System theory: A Hilbert space approach PDF
Preview System theory: A Hilbert space approach
SYSTEM THEORY A Hilbert Space Approach Avraham Feintuch Department of Mathematics Ben Gurion University of the Negev Beer Sheva, Israel Richard Saeks Department of Electrical Engineering Texas Tech University Lubbock, Texas 1982 ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers New York London Paris San Diego San Francisco SHo Paulo Sydney Tokyo Toronto COPYRIGH@T 1982, BY ACADEMICP RESS,T NC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. 111 Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NW1 7DX Library of Congress Catalogirq in Publication Data Feintuch, Avraham. System theory. (Pure and applied mathematics ; ) Bibliography: p. Includes index. 1. System analysis. 2. Hilbert space. I. Saeks, R. 11. Title. 111. Series: Pure and apolied mathematics (Academic Press) ; pA3.PB [PA4021 510s roo31 82-1816 ISBN 0-12-251750-4 AACR2 PRINTED IN THE UNITED STATES OF AMERICA 82 83 84 85 9 8 7 6 5 4 3 2 1 To our parents Preface Although one can trace the heritage of system theory back through the centuries, the field did not truly come into its own until the post-World War I1 era. By the mid-l960s, however, the field had expanded to the point where communication between its various branches had begun to break down, with some practitioners applying the new state space con- cepts while others employed the traditional frequency domain techniques. Moreover, the systems community was beginning to encounter new classes of distributed, time-varying, multivariate, and discrete time systems, and a multiplicity of new concepts associated therewith. As such, with the goal of bringing some order into this mounting chaos, a number of re- searchers began to search for a unified approach to linear system theory. Although many approaches were tried with varying degrees of success, the present work represents the culmination of one such search, in which Hilbert space techniques are used to formulate a unified theory of linear systems. xi xii PREFACE Unlike the classical applications to mathematical physics, however, the formulation of a viable theory of linear systems required the develop- ment of a modified Hilbert space theory in which a “time structure” is adjoined to the classical Hilbert space axioms. The present book thus represents an exposition of the resultant “theory of operators defined on a Hilbert resolution space” together with the formulation of a unified theory of linear systems based thereon. Interestingly, however, essentially the same theory evolved inde- pendently in the pure mathematics community. Indeed, the theory was developed simultaneously by two different research groups working independently and motivated by different problems. One such group developed the theory of nest algebras in the context of a study of non-self- adjoint operator algebras, while the second developed the theory of tri- angular operators models in the context of an effort to extend the Jordan canonical form to an infinite-dimensional setting. Although different terminology and notation were employed, the three theories proved to be essentially identical; as such, concepts developed in each of the three theories are consolidated in the present work within the Hilbert resolution space setting. In order to make it accessible to the system theory community at large, only a single course in Hilbert space techniques is assumed; the text is otherwise self-contained. In particular, the book can be employed in a second-year graduate course for students interested in either operator theory or system theory. Indeed, the authors have taught such courses to both mathematics and electrical engineering students. The text is divided into four parts dealing with I. operator theory in Hilbert resolution space, 11. state space theory. Ill. feedback systems, and IV. stochastic systems. As such, it can be used for a second course in operator theory in which Part I is covered in detail together with a sampling of topics from Parts 11-IV. Alternatively, one can gloss over Part I with the emphasis on the latter parts for a course on linear systems. Although it is impossible to acknowledge everyone who has contributed to this book in one way or another, the authors would like to express their sincerest thanks to the students who served as guinea pigs in our classes, to our colleagues who have served as sounding boards for our ideas, and to the numerous individuals who have read the several drafts of the manu- ... PREFACE Xlll script and commented thereon. To list but a few, we thank Gary Ashton, Roman DeSantis, John Erdos, Maria Fuente, Israel Gohberg, Dave Larson, Phil Olivier, Lon Porter, A1 Schumitzky, Leonard Tung, and George Zames. Finally, we would like to express our sincerest thanks to Mrs. Pansy Burtis, who painstakingly typed the several drafts of the manuscript and too many revision thereof to count. Introduction Intuitively, a system is a black box whose inputs and outputs are func- tions of time (or vectors of such functions). As such, a natural model for a system is an operator defined on a function space. This observation and its corollary to the effect that system theory is a subset of operator theory, unfortunately, proved to be the downfall of early researchers in the field. The projection theorem was used to construct optimal controllers that proved to be unrealizable, operator factorizations were used to construct filters that were not causal, and operator invertibility criteria were used to construct feedback systems that were unstable. The difficulty lies in the fact that the operators encountered in system theory are defined on spaces of time functions and, as such, must satisfy a physical realizability (or causality) condition to the effect that the operator cannot predict the future. Although this realizability condition usually takes care of itself in the analysis problems of classical applied mathe- matics, it must be externally imposed on the synthesis problems that are central to system theory. 