Analysis and Optimization of Systems: Proceedings of the 9th International Conference, Antibes, June 12-15, 1990 (Lecture Notes in Control and Information Sciences) (English and French Edition)

Lecture Notes in Control and Information Sciences Edited by M.Thoma and A.Wyner

144 A. Bensoussan, J. L. Lions (Editors)

Analysis and Optimization of Systems Proceedings of the 9th International Conference Antibes, June 12-15, 1990

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong

Series Editors M. Thoma • A. Wyner

Advisory Board L. D. Davisson • A. G. J. MacFarlane • H. Kwakernaak J. L. Massey • Ya Z. Tsypkin • A. J. Viterbi

Editors A. Bensoussan INRIA - Universite Paris IX Dauphine J. L. Lions College de France - CNES, Paris INRIA Institut National de Recherche en Informatique et en Automatique Domaine de Voluceau, Rocquencourt, B.P.105 78153 Le Chesnay/France

ISBN 3-540-52630-7 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-52630-7 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin, Heidelberg 1990 Printed in Germany The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printing: Mercedes-Druck, Berlin Binding: B. Helm, Berlin 2161/3020-543210 Printed on acid-free paper.

FOREWORD

These Proceedings of the 9th series of the I n t e r n a t i o n a l Conference "Analysis and Optimisation of Systems" gather more than 100 papers covering b o t h Theory and Applications in the field of Systems Analysis and Control (1 invited paper - 96 contributed papers - 10 papers presented at the two Invited Sessions).

This year again, we have noticed countries represented among the This vitality shows the complete continues to progress and allows promising results and applications

a large international participation through 18 speakers. maturity of this discipline which, regularly, scientists and engineers to expect the most for the future.

We would like to express our thanks to the Organizations which have sponsored this Conference : AFCET, CNRS, IEEE-CSS, IFAC, IIASA, INSERM, SEE, SIAM and SMAI.

We also would like to extend our gratitude to : the authors who have shown a real interest in this Conference the many reviewers who have accepted the hard task of selecting papers the chairpersons for having run with efficiency all the sessions of the Conference all the members of the Organization Committee the Scientific Secretaries who deserve a special mention the staff of the Public Relations Department for their excellent work i n making this Conference really happen. Professor M. THOMA and the Editor SPRINGER VERLAG who have accepted to publish this series in the Lecture Notes in Control and Information Sciences.

A. BENSOUSSAN

J.L. LIONS

THIS CONFERENCE IS UNDER THE SPONSORSHIP OF : AFCET

Association Franqaise pour la Cybern6tique Economiquc et Technique

CNRS

Centre National de la Recherche

IEEE-CSS

The Institute of Electrical and Electronics Enginecrs, Inc. Control System Society "Participating Societies"

IFAC

International Federation of

IIASA

International Institute for Applied

Scientifique

Automatic

Control

Systems

Analysis

INSERM

Institut National de la Sant6 et de la Recherche M6dicale

SEE SIAM

Soci6t6 des Electriciens et Electroniciens Society for Industrial and Applied Mathematics

SMAI

Socidt6 pour les Mathdmatiques Appliqu6es et Industrielles

ORGANIZATION COMMITTEE K.J. A. A. P. P. A. A. I.D. J.L. M. R. J.C.

ASTR6M BENSOUSSAN BENVENISTE BERNHARD FAURRE FOSSARD ISIDORI LANDAU LIONS THOMA VINTER WILLEMS

Lund Institute of Technology INRIA Universit6 Paris-Dauphine INRIA - IRISA - Rennes INRIA - Sophia-Antipolis SAGEM, Paris ENSAE, Toulouse Universita di Roma I.N.P.G., Grenoble Cotl6ge de France/CNES, Paris Technische Universit~t, Hannover Imperial College, London GrOningen U n i v e r s i t y

SCIENTIFIC SECRETARIES L. J.F. A.

BARATCHART BONNANS SULEM

INRIA INRIA INRIA

-

-

Sophia-Antipolis Rocqucncourt Rocquencourt

CONFERENCE SECRETARIAT Th. S. E.

BRICHETEAU GOSSET MANY

INRIA-France INRIA-France INRIA-France

AUTHORS

ABU EL ATA S AKIAN M. ANANTHARAM V. ANTOULAS A.C. AUBIN J.P. BAIRD C.R. BARATCHART L. BARBOT J.P. BARDI M. BEHAEGEL E. BENSOUSSAN D. BERGER W.A. BLANCHON G. BONGERS P.M.M. BONILLA E.M. BONNAN$ J.F. BOSE A.K. BRDYS M.A. BROGLIATO B. BULGAKOV A. YA. BYRNES C.I.

851 113 937 297 821 467 477 314 103 145 873 341 423 307 279 423 632 674 798 95 821-861

CANPONT D. CARVALHO M. TEIXEIRA. M. CHA I T.Y. CHANGYOU L. CHANG SHI-CHUNG CHANTRE P. CHAVENT G. CHEN CHUN-HUNG CHERNOUSKO F.L. COIC A. COMMAULT C.

37 900 684 985 133 851 452 133 580 85 t 361

DESCUSSE J. DE KEYSER R.M.C. DE LUCA A. DELFINI P. DI BENEDETrO M.D. DION J.M. DODU J.C. DONG CAO DULUC R.

288 77 833 623 843 361 423 755 549

EL-ANSARY M.

890

FALCONE M. FAYAZ M.A. FELIACHI A. FEVO"I'TE G. FLANDOLI F. FLIESS M. FLORCHINGER P . FONG I-KONG FRANKOWSKA H. GABASOV R. GAUBERT S. GAUVR1T M.

I 03 611 208 37 694 778-851 2 2 8 - 258 133 519 570 957 912

INDEX

GEORGES D. GOMES G. GOUZE J.L. GRIZZLE J.W. GRORUD A. GUGLIELMI M.

600 912 324 401-843 704 497

HAMAM Y. HAMMOURI H. HAN CHI-GEUN HANUS R. HARCAUT J.P HEJMO W. HEUBERGER P.S.C. HOANG NGOC MINH V. INSTALLE M. IOFFE A. ISIDORI A.

600 391 413 664 27 714 307 381 755 442 821-861

JACOB G. JODAR L.

381 87

KAMEN E.W. KARATZAS i. KEANE M.A. KINNAERT M. KIRILLOVA F.M. KOKOTOVIC P.V. KOWALEWSKI A. KOZA J.R. KULC"ZYCKIP.

655 3 47 880 570 861 174 47 714

LAKRORI M. LALL H.S. LAMNABHI-LAGARR1GUEF. LANARI L. LEBRET G. LEHOCZKY J.P. LEPSCH': A. LE GLAND F. LESCH K. LOBRY C. LOISEAU J.J.

37 123 611 833 37 1 3 218 228 590 623 37 1

MALABRE M. MARCHETTI C. MESSAGER F. MIAN G.A. MILHEIRO de OLIVEIRA P. MONACO S. MORAAL P.E. MORDUKHOVICH B.S. MYSLINSKI A.

279 187 778 218 198 788 401 539 164

NAVARRO E. NIKOUKHAH R. NORMAND-CYROT D.

87 67 788

OLIVI M. OTHMAN S.

477 391

OUSTALOUP A. OZVEREN C.M.

767 947

PARDALOS P.M, PENG S. PENG Y. PERCELL P.B. PEREZ A. PERRY R.J. POMET J.B. POMMARET J.F. POURTALLIER O. PRALY L.

413. 724 664 123 361 341 808 271 745 808

RENEKE J.A. ROCHA P. RUBIO J.E. RUNCHAL A.K.

632 332 154 145

SACHS G. SASTRY S.S. SHREVE S.E. SPIZZICHINO F.

590 861 3 238

TAKEUCHI Y. TALAY D. TAPIERO C.S. THEPOT J. TSIOTRAS G.

248 704 967 735 967

ULANICKI B. ULIVI G. URYAS'EV S.P.

674 833 432

VIARO U. VIGNERON C. V1NTER R.B.

218 549 529

WAGNEUR E. WAHNON E. WAYNE BEQUETTE B. WANG D. WANG P,K.C. WENBIN LIU WIELONSKY F. WILLEMS J.C. WILLSKY A.S. XU GAN-L1N XU HONG YAO.

977 487 57 600 145 154 477 297-332 947 3 467

YANG WE1LIN YE YINYU YONG J1ONGMIN YOU YUNCHENG

133-351-924 413 559 642

ZHANG SIY1NG ZI-QIANG LANG

351-924 507

TABLE OF CONTENTS

INVITED

CONFERENCE

Optimality conditions for utility maximization in an incomplete market I. K a r a t z a s , J.P. Lehoezky, S.E. Shreve, Gan-Lin

NUMERICAL

Xu

........................

ALGORITHMS

Nonlinear control of missiles through a geometric approach J.P. H a r c a u t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

Non linear control of a batch evaporative crystallization using an algorithm of "L/A" type G. Fevotte, D. Canpont, M. Lakrori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

Genetic breeding of non-linear optimal control strategies for broom balancing J.R. Koza, M.A. Keane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

Process control using nonlinear programming techniques B. W a y n e Bequette . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

Construction of autonomous boundary-value linear systems from acausal input-output functions R. Nikoukhah ...............................................................................

67

PC-TACT : personal computer tool for advanced control techniques R.M.C. De Keyser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

Solving singular regular systems A2X" + A1X' + AoX = F(t) without increasing the dimension of the problem L. J o d a r , E. Navarro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

Matrix spectrum dichotomy and generalized Lyapunov matrix cquation A. Ya. Bulgakov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

Discrete approximation of the minimal time function for systems with regular optimal trajectories M. Bardi, M. Faicone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103

Analyse de l'algorithme multigrille FMGH de rdsolution d'6quations d'Hamilton-Jacobi-B ellman M. Akian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113

A dynamic programming based gas pipeline optimizer H.S. Lali, P.B. Percell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123

Two effective approaches for hydroelectric generation scheduling

Shi-Chung I-Kong

Chang,

Fong,

Weilin

Chun-Hung

Yang, Chen

........................ . ..............................

133

VII

CONTROL OF DISTRIBUTED P A R A M E T E R S Y S T E M S Iterative solution of a free-boundary problem arising in microscopic particle manipulation inside a liquid layer P.K.C. W a n g , E. Behaegei, A.K. R u n c h a l .......................................... 145 Optimality conditions for elliptic variational inequalities Liu Wenbin, J.E. Rubio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

154

Minimax shape optimization problem for Von Karman system A. M y s l i n s k i ..............................................................................

164

Optimality conditions for a parabolic time delay system A. Kowalewski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

174

LINEAR

AND

NONLINEAR

FILTERING

A note about singular perturbations in the filtering of a Markov chain C. M a r e h e t t i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

187

Filtres approch6s pour un probl~me de filtrage non lindaire discret avec petit bruit d'obscrvation P. M i l h e i r o De Oliveira . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

198

;.................

Decentralized filtering for interconnected systems A. F e l i a c h i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

208

Split forms of z-domain algorithms for linear prediction and stability analysis A. Lepschy, G.A. Mian, U. Viaro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

218

Time-discretization of the Zakai equation for diffusion processes observed in correlated noise P. F l o r c h i n g e r , F. Le Gland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

228

Finite-dimensional stochastic filtering in discrete time : the role of convolution semigroups F. S p i z z i c h i n o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

238

On the decompositions of observations with non-Gaussian additive noise and their innovations processes Y. T a k e u c h i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

248

Nonlinear filtering with dependent noises. The case of unbounded coefficients P. F l o r c h i n g e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

258

VIII A L G E B R A I C AND

GEOMETRIC S Y S T E M T H E O R Y

Group theory and controllability of partial differential control systems J.F. Pommaret ..............................................................................

271

Algebraic characterization of invariant zeros at infinity for generalized systems E.M. Bonilla, M. Malabre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 The dynamic block decoupling problem : a minimal solution by precompensation J. Descusse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

288

Minimal rational interpolation and Prony's method A.C. Antoulas, J.C. Willems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

297

Discrete normalized coprime factorization P.M.M. Bongers, P.S.C. Heuberger

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

A forward accessibility algorithm for nonlinear discrete time systems J.P. B a r b o t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

314

Stabilisation globale de syst~mes non-lin6aires par un contr61e positif J.L. Gouz~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

324

Controllability of delay-differential systems P. Rocha, J.C. Wiilems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

332

Geometric approach to parametric sensitivity and gain suppression W.A. Berger, R.J. Perry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

341

A kind of nonlinear system and its reduction and structural control Weilin Yang, Siying Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

351

Rejet de perturbation dans les syst~mes structur6s C. Commault, J.M. Dion, A. Perez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

361

A new canonical form for descriptor systems with outputs J.J. Loiseau, G. Lebret . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Evaluation transform and symbolic calculus for nonlinear control systems V. Hoang Ngoc Minh, G. Jacob . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

381

Immersion in infinite dimension H. Hammouri, S. Othman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

391

On observers for smooth nonlinear digital systems J . W . Grizzle, P.E. Moraal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

401

NONLINEAR

PROGRAMMING

An interior-point algorithm for large-scale quadratic problems with box constraints P.M. P a r d a l o s , Yinyu Ye, Chi-Geun Han . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

413

IX Optimisation des r6seaux 61ectriques de grande taille G. Bianchon, J.C. Dodu, J.F. B o n n a n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

423

Adaptive variable metric methods for nondifferentiable optimization problems S.P. U r y a s ' e v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

432

Composite optimization : second order conditions, value functions and sensitivity A. Ioffe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

442

A new sufficient condition for the well-posedness of non-linear least square problems arising in identification and control G. Chavent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .; . . . . . . . . . . . .

452

SIGNAL PROCESSING An identification technique for adaptive systems in the case of poor excitation Hong Yao Xu, C.R. Baird . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

467

Asymptotic properties in rational L2-approximation L. Baratchart, M. Olivi~ F. Wielonsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 A min-max testing approach to failure detection and identification E. Wahnon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

487

On-line detection of minimal order for linear pieccwise stationary systems M. Guglielmi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

497

A simple method for identification of linear dynamics with hysteresis nonlinear input Lang Zi-Qiang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

507

DETERMINISTIC

CONTROL

A priori estimates for operational differential inclusions and necessary conditions for optimality H. F r a n k o w s k a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

519

Relaxed controls, for time delay systems R.B. Vinter ..................................................................................

529

Maximum principle for nonconvex finite difference control systems B.S. M o r d u k h o v i e h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

539

Functional R. Duluc,

viability constraints C. Vigneron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

549

Maximum principle of optimal controls for a nonsmooth semilinear evolution system J i o n g m i n Yong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

559

X New algorithms of solving extremal problems R. Gabasov, F.M. Kirillova .............................................................

570

Constrained controls in linear oscillating systems F.L. Chernousko ...........................................................................

580

Fuel savings by optimal aircraft cruise with singular and chattering control G. Sachs, K. Lesch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

590

Optimal path planning of manipulators with singular configurations, workspace and collision-free constraints D. Wang, D. Georges, Y. Hamam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

600

On singular output tracking in multivariable nonlinear systems M.A. F a y a z , F. L a m n a b h i - L a g a r r i g u e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

611

CONTROLLABILITY AND STABILIZATION DISTRIBUTED PARAMETER SYSTEMS

OF

Formal controllability and physical controllability of linear systems C. L o b r y , P. Deifini . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

623

Conditions yielding weak controllability for a class of linear hereditary systems J.A. Reneke, A.K. Bose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

632

Nonlinear exponential stabilization of Boussinesq equations Y u n e h e n g You . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

642

STOCHASTIC

AND

ADAPTIVE

CONTROL

Direct adaptive control in a state-space setting E.W. K a m e n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

655

Pole placement via generalized predictive control Y. Peng, R. H a n u s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

664

Separation principle in optimizing control of state-constrained dynamical systems under bounded uncertainty M.A. Brdys, B. Ulanicki ................................................................

674

Unification of some MIMO adaptive control algorithms and global convergence analysis T.Y. Chai .....................................................................................

684

Boundary control of a stochastic parabolic equation with nonsmooth final cost F. Flandoli ..................................................................................

694

XI Approximation of Lyapunov exponents of stochastic differential systems on compact manifolds A. Grorud, D. Talay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704 Synthesis of closed-loop system controlling a random object Hejmo, P. Kuiczycki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714

W.

Maximum principle for stochastic optimal control with non convex control domain S. Peng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

MULTIDECISION

CONTROL AND

724

GAMES

Nash vs Stackelberg strategies in a capital accumulation game J. Th~pot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

735

A two player dynamical game with imperfect information O. P o u r t a l l i e r .............................................................................

745

An interactive multiple criteria decision supporting tool with application to a simplified regional development problem Cao Dong, M. Installe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

755

CONTROL OF FINITE D I M E N S I O N A L S Y S T E M S CRONE control : principle synthesis, performances with non-linearities and robustness-input immunity dilemma A. O u s t a l o u p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vers une

767

stabilisation non-lingaire discontinue

M. Fliess, F. Messager . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

778

A combinatorial approach of the nonlinear sampling problem S. Monaco, D. N o r m a n d - C y r o t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

788

Non-linear MRAS in robots motion control B. Brogliato .................................................................................

798

Adaptive control of feedback equivalent systems J.B. Pomet, L. Praly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

808

NONLINEAR

ZERO DYNAMICS

AND

OUTPUT R E G U L A T I O N

Viability kernels, controlled invariance and zero dynamics for nonlinear systems J.P. Aubin, C.I. Byrnes, A. Isidori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

821

Output regulation of a flexible robot arm A. De Luca, L. Lanari, G. Ulivi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

833

An analysis of regularity conditions in nonlinear synthesis M.D. Di Benedetto, J.W. Grizzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

843

XII Discontinuous predictive control, inversion and singularities. Application to a heat exchanger M. Fliess, P. C h a n t r e , S. Abu El Ata, A. Coic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

851

The analysis of singularly perturbed zero dynamics A. I s i d o r i , P.V. Kokotovic, S.S. Sastry, C.I. Byrnes,

861

........................

ROB U S T N E S S Elargissement des objectifs de robustesse des syst~mes de commande D. B e n s o u s s a n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

873

Right half plane poles and zeros and robustness limitations in feedback systems M. Kinnaert .................................................................................

880

Stabilizing control of a singularly perturbed system driven by wide-band noises M. E I - A n s a r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

890

Control o f uncertain dynamical systems using strictly positive real systems M. Carvalho Minhoto Teixeira . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 900 Parameter robust control design based on parametric identification quality M. Gauvrit, G. Gomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

912

Rotation symmetric structure in control systems and related stability analysis Weilin Yang, Siying Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

924

DISCRETE EVENTS SYSTEMS A hydrodynamic limit for a lattice caricature of dynamic routing in circuit switched networks V. A n a n t h a r a m ............................................................................

937

Aggregation and multi-level control in discrete event dynamic systems C.M. O z v e r e n , A.S. Willsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

947

An algebraic method for optimizing resources in timed event graphs S. G a u b e r t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

957

WIP and CSP-1 quality control in a two-stages queue like production C.S. T a p i e r o , G. Tsiot~ras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

967

Modulo'fds and pseudomodules : 3 - the lattice structure problem E. Wagneur ..................................................................................

