Introduction to Adaptive Control

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Chapter 1

• Introduction to Adaptive Control
• Adaptive Control Identifier-Based
• Adaptive Control NonIdentifier-Based
• Gain Scheduling
• Why Adaptive Control
• A Brief History

2
Introduction

• Adapt means to "change (oneself) so that one's
  behavior will conform to new or changed
  circumstances."
• The words adaptive systems and adaptive control
  have been used as early as 1950.
• 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.

3
Introduction

• 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.
• 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.
• The design of autopilots for high-performance
  aircraft was one of the primary motivations for
  active research in adaptive control in the early
  1950s.

4
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.
5
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 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 .
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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 changes in the plant parameters.
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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
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 uncertainty.
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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
  introduced.

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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.

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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 small
  disturbances.

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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 with
  improved transient and steady-state performance.

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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