) from (15.16). Therefore, the
cost function in the maximization problem is piecewise affine with respect to both
a
the optimization vector wN −1 and the parameters uN −1 , xN −1 . By Lemma 15.2,
the worst case cost function (15.20) can be computed via mp-MILP, and, from
Theorem 6.10 we know that JN −1 (uN −1 , xN −1 ) is a piecewise affine function. Then,
a
since constraints (15.51) are linear with respect to wN −1 for each uN −1 , xN −1 , we
can apply Corollary 15.1 and Remark 15.5 by solving LPs of the form (10.54).
∗
Then, by Lemma 15.2, the optimal cost JN −1 (xN −1 ) of the minimization problem
338 15 Constrained Robust Optimal Control

(15.14) is again computable via mp-MILP, and by Theorem 6.10 it is a piecewise

affine function of xN −1 . By virtue of Theorem 6.10, XN −1 is a (possible non-convex)
polyhedral set and therefore the above maximization and minimization procedures
can be iterated to compute the solution (15.49) to the CROC-CL problem. 

15.5 Parametrizations of the Control Policies

From the previous sections it is clear that CROC-OL (15.5)–(15.11) is conservative
since we are optimizing an open-loop control sequence that has to cope with all
possible future disturbance realizations, without taking future measurements into
account. The CROC-CL (15.14)–(15.16) formulation overcomes this issue but it
can quickly lead to a problem so large that it becomes intractable.
This section presents an alternative approach which introduces feedback in the
system and, in some cases, can be more efficient than CROC-CL. The idea is
to parameterize the control sequence in the state vector and optimize over these
parameters. The approach is described next for systems with additive uncertainties.
Consider the worst case cost function as
P −1
J0 (x(0), U0 ) = maxw0a ,...,wN
a
−1
p(xN ) + N
k=0 q(xk , uk )

 xk+1 = Axk + Buk + Ewka (15.60)
subj. to wa ∈ W a ,
 k
k = 0, . . . , N − 1

where N is the time horizon and U0 = [u′0 , . . . , u′N −1 ]′ ∈ Rs , s = mN the vector of

the input sequence. Consider the robust optimal control problem

J0∗ (x0 ) = min J0 (x0 , U0 ) (15.61)

U0
 

 xk ∈ X , uk ∈ U 

 
xk+1 = Axk + Buk + Ewka ∀wka ∈ W a
subj. to (15.62)

 xN ∈ Xf 
 ∀k = 0, . . . , N − 1
 
k = 0, . . . , N − 1

Consider the parametrization of the control sequence

k
X
uk = Lk,i xi + gi , k ∈ {0, . . . , N − 1} (15.63)
i=0

with the compact notation:

U0 = Lx + g,
′
where x = [x′0 , x′1 , . . . , x′N ] and
   
L0,0 0 ··· 0 g0
 .. ..  ,  
L= . . .. ..
. . g =  ...  (15.64)
LN −1,0 · · · LN −1,N −1 0 gN −1
15.5 Parametrizations of the Control Policies 339

where L ∈ RmN ×nN and g ∈ RmN are unknown feedback control gain and offset,
respectively. With the parametrization (15.63) the robust control problem (15.60-
(15.62) becomes
PN −1
J0Lg (x(0), L, g) = maxw0a ,...,wN
a
−1
p(xN ) + k=0 q(xk , uk )


 xk+1 = Axk + Buk + Ewka
 wa ∈ W a , (15.65)
k P
subj. to

 u k = ki=0 Lk,i xi + gi

k = 0, . . . , N − 1

∗
J0Lg (x0 ) = min J0Lg (x0 , L, g) (15.66)
L,g
 

 xk ∈ X , uk ∈ U 


 Ax + Buk + Ewka 

 xk+1 = Pk k 
∀wka ∈ W a
subj. to uk = i=0 Lk,i xi + gi

 
 ∀k = 0, . . . , N − 1.

 x ∈ Xf 

 N 
k = 0, . . . , N − 1
(15.67)

We denote with X0Lg ⊆ X the set of states x0 for which the robust optimal control
problem (15.66)-(15.67) is feasible, i.e.
n o
X0Lg = x0 ∈ Rn : P0Lg (x0 ) 6= ∅
P0Lg (x0 ) = {L, g : xk ∈ X , uk ∈ U, k = 0, . . . , N − 1, xN ∈ Xf
∀ wka ∈ W a k = 0, . . . , N − 1,
Pk
where xk+1 = Axk + Buk + Ewka , uk = i=0 Lk,i xi + gi }.
(15.68)
Problem (15.65) looks for the worst value of the performance index J0Lg (x0 , L, g)
and the corresponding worst sequences wa ∗ as a function of x0 and the controller
gain L and offset g.
Problem (15.66)–(15.67) minimizes (over L and g) the worst performance sub-
ject to the constraint that the input sequence U0 = Lx + g must be feasible for
all possible disturbance realizations. Notice that formulation (15.65)–(15.67) is
based on a closed-loop prediction. Unfortunately the set P0Lg (x0 ) is non–convex,
in general [194]. Therefore, finding L and g for a given x0 may be difficult.
Consider now the parametrization of the control sequence in past disturbances

k−1
X
uk = Mk,i wia + vi , k ∈ {0, . . . , N − 1}, (15.69)
i=0

which can be compactly written as:

U0 = M w a + v
340 15 Constrained Robust Optimal Control

where
   
0 ··· ··· 0 v0
  
..
 M1,0 0 ··· 0   
.
M = .. .. .. ..  , 
v= . (15.70)
 
..
. . . .  .
MN −1,0 ··· MN −1,N −2 0 vN −1
Notice that since
Ewka = xk+1 − Axk − Buk , k ∈ {0, . . . , N − 1}.
the parametrization (15.69) can also be interpreted as a parametrization in the
states. The advantages of using (15.69) are explained next.
With the parametrization (15.69) the robust control problem (15.60-(15.62)
becomes
P −1
J0Mv (x0 , M, v) = maxw0a ,...,wN
a
−1
p(xN ) + N
k=0 q(xk , uk )



 xk+1 = Axk + Buk + Ewka
 wa ∈ W a , (15.71)
k Pk−1
subj. to
 uk = i=0
 Mk,i wia + vi

k = 0, . . . , N − 1

∗
J0Mv (x0 ) = min J0Mv (x0 , M, v) (15.72)
M,v
 

 xk ∈ X , uk ∈ U 


 Ax + Buk + Ewka 

 xk+1 = Pk−1k 
∀wka ∈ W a
subj. to u = i=0 Mk,i wia + vi
 k
  ∀k = 0, . . . , N − 1.


 x ∈ Xf 

 N 
k = 0, . . . , N − 1
(15.73)
The problem (15.71)-(15.73) is now convex in the controller parameters M and v.
As solution we obtain u∗ (0) = f0 (x(0)) = v0 (x(0)).
We denote with X0Mv ⊆ X the set of states x0 for which the robust optimal
control problem (15.72)-(15.73) is feasible, i.e.

X0Mv = x0 ∈ Rn : P0Mv (x0 ) 6= ∅
P0Mv (x0 ) = {M, v : xk ∈ X , uk ∈ U, k = 0, . . . , N − 1, xN ∈ Xf
∀ wka ∈ W a k = 0, . . . , N − 1, where xk+1 = Axk + Buk + Ewka ,
Pk−1
uk = i=0 Mk,i wia + vi }.
(15.74)
The following result has been proven in [130], the convexity property has also
appeared in [194][Section 7.4].

Theorem 15.8 Consider the control parameterizations (15.63), (15.69) and the
corresponding feasible sets X0Lg in (15.68) and X0Mv in (15.74). Then,
X0Lg = X0Mv
and P0Mv (x0 ) is convex in M and v.
15.5 Parametrizations of the Control Policies 341

∗
Note that in general X0Mv and J0Mv (x0 ) are different from the corresponding
∗
CROC-CL solutions X0 and J0∗ (x0 ). In particular X0Mv ⊆ X0 and J0Mv (x0 ) ≥
J0∗ (x0 ).
Next we solve the problem in Example 15.2 by using the approach discussed in
this section. The example will shed some light on how to compute the disturbance
feedback gains by means of convex optimization.

Example 15.3 Consider the system

xk+1 = xk + uk + wk (15.75)

where x, u and w are state, input and disturbance, respectively. Let uk ∈ [−1, 1] and
wk ∈ [−1, 1] be the feasible input and disturbance. The objective for player U is to
play two moves in order to keep the state x3 in the set Xf = [−1, 1].
Rewrite the terminal constraint as

x3 = x0 + u0 + u1 + u2 + w0 + w1 + w2 ∈ [−1, 1]
(15.76)
∀ w0 ∈ [−1, 1], w1 ∈ [−1, 1], w2 ∈ [−1, 1].

The control inputs u are parameterized in past disturbances as in equation (15.69)

u0 = v0
u1 = v1 + M1,0 w0 (15.77)
u2 = v2 + M2,0 w0 + M2,1 w1 .

Input constraints and terminal constraint are rewritten as

x0 + v0 + v1 + v2 + (1 + M1,0 + M2,0 )w0 + (1 + M2,1 )w1 + w2 ∈ [−1, 1]

u0 = v0 ∈ [−1, 1]
u1 = v1 + M1,0 w0 ∈ [−1, 1] (15.78)
u2 = v2 + M2,0 w0 + M2,1 w1 ∈ [−1, 1]
∀ w0 ∈ [−1, 1], w1 ∈ [−1, 1], w2 ∈ [−1, 1].

There are two approaches to convert the infinite number of constraints (15.78) into
a set of finite number of constraint. The first approach has been already presented
in this chapter and resorts to disturbance vertex enumeration. The second approach
uses duality. We will present both approaches next.

• Enumeration method

Constraint (15.78) are linear in w0 , w1 and w2 for fixed v0 , v1 , v2 , M1,0 , M2,0 ,

M2,1 . Therefore the worst case will occur at the extremes of the admissible dis-
turbance realizations, i.e., {(1, 1, 1), (1, 1, −1), (−1, 1, 1), (−1, 1, −1), (1, −1, 1),
(1, −1, −1), (−1, −1, 1), (−1, −1, −1)}. Therefore the constraints (15.78) can
342 15 Constrained Robust Optimal Control

be rewritten as:
x0 + v0 + v1 + v2 + (1 + M1,0 + M2,0 ) + (1 + M2,1 ) + 1 ∈ [−1, 1]
x0 + v0 + v1 + v2 + (1 + M1,0 + M2,0 ) + (1 + M2,1 ) − 1 ∈ [−1, 1]
x0 + v0 + v1 + v2 − (1 + M1,0 + M2,0 ) + (1 + M2,1 ) + 1 ∈ [−1, 1]
x0 + v0 + v1 + v2 − (1 + M1,0 + M2,0 ) + (1 + M2,1 ) − 1 ∈ [−1, 1]
x0 + v0 + v1 + v2 + (1 + M1,0 + M2,0 ) − (1 + M2,1 ) + 1 ∈ [−1, 1]
x0 + v0 + v1 + v2 + (1 + M1,0 + M2,0 ) − (1 + M2,1 ) − 1 ∈ [−1, 1]
x0 + v0 + v1 + v2 − (1 + M1,0 + M2,0 ) − (1 + M2,1 ) + 1 ∈ [−1, 1]
x0 + v0 + v1 + v2 − (1 + M1,0 + M2,0 ) − (1 + M2,1 ) − 1 ∈ [−1, 1] (15.79)
u0 = v0 ∈ [−1, 1]
u1 = v1 + M1,0 ∈ [−1, 1]
u1 = v1 − M1,0 ∈ [−1, 1]
u2 = v2 + M2,0 + M2,1 ∈ [−1, 1]
u2 = v2 + M2,0 − M2,1 ∈ [−1, 1]
u2 = v2 − M2,0 + M2,1 ∈ [−1, 1]
u2 = v2 − M2,0 − M2,1 ∈ [−1, 1].

A feasible solution to (15.79) can be obtained by solving a linear optimization

problem for a fixed initial condition x0 . If we set x0 = 0.8, a feasible solution
to (15.79) is
v0 = −0.8, v1 = 0, v2 = 0
(15.80)
M1,0 = −1, M2,0 = 0, M2,1 = −1
which provides the controller

u0 = −0.8
u1 = −w0 (15.81)
u2 = −w1

which corresponds to
u0 = −0.8
u1 = −x1 (15.82)
u2 = −x2 .
• Duality method

Duality can be used to avoid vertex enumeration. Consider the first constraint
in (15.78):

x0 + v0 + v1 + v2 + (1 + M1,0 + M2,0 )w0 + (1 + M2,1 )w1 + w2 ≤ 1,

∀ w0 ∈ [−1, 1], ∀w1 ∈ [−1, 1], ∀w2 ∈ [−1, 1].
(15.83)
We want to replace it with the most stringent constraint, i.e.

x0 + v0 + v1 + v2 +J ∗ (M1,0 , M2,0 , M2,1 ) ≤ 1 (15.84)

where
J ∗ (M1,0 , M2,0 , M2,1 )=maxw0 ,w1 ,w2(1 + M1,0 + M2,0 )w0 + (1 + M2,1 )w1 + w2
subj. to w0 ∈ [−1, 1], w1 ∈ [−1, 1], w2 ∈ [−1, 1].
(15.85)
For fixed M1,0 , M2,0 , M2,1 the optimization problem in (15.85) is a linear pro-
gram and can be replaced by its dual

x0 + v0 + v1 + v2 + d∗ (M1,0 , M2,0 , M2,1 ) ≤ 1 (15.86)

15.5 Parametrizations of the Control Policies 343

where

d∗ (M1,0 , M2,0 , M2,1 ) = minλu ,λl λu0 + λu1 + λu2 + λl0 + λl1 + λl2
i i
subject to 1 + M1,0 + M2,0 + λl0 − λu0 = 0
1 + M2,1 + λl1 − λu1 = 0 (15.87)
1 + λl2 − λu2 = 0
λui ≥ 0, λli ≥ 0, i = 0, 1, 2

where λui and λli are the dual variables corresponding to the upper- and lower-
bounds of wi , respectively. From strong duality we know that for a given v0 ,
v1 , v2 , M1,0 , M2,0 , M2,1 problem (15.84)-(15.85) is feasible if and only if there
exists a set of λui and λli which together with v0 , v1 , v2 , M1,0 , M2,0 , M2,1 solves
the system of equalities and inequalities

x0 + v0 + v1 + v2 +λu0 + λu1 + λu2 + λl0 + λl1 + λl2 ≤ 1

1 + M1,0 + M2,0 + λl0 − λu0 = 0
1 + M2,1 + λl1 − λu1 = 0 (15.88)
1 + λl2 − λu2 = 0
λui ≥ 0, λli ≥ 0, i = 0, 1, 2.

We have transformed an infinite dimensional set of constraints in (15.83) into

a finite dimensional set of linear constraints (15.88). By repeating the same
procedure for every constraint in (15.78) we obtain a set of linear constraints.
In particular,

x0 + v0 + v1 + v2 + (1 + M1,0 + M2,0 )w0 + (1 + M2,1 )w1 + w2 ∈ [−1, 1]

∀ w0 ∈ [−1, 1], w1 ∈ [−1, 1], w2 ∈ [−1, 1]

is transformed into

x0 + v0 + v1 + v2 + λu0 + λu1 + λu2 + λl0 + λl1 + λl2 ≤ 1

−x0 − v0 − v1 − v2 + µu0 + µu1 + µu2 + µl0 + µl1 + µl2 ≤ 1
1 + M1,0 + M2,0 + λl0 − λu0 = 0
1 + M2,1 + λl1 − λu1 = 0
1 + λl2 − λu2 = 0 (15.89)
−1 − M1,0 − M2,0 + µl0 − µu0 = 0
−1 − M2,1 + µl1 − µu1 = 0
−1 − µl2 − µu2 = 0
λui , λli , µui , µli ≥ 0, i = 0, 1, 2,

and
u1 = v1 + M1,0 w0 ∈ [−1, 1]
∀ w0 ∈ [−1, 1], w1 ∈ [−1, 1], w2 ∈ [−1, 1]
is transformed into
v1 + ν0u + ν0l ≤ 1
−v1 + κu0 + κl0 ≤ 1
M1,0 + ν1l − ν1u = 0 (15.90)
−M1,0 + κl1 − κu1 = 0
κu0 , κl0 , ν0u , ν0l ≥ 0,
and
u2 = v2 + M2,0 w0 + M2,1 w1 ∈ [−1, 1]
∀ w0 ∈ [−1, 1], w1 ∈ [−1, 1], w2 ∈ [−1, 1]
344 15 Constrained Robust Optimal Control

is transformed into
v2 + ρu0 + ρu1 + ρl0 + ρl1 ≤ 1
−v2 + π0u + π1u + π0l + π1l ≤ 1
M2,0 + ρl0 − ρu0 = 0
M2,1 + ρl1 − ρu1 = 0 (15.91)
−M2,0 + π0l − π0u = 0
−M2,1 + π1l − π1u = 0
ρui , ρli , πiu , πil ≥ 0, i = 0, 1

and the constraint

u0 = v0 ∈ [−1, 1] (15.92)
remains unchanged. The solution to the set of linear of equalities and inequali-
ties (15.89)-(15.92) for a given x0 provides the solution to our problem.

15.6 Example

Example 15.4 Consider the problem of robustly regulating to the origin the system
   
1 0.1 −0.5
x(t + 1) = x(t) + u(t) + wa (t)
−0.2 1 1

subject to the input constraints

U = {u ∈ R : − 3 ≤ u ≤ 1}

and the state constraints

X = {x ∈ R2 : − 10 ≤ x ≤ 10, k = 0, . . . , 3}.

The two-dimensional disturbance wa is restricted to the set W a = {v : kwa k∞ ≤ 2}.

We use the cost function
N−1
X
kP xN k∞ + (kQxk k∞ + |Ruk |)
k=0

with N = 3, P = Q = [ 10 11 ], R = 1.8 and we set Xf = X .

We compute three robust controllers CROC-OL, CROC-CL and CROC with distur-
bance feedback parametrization.
CROC-OL. The min-max state feedback control law u∗ (k) = fkOL (x(0)), k = 1, . . . , N −
1 is obtained by solving the CROC-OL (15.8), (15.9), (15.11). The polyhedral parti-
tion corresponding to u∗ (0) consists of 64 regions and it is depicted in Figure 15.2(a).
CROC-CL. The min-max state feedback control law u∗ (k) = fkCL (x(k)), k = 1, . . . , N −
1 obtained by solving problem (15.14)–(15.16), (15.20), (15.21) using the approach
of Theorem 15.5. The resulting polyhedral partition for k = 0 consists of 14 regions
and is depicted in Figure 15.2(b).
CROC-Parameterized. The min-max state feedback control law u∗ (k) = fkpar (x(0)), k =
1, . . . , N −1 obtained by solving problem (15.71)-(15.73) using the approach described
15.6 Example 345

in Example 15.3. The resulting polyhedral partition for k = 0 consists of 136 regions
and is depicted in Figure 15.3(a). Figure 15.3(b) shows the set of feasible initial
states X0OL , X0 and X0M v corresponding to the controllers CROC-OL, CROC-CL
and CROC-Parameterized, respectively.
x2

x2
x1 x1

(a) Example 15.4. Solution to the (b) Example 15.4. Solution to the
CROC-OL problem. Polyhedral CROC-CL problem. Polyhedral
partition of the state space. partition of the state space.

Figure 15.2 Example 15.4. Polyhedral partition of the state space corre-
sponding to the explicit solution of CROC-OL and CROC-CL at time t = 0.

X0 = X0Mv

X0OL
x2
x2

x1 x1

(a) Example 15.4. Solution to (b) Example 15.4. Sets X0OL

the CROC-Parametrized prob- (dashed line), X0 (continuous
lem. Polyhedral partition of the line) and X0M v (continuous line).
state space.

Figure 15.3 Example 15.4. (a) Polyhedral partition of the state space cor-
responding to the explicit solution of CROC-Parametrized at time t = 0.
(b) Sets X0OL , X0 and X0M v corresponding to the controllers CROC-OL,
CROC-CL and CROC-Parameterized, respectively.
346 15 Constrained Robust Optimal Control

15.7 Robust Receding Horizon Control

A robust receding horizon controller for system (15.1)-(15.4) which enforces the
constraints (15.2) at each time t in spite of additive and parametric uncertainties
can be obtained by setting
u(t) = f0∗ (x(t)), (15.93)
where f0∗ (x(0)) : Rn → Rm is the solution to the CROC-OL or CROC-CL problems
discussed in the previous sections.
If f0 is computed by solving CROC-CL (15.14)–(15.17) (CROC-OL (15.11)),
then the RHC law (15.93) is called a robust receding horizon controller with
closed-loop (open-loop) predictions. The closed-loop system obtained by control-
ling (15.1)-(15.4) with the RHC (15.93) is

x(k + 1) = A(wp )x(k) + B(wp )f0 (x(k)) + Ewa = fcl (x(k), wp , wa ), k ≥ 0 (15.94)

If the CROC falls in one of the classes presented in Section 15.1.1 then from the
theorems presented in this chapter we can immediately conclude that the robust
RHC law (15.93) is piecewise affine and thus its on-line computation comprises a
function evaluation.
As discussed in Chapter 12 convergence and persistent feasibility of the robust
receding horizon controller are not guaranteed for nominal receding horizon con-
trollers. In the robust RHC case it is desirable to obtain robust convergence to a
set O ⊆ Xf (rather than convergence to an equilibrium point) for all X0 . In other
words, the goal is to design a RHC control law which drives any feasible state in
X0 into the set O for all admissible disturbances and keeps the states inside the set
O for all future time and for all admissible disturbances. Clearly this is possible
only if O is a robust control invariant set for system (15.1)-(15.4).
We define a distance of a point x ∈ Rn from a nonempty set Y ⊂ Rn as:

d(x, Y) = inf d(x, y) (15.95)

y∈Y

The following theorem presents sufficient conditions for convergence and per-
sistent feasibility in the robust case. It can be proven by using the arguments
of Theorem 12.2. Slightly modified versions can be found with proof and further
discussion in [247, p. 213-217, p. 618] and in [23, Section 7.6].

