Feedback Control, Nonlinear Systems, and Complexity (Lecture Notes in Control and Information Sciences)

Lecture Notes in Control and Information Sciences Editor: M. Thoma

202

Bruce A. Francis and Allen R. Tannenbaum

Feedback Control, Nonlinear Systems, and Complexity

Springer London Berlin Heidelberg New York Paris Tokyo Hong Kong Barcelona Budapest

Series Advisory Board A. Bensoussan- M J . Grimble. P. Kokotovic. H. Kwakernaak J.L. Massey. Y.Z. Tsypkin

Editors Bruce Allen Francis Department o f Electrical and Computing Engineering, University o f Toronto, Toronto, Ontario M5S 1A4, Canada Allen Robert Tannenbaum Department o f Electrical Engineering, University o f Minnesota, Minneapolis, M N 55455, USA

ISBN 3-540-19943-8 Springer-Verlag Berlin Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the ease of reprographie reproduction in accordance with the terms of lieenees issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. © Springer-Verlag London Limited 1995 Printed in Great Britain The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Typesetting: Camera ready by contributors Printed and bound at the Athenaeum Press Ltd, Gateshead 69t3830-543210 Printed on acid-free paper

Preface This volume is the proceedings of a conference held May 6 and 7, 1994, at McGill University in Montreal to honour Professor George Zames on the occasion of his 60th birthday (January 7, 1994). The general chair of the conference was Dean Pierre Bfilanger, the local organizer Peter Caines, and the technical organizers Bruce Francis and Allen Tannenbaum. The speakers, all invited, were John Baras Roger Brockett Peter E. Caines Munther A. Dahleh Edward J. Davison John C. Doyle Tryphon T. Georgiou Keith Glover Pramod Khargonekar

Raymond Kwong Sanjoy Mitter A. Stephen Morse Kameshwar Poolla Michael G. Safonov Malcolm C. Smith Eduardo D. Sontag Le Yi Wang Jan C. Willems

Looking over George's list of publications, one sees the pioneering insights that he has had in the subject of feedback control. Indeed, we think it is fair to say that no other systems theorist has thought so deeply about feedback systems and has shaped the subject in so profound a manner. Those of us who have had the privilege of working with George Zames and becoming his friend must be impressed with his generosity, his depth, and his devotion to his subject. There is a Jewish saying that a truly full life is reached only at one hundred and twenty. In that case, we can only look forward to the surprises that, George will have for us in the next sixty years! H a p p y B i r t h d a y George! Bruce Francis and Allen Tannenbaum

A c k n o w l e d g e m e n t s Financial support for the conference was provided by the Natural Sciences and Engineering Research Council of Canada and by the Canadian Institute for Advanced Research. We thank Stephen Zucker of the CIAR for this support, Pierre B6langer for eloquently opening the conference, Peter Caines for his superb job of organizing the conference venue and banquet, and Jennifer Quinn and Mindle Levitt for all their help with correspondence and registration.

CONTENTS

Biographical Sketch of George Zames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Publications of George Zames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

H. BERCOVICI, C. FOIAS, AND A. TANNENBAUM On the Structured Singular Value for Operators on Hilbert Space . . . . . . . . .

11

R. B R O C K E T T Pulses, Periods, and Cohomological Terms in Functional Expansions . . . . . . 24 P.E. CAINES, T. MACKLING, C. MARTINEZ-MASCARUA, AND Y.a. W E I An Approach to the Problems of Complexity and Hierarchy with an Application to a Detection Problem . . . . . . . . . . . . . . .

36

M.A. DAHLEH AND M.M. LIVSTONE A Unified Framework for Identification and Control . . . . . . . . . . . . . . . . . . . . . .

56

E.J. DAVISON AND M. CHANG Intelligent Control: Some Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

T.T. G E O R G I O U AND M.C. SMITH Metric Uncertainty and Nonlinear Feedback Stabilization . . . . . . . . . . . . . . . . .

88

G. GU AND P.P. K H A R G O N E K A R Identification in Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

R.H. K W O N G AND L. ZItU Performance Analysis and Control of Stochastic Discrete Event Systems . 114 L.H. LEE AND K. P O O L L A Statistical Validation for Uncertainty Models . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131

L. LUBLIN AND M. ATHANS An Experimental Comparison of 7t2 and 7/oz Designs for an Interferometer Testbed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

150

A.S. MORSE Logic-Based Switching and Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

t73

M.G. SAFONOV AND T-C TSAO The Unfalsified Control Concept: A Direct Path from Experiment to Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

196

E.D. SONTAG State-Space and I / O Stability for Nonlinear Systems . . . . . . . . . . . . . . . . . . . .

215

L.Y. WANG AND W. ZHAN On Optimal Decentralized Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

236

J.C. W I L L E M S Control as Interconnection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

261

Biographical Sketch of George Zames George Zames was born on January 7, 1934. He was a child living with his parents in Warsaw, Poland, when the bombing of that city on September 1, 1939, marked the start of World War II. His family escaped Europe in an odyssey through Lithuania, whose occupation by Soviet tanks they witnessed, then through Russia, Siberia, followed by a triple crossing of the Sea of Japan, eventually reaching Kobe, Japan, early in 1941. (This episode, which involved the extraordinary help of the Japanese consul to Lithuania, Senpo Sugihara, is the subject of the book The Fugu Plan by M. Tokayer). Later that year they moved to the Anglo-French International Settlement in Shanghai, China, where they were stranded by the outbreak of the war in the Pacific. Despite the war, their sojourn in Shanghai was a happy one, and George was able to attend school without interruption. The family moved to Canada in 1948. In spite of losing a year in the move, George entered McGitl University at age 15. He graduated at the top of the Engineering Physics class, and won an Athlone Fellowship for study in England. He gravitated to Imperial College of London University, where his advisors included J.H. Westcott, Colin Cherry, and Denis Gabor. In 1956, George began his doctoral studies at the Massachusetts Institute of Technology. He was briefly associated with Doug Ross at the Servomechanisms Laboratory, where he was assigned the task of developing computer graphics for Gordon Brown's recently developed computer-controlled milling machine which, at the time, was the only one in existence. Eventually this program became the APT programming language. He later switched to the Communications Theory Group of Norbert Wiener, Y.W. Lee, and Amar Bose at MIT's Research Laboratory of Electronics. In 1959, while still working on his thesis, Zames introduced the first global linearization theory for nonlinear feedback systems. Key applications included the earliest versions of the Companding Theorem for the recovery of nonlinearly distorted bandlimited signals, the Small Gain Theorem, and the notion of "physical realizability," later renamed strict causality, for nonlinear systems. These results were described in the 1960 report Nonlinear Operators for Systems Analysis, which won wide recognition, and were repeated in his 1963-64 papers. The companding recovery involved the use of contraction-mapping based feedback, and presaged George's continuing interest in nonlinear feedback systems. In 1957, Niels Bohr arrived for a lecture tour of North America and asked for a "typical American" student to guide him around Cambridge. George was found to be appropriate for this task, and after being asked by Norbert Weiner for an introduction to Bohr, witnessed a remarkable argument between the two men on the merits of research into the natural sciences, such as Physics, versus the sciences which focus on man-made phenomena, notably Cybernetics. After receiving his doctorate of science degree, George was appointed Assistant Professor at MIT. The following summer he set out for a vacation in Greece. On his way he stopped in Israel and met Eva. He never got to Greece

