Stability of Nonlinear Control Systems
MATHEMATICS IN SCIENCE A N D E N G I N E E R I N G A Series o f Monographs a n d Textbooks
Edited by Richard Bellman
The RAND Corporation, Santa Monica, California 1.
2. 3. 4. 5.
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TRACY Y. THOMAS. Concepts from Tensor Analysis and Differential Geometry. 1961 TRACY Y. THOMAS. Plastic Flow and Fracture in Solids. 1961 ARIS. The Optimal Design of Chemical Reactors: A Study RUTHERFORD in Dynamic Programming. 1961 J O S E P H LA SALLEand SOLOMON LEFSCHETZ.Stability by Liapunov’s Direct Method with Applications. 1961 GEORGE LEITMANN (ed.) . Optimization Techniques: with Applications to Aerospace System. 1962 RICHARDBELLMANand K E N N E T HL. COOKE.DifferentialDifference Equations. 1963 FRANKA. HAIGHT.Mathematical Theories of Traffic Flow. 1963 Discrete and Continuous Boundary Problems. 1964 F. V. ATKINSON. NonLinear Wave Propagation: with AppliA. JEFFREY and T. TANIUTI. cations to Physics and Magnetohydrodynamics. 1964 JULIUS TOU. Optimum Design of Digital Control Systems. 1963 HARLEY FLANDERS. Differential Forms: with Applications to the Physical Sciences. 1963 SANFORD M. ROBERTS.Dynamic Programming in Chemical Engineering and Process Control. 1964 SOLOMON LEFSCHETZ.Stability of Nonlinear Control Systems. 1965
In preparation DIMITRIS N. CHORAFAS. Systems and Simulation A. A. PERVOZVANSKII. Random Processes in Nonlinear Control Systems V. E. B E N E ~Mathematical . Theory of Connecting Networks and Telephone Traffic WILLIAMF. AMES.Nonlinear Partial Differential Equations in Engineering A. HALANAY. Differential Equations : Stability, Oscillations, Time Lags R. E. MURPHY.Adaptive Processes in Economic Systems DIMITRISN. CHORAFAS. Control Systems Functions and programming Approaches J. ACZEL. Functional Equations MARSHALL C. PEASE,111. Methods of Matrix Algebra
STABILITY OF NONLINEAR CONTROL SYSTEMS Solomon Lefschetz PRINCETON UNIVERSITY THE NATIONAL UNIVERSITY OF MEXICO THE RESEARCH INSTITUTE FOR ADVANCED
STUDIES ( R I A S ) , BALTIMORE, MARYLAND
1965
ACADEMIC PRESS New York

London
COPYRIGHT @ 1965, BY ACADEMICPRESS INC. ALL RIGHTS RESERVED. NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.
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PREFACE The object of this monograph is to present a concise picture of control stability as it has developed from the direct method of Liapunov. The main impetus for this theory came some fifteen years ago from the work of the Soviet mathematician, A. Lurie, and for a decade or so remained practically a Soviet monopoly. Recently however this monopoly has been broken mainly through the work of V. M. Popov of Romania and of Kalman, LaSalle, and the author in this country. New developments in the stability of control are so rapid that interesting results are being attained even as this volume goes to press. Although it is impossible, for this reason, to be completely up to date, it is hoped that the interested scientist, mathematician, physicist, or engineer will be able through reading this work to navigate by himself the turbulent waters of nonlinear control theory. It is hoped even more that the engineering “controllist” will find here fruitful material for his operations. The outline of the monograph is as follows: The first chapter dealing exclusively with the dimensions one and two is quite elementary, and above all does not appeal to vectormatrix technique. Nevertheless, many of the important concepts already make their appearance in this early chapter. However, readers with a fair grasp of linear algebra may pass directly to the next chapters. These chapters, I1 to VI, present the theory in what may be referred to as the prePopov period. Here we lean heavily upon vectors and matrices, except in Chapter VI in which the emphasis is on the discontinuous characteristic. Popov’s striking contribution is dealt with in Chapter VII, which is decidedly arduous owing to Popov’s extensive use of Fourier transforms and rather advanced analysis. The last control chapterChapter VIIIdeals primarily with a theorem weaker than Kalman’s completion of Popov’s second theorem. Our theorem rests upon an important lemma due to Yacubovich. However, in its proof we follow from afar Kalman’s noteworthy treatment. The last c h q t e r consists virtually of a few appendices with which it did not seem appropriate to interrupt the main text. The author wishes to express his thanks to Dr. Robert Gambill who read most of the manuscript and made numerous corrections and valuable V
PREFACE
suggestions. He feels that he owes a good deal to discussions with various RIAS colleagues, notably Dr. Kalman, Dr. LaSalle, and Mr. Kenneth Meyer. The fact that this is the author's second monograph to appear in the Bellman Series is a strong indication of his high regard both for the Series and for the excellent work of the Academic Press. Finally the author takes pleasure in recognizing his debt to the U.S. Air Force, Office of Scientific Research [Contract A F 49(668)12421, thc U.S. Army Ordnance Missile Command (Contract DA36034ORD35 I4 Z), and the National Aeronautics and Space Administration (Contract NASw718) whose support of our research stimulated this monograph. Cross references are to the Bibliography at the end, or to chapters in the monograph. Thus (III,2) or (111, $2) refers to $2 of Chapter 111, (III,2.4) to a statement or a relation 2.4 of $2 in Chapter 111; LaSalle [2] refers to item 2 under LaSalle in the Bibliography. November, I964
vi
SOLOMON LEFSCHETZ
Special Abridged Notations n.a.s.c. : Necessary and sufficient conditions E : Unit matrix If A is a square matrix A , = ZE  A , so that [ A z /= 0 is the characteristic equation of A C.C. : Completely controllable, complete controllability C.O. : Completely observable, complete observability f * g : Convolution of the functionsf, g q(a) = characteristic function
vii
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CONTENTS PREFACE.
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SPECIALABRIDGEDNOTATIONS .
INTRODUCTION
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V
vii 1
Chapter One. Introductory Treatment of Dimensions One and Two The Characteristic Function . . . $2. Systems of Dimension Unity. Direct Control 53. System of Dimension Unity. Indirect Control &I System . of Order Two .
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17
$3. Comparison with a Recent Result of Yacubovich . . $4. On the Utilization of Certain Complex Coordinate Systems . $5. Special Cases . . . . . . . .
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18 22 23 24
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$1.
Chapter Two. Indirect Controls 51.
Vectors and Matrices
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$2. Indirect Control. General Type
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5 7 8 13
Chapter Three. Indirect Controls (Continued) $1. lnvariance under Change of Coordinates . $2. Reduction of the Number of Conditions on the 53. Luric's Method and a Variant . . $4. Application to Systems of Order Two .
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Control Parameters
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27 28 31 33
Chapter Four. Direct Controls. Linearization Multiple Feedback 61. Direct Control: General Case .
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82. Direct Control, Standard Example $3. Reduction of an Indirect Control to a Special Direct Control . . . . . $4. Linearization of Direct Controls . . . $5. Linearization of Indirect Controls . . 56. Direct Control with Matrices B or C of Rank < 11. . 87. Direct Controls with Matrices B, C , of Rank n  1. . $8. Direct Control Whose Matrix A Has Zero as a Characteristic
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.
. . . .
. . . .
39 41 42 43 46 48 50
Root (Kenneth Meyer) . $9. Direct Control Whose Matrix Has a Pair of Conjugate Pure Imaginary Characteristic Roots (Kenneth Meyer) . . . . . . . . . . 510. Multiple Feedbacks
51 55 56
ix
CONTENTS
Chapter Five. Systems Represented by a Set of Equations of Higher Order Generalities 92, A Digression on Linear Systems $3. Indirect Control . . . pt. Indirect Control: An Example . 95. Direct Control . . . $1.
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Continuous Approximation of Discontinuous Characteristics 92. Direct Discussion of Discontinuities . . . . 93. Some Examples . . . . . . . 9. Special Switching Lines . . . . . . 95. Multiple Feedback Switching Line . 96. Complementary Remarks . . . . .
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61 62 64 67 70
Chapter Six. Discontinuous Characteristics $1.
72 13 79 84 85 86
Chapter Seven. Some Recent Results of V. M. Popov Generalities. The Theorems of Popov . Preliminary Properties . . . Proof of Popov’s First Theorem . . The Generalized Liapunov Function of Popov Proop of Popov’s Second Theorem . Comparisons . . . . . . On the Function G ( z ) as Transfer Function. . . . . Direct Control. Conclusion .
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87 91 93 98 100 101
104 105
105
Chapter Eight. Some Further Recent Contributions Controllability and Observability . . . . . . Reduction of the System to One with a Completely Controllable Pair ( A , b) and Completely Observable Pair (c’, A ) . . 93. A Special Form for Systems with Completely Controllable . . . . Pair ( A . h ) . Main Lemma (Yacubovich and Kalman) . . . . . LiapunovPopov Function and Popov Inequality . . . Fundamental Theorem . . . . . . . A Recent Result of Morozan . . . . . . Return to the Standard Example . . . . . Direct Control . . . . . . . . Resume (Indirect Control: y > 0) Complement on the Finiteness of the Ratio cp(a)/a X
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.
107 109
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I I3 114 1 I8 1I Y
121 122 123 124 125
CONTENTS
Chapter Nine.
Miscellaneous Complements
The Jordan Normal Form for Real or Complex Matrices $2. On a Determinantal Relation . . . . . $3. On Liapunov’s Matrix Equation . . . . $4. Liapunov and Stability . . . 81.
Appendix A : An Application o f Multiple Feedback Control
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Appendix B : An Example from the Theory o f Nuclear Power Reactors (Kenneth Meyer) , .
BIBLIOGRAPHY INDEX
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128 132 133 136 139
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142 144 149
xi
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INTRODUCTION In the present monograph our concern will be solely with socalled real dynamical systems ; that is, with systems governed by a finite number of real parameters and whose performance is described by a finite set of ordinary differential equations. In order to present a full and clear picture, we propose to describe briefly in this Introduction the various simplifications which experience or practical considerations have imposed. Let x1,x2,...,x, be the parameters or coordinates and t the time. Time derivatives are written 1,3, ... . The equations of the system are then of the form (1)
&(xl )...)x,, x1)...,XI ,..., t ) = 0
( h = 1, 2 ,...)p).
As is well known, such a system is equivalent to another system of the general type
(2)
~XY,,...,ymrj1,...,jm, t ) = 0
(s = 1, 2,..., m).
Our first assumption is that the system (2) may be solved for the derivatives
)il,...,j,,thus giving rise to a new system
(3)
y j = F j ( y ,,..., y,, t )
j = 1, 2,..., m.
If we consider the y j and the F j as components of column vectors y , F , ( 3 ) may be given the simple form (4)
3 = F(Y, t). 1
INTRODUCTION
Let now y = q(t) be a particular solution of the system (4) and suppose that for some reason (e.g. stability) we are interested in the behavior of the solutions of (4) in its vicinity. Often the simplest way is to take as a new coordinate vector the vector difference z = y  q(t) and substitute in (4). This yields
i=3

Q(t) = F(y, t )  Q(t) = F(z
+ q(t),t )  ti(t),
which is of the form (5)
i = C(Z, t),
G(0, t ) = 0,
with the origin z = 0 corresponding to our earlier special solution y = q(t).The origin is a “pointsolution’’ or critical point of the new system (5). We are now merely interested in the neighborhood of the origin. Mainly for reasons of mathematical convenience we shall only consider systems “without t” at the right:
i = G(z),
(6)
G(0) = 0,
or autonomous systems. Main reason : they are so much more manageable than nonautonomous systems. Let C be an autonomous n vector system f = X(x).
(7)
One may be especially interested in what happens to the solutions in the neighborhood of the origin x = 0, which we assume to be a critical point : X ( 0 ) = 0. Suppose also that the system contains a control mechanism aiming to maintain the solutions as close as possible to the solution x = 0. Let it be assumed finally that one has been able to divide the components x l ,..., x , of x into two sets: the system components y l ,..., y , and the control components z l ,.... zq, where p q = n. The system components define the state p vector y (describing the state of the system) and the z,, define the control q vector z. The system assumes then the form
+
(a)
(8)
3 = Y(Y,Z),
(b)
Z(0,O) = 0.
Y(0,O)= 0, it, is
i = Z(Y,Z),
The system without control or the fundamental system, as we shall call
3 = Y ( y ,0).
(9) 2
INTRODUCTION
The purpose of the control mechanism may now be stated as aiming to secure or possibly to improve the asymptotic stability behavior of the origin, principally as regards the state vector y. Now in this general form the system is only rarely tractable and so one must have recourse to partial or complete linearization. Complete linearization has been applied most extensively in control theory and technique. In substance it replaces the system (8) by linear approximations. Linear equations with constant coefficients are easily dealt with, and under the linearity assumption one may proceed quite far, so this is a wellbeaten path only touched upon in Chapter V. In what follows the system but not the control vector will be linearized. As the fundamental system is generally quite stable, this is reasonable enough. The control, however, may involve a servomotor operating beyond a reasonable linearization of its characteristic. Our scheme will then assume the general form
3 = Ay
+ F(z)
i = C y + G(z).
Actually we shall deal almost exclusively with the case where the control variable z is a scalar (onedimensional) and with the numerous related problems that even this relatively simple case presents. There are two basic types: (a) direct control; (b) indirect control. They may be represented jointly by a vectormatrix system f = A X  0 for all o # 0, i.e., q(o)has the sign of o. 111. The integrals
.r:
q(o)do diverge. Ix
The first property states in substance that the graph of q(o)has no jumps; the second property states that the graph is situated in the first and third quadrants ; finally according to the third property the area under the graph tends to infinity at both ends.
FIG. I
As a matter of fact it will be shown in (111, $4) that property I11 may usually be dispensed with.
The function @(o). defined as
=r
This function, of constant occurrence later, is @(a)
q ( a )do.
0
I t is continuous, zero for o = 0, positive otherwise as a consequence of 11, and + cc as a , f 00 as follows from 111.
6
$2.
SYSTEMS OF DIMENSION UNITY. DIRECT CONTROL
$2. Systems of Dimension Unity. Direct Control Consider a system depending upon a single real variable. As described in the Introduction, one may select this variable, e.g., x , so that the desired position to be controlled corresponds to the value x = 0. Thus x represents the deviation that it is desired to minimize. Let the fundamental system be represented by a linear equation
(2.1)
X =
kx,
where k is a constant #O. The purpose of the control would be to accentuate the “return to zero” of the variable x . There are now essentially two distinct possibilities according to the manner of operating of the feedback. In a direct control it operates directly upon the system, in an indirect control it operates through one or more derivatives. In the former the operation is apt to be rather hard, through a ponderous mechanism. In an indirect control on the other hand, by making use of derivatives one may operate through a comparatively light scheme. For this reason indirect controls are often preferred to the other kind. At all events, mathematically, they are also much more interesting and for this reason they have been studied more extensively. In the present section we first deal with direct control. Physically speaking, a direct control regulating the system (2.1) is described by a system
(2.2)
i= kx
+ 0. C
There are now two possibilities; (a) k < 0. That is the basic system is already stable. To increase this feature one merely needs to increasef(x) for all x. Since ucp(u) > 0, and  kx2 > 0 it is sufficient to choose the parameter c negative. (b) k > 0. To havef(x) > 0, one must first select c negative and also such that the graph of cp(  cx) passes below the line y = kx when x < 0 and above it when x > 0. By this means the control will change the unstable system (2.1) into a stable system: all solutions + 0 as t + 00.
53. System of Dimension Unity. Indirect Control The system just treated could hardly be more elementary. Indirect controls of one dimensional systems however present many of the features and complications of higher dimensional systems. We begin by choosing as fundamental system i= kx,
(3.1)
k > 0,
so that the initial system is already stable. The effect of the control is to replace the system of order one (3.1) by a new system of order two,
where the constants b, c, p are the control parameters. It is evident that one must have b # 0, since b = 0 means that the control has no effect upon the system. But then one may replace the coordinate x by a new coordinate x* = x/b. This is merely a change of scale for x. The new system assumes the form X* = kx* + (
8
$3.
SYSTEM OF DIMENSION UNITY. INDIRECT CONTROL
Upon writing now x for x* and c for bc we have the system X =
e
(3.3)
(T
kx+5
= cp(4 = cx  p5,
which is like (3.2) but with b =  1. The control parameters are now c, p. Regarding the existence and uniqueness of solutions see the end of the Introduction. The fundamental system (3.1) is already asymptotically stable and one wishes to strengthen this properly. In fact, one would like to solve : Lurie’s problem. To find n.a.s.c. for the asymptotic stability of the system (3.3) regardless of the initial conditions and whatever the choice of admissible characteristic function cp(o). That is, one would like to have n.a.s.c. to guarantee that any solution (x(t),y 0
(
for all choices of y and ~ ( a )This . will happen, since pk > 0, provided that both roots of 2pku22 p +  u + p = O
(
3
11
1.
INTRODUCTORY TREATMENT OF DIMENSIONS ONE AND TWO
are complex, i.e. if
Let p l , p z be the two roots
[email protected]) 6 P1
+ Pz
= (C 
=
0 and 6 its discriminant. We have
2pk)’  C’
= 2Pk  c,
=
4pk(pk  c),
PIP2 =
C‘
4’
Now (3.9) requires that the roots off(p) = 0 be real and distinct and that p1 < p < p 2 . The roots will be real and distinct if, and only if, 6 > 0, or pk > c. Since one must be able to choose p > 0, one must have p 2 > 0. Since p1p2 > 0, both roots have the same sign and this sign is that of p1 + p 2 . Hence we must have pk > c/2, which holds if pk > c. Thus this last condition which is just (3.6) is sufficient to guarantee absolute stability. In the case just discussed, there is a certain unreality, for owing to k > 0, the fundamental system is already largely asymptotically stable. The effect of the control is then to make it more so. A more natural situation takes place when the fundamental system is actually unstable: i= kx,
k > 0,
and the effect of the control must be to make the system stable. This will occur only with a characteristic of a certain special type. To simplify matters, assume that for 101 small cp(o) has a power series expansion
do) = go + @(a) where g is a positive constant, and @ begins with terms of degree at least two, and otherwise has the properties 11, 111 earlier imposed on cp. As cp is not precisely fixed, we assume that g may vary within certain limits: 0 < u S g 5 8. Notice that in substance, the graph of cp is now a curve with positive slope g through the origin, and that this slope may vary between u and (Fig. 4). The equations with control are thus
3 = ky
+ go + @(a)
(5 = cy  pgo 
[email protected](o)*
For 101 sufficiently small, one may achieve asymptotic stability if the roots r l , r2 of the equation 12
$4.
SYSTEM OF ORDER TWO
=
r2
+ (gp  k)r  gc  pgk = 0
have negative real parts. (LaSalle and Lefschetz [ l , p. 481.) Now r l r 2 = g(c
+ pk),
rl
+ r2 = k  pg.
Whether r I ,r2 are both real or complex one must have r l r 2 > 0,
i.e. c
rl
+ pk < 0,
+ r2 < 0,
k  pg < 0,
and hence finally k C 0,
and let the system with control be xi = kixi  ( b , (4.2)
ri = d o ) a
=C]Xl
+
c2x2
i = 1,2
 p5
where q(a) is an admissible characteristic function. For existence and uniqueness of the solutions see the end of the Introduction. 13
1.
INTRODUCTORY TREATMENT OF DIMENSIONS ONE AND TWO
Since q(o)= 0 if and only if o = 0, the origin x i the only critical point provided that the determinant
=
0,
5
=
0 will be
Upon expanding A and dividing by k l k 2 # 0 this condition becomes P
(4.3)
+ b1c1 kl
b2c2 k2
+fO.
Assume for the present that it is fulfilled. As in the previous case it is convenient to apply a coordinate transformation from the variables x l , x 2 , 5 to new variables y , , y,, o defined by
(4.4) a = ClXl
+ CZX2  pt.
The determinant of the transformation is A, and so, under the assumption that (4.3)holds, the transformation is nonsingular. Thus again the stability properties will not be affected by the transformation. The differential equations in the new variables are found at once to be
3. =
 k . y1 .I  b Ida), .
(4.5) ci =
ClYl
+ CZY2

