THE UNIVERSITY OF MICHIGAN SYSTEMS ENGINEERING LABORATORY Department of Electrical Engineering College of Engineering Technical Report SEL-68-33 STRUCTURE OF SENSITIVITY REDUCTION by Charles L. Zahm November 1968 This research was supported by NSF Grant GK-1925.

This report was also a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the University of Michigan.

ACKNOWLEDGEMENTS I wish to acknowledge my debt of gratitude to Professor W. A. Porter for his guidance and enlightening discussions during the course of this research. I am also indebted to the other members of my committee for their constructive criticisms. For the preparation of this manuscript, I especially want to thank Mrs. Joanne Aichler who worked so carefully and diligently at this task, as well as Miss Sharon Bauerle. This work was supported by National Science Foundation Grant GK 1925. Finally, I shall be forever thankful to my wife whose patience and encouragement made this effort possible. ii

TABLE OF CONTENTS Page ACKNOWLE DGEMENTS ii LIST OF ILLUSTRATIONS v LIST OF APPENDICES vi ABSTRACT vii CHAPTER 1: INTRODUCTION 1 1. 1 Basic Concept of Sensitivity 1 1. 2 Review of the Literature 5 1. 3 Sensitivity Reduction of Linear Optimal Control Systems 12 1.4 Dissertation Objectives 15 CHAPTER 2: POSITIVE REAL OPERATORS 17 2.1 Introduction 17 2. 2 Integral Operators 17 2. 3 Linear Time-varying Systems 27 2. 4 Stationary Systems 36 2. 5 L -Positive Realness and Stability 41 2.6 Summary 44 CHAPTER. 3: SENSITIVITY REDUCTION IN LINEAR TIME-VARYING DYNAMICAL SYSTEMS 45 3.1 Introduction 45 3.2 Statement of Problem 45 3.3 Structure of the Sensitivity Operator 51 3. 4 Sensitivity Analysis Via the Inverse Sensitivity Operator 58 3. 5 Sensitivity Reduction with Plant Perturbations 61 3.6 Summary 64 111

TABLE OF CONTENTS (Continued) Page CHAPTER 4: SENSITIVITY REDUCTION IN LINEAR STATIONARY SYSTEMS 65 4.1 Introduction 65 4. 2 Conditions for Sensitivity Reduction in Stationary Systems 65 4. 3 Structure of the Stationary Sensitivity Operator 68 4. 4 Sensitivity Analysis Via the Inverse of the Stationary Sensitivity Operator 74 4. 5 Sensitivity Reduction with Plant Perturbations 79 4.6 Summary 83 CHAPTER 5: A COLLECTION OF RESULTS IN SENSITIVITY REDUCTION 85 5. 1 Sensitivity and Optimal Control 85 5. 2 Sensitivity Reduction in Linear Discrete Systems 88 5. 3 Sensitivity Reduction of a Linear System 93 5.4 Summary 98 APPENDICES 99 RE FERENCES 140 iv

LIST OF ILLUSTRATIONS Figure Page 1.1 A Closed-loop System 3 1.2 An Open-loop System 4 1.3 Two Nominally Equivalent Systems 6 1.4 Two Nominally Equivalent Systems 9 C.1 A Time-invariant Linear Feedback System 126

LIST OF APPENDICES Page APPENDIX A: A REVIEW OF FUNCTIONAL ANALYSIS CONCEPTS AND TERMINOLOGY 99 APPENDIX B: CONCEPTS IN MODERN CONTROL THEORY 110 APPENDIX C: A SUMMARY OF FOURIER ANALYSIS 130 vi

ABSTRACT For several years control theorists have been examining the problem of reducing the sensitivity of a control system containing a plant, a precorrpensator and a feedback compensator. The purpose of this study is to ascertain the underlying structure of a feedback control system irn order to reduce its sensitivity relative to a nominally equivalent open-loop system. This problem is initially formulated in Banach spaces in order to emphasize the operator theoretic point of view. Attention is then restricted to Hilbert spaces, to obtain more detailed results including interpretation, from a physical point of view. Finally, the problem is formulated in terms of plants described by linear timevarying differential equations of the form x(t) = A(t)x(t) + B(t)u(t), x(to)= x, t = [t0, tf y(t) = C(t)x(t). Here the tuplet x = (x1,.., xn) represents the state variables, the tuplet u = (u1,..., up) represents the independent plant inputs, and the tuplet y = (Y1,. * *,m) represents the independent plant outputs. The matrices A(t), B(t), and C(t) are real and continuous on the set C! and are of compatible dimensions. The pertinent question then answered in this thesis is: "Given a precompensator G and a feedback compensator M, what are the relationships between these compensators and the plant to ensure sensitivity improvement?" vii

To facilitate this study a concept of positive realness for timevarying kernels is introduced. This concept is demonstrated to indeed be a generalization of the standard definition given in the frequency domain for stationary systems. The concept of positive realness is then brought to bear on the pertinent issues, illustrating the natural relationship between positive realness and sensitivity improvement. viii

Chapter 1 INTRODUC TION 1.1 Basic Concept of Sensitivity There are basically two notions of sensitivity both of which have the distinction of being considered classical. The first notion considers a system as represented by a set of vector differential equations whose solutions are dependent upon the parameters of the equations. The second notion, which is perhaps better known to control engineers, was introduced by Bode [9] and is almost as old as the theory of feedback. The first method (see Kokotovic and Rutman [29]) considers the perturbations in the state vector due to disturbances in a parameter vector for systems, which can be described by x =f(x,u,It): x(to) =X (1. 1) where x is the nx 1 state vector, u is the mx 1 control vector, M is the px 1 time-invariant parameter vector and t is the independent variable. If the vector ji is perturbed such that - 1- = M/ + An, then the corresponding state vector becomes x =f(i,u,j,t); x(to) =xo (1.2) The perturbation in the state is then defined to be

Ax =x - x. (1. 3) Forming a Taylor series expansion of equation 1.3 in powers of A the perturbations in the state due to perturbations in Ai is straightforward. First define the trajectory sensitivity function crij (a z L 0=O To a first order approximation the perturbations in the state are then given by p Ax =.-.. (1. 4) j=1 If f is continuously differentiable and uniformly Lipschitz, the solution of 1.2 exists and is given by equations 1.3 and 1.4 where cor. satisfies the sensitivity equation ' i (afk afk '.+ ki-1 axi a~zj with the constraints akj(t)= 0 (k=1,... n; j=1,...,p). The second method, as devised by Bode, measures the system sensitivity by the ratio of the percentage change of the transfer function (transmission)T(p) to the percentage change of the parameter A. For differential variations, the sensitivity measure becomes $T= 8T(z)/T(/>) = [T )].

To illustrate an application of the sensitivity measure, consider the stationary single-variate feedback system of Figure 1.1. U -I P' Yc Figuzre 1. 1 A Clcsld-Locp SyaOi:;n The plant and feedback compensator transfer functions are represented by P and M where Yc and u respectively represent the output and input of the feedback system. The transfer function of the system is given by T() = P(s,)' ) 1 + P(s, i)M(s) where it is tacitly agreed that P and M are Laplace transformable. Applying the definition of sensitivity reduction, and interpreting the parameter /i. to be a parameter of the plant, the classical sensitivity measure becomes $T(SIO)= [l+P(s, l)M(s)]- (1.5) where lo is the nominal value of Al. Clearly, a good feedback design should insure that I $T(jw, ) I < 1 over the frequency band of interest. Two primary reasons for introducing feedback are to reduce the effects of parameters variations upon the system behavior, and to improve the rejection of disturbance signals. If the feedback design does not include sensitivity considerations, then it is certainly possible

that the effects of parameter variations could make the closed-loop system worse than the open-loop system. Consider, for example, the stationary single-input, singleoutput feedback system of Figure 1. 1 and the corresponding open-loop system of Figure 1.2 u I.p Yo Figure 1. 2 An Open-Loop System It is easy to verify that the nominal system equations for the openloop and closed-loop systems are given by Y (s) = P(s,1 o)U (s) (1. 6) and Y (s) = P(s, i) [U (s) -M(s)Y (s)] (1. 7) If the parameter changes from /Lo to Ali + jio, the deviations from nominal for the open and closed-loop system are given by 6Y (s) = AP(s, I)U(s) (1. 8) [1 + P(s, o0)M(s)] 6Y (s) = AP(s, 1)[U(S)-M(S)(Y (s)+6 YC(S))] (1. 9) where AP(s,) -= | P(s,i)\! A. From equations 1.8 and 1. 9, it follows that the percentage change in

the output of the closed and open-loop systems are related by 6Y = [F1 +P(s,u )M(s)] -1 $ ~Yo(S) Yc(s) Y (s) Y (s) C 0 0 Hence, if 1 +P(jw, )M(jwO) - < 1 for all ML and jw, then the percentage change of the output of the closed-loop system will be less than that of the open-loop system. Note that for small plant variations such that P(s, ) is approximately equal to P(s, o), $ is equal to the classical sensitivity given by equation 1. 5. The above discussion forms the beginning of the modern concepts of sensitivity reduction of multivariate control systems. That is, it serves as a foundation from which the modern concepts are constructed. The following section gives a review of some of the more important developments in the modern theory of sensitivity reduction, and illustrates how these results reduce to the classical concepts described previously for single-variate stationary systems. 1. 2 Review of the Literature Recently efforts have been directed to extend the concept of sensitivity as presented by Bode to linear multivariate systems (references [ 11], [ 39] and [ 43 ] serve as a introduction into this subject). In particular, the works of Perkins and Cruz 11 ], [39] were the first successful attempts to establish a sensitivity reduction criteria for multivariate systems. Although the purpose of these papers were not to establish all the elegant structure pertinent to the

sensitivity operator, they did serve to illustrate a methodology by which this objective could be achieved. The approach used by Perkins and Cruz established a criteria under which a closed-loop system would be less sensitive to disturbances than a nominally equivalent open-loop system. The criteria being that the closed loop system is said to be less sensitive than the open4oop system if for all tf e (0, co), t t I f 6 '(t)Qy (t)dt < Jo' (t)Q Yo(t)at Q > 0(1.10) providing, of course, both integrals exist. These authors considered the two system configurations illustrated in Figure 1. 3. u G P YC, I u M Figure 1. 3 Two Nominally Equivalent Systems 1 Throughout the thesis, prime is used to denote matrix transpose and the notation A > 0 is used to emply that A is a positive operator.

The plant, P, is linear, causal, and stationary and is represented by the matrix P(s, A) and the feedback compensator is a constant matrix M. The compensator G is chosen so that the two systems have equivalent transfer functions with no disturbances in the form of plant perturbations or additive noise r/. If the plant is perturbed, then the relationship between the output of both systems is given by 5Yc(s) = + P(s, )M] -16Y (s) (1.11) cO or 6 Yc(s) $ (s, p)6 Yo(s) (1.12) Using Parseval's theorem and equation 1. 12, it follows that the sensitivity criteria can be given a frequency domain interpretation via S Y*(j)[$ *(j)Q$(j) -Q] 6Yo(jw)dw < 0 (1.13) -00 where * denotes the complex conjugate transpose. If inequality 1. 10 exists and Q - $*(jw)Q$(j)> 0 for all WE (-00, 0), (1.14) the closed-loop system is said to be less sensitive then the open-loop system. If Q is replaced by the identity operator, we see that for single-variate systems inequality, 1.14 reduces to I1+P(jw,,)M 11 < 1, for all W e (-oo,o). For a definition of causality, see Appendix C.

Consequently, these results reduce to those discussed previously for single-variate systems. We now see that sensitivity reduction for multi-variate systems is given a new meaning via the performance index of inequality 1. 10, which is a generalization of the classical definition for the single-variate case. Having established this concept of sensitivity reduction for stationary multi-variate systems, Cruz and Perkins extended this concept to linear time-varying multi-variate systems [38 ]. This was accomplished by recognizing that 6yc(t) and 6yo(t) are related by a linear time-varying operator $, such that 6Yc(t) = $byo(t); $ = [I + PM]. (1.15) whenever I + PM is nonsingular. The sensitivity reduction criteria was then formulated by a natural generalization of inequality 1. 10. This criteria states that the sensitivity of the closed-loop system is less than the open-loop system if the norm of 5y is less than the norm of 6y.y It is evident this criteria is equivalent to the norm of $ less than unity. This fact led the above authors to establish the relationship between the mapping $, and contractionI mappings. These results although interesting for their own sake, were not sufficient to completely characterize the fundamental structure of $. Porter demonstrated, (see [41 ] through [45]), that the entire question of sensitivity reduction by means of feedback could be form - 1 For a definition of contraction mappings, see [34].

ulated in terms of Banach spaces. This was accomplished by considering the two nominally equivalent systems of Figure 1. 4 with 7j equal to zero. Letting B1 and B2 denote Banach spaces, the plant P is defined to be linear such that P: B1 B2. The compensators G and M are also linear and G: B1 - B1 as well as M: B2 - B1.,-' I PYc u — i P y0 Figure 1.4 Two Nominally Equivalent Systems As before when P is perturbed, the perturbations in the outputs of the two systems are related by a linear operator I, such that 6y =- 6y; = [I+PGM]-1. (1.16) The sensitivity was then said to be improved if and only if the norm of $ is less than one. Porter, then focusing attention on timevarying systems whose operators map between finite cartesian products of the Hilbert space, L2(-oo,oo), demonstrated that the See Appendix A for a definition of the Hilbert space L2(-oo, o).

perturbations of the open and closed-loop systems can be related by 2 2 y > 2 YcI= $oYoI <$6 yol$0 I < (I -* $ $)6yy o Consequently a necessary but not sufficient condition for sensitivity reduction is that the operator I-~*9 be positive. This condition is not sufficient in the sense that the inequality tf tf [6y(t),yc(t)dt [yt), Y6y(t)]dt, for all tf E (-o, o) -00o -00 will not be satisfied for noncausal $. The above author then illustrated that if P, G, M, and $ are bounded and stationary; I - *$ > 0 - I - $*(jw)$(jw) > 0 for all w e (-Co, o), where $(jw) is the frequency matrix representation of $. This result is identical to that given by Cruz and Perkins [11 ] with Q = I. It is emphasized that the results given by Porter are not restricted to the stationary case. In particular Porter extends the concepts of sensitivity reduction, in Hilbert spaces, to distributive systems (see [41 1 and [42]) as well as to time-varying systems [44]. Anderson and Newcomb [ 5 ], following the ideas of Cruz and Perkins, were the first to formalize the necessary as well as sufficient conditions for sensitivity reduction when the Hilbert space is L (-co, co). By means of distribution theory, these authors demonstrated that the relation between the perturbations of the open and closed loop response due to plant parameter variations is linear and describable by a

distributional1 kernel. For the systems of Figure 1. 3, the kernel of the sensitivity operator $ is given by - [6In~+pam] where p is the impulse response matrix of the perturbed plant P(i), and consequently a distributional kernel dependent on g, m is the impulse response matrix of the feedback compensator and 6In ande denote the unit impulse matrix and convolution respectively. The criteria for sensitivity reduction is not given by the usual norm on L 2n(-o, co), but instead by tf tf S [6ye(t), 6y (t)]dt < f [6yo(t), 6yo(t)]dt, for all tfe (-oo,oo). -00o -00 Employing this criteria, the necessary and sufficient conditions for sensitivity improvement are then demonstrated to be 1. $: L(-oo,oo) — Ln2(-ooo ) 2. $ causal (A1) 3. 11$I <1. For stationary systems the authors point out but do not prove that conditions (Al) are equivalent to 1. $(s) analytic for Re s >0 2. s(s) = $(s) for Re s >O (A2) 3. I-$*(s)$(s) > 0 for Re s >O 1 See Reference [50] for a comprehensive treatment of distribution theory.

where the overbar denotes complex conjugate and $(s) denotes the Laplace transform of the sensitivity kernel. 1. 3 Sensitivity Reduction of Linear Optimal Control Systems Since the primary reason for introducing feedback is to. reduce the effects of parameter variations upon the system behavior, the question of whether linear optimal regulator systems provide a closedloop sensitivity reduction becomes pertinent. Kalman r23 ] was the first to answer this question affirmatively on the basis of the analogy of his results with the classical return difference of Bode. The result can be stated as follows: consider the completely controllable plant, x(t) = Ax(t) + bu(t); x(O) = xo, (1. 17) with linear state feedback u(t) = -m'x(t), (where A,b, and m are nxn, nx 1, and nx 1 matrices respectively), and a performance index of the form 2 J(u) = [x' (t) L Lx(t)+u (t) ] dt; L'L > 0 o where {A, L} is completely observable. Then a necessary and sufficient condition for m to be an optimal control law is that m is a stable control law and I1+ m'4(jco)b12 = 1 + IL4(jw)b12 (1.18) hold for all w, where 4(jw) = (jwI-A). 1 The optimal regulator problem as well as associated concepts of modern control theory are discussed in Appendix B. Observability is discussed in Appendix B.

Cruz and Perkins [39] demonstrated that equation 1.18 was equivalent to the classical percentage-change sensitivity replacement I$(jv) I < 1 for all we (-oo,oo), provided D(jw)b f 0. Hence, Kalman's result demonstrated that for the single-input, multi-variate output, a necessary and sufficient condition for optimality is that the feedback control law reduce the sensitivity to disturbances. Anderson, [ 6 ] following arguments similar to Kalman, obtained analogous results for the multi-variate feedback system. In particular, Anderson considered the controllable plant described by the equations, x(t) = Ax(t) + Bu(t); x(0) = xo, (1. 19) where u(t) = -M'x(t) (such that A, B, and M are nx n, nx p, and nx p matrices respectively), is the optimal control law for a performance index of the form co J(u) = f [x' (t)L'(t)Lx(t) + u'(t)u(t)] dt. (1.20) 0 The following theorems follow from Anderson [ 6 ]: Theorem 1. 1 Let M be an optimal control law for the completely controllable plant 1. 19 with the performance 1. 20. If {A,L} is completely observable, then [I + B'?*(jwo)M] [I + M'?(jw) B] > I, for all woe (-co,oo). (1.21)

Theorem 1. 2 Let M be a stable control law for the completely controllable plant 1. 19 such that {A,M} is completely observable and [ I + B'?*(jco)M] [ I+M '~t(jw)B] > I for all w e (-oo, co), then, there exists a matrix L such that {A,L} is completely controllable such that M is an optimal control law for the performance index 1.20. Kreindler [32], who was also working in this area, demonstrated that inequality 1. 21 was equivalent to $*(jo)Q$ (jo) < Q, where Q=MM *> 0. Consequently, Anderson's result illustrated that for the multi-variate feedback system, a necessary and sufficient condition for optimality is that the feedback reduce the weighted system sensitivity. Generalizing these results to multi-variate time-varying systems [3 ], Anderson was again able to demonstrate that over the infinite interval, the optimally derived controller reduced the sensitivity to a particular weighting of the output errors. Kreindler, utilizing the notion of a trajectory sensitivity function, o, defined by (t) = ax(t) established a criteria under which the closed-loop system would be less sensitive to disturbances than a nominally equivalent open-loop

system. The criteria being that the closed-loop system is less sensitive than the open-loop system if there exists a positive selfadjoint matrix Q such that tf tf a' (t)Q (t)dt < f ' (t)Q ao(t)dt for all tf e (0,co), C 0 0 O O where ao and uc represent the open and closed-loop trajectory sensitivity. For the multi-variate control system described by 1. 19 where u(t) = -M'x(t) is the optimum control law for the performance index J(u) = f [x' (t)L'Lx(t)+u' (t)Ru(t)]dt; R > 0, then there exists a matrix Q namely, M'RM such that the sensitivity is reduced. Kreindler [32 ] also points out that in general it is not possible to reduce the sensitivity for arbitrary weighting matrices Q. To clarify the condition under which the optimal regulator reduces sensitivity, a necessary condition relating the matrices M and B is given in Chapter 5. 1. 4 Dissertation Objectives In the past, considerable effort has been devoted to identifying the structure of the sensitivity operator $. In the brief review given in section 1.2, it was seen that some of the tools used to tackle this problem owe their origin to functional aalsis and distribution theory. However, made obvious from its scarcity, is the application

of these tools to identify the structure of the compensators to achieve the aforementioned goal. The purpose of this thesis is to apply Banach and Hilbert space techniques, as employed by Porter and Newcomb, to determine the relationships between the plant and its associated compensators to ensure sensitivity improvement. That is, necessary conditions and sufficient conditions relating the plant matrices {A(t),B(t),C(t)} and the compensators G and M are given Mhich guarantee sensitivity reduction to noise and plant parameter perturbations. It is also shown that the concept of positive realness plays an important role in developing these relationships.

