generality, the authors also show much concern for concrete examples and ...... Gng. Gn â Gn. Z 1. G {Zâ}, where , the index of nonlinearity for separable ...
Frequency Domain for
Criteria
Absolute Stability
KUMPATI
S.
NARENDRA
Yale University
New Haven
JAMES
,
Connecticut
H.
TAYLOR
The Analytic Sciences Corporation Reading Massachusetts ,
TOg LTBRAOT
COLLEGE OF PETROLEUM & MtNBR* DHAHRAN, SAUDI ARABIA
45505 ACADEMIC PRESS New York A
and London
Subsidiary of Harcourt Brace Jovanovich Publishers ,
1973
N27
Copyright
©
1973, by
Academic Press,
Inc.
ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION
MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.
ACADEMIC
PRESS, INC. New York, New York
Ill Fifth Avenue,
10003
United Kingdom Edition published by
ACADEMIC
PRESS, INC.
24/28 Oval Road, London
NW1
(LONDON) LTD.
Library of Congress Catalog Card Number:
PRINTED IN THE UNITED STATES OF AMERICA
72-82641
To
BARBARA AND ANNE-MARIE
CONTENTS Foreword
xi
Preface
xiii
Acknowledgments Special Notation
.
xvii
Introduction
I.
1
xv
The System
2. Stability
1
of Motion
6
Lyapunov’s Direct Method The Quadratic Lyapunov Function
10
5.
Some Problems
1
6.
14
8.
The Conjectures of Aizerman and Kalman The Absolute Stability Problem The Criterion of Popov
9.
Synopsis
16
3.
4.
7.
in Stability
7
1
15
Problem Statement
II.
1.
System Definition
18
2.
Definitions of Stability
33
3.
Formal Problem Statement
38
III.
1
.
2.
Mathematical Preliminaries
Theorems The Absolute Lyapunov Function Candidates Sufficiency
40 44 vii
Contents
viii
4.
Restated Stability Theorems The Kalman-Yakubovich Lemma
5.
Positive Real Functions
48 57
6.
Existence Theorems
61
3.
48
Linear Time-Invariant Systems and Absolute Stability
IV. 1.
Relations between Linear Time-Invariant and Nonlinear Time-Varying
2.
The Existence of
3.
The Existence of
Systems
67 the Quadratic
Lyapunov Function x TPx and
the
Hurwitz Condition
8
the Quadratic
Lyapunov Function x TPx
-f-
kx t Mx and
the Nyquist Criterion
V. 1
.
84
Stability of Nonlinear
The Popov
Systems 91
Stability Criterion
2. Stability Criteria
for
Monotonic Nonlinearities
99
3.
Linear Systems
113
4.
Odd Monotonic Gains The General Finite Sector Problem
118
5.
VI.
Stability of Nonlinear
1.
The
2.
An
116
Time-Varying Systems 123
Circle Criterion
3.
—Point Conditions Stability Criteria for Restricted Nonlinear Behavior — Point Conditions
131
4.
The General
135
5.
Periodic Nonlinear Time-Varying Gains
6.
Extension of the Popov Criterion
7.
Integral Conditions for Restricted Nonlinear
8.
Integral Conditions for Linear Time- Varying Systems
VII.
Extension of the Popov Criterion Finite Sector
Problem
126
137
—Integral Conditions and Linear Gains
138 141
142
Geometric Stability Criteria
1.
Linear Time-Invariant Systems
149
2.
155
4.
The Circle Criterion The Popov Criterion Monotonic Nonlinearities: An Off-Axis
5.
Further Geometric Interpretations for Time-Varying Systems
3.
VIII.
158 Circle Criterion
166 175
The Mathieu Equation: An Example
1.
Solutions of the Mathieu Equation
186
2.
Linear Case (a
^ 1): A Perturbation Analysis Linear Case (a ^ 4): A Floquet Analysis
189
3.
187
Contents
ix
4.
Application of Stability Criteria to the Linear Case
190
5.
