SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS
Series A
Volume 14
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
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STABILITY OF STATIONARY SETS IN CONTROL SYSTEMS WITH
DISCONTINUOUS NONLINEARITIES
Copyright © 2004 by World Scientific Publishing Co. Pte. Ltd.
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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
set of n-dimensional real vectors
(n-dimensional Euclidean space)
C
set of complex numbers
n
C
set of n-dimensional complex vectors
rank M
rank of matrix M
igenvalues of a square matrix M
 j(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 a < b, [b, a] if b  a
A*
transposed matrix if a matrix A is real,
Hermitian conjugate matrix if A is complex
H>0
positive definiteness of a matrix H = H * (i.e.,
if H is n x n, then x*H x > 0 for all x  C*, x  0)
H≥0
nonnegative definiteness of a matrix H = H*
(i.e., x*Hx≥ 0 for all x  Cn)
1, x  0
sgn x  
 1, x  0
A square matrix is called Hurwitz stable if all its eigenvalues
have strictly negative real parts; a square matrix is called antiHurwitz if all its eigenvalues have strictly positive real parts.
Rn
Contents
Preface
v
List of Notations
ix
1.
Foundations of Theory of Differential Equations with Dis
continuous Right-Hand Sides
1
1.1 Notion of Solution to Differential Equation with
Discontinuous Right-Hand Side ......................................
2
1.1.1 Difficulties encountered in the definition of a solution.
Sliding modes ........................................................
2
1.1.2 The 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..3 Relation to some other definitions of a solution to a
system with discontinuous right-hand side ..........
14
1.1.4 Sliding modes. Extended nonlinearity. Example ...
20
1.2 Systems of Differential Equations with Multiple-Valued
Right-Hand Sides (Differential Inclusions) ......................
26
1.2.1 Concept 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
1.2.3 Sliding modes ........................................................
44
1.3 Dichotomy and Stability ..................................................
1.3.1 Basic definitions .....................................................
1.3.2 Lyapunov-type lemmas ..........................................
2. Auxiliary Algebraic Statements on Solutions of Matrix
Inequalities of a Special Type
55
55
57
61
2.1 Algebraic Problems that Occur when Finding Conditions for
the Existence of Lyapunov Functions from Some
Multiparameter Functional Class. Circle Criterion.
Popov Criterion ................................................................ 62
2.1.1 Equations of the system. Linear and nonlinear parts
of the system. Transfer function and frequency
response................................................................. 63
2.1.2 Existence of a Lyapunov function from the class of
quadratic forms. 5-procedure .............................. 64
2.1.3 Existence of a Lyapunov function in the class of
quadratic forms (continued). Frequency-domain
theorem ..................................................................
69
2.1.4 The circle criterion .................................................
71
2.1.5 A system with a stationary nonlinearity. Existence of
a Lyapunov function in the class "a quadratic form
plus an integral of the nonlinearity" .....................
75
2.1.6 Popov criterion . . ................ ...........................
79
2.2 Relevant Algebraic Statements.......................................
84
2.2.1 Controllability, observability, and stabilizability
84
2.2.2 Frequency-domain theorem on solutions of some matrix inequalities ......................................................
91
2.2.3 Additional auxiliary lemmas .................................. 101
2.2.4 The 5-procedure theorem .................................... 106
2.2.5 On the method of linear matrix inequalities in control
theory ..................................................................... 109
3. Dichotomy and Stability of Nonlinear Systems with Multiple Equilibria
111
3.1 Systems with Piecewise Single-Valued Nonlinearities ....
112
3.1.1 Systems with several nonlinearities.
Frequencydomain conditions for quasi-gradient-like behavior
and pointwise global stability. Free gyroscope with
dry friction .......................................................... 112
3.1.2 The case of a single nonlinearity and det P  0. Theorem 3.4 on gradient-like behavior and pointwise global
stability of the segment of rest. Examples .......... 120
3.1.3 The case of a single nonlinearity and one zero pole of
the transfer function. Theorem 3.6 on quasi-gradientlike behavior and pointwise global stability. The Bulgakov problem ........................................................ 124
3.1.4 The case of a single nonlinearity and double zero pole
of the transfer function. Theorem 3.8 on global stability 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
3.2 Systems with Monotone Piece wise Single-Valued
Nonlinearities.................................................................... 141
3.2.1 Systems with a single nonlinearity. Frequency-domain
conditions for dichotomy and global stability. Corrected gyrostabilizer with dry friction. The problem