2
= 132
Pd4.
In imitation of the treatment of the preceding case, consider the surfaces
V ( y ,a) = y 1 2
+ y z 2 + @(a) = u2.
They represent surfaces of revolution around the a axis generated by the curves y,Z @(a) = u2.
+
We have seen that these curves are concentric ovals around the origin. Hence our surfaces are likewise concentric ovals (of dimension two) around the origin. 14
$4.
SYSTEM OF ORDER TWO
We must find conditions under which the space ovals are all crossed inward by any path y of (4.5). Along such a path VY94
=
2Y1(k,Y,  cp(4bl) + 2YAkzYz

cp(4bZ)
+ ~ C J ) ( ~ I+Y cI z ~ z) PCP’(O) =  2 ( k i y i 2 
+ k,yz2)  pq2(a)
2dlY l c p ( 4  2dzYzcp(d
di = 6,  f c i ,
i = 1,2.
The expression V is to be negative for all y , , y , and not all zero. Since (P((T) = 0, if and only if, 0 = 0, this is the same as demanding that  V be positive for all y , , y , and cp(c) not all zero. This is the condition that all the paths penetrate every oval without exception. Now
P= 2P1(Yl +
$y
+ k,(Y, +
$y]
A sufficient condition for absolute stability is then d12 2k1
p>++.
(4.7)
dZ2 2k,
The righthand side is positive unless d l = d , = 0. In any case (4.7) implies that p > 0. It is interesting to compare the inequalities (4.3) and (4.7). Identically
( b  f~)’+ 2bc = ( b + f~)’2 0. Hence
(b1  fcl)’ 2k 1
+ ( 6 , 2kZ$c,)’
>   b1c1 
kl
bzc, kz .
That is the inequality (4.7) implies (4.3). Hence (4.7) alone i s a sgficient condition for absolute stability. This result is a special case of an analogous and more general property due to LaSalle and taken up later (111, §4). 15
1. INTRODUCTORY TREATMENT OF DIMENSIONS ONE AND TWO
Direct control. The controlled system is now
x . =  k . xI .,  b .
, 4 4 4 9
(4.8)
a
= ClXl
i = 172
+ c2x2.
In the absence of ( one will not need to change coordinates. However, a difficulty arises regarding the condition that the origin be the only critical point. This requires that the relations kixi
+ birp(clxl + c2x2)= 0,
i = 1,2
have the origin as their unique solution. In the absence of further information as to q(a)we can only assume that this is the case. Choose again V(x1,x2) = X I 2
+ x22 + @(a).
Since @(a) > 0 for all a # 0 and @(O) = 0, Vis positive for all x l , x 2 not both zero. From (4.8) there follows 6 = clf1
(4.9)
+ ~ 2 x 2=  k , c , x i
&r 
+
k 2 ~ 2 ~ 2(bici
By analogy then with the preceding case we find
V
= 2jk1( x 1
+
+ k2(x2 +
+ b2~2)~(0).
$y]
:;fI ):;fI
+ (p     V 2 ( 4 However, as shown in (IV, $1) one can only achieve here (4.10)
It will still guarantee absolute stability, but this is as far as we will proceed in this direction.
16
Chapter
2
INDIRECT CONTROLS With this chapter we initiate the general study of controls for n dimensional systems and it is continued in the next chapters. For the convenience of the reader some notions introduced in the previous chapter will be repeated. For the characteristic cp(a) however, we shall depend upon (1, $1). In dealing with a general dimension, vectors and matrices are obviously advantageous and they are discussed in $1.
$1. Vectors and Matrices By and large the notations and designations are those of LaSalle and Lefschetz [l, Chapt. I]. As a rule a, b, c,..., x, y will denote vectors and A, B, C,...,X ,Y will denote matrices, while small Greek letters will stand for scalars. In particular x denotes the column vector, one column matrix, whose components are xi, and A the matrix (a. ). !k The designation En, or E when n is obvious, represents the unitmatrix of order n. The transpose of a matrix A = (aij) is the matrix, written A’, whose elements are the symmetric of those of A relative to its principal diagonal. In particular the 1 x n matrix x’, corresponding to x above, is the onerow matrix: row vector, with the same elements as x. 17
2.
INDIRECT CONTROLS
If A = (ajk)is a complex matrix A* denotes the matrix ( a k j ) that is Abstractly ( )* = ()’. Note that if A is square and nonsingular so that its inverse A  ’ isdefined, the operation of inversion commutes with ()’ or ( )* as the case may be (proof elementary). That is (A I)’ = ( A ’ )  or (A ’)* = ( A * )  l , for A square and nonsingular ;also for any A : (A)’ = Note also the following property. If x, y are both n vectors, then (cijk)’.
’ v).
x’y
=
cxjyj
=
y’x.
Recall that xy’ represents the n x n matrix XY’ = ( X j Y d
The designation V ( x ) stands for a Liapunov type scalar function of the vector x . The gradient operator ( d / d x l ..., , d/dx,) is thought of as a onerow matrix operator d/dx. Thus dV/dx represents the row vector with components dV/dx,. For easy reference we repeat that an n x n matrix A, whose characteristic roots all have negative real parts, is said to be stable. If the quadratic form x’Fx or hermitian form x*Hx are > 0 [ < 01 for all x # 0 we write F or H > O [ 01. For further remarks on quadratic or hermitian forms see (IX, $1).
=
92. Indirect Control. General Type Recall that the difference between direct and indirect control is that the operation of the feedback is direct in the first and indirectthrough a derivativein the second. In our treatment indirect controls will serve as the predominant model and so they are fully discussed first. Let the fundamental (state) system be R = A x where x is an n vector and A a constant n x n matrix. Let the control depend upon a single coordinate 5. The indirect control system is (Lurie [l]) i= AX  ( b
g=
(2.1)
0
=
cp(4
c‘x  p5
where b , c are constant n vectors and cp is an admissible characteristic. This time we assume explicitly that A is nonsingular. 18
$2.
INDIRECT CONTROL. GENERAL TYPE
For the existence and uniqueness of the solutions see the end of the Introduction. Lurie's problem. To find n.a.s.c. to have (2.1) asymptotically stable in the large and this for all choices of an admissible cp(a). It implies that all solutions (x(t), 5(t)) of (2.1) , 0: x = 0, 5 = 0, as t , + 00 and this whatever an admissible cp(a). This is absolute stability. Owing to the special role of the parameter (T in relation to the system (2.1) it is manifestly desirable to choose coordinates with (T one of them. This is achieved by the transformation (2.2) given below. The important point is that the system in the new coordinates possess the same asymptotic stability properties as the initial system. To that end the only requirement is that the transformation be nonsingular. For it is then readily shown that if, in the initial system, the origin is asymptotically stable for all admissible cp, then the same property holds in the new system (see LaSalle and Lefschetz [l, p. 771). Take then the transformation (x, 5 ) + (y, (T) given by
(2.2)
y
=
AX  bl,
(T
= C'X  p 0. Hence the matrix of this linear system
(;4
 pPP b)
must have no characteristic roots with positive real parts. Now for p small, these roots are very near those of A and the root zero. Hence A must have no root with positive real part. Let a matrix whose characteristic roots all have nonpositive real parts be referred to as critical or semistable. Thus aJirst necessary condition for absolute stability is that the matrix A be semistable. Since semistability brings about many complications (discussed in IV) we assume for the present that A is stable. Now the only general method available for absolute stability rests upon Liapunov’s asymptotic stability theorem (IX, 4.5) plus the Barbashin and Krassovskii complement (IX, 4.7). Following Lurie and Postnikov we look for a Liapunov function of type V(y,a) = y’By
(2.6)
+ @(a),
and calculate its time derivative along the paths of (2.3):
V =YBy + yBj
Hence from (2.3):  V = y’Cy
(2.7)
d
+ cp(0)6.
+ pcp2(o)+ 2cp(a)d’y =
Bb  )C
and more importantly we have the Liapunov relation A‘B
(2.9)
+ BA = C.
Referring to Liapunov’s theorem we must first have V ( y ,a) positive defifor all values of y and a. This requires that B > 0. When nite (see IX, g), this condition is fulfilled we will have V > 0 if y # 0 and also > 0 if y = 0, a # 0 since then @ > 0 (property I1 of I, $1). This will actually hold for all admissible functions cp(a). Finally since B > 0 the expression Iy’Byl + 00 with IIyII, and owing to property 111 of (I, $1) @(a) ,co with 20
$2.
INDIRECT CONTROL. GENERAL TYPE
llyll + 101. Thus the requirement of the Barbashin and Krassovskii complement is actually fulfilled. That is the requirements on V(y,a) for absolute stability are satisfied through the mere fact that B > 0. There remains then to arrange matters so that  V is positive definite for all y, a. Here we have the fortunate circumstance that  contains a only through cp(a), and is a quadratic form in y and cp. (This is the great merit of the Lurie and Postnikov type of function V ) It suffices therefore to demand that  V be a positive definite quadratic form in y and cp. The n.a.s.c. due to Sylvester, is that the principal minors of the matrix
c 2)
all be positive. In particular this must hold for C and so C > 0, hence ICI # 0. Beyond this we still require that the determinant
Referring then to (IX, 52) this yields the fundamental inequality (Fi)
p
> d’C’d.
Notice incidentally that it implies that
(2.10)
P
> 0,
an inequality obvious enough since it merely states that  V(0,a) > 0. Since C > 0 implies B > 0, the only conditions left are C > 0 and (Fi). Hence
(2.11) Theorem. N.a.s.c. in order that the Liapunovfunction V(y,a) of (2.6) be positive definite for all y, a, and that  V =  v(y(t), a(t)) along the solutions of (2.3) be a positive definite quadratic form in y and cp (hence positive definite in y, a) are C > 0 and (Fi). When these conditions are fulfilled the system (2.3) (hence also (2.1)) is absolutely stable. PROOF OF NECESSITY. Since  V is a positive definite quadratic form in (y, cp) we must have C > 0 and (Fi).
PROOFOF SUFFICIENCY. If C > 0 and (Fi) hold  is a positive definite quadratic form in (y, cp). From C > 0 follows then by Liapunov’s relation B > 0, and hence by our earlier argument that V is positive definite for all y, a and all admissible functions cp(o). 21
2.
INDIRECT CONTROLS
(2.12) Properties C > 0 and (Fi)imply (2.5). They imply absolute stability and hence that the origin is the only critical point, from which (2.5) follows. Another proof (algebraic) of (2.12) will be given in the next chapter (111, 2.2). The preceding scheme will guide us throughout the sequel and we formulate : General rule. In seeking absolutestability we will usually lookfora Liapunov function V ( y ,a) positive definite in (y, a) such that  V is a positive definite quadratic form in (y, cp).
Of course any other state variable, for example x, could replace y.
93. Comparison with a Recent Result of Yacubovich The inequality (Fi) may be phrased as follows: (3.1) A n.a.s.c. in order that the quadratic form in (y, 9 )  V = y’Cy
+ pcpz + 240 d y
be positive definite is that C > 0 and (Fi)hold.
Now in Reference [2] Yacubovich obtained the following result : (3.2) A n.a.s.c. for the positive definiteness of the same form is that there exists a real vector g
=
df
f i such that C  gg‘ > 0.
Evidently the two statements are equivalent. To show that this is indeed the case one merely needs to “complete the squares” in two different manners. Incidentally there will result a new derivation of the inequality (Fi). The “square completing” corresponding to (3.1) is indicated by : 
V = (y‘ + cpd’C’)C(y
+ cpC’d) + (pd’C’d)cp2
and it is plain that, since C > 0,  V is positive definite if, and only if, (Fi)holds. The square completing corresponding to (3.2) is
22
$4.
ON THE UTILIZATION OF CERTAIN COMPLEX COORDINATE SYSTEMS
REMARKS.I. The only conditions imposed upon the control parameters are (2.5) and (Fi). 11. Actually a less stringent road to absolute stability is often available. Namely, according to a result of LaSalle (see IX, 4.8) it is often sufficient to have  V 2 0 provided that y = 0, a = 0 is the only solution of (2.3) in 0. However to prove this fact frequently requires considerable labor. It has therefore seemed best to adhere to the general rule.
v=
$4. On the Utilization of Certain Complex Coordinate Systems Referring to (IX, $1) one may have occasion to utilize a complex coordinate system in which a real point x is represented as follows: XI,...,X p ,
XI,..., 5,
X 2 p + I,..., X n
where the last q = n  2 p coordinates are real. In particular the real vectors b, c have components bl,...,b,,
h,...,Lp,
b2,+ l,..., bn
Cl,..., C p ,
C1,**., C p ,
C z p t I,..., C n
with the last 4 components real. The associated matrix A is in the Jordan normal form as described in (IX, $1). This time, if the matrices B, C are hermitian (positive definite), there takes place the Liapunov complex relation. A*B
+ B A = c,
and B is still unique. Upon introducing for convenience cp naturally V ( y , a ) = y*By
= @,
one takes
+ @(a).
Since now d = c*y  pcp(rJ),
one finds  V = y*cy
+
[email protected] + (@*y + qy*d), 23
2.
INDIRECT CONTROLS
where as before d
=
Bb