Application of Stability Criteria to the Nonlinear Case
198
IX.
Absolute Stability of Systems with Multiple Nonlinear Time- Varying Gains
1
.
2.
202 203 209 213 216 225
Introduction
Problem Statement
3.
Mathematical Preliminaries
4.
Linear Time-Invariant Systems and Absolute Stability
5.
Stability of
6. Stability
Nonlinear Systems
of Nonlinear Time-Varying Systems
APPENDIX.
Matrix Version of the Kalman-Yakubovich
Lemma
230
References
235
Index
243
FOREWORD To
write a highly interesting
book
in a field in
literature is certainly a difficult task.
performance.
Its
success
is
which there
The present book
exists a rich
fully realizes this
due mainly to the great care with which the authors aim of giving
selected the material included in the book, with the obvious
and broad understanding of the subject. While the main theorems are recent ones, the authors show very
the reader a deep
clearly
the strong connections of the theory with the classical results of Hurwitz,
Lyapunov, Nyquist, etc. Their reappraisal of the traditional engineering methods of control theory including the daring but often successful technique of “describing functions” provides an opportunity to point out another important source of the ideas that generated the contemporary view of the problem. While developing the theory with care for rigor and generality, the authors also show much concern for concrete examples and often illustrate the general theory by significant and illuminative applications,
—
—
treated in detail.
These little
qualities
make
the
book very
useful even for persons
who have
or no previous knowledge of the subject. These people will find this
book an
excellent introduction to the field.
On
the other hand, those already
some of the most which are harder to find elsewhere and which are due mainly to the outstanding research done in the field by the authors themselves. Books which successfully cover such a broad range of interests are rare. They are also very much needed, because they are bound to produce a familiar with the subject will find a detailed exposition of
advanced
results
favorable influence
upon
research.
V.
M. Popov
PREFACE This book presents some recent generalizations of the well-known Popov solution to the absolute stability problem proposed by Lur’e and Postnikov
and the results of problem are presented in detail in the excellent books of Lefschetz and of Aizerman and Gantmacher; the work that led to the formulation of the absolute stability problem and in 1944.
The Popov frequency domain
stability criterion
several earlier approaches to the Lur’e-Postnikov
the
first
The
solutions to
it
are not considered here.
success of Popov’s elegant criterion inspired
many
extensions of the
basic Lur’e-Postnikov problem. Studies of these related questions gave rise to a great
number of
stability criteria, derived using
both the direct method
of Lyapunov and the positive operator concept of functional analysis. The great interest in this area has resulted in a continuing state of rapid develop-
ment. The generation of this type of frequency domain stability criteria has
now
reached a relative state of completeness.
It is
also notable that the
two
seemingly disparate analytic approaches have led to stability criteria that are equivalent in most respects, and thus
it is
possible to present a unified picture
of the recent research in this area using only Lyapunov’s direct method. In
each of the two fundamental approaches there are several points of view which have been used to good effect by various groups of researchers. It should thus be noted that
this
book
is
founded on a
single set of techniques
based on the direct method of Lyapunov and developed first at Harvard University and then at Yale University and the Indian Institute of Science (Bangalore, India). This makes the
book
rather specialized in
its
overall
scope, but the techniques are found to be applicable to a wide range of
important questions regarding the
stability of nonlinear systems.
XIV
Preface
In view of the approach taken, several important results derived using a functional analysis viewpoint have either been omitted entirely or only mentioned in passing. Since the emphasis is on the application of Lyapunov’s direct
method
to generate frequency
domain
criteria for stability,
many
fine
results related to other aspects
bibliography
works
is
of the stability problem are omitted. The by no means complete for this reason and contains only
problems discussed. Continuous-time systems are considered here, although directly related to the
many
similar
results already exist for discrete systems. In the first eight chapters,
systems
with a single nonlinear function or time-varying parameter are treated. Systems with multiple nonlinearities or time-varying gains are considered in
Chapter IX; some criteria are derived in detail while others are presented in outline form as an indication of the state of current research. This book can serve very well as a reference for research courses concerning stability problems related to the absolute stability problem of Lur’e and Postnikov. Engineers and applied mathematicians should also find the results contained herein, particularly the geometric stability criteria, of use
Because of the diversity of the audience being developed with what we hope can be considered of mathematical formalism. Certain sections contain some quite
in practical applications.
