Table Of ContentControl Engineering
Series Editor
William S. Levine
University of Maryland
Editorial Advisory Board
Okko Bosgra William Powers
Delft University Ford Motor Company
The Netherlands USA
Graham Goodwin Mark Spong
University of Newcastle University of Illinois
Australia Urbana-Champaign
USA
Petar Kokotovic
University of California fori Hashimoto
Santa Barbara Kyoto University
USA Kyoto, Japan
Manfred Morari
ETH
ZOrich, Switzerland
Sergey Edward Lyshevski
Control Systems Theory with
Engineering Applications
With 169 Figures and a CD-ROM
Springer Science+Business Media, LLC
Sergey Edward Lyshevski
Department of Electrical and Computer Engineering
Purdue University at Indianapolis
Indianapolis. IN 46202-5132
USA
Library of Congress Cataloging-in-Publication Data
Lyshevski. Sergey Edward.
Control systems theory with engineering applications / Sergey Edward Lyshevski.
p. cm.
Includes bibliographical references and index.
Additional material to this book can be downloaded from http://extras.springer.com.
ISBN 978-1-4612-6636-5 ISBN 978-1-4612-0153-3 (eBook)
DOI 10.1007/978-1-4612-0153-3
1. Automatic control. 2. Control theory. 1. Title.
TJ213.L942000
629.8-dc21 00-060868
Printed on acid-free paper.
© 2001 Springer Science+Business Media New York
Originally published by Birkhlluser Boston in 2001
Softcover reprint ofthe hardcover Ist edition 2001
Ali rights reserved. This work may not be translated or copied in whole or in part without the written
permission ofthe publisher (Springer Science+Business Media, LLC), except for brief excerpts in
connection with reviews or scholarly analysis. Use in connection with any form of information sto
rage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology
now known or hereafter developed is forbidden. The use of general descriptive names, trade names,
trademarks, etc., in this publication, even ifthe former are not especially identified, is not to be taken
as a sign that such names, as understood by the Trade Marks and Merchandise Act, may accordingly
be used freely by anyone.
ISBN 978-1-4612-6636-5 SPIN 10772413
Production managed by Louise Farkas; manufacturing supervised by Jerome Basma.
Typeset by Archetype Publishing, Inc., Monticello, IL.
987654321
Contents
Preface vii
Acknowledgments xi
1. Introduction: Modeling, Identification, Optimization,
and Control 1
2. Mathematical Model Developments 7
2.1. Engineering Systems and Models 7
2.2. Basic Principles in Model Developments . 14
2.2.1. Newtonian Mechanics. . . . . . . 15
3. Modeling of Dynamic Systems using MATLAB and SIMULINK 37
3.1. Engineering Computations Using MATLAB . . . . . . . . . 37
3.2. Analysis and Modeling of Dynamic Systems Using MATLAB 44
4. Analysis and Control of Linear Dynamic Systems 97
4.1. Introduction: Analysis of Multivariable Continuous- and
Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . . .. 97
4.2. Continuous-Time Dynamic Systems and Analog Controllers. . 104
4.2.1. Control Bounds . . . . . . . . . . . . . . . . . . . . . 116
4.3. Control of Dynamic Systems Using Digital PID Control Laws. 123
4.4. Hamilton-Jacobi and Lyapunov Methods in Optimal Control of
Continuous-Time Systems . . . . . . . . . . . . . . . . . . . .. 142
4.4.1. The Hamilton-Jacobi Theory and Optimal Control 142
4.4.2. The Lyapunov Theory and Optimal Control . . . . . 179
4.5. Pole Placement Design by Using State Feedback. . . . . . 188
4.5.1. Control of a Servo-System with Permanent-Magnet
DC Motor . . . . . . . . . . . . . . . . . . . . . . . . . . 192
4.6. Control of Discrete-Time Systems Using the
Hamilton-Jacobi Theory . . . . . . . . . . . . . . . . . . . . . . 204
4.6.1. Linear Quadratic Regulator Problem .. . . . . . . 204
4.6.2. Constrained Optimization of Discrete-Time Systems 209
4.6.3. Tracking Control of Discrete-Time Systems. . . . . 212
v
VI Contents
5. Analysis, Identification, and Control of
Nonlinear Dynamic Systems 219
