2008 American Control Conference FrB13.

1
Westin Seattle Hotel, Seattle, Washington, USA
June 11-13, 2008

MPC and PID Control Based on Multi-objective Optimization

Adrian Gambier

Abstract— The design of sophisticated control systems have led in In the current paper, controller design for MPC and PID
the past ten years to the necessity of satisfying more than one design strategies based on MOO techniques is reviewed including the
criterion. Thus, it is natural to think that those criteria can be met in most used MOO algorithms and software implementations. In
an optimal manner. If several criteria have simultaneously to be opti- Section II, optimal PID and MPC based on a unique performance
mized, one is in presence of a Multi-Objective Optimization problem. index are shortly summarized. Section III is dedicated to present
In this paper, many efforts to design the most popular control the essentials of MOO, i.e. definitions and most relevant methods.
strategies, i.e. PID and MPC, by using multi-objective optimization Available software for MOO is the subject of Section IV. In
techniques are reviewed. Both control strategies have dissimilar Section V, some works on MOO-PID and MOO-MPC are
optimization characteristics and therefore, they can be considered as compared. Finally, Section VI is devoted to draw conclusions.
representative of two different multi-objective optimization problems,
which are described including definitions, possible solutions, II. OPTIMAL CONTROL SYSTEM DESIGN
algorithms and available software implementations.
Control design based on the minimization of performance indices
I.INTRODUCTION is an well established area in control engineering (see e.g. [6] and
references herein). Conventional optimal control systems are
I N control engineering, Multi-Objective Optimization (MOO)
has been used for a long time (pioneer works are for example
[24], [40], and the references herein). However, MOO has
obtained by optimization of a single performance index, which in
general is defined as
J = J [e(k0 )," , e(k ), u(k0 )," , u(k ), α ] , (1)
intensively been applied to obtain optimal control systems in the
last ten years. An excellent review of MOO applications in subject to constraints
control engineering can be found in [41]. A recently update is ge [e(k0 )," , e( k ), u( k0 )," , u( k ), α] = 0 or (2)
given in [19].
PID control or PI control (Proportional, Integral and Derivative) hie [e(k0 )," , e(k ), u(k0 )," , u( k ), α] ≤ 0 , (3)
and MPC (Model Predictive Control) can be considered the where e(·) is the control error, u(·) is the control or decision
most popular control strategies as one can infer from the vast vector (controller’s outputs) and α is a parameter vector (normally
available literature. In the case of PID control, a small sample of containing controller parameters).
books is for example [3], [13], [33], [62], [58] and [48]. Well- According to which argument J is optimized, two different
known books on model predictive control are for instance [9], optimization problems can be considered: (a) if J is optimized with
[27], [29], [39], [42] and [55]. However, none of these books treat respect to the parameter vector α, i.e.
the multi-objective control problem. Only in [9] and [42], MOO
ideas are shortly mentioned. In [41], it is treated the MOO-PID J (e(k ), u(k ), αo ) = min J [e(k0 ),", e(k ), u(k0 ),", u(k ), α] , (4)
α
control but not the MOO-MPC. one is in presence of a parameter optimization problem and (1) is
In spite of the absence of MOO in the literature, MOO has a cost function; (b) on the contrary, J can also be optimized with
been applied to design control systems based on PID as well as on respect to the function u(k), i.e.
MPC strategies in several opportunities. This is, for instance, the
case of PI and PID controller design described in [30] as a MOO J (e(k ), uo (k ), α) = min J [e(k0 ),", e(k ), u(k0 ),", u(k ), α] . (5)
u( k )
problem. A simplified goal-attainment formulation of MOO This is a problem of calculus of variations (or multi-stage optimi-
problem is used to tune PI and PID controllers in [37]. The design zation problem). That is a dynamic optimization problem, where (1)
problem of a robust PID controller with two degrees of freedom is now a cost functional and uo(k) is the optimal control law. Some-
based on the partial model matching approach is treated in [35]. times optimization problems can be solved analytically leading to
In [8], a design procedure for tuning PID controller parameters to an elegant closed solution, but in general optimization problems
achieve a mixed H 2/H ∞ optimal performance using genetic algo- with constraints can only be solved numerically. Numerical
rithms is described. All mentioned MOO problems deal with methods to solve parameter optimization problems can be used in
several objective functions that have to be satisfied by only one a repetitive fashion to solve problems of dynamic optimization.
controller (SISO or MIMO) in a mono-loop control system. MOO The most important limitation here is that the optimization must
methods for the simultaneous parameter optimization of PID normally be carried out on-line. In this case, the problem is also
controllers in several interacting control loops can be found for subject to time constraints given by the sampling time.
example in [20] and [63] for continuous-time systems and [21]
for the discrete-time case. The optimal PID controller design belongs to the first class of
optimization problems, while MPC is a typical case of the second
A MOO framework for MPC has been proposed in [36]. The one. In the following subsections, standard approaches for optimal
approach is based on a lexicographic algorithm taking advantage PID and MPC are shortly summarized.
about the fact that for this method, objective functions can be or-
dered according to pre-established priorities. Another MOO- A. Parameter-optimized PID Controllers
MPC approach is presented in [65], where the performance The idea of choosing PID controller parameters by minimizing
index is formulated as MOO problem and the goal attainment an integral square cost function is not new. It is already proposed in
method is used to obtain the solution. In [34], a similar approach is [54] for a discrete-time adaptive PID controller. The continuous-
used to solve a nonlinear MPC problem by using a NARX model. time case is analogous. The problem then is to find parameters for
the digital PID controller so that the performance index
Manuscript received September 21, 2007. ∞
J = ∑ e 2 ( k ) + λ Δu 2 ( k )
A. Gambier is with the Automation Laboratory, University of Heidelberg, B6,
No. 26, 68131 Mannheim, Germany (phone: +49 621 181-2783; fax: +49 621 (6)
181-2739; e-mail: agambier@ieee.org). k =1