addressed, rigorous theory
a
minimum
is
condensed technical material required as a foundation for the derivations; these may be omitted by those whose interest is limited to applications. It is assumed that the reader is familiar with matrix operations that are utilized in dealing with the state vector representation of dynamic systems. All definitions and theorems are developed as needed so that the derivations are independent of other works ; some acquaintance with the basic concepts
of stability and Lyapunov’s direct method would be helpful. The historical development of the work associated with the Lur’e-Postnikov problem has
been strongly linked to the theory of automatic control, so control systems is used sparingly wherever it is reasonable to expect that the
terminology
meaning
is
clear to all readers.
ACKNOWLEDGMENTS The authors are indebted to Roger M. Goldwyn, Charles P. Neuman, and Yo-Sung Cho (who were graduate students under the guidance of the first-named author), and M. A. L. Thathachar and M. D. Srinath, all of whose work forms an important basis for the material presented here. An early review of this effort by V. M. Popov was also significant; his suggestions and generous comments provided both encouragement and improvement in completing the final version of this work. It is a pleasure for the authors to acknowledge colleagues and graduate students sions
who have
and
in
generously given their assistance both in general discus-
recommending
of R. Viswanathan,
M. D.
specific
changes in the manuscript. The
Viswanadham, and
Srinath, N.
S.
efforts
Rajaram
are
an important factor in the completion of this effort was the able assistance provided by Mrs. Coralie Wilson, Mrs. Jean Gemmell, and Mrs. Anne-Marie Taylor, who typed many especially appreciated in this regard.
Finally,
and corrections. The support of various institutions has also been invaluable. The secondnamed author would like to acknowledge the support received from Yale
drafts
University while a graduate student and also the generosity of the Indian Institute of Science
One
where he was recently a Visiting Assistant Professor.
of the authors was in Bangalore, India while the other was in
Haven, Connecticut during the
entire period of preparation of this book.
New That
the sequence of corrections, additions, and revisions carried across several
continents finally converged authors.
It
may
is
in itself
an achievement
in the eyes of the
be safely said that without the patience, understanding, and
encouragement of our wives, Barbara Narendra and Anne-Marie Taylor, this book would not have been completed.
SPECIAL NOTATION
Throughout
the following symbol conventions are generally
book
this
adhered to (i)
t
(ii)
+
px
The
etc.).
principal exception
is
(p, r,
a0
=
the independent variable
(time).
Column
and
vectors
Latin characters that (iii)
Greek characters
Scalars are denoted by lower case
h Tx
x of the
elements
all
explicit functions are
t
denoted by lower case = 0 signifies
The notation x
(x, h; /(
x'
II
0 there exists a unique solution
id/dt)
then
it is
(4)
If
it
t,
that
,
t)
is,
to exist such that
x0
fix,
e
Whereas statements
M
t0)
t)
K = Kif)
K exists
all (x, t), (x', t)
,
1
1
0 there
Attractivity. T(rj,
x0
,
t0)
such that for
l|xO;*o>'o)ll7 for
0
.
all
t-t 0
^T
* 0 ||.
Definition 3 differential
Asymptotic Stability.
equation (1-3)
is
The equilibrium *
asymptotically stable
if it is
=0
of the
both stable and
attractive.
In this case, for any given e
>0
and
t0
,
there exists a constant
0)11
;* 0 ,O
3.
9
Lyapunov’s Direct Method
Definition differential
0 e
{.
K
},
and
for
some
t
>
t0
and
\\x 0
1|
asymptotically stable
is
>
p(t Q )
x 0 ,t 0 )
\\x(t;
for all
The equilibrium x
Asymptotic Stability.