977

Modelling and analysing a class of flexible manufacturing system L. C h a n g y o u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

985

I N V I T E D CONFERENCE

OPTIMALITY CONDITIONS F O I l UTILITY MAXIMIZATION IN AN INCOMPLETE MAltKET T Ioannis Karatzas Columbia University New York, NY 10027 **

John P. Lehoczky** Steven E. Shreve *#

Gan-Lin Xu Carnegie Mellon University Pittsburgh, PA 15213

ABSTRACT The problem of maximizing expected utility of final wealth in an incomplete market is investigatsd~ The incomplete market is modelled by a bond and a finite number of stocks, the latter being driven by a d--dimensional Brownian motion. The coefficients of the bond and stock price processes are adapted to this Brownian motion, and the number of stocks is less than or equal to the dimension of the driving Brownian motion. It is shown that there is a way to "fictitiously" complete this market so that the optimal po~tfoho for the resulting completed market coincides with the optimal portfolio for the original incomplete maxket. A number of equivalent characterizations of the fictitious completion are given, and examples are provided. SECTION 1. INTj~.ODUCTION This paper utudies the problem of an agent who receives a deterministic initial capital, and must invest it in an incomplete market so as to maximize the expected utihty from wealth at a prespecified terminal time. The market consists of a bond and m stocks, and stock prices are driven by a d---dimensional Brownian motion. Incompleteness in this market arises when m is strictly smaller than d. The market coefficients, i.e., the interest rate, the rates of stock appreciation, and the stock volatihty coefficients, are random processes adapted to the full

Research supported by the National Science Foundation under Grant DMS--87-2307g. Research supported by the National Science Foundation under Grant DMS-87-02537. Vpresented at the gth Intsr'natwnal Conference on Analysis and Optimization of Systems (Antibes, June 1990), and at the MS[ Workshop on the Mathematical Theory o/Modern Financial Markets (Cornell University, July 1989).

d---dimensionat Brownian motion. When m < d, it is impossible to construct a portfolio consisting of the bond and the m available stocks so as to completely hedge the risk associated with these coefficient processes. In Sections 2 through 5, we define the utility maximization problem faced by the agent. In Section 6 we present the solution when the market is complete (m = d), and complete hedging is possible. This solution proceeds in three steps. First, on the underlying probability space one determines a new measure P* which discounts the growth inherent in the market; under this measure, the expected value of the final wealth attained by any "reasonable" portfolio is equal to the initial capital. Secondly, among all random variables whose expectation under the new measure is equal to the initial endowment, a most desirable one is determined. Thirdly, it is shown that a portfolio can be constructed which obtains this most desirable random variable as terminal wealth; this portfolio is optimal. A complete market is one in which the agent can construct a portfolio which attains as terminal wealth any random variable whose P * - expectation is equal to the initial capital. Because such a construction is possible, it is said that the agent can hedge against the risk associated with this market. Mathematically, the construction of a portfolio uses the fact that any martingale with respect to a Brownian filtration can be represented as a stochastic integral with respect to the Brownian motion; the integrand in this representation leads directly to the portfolio that we are seeking, if m=d. However, if there are fewer than d stocks, (m < d) this line of argument fails. In Section 7 we introduce a convenient way of thinking about an incomplete market: fictitious completion. When m < d, then one introduces new, fictitious stocks so as to create a complete market. If these fictitious stocks have a high appreciation rate, then under an optimal portfolio the agent wiU hold a long position in them, but if they have a low (even negative) appreciation rate, then he will hold a short position. Thus one would expect to be able to adjust the appreciation rates of the fictitious stocks so that the agent, by optimal choice, does not invest in them at all. These judiciously chosen fictitious stocks allow us to write down the complete market solution for the utility maximization problem, but they are superfluous in the actual implementation of the optimal portfolio which must then also be optimal for the original incomplete market. The fictitious completion with the above property is the least favorable to the agent, because the portfolio which is optimal under this completion is available to him under every other fictitious completion. We thus have the notion of a ~inimax fictitious completion: for every fictitious completion we compute the portfolio which maximizes the expected utility of final wealth, and then choose the completion that makes this maximum expected utility as small as possible. As explained in Section 7, a convenient way to parametrize fictitious completions of an incomplete market is by a certain space of continuous local martingales, each of them being the Radon-Nikodym derivative process of the new measure P*, mentioned in the discussion of complete markets. One would like to be able to characterize the local martingale corresponding to the minimax fictitious completion, and to show that it gives rise to an optimal portfolio in the original incomplete problem. This program is carried out in Section 8. Section 9 provides two examples in which the minimax fictitious completion can be computed fairly explicitly. In the first example, the utility function is logarithmic, and it is discovered that the fictitious stocks in the minimax completion should have rates of appreciation

equal to the interest rate of the bond. This is a very general result, insensitive to the nature of the dependence of market coefficients on the driving Brownian motion. In the second example, it is assumed that the utility function is of the power form, and that the driving Brownian motion splits into two independent parts; the first part drives the stock price processes, whose coefficients are adapted solely to the second part. The minimax local martingale is exhibited as the solution to a martingale representation problem, and the optimal portfolio is found to be given by the formula already known to be correct for deterministic model coefficients. Our model for the financial market can be traced back to Merton (1969, 1971) and Samuelson (1969). The modern mathematical approach to portfolio management in complete markets, built around the ideas of equivalent martingale measures and the creation of portfolios from martingale representation theorems, began with Harrison & Kreps (1979) and was further developed by Harrison & Pliska (1981, 1983), in the context of option pricing. Phska (1986), Cox & Huang (1987a, 1987b) and Karatzas, Lehoczky & Shreve (1987) adapted the martingale ideas to problems of utility maximization. Much of this development appears in Section 5.8 of Karatzas & Shreve (1988), from which Section 6 of the present paper is drawn; see also the review article of Karatzas (1989) for a survey of financial economics problems in complete markets. A first step toward a martingale analysis of incomplete markets was taken by Pages (1987), who considered a Brownian model in which the number of stocks was strictly less than the dimension of the driving Brownian motion. However, the coefficients of the bond and stock prices in this model were allowed to depend on the underlying Brownian model only through the bond and stock prices themselves. Thus, the vector of bond and stock prices formed a Markov process. This specialization created an essentially complete market, and thus it avoided the more interesting case of a market with genuinely unhedgeable risk. However, Pages did characterize the class of equivalent martingale measures which could arise in an incomplete model, and this laid the groundwork for further developments. A more substantive step was taken by He & Pearson (1988) in a discrete-time, finite probability space model, where the authors proposed finding the optimal intermediate consumption and terminal wealth corresponding to each of the equivalent martingale measures, and then searching over those policies to find a pair yielding the minimum expected total utility. Using separating hyperplane arguments, they were able to show that the total utility obtained by this two-step "minimax" process is the optimal total value for the incomplete problem. He & Pearson have also studied the incomplete problem in a continuous-time, Brownian model. In an early version of He & Pearson (1989), the authors consider Pag~s's characterization of the family of equivalent martingales measures and search over this family for a "minimax" equivalent martingale measure which would lead them to the optimal consumption and portfolio processes just as in a complete market. The martingale associated with this measure would be the "Arrow-Debreu" state prices in the incomplete model. However, the continuous-time model is more subtle than one might expect, and although it is now clear that Azrow-Debreu state prices exist for the incomplete model under some assumptions, it is not clear that they are associated with a martingale. The present paper uses local martingales rather than martingales to address the issue of market incompleteness in continuous-time models. This work was motivated by the aforementioned previous version of He & Pearson (1989), and by the use of local martingale

methods introduced by Xu (1990) and Xu & Shreve (1990) in the study of incompleteness induced by a prohibition on the short--selling of stocks. Using the stochastic duality theory of Bismut (1973), Xu formulated a dual problem whose solution could be shown to exist and could then be used to obtain existence and characterization of the solution of the original problem. As this paper shows, such duality methods can also be used in the traditional incomplete Brownian market model. While we still do not know if the minimax equivalent martingale measure sought by He & Pearson exists in any generality, we show here that the solution to Bismut's dual problem is a "least favorable local martingale" which can be used to generate a sequence of equivalent measures. The existence of this least favorable local martingale is sufficient for the study of many models. A notable exception is the incomplete model in which the agent's endowment is a stochastic process; we do not know how to obtain existence and characterization of the optimal policy for such a model in terms of a least favorable local martingale, unless it is actually a martingale. tie & Pearson (1989) have incorporated Xu's local martingale techniques into their original work. He & Pearson (1989) report the existence of an optimal portfolio for the terminal wealth utility maximization problem when the index of relative risk aversion is everywhere greater than or equal to one, and they report similar results for the problem with intermediate consumption and consumption at the terminal time when the index of relative risk aversion is everywhere less than or equal to one. Our paper deals only with the case of terminal wealth utility maximization when the index of relative risk aversion is everywhere less than or equal to one; the generalization to also allow for intermediate consumption is straight-forward. Whereas He & Pearson (1989) assume that some augmentation of the market model will result in Markov prices, we allow general It5 price processes. He & Pearson (1989) do not provide the full set of equivalent conditions contained in our Theorem 8.4, nor do they use our assumption (8.2), which plays a fundamental role in our proof of Theorem 8.4. For a fuller account of this theory, and for proofs of the results which are not included hcre, we refer the interested reader to the paper Karatzas, Lehoczky, Shreve & Xu (1989). SECTION 2. 3?tIE MARKET MODEL We assume a model for the financial market consisting of one bond with price P0(t) given by (2.1)

dP0(t) = r(t)P0(t)dt,

P0(0) -- 1,

and m stocks with prices-per--share Pi(t), i = 1,...,m, satisfying the equations

(2.2)

d dPi(t) = Pi(t)[bi(t)dt + Z aij(t)dW(J)(t)],

j=1

i = 1,...,m.

ttere W = r~(W~U,...,W~dj)* _ t~ is a d-dimensional Brownian motion on a probabihty space ( f l , J , P ) , and we denote by {.~t} the P-augmentation of the filtration generated by W. We suppose that d > m, i.e., the number of stocks does not exceed the number of sources of uncertainty in the model.

7 The interest rate r(t), the vector bit ) = (bl(t),...,b~(t)) of stock appreciation rates, and the volatility matrix act ) = {crij(t)}l< m are the coefficients of the model. They are assumed to l< j< d be progressively measurable with respect to {~'t}, and it is also assumed that r(.) and o(.) are bounded uniformly in (t,w), and that /*

(2.3)

T

[ J

J]b(t)J[dt 0, let X x'~r denote the wealth process corresponding to a portfolio ~r defined by Xxjr(0) = x and (3.2)

dXX'~r(t) = r(t)xX'Tr(t)dt + xX'~r(t)Tr*(t)[(b(t) - r(t)l)dt + a(t)dW(t)]

= r(t)Xx3r(t)dt + xx'~r(t)~r*(t)a(t)dW0(t). In other words,

(3.3)

~rCs)a(s)dWo(s)+ 0

I t ( s ) - 8 9 I1~ * (s)~s)ll2]ds} '

SECTION 4. UTILITYFUNCTIONS We introduce a utility function U: (0,~) -~ ~, which is strictly increasing, strictly concave, twice continuously differentiable, and satisfies (4.1)

U'(0)-A l i m U ' ( x ) = | x~ 0

U'(|

limU'(x)=0. x-~ |

We denote by I: (0,| -~ (0,~) the (continuous, strictly decreasing) inverse of U ' , and consistent with (4.1) we note that (4.2)

I(~) A lira I(y) = 0, I(0) ~ lira I(y) = ~. y-~| y~ 0

We introduce the Legendre transform of U: (4.3)

Off) _Am i n [xy - U(x)] = yI(y) - U(I(y)), x>0

0 < y < ~.

The function lJ is strictly increasing, strictly concave and satisfies (4.4)

U'(y) = I(y),

0 < y < ~.

It is easily verified that (4.5)

U ( x ) = m i n { x y - lJ(y)} = x U ' ( x ) - 0 ( U ' ( x ) ) ,

y>0 From (4.3) and (4.5) we have the useful inequalities

0 < x < ~.

9 (4.6) (4.T)

u(I(y)) _> u(~) + y[I(y) - x], v x > 0, y > 0, 0 ( u ' ( x ) ) _> 0(y) + x[U' (x) - y], v x > 0, y > 0.

SECTION 5. THE UTILITY MAXIMIZATION P R O B L E M The problem at hand is, for a given initial wealth x > 0, to choose a portfolio a" that maximizes E u ( x X ' r ( T ) ) . If U(0) > - | this quantity is defined (but may be | for all portfolios r, whereas it may be undefined if U(0) = - | and U(co) = |

Thus, we define ,/#(x) to be the set

of portfolios ~r for which E max{0,-u(xX'Tr(T))) < |

(5.1) and set

V(x) =a

(5.2)

sup n~(x)

E u(xX'rr(T)).

We assume throughout that

(5.3)

v(,:) < |

v x > 0.

A portfolio process rr which attains the supremum in (5.2) is called optimal. In Section 8 we give conditions which ensure the existence of an optimal portfolio, and provide several characterizations for it. SECTION 6. THE COMPLETE M A R K E T SOLUTION The market model of Section 2 is complete if and only if a(t) is invertible for all t, i.e., d = m. Under this condition, the solution of the utility maximization problem has a simple solution, which we present below. Define the nonnegative local martingale

(6.1)

it , 1 I t z0(t ) =A e~p { - 0 O (s)dW(s) - , ~ 0 IIO(s)ll2ds},

a~d note from (3.3) that for any portfolio process ~r, flCt)Z0(t)xX'~(t) = x e~p { f t

( / C s) ~(s) - 0* (s))dWCs)- 21 f t

0

0

II~ . (s)~Cs) - 0(s)ll 2 ds}

is a nonnegative local martingale, hence a supcrmartingale. It follows that

10 (8.2)

E [Z(T) Z0(T ) xX'r(T)] 0,

so we may define a function ~o: (0,| -~ (0,~) by (6.4)

~o(Y) = E[~(T)Z0(T)I(y~(T)Z0(T))].

The function ~$0 inherits from I the property of being a continuous, strictlydecreasing mapping

of (0,| onto itself, and so ~'0 has a (continuous, strictly decreasing) inverse ]/o from (0,| itself. We define (6.5)

onto

~ a= I(]/o(x)fl(T)Z0(T)).

6.1 LSMMA. The random variable ~ satisfies (6.6)

E[fl(T)Z0(T)~] = x,

(6.7)

E ma~{0,-U(~)}
0

by analogy with (6.4). If

(7.7)

~V(y) < |

we may define ~ v to be the inverse of ~ ,

v y > 0,

and set

~x =~ I(~/~(x)fl(T)Zv(T))

(7.8)

by analogy with (6.5) If the fictitious stocks introduced in this section were really available, then EU(~,) would be the maximal expected utility of final wealth (Theorem 6.2). Since these stocks are not available, we have

(7.0)

v(x) ~= s u p

EU(XX'~(T)) _<E u ( c ) ,

and equality holds if there exists a portfolio process r such that

(7.10)

Xx'~(T) = C'

i.e., if the terminal wealth ~

can be financed without investment in the fictitious stocks. In light

of (7.9), such a z would be optimal for the problem of utility maximization in the incomplete market. In the next section, we discuss properties which 7r and v must have in order that they be related by (7.10). SECTION 8. ~

T I

0

CONDITION~ FOR OPTIMALITY IN THE INCOMPLETE

Let us define L2[0,T] to be the set of {~'(t)}-adapted ~d--valued processes r satisfying H~t)[I2dt < ~, a.s. We decompose L2[fl,T] into the orthogonal subspaces

15 K(c) A={v E L2[0,Tll a(t)u(t) = 0 V t E [0,T], a.s. P},

K~(a) ~ {~ e 1,2[0,T]I ~(t) e l~nge(a*(t)) V t e [0,T], a.s. P}. Note that the process # of (2.4) belongs to K• whereas the process u of (7.4) belongs to K(r On the other hand, if v E K((~) is given, then (7.4) can be solved for the fictitious stock appreciation rate process a. Thus, K(c) provides a parametrization space for fictitious completious of the incomplete market. We define KI(a) A_ {g E K(cr); v satisfies (7.7)}, and we recall definition (7.8). Then

E[#(T)

(8.1)

Z (t)r

= x.

We shall assume on occasion that the function I(.) has the property

(8.~.)

I ( ~ ) _ o

for some a E (0,1) and fl E(1,m). One consequence of (8.2) is that if JFv(y ) < m for some ~, E K(r and some y > 0, then ,.~ (y) < | for a]l y > 0, so that v E Kl(g). Let x > 0 and ~r E ,A(x) be given, and consider the statement that ~- is optimal for the incomplete market utility maximization problem: (A) Ootimalitv of ~r: Eu(xX,~'(T)) < Eu(xX,~(T)), V ~r E ~4(x). We shall characterize condition (A) with the help of the following conditions. For A E KI(c), consider the statements: (B) Finaneiliby of ~

There exists a protfolio ~r E ~ ( x ) such that Xx'~(T) = ~ ~, a.s.;

(C) Minimality of A: E U ( ~ ) < EU(~)

V L, E gl((r);,

(D) Dual optimality of A: For all v E K~((~), E0(~A(x)fl(T)Z~,(T)) < E0(~/A(x)fl(T)ZA(T)); (E) Parsimony of A: E[~7(T)Zy(T)~] _<x,

V ~, E Kl(a).

The relations among these conditions are expounded in the following two theorems. In the interest of brevity, we omit the lengthy proof of Theorem 8.1. The examples in Section 9 use only Theorem 8.2, whose proof we provide. 8.1 THEOREM. Assume that U(0) > ---m. Then conditions (B)--(E) are equivalent and imply (A) with the same ~- as in (B). Conversely, suppose that ~"E ..4(x) satisfies (A) and (8.2) holds; then there exists

A E Kl(r for which (B)--(E) hold.

The assumption U(0) > --m in Theorem 8.1 rules out the important special case of a logarithmic utility function. In the absence of this assumption, the following weaker rcsult can be established.

16 8.2 Tazo~r.M. Conditions (B) and (E) are equivalent, and imply both (C) and (A) with the same ~r as in (B). PIt00F: (B) ~ (E): The inequality E[fl(T)Zv(T)XX'~(T)] < x follows from (6.2) applied to the fictiously completed market corresponding to v. (B) v CA): The argument for this is given in the last paragraph of Section 7. (B) ~ (C): Under (B), we have (A), which means that V(x) = E U ( ~ ) . Now use

(7.9). (E) ~ (B): If ~r can be found for which xX'~r(T) = ~ , then the proof of (6.7) can be applied to show that ~r c M(x). Assume (E), and consider the Brownian martingale M(T) A=E[fl(T)ZA(T)~[~t]. From (8.1) we have M(0) = x. Imitating the argument preceding Theorem 6.2, we produce r e L2[0,T] such that t

M(t) = x + fl(T)~ =

= x +

f0

. r (s)dW(s),

- ' 3 . + M t O*.t + A 0

Decompose r as r = r + r where r E K• A E K((r), we may simplify the above equation to r 1 /~(T)~ -- x + J0 " g ~

+ IT ~ 0

(t))]

9 [dW(t) + O(t)dt + ,~(t)dt].

and r E K(cr). Because O E K• r) and

* * (~b'(t) + M(t)0 (t))dWo(t)

[r .

+ M(t) A. (t)][dW(t) + 0(t)dt + A(t)dt].

We will show that (8.3)

r

+ M(t)A(t) = 0 V t E [0,T], a.s.

We will then define

s

~

~t 1 * [x + 0 Z ~ -o(r

* + M(s)0 (s))dW0(s)].

Because r + M(t)0(t) e Kx(c), this d-dimensional vector can be written as r = ~ (t)a(t) for some m-dimensional a. Indeed,

+ M(t)0(t)

17 (8.4)

a(t) -- (a(t)a (t))'la(t)(r

§ M(t)0(t)).

We can define

~(t) A

(8.5)

~(t) f~(t)zA(t)x(t)

and it is then easily verified that 2(0) = x, :Xi(T) = ~ ,

'

and d•

= r(t)~:(t)dt +

X(t)~*(t)a(t)dWo(t) (cf. (3.2)). In other words, once we prove (8.3), then (8.4), (8.5) will define ~" e ..4(x) such that (B) holds. In order to establish (8.3), consider an arbitrary u E K((r) and introduce the sequence of stopping times {vn}:: l given by

rn = T h inf {t E [O,T]; M(t) + I fi~*(s)dW(s)l +

I t (llr

~ + IIr

0

+ IIA(s)ll ~ + 114s)ll~)ds _>n}. Denote un(t) A y(t) l[0,vn](t). Then for e E (-1,1),

rthrn , (8.6) ZA%eun(t) -- ZA(t)exp{- eJ0

e2 [tA~'n

u (s)(dW(s) + A(s)ds) - - 2 - j0

e--nlel < ZA§

(8.7)

ZA(T)

H4s)l[2ds},

(T) < en] el -

9

Because of (8.7) and the fact that A e K,(cr), we have A + e Vn e K,(~). It follows from (8.6), (8.7), the Dominated Convergence Theorem, and condition (E) that

(..8)

o = ~ E I~(T) Z~§176

;-E[~(T)Za(T) ~i

e=0 = ~E~(T) ~ ~ Z~§176

~=0

I?",, (s)(dW(s) + a(s)ds)] =-E[M(T~) s?", (s)(dW(s) + a(s)ds)].

According to ItS's rule, (8.9) M(rn) 0 u (s)(dW(s) + A(s)ds) =

v (t)(M(t)A(t) + r

+

(t)dW(t)

18 gTn

+ J|0

Ft ,

*

M(t)[|0uj(s)(dW(s) + A(s)ds)](C~(t)

*

§ r

From the definition of rn, we see that the two stochastic integrals on the right-hand side of (8.9) have expectation zero, so (8.8) implies that E is arbitrary, [M(t)A(t) +

r

u (t)(M(t)A(t) + r

= 0. Since u e K(cr)

= o. Letting n -~ | we obtain (8.3). []

8.3 Remark. Using Theorem 8.2, one can prove Theorem 8.1 by establishing the implications CA) (B), (C) v (D) and (D) v (E). In each of these, the hypothesis is a statement of optimality, and in each case, the proof of the implication is a variational argument similar to that just used for the implication (E) ~ (B). We have the following result concerning the existence of A E Kl(g) which satisfies (D), and by inference, the existence of an optimal portfolio process. 8.4 TlfE01tBM. Assume that (8.2) and U(0) > - co hold, as well as T

(i)

E I llO(t)ll~dt ) the second argument and returns -1 otherwise. The use of real-valued logic allows arbitrary composition of functions in the function set. The set of atoms for this problem consisted of velocity v, angle 0 , and angular velocity co. In addition, the set of atoms contained a smattering of random constants between -1.0 and +1.0. When a particular control slrategy (i.e. LISP S-expression) evaluates to any positive value for particular values of state variables v, 0 , and co at a particular time step, the force F is applied from the positive direction. Otherwise, the force is applied from the negative direction. Note that the control strategy can be viewed as defining a surface that separates the state space into parts. The environment consists of 10 randomly generated initial starting condition cases. Position x is chosen between -0.5 and +0.5 meters. Velocity v is chosen between -0.5 and +0.5 meters/ second. Angle 0 is chosen between -0.5 radians and +0.5 radians. Angular velocity co is chosen between -0.5 and +0.5 radians/second. We do not know of an analytic solution to this three dimensional broom balancing problem. However, an approximate optimal control strategy for a conventional linear approximation to

52 the solution has been computed by Keane. This analytic solution to the linear approximation calls for application of the positive force when co >if2, where A B (Sinh K(t + z ) - 2Sinh Kt)

I2=

K

and, otherwise, a negative force. Here t=z+

vA c

and

where the quantity A is computed as follows:

If

( v / C ) 2 ' then A = Sign O

>--- T --

Otherwise A : Sign (v/C). D, B, C, K are computed as follows: D= ~[4 B=

mc me +mp -F

D[m c + m,] C=

F [mo+mp]

K:~/D Since this solution to the linear approximation to the problem involved hyperbolic functions, we replaced the hyperbolic sine function by terms (up to the cubic level) from the Taylor series expansion to obtain an approximation to this approximation. The resulting "pseudo optimal strategy" serves as a benchmark for the genetically produced results below. Note that this

B3 "pseudo optimal strategy" is not used by the genetic algorithm in any way. Note also that the state transitions of the systems are controlled by Anderson's equations for 9 (t) and a(t) above. Time was discretized into 400 time steps of .02 seconds. The total time available before the system "times out" for a given control strategy is thus 8 seconds. If the square root of the sum of the squares of the velocity v, angle 0 , and angular velocity to (i.e. the "norm") is less than 0.07 (the "criterion"), the system is considered to have arrived at its target state. If a particular control strategy brings the system to the target state for a particular initial starting condition case in the environment, its fitness for that environmental case is the time required (in seconds). If a control strategy fails to bring the system to the target state before it "times out", its fitness for that environmental case is set to 8 seconds. The "fitness" of a control strategy is the average time for the strategy over all 10 initial starting condition cases in the environment. The process starts with the random generation of 300 random control strategies recursively composed from the available functions and atoms above. Examples of such random control strategies are linear control strategies such as v + 0 , co 0",/'~.57,and 0 > co and non-linear control strategies such as 02 + to. and 0 to31vl. As can be seen, the initial population of random control strategies in generation 0 includes many highly unfit control strategies, including strategies that totally ignore the state variables, strategies that repetitively apply the force in only one direction, strategies that are correct only for a particular few specific starting condition cases in the environment, strategies that are totally counter-productive, and strategies that cause wild oscillations and meaningless gyrations. In one run, the average time consumed by the initial random strategies in the initial random population averaged 7.79 seconds. In fact, many of these 300 random individuals "timed out" at 8 seconds for all 10 of the environmental cases (and very likely would have timed out even if more time had been available). However, even in this highly unfit initial random population, some control strategies are somewhat better than others. The best single control strategy for the initial random generation was the simple non-linear control strategy v 0 , which averaged 6.55 seconds. This control strategy correctly handled two (of the simpler cases) environmental cases and timed out for 8 of the environmental cases. Note that this control strategy is partially blind in that it does not even consider the state variable to in specifying how to apply the "bang bang" force. Nonetheless, in the land of the blind, the one-eyed man is king.