978-1-4244-2079-7/08/$25.00 ©2008 AACC. 4727

or equivalently (according to the Parseval’s theorem) case the optimization cannot be carried out analytically. The solu-
1 tion has numerically to be found on-line at every time step.
2π j v∫
J= [e ( z ) e( z −1 ) + λ Δu ( z ) Δu ( z −1 )]z −1dz (7)
III. MULTI-OBJECTIVE OPTIMIZATION
is minimized. A. General Definitions
The parameter λ is referred to a Lagrange multiplier and it can Multi-objective optimization (also known as multi-performance,
be exactly determined. However, it is usually assumed that λ is a multi-criteria or vector optimization) was introduced by V. Pareto
free design parameter by mean of which control signal amplitudes ([49]) and it can be defined as the problem of finding a vector of
can be softly limited. Thus, the index (6) can be seen as a weighted decision variables (or parameters), which satisfies constraints and
sum objective function. The digital PID control law is given by optimizes a vector field, whose elements represent objective func-
P( z −1 )u ( z ) = Q ( z −1 )e ( z ) , (8) tions. In general, the MOO problem can be formulated as follows:
Find u = [u1 … ul]T or α = [α1 … α np]T
where P and Q are the polynomials
to optimize J(u,α) = [J1(u,α) … Jnf(u,α)]T
P ( z −1 ) = 1 − z −1 and Q ( z −1 ) = q0 + q1 z −1 + q 2 z −2 . (9) with respect to u or to α
Variable e(z) is the control error e(z) = r(z) − y(z). Moreover, the subject to gi(u,α) ≤ 0 and hj(u,α) = 0,
coefficients qi must satisfy the inequality (see [31] for details) for i = 1,…, ng, j = 1, …, nh,
where J∈ϑ ⊂ \nf is the objective vector field, u∈U ⊂ \l is a
⎡ −1 0 0⎤ ⎡ q0 ⎤ vector of decision variables, α∈Λ ⊂ \np is a vector of fixed
⎢ 1 1 0⎥ ⎢q ⎥ ≤ 0 (10)
⎢ −1 parameters, nf is the number of objective functions, ng is the
− 1 − 1⎥ ⎢ 1⎥
⎢ ⎥ ⎢⎣ q 2 ⎥⎦ number of inequality constraints and nh is the number of
⎣ −1 0 1⎦
equality constraints. Optimize means here either minimize or
in order to hold the step response characteristic of a continuous- maximize depending on the application. If all Ji are convex, all
time PID controller. If it is assumed that the plant is modelled by inequality constraints gi are convex and all equality constraints hj
A( z ) y ( z ) = B ( z )u ( z ) (11) are affine, the vector optimization problem is also called convex.
and the controller design is carried out for a step setpoint r(z−1)=1/(1−z−1). Contrary to single-objective optimization (SOO), there is for
Hence, e(z) and Δu(z) are given by MOO no single global solution and it is often necessary to deter-
mine a set of points that all fit a predetermined definition for the
A(z) A(z)Q(z) optimum. Usually, it is accepted as multi-objective optimality the
e(z) = and Δu(z) = (12)
−1
(1− z )A(z) + Q(z)B(z) −1
(1− z )A(z) + Q(z)B(z) definition given by Pareto ([49]), which is stated as follows:
respectively. The unknown coefficients of Q, q = [q0, q1, q2]T Definition 1: A point, u°∈U ⊂ \l, is Pareto optimal with respect
are obtained by using a numerical optimization algorithm as e.g. to U iff there does not exist another point, u∈U, such that
Fletcher-Powell or Hooke-Jeeves. The index (7) can be evaluated J(u,α) ≤ J(u°,α) and Ji(u,α) < Ji(u°,α) for at least one function,
by using the algorithm given in [2] (see [18] for the evaluation of i.e. there is no way to improve upon a Pareto optimal point