on that trip. They were married two years later. They have two sons, Ethan and Jonathan. Between 1961 and 1965, he moved back and forth between MIT and Harvard, continuing his work on nonlinear stability. His version of the Circle Criterion, the Positive Operator Theorem and related multiplier methods appeared in a 1964 N.E.C. paper. In 1965, he won a Guggenheim Fellowship, which he spent at the NASA Cambridge Research Center, forming the nucleus of what was to become the Office of Control Theory, with which W.M. Wonham, S. Morse, P.L. Falb, W. Wolovich, and It. Kushner later became associated. At the Center he won the first of a half-dozen outstanding paper awards. In 1970, the NASA Center closed and was replaced by a Center for Transportation Systems. For a year he worked on Transportation planning with G. Kovatch, authoring the 1971 twenty-year Transportation Technology Forecast and the studies of Personalized Rapid Transit Systems. George then took an extended sabbatical at the Technion in Haifa, Israel. There he worked out a theory of structural stabilization of nonlinear systems by dither, which led to a seminal paper co-authored with his student at the time, N. Schneydor. He also hosted Claude Shannon during Shannon's visit to receive the Harvey Prize. George's interest in metric complexity theory was stimulated by this event as well as interactions with the Technion professors Jacob Ziv and Moshe Zakai. In 1974, George returned to McGill where he was appointed Professor of Electrical Engineering. At that point he had gotten the idea that the objective of complexity reduction could provide a basis for a unified theory of metric organizations, of which simple feedback is an example and which, more generally, take the form of feedback hierarchies. These powerful new ideas were described in a plenary lecture at the 1976 Conference on Decision and Control, which has been updated and repeated on many occasions. (It is one of George's best talks!) By then he had concluded that existing indicators of feedback performance were inadequate, and that without a better indicator a satisfactory synthesis theory could not be achieved. It was for this reason (and partly also in an attempt to show that feedback reduces metric complexity) that George introduced the H °° feedback synthesis theory in 1979-81. He subsequently collaborated on this subject with B. Francis, W. Helton, A. Tannenbaum, and S.J. Mitter among others. In 1981 together with his student A. E1-Sakkary, he also proposed the "gap" as a metric for unstable systems. George's work on H °° theory has had an enormous influence on control theory for the past decade. Besides becoming the dominant methodology in robust control, it has also influenced a number of world-class mathematicians to work in problems in feedback control. George was awarded the Macdonald Chair of Electrical Engineering at McGill in 1983 and the IEEE Field Award for Control Science in 1984. He is a Fellow of the Canadian Institute for Advanced Research and the Royal Society of Canada.

Publications of George Zames Papers

in Journals

G. Zames, L. Lin, and L.Y. Wang, "Fast Identification n-Widths and Uncertainty Principles for LTI and Slowly Varying Systems," [EEE Trans. Auto. Control, 39, Sept. 1994, pp. 1827-1838. G. Zames and J.G. Owen, "Duality Theory for MIMO Robust Disturbance Rejection," IEEE Trans. Auto. Control, 38, May 1993, pp. 743-751. G. Zames and J.G. Owen, "Metric Dimension and Feedback in Discrete Time," IEEE Trans. Auto. Control, 38, April 1993, pp. 664-667. J.G. Owen and G. Zames, "Duality Theory of Robust Disturbance Attenuation," Automatica, 29, 1993, pp. 695-705. J.G. Owen and G. Zarnes, "Robust H °° Disturbance Minimization by Duality," Systems and Control Letters, 19, 1992, pp. 255-263. G. Zames and L.Y. Wang, "Local-global Double Algebras for Slow H c~ Adaptation: Part I - Inversion and Stability," IEEE Trans. Auto. Control, 36, Feb. 1991, pp. 130-142. L.Y. Wang and G. Zames, "Local-global Double Algebras for Slow H c~ Adaptation: Part II - Optimization of Stable Plants, IEEE Trans. Auto. Control, 36, Feb. 1991, pp. 143-151. L.Y. Wang and G. Zames, "Local-global Double Algebras for Slow H ~° Adaptation: The Case of L 2 Disturbances," IMA Journal of Control and Optimization, 2, 1991, pp. 287-319. L.Y. Wang and G. Zames, "Lipschitz-continuity of H c~ Interpolation," Systems and Control Letters, 14, 1990, pp. 381-387. G. Zames and S.J. Mitter, "On the Essential Spectra and Norms of Hankel* Hankel plus Toeplitz* Toeptitz Operators," Systems and Control Letters, 10, 1988, pp. 159-165. C. Foias, A.Tannenbaum, and G. Zames, "Sensitivity Minimization for Arbitrary SISO Distributed Plants," Systems and Control Letters, 8, pp. 189-195, 1987. C. Foias, A.Tannenbaum, and G. Zames, "Some Explicit Formulae for the Singular Values of Certain tIankel Operators with Factorizable Symbols," SIAM J. Mathematical Analysis, 19, 1988, pp. 1081-1089. C. Foias, A.Tannenbaum, and G. Zames, "On the H°°-Optimal Sensitivity Problem for Systems with Delays," SIAM J. Control and Optimization, 25, 1987, pp. 686-705. C. Foias, A.Tannenbaum, and G. Zames, "Weighted Sensitivity Minimization for Delay Systems," IEEE Trans. Auto. Control, 31, August 1986, pp. 763766.

4 C. Foias, A.Tannenbaum, and G. Zames, "On Decoupling the H°%Optimal Sensitivity Problem for Products of Plants," Systems and Control Letters, 7, 1986, pp. 239-249. B. Francis, J.W. Itelton, and G. Zames, "H ~ - Optimal Multivariable Feedback Controllers," I E E E Trans. Auto. Control, 29, October 1984, pp. 888-900. B. Francis and G. Zames, "On Optimal Sensitivity Theory for SISO Feedback Systems," I E E E Trans. Auto. Control, 29, January 1984, pp. 9-16. G. Zames and D. Bensoussan, "Multivariable Feedback Sensitivity and Decentralized Control," I E E E Trans. Auto. Control, 28, November 1983, pp. i030-1035. G. Zames and B.A. Francis, "Feedback, Minimax Sensitivity, and Optimal Robustness," I E E E Trans. Auto. Control, 28, May 1983, pp. 585-601. G. Zames, "Feedback and Optimal Sensitivity: Model Reference Transformations, Multiplicative Seminorms and Approximate Inverses," I E E E Trans. Auto. Control, 26, April 1981, pp. 301-320. G. Zames, "On the Metric Complexity of Casual Linear Systems: c - Entropy and c - Dimension tbr Continuous Time," I E E E Trans. Auto. Control, 24, April 1979, pp. 222-230. G. Zames and N. Schneydor, "Structural Stability, Continuity and Quenching," t E E E Trans. Auto. Control, 22, June 1977, pp. 352-361. G. Zames and N. Schneydor, "Dither in Nonlinear Systems," IF~EE Trans. Auto. Control, 21, Oct. 1976, pp. 660-667. G. Zames and R.R. Kallmann "On Spectral Mappings, Iligher Order Circle Criteria and Periodically Varying Systems," I E E E Trans. Auto. Control, 15, December 1970, pp. 649-652. P.L. Falb, M. Freedman and G. Zames, "A Hilbert Space Stability Theory Over Locally Compact Abelian Groups," in S I A M J. on Control, 7, August 1969. G. Zames, "Stability of Systems with Sector Nonlinearities," 1EEE Trans. Auto. Control, 13, December 1968, pp. 709-711. G. Zames and M. Freedman, "Logarithmic Variations Criteria for the Stability of Systems with Time-Varying Gains," S I A M J. Control, 6, 1968, pp. 487-507. G. Zames and P.L. Falb, "On the Stability of Systems with Monotone and Odd Monotone Nonlinearities," I E E E Trans. Auto. Control, 12, April 1967, pp. 221-223. G. Zames and P.L. Falb, "On Crosscorrelation Bounds and the Positivity of Certain Nonlinear Operators," I E E E Trans. Auto. Control, April 1967. G. Zames, "On the Input-Output Stability of Time-varying Nonlinear Feedback Systems, Part I: Conditions Derived Using Concepts of Loop Gain, Conicity, and Positivity," I E E E Trans. Auto. Control, 11, April 1966, pp. 228-238.