1 2 INTRODUCTION Fig. 1. Block diagram for the optimal control problem The problem is readily illustrated by the optimal control problem, illustrated in Fig. 1. Here, the black box represented by the operator P is termed the plunf;i t could be an aircraft, an electric generator, or a chemical process, for example. To control the plant we desire to build a compensator, represented by the operator M,w hich generates a plant input r designed to cause the resultant plant output y to track a reference input u. Since large plant inputs are not acceptable, M is typically chosen to minimize the performance measure 4 M )= IlY - ullZ + llrIl2 which measures both the magnitude of the plant input and the deviation between plant output and reference input. From the block diagram of Fig. 1 r = Mu and y - u = (PM - 1)u Hence for any given M J(M) = IIY - ul12 + Ilrl12 = IVM - l)ul12 + IIMul12 + = ((PM - l)u, (PM - 1)) (Mu, Mu) + = ([(PM - l)*(PM - 1) M*M]w, u) + + = ([M*(1 P*P)M - M*P* - PA4 1]u, u) + + = ([(l P*P)l’ZM - (1 P*P)-1/2P*]* + + x [(i P*PrizM - (1 P*P)- 1/2P*]u,u ) + + ([P*(l PP*)-(cid:146)P]u,u ) where we have used the fact that the positive definite hermitian operator + 1 P*P admits a positive definite hermitian square root. Now, the term + ([P*(l PP*)P]u, u) is independent of M, while the term + + ([(l P*P)l’ZM - (1 P*P)-l/ZP*]* + + x [(l P*P)”’M - (1 P*P)-’”]u, U) is nonnegative. Hence the performance measure will be minimized by choosing an M that makes this latter term zero. That is, + M, = (1 P*P)-’P* INTRODUCTION 3 "CY FiR. 2. Feedback system Although this may at first seem to be a complete solution to our optimal control problem, a closer investigation will reveal that Mo may fail to be physically realizable. Indeed, consider the case where P is the ideal delay d defined on L2(- 00, 00) by L-~fI<=t> f (t - 1) Since d is unitary, its adjoint is its inverse, the ideal predictor P", defined on L2( - a,0 0) by Cm?) = f(t + 1) Thus M, = (1 + P*P)-’P* = +F which cannot be implemented. The difficulty lies not with our mathematics but with the formulation of the problem, which should have included some type of physically realiz- ability constraint. Although such a constraint can be readily formulated in L2(- CXaI, ),c ausality is not well defined in an abstract operator theoretic setting, so the methods of classical operator theory are not immediately applicable, as one might have expected, to the optimal control problem. As a second example consider the feedback system, illustrated in Fig. 2. Here r = u - Fy and y = Pr Hence when the appropriate inverse exists the operator mapping u to y defined by the feedback system takes the form y = Hu, where + H =(I PF)-'P Now consider the case where P = I is the identity on L2(- 00, 00) and F = d - I, where d is the ideal delay. Then H = [I + Z(d - [)]-'I = [d]-' = P 4 INTRODUCTION is once again the ideal predictor. It would seem that we have constructed a device for predicting the future using physically realizable components. In fact, the feedback system is unstable and cannot be implemented. The precise formulation of the concept of stability and its relationship to causality, however, once again requires additional structure not available in a classical operator theoretic setting. In an effort to alleviate these and similar problems encountered in the design of regulators, passive filters, and stochastic systems, the theory of operators defined on a Hilbert resolution space was developed in the mid-1960s. In essence, a Hilbert resolution space is simply a Hilbert space to which a time structure has been axiomatically adjoined, thereby allowing one to define such concepts as causality, stability, memory, and passivity in an operator theoretic setting. The present text is therefore devoted to an exposition of the theory of operators defined on a Hilbert resolution space and the formulation of a theory of system based thereon. Although the development of the theory of operators defined on a Hilbert resolution space was motivated by its potential applications to system theory, it is interesting to observe that the resultant theory is closely allied with two parallel developments in abstract operator theory, also dating to the mid-1960s: nest algebras and triangular operator models. Indeed, the algebras of causal operators we construct are in one-to-one correspondence with the nest algebras, and both theories define a class of triangular operator models. Accordingly, Part I, summarizing the theory of operators defined on a Hilbert resolution space, simultaneously serves as an exposition of these theories. The remaining three parts of the text are devoted to the formulation of a theory of systems defined in a Hilbert resolution space setting. In Part I1 the concept of memory is formalized by the development of a state space theory for an operator defined on a Hilbert resolution space. The design of feedback systems is the primary topic of Part 111, where stability is defined in a resolution space setting and algorithms for the optimal and/or asymptotic design of stable feedback systems are developed. Finally, stochastic systems are investigated in Part IV. A Hilbert resolution space-valued random variable is adopted as a model for a stochastic process and used to derive a theory for stochastic estimation and control.
The list of books you might like