this and other square criteria). Parameter λ is normally difficult without increasing the value of at least one of the other
to select and therefore chosen by trial and error. objective functions. Notice that the definition given above can be
applied to the parameter vector α instead of the vector of decision
B. Model Predictive Control variables u and this is also valid for the next definitions.
Model Predictive Control (MPC) comprises a family of control- Sometimes, it is useful to have a definition for a suboptimal
lers signed by the same design philosophy and similar characteris- point that is easier to be reached by the algorithms and simultane-
tics (see e.g. [9], [42] and [55]). The design is based on three ously is sufficient ‘good’ for practical applications. This is obtained
main steps: (i) the use of a model to predict the output vector y, N e.g. from the Weakly Pareto Optimality:
step in the future, (ii) the optimization of a quadratic performance
index during a finite period of time delimited for the control error Definition 2: A point, u°∈ U ⊂ \l, is weakly Pareto optimal iff
by the prediction horizon N and by Nu for the control action, i.e. there does not exist another point u∈U, such that J(u,α) < J(u°,α).
k + N −1 k + Nu In other words, a point is weakly Pareto optimal if there is
J =|| e( k + N ) ||S2 + ∑ || e(i) ||
i =k
2
Q + ∑ || Δu(i) ||
i =k
2
R (13) no other point that improves all of the objective functions simul-
taneously. Hence, Pareto optimal points are weakly Pareto optimal,
(where S = ST and Q = QT are positive semidefinite matrices and but weakly Pareto optimal points may not be Pareto optimal.
R = RT is a positive definite matrix; || mp ||2M = pT M pl , e(·) = r(·)-y(·), For any given problem, there may be an infinite number of Pareto
m
Δu(·) = u(·) – u(· -1). y ∈ \ , r ∈ \ and u ∈ \ are the output optimal points, which constitute the Pareto optimal set, i.e.
vector, reference vector and input vector, respectively) and ℘{u°∈U⊆\l | ¬∃u∈U,∀iJi (uα , ) < Ji (u°,α)} .(15)
, ) ≤ Ji (u°,α) ∧∃i : Ji (uα
(iii) the application of the receding horizon principle, i.e. all control
signals u(k) … u(k+Nu) are calculated and only u(k) is applied, in The Pareto optimal set ℘ ⊂ U has an image in the criterion
the next step the sequence is calculated again taking into account space ϑ, which is denoted here as ℘f and is called Pareto front.
new values for y and r and again, only the new u(k) is used and
so on. The solution of the unconstrained problem is given by Definition 3: For a given vector objective function J(u,α) and a
Pareto- optimal set℘, the Pareto front is defined as follows:
Δu(k ) = [I 0 " 0][R e + G T Q e G]−1 G T Q e ( r − f ) . (14)
℘f  {J ( ui , α ) ∈ ϑ ⊆ \ nf | ui ∈℘, ∀ i} , (16)
where Qe = diag (S, Q, …, Q), Qe = diag (R, …, R), G is a
Toeplitz matrix containing the step response coefficients of the An important point in the criterion space is the utopia point,
plant, r = [r (k ) " r( k + N )]T and f contains terms of the free which is defined in the following:
response of the predictor and depends on the used model.
Definition 4: A point, J° is a utopia point iff J i °= min J i (u, α)
One of the most important advantages of MPC is the ability to
handle explicitly constraints in the optimization process, but in this with respect to u for each i = 1,…, nf .