Chapter 2 POSITIVE REAL OPERATORS 2. 1 Introduction For several years, network theorists have been employing a concept of positive real functions in the analysis of time-invariant networks (references [17], [ 18] and [33] are recommended as entries to this literature). More recently this concept has become an important tool for the control systems analyst. Popov, for example, successfully employed this concept in his development of a stability criterion for nonlinear feedback systems [40 ]. Anderson, employing the matrix analog of positive real function r 4 ] explored the relationships between linear feedback control laws chosen to reduce system sensitivity, and linear feedback control laws derived on an optimal control basis from a quadratic loss function [ 6 ]. These and other applications provide in part the motivation of the present effort to formulate a notion of positive realness that can be fruitfully employed in the study of time-varying systems. 2. 2 Integral Operators We begin the discussion with the notion of an integral operator. Let k(t,T) be a complex valued m x p matrix of functions continuous in both variables on the set Qlx 2 where CQ r R denotes a closed 1 Henceforth, if a kernel is said to be continuous it is assumed to be continuous in both variables.

interval of the real line. Denote by K the multivariate integral operator (Ku)(t) = f k(t, T)u(T)dT, t eQ. (2.1) Let z = Ku, then in component form, each component of the vector z is given by n zi(t) = E f kij(t,T)uj(T)dT, i= 1,...,m. j=1 12 The matrix k(t, T) is called the kernel of the integral operator. Throughout this chapter, we will focus our attention on bounded operators mapping between finite cartesian products of the Hilbert space' L2(Q). The scalar product between any two vectors x, y L2 (Q) is given by n < xlY>= e, xi(t)Yi (t)dt i=1 consequently the Hilbert space norm is computed by jjxl12 = < x,x > = xi(t) i2dt = f [x(t),x(t)]dt, where [, denotes the inner product in the complex Euclidean n space E. The Hilbert space adjoint, K*, of the operator K, can be determined in the following manner: 1 Appendix A is recommended for a brief introduction into Hilbert spaces.

< y, Kx > = [y(t), f k(t, T)x(T)d7] dt =S Sf [y(t),k(t, r)x(T)]diTdt. By Fubini's theorem, we can then write < y,Kx> = f [I kt(t, T)y(t),x()]dtd=< K*y,x>. Hence, the adjoint operator is computed by (K*y)(t) = f k'(T,t)y(T)dT, t e Qo This serves to identify the kernel, k, of the adjoint operator namely k*(t, T) =k' (T,t), (t, )e c x.Q. Let w(t, -) be a continuous complex-valued nxm matrix defined on the set Qx Q, and let W denote the associated integral operator. The application of W to z of equation 2. 1 serves to identify the kernel of the composition WK (Wz)(t) = F w(t, ) f k(r, s)u(s)ds dr. Employing Fubini's theorem it follows that (Wz)(t) = f {f w(t, )k(T, s)d}u(s)ds; t e Q and thus the kernel of the composition N = WK is given by n(t, s) = w(t, T)k(r, s)d; (t,s) e x 1 See Halmos [20], P. 148.

20 Remark 2. 1 The composition of two operators may be used to introduce the idea of an inverse operator. That is, W is said to be the left inverse of K when x(t) = (WKx)(t) t 2, for all x in the domain of K. Using the unit matrix delta function, the identity can be viewed as an integral operator; namely Ix(t) = S I 6 (t -T)x(T)dT = x(t). Therefore the kernel of the inverse system, k (t, T), may be viewed formally as the solution of the integral equation I 6 (t-T) - ) k-(t, X)k(A, )dX. The following three lemmas (which are available in single variate form in the literaturel) are helpful in characterizing kernels of integral operators which are bounded on the function space L2 (Q) and in introducing the mathematical terminology. Because these kernels play an important role in the sequel, the proofs of these lemmas are included here. Lemma 2. 1 If the matrix kernel f(t, T) satisfies the condition2 n QxQ Ifij(t, 1 ) ddt < ij ~xi 1 See for example [28]. 2The notation e is equivalent to Z. ij i=1 j=l

21 it is called a Hilbert-Schmidt kernel, and its associated integral operator F is bounded on L2 n(Q). Proof: Let y = Fx such that x e L2'(Q) then in component form, each element of y is computed by n Yi(t) = 3 fij (t, T)X.(T)d7. j 2ij Employing the obvious inequality for I Yi(t) I we obtain n Iyi(t) l < I f Ifj(t, )I Ixj(T) Id. j Q By Schwartz's inequality for integrals lyi(t) I < [ If ij(t, ) 12 dr] [ Ixj(T) 12 d] n 2 n < [ s |f..(t, )12d7]2[ r s | (7)|2d ]2 < f Y f.(t, )12d7] Ilxll. j Qi Using this inequality we obtain for I l II 1n 1 5ly= [ lYi(t) l dt] < [ IS If (t 7)12 d dt] 2xl i QS! ij Q xQ Since Ixll < oo, the lemma is proved. Lemma 2. 2 If the matrix kernel f(t, T) satisfies the conditions ' If (t )I(tdT)dt<M < ordij = 1.. ft ii ' Mt ' " for 1 For further properties of Hilbert Schmidt operators see [12].

22 then, the associated operator F is bounded on L2(2). Proof: As in the previous lemma, for y = Fx, Yi is computed by n yi(t) = ffij (t, r)xj(T)dT. Taking absolute values n Iyi(t) < Z If (t, T)Xj(T) IdT, and using Schwartz's inequality rii ~ 1 1 Iy (t)t fl [If.j(t, 7) I IXi(T) ] If (t, T) 2dT j i 13 n < { [ i(t, ) I ]xj(T) 2dT} { f f i(t, 7) ]d}2 j 13 3 Q:ij n 1 _ M i1 { If1ij[(t, T) IIxj(T)12 dT}. We obtain for lyll ni 1 n I[Yl = n{ Z f Yi(t) 2dt}2< M { f fijf (t, T) Lxj () )1 2ddt} i QZ ij QxQ n < M1 { f ] xj(T) I 1 2 Ifij(t, T) IdtdT} Hence, 1 IlyIl < n M1M2 IIM, and since I lxll is finite, the lemma is established. If the matrix kernel f(t, T) is of the stationary type, that is f(t, T) = f(t- T), then the following lemma is pertinent.

23 Lemma2.3 Iffij e Ll( ) that is, fj lfij()ldr < Mfori, j= 1...,n, then the associated integral operator is bounded on L (Q) where Q = (-oo,CO). Proof: Let Fx = y such that x c L2(Q). Yi is computed by n n Yi (t) = C f fij (t -)xj(T)dT = fij (r)xj(t - T)dr. J J j Taking absolute values n lYi(t) I <. f Ifij(r) I Ixj(t- 7) d7, j 1j and using Schwartz's inequality n 1 y~(t)l < 3 f fij (r) 1 I xj(t- T)12] fij (T) I dr jIJ 1J n < If i(T)I Ixj(t-T)12dr} {fT fij.(T)IdT} ij3 f <M { f f.(T) Ix (t- )12dT}2 Computing Iy|II 1_1 Ilyll = { lf Yi(t) 12dt} < M { If.fij(T) Ix.(t-T) 2dTdt}i < M{ x (t- T)E i fij(T) |dTdt} 2 1 11 n < n M2 { ~ Ix.(t - T) 2dt}2 < n M2 Ilxll. HIence, the lemma is proved.

24 Definition 2. 1 A continuous self adjoint matrix kernel k(t,T), defined on Q x Q, is called a positive kernel if1 < x,Kx> = f [x(t),k(t, 7)x(T)]dTdt > 0 (2. 2) for all x e L2 (). To see that this class is nonvacuous, we consider, for example, the kernel defined by k(t, T) = f f'(X,t)f(X, T)dX, (t, T) e Qx (2. 3) where f(t, T) is a continuous complex valued m x m matrix. Since < x,Kx> = f [f(X,t)x(t),f(X, T)x(T)]dXdTdt = f [ f(X, t)x(t)dt, f (X, T)x(T)dT] dX g QQ5Q = IIFxI 12> 0 it is clear the k(t, v) is a positive kernel. The following two lemmas are helpful in further characterizing positive kernels. Let 1Q be the set to, tf], then Lemma 2. 4 If the kernel k(t, -) is continuous in both variables on the set Q x Q, then a necessary and sufficient condition for k(t,T) to be a positive kernel, is that for every finite uniform sequence {t,}m of distinct points belonging to Q and arbitrary complex vectors In short a kernel is positive if its associated operator is positive. See also Zaanen [58].

25 y,y2,...,y E E, the following relation holds m [ya,k(t,t )y] > 0. (2.4) a,3 Proof: Tilis proof is a straightforward generalization of the single variate case and follows that given in Widder [ 55]. Assume equation 2. 4 is valid, and let tf - t At= f o m+1 Set t =t + aAt, a= 1,...,m. a o Letting y a= x(ta) f O and substituting into 2. 4 yields m tf - t 2 E [x(ta) k(tat)x(tt)]( m 1 ) > o. (2. 5) a, f Equation 2. 2 then follows from 2. 5 by taking the limit m -o co. [56] To prove necessity we proceed as follows: suppose that there exist {ya} and {ta} for which equation 2. 4 is not positive. We shall then show that there exist continuous complex vectors x(t), such that equation 2. 2 is not satisfied. Define a continuous scalar function 0 (tt), such that t and t belong to Q. Let e and y be small enough so that to t - - y < t + + y < tf. 0 (t,t) is then defined to be zero in the interval (to < t < t - e - y) and (t + e + y < t < tf) and unity in (t - y < t < t + y). In the rest of the interval 0 (t,t) is defined to be linear in t. Now choose e > 0, and y > O so small that

26 no two intervals (-e - y + t, t + e + y) have a non-zero intersection. Set m 0(t) - y a0 (t,t ) and let m Km(t,7) = [ya,k(t+ta,+ t 3)y]. af Hence m <,KO>= fS Km(t,T)dtdT+ - ff [0(t),k(t,7)0(T)]dTdt -Y - a,f3 Aa where AA is the region between the square t -e-y < t < t +e+, t-e-7 < < t +e+y and the square ta -y< t t t +y, t1-y < 7 < t+ y. For any point (t, T) in the region Aaa n I[0(t),k(t, 7) () ] L < M lIyay, i where M = sup(t ) Ik ij(t, r) for i, j 1,.., n. Hence, m n I ff [O(t),k(t, )O(T)]dTdtl_< 4eM(2y+e)Y 3 yy I.Y a,3 ta a, i,j 1 Since we assumed that equation 2.4 was non-positive, we will let K (O, ) = - 6

27 and choose an y so small that K (t, T) < - 6/2; Itl < y, TI <y. Then, for such a y, 'Y 'Y 2 S S KM(t, T)dTdt < -26y -'Y -y and therefore, < 0,K>< -26y 2+4M(2y+eC) lyi y 1.,fii, j By choosing e small enough, the right-hand side of this inequality is negative and since 0(t) is a continuous vector function, we have a contradiction which completes the proof. Lemma 2. 5 If a continuous matrix kernel k(t, T), is positive, then k(t, t) is a continuous positive matrix. Proof: By lemma 2. 4, we have, for any point t in (to tf) [y,k(t, t)y] > 0 Hence, by definition, k(t,t) is a positive matrix, and since k(t, T) is continuous in Qx Q, k(t, t) is also continuous for t e Q. 2. 3 Linear Time-varying Systems Turning now to more physical considerations we will define a concept of L2 -positive realness for integral operators representing time-varying dynamical systems as well as demonstrate their necessary structure to ensure L2-positive realness. For time-invariant systems

28 it will be shown that these concepts are generalizations of the standard definition. 1 Definition 2. 2 An integral operator W: L2 (Q)- L2 (Q), whose kernel, w(t, v) is continuous in both real variables on Qx Q, is said to be L2 positive real if 1. w(t, T) is real 2. W is causal 3. W is bounded 4. W+W*> 0. Consider now the linear causal dynamical system with the mathematical model of the form x(t) = A(t)x(t) + B(t)u(t),x(to) = 0, t e Q = [to, tf] (2. 6) (S1) y(t) = C(t)x(t). (2. 7) Here the tuplet u =(ul,...,Um) represents the independent system inputs, the tuplet x = (xl,..,Xn) represents the state variables, and Y = ( 1,.. Ym) represents the output. The matrices A(t), B(t), and C(t) are real, and continuous on Q, (that is, each element of each matrix is real and continuous on Q), and are of compatible dimensions. The output of the linear system is related to the input by the integral operator (Wu)(t) = f w(t,)u(r)d (2. 8) where w is the impulse response matrix2 of the system described by S1 such tnat See definition B. 8 of Appendix B. 2See Zadeh [59].

29 w(t, 7) C (t) (t, ) B(-) t > w(t, ) = O t < and where c is the transition matrix of the system described by x(t) = A(t)x(t) + u(t), x(t ) = 0O t e Q. If the integral operator associated with S1, is L2-positive real, then S1 is said to be an L2- positive real system. The next theorem establishes a sufficient condition for L2 -positive realness in the context of S1. Theorem 2. 1 Let the kernel of the causal integral operator W, be bounded on L2 (92),andgiven by C(t) (t, T)B(T), If there exists a continuous, positive self -adjoint nx n matrix Q, and a continuous r x n matrix L such that 1. -Q(t) = Q(t)A(t) + A'(t)Q(t) + L'(t)L(t) (2.9) 2. Q(t)B(t) =C'(t) 3. (Fx)(t) = f L(t)4b(t,7)B(T)x(T)dT: IIFiI is finite where Q(to) is chosen to satisfy condition (2) at t = to, then W is L2- positive real. Proof: We shall make use of the identities dT d-T (Tit)Q(T)D(T, S)3 =C'(Tt){A!(T)Q(T)+Q( T7)A( 7)}'D(T S) (2. 10) = -(D'(T,t)L'(T) L(T) (7, s) tf T ttf 4 tftf Sf S f(t, T, s)ds d7 = Sf S f(t, T, s)dTds + f S f(t,, s)dTds. tt tt t s 0O

30 Employing the above condition, F"'F can be written as t tf (F*Fx)(t) = f f B'(t)4' (7,t) L'(T)L(T) (z, s)B(s)x(s)dTds t t tftf (2.11) + f f B'(t) V'( 7,t) L' (T)L(T)(TI, s)B( s)x(s)dTds. t s Now proceeding with the proof, it follows by substituting equation 2. 10 into equation 2. 11, t tf (F*Fx) (t) B'(t) a- - Bt (FFx)(t) = - f B(t) {'( T,t)Q(T)>(T, s)}B(s)x(s)dTds t t tf tf - f B'(t) d {'( 7, t)Q(T) b( 7, s)}B(s)x(S)dTds. t s Employing the fundamental theorem of integral calculus, it is clear that t (F*Fx)(t) = F B'(t)[Q(t)4(t, s)-4'(tf, t)Q(tf)( tf, s)]B(s)x(s) ds t 0 tf + f B'(t)[4' (s,t)Q(s)-P'(tf, t)Q(tf)(tfs)] B(s)x(s)ds. t Since 4 is the transition matrix of a causal system, it follows from the preceding equation that (F *Fx)(t) = f B' (t)[Q(t)4(t, s)+ '( s,t)Q(s)] B(s)x(s)ds - f B'(t)4' (tf, t)Q(tf)4(tf, s)B(s)x(s)ds. By employing condition (2) it is seen that (F*Fx)(t) = ([W +W*]x(t) - f B )4')/'( tf,t)Q(tf)4(tf, s)B( s)x(s) ds

and since F*F is positive and bounded and since the integral operator associated the last term of this equation is positive and bounded, the theorem follows. The following theorem gives a necessary condition for the L2 - positive realness of W in the context of the system described by S. Theorem 2. 2 Let the kernel of the integral operator W be given by C(t) (t, T)B(T) then a necessary condition for W to be L2 -positive real is C(t)B(t) + B'(t)C'(t) > 0, for all t e Q. Proof: From definition 2. 2 we see that a necessary condition for positive realness is that the kernel C(t) D(t, T)B(T) + B'(t)D?'(, t)C'(T) be positive. Employing lemma 2.5 it follows C (t) < (t, t)B(t) + B' (t) '(t, t)C'(t) > 0. Since (D(t,t) = I, the theorem follows. Before continuing the analysis of L2 -positive realness it is pertinent to examine some of the characteristics of the matrix differential equation given by condition (1) of theorem 2. 1 Remark 2.2 The solution of the differential equation -Q(t) = A'(t)Q(t) + Q(t)A(t) + L'(t)L(t) exists and is unique (see Porter [46]) and is given by

32 Q(t) = I'(t t) Q(to)-{ ( t'(7,t )L'(T)L(T)(Tt to)dT}((t t) te 1. Moreover, if Q(to) is self-adjoint and chosen so that tf Q(to) S> (D (to)L( ) L(T T) (T,t)dT 0 then Q(t) is positive and self-adjoint for all t e 2. It is also noted that if L 4 is the impulse response matrix of an exponentially asymptotically stable system, then lim I '(t, to)Q(t) C (t, to) II is finite. t ->O Remark 2. 3 Assume that; A(t) = A1 + A2(t) and Q(t) = Q1 + Q2(t) where A1 and Q1 are stationary and Re {oA(A1)} E < e < 0 where a (A) denotes the spectrum2 of A. Choose Q1 to be the unique, positive, self-adjoint solution of Q1A1 + A'1Q1 = -L'L. Then sufficient conditions (see Porter [44]) for the existence of a matrix L(t) and a bounded, uniformly positive matrix Q(t) satisfying equation 2. 9 are given by 1. for some scalarsk,X > 0, lD (t, to) 1< ke X(t -o),t (t, o) 2. for some scalars 6,/ > 0, [[ A2(t) II < he- 6 tt (t 0o) 3. for some scalars k',X' > 0, I (to t) I1< k' eX'(t tot e (to, co) 4. 3(kkD2I IQI I < y where Q1 > yI > 0. See Appendix B. 2 For discussion of the spectrum of an operator, see Appendix A.