3'
equation (1-3)
||
0 there exists a function
t0
)y/(t-
t0
;x 09
For autonomous systems
either set of definitions
e {L} such
y/
that
t 0)
is
to
3'
considerably simpler
p and T are not and the comparison funcare likewise unaffected by t 0
functions of the initial time in Definitions 1'
of the
< p.
both in concept and in application. The parameters tions of Definitions
=0
there exists a function
if
1
oo as and that the time derivative v values of p that v(x) ||*||
closely
,
be negative for
all
—
— x ^ 0.
For the system described dvjdt
in Eq. (l-3c),
Ai)A (V^)T * =
(Vv) T /0 (x)
(1-9)
10
I
where Vv
Introduction
the gradient:
is
A [dv/dx u dv/dx the condition v < then v(x) (Vw) T
If V satisfies
2,
.
0,
.
,
.
dv/dx„].
said to exist as a
is
Lyapunov
function for the specific system (l-3c) and the system is stable. If v(x) but not identically zero for any solution x(t) =£ 0, then (see Chapter Section
1)
the equilibrium
is
t)
is
positive definite
A dv/dt + C?v)T f (x, 0
and v
0,
v
is
(1-10)
t).
a Lyapunov function for the
nonautonomous system and the system is stable as before. For asymptotic stability somewhat stronger conditions have to be imposed on the Lyapunov function. These conditions are discussed in detail in Chapter HI. For systems, treated in Chapter VI, the theorem of Corduneanu
NLTV
applied in the development of one of the criteria for stability. In this case, v(x> t) has to satisfy the less restrictive condition that [1] is
v(x for stability where
y
equation. While v(x,
= m(y t)
,
9
1)
t) is
< m[v(x, t\
(1-11)
an asymptotically stable scalar
defined in this fashion
Lyapunov function, the Corduneanu theorem from Lyapunov’s method.
4.
t]
is
differential
is strictly speaking not a based on concepts derived
The Quadratic Lyapunov Function For
all
of the generality of Lyapunov’s direct method,
weakness when used in the definitive
There
has one important
is
sufficient conditions for stability except for the case
general,
it
no way of determining Lyapunov function candidate that would yield necessary and specific situations.
when a candidate
v(x
of LTI systems. In
chosen, the conditions that have to be imposed on the system (1-3), in order to guarantee that v(x t) satisfies the inequality v 0 or that (1-11) is satisfied, are conservative. For the example ,
t) is
,
,
p
h ns n
+r+
1
a ns n
+
~
l
•
+
•
H~ h 2 s ~h h
•
•
•
•
+
a 2s
x
+
(2-3)
a
h and a correspond to those of Eq.
A
y+A+a^+a^
A
ber of poles, or, in terms of frequency response,
The representation of the system Fig. 2- la. The nomenclature plant a matter of convenience; as
is
W(s)
may
A
(2 ‘ 4)
fl
where p. (h. + pa .) allows a more compact notation. have the special case where the number of zeros of W{s) is
dynamics
A, b)
{h,
second definition of parameters yields
^> = y+
ing
with
(1-4)
t
t
(2-2).
x
is
W
CJQ
in transfer function
A.
If
= 0,
p
less
lim^^
form
then
we
than the num-
is
W(ico)
= 0.
portrayed in
for W(s) and controller for g(*, t) is purely demonstrated in Fig. 2- lb, the controller
also be described by an
we have
NLTV differential equation.
By
defin-
a system of the form of Fig. 2- la that
equivalent to that of Fig. 2- lb for the purposes of this study.
The second
alternative representation of the system takes the
form of an
20
II
Problem Statement
(a)
Plant
(b) Fig. 2-1
(a)
The
general system model,
(b)
An
alternative form.
nth order ordinary scalar differential equation. The transfer function W(s) [Eq. (2-4)] plus the controller equation t g(a 0 t) corresponds to
=—
n~
1
n
2
n
1
•
n
where
in
eters a.,p.
related to
•
•
t]
2
,
and p are the same the state vector by *T
This description
however,
,
+ a D + ... +a D + aX ~ (2-5) + g[(pD* + p D + + p D + ptf, = 0, m m m differential operator notation D A d ldt and again the param[D
n
it is
is
of
=
as in previous formulations.