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 The set of Pareto-optimal solutions is also called non-inferior, However, the complexity due to the large number of objective
non-dominated, admissible, or efficient solutions. When the non- functions and temporal deadlines, within which the optimization
dominated vectors are collectively plotted in the criterion space, they must normally be accomplished, reduce the applicability of
constitute the Pareto front. Fig. 1 illustrates with a three/two dimen- classical game theory techniques to design optimal bargaining
models for decision making.
sional example all concepts introduced above.
u3 J2 C. Most Important MOO Methods
Criterion
space
At the present, a very huge number of methods to solve MOO
J:U→ϑ 2 problems can be found in the specialized literature. Broad reviews
Solution J1° ϑ ⊂ \ Threat can be found e.g. in [1] and in [43]. In this work, only the
space point c
U ⊂ \3 u1 most important methods, which have been used to solve
PID/MPC control problems, will be included for sake of space.
Utopia point J2° Methods to solve MOO problems can be classified according to
u2 J1 a wide spectrum of characteristics (see [44]). In Fig. 3, a classifica-
Pareto front
tion based on [14] is given. The two main groups are: (a)
Fig. 1. Evaluation mapping of the multiobjective problem Scalarization methods and (b) the Pareto methods.
Scalarization methods require the formation of an overarching
B. Decision Making and Control Performance objective function aggregating contributions from all components
All points of the Pareto front are equally acceptable solution of of the objective vector, normally by using coefficients, exponents,
the vector optimization problem. However, it is necessary to obtain constraint limits, etc. and then methods for single objective opti-
only one point in order to be able to implement the controller. This mization are used to find a unique solution. They are very efficient
selection is carried out by a decision maker. Although decision and fast to find a unique solution. On the other hand, they do not
making is a crucial aspect in the design, it seldom mentioned in the always give acceptable solutions because of interest conflicts
MOO control literature. However, decision making is a world of of design objectives. Moreover, these methods can converge to a
its own and therefore only a general idea is given here. local optimum and therefore they are not able to find the global
The decision making can be undertaken from two different solution. Finally, it is not always clear for the user, how to express
points of view: i) by including additional criteria such that at the the preferences for the scalarization process.
end only one point satisfies all of them, and ii) by considering one Pareto methods first find the optimal solutions space and then a
point that represents a fair compromise between all used criteria. unique optimal solution from the Pareto optimal set is chosen
The first option allows introducing additional criteria that could by a decision maker. Thus, these methods keep the elements of the
be oriented to improve the control performance. For example, if objective vector separate throughout the optimization process
the Pareto front corresponds to J = (JISE, JISC), eq. (19), the final and use the concept of dominance to distinguish between inferior
solution can be obtained for the point that leads to the minimum and non-inferior solutions. The advantage of Pareto methods con-
overshoot or the maximum rise time. It is also possible to start a sists in the fact that the preferences can be expressed once the
min-max optimization problem with domain in the Pareto set in optimization is already carried out, keeping the different objective
order to find the solution for the maximum rise time and the functions separately. Therefore, they are able to take care of all
minimum overshoot. In [64], it is selected the point whose parame- conflicting design objectives individually but compromising them
ters minimize the structured singular value μ such that the obtained concurrently. However, the search process requires a very high
controller is the most robust contained within the Pareto set. computation burden and the convergence can be very slow.
The second option does not introduce more information for the MOO
Methods
decision making and only a fair point for all indices is searched.
The ideal case would be to reach the utopia point. However, it is Scalarization Methods Non-Pareto, non-scalarization Pareto Methods
difficult to optimize all individual objective functions independently (a priori articulation of preference) Methods (a-posteriori articulation of preference)