G. Zames, "On the Input-Output of 'rime-Varying Nonlinear Feedback Systems, Part II: Conditions Involving Circles in the Frequency Plane and Sector Nonlinearities," IEEE Trans. Auto. Control, 11, July 1966, pp. 465-476. L. Stark, Y. Takahashi, and G. Zames, "Non-linear Analysis of Human Lens Accommodation," [EEE Trans. System Science and Cybernetics, SSC-1, November 1965, pp. 75-83. G. Zames, "Realizability Conditions for Nonlinear Feedback Systems," IEEE Trans. on Circuit Theory, CT-11, June 1964, pp. 186-194. G. Zames, "Functional Analysis Applied to Nonlinear Feedback Systems," IEEE Trans. on Circuit Theory, CT-10, September 1963, pp. 392-404. D.B. Brick and G. Zames, "Bayes Optimum Filters Derived Using Wiener Canonical Forms," IRE Trans. on Information Theory, IT-8, September 1962. P a p e r s in C o n f e r e n c e P r o c e e d i n g s G. Zames, "Adaptive Feedback, Identification and Complexity: An Overview," Proc. 32rid. IEEE Conf. Dec. Control, 1993, pp. 2068-2075. J.G. Owen and G. Zames, "Adaptation in the Presence of Unstructured Uncertainty," Proc. 32nd. IEEE Conf. Dec. Control, 1993, pp. 2110-2114. G. Zames and L.Y. Wang, "Adaptive vs. Robust Control: Information Based Concepts," IFAC Internat. Symposium on Adaptive Control and Signal Processing, Grenoble, July 1992, pp. 533-536. L. Lin, L.Y. Wang, and G. Zames, "Uncertainty Principles and Identification n-Widths for LTI and Slowly Varying Systems," Proc. 1992 Amer. Control Conf., Chicago, pp. 296-300. J.G. Owen and G. Zames, "Unstructured H¢~: Duality and Hankel Approximations for Robust Disturbance Attenuation," Proc. MTNS Congress, Kobe, June 1991, pp. 345-350. J.G. Owen and G. Zames, "Robust H ~ Disturbance Minimization by Duality," Proc. 30th. IEEE Conf. Dec. Control, Brighton, December 1991, pp. 206-209. G. Zames and L.Y. Wang, "What is an Adaptive-learning System?," Proc. 29th. IEEE Conf. Dec. Control, 5, Dec. 1990, pp. 2861-2864. G. Zames, "Feedback Organizations and Learning," NATO Workshop on Knowledge-based Robot Control, Bonas, France, 1988. G. Zames, "Feedback Organizations, Learning and Complexity in H °°, Plenary Lecture, American Control Conf., Atlanta., 1988. G. Zames, A. Tannenbaum, and C. Foias, "Optimal H ¢~ Interpolation: A New Approach," Proe. 25th. Conf. Dec. Control, I: pp. 350-355, 1986.

C. Foias, A. Tannenbaum, and G. Zames, "H °° Optimization Theory for Distributed Systems," Proc. 25th. I E E E Conf. on Decision and Control, 2: pp. 899-904, 1986. C. Foias, A. Tannenbaum, and G. Zames, "On Decoupling the H°°-Optimal Sensitivity Problem for Products of Plants," Proc. 1986 Amer. Contr. Conf., June 18-20, 1986. C. Foias, A. Tannenbaum, and G. Zames, "Weighted Sensitivity Minimization for Delay Systems," Proe. I E E E Conf. on Decision and Control, 1985, I, pp. 244-249. B.A. Francis and G. Zames, "Design of H°%Optimal Multivariable Feedback Systems," 22nd. I E E E Conf. on Decision and Control, Dec. 1983, pp. 103108. B. Francis and G. Zames, "On Optimal Sensitivity Theory for SISO Feedback Systems," Proc. 21st I E E E Conf. on Decision and Control, Dec. 1982, pp. 623-628. G. Zames and A. E1-Sakkary, "Uncertainty in Unstable Systems: The Gap Metric," Proc. 8th Congress, Kyoto, Japan, 1981, No. 5.6. G. Zames, "Feedback, Optimal Sensitivity, and Plant Uncertainty via Multiplicative Seminorms," Proc. 9th IFAC Congress, Kyoto, Japan, 1981, No. 38.7. G. Zames and B. Francis, "Feedback and Minimax Sensitivity," Proc. N A T O A G A R D , No. 117, October 1981, pp. 10.1-10.8. G. Zames and B. Francis, "A New Look at Classical Compensation Methods: Feedback and Minimax Sensitivity," Proc. 20th I E E E Conf. on Decision and Control, Dee. 1981, pp. 867-874. B. t~ancis and G. Zames, "On Minimax Servomechanisms," Proc. 2(]lh I E E E Conf. on Decision and Control, Dec. 1981, pp. 188-193. G. Zames and D. Bensoussan, "Multivariable Feedback and Decentralized Control," 20th. I E E E Conf. on Decision and Control, Dec. 1981. G. Zames, "The Tradeoff Between Identification and Feedback," 19th. I E E E Conf. on Decision and Control, Albuquerque, Dec. 1980. G. Zames and A. E1-Sakkary, "Unstable Systems and Feedback: The Gap Metric," Proc. 18th Allerton Conf., Oct. 1980, pp. 380-385. G. Zames, "Optimal Sensitivity and Feedback: Model Reference Transformations, Weighted Seminorms and Approximate Inverses," Proc. 17th. Allerton Conf., Oct. 1979, pp. 744-751. G. Zames, "On the Metric Complexity of Causal Linear Systems," Proe. 197~ I E E E Conf. on Decision and Control, New Orleans, Dec. 1977, pp. 807-810.

N. Schneydor and G. Zames, "On the Stabilization of Nonlinear Systems by Dither," Proc. 13th. Allerton Conf., 1975, pp. 560-579. G. Zames, "Feedback, Hierarchies and Complexity," Proc. 1976 IEEE Conf. on Decision and Control, Dec. 1976, Addenda. G. Zames, "Modeling Transportation Systems: An Overview," Proc. IEEE Decision and Control Conf., Miami, Dec. 1971, 294-304.