5.1. Nonlinear Analysis of Dynamic Systems . 219
5.2. State-Space Nonlinear Identification . . . 227
5.2.1. Least-Squares Identification. . . . 229
5.2.2. Time-Domain, Nonlinear, Mapping-Based Identification 251
5.3. Design of Stabilizing Controllers Using the Lyapunov Theory . 262
5.3.1. Lyapunov Stability Theory and Design of Control Laws 262
5.3.2. Design of the Constrained Controllers for Uncertain
Nonlinear Systems Using the Lyapunov Stability Theory 273
5.4. Optimization of Continuous-Time Systems. . . . . . . . . 288
5.4.1. Optimization of Time-Invariant Systems ...... 288
5.4.2. Constrained Optimization of Time-Varying Systems 313
5.4.3. Optimal Control of Nonlinear Continuous-Time
Systems: Design of Bounded Controllers Via
Generalized Nonquadratic Functionals ............ 326
5.4.4. Tracking Control of Continuous-Time Systems . . . . . 330
5.4.5. Robust Control of Nonlinear Continuous-Time Systems 337
5.4.6. Robust Tracking Control of Nonlinear Systems 342
5.5. Sliding Mode Control ...... 347
5.6. Control of Discrete-Time Systems 358
5.6.1. Tracking Control ..... 359
5.6.2. Constrained Optimization . 361
5.7. Nonlinear Control of Permanent-Magnet Synchronous Motors. 368
5.7.1. Nonlinear Motor Dynamics . . . . . 370
5.7.2. Feedback Linearization and Control. 373
5.7.3. The Lyapunov-Based Approach. . . 375
5.7.4. Analytical and Numerical Results. . 377
5.8. Case Study in Nonlinear Control of Multivariable Systems:
Motion Control of Induction Motors . . . . . . . . . . . . . . . . 378
5.8.1. Mathematical Models ofInduction Motors .. . . . . .. 380
5.8.2. Control of Induction Motors. . . . . . . . . . . . . . . .. 386
5.8.3. Control of a Three-Phase Squirrel Cage Induction Motor:
Analytical and Experimental Results ............. 395
5.9. Control of Servo-Systems With Permanent-Magnet DC Motors 401
5.9.1. Proportional-Integral Control . 402
5.9.2. Tracking Integral Control . . . . . . . . . 402
5.9.3. Relay (Time-Optimal) Control ...... 405
5.9.4. Sliding Mode Control with Soft Switching 407
References 411
Index 415
Preface
Dynamics systems (living organisms, electromechanical and industrial systems,
chemical and technological processes, market and ecology, and so forth) can be
considered and analyzed using information and systems theories. For example,
adaptive human behavior can be studied using automatic feedback control. As
an illustrative example, the driver controls a car changing the speed and steer
ing wheels using incoming information, such as traffic and road conditions. This
book focuses on the most important and manageable topics in applied multivariable
control with application to a wide class of electromechanical dynamic systems. A
large spectrum of systems, familiar to electrical, mechanical, and aerospace stu
dents, engineers, and scholars, are thoroughly studied to build the bridge between
theory and practice as well as to illustrate the practical application of control theory
through illustrative examples. It is the author's goal to write a book that can be
used to teach undergraduate and graduate classes in automatic control and nonlin
ear control at electrical, mechanical, and aerospace engineering departments. The
book is also addressed to engineers and scholars, and the examples considered
allow one to implement the theory in a great variety of industrial systems. The
main purpose of this book is to help the reader grasp the nature and significance
of multivariable control. To fulfill the goal, I prefer being occasionally imprecise
(but understandable and rigorous) than to building logically impeccable mathemat
ical setups in which definitions, lemmas, and theorems are welded together into a
formidable barrier to challenge the students and engineers. I have attempted to fill
a gap in the literature on multivariable control by presenting modem concepts of
control systems theory, covering the most viable approaches for continuous- and
discrete-time systems.