and simultaneously. Moreover, the utopia point J° is normally WSA

NBI Non-evolutionary
Algorithms
Evolutionary
Algorithms
unattainable and it is not in ϑ. Thus, it is only possible to find a
(Normal Boundary Intersection)
(Weighted Sum Approach)
MOEA
solution that is as close as possible to the utopia point. Such GAM
(Goal Attainment Method)
(Multi-Objective Evolutionary Alg.

solution is called compromise solution (CS) and is Pareto optimal. Lexicographic Method
MOGA
(Multi-Objective Gen. Alg.)
Two more efficient procedures to select a fair point are coop- NSGA, NSGA-II
erative negotiation ([22]) and bargaining games ([61]). Solutions (Non-dominated Sorting Gen. Alg.)

of a bargaining game lead to some practical procedure for choos- ε-constraint Method SPEA, SPEA2
(Strengthen Pareto Evolutionary Alg.)
ing a unique point (see Fig. 2). For example (see [61]), the Nash VEGA NPGA
solution of the game (NS) corresponds to a point of the Pareto set (Vector Evaluating Gen. Alg.) (Niched Pareto Gen. Alg.)

which yields the largest rectangle (c, B, NS, A), the Kalai- Fig. 3. General classification of MOO solving methods.
Smorodinsky solution (KS) is situated at the intersection of the
Pareto front and the straight line, which connects the threat point Remark: In addition, it is possible to find methods without
and the utopia point, and the egalitarian solution (ES) yields articulation of preference and methods with progressive articula-
the point given by the intersection of the Pareto front and a 45°- tion of preference, also called iterative methods (see [1] for details).
line through the threat point. In general, it is difficult to recommend a particular method and
J2
the choice mostly depends on the application. As a rule, some
Criterion space ϑ ⊂ \2 authors ([43]) suggest to select first methods, which guarantee nec-
J2* 45°
B Threat essary and sufficient conditions for Pareto optimality. Then, meth-
point c ods those guarantee only sufficient conditions and finally other ones.
KS
CS
1) Weighted Sum Approach
NS Pareto
A
ES front This is probably the most widely used MOO method. It consists
J2° in assigning a non-negative weight γi to each of the i objective
J1° Utopia point J1* J1 functions, so that the overarching scalar objective function can be
Fig. 2. Different criteria for the decision making expressed as

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 J = γ T J (u, α ) , (17) Evolutionary algorithms can be applied to solve SOO problems
as well as MOO problems. The MOO case was first studied in
where γ = [γ1 "γ nf ] is the weight vector. The other variables were
T

