Course/Course Code:

Adaptive Control System

(EE-504)

Course Teacher:

Dr.Tariq Rehman
 A BRIEF HISTORY
• Early 1950s, the design of autopilots for high-performance aircraft
motivated intense research activity in adaptive control.
• 1958, 1961, Model reference adaptive control (MARC) was
suggested by Whitaker and coworkers in to solve the autopilot
control problem.
• 1958, An adaptive pole placement scheme based on the optimal
linear quadratic problem was suggested by Kalman.
• The lack of stability proofs and the lack of understanding of the
properties of the proposed adaptive control schemes coupled with a
disaster in a flight test caused the interest in adaptive control to
diminish.
• The 1960s became the most important period for the development
of control theory and adaptive control in particular. State-space
techniques and stability theory based on Lyapunov were
introduced.
• Developments in dynamic programming, dual control and stochastic
control, and system identification and parameter estimation played a
crucial role in the reformulation and redesign of adaptive control.
 A BRIEF HISTORY
• By 1966, Parks and others found a way of redesigning the MIT rule-
based adaptive laws used in the model reference adaptive control
(MRAC) schemes using the Lyapunov design approach.
• The advances in stability theory and the progress in control theory
in the 1960s improved the understanding of adaptive control and
contributed to a strong renewed interest in the field in the 1970s.
• On the other hand, the simultaneous development and progress in
computers and electronics contributed in the implementation of
complex controllers, such as adaptive control.
• In 1970s, several breakthrough results in the design of adaptive
control. The concepts of positivity were used to develop a wide
class of MRAC schemes with well-established stability properties.
• The excitement of the 1970s and the development of a wide class of
adaptive control schemes for discrete-time plants with well
established stability properties were accompanied by several
successful applications.
 A BRIEF HISTORY
• However, in early 1979, it was pointed out by Egardt that the
adaptive schemes of the 1970s could easily go unstable in the
presence of small disturbances.
• 1980s, The non-robust behavior of adaptive control became very
controversial when more examples of instabilities were published by
loannou et al. and Rohrs et al.
• Rohrs's example of instability stimulated a lot of interest, and the
objective of many researchers was directed towards understanding
the mechanism of instabilities and finding ways to counteract them.
• By the mid-1980s, several new redesigns and modifications were
proposed and analyzed, leading to a body of work known as robust
adaptive control.
• An adaptive controller is defined to be robust if it guarantees signal
boundedness in the presence of "reasonable" classes of un-
modeled dynamics and bounded disturbances.
• The work on robust adaptive control continued throughout the
1980s and involved the understanding of the various robustness
modifications and their unification under a more general framework.
 A BRIEF HISTORY
• In discrete time, Praly was the first to establish global stability in the
presence of unmodeled dynamics. By the end of the 1980s several
results were published in the area of adaptive control for linear
time-varying plants.
• The focus of adaptive control research in the late 1980s to early
1990s was on performance properties and on extending the results
of the 1980s to certain classes of nonlinear plants with unknown
parameters.
• These efforts led to new classes of adaptive schemes, motivated
from nonlinear system theory as well as to adaptive control schemes
with improved transient and steady-state performance.
• New concepts such as adaptive backstepping, nonlinear damping,
and tuning functions are used to address the more complex
problem of dealing with parametric uncertainty in classes of
nonlinear systems.
• In the late 1980s to early 1990s, the use of Neural Networks as
universal approximators of unknown nonlinear functions led to the
use of online parameter estimators to "train" or update the weights
of the neural networks.
INTRODUCTION
• Adapt means to "change (oneself) so that one's behavior
will conform to new or changed circumstances."

• The design of autopilots for high-performance aircraft was one

of the primary motivations for active research in adaptive
control in the early 1950s.