54 The genetic crossover operation is then applied to parents from the current generation selected with probabilities proportionate to fitness to breed a new population of offspring control strategies. Although the vast majority of the new offspring control strategies are again highly unfit, some of them tend to be somewhat more fit than others. Moreover, some of them are slightly better than those that came before. For example, the average population fitness improved from 7.79 to 7.78, 7.74, 7.73, 7.70, 7.69, 7.66, 7.63, 7.62 seconds per environmental case in generations 1 through 8, respectively, This generally improving (but not necessarily monotonic) trend in average fitness is typical of genetic algorithms. In generation 6, the best single individual was the non-linear control strategy t9 > v 3 I 0 + v I. It required an average of 6.45 seconds. In generation 7, the best single individual in the population correctly handled (i.e. did not time out) 4 of the 10 environmental cases. It required an average of 5.72 seconds. In generation 8, the best single individual was the non-linear control strategy [0 + m + 2.762v + 03 ]3. It required an average of only 2.56 seconds. This individual is not partially blind and considers all three state variables. It correctly handles 10 of the 10 environmental cases. Note that the number of correctly handled environmental cases ("hits") is a statistic that we find useful for monitoring runs, but this statistic is not used by the genetic algorithm. In generation 14, the average time required by the best single individual in the population dropped below 2 seconds for the f'trst time. In particular, the non-linear control strategy 0 + co + v + v 2 [ co - 0.734] 2 > v 0)2 required an average of only 1.74 seconds. It, too, correctly handled all 10 of the environmental cases. Two histograms provide a graphical picture of the learning of the population from generation to generation. The "hits histogram" shows the number of individuals in the population that correctly handle a particular number of environmental cases. The "fitness histogram" shows the number of individuals in the population with whose fitness value lies in a particular docile of

55 the fitness values. Both of these histograms have an undulating left-to-right "slinky-like" motion as the population progressively learns. In generation 33, the best single individual in the population was the non-linear control strategy 3 v + 3 0 + 3o~+v 0 2 > v r After its discovery, this single best control strategy was retested on 1000 additional random environmental starting condition points. It performed in an average of 1.76 seconds. In another test, it averaged 3.20 seconds on the 8 comers of the cube, but could not handle 2 of the 8 comers of the cube. The benchmark pseudo optimum strategy averaged 1.85 seconds over the 1000 random environmental starting condition cases in the retest. It averaged 2.96 seconds for the 8 comers of the cube. It was unable to handle the 2 worst comers of the cube. These results (in seconds) are summarized in the table below: P E R F O R M A N C E FOR THREE DIMENSIONAL BROOM BALANCING PROBLEM CONTROL STRATEGY

WORST 2 CORNERS

1000 POINTS

Benchmark Pseudo Optimum

1.85

2.96

Infinite

3v+3 0+3r

1.76

3.20

Infinite

5.0 CONCLUSION In summary, we used the new "computing paradigm" to discover a non-linear control strategy for the three dimensional broom balancing (inverted pendulum) problem which slightly outperformed our benchmark "pseudo optimum" solution for the problem. We believe our nonlinear control strategy is a very close to the time-optimal solution to the problem.

6.0 REFERENCES Anderson, Charles W. Learning to control and inverted pendulum using neural networks. IEEE

Control Systems Magazine. 9(3). Pages 31-37. April 1989.

56 Barto, A. G., Anandan, P., and Anderson, C. W. Cooperativity in networks of pattern recognizing stochastic learning automata.In Narendra,K.S. Adaptive and Learning Systems. New York: Plenum 1985. Booker, Lashon, Goldberg, David E., and Holland, John I-I.Classifier systems and genetic algorithms. Artificial Intelligence 40 (1989) 235-282. Davis, Lawrence (editor) Genetic Algorithms and Simulated Annealing London: Pittman 1987. Goldberg, David Eo Computer-Aided Gas Pipeline Operation Using Genetic Algorithms and

Rule Learning. PhD dissertation. Ann Arbor: University of Michigan. 1983. Goldberg, David E. Genetic Algorithms in Search, Optimization, and Machine Learning. Reading, MA: Addison-Wesley1989. Holland, John H. Adaptation in Natural and Artificial Systems, Ann Arbor, MI: University of Michigan Press 1975. Koza, John R. Hierarchical genetic algorithms operating on populations of computer programs.

Proceedings of the 11th International Joint Conference on Artificial Intelligence (IJCAI). San Mateo, CA: Morgan Kaufman 1989. Koza, John R. Econometric modeling by genetic breeding of mathematical functions. Proceedings of International Symposiumon Economic Modeling. Urbino, Italy: 1990a. In Press. Koza, John R. Genetic computing: a paradigm for genetically breeding populations of computer program to solve problems. Machine Learning. In press 1990b. Koza, John R. and Keane, Martin A. Cart centering and broom balancing by genetically breeding populations of control strategy programs. Proceedings of International Joint

Conference on Neural Networks, Washington, January 1990. Volume I. Schaffer, J. D. (editor) Proceedings of the Third International Conference on Genetic

Algorithms (ICGA). San Mateo, CA: Morgan Kaufmann Publishers Inc. 1989. Widrow, Bernard. Pattern recognizing control systems. Computer and Information Sciences

(COINS) Symposium Proceedings. Washington, DC" Spartan Books, 1963. Widrow, Bernard. The original adaptive neural net broom balancer. 1987IEEE International

Symposium on Circuits and Systems. Volume2.

P R O C E S S C O N T R O L USING N O N L I N E A R P R O G R A M M I N G T E C H N I Q U E S

B. Wayne Bequette Department of Chemical Engineering, Rensselaer Polytechnic Institute Troy, NY 12180-3590, USA

ABSTRACT Some of the most difficult problems associated with process control are due to process nonlinearity, manipulated variable constraints, uncertain parameters and unmeasured variables. In this paper a nonlinear programming approach is developed to estimate process parameters, unmeasured state variables and process disturbances.

A constrained optimization-based

procedure is also used to maintain a desired output variable trajectory, similar to techniques that have proven successful for linear systems. The process model, characterized by a set of nonlinear differential equations, is transformed into algebraic equations using orthogonal collocation on finite elements. A system with inverse response characteristics and a bioreactor model are used as examples. INTRODUCTION Within the past 10 years there has been an increasing use of advanced multivariable control system techniques which are known as model predictive control methods. A survey of the field has been performed by Garcia e t al. (1989). One of the most common predictive control methods is Dynamic Matrix Control (DMC) (Cutler and Ramaker, 1980). DMC is based on selecting a set of L future manipulated variable moves (control horizon), to minimize an objective function based on a least squares minimization of the differences between model predicted outputs and a desired output variable trajectory over a prediction horizon, R, as shown in Figure 1. Although the DMC optimization is performed for a sequence of future control moves, only the next control move is implemented. Model uncertainty and process disturbances are handled by calculating an additive disturbance as the difference between the process measurement and the model prediction at the current time step. It is assumed that the future disturbances are equal to the current disturbance, and a new trajectory is calculated. Notice that predictive control is really an open-loop optimal control strategy with feedback provided by the re-optimizing procedure. Predictive control strategies have been well-received by industry because they are intuitive and explicitly handle constraints. One limitation to the existing methods is that they are based on linear systems theory and may not perform well on highly nonlinear systems.

58 desired trajectory (future)

Y a

c I

t I

1

u 1

I

a 1

I

~ I

I

I

1

I

1

I

I

R t k current time

Prediction Horizon

maximum I

H

. . . . . .

I

,

I .............. I ["-1 1.... 1........................................ minimum

I

1

1

I

I

1

[

1

1

~I~,~-----

t

I

past control moves

1

1

I

1

I

Control Horizon Figure 1. Predictive Control Approach For these types of processes we feel that it is important to use nonlinear predictive control techniques with nonlinear dynamic models that have a physical basis. In this paper we first discuss dynamic process models in general. Next, we present a strategy for the regulation of the system, using a predictive control approach that accounts for constraints on manipulated and state variables. It is important to have a model that is representative of the current state of the process, so we then present a method to perform parameter estimation of the nonlinear dynamic system. Nonlinear Process Models This paper is based on the control of process systems that are described by lumped parameter models, that is, systems that can be modeled by sets of first order ordinary differential equations. In addition, we include deadtime on process inputs or output variables. Deadtime can be due to transport (flow through pipes) or measurement delays (analytical instrumentation, etc.). Mathematically, the representation is

59 dx

dt - f(x,u(t-0),p,l)

dynamic modeling equations

(1)

gl(x,u,p) = 0

algebraic equilibrium relationships, etc.

y = g2(x(t-qS))

state-output relationships

(2) (3)

where state variables

X

tl

=

manipulated variables

p

=

parameters

1

=

load disturbances (measured or unmeasured)

y

=

0

=

output variables deadtime between manipulated and state variables

00

=

deadtime between state and measured variables

This type of formulation may also be used for distributed parameter systems that have been "lumped" by discretizing the partial differential equations and forming ordinary differential equations. In order to si,alplify the solution of this problem, we use orthogonal collocation on finite elements to reduce the differenti',d equations to a set of algebraic equations. The dynamic model can then be stated as Ax - f(x,u,p,l,0) = 0

(4)

where A is a matrix consisting of collocation weights (Finlayson, 1980) and x is now a vector of state variable values at each collocation point. NONLINEAR PREDICTIVE

CONTROL

(NLPC)

The linear predictive control techniques can be extended to handle nonlinear systems by formulating a nonlinear programming problem (Eaton et al., 1989).

A single input-single

output system is shown as an illustration.

rain

tk+Tp k+RN c q~ (u,x) = fe 2 dt = Y. Y. Wj(Ysp(i,j)-Ypred(i,j)) 2 tk i=k j=l

u(k),...,u(k+L- I)

subject to Ax - f (x,u,p,l,0) = 0 U mni _-tt ~

a

else /z ~ -

also

the

the

accuracy

condition of

input

possible accuracy give is

up

of

some t h r e s h o l d

condition

number,

i.e.

number the

very

error

important

of closure of

is

the

the

value

IIAx - fll

approximated

due

the

connected

together with

connected

solution x

(by

solution to

value

w i t h t h e n u m b e r o f d i g i t s in a ceil o f t h e c o m p u t e r used). T h e r e a t , a

the

computation

number data

(or

with

this

simply,

the

e r r o r o f c l o s u r e ) , It is t h i s p a i r w h i c h g u a r a n t e e s t h e e s t i m a t i o n : fix - xll/lIxll ~ /a(A)IIAx - fll/llfll. Not too simple is the question: and as t h e e r r o r try

to

what

must

be c o n s i d e r e d as c o n d i t i o n n u m b e r

o f c l o s u r e in o t h e r c o m p u t a t i o n p r o b l e m s .

i n d i c a t e such notions f o r t h e c l a s s i c a l p r o b l e m -

In t h i s r e p o r t

we will

the p r o b l e m on d i c h o t o m y

of a m a t r i x s p e c t r u m by a i m a g i n a r y a x i s , i m a g i n a r y in its ample m e a n i n g , when it

96 is n e c e s s a r y to m a k e s u r e t h a t the m a t r i x axis

(such

maximal

matrices

invariant

are

called

subspaces

A h a s no e i g e n v a l u e s on t h e a n i m a g i n a r y

exponent~ally

L § (A)

and

dichotomic),

L (A),

but

corresponding

also

to

the

to

indicate

parts

of

the

s p e c t r u m lying in t h e l e f t and r i g h t h a l f - p l a n e , r e s p e c t i v e l y .

Thus,

for

separation

of

example, a

in

maximal

the

theory

invariant

of

automatical

regulation,

L § (A) p l a y s

a

subspace

very

theproblem

impotent

of

role.

If

a r i s e s w h e n c o n s t r u c t i n g a n o p t i m a l c o n t r o l u(t), w h i c h m i n i m i z e s t h e v a l u e CO

S o

( llCx(t)ll2+

in t h e s o l u t i o n s of t h e s y s t e m d x / d t

llBu(t)ll z) d t

: Ax. It is w e l l - k n o w n

that

to c o n s t r u c t

such

u(t), a s u b s p a c e L,(W) f o r t h e m a t r i x

[ ;.]

W=

A

-C'C

is n e c e s s a r y to be found.

It would s e e m q u i t e n a t u r a l

to solve it by f i r s t

computing the eigenvalues

W

lying in t h e l e f t h a l f - p l a n e a n d t h e n by c o m p u t i n g e i g e n - a n d p r i n c i p a l v e c t o r s .

It c a n be e a s i l y s h o w n t h a t such a s c h e m e if r e a l i z e d on a c o m p u t e r c a n p r o v e non s o l v a b l e . eigenvalues dealing

The

is

with

thing

not

the

is t h a t one

a computer

the

which

problem

can

we m u s t

matrix

Ao,

where of

close

to

it.

We know

computation

be c o n s i d e r e d

consider

owing to w h i c h we c a n only g u a r a n t e e ,

of

its

the

computer.

such

Thereby,

about

instead

det(A-Al)=O we deal

for

which

there

with exist

of

this

co-called

matrix

eigenvalues

a spot a

correctly

A general

stated.

"indefiniteness

In f a c t , principle",

t h a t i n s t e a d o f a m a t r i x A we h a v e a c e r t a i n

c is a s m a l l p o s i t i v e n u m b e r c h a r a c t e r i z i n g

equation

as

of m a t r i x '

of

that

IIA - Aoll -< cllAoll,

the number of

its

matrix

only

the

of d i g i t s

matrix

"c-spectrum", B(A),

so

in a cell

A satisfying i.e.

that

with

the

a set

of

IIB(A)II < r

d e t ( A + B(X) - M)=O.

It spots means

of

is

well

known:

"e-spectrum" that

the

conditioned.

This

if

self-adjoint

possesses o f

problem is

A is

not

of the

matrix

a diameter

eigenvalues case w i t h

non

of

which the

of

(A=A'),

does n o t

each

of

e x c e e d cllAll,

self-adjoint

self-adjoint

then

matrices.

matrices There

its

which

is

well

exist

the

97 e x a m p l e s w h e n t h e s p o t s of t h e " c - s p e c t r u m "

can have rather

c a n lead to t h e v a g u e n e s s or t h e d i m e n s i o n s of s u b s p a c e s typical

example

can

be

taken.A

two-parameter

family

large diameters which

L § (A) a n d L (A). Here, of

matrices

A (t)

of

a the

twenty first order: -20

-20.5L

0

A

b)

-19

-19.5L

(t)= 0

-2

0

-2.5L

0

0

-i

0

0

0

0

i

has a characteristic equation 20

20

I IC-j-k) -

(l-~)(

19

2

5 ~t

I--I (-j-o.s}) = o. J:t

J=l Let E)

=

1.5/-19

2tz(A~(l))=0.5

Ptt(A~(t))=l, dimension

( a t l=4,

of

L (A-(L)) --

l e t ~ = 3 2 -39 = 3 . 6 c - I 0 , are

is not

eigenvalues less

than

of 2,

at

l=10 l e t E) = 1.5e-19) t h e n

the

matrix

A-(1),~

whereas

the

dimension

consequently of

'.~

-

e q u a l to 1. It is to be n o t e d t h a t

the

L (A (l))

is

0

A-(l)o

do

of

the

determine

the

A, F a d d e e v D.K.

and

t h e e l e m e n t s of t h e m a t r i c e s Ao(l) a n d

n o t d i f f e r m o r e t h a n by t h e value ~ ( a t t = i0, ~ = 1.5e-19).

The

existence

eigenvalues quality

of

problem"

such for

examples

arbitrary

of s t a b i l i t y of a s e p a r a t e

is

connected

with

non-self-adjoint

matrices.

e i g e n v a l u e 2, of t h e m a t r i x

F a d d e e v a V.N. (4) p r o p o s e d to use one n u m e r i c a l p a r a m e t e r b e t w e e n t h e e i g e n v e c t o r s of m a t r i c e s this

parameter

is too

large,

"the

instability To

connected with the angle

A a n d A" c o r r e s p o n d i n g to }t. If t h e v a l u e o f

the problem

of

the

determination

of t h i s

eigenvalues

is unstable. Unfortunately, this parameter is not used as a measure of stability of eigenvalues

in the concrete corriputations.

Probably,

it

is connected with the

fact, that no efficient algorithm to compute this parameter was advanced.

There

exist

ill-conditioned, conditioned.

the

some

matrices

subspaces

L + (A)

A

whose and

separate

L (A)

eigenvalues

corresponding

to

though them

are

being well

98 The p r e s e n c e of t h e like e x a m p l e s compute

the

bases

preliminarily

computed

(Kublanovskaya adjusted,

V.N.)

and

Balzer)

of

a

was

sign-functions

methods

subspaces.

all

necessary

in

of

certain

and

the

special

worked

In

invariant

led t o t h e d e v e l o p m e n t of a l g o r i t h m s

subspaces eigenvalues.

It

are

applications.

be

destined

In

method

must

likelihood,

For

it

is

this

the

in

compute

the

first

the

A.A,

QR-algorithm were

Kublanovskaya

V.N.,

that

on

having

V.V.)

theoretically

maximal

and

without

(Voevodin

mind

information

case,

purpose,

(Abramov

to

L (A)

method

born

only

and

this

orthogonal-power

sign-functions

out.

L + (A)

which

these

invariant subspaces

foremost

matrix that

problem

the

is

becomes

their stable computation.

Before

separating

non-self-adjoint

matrices

the

projectors

onto

let

us c o n s i d e r

maximal

a more

invariant

particular

subspaces

and important

of

Hurwitz

p r o b l e m , h e r e a s p e c i a l a t t e n t i o n m u s t be p a i d to t h e L y a p u n o v m a t r i x e q u a t i o n .

As a c o n d i t i o n n u m b e r of t h e H u r w i t z p r o b l e m i t w a s s u g g e s t e d in w o r k (5) t o use

a

stability

quality

parameter

~:(A).

K(A)=2-11AII-IIHII, if t h e Lyapunov m a t r i x matrix

of

dimensions

determined

H,

NxN),

otherwise

has

~(A)=o,.

s o l u t i o n s of t h e s y s t e m d x / d t

Its

algebraic

expression

looks

as:

e q u a t i o n A*H + HA + I = 0, (I b e i n g s i n g l e

only

one

This

solution

parameter

is

in

the

directly

class

of

positively

connected

with

the

= Ax by t h e e q u a l i t y (6): T

It is obvious, dx/dt

that

tlx(t)ll 2 dt ))

K(A) = 211All (Sup ( m a x T->O IIx ( 0 ) I1=1

J" o

t h e less is ~(A) t h e m o r e

rough

is t h e

stability of the

system

= Ax w i t h r e s p e c t t o t h e d i s t u r b a n c e s o f t h e e l e m e n t s in t h e m a t r i x A.