$c.
Since Sylvester's theorem holds for hermitian forms, or else utilizing the reduction of 93 leading to (3.1), we have this time instead of (Fi) p > d*C'd
Fi*)
with C > 0 as n.a.s.c. to have  positive definite hermitian in (y, cp). Since real quadratic forms are merely special hermitian forms, the present development includes the earlier theory.
$5. Special Cases Since after all C is a completely arbitrary matrix > O one may specialize it and obtain a simpler form of (Fi)or (Fi*).Take in particular C = E. Then (Fi)or (Fi*)reduces to p
(5.1)
> d*d
=
lld1I2.
This may be achieved by using a special coordinate system in which C = E. Of more interest is the case where the matrix A is relatively simple. Suppose that A diag(&,..., An), so that the 1, are the characteristic roots of A. Assume the coordinates so chosen that actually A = diag(Al,...,An). This will be called the standard example. Set once and for all

Re,& =
(5.2)
ph
> 0.
Referring to (IX,3.9) we have this time
Hence the components of d = Bb  3c are
and we obtain directly d*C ' d in terms of C. Take in particular C = diag(a,,...,an), ah > 0. As a consequence
B 24
=
(z'
diag ,...,
J;,

$5.
SPECIAL CASES
Thus (Fi) assumes the form
(5.3) The sum is a function of the variables P h , which are as yet purely arbitrary, except that they must be real and positive. If we obtain a minimum of this sum we will have a lower bound for p. To minimize the sum one merely has to minimize its individual terms, i.e. terms of the form
Here one must distinguish several cases.
A. 1 is real, hence b and c are also real. Then
;(
F(P) =
 ;)2.
If c = 0 the minimum of F(P) is obtained for = 0 and then F(0) = 0. If b = 0,c # 0, the lowest bound of F(P) corresponds to = co and is again zero. It is obviously zero if b = c = 0. Suppose bc # 0 and say bc < 0. One may assume that b < 0, c > 0. Now
F‘(P) = 2(;
);
+
(5 ;). 
The second factor is # 0, so that F’(B) = 0 only for P2 =  cp/b. Hence, FrniJP) = P b c / ~ l . Suppose now bc > 0. We may take b > 0, c > 0. Then in F’(P) the second factor alone may vanish and the minimum is zero. Note that in both cases
25
2.
INDIRECT CONTROLS
B. i, is complex, hence h, und ch ure both complex. This time one must deal directly with F ( P ) . Here
The cases h = 0 or c = 0 yield the same result as before. We may therefore assume that hc # 0. Now
for /I2
= CI = p
m = plc/bl (as before) and the minimum is
For the real case, this expression coincides with the one already obtained. Therefore, finally one obtains for p the lower bound (5.4)
26
Chapter
3
INDIRECT CONTROLS (continued) The general theory developed in the preceding chapter presents a number of side questions to be discussed in the present chapter.
81. Invariance under Change of Coordinates There have been obtained two inequalities involving expressions depending upon the coordinates, namely (11, 2.5) and (Fi).We shall show that these inequalities do not depend upon the coordinate system. To be precise, let y = Py0 be a transformation of coordinates and let A,, Bo,..., be the expressions corresponding to A , B,..., in the new coordinates. Let also (Fio),(11, 2.5)' correspond to (Fi)and (11, 2.5) in the new coordinates. We propose to prove: (1.1) (Fi)and (11, 2.5) are respectiuely equivalent to (Fio)and (11, 2.5)'.
This means that one is free to express these inequalities in any convenient coordinate system. The transformation y = P y , gives rise to a new system )" = P  ' A P y o  cp(a)P'b
6
=
c'Py,  pcp(a).
27
3.
INDIRECT CONTROLS (CONTINUED)
Hence A , = P'AP,
bo = P  ' b ,
c,'
=
c'P.
Consequently y'Py
= y,'P'BPy,
so that B, = P'BP, and also at once c,'A, b, = c'A ' b . This proves the invariance of the inequality (11, 2.5). We also have
do
Moreover
=
B,bo  *c, = P'BPP ' b  ~ P ' = c P'd.
c, 1 ( p ' c p )  l
and therefore
= plclpt1,
' '
do'C; 'do = d p p  C  p'  P'd = d C  ' d .
Since p is invariant we have p  d C  ' d = PO  do'C,'do
so that (FJ is likewise invariant.
52. Reduction of the Number of Conditions on the Control
Parameters
A variety of conditions have been imposed upon the control parameters b, c, p, cp(a). As we shall see, these conditions are far from independent and hence can be sharply reduced in number. They are: On cp(a): properties I, 11, 111 of (I, $1); notably 111 has been imposed to guarantee asymptotic stability in the large, i.e. in particular, that every solution (y(t),a(t))of (11, 2.3) 0 as t + 00. On b, c, p :
+
(2.1)
p # c'A'b,
likewise (Fi) for some C > 0. Actually it was shown in (11, 2.12) that, indirectly C > 0 plus (Fi)imply (2.1). A more complete result is the following proposition due to LaSalle [l]. (2.2) Theorem of LaSalle. Let A be stable and C (hence B ) > 0. Then d C  ' d > c'A'b.
(2.3) 28
52. REDUCTION OF THE NUMBER OF CONDITIONS ON THE CONTROL PARAMETERS Moreover theorem (11, 2.1 1) is still valid without imposing on q(a) property I11 of I , $1 (divergence o f t h e integral @(a) as a , f 00).
(2.3a) Corollary. C > 0 plus (Fi)imply (2.1) in the strongerform p > c'A'b. REMARK. A
weaker form of (2.3) was obtained earlier by Yacubovich.
PROOFOF (2.3). This is the crux of the matter and the more difficult part of the argument: we first show that if x , y are two n vectors then (Bx  y)'C ' ( B x  y ) 2 2y'A ' x .
(2.4)
By choosing x
=
b and y = fc, this yields
d C  ' d 2 c'A'b,
(2.5)
which is (2.3). Thus, we have to prove (2.4). This will be done through another relation. Incidentally the necessity of the relation (2.6)
p # c'A'b
for absolute stability was also proved in (11, $2).
Let, then, u, v be two arbitrary n vectors, and Q an n x n matrix such
that
Q
(2.7)
+ Q' > 0.
Note that in view of u'(Qu) = (Qu)'u = u'Q'u, we have (2.8)
u'Qu = iu'(Q
+ Q')u > 0
for all u # 0.
Moreover (2.8) implies that Q is nonsingular. For otherwise there would exist a vector u # 0 such that Qu = 0 and hence u'Qu = 0. It follows that Q' exists. Now in replacing in (2.8) u by Q  ' u # 0 if u # 0, there follows u'(Q ' ) ' Q . ( Q  ' u ) = u'(Q ' ) ' u if u # 0. Hence also u'*Q'u > 0
(2.9)
if u # 0. Thus actually Q ' likewise satisfies the relation (2.7) (Q replaced by Q '). Let now a = (u' v'XQ Q')(u v )  2u'(Q Q')Q'(Q Q')v,
+
+
+
/? = ~ ( u ' Q ' u'Q)Q'(Qu
+

+
Q'u). 29
3.
INDIRECT CONTROLS (CONTINUED)
From (2.9) one infers that fl >= 0. On the other hand by expanding both a and p one verifies that a = p and so a 2 0. Upon replacing now u and u in a by (Q + Q')'u and (Q + Q')'u there follows (u' + u'XQ + Q')'(u + U)  2 u ' Q  l ~2 0. Set now Q = BA, u =  y , u = Bx. This is legitimate since
Q There results then (Bx
+ Q' = (A'B + BA) = C > 0.
 y)'(A'B + BA)'(Bx
 y ) 2 2y'A'x
which is precisely (2.4). Thus (2.4) is proved and so is (2.3). Since p  c'A'b > 0, the transformation (x, 5 ) + (y, a) of (11, 2.2), is justified and so we are at liberty to keep on operating with the variables y, a,that is with the system (II,2.3).
PROOFTHAT J(y(t))(+ (a(t)J+ 0 as t + tions the function
v
+ co. Under
our assump
w ,4 = y'By + @(a)
is positive definite and is negative definite throughout the space (y, a). Hence (Liapunov, IX, 4.5) the origin is asymptotically stable. Referring to LaSalle and Lefschetz [l, p. 66, Theorem VIII], it is therefore sufficient to show that the solution (y(t),a(t)) is bounded. The first step is to show that it is defined in the future, that is, that there is no finite positive T such that (ly(t)(l + lcr(t)l + co as t + T from below. Take any 0 < to < 'I: Since 0 we have for t > t o :
v
V(y(t),m.
There is also an a > 0 such that Y'BY L alJYJJ2
for all y. Hence for t > to
vo > allAt)112 + @(o(t)). Therefore I(y(t)(l is bounded for t = T and so for all t. Moreover @(a(t)) is also bounded. This may happen through a(t) being bounded for all t, or else through la(t)(+ co with t. All that is required is to exclude this second possibility.
30
$3.
LURIE'S METHOD A N D A VARIANT
Suppose then that lo(t)l + m. As a consequence for some T and t 2 'I:o(t) will have a fixed sign, say a(t) > 0. (The case o(t) < 0 would be treated in the same way.) Now 6  C'JJ = C?  c'(A'j
+ A'bcp(o)) = pcp(o).
Hence in view of (2.3a) d 44)  c'A'y(t)) =  ( p  c'A'b)cp(a) < 0. dt
Since by hypothesis o(t) + + 00 with t, and since Ily(t)II remains finite we may choose T so large that for t 2 T we have o(t)  C'A  ' y ( t ) > 0.
Upon integrating then from T t o t we obtain [o(t)  C'A  'Y(t)]'T
< 0.
As a consequence, since Ily(t)II is bounded, so is the proof of the theorem.
o(t). This
completes
$3. Lurie's Method and a Variant By taking a more restricted type of matrix C and assuming that the characteristic roots of the matrix A are all distinct, Lurie obtained a narrower sufficiency condition for absolute stability than the inequality (Fi). In presenting these results we will operate at once with complex coordinates, that is take the situation of (11, $4). Now Lurie's choice of B and C is as follows: C = ad' diag(a, ,..., an), ah 0
+
B= 
(*) + Aj
where
lj =  p j
=
jlk
+ diag
+ ivj,
(2,$) ...,
p j > 0.
Note that
JJ *cY = JJ * a * y
+ EphYhyh 31
3.
INDIRECT CONTROLS (CONTINUED)
Hence likewise B > 0 (mere verification). Then Lurie completes the squares in the expression of and obtains cross products of type y*y, whose coefficients, set equal to zero, give suficient conditions for absolute stability. The computations are definitely involved and we merely reproduce the final result with mild changes of notation. Observing that the only condition on the ah are that they be positive, we find for the Lurie system under the general Lurie assumption that every b j =  1 :
v
( j = 1,2,..., n).
(3.1)
His conclusion is that ifone canfind a real point a such that all the inequalities (3.1) are satisfied then the control is absolutely stable.
A variant. Observe that a simpler way to guarantee that positive definite is to take d = Bb  fc
(3.2)
=
v be
0.
With B, C and the general situation unchanged, the system (3.2) yields akbk
(3.3)
+
djx+k l j
ajbj 2pj
0,
2 j 
j = 1 , 2 ,..., n.
If one assumes, as done throughout in Lurie's equation that every bj =  1 then (3.3) reduces to (3.2), except of course, for the additional term in Our system is thus mildly more complicated, but the difference is not really significant. However, its derivation is altogether simpler than that of Lurie.
fi.
REMARKS. I. As observed by the author in a recent paper (Lefschetz [3]) the relation (3.2) was utilized earlier and independently by Mufti. 11. The number of parameters in the matrix C of Lurie's special type is 2n. Since it is not certain that they are independent, one may say that his system depends on at most 2n parameters. On the other hand, the general system: C arbitrary positive, depends upon n(n + 1)/2 parameters and this number >2n for n 2 4. In point of fact for n = 2 the two systems depend upon three parametersnot the apparent four since three is the maximum possible. Thus, for n 2 4,the class of Lurie matrices C is definitely special.
32
54. APPLICATION
TO SYSTEMS OF ORDER TWO
54. Application to Systems of Order Two We shall take advantage of the low dimension to study our system ($3, variant) more fully. And specifically, we shall discuss at length the properties of the relation (3.2). First, a preliminary remark. We will assume the coordinates y , , y , so chosen that the matrix A is in one of the Jordan normal forms. Moreover, we will also suppose that under these coordinates the vector h has both components b,, h, # 0. Observe that from the control point of view, if e.g. b , = 0, the first equation of the system reads Y1
=
IlYl
which indicates that y , is not affected by the operation of the control. In other words, practically speaking the control is only partly effective. Thus, our assumption that both h,, h, # 0 means that the control is completely effective. Under our assumption then, the change of coordinates y j + hjyj, j = 1 , 2 will yield the same system but with
This scheme will somewhat simplify the calculations. We continue to assume, of course, that the matrix A is stable. There are then the following three possible normal forms for A : I. 11. 111.
(:
diag(A,,A,),
diag(1, I),
A, and I , real and A
real and
I
complex.
0.
v2/p2 this inequality reduces to
~j(q0= ) CC;
+
+
40~;
~ O C 
We also have from (4.14) and (4.16) that c Hence we may write c =M
+ ig,
(70
~9,340> 0.
+ q,, must be real and negative.
=y 
ig,
where (4.18)
a+y
0.
Since we dispose of 4 let us choose it so that y # 0. Denote by a l , u2 the two roots of h(cc).Since a 1 a 2 < 0 both are real and of opposite sign, and M must be between them. Let u1 < 0. Since h (  y ) < 0, y is between the roots. Hence in order that (4.18) hold, one must take M < al. If that is done the present scheme based on (3.2) will effectively guarantee absolute stability.
38
Chapter
4
DIRECT CONTROLS, LINEARIZATION, AND MULTIPLE FEEDBACK $1. Direct Control: General Case The basic system is (1.1)
i= AX  q(0)b 0 =
c’x
where A continues to be a stable matrix and q(a)is the usual characteristic function. The system is of the same order n as the state system i= A x . We refer again to the Introduction for existence and uniqueness of solutions. The problem continues to be to find sufficient conditions for absolute stability. At all events for absolute stability the origin must be the only critical point. That is, the only solution of
(1.2)
A X  q(0)b = AX  q(c’x)b = 0
must be x = 0. In the absence of more information about q(a),all that one can do is to assume explicitly that this condition is fulfilled. In a moment 6 will be required. Here
(1.3)
& = c ’ i = c‘Ax  q(0)c’b.
39
4. DIRECT CONTROLS, LINEARIZATION, A N D MULTIPLE FEEDBACK
Upon setting (1.4)
CO =
A'c,
po
=
c'b
the relation (1.3) assumes the more familiar form 6
= co'x 
poq(a).
In the search for absolute stability the temptation is great to follow the same path as for indirect controls. Unfortunately this turns out to be only partly possible. Take the same Liapunov function as in (11, § 2 ) : V ( X )= X'BX
(1.5) Here again
 V = X'CX
(1.6)
A'B d
+ @(a).
+ 2 d x q + c'brp2
+ BA =  C
=
Bb  ~ A ' c .
Choose once more C > 0, and as a consequence B > 0. Hence V ( x ) is positive definite for all x and +m with IIxII. Thus V still behaves satisfactorily. Not so, however, regarding F! We still have (as in 11, $3):  V = (X
(1.7)
+ C'
drp)'C(x
+ C'
dq)
+ (c'b  d C  '
d)rp2,
and one would like to adopt c'b > d C  ' d
(1.7a) as a condition. However
V = (2Bx  crp)'.i
(1.8)
=
(2Bx  c ~ ) ' ( Ax bq).
This shows that V cannot be formally a definite quadratic form in the independent variables x and rp since, for example, one may have Ax  brp = 0 for x , cp not both zero. The best that one may achieve is V 0, and this will follow if we impose c'b = d C  ' d.
(1.9)
since then by (1.7) (1.10)