1971

P.L. Falb, M.J. Freedman, and G. Zames, "Input-Output Stability: A General Viewpoint," If'AC Congress, Warsaw, June 1969. G. Zames and P. FMb, "On the StabiIity of Systems with Monotone and Odd Monotone Nonlinearities," Conference on Mathematical Control Theory, USC, January 1967. G. Zames and M.I. Freedman, "Logarithmic Variation Criteria for the Stability of Systems with Time-Varying Gains," Proc. Joint Auto. Contr. Conf., 1968, 769-786. G. Zames, "On the Input-Output Stability of Nonlinear Systems-Conditions for Loo - Boundedness Derived Using Conic Operators on Exponentially Weighted Spaces," Proc. 1955 Allerton Conference, Urbana, October 1965, pp. 460-471. G. Zames, "On the Stability of Nonlinear Feedback Systems," Proc. National Electronics Conference, Chicago, Illinois, Vol. 20, October 1964, pp. 725-730. G. Zames, "Contracting Transformations-A Theory of Stability for Nonlinear, Time Varying Systems," International Conference on Microwaves, Circuit Theory and Information Theory, Tokyo, September 1964. Book

Chapters

J.G. Owen and G. Zames, "Unstructured Uncertainty in H a,'' Control of Uncertain Dynamical Systems, S.P. Bhattacharyya and L.H. Keel, Eds., CRC Press, 1992, pp. 3-21. M. Verma and G. Zames, "An Energy and 2-port Framework for H °~ Control," Control of Uncertain Dynamical Systems, S.P. Bhattacharya and L.H. Keel, Eds., CRC Press, 1992, pp. 39-51. J.G. Owen and G. Zames, "Unstructured H °° : Duality and ttankel Approximations for Robust Disturbance Attenuation". Recent Advances in Mathematical Theory of Systems, Control, Networks and Signal Processing t, It. Kimura and S. Kodama, Eds., MITA Press, Tokyo, 1991, pp. 345-350. L.Y. Wang and G. Zames, "Local-Global Double Algebras". Advances in Control and Dynamical Systems, 50, Academic Press, 1991. G. Zames and D. Bensoussan, "Multivariable feedback and decentralized control," Advances in Control and Dynamical System, 22, Part I, Academic Press, 1985.

G. Zames, "Biological Control Mechanisms: Human Accommodation as an Example of a Neurological Servomechanisms," Theoretical and Experimental Biophysics, Arthur Cole Ed., Marcel Dekker, New York, 1967, with L. Stark and Y. Takahashi, pp. 130-160.

Reports or Monographs G. Zames, Transportation Systems Technology: A Twenty-):'ear Outlook, with G. Kovateh, F. Casey, and J. Barber. Report for the Secretary of Transportation, DOT, Washington, D.C.; also presented at the Conference on Personalized Rapid Transit, University of Minnesota, Nov. 1972. G. Zames, Personalized Rapid Transit System: A First Analysis, with G. Kovatch, (Report No. DOT-TSC-ST-71-11, Transportation Systems Center, Cambridge, Mass.) prepared for the Office of the Secretary of Transportation, DOT, Washington, D.C., Proc. Second Conference on Personalized Rapid Transit, University of Minnesota, Minneapolis, Minn., Nov. 1972. G. Zames, "On the Mapping of Random Processes into Random Variables on the Unit Interval and their Characterization by Probability Distribution Operators," ARM No. 258, July 21, 1961; Applied Research Laboratory, Sylvania Electronic Systems, Waltham, Massachusetts, 02154. G. Zames, "On the Transmission of Signals through Randomly Time Varying Media; Some Preliminary Studies," ARM No. 213, June 15, 1960; Applied Research Laboratory, Sylvania Electronic Systems, Waltham, Massachusetts, 02154. G. Zames, "Nonlinear Operators for Systems Analysis," Technical Report No. 370, Research Laboratory of Electronics, MIT, September 1960.

Research Laboratory of Electronics QPR's G. Zames, "Conservation of Bandwidth in Nonlinear Operations," Quarterly Progress Report No. 55, Research Laboratory of Electronics, MIT, October 15, 1969, pp. 98-109. G. Zames, "Realizability of Nonlinear Filters, and Feedback Systems," Quarterly Progress Report No. 56, Research Laboratory of Electronics, MIT, Jannary 15, 1960, pp. 137-143. G. Zames, "Canonical Forms for Nonlinear Statistical Estimators," Quarterly Progress Report No. 52, Research Laboratory of Electronics, MIT, January 15, 1959.

Other Presentations G. Zames, "On Learning, Adaptation and Complexity," Workshop on Intelligent Control, Cortona, Italy, April 1993.

G. Zames, "Unstructured Uncertainty in H ~,'' International Conf. on Robust Control, San Antonio, March 1991. G. Zames, "Visions of Tomorrow in Communication and Control," distinguished panel, MIT Symposium "From Servo-loops to Fiber Nets: 50 years of Systems, Communications and Control," Cambridge, Oct. 1990. G. Zames, "Feedback organizations and complexity". Plenary paper delivered to the Conference on Operator Theory, College Park, Maryland, Dec. 1985. G. Zames, "Feedback organizations and complexity in H °~ , plenary paper delivered to the International Symposium on the Mathematical Theory of Networks and Systems, Stockholm, June 1985. G. Zames, "Feedback and Nonparametric Modelling Uncertainty in H °~,'' plenary session NSF Workshop on PDE's and Flexible Structures, Tampa, March 1985. G. Zames, "Feedback Organizations and Complexity," keynote lecture, ONR Symposium on Multivariable Design, Minneapolis, Oct. 1984. G. Zames, "H °° Feedback Theory," Workshop on Adaptive systems, Grenoble, France, June 1984. G. Zames, "H~-Methods in Optimal Design," Amer. Diego, June 1984.

Control, Conf., San

G. Zames, "Optimally Robust Feedback Theory: Recent Developments," Internat. Syrup. Circuits and Systems, Montreal, May 1984, $10. B.A. Francis, G. Zames, J.W. Helton, "H~-Optimal Feedback Controllers for Linear Multivariable Systems," Internat. Symposium Math. Theory of Networks and Systems, Beersheva, Israel, June, 1983. G. Zames, "The Optimally Robust Servomechanism Problem," IFA C Workshop on Singular Perturbations, Ohrid, Yugoslavia, July 1982. G. Zames, "Robustness of Feedback Systems," Internat. Symposium on Circuits and Systems, Rome, May 1982, $2.2. G. Zames, "Multivariable Feedback Synthesis," Bilateral Meeting on Control Systems, Shanghai, China, August 10-12, 1981, No. 1.3. G. Zames, "Multivariable Feedback Under Plant Uncertainty: Explicit InputOutput Methods based on Approximate-Inverse Optimization," Parts I and II, NATO distinguished Lecturer Series, Ankara, Turkey; Bolkesjo, Norway; and Delft, the Netherlands; October 1981. G. Zames,"Feedback and Optimal Sensitivity," Math. Theory of Networks and Systems Workshop, Virginia Beach, May 1980. G. Zames, "Feedback, Optimal Sensitivity, and Plant Uncertainty," IEEE International Symposium Circuits and Systems, Houston, April 1980.

l0 G. Zames, "Feedback Organizations and Complexity," Mathematical Theory of Networks and Systems Conf., Delft Netherlands, July 1979. G. Zames,"Decentralized Control of Large Transportation Networks," Transportation Workshop, 1972 IFAC Congress, Paris, June 1972.

On the Structured Singular Value for Operators on Hilbert Space Hari Bercovici a n d Ciprian Foias D e p a r t m e n t of M a t h e m a t i c s Indiana University B l o o m i n g t o n , I n d i a n a 47405 Allen T a n n e n b a u m D e p a r t m e n t of E l e c t r i c a l E n g i n e e r i n g U n i v e r s i t y of M i n n e s o t a Minneapolis, M i n n e s o t a 55455

Abstract In this note, we discuss some new results concerning a firing method introduced by the authors in order to study the structured singular value applied to i n p u t / o u t p u t operators of control systems. We moreover give a new criterion which guarantees that the structured singular value equals its upper bound defined by D-scalings.

With great admiration, this paper is dedicated to Professor George Zames on the occasion of his 60-th birthday.