It is known that linear dynamic systems are systems in which the input-output
relationships satisfy the principle of superposition. These systems are modeled
by linear differential or difference equations, and transfer function in s- and z
domains can be found. Different approaches in analysis and control of linear sys
tems can be researched using frequency-domain and state-space (time-domain)
techniques. The duality of these concepts is well known, e.g., one can find trans
fer functions from differential equations and vise versa. Frequency-domain and
state-space concepts are covered in the junior-level classes Signals and Systems
(at electrical- and computer-engineering departments) and Dynamic Systems and
vii
viii Preface
Automatic Control (at mechanical- and aerospace-engineering departments). The
state-space methods must be prioritized for multivariable systems. Furthermore,
the state-space concept is not only useful in analysis and design of linear systems,
but also is an important starting point for advanced optimal and nonlinear control.
It is difficult to find linear real-world dynamic systems; that is, the system that
is modeled using linear differential or difference equations. Electric circuits and
electric motors are commonly used to introduce the students to the concepts of
transfer functions and differential equations. In general, simple circuits can be
described by linear differential equations. However, in studying power converters
or nonlinear filters, one finds that nonlinear differential equations must be used.
From a great variety of electric machines, only the permanent-magnet, direct
current motor is a nonlinear dynamic system. However, nonlinear friction and the
maximum allowable applied voltage to the armature winding (specified by manu
facturers) lead to non-linearities. Most electrical, electromechanical, mechanical,
and aerospace systems are nonlinear, and assumptions and simplifications, used
to perform linearization, cannot be viewed as a constructive way in analysis and
design of high-performance real-world systems. For example, it is impossible to
linearize not one-to-one functions (sin or cos) in nonlinear differential equations.
All systems exhibit some degree of nonlinearity, and if the control system is ana
lyzed and designed using linear models, performance objectives and criteria may
not be met. However, linear systems shall be studied first to attain readability and
avoid anxiety. The focus of this book is to analyze and design linear and nonlinear
multivariable systems. The keys to these developments are analysis, identification,
and optimization, with a major emphasis on nonlinear phenomena.
The three major modeling paradigms are
• Linear and multilinear models (multiple linear models can be used to describe
nonlinear system dynamics at the operating regions)
• Hammerstein-Wiener models (mathematical models consist of linear dy
namic elements in sequence with static nonlinearities) that map the transient
behavior of a wide variety of real-world dynamic systems
• Functional expansion models (block-oriented models that provide excellent
approximation capabilities when compared with contestant expansion mod
els, such as Volterra series)
Nonlinear analysis and controllers design must be performed on the basis of
complete mathematical models. Many methods can be used for analyzing, de
signing, and optimizing control systems. Analysis and feedback control of linear
systems are straightforward and well understood. Manageable analytic meth
ods, largely based on the eigenvalue analysis and the state-transition matrix, can
be straightforwardly applied to multi-input/multi-output systems. In contrast,
for nonlinear systems, control and optimization problems must be researched,
and further developments must be made. Even though enormous computational
power (tera-scale computing) is becoming available to attain analysis and control
through numerical analysis, analytical methods must be developed and thoroughly
studied because in the most practical scenarios, the designer must perform mul-
Preface ix
tidimensional numerical-analytical studies. Pure computer-oriented approaches,
as applied to nonlinear systems, frequently lead to inadequate, confusing, and
conflicting results. There are increasing demands for further developments of
a nonlinear control theory to facilitate thorough research in inherently difficult
nonlinear problems, analysis of nonlinear phenomena and system behavior, and
finally, design of control algorithms. The Hamilton-Jacobi theory and Lyapunov
methods can be applied using the state-space concept in modeling of dynamic
systems. The state-space concept in analysis of dynamic systems emerged more
than 100 years ago, when A. Lyapunov (professor of mechanical engineering at
the Kharkov University, Ukraine) developed the pioneering results in analysis of
stability. In the 1950s, the state-space concept was implemented primarily for
multi-inputlmulti-output aircraft and spacecraft. Today, owing to availability of
state-of-the -art microcontrollers and digital signal processors (DSPs), complex
control algorithms can be easily implemented to attain the desired system perfor
mance. This book is intended to introduce important methods and algorithms in
the study, analysis, design, and optimization of multi-inputlmulti-output dynamic
systems, and a great number of practical examples are covered, particularly those
that are relevant to electrical, mechanical, and aerospace engineering.