Theorem 15.9 Consider system (15.1)-(15.4), the closed-loop RHC law (15.93)
where f0 (x) is obtained by solving the CROC-CL (15.14)–(15.16) with cost (15.20), (15.21)
for x(0) = x and the closed-loop system (15.94). Assume that
(A0) There exist constants c1 , c2 , c3 , c4 > 0 such that

c1 d(x, O) ≤ p(x) ≤ c2 d(x, O) ∀x ∈ X0 (15.96)

c3 d(x, O) ≤ q(x, u) ≤ c4 d(x, O) ∀(x, u) ∈ X0 × U (15.97)

(A1) The sets X , Xf , U, W a , W p are compact.

(A2) Xf and O are robust control invariants, O ⊆ Xf ⊆ X .
 15.8 Literature Review 347

(A3) J p (x) ≤ 0 ∀x ∈ Xf where

J p (x) = minu∈U maxwa ,wp p(x+ ) − p(x) + q(x, u)
 (15.98)
wa ∈ W a , wp ∈ W p
subj. to
x+ = A(wp )x + B(wp )u + Ewa

Then, for all x ∈ X0 , limk→∞ d(x(k), O) = 0.

Compare Theorem 15.9 with the nominal MPC stability results in Theorem 12.2.
• Assumption (A0) characterizes stage cost q(x, u) and terminal cost p(x) in
both the nominal and the robust case. In the robust case q and p need to be
zero in O, positive and finitely determined outside O and continuous on the
border of O. Assumption (A0) of Theorem 15.9 guarantees these properties.
Note that this cannot be obtained, for instance, with the cost (15.22) in (15.8).
• Assumption (A1) characterizes the initial sets in both the nominal and the
robust case. Notice that the compactness of W a and W p is required to have
a finite cost in the inner maximization problem. In unconstrained min-max
problems often the stage cost is augmented with the term −ρ2 kwk to have a
well defined maximization problem [247][Sec. 3.3.3].
• Assumption (A2) characterizes the terminal set in both the nominal and the
robust case. Recall that, as discussed earlier, the set O takes the role of the
origin for the nominal case. The set O must also be a robust invariant.
• Assumption (A3) characterizes the terminal cost in both the nominal and the
robust case. The minimization problem in the nominal cases is replaced by
the min-max problem in the robust case. Robust control invariance of the
set Xf in assumption (A2) of Theorem 15.9 implies that problem (15.98) in
assumption (A3) is always feasible.

Remark 15.10 Robust control invariance of the set O in assumption (A2) of Theo-
rem 15.9 is also implicitly guaranteed by assumption (A3) as shown next.
Because of assumption (A0), assumption (A3) in O becomes
min max
a p
(p(A(wp )x + B(wp )u + Ewa ) ≤ 0 ∀x ∈ O.
u w ,w

From assumption (A4), this can be verified only if it is equal to zero, i.e., if there
exists a u ∈ U such that A(wp )x + B(wp )u + Ewa ∈ O for all wa ∈ W a wp ∈ W p
(since O is the only place where p(x) can be zero). This implies the robust control
invariance of O.

15.8 Literature Review

An extensive treatment of robust invariant sets can be found in [57, 58, 51, 59].
The proof to Theorem 10.2 can be fond in [172, 100] For the derivation of the algo-
348 15 Constrained Robust Optimal Control

rithms 10.4, 10.5 for computing robust invariant sets (and their finite termination)
see [10, 49, 172, 124, 164].
Min-max robust constrained optimal control was originally proposed by Witsen-
hausen [289]. In the context of robust MPC, the problem was tackled by Campo
and Morari [78], and further developed in [5] for SISO FIR plants. Kothare et
al. [180] optimize robust performance for polytopic/multi-model and linear frac-
tional uncertainty, Scokaert and Mayne [259] for additive disturbances, and Lee
and Yu [189] for linear time-varying and time-invariant state space models depend-
ing on a vector of parameters θ ∈ Θ, where Θ is either an ellipsoid or a polyhe-
dron. Other suboptimal CROC-Cl strategies have been proposed in [180, 29, 181].
For stability and feasibility of the robust RHC (15.1), (15.93) we refer the reader
to [43, 197, 204, 23].
The idea of the parametrization (15.69) appears in the work of Gartska & Wets
in 1974 in the context of stochastic optimization [119]. Recently, it re-appeared in
robust optimization work by Guslitzer and Ben-Tal [136, 46], and in the context
of robust MPC in the work of van Hessem & Bosgra, Löfberg and Goulart &
Kerrigan [281, 194, 130].
 Part V

Constrained Optimal Control of

Hybrid Systems
 Chapter 16

Models of Hybrid Systems

Hybrid systems describe the dynamical interaction between continuous and discrete
signals in one common framework (see Figure 16.1). In this chapter we focus our
attention on mathematical models of hybrid systems that are particularly suitable
for solving finite time constrained optimal control problems.

16.1 Models of Hybrid Systems

The mathematical model of a dynamical system is traditionally expressed through
differential or difference equations, typically derived from physical laws governing
the dynamics of the system under consideration. Consequently, most of the con-
trol theory and tools address models describing the evolution of real-valued signals
according to smooth linear or nonlinear state transition functions, typically dif-
ferential or difference equations. In many applications, however, the system to be
controlled also contains discrete-valued signals satisfying Boolean relations, if-then-
else conditions, on/off conditions, etc., that also involve the real-valued signals. An
example would be an on/off alarm signal triggered by an analog variable exceeding
a given threshold. Hybrid systems describe in a common framework the dynamics
of real-valued variables, the dynamics of discrete variables, and their interaction.
In this chapter we will focus on discrete-time hybrid systems, which we will
call discrete hybrid automata (DHA), whose continuous dynamics is described by
linear difference equations and whose discrete dynamics is described by finite state
machines, both synchronized by the same clock [276]. A particular case of DHA
is the class of piecewise affine (PWA) systems [266]. Essentially, PWA systems
are switched affine systems whose mode depends on the current location of the
state vector, as depicted in Figure 16.2. PWA and DHA systems can be translated
into a form, denoted as mixed logical dynamical (MLD) form, that is more suitable
for solving optimization problems. In particular, complex finite time hybrid dy-
namical optimization problems can be recast into mixed-integer linear or quadratic
programs as will be shown in Chapter 17.
In Section 16.7 we will introduce the tool HYSDEL (HYbrid Systems DEscrip-
352 16 Models of Hybrid Systems

binary binary
inputs discrete outputs
dynamics
and logic

mode
events
switches

continuous
dynamics
real-valued real-valued
outputs inputs

Figure 16.1 Hybrid systems. Logic-based discrete dynamics and continuous

dynamics interact through events and mode switches

tion Language), a high level language for modeling and simulating DHA. Therefore,
DHA will represent for us the starting point for modeling hybrid systems. We will
show that DHA, PWA, and MLD systems are equivalent model classes, and in par-
ticular that DHA systems can be converted to an equivalent PWA or MLD form
for solving optimal control problems.
After introducing PWA systems, we will go through the steps needed for model-
ing a system as a DHA. We will first detail the process of translating propositional
logic involving Boolean variables and linear threshold events over continuous vari-
ables into mixed-integer linear inequalities, generalizing several results available in
the literature, in order to get an equivalent MLD form of a DHA system. Finally, we
will briefly present the tool HYSDEL that allows to describe the DHA in a textual
form and to obtain equivalent MLD and PWA representations in MATLAB R
.

16.2 Piecewise Affine Systems

Piecewise Affine (PWA) systems [266, 145] are defined by partitioning the space
of states and inputs into polyhedral regions (cf. Figure 16.2) and associating with
each region different affine state-update and output equations:

x(t + 1) = Ai(t) x(t) + B i(t) u(t) + f i(t) (16.1a)

y(t) = C i(t) x(t) + Di(t) u(t) + g i(t) (16.1b)
H i(t) x(t) + J i(t) u(t) ≤ K i(t) (16.1c)
16.2 Piecewise Affine Systems 353

x-space
C3
C4
C7

C1

C6

C2 C5

Figure 16.2 Piecewise affine (PWA) systems. Mode switches are triggered
by linear threshold events. In each one of the shaded regions the affine mode
dynamics is different.

where x(t) ∈ Rn is the state vector at time t ∈ T and T = {0, 1, . . .} is the set
of nonnegative integers, u(t) ∈ Rm is the input vector, y(t) ∈ Rp is the output
vector, i(t) ∈ I = {1, . . . , s} is the current mode of the system, the matrices Ai(t) ,
B i(t) , f i(t) , C i(t) , Di(t) , g i(t) , H i(t) , J i(t) , K i(t) are constant and have suitable
dimensions, and the inequalities in (16.1c) should be interpreted component-wise.
Each linear inequality in (16.1c) defines a half-space in Rn and a corresponding
hyperplane, that will be referred to as guardline. Each vector inequality (16.1c)
defines a polyhedron C i = {[ xu ] ∈ Rn+m : H i x + J i u ≤ K i } in the state+input
space Rn+m that will be referred to as cell, and the union of such polyhedral cells as
partition. We assume that C i are full-dimensional sets of Rn+m , for all i = 1, . . . , s.
A PWA system is called well-posed if it satisfies the following property [40]:

Definition 16.1 Let P be a PWA system of the form (16.1) and let C = ∪si=1 C i ⊆
Rn+m be the polyhedral partition associated with it. System P is called well-posed
if for all pairs (x(t), u(t)) ∈ C there exists only one index i(t) satisfying (16.1).

Definition 16.1 implies that x(t+1), y(t) are single-valued functions of x(t) and u(t),
and therefore that state and output trajectories are uniquely determined by the
initial state and input trajectory. A relaxation of definition 16.1 is to let polyhedral
cells C i share one or more hyperplanes. In this case the index i(t) is not uniquely
defined, and therefore the PWA system is not well-posed. However, if the mappings
(x(t), u(t)) → x(t + 1) and (x(t), u(t)) → y(t) are continuous across the guardlines
that are facets of two or more cells (and, therefore, they are continuous on their
domain of definition), such mappings are still single valued.
354 16 Models of Hybrid Systems

16.2.1 Modeling Discontinuities

Discontinuous dynamical behaviors can be modeled by disconnecting the domain.

For instance, the state-update equation
 1
x(t + 1) = 2 x(t) + 1 if x(t) ≤ 0 (16.2a)
0 if x(t) > 0

is discontinuous across x = 0. It can be modeled as

 1
x(t + 1) = 2 x(t) + 1 if x(t) ≤ 0 (16.2b)
0 if x(t) ≥ ǫ

where ǫ > 0 is an arbitrarily small number, for instance the machine precision.
Clearly, system (16.2) is not defined for 0 < x(t) < ǫ, i.e., for the values of the state
that cannot be represented in the machine. However, the trajectories produced
by (16.2a) and (16.2b) are identical as long as x(t) > ǫ or x(t) ≤ 0, ∀t ∈ N.
As remarked above, multiple definitions of the state-update and output func-
tions over common boundaries of sets C i is a technical issue that arises only when
the PWA mapping is discontinuous. Rather than disconnecting the domain, an-
other way of dealing with discontinuous PWA mappings is to allow strict inequal-
ities in the definition of the polyhedral cells in (16.1), or by dealing with open
polyhedra and boundaries separately as in [266]. We prefer to assume that in the
definition of the PWA dynamics (16.1) the polyhedral cells C i(t) are closed sets.
As will be clear in the next chapter, the closed-polyhedra description is mainly
motivated by the fact that numerical solvers cannot handle open sets.

Example 16.1 The following PWA system

    

 cos α(t) − sin α(t) 0

 x(t + 1) = 0.8 x(t) + u(t)
  sinα(t) cos α(t) 1
y(t) =  0 1 x(t)  (16.3)

 π

 3
if  1 0  x(t) ≥ 0
 α(t) =
− π3 if 1 0 x(t) < 0
 
0
is discontinuous at x = , ∀x2 6= 0. It can be described in form (16.1) as
x2
   √   

 
 1 − 3 0  

 
 0.4 √ x(t) + u(t) if 1 0 x(t) ≥ 0

  3 1 1

x(t + 1) =  √   

 1 3 0  

 
 0.4 √ x(t) + u(t) if 1 0 x(t) ≤ −ǫ

  1


  − 3 1
y(t) = 0 1 x(t)
(16.4)
for all x1 ∈ (−∞, −ǫ] ∪ [0, +∞), x2 ∈ R, u ∈ R, and ǫ > 0.
Figure 16.3 shows the free response of the systems (open-loop simulation of the system
for a constant input u = 0) starting from the initial condition x(0) = [1 0] with
sampling time equal to 0.5s and ǫ = 10−6 .
 16.2 Piecewise Affine Systems 355

x1
time

mode

time
Figure 16.3 Example 16.1. Free response of state x1 for x(0) = [1 0].

In case the partition C does not cover the whole space Rn+m , well-posedness
does not imply that trajectories are persistent, i.e., that for all t ∈ N a successor
state x(t + 1) and an output y(t) are defined. A typical case of C 6= Rn+m is
when we are dealing with bounded inputs and bounded states umin ≤ u(t) ≤ umax ,
xmin ≤ x(t) ≤ xmax . By embedding such ranges in the inequalities (16.1c), the
system becomes undefined outside the bounds, as no index i exists that satisfies
any set of inequalities (16.1c).
As will be clearer in the next chapter, when model (16.1) is used in an opti-
mal control formulation, any input sequence and initial state that are feasible for
the related optimization problem automatically define unique trajectories over the
whole optimal control horizon.
PWA systems can model a large number of physical processes, as they can
model static nonlinearities through a piecewise affine approximation, or approxi-
mate nonlinear dynamics via multiple linearizations at different operating points.
Moreover, tools exist for obtaining piecewise affine approximations automatically
(see Section 16.8).
When the mode i(t) is an exogenous variable, condition (16.1c) disappears and
we refer to (16.1) as a switched affine system (SAS), see Section 16.3.1.

16.2.2 Binary States, Inputs, and Outputs

When dealing with hybrid systems, quite often one encounters some signals that
can only assume a binary value, namely either 0 or 1. In the most general form, let
us assume that the state vector x = [ xxcℓ ] where xc ∈ Rnc are the continuous states,
xℓ ∈ Rnℓ are the binary states, and n = nc + nℓ . Similarly, let y ∈ Rpc × {0, 1}pℓ ,
p = pc + pℓ , u ∈ Rmc × {0, 1}mℓ , m = mc + mℓ . By defining a polyhedral partition
s−1
{C i }i=0 of the sets of state and input space Rn+m , for any xℓ ∈ {0, 1} and uℓ ∈
{0, 1} a sufficient condition for the PWA system (16.1) to be well posed is that
the rows and columns of matrices Ai , B i , C i , Di corresponding to binary states
and binary outputs are zero and that the corresponding rows of matrices f i , g i are
either 0 or 1, i.e., that the binary state update and output equations are binary
piecewise constant functions.
In the following sections we will treat 0-1 binary variables both as numbers
356 16 Models of Hybrid Systems

(over which arithmetic operations are defined) and as Boolean variables (over which
Boolean functions are defined, see Section 16.3.3). The variable type will be clear
from the context.
As an example, it is easy to verify that the hybrid dynamical system

xc (t + 1) = 2xc (t) + uc (t) − 3uℓ (t) (16.5a)

xℓ (t + 1) = xℓ (t) ∧ uℓ (t) (16.5b)

where “∧” represents the logic operator “and”, can be represented in the PWA
form
  
 2xc (t) + uc (t)

 if xℓ ≤ 12 , uℓ ≤ 1

 0 2



  

 2xc (t) + uc (t) − 3

 if xℓ ≤ 21 , uℓ ≥ 1
+ǫ
  
 0 2
xc
(t + 1) =   (16.5c)
xℓ 
 2xc (t) + uc (t)

 if xℓ ≥ 1
+ ǫ, uℓ ≤ 1

 0 2 2





  

 2xc (t) + uc (t) − 3 1 1
 if xℓ ≥ 2 + ǫ, uℓ ≥ 2 +ǫ
1

by associating xℓ = 0 with xℓ ≤ 12 and xℓ = 1 with xℓ ≥ 12 + ǫ for any 0 < ǫ ≤ 21 .

Note that, by assuming xℓ (0) ∈ {0, 1} and uℓ (t) ∈ {0, 1} for all t ∈ T, xℓ (t) will be
in {0, 1} for all t ∈ T.

Example 16.2 Consider the spring-mass-damper system

M ẋ2 = u1 − k(x1 ) − b(u2 )x2

where x1 and x2 = ẋ1 denote the position and the velocity of the mass, respectively,
and u1 a continuous force input. The binary input u2 switches the friction coefficient

b1 if u2 = 1
b(u2 ) =
b2 if u2 = 0.

The spring coefficient switches to a different value at position xm


k1 x1 + d1 if x1 ≤ xm
k(x1 ) =
k2 x1 + d2 if x1 > xm .

Assume the system description is valid for −5 ≤ x1 , x2 ≤ 5, and −10 ≤ u1 ≤ 10.

The system has four modes, depending on the binary input u2 and the position x1 .
Assuming that the system parameters are M = 1, b1 = 1, b2 = 50, k1 = 1, k2 = 3,
d1 = 1, d2 = 7.5, xm = 1, after discretizing the dynamics in each mode with a
sampling time of 0.5 time units we obtain the following discrete-time PWA system
16.2 Piecewise Affine Systems 357

position x1

position x1
time
mode

mode
time time
(a) Large damping (u2 = 0). (b) Small damping (u2 = 1).

Figure 16.4 Example 16.2. Open-loop simulation of system (16.6) for u1 = 3

and zero initial conditions.


  0.8956   −0.0096  Mode 1

 0.0198


0.1044
x(t) + [ 0.0198 ] u1 (t) + if x1 (t) ≤ 1, u2 (t) ≤ 0.5


−0.0198 −0.0004 −0.0198







 Mode 2

  0.8956   −0.0711 

 0.0195
x(t) + [ 0.1044 if x1 (t) ≥ 1 + ǫ, u2 (t) ≤ 0.5

 −0.0584 −0.0012 0.0195 ] u1 (t) + −0.1459

x(t+1) =



 Mode 3

    −0.1044 

 0.8956 0.3773
x(t) + [ 0.1044 if x1 (t) ≤ 1, u2 (t) ≥ 0.5

 −0.3773 0.5182 0.3773 ] u1 (t) + −0.3773









   Mode 4  −0.7519 
 0.8956 0.3463
−1.0389 0.3529 x(t) + [ 0.1044 if x(t) ≥ 1 + ǫ, u2 (t) ≥ 0.5
0.3463 ] u1 (t) + −2.5972
(16.6)
for x1 (t) ∈ [−5, 1] ∪ [1 + ǫ, 5], x2 (t) ∈ [−5, 5], u1 (t) ∈ [−10, 10], and for any arbitrary
small ǫ > 0.
Figure 16.4 shows the open-loop simulation of the system for a constant continuous
input u1 = 3, starting from zero initial conditions and for ǫ = 10−6 .

Example 16.3 Consider the following SISO system:

x1 (t + 1) = ax1 (t) + bu(t). (16.7)
A logic state x2 ∈ {0, 1} stores the information whether the state of system (16.7)
has ever gone below a certain lower bound xlb or not:
_
x2 (t + 1) = x2 (t) [x1 (t) ≤ xlb ]. (16.8)

Assume that the input coefficient is a function of the logic state:


b1 if x2 = 0
b= (16.9)
b2 if x2 = 1.
358 16 Models of Hybrid Systems

10 10

x1

x1
-10 -10
x2 1 1

x2
0 0
10 10
u

u
-10 -10
time steps time steps
(a) Scenario 1. (b) Scenario 2.

Figure 16.5 Example 16.3. Open-loop simulation of system (16.10) for dif-
ferent sequences of the input u.

The system can be described by the PWA model:

      
 a 0 b2 0  

 x(t) + u(t) + if 1 0 x(t) ≤ xlb

 0 0 0 1





        
a 0 b1 1 0 xlb + ǫ
x(t+1) = x(t) + u(t) if x(t) ≥

 0 1 0 0 −1 −0.5



      



 a 0 b2 xlb + ǫ
 x(t) + u(t) if x(t) ≥
0 1 0 0.5
(16.10)
for u(t) ∈ R, x1 (t) ∈ (−∞, xlb ] ∪ [xlb + ǫ, +∞), x2 ∈ {0, 1}, and for any ǫ > 0.

Figure 16.5 shows two open-loop simulations of the system, for a = 0.5, b1 = 0.1,
b2 = 0.3, xlb = −1, ǫ = 10−6 . The initial conditions is x(0) = [1, 0]. Note that when
the continuous state x1 (t) goes below xlb = −1 at time t, then xℓ (t + 1) switches to
1 and the input has a stronger effect on the states from time t + 2 on. Indeed, the
steady state of x1 is a function of the logic state x2 .