1

Introduction

This paper is concerned with the study of the structured singular value for very general linear operators. More precisely, let A be a linear operator on a Hilbert space g, and let A be an algebra of operators on g. The structured singular value of A (relative to A) is the number pa(A) = 1/inf{llxl] : x E A, - 1 e ~r(AX)}. This quantity was introduced by Doyle and Safonov [7, 13] under a more restrictive context, and it has proved to be a powerful tool in robust system *This work was supported in part by grants from the Research Fund of Indiana University, by the National Science Foundation DMS-8811084, ECS-9122106, by the Air Force Office of Scientific Research AFOSR AF/F49620-94-1-00SSDEF, and by the Army Research Office DAAH04-94-G-0054 and DAAH04-93-G-0332.

12 analysis and design. In system analysis, the structured singular value gives a measure of robust stability with respect to certain perturbation measures. Unfortunately, pa(A) is very difficult to calculate, and in practice an upper bound for it is used. This upper bound is defined by

~A(A) : =

inf{iiXAX-ll[ : X ~ A ' , X

invertible},

where A ~ is the commutant of the algebra A. In [1, 6], we formulated a lifting technique for the study of the structured singular value. The basic idea is that ~a(A) can be shown to be equal to the structured singular value of an operator on a bigger Hilbert space. (In [1] this was done for finite dimensional Hilbert spaces, and then in [6] this was extended to the infinite dimensional case.) The problem with these results is that the size of the ampliation necessary to get ~a(A) equal to a structured singular value, was equal to the dimension of the underlying Hilbert space. Hence in the infinite dimensional case we needed an infinite ampliation. In this work, we will show that in fact, one can always get by with a finite lifting. (Note that in this paper we will be using the terms "ampliation" and "lifting" interchangeably.) For the block diagonal algebras of interest in robust control, the ampliation only depends on the number of blocks of the given perturbation structure. (See Theorem 2 below.) We moreover, give a new result when ~a(A) = #a(A), that is, when no lifting is necessary and so ~a(A) gives a nonconservative measure of robustness. (See Theorem 3.) This is then used to derive an elegant result of Shamma [14, 15] on Toeplitz operators. See also [8, 10, 11] for related work in this area. Complete details of the results in this note may be found in [5].

2

Background Results

Denote by L:(£) the algebra of all bounded linear operators on the (complex, separable) Hilbert space g. Fix an operator A E £(£) and a subalgebra A C £(g). The quantities I-tzx(A) and "fia(A) have already been defined in the Introduction. Note that A C A" and A m = ( A ' y = A' so that we have the inequalities pa(A) 1.

L e m m a 5 ([4]) . Let TI be a Hilbert space, T E f~(7-l), and Dj E f~(7"l) invertible so that To =

lim

j--*oo

DiTD I.

If the set {Dj, Dj -1 : j = 1, 2,...} is contained in a finite dimensional subspace, then IITollsp = IITIIsp-

1l+

3

R e l a t i v e N u m e r i c a l Range

We will also need some results in what follows about the relative numerical range. Let 7 / b e a complex separable Hilbert space, and let £(7/) denote the set of bounded linear operators on 7/. Let T 1 , . . . , T m , Q E £(7/). Then we define the following relative numerical ranges: lim , h n))j=l: m WQ(T1 , . . . , Tin) := {A E C '~, A = n_,oo((Tjh,~ h~ E 7/, llh~ll = 1, lim HQh,~[I = 0}, and W~(T1 , . . . ,

Tin) := {A e C n, A = l i m ((Tjhn, h n ) ) rjn= l :

h , e 7/, I[hnll = 1, lim IIQhnl[ = O, hn --* 0 weakly}. n "-~ O0

L e m m a 6 ([5]) W ~ ( T 1 , . . . , Tin) is a compact convex subset of C m. L e m m a 7 ([5]) Wo.(T1,... , Tin) is the union of all segments {0A + ( 1 - 0)#: 0 < 0 < 1}, where A E W ~ ( T 1 , . . . , T,~) and It = ((Tjh, h))~= 1 for some h E ker Q, IIhll = t.

C o r o l l a r y 1 ([51) For all T, Q • £(7/), the set WQ(T) = {A = lim (Thn, h,~): hn E H , IIh~ll -n---+ O0

1,

lim

n - - + O0

IIQhnll --

0}

is a compact convex set.

Finally, for the proof of our lifting theorem (to be given in Section 4), we will need the following elementary fact: L e m m a 8 ([51) Let Z denote a finite dimensional normed space, and let S be a set of linear functionals on Z. Suppose that for every z E Z there exists a sequence ~, E S such that limn--+ooin(Z) = O. Then there exists a sequence £n in the convex hull o r s such that lirn,-.oo Ilgnll--- 0.

4

Liftings of Perturbations

In this section, we will formulate and prove a new lifting result relating Itzx(A) and ~,~ (A). For finite dimensional E, a lifting result of this type was first proven in [1]. The result was then generalized to the infinite dimensional ease in [6]. (For another proof of this type of lifting result in finite dimensions, see [8].) In these theorems, the lifting or ampliation of the operator A and perturbation structure A depends on the dimension of $. Thus if $ is infinite dimensional, we get an infinite lifting. In the new result proven below, we only have to lift up

15 to the dimension of A' which in the cases of interest in the control applications of this theory only depends on the number of blocks of the given perturbation structure. We now state the original Lifting Theorem which turns out to be very useful in analyzing the structured singular value. In the finite dimensional case, i.e., g being finite-dimensional, it was employed in [1] to show that #zx(A) = ~s,(A) when the relevant diagonal algebra has three or fewer blocks, and so gave an alternative proof of a result due to Doyle [7]. We therefore briefly review the set-up of [1, 6]. Let H S ( g ) denote the space of all Hilbert-Schmidt operators on g equipped with the Hilbert space structure

(T1, T2) := Tr(T~T, ), where T r denotes the trace. Define the operator LA : H S ( g ) ~ H S ( g ) by LA := AX. Now we set pa(A) := IJA(LA) where

£ := {Lx : x ~ A'}'. In what follows, we will assume that A ~ is finite dimensional *-algebra, but that g is arbitrary. We can now state the following lifting result: T h e o r e m 1 ([1, 6]) Let A' be a finite dimensional *-algebra. Then ~A(A) := p a ( A ) . We can now state our new lifting theorem: T h e o r e m 2 Assume that/V is a .-algebra of finite dimension n. Then ~A(A) =/.tA-(A ('~))

for every A E L(£). P r o o f . The argument starts as in the proof of Theorem 3 in [1], and of Theorem 1 of [6]. Without loss of generality, we may assume that ~zx (A) = 1. We must show that #A,,(A(n)) > 1. Choose a sequence of invertible operators Xj E A' such that ItXjAX~111 --+ ~a(A). Since X j A X ~ -1 belongs to the finite dimensional space generated by A'AA ~, we may assume that the sequence X j A X ~ 1 converges to some operator A0 such that IIAoH = 1. Obviously IIXAoX -ill > I]A0]] for every invertible operator X E A'. In particular we have

I1(I -

X)Ao(I + x + x 2 + . . )11 > 1

for X E A' with IIXll < 1. Fix an operator X E A' and a sequence ej > 0 converging to zero. There exist vectors hj E £ with lih¢ll = 1, such that I1(I-

2 2 ejX)Ao(I + ejX + cjX +...)hill 2

2

> 1 - ej.

16 This can be rewritten as,

(A~Aohj, hi) + 2¢j~(A~(AoX - XAo)hj, hi) + 0(¢~) >_ 1 - ¢j2 or equivalently,

2 Q ~ ( A ; ( A o X - XAo)hj, hi) + O(c~) >_ ((I - A~Ao)hj, hi) - ¢~ >_ - ~ . Dividing by ¢j and letting Q -~ 0 as j --* oo, we see from the last equation that

((I - A~Ao)hj, hi) lim i n f ~ ( A ~ ( A o X - XAo)hj, hi)

--~ 0, > 0.