40
P
= (X
+ C'dq)'C(x + C  ' d q )
$2.
DIRECT CONTROL, STANDARD EXAMPLE
It is clear from the above that in the (n + 1)dimensional space x, cp (cp considered as independent of x)  V is positive semidefinite and furthermore (since C > 0)  is zero only on a onedimensional subspace of the space of x and cp. Now we shall make  V positive definite in x alone. By looking at (1.8) we can see that  is zero when Ax  bip is zero and since A is stable this system of equations has a onedimensional subspace of solutions. Thus  V is zero if and only if A x  bcp = 0. Now we take into account that x and cp do not in fact vary independently. We have already assumed that the only critical point for (1.1) is the origin, hence A x  bcp(c'x) is not zero and  V is positive definite in x alone. Thus the system (1.1) is absolutely stable. An easy way to find that the origin is the only critical point is this: let A  pbc' be nonsingular for all p > 0, but that for some x,: Ax,  bcp(c'x,) = 0. Now cp(c'x,) = poc'xo, for some p o > 0. Hence (A  pobc')x, = 0 which implies x , = 0. We shall show later ($4) that [ A  pbc'l = IA((1 pc'A'b) # 0. Hence [ A  pbc'l # 0 for all p > 0 if and only if c'A ' b 5 0. (1.1) Theorem. If (1.9) holds, C > 0, and c'A'b absolutely stable.
5 0 the system (1.1) is
A more flexible situation is discussed in $$6,7. In my paper [3] I wrongly used (1.7a). This was recently indicated in Aizerman and Gantmacher [l, p. 181.
42. Direct Control, Standard Example An interesting special case is the standard example of (11, $5). In this case A is taken as diag(Al,A2,...,A,,) where the Ah = ph are real and negative. As in (11, $5) take C = diag(a,, ..., a,,), ah > 0 and as a consequence B
and
=
';;(
diag ,...,

41
4.
DIRECT CONTROLS, LINEARIZATION, AND MULTIPLE FEEDBACK
Thus (1.9) assumes the form
or
The above equality can be satisfied if and only if ah = ph2(ch/bh)if we make the physically reasonable assumption that the bh and ch are not zero. The inequality ah > 0 implies Chbh > 0. Since c'A'b = Z(chbh/ph) the condition that every chbh> 0 implies c'A'b < 0 and hence the origin is the only critical point (11, 2.5 and 1.4). Thus a suficient condition for absolute stability of (1.9) when A = diag(ll, ,..., An), A h < 0, is that every Chbh > 0. One should not be deceived by the greater mathematical simplicity of the equations for direct control. If one tries to use the Lurie variant for case I of (111, $4) one finds that the conditions cannot be satisfied. In this case (1.9) becomes c1
+ c2 = 0
and the inequality (4.6)of 111 becomes (E
MO2
 8PlCl
+ PZCZ)40
 4PlP2CICZ < 0.
But the discriminant of this quadratic qo is zero and hence the inequality can never be satisfied.
$3. Reduction of an Indirect Control to a Special Direct Control Take the initial basic system (II,2.1) of indirect control and introduce new ( n + 1) vectors z
=(;I,
b,
=(o), 0 1
42
c0 =
(:j.
w. LINEARIZATION OF DIRECT CONTROLS Define also an (n x 1) x (n + 1) matrix A , to take the place of A as A,
=
(:3,
Then (11, 2.1) is equivalent to the system z = Aoz 0
 cp(o)b,
= CO‘Z
which is of the direct control type but with the peculiarity that the matrix A, is not stable since it has one characteristic root zero. The order of the new system is n + 1.
54. Linearization of Direct Controls Since po, p > 0, is an admissible cp function, its substitution for cp in a control equation gives rise to a linear system. In order to have absolute stability this linear system must be asymptotically stable. From this there will arise certain absolute stability conditions. It may be said that this question has been fully investigated by Yacubovich notably in references [a] and [4]. The linearization problem is closely related to the wellknown problem of Aizerman. The latter author inquired essentially into the extent to which one could substitute for a function cp restricted by a d
< acp(a) < Po’
a linear function p~b,a < p < /I. Ample details and references on this problem will be found in Pliss [l]. Take first the direct control (1.1). The linearized system is (4.1)
i= ( A  pbc’)x.
A n.a.s.c. to have (4.1) asymptotically stable is that the matrix A  pbc’ be stable, i.e., that its characteristic roots have negative real parts. The characteristic equation of this matrix is (zE  A + pbc‘l = 0. It is convenient to write zE  A = A, so that 1A.l = 0 is the characteristic 43
4.
DIRECT CONTROLS, LINEARIZATION, AND MULTIPLE FEEDBACK
equation of A itself. The above relation is then (A, + pbc'( = 0.
(4.2)
As in (11, $2) one shows that absolute stability does impose the condition that A possess no characteristic roots with positive real parts. This does not exclude that A possess characteristic roots with zero real parts: zero or pure imaginary. Let it be assumed then that A is semistable or stable and examine more closely the characteristic equation (4.2). We are concerned with p small and briefly assume that z is not a characteristic root of A so that A,' exists. Then (4.2) is equivalent to IE pbc'Az'I = 0. A closer look at this last expression will yield a simpler expression for (4.2). Consider the characteristic roots of the n x n matrix pq' where p and q are n vectors. Since each row of pq' is a multiple of the row vector q' the matrix has rank one and hence one nonzero characteristic root. Since the trace of a matrix is the sum of its characteristic roots it follows that the one nonzero characteristic root of pq' is trace pq'. But trace pq' = q'p and so IzE  pq'1 = Z "  ' ( Z  q'p). This reduces to IE  pq'1 = (1  q'p) by setting z = 1. For q' = c'A;' and p =  p b , (4.2) becomes
+
(4.2a)
lAzl (1
+ pc'A,'b)
= 0.
The solutions z ( p ) of (4.2a) for p small are very near to, but not identical with, the characteristic roots of A. Since A is semistable some are then very near to zero or to some points on the complex axis. We first investigate z(p) which + 0 with p. Since for such a solution IA,l # 0, it satisfies the relation (4.3) Let zero be a root of order k of [All = 0, and so a pole of order at most k of c'A; 'b. Thus near zero
The coefficients a, p,... are all real for they are merely the coefficients of the McLaurin expansion of zkc'A;'b. Moreover a # 0 since otherwise (4.2) would have the fixed solution z(p) = 0 and (4.1) would not 44
$4. LINEARIZATION OF DIRECT CONTROLS
be asymptotically stable for p > 0 and small. Thus zero is actually a pole of order k of c'A;'b. From the above series expansion there follows in succession C'A; ' b = 5 Zk
(I +
12
+ ...),
1
Hence setting p = vk, v > 0, one finds
( ka, + .)
= (a)llkv.
z 1  z
By the implicit function theorem this yields for v small
(4.4) This expression provides k determinations of z(p) for p small which are very near to the vertices of a regular polygon centered at the origin. Hence if k > 2 some determination will fall to the right of the imaginary axis thus contradicting the assumption that the linearized system is asymptotically stable. Therefore one requires that k 5 2, i.e. k = 0, 1, 2. The value k = 0 offers no interest; no characteristic root comes near zero for p small. There remains k = 1 or 2 and we examine these values separately. Letfirst k = 1. Then from (4.4) z = av + ... which must be to the left of the imaginary axis. This requires that a > 0. Take now k = 2. This time z =
 v2 P
k
+ ....
If a < 0 one of the solutions will fall to the right of the imaginary axis. 45
4. DIRECT
CONTROLS, LINEARIZATION, AND MULTIPLE FEEDBACK
Hence one must have u > 0. As a consequence
Re z
=
P v2
+ ..
k Thus to have z fall to the left of the complex axis for p > 0 and small, one must have fl 2 0. To sum up when zero is a double characteristic root of A necessary absolute stability conditions for the system (1.1) are u > 0, 13 2 0. Quite similarly one may obtain the following results for absolute stability and a complex characteristic root iw : (a) i o may have at most multiplicity two. (b) I f i o is simple and in its vicinity
then R e a 2 0. (c) If i o is double and in its vicinity CIAz b
=
u
(z  io)2
+( z P iw) + y + d(z  i o ) + *,
then u > 0, R e g 2 0.
95. Linearization of Indirect Controls The linearized system (11, 2.3) is (5.1)
with characteristic equation A, c'
Pb z
=
+ pp
0.
As before it is equivalent to 1+P
46
p
+ c'A; ' b
=
1
+ pg(z) = 0.
$5.
LINEARIZATION OF INDIRECT CONTROLS
If one considers (5.1) as a direct control then g(z) plays the role of the expression c'A;'b. In terms of the present A , we have: zero is at most a simple, and iw, w # 0, at most a double characteristic root of A . The special property of the zero root is caused by the fact that z = 0 is already a pole of g(z). The detailed analysis follows. (a) Zero is not a characteristic root of A . Then near z = 0 U
g(z) = Z
where u = p at once A;' which yields
+ p + yz + ...
+ c'A, 'b. Now A , = 
=  A and the calculation of A,' yields A  ' . Hence, referring to $4 one must have u > 0
p > c'A'b,
an inequality already obtained by another method in (111, $2). (b) Zero is a simple characteristic root of A. Let then dA, ' b =  + U Z
a + YZ + ....
As a consequence
and from tj4 one obtains the conditions
p 2 p.
U>O,
(c) iw(w # 0) is a simple characteristic root o f A . Here then
Hence
Hence from (tj4), U
Im > 0. w
47
4.
DIRECT CONTROLS, LINEARIZATION, AND MULTIPLE FEEDBACK
(d) iw(o # 0) is a double characteristic root of A . This time c'AZ'b =
+
U
( z  io)2 ( z  io)
+ y + ...
and so from $4 Im7P 5 0.
ImU c 0, w
1 0
$6. Direct Control with Matrices B or
C of Rank < n
Absolute stability has rested in $1 upon matrices B, C > 0. Still preserving the form of the Liapunov function V of $1, it is of interest to discuss to what extent one may be able to reach the desirable goal of absolute stability with critical matrices B or C. Recall these two properties $3):
(1x9
(6.1) If A is stable and C > 0 then B > 0. (6.2) If B and C > 0 then A is stable. Observe now that if c = 0 then the system (1.1) is uncontrolled. As this is without interest we may assume that c # 0. Hence one may select the coordinates such that c'x = x, = fs. This suggests the following convention : If F is an n x n matrix or f an n vector, denote by Fo or fo the result of deleting the last row and column from F and the last component fromf: In particular write A = l A0 g , u fl .
B = l Bo h
h'
P
1,
c = c/o k'
k
1.
Y
Heref, g , h are (n  1) vectors and u, P, y are scalars. With these designations and coordinates the system (1.1) assumes the form
+.f  bocp(o) 6 = g'x, + ufs  pcp(a).
i o = Aoxo
(6.3)
48
56.
DIRECT CONTROL WITH MATRICES
B
OR
C
OF RANK
0. Since B, > 0 the only additional Sylvester condition is
or, according to ( I X , 52) (6.5)
(B + p) > h'B, ' h .
Now
fl > h'B; ' h implies that B > 0 and so it is excluded. If
B = h'B, ' h thcn
+ h'B, ' ~ ) B , ( x ,+ hB; + paZ. Hence if one chooses as coordinates (x, + hB, la,a)our situation will x'Bx
=(x~'
'0)
be unchanged save that with the new coordinates /J = 0. This is assumed henceforth. As a consequence (6.5) becomes p > h'B, 'h.
Since this must hold for p arbitrarily small positive and B , > 0, necessarily h = 0. Thus under our stringent conditions we have B = diag(B,,O). Upon taking into account the properties of the admissible class cp it is easily verified that V fulfills all its expected requirements regarding absolute stability. 49
4. DIRECT CONTROLS, LINEARIZATION, AND MULTIPLE FEEDBACK
The situation regarding  p is less simple. There are again the relations (1.5) and (1.6). One faces then two alternatives: (a) as in $1 obtaining absolute stability through A stable, C > 0, the relation (1.9) and a positive semidefinite quadratic form in x and cp; (b) accept that A may be unstable, preserve B, > 0 and look for p(xo, a) a positive definite function of xo,a. Since then  p(xo,0) = x,'C,x, > 0 for x, # 0, one will have Co > 0. We do not return to (a) and in the next section discuss B,C both of rank n  1. Coupled with our desire to have V ( x , , a) and p(x,, a) positive definite this is equivalent to B,, C , > 0.
97. Direct Controls with Matrices B, C, of Rank n  1 As just observed this is equivalent to B,,C, > 0. As in $6 let the coordinates be so chosen that B = diag(B,, 0). Now from the Liapunov relation for A , B,C there follows
(7.1) Hence (7.2)
Ao'Bo
+ BOA,
=
Co,
k
=
BOA
7
=
0.
Now the determinant of the quadratic form x'Cx must be zero, as rank C = n  1. Hence
co k lkl
J= O.
Since Co > 0 and referring to (IX, $2) this yields k'C; ' k = 0. Since C'; is likewise > O necessarily k = 0, hence alsof = B; ' k = 0. Thus C = diag(Co,0), and (6.3) looks like this : (7.3)
X o = Aoxo  bocp(a)
6 = g'x, + cia  pcp(0). Note that ci is now a real characteristic root of A . Hence c1 g 0. Moreover from (6.2) and (7.2) there follows that A , is stable and so if ci < 0, A itself
$8. DIRECT CONTROL WHOSE MATRIX A
HAS ZERO AS A CHARACTERISTIC ROOT
is stable. The system (7.3) is then a standard direct control of the type of (1.1). Suppose that u = 0. Then the system (7.3) may be identified with (11, 2.3) if p # 0: an indirect control, or with (1.1) if p = 0: direct control, both of dimension n  1 and basic matrix A,. In the first case A , b, c, y of (II,2.3) correspond here to A,, b,, g, x while in the second A , b, c of (1.1) correspond to A,, b,, A ; 'go. Both cases are covered by our earlier arguments. Suppose now u # 0. With V as before we can calculate V simply by observing that it is the sum of its value for u = 0 plus u times the coefficient of a.Hence:
P= {x,'B,x,
+ 2d9'xocp(a) + pcp2(a))  2aocp(o),
do = Bobo  ig. Since a 5 0, and acp(a) > 0 for a # 0, in order to have  positive definite as a function of xo,a it is sufficient that the quadratic form in x,, cp in the bracket be positive definite if a = 0 positive semidefinite if u < 0. Hence the final condition (7.4)
Pp >=>{
do'C, 'do
for}
u=o u
< 0.
When (7.4) holds we will have  positive definite in x,, cr, that is, in x and this for all admissible cp. Since Vco with (Ix(1,absolute stability will have been achieved in both cases. The preceding discussion leads to the following result :
(7.5) Let both B, C be of rank n  1. Then n.a.s.c. to have V and  V both positive deJinite as functions of x or x,, cp, are that in appropriate coordinates B = diag(B,, 0), C = diag(C,, 0) with B,, C,, > 0 (hence A , stable) plus the property (7.4). When this holds the system (1.1) is absolutely stable.
$8. Direct Control Whose Matrix A Has Zero as a Characteristic Root (Kenneth Meyer) In this section and the next we fully discuss several critical direct controls. In the first case the matrix of the linear part of the system has 51
4.
DIRECT CONTROLS, LINEARIZATION, AND MULTIPLE FEEDBACK
a simple zero characteristic root. By a suitable change of coordinates the system is then f = A x  bq(a)
g=
(8.1)
Bda)
a = c‘x  y(
where x , b and c are n vectors; 5, B, y are scalars and A is an n x n stable matrix. Now /3 = 0 implies that 5 = constant and therefore 5 = 0 to achieve stability. Hence the system is just (1.1). We may therefore assume /I# 0. Then replacing 5 by 0. In this case Bo = diag(a, 0, B) and from the Liapunov relation Ao’Bo
+ BOA0 =  Co
one finds Co = diag(0, 0, C),
C =
A’B
+ BA.
Also do’ = (Bobo  +AO’cO)’ = ((aB,  + y 2 ) , 0. d ) d = Bb  )A‘c
Hence
co’bo = BlYl
+ B 2 Y 2 + c‘b.
 V = xO’COXO = x’Cx
+ 2do’xOcp + co’b0cp2
+ (as1  y2)t1(p + 2d‘xcp
+ (BlYl + P 2 Y 2 + c’b)cp2. Let us suppose that there exists an a > 0 such that uB1  t Y 2 or what is equivalent (8.4)
B1Y2
=
0
’0.
Then we can choose C > 0 and thus B > 0. Note now that Bo is not positive definite in t l , t 2and x but only positive definite in t1 and x.
53
4.
DIRECT CONTROLS. LINEARIZATION, A N D MULTIPLE FEEDBACK
c2,
Since y2 # 0 (8.4) the term @(a) makes V positive definite in (,, x and moreover V + cc as IlxII + Itl[+ It2[+ x . We cannot achieve as much for  v. Let us, however, endeavor to make it a positive definite quadratic form in x and cp. This merely requires the inequality (Fd)or (8.5)
/jly,
+ P 2 y 2 + c‘h > d‘C Id.
0 This only guarantees that  2 0 in the space of in the space X, u. At this point we have recourse to the theorem of LaSalle (IX, 4.8). In our case = 0 if, and only if, x = 0, u = 0. This implies that if a solution of (8.3) is to remain in the set where = 0 it must satisfy
+ Y252. But the general solution to the above equation is t l = 6,, 5, = 6,t + 6,, 0 = y16, + ~ , ( 6 ~+t 6,) where 6, and b2 are arbitrary constants. Since lil
=
0,
li2
=
51,
0
= Y151
y2 # 0 this implies that 6, = S2 = 0. Thus the LaSalle conditions are satisfied and so absolute stability has been established under the condition C > 0, (8.4) and (8.5). It will be of some interest to compare the above conditions with the necessary conditions found in tj4. The function c’A;’b found in $4 is now
PlY2 2,
+
PlYl
+ P2Y2
+
...
2
Thus we see that the conditions found in $4 are simply /J1y2 > 0 and plyl + P 2 y 2 2 0, and are compatible with (8.4) and (8.5). Indirect control whose matrix has zero as simple characteristic root. In accordance with the scheme of (33, this case may be identified with 1 and the one just considered. The indirect control is of dimension n has for “state” matrix diag(A,O) the state variables being x and t2.The preceding treatment : case O 2 may thus be directly applied here.
+
54
$9.
CONJUGATE PURE IMAGINARY CHARACTERISTIC ROOTS
99. Direct Control Whose Matrix Has a Pair of Conjugate Pure Imaginary Characteristic Roots (Kenneth Meyer) It is again convenient to take a system of dimension n coordinates the system will be represented by
(9.1)
+ 2. In suitable
f = AX  bq(o)
where the Greek letters are scalars with w > 0; x, b and c are n vectors and A is a real stable n x n matrix. Let A,, ... be as in the previous section. Thus A,
=
diag(io,  io, A),
b,’
=
(8,p, b’),
c,’ = (y,?, c’).
Choose as a Liapunov function
v=
+ x’Bx + @(o)
2u5g
where u will be specified in a moment and u will be positive. Here then
B,
=
diag(u, u, B).
From the Liapunov relation A,*B,
+ BOA, = c,
one finds at once
C,
= diag(0, 0, C),
C =
A’B + BA,
55
4.
DIRECT CONTROLS, LINEARIZATION, AND MULTIPLE FEEDBACK
Hence
V
+ 2do'xOcp + co'bocp' = X'CX + 2 ( ( 4  3iwy)t; + (a/?+ $ion[+ d'x)cp + (By + pp + c'b)cp2. =
xO'COXO
Let us assume that there is a positive a such that (a13  $my) = 0 or what is equivalent
These conditions may clearly be satisfied. We also choose C > 0 hence also B > 0, and as a consequence I/ is positive definite in x, t;, (,c for every admissible cp and moreover V  r 00 with llxll + 151 + Aiming again toward the theorem of LaSalle we require  V to be positive definite in x and cp. This merely requires here
[GI.
By
(9.3)
+ bT + c'b > d'C'd.
Under the condition (9.3) V is a positive definite quadratic form in x and cp. Thus  V = 0 if, and only if, x = 0, c = 0. Thus if a solution of (9.1) is to remain in the set where = 0 it must satisfy the equations
[ = iot;, f
=  iot,
yt;
+ y[
=0
That is, there must exist a constant 6 such that yde'"'
+ y&'"'
= 0.
Since the two exponentials are linearly independent y6 = 0 and since y # 0,6 = 0. Thus LaSalle's conditions are satisfied and absolute stability is assured by C > 0, (9.2) and (9.3).
$10. Multiple Feedbacks In various practical situations one may have a control depending on several parameters. This case has already been dealt with by Letov [l, Chapter IV] and Popov [4]. 56
$10.
MULTIPLE FEEDBACKS
To conform with our general notations designate the analog
by the r vector
By analogy with the earlier situation, the following conditions are imposed upon the vector function f ( v ):
I. f ( v ) is continuous;
11. fh(v)’ u h > 0 if o h # 0, andfh(o) = 0 if Uh = 0 ; 111. the integral j f ( v ) d v along any ray from the origin v
divergent.
=0
is
This last condition plays the same role as its analog: to make all solutions tend to the origin as t + + 03. An example of a function such as f ( o ) (but not the only example) is one in which every fh = fh(uh) (scalar function) where fk satisfies the conditions of I, namely,
I.
is continuous for all oh; > 0 for uh # 0 ; fh(0)= 0 ; HI. j’ mfh(vh) doh is divergent. 11.
fh(vh)
vhfh(vh)
However, for instance I(v(lf(v),( f ( v )as just defined), is likewise a function of the general class. We have now the possibility of associating this new situation with indirect and direct controls. Indirect control. The system will be X
(10.1)
= AX
ti = f ( v )
u
=
HX
+ GU + Ru,
where A is our usual stable n x n matrix and G, H , R are constant n x r, r x n and r x r matrices. We proceed in full analogy with the earlier case. In the first place the origin will be the only critical point if, and only if, the determinant ( 10.2)
:,I 57
4.
DIRECT CONTROLS, LINEARIZATION, A N D MULTIPLE FEEDBACK
This is assumed henceforth. We are then justified in applying the change of coordinates ( x ,u) + ( y , u ) defined by y = A x + Gu, u = H x + Ru. The new system is (10.3)
The goal is still to make the system absolutely stable: asymptotically stable in the large and regardless of the choice of the functionf(u) within its class. The same theorem of Liapunov, with BarbashinKrassovskii complement is to be applied. First define
@(u)
=
sa
f ( 0 ) .du
where the integral is taken along the ray Lfrom the origin to the point u. That is, if, say s is a parameter along the ray, with value s at u then
Sincef,(u) always has the sign of u,, (except that f h = 0 for uh = 0) we see that @(u), for u # 0, is a sum of positive terms if u # 0, and only zero for u = 0. Thus @(u) > 0 for every u # 0. It is, of course, continuous in u, and + + 00 with IIuIJ as the latter + 00. Choose now (10.4)
V Y , 0) = Y’BY
+Wu)
where B is selected as before: one takes an arbitrary n x n matrix C > 0 and defines B as the unique solution of Liapunov’s equation A’B
+ B A = C.
One finds now (10.5)
 P(Y, u) = Y’CY  .f’(u)Rf(u)  f ’ ( W Y  Y ‘ K f
K
=
G’B
+ )H.
In writing this relation we have utilized the property that since f ’ H y = y ’ H f , we have
f ’ H y = 3f ’ H y + y’H’f).
58
9 10. MULTIPLE
FEEDBACKS
Since f ' Rf = f 'R'J we observe at all events that the positive definiteness of  implies that of  f ' R f = f'(*(R + R'))f;and hence that
v
(10.6)
+ R' < 0.
R
Basically, however, the positive definiteness of  v is equivalent to the classical Hurwitz conditions for the positive definiteness of the quadratic form in (10.5) in the variables y,f: If we set K = (kij), R = (rij) and define
K,
=
A,
=
.,"I
(kij);
R,
=
(rij);
i,j
S
s,
Ks
R]'
then the Sylvester conditions not yet fulfilled are (10.7)
A,>O,
A , > O ,..., A , > O .
These are the conditions which correspond to the unique inequality (Fi)for r = 1. Of course, the inequalities (10.7) imply (10.6). Indeed if one reverses the orders of the variables from y,.f to f ; y one obtains, from Sylvester's inequalities, as first conditions for the positive definiteness of  those which express that f ' ( g R + R'))f is positive definite.
Direct control. This time the system is
+ Gu
X
=
u
=fb)
AX
v = Hx,
or with u eliminated (10.8)
X
=
AX
u
=
Hx.
+ Gf(v)
It may also be written as a single vector equation X =
AX
+ Gf(Hx).
The origin will be the only critical point provided that, as assumed henceforth, A x + G f ( H x ) = 0 has x = 0 as its sole solution. One may conveniently complete (10.8) with i, = H i = H A x
+ HG(u) 59
4.
DIRECT CONTROLS, LINEARIZATION, A N D MULTIPLE FEEDBACK
or setting H A = H,, H G
=
fi
R,, by =
H,x
+ R,f(u).
Take now as before V ( X )= X’BX
+ @(u).
With B, C related as before, one obtains  V(X)= X’CX  f ’ ( ~ ) R o f ( ~) f ’ K 0 x 
K O = G‘B
+ iH,.
~’Kof
Set now
K O = (k;);
R,
(r;);
Kso = (kij);
Kso = (k;);
i , j 5 s;
R,’ = (r:); Aso =
=
C
Kso
K,O’
Rs
The difficulty found in $1 occurs here also. It is settled again by reference to LaSalle’s theorem (IX, 4.8) and the sufficient conditions for absolute stability are A , = 0 , A 2 = 0 ,..., A.,=O.
60
Chapter
5
SYSTEMS REPRESENTED BY A SET OF EQUATIONS OF HIGHER ORDER In practice one has often to deal with fundamental systems composed of several equations of any order. In theory this offers no novelty since by introducing more variables such a system may be reduced to the standard type of a number of equations of order unity. Practically however it is decidedly advantageous to be able to deal with these systems as one finds them and not after a more or less artificial reduction to another type. Our purpose in the present chapter is to examine some of the problems without undue change of type. This does not mean, of course, that we forego the theoretical convenience of the reduction to the standard set of equations of order one, but only that we will endeavor as much as possible to deal with the equations as they stand.
$1. Generalities The variety of the systems that may arise is virtually endless. For the sake of orientation we discuss rapidly a fundamental system consisting of a single equation of order n with constant coefficients: (1.1)
t+")
+ a,rfn + ... + a,q ')
=
0. 61
5.
SYSTEMS REPRESENTED BY A SET OF EQUATIONS OF HIGHER ORDER
Introduce the new variables xl, x2,..., x, components of a vector x, and defined by x 1 = rl, x2 = ti,..., x, = q'" l'.As a consequence the unique equation (1.1) becomes equivalent to the system x, = x2,
(1.2)
x, =
a,x,
1, = x j,..., in1 = x, 
CL~X,,