16.3 Discrete Hybrid Automata

As shown in Fig. 16.6, a discrete hybrid automaton (DHA) is formed by generating
the mode i(t) of a switched affine system through a mode selector function that
depends on (i) the discrete state of a finite state machine, (ii) discrete events
generated by the continuous variables of the switched affine system exceeding given
linear thresholds (the guardlines), (iii) exogenous discrete inputs [276]. We will
detail each of the four blocks in the next sections.
 16.3 Discrete Hybrid Automata 359

EEvent xyc(k)
Generator
(EG)
!ed(k) uvc(k)

FFinite State
!ed(k) Machine y!(k) 1
u!u(k) (FSM) uvc(k)
yc(k)

...
Switched s
Affine
System
(SAS)
S

M
Mode
x!(k)
x Selector
(MS) i(k)
u!(k)
u i
!e(k)
d

Figure 16.6 A discrete hybrid automaton (DHA) is the connection of a finite

state machine (FSM) and a switched affine system (SAS), through a mode
selector (MS) and an event generator (EG). The output signals are omitted
for clarity

16.3.1 Switched Affine System (SAS)

A switched affine system is a collection of affine systems:

xc (t + 1) = Ai(t) xc (t) + B i(t) uc (t) + f i(t) (16.11a)

i(t) i(t) i(t)
yc (t) = C xc (t) + D uc (t) + g , (16.11b)

where t ∈ T is the time indicator, xc ∈ Rnc is the continuous state vector, uc ∈ Rmc
is the exogenous continuous input vector, yc ∈ Rpc is the continuous output vector,
{Ai , B i , f i , C i , Di , g i }i∈I is a collection of matrices of suitable dimensions, and the
mode i(t) ∈ I = {1, . . . , s} is an input signal that determines the affine state update
dynamics at time t. An SAS of the form (16.11) preserves the value of the state
when a mode switch occurs, but it is possible to implement reset maps on an SAS
as shown in [276].

16.3.2 Event Generator (EG)

An event generator is an object that generates a binary vector δe (t) ∈ {0, 1}ne
of event conditions according to the satisfaction of a linear (or affine) threshold
n
condition. Let h : Rnc × Rnc → {0, 1} e be a vector function defined as

1 if Hi xc + Ji uc + Ki ≤ 0
hi (xc , uc ) =
0 if Hi xc + Ji uc + Ki > 0
360 16 Models of Hybrid Systems

where the lower index i denotes the i-th component of a vector or the i-th row of
a matrix, and H, J, K are constant matrices of suitable dimensions. Then events
are defined as
δe (t) = h(xc (t), uc (t)). (16.12)

In particular, state events are modeled as [δe (t) = 1] ↔ [a′ xc (t) ≤ b]. Note that
time events can be modeled as in (16.12) by adding the continuous time as an
additional continuous and autonomous state variable, τ (t + 1) = τ (t) + Ts , where
Ts is the sampling time, and by letting [δe (t) = 1] ↔ [tTs ≥ τ0 ], where τ0 is a given
time. By doing so, the hybrid model can be written as a time-invariant one. Clearly
the same approach can be used for time-varying events δe (t) = h(xc (t), uc (t), t), by
n
using time-varying event conditions h : Rnc × Rm × T → {0, 1} e .

16.3.3 Boolean Algebra

Before we deal in detail with the other blocks constituting the DHA and introduce
further notation, we recall some basic definitions of Boolean algebra. A more
comprehensive treatment of Boolean calculus can be found in digital circuit design
texts, e.g. [86, 142]. For a rigorous exposition see e.g. [208].
A variable δ is a Boolean variable if δ ∈ {0, 1}, where “δ = 0” means something
is false, “δ = 1” that it is true. A Boolean expression is obtained by combining
Boolean variables through the logic operators ¬ (not), ∨ (or), ∧ (and), ← (implied
by), → (implies), and ↔ (iff). A Boolean function f : {0, 1}n−1 7→ {0, 1} is used
to define a Boolean variable δn as a logic function of other variables δ1 , . . . , δn−1 :

δn = f (δ1 , δ2 , . . . , δn−1 ). (16.13)

Given n Boolean variables δ1 , . . . , δn , a Boolean formula F defines a relation

F (δ1 , . . . , δn ) (16.14)

that must hold true. Every Boolean formula F (δ1 , δ2 , . . . , δn ) can be rewritten in
the conjunctive normal form (CNF)
   
m
^ _ _ _
(CNF)  δi   ∼ δi   (16.15)
j=1 i∈Pj i∈Nj

Nj , Pj ⊆ {1, . . . , n}, ∀j = 1, . . . , m.

As mentioned in Section 16.2.2, often we will use the term binary variable
and Boolean variable without distinction. An improper arithmetic operation over
Boolean variables should be understood as an arithmetic operation over corre-
sponding binary variables and, vice versa, an improper Boolean function of binary
variables should be interpreted as the same function over the corresponding Boolean
variables. Therefore, from now on we will call “binary” variables both 0-1 variables
and Boolean variables.
 16.3 Discrete Hybrid Automata 361

∼ δ1

Red

∼ δ3 δ1 ∧ ∼ uℓ2
δ1 ∧ uℓ2
δ2
Green Blue

δ3 ∧ uℓ1

∼ uℓ1 ∧ δ3 ∼ δ2

Figure 16.7 Example of finite state machine

16.3.4 Finite State Machine (FSM)

A finite state machine (or automaton) is a discrete dynamic process that evolves
according to a Boolean state update function:
xℓ (t + 1) = fℓ (xℓ (t), uℓ (t), δe (t)), (16.16a)
n m
where xℓ ∈ {0, 1} ℓ is the binary state, uℓ ∈ {0, 1} ℓ is the exogenous binary input,
n
δe (t) is the endogenous binary input coming from the EG, and fℓ : {0, 1} ℓ ×
mℓ ne nℓ
{0, 1} × {0, 1} → {0, 1} is a deterministic Boolean function. In this text
we will only refer to synchronous finite state machines, where the transitions may
happen only at sampling times. The adjective synchronous will be omitted for
brevity.
An FSM can be conveniently represented using an oriented graph. An FSM
may also have an associated binary output
yℓ (t) = gℓ (xℓ (t), uℓ (t), δe (t)), (16.16b)
pℓ nℓ mℓ p
where yℓ ∈ {0, 1} and gℓ : {0, 1} × {0, 1} × {0, 1}ne 7→ {0, 1} ℓ .

Example 16.4 Figure 16.7 shows a finite state machine where uℓ = [uℓ1 uℓ2 ]′ is
the input vector, and δ = [δ1 . . . δ3 ]′ is a vector of signals coming from the event
generator. The Boolean state update function (also called state transition function)
is: 

 Red if ((xℓ (t) = Green)∧ ∼ δ3 )∨

 ((xℓ (t) = Red)∧ ∼ δ1 ),





 Green if ((xℓ (t) = Red) ∧ δ1 ∧ uℓ2 )∨

((xℓ (t) = Blue) ∧ δ2 )∨
xℓ (t + 1) = (16.17)

 ((xℓ (t) = Green)∧ ∼ uℓ1 ∧ δ3 ),



 Blue if ((xℓ (t) = Red) ∧ δ1 ∧ ∼ uℓ2 )∨



 ((xℓ (t) = Green) ∧ (δ3 ∧ uℓ1 ))∨

((xℓ (t) = Blue)∧ ∼ δ2 )).
362 16 Models of Hybrid Systems

By associating a binary vector xℓ = [ xxℓ1

ℓ2
] to each state (Red = [ 00 ], Green = [ 01 ], and
1
Blue = [ 0 ]), one can rewrite (16.17) as:
xℓ1 (t + 1) = (∼ xℓ1 ∧ ∼ xℓ2 ∧ δ1 ∧ ∼ uℓ2 ) ∨
(xℓ1 ∧ ∼ δ2 ) ∨ (xℓ2 ∧ δ3 ∧ uℓ1 )
xℓ2 (t + 1) = (∼ xℓ1 ∧ ∼ xℓ2 ∧ δ1 ∧ uℓ2 ) ∨
(xℓ1 ∧ δ2 ) ∨ (xℓ2 ∧ δ3 ∧ ∼ uℓ1 ),
where the time index (t) has been omitted for brevity.

Since the Boolean state update function is deterministic, for each state the
conditions associated with all the outgoing arcs are mutually exclusive.

16.3.5 Mode Selector

In a DHA, the dynamic mode i(t) ∈ I = {1, . . . , s} of the SAS is a function of the
binary state xℓ (t), the binary input uℓ (t), and the events δe (t). With a slight abuse
of notation, let us indicate the mode i(t) through its binary encoding, i(t) ∈ {0, 1}ns
where ns = ⌈log2 s⌉, so that i(t) can be treated as a vector of Boolean variables.
d(j)
Any discrete variable α ∈ {α1 , . . . , αj } admits a Boolean encoding a ∈ {0, 1} ,
where d(j) is the number of bits used to represent α1 , . . . , αj . For example,
α ∈ {0, 1, 2} may be encoded as a ∈ {0, 1}2 by associating [ 00 ] → 0, [ 01 ] → 1,
[ 10 ] → 2.
Then, we define the mode selector by the Boolean function fM : {0, 1}nℓ ×
m n
{0, 1} ℓ × {0, 1} e → {0, 1}ns . The output of this function
i(t) = µ(xℓ (t), uℓ (t), δe (t)) (16.18)
is called the active mode of the DHA at time t. We say that a mode switch occurs
at step t if i(t) 6= i(t − 1). Note that, in contrast to continuous-time hybrid models
where switches can occur at any time, in our discrete-time setting, as mentioned
earlier, a mode switch can only occur at sampling instants.

16.3.6 DHA Trajectories

h i h i
xc (0)
For a given initial condition xℓ (0) ∈ Rnc × {0, 1}nℓ , and inputs uc (t)
uℓ (t) ∈ Rm c ×
mℓ
{0, 1} , t ∈ T, the state x(t) of the system is computed for all t ∈ T by recursively
iterating the set of equations:
δe (t) = h(xc (t), uc (t), t) (16.19a)
i(t) = µ(xℓ (t), uℓ (t), δe (t)) (16.19b)
yc (t) = C i(t) xc (t) + Di(t) uc (t) + g i(t) (16.19c)
yℓ (t) = gℓ (xℓ (t), uℓ (t), δe (t)) (16.19d)
xc (t + 1) = Ai(t) xc (t) + B i(t) uc (t) + f i(t) (16.19e)
xℓ (t + 1) = fℓ (xℓ (t), uℓ (t), δe (t)). (16.19f)
 16.4 Logic and Mixed-Integer Inequalities 363

A definition of well-posedness of DHA can be given similarly to Definition 16.1

by requiring that the successor states xc (t + 1), xℓ (t + 1) and the outputs yc (t),
yℓ (t) are uniquely defined functions of xc (t), xℓ (t), uc (t), uℓ (t) defined by the DHA
equations (16.19).
DHA can be considered as a subclass of hybrid automata (HA) [7]. The main
difference is in the time model: DHA are based on discrete time, HA on continuous
time. Moreover DHA models do not allow instantaneous transitions, and are de-
terministic, contrary to HA where any enabled transition may occur in zero time.
This has two consequences: (i) DHA do not admit live-locks (infinite switches in
zero time), (ii) DHA do not admit Zeno behaviors (infinite switches in finite time).
Finally, in DHA models, guards, reset maps and continuous dynamics are limited
to linear (or affine) functions.

16.4 Logic and Mixed-Integer Inequalities

Despite the fact that DHA are rich in their expressiveness and are therefore quite
suitable for modeling and simulating a wide class of hybrid dynamical systems,
they are not directly suitable for solving optimal control problems, because of their
heterogeneous discrete and continuous nature. In this section we want to describe
how DHA can be translated into different hybrid models that are more suitable
for optimization. We highlight the main techniques of the translation process, by
generalizing several results that appeared in the literature [126, 244, 287, 213, 83,
42, 81, 277, 286, 219].

16.4.1 Transformation of Boolean Relations

Boolean formulas can be equivalently represented as integer linear inequalities.

For instance, δ1 ∨ δ2 = 1 is equivalent to δ1 + δ2 ≥ 1 [287]. Some equivalences are
reported in Table 16.1. The results of the table can be generalized as follows.

Lemma 16.1 For every Boolean formula F (δ1 , δ2 , . . . , δn ) there exists a polyhedral
set P such that a set of binary values {δ1 , δ2 , . . . , δn } satisfies the Boolean formula
F if and only if δ = [δ1 δ2 . . . δn ]′ ∈ P .

Proof: Given a formula F , one way of constructing a polyhedron P is to rewrite

F in the conjunctive normal form (16.15), and then simply define P as
 P P 

 1 ≤ i∈P1 δi + i∈N1 (1 − δi )  
P = δ ∈ {0, 1}n : .. . (16.20)
 P . P 
 
1 ≤ i∈Pm δi + i∈Nm (1 − δi )


The smallest polyhedron P associated with formula F has the following ge-
ometric interpretation: Assume to list all the 0-1 combinations of δi ’s satisfying
364 16 Models of Hybrid Systems

relation Boolean linear constraints

AND δ1 ∧ δ2 δ1 = 1, δ2 = 1 or δ1 + δ2 ≥ 2
OR δ1 ∨ X2 δ1 + δ2 ≥ 1
NOT ∼ δ1 δ1 = 0
XOR δ1 ⊕ δ2 δ1 + δ2 = 1
IMPLY δ1 → δ2 δ1 − δ2 ≤ 0
IFF δ1 ↔ δ2 δ1 − δ2 = 0
ASSIGNMENT δ1 + (1 − δ3 ) ≥ 1
δ3 = δ1 ∧ δ2 δ3 ↔ δ1 ∧ δ2 δ2 + (1 − δ3 ) ≥ 1
(1 − δ1 ) + (1 − δ2 ) + δ3 ≥ 1

Table 16.1 Basic conversion of Boolean relations into mixed-integer inequal-

ities. Relations involving the inverted literals ∼ δ can be obtained by sub-
stituting (1 − δ) for δ in the corresponding inequalities. More conversions
are reported in [210], or can be derived from (16.15)–(16.20)

F (namely, to generate the truth table of F ), and think of each combination as

an n-dimensional binary vector in Rn , then P is the convex hull of such vec-
tors [83, 153, 211]. For methods to compute convex hulls, we refer the reader
to [111].

16.4.2 Translating DHA Components into Linear Mixed-Integer Relations

Events of the form (16.12) can be expressed equivalently as

hi (xc (t), uc (t), t) ≤ M i (1 − δei ), (16.21a)

i
h (xc (t), uc (t), t) > mi δei , i = 1, . . . , ne , (16.21b)

where M i , mi are upper and lower bounds, respectively, on hi (xc (t), uc (t), t). As
we have pointed out in Section 16.2.1, from a computational viewpoint it is con-
venient to avoid strict inequalities. As suggested in [287], we modify the strict
inequality (16.21b) into

hi (xc (t), uc (t), t) ≥ ǫ + (mi − ǫ)δei (16.21c)

where ǫ is a small positive scalar (e.g., the machine precision). Clearly, as for the
case of discontinuous PWA discussed in Section 16.2.1, Equations (16.21) or (16.12)
are equivalent to (16.21c) only for hi (xc (t), uc (t), t) ≤ 0 and hi (xc (t), uc (t), t) ≥ ǫ.
Regarding switched affine dynamics, we first rewrite the state-update equa-
tion (16.11a) as the following combination of affine terms and if-then-else condi-
 16.5 Mixed Logical Dynamical Systems 365

tions:

A1 xc (t) + B 1 uc (t) + f 1 , if (i(t) = 1),
z1 (t) = (16.22a)
0, otherwise,
..
.

As xc (t) + B s uc (t) + f s , if (i(t) = s),
zs (t) = (16.22b)
0, otherwise,
s
X
xc (t + 1) = zi (t), (16.22c)
i=1

where zi (t) ∈ Rnc , i = 1, . . . , s. The output equation (16.11b) admits a similar

transformation.
A generic if-then-else construct of the form
′ ′ ′ ′
IF δ THEN z = a1 x + b1 u + f 1 ELSE z = a2 x + b2 u + f 2 , (16.23)

where δ ∈ {0, 1}, z ∈ R, x ∈ Rn , u ∈ Rm , and a1 , b1 , f 1 , a2 , b2 , f 2 are constants of

suitable dimensions, can be translated into [45]
′ ′
(m2 − M1 )δ + z ≤ a2 x + b 2 u + f 2 , (16.24a)
2′ 2′ 2
(m1 − M2 )δ − z ≤ −a x − b u − f , (16.24b)
1′ 1′ 1
(m1 − M2 )(1 − δ) + z ≤ a x+b u+f , (16.24c)
1′ 1′ 1
(m2 − M1 )(1 − δ) − z ≤ −a x − b u − f , (16.24d)

where Mi , mi are upper and lower bounds on ai x + bi u + f i , i = 1, 2.

Finally, the mode selector function and binary state-update function of the au-
tomaton are Boolean functions that can be translated into integer linear inequalities
as described in Section 16.4.1. The idea of transforming a well-posed FSM into a
set of Boolean equalities was also presented in [229] where the authors performed
model checking using (mixed) integer optimization on an equivalent set of integer
inequalities.

16.5 Mixed Logical Dynamical Systems

Given a DHA representation of a hybrid process, by following the techniques de-
scribed in the previous section for converting logical relations into inequalities we
obtain an equivalent representation of the DHA as a mixed logical dynamical (MLD)
system [42] described by the following relations:

x(t + 1) = Ax(t) + B1 u(t) + B2 δ(t) + B3 z(t) + B5 , (16.25a)

y(t) = Cx(t) + D1 u(t) + D2 δ(t) + D3 z(t) + D5 , (16.25b)
E2 δ(t) + E3 z(t) ≤ E1 u(t) + E4 x(t) + E5 , (16.25c)
366 16 Models of Hybrid Systems

where x ∈ Rnc × {0, 1}nℓ is a vector of continuous and binary states, u ∈ Rmc ×
{0, 1}mℓ are the inputs, y ∈ Rpc × {0, 1}pℓ the outputs, δ ∈ {0, 1}rℓ are auxiliary
binary variables, z ∈ Rrc are auxiliary continuous variables which arise in the
transformation (see Example 16.5), and A, B1 , B2 , B3 , C, D1 , D2 , D3 , E1 ,. . . ,E5
are matrices of suitable dimensions. Given the current state x(t) and input u(t),
the time-evolution of (16.25) is determined by finding a feasible value for δ(t) and
z(t) satisfying (16.25c), and then by computing x(t + 1) and y(t) from (16.25a)–
(16.25b).
As MLD models consist of a collection of linear difference equations involving
both real and binary variables and a set of linear inequality constraints, they are
model representations of hybrid systems that can be easily used in optimization
algorithms, as will be described in Chapter 17.
A definition of well-posedness of the MLD system (16.25) can be given similarly
to Definition 16.1 by requiring that for all x(t) and u(t) within a given bounded
set the pair of variables δ(t), z(t) satisfying (16.25c) is unique, so that the succes-
sor state x(t + 1) and output y(t) are also uniquely defined functions of x(t), u(t)
through (16.25a)–(16.25b). Such a well-posedness assumption is usually guaranteed
by the procedure described in Section 16.3.3 used to generate the linear inequali-
ties (16.25c). A numerical test for well-posedness is reported in [42, Appendix 1].
For a more general definition of well-posedness of MLD systems see [42].
Note that the constraints (16.25c) allow one to specify additional linear con-
straints on continuous variables (e.g., constraints over physical variables of the
system), and logical constraints over Boolean variables. The ability to include
constraints, constraint prioritization, and heuristics adds to the expressiveness and
generality of the MLD framework. Note also that despite the fact that the de-
scription (16.25) seems to be linear, clearly the nonlinearity is concentrated in the
integrality constraints over binary variables.

Example 16.5 Consider the following simple switched linear system [42]

0.8x(t) + u(t) if x(t) ≥ 0
x(t + 1) = (16.26)
−0.8x(t) + u(t) if x(t) < 0

where x(t) ∈ [−10, 10], and u(t) ∈ [−1, 1]. The condition x(t) ≥ 0 can be associated
to an event variable δ(t) ∈ {0, 1} defined as

[δ(t) = 1] ↔ [x(t) ≥ 0] . (16.27)

By using the transformations (16.21a)–(16.21c), equation (16.27) can be expressed

by the inequalities

−mδ(t) ≤ x(t) − m (16.28a)

−(M + ǫ)δ(t) ≤ −x(t) − ǫ (16.28b)

where M = −m = 10, and ǫ is an arbitrarily small positive scalar. Then (16.26) can
be rewritten as
x(t + 1) = 1.6δ(t)x(t) − 0.8x(t) + u(t). (16.29)
 16.6 Model Equivalence 367

By defining a new variable z(t) = δ(t)x(t) which, by (16.23)–(16.24) can be expressed

as

z(t) ≤ M δ(t) (16.30a)

z(t) ≥ mδ(t) (16.30b)
z(t) ≤ x(t) − m(1 − δ(t)) (16.30c)
z(t) ≥ x(t) − M (1 − δ(t)) (16.30d)

the evolution of system (16.26) is ruled by the linear equation

x(t + 1) = 1.6z(t) − 0.8x(t) + u(t) (16.31)

subject to the linear constraints (16.28) and (16.30). Therefore, the MLD equivalent
representation of (16.26) for x ∈ [−10, −ǫ]∪[0, 10] and u ∈ [−1, 1] is given by collecting
equations (16.31), (16.28) and (16.30).

16.6 Model Equivalence

In the previous chapters we have presented three different classes of discrete-time
hybrid models: PWA systems, DHA, and MLD systems. For what we described
in Section 16.3.3, under the assumption that the set of valid states and inputs
is bounded, DHA systems can always be equivalently described as MLD systems.
Also, a PWA system is a special case of a DHA whose threshold events and mode
selector function are defined by the PWA partition (16.1c). Therefore, a PWA
system with bounded partition C can always be described as an MLD system (an
efficient way of modeling PWA systems in MLD form is reported in [42]). The
converse result, namely that MLD systems (and therefore DHA) can be represented
as PWA systems, is less obvious. For any choice of δ, model (16.25) represents an
affine system defined over a polyhedral domain. Under the assumption of well
posedness these domains do not overlap. This result was proved formally in [36,
30, 122].
Such equivalence results are of interest because DHA are most suitable in the
modeling phase, but MLD systems are most suitable for solving open-loop finite
time optimal control problems, and PWA systems are most suitable for solving
finite time optimal control problems in state feedback form, as will be described in
Chapter 17.