(1) (2)

3.-.*00

We easily conclude that lim i n f ~ ( ( X - A~XAo)hj, hi) >_ O. 3--.+oo

(3)

Set

Q = I - A~Ao, T = X - A~XAo. Then from (1,3), we see that

Qhj ---+O, lira infN(Thj, hi) >_ O. j --* co

(4)

Applying the above argument to ( X for any ¢ E 0D (the unit circle), we IIh Oll = 1 such that

see that there exists a sequence

Qh~ ¢) -+ O, l i m i n f ~ ¢ ( T h j , hi) >_ O.

(5)

We claim that 0 E WQ,o(T). Indeed, if this were not the case, Corollary 1 would imply the existence of ¢ E 0 D such that lim i n f ~ ¢ ( T h j , hi) < 0 3---+oo

for all sequences of unit vectors hj such that Qhj ---+0 contradicting (5). Thus, we have shown that for each X E A ~, there exists a sequence of unit vectors hj E E such that

(I - A;Ao)hj -+ O, and ( ( Z - A~ZAo)hj, hi) -+ O.

(6)

Let A's~ := { x = x * : x E A ' } .

Consider now a subspace D C A~a of real dimension n - 1 such that A'sa = D T R I . Set Z = { X - A ~ X A o : X E D}, and for every unit vector h E C define a linear functional t(h) on Z by £(h)(T) = (Th, h), T E Z. Then L e m m a 8 applied to the set Sk = { / ( h ) : H(I - AGAo)hll _ O, ~ j = and the h~k) are unit vectors in C, such that II(I - A;A0)h~k)ll < 1/k. Let us define unit vectors vectors uk E E ('~) by uk=

~n~=i~Ca(k)~l/~h(k) ~ J i '

and observe that lim~--,oo ( ( X ( " ) Taking X = Y*Y we obtain lim

k--* oo

(llY(")A~")ukll-

(7)

A*o('OX('~)A(n))uk, uk) = 0 for every X e A'.

IIV(")ukll)

= 0

(8)

for every Y E At. Consider now the spaces 7/~ = AInA~n)uk and /Ck = Anuk. ' Lemma 3 implies that, by passing to appropriate subsequences, we may assume that all the representations X --* x(n){7/k (resp. X --* X(n){ICi) are unitarily equivalent. It follows that we can find partial isometries Ok, Vk in A~ such that UkT/k = 9/1 and Vk/Ck =/C1. Dropping again to appropriate subsequences, we may assume that the limits u = limk--.oo UkA~'~)uk and v = limk~oo Vkuk exist. Then (8) implies that

IlV(-)~ll = llY(-)~ll for every Y E A~. Therefore there exists a partial isometry W E A~ such that

Wy(n)u = y(n)v for every Y E A t. Of course, W can be chosen equal to zero on the orthogonal complement of A~u and thus to have finite rank at most n. The partial isometries Rk := Vk*WUk are in A~, they have uniformly bounded rank, and lim

(RkA(o'0

-

I)uk

=

k ---* o o

lim

V~(WUkA~n)uk

-- Vkuk)

k -.-* o o

Therefore L e m m a 4 implies that

liminf[[RkA(on)[]~p >_ 1. k -.~, o o

Finally, since

Rk commutes with X(n), X E At, and we have (-)

, a(~)v(~)

-1

in norm as j --, oo. L e m m a 5 shows that IIRkAgn)[[sp = HRkA(n)Hsp.

= O.

18 Consequently, we have liminftlR~A(')llspk~¢~ = likm~fllRkA~n)ll > - 1. Thus,

pa~(A (n)) > liminf[[A(n)Zkllsp > 1 = ~ a ( A ) , --

k ---+ o o

which completes the proof of the theorem. [] R e m a r k . In the cases of interest in control, A II = A, and so one has from Theorem 2 that tt~, (A) = pA(A).

5

C o n d i t i o n s for # =/2

In this section, we will discuss some new conditions when # =/~ without any need for lifting. In the finite dimensional case, there have been some results of this kind, the most famous of which is that of Doyle [7], who showed that no lifting is necessary for perturbation structures with three or fewer blocks. We begin by noting that in the proof of Theorem 2, we established a useful property of the critical operators A0 in the closed A' similarity orbit

Oa,(A) = { X A X -1 : X • A'} of A. Namely, if we call critical any A0 E Oz~,(A) satisfying limsupll(l

-

cX)do(l

-

ex)- ll > llAoll, VX • ±',

then the first part of the proof of Theorem 1 establishes the following: L e m m a 9 If Ao is a critical operator in OA,(A), then it enjoys the following property (0): 0 • WQ(IIAo[I2X - A~XAo), X • A',

where Q =

IlAol]2I -

A;Ao.

Indeed, property (O) is a reformulation of equation (6) in the case in which the norm of A0 may be different from 1. The next lemma is the key step in adapting the proof of Theorem 2 in order to show that #a(A) = ~a(A) in several interesting cases.

19

L e m m a 10 Let Ao be an operator on E which satisfies the essential version of property (0), property (0°), namely

o E Wga(llAoll2X - A S X A o ) , X • ~X', where Q =

IIAol12I- A~Ao.

Then there exists a sequence {hk}~=l C E, {{h~ll =

such that

1, k = 1 , 2 , . . . ,

Qhk --+ 0 strongly and ((IIA011=X - A~XAo)hk, h~) --+ O, for all X • A I. P r o o f , W i t h o u t loss of generality we can assume t h a t IIA011 = 1. Let be an algebraic basis of A ~. (Note t h a t A ~ is finite dimensional.) A~ - A ~ X j A o , j = 1 , . . . , n. T h e n by virtue of L e m m a 6, I 4 ~ ( T 1 , . . . , vex and c o m p a c t . If 0 ~ Wffd(T1,..., T,~), there exists ¢ = ( ¢ 1 , . - - , and e > 0 such t h a t

X1,...

,X

n

Set ~ := T,~) is con¢~) E C '~

n

V)~ = ('~1,---, ~-n) ~ [/V~)(T1,-" j=l

.,Tn).

Set

T:

X: ~./T,.. ./=1

P r o p e r t y (O °) implies t h a t there exists a sequence {gk}k°°__l C E, llgkll = 1, gk --+ 0 weakly such t h a t (Tgk, gk) --+ O. W i t h o u t loss of generality (by passing to a subsequence if necessary), we can assume t h a t

(Tjg~, g~,) -~ ),j, (j = 1,..., ,), for k --~ oo. T h u s

= (A1, . . . , ~ ° ) e W~(T~,..., T~). Hence

n

0 ~- ~(Tg~, g~) = ~

n

¢./(Tjg~, g~) -~ ~ j=l

¢./~./ >_ c, j=l

a contradiction. We therefore conclude t h a t 0 E W~(T1,...,T,~), i.e., there exists a sequence {hk}~=l C E satisfying the properties IIh~li = 1, k = 1, 2,..., ]lQh~ll ~ O, hk --* 0 weakly, and

((Xj - A;X./Ao)hk, hk) = ( ~ h k , hk) --* 0 for all j = 1 , 2 , . . . , n. This implies t h a t

{(X - A~XAo)hk, hk) --~ O, for a l l X E A ~. D We can now state the second m a i n result of this paper:

20 T h e o r e m 3 If there exists a critical operator Ao satisfying property 0 ° in the closed A ~-orbit of A, then

I~Lx,,(A) = f*a (A). P r o o f . We only have to note that because of Lemma 10, in the proof of Theorem 1, we need not take direct sums. More precisely, referring to equation (7) in the proof of Theorem 1, we can take uk "-" hk, where 00

