SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS
Series A Volume14
Founder & Editor: Ardeshir Guran
Co-Editors:M. Cloud & W. B. Zimmerman
Stability of Stationary Sets
in Control Systems with
Discontinuous Nonlinearities
V. A. Yakubovich
G. A. Leonov
A. Kh. Gelig
St. Petersburg State University, Russia
'World Scientific
NEW JERSEY • LONDON • SINGAPORE • SHANGHAI • HONGKONG • TAIPEI • CHENNAI
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
STABILITY OF STATIONARY SETS IN CONTROL SYSTEMS WITH DISCONTINUOUS NONLINEARITIES
Copyright © 2004 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN 981-238-719-6 Printed in Singapore by World Scientific Printers (S) Re Ltd
Preface
Many technical systems are described by nonlinear differential equations with discontinuous right-hand sides. Among these are relay automatic control systems, mechanical systems (gyroscopic systems and systems with a Coulomb friction in particular), and a number of systems from electrical and radio engineering.
As a rule, stationary sets of such systems consist of nonunique equilibria. In this book, the methods developed in absolute stability theory are used for their study. Namely, these systems are investigated by means of the Lyapunov functions technique with Lyapunov functions being chosen from a certain given class. The functions are constructed through solving auxiliary algebraical problems, more precisely, through solving some matrix inequalities. Conditions for solvability of these inequalities lead to frequency-domain criteria of. one or another type of stability. Frequently, such criteria are unimprovable if the given class of Lyapunov functions is considered.
The book consists of four chapters and an appendix.
In the first chapter some topics from the theory of differential equations with discontinuous right-hand sides are presented. An original notion of a solution of such equations accepted in this book is introduced and justified. Sliding modes are investigated; Lyapunov-type lemmas whose conditions guarantee stability, in some sense, of stationary sets are formulated and proved.
The second chapter concerns algebraic problems arising by the construction of Lyapunov functions. Frequency-domain theorems on solvability of quadratic matrix inequalities are formulated here. The so-called S-procedure, which is a generalization of a method proposed by A.I. Lur'e [Lur'e (1957)], is also justified in this chapter. The origin of these problems
is elucidated by the examples of deducing well-known frequency-domain conditions for absolute stability, namely, those of the Popov and circle criteria. The chapter also contains some basic information from the theory of linear control systems, which is used in the book. The proofs of the algebraic statements formulated in Chapter 2 are given in the Appendix.
The third chapter is devoted to the stability study of stationary sets of systems with a nonunique equilibrium and with one or several discontinuous nonlinearities, under various suppositions concerning the spectrum of the linear part. Systems whose discontinuous nonlinearities satisfy quadratic constraints, monotonic, or gradient-type are studied. Frequency-domain criteria for dichotomy (non-oscillation) and for various kinds of stability of equilibria sets are proved.
With the help of the results obtained, dichotomy and stability of a number of specific nonlinear automatic control systems, gyroscopical systems with a Coulomb friction, and nonlinear electrical circuits are investigated.
In the fourth chapter the dynamics of systems with angular coordinates (pendulum-like systems) is examined. Among them are the phase synchronization systems that occur widely in electrical engineering. Such systems are employed in television technology, radiolocation, hydroacoustics, astri-onics, and power engineering. The methods of periodical Lyapunov functions, invariant cones, nonlocal reduction, together with frequency-domain methods, are used to obtain sufficient, and sometimes also necessary, conditions for global stability of the stationary sets of multidimensional systems. The results obtained are applied to the approximation of lock-in ranges of phase locked loops and to the investigation of stability of synchronous electric motors.
The dependence diagram of the chapters is the following:
I II
III------>IV
The authors aimed to make the book useful not only for mathematicians engaged in differential equations with discontinuous nonlinearities and the theory of nonlinear automatic control systems, but also for researchers studying dynamics of specific technical systems. That is why much attention has been paid to the detailed analysis of practical problems with the help of the methods developed in the book.
A reader who is interested only in applications may limit himself to
reading Sections2.1 and 1.1, and then pass immediately to Capter 3.
The basic original results presented in the book are outgrowths of the authors' cooperation; they were reported at the regular seminar of the Division of Mathematical Cybernetics at the Mathematical and Mechanical Department of Saint Petersburg State University.
Chapter 2, Appendix, and Section 1.1 of Chapter 1 were written by V.A. Yakubovich; the rest of Chapter 1 and Chapter 3 were contributed by A.Kh. Gelig; Chapter 4 was written by G.A. Leonov. The final editing was performed by the authors together.
We are greatly indebted to Professor Ardeshir Guran for inviting us to publish this book in his series on Stability, Vibration and Control of Systems. We would like to express our profound gratitude to Professor Michael Cloud for his patient work of bringing the language of the book into accord with international standards and improving a lot of misprints. Our sincere thanks are due to Professor Alexander Churilov for his assistance in typesetting and copyediting and to doctoral student Dmitry Altshuller, whose numerous comments helped us to improve English of the book. We thank the reviewers for their relevant and helpful suggestions.