16.7 The HYSDEL Modeling Language

A modeling language was proposed in [276] to describe DHA models, called HYbrid
System DEscription Language (HYSDEL). The HYSDEL description of a DHA is
an abstract modeling step. The associated HYSDEL compiler then translates the
description into several computational models, in particular into an MLD model
368 16 Models of Hybrid Systems

SYSTEM sample {
INTERFACE {
STATE {
REAL xr [-10, 10]; }
INPUT {
REAL ur [-2, 2]; }
}
IMPLEMENTATION {
AUX {
REAL z1, z2, z3;
BOOL de, df, d1, d2, d3; }
AD {
de = xr >= 0;
df = xr + ur - 1 >= 0; }
LOGIC {
d1 = ~de & ~df;
d2 = de;
d3 = ~de & df; }
DA {
z1 = {IF d1 THEN xr + ur - 1 };
z2 = {IF d2 THEN 2 * xr };
z3 = {IF (~de & df) THEN 2 }; }
CONTINUOUS {
xr = z1 + z2 + z3; }
}}

Table 16.2 Sample HYSDEL list of system (16.32)

using the technique presented in Section 16.4, and a PWA form using either the
approach of [122] or the approach of [30]. HYSDEL can generate also a simulator
that runs as a function in MATLAB R
. Both the HYSDEL compiler and the Hybrid
Toolbox can import and convert HYSDEL models.
In this section we illustrate the functionality of HYSDEL through a set of
examples. For more examples and the detailed syntax we refer the interested
reader to [149].

Example 16.6 Consider the DHA system:


 xc (t) + uc (t) − 1, if i(t) = 1,
SAS: x′c (t) = 2xc (t), if i(t) = 2, (16.32a)

2, if i(t) = 3,

δe (t) = [xc (t) ≥ 0],
EG: (16.32b)
δf (t) = [xc (t) + uc (t) − 1 ≥ 0],
 h i
 δe (t) 0
 1, if
 δf (t) = [ 0 ] ,
MS: i(t) = 2, if hδe (t) =i 1, (16.32c)

 δe (t)
 3, if = [0].
δf (t) 1

The corresponding HYSDEL list is reported in Table 16.2.

The HYSDEL list is composed of two parts. The first one, called INTERFACE,
contains the declaration of all variables and parameters, so that it is possible to
16.7 The HYSDEL Modeling Language 369

/* 2x2 PWA system */

SYSTEM pwa {

INTERFACE {
STATE { REAL x1 [-5,5];
REAL x2 [-5,5];
}
INPUT { REAL u [-1,1];
}
OUTPUT{ REAL y;
}
PARAMETER {
REAL alpha = 60*pi/180;
REAL C = cos(alpha);
REAL S = sin(alpha);
REAL MLD_epsilon = 1e-6; }
}

IMPLEMENTATION {
AUX { REAL z1,z2;
BOOL sign; }
AD { sign = x1<=0; }

DA { z1 = {IF sign THEN 0.8*(C*x1+S*x2)

ELSE 0.8*(C*x1-S*x2) };
z2 = {IF sign THEN 0.8*(-S*x1+C*x2)
ELSE 0.8*(S*x1+C*x2) }; }

CONTINUOUS { x1 = z1;
x2 = z2+u; }

OUTPUT { y = x2; }
}
}

Table 16.3 HYSDEL model of the PWA system described in Example 16.1

make the proper type checks. The second part, IMPLEMENTATION, is composed
of specialized sections where the relations among the variables are described.
The HYSDEL section AUX contains the declaration of the auxiliary variables used
in the model. The HYSDEL section AD allows one to define Boolean variables from
continuous ones, and is based exactly on the semantics of the event generator (EG)
described earlier. The HYSDEL section DA defines continuous variables according
to if-then-else conditions. This section models part of the switched affine system
(SAS), namely the variables zi defined in (16.22a)–(16.22b). The CONTINUOUS
section describes the linear dynamics, expressed as difference equations. This section
models (16.22c). The section LOGIC allows one to specify arbitrary functions of
Boolean variables.

Example 16.7 Consider again the PWA system described in Example 16.1. Assume
that [−5, 5] × [−5, 5] is the set of states x(t) of interest and u(t) ∈ [−1, 1]. By using
370 16 Models of Hybrid Systems

HYSDEL the PWA system (16.3) is described as in Table 16.3 and the equivalent
MLD form is obtained
x(t + 1) = [ 10 01 ] z(t) + [ 01 ] u(t)
y(t) = [ 0 1 ] x(t)
 −5−ǫ   0 0   1 0   −ǫ 
5 0 0 −1 0
5
 c1   1 0 
−1 0  −0.4 −c 2 
 cc1 
 c1     0.4 c2   1 
 −c1   −0.4 c2 
 −c1  δ(t) +  −1 0 
 1 0  z(t) ≤ 
 0 
0.4 −c2  x(t) +  0 
 c1   0 −1     c1 
 c1  0 1  c2 −0.4 
−c2 0.4 c1
−c1 0 −1 −c2 −0.4 0
−c1 0 1 c2 0.4 0

√ √
where c1 = 4( 3 + 1), c2 = 0.4 3, ǫ = 10−6 . Note that in Table 16.3 the OUTPUT
section allows to specify a linear map for the output vector y.

Example 16.8 Consider again the hybrid spring-mass-damper system described in

Example 16.2. Assume that [−5, 5] × [−5, 5] is the set of states x and [−10, 10] the set
of continuous inputs u1 of interest. By using HYSDEL, system (16.6) is described as
in Table 16.4 and an equivalent MLD model with 2 continuous states, 1 continuous
input, 1 binary input, 9 auxiliary binary variables, 8 continuous variables, and 58
mixed-integer inequalities is obtained.

Example 16.9 Consider again the system with a logic state described in Example 16.3.
The MLD model obtained by compiling the HYSDEL list of Table 16.5 is

x(t + 1) = [ 00 01 ] δ(t) + [ 10 ] z(t)

 −9 0   0   0   1 0   1

11 0 0 0 −1 0 10
 00 0   −1   −0.3   −0.5 −14   14 
 0 0   1   0.3   0.5 −14   14 
 0 0  δ(t) +  −1  z(t) ≤  −0.1  u(t) +  −0.5 14  x(t) + 0
 0 0 
−1
 1   0.1   0.5 14  0
0 0 0 −1 0
1 −1 0 0 0 0 0
−1 1 0 0 0 1 0

where we have assumed that [−10, 10] is the set of states x1 and [−10, 10] the set of
continuous inputs u of interest. In Table 16.5 the AUTOMATA section specifies the
state transition equations of the finite state machine (FSM) as a collection of Boolean
functions.

16.8 Literature Review

The lack of a general theory and of systematic design tools for systems having
such a heterogeneous dynamical discrete and continuous nature led to a consid-
erable interest in the study of hybrid systems. After the seminal work published
in 1966 by Witsenhausen [288], who examined an optimal control problem for a
class of hybrid-state continuous-time dynamical systems, there has been a renewed
interest in the study of hybrid systems. The main reason for such an interest is
16.8 Literature Review 371

/* Spring-Mass-Damper System
*/

SYSTEM springmass {

INTERFACE { /* Description of variables and constants */

STATE {
REAL x1 [-5,5];
REAL x2 [-5,5];
}

INPUT { REAL u1 [-10,10];

BOOL u2;
}

PARAMETER {
/* Spring breakpoint */
REAL xm;

/* Dynamic coefficients */
REAL A111,A112,A121,A122,A211,A212,A221,A222;
REAL A311,A312,A321,A322,A411,A412,A421,A422;
REAL B111,B112,B121,B122,B211,B212,B221,B222;
REAL B311,B312,B321,B322,B411,B412,B421,B422;
}
}

IMPLEMENTATION {
AUX {
REAL zx11,zx12,zx21,zx22,zx31,zx32,zx41,zx42;
BOOL region;
}

AD { /* spring region */
region = x1-xm <= 0;
}

DA {
zx11 = { IF region & u2 THEN A111*x1+A112*x2+B111*u1+B112};
zx12 = { IF region & u2 THEN A121*x1+A122*x2+B121*u1+B122};
zx21 = { IF region & ~u2 THEN A211*x1+A212*x2+B211*u1+B212};
zx22 = { IF region & ~u2 THEN A221*x1+A222*x2+B221*u1+B222};
zx31 = { IF ~region & u2 THEN A311*x1+A312*x2+B311*u1+B312};
zx32 = { IF ~region & u2 THEN A321*x1+A322*x2+B321*u1+B322};
zx41 = { IF ~region & ~u2 THEN A411*x1+A412*x2+B411*u1+B412};
zx42 = { IF ~region & ~u2 THEN A421*x1+A422*x2+B421*u1+B422};
}

CONTINUOUS { x1=zx11+zx21+zx31+zx41;
x2=zx12+zx22+zx32+zx42;
}
}
}

Table 16.4 HYSDEL model of the spring-mass-damper system described in

Example 16.2. The A and B values are set according to equation (16.6).
372 16 Models of Hybrid Systems

/* System with logic state */

SYSTEM SLS {
INTERFACE { /* Description of variables and constants */
STATE {
REAL x1 [-10,10];
BOOL x2;
}
INPUT { REAL u [-10,10];
}
PARAMETER {

/* Lower Bound Point */

REAL xlb = -1;

/* Dynamic coefficients */
REAL a = .5;
REAL b1 =.1;
REAL b2 =.3;
}
}

IMPLEMENTATION {
AUX {BOOL region;
REAL zx1;
}
AD { /* PWA Region */
region = x1-xlb <= 0;
}
DA { zx1={IF x2 THEN a*x1+b2*u ELSE a*x1+b1*u};
}
CONTINUOUS { x1=zx1;
}
AUTOMATA { x2= x2 | region;
}
}
}

Table 16.5 HYSDEL model of the system with logic state described in Ex-
ample 16.3
16.8 Literature Review 373

probably the recent advent of technological innovations, in particular in the domain

of embedded systems, where a logical/discrete decision device is “embedded” in a
physical dynamical environment to change the behavior of the environment itself.
Another reason is the availability of several software packages for simulation and
numerical/symbolic computation that support the theoretical developments.
Several modelling frameworks for hybrid systems have appeared in the litera-
ture. We refer the interested reader to [8, 127] and the references therein. Each
class is usually tailored to solve a particular problem, and many of them look largely
dissimilar, at least at first sight. Two main categories of hybrid systems were suc-
cessfully adopted for analysis and synthesis [66]: hybrid control systems [195, 196],
where continuous dynamical systems and discrete/logic automata interact (see
Fig. 16.1), and switched systems [266, 67, 155, 292, 261], where the state space
is partitioned into regions, each one being associated with a different continuous
dynamics (see Fig. 16.2).
Today, there is a widespread agreement in defining hybrid systems as dynamical
systems that switch among many operating modes, where each mode is governed
by its own characteristic dynamical laws, and mode transitions are triggered by
variables crossing specific thresholds (state events), by the lapse of certain time
periods (time events), or by external inputs (input events) [8]. In the literature,
systems whose mode only depends on external inputs are usually called switched
systems, the others switching systems.
Complex systems organized in a hierarchial manner, where, for instance, dis-
crete planning algorithms at the higher level interact with continuous control algo-
rithms and processes at the lower level, are another example of hybrid systems. In
these systems, the hierarchical organization helps to manage the complexity of the
system, as higher levels in the hierarchy require less detailed models (also called
abstractions) of the lower levels functions.
Hybrid systems arise in a large number of application areas and are attracting
increasing attention in both academic theory-oriented circles as well as in industry,
for instance in the automotive industry [16, 157, 62, 125, 74, 214]. Moreover,
many physical phenomena admit a natural hybrid description, like circuits involving
relays or diodes [28], biomolecular networks [6], and TCP/IP networks in [150].
In this book we work exclusively with dynamical systems formulated in dis-
crete time. Therefore this chapter focuses on hybrid models formulated in discrete
time. Though the effects of sampling can be neglected in most applications, some
interesting mathematical phenomena occurring in hybrid systems, such as Zeno be-
haviors [158] do not exist in discrete time, as switches can only occur at sampling
instants. On the other hand, most of these phenomena are usually a consequence of
the continuous-time switching model, rather than the real behavior. Our main mo-
tivation for concentrating on discrete-time models is that optimal control problems
are easier to formulate and to solve numerically than continuous-time formulations.
In the theory of hybrid systems, several problems were investigated in the last
few years. Besides the issues of existence and computation of trajectories described
in Section 16.1, several other issues were considered. These include: equivalence of
hybrid models, stability and passivity analysis, reachability analysis and verifica-
tion of safety properties, controller synthesis, observability analysis, state estima-
tion and fault detection schemes, system identification, stochastic and event-driven
374 16 Models of Hybrid Systems

dynamics. We will briefly review some of these results in the next paragraphs and
provide pointers to some relevant literature references.

Equivalence of Linear Hybrid Systems

Under the condition that the MLD system is well-posed, the result showing that
an MLD systems admits an equivalent PWA representation was proved in [36]. A
slightly different and more general proof is reported in [30], where the author also
provides efficient MLD to PWA translation algorithms. A different algorithm for
obtaining a PWA representation of a DHA is reported in [122].
The fact that PWA systems are equivalent to interconnections of linear systems
and finite automata was pointed out by Sontag [267]. In [148, 40] the authors proved
the equivalence of discrete-time PWA/MLD systems with other classes of discrete-
time hybrid systems (under some assumptions) such as linear complementarity
(LC) systems [146, 279, 147], extended linear complementarity (ELC) systems [94],
and max-min-plus-scaling (MMPS) systems [95].

Stability Analysis

Piecewise quadratic Lyapunov stability is becoming a standard in the stability

analysis of hybrid systems [155, 96, 102, 233, 234]. It is a deductive way to prove
the stability of an equilibrium point of a subclass of hybrid systems (piecewise
affine systems). The computational burden is usually low, at the price of a con-
vex relaxation of the problem, that leads to possibly conservative results. Such
conservativeness can be reduced by constructing piecewise polynomial Lyapunov
functions via semidefinite programming by means of the sum of squares (SOS) de-
composition of multivariate polynomials [230]. SOS methods for analyzing stability
of continuous-time hybrid and switched systems are described in [238]. For the gen-
eral class of switched systems of the form ẋ = fi (x), i = 1, . . . , s, an extension of the
Lyapunov criterion based on multiple Lyapunov functions was introduced in [67].
The reader is also referred to the book by Liberzon [191].
The research on stability criteria for PWA systems has been motivated by the
fact that the stability of each component subsystem is not sufficient to guarantee
stability of a PWA system (and vice versa). Branicky [67], gives an example where
stable subsystems are suitably combined to generate an unstable PWA system.
Stable systems constructed from unstable ones have been reported in [278]. These
examples point out that restrictions on the switching have to be imposed in order
to guarantee that a PWA composition of stable components remains stable.
Passivity analysis of hybrid models has received little attention, except for
the contributions of [77, 199, 295] and [236], in which notions of passivity for
continuous-time hybrid systems are formulated, and of [31], where passivity and
synthesis of passifying controllers for discrete-time PWA systems are investigated.
16.8 Literature Review 375

Reachability Analysis and Verification of Safety Properties

Although simulation allows to probe a model for a certain initial condition and
input excitation, any analysis based on simulation is likely to miss the subtle phe-
nomena that a model may generate, especially in the case of hybrid models. Reach-
ability analysis (also referred to as “safety analysis” or “formal verification”), aims
at detecting if a hybrid model will eventually reach unsafe state configurations
or satisfy a temporal logic formula [7] for all possible initial conditions and in-
put excitations within a prescribed set. Reachability analysis relies on a reach set
computation algorithm, which is strongly related to the mathematical model of the
system. In the case of MLD systems, for example, the reachability analysis problem
over a finite time horizon (also referred to as bounded model checking) can be cast
as a mixed-integer feasibility problem. Reachability analysis was also investigated
via bisimulation ideas, namely by analyzing the properties of a simpler, abstracted
system instead of those of the original hybrid dynamics [182].
Timed automata and hybrid automata have proved to be a successful modeling
framework for formal verification and have been widely used in the literature. The
starting point for both models is a finite state machine equipped with continuous
dynamics. In the theory of timed automata, the dynamic part is the continuous-time
flow ẋ = 1. Efficient computational tools complete the theory of timed automata
and allow one to perform verification and scheduling of such models. Timed au-
tomata were extended to linear hybrid automata [7], where the dynamics is modeled
by the differential inclusion a ≤ ẋ ≤ b. Specific tools allow one to verify such mod-
els against safety and liveness requirements. Linear hybrid automata were further
extended to hybrid automata where the continuous dynamics is governed by differ-
ential equations. Tools exist to model and analyze those systems, either directly or
by approximating the model with timed automata or linear hybrid automata (see
the survey paper [265]).

Control

The majority of the control approaches for hybrid systems is based on optimal
control ideas (see the survey [291]). For continuous-time hybrid systems, most
authors either studied necessary conditions for a trajectory to be optimal [235,
271], or focused on the computation of optimal/suboptimal solutions by means
of dynamic programming or the maximum principle [128, 245, 143, 144, 80, 292,
192, 261, 264, 266, 273, 70, 141, 196, 128, 69, 252, 71] The hybrid optimal control
problem becomes less complex when the dynamics is expressed in discrete-time,
as the main source of complexity becomes the combinatorial (yet finite) number
of possible switching sequences. As will be shown in Chapter 17, optimal control
problems can be solved for discrete-time hybrid systems using either the PWA
or the MLD models described in this chapter. The solution to optimal control
problems for discrete-time hybrid systems was first outlined by Sontag in [266].
In his plenary presentation [202] at the 2001 European Control Conference Mayne
presented an intuitively appealing characterization of the state feedback solution to
optimal control problems for linear hybrid systems with performance criteria based
376 16 Models of Hybrid Systems

on quadratic and piecewise linear norms. The detailed exposition presented in the
first part of Chapter 17 follows a similar line of argumentation and shows that the
state feedback solution to the finite time optimal control problem is a time-varying
piecewise affine feedback control law, possibly defined over non-convex regions.
Model Predictive Control for discrete-time PWA systems and their stability and
robustness properties have been studied in [68, 60, 32, 188, 185, 186]. Invariant
sets computation for PWA systems has also been studied in [187, 4].

Observability and State Estimation

Observability of hybrid systems is a fundamental concept for understanding if a
state observer can be designed for a hybrid system and how well it will perform.
In [36] the authors show that observability properties (as well as reachability prop-
erties) can be very complex and present a number of counterexamples that rule out
obvious conjectures about inheriting observability/controllability properties from
the composing linear subsystems. They also provide observability tests based on
linear and mixed-integer linear programming.
State estimation is the reconstruction of the value of unmeasurable state vari-
ables based on output measurements. While state estimation is primarily required
for output-feedback control, it is also important in problems of monitoring and
fault detection [41, 17]. Observability properties of hybrid systems were directly
exploited for designing convergent state estimation schemes for hybrid systems
in [103].

Identification
Identification techniques for piecewise affine systems were recently developed [104,
255, 163, 39, 165, 282, 227], that allow one to derive models (or parts of models)
from input/output data.

Extensions to Event-driven and Stochastic Dynamics

The discrete-time methodologies described in this chapter were employed in [73] to
tackle event-based continuous-time hybrid systems with integral continuous dynam-
ics, called integral continuous-time hybrid automata (icHA). The hybrid dynamics
is translated into an equivalent MLD form, where continuous-time is an additional
state variable and the index t counts events rather than time steps. Extensions
of DHA to discrete-time stochastic hybrid dynamics were proposed in [33], where
discrete-state transitions depend on both deterministic and stochastic events.
 Chapter 17

Optimal Control of Hybrid

Systems

In this chapter we study the finite time, infinite time and receding horizon optimal
control problem for the class of hybrid systems presented in the previous chapter.
We establish the structure of the optimal control law and derive several different
algorithms for its computation. For finite time problems with linear and quadratic
objective functions we show that the time varying feedback law is piecewise affine.

17.1 Problem Formulation

Consider the PWA system (16.1) subject to hard input and state constraints

Ex(t) + Lu(t) ≤ M (17.1)

for t ≥ 0, and rewrite its restriction over the set of states and inputs defined
by (17.1) as h i
x(t)
x(t + 1) = Ai x(t) + B i u(t) + f i if u(t) ∈ C˜i (17.2)

where {C˜i }i=0

s−1
is the new polyhedral partition of the sets of state+input space
n+m
R obtained by intersecting the sets C i in (16.1c) with the polyhedron described
by (17.1). In this chapter we will assume that the sets C i are polytopes.
Define the cost function
N
X −1
J0 (x(0), U0 ) = p(xN ) + q(xk , uk ) (17.3)
k=0

where xk denotes the state vector at time k obtained by starting from the state
x0 = x(0) and applying to the system model

xk+1 = Ai xk + B i uk + f i if [ uxkk ] ∈ C˜i (17.4)

378 17 Optimal Control of Hybrid Systems

the input sequence u0 , . . . , uk−1 .

If the 1-norm or ∞-norm is used in the cost function (17.3), then we set p(xN ) =
kP xN kp and q(xk , uk ) = kQxk kp + kRuk kp with p = 1 or p = ∞ and P , Q, R full
column rank matrices. Cost (17.3) is rewritten as
N
X −1
J0 (x(0), U0 ) = kP xN kp + kQxk kp + kRuk kp . (17.5)
k=0

If the squared Euclidian norm is used in the cost function (17.3), then we set
p(xN ) = x′N P xN and q(xk , uk ) = x′k Qxk + u′k Ruk with P  0, Q  0 and R ≻ 0.
Cost (17.3) is rewritten as
N
X −1
J0 (x(0), U0 ) = x′N P xN + x′k Qxk + u′k Ruk . (17.6)
k=0

Consider the constrained finite time optimal control problem (CFTOC)

J0∗ (x(0)) = min J0 (x(0), U0 ) (17.7a)

U0

 xk+1 = Ai xk + B i uk + f i if [ uxkk ] ∈ C˜i
subj. to x ∈ Xf (17.7b)
 N
x0 = x(0)

where the column vector U0 = [u′0 , . . . , u′N −1 ]′ ∈ Rmc N × {0, 1}mℓN , is the opti-
mization vector, N is the time horizon and Xf is the terminal region.
In general, the optimal control problem (17.7) may not have a minimizer for
some feasible x(0). This is caused by discontinuity of the PWA system in the input
space. We will assume, however, that a minimizer U0 ∗ (x(0)) exists for all feasible
x(0). Also the optimizer function U0∗ may not be uniquely defined if the optimal
set of problem (17.7) is not a singleton for some x(0). In this case U0∗ denotes one
of the optimal solutions.
We will denote by Xk ⊆ Rnc × {0, 1}nℓ the set of states xk that are feasible
for (17.7):
  
  ∃u ∈ Rmc × {0, 1}mℓ , 

  

nc

nℓ  ∃i ∈ {1, . . . , s}
Xk = x ∈ R × {0, 1}  x ˜i ,


  [ u ] ∈ C and 
 (17.8)
 Ai x + B i u + f i ∈ Xk+1 
k = 0, . . . , N − 1
XN = Xf .