of Vyshnegradskh .................................................. 142
3.2.2 Systems with several nonlinearities.
Frequencydomain criteria for dichotomy. Noncorrectable gyrostabilizer with dry. friction .................................. 160
3.3 Systems with Gradient Nonlinearities ............................ 167
3.3.1 Dichotomy and quasi-gradient-likeness of systems
with gradient nonlinearities ................................... 167
3.3.2 Dichotomy and quasi-gradient-like behavior of
nonlinear electrical circuits and of cellular neural
networks ................................................................. 171
Stability of Equilibria Sets of Pendulum-Like Systems
175
4.1 Formulation of the Stability Problem for Equilibrium Sets of
Pendulum-Like Systems .................................................. 175
4.1.1 Special features of the dynamics of pendulum-like systems. The structure of their equilibria sets ........ 175
Canonical forms of pendultim-like systems with a sin
gle scalar nonlinearity
183
4.1.2 Canonical forms of pendultim-like systems with a sin
gle scalar nonlinearity ........................................... 183
4.1.3 Dichotomy. Gradient-like behavior in a class of nonlinearities with zero mean value
189
4.2 The Method of Periodic Lyapunov Functions ................ 192
4.2.1 Theorem on gradient-like behavior ...................... 192
4.2.2 Phase-locked loops with first- and second-order lowpass filters ............................................................. 201
4.3 An Analogue of the Circle Criterion for Pendulum-Like Systems ................................................................................. 203
4.3.1 Criterion for boundedness of solutions of pendulumlike systems ........................................................... 204
4.3.2 Lemma on pointwise dichotomy ........................... 210
4.3.3 Stability of two- and three-dimensional pendulum-like
systems. Examples ............................................... 212
4.3.4 Phase-locked loops with a band amplifier ........... 216
4.4 The Method of Non-Local Reduction ............................. 218
4.4.1 The properties of separatrices of a two-dimensional
dynamical system ................................................. 219
4.4.2 The theorem on nonlocal reduction ..................... 222
4.4.3 Theorem on boundedness of solutions and on
gradient-like behavior ............................................. 223
4.4.4 Generalized Bohm-Hayes theorem ................... 228
4.4.5 Approximation of the acquisition bands of phaselocked loops with various low-pass filters .............. 229
4.5 Necessary Conditions for Gradient-Like Behavior of
Pendulum-Like Systems
235
4.5.1 Conditions for the existence of circular solutions and
cycles of the second kind ....................................... 236
4.5.2 Generalized Hayes theorem ............................. 244
4.5.3 Estimation of the instability regions in searching PLL
systems and PLL systems with 1/2 filter ............. 245
4.6 Stability of the Dynamical Systems Describing the
Synchronous Machines ..................................................... 251
4.6.1 Formulation of the problem ................................... 252
4.6.2 The case of zero load ............................................ 253
4.6.3 The case of a nonzero load .................................. 258
5.
Appendix. Proofs of the Theorems of Chapter 2
269
5.1 Proofs of Theorems on Controllability, Observability,
Irreducibility, and of Lemmas 2.4 and 2.7 ...................... 269
5.1.1 Proof of the equivalence of controllability to proper
ties (i)-(iv) of Theorem 2.6 .................................... 269
5.1.2 Proof of the Theorem 2.7 ................................. 273
5.1.3 Completion of the proof of Theorem 2.6 .............. 274
5.1.4 Proof of Theorem 2.8 ............................................ 275
5.2
5.3
5.4
5.5
5.1.5 Proof of Theorem 2.9 in the scalar case m = l = 1 275
5.1.6 Proof of Theorem 2.9 for the case when either m > 1
or l > 1 and proof of Theorem 2.10 ...................... 277
5.1.7 Proof of Lemma 2.4 ............................................... 279
5.1.8 Proof of Lemma 2.7 ............................................... 281
Proof of Theorem 2.13 (Nonsingular Case). Theorem on
Solutions of Lur'e Equation (Algebraic Riccati Equation) 283
5.2.1 Two lemmas. A detailed version of frequency-domain
theorem for the nonsingular case ........................ 283
5.2.2 Proof of Theorem 5.1 The theorem on solvability of
the Lur'e equation ................................................. 289
5.2.3 Lemma on J-orthogonality of the root subspaces of a
Hamiltonian matrix................................................ 295
Proof of Theorem 2.13 (Completion) and Lemma 5.1
297
5.3.1 Proof of Lemma 5.1............................................... 297
5.3.2 Proof of Theorem 2.13 .......................................... 298
Proofs of Theorems 2.12 and 2.14 (Singular Case) ...... 301
5.4.1 Proof of Theorem 2.12 .......................................... 301
5.4.2 Necessity of the hypotheses of Theorem 2.14 .... 306
5.4.3 Sufficiency of the hypotheses of Theorem 2.14.
309
Proofs of Theorems 2.17-2.19 on Losslessness of 5-procedure
316
5.5.1 The Dines theorem ............................................... 316
5.5.2 Proofs of the theorems on the losslessness of the 5procedure for quadratic forms and one constraint 318
Bibliography
323
Index
333