Chapter 1
Introduction to Adaptive Control
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Adaptive Control: Identifier-Based
Adaptive Control: Non–Identifier-Based
Gain Scheduling
Why Adaptive Control
A Brief History
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Introduction
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Adapt means to "change (oneself) so that one's behavior will
conform to new or changed circumstances."
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The words adaptive systems and adaptive control have been used
as early as 1950.
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We use the following specific definition of adaptive control:
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.
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Introduction
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The choice of the parameter estimator, the choice of the control
law, and the way they are combined leads to different classes of
adaptive control schemes.
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Adaptive control as defined above has also been referred to as
identifier-based adaptive control in order to distinguish it from
other approaches referred to as non-identifier-based, where
similar control problems are solved without the use of an online
parameter estimator.
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The design of autopilots for high-performance aircraft was one of
the primary motivations for active research in adaptive control in
the early 1950s.
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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.
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Adaptive Control: Identifier-Based
The class of adaptive control schemes studied in this course is
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:
1- In the first approach, referred to as 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.
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Adaptive Control: Identifier-Based
2- In the second approach, referred to as 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.
The basic structure of indirect adaptive control is shown in
following Figure. The plant model G(*) is parameterized with
respect to some unknown parameter vector *.
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Adaptive Control: Identifier-Based
Indirect adaptive control structure.
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Adaptive Control: Identifier-Based
Direct adaptive control structure.
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Adaptive Control: Identifier-Based
In general, direct adaptive control is applicable to SISO linear plants
which are minimum phase, 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 a problem known as the 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.
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Adaptive Control: Identifier-Based
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.
So, solutions to this stabilizability problem are possible at the
expense of additional complexity. Efforts to relax the minimumphase 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 .
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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 at least one is
stabilizing or uses multiple fixed models for the plant covering all
possible parametric uncertainties or consists of a combination of
these methods.
We briefly describe the main features, advantages, and limitations
of these non-identifier-based adaptive control schemes. Some of
these approaches are relatively recent and research is still going on.
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Adaptive Control: Non-Identifier-Based
• 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. The advantage of gain scheduling is that
the controller gains can be changed as quickly as the auxiliary
measurements respond to parameter changes. Frequent and rapid
changes of the controller gains, however, may lead to instability;
therefore, there is a limit to how often and how fast the controller
gains can be changed.
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Adaptive Control: Non-Identifier-Based
• Gain Scheduling
Gain scheduling structure.
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Adaptive Control: Non-Identifier-Based
• 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.
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
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changes in the plant parameters.
Adaptive Control: Non-Identifier-Based
• 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.
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Adaptive Control: Non-Identifier-Based
• 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.
Without going into specific details, the general structure of this
multiple model adaptive control with switching, as it is often called,
is shown in next Figure.
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Adaptive Control: Non-Identifier-Based
Multiple models adaptive control with switching
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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
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.
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Why Adaptive Control
x  0 as t  
In the absence of an upper bound for the plant parameter no linear
controller could stabilize the plant and drive the state to zero.
As we will establish later , 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. This simple example
demonstrates that adaptive control is a potential approach to use in
situations where linear controllers cannot handle the parametric
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uncertainty.
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 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
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introduced.
A Brief History
• 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.
• 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 that made the implementation of
complex controllers, such as the adaptive ones, feasible contributed
to an increased interest in applications of adaptive control. 21
A Brief History
• The 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.
• At the same time several classes of adaptive control schemes
produced for discrete-time plants.
• The excitement of the 1970s and the development of a wide class of
adaptive control schemes with well established stability properties
were accompanied by several successful applications.
• The successes of the 1970s, however, were soon followed by
controversies over the practicality of adaptive control.
• As early as 1979 it was pointed out by Egardt that the adaptive
schemes of the 1970s could easily go unstable in the presence of
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small disturbances.
A Brief History
• 1980s, The nonrobust 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 unmodeled
dynamics and bounded disturbances
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A Brief History
• 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.
• 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 unknow
parameters.
• These efforts led to new classes of adaptive schemes, motivated
from nonlinear system theory as well as to adaptive control schemes
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with improved transient and steady-state performance.
A Brief History
• 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.
• Adaptive control has a rich literature full of different techniques for
design, analysis, performance, and applications. Several survey
papers and books and thesis have already been published.
• Despite the vast literature on the subject, there is still a general
feeling that adaptive control is a collection of unrelated technical
tools and tricks.
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THE END
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