The definition of Xi requires that for any initial state xi ∈ Xi there exists a feasible
sequence of inputs Ui = [u′i , . . . , u′N −1 ] which keeps the state evolution in the
feasible set X at future time instants k = i + 1, . . . , N − 1 and forces xN into Xf
at time N . The sets Xk for i = 0, . . . , N play an important role in the solution of
the optimal control problem. They are independent of the cost function and of the
algorithm used to compute the solution to problem (17.7). As in the case of linear
systems (see Chapter 11.2) there are two ways to rigourously define and compute
 17.2 Properties of the State Feedback Solution, 2-Norm Case 379

the sets Xi : the batch approach and the recursive approach. In this chapter we
will not discuss the details on the computation of Xi . Also, we will not discuss
invariant and reachable sets for hybrid systems. While the basic concepts are
identical to those presented in Section 10.1 for linear systems, the discussion of
efficient algorithms requires a careful treatment which goes beyond the scope of
this book. The interested reader is referred to the work in [131, 121] for a detailed
discussion on reachable and invariant sets for hybrid systems.
In the following we need to distinguish between optimal control based on the
squared 2-norm and optimal control based on the 1-norm or ∞-norm. Note that
the results of this chapter also hold when the number of switches is weighted in the
cost function (17.3), if this is meaningful in a particular situation.
In this chapter we will make use of the following definition. Consider sys-
tem (17.2) and recall that, in general, x = [ xxcℓ ] where xc ∈ Rnc are the continuous
states and xℓ ∈ Rnℓ are the binary states and u ∈ Rmc × {0, 1}mℓ where uc ∈ Rmc
are the continuous inputs and uℓ ∈ Rmℓ are the binary inputs (Section 16.2.2). We
will make the following assumption.

Assumption 17.1 For the discrete-time PWA system (17.2) the mapping (xc (t), uc (t)) 7→
xc (t + 1) is continuous.
Assumption 17.1 requires that the PWA function that defines the update of the
continuous states is continuous on the boundaries of contiguous polyhedral cells,
and therefore allows one to work with the closure of the sets C˜i without the need
of introducing multi-valued state update equations. With abuse of notation in the
next sections C˜i will always denote the closure of C˜i . Discontinuous PWA systems
will be discussed in Section 17.7.

17.2 Properties of the State Feedback Solution, 2-Norm

Case
Theorem 17.1 Consider the optimal control problem (17.7) with cost (17.6) and
let assumption 17.1 hold. Then, there exists a solution in the form of a PWA state
feedback control law

u∗k (x(k)) = Fki x(k) + gki if x(k) ∈ Rik , (17.9)

where Rik , i = 1, . . . , Nk is a partition of the set Xk of feasible states x(k), and the
closure R̄ik of the sets Rik has the following form:
n
′
R̄ik = x : x(k)′ Lik (j)x(k) + Mki (j) x(k) ≤ Nki (j) ,
(17.10)
j = 1, . . . , nik , k = 0, . . . , N − 1,

and
x(k + 1) = Ai x(k)
h + B ii u∗k (x(k)) + f i
x(k) (17.11)
if u∗ (x(k)) ∈ C˜i , i = {1, . . . , s}.
k
380 17 Optimal Control of Hybrid Systems

Proof: The piecewise linearity of the solution was first mentioned by Sontag
in [266]. In [202] Mayne sketched a proof. In the following we will give the proof
for u∗0 (x(0)), the same arguments can be repeated for u∗1 (x(1)), . . . , u∗N −1 (x(N −1)).

• Case 1: (ml = nl = 0) no binary inputs and states

Depending on the initial state x(0) and on the input sequence U = [u′0 ,. . .,
u′k ], the state xk is either infeasible or it belongs to a certain polyhedron
C˜i , k = 0, . . . , N − 1. The number of all possible locations of the state
N
sequence x0 , . . . , xN −1 is equal to sN . Denote by {vi }si=1 the set of all possible
k
switching sequences over the horizon N , and by vi the k-th element of the
sequence vi , i.e., vik = j if xk ∈ C˜j .
Fix a certain vi and constrain the state to switch according to the sequence
vi . Problem (17.3)-(17.7) becomes

Jv∗i (x(0)) = min J0 (U0 , x(0)) (17.12a)

{U0 }
 k k k

 xk+1 = Avi xk + B vi uk + f vi

 k
 [ uxkk ] ∈ C˜vi
subj. to k = 0, . . . , N − 1 (17.12b)



 xN ∈ Xf

x0 = x(0)

Problem (17.12) is equivalent to a finite time optimal control problem for a

linear time-varying system with time-varying constraints and can be solved
by using the approach described in Chapter 11. The first move u0 of its
solution is the PPWA feedback control law

ui0 (x(0)) = F̃ i,j x(0) + g̃ i,j , ∀x(0) ∈ T i,j , j = 1, . . . , N r i (17.13)

SN r i
where Di = j=1 T i,j is a polyhedral partition of the convex set Di of feasible
states x(0) for problem (17.12). N r i is the number of regions of the polyhedral
partition of the solution and it is a function of the number of constraints in
problem (17.12). The upper index i in (17.13) denotes that the input ui0 (x(0))
is optimal when the switching sequence vi is fixed.
SsN
The set X0 of all feasible states at time 0 is X0 = i=1 Di and in general
it is not convex. Indeed, as some initial states can be feasible for different
switching sequences, the sets Di , i = 1, . . . , sN , in general, can overlap. The
solution u∗0 (x(0)) to the original problem (17.3)-(17.7) can be computed in
the following way. For every polyhedron T i,j in (17.13),
1. If T i,j ∩ T l,m = ∅ for all l 6= i, l = 1, . . . , sN , and for all m 6= j,
m = 1, . . . , N r l , then the switching sequence vi is the only feasible one
for all the states belonging to T i,j and therefore the optimal solution is
given by (17.13), i.e.

u∗0 (x(0)) = F̃ i,j x(0) + g̃ i,j , ∀x(0) ∈ T i,j . (17.14)

17.2 Properties of the State Feedback Solution, 2-Norm Case 381

2. If T i,j intersects one or more polyhedra T l1 ,m1 ,T l2 ,m2 , . . ., the states

belonging to the intersection are feasible for more than one switching
sequence vi , vl1 , vl2 , . . . and therefore the corresponding value functions
Jv∗i , Jv∗l ,Jv∗l , . . . in (17.12a) have to be compared in order to compute
1 2
the optimal control law.
Consider the simple case when only two polyhedra overlap, i.e., T i,j ∩
T l,m = T (i,j),(l,m) 6= ∅. We will refer to T (i,j),(l,m) as a double feasibility
polyhedron. For all states belonging to T (i,j),(l,m) the optimal solution
is:
 i,j

 F̃ x(0) + g̃ i,j , ∀x(0) ∈ T (i,j),(l,m) :

 Jv∗i (x(0)) < Jv∗l (x(0))




 F̃ l,m x(0) + g̃ l,m , ∀x(0) ∈ T (i,j),(l,m) :
u∗0 (x(0)) =  i,j Jv∗i (x(0)) > Jv∗l (x(0)) (17.15)



 F̃ x(0) + g̃ i,j or (i,j),(l,m)

 ∀x(0) ∈ T :


 F̃ l,m x(0) + g̃ l,m
Jv∗i (x(0)) = Jv∗l (x(0))

Because Jv∗i and Jv∗l are quadratic functions of x(0) on T i,j and T l,m
respectively, we find the expression (17.10) of the control law domain.
The sets T i,j \T l,m and T l,m \T i,j are two single feasibility P-collections
which can be partitioned into a set of single feasibility polyhedra, and
thus be described through (17.10) with Lik = 0.
In order to conclude the proof, the general case of n intersecting polyhedra
has to be discussed. We follow three main steps. Step 1: generate one
polyhedron of nth -ple feasibility and 2n − 2 P-collections possibly empty and
disconnected, of single, double, . . ., (n−1)th -ple feasibility. Step 2: the ith -ple
feasibility P-collection is partitioned into several ith -ple feasibility polyhedra.
Step 3: any ith -ple feasibility polyhedron with i > 1 is further partitioned
into at most i subsets (17.10) where in each one of them a certain feasible
value function is greater than all the others. The procedure is depicted in
Figure 17.1 when n = 3.

• Case 2: binary inputs, mℓ 6= 0

The proof can be repeated in the presence of binary inputs, mℓ 6= 0. In this
case the switching sequences vi are given by all combinations of region indices
and binary inputs, i.e., i = 1, . . . , (s · mℓ )N . For each sequence vi there is an
associated optimal continuous component of the input as calculated in Case
1.
• Case 3: binary states, nl 6= 0
The proof can be repeated in the presence of binary states by a simple enu-
meration of all the possible nNℓ discrete state evolutions.


382 17 Optimal Control of Hybrid Systems

1 1

1,2
1,3 2 2

1,2,3

3 3

(a) Step 1. (b) Step 2 and Step 3.

Figure 17.1 Graphical illustration of the main steps for the proof of Theo-
rem 17.1 when 3 polyhedra intersect. Step 1: the three intersecting poly-
hedra are partitioned into: one polyhedron of triple feasibility (1,2,3), 2
polyhedra of double feasibility (1, 2) and (1, 3), 3 polyhedra of single fea-
sibility (1),(2),(3). The sets (1), (2) and (1,2) are neither open nor closed
polyhedra. Step 2: the single feasibility sets of (1), (2) and (3) are parti-
tioned into six polyhedra. Step 3: value functions are compared inside the
polyhedra of multiple feasibility.

From the result of the theorem above one immediately concludes that the value
function J0∗ is piecewise quadratic:
′
J0∗ (x(0)) = x(0)′ H1i x(0) + H2i x(0) + H3i if x(0) ∈ Ri0 . (17.16)
The proof of Theorem 17.1 gives useful insights into the properties of the sets
Rik in (17.10). We will summarize them next.
Each set Rik has an associated multiplicity j which means that j switching
sequences are feasible for problem (17.3)-(17.7) starting from a state x(k) ∈ Rik .
If j = 1, then Rik is a polyhedron. In general, if j > 1 the boundaries of Rik
can be described either by an affine function or by a quadratic function. In the
sequel boundaries which are described by quadratic functions but degenerate to
hyperplanes or sets of hyperplanes will be considered affine boundaries.
Quadratic boundaries arise from the comparison of value functions associated
with feasible switching sequences, thus a maximum of j − 1 quadratic boundaries
can be present in a j-ple feasible set. The affine boundaries can be of three types.
• Type a: they are inherited from the original j-ple feasible P-collection. In
this case across such boundaries the multiplicity of the feasibility changes.
• Type b: they are artificial cuts needed to describe the original j-ple feasible
P-collection as a set of j-ple feasible polyhedra. Across type b boundaries
the multiplicity of the feasibility does not change.
• Type c: they arise from the comparison of quadratic value functions which
degenerate in an affine boundary.
17.2 Properties of the State Feedback Solution, 2-Norm Case 383

In conclusion, we can state the following lemma

Lemma 17.1 The value function Jk∗

1. is a quadratic function of the states inside each Rik

2. is continuous on quadratic and affine boundaries of type b and c

3. may be discontinuous on affine boundaries of type a,

and the optimizer u∗k

1. is an affine function of the states inside each Rik

2. is continuous across and unique on affine boundaries of type b

3. is non-unique on quadratic boundaries, except possibly at isolated points.

4. may be non-unique on affine boundaries of type c,

5. may be discontinuous across affine boundaries of type a.

Based on Lemma 17.1 above one can highlight the only source of discontinuity
of the value function: affine boundaries of type a. The following corollary gives a
useful insight on the class of possible value functions.

Corollary 17.1 J0∗ is a lower-semicontinuous PWQ function on X0 .

Proof: The proof follows from the result on the minimization of lower-semicontinuous
point-to-set maps in [47]. Below we give a simple proof without introducing the
notion of point-to-set maps.
Only points where a discontinuity occurs are relevant for the proof, i.e., states
belonging to boundaries of type a. From assumption 17.1 it follows that the feasible
switching sequences for a given state x(0) are all the feasible switching sequences
associated with any set Rj0 whose closure R̄j0 contains x(0). Consider a state x(0)
belonging to boundaries of type a and the proof of Theorem 17.1. The only case of
discontinuity can occur when (i) a j-ple feasible set P1 intersects an i-ple feasible
set P2 with i < j, (ii) there exists a point x(0) ∈ P1 , P2 and a neighborhood
N (x(0)) with x, y ∈ N (x(0)), x ∈ P1 , x ∈ / P2 and y ∈ P2 , y ∈ / P1 . The proof
follows from the previous statements and the fact that J0∗ (x(0)) is the minimum of
all Jv∗i (x(0)) for all feasible switching sequences vi . 
The result of Corollary 17.1 will be used extensively in the next sections. Even
if value function and optimizer are discontinuous, one can work with the closure
R̄jk of the original sets Rjk without explicitly considering their boundaries. In fact,
if a given state x(0) belongs to several regions R̄10 ,. . . ,R̄p0 , then the minimum value
among the optimal values (17.16) associated with each region R̄10 , . . . , R̄p0 allows us
to identify the region of the set R10 , . . . , Rp0 containing x(0).
Next we show some interesting properties of the optimal control law when we
restrict our attention to smaller classes of PWA systems.
384 17 Optimal Control of Hybrid Systems

Corollary 17.2 Assume that the PWA system (17.2) is continuous, and that E =
0 in (17.1) and Xf = Rn in (17.7) (which means that there are no state constraints,
i.e., P̃ is unbounded in the x-space). Then, the value function J0∗ in (17.7) is
continuous.
Proof: Problem (17.7) becomes a multi-parametric program with only input
constraints when the state at time k is expressed as a function of the state at time 0
and the input sequence u0 , . . . , uk−1 , i.e., xk = fP W A ((· · · (fP W A (x0 , u0 ), u1 ), . . . , uk−2 ), uk−1 ).
J0 in (17.3) will be a continuous function of x0 and u0 , . . . , uN −1 since it is the
composition of continuous functions. By assumptions the input constraints on
u0 , . . . , uN −1 are convex and the resulting set is compact. The proof follows from
the continuity of J and Theorem 5.2.

Note that E = 0 is a sufficient condition for ensuring that constraints (17.1)
are convex in the optimization variables u0 , . . . , un . In general, even for continuous
PWA systems with state constraints it is difficult to find weak assumptions ensuring
the continuity of the value function J0∗ . Ensuring the continuity of the optimal
control law u(k) = u∗k (x(k)) is even more difficult. A list of sufficient conditions for
U0∗ to be continuous can be found in [106]. In general, they require the convexity (or
a relaxed form of it) of the cost J0 (U0 , x(0)) in U0 for each x(0) and the convexity
of the constraints in (17.7) in U0 for each x(0). Such conditions are clearly very
restrictive since the cost and the constraints in problem (17.7) are a composition
of quadratic and linear functions, respectively, with the piecewise affine dynamics
of the system.
The next theorem provides a condition under which the solution u∗k (x(k)) of
the optimal control problem (17.3)-(17.7) is a PPWA state feedback control law.

Theorem 17.2 Assume that the optimizer U0 ∗ (x(0)) of (17.3)-(17.7) is unique

for all x(0). Then the solution to the optimal control problem (17.3)-(17.7) is a
PPWA state feedback control law of the form

u∗k (x(k)) = Fki x(k) + gki if x(k) ∈ Pki k = 0, . . . , N − 1, (17.17)

where Pki , i = 1, . . . , Nkr , is a polyhedral partition of the set Xk of feasible states

x(k).
Proof: In Lemma 17.1 we concluded that the value function J0∗ (x(0)) is con-
tinuous on quadratic type boundaries. By hypothesis, the optimizer u∗0 (x(0)) is
unique. Theorem 17.1 implies that F̃ i,j x(0) + g̃ i,j = F̃ l,m x(0) + g̃ l,m , ∀x(0) be-
longing to the quadratic boundary. This can occur only if the quadratic boundary
degenerates to a single feasible point or to affine boundaries. The same arguments
can be repeated for u∗k (x(k)), k = 1, . . . , N − 1. 

Remark 17.1 Theorem 17.2 relies on a rather strong uniqueness assumption. Some-
times, problem (17.3)-(17.7) can be modified in order to obtain uniqueness of the
solution and use the result of Theorem 17.2 which excludes the existence of non-
convex ellipsoidal sets. It is reasonable to believe that there are other conditions
under which the state feedback solution is PPWA without claiming uniqueness.
 17.3 Properties of the State Feedback Solution, 1, ∞-Norm Case 385

Example 17.1 Consider the following simple system

       

 
 0 −1 0.2 −0.2

 
 x(t) + u(t) +

 
 1 0 0 0.2

 
 1

  if x(t) ∈ C = {x : [0 1]x ≥ 0}


 x(t + 1) =      



 0 −1 0.2 0.2 (17.18)

 
 x(t) + u(t) −

 
 1 0 0 −0.2

  2

 if x(t) ∈ C = {x : [0 1]x < 0}



 x(t) ∈ [−0.5, −0.5] × [0.5, 0.5]

u(t) ∈ [−1000, 1000]
 
1 −0.1
and the optimal control problem (17.7) with cost (17.6), N = 2, Q = ,
−0.1 1
R = 10, P = 10Q, Xf = X .
The possible switching sequences are v1 = {1, 1}, v2 = {1, 2}, v3 = {2, 1}, v4 = {2, 2}.
The solution to problem (17.12) is depicted in Figure 17.2. In Figure 17.3(a) the
four solutions are intersected. All regions are polyhedra of multiple feasibility. In
Figure 17.3(b) we plot with different shadings the regions of the state space partition
where the switching sequences v1 = {1, 1}, v2 = {1, 2}, v3 = {2, 1}, v4 = {2, 2} are
optimal.

17.3 Properties of the State Feedback Solution, 1, ∞-Norm

Case
The results of the previous section can be extended to piecewise linear cost func-
tions, i.e., cost functions based on the 1-norm or the ∞-norm.

Theorem 17.3 Consider the optimal control problem (17.7) with cost (17.5), p =
1, ∞ and let assumption 17.1 hold. Then there exists a solution in the form of a
PPWA state feedback control law

u∗k (x(k)) = Fki x(k) + gki if x(k) ∈ Pki , (17.19)

where Pki , i = 1, . . . , Nkr is a polyhedral partition of the set Xk of feasible states

x(k).
Proof: The proof is similar to the proof of Theorem 17.1. Fix a certain switch-
ing sequence vi , consider the problem (17.3)-(17.7) and constrain the state to switch
according to the sequence vi to obtain problem (17.12). Problem (17.12) can be
viewed as a finite time optimal control problem with a performance index based on
1-norm or ∞-norm for a linear time varying system with time varying constraints
and can be solved by using the multi-parametric linear program as described in
Chapter 11.4. Its solution is a PPWA feedback control law

ui0 (x(0)) = F̃ i,j x(0) + g̃ i,j , ∀x ∈ T i,j , j = 1, . . . , N r i , (17.20)

386 17 Optimal Control of Hybrid Systems

x2

x2
x1 x1
(a) v1 = {1, 1} (b) v2 = {1, 2}
x2

x2

x1 x1
(c) v2 = {2, 1} (d) v4 = {2, 2}

Figure 17.2 Example 17.1. Problem (17.12) with cost (17.6) is solved for
different vi , i = 1, . . . , 4. Control law u∗0 (x0 ) is PPWA. State space partition
is shown.

and the value function Jv∗i is piecewise affine on polyhedra and convex. The rest
of the proof follows the proof of Theorem 17.1. Note that in this case the value
functions to be compared are piecewise affine and not piecewise quadratic. 

17.4 Computation of the Optimal Control Input via Mixed

Integer Programming
In the previous section the properties enjoyed by the solution to hybrid optimal
control problems were investigated. Despite the fact that the proofs are construc-
tive (as shown in the figures), they are based on the enumeration of all the possible
switching sequences of the hybrid system, the number of which grows exponen-
tially with the time horizon. Although the computation is performed off-line (the
on-line complexity is the one associated with the evaluation of the PWA control
17.4 Computation of the Optimal Control Input via Mixed Integer Programming 387

v1 v3
x2

x2
v2 v4

x1 x1
(a) Feasibility domain obtained as (b) Regions of the state space
the union of the domains shown partition where the switching se-
in Figure 17.2. All regions are quences v1 = {1, 1}, v2 = {1, 2},
polyhedra of multiple feasibility. v3 = {2, 1}, v4 = {2, 2} are opti-
mal.