...  %lX
1
with coefficient matrix 0
1
0
0
.
.
1
The characteristic equation is
IrE  A J = r"
(1.3)
+ a,r"' + *.. + u, = 0.
As before A is assumed to be stable. Part of our problem is that we will have to deal directly with the matrix A , that is with the coordinates x, since they have a particular significance for the problem under consideration. Or more accurately, one will be free to change coordinates provided that the system (1.2) retains its form and the matrix A is unchanged. This will become clear in a moment. This is as good a place as any to make a few rapid observations regarding linear systems. After that we turn our attention to indirect, then to direct controls.
52. A Digression on Linear Systems There has been developed around linear systems a widely used technique based upon the simple and wellknown device of designating time differentiation by an operator, usually D.Thus x is written Dx,X is 62
$2.
A DIGRESSION O N LINEAR SYSTEMS
written D'x, etc. As an example the standard indirect control system assumes the form ( D  A)x Da
=
5b
= cp(o)
a = c'x  p 5 .
If one has a nonlinear characteristic cp(a), this notation offers little advantage. Suppose, however, that the characteristic cp is linear, or say that we limit cp to a sufficiently small neighborhood of the origin to make it reasonable to linearize cp, i.e., to replace it by a linear approximation. Let this also be combined with a fundamental system such as (1.1) consisting of a single nth order equation. The resulting system assumes then the form (D"
+ alDn' + ... + a,)q
=
5
where the a,, and yk are constants. If we set g(D) = D"
+ a1Dn' + ... + a,
then the system assumes the form
Now the operators g(D) form what is known in algebra as a ring ofpolynomials and one may apply to such a collection the usual operations of rational algebra : addition, substraction and multiplication (but not division). As a consequence one may solve the system (2.1) in the usual way and obtain the relation
{(D+ P)@)

W))V
=
0.
The bracket will be denoted by k(D),sothat one has to deal with the equation (2.2)
k(D)q = 0. 63
5.
SYSTEMS REPRESENTED BY A SET OF EQUATIONS OF HIGHER ORDER
The characteristic equation is
k(A) = 0.
(2.3)
The function l/k(s) is the wellknown transferfunction of the linear theory. Let rn be the degree of k ( l ) and let the roots IZ1,...,Am of k(A) all be distinct. The general solution of (2.2) is then
+ ... + Cme'mr.
q(t) = Cle'tr
Even if the Ah are not all distinct n.a.s.c. that q(t) and all its derivatives +O as t + + co is simply that every Re A h < 0. This is the full solution of the absolute stability problem in the present instance. A more complicated situation would correspond to e.g., r parameters ql,..., qr (fundamental parameters) and r + 1 equations k
gjk(D)qk
Let
 h,iiD)C
0' = 1, 2,...,r + 1).
=0
A(D) = JgjXD),...,gjr(D), hi(D)I*
The q's and also
< are solutions of
[email protected])[ = 0.
The characteristic equation is again A(A)
(2.4)
=
0,
and the stability condition: the q's, 5 and all their derivatives 4 0 as t + + 00 is again : the real parts of all the roots of (2.4) must be negative.
$3. Indirect Control Taking (1.1) as the fundamental (state) system, the indirect control system has naturally the form $1
+ qq(n1) +
ri = d o )
(3.1)
d =
64
ylq'""
**a
+ m,q
=
5
+ y z q ( n  2 ) + ..* + ynq  pc.
$3.
INDIRECT CONTROL
Such a system may arise under the following conditions. Consider a system S made up of a chain of subsystems S1,..., Sn where Sh depends upon the variable q h and Sh acts upon Sh+ in accordance with a relation h
gh(D)?h = ?h+1,
1, Z Y . . ~n  1
=
except that at the last step there appears the control variable gfl(Wfl
=
r.
Here gh(D) is a polynomial with constant coefficients in the differential parameter D. One may easily realize such a scheme in which for instance the gh(D) are linear or quadratic. By elimination and setting q 1 = q, one obtains the first equation (3.1): g1(D)gz(D)..*gn(D)q =
5.
Returning to the system (3.1) observe that here as in (11, $2) and for the same reason it is advantageous to change from the variables q,t to new variables c,a so that a becomes one of the basic variables of the system. This change is defined by tj = c and the new system is
p)+
(3.2)
=
+ ... +
ylc(nl)
Cc r,
= Cp(0)
+ ... + Y n l  PCP(O).
We must prove however the equivalence of the (q,5 ) and in the sense that the conditions (4i,..., 4  l), 5 )
+
(c, a) systems
0
(C, [)...)pI), a)+ 0 are equivalent. For that purpose it is convenient to use the equivalent vector forms e.g. x, and y, B. They are
dC'd.
(Fi)
One will merely recall that (3.4) is a consequence of (Fi)(see 111, 9) so that C > 0 and (Fi)remain as the only required inequalities. The remaining considerations regarding an indirect control are the same as before. One must bear in mind, in applying the optimal inequality (111, $2) that the coordinate vector y hence also the vectors b, c are not necessarily the same as the initial vectors y, b, c.
$4. Indirect Control: An Example We will borrow an example from Letov [l, Chapt. 11, 55Jwhich Letov refers to as the second Bulgakov problem. Using at first the same designation for the constants as Letov, the system is (a) T2ij + Utj
+ k q = T25 t = cp(4
(4.1) (b)
0
= uv
1 + Etj + G2ij  5. 1
Here T 2 and G2 are inertial constants, U and E are dissipative constants and k is a restoring force. At all events U and k are positive.
67 \
5.
SYSTEMS REPRESENTED BY A SET OF EQUATIONS OF HIGHER ORDER
One may change at once the expression of a, using (4.la) to
(4.2)
I
p =   G2.
1 We thus see at once that absolute stability will require that p > 0 and hence that 1 > G2. 1

(4.3)
We may now change the system to the standard form (4.4) with the following values of the constants a1
U
=p
u2
G2U Y ~ = E  T2 '
(4.5)
=
k
T2 ~
~
kG2 = T
a