33 Remark 2. 4 If the matrices B and C are stationary then equation 2. 9 becomes algebraic, namely; QA(t) + A'(t)Q = -L'(t)L(t) (2. 12) Porter has shown (see [44]) that for the matrix A(t) such that A(t) = Al + yA2(t); Re [o(A1)] < e < 0, it may still be possible to satisfy equation 2. 12 with a constant Q. Following Porters development it follows that if II A2(t) II < 1 for t [to,oo) then equation 2. 12 can be expressed by QA(t) + A'(t)Q= AlQ+QA1 + Y[A (t)Q + QA2(t)] = L'(t)L(t). Hence it is sufficient that Q satisfies AQ + QA = -L'L where L'L > 2y 1IQIII. The following three examples serve to illustrate the previous theorems. Example 2. 1 To illustrate the use of theorem 2. 1, consider a single variate system. In equations 2. 6, 7 let x, y and u be scalar valued functions. The matrices A, B, and C are replaced by the scalar functions -a(t), b, and c respectively. Equation 2. 6 reduces to k(t) + a(t)x(t) = bu (t), x(O) = 0, t E Q = [0, co], (2. 13. 1) y(t) =c x(t) (2.13.2)

34 where it is assumed that a(t),b, c are real and a(t) =t+ 6, and 6,b,c> 0. This system defines an integral operator given by t (Wu)(t) = S c 4(t,7)bu(T)d7, 0 whe re t &1(t, T) =exp { - a(s)ds} t > - T =0 t< T. Clearly (1) and (2) of definition 2. 2 are satisfied. It is evident that W is bounded on L2(Q), since t(t, ) < e 6(t 7) =e 6X=t-7; therefore f Ic4(X)bIdX < oofor tE(O, oo). Hence, from lemma 2.3, 4 is an L1 kernel and the boundedness of W follows. The operator W + W* is positive since there exists an (.(t) and a q(t), namely d2 a(t)q = &(t), q = C/b, such that theorem 2. 1 is satisfied. Consequently, the system described by equations 2.13 is L2 -positive real. Example 2. 2 The purpose of this example is to illustrate a kernel which doesn't satisfy theorem 2.2 and demonstrate that it is not L2-positive real.

Consider the single-variate system represented by x(t) + x(t) = Sin t u(t), x(O) = 0, t E [o, x] y(t) = tx(t). This system defines the integral operator given by t (Wu)(t) = te -(t- T)Sin Tu(T)dT. Let u be given by u(t) = 1 r < t < 21r u(t) = 0 otherwise, hence, u e L2(o, co). It will now be demonstrated that < u, (W + WS)u > is non- positive, hence the single-variate system is not L2 - positive real. It is clear that 21 2at 27r 2It < u,(W+W )u > = f jte S(in dTSdt + (S it)e (7d rdt 2WT I T t -47 - 3 + e 7 (1+ 2r) < 0. Example 2. 3 The purpose of this example is to illustrate an operator that satisfies theorem 2. 2 and is not L2 -positive real. The kernel under consideration is given by w(t, ) = - e-4(t)+ 1 e-(t) t> 3 3 w(t, ) =0 t <z and QZ is the set (-co, 1). Theorem 2. 2 is satisfied since the matrix BC is the zero matrix. Let u be given by

36 (1 +j4)t u(t) =, -4 00< t < 1 u(t) = 0 otherwise, hence u a L2( —o, 1). Substituting u into the following inner product, it follows that u,(W+W*)u> ( t 1 (e-(t-T) -4(t-T), (l+j4) e(1 j4)t -00 -00 + 1 (e-(T-t) e-4(T-t) (1+j4)T + f f — W-(e )e e tdzdt -0o t = - 1. 04. consequently W is not L2 -positive real. 2. 4 Stationary Systems In this section, we will focus our attention on the dynamical system S1 when A, B, and C are constant real matrices, and demonstrate that definition 2. 2 is a generalization of the usual concept of positive realness given by definition B.8, when the integral operator representing S1 is defined in Lm(-oo,o). The first order of business is that of establishing the relationship between L - positive-realness in the time and frequency domains. Lemma 2. 6 If the integral operator W, defined on L2 (-~o, oo), is L2 positive real and its kernel, w(t-T) is continuous and stationary, then for Re s> 0 1. W(s) is analytic

37 2. W(s) = W(s) 3. W(s) + W(s) > 0 where W(s) is the Laplace transform of w(t-T). Proof: 1. Since the kernel w(t, T) is stationary and causal and since its associated integral operator is bounded on L2(-o, cO), then it follows from theorem C. 5 of Appendix C, that W(s) exists and is analytic for Re s > 0. 2. Condition (2) is equivalent to w(,t-T) being real. 3. Since W + W is positive, it is also real and consequently we need only concern ourselves with Re < x,Wx >. Let e stx, for t < tf where s = - + jw (a > 0) 0 for t >tf, such that x cEE, hence x e Lm (-o, co). Since x(t) = 0 for t> tf tf ~'t Re < x,Wx> = Re f [x(t), S w(t-T)x(T)dT] dt -o00 -o0 tf 00 = Re S [x(t), S W(T)x(t-)dT ] dt. -co O Consequently, tf 00 Re<x,Wx>=Re S [est x w(-)e- Ste x ]dt = Re [x,w(s)x ] f eadt. -0o Since

38 tf Se2a dt > 0 -00 and since Re < x,Wx >is positive, we obtain the desired result; namely Re [x,W(s)x] > 0, Re s > 0. (2. 14) Hence the definition for L2 -positive realness of time-varying system is a natural one, since it is a generalization of the well established definition for time-invariant systems, when restricted to operators defined on L (-O Coo). It is for precisely this reason that the definition is given by "L2 -positive real" instead of "positive real. " If W(s) is represented by rational polynominals in s and is analytic for Re s > 0, then the two definitions are equivalent. Corollary 2. 1 If, in addition to the three conditions describing W(s) in lemma 2. 6, W(s) is absolutely convergent for Re s >0 and W(jo) is essentially bounded for wo(-c,oo,), then the converse of lemma 2.6 is easily established. The proof follows from theorem C. 6 and lemma 2. 6. Consider now the stationary representation of the system described by S1, which is given by x.(t) = Ax(t) + Bu(t), x(O) = 0 (S2) y(t) =Cx(t) where it is assumed that x, u e L2(o, o). Taking Laplace transforms of the above system, it is easily seen that

39 Y(s) = C(sI -A) B U(s) = W(s) U(s). Since y is given by the convolution integral t y(t) = f Cb(t - T)Bu(T)dT, o it follows easily that the transform of the kernel CbB, is given by the matrix C (s I -A) -B. A pertinent question to ask at this point is '"What relationships must exist between the matrices A, B, and C to ensure that W(s) is L2- positive real?" The following two theorems are useful in resolving this question. Theorem 2. 3 A necessary condition for C(sI -A) 1B to be L2 positive real1 is that CB = B'C' > 0 Proof: Since we are concerned with a real-rational matrix, equation 2. 14 is valid for Re s > 0, and we can write the following chain of identities: 0 < Re [x,W(s)x]# => W(s) + W*(s) > 0 - -- C(sI-A) 1B+B'(sI-A')-IC' > 0, for Re s > 0. (2. 15) For large s inequality 2. 15 reduces to CB + B'C' > 0, Re s > 0. (2. 16) S s 1 This theorem also applies for positive realness.

40 If s = re j for -7/2 < 0 < 7/2 then 2. 16 becomes [CB+ B'C'] COSO + jB'C' -CB] sin > 0. (2.17) r r - Evaluating inequality 2. 17 at 0 = 0; 2/2; and -71/2 we obtain 0 = 0: CB + B'C' > O 0 = /2: j(B'C' -CB) > 0 }#B'C' =CB 8= -/2: -j(B'C'-CB) > 0 Hence the theorem is proved. It is to be noted that these conditions are analogous to those of theorem 2. 2. Theorem 2.4 If the matrix A is nonsingular, a necessary condition for C(s I-A) 1B to be positive real is: CA IB + B'(A') C' < 0. This is easily seen by evaluating the inequality 2. 15 as s - 0. Returning now to the matrix differential equation Q + A'(t)Q(t)+ Q(t)A(t) + L'(t)L(t) = 0. (2. 18) it is apparent that for stationary systems, equation 2. 18 reduces to A'Q + QA = -L'L. Hence from theorem 2. 1, sufficient conditions for positive realness can be stated by the following: Corollary 2.2 If there exists a positive self adjoint matrix Q and a matrix L such that

1. QA + A'Q = -L'L, Re {o(A)} < 0, 2. QB = C' then the real time-invariant dynamical system S2 is L2 -positive real. As pointed out in lemma B.5 of Appendix B, Anderson was able to sharpen this result for positive realness by showing that it was necessary as well as sufficient for Q >O. His result, however, relied heavily on concepts developed in the complex frequency domain. At the present time, this Author has been unable to obtain the analogous necessary conditions for L2 -positive realness in the time domain. 2. 5 L - Positive Realness and Stability As a direct application of the preceding section, we shall demonstrate the utility of the concept of L - positive realness as applied to stability in time-varying systems. The application of positive realness is not new to stationary plants (see for example [ 1 ] and [10]); however it is refreshing to see that a generalization of these concepts can be extended to time-varying systems. Theorem 2. 5 If the kernel of the composition of the operators P and M is given by C(t)@4 (t, T) B(r)M(T) and is L2-positive real such that theorem 2. 1 is satisfied with Q(t) > 0, then the closed-loop system illustrated in Figure.. 1 is stable. Proof: Since theorem 2. 1 is satisfied we have 1. -Q(t) = A'(t)Q(t) + Q(t)A(t) + L'(t)L(t); Q(t) > 0 (2. 19)

42 2. Q(t)B(t)M(t) = C'(t) (2.20) The tentative Lyapunov function V is defined by V(x, t) = [x(t), Q(t)x(t)], (2. 21) where x(t) = [A(t) - B(t)M(t)C(t)]x(t). (2. 22) Differentiating equation 2. 21 V(x,t) = [x(t), { [A(t)-B(t)M(t)C(t)] 'Q(t) +Q (t)+Q(t)[A(t)-B(t)M(t)C(t) ]}x(t)] (2. 23) and writing equation 2. 19 in the form -Q(t)-Q(t) B(t)M(t)C (t)-C '(t)M' (t)B' (t)Q(t) = Q(t) [A(t)- B(t)M(t)C (t) ] + [A(t) - B(t)M(t)C(t)]'Q(t) +L'(t)L(t), (2. 24) and by employing (2) it is clear that equation 2. 23 becomes V(x, t) = [x(t), (-2C'(t)C(t)-L'(t)L(t))x(t)] (2. 25) = -[L(t)I (t, to)x(t0),L(t)' -(t,to)x(to)] — 2[C(t)J (t,to)x(t0),C (t)IC (t,to)x(to)] = -I L(t)L (t, t)x(t0)1-2IC(t),I(tt x(t to)X(to) 12 < 0 where E(t, to) satisfies the equation I (t, to) = [A(t) - B(t)M(t)C(t)]*(t, to), (toto) = I Therefore the closed-loop system is stable but not necessarily asymptotically stable. The following corollary gives a sufficient condition for the closed-loop system to be asymptotically stable. See Appendix B for a short review of Lyapunov functions.

43 Corollary 2. 2 If either of the matrix pairs {A(t)-B(t)M(t)C(t),L(t)} or {A(t)-B(t)M(t)C(t),C(t)} are observable, and IIQ(t) is finite for all tc (to o), then the closed-loop system is asymptotically stable. Another version of the stability criteria is given by the following theorem. Theorem 2. 6 If the kernel C(t)4(t, T)B(T) is L2 -positive real such that theorem 2. 1 is satisfied and Q(t) > 0, M(t) > 0, then the closedloop system illustrated in Figure 1 is stable. Proof: Since theorem 2. 1 is satisfied it follows that 1. -Q(t) = A'(t)Q(t) + Q(t)A(t) + L'(t)L(t) Q(t) >0 2. Q(t)B(t) = C'(t) The body of the proof is identical to the preceding theorem with the exception that V(x,t) = [x(t),(-2C'(t)M(t)C(t)-L'(t)L(t))x(t)] (2.26) = - L(t)t(t, to)X(t) 11 - 2 - |1C(t)M(t)2~(t, to)x(to) |2 < 0, 1 2 where M(t)2 denotes the unique positive square root of M. For completeness we shall add the following corollary which is analogous to corollary 2. 1. See Appendix B for the definition of observability. 2 This terminology is discussed in Appendix A.

44 Corollary 2. 3 If either of the matrix pairs {A(t)-B(t)M(t)C(t), L(t)} or {A(t)-B(t)M(t)C(t), C (t)M-2(t) } is observable and I I Q(t) I is finite for all te (to,oo), then the closed-loop system is asymptotically stable. 2. 6 Summary This chapter has served to define a generalization of the concept of positive realness to time-varying linear systems. A natural consequence of this generalization has led directly to, what network theorists call, passivity. That is, if the integral operator, W, defined in definition 2. 2, is viewed as representing an n-port such that W: v(t) - i(t) where a = (- co, co), then it is easily seen that conditions 2 and 4 of this definition are equivalent to Re S v*(t)i(t)dt > 0 for -09 all ' e (-oo, oo). Hence, the material set forth in this chapter have their obvious implications in the area of linear passive network theory. Starting with the definition of L2-positive realness, necessary conditions and sufficient conditions were then formulated in the context of time-varying linear systems described by S1 and time-invariant systems described by S2 to illustrate the relationships between the matrices A(t), B(t), and C(t). Finally, an application of the concept of L2-positive realness was given to illustrate its usefulness in the study of stability.

Chapter 3 SENSITIVITY REDUCTION IN LINEAR TIME-VARYING DYNAMICAL SYSTEMS 3. 1 Introduction One of the classical problems of control theory is to reduce, by means of feedback, the sensitivity of single variate systems to parameter variations of the plant and other disturbances. A lot of effort has been devoted to the pertinent aspects of this problem resulting in a considerable wealth of literature (see [9 ] and [22]) More recently efforts have been directed toward generalizing these concepts to include multivariate systems (references [11], [39], and [45i serve as a good introduction to this literature). This chapter and the following chapters establish the fundamental relationships between the plant matrices {A(t), B(t), C (t)} and the compensators G and M to ensure sensitivity reduction of multivariate systems. This is clearly the first step which must be dealt with before an effective design procedure can be established. 3. 2 Statement of the Problem To begin the discussion of sensitivity reduction of linear systems, the problem is formulated in Banach spaces in order to emphasize the operator theoretic point of view. Attention is then restricted to Hilbert spaces, in order to obtain more detailed results including interpretations, from a physical point of view. 45

The problem of sensitivity reduction is based on the comparison of the perturbations of the outputs of two nominally equivalent systems under the influence of a disturbance. The system whose perturbation, in some sense, is the smallest, is said to be less sensitive. The disturbances may take the form of parameter variations or noise. For this analysis, attention is focused on the two nominally equivalent closed-loop and open-loop systems of Figure 1. 3. The compensators G and M are to be chosen so that the closed-loop system is less sensitive, in some sense, than the open-loop system. Let B1 and B2 denote Banach spaces. P is a linear plant with parameters{ a1,..., an} such that P: B1 - B2. The compensators, G and M are such that G: B1- B1 and M: B2 -B1. The symbols u and y denote the system input and output respectively while r denotes a system disturbance. The two systems are said to be nominally equivalent when the terminal mapping u -p y is identical for both systems with no disturbances. The equations describing the systems of Figure 1.3 are given by Y =P [Gu - MY] (3. 1) and = Pu. (3. 2) It is easy to verify that for linear transformations such that (I + PM) is invertible, the above equations reduce to Y = [I + PM] -PGu (3.3)

47 and y, =Pu, (3. 4) which are defined to be the nominal system equations. To establish a terminal equivalence between them, it is necessary that (I + PM) PG = P or equivalently PG = P(I + MP). Therefore, if G is chosen so that G = I + MP, (3.5) the requirement is satisfied. Equation 3. 5 gives the explicit relationship for G in terms of the plant and the feedback compensator M, which is chosen to reduce the relative sensitivity of the closed-loop system. If there are plant parameter perturbations, that is {a1, a2....} {a(1'a2...} such that the perturbations in P are bounded additive and linear, then P -p where P =P+ 6P. The system equations are then given by Yo + 6yo = (P + 6P)u

48 and Yc + bye (P + 6 P)[(I + MP)u - M(yc + 6Yc)] where 6y represents the perturbation in the output. The perturbations in the open and closed-loop can then be shown to be related by the linear operator $: 6yc =$ 6y (3.6) where -[I +PM] - (3.7) If the closed-loop system is to be less sensitive than the openloop system with respect to the norm of the perturbations, then M must be chosen so that: 16yc I (< I 6yo 11 (3.8) where the norm is taken on B2. Hence, it is necessary and sufficient that $ satisfy 11$11< 1. (3.9) Clearly, M cannot be chosen a priori to reduce the sensitivity for all perturbations. However, it is possible to choose M to reduce the sensitivity for perturbations belonging to a specified class. This topic will be pursued in the sequel. The aspect of sensitivity reduction which we shall discuss first, is that of reducing the perturbations of the output of the closedloop system to additive noise r/. The output of the closed4oop and

open-loop systems due to the presence of noise, r7, is given by YC Y + (I + PM) and Yo = Yo + qr. where y represents the perturbed output. 'he perturbations in the closed-loop and open-loop systems are related by the linear operator $p. where 6y = y - y, and evidently $ = (I + PM) (3. 10) Therefore, if the closed-loop system is to be less sensitive than the open-loop system, $ must also satisfy equation 3. 9. From equation 3. 9, it is seen that the formulation of the sensitivity criterion in Banach spaces is not very rewarding, since it gives only a condition on the norm of the operator $ and consequently doesn't lend itself to easy evaluation. If the problem is formulated on Hilbert spaces, additional structure on $ can be obtained. Since the perturbations are related by 6ye = $6yo, the norm 6 Yc!} can be written as 11Y 12 =<6y,Y >=<$6y $6yo>=<6y,$*$6y > I6Cy6yol2- -<6c c(I-$ $6Y>. (.