List of Notations
R1 (R)set of real numbers
Rnset of n-dimensional real vectors
(n-dimensional Euclidean space)
Cset of complex numbers
Cnset of n-dimensional complex vectors
rank Mrank of matrix M
j(M)igenvalues of a square matrix M
On(n x n) zero matrix
In(n x n) identity matrix (order n may be omitted
if implied by the text)
(a, b)[a, b] if ab, [b, a] if ba
A*transposed matrix if a matrix Ais real,
Hermitian conjugate matrix if Ais complex
H > 0positive definiteness of a matrix H = H* (i.e.,
if H is n x n, then x*H x > 0 for all x C*, x0)
H ≥ 0nonnegative definiteness of a matrix H = H*
(i.e., x*Hx≥ 0 for all x Cn)
A square matrix is called Hurwitz stable if all its eigenvalues have strictly negative real parts; a square matrix is called anti-Hurwitz if all its eigenvalues have strictly positive real parts.
Contents
Prefacev
List of Notationsix
1. Foundations of Theory of Differential Equations with Dis
continuous Right-Hand Sides1
1.1Notion of Solution to Differential Equation with
Discontinuous Right-Hand Side...... 2
1.1.1Difficulties encountered in the definition of a solution.
Sliding modes...... 2
1.1.2The concept of a solution of a system with discon
tinuous nonlinearities accepted in this book. Con
nection with the theory of differential equations with
multiple-valued right-hand sides...... 6
1.1..3Relation to some other definitions of a solution to a
system with discontinuous right-hand side..... 14
1.1.4Sliding modes. Extended nonlinearity. Example ... 20
1.2Systems of Differential Equations with Multiple-Valued
Right-Hand Sides (Differential Inclusions)...... 26
1.2.1Concept of a solution of a system of differential
equations with a multivalued right-hand side, the
local existence theorem, the theorems on continuation
of solutions and continuous dependence on initial
values...... 27
1.2.2"Extended" nonlinearities...... 37
1.2.3Sliding modes...... 44
1.3 Dichotomy and Stability...... 55
1.3.1Basic definitions...... 55
1.3.2Lyapunov-type lemmas...... 57
2.Auxiliary Algebraic Statements on Solutions of Matrix
Inequalities of a Special Type61
2.1Algebraic Problems that Occur when Finding Conditions for
the Existence of Lyapunov Functions from Some
Multiparameter Functional Class. Circle Criterion.
Popov Criterion...... 62
2.1.1Equations of the system. Linear and nonlinear parts
of the system. Transfer function and frequency
response...... 63
2.1.2Existence of a Lyapunov function from the class of
quadratic forms. 5-procedure...... 64
2.1.3Existence of a Lyapunov function in the class of
quadratic forms (continued). Frequency-domain
theorem...... 69
2.1.4The circle criterion...... 71
2.1.5A system with a stationary nonlinearity. Existence of
a Lyapunov function in the class "a quadratic form
plus an integral of the nonlinearity"...... 75
2.1.6Popov criterion ...... 79
2.2Relevant Algebraic Statements...... 84
2.2.1Controllability, observability, and stabilizability84
2.2.2Frequency-domain theorem on solutions of some ma-
trix inequalities...... 91
2.2.3Additional auxiliary lemmas...... 101
2.2.4The 5-procedure theorem...... 106
2.2.5On the method of linear matrix inequalities in control
theory...... 109
3.Dichotomy and Stability of Nonlinear Systems with Mul-
tiple Equilibria111
3.1 Systems with Piecewise Single-Valued Nonlinearities .... 112
3.1.1Systems with several nonlinearities. Frequency-
domain conditions for quasi-gradient-like behavior
and pointwise global stability. Free gyroscope with
dry friction ...... 112
3.1.2The case of a single nonlinearity and det P 0. Theo-
rem 3.4 on gradient-like behavior and pointwise global
stability of the segment of rest. Examples...... 120
3.1.3The case of a single nonlinearity and one zero pole of
the transfer function. Theorem 3.6 on quasi-gradient-
like behavior and pointwise global stability. The Bul-
gakov problem...... 124
3.1.4The case of a single nonlinearity and double zero pole
of the transfer function. Theorem 3.8 on global stabil-
ity of the segment of rest. Gyroscopic roll equalizer.