• Adaptive Control is the combination of a parameter

estimator, which generates parameter estimates online,
• with a control law in order to control classes of plants,
whose parameters are completely unknown and/or could
change with time in an unpredictable manner.
INTRODUCTION
• The controller structure consists of a feedback loop and a
controller with adjustable gains, as shown in following Figure.

General adaptive control structure for aircraft control.

CLASSIFICATION OF ADAPTIVE CONTROL
1. Identifier-based Adaptive Control:

The choice of the parameter estimator, control law, and the

way they are combined leads to different classes of adaptive
control schemes.

2. Non-identifier-based Adaptive Control:

The similar control problems are solved without the use of

an online parameter estimator.
 ADAPTIVE CONTROL: IDENTIFIER-BASED

• The class of adaptive control schemes characterized by the

combination of an online parameter estimator, with a control
law.

• The way the parameter estimator, also referred to as adaptive

law, is combined with the control law gives rise to two
different approaches:
 ADAPTIVE CONTROL: IDENTIFIER-BASED
1. Indirect Adaptive Control:
• The plant parameters are estimated online and used to
calculate the controller parameters.
• In other words, at each time t, the estimated plant is formed
and treated as if it is the true plant in calculating the controller
parameters.
• This approach has also been referred to as explicit adaptive
control, because the controller design is based on an explicit
plant model.

Indirect adaptive control structure

 ADAPTIVE CONTROL: IDENTIFIER-BASED
2. Direct Adaptive Control:
• The plant model is parameterized in terms of the desired
controller parameters, which are then estimated directly
without intermediate calculations involving plant parameter
estimates.
• This approach has also been referred to as implicit adaptive
control because the design is based on the estimation of an
implicit plant model.

Direct adaptive control structure

 ADAPTIVE CONTROL: IDENTIFIER-BASED
• “Parameterize" means "to express in terms of parameters".

• Parametrization is a mathematical process consisting of

expressing the state of a system, process or model as a
function of some independent quantities called parameters.

• The basic structure of indirect adaptive control is shown in

Figure. The plant model G(*) is parameterized with respect
to some unknown parameter vector *.
ADAPTIVE CONTROL: IDENTIFIER-BASED
Applications:
• In general, direct adaptive control is applicable to SISO linear
plants, since for this class of plants the parameterization of
the plant with respect to the controller parameters for some
controller structures is possible.

• Indirect adaptive control can be applied to a wider class of

plants with different controller structures, but it suffers from
stabilizability problem explained as follows:
• The controller parameters are calculated at each time, t based on
the estimated plant. Such calculations are possible, provided that
the estimated plant is controllable and observable or at least
stabilizable and detectable.
• Since these properties cannot be guaranteed by the online
estimator in general, the calculation of the controller parameters
may not be possible at some points in time, or it may lead to
unacceptable large controller gains.
 ADAPTIVE CONTROL: IDENTIFIER-BASED
• Solutions to this stabilizability problem are possible at the
expense of additional complexity.

• Efforts to relax the minimum-phase assumption in direct

adaptive control and resolve the stabilizability problem in
indirect adaptive control led to adaptive control schemes

• Where, both the controller and plant parameters are

estimated online, leading to combined direct/indirect
schemes that are usually more complex .
 ADAPTIVE CONTROL: NON-IDENTIFIER-BASED
• Another class of schemes that do not involve online parameter
estimators is referred to as non-identifier-based adaptive
control schemes.

• In this class of schemes, the online parameter estimator is

replaced with search methods for finding the controller
parameters in the space of possible parameters,

• Or, it involves switching between different fixed controllers,

assuming that atleast one is stabilizing

• Or, uses multiple fixed models for the plant, covering all
possible parametric uncertainties

• Or, consists of a combination of these methods.

 ADAPTIVE CONTROL: NON-IDENTIFIER-BASED
1. Gain Scheduling
• The gain scheduler consists of a lookup table and the
appropriate logic for detecting the operating point and
choosing the corresponding value of control gains from the
lookup table.