~
The matrix A is
and the characteristic equation is s2
+ u,s + u2 = 0.
Since ulru2 are positive, the roots have their real parts negative: A is stable. Let the matrices B, C be given by
68
$4.
INDIRECT CONTROL: AN EXAMPLE
The conditions that C > 0 are (4.6)
> 0,
p
The relation A'B
i
+ BA =  C
pr  q 2 > 0.
yields here
2u2qo7
Po  u1qo  u 2 r 0 )
2(qo  %ro)
P o  u1qo  u2r07
or (4.7)
P
=
2u2q0,
4
=
u1qo
+ u2ro
 _ 
i"
4)
4 r
 Po
r = 2(u,r0  qo).
We also know from general theory that with A stable, C > 0 implies B > 0. Choose as sufficient conditions for stability in the large, the vector relation d = 0 (111,3.2). Here
d
=
+ b2
i, + 40
+y).
The determinant of the linear system (4.7) in po,qo,ro is 4u1u2 > 0. Hence (as we know already) there is a unique solution for po,qo,ro. We only need qo and ro and we find P qo = , 2a2
r ro = 2a1
P + 2u,a, I
Therefore
Hence d (4.81
=
0 yields the relations
+ u,y, = 0, p + ru2 + u1u2y1= 0. p
69
5.
SYSTEMS REPRESENTED BY A SET OF EQUATIONS OF HIGHER ORDER
The first relation already yields y 2 < 0. Moreover (4.6) implies r > 0 and hence the second of (4.8) yields also y1 < 0. Thus both control constants yl, y 2 must be negative. Upon eliminating p between the two relations of (4.8) there follows r = y 2  a 1 y l . Since r > 0 one must have y 2 > a l y l . Once this is satisfied one takes p
=
a2727
= 7 2  a1y1,
141 =
Jpr.
Thus, both y1,y2 negative and y 2 > a1y2 are sufficient conditions for absolute stability of the system. In terms of the initial constant, the sufficient conditions are G 2u E 0 andf(t) 1
For
92
+0
as t + co then likewise
$3.
PROOF OF POPOV'S FIRST THEOREM
and by the rule of 1'Hospital as t
, + 0 0 :
lim h(t) = limlf(t)l
=
0.
IV. By convolution of two functionsf(t), g(t) is meant the operation
* defined by
f *g
=I'
f(r)g(t
2 ) dr
0
and we note that f * g = g *f.With this notation one may write (2.1) and (2.2) as x
=
X(t)xo X(t)*bq(t)
CT
=
c ' X ( t ) x ,  v ( t ) * ~ ( t) ~ ( ( t ) .
The basic properties are (2.5)
1'
Wfl
*fi) =
f ( t ) g ( t )dt =
03
1'
m
Wfl)F(fA F(f)@(g) dw,
and if g is real (2.6)
/+O3/(t)g(t)dt = m
m
5 ( f ) @ ( g ) dw.
(For these properties consult Doetsch [l, pp. 1571631 and Titchmarsh [I, P. 501.) Of course one must assume that the integrals just written do exist.
1;')
53. Proof of Popov's First Theorem Introduce the following functions (PT(t) =
for 0 5 t 5 T otherwise,
93
7.
SOME RECENT RESULTS OF V . M. POPOV
i(t)= c ' ( ~ ( t +) qX(t))x,. Now (2.2) yields
=
c'X(t)xo
From this and (2.2) follows (3.1)
A(t) = a(t)
Sb
i(t  z)cp(z)dz
+ 4 4 t ) + yt(t)  i(t),

pcp(t).
0 5 t 5 T.
Another required property of A(t) is : (3.2) There exist positive constants k , C such that for t 2 T: (3.2a)
I,l(t)l < C e  k r .
Since both v(t) and i(t) are quasi exponential so is p(t). Hence for t 2 T a n d some k,C > 0
IA(t)l < C[ ek"r)IcpT(z)Idz
T
=
C e  k j ekrlcpT(z)ldz
0
0
which is (3.2a) with suitable C. Taking into account (3.2) or the quasi exponential property or continuity of the various functions integrated in the sequel, all our integrals will be meaningfula property not emphasized later. Set now L(iw) = 9(/?), F(iw) = p ( ( P T ) .
By a basic property of Fourier integrals
9 ( t )= =
s,'"
eiw'b(t)dt
=
i o N ( i o )  c'b
ioiV(io) + y  p,
9 ( A ) = L(io)=
 {N(io)(l
+ i o q ) + qy}F(iw).
Define now the functions O(T) =
l ( t ) ( P T ( t ) dt. 0
94
$3.
PROOF OF POPOV’S FIRST THEOREM
(3.3) The inequality (1.7) (hypothesis of Theorem 1.6) implies that B(T) 5 0.
s
In fact referring to (2.6) + W
B(T) =
L(io)F(io) d o
m
1
+m
=
m
+m
{N(iw)(l + iwq) + qy}F(io)F(io)d o IF(iw)12(N(iw)(1+ ioq)
+ qy) d o
 W
1
s
=
2n
IF(iw)12{ReN ( i o ) ( l + ioq) + q y } d o
+m m
IF(io)12 Re(1
+ ioq)G(io)d o
As a consequence of (1.7) the last integral is 2 0 and so (3.3) follows.
Upon combining (3.1) and (3.3) there follows the important inequality
(3.4)
ST 0
a(t)cp(t) dt
+q
1
T
0
W)cp(t)dt + Y
T 0
t(t)cp(t)dt 5
1 T
0
Wcp(t)dt.
One must analyze now all these integrals. The lefthand side of (3.4):
s:
Since acp(o) > 0 for all u # 0, for all T : (3.5)
s:
a(t)cp(t)d t > 0.
Regarding the second integral (3.6)
&(t)cp(t)dt = @(T) @(O).
Finally the third integral yields
95
7.
SOME RECENT RESULTS OF V. M. POPOV
A t the righthand side of(3.4):
I’ 0
1
T
C(t)cp(t) dt =
0
i ( t ) r i ( t )dt
1
‘7
= [C(t)m];
(If.
S(t)&) 0
Let t1= sup(t(t)(in 0 5 t 5 T Since [ is quasi exponential, and proportional to llxoI),if u = llxoII to,then
+
Since [ ( t )is quasi exponential of the same nature as
[(t)
Hence with C > 0 and independent of T :
Upon combining the inequalities (3.4), (3.6), (3.7), (3.8) there follows
1:
a@)&)
dt
+ q@(T)+ i t 2 ( T )5 Cut1 +,qO0+ +yu2.
Referring now to ( 1 . 1 ~the ) initial value oo of o(t) tends to zero with l(ol = u. Since @(a) is a continuous positive function of o and vanishes for CT = 0 one may take u so small that qQ0 < yu, where u is a given positive quantity. Hence the above inequality yields the following two special cases: llxoI(
+
Jl
(3.9)
a(t)cp(t) dt
< Cut,
+ $yu2 + yu.
Then if t1 = ((T,) and since we may take T @(T)2 0: (3.10)
f(ll)= t 1 2  2CU(,

y(u2
=
+24
T,, and also because 0.
+
Let l,’, tl”,be the roots o f f = 0. Since tl‘tl”=  ( u 2 20) S 0, the roots are of opposite sign and (3.10) requires that t, be between 96
93.
PROOF OF POPOV’S FIRST THEOREM
them. Since t1 > 0 it must be below the largest root or
t1 < c u
+ J(C’ + y)u2 + 2v.
Since this upper bound of It(t)l for 0 5 t 5 T is independent of T we conclude at once that for all t 2 0: (3.11)
It(t)l < c u
+ J(C’ + y)u’ + 2v.
Since t1is bounded we also have from (3.9) that a(t)cp(t)dt< co
(3.12)
that is: the integral is bounded. Note that the bounds in both cases depend solely upon the initial values x o and to. Going back to (2.1) we have X ( t  r ) b ~ (dr~ = )
Referring again to (2.1) the quasiexponential property of X (t), X ( t ) and (3.11) we obtain (3.13)
IIx(t)ll < C(U
+ tl),
c > 0.
In view of (3.11) and (3.13) we conclude that given E > 0, one may select q > 0 such that if u < q then Ilx(t)ll 1t(t)l < E for all t 2 0. In other words the origin is stable for the system (1.1). To complete the proof of Popov’s first theorem we still need:
+
(3.14). Every solution (x(t),( ( t ) )of (1.1) tends to the origin x t + + 00, and this whatever the admissible function cp(a).
=
0, 5
=
0 as
As a preliminary step we require :
+
(3.15) Both o(t) and cp(t)+ 0 as t + co. The boundedness of IIx(t)ll and It(t)l, coupled with (1.1~)already show that a(t) is bounded. Since cp(a) is continuous and cp(0) = 0, cp(t)is 97
7.
SOME RECENT RESULTS OF V. M. POPOV
likewise bounded. From (1.3b) and the boundedness of (1x11 and q ( t ) follows that qt) is likewise bounded. Hence one may select an M > 0 such that la(t)l and Ib(f)l < M for all t 2 0. Suppose now that (3.15) does not hold for a.There exists then a S > 0, and a divergent positive sequence t , < t2 < ... such that la(tk)l > 6 for all k. We may actually assume that tk  tkl > S/M, t , > S/2M and we will have la(t)l = la(tk)
+
s’
q?)dzl > la(t,)l
 /M(t
fk)l
> 6  )a
=
$6.
tk
Since q(a)is continuous and cp(0) = 0 if and only if a = 0, for M > 161 > T < tN+l, necessarily we will have Iq(a)l > R > 0. Hence if tN
with T. Since this contradicts (3.12), a ( t ) 4 0 and so does q(t).
PROOFOF (3.14). Since X ( t ) is quasi exponential in the sense that all its terms are, Ilx(t)ll , 0 as a consequence of (2.1), of q ( t ) + 0 and of property 111 of $2. Referring now to (1.1~)since )Ix(t)I(+ 0, a(t) , 0 and y # 0, l ( t ) also +O. This completes the proof of (3.14) and likewise of Popov’s first theorem.
$4. The Generalized Liapunov Function of Popov In his work Popov considered a more general Liapunov function than the one of the previous chapters. Before attacking the proof of Popov’s second theorem it will be profitable to discuss this generalization. Its general type is: quadratic in x, a plus j?@(a),i.e. :
+ on2 + a f ’ x + pD(0).
V(x, a) = x‘Bx
This function may immediately be put in the form
v = x’Bx + cr(a  c’x)2 + B@(a)+ o f ’ x .
Then
 V = X’CX
d + @p2 + 2d0‘xq + 2ayaq  (af’x) = dt
do = Bb  ($A’c
A’B
98
+ C~YC),
+ B A = c.
$4. THE GENERALIZED LlAPUNOV FUNCTION OF POPOV
The E method. Upon making a substitution x + cPx,a + E%, cp + f c p , q r even (to preserve ocp > 0), V or V is turned into a polynomial P(E).Let As” be its lowest degree term. We shall then write P A AE”, and P will have the sign of AE“ for E small. Now
+
d dt
(~f‘x)
=
+
(x’A’c  p ~ p ) f ’ ~a f ’ ( A ~ qb).
Hence the substitution x
a + a,
+ EX,
yields iff # 0 :
VA
cp
+ E’cp
&Of‘AX.
Since A is stable it is nonsingular. Hence iff # 0 likewise o f ’ A x # 0 for x arbitrary and a # 0. Hence the sign of V for E small is then that of ~ a f ’ A xi.e. , it may be + or  according to the sign of E. Since this is ruled out necessarily f = 0. Thus finally (with repetition and to keep everything together) (4.1)
(4.2)
(4.3) (4.4)
+ a(a  c’x)’ + p@(a), + /3pcp2 + 2do‘cpx + 2ayacp,
V = x‘Bx
V
= X‘CX
A‘B + BA =
c, do = Bb  (;/3A’c + ayc).
(4.5) If absolute stability is to be determined by V of (4.1) positive deJinite one requires that u 2 0, p 2 0, a /3 > 0.
+
Let a # 0. The substitution x + E X , CT + a,cp + c3cp yields V A aa’, hence a > 0. Let p # 0. Then the substitution x + E ’ X , a + 8’0, cp + cp yields V A E ’ ~ @ ( oand ) , since @ > 0 one must have p > 0. If both a, p = 0, V = 0 for x = 0 and a # 0, hence a + /3 > 0. To sum up we are left with the following two types of Liapunov functions : The LuriePostnikov function. (4.6)
V = x’Bx
+ /3@(a)
7.
with
SOME RECENT RESULTS OF V. M. POPOV
+ ppcp2 + 2d’cpx Bb  + / ~ A ‘ c ; A‘B + BA =  C .
 V = X’CX
d
3
The Popov function. V = X‘BX + ~ ( a c’x)’ (4.7) (4.8)
+ pCD(0)
+ Bp’p2 + 2d0’xcp + 2uyocp = Bb  (#A’c + U Y C ) , A’B + BA =  C ,
 V = X’CX
do together with property (4.5).
$5. Proof of Popov’s Second Theorem Let us modify, with Kalman, Popov’s expression in (1.7) through replacing (1 ioq) for some 4 2 0 by 2uy imp for some nonnegative pair u , p such that u + p > 0 (a and fi do not vanish simultaneously). For u # 0 the two expressions are clearly equivalent as far as (1.7) goes. However the new expression for u = 0 corresponds to q very great in (1.7). The modified Popov inequality is :
+
+
(5.1)
P(u, p, w) = /3y
+ Re((2uy + iwp)c’A,’b}
20
for all real w and some pair u, p such that (5.la)
a20,
/320,
u+/l>O.
Thus Popov’s first theorem reads now: P 2 0, under (5.la) is a sufficient condition for absolute stability of the system (1.1). To prove Popov’s second theorem we must show that for the function I/ of (4.7), the double property V and  both positive definite for every admissible ‘p implies (1.7) for some q 2 0. We shall show, as does Popov, that it implies (5.1) with u, p the same constants as in (4.7). If a real bilinear form u’Fu > 0 for u # 0 we will write F > 0 as if it were a quadratic form. We will require presently the property: (5.2) Let u be allowed complex vector values. Then F > 0 and the property Re u*Fu > 0 for u # 0 are equivalent.
+
Here, if u = u1 iu, then Re u * Fu = ui‘Fu, + u,‘Fu, > 0 for u , , u2 not both zero if F > 0. Conversely Re u * Fu > 0 for u # 0 (complex) yields F > 0 for u real and #O.
$6.
COMPARISONS
Regarding the theorem itself, absolute stability has already led to the proof of (4.5) which is (5.la). Regarding (5.1) itself, begin with V =
2x'B(Ax  b q ) + 2aycp(uo 
Hence by (5.2):
C'X)
 P~[c'(Ax b q )  yep] > 0.
Re{  2x * B(Ax  b q )
+ 2ayq(a  c'x)
This inequality is the starting point of the proof. By definition i o E = A + Aim. As already observed A,i' exists for all real w and so the preceding relation yields iwAG'b = A . A G ' b + b.
The hypothesis b = 0 is unrealistic since it means that the control is not operating. We assume then b # 0. As a consequence m(iw) = A i ' b # 0 for all w. Upon substituting in (5.3)x =  m , cp = puo, p > 0 and r~ = l / p ; the inequality must hold since it must hold for all complex x , real r~ and all admissible cp (in particular for cp = puo). Thus we find 2m*Biwm
+ B[(c'iwm) + y]
+ 2ay
Since m*Bm is real Re iwm*Bm
=
0. Hence
for all real o.Since the sum must be positive for arbitrarily large positive p, if a # 0 we must have P 2 0, while if a = 0 (and y # 0) we require P > 0, all this for all real o.Thus the Popov inequality (5.1) holds, with a, B the same constants as in V . This proves the theorem.
96. Comparisons It is interesting to compare what may be accomplished by our comparatively simple methods using the LiapunovPopov function (4.7) and the earlier type (4.5). 101
7.
SOME RECENT RESULTS OF V . M. POPOV
From the function (4.5) one obtained the inequality of (11): p > d'C 'd,
(Fi)
where in reference to the system (1.1) d = Bb

)A'c.
Passing now to the function (4.8) since GI 2 0 and acp > 0 for o # 0, we see that to have  positive definite it is sufficient that the quadratic form in x, cp: W
= X'CX
+ pcp2 + 2dO'cpx
be positive definite. The same reasoning as for the proof of (Fi)yields the inequality (6.1)
p > do'C 'do.
Let for the present ccy
= u,
and set
$(u) = d0'Cl do do = d  U ~ C ,
2
=
=
Iu'
c'C 'c,
+
 2 p ~ V,
p
=
'
c'C d,
'
v = d'C d,
\
One must bear in mind that u must be 2 0. The righthand side of (6.1), compared with that of (Fi)contains the additional parameter u and we may dispose of it to optimize the1 inequality (6.1). To be precise the least value of p afforded by (Fi)is ~
pm = d'C Id,
while the least value pm* corresponding to (6.1) is to be obtained as the positive minimum of $(u) for u 2 0. One must then discuss this minimum. If one finds that pm > pm* the LiapunovPopov function V will have been proved more advantageous, if not, the earlier function may as well be used. Observe now that c = 0 would mean that the feedback variable 0 is independent of the system variable x . Since this is entirely unrealistic and uninteresting, one may assume that the vector c # 0. Since C > 0, hence also C  ' > 0, the coefficient I > 0. It follows that the minimum of $(u) occurs for u = p / 2 . However this minimum is only admissible if p > 0. If p 5 0 the minimum of $(u), for u 2 0 occurs at u = 0. We discuss separately the two possibilities. 102
$6. COMPARISONS
(a) p > 0. The minimum $, of $(u) occurs for u value is p2
*m=v=
A
VA
p2 
= u, =
p / A and its
6