To avoid possible obstacles, the material is presented in sufficient detail. Basic
introductory theory, related to the linear state-space concept, is covered to help
one to fully understand, appreciate, and apply the knowledge. A wide range
of worked-out examples and qualitative illustrations, which are treated in-depth,
bridge the gap between theory and practice. Step-by-step, Control Systems Theory
With Engineering Applications guides one through the most important aspects in
analysis and control: from a rigorous theoretical foundation to applications of
results. Some results from linear signals and systems theory, as well as modeling
in the MATLAB environment (which promotes enormous gains in productivity and
creativity), are given in the introductory chapters.
Simulation is a critical and urgently important aspect of analysis and control,
development and prototyping, as well as stabilization and optimization of dynamic
systems. To speed analysis and design, facilitate enormous gains in productivity
and creativity, integrate control laws using advanced microprocessors and DSPs,
accelerate prototyping features, generate real-time C code, and visualize the re
sults, MATLAB® can be efficiently used. The MATLAB is a computational environ
ment that integrates a great number of toolboxes, such as SIMULINK, Real-Time
WorkshopT M, Control System, Nonlinear Control Design, Optimization, Robust
Control, Signal Processing, Symbolic Math, System Identification, and so on. A
flexible, high-performance modeling and design environment, MATLAB has be
come a standard cost-effective tool within the engineering community. This book
demonstrates MATLAB's capabilities and helps one to master this user-friendly
environment to attack and solve different problems. The application of MATLAB
increases designer productivity as well as shows how to use the advanced software.
The MATLAB environment offers a rich set of capabilities to efficiently solve a va
riety of complex analyses and control, stabilization, and optimization problems,
The MATLAB files and SIMULINK models, which are given in this book, can be
x Preface
easily modified to study application-specific problems encountered in engineering
practice. The examples, modeled and simulated in this book, consist of a wide
spectrum of practical electrical, electromechanical, and aerospace dynamic sys
tems. A variety of complex nonlinear mathematical models are thoroughly studied,
and SIMULINK diagrams to simulate dynamic systems and numerical results are
reported. Users can easily apply these results as well as develop new MATLAB
files and SIMULINK block-diagrams using the treated enterprise-wide practical ex
amples. Through these examples, the most efficient and straightforward analysis
and design methods to approach and solve motion control problems are docu
mented. The developed scripts and models are easily assessed, and they can be
straightforwardly modified.
The major objectives of this readable and user-friendly book are to give students
and engineers confidence in their ability to apply advanced theoretical concepts, to
enhance learning, and to provide a gradual progression from versatile theoretical to
practical topics to apply the results. This book is written for engineers and students
interested in nonlinear control of real-world systems. Students and engineers are
not primarily interested in theoretical encyclopedic studies. They need straightfor
ward and practical instructions on how to approach and solve the specific practical
challenging problems in control of electrical, electromechanical, and aerospace
systems. It is the author's goal to stress the practical aspects in order to help
one to understand and apply the advanced developments in analysis and design of
multivariable dynamic systems. In analysis and control of multivariable systems,
modem theory and innovative methods, as well as advanced algorithms and soft
ware, are of great importance. This book presents a well-defined theoretical base
with step-by-step instructions on how to apply it by thoroughly studying a great
number of practical real-world problems and using numerous examples. These
worked-out examples prepare one to use the analysis, identification, control, and
optimization methods presented. This helps one to fully understand, appreciate,
visualize, grasp, use, and finally apply the results.
Indianapolis, Indiana Sergey Edward Lyshevski