Figure 17.3 Example 17.1. State space partition corresponding to the opti-
mal control law.

law (17.17)), more efficient methods than enumeration are desirable. Here we show
that the MLD framework can be used to avoid the complete enumeration. In
fact, when the model of the system is an MLD model and the performance index is
quadratic, the optimization problem can be cast as a Mixed-Integer Quadratic Pro-
gram (MIQP). Similarly, 1-norm and ∞-norm performance indices lead to Mixed-
Integer Linear Programs (MILPs) problems. In the following we detail the trans-
lation of problem (17.7) with cost (17.5) or (17.6) into a mixed integer linear or
quadratic program, respectively, for which efficient branch and bound algorithms
exist.
Consider the equivalent MLD representation (16.25) of the PWA system (17.2).
Problem (17.7) is rewritten as:

J0∗ (x(0)) = minU0 J0 (x(0), U0 ) (17.21a)



 xk+1 = Axk + B1 uk + B2 δk + B3 zk


 yk = Cxk + D1 uk + D2 δk + D3 zk + D5
subj. to E2 δk + E3 zk ≤ E1 uk + E4 xk + E5 (17.21b)



 xN ∈ Xf

x0 = x(0)

Note that the cost function (17.21a) is of the form

N
X −1
J0 (x(0), U0 ) = kP xN kp + kQxk kp + kRuk kp + kQδ δk kp + kQz zk kp (17.22)
k=0
388 17 Optimal Control of Hybrid Systems

when p = 1 or p = ∞ or

N
X −1
J0 (x(0), U0 ) = x′N P xN + x′k Qxk + u′k Ruk + δk′ Qδ δk + zk′ Qz zk (17.23)
k=0

when p=2.
The optimal control problem (17.21) with the cost (17.22) can be formulated as
a Mixed Integer Linear Program (MILP). The optimal control problem (17.21) with
the cost (17.23) can be formulated as a Mixed Integer Quadratic Program (MIQP).
The compact form for both cases is

min ε′ H1 ε + ε′ H2 x(0) + x(0)′ H3 x(0) + c′1 ε + c′2 x(0) + c

ε
(17.24)
subj. to Gε ≤ w + Sx(0)

where H1 , H2 , H3 , c1 , c2 , G, w, S are matrices of suitable dimensions, ε = [ε′c , ε′d ]

where εc , εd represent continuous and discrete variables, respectively and H1 , H2 ,
H3 , are null matrices if problem (17.24) is an MILP.
The translation of (17.21) with cost (17.23) into (17.24) is simply obtained by
substituting the state update equation

k−1
X
k
xk = A x0 + Aj (B1 uk−1−j + B2 δk−1−j + B3 zk−1−j ) (17.25)
j=0

The optimization vector ε in (17.24) is ε = {u0 , . . . , uN −1 , δ0 , . . . , δN −1 , z0 , . . . , zN −1 }.

The translation of (17.21) with cost (17.22) into (17.24) requires the introduc-
tions of slack variables as shown in Section 11.4 for the case of linear systems. In
particular, for p = ∞, Qz = 0 and Qδ = 0, the sum of the components of any
vector {εu0 , . . . , εuN −1 , εx0 , . . . , εxN } that satisfies

−1m εuk ≤ Ruk , k = 0, 1, . . . , N − 1

−1m εuk ≤ −Ruk , k = 0, 1, . . . , N − 1
−1n εxk ≤ Qxk , k = 0, 1, . . . , N − 1
(17.26)
−1n εxk ≤ −Qxk , k = 0, 1, . . . , N − 1
−1n εxN ≤ P xN ,
−1n εxN ≤ −P xN ,

represents an upper bound on J0∗ (x(0)), where 1k is a column vector of ones of

length k, and where x(k) is expressed as in (17.25). Similarly to what was shown
in [78], it is easy to prove that the vector ε = {εu0 , . . . , εuN −1 , εx0 , . . . , εxN , u(0), . . . , u(N −
1)} that satisfies equations (17.26) and simultaneously minimizes

J(ε) = εu0 + . . . + εuN −1 + εx0 + . . . + εxN (17.27)

also solves the original problem, i.e., the same optimum J0∗ (x(0)) is achieved.
Therefore, problem (17.21) with cost (17.22) can be reformulated as the follow-
17.4 Computation of the Optimal Control Input via Mixed Integer Programming 389

ing MILP problem

min J(ε)
ε
subj. to −1m εuk ≤ ±Ruk , k = 0, 1, . . . , N − 1
Pk−1
−1n εxk ≤ ±Q(Ak x0 + j=0 Aj (B1 uk−1−j +
B2 δk−1−j + B3 zk−1−j )) k = 0, . . . , N − 1
PN −1
−1n εxN ≤ ±P (AN x0 + j=0 Aj (B1 uk−1−j + (17.28)
B2 δk−1−j + B3 zk−1−j ))
xk+1 = Axk + B1 uk + B2 δk + B3 zk , k ≥ 0
E2 δk + E3 zk ≤ E1 uk + E4 xk + E5 , k ≥ 0
xN ∈ Xf
x0 = x(0)

where the variable x(0) in (17.28) appears only in the constraints as a parameter
vector.
Given a value of the initial state x(0), the MIQP (17.24) or the MILP (17.28)
can be solved to get the optimizer ε∗ (x(0)) and therefore the optimal input U0∗ (0).
In the next Section 17.5 we will show how multiparametric programming can be
also used to efficiently compute the piecewise affine state feedback optimal control
law (17.9) or (17.19).

Example 17.2 Consider the problem of steering in three steps the simple piecewise
affine system presented in Example 16.1 to a small region around the origin. The
system state update equations are
   √   

 
 1 − 3 0  

 
 0.4 √ x(t) + u(t) if 1 0 x(t) ≥ 0

  3 1 1

x(t + 1) =  √   

 1 3 0  

 
 0.4 √ x(t) + u(t) if 1 0 x(t) ≤ −ǫ

  1

  − 3 1

y(t) = 0 1 x(t)
(17.29)
subject to the constraints

x(t) ∈ [−5, 5] × [−5, 5]

(17.30)
u(t) ∈ [−1, 1].

The MLD representation of system (17.29)-(17.30) was reported in Example 16.7.

The finite time constrained optimal control problem (17.7) with cost (17.5) with
p = ∞, N = 3, P = Q = [ 10 01 ], R = 1, and Xf = [−0.01, 0.01] × [−0.01, 0.01], can be
solved by considering the optimal control problem (17.21) with cost (17.22) for the
equivalent MLD representation and solving the associated MILP problem (17.24).
The resulting MILP has 27 variables (|ε| = 27) which are x ∈ R2 , δ ∈ {0, 1}, z ∈ R2 ,
u ∈ R, y ∈ R over 3 steps plus the real-valued slack variables εu0 , εu1 , εu2 and εx1 , εx2 ,
εx3 (note that εx0 is not needed). The number of mixed-integer equality constraints
is 9 resulting from the 3 equality constraints of the MLD systems over 3 steps. The
number of mixed-integer inequality constraints is 110.
The finite time constrained optimal control problem (17.7) with cost (17.6), N = 3,
P = Q = [ 10 01 ], R = 1, and Xf = [−0.01 0.01] × [−0.01 0.01], can be solved by
390 17 Optimal Control of Hybrid Systems

considering the optimal control problem (17.21) with cost (17.23) for the equivalent
MLD representation and solving the associated MIQP problem (17.24).
The resulting MIQP has 21 variables (|ε| = 21) which are x ∈ R2 , δ ∈ {0, 1}, z ∈ R2 ,
u ∈ R, y ∈ R over 3 steps. The number of mixed-integer equality constraints is
9 resulting from the 3 equality constraints of the MLD systems over 3 steps. The
number of mixed-integer inequality constraints is 74.

17.4.1 Mixed-Integer Optimization Methods

With the exception of particular structures, mixed-integer programming problems

involving 0-1 variables are classified as N P -complete, which means that in the
worst case, the solution time grows exponentially with the problem size.
Despite the combinatorial nature of these problems, excellent solvers for MILPs
and MIQPs have become available over the last 10-15 years. As discussed in Chap-
ter 2.4, the most obvious way to solve an MIQP (MILP) is to enumerate all the
integer values and solve the corresponding QPs (LPs). By comparing the QPs
(LPs) optimal costs one can derive the optimizer and the optimal cost of the MIQP
(MILP). This is far from being efficient. The book [108] is a good reference for a
long list of algorithmic approaches to efficiently solve mixed integer programming
problems.
One simple idea called “Branch and Bound” has become the basis of many
efficient solvers. We will explain the basic ideas behind Branch and Bound solvers
with the help of a simple example.

Example 17.3 Illustrative Example for Branch and Bound Procedure We

refer to Figure 17.4. Assume that we are solving an MILP with three 0-1 integer
variables xd,1 , xd,2 , xd,3 .

1. In a first step we relax the integer constraints, i.e., we allow the variables
xd,1 , xd,2 , xd,3 to vary continuously between 0 and 1. We find the minimum
of the Relaxed LP to be 10, which must be a lower bound on the solution.
2. We fix one of the integer variables xd,1 at 0 and 1, respectively, and leave the
others relaxed. We solve the two resulting LPs and obtain 13 and 12, which are
new improved lower bounds on the solutions along the two branches.
3. Heuristically we decide to continue on the branch with xd,1 = 1, because the
lower bound is lower than with xd,1 = 0. We now fix one of the remaining
integer variables xd,2 at 0 and 1, respectively, and leave xd,3 relaxed. We solve
the two resulting LPs and obtain 16 and 14, which are new improved lower
bounds on the solution along the two branches.
4. Heuristically we decide again to continue on the branch with xd,2 = 1, because
the lower bound is lower than with xd,2 = 0. We now fix the remaining integer
variable xd,3 at 0 and 1, respectively. We solve the two resulting LPs and obtain
15 and 17. We can conclude now that 15 is an upper bound on the solution and
the integer choices leading to 17 are non-optimal. We can also stop the further
exploration of node 16 because this lower bound is higher than the established
upper bound of 15.
17.4 Computation of the Optimal Control Input via Mixed Integer Programming 391

relaxed LP
LB=10
10
xd,1 = 0 xd,1 = 1

13 12
xd,2 = 0 xd,2 = 1 xd,2 = 0 xd,2 = 1

∞ 19 16 14
infeasible 19 > UB 16 > UB
xd,3 = 0 xd,3 = 1

optimal
solution 15 17
UB=15 15 < UB = 17

Figure 17.4 Illustration of Branch and Bound procedure.

5. We explore node 13 and find the integer choices either to be infeasible or to lead
to a lower bound of 19 which is again higher than the established upper bound
of 15. Thus no further explorations are necessary and 15 is confirmed to be the
optimal solution.

More generally, the Branch and Bound algorithm for MILPs involves solving
and generating a set of LP subproblems in accordance with a tree search, where
the nodes of the tree correspond to LP subproblems. Branching generates child-
nodes from parent-nodes according to branching rules, which can be based, for
instance, on a-priori specified priorities on integer variables, or on the amount by
which the integer constraints are violated. Nodes are labeled either as pending, if
the corresponding LP problem has not yet been solved, or fathomed, if the node
has already been fully explored. The algorithm stops when all nodes have been
fathomed.
The success of the branch and bound algorithm relies on the fact that whole
subtrees can be excluded from further exploration by fathoming the corresponding
root nodes. This happens if the corresponding LP subproblem is either infeasible
or an integer solution is obtained. In the second case, the corresponding value of
the cost function serves as an upper bound on the optimal solution of the MILP
problem, and is used to further fathom any other nodes having either a greater
optimal value or lower bound.
392 17 Optimal Control of Hybrid Systems

17.5 State Feedback Solution via Batch Approach

Multiparametric programming [115, 101, 44, 63] has been used to compute the
PWA form of the optimal state feedback control law u∗ (x(k)) for linear system.
By generalizing the results of the previous chapters to hybrid systems, the state
vector x(0), which appears in the objective function and in the linear part of
the right-hand side of the constraints (17.24), can be considered as a vector of
parameters. Then, for performance indices based on the ∞-norm or 1-norm, the
optimization problem can be treated as a multiparametric MILP (mp-MILP), while
for performance indices based on the 2-norm, the optimization problem can be
treated as a multiparametric MIQP (mp-MIQP). Solving an mp-MILP (mp-MIQP)
amounts to expressing the solution of the MILP (MIQP) (17.24) as a function of
the parameters x(0).
In Section 6.4.1 we have presented an algorithm for solving mp-MILP problems,
while, to the authors’ knowledge, there does not exist an efficient method for solving
general mp-MIQPs. In Section 17.6 we will present an algorithm that efficiently
solves the specific mp-MIQPs derived from optimal control problems for discrete-
time hybrid systems.

17.6 State Feedback Solution via Recursive Approach

In this section we propose an efficient algorithm for computing the solution to
the finite time optimal control problem for discrete-time linear hybrid systems.
It is based on a dynamic programming recursion and a multiparametric linear
or quadratic programming solver. The approach represents an alternative to the
mixed-integer parametric approach presented earlier.
The PWA solution (17.9) will be computed proceeding backwards in time using
two tools: a linear or quadratic multi-parametric programming solver (depending
on the cost function used) and a special technique to store the solution which will be
illustrated in the next sections. The algorithm will be presented for optimal control
based on a quadratic performance criterion. Its extension to optimal control based
on linear performance criteria is straightforward.

17.6.1 Preliminaries and Basic Steps

Consider the PWA map ζ defined as

ζ : x ∈ Ri 7→ Fi x + gi for i = 1, . . . , NR , (17.31)

where Ri , i = 1, . . . , NR are subsets of the x−space. Note that if there exist

l, m ∈ {1, . . . , NR } such that for x ∈ Rl ∩ Rm , Fl x + gl 6= Fm x + gm the map
ζ (17.31) is not single valued.

Definition 17.1 Given a PWA map (17.31) we define fP W A (x) = ζo (x) as the
17.6 State Feedback Solution via Recursive Approach 393

ordered region single-valued function associated with (17.31) when

ζo (x) = Fj x + gj | x ∈ Rj and ∀i < j : x ∈

/ Ri ,
j ∈ {1, . . . , NR },

and write it in the following form


 F1 x + g1 if x ∈ P1

 ..
ζo (x) =  .

y FN x + gN if x ∈ PN R .
R R

Note that given a PWA map (17.31) the corresponding ordered region single-valued
function ζo changes if the order used to store the regions Ri and the corresponding
affine gains change. For illustration purposes consider the example depicted in
Figure 17.5, where x ∈ R, NR = 2, F1 = 0, g1 = 0, R1 = [−2, 1], F2 = 1, g2 = 0,
R2 = [0, 2].
In the following, we assume that the sets Rki in the optimal solution (17.9) can
overlap. When we refer to the PWA function u∗k (x(k)) in (17.9) we will implicitly
mean the ordered region single-valued function associated with the map (17.9).

Example 17.4 Let J1∗ : P1 → R and J2∗ : P2 → R be two quadratic functions,

J1∗ (x) = x′ L1 x + M1′ x + N1 and J2∗ (x) = x′ L2 x + M2′ x + N2 , where P1 and P2
are convex polyhedra and Ji∗ (x) = +∞ if x ∈ / Pi , i ∈ {1, 2}. Let u∗1 : P1 → Rm ,
u∗2 : P2 → Rm be vector functions. Let P1 ∩ P2 = P3 6= ∅ and define

J ∗ (x) = min{J1∗ (x), J2∗ (x)} (17.32)

 ∗
u1 (x) if J1∗ (x) ≤ J2∗ (x)
u∗ (x) = (17.33)
u∗2 (x) if J1∗ (x) ≥ J2∗ (x)

where u∗ (x) can be a set valued function. Let L3 = L2 − L1 , M3 = M2 − M1 ,

N3 = N2 − N1 . Then, corresponding to the three following cases

(i) J1∗ (x) ≤ J2∗ (x) ∀x ∈ P3

(ii) J1∗ (x) ≥ J2∗ (x) ∀x ∈ P3
(iii) ∃x1 , x2 ∈ P3 |J1∗ (x1 ) < J2∗ (x1 ) and J1∗ (x2 ) > J2∗ (x2 )

the expressions (17.32) and a real-valued function that can be extracted from (17.33)
can be written equivalently as:

(i)

 J1∗ (x) if x ∈ P1
J ∗ (x) =  (17.34)
y J2∗ (x) if x ∈ P2

 u∗1 (x) if x ∈ P1
u∗ (x) =  (17.35)
y u∗2 (x) if x ∈ P2

(ii) as in (17.34) and (17.35) by switching the indices 1 and 2

394 17 Optimal Control of Hybrid Systems

F2 x + g2

ζ(x)
F1 x + g1

x
(a) Multi valued PWA map ζ.

 
 F1 x + g1 if x ∈ R1  F2 x + g2 if x ∈ R2
ζ12 (x) = 
y F2 x + g2 if x ∈ R2 ζ21 (x) = 
y F1 x + g1 if x ∈ R1

x x
(b) Ordered region single valued (c) Ordered region single valued
function ζ12 . function ζ21 .

Figure 17.5 Illustration of the ordered region single valued function.

(iii)

 min{J1∗ (x), J2∗ (x)} if x ∈ P3
 ∗
J ∗ (x) =  J1 (x) if x ∈ P1 (17.36)
 ∗
y J2 (x) if x ∈ P2
 ∗ T
 u1 (x) if x ∈ P3 {x | x′ L3 x + M3′ x + N3 ≥ 0}
 ∗ T
 u2 (x) if x ∈ P3 {x | x′ L3 x + M3′ x + N3 ≤ 0}
u∗ (x) =  ∗ (17.37)
 u1 (x) if x ∈ P1
 ∗
y u2 (x) if x ∈ P2

where (17.34), (17.35), (17.36), and (17.37) have to be considered as PWA and
PPWQ functions in the ordered region sense.

Example 17.4 shows how to

• avoid the storage of the intersections of two polyhedra in case (i) and (ii)
 17.6 State Feedback Solution via Recursive Approach 395

• avoid the storage of possibly non convex regions P1 \ P3 and P2 \ P3

• work with multiple quadratic functions instead of quadratic functions defined

over non-convex and non-polyhedral regions.

The three points listed above will be the three basic ingredients for storing and
simplifying the optimal control law (17.9). Next we will show how to compute it.

Remark 17.2 In Example 17.4 the description (17.36)-(17.37) of Case (iii) can always
be used but the on-line evaluation of the control u∗ (x) is rather involved requiring
a series of set membership and comparison evaluations. To assess if the simpler
description of u∗ (x) of Case (i) or Case (ii) could be used instead, one needs to solve
indefinite quadratic programs of the form

minx x′ L3 x + M3′ x + N3
(17.38)
subj. to x ∈ P3

which are nontrivial, in general.

17.6.2 Multiparametric Programming with Multiple Quadratic Functions

Consider the multi-parametric program

J ∗ (x) = minu l(x, u) + q(f (x, u))

(17.39)
subj. to f (x, u) ∈ P,

where P ⊆ Rn is a compact set, f : Rn ×Rm → Rn , q : P → R, and l : Rn ×Rm → R

is a convex quadratic function of x and u. We aim at determining the region X
of variables x such that the program (17.39) is feasible and the optimum J ∗ (x) is
finite, and at finding the expression u∗ (x) of (one of) the optimizer(s). We point
out that the constraint f (x, u) ∈ P implies a constraint on u as a function of x
since u can assume only values where f (x, u) is defined.
Next we show how to solve several forms of problem (17.39).

Lemma 17.2 (one to one problem) Problem (17.39) where f is linear, q is

quadratic and strictly convex, and P is a polyhedron can be solved by one mp-QP.

Proof: See Chapter 6.3.1.

Lemma 17.3 (one to one problem of multiplicity d) Problem (17.39) where

f is linear, q is a multiple quadratic function of multiplicity d and P is a polyhedron
can be solved by d mp-QP’s.

Proof: The multi-parametric program to be solved is

J ∗ (x) = minu l(x, u) + min{q1 (f (x, u)), . . . , qd (f (x, u))

(17.40)
subj. to f (x, u) ∈ P
396 17 Optimal Control of Hybrid Systems

and it is equivalent to
 

 minu l(x, u) + q1 (f (x, u)), 


 


 subj. to f (x, u) ∈ P, 

∗
J (x) = min .. . (17.41)
 . 

 

 minu l(x, u) + qd (f (x, u))
 

 
subj. to f (x, u) ∈ P

The ith sub-problems in (17.41)

Ji∗ (x) = min l(x, u) + qi (f (x, u)) (17.42)

u
subj. to f (x, u) ∈ P (17.43)

is a one to one problem and therefore it is solvable by an mp-QP. Let the solution
of the i-th mp-QPs be

ui (x) = F̃ i,j x + g̃ i,j , ∀x ∈ T i,j , j = 1, . . . , N r i , (17.44)

SN r i
where T i = j=1 T i,j is a polyhedral partition of the convex set T i of feasible x for
the ith sub-problem and N r i is the corresponding number of polyhedral regions.
The feasible set X satisfies X = T 1 = . . . = T d since the constraints of the d
sub-problems are identical.
The solution u∗ (x) to the original problem (17.40) is obtained by comparing
and storing the solution of d mp-QP subproblems (17.42)-(17.43) as explained
in Example 17.4. Consider the case d = 2, and consider the intersection of the
polyhedra T 1,i and T 2,l for i = 1, . . . , N r 1 , l = 1, . . . , N r 2 . For all T 1,i ∩ T 2,l =
T (1,i),(2,l) 6= ∅ the optimal solution is stored in an ordered way as described in
Example 17.4, while paying attention to the fact that a region could be already
stored. Moreover, when storing a new polyhedron with the corresponding value
function and optimizer, the relative order of the regions already stored must not
be changed. The result of this Intersect and Compare procedure is
u∗ (x) = F i x + g i if x ∈ Ri ,
(17.45)
Ri = {x : x′ Li (j)x + M i (j)′ x ≤ N i (j), j = 1, . . . , ni },
SNR
where R = j=1 Rj is a polyhedron and the value function

J ∗ (x) = J˜j∗ (x) if x ∈ Dj , j = 1, . . . , N D , (17.46)

where J˜j∗ (x) are multiple quadratic functions defined over the convex polyhedra
Dj . The polyhedron Dj can contain several regions Ri or can coincide with one of
them. Note that (17.45) and (17.46) have to be considered as PWA and PPWQ
functions in the ordered region sense.
If d > 2 then the value function in (17.46) is intersected with the solution of
the third mp-QP sub-problem and the procedure is iterated by making sure not
to change the relative order of the polyhedra and the corresponding gain of the
solution constructed in the previous steps. The solution will still have the same
form (17.45)– (17.46). 
 17.6 State Feedback Solution via Recursive Approach 397

Lemma 17.4 (one to r problem) Problem (17.39) where f is linear, q is a

lower-semicontinuous PPWQ function defined over r polyhedral regions and strictly
convex on each polyhedron, and P is a polyhedron, can be solved by r mp-QP’s.