As

an

error

of

closure

in

the

numerical

solution

of

the

Hurwitz

closure

of

the

Lyapunov

problem

o

IIA X + XA + Ill

is

used

equation.

is

the

True

which

is

theorem

the on

error the

of

error

of

closure

(7):

if

matrix

X=X" > O,

= IIA'X + XA + Ill < 1 t h e n t h e H u r w i t z ' A a n d

IIH-XJI/IIHII ~ A,

It Hurwitz

is

the

problem

theorem

which

Im(A)-2-11AIJ.IIXIIJ/~(A) "< /~.

allowed

(6) up to t h e a l g o r i t h m

to

close

the

algorithm

with a guaranteed

a n d d e s c r i b e d in i t s f i n i t e f o r m in w o r k (7).

of

solution

accuracy,

of

the

substantiated

99 Here

we

errors

in t h e

played

a

like

to

note

that

computation algorithm of

very

computation equation

would

important

algorithm

role

of

in

as

detailed

analysis

of

the

matrix

function

exp(tA) a t

estimating

positively

A ' H + HA + I = 0,

a

it

the

determined was

shown

influence

solutions in

(8).

the

of

of

computational

these

the

Thereat

Hurwitz A

errors

Lyapunov

in

a

matrix

estimation

of

the

m a t r i x e x p o n e n t i a l exp(tA) w a s amply m a d e use of:

Ilexp(tA)l] -< v/K(A)'.exp(-tllAII/~(A)),

t>--0.

Now, let us p a s s over to t h e s e p a r a t i o n o f m a x i m a l

invariant subspaces of a

non-self-adjoint matrix.

If

A is

the

exponentially dichotomic,

then

for

the

sought

for

projector

G

o v e r L + (A) a w e l l - k n o w n i n t e g r a l p r e s e n t a t i o n 00

1 G = ~-~ S (izI - A ) - l d z -CO

is valid. The value of the n o r m IIH;I o f t h e s y m m e t r i c a l m a t r i x 1 co H = ~-~ J" (A ~ + i z I ) - ! ( A - i z I ) - i d z

(1)

-00

can

be

there If

used

exists

as

a

only

dichotomy one

numerical

eigenvalue,

IIHII is n o t too large,

characteristic.

then

IIHll = ~,

If

since

the

i t can be e a s i l y o b t a i n e d t h a t

and H c h a n g e v e r y l i t t l e a t

small v a r i a t i o n s

of

A for

on

the

imaginary

integral

(1)

no

axis sense.

estimation showing that

G

w h i c h t h e p r o j e c t o r G and

t h e m a t r i x H have been built. This c o n t i n u i t y is c o n n e c t e d : i) w i t h t h e g u a r a n t e e of c o r r e c t n e s s o f t h e c o m p u t i n g p r o c e s s ; 2) w i t h t h e n e c e s s i t y t h a t t h e s o l u t i o n a d m i t o f a p r a c t i c a l u s a g e in p r o b l e m s .

It

is

interesting

to

note

that

if

A is

exponentially dichotomic,

then

for

H

f r o m (1) t h e p r e s e n t a t i o n C0

H = J" G ' ( t ) G(t) d t -00

is

valid.

Here

G(t)

is

d / d t ( G ( t ) ) = AG(t) + 6(t) determined

the

the

parameter

~(A) = 2.1IAII.IIHII.

Green

bounded of

matrix at

the

all

determined t.

Using

exponential

as

this

the

solution

equation,

dichotomy

of

of

equation

S.K.Godunov

(9)

the

A:

matrix

100 So, S.K.Oodunov proposed to

simultaneously compute the matrix

H and the

p r o j e c t o r O on L § (A) while s t u d y i n g t h e e x p o n e n t i a l d i c h o t o m y of A. In (10) i t

is s h o w n t h a t

to e s t i m a t e

matrices

G and

Ft, a s y s t e m

of

matrix

e q u a t i o n s c a n be used Gz-

G = O, GA - AG = 0, ~

HA

+

A

(GH) ~

~

H

+

GH = 0, (2)

9

G CG

w h i c h g e n e r a l i z e s t h e Lyapunov m a t r i x

(I-G)

-

C(I-G)

= 0

e q u a t i o n f o r t h e c a s e of t h e n o n - H u r w i t z A.

We h a v e p r o v e d t h e f o l l o w i n g t h e o r e m .

THEOREM: a) i f f o r some C=C'> 0 t h e r e e x i s t m a t r i c e s G a n d H(C)=(H(C)) ~ > 0 satisfying

system

if a m a t r i x

(2),

A has

then matrix

A h a s no e i g e n v a l u e s

no e i g e n v a l u e s

on t h e

imaginary

on t h e i m a g i n a r y

axis,

then

system

~

axis;

(2)

at

b) any

9

C=C > 0 h a s t h e only one s o l u t i o n w i t h r e s p e c t to G a n d H(C)=(H(C)) > 0 t h e r e a t 0O

G = l / ( 2 n i ) s (zI-A)- ldz;

where

~- is

eigenvalues

a A

closed

contour

having

located

strictly

in

theorem

presents

a

equation.

In t h e

same

error

of c l o s u r e

the algorithm control

and

the

computations

determination

of

of

either

correct

of

left

a

half-plane.

known

problem,

theorem

is o b t a i n e d ,

occurring

projectors

error

of c l o s u r e

In (11) t h e r e errors

over

the

was

exercise

assertion

in t h e

carried

clear

is the

to

important theorem

L (A)

and

based

(9). T h e

that

role

of

of

work

~:(A)>~:~

i.e.

that

matrix on t h e

in t h e in for

projectors

the

over

analysis

destined variant

of

for

the

of

the

in t h i s a l g o r i t h m

H a n d t h e v a l u e ~(A) w i t h t h e that

this

computations

a complete

on a

the

that

is u s e d

results

out

all

Lyapunov

in one a l g o r i t h m

by S.K.Godunov

G, m a t r i x

It

in t h e c o m p u t a t i o n s

L § (A) a n d

method suggested

in t h e p r o j e c t o r

contains

analogous

of

the

which

on

which plays an v a l u e ~(A). T h i s

subspaces.

or

the

self-crossings

G, H a n d t h e

computational

signs,

no

(lO) a t h e o r e m

of t h e e r r o r

the

orthogonal-power results

work

of matrices

invariant

influence

generalization

in t h e H u r w i t z

and play the role maximal

H = l / ( 2 ; r ) J ' ( A ' + i z l ) - l C ( A - i z I ) - 1dz

there

indication exist

no

" p r a c t i c a l " d i c h o t o m y of t h e s p e c t r u m A by t h e i m a g i n a r y a x i s .

L e t us p r e s e n t

one of t h e p o s s i b l e a l g o r i t h m s

of G a n d H s u g g e s t e d by A.N.Malyshev (12).

of a simultaaleous d e t e r m i n a t i o n

101 i) W i t h t h e h e l p o f t h e T a y l o r s e r i e s

A = exp(xA'),

B = -I,

0

and using

the

Choletsky

LL 9 =~. T h e r e a f t e r ,

2). AdBI;

We

Az, B2; ...;

a

we find

recurrent

by m e a n s

I

Aj_IO

0

X - -

m

A

)-ill = ~

approximations inequalities

) C,

m

invertability m

m

- B

= r to

O

~.

L by d e c o m p o s i n g

procedure

of

which

the

is r e d u c e d

transformation

0

A

matrix

pairs

at

every

P, s u c h a s

B

J

J

m --~

B*(A* - B') -t

m

determination

]I

" 9• !

Bj_I

It turns out that at

II(A - B

the

•

]=

to

Thereat

of

triangular

B o= L-XBo = L -I

to t h e c h o i c e o f t h e o r t h o g o n a l

P[Bj-tAj-IO

m

a lower

A 0 = L-IAo ,

successive

Are,Bin; ...

dt

0

decomposition

out

step of the recursion

e) = 211All f e x p ( t A * ) . C . e x p ( t A )

0

we s u p p o s e t h a t

carry

a t 9 = 1/(211AII)

G,

n~

(A - B )-ICA* - B* )-I m

at

m

rather

IIAII.IIHII a n d

"f2-'-IIAII.H

is

rn

large the

estimated

) 2UAII.H.

m

m

follows

velocity

of

beginning

from the

the

fact

convergency

from

some

m

that of

the

by

the

of the form:

II El" (A" - B * )-i in m tn

II (A - B rn

rn

G II :i C o n s t V/KCA)' exp(-2m/~c(A)),

)(A -B ) - 2 - t l A I I . H Ill

EP.

II --~:" providing a "actual"

102 absence

of the dichotomy of the m a t r i x

computation

of

the

matrices

G,

spectrum

H and

the

by the

value

imaginary

~:(A) with

axis,

indicated

or

the

the true

signs.

BIBLIOGRAPHY:

1. T u r i n g A.M. Rounding-off e r r o r s in m a t r i x processes. Quart.

J. Mech., 1, 1948,

287-308. 2. J.von Neuman and H.H.Goldstine. Numerical inverting of matrices of high order. - B u l l . Amer. Math. Soc., 1947, v.S3, no.ll, 1021-1099. 3.

Guaranteed

accuracy

Godunov S.K.,

for

the solving of

Antonov A.G.,

linear

K i r i l j u k O.P.

systems

and

in Euclidean

Kostin V.I.

-

spaces./

Novosibirsk:

"Nauka", 1988. (Russian). 4.

F ad d eev

D.K.

and

Faddeeva

V.N. Computational

methods

of

linear

algebra.-

Gos.lzdat.Fiz.Mat.Lit., Moscow, 1963. (Russian). 5. Bulgakov A.Ya. - Sibirsk.

Math. Zh.21(1980),

No 3, (32-41); English Transl.

in

Siberian Math. J. 21 (1980). 6.

S.K.Godounov

(Godunov)

calculatives dans calculatif (Proc.

du

Fifth

A.J.Boulgakov

(A.Ya.Bulgakov),

le probleme de Hurwitz et methodes a l e s

probleme Intcrnat.

Information

and

Sci.,

de

Hurwitz),

Conf.

vol.

Analysis

(Versailles,

44,

1982)),

and

surmonter

Optimization

Lecture

Springer-Verlag,

Difficultes

Notes

1982,pp.

(aspect

of

Systems

in Control

846-851.

and

(English

ab s t ract , p.845.) 7.

A.Ya.Bulgakov and Lyapunov's

S.K.Godunov,

equation,

Calculation

Computational

of

Methods

positive definite of

linear

solutions

algebra,

of

"Nauka",

Novosibirsk, 1985,pp. 17-38. (Russian). 8. Bulgakov A.Ya. Computation of the stable

matrix,

Computational

exponential

Methods of

linear

function of an asymptotically algebra,

"Nauka",

Novosibirsk,

1985, pp.4-17. (Russian). 9. Godunov S.K. - Sibirsk.Math. Zh. 27 (1986), no.5, pp.24-37. 10. Bulgakov A.Ya. - Sibirsk.Math. Zh. 30 (1989), no. 4, pp.30-39 11.

Bulgakov

A.Ya.

non-self-adjoint

Guaranteed

accuracy

matrix,Numerical

of

calculating

analysis,"Nauka",

or

invariaat

subspaces

Novosibirsk,

1989,pp.

12-93, (Russian). 12.

Malyshev

A.N.

Preprint

Novosibirsk, 1988 (Russian).

no.6,

Inst.Math.,

Siberian

Branch

Acad.

Sci.

USSR,

Discrete approximation of the minimal time function for systems with regular optimal trajectories

M. Bardi

M. Falcone

and

Dipartimento di Matematica P. e A. Universi~ di Padova via Belzoni 7 35131 Padova - Italy

Dipartimento di Matematica Unlversi~ di Roma "La Sapienza" P. Aldo Moro 2 00185 Roma - Italy

Abstract. In this paper we prove an estimate of the rate of convergence of the approximation scheme for the nonlinear minimum time problem presented in [2]. The estimate holds provided the system have time-optimal controls with bounded variation. This estimate is of order

v with respect to the

discretization step in time, if the minimal time function is H61der continuous of exponent v. The proof combines the convergence result obtained in [2] by PDE methods, with direct control-theoretic arguments.

1. I n t r o d u c t i o n . In this paper we continue the study of an approximation scheme for the classical minimum time problem for nonlinear systems which we began in [2]. We consider the continuous-time controlled dynamical system in IRN (1.1)

~ y' = b(y, (x) t y(0) = x

and the corresponding discrete-time system, with time step h > 0,

(I .2)

xj+ X0

= xj + h b(xj,aj) =

X

where the controls are taken in a given set A~IR M. For a given compact target set T we are interested in the minimum times T(x) and hNh(x) taken respectively by systems (1.1) and (1.2) to reach T , where Nh(x ) indicates the minimum number of discrete steps. Note that T is finite only on the set R of points controllable to the target T in finite time, that it tends to + ~ near D R and that R is not known a priori. We recall that the Dynamic Programming method provides time-optimal controIs in feedback form for the discrete-time problem, once the discrete Bellman equation for N h is solved. In [2] we considered the new unknown functions

104

(1.3)

v(x) :=

1 - e -T(x), if T(x)0. The general structural assumptions on b and T we need are the following b:]RNxA---~IR N is c o n t i n u o u s , l b ( x , a ) - b ( y , a ) ] < Llx-y I

(At)

and lb(y,a)l < K(l+lyl), V x , y e IR N and "v' ae A ; T i s compact ;

(A2)

there exists 7, 0 0 there e~-tsts an a h ~ Ph such that R

(2.2)

~

c~(s) - (Xh(S) ds < h V(a,[O,R])

q

0

Proof. Set Ik:=[kh, (k+l)h[ for k=0,1 ..... n , n such that (n+l)h < R < (n+2)h and In+l:=[(n+l)h,R]. We can define c~h in Ik as any value taken up by (x in the same interval, or for instance

ah(S):= lim a(t) forse Ikt--.->kh+ Then f Ik

~(s) _ O~h(S) ds < h V(a,I k) ,

and (2.2) follows easily. In the previous paper [2] we have proved the following result (the definition of v and v h is (1.3)):

108

Theorem 2,2 Assume (A1), (A3), (A4). Then v h converge to v uniformly on compact subsets of IRN. Corollary 2.3 Under the assumptions of Theorem 2~2 for any given compact subset K of R there exist -h and T, such that K~P,h

a n d hNh(x) The spatial domain occupied

by a spherical particle with normalized radius rp centered at xp is specified by B(xp) = {x 9 R 3 : [lx-xp[[ _< rp}, where [1. [[ denotes the Euclidean norm. Let S = {x 9 R 3 : 0 < x3 < ~3,(Xl,X2) 9 FB}, where ~3 is a specified positive number. For the single particle case, the spatial domains occupied by the liquid layer and the gas are given by s

= (f/-NS)\B(xv)

and ~2g = f~+ f3 S respectively. In f~g, the steady-state motion of the gas is described by the following dimensionless Navier-Stokes equations: 3

~--~uiDiu+ V p g - v a V 2 u = O ,

divu=O,

(1)

i=1.

where u = (ul,u~, u3) denotes the gas velocity (normalized by pgv~/h~); and D~ = O/Ozi. IIere, we have neglected the gravitational effects. Similarly, in fl/, the steady-state motion of the liquid is described by 3

C viDiv + V p / -- V / V 2 V = --gVX3,

divv = O,

(2)

i=1

where v = (vl, v2, v3) denotes the liquid velocity normalized by p/v//ho, p~ the liquid pressure (normalized by plv~/h~); and g the acceleration due to gravity (normalized by vl/ho).2 3 Assuming no slip at the rigid bottom plane $ 8 and at the particle surface, we have the following boundary conditions: v

where O ~ ( x . ) = { x ~ ~

= o

on

SsuOt~(xp),

(3)

: IIx - x~ll = ~ } .

When the liquid is confined to a container, we may also impose no slip conditions at the lateral surface SLy" = { x 9 /~3 : 0 < ~3 < h ( ~ l , ~ 2 ) , ( ~ , ~ )

9 ors}, V = 0

i.e. On SLF.

(4)

At the top boundary surface ST = {x E R 3 : x3 = k3,(xl,x2) 9 FB}, the gas velocities at the jet entrance and suction port are specified, i.e. u = (O,O,~,s)

on Ss,

u = (O,O,-~,s)

o,, Ss,

(5)

148 where u j and us are given positive constants; S j and S s are subsets of ST representing respectively the jet and suction port surfaces given by

s, = {x

-

, , , i i , , , r l l , i i

U

7 "C

"'bt?" ((t \ !if ,J ii

GAS JET

? i

/ i

/ i

i

i

i

i

i

i

LIQUID LAYER

k "" i

I

PART lCLE

"ql"]

Xi

J I m

P

Fig.la Sketch of streamlines for the gas and liquid flows induced by the gas jet and the suction port.

SUCTION PORT

GAS JET

GAS JET

U INTERFACE ,. - 7 ~ r ~ ' = - - - . ~

,

~

,,'/ / .................

BOTTOM PLANE

.

- - - - -4- - - L

LAYER

......

I

Q

.

-- _

~4~S~-i.

,t~,~

../4,t

...

---

~,,-"

'~

;T ......

PARTICLE 0

' ....

J ..,'T

t

.............

X! ~

Fig.lb Sketch of the streamlines for the gas and liquid flows induced by the displaced gas jet and suction port.

1,52

GA.S Js

"''Li/I

~V/II Ill/ l/

ID./#1

~9.. 49

/,[/tl

1,~',';_~.,./III/

t'/ ~t i :

, l

i\ ~''--

/11/#

~ +l~'''" ......

~ I t ~ # ', *

l l ~ . -

-#

lit

I

........

-~-------,,',,\\\~..lz,[ . x \\

\x-'--..--

t

~\\\\'~'-'~,

'

/

-%%\~'~'--#a

i 39, 8E

39.165

3~

1 39.99

t

I i

I (JO. 1S

I qO. 32

]

I

gO. ~i9

VEUTORSI~II,.IEI 5.00E-Ol UNITS i

i

I

2?~''

-7, ; : . . . .

tt

"

I

~..-//t

. . . . . . .

"~

"r

.~,,

,

-411,.._" .

O

GAS ,/ET

~'IOR SOtL.~i ~'.OOE+O0UNITS SUC'CZON PORT

,~ , ''"

I - - 1

.,~\X..,

. . . . . .

'

i"

' ~"\

i "

"

,

"

Fig.2 Velocity vector field of liquid and gas with particle along the suction port axis.

153

G~

JET

~'TOR

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Fig.3 Velocity vector field of gas and liquid with displaced particle.

OPTIMALITY CONDITIONS FOR ELLIPTIC VARIATIONAL INEQUALITIES

Liu Wenbin And J.E. Rubio School of Mathematics, University of Leeds Leeds LS2 9JT, U.K.

Suwa~ary. By

the

penalty method,

Ekeland's

Variational Principle

and

necessary conditions

of

lower-semicontinuity of some set-valued mappings, optimal

control for some abstract elliptic

obtained

in

the

conditions.

The

cases idea

where is

to

the find

convex

variational sets

optimality

inequalities are

satisfy

conditions

some

smoothness

first

for

some

penalized problems by Ekeland's Variational Principle then to pass limits to obtaln the optimality conditions. some

known

optimality

It is shown that these conditions lead to

conditions

in

necessary conditions for some problems.

many

cases.

They

also

yield

new

Our results give uniform forms for

several known optimality conditions for elliptic variational inequalities.

1.Introduction.

Optimal control problems for elliptic variational inequalities have been much

studied.

conditions,

One

of

the

main

difficulties

even though optimality

important cases (see for example,

conditions

is

to

obtain

optimality

have been obtained in some

[3],[4],[6],[7],[I0],[12],[13])o Recently,

Shi Shuzhong got some optimality conditions for abstract variational inequalities [14] [IS], which covers very wide range of problems. seem very complicated to apply.

But his results

In this paper we will, by using the

penalty

method, Ekeland's Variational Principle and lower semicontinuity of some set-valued

mappings,

problems. Then

establish

we show that

general

optimality

in many cases,

conditions

the conditions

for

this

lead to

known

results. We also show that they give new necessary conditions for a problem posed in [7] and [13]. It seems that our optimality conditions tend to unify known results. The methods used in this paper can be applied to obtain some necessary conditions for abstract strongly monotone variational inequalities

155 (see [14],[15]) without essential changes and we shall discuss this

problem

in some later papers.

2. Some studies about the lower limits of set sequences.

m

let V be a Banach space and V

its dual space.

As in iS], we give the

following definition: DEFINITION 2.1. Let A cV for n=l,2,..-. The lower limit of {A } is the set, n n denoted as lim A n-~

n=l,2,-- and

n'

given by lim A ={ y~V: there is {yn } such that yn ~ A n for n n-~

yn~y strongly in V}.

A well-known result is that lim A is always a closed set iS]. Let K be n n-~m a closed nonempty convex set in V; we will see later that lim(K-Yn)~(Yn-K), n-~ where Yn~K for n=l,2,-- and llyn-YoiIv~O,plays an important role. We note that li__mm(K-Yn)n(Yn-K] is not empty. The problem whether llm(K-Yn)n(Yn-K)=(K-Yo)n n-~ n-J

O, there is P>O such that IIUlIu~P if (y,u)gKXUad and g(y)+h(u)~Q. TI~OREM 3.1.

[4] (OCV) has at least one solution.

[]

We consider now the optimality conditions for (OCV). hong

In [14], Shi Shuz-

gave some optimality conditions for it but unfortunately,

conditions complicated

one of his

is wrong as shown by V. Barbu and his corrected results seems very to use

[IS]. Nevertheless,

the

idea in [14]

seems applicable.

157 By using a different treatment,

we give the following conditions for (0CV).

THEOREM 3.2. Let (Y0,U0) be a solution of (OCV).If the mapping y~(K-y)n(y-K) is l.s.c at Y0'

then there are P0~V and ~e 0g(y0)cH

(ag is the generalized

gradient in [8]) such that for any (z,v)~K>4Jad,

(3.3)

b(Yo,Z-yO)-(f+Buo,Z-yo)ZO,

(3.4)

b(s,PO)=(~,s),

(3.s)

-(B(V-Uo), p o ) - h (Uo, V-Uo),

(3.6)

b(Yo,Po)-(Po,f+BUo)=O,

(3.7)

b(Po,Po)~(~,po),

f o r any se U k(K-Yo)n(Yo-K)H,




Ca. 89

b3J : < c I a.e.