50 Hence, equation 3.8 is satisfied if I-$*$ > o 11 < 1. Therefore, in Hilbert spaces the necessary and sufficient condition for sensitivity reduction is I - $*$ >. (3. 12) This equation, although abstract in nature, has proved very fruitful in understanding the nature of sensitivity reducing compensators. (See for example the papers of Porter [43] through [45].) Throughout this discussion the sensitivity problem has been discussed in an abstract function space setting. A framework is now established which is helpful in bringing these abstract ideas to a concrete form. The analysis of linear time-varying systems will be the method by which this will be accomplished. Attention is focused on those transformations which act between finite cartesian products of L2(to, tf) equipped with the usual inner product defined by < x, y > = [x(t),y(t)]dt, Q = (to tf). For time-varying systems, the closed-loop system is then said to be less sensitive than the open-loop system if ftfo 6yo(t), 6yo(t)]dt> > Yc(t), 6by(t)]dt, for all TE Q, (3. 13) where it is tactitly agreed that and belong to complex E where it is tactitly agreed that 6y and. y belong to complex F Inequality 3. 13, is a stronger condition than 3. 12 since it not only implies that the norm of $ must be less than unity but moreover, $ must be causal on the interval S~. Causality is easily established

by selecting 6yo(t) = 0 for t< z and for all TE fQ consequently, from inequality 3. 13 it must be the case that 6yc(t) = 0 for t <z and for all TE 52. Therefore, for time-varying systems, the generalization of the necessary and sufficient condition for sensitivity improvement given by 3. 12 can be formulated by the following 1. $: L (5) -L2 (Q), Q = (to,tf) (3. 14) 2. $ causal (3. 15) 3. I-$ $> 0. (3.16) Now that the criteria for the sensitivity operator has been established, we shall investigate the implicit constraints that it imposes upon a plant which is described by S1, (where A(t), B(t), and C(t) are real continuous nxn, nxp, and mxn matrices respectively), and a feedback compensator, M(t), which is a real continuous p< m time -varying matrix. 3. 3 Structure of the Sensitivity Operator To understand the physical implications of conditions 3. 14, 15, 16, it is necessary to identify the kernel of the operator $. From equation 3. 10, recall that ($y)(t) =([I+ PM] 1y)(t), (3.17)

and (Pz)(t) = f C(t) (t, r)B(r)z(T)dr, t e Q (3. 18) To manipulate equation 3. 17 to a more meaningful form, consider the following differential system a(t) = A(t)a(t) + B(t)M(t)3 (t): 3(t) = ($y)(t) (3. 19) y (t) = C (t)a(t) + 3(t) (3.20) The solution of this system is given by: y (t) = f {I 6 (t-T) + C(t) (t, T)B(T)M(T)} (T)dT, t e, which is an explicit form of $-. The mapping $, is identified as follows: 3(t) y(t)-C(t)(t) =(t)- C(t) I (t, 7)B(T)M(T)y (T)dr f {I (t-T) -C(t)I(t, T)B(T)M(T)}y(T) d, (3. 2 1) where ' (t, T) satisfies: d(tT) = [A(t) - B(t)M(t)C(t)]I(t, T), '(t,t) = I The mapping $: y - /, given by equation 3.21, identifies the kernel of the sensitivity operator as: (t, ) I (t -C(t, T) =I 6 (t-) -C(t) (t, T)B(T)M(T). (3. 22) = I 6 (t —) -h(t, ) (3.23)

where the m x m matrix kernel h(t, T) is defined in the obvious way. It is therefore evident that the sensitivity operator is equivalent to $ I - H, (3.24) where H is the integral operator associated with the kernel h(t, 7). The following theorem is helpful for establishing the sensitivity conditions 3. 14, 15, and 16, in terms of the integral operator H. Theorem 3. 1 A necessary and sufficient condition for sensitivity reduction is that the operator H be L2 -positive real and H + H*- H*H > 0. Proof: The proof is made obvious from the following four conditions and definition 2. 2: 1. $: L2 (Q) L2 () H: L(Q) L (), = (ttf) 2. $ causal 4 — > H causal 3. I-$$ > 0 -H +~H* -H*H > 0, hence H+H* > 0. 4. Since A(t), B(t),C(t), and M(t) are real time varying matrices h(t,T) is real. In the theorems that follow, the concept of L2 -positive realness is brought to bear on theorem 3. 1 thus establishing the relationships between the parameters of the plant described by S1 and the feedback compensator M(t). For example, theorems 2. 1 and 2.2 supply sufficient conditions and necessary conditions for L2 - positive realness of H, by replacing the kernel C(t)4i(t, T)B(T) with C(t)4 (t, T)B(T)M(T).

54 The following three theorems help to identify the conditions under which sensitivity is improved in the context of S1. Theorem 3.2 Let W be a integral operator bounded on L2 (Q) whose kernel is given by [C(t) -M'(t)B'(t)K(t)]l'(t, T)B(T)M(T) where K(t) is the unique self-adjoint solution of -K(t) = K(t)[A(t)-B(t)M(t)C(t)] + [A(t)-B(t)M(t)C(t)] '?(t)+C '(t)C(t), t e SQ, and where K(to) is given by tf K(t ) = f,'(T,to)C'(T)C(T)J(T, tO)dT. If H is a bounded, causal and real operator defined on L 2(Q), then the sensitivity is reduced if W + W* is positive. Proof: The first step of the proof will be that of identifying the kernel of H*H. Following the arguments of theorem 2.1, it is seen that t tf (H*Hx)(t) =S S M' (t)B'(t)'( T, t)C'(T)C' (T)C(7T, s)B(s)M(s)x(s)dTds t t 0 tf tf + f M' (t)(t)t)*I(T.)'(,t)C'(T)C(T)(T,s)B(s)M(s)x(s)dTds. t s Hence, substituting the differential equation for K into (H Hx) (t) t tf (H *Hx)(t)=-f I M?(t)B(t) d {4"'(T,t)K(T)i(T,s)}B(s)M(s)x(s)dTds (H*Hx) (t) - M'(t)B' (t) dT tf tf - f fMO)(t)B d ({'(',t)K(T)I1((, s)}B(s)M(s)x(s) dTds. t s dT

Since l is thq transition matrix of a causal system, it follows that tf (H*Hx)(t) = f M' (t)B' (t)[ K(t)[(t, s)+I'(s, t)K(s)] B(s)M(s)x(s)ds t t to - S M'(t)B'(t)I '(tf, t)K(tf) (tf, s) B(s)M(s)x(s)ds to Therefore, the kernel of H*H is given by M?(t)B'(t)[K(t)'?(t,T) + *J'(r, t)K(T)] B(T)M(T) - M'(t)B'(t)1'(tf,t)K(tf)4i(tf, 7)B(T)M(T). (3. 25) The kernel of H + H* is clearly given by C(t)yI(t, T)B(T)M(T) +M'(t)B'(t)1'(r, t)C(T), (3. 26) and by subtracting equation 3. 25 from 3.26 it follows that the kernel of H+ H -H His C (t) -M' (t) B'(t)K(t) ] (t, T)B(T)M(T)+M' (t) B'(t) '((7, t) [C'(T) -K(T)B( T)M(T)] + M'(t)B'(t)*'(tf, t)K(tf)Ii(tf, 7)B(T)M(T). The theorem follows by recognizing that the first two terms of the preceding expression correspond to the kernel of W + W*, and the last term corresponds to the kernel of a positive operator. If Q is the set (to, oo), and C' is the impulse response matrix of an exponentially asymptotically stable system, then I'(0o0,t)K(oo)4(Co, T) is the zero matrix and theorem 3. 2 is necessary as well as sufficient.

56 Theorem 3. 3 A necessary condition for W + W* to be positive is that C(t)B(t)M(t) +M'(t)B'(t)C'(t)> 2M'(t)B'(t)K(t)B(t)M(t) where K is defined in theorem 3. 2. The proof follows directly from theorem 2. 2. A sufficient condition for positivity of W + W* can be obtained via theorem 2. 1, thus ensuring sensitivity reduction. This is established by the following Theorem 3. 4 A sufficient condition for positivity of W + W* is the existence of a continuous, positive, self -adjoint nxn matrix Q and a continuous r x n matrix L such that 1. -Q(t) =Q(t)[A(t)-B(t)M(t)C(t)] + [A(t)-B(t)M(t)C(t)]'Q(t)+L'(t)L(t), tE e 2. [Q(t)+K(t)] B(t)M(t) = C'(t) 3. (Fx) (t) -A f L(t)I(t, T)B(T)M(T)x(T)dT: UF l is finite, where Q(to) is chosen to satisfy condition (2) at t = to The proof follows from theorem 3. 2 and theorem 2. 1. Example 3. 1 To illustrate the use of previous theorems we will consider a single-variate plant. In equations 2. 6, 7, let x, y and u be scalar valued functions. The plant matrices A(t), B(t), C(t) and the feedback compensator M(t) are replaced by the scalar functions: -a(t), b, c and m respectively.

57 The plant equations reduce to x(t) + a(t)x(t) = bu(t), x(to) = 0, t e (to, 0o) y(t) = cx(t) where it is assumed that a(t),b, c, and m are real and 3 > a(t) > 6 > 0 and b, c, m > 0. The sensitivity kernel is then given by ~(t, 7) = 6 (t-v) - c' (t, T)bm and I is seen to be an L1 kernel since t '(t, -) = exp {- S [a(s) + bmc]ds} t > T *(t, T) = O t< T. The operator k(t) is the solution of the differential equation -k(t)= -2 [ a(t) + bmc] k(t) + c2 which is given by k(t) =2(to, t) f c (7, to)dT, and is positive and finite for all t and is bounded by 0 < k(t) < c < c 2 (6 + bmc) bm Define the scalar q(t) by q(t) c -k(t) bm then substituting this expression into the differential equation for k(t) gives q(t)= -2 [a(t)+bmc] [ r - q(t) ] + c2,

58 hence, 2 2a(t)c -q(t) = -2 fa(t) + bmc]q(t)+c + bm) Consequently, if &(t) is defined by t)= 2 + 2a(t)c 2 then l(t) '(t, T)bm is an L1 kernel and theorem 3.4 is satisfied. Throughout this section, structure of the sensitivity operator in the context of S1I has been identified and the necessary and sufficient conditions for sensitivity improvement have been interpreted in terms of the matrices A(t), B(t),C (t) and M(t). The relationship between positive realness and sensitivity reduction was also established. The following section serves to supplement the preceding conditions relating the structure of P and the compensator M to insure sensitivity improvement. 3. 4 Sensitivity Analysis Via the Inverse Sensitivity Operator The purpose of this section is to demonstrate that insight into sensitivity reduction can be obtained by employing the inverse of the sensitivity operator. As noted previously, the inverse operator for the system under discussion exists and is given by 1$ I+PMP (3.27) where P is a real causal plant described by S1, and M is a continuous time-varying matrix. The necessary condition for sensitivity reduction given by inequality 3.16 can be reformulated via $-1 by

59 the following: o < I-$*$ - $* [(($ )- l - I] $I-> ($*)-1$-1 I> 0. The necessary and sufficient conditions for sensitivity reduction can then be written in the form 1.: L2 (s2) - L2 (), Q = (to tf) 2. $ causal 3. ($)- $- -I > 0. (3.28) Since the implications of (1) and (2) have previously been examined, the pertinent question at this point is: "What is the sufficient structure of PM to insure that inequality 3. 28 is satisfied?". The answer to this question is readily obtained by substituting equation 3. 27 into 3.28 which results in 0 < ($*) 1$-1-I -i PM + (PM)* +(PM)* PM > 0. (3.29) It is clear that if the operator PM + (PM)* is positive, then inequality 3. 29 is satisfied. rTranslating this result into the context of the system described by S1, the following theorem is formally stated. Theorem 3.5 A sufficient condition for PM + (PM)* > O0 is the existence of a continuous positive self-adjoint nx n matrix Q a continuous rx n matrix L such that 1. -Q(t) =Q(t)A(t) +A'(t)Q(t) +L'(t)L(t) 2. Q(t)B(t)M(t) = C'(t) 3. (Fx)(t) - f L(t)m(t,T)B(i)M(T)x(')d.T: I IFI is finite, -Q

60 where Q(to) is chosen to satisfy condition (2) at t = to The proof follows that of theorem 2. 1. Remark 3. 1 If the matrices B(t) and C(t) are equal to the identity and M(t) is invertible, then the above theorem for sensitivity reduction is equivalent to a condition established by Porter [44]. Theorem 3. 5 is overly strong since the operator (PM) PM was not considered to aid in the positiveness of 3. 29. The following theorem provides a split of (PM)*PM and utilizes this to give a weaker sufficient condition for ($)- 1$ 1-I> 0. Theorem 3. 6 Let V be a integral operator bounded on L"m(9) whose kernel is given by [C(t)+M'(t)B'(t)K(t)]4(t, T)B(T)M(T) where K(t) is the unique self-adjoint solution of -K(t) = K(t)A(t) + A'(t)K(t) + C'(t)C(t), t e 2, and where K(to) is given by t K(tO) = f?(Ttm)C'(T)C(T) (T, t )dT. to If PM is a bounded operator on L(Q), and V + V* > 0, then PM- + (PM)* + (PM)*PM > 0. The proof is straight forward and follows theorem 3. 2. It is clear that if 52 is the set (to, Co) and if C: is the impulse response matrix of an exponentially asymptotically stable system, then PM + (PM)* + (PM)*PM is positive if and only if V + V* is positive.

61 Moreover, it is also noted that by employing the inverse sensitivity operator, the sufficient conditions for sensitivity reduction impose a more severe restriction on the plant matrix A(t) then previously. This is easily seen by noting that the operators associated with the kernels C(t)I(t, T)B(T)M(T) and C(t)4(t, T)B(T)M(T) must both be bounded on L2 (Q)' 3. 5 Sensitivity Reduction with Plant Perturbations In the previous sections the structure of the plant and it 's associated feedback compensator has been discussed from the point of view of reducing the sensitivity to additive noise r7. In this section attention is directed to the question of sensitivity reduction when the parameters of the plant matrix A(t) are perturbed, such that A(t) — > A(t) + 6A(t) = A(t), correspondingly, H -H. For such disturbances, the necessary and sufficient conditions for sensitivity reduction are given by theorem 3. 1 by replacing H with H. The starting point for this discussion is the assumption that the closed-loop system does indeed have a reduced sensitivity to noise compared with the nominally equivalent open-loop system. It is also assumed that theorem 3. 4 is satisfied. In view of these assumptions, the following theorem is presented.

62 Theorem 3. 7 If H is a real, causal, bounded integral operator on L2m(2) and if there exists positive, self-adjoint nxn matrices Q and K, a continuous rxn matrix L, such that 1. -Q(t) = Q(t) [A(t) - B(t)M(t)C (t)] + [A(t) -B(t)M(t)C (t)] 'Q(t) + L'l(t)L(t) 2. -K(t) = K(t)[A(t)-B(t)M(t)C(t)] + [A(t) - B(t)M(t)C (t) ] 'K(t) + C '(t)C (t) 3. [Q(t) + K(t)] B(t)M(t) = C'(t) 4. (Fx)(t) L(t)I(t, T)B(T)M( T)x(-)dT: IIFII is finite where Q(to) is chosen to satisfy condition (3) at t = t and where K(to) is given in theorem 3. 2, then a sufficient condition for sensitivity reduction is [Q(t) + K(t)] 5A(t) + 6 A'(t) [Q(t) + K(t)] < 0. Proof: Following the proof of theorem 2. 1, we shall make use of the integral operator F: _ t _t (Fx)(t) = L(t)*I(t, T)B(T)M(T)x(T)dr t ecQ t and its adjoint, F* given by (F*z)(t)= f M'(t)B'(t)f'(T,t)L(T)z(T)dT, te 2. We shall also make use of the identity. dT {(7,t)Q(7) (7, s)} =-~'(7,t){6A'(f)Q ( ) +Q()6SA() - L'(T)L(T)}q)(, S dz

63 Hence F F can be written as t tf (F*Fx) (t) =- f F M'(t)B'(t) d {I"(T, t)Q(-T)J(T, s)}B(s)M(s)x(s)dTds tf tf - f M 'f'(t)'(t) B d { '(T, t)Q(T)+(T, s)}B(s)M(s)x(s)dTds t s dT tf - + f f 4'(T, t)[ 6A'(T)Q(T) + Q(T)6A(T) ](T, s)x(s)ds d7. t t 0 Since ' is the transition matrix of a causal system, it is clear that tf (F Fx)(t) = M'(t)B' (t)[Q(t)O(t, s) + '(st)Q(s)] B(s)M(s)x(s)ds t 0 tf tJo M'(t)B'(t)'(tf, t)Q(tf) I(tf, s)B(s)M(s)x(s)ds tf T + f f JI'(7,t)[ 6A'(T)Q(T)+Q(T)6A(T)]A(T-,s)x(s)dsd7. t t 0 ~* * ~*Leaving this expression for the moment, and computing H+H -H H, it follows that tf ([H +H - H H]x)(t)= f [C(t) - M'(t)B'(t)K(t)]4(t, s)B(s)M(s)x(s)ds t 0 tf + jt M'(t)B'(t)*'(s,t)[C'(s)-K(s)B(s)M(s)]x(s)ds 0 + ftM'(t)B'(t)4'(tf, t)K(tf)4 (tf, S)B(s)M(s)x(s)ds t -f 1 Io I(7,t)[ 6A'(T)K(T)+K(T)6A (T)],(T,s)x(s)dsdT. Using condition (3), it is seen that

64 ([H + H- H H]x)(t) = (F Fx)(t) t-_ + f fM'(t)B'(t)?' (tf, t)[Q(tf)+K(tf) ](tf, s)B(s)M(s)x(s)ds t 0 tf f -S fr f '(-,t){[Q(-)+K(7-)]6A(-)+bA'(r)[Q(T)+K( )]},I(Ts)x(s)ds, t t 0 The theorem follows since H + H -H H is positive. It is remarked that in this last theorem, the assumption was made that JJi II < 1 for all 6yo e L2(9 ). This assumption is not necessary for sensitivity reduction when the plant is perturbed but it is necessary that I I J < 1 when its domain is restricted to the range of 6 P. For elements belonging to the complement of the range of 6 P, ( $ JI may be greater than one. 3. 6 Summary In this chapter, the basic problem of sensitivity reduction was formulated for multivariate systems in terms of the sensitivity operator $. The structure of its kernel was given for the class of systems described by S1. The concept of L2 -positive realness then was brought to bear on the problem to establish fundamental relationships between the plant matrices {A(t),B(t),C(t)} and the compensators G and M to ensure sensitivity improvement for disturbances in the form of noise and plant parameter perturbations.