{hk}k=l is the sequence provided by Lemma 10. The proof then proceeds exactly as in Theorem 1 with A0 replacing A (n), X replacing X('~), and Y replacing y(n). [] R e m a r k . Under the hypotheses of Theorem 3, when A" -- A (which happens in all cases of interest in control), we have that #A(A) =/~a(A). Let L ( A I A A ~) denote the linear space generated by AIAA I = { X A Y : X, Y E A'}. Obviously L ( A ' A A ' ) is finite dimensional, and therefore closed. Hence OA, (A) C L(A~AA'). C o r o l l a r y 2 If for every B G L(A~AA~), B • O, the norm o r b is not attained (that is, there is no h E 7~ such that [IBhH = [ISlllIhll # 0), then

#Lx,, (A) = f ~ (A). Proof. The critical operator A0 constructed in the first part of the proof of Theorem 1 belongs to L(A~AA~), and therefore its norm is not attained. However in equation (6), we can assume that the sequence {hi}~°=l is weakly convergent, say hj ~ h weakly. Without loss of generality, we may assume that IIA011= 1. Then (6) shows that

(I - A ; A o ) h = O. Therefore if h # O, we would have

IIAohi[ 2 = {lhlf2 = [IAoll2flhll 2 # o, and so the norm of A0 would be attained. We conclude that hj --~ 0 weakly, and so A0 satisfies property (O°). The required result now follows by Theorem 3. D R e m a r k . Note that Corollary 2 applies only to infinite dimensional Hilbert spaces g.

21

6

LTI S y s t e m s

In this section, we want to use our lifting methodology in order to derive a beautiful result of Shamma [14, 15] on the structured singular value of a linear time invariant system, i.e., a Toeplitz operator. Accordingly, set g = H2(C n) and let A denote the multiplication (analytic Toeplitz) operator on g defined by

Izl < 1, h E g,

(Ah)(z) = A(z)h(z),

where a

A(z) = [

jk]j,k:l, n

Izl < 1,

has H °° entries. Let A ~ be any *-subalgebra o f / : ( C n ) , the elements of which are regarded as multiplication operators on g. Note that in this case, A " = A is the algebra generated by operators of the form (Bh)(z) = B(z)h(z),

with B ( z ) Z

= XS(z),

Izl < 1, h E g

[z I < 1, X E A' as well as of the form

B

hi h2 :

:

hn

Yhl Yh2 .

,

Yho

with Y E / : ( H ~ ( C ) ) arbitrary. We can now state: L e m m a 11 Let Ao be an analytic Toeplitz operator. erty ( 0 ) , it also has property ( 0 ° ) .

Then if Ao has prop-

P r o o f . Without loss of generality we may assume IIA011 = 1. Let X E A' and let hi, j = 1, 2 , . . . be a sequence of unit vectors satisfying [ [ ( I - A*oAo)hj[[ 2 --+ O,

((Z - A~XAo)hj,hj)

(9)

-+ O.

Note that since I - A~Ao > 0 the first condition in (9) is equivalent to ((I - A ~ A o ) h j , hi) --+ O.

Let U denote the canonical unilateral shift on • = H ~ ( c n ) , that is, ( U h ) ( z ) := z h ( z ) ,

]z I < 1 ,

h E g.

As is well-known, we can view H2(C ") as a subspace of L 2 ( c n ) . In particular, in this representation the relations (9) are equivalent to o2r(llhj(eit)ll 2 - I I A o ( e i t ) h j ( e i t ) n 2 ) d t

fo

~r[(Xhj(eit),hj(eit)) - (XAo(eit)hj(eit),hi(eit))]dt

--+

0

~

O.

(10)

22 Note that X is an n x n matrix with constant coefficients. Therefore in (10), hj can be replaced by Ukhj for any k > 0 without changing the values of the integrals. We infer that

II(I - A~Ao)UkJhjll 2 ~ O, ( ( X - A ~ X A o ) U k J h j , UkShj) --~ O, for any sequence {kj}~= 1 of natural numbers. Since for any g, h E g

[(Vk g, h)[ = [(g,U*kh)[ independent condition (see l e m m a 4 of [5]) k ~_~(_1) j dJ [)3¢ dtJ OUj -'~ 0 j=O

For example, both 4u and u(4)u (1) are independent of path; 4 2 is not. A t e r m of the form

with the yy being closed first order integral terms and ¢ being a closed function of the u's and the y's will be called a closed second order integral term. Continuing in this way,

will be said to be a closed k th order integral term if q5 is closed and each of the yi are closed integral terms of order k - 1. We characterize sudt terms as being "cohomological" for the following reason. A standard device for investigating the geometry of a differentiabte manifold is to look for differentials such as

that satisfy the integrability condition e~(~i

(~j

Orn5 = &h~ and yet can not be expressed as the derivative of a smooth flmction define everywhere on the manifold. These closed but not exact differentials form a real vector space and constitute the first de R h a m cohomology class of the manifold. If "y defim~ a closed p a t h in the manifold the differential can be integrated along the p a t h to get a number

Because of the condition on the partial derivatives, continuous deformations of the closed p a t h will not change p. We m a y think of p as being a period of

28

the differential associated with an equivalence class of closed paths. For our purposes it is convenient to divide dO by dt and use forms involving u and its derivatives. Here, as in [7], we are interested in situations where derivatives of higher order occur. In this case the above exactness condition is replaced by the condition involving the higher order Euler-Lagrange operator as indicated above. There are two new aspects associated with this type of expansion. Not only does the expansion involve both u and 4, but the expansion is only defined for those u~s having the property that u(t) and certain of its derivatives belong to a domain having particular topological features. In the situation just considered, we needed to ask that it2(t) + (u(t) - a) 2 never vanish. The line integral terms in these expansions capture effects that depend on the remote past. In the development given here, such forms are chosen to match certain structural properties of the differential equation. Having made a choice of these terms, the remainder of the expansion is uniquely determined. To use these techniques we must assume that u is differentiable and, even more, t h a t u and its time derivative always belong to some set where the closed form is well defined. However, by restricting the input space and then accounting for the effect of such terms separately we can significantly influence the domain of convergence.

3

First E x a m p l e s

We will develop expansiot~ that involve not only the input, but also its time derivatives. This is not standard but it is essential in our context. In order to build some intuition about this we point out how it can be useful even in the case of linear systems. E x a m p l e 1: Consider the linear system

~(t) = A x + Bu(t) Assume t h a t A is invertible and that u is differentiable. Make the change of variables z = x + A - l b u . Then z satisfies the equation

k(t) = Az(t) + A - 1 B i t ( t ) If A has its eigenvalues in the right half-plane and if u is slowly varying then z is small. In any case

z(t) = eAtz(O) +

cA(t-C~)A-1Bit(a)da

and so x is given by

x(t) = - A - l b u ( t ) + eAtx(o) + .

.eA(t-°) A-tbi~(~)&r

This shows that x can be expanded as a sum of a "slowly varying" t e r m - A - 1 B u ( t ) and a continuous functional of/~. Notice t h a t instead of making the change of variable we could have, in this case, arrived at the same result using integration by parts.

