Linear Control Systems: With solved problems and MATLAB examples

384 Pages·2001·8.33 MB·English

by  Branislav Kisačanin Ph.D.

Linear Control Systems
THE UNIVERSITY SERIES IN MATHEMATICS
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AN INTRODUCTION TO ALGEBRAIC NUMBER THEORY
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Linear Control Systems
with solved problems and
MATLAB examples
Branislav Kisacanin, Ph.D.
Delphi Delco Electronics Systems
Kokomo, Indiana
Gyan C. Agarwal, Ph.D.
University of Illinois at Chicago
Chicago, Illinois
Springer Science+Business Media, LLC
Library of Congress Cataloging-in-Publication Data
Kisacanin, Branislav, 1968-
Linear control systems: with solved problems and MATLAB exampleslby Branislav
Kisacanin, Gyan C. Agarwal.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-4613-5129-0 ISBN 978-1-4615-0553-2 (eBook)
DOI 10.1007/978-1-4615-0553-2
1. Linear control systems. 2. Linear control systems-Problems, exercises, etc. 3.
MATLAB. 1. Agarwal, Gyan C. II. Title.
TJ220 .K57 2002
629.8'32-dc21
2001053984
ISBN 978-1-4613-5129-0
©2001 Springer Seience+Business Media New York
OriginalIy published by Kluwer Academic/Plenum Publishers, New York in 2001
Softcover reprint of the hardcover 1 st edition 2001
http://www.wkap.nl/
10 9 8 7 6 543 2 1
A C.I.P. record for this book is available from the Library of Congress
AII rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted in any
form or by any means, electronic, mechanical, photocopying, microfilming, recording, or
otherwise, without written permission from the Publisher
To Baska €3 To Badhna
Contents
Foreword ix
Preface xi
I Theory of linear control systems 1
1 Historical overview of automatic control 3
1.1 Automatic control before the 1930's . 5
1.2 Classical period of automatic control . 12
1.3 Beginnings of modern control theory 18
2 Modern control theory 23
2.1 State-space representation .. 25
2.2 System properties ...... . 34
2.3 State feedback and optimal control 62
2.4 State observers and estimators .. 68
II Solved problems 71
3 Continuous linear systems 73
3.1 Simple differential equations . . . . . . . 75
3.2 Matrix theory . . . . . . . . . . . . . . . 83
3.3 Systems of linear differential equations . 95
3.4 Input-output representation 98
3.5 State-space representation . . . . 113
3.6 Stability.............. 128
3.7 Controllability and observability 134
3.8 Canonical realizations 152
3.9 State feedback . 166
3.10 Optimal control. . . 178
3.11 State observers . . . 186
3.12 Kalman-Bucy filter . 194
3.13 Reduced-order observers 199
vii
viii CONTENTS
4 Discrete linear systems 207
4.1 Simple difference equations ..... . 209
4.2 More matrix theory ......... . 217
4.3 Systems of linear difference equations . 220
404 Input-output representation 222
4.5 State-space representation . . . . 235
4.6 Stability.............. 239
4.7 Controllability and observability 244
4.8 Canonical realizations 251
4.9 State feedback . 255
4.10 Optimal control. 259
4.11 State observers . 263
4.12 Kalman filter .. 265
4.13 Reduced-order observers 272
5 Exercise problems 273
III Appendixes 287
A A quick introduction to MATLAB 289
A.l Introduction. . . . . . . 289
A.2 Basic matrix operations 290
A.3 Plotting graphs . . . . . 294
Ao4 Data analysis . . . . . . 295
A.5 Data management and I/O operations 296
A.6 Exercises ............... . 296
B Mathematical preliminaries 305
B.l Introduction ........ . 305
B.2 Differential and difference equations 306
B.3 Laplace and z-transforms . 312
Bo4 Matrices and determinants. . . . . . 318
C Results from advanced matrix theory 325
C.l Eigenvectors and eigenvalues 325
C.2 Diagonal and Jordan forms ... . 330
C.3 Similarity of matrices ...... . 334
Co4 Symmetric and Hermitian matrices 340
C.5 Quadratic forms and definiteness . 345
C.6 Some special matrices .... . . , 353
C.7 Rank, pseudoinverses, SVD, and norms 355
C.8 Problems ................ . 365
Bibliography 371
Index 375
Foreword
In science and engineering, a proper way to master theory is to solve relevant
and meaningful problems that provide a bridge between theory and applications.
Problem solving is necessary not only as a stepping stone towards the design
of real systems and experiments, but also to reveal the scope, flexibility, and
depth of the theoretical tools available to the designer. In this book, the authors
present an excellent choice and a lucid formulation of a wide variety of problems
in control engineering. In particular, their constant reliance on MATLAB in the
problem-solving process is commendable, as this computational tool has become
a standard and globally available control design environment.
The chapter on theoretical elements of control theory, which precedes the
problem-solution part of the book, sets a proper background for problem-solving
tasks. In their presentation, the authors struck the right balance in achieving a
self-contained text without overwhelming the reader with detailed and exhaus
tive theoretical arguments.