Uad

functional This

intensity

optimization

in

~Cv9

1 X vC• 0 C2..79, cI

satisfies

The

[

solution.

Following

Id v / d x 2 1

].

symmetric

condition

v ,~tntmtztn&

cost

ape

respectively

+ 1C@~

the

lwCx,v) l

:

is

lot

C~.4]

and

HIcc/)

the

a unique

the

constants 9 the

Ca. Sb

]:

on

has

function

that

and

indicating

to v ~ U a d

the

C~.39

suitable

Ca. 79

consider

stlbject

Ce. 5]

V rl ~ H ~ C CD

acting

satisfies

problem

shall

forms

+ kbCw,r

parameter

forces and

- UllZ~Dro.]dx

< Cx I , x~]

=

the

described

aCf,D] R

+ CUl~.Z I

~;s i s

H~CnD

form

= bCf,w,r

~

C2.33

dx

C~..'~)

~)a

formulation

aCw,r

tensile

+ ~Ul~Zlyb

dx

element,

well

pl a t e

equations

:

- u2.2zl]r I

} where

variational

~ar man

as

constant.

as

The

[~

~

Ca. 6D

a.~ > ~

that

].

In

: [ Hoc)CCZ) ]3

dx

a given

a given

continuous

+ u2,~z22

s [CUl~Z ~ QC v3

I dz QC vD

bC . . . . . b

~ R give n by

bCu,zD

x~

the

,

I CUllZll L%C v)

d 6 L2C{)a ) i s

We

~

=

ICz9

k

~

aCu,zD

bCu,z,r)

a~

x. H I CCD

is

assumed

Ca.89

functional

to

,

= c~ and be

cs

are

given

nonempty.

approximates is

>

stiffness

nondifferentiable.

of

167 Lemma

2.1

C DE. l E D

]

optimal Proof

For

X

does

solution

shal i

v

given

section

more

3

we

t.

ping

of

C)t

[

].

Let

us

shape

vector

field

E

~- C C [ O , ~ , )

:

TtCVD

*%

f)

for

there

order

C)t

}

=

an

an

condition to

do

it

we

depending image

DO .

f)t w e

To

of

a

descri

shall

use

problem

this

exists

(2".g).

In

as

variational

and

:


0 such

implicit

of

~

Ca'q-.ir.~).

= ft ~ Tt

two

canonical

employing

C w t , f t)

parameter

C w t , f t)

system ft

derivative

solution

< e t HeC O D -

consists

material

to

constant

differentiability

this

pair

the

exists

- w

:

Ca. iK~)

o%" t h e

the

of

the

respect

= w 0 E H 0 Cfl),

there

Proof

calculate

continuity

Ca. 169

system

be

to

~-

write

ft we

t

C3.3D

are can

--~ w t

problem

uncoupled.

show ~

Frechet

H 02 C O b

with

differentiability

of

(3.3). D

Lemma

3. S

: The

functions (a.

16)

wt

are

aC~,r

material E

given

derivatives

HOOCh%t) by

+ a'Cw,cb)

and

ft

~

w

~

Ho2Cf~L D

HaCOO

and

~" ~

satisfying

HaCOD

the

~y~tem

:

= bCf,w,r

+ bC~',w,r

+ b'Cf,w,r

+ kbCw,r

0 a C f , O) where

Cd,r

+ a'Cf,~))

of

= - b C 'w,w, D) - b C w, w, o) - b 'C w, w, D)

= Idiv

[dVCO)]Cdx

, VCO)

= VCO,X),

C3.4) 2 V D ~ HoC~'I3 a'C. , . ) , b ' C .

,.),

Q

b'C ..... D bC . . . . . ) = O.

These

are

Frechet

derivatives

,respectively, forms

are

with

given

by

respect :

of

the to

forms

parameter

aC. ,.), t at

a

bC. ,.), point

t

170

a'Cw,~D

= J

[ div

VCO)

- ~C

DVCOD

+

DVCODD]AwA~

dx

+ C3.5)CaD

f Q b'Cw,~D

dim

Cdiv

= g [ dim f~

VCOD

- CDVCOD

VC03

+ ~DVCODD]CAw~r

- CDVCOD

+ ~DVCODD]

+ X/wAr

~w~

dx

C3. S D C b D

C3. 6)Cc9 ~ZC~DVCOD~Zf

CDVCO) denotes

Proof

: Subtracting

domain t,

the

the

~] [ 1 9

passing

account

VC~fD~DVCODI~x

+ ~DVCODD

DVCOD

transforming

-

jacobian

from

] ,dividing

to

the

]Vf

+ ATV~w

matrix

of

the

integrals

both

the

t

C2. i O D - C ~ - . i ~ D , C 3 . 2 D , C 3 .

the as

SD

we

div

VCOD,

the

f]t i n t o

0

[

>

A~

as

+

C2.7D over

equations taking

C3.4D

dx

integrals

resulting

-

= ~ii

equations

the

well

obtain

VCOD

+ ~DVCOD)V~

matrix

of

--+

C div

C2.16)

domain

sides

with

~

CDVCO)

equations

over

limit

-

[ i~

by

into

]. o

Next

we

H o C ~ t)

determine

• H C Q t)

detern~nes the

the

increment

Definition function Cw t)

where

~

= w

II

H 2 C R ~'9 i s

shape

the [

system

the

parameter

19

]

HoCQtD

:

is

function

t value

The

shape

determined

the

C~.7).

The

pair

I IHeCC e

prolongation

~-

Cw t,ftD

shape

derivative

with

respect

value

to

only. W

derlvative

by

E

H2Cf)O

of

the

:

t W + oCtD

+

oCt)

of

of

derivative

the

increment of

3.3 wt

a

~atisfying

C~.~)

/t

-~

of

the

0

for

t

--~ O,

function

wt

w0

=

w

~ H ~ C Q t)

~

H COD,

into

wt

the

space

H'~ ~ 2 D . Let the

us

recall

functlon

holds W

[ l~

wt

~

] that

Ho~CQt D

if

the

exists,

shape then

derivative the

W ~ H~CC/]

following

of

condition

: = w

- ~w

VCOD

C3. VD

9

w

~

and

H~CCO

ia

~w

a

is

Integrating

the

,nate~'ial

gradient by

of

pa~ts

derivative

function the

w

system

of with C3.4D

the

function

respect two

to

wt

~

H

C~tD

xo

times,

taking

into

171 account

C3. VD

obtain

the

CW,FD of

~

the

and

eliminating

system

H~CC/9

x

system

of HeC{iD

CS. VD

derivative~

equations of

the

of

determining solution

the

shape

Cwt,ftD

VCOD

matrix

the ~

we

derivative

HO2Cs

D

x

HO2Cs

D

:

aCW,~D

= bCF,w,~D

+ bCf,W,~)

aCF,))>

= -~bCW,w.'Q)

+ kbCW,,~D

+ IICW, w,~D

HO2C~D

V 9~ ~

C3.83 I

li ( w ' w ' ~ j Taking that

into W and

W

= F

OW/"On where ing

of

point

[W

~-~A~

account

=

the

5

W

as

+ AwVCODnDA~

well

following

as

] dx

C3.7D

boundary

it

is

easy

conditions

[-O~'w.'3n a

]VCODn

,

OCOw."gh'~D/On.

integral

f] i n

the

OF/'On From

i i in

CS. 89.

dJff~,%O

of

direction

V,

=

[-@~'f/On 8

C3.~),CS.

the

cost

defined

IO>

]VCODn

it

functional

by

= max



+ I F

:

(3.113COD

C3.11DCaD by

we

introduce

an

- -~

.[i,

adjoint

: - $ n

6C•

dx

V )) a

ill,

+

C3. i ~ 9

A q ~h9 d y an

measure

llgCa)

C3.1iJCbD

V ,~ ~ HCfI)

d~

Dirac

[wC'~b [

CS.

>

x > 0 x = O x < 0.

eliminate [HoCCD]~-

max

defined

if if if

= bCp,w,)))

is

=


d J C Q , VD

in in

dx

v

E

U

Remark

2.. i a s

derivative at

a

point

of v

~

well

the U

in

as

cost

C3. 1 5 )

we

functional

direction

~

~

U

:

< C sgn

wCx))C-

I

WlCX)

XlVCXa)~b

]

CAwZkp

+ ZlfAq)XlVCX2D~n

[ x E

Q~

I

dx

+

>

References

[1]

Adams

(2]

Begis D. , G l o w i n s k i R. c I g ? J ) , Application de la elementes finis a l'approximation d'un probleme optimal. Applied ~thematics and Optimization ~

[3]

Bendsoe M.P. , O l h o f f N. , S o k o l o w s k i J.Clg853, Sensitivity analysis of problems of elasticity with unilateral constraints. Journal of Structural Mechanics 13 : 2-Ol-a~.

[4]

Cea J.Clg71), Paris.

R.A. c I g T J ) ,

Sobolev

Optimisation.

Spaces,

Theorie

Academic

et

Press,

algorithmes.

New

York.

methode des de domaine : 130-16g.

Dunod,

Chenais D. C l g S 7 ) , Optimal design of midsurface of shells : di f f e r e n t i a b i l i ty proof and sensitivity computation. Applied Mathematics and Optimization 16 : g3-133.

173 [6]

Ciarlet Lecture

[73

Delfour M. , Z o l e s i o 5. P . ( 1 0 8 8 3 , v i a m i n m a x differentiability. Optimization a 6 : 834-86~..

[83

F u j i N. C 1 0 8 6 3 , N e c e s s a r y conditions zation problem in elliptic boundary Journal on Control and Optimization

[0]

H a u g E.J. , C e a ]. e d s C l g 8 1 ) , Optimization oF distributed parameter systems. NATO A d v a n c e d Study Institute Series, S e r i e E, 40, S i j t h o f f and Noordhoff, Alphen aan den ~ijn, Amsterdam.

[I0]

Hlavacek I., N e c a s J. Clg82.3, elliptic unilateral boundary sent method. RAIRO Numerical

[11]

bILchlin S.G. C l g 7 O D , physics. Mir, M o s c o w

[1~]

Miyoshi T. , ( 1 0 7 6 ) , A m i x e d f i n i t e e l e m e n t method solution oF the Karman equations. Numerische $5 : 255-360.

[13]

M u r a w i e O L.A. C I g 8 4 5 , On the existence of the solutions to variational problems in domains with free boundary. Doklady of the Academy os ~ciences os the USRR, Mathematics 248 : 541-S44 (in Russian).

[14]

Myslinski A. , S o k o l o w s k i 5. 0

Q) .

The

equations of

V

(x,t)

~: Q

9

V

{~

i

~

R

(7)

i=i are

a .lJ .(x,t)

of

side

condition

n a

where:

the

(6)

(4]

(i)

+

real

(5)

C OO

functions

constitute

is w r i t t e n

in

the

aij(x't)

c~

defined

a Neumann

following

on

[closure

The

problem.

left-hand

form

n Oy

_

O))A

Oy(x,t)

-

Ox.

i,j= 1

q[x,t)

C8)

3

a where:

is

a normal

derivative

at

r

directed

towards

the

0)? A

exterior being

of

the q(x,t)

First

we

unique

, c o s ( n , x i)

=

the

shall

For

this

introduce

case

the

F

prove of

Sobolev =

a Hilbert

an

y(x,t-r)

mixed

any

direction

~]

cosine

of

n

n

-

and (9)

conditions

for

initial-boundary

the value

existence problem

of

(i)

+

a (5)

L2(Z).

pair

space

of

Hr'S(Q)

HOc0,T;Hr(~]))

space

i-th to

+ v(x,t]

sufficient

v 6

for

is

exterior

the

where

purpose

Hr'S(Q)

Js

at

c(x,t)

solution

for

which

Q

normal

normed

n

real

numbers

([I0}o

p.

HSc0,T;H0(~]))

by

6)

r,s defined

~

0

we

by (I0]

176

II y ( t )

I[ 2

dt

+ II y II 2

H r (~]) where:

the

Chapter

The

spaces

i

[9]

of

problem

the

(5)

the

we

H s ( 0 , T ; H ~ (~]))

are

for

initial-boundary

defined

in

on

cylindeer first,

on

Q2

. In t h i s

way

introduce

15.2

the

=

Qj

Theorem

the

the Q

mixed can

solving ' etc. the

be

proved

(i)

until

+

the

solution

using

(5)

on

procedure

in t h e

a the

covers

previous

step

one.

((j-l)r.jr)

the

solution

in t u r n Q

next

simplicity,

Using

and

i.e.,

and

cylinder

determines

Ej

+

method, Q1

whole

For

a unique

(i)

constructive subcylinder

Hr(~2)

respectively.

existence

value

(11)

H s ( 0 . T ; H ~ (~]))

following

Q x Ej

([I0],

p.

, Ej

811)

notations: = F

• Ej

can

we

for

prove

j =

the

1 ..... K

fol l o w i n g

result. Lelrm~a i Let u ~ H -I/2'-I/4

f]

(Q)

~ H -I/2,-I/4

v e

L 2 (E)

(12)

(Q])

(13)

where f.(x,t) J yj_l(', qj

u(x,t)

(j-l)~)

- b(x,t)yj_l(x,t-r)

~ H I/2

(~])

(14)

e L2(Zj)

(15)

where qj(x,t)

=

Then,

there

exists

mixed

initial-boundary

+ v(x,t)

c(x,t)yj_l(x,t-r

a unique

solution

value

problem

~ H3/2"3/4(Qj)

yj (i),

(4),

for

the

(14).

Proof We

observe

that

for

j = 1

,

yj_llQo

(x,t--r)

=

9

o

(x,t--T)

and

177

yj_iIE,

(x,t-r)

respectively

= 9o(X't-r)

Then

the

assumptions

O

(13) 9 O

(14)

,

H3/2

E

and

3/4(Qo)

assumptions solution

are

(15)

are

Yo

HI/2(Q)

Yl ~ H 3 / 2 " 3 / 4 ( Q I )

f2

H-I/2"-I/4(Q2)

,

easy

to n o t i c e

the

that

Yl 6 H3/2"3/4(QI) 3.1

([9],

the

p.19)

we

[0,r] Trace

that

~

exists the

a unique

Theorem Let

,

for

that

and

ylIXI

follows

2 1

p.

9)

Qj

E

2)

a

uniqe

to v e r i f y L2

(X 1 )

the

using

that

fact

that

Theorem

implies

that

continuous HI/2(~).

(Ql)

. We

implies

Then,

shall

from

mapping

L 2 ( X I)

j = 3 ..... K

from

Again

H 3/2"3/4

is

It

the

continuous

ylIXl

Y2 e H 3 / 2 " 3 / 4 ( Q

~ yl ~

linear,

Thus,

for a n y

from

is yl(-,T)

,

[I0] a

~-

These

of

have

Then

that

L 2 ( X o)

Yl ~ H S / 2 " S / 4 ( Q I )

hence .

~

assume

existence we

HI/2(~) (13)

we

j = 2

yl(-,t)

is

solution

result

now

of there

sun~narize

.

1 and

9o ~ H3/2"3/4(Qo) there

exists

a unique

initial-boundary

value .

j = 1 ..... K

2. P r o b l e m

shall

Neumann

formulation.

now

formulate

problem.

horizon

The

time

The

performance

Icy)

:

u

be

" 9o ~ L 2 ( X o)

for

We

prove

(Theorem

yo,9o,9o,V

Then.

can

, H I ' I / 2 ( X I)

foregoing

~o

the

u ~ H-I/2"-I/4(Q)

YlIZ 1

H3/2"3/4(Q I)

~-

c HI/2(Q)

Theorem

Yl

and

ensure

condition

t

~ H3/4(~)

if

" Next

yl(-,r)

and

mapping

the

to

sufficient

fulfilled

problem

the

(I) +

optimal

T

is f i x e d

functional

by

t,v)

and

u ~

(5).

Moreover,

~

HI/2(Q)

H-I/2"-I/4(Q). for

the

y(-,jr)

mixed

~ HI/2(~)

Theorems

boundary control problem U = L2(X)

in o u r

is g i v e n

Yo

y ~ H3/2"3/4(Q)

Optimization

us d e n o t e

with

, v ~ L2(E)

solution

Let

SQly(x

given

the

space

of

Iz

(Nv) vat dt

for

the

controls.

problem.

by

- Zdl 2 dxdt

+

(16)

178 where: N

k I k

is

0

and

a strictly

Finally,

we

kl+k 2

positive

assume

the

> 0)

;

linear

zd

is

a

operator

following

on

constraint

given element L 2 (E) into on

controls

in L~(Q); L 2 (Z) . v

e

Uad

where Uad

Let

is

a

closed,

y(x,t;v)

value v E

denote

problem

Uad

(i)

We

~

control

problem

I (v o)

I (v)

Then

from

a unique by

the

from

the

c is

V

the

(x,t)

the

-

(17)

of

the

mixed

1 that

initial-boundary

to

for

a

any

is

given v

e

v 9

simplify Uad

solution

, we of

8p(v) at

p(:

0

characterized

condition

of

the

Z

0

V

performance

v 6

Uad

(18)

functional

(16)

we

can

express

(18)

form

Zd) (y(v)

(19),

we

define the

- Y(Vo))

dxdt

+ k 2 LNVo(Vg

- v o)

introduce the

the

adjoint

adjoint

variable

equation p

= p(v)

dr

dt

Uad

and

h

0 (19)

for

= p(x,t;v)

every as

the

equation

+ A

(t)

=

ap (v) at

Uad

wel l-defined

V v 9 To

control

p

(v - v o)

form

U

(16)

control

following

k I ~Q(y(v ~

Uad

of

corresponding

Theorem

L2(Q)

Theorem

following

i17 t h e

solution

at

equivalent

v 9

optimal

I' (v o) Using

(5)

subset

functional

H3/2"S/4(Q)

~

the

+

note

performance y(v)

convex

p(v)

kl(Y(v)

+ b(x,t+T) -

p(x,t+r);v)

zd)

x

= ~

~

,

t

9

(0,T-T)

(20)

(T-T,T)

(21)

*

+ A

=

(t)

0

p(v)

=

kl(Y(v)

-

z d)

x 9 r~ , t e

x

,-s_- f )

(22)

179 ap(v)

(x,t)

=

c(x,t+'r)

(x,t)

=

0

p(x,t+T;V)

X ~ F

, t ~

(0,T-r)

(23)

x

,

(T-r,T)

(24)

A ap(v)

E

F

t

~

A where n

A

(t)p

ap(v)

00

__ ~ i,j=l

(x,t)

,

existence

=

apply

the

Lemma

1

solution

zd

(with

v

t = T

and

in

of

Q

change time,

(20)

on

(20)

+

first,

this

t'

on

be

to

solved

solving

(20)

until

purpose, to

the

is e a s y

can

etc.

of v a r i a b l e s )

i.e.,

(24)

It

(24)

QK-I

. For

+

method.

, i.e.,

turn

cylinder

sense

problem

. problem

an o b v i o u s

reversed

the

a constructive

from

QK

whole

for

using and

startffng

the

(x,t)

+ tile

we

may

problem

(20)

= T-t)

2 hypothesis

z d ~ L2(Q)

We

given

covers

(with

the

p(v)

c o s ( n , x i) --~--]

proved

subcylinder

Theorem

(24)

Let

for

be

in t i m e

on

Op (v)

aji(x,t)

a unique

can

that

procedure

+

of

Q

backwards (24)

(25)

i,j= 1

cylinder notlce

[ a i j ( x , t ) a--~i lap

n ]~

A

The

~a

and

of

any

~ HS/2"3/4(Q)

simplify

purpose (20),

(19)

setting (21)

and

Q

(21)

we

x

get

X1

f

for

using

(y(v)

(T-T,T)

(Y(Vo)

1

v ~ L2(Z)

v = v

by

Theorem

the

the

satisfied.

there

exists

problem

adjoint

in

o - Y(Vo))

respectively

- z d) (y(v)

be

(20)

+

and

- Y(Vo))

unique

+

(24).

multiplying

both

+

(20)

integrating

then

adding

dxdt

glven

solution

(24).

equation

then

for

a

(20)

(24),

Then,

over both

~ sides

For

this

sldes x

of

(0,T-T) of

(20),

=

Q aP(v o ) . ] 0t + A ( t ) p ( v o) (y(v)

[ =

IQ

Y(Vo))

dxdt

+

T--T + f

f o

b(x,t+~)

p(x,t+T;Vo)

(y(x,t;v)

- y(x,t;Vo))

dxdt

=

180 =

#

p ( v o) ~

(y(v)

-Y(Vo))

dxdt

+

- Y(Vo))

dxdt

+

q + ~

A*(t)

Q

P[Vo)(Y(v)

T--T + f

f o

Using

the

(26)

b(x,t+r)

p(x,t+T;v

) (y(x,t;v)

f) equation

can

be

(i),

rewritten

~ Q p ( v o) - ~

-y(x,t:Vo))

dxdt

(26)

o

(y(v)

the

first

integral

on

the

right-hand

side

of

- Y(Vo))dxdt

-

as - Y(Vo)]dxdt

=

-~QP(Vo)A(t)

(y(v)

T - I

# P ( x ' t ; v o) b(x,t) ( y ( x , t - T . v ) - y(x,t-r;Vo)) dxdt = o ~] T-r = - f P ( V o ) A ( t ) (u - Y(Vo))dxdt -I J" P(x'L'+~" ;Vo)b(x't'+r) Q --T ~ (y(x,t' ;v)

The

second

Green's

- y(x.t' ;Vo]]

integral

formula,

A

on the

right-hand

can be e x p r e s s e d

(t)p(Vo](Y(V)

(27)

dxdt

side

- u

=

lay (v)

oy (Vo) ]

o Using

the

boundary

right-hand

side

T

of

condition (28)

[~y (v)

0 optimal

problem (34)

condition

following

Theorem

exists

to

and

(35)

be

+

(39).

also

performance

with

constraints

v

in

the

form

(33)

wit]]

(17),

there

fulfilled.

the

control

rewritten

o

which

functional on

control

satisfies

the

maximum

(32).