Chapter 4 SENSITIVITY REDUCTION IN' LINEAR STATIONARY SYSTEMS 4. 1 Introduction The last chapter discussed the problem of sensitivity reduction in the time domain and established the relationships between the time-varying plant matrices {A(t), B(t), C(t)} and the feedback compensators G and M. Our present objective is to examine these previous results with the additional assumption of stationarity, and employ frequency domain techniquesl as an added tool with which to work. It will become evident that many of the previous theorems, which proved to be only sufficient in the time domain, are necessary as well in the frequency domain. This is to be expected since as the definition of the system is sharpened, it is possible to obtain more detailed results about its structure. In short, the relationships between the constant plant matrices {A, B, C} and the compensators G and M which ensure sensitivity reduction are established. 4. 2 Conditions for Sensitivity Reduction in Stationary Systems To begin the analysis of sensitivity reduction in stationary systems, it will first be helpful to translate the necessary and sufficient conditions for sensitivity improvement given by 3. 14, 15, 16, to the complex-frequency domain when Q is given by [-oo, co]. The following lemma formalizes these conditions. See Appendix C. 65

66 Lemma 4. 1 Let B(t) be the kernel of a stationary sensitivity operator, satisfying 3. 14, 15, 16, then its Laplace transform, $(s), is such that for Re s > 0 1. $(s) is analytic 2. I - $* (s)$(s) > 0. Proof: 1. Employing the conditions given by statements 3.14 and 3.15 and using theorem C.5 of Appendix C, it follows that S(s) exists and is analytic for Re s > 0. 2. Inequality 3.16 implies condition (2) of this lemma by the following arguments: I-*$ > - < x, (I - $*$)x > > 0 for all x L (-oo,) (4. 1) where t 0o ($x)(t) =f f(t,7)x(T)dT= S (T)x(t-T)dT. Let st x(t) =e x, = +jw, a > 0, t < tf x(t) =0, t >tf where x e Em hence x e L2 (-oo, oo). Therefore, o ($x)(t)= f 5(t TdT))dr = $(s) etx. Inequality 4. 1 can now be written as

67 st st- st st > <x, (I-$*$)x > =exex> < eSt$()x e$( ) 2at X, 2$*(s)$(s)x, > = <xe X> <x,e x>,(s)(s)x> o>0 tf = e2t[x, (I - $*(s)$(s))x] dt = [x(I-$*(s)$(s))x] F e2(tdt a >0. -0o tf Since j e dt is positive, the lemma follows. -co The converse of this lemma is easily established from theorem C. 6 if $(s) is absolutely convergent for Re s >O and $(jw) is essentially bounded for Ce(O o, -o). It is pointed out that Newcomb, concerned with stationary sensitivity operators mapping between real Hilbert spaces, stated that conditions (1) and (2) of lemma 4. 1 as well as the condition i(s) = $(s) (for Re s > 0) were equivalent to conditions 3. 14, 15, and 16. These results were made possible by the additional structure employed in the use of distribution theory. Techniques in this direction are not within the intended scope of this thesis. It is convenient to translate conditions 3. 14, 15, 16 to an equivalent form along the jw axis. Lemma 4. 2 Let s(t) be the kernel of a stationary sensitivity operator satisfying 3. 14, 15, 16 such that $(s) is absolutely convergent for Re s > 0 and is described by a rational matrix then, the sensitivity is reduced if and only if 1. $(s) analytic for Re s > O 2. I- $*(jw)$(jw) > 0 for all oi

68 The proof of this lemma is well known and can be found in most network synthesis textbooks (for example Newcomb [36 ]). Having now established the criteria that the stationary sensitivity operator must satisfy, the constraints that it imposes upon plants described by S2 and feedback compensators described by stationary matrices will be investigated. It will be noted that as we proceed through the development of sensitivity reduction for stationary systems, we shall heavily rely upon the results developed in chapter 3. 4. 3 Structure of the Stationary Sensitivity Operator For stationary systems, the sensitivity kernel given by equation 3. 22 becomes $(t-T) = I 6 (t-C(t-) CI BM, where I (t-T ) satisfies di(t-) = [A-BMC ]J(t-T), (to,to) =I. (4. 2) dt 0 0 Since the sensitivity kernel is time invariant, its Laplace transform is given by $(s) =I-C [sI-(A- BMC)] BM I-C(s)BM. (4.3) It follows that $(s) is analytic and converges absolutely for Be s > 0 if and only if the irreducible realization of C [ s I - (A - BC) BM is asymptotically stable, or if Re{(c(A-BMC)} < e < 0, For a definition of irreducible realization, see Appendix B.

69 hence closed4oop stability. As demonstrated in the preceding chapter, condition (2) of lemma 4. 1 is equivalent to < - *( jco$(j) -- H(j)+H*(jwcot)-H*(jc)H(jco)> 0 (4.4) where H(jw) is identified by H(jco) = C [ j coI -(A - BMC) ] BM, and is L2- positive real. The following theorem consolidates these results. Theorem 4. 1 A necessary and sufficient condition for sensitivity reduction is that 1. The irreducible realization of H(s) is asymptotically stable 2. H(jo) +H (jco) - H (jco)H(jw) > 0 co e (-co, oo) The proof follows directly from the preceding discussion, theorem 3.1 and lemma 4. 2. Hence, as expected from the preceding chapter, positive realness plays a dominant role in the basic structure of sensitivity reduction. Because of this, we are able to obtain additional structure on the matrices B, C, and M via theorem 2. 3. Theorem 4. 2 A necessary condition for sensitivity reduction is CBM = M'B'C' > 0 1 Since H(s) is analytic in Re s > 0, L2-positive realness is equivalent to positive realness.

70 Remark 4. 1 Since it is necessary that H(s) be positive real, theorem 4. 2 can be supplemented by utilizing the concepts developed by Falb and Wolovich 13 ]. It is demonstrated in section B.6 of Appendix B that a necessary condition for H(s) to be positive real is that CiBM # 0 for i = 1,2,...,m, where Ci denotes the ith row of the matrix C. The following theorem serves to further clarify theorem 4. 1 in the context of S2. Theorem 4. 3 If Re {((A - BMC)} < E < 0 then the necessary and sufficient condition for sensitivity reduction is that W(s) be positive real where W(s) = (C -M'B'K)'(s)BM, (4.5) and where K is the unique positive self-adjoint solution of K(A- BMC) + (A- BMC)'K = -C'C Proof: The proof follows from the definition of positive realness and by equation 4. 4: o < C~(jw)BM+M'B'V* (jw)C'- M'B'* (jco)C'Cr(jco)BM - C (jw) BM+M'B'e* (jco)C'- M'B'e*(jco)K[ jco I-A+BMC ] * (jco)BM -M'B'* (jw) [ jw I-A+BMC]* (jco)BM = C (jco)BM+M'B'KJ (Wc)C'- M 'B' (jw)KBM - M'B'K (jco)BM (C- M'B'K')9(jw)BM + [(C - M'B'K)~ (jw)BM]* > 0.

71 The following two theorems give necessary conditions and sufficient conditions for (C - M'B'K)*I(s)BM to be positive real. Theorem 4. 4 If Re o(A - BMC)} < e < 0 then a sufficient condition for (C - M'B'K)*1(s)BM to be positive real is the existence of a positive self-adjoint matrix Q and a matrix L such that 1. Q(A- BMC) + (A - BMC)'Q = -L'L 2. (Q + K)BM = C' The proof follows from theorem 3. 4. Theorem 4.5 Necessary conditions for (C - M'B'K)*I(s)BM to be positive real are 1. CBM = M'B'C' 2. CBM> M'B'KBM > 0. This sharpens the results of theorem 4. 2 and is proved by application of theorem 2. 3. Example 4. 1 To illustrate the previous theorems consider the plant described by.52 1.89 1 1 X= X + U -.74 -1. 94 0 1 o 0 1.= x with the feedback matrix

72.89.26 M = -1. 51.26 A necessary condition for sensitivity reduction given by theorem 4. 2, is satisfied since 17 5 CBM = M'B'C'= 115 | i > 0. 5 15 It is also seen that the eigenvalues of A - BMC are given by X! = -1, x2 = -2, hence the closed-loop system is asymptotically stable and $(s) is analytic for Re s > 0. For sensitivity reduction, it is then necessary and sufficient that W(s) = (C -M'B'K)[s I -(A-BMC)] BM be positive real where K is the unique positive, self-adjoint solution of K(A - BMC) + (A - BMC)'K = -C'C. Solving the preceding equation, K is given by = 7/12 1/2 1/2 2/3 hence W(s) can be written as -18 166 s+3 2 -12 W(s) 15)(1)(2) -1 3

73 Since 2 Sine 436+~1171c2 -180+415c2+920j ' W(jw)+W*(jw) = (6/115)2 436+1171 (2 w2)2+9w,2 [ -180+415o2-920jw 1300+325w2 is a positive matrix for all w, the sensitivity is reduced. This fact could also have been established from theorem 4. 4 by establishing the existence of positive self-adjoint matrix Q and a matrix L, namely; 3 7 1-i 1 0 2 such that theorem 4. 4 is satisfied. By using the concept of irreducibility introduced in Appendix B, it is possible to obtain additional structure on the matrices B, C, and M. Theorem 4. 6 If Re {(x (A-BMC)} < e < 0 and C*I(s)BM is irreducible then a necessary condition for sensitivity reduction is the existence of a positive definite self adjoint matrix Q and a matrix L such that 1. Q(A-BMC) + (A-BMC)'Q = -L'L 2. QBM = C' which implies Rank (BM) = Rank (C) The proof follows theorem 4. 1 and lemma B.5.

74 Theorem 4. 7 If Re {c(A-BMC)} < e < 0 and (C - M'B'K)*J(s)BM is irreducable then a necessary and sufficient condition for sensitivity reduction is the existence of a positive definite self-adjoint matrix Q and a matrix L such that 1. Q(A - BMC) + (A - BMC)'Q = L'L 2. (Q+K)BM = C'. The proof follows from theorem 4.3 and lemma B.5. Throughout this section we have developed the necessary and sufficient conditions, in the complex frequency domain, that the sensitivity operator must satisfy to insure sensitivity reduction. These conditions illustrate the strong ties that exist between the relatively old concept of positive realness and the relatively new concept of sensitivity reduction. Also exhibited were the necessary conditions relating the matrices B, C, and M, to ensure sensitivity improvement. The following section will employ the inverse of the sensitivity operator to supplement the conditions already developed. 4. 4 Sensitivity Analysis Via the Inverse of the Stationary Sensitivity Operator To begin this discussion we will borrow freely from lemma 4. 2 and inequality 3. 28 to formulate the sensitivity conditions as Lemma 4.3 Let $(s) be described by equation 4. 3, then the necessary and sufficient conditions for sensitivity reduction are 1. $(s) analytic in Re s > 0

75 2. $* (jw) l$(jw) -I> 0 for all we (-oo,0o). (4.6) Since $-1(jc) = I+P(jw)M, condition (2) is equivalent to 0 < $*(j)-$- l(jw) -I ----- P(jc)M+ [P(j)M]*+ [P(jc)M] P(jw)M> O. (4.7) To gain understanding of the physical implications of inequality 4. 7, consider a single-variate time invariant system. Let P and M be represented by P(s) = kr (s) /y (s), M = m, where 'r(s) and 4p(s) denote two polynominals in the complex variable s with real coefficients. The constants k and m are given by positive scalars. Inequality 4. 7 then takes the form 0 < P(jw)m+P* (jw)m+mP*(jwo)P(jc)m = (4. 8) mk [, (jw)+ 7r* (jw)] + m2k _> 1. (4.9) By denoting (-,)(jw) + (-)(jco) = 2 Re [( jc)], a sufficient condition for I - $*(jw)$(jw) to be positive is for 2mk Re [(- )(jw)] > 0. rhis condition can be substantially relaxed by rearranging inequality 4. 8 to the following P(jw)m+mP*(jw) +mP*(jcO)P(jc)m = 7L (jw6) {mk[(~ )(jw) +,4' (jw)]+m k }fr(jw2c).

76 It follows that 0 < {mP*+Pm+mP*Pm}(jCw) -=_> {mk( -' + - )+m k }(jw) > O. -- 7 -Hence the necessary and sufficient conditions for $*(j) 1-$(j) 1- I > 0 for all co is equivalent to 0<{mk( + /*1 22 < mk( + )+ mk }(jO) 2mkRe [()(jw)] +m 2k2 > 0 for all we(-co,o). (4. 10) It can be seen from 4. 10 that a positive scalar m can be found to satisfy this inequality provided that -c < inf Re [(-T)(jco)]. This is formalized by the following theorem. Theorem 4. 8 If P and M are represented by kmr(s)/4(s) and m respectively where 7T(s) and 4I(s) denote two polynomials in the complex s with real coefficients, and if -oo < inf Re[(~)(jw)] then there exists a positive scalar m such that I-S* (jw)S(j w) > 0. Proof: Writing 4. 10 in the form v:(mk~C/+-"_ 2. m ~mk > O and Re t '(i 'k) 0 {mk + )m k }( - (4. 11).mk < 0 and Re [( r )(jOw)] < 2 and since the two conditions on the right side of 4. 11 differ only by a change in equality sign, we will without loss of generality, consider only the case mk > O. Let a = inf Re [(~-)(jw)], then for -2a 4, -ink mcE(max f0, k ]~)~ theconditions: mk > OandRe [( )(jw)] > 2 hold.

77 Note, that the difference between the orders of the polynominals w and 4 play an important role in the existence of an m to satisfy theorem 4. 8. this point is exemplified by the following: Example 4.2 Consider a second order plant represented by k P(s) = 2 = k (w/g/)(s); k, a, b > 0. s +as+b Equation 4. 10 becomes 2mk (b -co) + m2 k> 0, which can't be satisfied for any finite m as w -> oo. From this example, it is clear that not all types of plants are suitable for satisfying the condition $*(jw)-l$(jw) -1-I > 0. This point is well borne out in the multidimensional case, as we shall demonstrate in the following theorem. Theorem 4. 9 If $(s) is analytic in Re s > 0, and r(A) < e < 0, then the necessary and sufficient condition for sensitivity reduction is that the matrix V(s) be positive real where V(s) = (C + M'B'K)4(s)BM and where K is the positive self adjoint solution of KA + A'K = -C'C. Proof: The following chain of conditions follow from inequality 4. 7:

78 C@ (j w)BM+M'B'?4* (jw)C '+M'B'4* (jco)C 'C ~(j ) BM = C@ (j)BM+M'B'Ie~(jo)C'+M'B'e*(jo)[ [(jwI-A)+(jwI-A)*K]4P(jw)BM = C4(jco)BM+M'B' *(jw)C'+M'B?*(jB)KBM+M'B'K (jc)BM = [(C+M'B'K)4(jwc)BM] + [(C+M'B'K) (jcw)BM]* > O0 and the theorem follows from the definition of positive realness. From the above theorem it is seen that if it is necessary to ascertain if the closed-loop system is less sensitive than the nominally equivalent open-loop system, (given that both are asymptotically stable), all that has to be done is to verify the positive realness of (C + M'B'K)b (s)BM. Example 4. 3 This example serves to illustrate theorem 4. 9. Consider the multivariate plant given by -2/3 0 0 1 ' y 0 1K with the feedback matrix 2 2 A necessary condition for sensitivity reduction is CBM = M'B'C' > 0

79 which is satisfied since 1 0 CBM=- > 0. 3 0 1 To establish sensitivity reduction, it is seen that {cr(A-BMC) = {-1} < 0 hence it is necessary and sufficient to establish the positive realness of V(s) = (C +M'B'K)(D(s)BM. Performing the indicated matrix multiplications it is seen that V(j) + V*(jw) 12 I > O for allE co(-o, o) o2+ 4/9 - hence, sensitivity reduction. This result is expected from corollary 2. 2, since there exists a positive self adjoint matrix Q and a matrix L, namely; 3.75 3 75 2 1 Q =,and L = 3.75 7 5 1 3 such that QA + A'Q = -L'L and (Q - K)BM = C'. 4. 5 Sensitivity Reduction with Plant Perturbations The previous sections have been discussing the structure of the plant and the control law to insure that the sensitivity of the closedloop system to additive noise, r/, is less than the nominally equivalent open-loop system. In this section attention is directed to the question

80 of sensitivity reduction when the parameters of the plant matrix A are perturbed. As a logical starting point for this discussion the assumption is made that the closed-loop system does indeed have a reduced sensitivity to noise -7 compared with the open-loop system. Hence, it is assumed that the operator H(s) is positive real, and, without loss of generality, it is also assumed that it is irreducible. For notational convenience the perturbation in A is defined to be the mapping A-DA+ A=A consequently, H -H, W - W. The necessary and sufficient conditions for sensitivity reduction with plant parameter variations are interpreted from lemma 4.2 and are formalized by Lemma 4.4 The necessary and sufficient conditions for sensitivity reduction with perturbation in the matrix A are given by 1. $() analytic in Re s > 0 (4. 12) 2. I - $* (jw)$(jc) > 0 for all ce (-co, co) (4. 13) where $(jc) =I-C [jcoI-(A-BMC)]- BM. Because of the perturbations, theorem 4. 1 is written in the following form

Lemma 4.5 A necessary and sufficient condition for sensitivity reduction with plant perturbations is 1. Re {a(A-BMC)} < e <O 0 2. H(jcw) + H*(jco)-H*(jw)H(jw) > 0 for all w e (-co, o). (4. 14) Employing lemmas B.5, 4. 4 and 4. 5, sufficient conditions can be obtained on the perturbations, 6A, to ensure sensitivity reduction. Theorem 4. 10 If Re {c (A- BMC)} < e < 0 and if -Q6A - 5A'Q + L'L -C'C > 0, where Q and L are given by lemma B.5, then the sensitivity of the closed-loop system is reduced. Proof: Since H(s) is positive real, then by lemma B. 5 there exists a positive definite self adjoint matrix Q and a matrix L such that 1. Q(A- BMC) + (A - BMC)'Q -L -'L. (4. 15) 2. QBM = C'. (4.16) Writing inequality 4. 14 in the form C4(j w)BM+M'B'Y*(jw)C'- M'B'* (jw)C'C (j w)BM > 0 and employing equation 4. 16, we obtain M'B'Q](4jw)BM+M'B'B* (jic)QBM-M'B'* (j c)C 'CQ (jcw)BM = M'B'k(jco) [ jwcI-(A-BMC+6A)]* Q(jw)BM + M'B'?r*(jco){Q[jcoI-(A-BMC+ 5A)] -C'C}) (j~C)BM = M'B'* (j ){-(A-BMC)'Q-Q(A-BMC)-C 'C - A'Q-Q6 A)(jwc)BM = M'B'*(jwc) {L'L-C 'C - A'Q-Q6A)s(jco)BM > 0. Hence, the theorem is proved.