Finally, the opening chapter on the history of automatic control is a welcome
part of the book. The authors admirably describe the events and concepts that
have evolved over the centuries to the present-day control theory and technol
ogy. Theoretical analysis and problem-solving processes are not only useful in
understanding what the world is, but also what it can become. In attempting
to predict future development in science and technology we are greatly aided .by
the history of scientific innovations and discoveries.
The present book is an excellent piece of work and should be on the shelf of
every student and practicing engineer of automatic control.
May 2001 Dragoslav D. Siljak
Santa Clara, California
ix
Preface
This book is a self-contained exposition of the theory of linear control systems
and the underlying mathematical apparatus. It has more than 250 solved prob
lems, numerous illustrative examples, and over 70 figures and diagrams. In
addition, MATLAB examples provide a good introduction to this powerful de
sign and simulation tool. A historical overview of the original ideas in control
theory is provided to describe the evolution of the theory from its early stages
of development.
For whom is this book?
It was written for students and engineers interested in Control Systems and Sig
nal Processing, typically the first-year graduate students of Engineering. There
is more than one way this book can be used: it contains a sufficient amount of
theory to allow its use as a textbook, it has many solved problems so it can be
a good supplement to a more advanced text, and it has enough of both to be
used as a self-study guide. A suitable choice of material can be made to fit the
format of the course and preferences of the instructor and the audience.
What is in the book?
The book has three major parts:
Part I is an overview of the history and theory of the subject. It consists of
Chapters 1 and 2:
• In Chapter 1 we investigate the historical development of the automatic
feedback control from the Antiquity to the present day.
• In Chapter 2 we present the fundamental concepts of control theory. We
start from the representation of systems using the states and continue
with a discussion of the most important system properties: stability, con
trollability, and observability. We end that Chapter by discussing the
most important design techniques: state feedback, optimal control, state
observation, and state estimation (Kalman filtering).
xi
xii PREFACE
Part II consists of Chapters 3, 4, and 5, which contain solved problems
categorized by the basic type of systems (continuous and discrete) and by the
topic (system representations, system properties, and design techniques):
• In Chapter 3 the reader will find solved problems about continuous-time
systems. It starts with simple differential equations and some matrix the
ory and continues with matrix representation of simultaneous differential
equations and input-output representation of control systems. Then it
moves on to problems of state representation of systems, stability, con
trollability and observability, canonical forms, and finishes with design
techniques for state feedback, optimal control, and state estimation and
observation.
• Chapter 4 is very similar in format to Chapter 3 and contains solved prob
lems on discrete-time systems. It starts with simple difference equations
and input-output representation of discrete-time systems. Then, just like
Chapter 3, it discusses problems of state representation, system properties,
and design techniques.
• Chapter 5 contains several exercise problems.
Part III consists of three Appendixes:
• Appendix A is a quick introduction to the basic syntax and functionality of
MATLAB, a powerful numerical and simulation tool for many engineering
disciplines, including Control Theory.
• In Appendix B we review the mathematical tools and notation used in
the book: differential and difference equations, Laplace and z transforms,
and matrices and determinants. To make the exposition more interesting
we also trace the origins of these branches of mathematics.
• Appendix C is a compilation of the basic notions and results in matrix
theory, the most important mathematical tool used in this book. It covers
similarity of matrices, important classes of matrices and their proper
ties, and some important techniques such as singular value decomposition
(SVD). Most of the results are given with proofs and some are illustrated
by examples.
To make the book easier to read
To help the reader we included a detailed Index at the end of the book and
used standard textbook conventions. The asterisk in a subsection name de
notes advanced material that can be skipped during the first reading. When
we reference a book or an article, we use a bracketed number, e.g., [22). All
references are listed alphabetically in the Bibliography section towards the end
of the book. All definitions, theorems, problems, examples, and exercises are