29

Our second example deals with tile highly nonlinear model mentioned above. It plays a role in our earlier papers [7,8] and, in the present context, allows us to introduce one of the important aspects of tile representation we seek to establish. E x a m p l e 2: Consider x(t) = - sin 27rx(t) + u(t) The system is invariant with respect to the change of variables x ~-, x + k if k is an integer. There is a countable infinity set of asymptotically stable equilibrium points located at the integer nmltiples of 2~r. Of course the behavior of this system is highly nonlinear but for certain types of inputs we can explain its behavior in qualitative terms. If lx(O)I is less ~han 1/4 and u is less than one then the system behaves more or less like tile linear system x = -2~rx + u. On the other hand, if Ix(0)l is small and u is a function t h a t is close to zero except for an occasional tall, narrow, positive pulse whose area is approximately 1 then x will stay near 0 until the first such pulse occurs, at which time it will move rapidly to a neighborhood of 1, stay near there until the next pulse occurs, at which time it will move to a neighborhood of 2, etc. It is i m p o r t a n t to point out that because of the asymptotic stability of the equilibria at integer values of x it is not necessary for the area under the pulses to be exactly 1. More specifically, it is quite possible for the difference between x(t) and the integral

I(t) =

//

u(e)da

to become large. In order to get an expansion for x valid for such inputs it is necessary to incorporate the effect of a pulse in such a way as to reflect the "quantization" of x caused by its attraction to the periodically spaced asymptotically stable equilibria. Simple integration of u does not do this and we must replace the integral of u by a functional t h a t counts pulses accurately. In [7,8] we discussed the characterization bf pulses in terms of the phase space representation of u. We noted t h a t if u is pul~-tike then in (~i, u) space the trajectory of a pulse avoids some disk centered on the positive half of the u axis.(See figure 2.) With this in mind, we introduce cylindrical coordinates in (it, u)-spaee, centered at it, u) = (0, a) u(t) = a + r(t) cos 0(t)

it(t) = r(t) sin O(t) these equations imply that r(t)

=

- a)2 +

and t h a t e(t) = t a n

it(t) - a

30 Thus ~)(t) = (/i(t)(u(t) - a) - / ~ 2 ( t ) ) ( ~ ( t ) + ~(t) - ~)2 T h e variable 0 does not necessarily decrease by exactly 27r when u executes a pulse.like excursion. Its decrease will be approximately 2~r but, and this is of crucial importance, after n pulses it will have decreased by an a m o u n t close to 2nTr and the difference between its decrease and 2~tTr does not grow with n. This m e t h o d of counting pulses is superior to integration of u for the purpose of determining their effect on the s y s t e m being discussed. To convert these remarks into mathematics, we proceed b y analogy with the approach used in example 1. Make a change of variable,

z(t) = ,v(t) + ~

O(t)dt

It is easy to see t h a t z satisfies the differential equation E(t) = - sin(27rz(t) - O(t)) + u(t) +

O(t)

E x p a n d i n g the sine function we can rewrite tiffs as :?(t) = - sin 27rz(t) cos O(t) - cos 21rz(t) sin O(t) + u(t) +

O(t)

This s y s t e m can be written as ~(t) = - sin 27rz(t) - sin 27rz(t)ul (t) - cos 27rz(t)u2(t) + ua(t) with

~ ( t ) = (-1 + cosO(t)) u2(t) = sin

o(t)

~3(t) = ~(t) + O(t) Of course

1//

x(t) = z(t) - 2-7 and so

O(t)dt

1Lt x(t) = -2-7 O(t)dt 4 V(~,~)

T h e Volterra series for z, as defined by the three input system, converges provided t h a t the L1 n o r m s of the input terms are sufficiently small. Of course Ilulll and IIu211 are small if ~t(t) is close to 0 most of the time. T h e situation with respect to ua is more complex. For some purposes, however, one can define a pulse train to be a function such t h a t u3 has a small L1 norm. Figure 2 shows how u and t} c o m p a r e for a Gaussian pulse.

31 4

/

0.6

-11.6

Figure 2: The phase space plot 5 versus u for u a Gaussian pulse and the corresponding plots of u(t) and 0(t).

4

P u l s e s , P e r i o d s and A s s o c i a t e d A u t o m a t a

Suppose x(t) and u(t) take on vahms in ]R~, and ]Rm, respectively, and that we are given a system of the fornl

x(t) = f ( x ( t ) ) + G ( x ( t ) ) u ( t ) Suppose, further, that f and (7, are real analytic and that f is periodic in the sense that f ( x + ei) = f ( x ) for each standard basis element of IR'~. We let 2~n denote the integer lattice in IRn Cmd let V " denote the set of unit vectors in IR'~. Assume that x = 0 is an asymptotically stable equilibrium point for f, and hence that f ( e i ) is also an ~tsymptotically stable equilibrium point for each ei E V n. With respect to G, we assume that the vector fields G ( x ) e i are such that for x0 E 2~n, the trajectory starting at x0 at t = 0 pa~sses through a point in 2~n at t = 1. SpecificMly, we assume t!lat, the solution of the differential equation

~(t) = C;(x(t))e, ; x(O) = p j e Z ~ passes through [Ti(Pj) E Z '~ at t = 1. Systems of this form will be said to be

control periodic. We will now associate an automaton with such systems. The state space of this automaton is E'~; its input space V'~; and its evolution is given by

with G(~?) being a matrix whose columns are the vectors gi(Pj) just introduced. We will call this the associated automaton. We now turn to the problem of specifying what we mean by a pulse. Consider the k + l-parameter family of scalar valued functions given by

u(a, t l , t 2 ,

tk)(t) = E

1---J--e-(t-t~)2/2° " Iti - ti+ll > a

If 0 < a : X x U --+ X the transition function generally a partial function. Every partition 7r of X, has an associated partition machine Ad ~ that will have X ~ as its state space. Given a partition 7r, and Xi E re, X i denotes the set X taken as an atomic entity (singleton). X ~ denotes the set { X i : Xi E re}. A* is the set of all finite sequences of elements of a set A. For c~,c~r E A*, the notation c~~ _< o- (respectively ~r~ < cQ means that cr~ is the initial segment (proper initial segment) of cr. I~1 denotes the length of or. D e f i n i t i o n 3. Given a machine Ad = (X, U, ~), a sequence cr = uo. •. uk E U* is called a legal c o n t r o l sequence (or simply, control sequence) of Ad, if there is x C X such that q)(x, u o . . . uz), 0 < l < k is defined. The state sequence generated by ~r is called a t r a j e c t o r y . T r a ( x , y) will denote the set of all possible

trajectories fro,, x to y. Co~(:~) c u* (respectively, T r a ( ~ ) c x * ) will denote the set of all possible finite control sequences(respectively, trajectories)

gAt.

[]

4O Given a partition 7r of X, the elements of T r a ( M ) fall into two categories: (1) those trajectories lying within a single block; (2) those trajectories lying in more than one block which can be decomposed into a sequence of subtrajectories, say t i t 2 . . , tin, such that each segment ti is either an internal or a direct trajectory as defined below. D e f i n i t i o n 4. Given M = (X, U, O) and rr E P A R ( X ) , a sequence of states t = x l . . . x l E T r a ( M ) is called an i n t e r n a l t r a j e c t o r y with respect to 7r if t E 2[* for some Xi @ 7r; t is called an direct t r a j e c t o r y from Xi to X j (Xi, X j E 7r, i # j ) if xl . . . . xl-1 E X [ and xl E X j . [] Tra(r)~ (respectively, Tra(~r)~ for i # j) shall denote the set of all internal (respectively, direct) trajectories of M with respect to 7r. A trajectory is specified entirely by its initial state and the control sequence applied to it; hence, there exists a mapping O: X x C o n ( M ) ~ T r a ( M ) : O(x,g) = xo . . . x v E T r a ( M ) , where cr = u o . . . u p - 1 E C o n ( M ) and x0 = x, xi = O ( x i - l , u i - 1 ) , 1 < i