[26]. Today, there are many MOEAs distinguished mainly by the

already defined in Subsection A. Weights are chosen in such a algorithms for the population ranking in the fitness assignment.
form that ∑| γ i | = 1. Moreover, the objective functions have to The most important are: MOGA (Multiple Objective Genetic
be normalized since not all objectives have the same range of Algorithm, [17]), NSGA-II (Non-dominated Sorting Genetic
values. The optimal solution is given by Algorithm, [15]), SPEA2 (Strength Pareto Evolutionary Algo-
J ° = J (u°, α) = min γ T J ( u, α) . (18) rithm, [67]), NPGA-II (Niched Pareto Genetic Algorithm, [16]).
u
The most important advantage of this method is the transforma- A very important advantage of the MOEAs is that they do not
tion of the vector objective function in a single-objective function, need information about the objective-functions derivatives and
such that traditional optimization methods can be used. The prob- they can solve non-convex problems. Moreover, the algorithms are
lem is the setting of the weights: On one hand, results are sensitive relatively robust and they do not require solving a sequence of
to weights ratio and they are difficult to be chosen. On the other single-objective problems and it is a parallel search technique.
hand, weights indicate the relative importance of the corresponding However, Pareto optimality is not a concept embedded in the
objective function but they do not mean priorities. Hence, if the fundaments of evolutionary algorithms. Consequently, it could be
optimization process cannot be completed for all objective that a Pareto optimal solution born and then, by chance, also it dies
functions, the method does not indicate, in which sequence out. An additional drawback is the intensive computational burden
objective functions may be discarded. Moreover, the method required until the solution is obtained.
presents difficulties in case of non-convex problems.
5) Other Algorithms
2) Goal Attainment Method ([23]) Finally, the NBI method (Normal Boundary Intersection) was
This method is formulated as follows: proposed in [12]. It is a very fast Pareto method that does not
find u, ξ belong to the family of evolutionary algorithms. The VEGA
to minimize ξ (Vector Evaluated Genetic Algorithm, [56]) is also an algorithm
subject to Ji(u,α) − γi ξ ≤ gi, for i = 1, …, nf, for MOO based on a genetic algorithm but it is neither a scalari-
where Ji(u,α) are objective functions, γi are weights indicating zation nor Pareto based algorithm.
the relative importance of each objective function, gi are the
goals, which have to be reached, and ξ is an unrestricted scalar. IV. AVAILABLE SOFTWARE IMPLEMENTATIONS
This method presents similar properties as the weighted sum
approach, where the most important difficulty is to select the In order to be able to implement multi-objective optimal
weights, which, in turn, do not correspond to priorities. control systems it is reasonable to ask about the state of the
art of the development of software for such task. A short review
3) Lexicographic method ([4]) about available software can be found in [44]. Updated informa-
In this method, objective functions are arranged in order tion is also given in the web (for example in [11] and [45]).
of importance by the user. The optimum solution u° is then Because the MATLABTM software system is a very common
obtained by minimizing the objective functions, starting with environment for control engineers and students and also for
the most important one and proceeding according to the order of the sake of space, this review is limited to code available for
importance assigned to the objectives. Thus, the following opti- this package. Available toolboxes for MOO can be divided
mization problems are solved one at a time: in two groups: commercial code and free code. Both will be
minimize Ji(u,α) respect to u∈U presented in the following two subsections without intending
subject to Jj(u,α) ≤ Jj(uj°,α), to be exhaustive because this field changes dynamically.
for j = 1, 2, …, i-1, i > 1, i = 1, 2, …, nf.
A. Commercial Code for MOO Problems
The subscript i represents here not only the objective function Matlab itself bring support for MOO by means of its Optimi-
number but also the priority assigned to the objective, and zation ToolboxTM ([5]). It includes the weighted sum method, the
Jj(uj°,α) represents the optimum of the jth objective function, ε-constraint method and the goal attainment method. Another
found in the jth iteration. Notice that after the first iteration
Jj(uj°,α) is not necessarily the same as the independent minimum general purpose development environment in Matlab for optimiza-
of Jj(u,α), because of the introduction of new constraints. tion problems is TOMLAB ([28]). Its support for MOO is given
through FORTRAN routines that are acceded by means of a
The advantage of this method is that there is only one optimum MEX interface. However, this software does not support Pareto
for a given lexicographic order and it is very easy to implement
because of the sequential optimization. Its disadvantage is that methods. Implemented methods are e.g. weighted sum, ε-con-
normally objectives with lower priorities will not be satisfied. straint, hierarchical optimization, min-max and global criterion.
Thus, priority assignment, which has to be done a priori, is crucial In [32], the software package MOPS (Multi-Objective Parameter
in order that the methods would be successful. Synthesis) is presented. The MOO support is based on min-max
MOO, which is solved by reformulating it as a non-linear pro-
4) Multiobjective Evolutionary Algorithms gramming problem. Matlab is also supported by a MEX interface.
Evolutionary optimization algorithms simulate the survival A comprehensive implementation of evolutionary algorithms
of the fittest in biological evolution by means of algorithms. The in Matlab is given by the GEA Toolbox (GEATbx, [51]),
renewal of a population (entire set of variables that represent where multi-objective ranking of MOGA ([17]) and goal attain-
a group of potential solution points) is based on the so called ment are completely implemented.
‘genetic operators’: recombination (out of two points of the popu-
lation picked out so that a new point is generated, e.g. by averag- B. Free Code for MOO Problems
ing), mutation (single, randomly selected digits of a newly gener-
ated point are substituted by a realization of a random variable) and Free Matlab code for MOO problems, which is not genetic-
selection (out of the union of the original population and the newly algorithm based, is difficult to find. In fact, only the NBI Toolbox
generated points, which are taken over into the new population ([12]) could be found by the author. However, this toolbox
with the best fitness). There are many algorithms to implement this needs the Optimization Toolbox and therefore it cannot be
functions (see [38] and [66] for comparative reviews). considered completely free.

