Loosely speaking, adaptive
systems are designed to deal with,
changing environmental conditions whilst maintaining performance
objectives. Over the years, the theory of
adaptive systems evolved from relatively
simple and intuitive concepts to a complex multifaceted theory
dealing with stochastic, nonlinear and infinite dimensional systems.
This book provides a first introduction to the theory of adaptive
The book grew out of a graduate course that the authors taught
several times in Australia, Belgium, and The Netherlands for students
with an engineering and/or mathematics background. When we
taught the course for the first time, we felt that there was a need
for a textbook that would
introduce the reader to the main
aspects of adaptation with emphasis on clarity of presentation and
precision rather than on comprehensiveness. The present book
tries to serve this need.
We expect that the reader will have taken a basic course in
linear algebra and multivariable calculus. Apart from the basic
concepts borrowed from these areas of mathematics,
the book is intended to be self contained.
The main body is formed by the treatment
of three major streams in the theory of deterministic
adaptive control systems:
adaptive pole placement (Chapter 4), model
reference adaptive control (Chapter 5), and the
theory of universal controllers (Chapter 6).
Clearly, this does not
cover the whole area of adaptive control;
most notably optimal control and stochastic systems are missing.
However, we wanted to present an introductory text, suitable to
be covered in a one trimester course at the graduate level.
Alternatively part of the book is well suited for a more
modest yet advanced undergraduate level course.
Apart from Chapter
6, dealing with universal controllers,
the text treats the subject
in a discrete time setting.
Even in the approach of the three major subjects, we have not pursued
generality. We strongly feel
that the intended audience is better served with a fairly
complete and precise treatment of more or less simple algorithms
rather than being subjected to an as-general-as-possible
treatment. Even if we had tried, we would inevitably have produced
a less transparent exposition. Anyway, encyclopedic treatments of adaptive
control are already available in the literature.
As a general point of view we have
chosen to consider situations that are as simple as possible, yet sufficient
to illustrate the main issues of adaptation. Upon completing the
book, it is our hope that the reader
will find it relatively easy to access the more specialized
Although the selection of material reflects tradition and the historical
development of adaptive systems theory, our presentation of the ideas does
not. The different adaptive algorithms are approached from a unified
plays a key
role in gaining an understanding of adaptive algorithms. In essence
the same technique, equilibrium analysis followed by transient analysis,
is used for all algorithms presented in the text. The equilibrium
or no-adaptation analysis is truly a
throughout the text.
Whereas the main part of the book is concerned with the three subjects
that we just mentioned, we have included several chapters on both
prerequisites and extensions. The prerequisites consist of a
chapter on representations of linear deterministic systems (Chapter
2) and a
chapter on identification (Chapter 3).
Representations play an important role in systems theory in general
and in adaptive control in particular. The classical transfer
function approach is in our view not quite the appropriate tool
since transfer functions cannot take into account autonomous parts,
or, almost equivalently, the initial conditions of a system.
Therefore we have chosen to use elements of the
we benefit from the theory of equivalent representations
and elimination of auxiliary variables within this framework, both
of which are of great importance in the exposition of adaptive system
We have hesitated as to whether this material should be the
contents of a chapter or should be part of the appendix. Since we
make essential use of the behavioral approach at several points,
and since it is still not quite standard theory, we finally felt
that we should provide it right after the introductory first
The chapter on identification introduces the reader to the
basic gradient or steepest-descent-like algorithms that
minimize some prediction error criteria,
such as recursive least squares, normalized least mean square,
and projection. The treatment is kept to a minimum,
sufficient to introduce the
adaptive algorithms. The properties of the identification
algorithms are studied with as few
restrictions imposed on the input signals as feasible,
or, better stated, we derive
as many properties as possible without assuming anything about the
nature of the inputs at all. The rationale is, of course, that in
adaptive control systems, the input is generated in a highly
complicated, practically unpredictable way so that it would be
difficult to guarantee particular conditions on the input signal.
Chapter 7 revisits the basic algorithms and addresses
the validity of one of the underlying assumptions in the adaptive
system analysis, namely that the control problem is well posed
during adaptation. It is shown how the well-posedness can be ensured
at the expense of more complicated adaptive rules.
Chapter 8 presents
averaging techniques for the analysis of adaptive systems.
This way we aim to address concerns about another important
simplifying assumption in the analysis of adaptive systems,
that the plant belongs to the model class.
Averaging techniques are a powerful tool in the analysis
of the performance of adaptive systems. The basic concepts introduced
should enable the reader to gain some insight into the complexity
of adaptive systems and get some intuition about the behavior of
adaptive systems in general. No prior knowledge of averaging techniques
is required. The minimal tools are developed and immediately illustrated
in the context of some particular examples of adaptive systems.
It transpires from the examples how the averaging analysis actually
leads to design guidelines for adaptive systems.
Chapter 9 introduces the idea of a global
dynamical analysis of adaptive systems operating in a plant-
model mismatch situation. This topic is by necessity treated
Finally, we have referred some of the technical and preliminary
material to the appendix.
Every chapter, except Chapter 2 and the appendix,
ends with a list of exercises. Some of them are fairly
straightforward; others require more work and serve
to extend the material covered. We have also included
some simulation exercises to illustrate the theory and to allow one
to venture beyond the theory.
The book is clearly intended as a text book for teaching. It can be
used at different levels. Suggestions of
sensible selections, for a short course, are:
Chapters 1, 2, 3,
4, 5, and
6 for a fairly complete coverage of the basic
theory. Chapters 1, 2, 3,
4, 5, 6, and 7
for a slightly extended course. Adding Chapters 8 and
9 would probably not be feasible for a one trimester
course. However, skipping one or two of the Chapters
4, 5, 6, and
7 would solve
that problem, although skipping both 4 and
5 is strongly dissuaded.
We have covered most of the book, say more than eighty percent, in
a sixteen hour lecture course. The book can be covered
in its entirety in about twenty class room hours.
In our course we actively involve the students by requiring them
to work out substantial
exercises in take home exams. Most of these exams
have found their way into the
As a virtual appendix to the book we provide a homepage,
containing additional exercises, simulation problems with Matlab
and Mathematica or Maple codes and a regularly updated list of
errata. Also, readers are invited to leave comments, remarks and
the like. Solutions to all exercises will be made available through a
protected part of the homepage; access will be granted on request
to teachers only.
Finally, it is our sincere wish that the reader, both the
practitioner and the theoretician, will find the book useful as an
introduction to the intriguing field of adaptive systems.
audience that we have in mind, graduate students as well as
undergraduates, enjoy reading the book, we have achieved our goal.
We hope that the learning experience will be a pleasant
It would be a delight to know that we enticed a few to venture
beyond the known.
Table of contents
DISC Course 1996
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