A
A
where 6 is the discriminant of the quadratic $(u). That is
6
=
(c'C 1d)2 (c'C 'c)(d'C'4.
In order to have pm* = $,, the latter must be positive and so one must have 6 < 0. Thus if 6 > 0 one can only have pm* arbitrarily small. At all events since v = pm we have
Hence the LiapunovPopov function V is certainly more advantageous if 6 0 and p > 0, that is c'C 'd > 0. If 6 > 0 and d # 0 we have pm > 0 and pm* arbitrarily small, hence again the LiapunoyPopov function is more advantageous. On the other hand whenev d = 0, one cannot claim any advantage for it. (b) p S 0. This ime u, = 0 and so pm = v = pm*. Hence no advantage is afforded by the LiapunovPopov function.
=
r
Application. Consider the standard example (111, $2) under the assumptions that b,c have no zero components, and also that C corresponds to optimum. In all cases for optimum
and
3.
C'
=
diag(6, ...,
0
=
(A'c)'x  p 0.
Upon applying (4.6) there follows (4.7)
2 Re k'rn.= Iq'rnl'  2& Re q'm
+ 6. 115
If q'm
=
8.
SOME FURTHER RECENT CONTRIBUTIONS
7
+ 2Rek'm = (1 &)' + pz + 6 > 0
1 + ip, (4.7) yields
which is (4.3). This proves necessity.
PROOFOF
SUFFICIENCY.
We first establish a preliminary result.
(4.8) If u is a real constant uector such that Re u'm (iw) = 0 whatever w then u = 0. Suppose that u # 0 and let we have I)~(Z)
=
u'm(z) =
uh
u1
be its components. Referring to (3.7)
+ u2z + ... + u,znl lAzl
This function has the following properties : (a) It is rational in z and not identically zero. (b) Its poles are among the characteristic roots of A and hence, since A is stable, they are all to the left of the complex axis. (c) Since the numerator of I),&) is of smaller degree than the denominator there is at least one such pole. (d) $,,(z) takes only complex values on the complex axis. It follows that +(z) = ii,b0(iz) is a rational function of z which takes only real values on the real axis and hence it is real. Moreover it has one or more poles and they are all to one side of the real axis. Now if a is such a pole so is d and the two are separated by the real axis. This contradiction shows that u = 0. Passing now to the sufficiency proof proper since both (4.9)
~ ( w=) m*k
+ km,
n(w) = m*Dm
are real rational functions of w with numerators of degree 5 n  1 and denominator of degree n, both ,0 as w + + 00. Furthermore they are continuous for w finite. Hence they have finite upper and lower bounds. Let p be the upper bound of n(w) and v the lower bound of ~ ( w )Since . n(w) 0 for all finite w, we have p > 0. Hence
=
+ m*k + k'm  Em*Dm 2 z + v  ep. Moreover owing to (4.3) 7 + v > 0. Hence if one chooses E < *[(z + v)/p], 7
we have (4.10)
7
116
+ m*k + k'm  Em*Dm = 0.
$4.
MAIN LEMMA (YACUBOVICH AND KALMAN)
Let now $(z) = IAZI. Thus $(z) is a real polynomial with leading coefficient unity. Now the left hand side of (4.10)may be written z
Him) + k'm(io) + m*(io)k  Em*Dm = 
Here q(z) is a polynomial of degree 2n with leading coefficient t. Since q(io) is real and > O (4.10),q(iw) = q l ( 0 2 ) , q l a real polynomial without real roots. Hence q1(w2)= O(io)O( io),
where O(z) is a real polynomial. Since the leading coefficient of O(z)O(z) is t , that of O(z) is and the degree of O(z) is n. By division and since the leading coefficient of $(z) is unity
&,
where v(z) is a polynomial of degree at most n  1. If q l , q2,..., are its (real) coefficients, define q by q' = (q1,
..., 4").
Once q is known one obtains the matrix B from (4.2)and as we know B > 0. The above leads to the relation (4.1 1) z
+ k'm + m*k  &m*Dm=
v(io)
io)
Referring on the other hand to (2.2) and recalling the meaning of m we have
117
8.
SOME FURTHER RECENT CONTRIBUTIONS
Hence the relation (4.1 1) yields for the chosen q
+ m*k  Em*Dm = (m*q  &)(q'm
k'm
 &)
= m*qq'm  Jz(q'm = 
 T
+ m*q)
(m*Bb + b'Bm)  Em*Dm  &(q'm
+ m*q),
the last step by (4.5). Hence whatever w : m*(Bb  k  &q) =
+ (Bb  k  &q)'m
2Re(Bb  k  &q)'m
=
0.
Since the vector in parentheses is real it must vanish, showing that (4.2b) is satisfied. That is, a solution ( B , q) has been found for the system (4.2). Thus sufficiency of (4.3) is proved. This completes the proof of (4.1).
$5. LiapunovPopov Function and Popov Inequality Their connection has already been emphasized (VII, §§4,5) and we recall that the function V ( x ,a), the related and the Popov inequality are
v
+ a(a  c ' x ) ~+ B@(o);  3 = x'Cx + Bpcp2(a)+ 2d0'xcp(cr)+ 2ayacp(o); do = Bb  (@A'c + UYC). P(a,b,w) = f l y + Re((2ay + iwfl)c'&'b} 2 0. V ( X ,a) = X'BX
(5.1) (5.2)
(5.3) We also state the following generalization of the Lurie problem resembling one due to Kalman: Generalized Lurie problem. To3nd n.a.s.c. to assure absolute stability by means of the function V ( x ,a) of (5.1), i.e., through Vand  both positive for all x, a not both zero, and all admissible characteristic cp(a).
v
This problem is solved by the fundamental theorem ($6). Reduction of Kalman's relation (4.3) to Popov's (5.3). At first glance, although quite similar, they seem to deal with two different problems. 118
$6.
FUNDAMENTAL THEOREM
Actually by a specialization of the constant k appearing in (4.3) one obtains (5.3). Referring to the expression (5.2) of
+ UYC relation let z = /?p = /?(y + c'b). As a consequence (4.3) k
and in Kalman's yields (5.4)
let
/?p
=
Bb  do = @A'c
+ 2 Re{(&'A + uyc')A,'b}
The bracket may be written
> 0.
+ uyc'AG'b + i/?c'ioA,'b + uyc'A,'b.
@c'(ioE  Ai,)A,'b =  &'b
Since p  c'b = y, (5.4) reduces to (4.3). That is, with the substitutions indicated for the constants (5.3) reduces to (4.3).
56. Fundamental Theorem With the lemma behind us we are in position to prove : (6.1) Theorem. N.a.s.c. for both V and  V of 45 to be positive dejinite for all ( x , ~ )and choice of an admissible cp are the PopovKalman inequality (4.3) together with (6.2)
(a) u 2 0 , p 2 0 , u + / ? > O ; (b) z > 0 or z
=
0, do = 0, u > 0.
When these properties are satisfed the system (VII, 1.1) is absolutely stable.
(6.3) REMARK.It is quite instructive to compare the above theorem and the apparently similar theorem (11, 2.1 1). The earlier theorem refers to the LuriePostnikov function V and in its conditions there enter the matrix C and the control parameters b, c, p. In the present theorem the V function is the more general Popov type and in its conditions there enter merely the scalars u, /? and the control parameters. The difference is due, of course, to the appearance of the powerful Popov condition.
PROOF OF NECESSITY. The necessity of (6.2a) has already been proved. Regarding (6.2b) let 7 # 0. Then the substitution x + E X , cp ,cp, 0 + E% 119
8.
SOME FURTHER RECENT CONTRIBUTIONS
yields  V A zcp’ and so one must have z > 0. Suppose now z = 0, do # 0. Then the same substitution yields  p A 2c(p(o)d 0 ’ x : the sign of pchanges with that of E hence one must have do = 0. Then however u # 0, hence u > 0, since otherwise V = 0 for x = 0, CT 0. Thus (6.3b) must hold.
+
Consider now separately z > 0 and z
=
0.
I. z > 0. One may then write  V = X’(C  qq’)x
(6.4)
+ (&q + q’x)2 + 2uyacp,
where q is defined by (4.2b). Choose llxll large, E small, oo # 0 and fixed q’x = 0 if and o = E ’ O ~ ,q(o)= p 2 0 0 with p such that E~,uo,& q’x # 0, and any p > 0 if q’x = 0. Then  V A x ’ ( C  qq’)x, hence C  qq’ = D > 0. Thus (4.2) holds, hence by the lemma (4.3) is satisfied. Thus necessity is proved in this case. 11. z = 0, do = 0, u > 0. Taking o = 0 and any x , we have  V A x ‘ C x , hence C > 0 and so by (6.2~)B > 0. Since do = 0, (4.2) holds with q = 0. E = 1. Therefore the PopovKalman inequality is satisfied, and necessity is completely proved.
+
E
PROOFOF SUFFICIENCY. Since (4.3) holds given D > 0 there exists > 0 such that the system (4.2) has a solution (B, q). Note that C = ED
+ 44’ > 0,
hence also B > 0. It is convenient now to deal separately with Vand V. Take first K Ifu = 0 then B > 0 and so V > 0 for x # 0 or i f x = 0 for o # 0. Hence Vis positive definite for all X , C Tand admissible cp. On the other hand if u # 0 the sum of the first two terms in (5.1) is a positive definite quadratic form in x , CT and the conclusion is the same. Consider now V. If z # 0 (5.2) becomes
+
 V = EX’DX (&cp(a)
+ q’x)’ + 2~yocp(~).
Hence the sum of the first two terms is positive definite for all x , o and admissible cp and so  V has the same property. If z = 0 then do = 0, u > 0. Hence (5.2) reduces to  V = X’CX
+ 2uyocp(o).
Since C > 0 and u > 0 this expression is likewise positive definite for all x , o and admissible cp. 120
$7.
A RECENT RESULT OF MOROZAN
Since both V and  are, always, positive definite for all x, t~ and admissible cp, sufficiency is proved. PROOFOF ABSOLUTE STABILITY. All that is now needed is to show (BarbashinKrassovskii complement, IX, 4.7). that V + 00 with llxll + Owing to B > 0 and property I11 of (I, $1) for cp this is true if a = 0 since then B > 0. It holds also when a > 0 since the first two terms in the expression (5.1) of Vmake up a positive definite quadratic form in x and 0.
$7. A Recent Result of Morozan It is interesting to return to the inequality (Fi)of (11, $2) for the number
p. In our present notations and since c loc. cit. is here A'c, and hence k for a = 0, /l= 1 is fA'c, we have (Fi)
p > (Bb
 k)'C'(Bb

k).
At a meeting in Kiev in September 1961, the author raised the question of finding the minimum of p for all choices of the basic matrix C > 0. This question has recently been solved by Morozan [l]. We have however all that is required to obtain an answer here. Namely when a = 0, taking B = 1, (4.3)yields p
+ 2Rek'A,;'b
> 0.
Here, however, k = $4'~. Hence p
+ Re(c'A
*
Ak'b) > 0.
From Aim= i o E  A there follows p  c'b
+ Re iwc'Ai,'b
> 0.
Since this last inequality must hold for all real o we find (7.1)
p > c'b
+ sup Im oc'Ai,'b. m
Owing to the n.a.s.c. of the fundamental theorem the righthand side represents the true least value of p. Owing no doubt to differences in notations this result does not coincide with that of Morozan. 121
8.
SOME FURTHER RECENT CONTRIBUTIONS
$8. Return to the Standard Example In the preceding chapter we have already made a comparison between the two types of function V ( x ,0 ) : (VII, 4.6), form of LuriePostnikov (a = 0), and Popov form (VII, 4.Q (a > 0). We return to the same question here and arrive at comparisons based upon rather simple estimates obtained from Popov's inequality. For simplicity the discussion will be restricted to the case /3 > 0. Thus Popov's inequality may be written
y
(8.1)
+ Re(6 + iw)c'A,'b
>0
where 6 = 2ay//3. In our earlier notation m(iw) = c'AL'b one may write (8.1) as
y > os,
(8.2)

=
Sl(o) + iS,(w)
6S1,
which is to hold for some 6 2 0 and all real o. It is evident that one may take /3 = 1. Then 6 = 0 corresponds to the LuriePostnikov type of function V ( x , a ) and 6 > 0 to the Popov generalization. Since (8.1) is independent of the choice of coordinates (a fact readily established) we may assume that A = diag(Al,..., An). As a consequence
and therefore
The relation (9.1) will only yield rather simple estimates when all the & are real. We confine our attention to this case. Thus = ph < 0. Hence C'AL'b
Hence
122
=
1ph + iw' bhch
$9.
DIRECT CONTROL
Taking into account the relation p p >
(8.3)
c
+ c‘b we obtain from (8.1):
=y
Ph(Ph Ph2
+
6)bhCh
oz
.
It is now necessary to distinguish between the signs of the products bhCh. Let bh’Ch‘ denote the positive products and Ph‘ their P h , and bit;, p i the negative products and their p,,. Suppose also that PI’
5 Pz‘
s ... 5 PP”
P;
s Pi s ... s P;.
Now it is clear that (8.3) will hold if one merely preserves the bh’Ch‘, chooses 6 = p l ’ and o = 0. Let
Similarly set
It is evident that if p p is the least of the numbers pp’, p: then p p is a suitable lower bound for the number p. Now let us see what one obtains as lower bound for p from our inequality (Fi). Referring to (II,5) its vector c, now written c,,, has for components p,,c,,. Hence the inequality (Fi)yields here p >
1bh‘ch‘ = Pm.
It is clear that if the products bhch are not all negative p p < pm, hence the Popov type of Liapunov function, i.e. with a suitable u > 0, is then more advantageous than the LuriePostnikov type with u = 0 (our earlier type).
99. Direct Control The most interesting direct control of order n is the one which reduces to an indirect control of order n  1 and is fully discussed in (IV, §§6,7). The state matrix A of the direct control has zero as simple characteristic root. All that we propose to do here is to adapt the theorem of $6 to that case.
123
8. SOME FURTHER
RECENT CONTRIBUTIONS
In the notations loc. cit. the system is Xo = A O X O
c i = g’x,
 boV(0)  pCp(0)
where A, is a stable ( n  1) x (n  1) matrix. One may apply directly the fundamental theorem of $6 under the following identifications : A, corresponds to A ;x, to x , bo to b, g’ = co’A, to c’A,co to c. Here also y = p  c,’A,b,. Finally p and q(a) have the same meaning as in $6.
$10. RQumC (Indirect Control: y > 0) The variety of results on the Popov expression P ( a , p , o ) and the Liapunov function V ( x , 6)of (5.1) as related to absolute stability, may be summarized as follows :
I. Popou’s first theorem. A sufficient condition for absolute stability is (10.1) P(cr,p,o) 2 0 for some a,B 2 0, a B > 0, and all real o.
+
11. Popou’s second theorem. (10.1) is a necessary condition to have absolute stability via Vand  p positive definite, with a and /?the same in P and K 111. Kalman’s theorem. (10.1) plus another (complicated) condition is a n.a.s.c. to have absolute stability through V > 0, p 4 0, with same a, fi in P and K
IV. Theorem of $6.(10.1) with P > 0 plus Bp > 0 or Bp = 0, do = 0: a > 0 are n.a.s.c. to have absolute stability secured through Vand V both positive definite. V. However, in 111 and IV the pair (A, b) is assumed completely controllable. In both also a certain theorem of Yacubovich plays a major role. Suppose now that in the initial system (VII, 1.1) the pair ( A ,b) is not completely controllable. As we have shown in $2, one may choose coordinates, and select a reasonable vector cz such that the initial system is replaced by (2.2) together with (2.3) where now (Al, b) is completely controllable. Popov’s first theorem provides a sufficient condition for the 124
$11.
COMPLEMENT ON THE FINITENESS OF THE RATIO cp(0)/0
absolute stability of the full system (2.2, 2.3). It would evidently be most desirable to prove that P 2 0, supplemented, perhaps, by some simple inequality is also a necessary condition for absolute stability. Since (2.3) already has this property, it might suffice to obtain this result for a completely controllable pair ( A , b). Up to the present, however, this remains an open question.
$11. Complement on the Finiteness of the Ratio cp(a)/a Two recent publications led to this complement : (a) a noteworthy paper by Yacubovich [4] in which he deals not only with the restriction in the title but even with a possible isolated function cp(a); (b) an extensive monograph by Aizerman and Gantmacher [l] where the restriction in question is accepted throughout. This has induced the author to examine the possible modifications in the results of the chapter presented by the added condition (11.1)
0
# 0:
to our admissible class. As indirect and direct controls proceed along entirely distinct lines, the two cases are separated. Indirect control. Take I/=
and modify
X’BX+ U(O  c ‘ x ) ~+ /3@(~)
v by adding and subtracting A(0) = 2ay
Thus A(o)> 0 for
0
# 0, or cp(o)
 V = X’CX
A’B
(.q) 
cp(0).
= KO and A(0)= 0 if q(a) = KO. Then + 2dO’xcp(0)+ A ( 0 )
+ z0cp2(0)
+ B A = c,
do = Bb  ~ B A ‘ c UYC. Replace now Popov’s initial expression by the KPopov expression :
P(a, B, w, K)
=
P(a, p, w )
+ 2UKY ~
=
By
+ 2UY + Re((2a + iw/?)c’A,;’b}. K 
125
8.
SOME FURTHER RECENT CONTRIBUTIONS
Under the same modifications as before in $5 (expression of k ) it is identical with the KKalman expression
K(a, b, w,
K) =
zo
+ 2 Re k'AZ'b.
The new fundamental theorem is : (11.2) &Theorem for indirect control. N.a.s.c., in order that, with V as above, both V and  V be positive definite for all x, o and all Kadmissible functions p (p restricted by 11.1) is that the KPopovKalman inequality:
P(a, b, w, 4 > 0 hold for all real w together with
(1 1.3)
zo
> 0.
When these properties are satisfied the system is absolutely stable in the sense that cp is restricted by (11.1).
PROOFOF
NECESSITY. To
V
prove (11.3) take cp(a) = K O so that
= X'CX
+ zocp2 + 2d0'px.
For x = 0 and zo # 0 then  = Z ~ K ~ hence O ~ , zo > 0. On the other hand zo = 0 is ruled out since then  V cannot be positive definite in x, O. Thus (1 1.3) holds. Write now  V = x'(C  qq')x
+ (&p + q'x)2.
Take any x # 0 and determine a by J z o ~ a = q'x. As a consequence  V = x'(C  qq')x > 0 for all x # 0. Hence C  qq' = D > 0. Thus all the necessary conditions of the main lemma are fulfilled with z = zo. Hence the KPopovKalman inequality holds and necessity is proved. PROOFOF SUFFICIENCY.It is practically the same as in (6.3) save that one need not consider zo = 0. The proof of absolute stability with the K restriction added is the same as in $6, with the modification
+
 V = EX'DX (&&a)
which does not affect the proof. 126
+ q'x)2 + 2ay
(o
 q(a)
?:p
$1 1.
COMPLEMENT O N THE FINITENESS OF THE RATIO cp(O)/U
Direct control. This time the system is (11.4) X’ = A X  bq(a), CT = C’X.
As Liapunov function take V ( X )= X’BX (1 1.5) hence
+ P@(D),
+ 2 d ’ x ~ ( o+) 7cp2(a)
 V(X)= X‘CX
(11.6)
+ B A =  C , d = Bb  &?A’c, T = fic‘b. Actually the role of v is really played by the function W ( x )=  v  s 0 ( A’B
9
 cp(0)
=
X‘CX
+ 2(d  ~SC)’X(P +
( +3 7