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 On the contrary, several free toolboxes for MOO are available Finally, a PI controller is optimized in [37] by using special
for evolutionary algorithms. In [57], a small but complete objective functions (Weighted Integral Square Error, WISE,
package that implements the NSGA-II algorithm and some gain and phase margins), which are derived for first order
examples is available. Moreover, a complete package named plus delay time (FOPDT) models and integrator plus delay
SGALAB including many algorithms (SPEA2, NSGA-II, VEGA, time (IPDT) models. The MOO problem is solved by using
MOGA, etc.) is currently being developed. A beta version can be the Goal Attainment Method.
downloaded from [10]. The drawback of this software consists
in the fact that the Matlab source code is not available with B. Multi-objective Predictive Control
exception of the examples. In addition, a small package A MOO framework for MPC has been proposed in [36] and
based on NSGA is available in [53]. applied in [47]. The approach is based on a lexicographic algorithm
Finally, the MOEA toolbox described in [60] is no longer taking advantage about the fact that for this method, objective
functions can be ordered according to pre-established priorities. As
available either on the web or by writing to the author. example, three objective functions are given in addition to the
standard cost index J4 given by (13): the size of the largest
V. MULTI-OBJECTIVE CONTROL SYSTEM DESIGN constraint violation
The controller design based on the optimization of performance J 1 (g )  max{0; g1 ( u, x ); g 2 ( u, x ); " ; g ng ( u, x )} , (21)
indices (6) and (13) is actually the solution of a MOO problem
since (6) and (13) can be considered as weighted sums of objective where g models the constraints, x is the vector of state variables
functions. The a-priori selection of weights yields a unique and u the control vector, the weighted sum of constraint violations
solution like a method with a-priori articulation of preferences. If J 2 ( g )  g + ( u, x ) T M g + ( u , x ) + ν T g + ( u, x ) , (22)
the weighted sums are decomposed in its components, the vector
performance indices with g+(u, x)  max{0, gi (u, x)} , M > 0 and ν > 0, and the largest
T
⎡∞ ∞ ⎤ element in index set of violated constraints
J = ⎢ ∑ e 2 (k ) ∑ Δu (k )⎥⎥
2
and (19)
⎢⎣ k =1 k =1 ⎦ ⎧0 if g( u, x ) ≤ 0
J 3 (g )  ⎨ . (23)
⎩ max i {i | gi ( u, x ) > 0} otherwise
T
⎡ k + N −1 k + Nu
⎤
J = ⎢|| e(k + N ) ||S2 + ∑ || e(i ) ||Q2 ∑ || Δu(i ) ||2R ⎥ (20)
⎣ i=k i =k ⎦ This MOO approach for MPC contributes to improve the
feasibility of the algorithm since constraints can be relaxed
are obtained, revealing the multi-objective nature of the problem. according to on-line assignable priorities when their satisfaction is
Moreover, the MPC with constraints can also be analyzed as a not strictly necessary. The cost, which must be paid for this
MOO problem because constraints can normally be formulated advantage, is an increased computational burden. This topic has
as additional objective functions with priorities. The aim of this still to be studied. Other approaches of MOO-MPC are given
