Stability of Stationary Setsin Control Systems withDiscontinuous Nonlinearities

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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

NEW JERSEY


LONDON


SINGAPORE


SHANGHAI


HONGKONG


TAIPEI


CHENNAI


Published by

World Scientific Pub
lishing 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 b
ook 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 reproduc
ed 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 vol
ume, 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 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