Re#erences [I]

[2]

[3]

[4]

[5]

[6]

[7]

[8] [9]

[I0]

[ii]

[12]

Knowles. G . , "Time-Optimal Control of Parabolic Systems with Boundary Conditions Involving Time Delays". J. Optim. Theory AppI., Vol. 25. 1978, pp. 5 6 3 - 5 7 4 . I I,

(ek)

P k + /JkC'+ Pk+IG'B = (Pk+ PkO')'A+ (Pk+i+ Pk+IC')B + (pk+ PkC')H~

I t e m ark

If A be the s u p r e m u m of the real p a r t s of the n o n z e r o eigenvalues (AI) of the m a t r i x B , then, for any t,

IlCz - P~)s'~ll

_< c, ~'

w h e r e G is a p o s i t i v e c o n s t a n t , and k < O. T h e v e c t o r P k ( f . ) ( I -- PB)r ~B belongs to R ( B ) , and is called t h e k-th o r d e r b o u n d a r y layer term.

T e n d i n g 9 to O, w e get

(S_~)

0 = po B po(o) + Po(o)

(so)

Po

PoG"

= p

= poA + p i B

+ p o H ~I

= PoC'-A+ P o C ' H 9

a n d , for k > 1, p~(o) + pk(o)

= o

Pk = P k ' A + Pk+1 B + P k H ~

(Sk)

/6k(7" = Pk6"'A-+ P ~ G ' H 9 Using the decomposition of ~ n

p = p+ p w e get finally

with

for every p, w e have

~'B = 0

and

PPB = PPB = "P,

193

(s_~)

0

po(O) + ?o(O) (So)

= /3oB =

p

/~o = poA + ,6tB + PoHi)

? oC" = ['oc'-E + PoC "H a n d , for k >_ 1,

p~(o) + ?k(o)

(sk)

Pk

= 0 =

pkA

+

Pk+lB

+

pkH~l

?~c" = ?kC"T+ i'kC'U~ in I t b - f o r m , we get

And

po(0) + 2;0(0) (So)

=

p

dpo

= poAdt + p t B d t + poHdy

dPo

=

P o C ' A C -tdt + P o C S l t C - ' d Y

and, for k >_ 1,

pk(o)

+ ?k(o)

dpk

(Sk)

= o = pkAdt + @k+iBdt + pkHdy

di'k = @kC'AC-'dt + ['kC'~C-'dy Remazks

9

C o n s i d e r t h e s y s t e m ( S _ I ) ; a n d , we h a v e , for e v e r y t

poCO = FoCt) and,

d'fo

= ~oAdt + ~lBdt +'PoHdY

a n d , if we a p p l y t h e o p e r a t o r Pn~ we o b t a i n

d~o = "~oAPBdt + poH PBdy Now,

by the unicity of the decomposition

of semimartingales, w e have

~'oH = FoH pn. This m e a n s

that w e m u s t verify the following equality

PDH

= PBHPB

or

MIT

= MHPB.

1 94 F o r i n s t a n c e , t h i s e q u a l i t y is v e r i f i e d if t h e m a t r i x H is d i a g o n a l c o n s t a n t on e v e r y b l o c k of the matrix

B:

A s PB a n d B h a v e t h e s a m e b l o c k - s t r u c t u r e ,

we o b s e r v e e v e r y b l o c k of B .

P~ and H commute.

In the sequel, we shall assume that. 9

We

need to a s s u m e the invarlance of k e r B by the operator A in order to justify the

existence of the equations in ]~k. 9

T h e c o m p o n e n t Pk is obtained by solving the Poisson-equation

d (pk-l-- fo'pk-l(s)Hdy)--pk-tA Let J~fn be the inverse of B in the subspace R ( P n -- I) :

HBB

= B I I B = P D -- I

and

HBPD

= P D I I B = O.

Then

~k=--~d( P k - 1 - - fo'Pk-l(S)Hdy)~ + pk-l.4~v. 9

T h e c o m p o n e n t P'k satisfies the following equation

d'~k = p k A P B d t + ~'ktt P B d y , obtained by applying the operator PD.

5

Asymptotic

Result

In this section, we give an approximation result for the (m + 1) th order. Towards this end, we consider the following system of equations

p0(0)+ P0(0) (So)

a n d , for 1 < k < m ,

=

p

dpo

ffi poAdt + p l B d g .4- p o H d y

dPo

= P o G * A G - S d t + PoG*HG-Sd~/

195

pk(o) + e~(o) ( s k)

=o

tips, = p k A d t + p k + t B d t + p k H d y = PkC 9149

dPk We choose

Pm+ls u c h

+ PkC*HC- 9

that

p.~+~(0) dpm+t Proposition

= pm+J.Adt + Pm+lHdy

G.I

F o r a n y t E [O, T1, w e h a v e

E~Clp'(~) - ~ "kCPkCt) + PkC~)c'C~))123 = oC.~I",+~)) k=O

Proof We d e f i n e , for a n y t E [0, T],

r~.C~) =

p'CO - ~ ~kCp~C~) + PkCOc'ctl)) - ~ ' + I p . , + I C O k=O

It can be s h o w

that r m9 is solution of the following equation B

dr" =

r'~(O) = Thus

, ~ C - + Ale~+ r~,Uey

0

we have

(t(r:(s)(B+ A), B u t , for every p, we have C P ( ~ + A L p )

fot

fot

< 0, so we obtain

~(Id.(~)l ~) -< ~'('+~)o~ + I1~ 11'

/: ~ClrkCs)l~d.

Using proposition 4.3, and then the G r o a w a l t - B e llman inequality, we obtain, for any t E I0, T]

~.(l~c,)l ~) _< ~(..+,o and we get the desired result.

196

Decentralization and aggregation

6

We consider the systems (SO) and (Sk)

vo(O)+

?o(O)

(So)

= p

dpo

= poAdt + P t B d t + p o H d y

d~'o

= fioC*AG-*dt + PoC*HC-*dy

and, for k > 1,

pk(0) + ?k(0) (Sk)

dpk

=0 = p~Adt + ~k+xBdt + p ~ H d y

dPk = P k C * A G - C d t + P k G * H C - * d y If we apply the o p e r a t o r PB to the different equations of the systems (So) and (Sk), we get (S'o)

P~176 d~o

=F = p o A P B d t + poHPl3dy

and, for k _> I,

(sk)

LCo)

=0

d~k = pkAPmdt + p k H P B d Y B y subtraction, we have Po(o)

--

d]30 = p o G C A G - ' d t + ~6oC~HC-Cdy and, for k > 1,

~(o) + ?k(0) -- 0 (Sk)

d~k

= ~ k A ( I - - P B ) d t + Pk+IB d t + p k I t ( I -

dP k

= PkC*AC-'dt + Pk_xC*HC-*dY

PD)dy

Thus, by the block-diagonal structure of B, /5k and /Sk m a y be calculated by solving all equations on every block of B. The dimension of these systems is reduced.

Now, consider the systems (~'o) and (S'k) ~o(O)

=

CSo) d'fo = "foAPBdt + "foH PBdy

197 a n d , for k > 1, ~k(o)

CTk)

= o

d'pk = p k A P f l d t + pkH PBdy or,

cT,~)

~k(O)

P u t p ~ = PkQ , P ~ =

PkQ a n d A ~ = M A Q .

d'~

= 0

= "fkAPBdt + ~ k A P n d t + "fkH PBdy + ~kH PBdy T h e n we o b t a i n t h e a g g r e g a t e s y s t e m s

(s~) dp a = paAadt + paHdy a n d , for k > 1, p~(0)

dp~

= 0

= p~Aadt + ~ k A Q d t + "pkHdy

Remark T h e s e d i f f e r e n t e q u a t i o n s a r e of d i m e n s i o n

r, w h e r e r is t h e n u m b e r of t h e e r g o d i c s e t s

of t h e m a t r i x B . References [1] C o d e r c h - W i l l s k y - S a s t r y - C a s t a n o n ,

Hierarchical Aggregation of Linear S y s t e m s with Mttl-

tiple T i m e Scales , I E E E T r a n s . on A u t . C o n t r o l , A C - 2 8 , 1 0 1 7 - 1 0 2 9 , 1983. [2] C o d e r c h - W i l l s k y - S a s t r y - C a s t a n o n ,

Hierarchical Aggregation o/ Singularly Perturbed Fi-

nite Stats Markoe P r o c e s s e s , S t o c h a s t i c s , 8, 2 5 9 - 2 8 9 , 1983. [3] C o u r t o i s , Decomposability, A C M M o n o g r a p h [4] D e l e b e e q u e ,

S e r v i c e s , 1977.

A Reduction Process for Perturbed Markoe Chains , J. S i a m A p p l .

Math,

43, 2, 325-350, 1983. [5] D e l e b e c q u e - Q u a d r a t ,

actions , A u t o m a t i o n ,

Optimal Control Markoe Chains A d m i t t i n g Strong and Weak Inter17, 2 8 1 - 2 9 5 , 1981.

[6] K a t o , Perturbation T h e o r y / o r Linear Operators, S p r i n g e r - V e r l a g , 1966. [7] M a r c h e t t i ,

Mgthodss de Perturbations Singuli~res en Filtrage non Lingaire , T h ~ s e , U n i -

v e r s i t 6 d e P r o v e n c e , 1986.

FILTRES APPROCHI S POUR UN PROBLI ME DE FILTRAGE NON LINI AIRE DISCRET AVEC PETIT BRUIT D'OBSERVATION P a u l a Milheiro de Oliveira * INRIA centre de Sophia Antipolis BP 109 06561 Valbonne Cedex (France)

Abstract W e study the ~ymptotic behaviour of a nonlinear one-dimensional filtering discrete time problem, as

some parameter r tends to O. We treate the case of a nonlinear discrete time problem coming from a continuous time one with small observation noise. Finite dimensionM aproximate filters are proposed and results concerning estimations of their performance are stated and proved. For the result concerning the error between the approximate filter and the optimal one, only a sketch of the proof will be given. It makes use of probability changes and differentiation with respect to the initial condition. FinMly, we present the results obtained when applying those filters to an example and we notice that the propositions previously stated are verified numerically.

1

Introduction

On consid~re le prohi~me de filtrage non lin6aire unidimensionnel, eu t e m p s discret, suivant: On a un signal {X~} v6rifiant

x~+, = x~ + b (x~) zx, + . v ~

~ + , , x0 = ~

(1)

et on dispose d ' u n e observation y donn6e par g

o~ {wk} et { ~ } sont des bruits blancs gaussiens s t a n d a r d s ind6pendants et ~ une v.a. ind6pendante de {w~} et {~h~}. Le param~tre e eat suppos6 petit et A t = ~ , cz > 0. Le probl~me de filtrage consiste b. calculer, pour toute fonction mesurable r l'esp6rance conditionnelle de r 6taut donn6e l'observation yt j u s q u ' k l'instant k E {0, 1 , . . . , K}. On remarque qu'on peut regarder ce probl~me comme 6tant la discr6tisation d'un probl~me en temps continu, n s'agit, k cheque instant t E [0, T], d'estimer le signal {X,} solution de l'6quation diff6rentieUe stochastique dXt = b ( X t ) d t + a d w : , X0 = ~ (3) au vu de l'observation {Yt} donn6e par

dY~ = h(Xt) dt + ~dw~ , Y0 = 0 "et Faculdade de Eugenlmria da Universidade do Porto, Run dos Bragas, 4099 Porto Codex (Portugal).

(4)

199

oh wl,w 2 sont des processus de Wiener standards ind6pendants et ~ est une variable al6atoire ind6pendante de w 1 et w 2. (cf. [Le Gland], [Picard]). I1 est naturel de penser que si a est grand (donc At petit) le comportement de (1)-(2) est proche de (3)-(4) alors que, si a est petit, il s'en 6carte. I1 se trouve en falt que le cas critique est a = 1. Plus pr6cis6ment on s'int6ressera au calcul de l'estimation du signal {Xe} sachant l'observation j usqu'~, l'instant k, )(k = E[Xe[Y0k], laquelle minimise l'erreur quadratique moyenne. Vu les difficult6s num6riques qu'un tel calcul pr4sente, on cherche des approximations de {)(a} (filtre optimal) donn4es par des sch6mas r6cursifs. Le comportement asymptotique de ces filtres lorsque r tend vers 0 fern objet de notre 6tude.

2

Construction

des filtres approch6s

Pour des raisons li6es g l'application d'une m6thode de changement de probabilit6s, pour obtenir les estimations de l'erreur quadratique moyenne, il sera plus commode de consid6rer notre observation sous la forme suivante:

~+~ = h(x~) + ~

~+,,

(5)

off {va} est un bruit blanc gaussien standard ind6pendant de {wk}. Formellement, l'observation ffk+~ contient la m6me information que l'observation yk. On suppose que ~'~ est la tribu des observations jusqu'~ l'instant k:

Alors 17d = y0~-I ce qui signifie que le probl~me de filtrage pour 17"~ correspond en fait ~t une pr6diction pour Y0k. On consid6re le processus Me+t=Me+b(Me)At+0(~e+l-h(Me)),

Mo=mo

(6)

et on veut 6tudier la fa~on dont il approche Xk = E[XklY~]. Des rdsultats qu'on obtiendra sur l'erreur de pr4diction Xk - Me on d6duira immediatement les r6sultats 6quivalents sur l'erreur de filtrage, puisque, voir le paragraphe 2.1, on a

I s - .eel < c a t , o~ Re = E[Xe_d~?]. Pour la construction du flltre approch6 on peut s'inspirer de ce qui se passe dans le cas lin6aire et ainsi, par g6n6ralisation, choisir une approximation du gain. L'estimation de Xe - Me sera obtenue b. partir de l'6tude d'une suite r$currente. Par contre, pour estimer )~k - Mk, on introduira d'abord deux changements de probabilit6s et on obtiendra une expression pour Xe - M~ qui fait intervenir les d4riv6es de Xk par rapport ~ la condition initiale. Sur cette expression, on conditionnera les diff6rents termes par rapport ~ la tribu ~k des observations et on proc~dera ~ des estimations asymptotiques. Selon la valeur de c~, les changements de probabilitds choisis seront diffdrents et on aboutira, naturellement, ~. des expressions asymptotiques diff~rentes pour Xe-Me. Darts une premiere section on 6tudie le cas a > 1 (section 2.1) et ensuite le cas a = 1 (section 2.2). Le cas cr < 1 offrant certaines difficult6s dans l'application de cette m.4thode de d6monstration, on se limitera k faire une remarque proposant un filtre approch6 "performant".

200 HypothSses:

On suppose que

(H1) ( est une variable aJdatoire de loi de probabilitd po telle que I

poet', fl ol'Po( )d


_ eh > O, Vz. (It3) best une fouction C 2 ~. d~riv&s born~es, [b'(z)[ _< lib'I[, Vz. (H4) A t = ~ , 2.1

Le cas

a>_ l e t t , . = k A t . a>

1

On considSre d'abord le cas oh At = ~', avec a > 1. On propose un filtre approch~ unidimensionnel et, pour ce filtre, on estime XL. - M~ et )(~ - M,.. Le filtre a p p r o c h d On prendra 8 = a At darts le sch+ma (6), i.e. M,.+~ = Mk + b(M~) A t + a A t (Ok+x

h(M,.))

Mo = mo

(7)

On obtient facilement le r6sultat suivant: P r o p o s i t i o n 2.1 Mk dtant donng par (7), on a l'estimas r

tk+l 1

E[(Zt.+t - M,+,)2I 5 c e x p t - c T ~

+ ce.

(8)

Preuve On peut utiliser les d~veloppements de Taylor des fonctions b e t h, b(X~) = b(M,.) + b'((X)(x~ - M~) et h(X~) = h(M~) + h'((~)(X~ - Mk) , pour obteuir l'~galitd Xk+l - M,.+, = [1 - ~r At h,({x) + b,(c.x) At] (Xk - M~) + a v / ~ { w k + , - v k + 0 .

(9)

E

On utilise les hypothb.ses pour dfimontrer que, pour v "assez petit" (e _< (ah)-~l{"-l)), B EI(X%+, - M~+,) 21 _ (1 - d)k+'E[(Xo - M0) 21 + ~ - , o~t

At A --a- 1 - ( 1 - c h - - + t l b ' l l A t ) 2

qi.

G i has Then

no

free

D i can

be

A compatible partitioning

Gi2]0

G i D i -- Gil Dil + Gi2 Di2 -- M i

(22)

Since Dil is invertible: Gil = [Mi-GizDi2] Di}

(23)

Gil is used to satisfy GiD i = Mi, a n d Gi2 is obtained by optimizing Ji.

The

matrix G i can be written: G i -- Gil E l + Gi2 E 2

(Z4)

where E 1 and E 2 are selection matrices. Substitute

(23) into (24):

G i -- H i + Gi2 T i where

(25)

H i = M i Di~ 1 E l T i = (E 2 - Di2 Di~ 1 E 1 ) It remains

propagates

as:

to determine

K i and

Gi2 that minimize

Ji"

The

error

covariance

214

# = FiP + PF~ + BiQB I + KiRiK ~ + (Hi + Gi2T i) Si(Hi + Gi2Ti) T

(26)

P(o} = Po The Hamiltonian corresponding to the optimization problem defined by equations 17, 19, 26 is given by: B = trace (UiP + [FiP + PF[ + BiQB l + KiRiK ~ + (Hi+Gi2Ti)Si(Hi+GizTi)T]A T where

A, m a t r i x of L a g r a n g e

multipliers, is determined

(27)

by:

(28)

=- aB =- [Ui + F i' A + ,%Fi]

A(Tf) = V i

(Z9)

K i and Gi2 are obtained from the necessary optimality conditions (30)

al~/aK i = 0 a n d a~/aGi2 = 0

These conditions give: K i = p R~ ~

(31)

Gi2 =

(32)

- H i S i TiT(Ti S i T~)-*

The gain matrix, given by equation (31) is the Kalman gain associated with the following subsystem:

z i = A i zi + [Bi

Gi][ wvi ]

(33)

Yli = zi + ;Ji S i n c e (Ai, I i) i s o b s e r v a b l e

(34) f o r a n y A i t h e n F i : [ A i - K i] i s s t a b l e . Q.E.D.

S u f f i c i e n t c o n d i t i o n f o r t h e s t a b i l i t y o f t h e l o c a l f i l t e r c a n be s t a t e d in t e r m s o f the observabilty

~t

of the original system.

Ei = [ Li

Theorem 2:

0]

If (A, L i) is observable then the local filter is stable.

Proof: Consider equation (6). Define :

=

[ -Alp

-MN] ;

~i = [li

O]

215 where

(A, Li)

is observable if and only if

(F, C i) is observable since they are

related by a similarity transformation (T i in eq. 7).

p( [ ~ Write

-T-r F Ci

(~I )n-l~lr

...

Therefore:

) = n

[Ai o]+[o

F =

0

0

Then:

ci~k [li~ where

•k(1,2)

o]

+

[o

li.k(1,21]

is t h e b l o c k (1,2) o f n k c o m p a t i b l e

with the partition

used

here.

The observability matrix corresponding to the system (F,Ci) is:

Ci F

[ Ii

0

i Ai

0 :

:

i

Am-I

0

0

0

a1(1,2)

o

0

am_l(1,2)

o

LG

nn_l(1,2)

+

J: : ".11-I

~i ~ n-I

The Therefore

n

.

columns

Ai

of

the

above

observability

matrix

are

t h e m c o l u m n s of t h e f o l l o w i n g m a t r i x a r e l i n e a r l y

[ I i T AI

""

independent.

(A~I~-I]T

This corresponds local filter designed

linearly independent.

to t h e o b s e r v a b i l i t y using

equations

m a t r i x o f t h e local s u b s y s t e m .

(33) a n d

(34) is s t a b l e .

Q.E.D. V.

Consider

the 3rd order

EXAMPLE

system:

[120] ill

x=

2

4

5

1

I

3

x

+

2

1

w

Hence the

216 Let x I be the observed variable at Station variables at Station

(I), a n d x 2 and x 3 be the observed

(2), i.e.,

Zl = Xl

Therefore,

it is s o u g h t , to e s t i m a t e z i a t S t a t i o n S i.

C o n s i d e r S t a t i o n (1).

The measurement

is g i v e n

by:

Ytl = zl + Pl Let the required

additional

m e a s u r e m e n t on t h e i n t e r f a c e

variable

z I = x 2 be

given by:

Y12 = [1 Let:

2] r z{ +

v1

Q = R 1 = I; S 1 = Zx2 identity matrix Using the notations of the paper: A 1 = I,

M I = 2,

Then:

GIDI=M1

gives

Let

Dll = I

Then

G l l = (2 - 2 G12) G 1 = [GII

=

E 1 = 2 [1 1]

-.