Section is to describe some works, where MOO techniques are in [7] and in [65] in order to implement a control system for
used for a particular goal. The summery is presented in Table I. nonlinear system; in [25] to obtain robustness and in [46] to
satisfy objectives with different priorities.
A. Multi-objective PID Control
Maybe one of the most important contributions in the field of VI. CONCLUDING REMARKS AND FINAL DISCUSSION
MOO-PID is presented in [50], where several classic measures in In this paper, a short overview about MOO methods and their
time domain (overshoot, rise time, settling time) together with application to PID and MPC is presented. In general, the
multi-objective formulation of control problems seems to be a
IAE index (Integral absolute error) are simultaneously optimized very attractive approach in order to improve control systems
by using an evolutionary algorithm. A similar approach is pre- in many directions and it appears as a promising methodology
sented in [30] but the solution is obtained by using a gradient for the near future. However, the state of the art of algorithms
MOO algorithm. In [59], the idea was just to eliminate weighting for MOO allows concluding that at the present time not all optimal
factors by using a two-objective optimization problem for the control problems can be formulated as a MOO problem: Multi-
objective parameter optimization problems can efficiently be
continuous time case. The solution is then found by using a genetic solved by the existing algorithms since the optimization is
algorithm. In [52], a similar approach for the continuous-time case habitually carried out off-line. This is, for example, the case of
but for an objective function based on |u| is proposed. parameter optimization of PID controllers.
TABLE I
SHORT SUMMERY ABOUT THE MOST IMPORTANT APPLICATION OF MOO IN CONTROL ENGINEERING
MOO-PID
REFERENCES AIM FOR USING MOO MOST IMPORTANT OBJECTIVE FUNCTIONS USED MOO REAL-TIME
[50] Stabilize the system + minim. performance measures Overshoot, rise time, settling time, IAE NSGA-II yes
[52] Control of a non-linear plant with a fixed-gain PID ISE + ISC SPEA2 yes
[30] Simultaneously satisfaction of many objectives Overshoot, rise time Gradient No
[37] Simultaneously satisfaction of many objectives Gain and phase margins, WISE Goal Attainment No
[59] Elimination of weighting factors ISE + ISDC MOGA yes
MOO-MPC
Duration of constraint relaxations, square norm of
[36], [47] Fault tolerant control Lexicographic No
deviations, size of largest constraints violation
[7] Alternative to a multi-model control scheme Eq. (13) for each neural network (nonlinear system) WARGA, NSGA No
[46] To satisfy objectives with different priorities Absolute values of deviations of outputs and control signals Not mentioned No
[65] Control of nonlinear systems with linear controllers Sum of square errors, sum square of Δu Goal Attainment No
[34] Solving a problem of application Energy consumption, filtration time of pulse jet fabric filters Goal Attainment No
[25] Robust control system Eq. (13) for each linearized model of the nonlinear system Weighted sum No

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