[£,
The
variable £
is
D(,
less utility
than Eqs. (1-4) or (2-1) for our purposes;
useful in proving certain subsidiary results.
In completing the formal definition of the system, various other properties
of the plant and controller must be specified. In the sequel, the triple
(/z,
A
,
b)
form [Eq. (2-2)], the A matrix possesses certain stability properties, denoted by A e {A.}, and the nonlinear time-varying controller is specified as to its separability, nonlinear behavior and time variation (denoted in the aggregate by g(
0
Tc
= b,
where
Wonham
(Johnson and
=P Ty + P¥
may be transformed
into the phase variable canonical
TDT~ = A and
(2-2)
is
1,
completely controllable
= Ty
that this
triangular with diagonal terms equal to unity.
system
y that
To demonstrate
is
[1]),
form
A and b and T is
this viewpoint, then, this useful
using the transformation
Eq. (2-2), that have the forms indicated in Eq. set forth in
a constant nonsingular matrix.
form may be considered with no
loss
of generality. (2)
Observability
(a)
The necessary and
observable
is
sufficient condition that
Eq. (2-7)
is
completely
that
CA
[(A
T y~ l
h (A T y~ 2 h j
i
•••\A T h
j
h]
(2-9)
System Definition
1.
is
if
25
a nonsingular matrix. (A
T
/z)
,
A
pair (h T , A)
completely observable
is
completely controllable (Kalman
Since the observability of Eq. (2-7)
(b)
the
is
The
if
and only
[3]).
determined by the h vector and
is
matrix, and the elements of h are not specified in the phase-variable
canonical form
use of this form does not guarantee complete
(2-2), the
observability.
A lack of either complete observability, or complete controllability, or both results in a
degeneracy of the impulse response of the system. This response,
AL
w(t)
~
1
[W(s)l
depends only on the controllable and observable states of the system (Kalman [3]). For the examples of systems that are not completely controllable and observable given earlier, we have ncc:
=
W(s)
(h
nco:
W(s)
x
j- h 2 )(s 4~
4~ /?)
(X
+
w(t)
^
- /i)(s + s + a = (s + a)(y + (s
a
a
+
=
(/*!+ h 2 )e
(a_/?) ',
fi)
w(t)
— e~
pt .
/?)
The
and completely observand the corresponding impulse response are those of a first-order
transfer function of the completely controllable
able part
system. In general,
W
if
3
(s)
A.h 3 T (sI
— A 33 )~
1
b 3 (Fig. 2-2), then w(t)
L~Ws(s)l Any function W(s) which is degenerate, that is, that tions, may be described by a state vector differential
=
has pole-zero cancella-
equation that
is
either
not completely controllable, or not completely observable, or neither. Thus,
from our viewpoint, we may consider only systems that possess one or the other of these forms of degeneracy, and hence the phase variable canonical form is used throughout this book to guarantee complete controllability with no loss in generality. The stability properties of the total or closed-loop system (Fig. 2-2) are always specified for at least one value of LTI feedback gain. In terms of the lower bound
GN
of the
Section IB),
we
take r
x
NLTV
function g(a 09
1)
(usually this
= —G n g in Eq. (2-6); eliminating = [A — (G n/( + pG N))bh T]x A A Qnx q
1
must have nonpositive
2/
x
}:
yields
(2-10)
^ — 1)
The case most frequently
treated
is,
0
|
real parts,
{A
0; see
pG N
— A Qn =
Re X
(the principal case; see Section 1C)
A Qn e
GN = and t
.
The elements of A Qn are always assumed to be bounded (that and the roots of the characteristic equation 1
is
) CM