• With this approach, plant parameter variations can be

compensated by changing the controller gains as functions of
the input, output, and auxiliary measurements.

Gain scheduling structure

 ADAPTIVE CONTROL: NON-IDENTIFIER-BASED
1. Gain Scheduling
• The advantage of gain scheduling is that the controller gains
can be changed as quickly as the auxiliary measurements
respond to parameter changes.

• However, frequent and rapid changes of the controller

gains may lead to instability;

• Therefore, there is a limit to how often and how fast the

controller gains can be changed.
 ADAPTIVE CONTROL: NON-IDENTIFIER-BASED
1. Gain Scheduling
• One of the disadvantages of gain scheduling is that the
adjustment mechanism of the controller gains is precomputed
offline and provides no feedback to compensate for incorrect
schedules.

• A careful design of the controllers at each operating point to

meet certain robustness and performance measures can
accommodate some uncertainties in the values of the plant
parameters.

• However large unpredictable changes in the plant parameters,

may lead to deterioration of performance or even to complete
failure.
 ADAPTIVE CONTROL: NON-IDENTIFIER-BASED
1. Gain Scheduling
• Despite its limitations, gain scheduling is a popular method for
handling parameter variations in flight control and other
systems.

• While gain scheduling falls into the generic definition of

adaptive control, we do not classify it as adaptive control due
to the lack of online parameter estimation, which could track
unpredictable changes in the plant parameters.
 ADAPTIVE CONTROL: NON-IDENTIFIER-BASED
2. Multiple Models, Search Methods, and Switching Schemes

• A class of non-identifier-based adaptive control schemes

emerged over the years which do not explicitly rely on online
parameter estimation.

• These schemes are based on search methods in the controller

parameter space until the stabilizing controller is found or the
search method is restricted to a finite set of controllers, one of
which is assumed to be stabilizing.

• In some approaches, after a satisfactory controller is found it

can be tuned locally using online parameter estimation for
better performance.
 ADAPTIVE CONTROL: NON-IDENTIFIER-BASED
2. Multiple Models, Search Methods, and Switching Schemes

Multiple models adaptive control with switching

 ADAPTIVE CONTROL: NON-IDENTIFIER-BASED
2. Multiple Models, Search Methods, and Switching Schemes

• Since the plant parameters are unknown, the parameter

space is parameterized with respect to a set of plant models,
which is used to design a finite set of controllers so that each
plant model from the set can be stabilized by at least one
controller from the controller set.

• A switching approach is then developed so that the stabilizing

controller is selected online based on the I/O data
measurements.
 WHY ADAPTIVE CONTROL
• The choice of adaptive control as a solution to a particular
control problem involves understanding of the plant
properties as well as of the performance requirements.

• The following simple example illustrates situation where

adaptive control is superior to linear control.
• Consider the scalar plant:
𝑥=
ሶ 𝑎𝑥 + 𝑢
• where, u is the control input and x the scalar state of the
plant. The parameter a is unknown.
• We want to choose the input u so that the state x is bounded
and driven to zero with time.
• If a is a known parameter, then the following linear control
law can meet the control objective.
𝑢 = −𝑘𝑥 𝑘> 𝑎
 WHY ADAPTIVE CONTROL
𝑥ሶ = 𝑎𝑥 + 𝑢 𝑎ഥ ≥ 𝑎
𝑥 → 0 as 𝑡 → ∞
𝑢 = −𝑘𝑥
𝑘 > 𝑎ഥ
• In the absence of an upper bound for the plant parameter no
linear controller could stabilize the plant and drive the state to
zero.
• The adaptive control law guarantees that all signals are
bounded and x converges to zero no matter what the value of
the parameter a is.
𝑢 = −𝑘𝑥 ሶ
𝑘=𝑥 2

• This simple example demonstrates that adaptive control is a

potential approach to use in situations where linear controllers
cannot handle the parametric uncertainty.
thank you !!!