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.
All rights reserved. This book, or parts thereof, may not be reproduc
ed in any form or by any means,
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ISBN 981

238

719

6
Printed in Singapore by World Scientific P
rinters (S) Re Ltd
Preface
Many technical systems are described by nonlinear differential equations
with
discontinuous right

hand sides. Among these are relay automatic con
trol
systems, mechanical systems (gyroscopic systems and systems with a
Coulomb
fr
iction in particular), and a number of systems from electrical
and radio
engineering.
As a rule, stationary sets of such systems consist of nonunique equilib
ria. In
this book, the methods developed in absolute stability theory are
used for
their study. Na
mely, 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 solv
ing auxiliary
algebraical problems, more precisely, through solvin
g 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 con
struction
of Lyapunov
functions. Frequency

domain theorems on solvabil
ity 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 cri
teria.
The chapter also contains some basic information from the theory
of linear
contr
ol 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 di
chotomy (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 num
ber
of specific nonlinear automatic control systems, gyroscopical systems
with a
Coulomb fri
ction, 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 synchro

nization systems that occur widely in electrical enginee
ring. Such systems
are
employed in television technology, radiolocation, hydroacoustics, astri

onics,
and power engineering. The methods of periodical Lyapunov func
tions,
invariant cones, nonlocal reduction, together with frequency

domain
methods,
are use
d to obtain sufficient, and sometimes also necessary, condi
tions 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 investigatio
n of stability of synchronous electric motors.
The dependence diagram of the chapters is the following:
I II
III

>I
V
The authors aimed to make the book useful not only for mathemati
cians
engaged in differential equations with discontinuo
us nonlinearities and
the
theory of nonlinear automatic control systems, but also for researchers
studying dynamics of specific technical systems. That is why much atten
tion
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' cooper
ation; 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.
Yakubovi
ch; 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
R
1
(R)
set of real numbers
R
n
set of n

dimensional real vectors
(n

dimensional Euclidean space)
C
set of complex numbers
C
n
set of n

dimensional complex vectors
rank M
rank
of matrix M
j
(M)
igenvalues of a square matrix
M
O
n
(
n
x
n)
zero matrix
I
n
(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*H
x
≥ 0 for all
x
C
n
)
A square matrix is called
H
urwitz stable
if all its eigenvalues
have strictly
negative real parts; a square matrix is called
anti

Hurwitz
if all its eigen
values have strictly positive real parts.
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
................................
..................
55
1.3.1
Basic definitions
................................
.....................
55
1.3.2
Lyapunov

type lemmas
................................
..........
57
2.
Auxiliary Algebraic St
atements on Solutions of Matrix
Inequalities of a Special Type
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
Equatio
ns 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 ma

trix 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 Mul

tiple Equilibria
111
3.1 Systems with Piecewise Single

Valued Nonlinearities ....
112
3.1.1
Systems with several
nonlinearities. Frequency

domain 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. Theo

rem 3.4 on gradie
nt

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

gradient

like behavior and pointwise global stability. The Bul

gak
ov problem
................................
........................
124
3.1.4
The 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. Probl
em 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. Cor

rected gyrostabilizer with dry friction.
The problem
of Vyshnegradskh
................................
..................
142
3.2.2
Systems with several nonlinearities. Frequency

domain criteria for dichotomy. Noncorrectable gy

rostabilizer 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 Form
ulation of the Stability Problem for Equilibrium Sets of
Pendulum

Like Systems
................................
..................
175
4.1.1
Special 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 sca
lar nonlinearity
1
83
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 non

linearities 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 low

pass filters
................................
.............................
201
4.3
An Analogue of the Circle Criterion for Pendul
um

Like Sys

tems
................................
................................
.................
203
4.3.1
Criterion for boundedness of solutions of pendulum

like 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
Generalize
d Bohm

Hayes theorem
...................
228
4.4.5
Approximation of the acquisition bands of phase

locked 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 Contr
ollability, 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 Theo
rem 2.8
................................
............
275
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
5.2
Proof of Theorem 2.13 (Nonsingular Case). The
orem 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

ortho
gonality of the root subspaces of a
Hamiltonian matrix
................................
................
295
5.3
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
5.4
Proofs of Theorems 2.12 and 2.14 (Singular Case)
......
301
5.4.1
Proof of Theorem 2.12
................................
..........
301
5.4.2
N
ecessity of the hypotheses of Theorem 2.14
....
306
5.4.3
Sufficiency of the hypotheses of Theorem 2.14
.
309
5.5
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 5

procedure for
quadratic forms and one constraint
318
Bibliography
323
Index
333
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