Proof: Let q(x) = qi if x ∈ Pi the PWQ function where the closures P̄i of
Pi are polyhedra and qi strictly convex quadratic functions. The multi-parametric
program to solve is
 

 minu l(x, u) + q1 (f (x, u)), 


 


 subj. to f (x, u) ∈ P̄1 


 


 f (x, u) ∈ P 

∗
J (x) = min .
.., . (17.47)

 

 
 minu l(x, u) + qr (f (x, u))} 

 

 
 subj. to f (x, u) ∈ P̄r





f (x, u) ∈ P

The proof follows the lines of the proof of the previous theorem with the excep-
tion that the constraints of the i-th mp-QP subproblem differ from the j-th mp-QP
subproblem, i 6= j.
The lower-semicontinuity assumption on q(x) allows one to use the closure of the
sets Pi in (17.47).The cost function in problem (17.39) is lower-semicontinuous since
it is a composition of a lower-semicontinuous function and a continuous function.
Then, since the domain is compact, problem (17.47) admits a minimum. Therefore
for a given x, there exists one mp-QP in problem (17.47) which yields the optimal
solution. The procedure based on solving mp-QPs and storing the results as in
S R j
Example 17.4 will be the same as in Lemma 17.3 but the domain R = N j=1 R of
the solution can be a P-collection. 
If f is PPWA defined over s regions then we have a s to X problem where X
can belong to any of the problems listed above. In particular, we have an s to
r problem of multiplicity d if f is PPWA and defined over s regions and q is a
multiple PPWQ function of multiplicity d, defined over r polyhedral regions. The
following lemma can be proven along the lines of the proofs given before.

Lemma 17.5 Problem (17.39) where f is linear and q is a lower-semicontinuous

PPWQ function of multiplicity d, defined over r polyhedral regions and strictly
convex on each polyhedron, is a one to r problem of multiplicity d and can be solved
by r · d mp-QP’s.
An s to r problem of multiplicity d can be decomposed into s one to r problems of
multiplicity d. An s to one problem can be decomposed into s one to one problems.

17.6.3 Algorithmic Solution of the Bellman Equations

In the following we will substitute the PWA system equations (17.2) with the
shorter form
x(k + 1) = f˜P W A (x(k), u(k)) (17.48)
398 17 Optimal Control of Hybrid Systems

˜ ˜ n ˜ i i i x ˜i
wheren fPoW A : C → R and fP W A (x, u) = A x + B u + f if [ u ] ∈ C , i = 1, . . . , s,
and C˜i is a polyhedral partition of C.˜
Consider the dynamic programming formulation of the CFTOC problem (17.6)-
(17.7),

Jj∗ (x(j)) = minuj x′j Qxj + u′j Ruj + Jj+1

∗
(f˜P W A (x(j), uj ))
(17.49)
subj. to f˜P W A (x(j), uj ) ∈ Xj+1 (17.50)

for j = N − 1, . . . , 0, with terminal conditions

XN = Xf (17.51)
∗ ′
JN (x) = x P x, (17.52)

where Xj is the set of all states x(j) for which problem (17.49)–(17.50) is feasible:

Xj = {x ∈ Rn | ∃u, f˜P W A (x, u) ∈ Xj+1 }. (17.53)

Assume for the moment that there are no binary inputs and binary states,
mℓ = nℓ = 0. The Bellman equations (17.49)–(17.52) can be solved backwards in
time by using a multi-parametric quadratic programming solver and the results of
the previous section.
Consider the first step of the dynamic program (17.49)–(17.52)
∗ ∗ ˜
JN −1 (xN −1 ) = min{uN −1 } x′N −1 QxN −1 + u′N −1 RuN −1 + JN (fP W A (xN −1 , uN −1 ))
(17.54)
subj. to f˜P W A (xN −1 , uN −1 ) ∈ Xf . (17.55)
∗
The cost to go function JN (x) in (17.54) is quadratic, the terminal region Xf is
a polyhedron and the constraints are piecewise affine. Problem (17.54)–(17.55) is
an s to one problem that can be solved by solving s mp-QPs. From the second
∗
step j = N − 2 to the last one j = 0 the cost to go function Jj+1 (x) is a lower-
semicontinuous PPWQ with a certain multiplicity dj+1 , the terminal region Xj+1 is
a P-collection and the constraints are piecewise affine. Therefore, problem (17.49)–
r r
(17.52) is an s to Nj+1 problem with multiplicity dj+1 (where Nj+1 is the number of
∗ r
polyhedra of the cost to go function Jj+1 ), that can be solved by solving sNj+1 dj+1
mp-QPs (Lemma 17.5). The resulting optimal solution will have the form (17.9)
considered in the ordered region sense.
In the presence of binary inputs the procedure can be repeated, with the dif-
ference that all the possible combinations of binary inputs must be enumerated.
Therefore a one to one problem becomes a 2mℓ to one problem and so on. In the
presence of binary states the procedure can be repeated either by enumerating
them all or by solving a dynamic programming algorithm at time step k from a
relaxed state space to the set of binary states feasible at time k + 1.
Next we summarize the main steps of the dynamic programming algorithm
discussed in this section. We use boldface characters to denote sets of polyhedra,
i.e., R = {Ri }i=1,...,|R| , where Ri is a polyhedron and |R| is the cardinality of
17.6 State Feedback Solution via Recursive Approach 399

the set R. Furthermore, when we say SOLVE an mp-QP we mean to compute and
store the triplet Sk,i,j of expressions for the value function, the optimizer, and the
polyhedral partition of the feasible space.
400 17 Optimal Control of Hybrid Systems

Algorithm 17.1
Input: CFTOC problem (17.3)–(17.7)
Output: Solution (17.9) in the ordered region sense
RN ← {Xf }
∗
JN,1 (x) ← x′ P x
For k = N − 1, . . . , 1
For i = 1, . . . , |Rk+1 |
For j = 1, . . . , s
Sk,i,j ← {}
Solve the mp-QP

Sk,i,j ← minuk x′k Qxk + u′k Ruk + Jk+1,i

∗
(Aj xk + Bj uk + fj )

Aj xk + Bj uk + fj ∈ Rk+1,i
subj. to
[ uxkk ] ∈ C˜j

End
End
∗
Rk ← {Rk,i,j,l }i,j,l . Denote by Rk,h its elements, and by Jk,h and u∗k,h (x)
the associated costs and optimizers, with h ∈ {1, . . . , |Rk |}
∗
Keep only triplets (Jk,h (x), u∗k,h (x), Rk,h ) for which ∃x ∈ Rk,h : x ∈ /
∗ ∗
Rk,d , ∀d 6= h OR ∃x ∈ Rk,h : Jk,h (x) < Jk,d (x), ∀d 6= h
Create multiplicity information and additional regions for an ordered re-
gion solution as explained in Example 17.4
End

In Algorithm 17.1, the structure Sk,i,j stores the matrices defining quadratic
∗
function Jk,i,j,l (·), affine function u∗k,i,j,l (·), and polyhedra Rk,i,j,l , for all l:
[

∗
Sk,i,j = Jk,i,j,l (x), u∗k,i,j,l (x), Rk,i,j,l , (17.56)
l

where the indices in (17.56) have the following meaning: k is the time step, i
indexes the piece of the “cost-to-go” function that the DP algorithm is considering,
j indexes the piece of the PWA dynamics the DP algorithm is considering, and l
indexes the polyhedron in the mp-QP solution of the (k, i, j)th mp-QP problem.
The “KEEP only triplets” Step of Algorithm 17.1 aims at discarding regions
Rk,h that are completely covered by some other regions that have lower cost. Obvi-
ously, if there are some parts of the region Rk,h that are not covered at all by other
regions (first condition) we need to keep them. Note that comparing the cost func-
tions is, in general, non-convex optimization problem. One might consider solving
the problem exactly, but since algorithm works even if some removable regions are
kept, we usually formulate an LMI relaxation of the problem at hand.
The output of Algorithm 17.1 is the state feedback control law (17.9) considered
in the ordered region sense. The online implementation of the control law requires
simply the evaluation of the PWA controller (17.9) in the ordered region sense.
 17.6 State Feedback Solution via Recursive Approach 401

17.6.4 Examples
Example 17.5 Consider the control problem of steering the piecewise affine sys-
tem (17.29) to a small region around the origin. The constraints and cost function
of Example 17.2 are used. The state feedback solution u∗0 (x(0)) was determined by
using the approach presented in the previous section. When the infinity norm is used
in the cost function, the feasible state space X0 at time 0 is divided into 84 polyhedral
regions and it is depicted in Fig. 17.6(a). When the squared Euclidean norm is used
in the cost function, the feasible state space X0 at time 0 is divided into 92 polyhedral
regions and is depicted in Fig. 17.6(b).
x2

x2

x1 x1
(a) Partition of the feasible state (b) Partition of the feasible state
space X0 when the infinity norm space X0 when the squared Eu-
is used in the cost function clidian norm is used in the cost
(N r 3 = 84) function (N r 3 = 92)

Figure 17.6 Example 17.5. State space control partition for u∗0 (x(0))

Note that as explained in Section 17.6.2 for the case of the squared Euclidian norm,
the optimal control law is stored in a special data structure where:

1. The ordering of the regions is important.

2. The polyhedra can overlap.

3. The polyhedra can have an associated value function of multiplicity d > 1.

Thus, d quadratic functions have to be compared on-line in order to compute
the optimal control action.

Example 17.6 Consider the hybrid spring-mass-damper system described in Exam-

402 17 Optimal Control of Hybrid Systems

ple 16.2
  0.90 0.02  0.10
 −0.01 

 −0.02 −0.00 x(t) + [ 0.02 ] u1 (t) + −0.02 if x1 (t) ≤ 1, u2 (t) ≤ 0.5





  0.90 0.02   −0.07 


 0.10
if x1 (t) ≥ 1 + ǫ, u2 (t) ≤ 0.5
 −0.06 −0.00 x(t) + [ 0.02 ] u1 (t) + −0.15

x(t+1) =

    −0.10 

 0.90 0.38
x(t) + [ 0.10 if x1 (t) ≤ 1, u2 (t) ≥ 0.5

 −0.38 0.52 0.38 ] u1 (t) + −0.38





    −0.75 
 0.90 0.35
−1.04 0.35 x(t) + [ 0.10
0.35 ] u1 (t) + −2.60 if x(t) ≥ 1 + ǫ, u2 (t) ≥ 0.5
(17.57)
subject to the constraints

x(t) ∈ [−5, 5] × [−5, 5]

(17.58)
u(t) ∈ [−1, 1].

We solve the finite time constrained optimal control problem (17.7) with cost (17.6)
with N = 3, P = Q = [ 10 01 ], R = [ 0.2 0
0 1 ]. The state feedback solution was de-
termined by using the approach presented in the previous section in two cases:
without terminal constraint (Figure 17.7(a)) and with terminal constraint Xf =
[−0.01 0.01] × [−0.01 0.01] (Figure 17.7(b)).
x2

x2

x1 x1
(a) Partition with no terminal (b) Partition with terminal con-
constraint (N r 3 = 183). straint (N r 3 = 181).

Figure 17.7 Example 17.6. State space optimal control partition for u∗0 (x(0))

17.7 Discontinuous PWA systems

Without assumption 17.1 the optimal control problem (17.3)-(17.7) may be feasible
but may not admit an optimizer for some x(0) (the problem in this case would be
to find an infimum rather than the minimum).
 17.8 Receding Horizon Control 403

Under the assumption that the optimizer exists for all states x(k), the ap-
proach explained in the previous sections can be applied to discontinuous systems
by considering three elements. First, the PWA system (17.2) has to be defined on
each polyhedron of its domain and all its lower dimensional facets. Secondly, dy-
namic programming has to be performed “from” and “to” any lower dimensional
facet of each polyhedron of the PWA domain. Finally, value functions are not
lower-semicontinuous, which implies that Lemma 17.4 cannot by used. Therefore,
when considering the closure of polyhedral domains in multi-parametric program-
ming (17.47), a post-processing is necessary in order to remove multi-parametric
optimal solutions which do not belong to the original set but only to its closure.
The tedious details of the dynamic programming algorithm for discontinuous PWA
systems are not included here but can be immediately extracted from the results
of the previous sections.

In practice, the approach just described for discontinuous PWA systems can
easily be numerically prohibitive. The simplest approach from a practical point of
view is to introduce gaps between the boundaries of any two polyhedra belonging
to the PWA domain (or, equivalently, to shrink by a quantity ε the size of every
polyhedron of the original PWA system). This way, one deals with PWA systems
defined over a disconnected union of closed polyhedra. By doing so, one can use
the approach discussed in this chapter for continuous PWA systems. However, the
optimal controller will not be defined at the points in the gaps. Also, the computed
solution might be arbitrarily different from the original solution to problem (17.3)-
(17.7) at any feasible point x. Despite this, if the magnitude ε of the gaps is close
to the machine precision and comparable to sensor/estimation errors, such an ap-
proach can very appealing in practice. In some cases this approach might be the
only one that is computationally tractable for computing controllers for discontin-
uous hybrid systems fulfilling state and input constraints that are implementable
in real-time.

Without assumption 17.1, problem (17.3)-(17.7) is well defined only if an op-

timizer exists for all x(0). In general, this is not easy to check. The dynamic
programming algorithm described here could be used for such a test but the de-
tails are not included in this book.

17.8 Receding Horizon Control

Consider the problem of regulating the PWA system (17.2) to the origin. Receding
Horizon Control (RHC) can be used to solve such a constrained regulation problem.
The control algorithm is identical to the one outlined in Chapter 12 for linear
systems. Assume that a full measurement of the state x(t) is available at the
404 17 Optimal Control of Hybrid Systems

current time t. Then, the finite time optimal control problem

Jt∗ (x(t)) = min Jt (x(t), Ut→t+N |t ) (17.59a)

Ut→t+N |t
  x 
 xt+k+1|t = Ai xt+k|t + B i ut+k|t + f i if ut+k|t
t+k|t
∈ C˜i
subj. to x ∈ Xf (17.59b)
 t+N |t
xt|t = x(t)

is solved at each time t, where Ut→t+N |t = {ut|t , . . . , ut+N −1|t }.

Let Ut∗ = {u∗t|t , . . . , u∗t+N −1|t } be the optimal solution of (17.59) at time t. Then,
the first sample of Ut∗ is applied to system (17.2):

u(t) = u∗t|t . (17.60)

The optimization (17.59) is repeated at time t+1, based on the new state xt+1|t+1 =
x(t + 1), yielding a moving or receding horizon control strategy.
Based on the results of previous sections the state feedback receding horizon
controller (17.59)–(17.60) can be immediately obtained in two ways: (i) solve the
MIQP/MILP (17.24) for xt|t = x(t) or (ii) by setting

u(t) = f0∗ (x(t)), (17.61)

where f0∗ : Rn → Rnu is the piecewise affine solution to the CFTOC (17.59)
computed as explained in Section 17.6. The explicit form (17.61) has the advantage
of being easier to implement, and provides insight on the type of action of the
controller in different regions of the state space.

17.8.1 Stability and Feasibility Issues

As discussed in Chapter 12 the feasibility and stability of the receding horizon

controller (17.59)–(17.60) is, in general, not guaranteed.
Theorem 12.2 in Chapter 12 can be immediately modified for the hybrid case:
persistent feasibility and Lyapunov stability are guaranteed if the terminal con-
straint Xf is a control invariant set (assumption (A2) in Theorem 12.2) and the
terminal cost p(xN ) is a control Lyapunov function (assumption (A3) in Theo-
rem 12.2). In the hybrid case the computation of control invariant sets and control
Lyapunov functions is computationally more involved. As in the linear case, Xf = 0
satisfies the aforementioned properties and it thus represents a very simple, yet re-
strictive, choice for guaranteeing persistent feasibility and Lyapunov stability.

17.8.2 Examples

Example 17.7 Consider the problem of regulating the piecewise affine system (17.29)
to the origin. The constrained finite time optimal control problem (17.7) is solved
with N = 3, P = Q = [ 10 01 ], R = 1, and Xf = [−0.01 0.01] × [−0.01 0.01]. Its state
17.8 Receding Horizon Control 405

feedback solution (17.9) u∗ (x(0)) = f0∗ (x(0)) at time 0 is implemented in a receding

horizon fashion, i.e., u(x(k)) = f0∗ (x(k)).
The state feedback control law with cost (17.5) with p = ∞ was computed in Exam-
ple 17.5 and consists of 84 polyhedral regions. None of them has multiplicity higher
than 1. Figure 17.8 shows the corresponding closed-loop trajectories starting from
the initial state x(0) = [−2 2]′ .
The state feedback control law with cost (17.6) was computed in Example 17.5 and
consists of 92 polyhedral regions, some of which have multiplicity higher than 1.
Figure 17.9 shows the corresponding closed-loop trajectories starting from the initial
state x(0) = [−2 2]′ .
x1 and x2

time time
(a) State trajectories (x1 dashed (b) Optimal input.
line and x2 solid line).

Figure 17.8 Example 17.7. MPC control of system (17.29) when the infinity
norm is used in the objective.
x1 and x2

time time
(a) State trajectories (x1 dashed (b) Optimal input.
line and x2 solid line).

Figure 17.9 Example 17.7. MPC control of system (17.29) when the squared
Euclidian norm is used in the objective.
406 17 Optimal Control of Hybrid Systems

Example 17.8 Consider the problem of regulating the hybrid spring-mass-damper

system (17.57) described in Examples 16.2 and 17.6 to the origin. The finite time
constrained optimal control problem (17.7) with cost (17.6) is solved with N = 3,
∗ ∗
P = Q = [ 10 01 ], R = [ 0.2 0
0 1 ]. Its state feedback solution (17.9) u (x(0)) = f0 (x(0)) at
∗
time 0 is implemented in a receding horizon fashion, i.e., u(x(k)) = f0 (x(k)).
The state feedback solution was determined in Example 17.6 for the case of no termi-
nal constraint (Figure 17.7(a)). Figure 17.10 depicts the corresponding closed-loop
trajectories starting from the initial state x(0) = [3 4]′ .
The state feedback solution was determined in Example 17.6 for terminal constraint
Xf = [−0.01, 0.01] × [−0.01, 0.01] (Figure 17.7(b)). Figure 17.11 depicts the corre-
sponding closed-loop trajectories starting from the initial state x(0) = [3 4]′ .
Comparing the two controllers, we find that the terminal constraint leads to a more
aggressive behaviour. In addition, in the case with terminal constraint we observe the
coordination of the binary and continuous inputs. The small damping coefficient is
switched on to increase the effect of the continuous input. The damping is increased
again once the steady-state value is approached.

u1 and u2
x1 and x2

time time
(a) State trajectories (x1 dashed (b) Optimal inputs (u1 solid line
line and x2 solid line). and binary input u2 dashed line)

Figure 17.10 Example 17.8. MPC control of system (17.57) without terminal
constraint
17.8 Receding Horizon Control 407

u1 and u2
x1 and x2

time time
(a) State trajectories (x1 dashed (b) Optimal inputs (u1 solid line
line and x2 solid line). and binary input u2 dashed line)

Figure 17.11 Example 17.8. MPC control of system (17.57) with terminal
constraint
408 17 Optimal Control of Hybrid Systems
List of Figures

1.1 Example 1.1. Constraints, feasible set and gradients. The feasible set is
the point (1, 0) that satisfies neither Slater’s condition nor the constraint
qualification condition in Theorem 1.7. . . . . . . . . . . . . . . . . . . 16
1.2 Geometric interpretation of KKT conditions [27]. . . . . . . . . . . . . . 17

2.1 Polytope. A polytope is a bounded polyhedron defined by the intersection

of closed halfspaces. The planes (here lines) defining the boundary of the
halfspaces are a′i x − bi = 0. . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Linear Program. Geometric Interpretation, level curve parameters ki <
ki−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Primal Degeneracy in a Linear Program. . . . . . . . . . . . . . . . . . 24
2.4 Example 2.1. LP with Primal and Dual Degeneracy. The vectors max and
min point in the direction for which the objective improves. The feasible
sets are shaded. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 Convex PWA function described as the max of affine functions. . . . . . . 27
2.6 Geometric interpretation of the Quadratic Program solution. . . . . . . . 28

3.1 Upper and lower bounds on functions: The quadratic upper bound stems
from L-smoothness of f (Definition 3.2), while the quadratic lower bound
is obtained from strong convexity of f (Theorem 3.1). In this example,
f (z) = z 2 + 12 z + 1 and we have chosen L = 3 and µ = 1 for illustration. . 37
3.2 Example 3.1. First 10 iterations of the gradient method (GM) with
constant step size 1/L using the tight, a slightly underestimated and a
strongly underestimated Lipschitz constant L. Also shown are the iter-
ates of the fast gradient method (FGM) using the tight Lipschitz constant.
The contour lines indicate the function value of each iterate. Note that
the gradients are orthogonal to the contour lines. . . . . . . . . . . . . . 40
3.3 Classic gradient method (left) and fast gradient method (right) for
the constrained case. After computing the standard update rule
from the unconstrained case, a projection onto the feasible set is
applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4 Example 3.4. Illustration of interior point method for example prob-
lem (3.37). Black dots are the solutions z ∗ (µ). . . . . . . . . . . . . . 55
410 LIST OF FIGURES