82 Example 4. 4 [he purpose of this example is to illustrate theorem 4.10. We will again consider the system given by example 4. 1 which is described by 52 1.89 1 1 x=~ I ~ x + u ~74 - 1.94 0 1 0 1 1 1 with the feedback matrix 89. 26 M= =.2j -1. 51. 26 The matrices Q and L, computed from equations 4. 15, 16, are: 1.41 1.41 1.4 1.83' 2.23 Let the entries of the matrix A be given a 10% perturbation, so that.52 1.89 -. 05 -. 19 A- + L-. 74 -1. 94 0.7.19 The perturbed matrix W is then given by -18 1667l s+2. 9 1. 3 12 10 W(s)= - (115)2(s +2.95s+l.82)r130 130 -.93 s.052 17 15

83 Since * - W(jw) + *(j ) = r, L7520 + 13798w2 2-2120+5000w 2-j9440w1 ______ 6 ___ (115)21. 82 2)2+ 2. 95 -2120 + 50002w +j9440wc 13720+ 390co2J is a positive matrix for all w, the sensitivity is still reduced. Employing theorem 4. 10, it is seen that ~1.( 1.0 L'L - C'C -Q6A- 6A'Q = 1.0 4.68 is also positive hence, this result also ensures sensitivity improvement. 4. 6 Sum mary In this chapter the problem of sensitivity reduction was formulated for stationary multivariate systems in terms of the sensitivity operator $. Tae results of the previous chapter were then translated via frequency domain techniques to establish the relationships between the constant plant matrices {A, B, C} and the compensators G and M to ensure sensitivity improvement for disturbances in the form of additive noise and plant parameter perturbations. It was also demonstrated that the concept of positive realness plays an important role for sensitivity operators of the form $(s) = I-H(s). This concept is a very important one since it brings to bear on H(s), all the structure associated with positive real matrices. For example, for plants described by S2 and feedback compensators given by constant matrices, it can

84 be shown by using the concepts developed in section B. 5 that every entry of H(s) must be a proper rational fraction of s and moreover, the order of the numerator and denominator can not differ by more than one. This is clearly a very severe limitation which was well illustrated by example 4. 2. This restriction could be reduced if it were desired to improve the sensitivity reduction only over some finite frequency band and not the entire frequency band. This is indeed a fruitful area for future research.

Chapter 5 A COLLECTION OF RESULTS IN SENSITIVITY REDUCTION 5. 1 Sensitivity and Optimal Control In Chapter 1, it was illustrated that there is a very definite relationship between sensitivity reduction and optimal control of regulator systems. In particular, Kalman [23] has shown that for single-input, single-output systems, the optimal control law does indeed reduce the sensitivity. More recently, Anderson F 6 ] and Kreindler [ 32] extended the results of Kalman to include multi-variate systems. They have shown that the optimal control law does reduce sensitivity if a particular weighting is employed on the output errors. This weighting is not arbitrary, but follows from the optimization. Kreindler, points out that for an arbitrary weighting matrix, the optimum control law does not always reduce the sensitivity. Therefore, from previous discussions, it is germain to answer the question: "Under what conditions does the optimum control law reduce system sensitivity for a unity weighting matrix?" This question is not completely answered here, but a necessary condition is given to ensure reduction. In this section, the optimal linear regulator problem, described in Appendix B is analyzed from the point of view of sensitivity reduction, and the salient features are presented.

To review, the linear dynamical system is characterized by x(t)= A(t)x(t) + B(t)u(t); y(t) = x(t), and the optimal control law is given by u(t) = -M(t)x(t) = -R B' (t)E(t)x(t), where R(t) is a positive, definite self adjoint matrix, and E(t) is the unique positive self adjoint solution of the matrix Riccati equation discussed in section B. 3. To discuss the behavior of the optimal regulator system from the point of view of sensitivity reduction, it is pertinent to establish the sensitivity kernel s(t,r). From equation 3. 22, it is clear that (t, T) = I 6 ( (-)- (t, T)B(T)R 1T)B'( T) E(T), (5.1) where If(t, T) satisfies dd (t T) = [A(t) - B(t)R l(t)B'(t)E(t)]*(t, r), q(tto) I It is evident from equation 5. 1 that the sensitivity kernel can be written as &(t, T) = I6 (t —)- h(t, T), (5. 2) where h(t, T) is defined in the obvious way. Employing theorem 3. 1 we see that it is necessary for h(t, T) to be a positive real kernel. Hence, from theorem 2. 2, a necessary condition for the optimal control law to reduce sensitivity is B(t)R (t)B')E(E(t)+E(t)B(t)R (t)B'(t) > 0. (5.3)

87 This is clearly a pertinent result since it illustrates the relationship which must exist between B, R, and E for sensitivity reduction. For time invariant systems with regulation on the infinite interval, the necessary conditions for sensitivity reduction become more clearly defined. Theorem 5.1 Necessary conditions for the optimal control law on the infinite interval to reduce sensitivity are -1 -1 1. E BR B' = BR B E -1 -1 -1 2. BR B'E > EBR B'KBR B'E> O where K is the positive definite self adjoint solution K(A - BR B'E) + (A- BR B'E)'K= -I. Proof: Since, from theorem B. 6, Re fa(A-BR 1B'E)} < e < 0, the theorem follows from theorem 4.5. Hence, the solution of the algebraic form of the matrix Riccati equation must commute with BR B'. A stronger condition relating E, B, and R can be obtained via theorem 4. 3 if (I -EBR B'K)4'(s)BR B'E (5. 4) is irreducible. Theorem 5.2 If 5.4 is positive real and irreducible, then a necessary condition for sensitivity reduction is BR B'> 0.

88 Proof: From theorem 4. 7 there exists a positive definite self adjoint matrix Q, such that (Q+K)BR B'E =I or Q = -K+ (BR B'E) Therefore, it is necessary that BR B' be nonsingular, consequently positive definite. It is also to be noted that B' must be one to one. this section has demonstrated the necessary conditions which must exist for the optimal regulator problem to have improved sensitivity as defined in Chapter 3. rhe next section shall discuss linear discrete systems and give sufficient conditions for sensitivity reduction. 5.2 Sensitivity Reduction in Linear Discrete Systems The discussion of Chapter 3 and 4 illustrated the natural relationship between positive realness and sensitivity reduction of linear systems. In this section, we shall study the question of sensitivity reduction for discrete systems but, as we shall see, there is no relationship similar to positive realness to facilitate this study. Consider the two nominally equivalent systems of Figure 1. 3. The plant is described by the set of linear vector difference equations x [ (k + 1)T] = Ax(kT) + Bu(k7) (5.5) y(kT) = Cx(kTr), (5.6)

89 where the mapping u(kT) - y(kT) acts between finite cartesian products of k2(Q2) equipped with the usual inner product. The feedback operator M i.s given by a2constant matrix. If the plant P is bounded linear and stationary, then P has a multiplicative form and the mapping y = Pu can be written as Y(z) = P(z)U(z), (5. 7) where z is the complex variable associated with the z-transform. Taking z-transform of equations 5. 5, 6, P(z) can be identified by the following: Y(z) = C4(z)BU(z), whe re (z) = (z I-A) The discrete-time analog to the necessary and sufficient conditions for sensitivity reduction given by lemma 4.2 can be written as 1. $(z) analytic Izl > 1 (5.8) 2. I-$*(z)$(z) > 0 for all z such that Izl = 1, (5. 9) where $(z) = [I+P(z)M] -1. (5. 10) Rearranging equation (5. 10) the sensitivity operator takes the now familiar form $(z) = I-C* (z)BM, 1 See also Perkins and Cruz [39].

90 where 'I'(z) is given by I(z) = [zI -(A-BMC)] Previously it was shown that the analysis of the single-variate case was fruitful for establishing insight into the multivariate case. With this in mind, consider the operator H(z) H(z) = C4(z)BM = k (z) m 4i(z) where 7T(z) and 41 (z) are polynominals in z, with real coefficients. The gain k and the feedback operator m are real scalars. Condition (2) can then be written as V (z) 7T 2 2 2r k Z)m+k rTm-km2 (z) > 0, for all z iz =1 (5. 11) Clearly a necessary condition for inequality 5. 8 to be satisfied is for mk [ - + -](z) > 0; for all z~ i zl = 1. Under general pole zero configuration, the above inequality cannot be satisfied. For example, consider the operator H(z) described by cm H(z) - = mk (-$ )(z): c,m > 0, lal <1. Then, 19 Iff 2mc(a -a) mk [ v ](z) -=2c; z = a + j. (5. 12) C *(a-a)2+;z. (5.12) It is clear that equation 5.12 is negative for all c less than a. Hence, the positive realness condition which was used so fruitfully in Chapter

4 does not seem to have an analogous counterpart in the z-plane. Even though positive realness doesn't carry over, it is still possible to reduce the sensitivity. Representing P(z) by P(z)- k(z) a(z) it is clear that 1 1 Z a*(z) a a2 (z) < $(z) $ (Z) -I a —Y {mk[( )(z)+ ( -)(z)] +m2k2 a(z) > 0. Hence we can formulate the following theorem. Theorem 5. 2 If H(z) is analytic for I z > 1, and 7(z) has no a(z) poles on the unit circle, then an m can be found so that I-$ (z)$(z) is positive. The proof is trivial, and follows that given in theorem 4. 8. Turning now to the multivariate case we will again consider the question of sensitivity reduction when perturbations take the form of additive noise rl and variations in the plant matrix A. If A, B, C and M are real matrices then sufficient conditions for sensitivity reduction in the presence of additive noise is given by Theorem 5. 3 If the absolute value of the eigenvalues of A-BMC are less then unity, then a sufficient condition for sensitivity reduction is the existence of a self adjoint matrix Q and a matrix W such that 1. QBM = C' 2. -Q(A-BMC)-(A-BMC)'Q + 26Q-C'C - W'W, for all 6 E [-1, 1].

92 Proof: I-$*(z)$(z) = C4 (z)BM+M'B't*(z)C '-M'B'I*(z)C 'C'I,(z)BM = M'B'QJ (z)BM+M'B'+ (z)QBM-M'B'I*(z)C 'C4 (z)BM = M'B't4*(z)[-Q(A- BMC)-(A-BMC)'Q+2Re zQ-C'C] 'C( IBM = M'B',* (z)W'W4 (z)BM. which is clearly positive. As seen in Chapter 4, the problem of insuring sensitivity reduction in the presence of perturbations of the plant parameters is very difficult to solve satisfactorily. This problem is compounded since the concept of positive reality doesn't carry over in discrete systems. We can however obtain sufficient conditions on &A which guarantee sensitivity reduction. Theorem 5. 4 If the absolute value of the eigenvalues of A-BMC are less than unity and the nominal closed-loop system satisfies conditions (1) and (2) of theorem 5. 3, then the sensitivity of the perturbed closed-loop system is less than the open-loop system if -Q6A-6A'Q+W'W is positive. The proof follows directly from the previous theorem by replacing A with A. From this section it is seen that the results consistent with the preceding chapter, are not very conclusive and are left as an area of future research. In the next section we will study a different closedloop configuration and discuss its structure to ensure sensitivity reduction.

93 5.3 Sensitivity Reduction of a Linear System This section will discuss the question of sensitivity reduction for the nominally equivalent systems of Figure 1. 4. It is to be noted that the only difference between the closed-loop system shown here and the one discussed in Chapters 3 and 4, is that the compensator G is inside the loop. We will see, however, that this makes a substantial difference in the structure of the plant and the associated compensators. As before: P: B1 B2 represents the response of the plant, G: B1 - B1 and M: F2 - B1 are compensation transformations. The system equations for the closed-loop and open-loop systems are given by = [I + PGM] PGu c and Y = Pu which are defined to be the nominal system equations. To insure terminal equivalence between the open and closed-loop systems, G is given by G = (I - MP) provided that I - MP is nonsingular. It can easily be shown that the perturbations in the closed-loop and open-loop system in the presence of additive noise are related by 6yc =(I+PGM) 6yo.

94 Hence by the following sequence of identities (I+PGM)1= [I + P(I - MP)- M] = I+P(I - MP) M(I-P I-PM ) -1] -1 =[I+PM(I-PM) ] = [(I-PM + PM)(I-PM)] =I - PM, the sensitivity operator $, is given by $= (I - PM). (5.13) Now focusing attention on plants which are described by S1 and feedback compensators M which are real and continuous, it is clear from the discussion in Chapter 3 the kernel of $ is given by (t,T) = I6 (t-T)-C(t)4l(t, 7)B(T)M(T). The necessary and sufficient conditions for sensitivity improvement given by 3. 14, 15, 16 can be formulated in terms of the composition PM by the following 1. S: L2 (Q) — L (Q) G PM: L () - L (Q), Q = (to,t) 2. $ causal K — > PM causal 3 I - 0$*$ > 0 -. PM + (PM)* - (PM)*PM > 0. Since the system described by S1 is real, the following theorem is given;

95 Theorem 5. 4 A necessary condition for sensitivity improvement is that the kernel associated with PM be L2 -positive real. The proof follows from the above conditions and definition 2. 2. From the preceding theorem, it is clear that the necessary conditions for sensitivity reduction are similar to those of Chapter 3 rhe primary difference being the L2 -positive realness of PM instead of H. Theorem 5.5 Let G be an integral operator, bounded on L2 (Q) whose kernel is given by [C (t) -M (t)B' (t)K(t)] (1 (t, 7)B(T)M(T) where K(t) is the unique, positive, self adjoint solution of -K (t) = K(t)A(t) + A'(t)K(t) + C '(t)C(t) such that K(t ) is given by tf K(to)= f O?(7,to)C'(T)C(T)b (7,t )d7. o '0 If PM is a bounded, causal and real operator defined on L2 (Q) then the sensitivity is reduced if G + G* is positive. The proof is straight forward and follows that of theorem 3. 2. Several theorems, similar to those following theorem 3. 2, could be given here but because of their similarity, they will be omitted. Instead, attention is focused to the analysis of sensitivity improvement of stationary plants described by S2 with constant

96 feedback matrices. The following theorem helps in establishing the necessary and sufficient conditions for sensitivity improvement for this system configuration. Theorem 5. If a(A) < e < 0, then the necessary and sufficient condition for sensitivity improvement is that (C-M'B'K)4(s)BM be positive real, where K is the positive, self adjoint solution of KA+A'K = -C'C. Proof: the proof follows from definition B. 8, and theorem 5. 4 0 < I-S*(jw)S(jw)=C4-(jw)BM+M'B'"c*(jw)C?'-M'B'*(jw)C?'C(jwo)BM = C (joo)BM+M'B'4?*(j co)C '-M' B'*(j o) [K(j co I-A)+(jco I-A)*K] (I(j co)BM = C4(jwo)BM+M'B' (jco)C '-M'B' *(jco)KBM-M'B'K4(jco)BM = (C-M'B'K),BM+[(C-M'B'K)eb(jw)BM]* > O for all wo E (-c, cx). To gain a better understanding of the restrictions placed in the plant, consider a single variate system with the plant represented by P(s) = k (s) P(s) The function 7w and 4' denote two polynominals ot a complex variable s with real coefficients such that all the zeros of 4' are in the left half plane. rhe operators k and m are also real scalars. Hence O < {m'P*+Pm-m'P*Pm}(jwc) <-. mk(,(j)-mk> 0. -+m(jw)-m > 0. -- ~~~~ ~ ~~~~~7[ 7[ -

97 Noting that (q//r + =*/*( j) - 2)Re [( -2Re )(jc) ], the preceding inequality, can be written in the form mk > 0 and Re [(4')(j w)] > mk/2 0 < mk (4/7r + /*)(jow)-m — 2k2 (5.14) mk < 0 and Re F(/v7)(jcw)] < mk/2 Without loss of generality we will consider only the case; mk > 0. Theorem 5.7 A scalar m can be found which satisfies 5.14 if and only if the frequency plot of (4'7/)(jW) lies in either the left half or the right half of the complex plane. Proof: (SufficiJency) Suppose that 0 < a =inf {Re[(//r/)]}. Then for m e [ 0, 2 a/k] the conditions; mk> 0 and Re [(4//r)(jw)] > mk/2 hold. Similarly if sup {Re [(4,/ir)(jo)] } -= / < 0 then for m e [ 20/k, 0 ] the condition; mk < 0 and Re [( 4//r)(jwr)] < mk/2 hold. Necessity is obvious. Corollary 5. 1 The value of m which maximizes I-$*(jwo)$(jco) is given by m = 1 inf Re[(,/r)(jco)] if the frequency plot of (~/1r)(jo) lies in the right half plane. Corollary 5.2 The value of m which maximizes I-$*(jwo)$(jo) is given by m = k sup Re [(i/r)(jwo)] if the frequency plot of (/,rf)(jco) lies in the left hand plane.

98 Hence, by the preceding corollaries it is possible to select the feedback compensator M, to minimize the sensitivity. Thris feasibility was demonstrated for single-variate systems, however, it is felt that there may be a generalization of this result for the multivariate case. This problem is left open as an area of future research. 5.4 Summary In summary, this chapter has discussed three problems pertinent to sensitivity reduction. First, a necessary condition relating the solution of the matrix Riccati equation and the plant matrix B was given to ensure that the optimal regulator does indeed reduce sensitivity. Second, in the study of discrete systems, a sufficient condition for sensitivity improvement was established relating the plant matrices {A, B, C} and the feedback compensators G and M. Finally, a variation of the closed4oop system was studied resulting in necessary and sufficient conditions relating the plant matrices and its associated compensators.