The problem of Lur'e and Postnikov. Control system
for a turbine. Problem of an autopilot...... 130
3.2Systems with Monotone Piece wise Single-Valued
Nonlinearities...... 141
3.2.1Systems with a single nonlinearity. Frequency-domain
conditions for dichotomy and global stability. Cor-
rected gyrostabilizer with dry friction. The problem
of Vyshnegradskh...... 142
3.2.2Systems with several nonlinearities. Frequency-
domain criteria for dichotomy. Noncorrectable gy-
rostabilizer with dry. friction...... 160
3.3Systems with Gradient Nonlinearities...... 167
3.3.1Dichotomy and quasi-gradient-likeness of systems
with gradient nonlinearities...... 167
3.3.2Dichotomy and quasi-gradient-like behavior of
nonlinear electrical circuits and of cellular neural
networks...... 171
Stability of Equilibria Sets of Pendulum-Like Systems175
4.1 Formulation of the Stability Problem for Equilibrium Sets of
Pendulum-Like Systems...... 175
4.1.1Special features of the dynamics of pendulum-like sys-
tems. The structure of their equilibria sets...... 175
Canonical forms of pendultim-like systems with a sin
gle scalar nonlinearity 183
4.1.2Canonical forms of pendultim-like systems with a sin
gle scalar nonlinearity...... 183
4.1.3 Dichotomy. Gradient-like behavior in a class of non-
linearities with zero mean value 189
4.2The Method of Periodic Lyapunov Functions...... 192
4.2.1Theorem on gradient-like behavior...... 192
4.2.2Phase-locked loops with first- and second-order low-
pass filters...... 201
4.3An Analogue of the Circle Criterion for Pendulum-Like Sys-
tems ...... 203
4.3.1Criterion for boundedness of solutions of pendulum-
like systems...... 204
4.3.2Lemma on pointwise dichotomy...... 210
4.3.3Stability of two- and three-dimensional pendulum-like
systems. Examples...... 212
4.3.4Phase-locked loops with a band amplifier...... 216
4.4The Method of Non-Local Reduction...... 218
4.4.1The properties of separatrices of a two-dimensional
dynamical system...... 219
4.4.2The theorem on nonlocal reduction...... 222
4.4.3Theorem on boundedness of solutions and on
gradient-like behavior...... 223
4.4.4Generalized Bohm-Hayes theorem ...... 228
4.4.5Approximation of the acquisition bands of phase-
locked loops with various low-pass filters...... 229
4.5Necessary Conditions for Gradient-Like Behavior of
Pendulum-Like Systems 235
4.5.1Conditions for the existence of circular solutions and
cycles of the second kind...... 236
4.5.2Generalized Hayes theorem ...... 244
4.5.3Estimation of the instability regions in searching PLL
systems and PLL systems with 1/2 filter...... 245
4.6Stability of the Dynamical Systems Describing the
Synchronous Machines...... 251
4.6.1Formulation of the problem...... 252
4.6.2The case of zero load...... 253
4.6.3The case of a nonzero load...... 258
5. Appendix. Proofs of the Theorems of Chapter 2269
5.1 Proofs of Theorems on Controllability, Observability,
Irreducibility, and of Lemmas 2.4 and 2.7...... 269
5.1.1Proof of the equivalence of controllability to proper
ties (i)-(iv) of Theorem 2.6...... 269
5.1.2Proof of the Theorem 2.7 ...... 273
5.1.3Completion of the proof of Theorem 2.6...... 274
5.1.4Proof of Theorem 2.8...... 275
5.1.5Proof of Theorem 2.9 in the scalar case m = l = 1 275
5.1.6Proof of Theorem 2.9 for the case when either m > 1
or l > 1 and proof of Theorem 2.10...... 277
5.1.7Proof of Lemma 2.4...... 279
5.1.8Proof of Lemma 2.7...... 281
5.2Proof of Theorem 2.13 (Nonsingular Case). Theorem on
Solutions of Lur'e Equation (Algebraic Riccati Equation) 283
5.2.1Two lemmas. A detailed version of frequency-domain
theorem for the nonsingular case...... 283
5.2.2Proof of Theorem 5.1 The theorem on solvability of
the Lur'e equation...... 289
5.2.3Lemma on J-orthogonality of the root subspaces of a
Hamiltonian matrix...... 295
5.3Proof of Theorem 2.13 (Completion) and Lemma 5.1 297
5.3.1Proof of Lemma 5.1...... 297
5.3.2Proof of Theorem 2.13...... 298
5.4Proofs of Theorems 2.12 and 2.14 (Singular Case)....301
5.4.1Proof of Theorem 2.12...... 301
5.4.2Necessity of the hypotheses of Theorem 2.14....306
5.4.3Sufficiency of the hypotheses of Theorem 2.14.309
5.5Proofs of Theorems 2.17-2.19 on Losslessness of 5-procedure 316
5.5.1The Dines theorem...... 316
5.5.2Proofs of the theorems on the losslessness of the 5-
procedure for quadratic forms and one constraint 318
Bibliography323
Index333