G1 = [2

0]

+ G12 [ - 2

2 [1

0] 0]

=

0]

+

=

[2 [-2

G12 [0

I]

0] 1]

1]

[:][ [:]1

0l - [Z 01 -

G 1 = [.4

It remains

2]

GII [I

T 1 = [0

GI= [2

Finally:

= [I

GIIDII + GI2DI2 = GII + 2G12 = Z

GI2]

H1 = M1 D [ }

And :

DII r

[ - 2 1] -

[ - 2 11

.S]

to s o l v e f o r K 1. U s i n g e q u a t i o n

(31}, t h e

T h e n , t h e local f i l t e r a t s t a t i o n (1) is g i v e n by:

z 1 = - 1 . 6 7 z 1 + 2 . 6 7 Y l l + [ .4 S i m i l a r l y , a local f i l t e r is d e s i g n e d

. 8 ] Y12

for station

{2).

steady-state

g a i n is 2.67.

217 VI.

The p a p e r

a d d r e s s e s t h e d e s i g n o f d e c e n t r a l i z e d f i l t e r s for l a r g e - s c a l e i n t e r -

connected systems. models.

The

variables. unbiased

These filters are

decoupling

These and

CONCLUSION

is

independent

stable.

designed

achieved filters

use

measurements

local

Sufficient condition for

o b s e r v a b i l t i y of original s y s t e m .

u s i n g i n d e p e n d e n t local dynamical

through

of

the

information only

and

interface they

are

s t a b i l i t y is s t a t e d in t e r m s of t h e

An example is g i v e n .

ACKNOWLEDGEMENT

This research was sponsored in par$ by the National Science Foundation under grant

ECS-8707139,

Research

Center

in part

under

by the

grant

West

PEL-21-88,

virginia University Energy and

in part

by

ONR

and

under

;r

contract

NO0014-K-0651.

REFERENCES

[1]

J.L. S p e y e r , "Computation a n d T r a n s m i s s i o n R e q u i r e m e n t s f o r D e c e n t r a l i z e d LQG C o n t r o l P r o b l e m s , " IEEE T r a n s . A u t o . Control, Vol. AC-24, No. 2, p p . 266-269, 1979.

[2]

C.Y. Chong, " H i e r a r c h i c a l E s t i m a t i o n , " P r o c e e d i n g o f 2 n d I~IIT/ONR C3 W o r k s h o p , M o n t e r e y , CA, J u l y 1979.

[3]

A.N. Willsky, et.al., "Combining a n d U p d a t i n g of Local E s t i m a t e s a n d Regional Maps Along S e t s of O n e - D i m e n s i o n a l T r a c k s , " IEEE T r a n s . Aurora. Control, Vol. AC-27, No. 4, pp. 799-813, 1982.

[4]

A.T. Alouani, J.D. Birdwell, " D i s t r i b u t e d Estimation: C o n s t r a i n t s on t h e Choice of t h e Local Models," IEEE T r a n s . A u t o m a t i c Control, Vol. AC-33, No. 5, pp. 503-506, 1988.

[5]

C.W. S a n d e r s , E.C. T a c k e r , T.D. L i n t o n , "A New Class of D e c e n t r a l i z e d F i l t e r s f o r I n t e r c o n n e c t e d S y s t e m s " , IEEE T r a n s . A u t o m a t i c Control, Vol. AC-19, pp. 259-262, 1974.

[6]

A. Feliachi, R.K. S p e r r y , " D e c o u p l e d F i l t e r s f o r I n t e r c o n n e c t e d S y s t e m s , " P r o c e e d i n g s o f 27th IEEE-CDC, A u s t i n , TX, p p . 2367-2368, December 1988.

[7]

E.C.Y. Tse, J.V. Medanic, W.R. P e r k i n s , " G e n e r a l i z e d H e s s e n b e r g T r a n s f o r m a t i o n f o r R e d u c e d - O r d e r Modelling of L a r g e - S c a l e S y s t e m s " , I n t . J. C o n t r o l , Vol. 27, No. 4, pp. 493-512, 1978.

SPLIT FORMS OF z - D O M A I N A L G O R I T H M S FOR LINEAR PREDICTION A N D S T A B I L I T Y A N A L Y S I S

Antonio Lepschy, Gian Antonio Mian and Umberto Viaro Department of Electronics and Informatics, University of Padova via Gradenigo 61A - 35131 Padova, Italy

A b s t r a c t - Many algorithms in linear prediction and s t a b i l i t y analysis may be expressed by the same general t w o - t e r m recursion involving two polynomials of consecutive degrees and the reciprocated polynomial of one of them. The properties o f these algorithms are interpreted from a geometrical point of view that refers to the loci described by the zeros of the polynomials in the related sequences as a characteristic real p a r a m e t e r varies. An analysis of all possible threeterm (split) forms involving only the symmetric and/or the a n t i s y m m e t r i c parts of three consecutive polynomials generated by the same t w o - t e r m recursion, is carried out. 1. I N T R O D U C T I O N A number of algorithms in linear prediction and s t a b i l i t y theory can be expressed by a recurrence relation for generating a sequence of polynomials of either ascending or descending degree [ I , 2] . The Levinson algorithm is perhaps the most popular of these. As is known, it was conceived in the c o n t e x t oF mean square estimation to find in a c o m p u t a t i o n a l l y e f f i c i e n t way an all-pole predictor. However, it may be used for d i f f e r e n t purposes; for instance, a tight relation exists between this algorithm and the s t a b i l i t y c r i t e r i a of Lehmer, Schur, Cohn, Marden, and Jury [ 3 ] , as well as the Nevanlinna-Pick problem ELI) 9 Another f a m i l y of z-domain algorithms comprises the so-called Routh-type methods, obtainable through a suitable variable transformation from the original s-domain three-term Routh procedure or from its t w o - t e r m form [ 5 ] . In particular, if the usual bilinear transformation is adopted, the algorithms for discrete-time systems modelling presented b y B i s t r i t z in E6] are obtained, whereas if the transformation s=(z +z- )12 is used, the B i s t r i t z s t a b i l i t y - t e s t procedure ~ 7 , 8 ] is obtained. Other families of z-domain algorithms may be found by transforming the s-domain Euclid-type a l g o r i t h m presented in ~ 9 ] . Much interest has been paid in the recent l i t e r a t u r e to unifying interpretations of some of the mentioned procedures. This has been done, for instance, by Benidir and Picinbono [ 10], by Delsarte et al. E 4 ] , and by Vaidyanathan and M i t r a [ I I ] according to either a network-synthesis approach or to an interpolation approach. Another a t t e m p t has been made in [ 12] for s-domain algorithms by adopting a different point of view; specifically, it has been shown that many algorithms can be regarded as particular cases of a unique general recursive procedure from which new interesting algorithms may be generated as well. A similar approach can be applied to zdomain algorithms as will be briefly outlined in Section 2 . In this way, it is also possible to give a geometrical c h a r a c t e r i z a t i o n of the various procedures along lines analogous with those followed in E13-] for the s-domain. The considered algorithms may be given either of the following recursive forms: ( i ) the t w o - t e r m form by which every polynomial in the related sequence is expressed by a suitable combination o f the preceding polynomial and its r e c i p r o c a t e d one, and [ i i ] the t h r e e - t e r m form which relates the s y m m e t r i c or a n t i s y m m e t r i c part of three consecutive polynomials in the sequence. The latter have been called in the literature split forms E14, 15], or immitance-domain forms

219

[ 16J as opposed to s c a t t e r i n g - d o m a i n ones. Such f o r m s are a t t r a c t i v e because they e n t a i l c o m p u t a t i o n o f a s m a l l e r n u m b e r o f c o e f f i c i e n t s and a l l o w one t o i n t e r p r e t the a l g o r i t h m s terms of reactance functions.

the in

In S e c t i o n 3 we c l a s s i f y and analyse f r o m the p o i n t o f v i e w o f the c o m p u t a t i o n a l c o m p l e x i t y all possible split r e l a t i o n s a s s o c i a t e d w i t h the same g e n e r a l t w o - t e r m r e c u r s i o n d e r i v e d in Section 2.

2. G E N E R A L

TWO-TERM

FORM

L e t us d e n o t e the i - t h degree p o l y n o m i a l in a s e q u e n c e by i

P.(z] = ~, a.. z j i iI j=0 and its r e c i p r o c a t e d p o l y n o m i a l by i P.(z) = z i P.(z -1] = ~ a.. z i-j . I

I

(1]

(2]

U

j=O The Common f o r m o f the t w o - t e r m

s t e p d o w n ( b a c k w a r d ) r e c u r s i o n s m e n t i o n e d in S e c t i o n 1 is

Ri(z) Pi I (z) = F.(z) P . [ z } + G.{z) P.(z) -

I

I

I

(3}

1

w h e r e Ri(z), Fi(z), Gi(z) are p o l y n o m i a l s o f s u i t a b l e d e g r e e (which is, in g e n e r a l , d i f f e r e n t From i and m a y d i f f e r From step to step) such t h a t Pi(z) is a Schur p o l y n o m i a l (all its zeros are inside the unit c i r c l e ) i f and o n l y i f Pi_l(Z) is Schur and the p a r a m e t e r s belong to c e r t a i n d o m a i n s . In p a r t i c u l a r , i f R.(z) = r

(4) = - ai0/aii w h e r e r i m a y be v i e w e d as a scaling f a c t o r ( w h i c h is equal to I - k 2 in the s t a n d a r d f o r m ) , r e c u r sion (3) r e d u c e s to the ( b a c k w a r d ) L e v i n s o n r e l a t i o n : i

z

i

'

F.Cz} = 1 I

"

Gi(z) = k i

=

-

PiC0)/Pi(O)

r z P (z) = P.(z} + k . P . ( z ) (S) i i-1 I I ; in which k i is the s o - c a l l e d i - t h r e f l e c t i o n c o e f f i c i e n t . As is k n o w n Pi(z) is Schur i f and o n l y i f P i _ l ( z ] is Schur and I k i I < 1 . A g e o m e t r i c a l insight i n t o this a l g o r i t h m can be g a i n e d by c o n s i d e r i n g t h a t the zeros o f both sides o f iS) b e l o n g t o the r o o t locus f o r the e q u a t i o n : P.(z) + c P.(z) = 0 1

(6)

i

f o r c = k i . This locus is s y m m e t r i c w i t h r e s p e c t t o the unit c i r c u m f e r e n c e and its " i " b r a n c h e s [which include the real axis) i n t e r s e c t this c i r c l e f o r ] c I = I . C l e a r l y , f i x i n g a r o o t o f (6) o,1 this locus c o r r e s p o n d s to assigning the v a l u e o f c and, thus, l o c a t i n g all o t h e r r o o t s too. The l e f t - h a n d side o f (5) has a zero at the o r i g i n and, t h e r e f o r e , the z e r o s o f Pi_l(ZJ lie on the o t h e r " i - 1 " b r a n c h e s o f the locus f o r the same v a l u e o f p a r a m e t e r c, i.e., ki, as the o r i g i n . The a l g o r i t h m m a y be e x t e n d e d t o the m o r e g e n e r a l case [4"] o f R.(z) : r .

(z-q),

I

q real,

] q I < I .

(7)

I

C o n s e q u e n t l y , the zeros o f Pi_1(z) (which c o r r e s p o n d t o the same v a l u e o f c as q) lie inside the unit c i r c l e i f Pi(z) is a Schur p o l y n o m i a l . The s i t u a t i o n is d e p i c t e d in Fig. I f o r a Schur p o l y n o m ial Pi(z) w i t h i = L I . If, instead, R.(z) = r. ( z + l } 2 , I

where

i

F.(z) = l+h. + (1-h.) z , I

I

a g a i n r. is a s c a l i n g f a c t o r and !

I

G . ( z ) = ( - I ) i h. ( z - l ) I

I

(8)

220 Im z

/

/

\

!

t

\

i

)o

9 o"~o

O

Sk

Ak

9

O

O

If, instead, only s y m m e t r i c or only a n t i s y m m e t r i c parts a r e o f interest, the same split relation (either that corresponding to column 1 or that corresponding to column 2, respectively) will be used in all steps ( s t r i c t l y recursive split procedures). On the other hand, it may be shown that the relations corresponding to columns 3 and 4 (as well as those corresponding to columns 1 and 2, to columns 5 and 6, and to columns 7 and 8) can be given a common structure so that one might consider as being s t r i c t l y recursive also the procedure involving s y m m e t r i c and a n t i s y m m e t r i c parts alternately. Observe that the split algorithms must be p r o p e r l y i n i t i a l i z e d by computing the s y m m e t r i c or a n t i s y m m e t r i c part of Pn-I from the given original polynomial Pn" To this purpose, use can be made o f equations (22a) and (22b). Also, if the complete polynomial Pr' r < n , is to be recovered, as is the case in model reduction, both Sr and A r are to be computed from the same pair of parts of Pr+l and Pr+2 " These considerations must be taken into account when evaluating the computational c o m p l e x i t y of the algorithm. For b r e v i t y , we shall limit attention to the split relations corresponding to columns 1 and3,respectively. They turn out to be of the following form: Li Si-2 + Mi Si-1 + N.J S.z = 0

(23)

L'., Si-2 + M'.n Ai-1 + N'.z Si = 0

(2q)

and

where tile combination polynomials L., M., N. and L'., M'., N~ are as specified in the Appendix. I

I

I

I

I

I

Clearly, the computational complexity, i.e., the number of a r i t h m e t i c a l operations (additions/ subtractions and multiplications/divisions) necessary to compute every coefficient of Si_2, d i r e c t ly corresponds to the number of addenda in the combination polynomials and to the number of their coefficients d i f f e r e n t from + 1, as will be shown with reference to some specific case. The simplest forms of equation (23) are therefore obtained (possibly, after the e l i m i n a t i o n of factors common to L i, M i and N i) when Ni(z) = n i = const., and Li(z) and Mi(z) are (symmetric) polynomials of degree not greater than 2 and 1, r e s p e c t i v e l y . In particular, L i may also be o f degree 1; in this case Li(z) = I i z (which may be regarded as a degenerate second-degree polynomial) and Mi(z) = m i ( z + l ) . Indeed, this is the case for the split Levinson algorithm and for the B i s t r i t z algorithm presented in ]'8~ . Specifically, by r e c a l l i n g (5), the split Levinson algorithm takes the form: r i ri_ 1 z Si_ 2 - r i (z+l) Si_ 1 + n.z S.s = 0

(25)

with n i = (l-ki_1)(1+k i} and, by recalling (17) and the fact that R i = Ria i f R i is considered as a (degenerate) third-degree polynomial , the B i s t r i t z a l g o r i t h m is:

225 1

1

r i r i _ I z Si_ 2 -

~ r i fi (z+1)Si_ 1 +S. = 0 . I

(26)

Of course, one can exploit the scaling factors to further s i m p l i f y the computations, e.g., by f o r c ing to + I two of the coefficents I i, m i and n i. In particular, it is possible to set I i = I and m i = -I in (25) (i.e., r k = I, V k ), thus obtaining the simplest version of the split Levinson algorithm } 15~; the simplest version of (26) is obtained by choosing I i = n i = I (i.e., r k = 2, V k ). The number of a r i t h m e t i c a l operations that are needed to compute a c o e f f i c i e n t of Si_ 2 from those of Si_ I and Si is the same in the two cases: 2 additions and I m u l t i p l i c a t i o n , in g e n e r a l . In fact, referring for instance to the simplest split version of (26) (Bistritz algorithm) and denoting by bh, k = bh,h_ k the c o e f f i c i e n t of z k and z h-k in Sh(Z), h = i-2, i - I , i , we have: bi_2, 0 = fi.(bi_1,0 + bi_1, I) bi_2, k = fi.(bi_1,k + bi_1,k+ I) - bi,k+ I ,

(27a) 0 < k < i-2 .

(27b)

Notice that, although the computational c o m p l e x i t y of the simplest versions of (25) and (26) is the same, they do correspond to d i f f e r e n t t w o - t e r m recursions and, thus, to different sequences of complete polynomials Pk ' k < n, starting from the same original polynomial Pn" In other words they d i f f e r in the i n i t i a l i z a t i o n rule. From the previous considerations, it is also apparent that the s i m p l i c i t y o f a specific split form is not s t r i c t l y related to the s i m p l i c i t y of the corresponding non-split algorithm. The main feature of the z-domain Routh-type a l g o r i t h m (8) is to retain, except for the factor (z+l), either the symmetric part or the a n t i s y m m e t r i c part, in the same way as in the s-domain version either the even or the odd part is retained. It follows that equation (23) and the corresponding equation relating the a n t i s y m m e t r i c parts only, are a restatement of this simple rule. Equation (24) becomes instead r i ri_ I (z+1) 2 Si_ 2 + r i { hi_ I ~I - (-I) i'] + h.i ~1 + (-1)i~}(z-1) Ai_ I - S.j = 0

(28)

I

in which L i is a non-degenerate polynomial o f degree 2 . To construct the entire sequence, one also need the companion relation corresponding to column 4 of Tab. I, which allows one to obtain Ai_ 2 from Si_ I and A i. Obviously, the simplest form of (29,1 and of its companion relation is obtained for r k = I, V k . As a consequence of the bilinear transformation, starting from a Schur polynomial Pn' the zeros of polynomials Sk and (z+1)Ak_ I , V k , a l t e r n a t e along the unit circle, like the zeros of Sk and A k. It could be shown that a similar property also holds for the s y m m e t r i c and a n t i s y m m e t r i c parts of t w o consecutive polynomials generated according to (2S) and (26). ~. CONCLUSIONS A common form has been considered for the t w o - t e r m recurrence relations corresponding to: ( i ) the Levinson (Nevantinna-Pick) algorithm, ( ii ~ the z-domain algorithm obtained from the classical Routh algorithm via the usual bilinear transformation, ( i i i ) t h e B i s t r i t z algorithm,and ( iv ) other Euclid-type s t a b i l i t y - t e s t algorithms, A geometrical interpretation of their properties has been given. An analysis o f the eight possib|e t h r e e - t e r m (spiitJ forms associated with the general t w o t e r m form, has been carried out, Particular a t t e n t i o n has been paid to the relation involving the s y m m e t r i c (antisymmetric) parts only and to the relations involving the s y m m e t r i c and a n t i s y m m e t r i c parts, alternately. The conditions leading to the c o m p u t a t i o n a l l y most e f f i c i e n t formulae have been p o i n t e d out and discussed,

226 APPENDIX With reference to the notation adopted in (22), the combination polynomials in (23) and (24) take the forms: Li = F(Fis- Gis) R i a - ( F i a - G i a )

Ris] Ri-I Ri-1

(At)

Mi = E(Fi-l,s - Gi-l,s) Ri-l,a - {Fi-l,a - Gi-l,a) R i - l , s ] (Fi ~'i - Gi (~i) Ni = ~(Fi-1 ,a + Gi-l,a) R i - l , a - ( F i - l , s

(A2)

+ Gi-l,s) Ri-l,s'][-(Fis-Gis) R i a - ( F . , a - G.la } R.zs] +

+ E(F.. . ) R l.-.l , a - { F .l -. l , a - G .l .- l , a ) R .l--l,S . ] ~Fia-Gia)Ria-(Fis-Gis)Ris] I--I,S - G .I--I,S

(A3)

and L'., = [(F.la - G.,a) R.la - {Fis - Gis) Ris] Ri_ I Ri_ I-

(A4}

M'i = [(Fi_1, s + Gi-l,s) Ri-l,s - (F i-l,a + Gi-1, a } R i - l , a ] (Fi F:i - G i (]i)

(AS}

N'i = [{F.,s - G.,s) R.,s - (F.,a-G ia)Ria] [{F i-I ,a- G i- I ,a) R i - l , s - {Fi-i ,s- G i-1 ,s) Ri-l,a -] + +[{Fi i -

,a

+G

i-l,a

)R

i-l,a

-(F

i-l,s

+G

i-I,s

)R

. ] [ ( F . - G )R. - { F . - G . ) R . ] . i-~,s is is la Ja la ts

(A6)

REFERENCES

['1 ] T. Kailath, "A view of three decades of linear filtering theory", IEEE Trans. Inf. Theory, vol. IT-20, No. 2, pp. lZ16-181, 1974. [-2] J. Makhoul, "Linear prediction: A tutorial review", Proc. IEEE, vol. 63, No. 4, pp. 561-580, 1975. ~3] E. Jury, Theory 1964.

and Application

of the z-Transform Method. J. Wiley & Sons., New York,

F4] P. Delsarte, Y. Genin and Y. Kamp, "On the role of the Nevanlinna-Pick problem in circuit and system theory", Int. J. Circuit Theory Appl., vol. 9, pp. 177-187, 1981. [5]

K. ~striSm, Introduction to Stochastic Control Theory. Academic Press, New York, 1970.

[6]

Y. Bistritz, "Direct bilinear Routh stability criteria for discrete systems", Systems Lett., vol. 4, No. 5, pp. 265-271, 1984.

Control

]-7] Y. Bistritz, "A new unit circle stability criterion", in Mathematical Theory of Networks and Systems ( Proc. MTNS-83 Int. Symp., Beer-Sheva, Israel, June 20-24, 1983 ), pp. 69-87. [8]

Y. Bistritz, "Zero location with respect to the unit circle of discrete-time linear system polynomials", Proc. IEEE, vol. 72, No. 9, pp. 1131-1142, 1984.

Eg] A. Lepschy, G.A. Mian and U. Viaro, "Euclid-type algorithm and its applications", Int. J. Systems Sci., vol. 20, No. 6, pp. 945-956, 1989. [-10] M. Benidir and B. Picinbono, "Comparison of some stability criteria of discrete-time filters", IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-36, No. 7, pp. 993-1001, 1988. [11] P. VaidyanathanandS. M i t r a , " A unified structural interpretation of some well-known stability-test procedures for linear systems", Proc. IEEE, vol. 75, No. 4, pp. 478-496, 1987. ['12] W. Krajewski, A. Lepschy, G.A. Mian and U. Viaro,"A unifying frame for stability-test algorithms for continuous-time systems", IEEE Trans. Circuits Syst., vol. CAS-37, No. 2, 1990.