Appendix A A REVIEW OF FUNCTIONAL ANALYSIS CONCEPTS AND ITERMINOLOGY The books by Porter [48], Liusternik and Sobolev [34], Kolmogorov and Fomin [30], Simmons [52], Taylor [53], Riesz and Nagy r49], and Bachman and Narici [ 7 ] are the principal sources of definitions. A. 1 Linear Space A set X of elements {x, y,...,...} is called a linear space if the following conditions are satisfied. A. For any two elements, x,y e X (read "x and y belong to the set X"), there is a uniquely defined third element z = x + y, z e X, called their sum, such that 1. x+y=y+x 2. x + (y + z) = (x + y) + z 3. There exists an element 0 having the property that x + 0 = x for all xeX 4. For every x e X there exists an element -x such that x + (-x) = 0. B. For arbitrary scalars a-, 3 E F and any element x e X, there is defined an element a xeX such that 1. a(Ix) = (oa~)x 2. 1 x=x. 99

100 A linear space is said to be normed if to each element x e X there exists a mapping II II called a norm such that II: X-R x- IIxll sati sfying the conditions 1. Ilxll > 0 2. | Ix I I= 0 if and only if x = 0 3. Ilaxll = la Ilxll 4. IIx+yII < IIxI + Ilyll. x,ycX where R is the space of real numbers. When applied to the difference between two elements x,y e X, the norm I x - y I has the geometric interpretation of being the distance between x and y in the space X ( Ix -y I I defines a metric for the space X). A sequence of elements xn } of a normed linear space X is called a fundamental sequence or Cauchy sequence if for every number e > 0 there exists an index number N such that I Ixm -xn I < e for all m, n > N. An element x of a normed linear space X is called the limit of a sequence {xn } of X (written xn -x or lim xn = x) if I Ixn - x O -0and n ->. A normed linear space X is called a complete space if every fundamental sequence of this space has a limit in X. Complete normed linear spaces are by convention called Banach spaces. We now describe some of the main examples of

Banach spaces. Example 1 Let X be an n-dimensional real Fuclidean space with the norm defined by I X I = I x1 2 +...+ Ixn 2 for allxeX. Example 2 Let M be the set of all measurable functions on the interval fQ, then the linear space defined by X = L2(Q) = {x M: f Ix(t) 2dt < } where IIxHI = {f Ix(t) dt}2 is a Banach space. Example 3 Let X = L1(Q) = {x E M: f Ix(t) I dt < o} where Ilx l = f Ix(t) Idt, then X is a Banach space. Example 4 Let X = L = {x E M: ess sup Ix(t) I < co} where IIx HI = ess sup Ix(t) i, then X is a Banach space. It is to be noticed that all the definitions given up to now do not say anything about the 'angle' between two elements. The concept of the 'angle' helps to determine when two elements are orthogonal (or perpendicular). Precisely to take care of this need the definition of an inner product space is introduced. A linear space X is said to be an inner product space if there exists a function (denoted by <, > ) defined on X which maps X x X the cartesian product space, into the scalar field F such that Xx X -F (x,y) - < x,y > (where (, ) represents an order pair in Xx X) with the following

102 properties: 1. < x,y > = < y,x >, x,y cX (where bar denotes complex conjugate) 2. < ax + fy,z > = a < x,z > + f < y,z >, a, 0cF and x, y, z cX 3. < x,x > > O ifx = 0 4. < x,x > = O if and only if x is the zero vector. Any two elements x, y c X are said to be orthogonal if < x, y > = 0. Because of conditions (3) and (4) we can define |Ix | = < x,x >2 and it is a simple exercise to check that this defines a norm on X. In this case it is said that the norm is induced by the inner product. A Hilbert space is an inner product space which is complete with respect to the norm induced by the inner product. The following examples illustrate two of the more important Hilbert spaces. Example 1 An n-dimensional real Euclidean space is a real Hilbert space. Example 2 The space L2(Q) is a complex Hilbert space where < x,y > = x(t)y(t)dt. Many Banach spaces are not Hilbert spaces since an inner product does not exists which generates a particular norm. An example of two such spaces are 1. L1() 2. L (Q). cc

103 A. 2 Functions If there exists a correspondence between the elements of one space X and the elements of another space Y, then the mechanism by which the relationship is established is called a transformation or mapping. The relation between an element x c X and its image element y c Y may be denoted by y = F(x) or y = Fx. The space X is called the domain of the operator F and Y the range. It is assumed here that both X and Y are normed linear spaces. If the domain and range of F are both in X then F is said to be an operator on X. At times it so happens that Y is a subset of the real line, i. e., the image elements y are simply real numbers. Then the operator is called a functional. A transformation is said to be bounded if there exists a constant c such that IIFxII < c Ilxll for all x E X. To site two examples of bounded operators consider the following: Example 1 Let x c L1(Q) and let y c L0(Q) be a fixed element, then Fx= -fQx(t)y(t)dt

104 is a bounded linear operator and IIFxII = IFxI < J[x(t) I I y(t)ldt < I ly II 00 Ix(t)Idt = Ilyll Ilxll Example 2 Let x e L2(9) and let y e L2(Q) be fixed then Fx =S x (t) (t)dt is a bounded linear operator and by the Schwartz inequality [IFx2< _[fx(t)I' ly(t) ldt]2_ f Ix(t)I2dt~ fly(t)<2dt = Ixll2 Ily2II Unbounded operators often arise in control problems and the following examples illustrate two such operators. Example 1 Let X = {x L2(o, o): x is absolutely continuous and x e L2(o, co)}. If Fx = x, then F is an unbounded operator. Example 2 Let X = {x e L2(o, co): t x(t) e L2(o, o)}. If (Fx)(t) = t (t), then F is an unbounded operator. A transformation F is said to be continuous if for every number e > 0 there exists a number 6 >0 such that iIFXl-FX2FI < e whenever IXl-x211 < 6 for all x1,x2 e X. Note that the norms in these expressions can be interpreted according to whether the domain X is involved or the image space Y. That is I IFx ll refers to the norm in the range space while Ixll refers to the norm in the domain space.

105 A. 3 Linear Operators An operator F is said to be linear if it satisfies the condition: F( x1 + ox2) = o F(xl)+ 3F(x2) for any two elements xl,x2 e X and arbitrary scalrs co and y. If a linear operator is continuous, then it is also bounded. The reverse is also true. The norm of a linear operator F, (denoted by IIFII), is defined as the greatest lower bound of the numbers c which satisfy the boundedness condition: I Fx II iIFll = sup = sup IIFxl = sup IIFxlI. Ilxll=0o Ilxl Ilxll < 1 Ilx l=1 If F1 and F2 are two linear operators on a normed linear space X, then the inequality IIF1 < IIF1ll + IIF211 holds for their sum F1 + F2 = F. If F is a linear transformation from the space X into space Y and G is a linear transformation from Y to Z, thenthe composite operator H = GF defined by Z = H(x) = G(F(x)), xe X, z E Z is called the product of the transformations F and G. The norm IIHII satisfies the inequality i l i i l _ I. l l lI

106 Some Additional Definitions In dealing with operators on a Hilbert space several additional concepts occur with sufficient frequency to merit specific mention in this Appendix. A. 4 Adjoint Operators Before defining an adjoint operator, it is worthwhile to introduce the concept of a conjugate space. It is kn-)own that the set of all bounded linear functionals defined on a normed linear space X forms itself a normed linear space called the conjugate space of X and denoted by X*. It is true that for any arbitrary bounded functional J defined on the Hilbert space H there is a unique element y c H such that J(z) = < z,y > for every z e H. Letting z = Tx where T is a linear bounded operator on H, the definition of the adjoint of T (denoted by T*) is arrived at by letting < Tx,y > = < x,T y > for every x, y e H. T maps into H, is linear, bounded and the equality IIT*ll = lITll holds. A linear, bounded operator T on H is said to be self-adjoint if T=T*.

107 Four basic properties of adjoint operators are the following: If T1 and T2 are two linear operators on H, then 1. The adjoint of their sum is equal to the sum of their adjoints, (T1+T2) =T + T2 2. The adjoint of their product is equal to the product of their adjoint in reverse order, (T1T2) = T *T1 3. The identity operator is self-adjoint I* = I 4. (T*)*= T**= T. A.5 Positive Operators on a Hilbert Space A linear, bounded operator T different from zero defined on a real Hilbert space H is said to be positive if for every f c H the condition < Tf,f > >0 holds. The operator T is said to be positive definite if 0 in this equation is replaced by 6 < f,f > (6 > 0). It is customary to normalize f in which case the inequality becomes < Tf,f > >6 for all f e H with ifl = 1. Symbolically the notation T > 61I is used.

108 It is true that a positive definite, self-adjoint operator T on H 1 possesses a unique, self-adjoint, positive definite operator T 2 welldefined on H called the square root of T such that 1 i T2T2 = T. Five basic properties of positive and self-adjoint operators are the following: If T1 and T2 are operators mapping a complex Hilbert space H into itself then 1. T! is self adjoint if and only if < Tlx,x > is real for all x. Hence every positive operator on H is self adjoint. 2. If T1 and T2 are bounded and if T1 and T2 commute, then T > 0 and T > 0 implies that T1T2 > 0. 3. If T1 and T2 are bounded and if T1 >, T2 > 0, and T 1T > 0 then T1 and T2 commute. 4. If T1 is positive definite, then T1 is non-singular. 5. If T1 is one to one then T*T > 0 otherwise T > 0. 1 1 1 1 1 -A. 6 Unitary Transformations The linear transformation U of one Hilbert space H1 into another Hilbert space H2 is said to be isometric if it leaves the inner products invariant, i. e., for every f, g e H1 the equality < Uf, Ug>=<f,g> holds, It is to be noted that the inner products are on different spaces

109 and have to be interpreted accordingly. If the image of the space H1 under the transformation coincides with H2, in other words, the mapping under U is 'onto', then the transformation U is said to be unitary. Equivalently, the relation * -1 U =U holds on H1. A. 7 Eigenvalues and Eigenvectors If T is an operator on a complex Hilbert space, and if Tx = Xx for some non-zero x and for some scalar X, then X is called an eigenvalue of T and x is called an eigenvector of T. The set of all eigenvalues of T is called the spectrum of T and is denoted by a(T). If Re { (or(T) } < O ( < O) then T is asymptotically stable (stable).

Appendix B CONCEPTS IN MODERN CONTROL THEORY B. 1 Stability Theory A fundamental aspect of the study of dynamic systems is the determination of their stability. There are several powerful methods for the study of stability but perhaps the most general method is that of Lyapunov which was developed 70 years ago in Russia [35] and which has become an invaluable tool in optimal control theory. Intuitively the problem of stability is that of determining the behavior of a physical system in the neighborhood of an equilibrium state. If the system returns to this state after being perturbed it is called stable; if not it is unstable. Unfortunately, this intuitive concept excludes many forms of motion which are also recognized as being stable. It is for this reason that terms like "asymptotically stable", "uniformly stable", "stable in the large", etc. have arisen. For the purposes of this thesis, the stability concepts introduced are related to linear systems which are characterized by x(t) = A(t)x(t); x(to) = x (B. 1) where A(t) is an nxn matrix whose elements are continuous functions of time on (-co, co). Definition B. 1 The state x = 0 is said to be Lyapunov stable, if for any t and any e > 0, there is a 6 >0 depending on e and to such 110

that lxo i < implies I x(t:x,t) l1 < e for all t > to where x(t;x, to) is the response of time t to the perturbations x =x(t ), and Ilxil is the Euclidean norm defined by Iixi 2= 1x12.. + xn 2. Definition B. 2 The state x =0 is said to be asymptotically stable if (1) it is Lyapunov stable and (2) for any xo sufficiently close to 0, x(t;xo, to)- 0 as t ->o. Theoreml B. 1 The system described by equation B. I is Lyapunov stable if and only if there exists a constant M, which may depend on to, such that I Ib(t, to) 1 <M for all t > to Theorem1 B. 2 The system described by equation B.1 is asymptotically stable if and only if there is a constant M such that I (t, to) i < M for t > to; lim Ii i(t, to) ) =O for all to, t --- O hence I Ix(t;x, to) ii - 0 as t - co. Definition B. 3 The system described by B. 1 is said to be exponentially asymptotically stable if there exists scalars X, k >0 such that II (t, T)II <ke (tT) for all t > e [t, co). Lyapunov devised a method of applying quadratic functionals to discuss the questions of asymptotic behavior of solutions of linear differential equations. Before proceeding with the method of Lyapunov, the following terminology is introduced. 2 See Zadeh [59]. 2See Hahn [19].

Function of Class K: A function 0 (r) which is a continuous real function defined on the closed interval [0,h] such that ~,(r) = 0 when r = 0 and increasing strictly monotonically with r is called a function of class K. Decrescent function: A function V(x, t) is said to be a decrescent function if IV(x,t) I < 0 (lIxll) in the region I xll < h for allt > to where (( xl ) is afunction of class K. Radially Unbounded function: A scalar function V(x, t) is said to be radially unbounded if V(x,t) > 4(I xll I) in the region Ilxll < h for all t > t for arbitrarily large hwhere ( Ilxll) is a function of class K. Positive (Negative) Definite function: A scalar function V(x, t) is said to be positive (negative) definite if V(O,t) = 0 and V(x,t) > IY(lixMl) ( < - V(Ijxll)) in the region Ilxll < h for all t > t where ( xll ) is a function of class K. In studying the stability of the system characterized by equation B. 1, the following lemmas are very important. 1 Lemma B. 1 The origin is stable in the sense of Lyapunov if there exists a positive definite function V(x, t) such that -d V(x, t) evaluated along the solution of equation B.1 is not positive. Ibid.

Lemma B. 2 The origin is asymptotically stable if there exists a function V(x, t) which is everywhere positive definite, radially unbounded, descrescent, and -d V(x, t) evaluated along the solution of equation B.1 is negative definite. Definition B. 4 A function V which satisfies lemmas 3. 1 and 3. 2 is called a Lyapunov function for the differential equation B. 1. One of the important uses of Lyapunov functions other than for purposes of stability, is to establish the nature of the solution of the algebraic equation A'E + EA = Q (B.2) Lemma B. 3 Consider equation B. 2 where E, A and Q are nxn matrices and Q < 0, then there exists a unique solution E >0 if and only if Re {or(A)} < e < 0. Proof: Define the solution of equation B. 2 by the integral co A't At E=-f e Qe dt. (B.3) 0 E is clearly positive definite since e is never singular and X, = S At eAt X, Ex= - f [xe Qe x] dt 0 -feAtx,QeAt]dt. Integrating B. 3, premultiplied by A, by parts it follows that Ibid.

r0 At At Alt At0 0Alt At AE - A'e Qe Atdt= [e QeAt] + E Qe Adt =Q - EA. Conversely assume E > 0 and consider the differential equation x = Ax. A Lyapunov function V(x) is defined by V(x) = [x,Ex] then V(x) = [x, Ex] + [x, Ex] =[Ax, Ex]+ [x,EAx] = [x, (A'E + EA)x]. Hence V(x) = [x, Qx] < 0 and x(t)->O which implies Re {a(A)} < e < 0. To prove uniqueness consider the existence of a solution E1 such that A'E1 +EA=Q. (B. 4) Subtracting B. 4 from B. 2 we obtain A'(E -E 1) =- (E -E 1)A. Since a(A)(n(-A') = X, E = E1 and the lemma is proved. B. 2 Controllability and Observability The fundamental concepts of controllability and observability of linear systems were first introduced by Kalman and play an 1 See Gantmacher Ref [14 ] page 220.

115 important role in modern control theory. In this section we will restrict our attention to continuous and discrete, linear multi-variate control systems. However, since most of the results for discretetime systems can be obtained from the continuous-time case by replacing the variable t with discrete points tk, and by replacing integrals with summations, it will be sufficient to study only the continuous time case. Consider the multi-variate linear system described by x(t) = A(t)x(t) + B(t)u(t) (B. 4. 1) y(t) = C(t)t)x(t) + D(t)u(t) (B. 4. 2) where: x is an n-vector, the state; u(t) is an m-vector, the input; y(t) is a p-vector, the output; A(t), B(t),C(t),D(t) are nxn, nx m, pxn, pxm matrices, respectively. Definition B. 5 A plant is said to be completely state-controllable if for each pair of points x and xf in R, there exists a bounded 0 measurable controller u(t) Rm on some finite interval t < t < tf to f Definition B. e A plant is said to be completely output-controllable if for each pair of points y and yf in RP, there exists a bounded measurable controller u(t) e R on some finite interval to ' t < tf It has been shown [31] that the plant is completely output-controllable on [t, tf], if and only if, the Gramian matrix

P(to, tf) + D(tf)D*(tf) where tf P(to, tf) = St C(tf)(tf, T)B(T)B*(T)(*(tf T)C(t)d, is nonsingular. The necessary and sufficient conditions for the doublet {A, B} identified with equation B. 4.1 to be completely state-controllable is that the Gramian matrix tf W(t0 - f ~ (to, t) B(r) B*(r)qD*(to, T) dT, be nonsingular. From the above equation it is clear that the necessary and sufficient conditions for controllability depend explicitly on the state transition matrix ~f(to,t). Consequently, it is necessary to obtain a solution of the time-varying differential equations. To overcome this difficulty Silverman and Meadows [ 51] define state controllability matrices characterized in terms of A(t), B(t), and C(t) and their derivatives. The state controllability matrix of the system described by equation B. 4 is Qc(t) = [Po(t), P(t) Pn (t)] (B.5.1) where Pk+l(t) = -A(t)Pk(t) + Pk(t), Po (t) = B(t). (B. 5. 2) It can be shown that the system is completely state controllable if and only if Qc(t) has rank n on the interval fto0tfJ.

For stationary systems, the preceding conditions for state controllability reduce the fact that Qc =BAB,.,A 1n-B] has rank n. A necessary and sufficient condition for complete outputcontrollability is that the px (n+1) m matrix r1 = [CB, CAB,..., CAn- B, D] have rank p. Note that although output and state -controllability are conceptually similar, they do not imply each other. 'This is easily seen by setting D = 0. The rank of F is less than or equal to the rank of C or Qc. Thus if the rank of r1 < p the system would not be completely output-controllable. Conversely, the rank of rl = p does not imply rank Qc = n. Definition B. 7 An unforced plant is said to be completely observable on [to tf] if for a given to and tf every x(t ) in R can be determined from the knowledge of y(t) on [to, tf]. It is well known [48] that the necessary and sufficient conditions for the doublet {A,C} identified with the plant, to be completely observable is that t f M(t o,tf) = (t,to)C* (t)C(t) (t, t0)dt 0 be non-singular. The corresponding observability matrix due to Silverman and Meadows, is given by

Qo (t) = [R (t), Rl(t)..., Rn-(t) ] where Rk+l(t) = A'(t)Rk(t) + Rk(t), Ro(t) = C'(t). Hence the system described by equations B. 4 is completely observable on the interval Ito tf] if and only Qo(t) has rank n on the interval. For stationary systems this condition is equivalent to the nx np matrix [C',A'C',...,A(n1)C'] having rank n. Further simplification of these conditions is possible for special forms of the constant matrix A. For example, if A has distinct eigenvalues and if A is a diagonal matrix then 1. The stationary system is completely controllable if and only if B has no all-zero rows. 2. The stationary system is completely observable if and only if C has no all-zero columns. B. 3 Optimal Regulator Problem In this section the optimal regulator problem is formulated and the salient features of the optimal solution are presented. Consider the linear dynamical system characterized by x(t) = A(t)x(t)+ B(t)u(t), X(to) = x. 0'

119 where x(t) is the n dimensional state vector, u(t) is the m-dimensional control input and the matrices A(t) and B(t) are locally measurable and of compatable dimensions. The optimal linear regulator problem is to determine the control u on the interval (to tf) which minimizes the cost functional: tf J(u) = x'(tf)Fx(tf) + [x'(t)L'(t)L(t)xL(t(t) +u'(t)Ru(t)]dt t 0 where the terminal state x(tf) is unconstrained, F is a positive matrix and R(t) is a continuous positive definite matrix. The optimal control u is given by the linear feedback control law -1 u(t) = -R (t)B'(t)E(t)x(t) = -M(t)x(t) where E(t) is the unique positive solution of the matrix Riccati equation -E(t) -A'(t)E(t) + E(t)A(t) -E(t)B(t)R (t)B'(t)E(t) + L'(t)L(t) satisfying the boundry condition E(tf) = F. rhe matrix E(t) has the property that for arbitrary te [to tf] J(u)= minJ(u) = [x(t),E(t)x(t)]. u Restricting our attention to the case when the matrices A, B, R, and W are stationary and tf -> o, the cost functional takes the form tf J(u) = lim f {[x(t),L'Lx(t)] +[u(t),Ru(t)]}dt tfoo o