q2
where S > 0. In the presence of the restriction ( 1 1.1) the adequate theorem here is : (11.7) KTheorem f o r a direct control. Suficient conditions for V positive definite as a function of x for all admissible functions cp satisfying (11.1) and W as a quadratic form in x and cp (unrestricted) is the K  P O ~ Oinequality. V ( 1 1.8)
S
P(S, /3, o,K) = K
+ Re{(S + io/?)c’A,’b}
>0
for some # 0, some positive 6, and all real w. When these conditions are fuljilled both V and  V are positive definite and we have absolute stability. REMARK. This theorem does not really differ in substance from a theorem of Aizerman and Gantmacher [2, p. 781. They give conditions referring to a Liapunov function V with the property that if V is positive definite under the restriction (11.1) then W is positive definite in x, cp without restriction. The proof of sufficiency can be carried out by a slight modification of the argument of the sufficiency proof of our fundamental theorem ($6). It is also obvious that when the given conditions hold
 v = w+s
(
0
V:))
cp(0)
is positive definite in x (arbitrary) and cp (restricted by 11.1). Absolute stability is then established as in $6. 127
Chapter
9
MISCELLANEOUS COMPLEMENTS $1. The Jordan Normal Form for Real or Complex Matrices In the sections on vectors and matrices the notations are those of (11, 01). In connection with the reduction to the normal form, it is particularly convenient to have recourse to bases for vector spaces. Changing slightly our previous point of view, consider the symbol x to represent a certain vector u in the coordinate system x. Thus the transformation of coordinates y = Px, P nonsingular, does not change the vector u but merely provides a new representation of u in the y coordinate system. In other words the vector u is independent of the coordinate system but is merely endowed with various representations in various coordinate systems. Let eh denote the vector which in the x system has coordinates 6 h k (Kronecker deltas). The system {eh}is a base in the sense that the vector u may be written uniquely
u
(1.1)
=
elxl
+ ... + e,x,
(it is convenient to put the xi after the factors ej). A useful convention is to think of the eh as the components of a onecolumn matrix (vector) designated by e. Then in an evident sense, 128
5 1.
THE JORDAN NORMAL FORM FOR REAL OR COMPLEX MATRICES
under our matrix conventions (1.1) may be written (1.2) u = e‘x. Now if one applies the transformation x u
=
e’Py = f’y,
=
P y one finds
f ’ = e’P
or equivalently f = P‘e. Thus the coordinate transformation x = P y is associated with the “base transformation” f = P‘e. We have already observed on repeated occasions that if in the differential equation f = Ax, A constant, one applies the above transformation of coordinates, the system is replaced by A, = P’AP. j = A,y, That is, the effect on the matrix A is to replace it by the similar also called equivalent matrix A,. The relationship is written with the standard equivalence symbol A, A. Furthermore A , is any matrix A. In order to describe the Jordan normal form we will utilize two special designations. We will write A = diag(A,, ..., Ar), where the A,,, are, like A, square matrices and follow one another in the principal diagonal as if they were elements, the rest being zero matrices. We shall also write

9
. . . . I 1 I
where Cs(i) is n x n and all missing terms are zero. Actually one may also assume that the diagonal 1, 1,... is below the main diagonal. We now state without proof (for the proof see Lefschetz [1, Appendix 11):
(1.3)Theorem. Every complex matrix A is equivalent in the complex domain (that is by means of a complex transformation matrix P ) to a matrix A,
=
diag(CrI(Il),., crs(&)) 129
9.
MISCELLANEOUS COMPLEMENTS
where the A h are the characteristic roots of A (and also of any matrix .A), all included, some perhaps repeated. The order of the “blocks” C(A) is immaterial.
The proof loc. cit. is based on a systematic selection of base elements, one set e.g. ehlr...,ehrhcorresponding to the block Crh(&). Now when the matrix A is real, the characteristic roots appear in is a block conjugate pairs Ah, 1,. It is also shown loc. cit. that if crh(&) associated with Ah, then there appears also the block cr,,(&,). Moreover is the subbase associthe base vector f may be so chosen that if (hl,...,h*) then one may choose a subbase (fh1, . . . , f h r h ) corresponding ated with crh(Ah) as part off: As a consequence, if the vector u (the point x) is real to cr,(&) and if yj is its coordinate corresponding to f h j then its coordinate corresponding tofhj will beyj. This is expressed by the statement : real x points will be represented in the complex y coordinates by pairs of conjugate coordinates. Thus the y coordinates may be ordered like this :
91,...,Pp, Y2,+
y193 Y,,
l,...?
Y,
where the y2,+h are real and correspond to the blocks with real Ah. Letfj,A be associated as above to yj, pj. One may replace them in the base vector f by the real elements
which are both real. If yj, y; are the real coordinates corresponding to then identically
fi’,f
=
From this follows (since
Yj’
Yi.(T) fj
fi,&
+A
+ Y;(li). fj
A
are independent)
y j + iy; 2
7
y.= J
y .‘  iyl 2 . J
By examining the effect on the bases e (real) and f (complex) it is readily found that the transformation matrix P from the real coordinates
130
$1.
THE JORDAN NORMAL FORM FOR REAL OR COMPLEX MATRICES
x to the complex coordinates y has the form
where Q and Q are p x n matrices and R is a real q x n matrix, q
=
n  2p.
Consider a general system
3
=X(x),
X(0) = 0
whereX is e.g. of class C' in a neighborhood SZ of the origin. We merely assert that a change of coordinates x = Py does not apect the stability properties ofthe origin. This is readily proved for instance by means of the inequality on p. 20 of LaSalle and Lefschetz [11. For additional information (especially proofs) of the properties just considered, see the following wellknown books : Bellman [13, Gantmacher [ l ] , also in part Lefschetz [ l , Appendix I ] . We recall the following : (1.4) Theorem. Every positive definite quadratic [hermitian] form may be
reduced by a real [complex] affine transformation of coordinates to the form
The transformation, for example, for a quadratic form may be made in two steps. A first orthogonal transformation x + Px, P' = P', reduces the form to xa&h2,
ah > 0.
Then the affine transformation xh + xh/\& brings it to the required type. We also recall the following property utilized earlier.
(1.5) If the real symmetric [hermitian] matrix C > 0 then likewise C  ' is real symmetric [hermitian] and >O. The treatment of the hermitian case is the same as the other [with )'I, so we confine our attention to the real case.
( )* instead of (
131
9.
MISCELLANEOUS COMPLEMENTS
Since C' = C we have C' = (C')' = (CI)', and so C  ' is symmetric. Let now P be an orthogonal transformation reducing C to the diagonal form P'CP
=
P'CP = D = diag(a,, ...; a,)
where every a,, > 0. Then
D1 = P  l C  l ( P r )  l
=
PrC1(P1)#= P ! c  ' ( P ' ) ' = p'C'P,
Since D' = diag(l/a,,..., l/a,), we have D' > 0, and since it is the transform of C  ' by the orthogonal transformation P, likewise C  ' > 0.
92. On a Determinantal Relation There has occurred in the previous chapters a determinantal relation which we propose to derive in the present section. Let M be an (n + 1) x (n f 1) matrix and suppose that
M=(;
:j
where N is a nonsingular n x n matrix, p and 4 are n vectors and a is a scalar. The relation in question is IM( = INl(a  4 ' W ' p ) .
(2.1)
To prove this relation multiply M (lefthand side) by diag(N', 1). As a consequence diag(N', 1). M = Let N'p
132
=
r, so that the last matrix is
$3. ON
LIAPUNOV’S MATRIX EQUATION
and
From wellknown rules on determinants it is found that rl and q1 appear in the expansion as the product qlrl and nowhere else. By permuting rows and columns it is seen that similarly qh and i h appear in the product qhrh and nowhere else. Hence IS1 = a  q‘r and from this to the desired relation (2.1) is but a step. The relation (2.1) may be generalized as follows. Let M=JQN
j:
where N is n x n, R is s x s and P , Q are n x s matrices, with N nonsingular. Then
IMI
=
JNI(IR Q’NIPI).
The proof is practically the same as for (2.1).
$3. On Liapunov’s Matrix Equation Take the n vector equation (3.1)
X =
AX
where A is a constant stable matrix. Let V(x)= x’Bx be a quadratic form. Its time derivative along the paths of (3.1) is (3.2)
v = av . Ax =  W(X) = X’CX ax
where we have the Liapunov relation
(3.3)
A’B
+ B A = c.
This relation has frequently occurred in the previous chapters, the following property being repeatedly utilized : 133
9.
MISCELLANEOUS COMPLEMENTS
(3.4) If the matrix A is stable and C is any given matrix > 0 then (3.3) has a unique solution B and B > 0. The same propertyholds for a matrix A with some complex characteristic roots, but still stable, save that (3.3) is replaced by
+ BA = C
A*B
(3.5)
where B, C are now hermitian and >O. There is so little difference between the treatments of the two cases that it will suffice to deal with (3.4). As a matter of fact, the result has often been dealt with in the literature (see for example LaSalle and Lefschetz [ l ; 3, $171) but we believe that our present attack is particularly direct and simple. Observe first that a transformation of coordinates x = P y will replace (3.1) by
3 = Aoy,
A,
=
P’AP
and x’Bx, x’Cx by y‘B,y, y’C,y, where B , = P’BP, Co = P’CP. Hence (3.3) yields PA’P‘’P’BP
+ P’BPP’AP
=
P’CP
which is AO’BO
(3.6)
+ BOA0 = Co.
Thus (3.3) is unaffected by a transformation of coordinates. Since this transformation may be inverted, it is sufficient to prove the asserted property for (3.6). Now one may choose the transformation of coordinates so that A , is triangular, that is, only with zeros above the main diagonal. Indeed the Jordan normal form is already of this form. The terms in the main diagonal are then the characteristic roots of A , (the same as those of A). Thus A,
=
diag(L, ,..., An)
+ D,
where D = (djk)is triangular with zeros in the main diagonal. Let bykk,cjq, be the terms of B,, C,. The equation (3.6) gives rise to a set of relations (3.7)
+
+
where djk is a linear form in the bhk such that h k > j k with coefficients in the drr Since every Re l j < 0, the system (3.7) can be solved step by step 134
$3.
ON LIAPUNOV’S MATRIX EQUATION
by an induction downward from j + k = 2n, and the solution is unique. Hence (3.6) has a unique solution B, and (3.3) has a unique solution B. If B is that solution (3.3) yields flA
+ A’#
= C‘ = C.
Hence, B’ is likewise a solution of (3.3)and so B’ = B. The treatment of (3.5)is the same with A’ replaced by A*. REMARK. There are two noteworthy special cases, repeatedly considered in the previous chapters as the standard example, when the solutions of (3.3) or (3.5) are especially simple. (a) The A,, are all real, Ah =  , u h < 0, and A = diag(pl, ..., p,,). (b) Some A,, are complex, Re Ah =  p h < 0 and A = diag(Al,...,A,,). In the case (a) the relation (3.1) yields at once h j
+ pklbjk
= cjk,
hence (3.8) In the case of (b) (3.2) yields hence now (3.9) We shall now prove: (3.10) T h e unique solution B of (3.3)just obtained is >O. Notice that with the earlier result this property is equivalent to this proposition : (3.1 1) Theorem. (Liapunov). Given the positive definite quadraticform W(x), the partial dgerential equation (3.2) for V ( x ) has a unique solution a s a positive definite quadratic form. At all events there is a solution V ( x ) = x‘Bx as a quadratic form. Suppose that V(x) is not positive definite. If it can take negative values, it is known from Liapunov’s instability theorem that the origin must be 135
9.
MISCELLANEOUS COMPLEMENTS
unstable for the system (3.1). However we know from the form of the solutions that the origin is asymptotically stable. Hence V(x)2 0. Suppose that V ( x o )= 0 for some x o # 0. Let y be the path issued from xo at time t = 0. Along y we have =  W ( x ) < 0, hence V ( x ) will become and remain < O along y. Since this contradicts the result already obtained V ( x ) 0 for x # 0, hence B 0. Thus both (3.10)and (3.11) are proved. We have actually all the elements for the proof of the following noteworthy proposition.
=
=
(3.12) Theorem. A necessary and suficient condition for the stability of the real matrix A is that there exist two real matrices B, C > 0 which satisfy the relation (3.3). Necessity has already been proved. To prove sufficiency let 1 be a characteristic root of A . There exists then a vector u # 0 such that Au = lu. Since A is real A’ = A*. Thus u*A* = lu*. Now from (3.6) follows u*(A*B + BA)u = 2(1 + X)U*BU< 0.
Since B > 0 this implies 2
+ 1= 2 Re 1 = 0 and so A is stable.
$4. Liapunov and Stability As Liapunov’s theory has been referred to many times in the previous chapters, we shall give a rapid resume of it insofar as it applies to an autonomous system i= X ( x ) ,
(4.1)
X ( 0 ) = 0.
Let S R denote the spherical region llxll < R and H R the boundary sphere of the region. The system (4.1) is supposed to be of class C’ in a certain region S,. Liapunov defines the origin as: stable for (4.1) whenever given any 0 < E < A there corresponds to it a 0 < V ( E ) 5 E such that if x(t) is a solution whose initial position x o = x(0) lies in S, then x(t) lies in S, ever after ; Asymptotically stable whenever the origin is stable, and furthermore for some E every solution x(t) as above +O as t + + co ; Unstable whenever given any 0 < E < A and no matter what 0 < q < E there is always an x ( t ) as above reaching H,at some time t > 0. 136
@. LIAPUNOV AND STABILITY
The stability theorems of Liapunov given below rest upon this concept : A scalar function V ( x )is positive [negative] definite in the region S , whenever V ( x )is of class C' in S,, V(0)= 0, and V ( x )> O [