3.5 Affine-scaling (pure Newton) and centering search directions of classic

primal-dual methods. The Newton direction makes only progress towards
the solution when it is computed from a point close to the central path,
hence centering must be added to the pure Newton step to ensure that
the iterates stay sufficiently close to the central path. . . . . . . . . . . . 59
3.6 Search direction generation in predictor-corrector methods. The affine
search direction can be used to correct for linearization errors, which re-
sults in better performance in practice due to closer tracking of the central
path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.1 Example 4.2. Illustration of the convex hull of three points x1 = [1, 1]′ ,
x1 = [1, 0]′ , x3 = [0, 1]′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2 Example 4.3. Illustration of a cone spanned by the three points x1 =
[1, 1, 1]′ , x2 = [1, 2, 1]′ , x3 = [1, 1, 2]′ . . . . . . . . . . . . . . . . . . . . . 73
4.3 H-polyhedron. The planes (here lines) defining the boundary of the half-
spaces are a′i x − bi = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4 H-polytope. The planes (here lines) defining the halfspaces are a′i x − bi =
0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.5 Illustration of a polytope in H- and V- representation. . . . . . . . . 76
4.6 Illustration of the convex hull operation. . . . . . . . . . . . . . . . 80
4.7 Illustration of the envelope operation. . . . . . . . . . . . . . . . . . 80
4.8 Illustration of the Chebychev ball contained in a polytope P. . . . . . . 82
4.9 Illustration of the projection of a 3-dimensional polytope P onto the plane
x3 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.10 Two dimensional example illustrating Theorem 4.2. Partition of the
rest of the space Y\R0 . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.11 Illustration of the Pontryagin difference and Minkowski sum opera-
tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.12 Illustration that the Minkowski sum is not the complement of the
Pontryagin difference. (P ⊖ Q) ⊕ Q ⊆ P. . . . . . . . . . . . . . . . 87
4.13 Illustration of Algorithm 4.3. Computing the set difference G = C⊖B
between the P-collection C = C1 ∪ C2 and the polytope B. . . . . . . 90

5.1 Example 5.1. Optimizer z ∗ (x) and value function J ∗ (x) as a function of
the parameter x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2 Example 5.3. Point-to-set map and value function. . . . . . . . . . . . . 100
5.3 Example 5.4. Problem (5.10). (a)-(c) Projections of the point-to-set map
R(x) for three values of the parameter x: x̄1 < x̄2 < x̄3 ; (d) Value function
J ∗ (x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.4 Example 5.5. Open and not closed point-to-set map R(x). . . . . . . . . 102
5.5 Example 5.6. Closed and not open point-to-set map R(x). . . . . . . . . 102
5.6 Example 5.8. Point-to-set map and value function. . . . . . . . . . . . . 105

6.1 Example 6.2. Optimizer z ∗ (x) and value function J ∗ (x) as a function of
the parameter x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.2 Example 6.3. Polyhedral partition of the parameter space. . . . . . . . . 117
6.3 Example 6.4. Polyhedral partition of the parameter space corresponding
to the solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
LIST OF FIGURES 411

6.4 Example 6.4. A possible sub-partitioning of the degenerate region CR2

where the regions CRd {2,5} (Fig. 6.4(a)), CR d {2,4} (Fig. 6.4(b)) and
d
CR{2,1} (Fig. 6.5(a)) are overlapping. Note that below each picture
the feasible set and the level set of the value function in the z-space are
depicted for a particular choice of the parameter x indicated by a point
marked with ×. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.5 Example 6.4. A possible sub-partitioning of the degenerate region CR2
where the regions CRd {2,5} (Fig. 6.4(a)), CR d {2,4} (Fig. 6.4(b)) and
d
CR{2,1} (Fig. 6.5(a)) are overlapping. Note that below each picture
the feasible set and the level set of the value function in the z-space are
depicted for a particular choice of the parameter x indicated by a point
marked with ×. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.6 Example 6.4. A possible solution where the regions CR d {2,5} and CR
d {2,3}
are non-overlapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.7 g {2,4} is obtained by intersect-
Example 6.4. A possible solution where CR
d {2,4} with the complement of CR
ing CR d {2,5} , and CR
g {2,1} by intersecting
d {2,1} with the complement of CR
CR d {2,5} and CRd {2,4} . . . . . . . . . . 123
6.8 Example of a critical region explored twice . . . . . . . . . . . . . . . . 125
6.9 Example 6.5. Optimizer z ∗ (x) and value function J ∗ (x) as a function of
the parameter x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.10 Example 6.6. Polyhedral partition of the parameter space. . . . . . . . . 132
6.11 Example 6.8. Solution to the mp-MIQP (6.84). . . . . . . . . . . . . . . 143

7.1 Lyapunov function (7.41). . . . . . . . . . . . . . . . . . . . . . . . . . 159

9.1 Example 9.1. ∞-norm objective function, ∞-horizon controller. Piecewise

linear optimal cost (value function) and corresponding polyhedral partition. 180

10.1 Example 10.1. One-step controllable set Pre(X ) ∩ X for system (10.8)
under constraints (10.9). . . . . . . . . . . . . . . . . . . . . . . . . . 186
10.2 Example 10.1. Successor set for system (10.8). . . . . . . . . . . . . . . 187
10.3 Example 10.2. One-step controllable set Pre(X ) ∩ X for system (10.15)
under constraints (10.16). . . . . . . . . . . . . . . . . . . . . . . . . . 188
10.4 Example 10.2. Successor set for system (10.15) under constraints (10.16). 189
10.5 Example 10.3. Controllable sets Kj (S) for system (10.15) under con-
straints (10.16) for j = 1, 2, 3, 4. Note that the sets are shifted along the
x-axis for a clearer visualization. . . . . . . . . . . . . . . . . . . . . . 189
10.6 Example 10.3. Reachable sets Rj (X0 )) for system (10.15) under con-
straints (10.16) for j = 1, 2, 3, 4. . . . . . . . . . . . . . . . . . . . . . . 190
10.7 Example 10.5. Maximal Positive Invariant Set of system (10.8) under
constraints (10.9). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
10.8 Example 10.6. Maximal Control Invariant Set of system (10.15) subject
to constraints (10.16). . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
10.9 Example 10.8. One-step robust controllable set Pre(X , W) ∩ X for system
(10.34) under constraints (10.35)-(10.36). . . . . . . . . . . . . . . . . . 199
10.10Example 10.8. Robust successor set for system (10.34) under constraints
(10.36). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
412 LIST OF FIGURES

10.11Example 10.9. One-step robust controllable set Pre(X , W) ∩ X for system

(10.44) under constraints (10.45)-(10.46). . . . . . . . . . . . . . . . . . 201
10.12Example 10.9. Robust successor set for system (10.44) under constraints
(10.45)-(10.46). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
10.13Example 10.10. One-step robust controllable set Pre(X , W)∩X for system
(10.55) under constraints (10.56). . . . . . . . . . . . . . . . . . . . . . 204
10.14Example 10.11. Robust successor set Suc(X , W) for system (10.62) under
constraints (10.63). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
10.15Example 10.12. Maximal Robust Positive Invariant Set of system (10.67)
subject to constraints (10.68). . . . . . . . . . . . . . . . . . . . . . . 207
10.16Example 10.13. Maximal Robust Control Invariant Set of system (10.44)
subject to constraints (10.45). . . . . . . . . . . . . . . . . . . . . . . 209

11.1 Example 11.1. Propagation of the feasible sets Xi to arrive at C∞ (shaded

dark). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
11.2 CLQR Algorithm. Region Exploration. . . . . . . . . . . . . . . . . . . 225
11.3 Example 11.2. Double Integrator, 2-norm objective function, horizon N =
6. Partition of the state space for the time-varying optimal control law.
Polyhedra with the same control law were merged. . . . . . . . . . . . . 228
11.4 Example 11.3. Double Integrator, 2-norm objective function, horizon N =
∞. Partition of the state space for the time invariant optimal control law. 229
11.5 Example 11.4. Double Integrator, ∞-norm objective function, horizon
N = 6. Partition of the state space for the time-varying optimal control
law. Polyhedra with the same control law were merged. . . . . . . . . . 241
11.6 Example 11.5. Double Integrator, ∞-norm objective function, horizon
N = ∞. Partition of the state space for the time invariant optimal control
law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
11.7 Example 11.6. Double Integrator, minimum-time objective function. Par-
LQR
tition of the state space and control law. Maximal LQR invariant set O∞
is the central shaded region in (a). . . . . . . . . . . . . . . . . . . . . 242

12.1 Receding Horizon Idea. . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

12.2 Example 12.1. Closed-loop trajectories realized (solid) and predicted
(dashed) for two initial states x(0)=[-4.5,2] (boxes) and x(0)=[-4.5,3] (cir-
cles). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
12.3 Example 12.1. Double integrator with RHC. . . . . . . . . . . . . . . . 249
12.4 Example 12.2. Closed-loop trajectories for different settings of horizon N
and weight R. Boxes (Circles) are initial points leading (not leading) to
feasible closed-loop trajectories. . . . . . . . . . . . . . . . . . . . . . . 250
12.5 Example 12.2. Maximal positive invariant sets O∞ for different parameter
settings: Setting 1 (origin), Setting 2 (dark-gray) and Setting 3 (gray and
dark-gray). Also depicted is the maximal control invariant set C∞ (white
and gray and dark-gray). . . . . . . . . . . . . . . . . . . . . . . . . . 251
12.6 Example 12.3. Double integrator. RHC with 2-norm. Region of attraction
X0 for different horizons N and terminal regions Xf . . . . . . . . . . . . 259
12.7 Example 12.4. Double integrator. RHC with ∞-norm cost function, be-
haviour for different terminal weights. . . . . . . . . . . . . . . . . . . 261
LIST OF FIGURES 413

13.1 Examples of different barycentric interpolations. . . . . . . . . . . . . . 282

13.2 Illustration of the Convex-Hull Property of Barycentric Interpolation. The
first image shows a number of samples of a function, the second the convex
hull of these sample points and the third, a barycentric interpolation that
lies within this convex hull. . . . . . . . . . . . . . . . . . . . . . . . . 283
13.3 Illustration of the relationship between the optimal value function J ∗ (x),
the value function evaluated for the sub-optimal function z̃(x) and the
convex hull of the optimal value at the sample points. . . . . . . . . . . . 285
13.4 Polyhedron P constructed to be an outer approximation of epi J ∗,γ and
an inner approximation of epi J ∗ . . . . . . . . . . . . . . . . . . . . . . 288
13.5 Example of one iteration of the implicit double-description algorithm.
Square: Point vmax at which the worst-case error occurs. Circle: Point
z ∗ (vmax ); Projection of vmax onto the set. Line: Separating hyperplane. . 289
13.6 Several steps of the Double Description algorithm with the termination
criterion extreme(P ) ⊂ epi J ∗ (x) satisfied in the last step. . . . . . . . . . 294
13.7 Plot of the second order interpolant in 2D centered at zero, with a width
of one. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
13.8 Example 13.1. Optimal cost J ∗ (x) and approximation stability bound
J ∗ (x) + γ(x) above it. . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
13.9 Example 13.1. Error function −δV (top). Points marked are the vertices
with worst-case fit. Partition (triangulation) of the sampled points (mid-
dle). Upper-bound on approximate value function (convex hull of sample
points) (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
13.10Example 13.1. Error function −δV (top). Points marked are the vertices
with worst-case fit. Partition (triangulation) of the sampled points (mid-
dle). Upper-bound on approximate value function (convex hull of sample
points) (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
13.11Example 13.1. Error function −δV (top). Points marked are the vertices
with worst-case fit. Partition (triangulation) of the sampled points (mid-
dle). Upper-bound on approximate value function (convex hull of sample
points) (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
13.12Example 13.1. Approximate control law using triangulation. . . . . . . . 299
13.13Example 13.1. Error function −δV (top). Points marked are the vertices
with worst-case fit. Partition (polyhedral approximation) of the sampled
points (middle). Upper-bound on approximate value function (convex hull
of sample points) (bottom). . . . . . . . . . . . . . . . . . . . . . . . 300
13.14Example 13.1. Error function −δV (top). Points marked are the vertices
with worst-case fit. Partition (polyhedral approximation) of the sampled
points (middle). Upper-bound on approximate value function (convex hull
of sample points) (bottom). . . . . . . . . . . . . . . . . . . . . . . . 301
13.15Example 13.1. Partition generated by the second-order interpolant method.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
13.16Example 13.1. Approximate control law using second-order interpolants. . 302
13.17Example 13.1. Approximate value function J˜(x) using second-order in-
terpolants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
414 LIST OF FIGURES

14.1 Example for Algorithm 14.3 in one dimension: For a given point x ∈ P3
(x = 5) we have J ∗ (x) = max(T 1 x + V 1 , . . . , T 4 x + V 4 ). . . . . . . . . . 306
14.2 Example for Algorithm 14.4 in one dimension: For a given point x ∈ P2
(x = 4) we have O2 (x) = [−1 1]′ = S2 , while O1 (x) = 1 = 6 S1 = −1,
O3 (x) = [−1 − 1]′ 6= S3 = [1 − 1]′ , O4 (x) = 1 6= S4 = −1. . . . . . . . . 308
14.3 Illustration for Algorithm 14.5 in two dimensions. . . . . . . . . . . . . . 311

15.1 Batch Approach with Closed-Loop Prediction. State propagation and

associated input variables. The disturbance set has two vertices w1 and
w2 (nW a = 2) and the horizon is N = 4. The continuous line represents
the occurrence of w1 , while the dashed line represents the occurrence of
w2 . Over the horizon there are eight possible scenarios associated to eight
extreme disturbance realizations. The state x̃j3 denotes the predicted state
at time 3 associated to the j-th disturbance scenario. . . . . . . . . . . . 330
15.2 Example 15.4. Polyhedral partition of the state space corresponding to
the explicit solution of CROC-OL and CROC-CL at time t = 0. . . . . . 345
15.3 Example 15.4. (a) Polyhedral partition of the state space corresponding to
the explicit solution of CROC-Parametrized at time t = 0. (b) Sets X0OL ,
X0 and X0Mv corresponding to the controllers CROC-OL, CROC-CL
and CROC-Parameterized, respectively. . . . . . . . . . . . . . . . . 345

16.1 Hybrid systems. Logic-based discrete dynamics and continuous dynamics

interact through events and mode switches . . . . . . . . . . . . . . . . 352
16.2 Piecewise affine (PWA) systems. Mode switches are triggered by linear
threshold events. In each one of the shaded regions the affine mode dy-
namics is different. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
16.3 Example 16.1. Free response of state x1 for x(0) = [1 0]. . . . . . . . . . 355
16.4 Example 16.2. Open-loop simulation of system (16.6) for u1 = 3 and zero
initial conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
16.5 Example 16.3. Open-loop simulation of system (16.10) for different se-
quences of the input u. . . . . . . . . . . . . . . . . . . . . . . . . . . 358
16.6 A discrete hybrid automaton (DHA) is the connection of a finite
state machine (FSM) and a switched affine system (SAS), through a
mode selector (MS) and an event generator (EG). The output signals
are omitted for clarity . . . . . . . . . . . . . . . . . . . . . . . . . . 359
16.7 Example of finite state machine . . . . . . . . . . . . . . . . . . . . . 361

17.1 Graphical illustration of the main steps for the proof of Theorem 17.1
when 3 polyhedra intersect. Step 1: the three intersecting polyhedra are
partitioned into: one polyhedron of triple feasibility (1,2,3), 2 polyhe-
dra of double feasibility (1, 2) and (1, 3), 3 polyhedra of single feasibility
(1),(2),(3). The sets (1), (2) and (1,2) are neither open nor closed poly-
hedra. Step 2: the single feasibility sets of (1), (2) and (3) are partitioned
into six polyhedra. Step 3: value functions are compared inside the poly-
hedra of multiple feasibility. . . . . . . . . . . . . . . . . . . . . . . . . 382
17.2 Example 17.1. Problem (17.12) with cost (17.6) is solved for different vi ,
i = 1, . . . , 4. Control law u∗0 (x0 ) is PPWA. State space partition is shown. 386
LIST OF FIGURES 415

17.3 Example 17.1. State space partition corresponding to the optimal control
law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
17.4 Illustration of Branch and Bound procedure. . . . . . . . . . . . . . . . 391
17.5 Illustration of the ordered region single valued function. . . . . . . . . . 394
17.6 Example 17.5. State space control partition for u∗0 (x(0)) . . . . . . . . . 401
17.7 Example 17.6. State space optimal control partition for u∗0 (x(0)) . . . . . 402
17.8 Example 17.7. MPC control of system (17.29) when the infinity norm
is used in the objective. . . . . . . . . . . . . . . . . . . . . . . . . . 405
17.9 Example 17.7. MPC control of system (17.29) when the squared Eu-
clidian norm is used in the objective. . . . . . . . . . . . . . . . . . . 405
17.10Example 17.8. MPC control of system (17.57) without terminal constraint406
17.11Example 17.8. MPC control of system (17.57) with terminal constraint 407
Index

N -Step Controllable Set, 186 Dual variables, 11

N -Step Reachable Set, 187
N -Step Robust Controllable Set, 199 Efficient On-Line Algorithms, 306
descriptor function, 309
Active Constraints, 5
Algebraic Riccati Equation, 170 Fast Gradient Method, 39
Approximate Receding Horizon Control
Barycentric Interpolation, 283 Gradient Method, 36
Partitioning and Interpolation Meth- Gradient Projection Methods, 47
ods, 288
Second-Order Interpolants, 293 Hybrid System
Stability, 280 2-norm Optimal Control, 381
∞-norm Optimal Control, 387
Barycentric Interpolation, 283 Optimal Control, 379
Optimal Control via Mixed-Integer
Complementary Slackness, 13 Programming, 388
Constrained Least-Squares Problems, 30 Hybrid Systems
Constrained LQR, 213 Binary States, Inputs, and Outputs,
algorithm, 226 357
Constrained Optimal Control, 149 Discrete Hybrid Automata, 360
Batch Approach, 151 Finite State Machine, 363
Dynamic Programming, 152 Hysdel Modeling Language, 369
Infinite Horizon, 154 Logic and Mixed-Integer Inequali-
Recursive Approach, 152 ties, 365
Value Function Iteration, 155 Mixed Logical Dynamical Systems,
Constraint Qualifications, 12 367
Control Lyapunov Function, 257 Model Equivalences, 369
Convergence Rates, 34 Modeling Discontinuities, 356
Convex Hull, 79 Models, 353
Convex Optimization Problems, 8 Piecewise Affine Systems, 354
Convex Piecewise Linear Optimization, Hysdel, 369
26
Convexity, 6 Interior Point Methods, 52
Operations Preserving, 7 Invariant Set, 185
Critical Region, 110 Control Invariant, 195
Max Control Invariant, 195
Descent Method, 36 Max LQR Invariant, 225
Discrete Hybrid Automata, 360 Max Positive Invariant, 192
Discrete Time Riccati Equation, 168 Max Robust Positive Invariant, 207
Index 417

Positive Invariant, 192 Properties, 114, 121

Robust Control Invariant, 209 Set of Active Constraints, 117
Robust Positive Invariant, 207 mp-QP, 128
Algorithm, 137
Karush-Kuhn-Tucker Conditions, 14 Formulation, 128
Properties, 130, 134
Lagrange Dual Problem, 11 Set of Active Constraints, 133
Lagrange Duality, 10 Multiparametric Mixed-Integer Program,
Lagrange Multipliers, 11 140
Lagrangian, 10 Multiparametric Program, 97
Line Search Methods, 46 Example, 101
Linear Constrained Optimal Control, 213 General Results, 100
Feasible Sets, 216 Main Properties, 105
State Feedback Solution, 223
Linear Program, 20 Newton’s Method, 43
Dual of, 22
KKT Conditions, 23 Optimization Problem, 3
Linear Quadratic Regulator, 165
Batch Approach, 166 Point-to-Set Maps, 102
Infinite Time Solution, 170 Continuity, 102
Recursive Approach, 167 Continuous Selections, 105
Logic and Mixed-Integer Inequalities, 365 Polyhedron, 73
LQR, 165 Affine Mappings of, 88
Lyapunov Equation, 162 Chebychev Ball, 81
Lyapunov Function, 160 Envelope, 79
1/∞ norm, 163 Minimal Representation, 78
Quadratic, 162 Minkowski Sum, 86
Radially Unbounded, 161 Open, 76
Lyapunov Inequality, 163 P-collection, 76
Lyapunov Stability, 158 Projection, 82
Definition, 158 Ray, 74
Global, 160 Set-Difference, 82
Union, 86
Maximal Controllable Set, 196 Union of P-collections, 92
Maximal Robust Controllable Set, 211 Valid Inequality, 75
Minimum-Time Control, 240 Vertex Enumeration, 80
Minkowski Difference Polytopal Complexes, 76
of P-collections, 89 Polytope, 73
of Polytopes, 85 Pontryagin Difference
Mixed Integer Linear Programming, 31 of Polytopes, 85
Mixed Integer Quadratic Programming, Pontryagin Difference¿of P-collections, 89
31 PWA System
Mixed Logical Dynamical Systems, 367 2-norm Optimal Control, 381
Model Predictive Control, 246 ∞-norm Optimal Control, 387
mp-LP, 112
Algorithm, 125 Quadratic Program, 27
Formulation, 112 Dual of, 29
Nonunique Optimizer, 119 KKT Conditions, 29
418 Index

Receding Horizon Control, 246

receding horizon control, 277
RHC
Feasibility Issue, 253
Implementation, 246
Observer Design, 270
Offset-Free Reference Tracking, 269
Stability Issue, 256
Robust Pre
Computation for Linear Systems with
Inputs, 201
Computation for Linear Systems with
Parametric Uncertainty, 204
Computation for Linear Systems with-
out Inputs, 199
Definition, 198
Robust Suc
Definition, 198
Computation for Linear Systems with
Inputs, 201
Computation for Linear Systems with
Parametric Uncertainty, 204
Computation for Linear Systems with-
out Inputs, 199
Robust Invariant Set, 198
Robust Reachable Set, 199

Strong Duality, 12

Triangulation, 288

Unconstrained Optimal Control

1/∞ Norm, 173

Wolfe Conditions, 46
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