120 where an admissible controller u is measurable on (0< t<oo) and is such that the cost functional converges to a finite value. By a suitable normalization of the input vector, it is possible to take R = I, the unit matrix; thus the cost function takes the -form tf J(u) = lim J {[x(t),L'Lx(t)] + [u(t),u(t)]}dt (B. 6) tf - -oo where u = R2u. Now stating the results due to Kalman [23], the problem may be solved as follows: Let r(t: tf,0) = E(t) be the unique self adjoint solution of -E(t) = E (t)A + A'E (t) -E ' (t) BB'E (t) + L' L (B. 7) such that 7r(tf;tf, 0) = E(tf) = 0. If the plant is completely controllable, then lim '7(t; tf, 0) = E(t) exists for all t and lim r (0;tf, 0) =E E (B. 8. 1) t f-> OO 00 tf O is the steady state solution of equation B. 7. Let the control law be given by M = B'E (B. 8. 2) then as given by Kalman:

121 Theorem B.3 If {A, B} is completely controllable and the cost functional is given by B. 6 then the optimal control law is given by B. 8. 2 where E is obtained by evaluating the limit of B.8. 1. Theorem B. 4 If {A, B} is completely controllable and {A,L} is completely observable, then E is positive definite and self adjoint and the optimal control law is asymptotically stable. Theorem B. 5 If fA, B} is completely controllable, a necessary and sufficient condition for the stability of the optimal control law is that all the eigenvalues of A restricted to uncontrollable states have negative real parts. Theorem B.6 Let {A, B} be completely controllable and {A, L} be completely observable, then a necessary and sufficient condition for M to be a stable optimal control law is that there exists a matrix E such that E >0 and E = E' M = B'E -E(A -BM) - (A - BM)'E = L'L + M'M B. 4 Irreducible Dynamic Systems Consider the linear, time-invariant, finite dimensional plant, described by the following equations x = Ax + Bu (B. 9. 1) y = Cx (B. 9.2) where u,x, and y are m, n, and p-vectors respectively. The number

122 n is defined to be the dimension of the system. This description is isomorphic with the triple of matrices {A, B, C} which is called a realization of the transfer function matrix T(s) =C(sI-A) B. (B. 10) Any matrix triple {A, B, C} satisfying B. 10 is a realization of T(s) but this realization is not unique. A realization of T(s) is called irreducible if the dimension of A is as small as possible. Kalman [26] has shown that every rational, proper matrix T(s) possesses an irreducible realization. Moreover if {AT, BT, CT} and AT, BT, CT} are two irreducible realizations of T(s) then there exists a constant nonsingular matrix F such that A = rA r1 BT = BT CT CTr Hence the two realizations represent the same system but with respect to two different basis. This is easily seen since T(s) =CT(sI-AT) B=CTr r(sI-AT) r rBT CT(SI- AT F)lBT CT(sI-AT) BT Kalman [26] has shown that if T(s) is a transfer function matrix satisfying T(s) =C(sI-A) B

123 then, 1. T(s) is irreducible if and only if it is completely state-controllable and completely observable. 2. If {A, B, C} is a non-minimal realization, then there exists a nonsingular matrix r such that the triple {A, B, C} is brought into the canonical form (A B C } with AA AB AU A A A A = rAr 0 ABB ABU B = FB= r C =Cr-1 [o0 CBCUJ c operator I such that xc =rx=(x,x,x ), The superscript letters have the following meanings A: controllable and unobservable B: controllable and observable

124 U: uncontrollable. The triple {ABB, B, C is irreducible and its realization T(s) = C B(sI -AB) 1BB is equivalent to T(s). B.5 Positive Realness For several years, network theorists have been employing the concept of positive realness. More recently this concept has become an important tool for control systems analyst. This section reviews the concept of positive real matrices and gives an important lemma due to Anderson [ 4 ]. The definition of a positive real matrix can be found in several books on network synthesis (for example Newcomb [36]) and is given by Definition B. 8 An nx n matrix r(s) is called positive real if for Re s > 0 1. T(s) is analytic 2. T(s) = T(s) 3. T*(s) + T(s) > 0 where the overbar denotes complex conjugate and * denotes complex conjugate transpose. If T(s) is composed of rational polynominals in s then the above statements are equivalent to 1. T(s) is real-rational 2. T(s) analytic in Re s > O

125 3. Poles of T(s) on Re s = 0 are simple 4. For each pole on Re s = 0, the residue matrix is positive and self-adjoint 5. T (j w) + T(j C) > 0 for all ce (-oo,co). It is of interest to see what necessary conditions are placed on the entries of a matrix described by equation B. 10 to ensure positive realness. To facilitate this, it is helpful to write equation B. 10 in the form all(s) al2(s)... alp(s) a2 1(s) T(s) = apl(s).. app(S) Consider an arbitrary entry aij(s) of T(s), given by aij (s) = nij (s)/ /ij (s ) Let frij and 4ij be the nth and mth order polynominals in s respectively. It is well known that for the given system of differential equations, described by equations B. 9, n < m for all i, j and since the poles of T(s) must be simple on Re s = 0, it must be the case that m = n + 1 for all i, j. A little reflection also shows that all of the entries along the main diagonal must be positive real functions.

126 Lemma B. 5 (Anderson [4 ]) Let T(s) be a matrix of proper rational functions such that T(s) has poles which lie in Re s < O and are simple on Re s = 0. Let {A, B, C} be an irreducible realization of T(s). Then r(s) is positive real if and only if there exists a positive definite self adjoint matrix Q and a matrix L such that 1. QA + A'Q = -L'L 2. QB = C'. B. 6 Decoupling and the Inverse System In the recent months there have been several studies (see [13], [15] and [47]) concerned with the problem of decoupling multi-variate systems by state feedback. In particular the paper by Falb and Wolovich [13] give the necessary and sufficient for this characterization for stationary systems, as well as conditions relating to the inverse system. Consider the time-invariant linear feedback system shown in Figure C. 1. Figure C. 1 A Time-invariant Linear Feedback System

127 The plant is described by equations B. 9 with the control vector u, given by u = Fx + Gv; G is nonsingular. The closed-loop form is then given by x = (A+ BF)x + BGv; y = Cx. th Definition B. 9 The system is said to be decoupled if the i input affects only the ith output. Let a1, a2,..., am be given by ai =min{ j: CiAjB / O j = O,1,...,n-1} or ai = n-1 if C iAB = 0 for all j, th where Ci is the i row of the matrix C. Let A, B, D denote the matrices at a 1 a I ALlg~l1+1 C1A C1A BA=l l; B=AB=; D= (~ "a +1 The next two lemmas follow from [13 Lemma B. 5 There exist a pair of matrices F and G which decouple The next two lemmas follow from [13 ]. Lemma B. 5 There exist a pair of matrices F and G which decouple the system if and only if B is nonsingular. In particular the matrix pair F=B AA, G=B satisfies the requirement of decoupling the system.

128 Lemma B. e If the matrix B is nonsingular then an inverse system exists which can be represented by -1 - (S3) V=(BG) {Dy-A(A+BF)z} z (I - A)A + B Dy. A question of interest is "What are the restrictions on the plant described by B.9, 1, 2 in the context of the previous discussion, to be positive real?" To help answer this question, the representation of the plant in the context of (S3) with G = I and F = O (in the transformed domain) is given by D(s)Y(s) = AAX(s) + BU(s) X(s) = (sI-A) 1BU(s), which combine to the form D(s)Y(s) = {AA(s I - A) B + B}U(s). With some manipulation the plant transfer function simplifies to the form P(s) D-I~s)A(I - A/s) l for all values of - for which the inverse exists. Since D (s) is for all values of s for which the inverse exists. Since D (s) is diagonal it follows that m -(<xei+l) D (s)=, s Ei i=1 where E, is the orthogonal projection on the pith coordinate subspace. Hence, we have the expression

129 m -(cr+l) 1 P(s)-E s Ei A(I- A/s) Bi (B.11) i=1 1 For large s, equation B. 11 becomes m _(ci+1) _ lim p(s) Z s E.AB. S - 6 i=l For P(s) to be positive real, it is necessary that all the poles on Re s = 0 be simple, hence ai = 0 for all i. Therefore, CiB f 0 for any i. We can formulate the following: Theorem B. 7 A necessary condition for the plant described by equations B. 9 to be positive real is that CiB / 0 for all i. 1

Appendix C A SUMMARY OF FOURIER ANALYSIS C. 1 Introduction Throughout this Appendix Lp will denote the usual Lebesque space on the infinite interval (-co, co). That is L consists of all measurable complex valued functions, f, of a real variable such that -00 holds (with integration in the Lebesque sense). The cases p = 1, 2 will be of primary interest. The Hilbert space L2 is equipped with the innerproduct (, ) where 00 (x,y) = f x(s) y(s)ds, x, yeL2. The following discussion extends easily to finite products of L2. The notation L 2 will denote the space of all tuplets f = (f,. '.. ), f EL2 being finite with respect to the norm n co A lfil = [ f lf(t)l2dt]2 i=l -oo n Again L 2 is a Hilbert space with the innerproduct <, > where n oo < f,g> = E f f(t)gi(t)dt, f,geL.2' i=l -o L2 is equipped with the usual algebraic operations. 130

131 In dealing with physical systems an important notion is that of stationarity. Let D(T) denote the domain of T and Ka, for real a, denote the time translation operator (Ka x)(t) = x(t-a), x e L Then T is stationary whenever 1. D(T) = KaD(T) all real a 2. TK =K T all real a In other words if x e D(T) and if y = Tx then Kax e D(T) and Kay = TKax. In dealing with stationary systems Fourier transforms play an important role. The Fourier transform 'F of a function x e L2 is to be defined by the expression N (Tx)(cW) = 1. i. m (12S ejt(t)dt. (C. 1) N -oo Here 1. i. m. denotes limit in the mean. Some salient properties of F are summarized in the following theorem of Plancherel (See [ 54], page 51). Theorem C. 1 As an operator on L2, TF is one-to-one, onto, and norm preserving. The inverse of IF is determined by -N ( y)(t) = 1 2 -jWt Yit m.(-) f e y(w)dw. (C.2) N -Woo -N In other words 'F is an isometric isomorphism of L2 onto itself. A consequence of this is that for x,y e L2 (Fx, 'Fy) = (x, y)

132 * -1 holds, moreover F = F. The Fourier transform is also extended to the space L2 in the natural way; the Fourier transform of a tuplet being the tuplet of component Fourier transforms. We remark only that the innerproduct relationship takes the form n 0o < 'Fx,Fy > =E f= ('X)i(w)(FY)i(c) do i=1 -cO n -Q =, f x.(t)y.(t)dt = < x,y > i= -oo 1 The next result which is apparently due to Bochner (see [21]) is a cornerstone in the study of stationary systems. In this theorem m n y = Tx is mapping from L into L2 while y and x denote the 2 2 2L Fourier transform of y and x respectively. T denotes an nx m matrix of measurable functions. Theorem C. 2 A necessary and sufficient condition for T to be linear bounded and stationary is that y(w) = T(w)x(w), we (-oo, oo). where each component of the matrix T is a uniformly bounded measurable function. This theorem may be paraphrased as stating that the transformation T from L m into Ln is linear bounded and stationary, if and only if, T has a multiplicative form. Since y = Tx = y = Tx the matrix T, which represents T in its multiplicative form can, as we see from the equality chain

133 y =Fy + 'FTx = 9FT9F-5eFx be represented as T = FTTF C. 2 Convolutions The examples to be presented later come from the class of systems which may be identified with convolution operators. The convolution operator "fe" is defined by o0 (feg)(t) = f (t-s)g(s)ds, tE(-0o,oo) (C. 3) - 00 The well known operational properties: fog = gek and fe(gak) = (fog)ek are easily verified. The convolution of a nx m matrix and a mxk matrix is defined in the obvious manner. In particular the case mx 1 is of interest for, as we shall see, such convolutions can define the type of linear transformation under discussion. The next theorem (see [12], pg. 951) isolates several important facts concerning convolutions on L1 and L2. Theorem C. 3 For f, x e L the convolution fox is well defined and satisfies ilfxill < Ilfll " IIxII. For f E L1, x e L2 the convolution fox exists in L2 and satisfies IIfexll2 < lflll1 Ilxl12 If f, x L2 the convolution fox defines a continuous function with norm (sup norr) at most lIfIl2 Ilxl12.

134 In view of equation C. 3 and this theorem it is clear that for f e L the convolution fe defines a bounded linear transformation on both L1 and L2. With the domain of fe being the entire space L2, it is easily verified that this operator is stationary. The theorem also generalizes easily to the multivariate setting. For instance if W is a nx m matrix of functions Wij e L1 such that I W e L1 where IWI (t)= IW(t) I denotes the norm of W(t) as a mapping from {l(m) into Pl(n) then We is a bounded linear stationary transformation sending L into L with norm satisfying 2 2 00 Finally it is noted that fe, as an operator on L2, has a Hilbert space adjoint (fe) This adjoint is itself a convolution namely (fe)* = f~ where f(t) = f(-t), t e (-oc, oo). More generally for any bounded linear stationary system T, acting between finite products of L2 with T the matrix multiplicative representation T, the identity chain < T*z,x > = < z, T x > = < z, Tx > = < zx > shows that (T), the conjugate transpose of T1, is the multiplicative matrix representation of T C. 3 Causal Systems Heuristically a noncausal system is one in which present values of the output are not influenced by future values of the input. To sharpen this somewhat let PT for real T, denote the projection

135 operator defined by x(t) -co < t-< (.P x)(t) = In other words P is computed by multiplication with the characteristic T function of the interval (-oo, T]. A function T is said to be causal if for every xl,x2 e D(T) such that P x1 = PTx2, for any real T, then PTxl = PTx2. A convolution fe causal if and only if f(t) = 0 for t < 0. In this case with y = fox we have t co y(t) = f(t-s)x(s) ds = f(s)x(t-s) ds, t e (-0o, so) -00 and the output y at any t e (-co, co) clearly depends only on past values of the input. Similarly a convolution fe, where f(t) = 0 for t > 0, is called purely anticausal. Evidently 00 (fox)(t) = f (t-s)x(s)ds, tE (-o,cO) t and hence present output values depend only on future values of the input. The Fourier transform representation of a causal convolution has a certain familiar and important property. To state this result we introduce the complex Fourier transform on L2 by means of definition 1 )o x(s) = (Fx)(s) = ( -1 ) S x(t)e stdt, s = C + jw. -cc

136 From the definition it is not difficult to show that if x(t) = 0, t < 0 then x(s) exists for a > 0 and is analytic in the right half (i.e. a >0) of the complex plane. Furthermore for ac >0 -oO O fx( + jw)12dw 1x(t) l2e-2odt < II < coo-tIe dt< xi oo. The converse of this result is also true as is stated in the next theorem (see [37] section 1.4). Theorem C. 4 The two subsets of L2(-co, co) E-= { x: x(t) = 0, t < 0} E = {x: x(s) is analytic in ( > 0, l Ix(r) || < I Ixl. a > 0} are identical. Moreover for x 6 E x(w) = 1. i.m. (s). a-0+ where the limit exists for almost all we (-oo, oc). Theorem C. 4 extends to L2 with norms replacing absolute values where necessary. Suppose now that xE L2 satisfies x(t) = 0 for t > 0 and that y is the function defined by y(t) = x(-t). Then y e L2 and y(t) = 0 for t < 0. The equality chain 00 -st o t x(s) = f e x(t)dt = f e x(-t)dt O O o -st =- fe y(t)dt = y(-s) -co 1SeeWidder [55] pp. 80-81.

137 then shows that the properties set forth in Theorem C. 4 for y in the half space rA > 0 hold also for x in the half space a < 0. This observation establishes the corollary: the conjugate of a causal convolution is a purely anticausal convolution. The proof of this statement follows from the above remarks and the fact that if 4(s) 0 then b (s) -( 0().* Employing the preceding theorems it follows that stationary, causal convolutions associated with operators bounded on L2(-oc, cO) have complex Fourier transforms that are analytic for a> 0. This is formalized by the following. Theorem C. 5 Let g(t) be a continuous function for t e (-co, co), such that ge is a causal stationary convolution bounded on L2(-cc, o), then g(s) exists and is analytic for a > 0. Proof: Let y = gox where x is chosen so that x(t) = est for t < tf: s = +jw, > 0 x(t) = 0 for t > tf. Then, since go is causal and bounded on L2(-oc, co), y(t) = e tf g(t-T)e (t-T)dT -00 exists for almost all t and belongs to L2(-oc, co). Making the change in variable; X = t-T, y(t) = est Sg()e-S dX, C > O. - '>O.

138 Since y(t) exists for almost all t, it follows that g(s) exists, and converges for a > 0. Moreover, g(s) is analytic for c > 0. With a few additional assumptions placed on g(s), the converse of theorem C.5 is demonstrated. Theorem C. 6 If g(s) is absolutely convergent for a > 0 such that g(w) is essentially bounded for almost all wc(-o, co), then go is a causal stationary convolution bounded on L2(-xc, xo). Proof: Let x e L2(o, co), then x(s) exists and converges absolutely for a> 0. Hence, g(s)x(s) = {g*x }(s) = y(s). Since the 1. i. m. as a -O 0 of g, x, and y exist, then by theorems C. 2 and C.4, g@ is linear, bounded, causal, and stationary on L2(-oC, co). Theorem C. 7 If go is a causal stationary convolution bounded on L2(-CO, C) such that g(t) is continuous on t e (0, cc) and if g(s) consists of rational polynomials in s with real coefficients such that g(co) = a, then g(s) is analytic for a > 0. Proof: By hypothesis, the inverse complex Fourier transform of g(s) consists of finite linear combinations of terms of the form g(t) a 6 (t) + tne-t, t > O =O,t<O where n is a nonnegative interger and y is a complex constant. If there See Widder [55], PP. 57. 2 See Widder [56], PP. 453.

139 exist any poles on the jco axis, then g(t) can be written in the form g(t) = g l(t) + g2(t) where g (t) consists of finite linear combinations of terms of the form g1(t) tnej (Omt+7m) Let y = g6x be the response when g(t) = t e(t+o) where x(t) = 1 for t e 0, 1] and zero otherwise. Then y(t) f tn+ntn-l T+... +n}e e 0 dT+ t> 1 o j(cot + 7o) [antn+...+ao]e t where an,... a are constants resulting from the integration. Consequently, the total response at wm = co is the form y(t) = [Antn+..Ao]e 0 where An,...,A are complex constants. Since y(t) must belong to L2(1, oo), it is the case that An,...,A 0 and the theorem follows. Corollary C. 1 If go is a causal stationary convolution bounded on L2(-cx, xo) such that g(t) is continuous on tE (0, co) and if g(s) consists of rational polynomials in s with real coefficients such that g(co) = 0, then g(s) is analytic for u > O and g(t) e L1(0, oo)n L2(0, xo).

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