227 E13] A. Lepschy, G.A. Mian and U. V i a r o , " A s t a b i l i t y test for continuous systems", Systems Control Lett., vol. 10, No. 3, pp. 175-179, 1988. ~14] P. Delsarte and Y. Genin, "The split Levinson algorithm", IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, No. 3, pp. 470-477, 1986, [15]

P. Delsarte and Yo Genin,"On the splitting of classical algorithms in linear prediction theory", IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, No~ 5, pp. 645653, 1987.

[16]

Y. Bistritz, H. Lev-Ari andT. K a i l a t h , " I m m i t a n c e - d o m a i n Levinson algorithms", Proc. IEEE ICASSP-86, Tokyo, pp. 253-256, 1986.

[17]

Y. B i s t r i t z , " C o m m e n t on 'Zero location with respect to the unit circle of discrete-time linear system polynomials' '% Proc. IEEE, vol. 74, No. 12, pp. 1802-1803, 1986.

[18]

A. Lepschy, G.A~ Mian and U. V i a r o , " A common setting for some classical z-domain algorithms in linear system theory", Int. J. Systems Sci., vol. 21, ~990.

TIM E - D I S C R E T I Z A T I O N OF THE ZAKAI EQUATION FOR DIFFUSION PROCESSES OBSERVED IN CORRELATED NOISE* Patrick FLORCHINGER Universitd de Metz D~parternent de Math~matiques Ile du Saulcy F-57045 METZ C5dex

Franqois LE GLAND INRIA Sophia-Antipolis Route des Lucioles F-06565 VALBONNE C~dex

Abstract A time dlscretization scheme is provided for the Zakai equation, a stochastic PDE which gives the conditional law of a diffusion process observed in white-noise. The case where the observation noise and the state noise are correlated, is considered. The numerical scheme is based on a Trotter-like product formula, which exhibits prediction and correction steps, and for which an error estimate of order ~f is proved, where 8 is the time discretization step. The correction step is associated with a degenerate second-order stochastic PDE, for which a representation result is available in terms of stochastic characteristics [8,9,11]. A discretization scheme is then provided to approximate these stochastic characteristics. Under an additional assumption on the correlation coefficient, an error estimate of order v ~ is proved for the overall numerical scheme. This has been proved to be the best possible error estimate by ElllottGlowinski [5].

1

Introduction

T h e purpose of this paper is to present a computable time discretization scheme for the Zakai equation of nonlinear filtering with correlated noises, and to provide an estimate of the rate of convergence. In the case of independent noises, the problem has been studied by Newton [13], KorczllogluMazziotto [6], B e n n a t o n [1], D i M a s i - P r a t e l l i - R u n g g a l d i e r [4], e i c a r d [14], Bensoussan-GlowinskiRascanu [2] and LeGland [12]. Some of these authors have actually considered the associated Zakai equation. T i m e discretization schemes have been provided with a rate of convergence of order 6, where 6 is the time discretization step. In the case of correlated noises, the problem has been studied by Elliott-Glowinski [5]. T h e bcst approximation of the continuous filter based on the values of the observation process at a regular

9Rcseareh partially supported by USACCE under Contract DAJA45-87-M-0296.

229 partition (with mesh 5) hm been considered, and it has been proved that the rate of convergence is of order V"~. However, no algorithm is provided to actually compute this approximation. The paper is organized as follows. In Section 2, the nonlinear filtering problem is presented. Some results are recalled in Section 3, on the Zakai equation, and on a related degenerate secondorder stochastic PDE. A Trotter-like product formula is then considered, with aa error estimate of order 5. However, this numerical scheme is not computable. In Section 4, a representation result in terms of stochastic characteristics is presented for the degenerate second-order stochastic PDE. This part follows mainly the work of Krylov-Rozovskii [7,8] - see also Kunita [11]. Under an additional assumption on the correlation coefficient, a time discretization scheme is presented in Section 5 based on an approximation of the stochastic characteristics - with an error estimate of order v/~. In addition, this numerical scheme is actually computable, as far as time discretization is concerned.

2

The

filtering

problem

On the probability space (~, F , P), consider the stochastic differential system dXt = b( X,) dt + a( X,) dW, + p(Xt) dV, dY~ = h(X,) dt + dVt

where {IV,, t _> 0} and {Vt, t > 0} are independent Wiener processes, with covariance matrix I (identity) and r respectively. For the clarity of exposition, it is assumed throughout the paper that r=I. Throughout the paper, the coefficients will satisfy the following hypotheses (i) b is a measurable and bounded function from R"* to Iq.'~, (ii} cr is a continuous and bounded[unction on R"*, with a ~=a n ' , such that 9 a is a uniformly elliptic m x m matriz, i.e. a(x) > cd, Oa~,~ 9 ~ ~= j=a ~ is a measurable and bounded function on R"*, (ii O p is a continuous and bounded[unction from I t " to the space of m x d matrices, with A C = pp', such that

i, ~ Op'k

9 ak =

is a measurable and bounded [unction on R'*,

9 ~ =~j=l ~Od,i

is a measurable and bounded function on R'*,

(in) h is a measurable and bounded function from It. '~ to R d.

With the diffusion signal {Xt, t _ O} are associated the two partial differential operators L =

' g i,j=l

L o A=1 ~

i,j=l

+ c"q

02 OxiOxi

+

.. i=l

a '." ' J ~ + ~ b -i O. OxiOxj i=a Oxl

Ozi '

230 An other family of partial differential operators to be considered is m

9 0

- ' B~ =z~ h~ + ~ P'~-Oxl

l 0 the original probability measure P is equivalent on [0, T] to the reference probability measure p t with Radon-Nikodym derivative ZT, so that under p t

dX,

= [b(X,) -

p(X,)h(X,)] dt + ~(X,) dW~ + a(X,) dY~

(2.1)

where {Wt, t >__ 0} and {Yt, t _> 0} are independent Wiener processes, with covariance matrix I (identity). By the Bayes formula E t ( f ( X , ) Z, l Y,)

E(f(S,)[y~)

=

In addition

Et(Z, [Yd f

E t ( f ( X ~ ) Z , l Y,) = J f ( z ) p t ( x ) dx ,

where the unnormalized conditional density {pt, t > O} satisfies the Zakai equation

d p =$ L * p~dt

+B~p~dYt k

The purpose of the next section is to recall existence, uniqueness and regularity results for this equation and related stochastic partial differential equations.

3

T h e Zakai e q u a t i o n , a n d r e l a t e d s t o c h a s t i c P D E ' s

For all n_> 0, let H " denote the usual Sobolev space, with norm [[. [[,. The following shorthand notations will be used throughout the paper: I" 1 -~ t[" ][0 and ![" I[ ~ I]" Ill 9 Consider the Zakai equation dpt = L*p~ dt + B~p, dY~~ .

(3.1)

The following existence, uniqueness and regularity result is proved in Krylov-Rozovskii [7] and Pardoux [15]. T h e o r e m 3.1 Let n>_O be fixed. Assume that the coefficients satisfy 9 a, c and p have bounded derivatives up to order (n+l), 9 b and h have bounded derivatives up to order n, and that the initial condition satisfies po E H". Then the equation (3.1) has a unique solution p E M2(O, T ; H"+I). In addition

9 pC L2(~; C([O,T]; H")),

231 . the following estimate holds Et[ sup [Iptll~] -< cllpoll~ 9 o 0}, using some classical approximation scheme. For instance, using the backward Euler scheme would result in the following global approximation scheme --t i

with same error estimate.

6

Conclusion

A time-discretizatiou scheme of the Zakai equation for diffusion processes observed in correlated noise has been proposed, based on the stochastic characteristics introduced in [8] and [11]. Under an additional assumption on the correlation coefficient, it has been shown that the rate of convergence of this approximation is of order v~, where 6 is the time discretization step. The same rate of convergence has been obtained in EUiott-Glowinski [5] for a different approximation 9 on one hand, the approximation considered in [5] has a probabilistic interpretation, which is not the case for the time discretization scheme presented here, 9 on the other hand, the latter is actually computable, whereas no numerical algorithm is provide to compute the approximation considered in [5]. Another point of interest would be to study some particle approximation for the degenerate second-order stochastic PDE, adapting the results presented in Raviart [16] for deterministic firstorder PDE's.

237

References [1] J.F. BENNATON, Discrete time Galerkin approximations to the nonlinear filtering solution, J.Math.Anal.Appl. 110 (2) 364-383 (1985). [2] A. BENSOUSSAN, R. GLOWINSKI and A. RASCANU, Approximation of Zakai equation by the splitting-up method, in: Stochastic Systems and Optimization (Warsaw-1988), (ed. J.Zabczyk) 257-265, Springer-Verlag (LNCIS-136) (1989). [3] Yu.N. BLAGOVESCHENSKII and M.I. FREIDLIN, Certain properties of diffusion processes depending on a parameter, Soviet Math. 2 633-636 (1961). [4] G.B. DI MASI, M. PRATELLI and W.J. RUNGGALDIER, An approximation for the nonlinear filtering problem with error bound, Stochastics 14 (4) 247-271 (1985). [5] R.J. ELLIOTT and R. GLOWINSKI, Approximations to solutions of the Zak~i filtering equation, Stoch.Anal.Appt. 7 (2) 145-168 (1988). [6] H. KOREZLIOGLU and G. MAZZIOTTO, Approximations of the nonlinear filter by periodic sampling and quantization, in: Analysis and Optimization of Systems, Part 1 (Nice-198.~), (eds. A.Bensoussan and J.L.Lions) 553-567, Springer-Verlag (LNCIS-62) (1984). [7] N.V. KRYLOV and B.L. ROZOVSKII, On the Cauchy problem for linear stochastic partial differential equations, Math. USSR Izvestija 11 (6) 1267-1284 (1977). [8] N.V. KRYLOV and B.L. KOZOVSKII, Characteristics of degenerating second-order parabolic Ito equations, J.Soviet Math. 32 (4) 336-348 (1982). [9] H. KUNITA, Stochastic partial differential equations connected with nonlinear filtering, in: Nonlinear Filtering and Stochastic Control (Cortona-1981) (eds. S.K.Mitter and A.Moro) 100169, Springer-Verlag (LNM-972) (1982). [10] H. KUNITA, Stochastic differential equations and stochastic flows of diffeomorphisms, in: Ecole d'Ete de Probabilites de Saint-Flour XII (1982) (ed. P.L.Hennequin) 144-303, Springer-Verlag (LNM-1097) (1984). [11] H. KUNITA, First order partial differential equations, in: Stochastic Analysis (Katata and Kyoto-1982) (ed. K.Ito) 249-269, North-Holland (1~984). [12] F. LE GLAND, High order time discretization of nonlinear filtering equations, in: #Sth IEEE COG (Tampa-1989) 2601-2606 (1989). [13] N.J. NEWTON, Discrete approximations for Markov-chain filters, Ph.D Thesis, Imperial College (1983). [14] J. PICARD, Approximation of nonlinear filtering problems and order of convergence, in: Filtering and Control of Random Processes (ENST/CNET-1983), (eds. H.Korezlioglu, G.Mazziotto and J.Szpirglas) 219-236, Springer-Verlag (LNCIS-61) (1984). [15] E. PARDOUX, Stochastic partial differential equations and filtering of diffusion processes, Stochastics 3 (2) 127-167 (1979). [16] P.A. RAVIART, An analysis of particle methods, in: Numerical Methods in Fluid Dynamics (Como-1983) (ed. F.Brezzi) 243-324, Springer-Verlag (LNM-1127) (1985).

F I N I T E - D I M E N S I O N A L S T O C H A S T I C F I L T E R I N G IN D I S C R E T E T I M E : TIIE ROLE OF CONVOLUTION SEMIGROUPS Fabio Spizzichino Dept. of Mathematics, University " La Sapienza " Piazzale A. Moro, 5 - 00185, R o m e

Summary.

We consider a stochastic dynamic system in discrete time with an observable

component Yn C ~ C ~Rw (observation), and a non-observable component X n E ~ C ~}~v (signal). The signal is a Markov chain and, at any step, the conditional density of the observation given the signal is fixed in the exponential class. We face the problem of finding transition kernels for X n for which the system admits filters of dimension v; we reduce this problem

to an analytical procedure in which the first step can be solved in terms of

convolution semigroups of probability distributions.

1. Introduction W e consider a stationary discrete-time stochastic process ({Xn}, {Yn})n=l,2,...(Xn E :~ C ~v, Yn e qJ C ~w) whose dynamics is described by the following properties:

{Xn} is a Markov chain with given

transition densities pn(x [ x'), Yn is conditionally independent on X (n-l) = (Xl, X2, ... , Xn-1) and y(n-1) _~ ( y t , y 2 ..... Yn-I). given X n (n = 2 , 3, ...), the conditional densities fn(Y 1 x) ----fyn(y [ X (n-l) = x(n-1);

y(n-l)=

y(n-1); Xn = x)

are given, n = 1, 2 , . . . . Let Pl be the probability density function of X1, pn(x [ y(n-1)) be the conditional density of X n given y(n-z) = Y (n-l) conditional density of X n given y(n) = y(n)

(prediction density),

(filtering density).

7rn(x ] y(n)) be the

The sequences of filtering

densities and prediction densities obey the "Dynamical Bayes Formula": 7rl(x ]Yl) oc Pl(X) 9 f l ( y l [ x),

...,

zrn( x [ y(n))

c< pn(X [ y(n-1)) . fn(Yn I x )

(i.i) pn(X [ y (n-l)) = J

Zr(n_l)(x' [ y (n-l)) 9 pn_l(x] x') dx'

the symbol cx indicates that two functions differ for a multiplicative quantity which does not depend on tile a r g u m e n t x E ~'.

239 In this paper we consider the time-homogeneous case in which pn(x t x') and fn(Y }x) do not depend on n. One is usually interested in finding out conditions on the triple Pl, p(x [ x'), f(y [ x), under which the recursive formula (1.1) actually allows for feasible computation of rrn and Pn. For this problem the following definition is of interest. Let cO. be an index set and B (~ be a ~r-algebra of subsets of ql.

Definition A family ~2 ---- {~u(x); x E E) of probability density functions on :s indexed by uE~

filter-conjugate

is

if there exists a measurable function T : ~ x ctJ --* % such that

[[- ~u(X') 9 p(x [ x') d x ' ] - f(y [ x) c< ~o(u.y)(x)

V u E ~

V y E ctJ.

(1.2)

E As it can be immediately checked by a direct application of (1.1): ( p l ( x ) = f ~uo(X') . p ( x ] x ' ) dx' , E imply

~Uo E ~ ,

~ filter-conjugate)

~rn(x [ y(n)) = ~Un(Uo;y(n))(x ) e ~, '~ y(n) e cLjn where u n : ~ x ~n .., cU. are recursivelly defined by ut(uo; Y,) = ~~

The ~

Yl), u2(uo; y(2)) = ~(ul ( uo, y,), y~), ...

(1.3)

stochastic process ul(u0,Y1) , u2(u0,Y1,Y2), ... is the "filter-process" and it

results to be a Markov chain as can be easily checked. The definition of a filter-conjugate family is the suitable translation of the concept of "conjugate family" in bayesian Statistics (see e.g. [2]) to the present case, where we must replace the usual Bayes formula with the dynamic Bayes formuta (1.1); see also the discussion presented in [6]. Filtering theory motivates the search for a filter-conjugate family with the following properties:

a) dim ~ < co,

b) There exists an algorithm for obtaining ~ and (u from the

knowledge of the elements E, ~, f(ylx), p(x[x'). W e consider here an exponential family of "observation densities" {f(y [ X)}xE t

(see

Sect. 2). Our aim is to build "compatible" transition kernels p(x ] x') which allow for the existence of filter-conjugate families with dim ~ = dim E = v and, in particular to show the role that convolutions semigroups of probability distributions can have in the problem. The results we obtained can be extended to the case in which p(x [ x') and f ( y [ x) depend 011 n. It has been preferred to treat the time-homogeneous case so as not to obscure the substancial aspects of the problems.

240 2. Sufficient conditions for the existence of a v-dimensional filter-conjugate family. W e assume that f (y ] x) is of the form f(Ylx) = a(x) 9 b(y) 9 exp { < x , d ( y ) > } x 6 ~" c ~v, y E ~ c ~v

(2.1)

with suitable functions a : t ~ RA', b : ~ -~ N4-, d : ~ -~ Nv and < 9 , 9 >

denoting the scalar

product. As shown in [1], [4], [7], under different kinds of conditions, the exponentiality of the family {f(y [ x), x 6 ~} is a necessary condition for the existence of a filter-conjugate family with % contained in a Euclidean space. Consider a quadruple ( ~ , %c~V,K:~xt~

K, c, p) with

+ U {0},c:~

and satifying the following conditions: I

e x p { < u , x ' > } 9 K(x;x') dx' cr e x p { < c ( u ) , x > } c(u) + d ( y )

6 ~

e(x') = ~ 9

V u ~ ~,

V x 6 t , V u 6 91.

Vy 6 ~

(2.2) (2.3)

I p(x) ~ 9 K (x;x') dx

(2.4)

t

V x' 6 :E and for some constant 7 > 0. I

p(x') e x p { < u , x ' > } dx' < ~o

u u 6 %

(2.5)

The elements t , y, a, d are those appearing in the representation (2.1). Remark

If K(x;x') satisfies (2.2) and xl, x 2 6 X are such t h a t xl-Fx 2 q t , it follows

V u 6 ~

J'~ e x p { < u , x ' > } 9 K(x I -4-x2 ; x ' ) d x ' =

= r

9 [ ; ~ e x p { < u , x ' > } 9 K(xl;x' ) dx'] 9 [ ; ~ e x p { < u , x ' > } - K(x2';x' ) dx]

where r

is a positive quantity depending on u, ~, c but non depending on xl,x 2.

V x, x' 6 t ,

define

p(x Ix')

7.

-

)

X(x; x,)

By the condition (2.4), it results

(2.6)

f t p(x I x ' ) dx = 1

V x' 6 it~ and so {p(x I x')}xE % can

be seen as a stationary transition kernel for the signal process {Xn}. Define, moreover, p(x) 9 e x p { < u , x > } ~~215 -- I ~ p(0) 9 e x p { < u , 0 > }

de

V x e ~,

~ -- {~.(x); x e ~ } , , ~

(2.7)

241 The

following result

extends

and

summarizes

arguments

contained

in the

paper

by Bather [1] and it is the starting point for the subsequent discussion.

2.1 T h e o r e m ~P is filter-conjugate with respect to the pair {f(y ] x)}, {p(x I x')}. Proof Our task is simply to check the validity of Eq. (1.2). Fix u E qJ., y E o~. By taking into account (2.1), (2.6) and (2.7), we have [J'a~ ~u(x') 9 p(x]x') d x ' ] 9 f(y I X) r [.rE p ( x , ) , e x p { < u , x ' > } 9 a(•

g ( x ; x') dx'] a ( x ) . b ( y ) . e x p { < d ( y ) , x > } or

i r e e x p { < u , x ' > } - K(x; x')dx'] p(x) 9 e x p { < d ( y ) , x > } cr p(x). e x p { < c ( u ) + d ( y ) , x > } c< r

~o(u,y) = c(u) + d(y) E ~

by the condition (2.3) and then ~o(u,y)(X) E 9.

[] For

a given f(y[x)

of the

form

(2.1),

the

problem

of finding

a filter-conjugate

family with dim ~ = v can then be solved by the following procedure: -

find ~ c Nu, c : q.t --* N v, K : E x ~: -* R + U { 0 } , satisfying the conditions (2.2) and (2.3) find p : t --* R-{- satisfying the conditions (2.4) and (2.5)

-

-

obtain the transition kernel p(xlx') by means of (2.6) and the conjugate family 9 by means of (2.7). In the terminology introduced by Bather, p(x) is an

invariant function and c(u) is a

connecting function. From a fixed K(x ; x') satisfying (2.2) we can obtain respectively one or infinitely many transition kernels p(x I x') depending on the sum of multiplicities of positive eigenvalues with positive eigenfunctions for the integral operator in (2.4) being one or greater than one. The procedure described just above will be illustrated in detail in the example shown in the last Section. Tile first step of the procedure asks for finding qJ., c, K satisfying the conditions (2.2) and (2.3). The form of the solutions for the Eq. (2.2), was already studied in detail by Bather [1] in the case with v = l , c linear. In terms of the symbols introduced here, one of his results can be stated as follows: if K(x;x') solves Eq. (2.2) with ~, ~

R open intervals and with c(u) = r.u

+ k (r > 0, k E ~), then it must result K(x;x') = g(x'-r-x)-exp{k.x} for some function g: ~

-*

~+ u

{o}.

W e are rather interested in obtaining sufficient conditions for (~

to satisfy Eq. (2.2).

This will be made in general in the Sect. 3; now we concentrate attention on the particular case c ( u ) = r. n (r e ~ ) .

242

For functions g, g: Nv _, R + U { 0 } , we shall use the notation Zg _= {x 9 Rv[ g ( x ) > 0 } . If Z1, Z 2 are two subsets of Nv, the symbol Z I + Z 2 denotes the set {x 9 ~v[ x

=

z 1--I- z2, zl 9 Z 1,z 2 9 Z2}

r . Z - - {z' 9 ~ V l z ' = r . z , z 9

and, for r 9 ~ , x 9 ~v, Z C ~v,

ZWx-

{z' 9 ~Viz'=z-J-x,z

9 Z}

2.2 L e m m a Fixr > 0andg:

~v._,~+U

{0} such t h a t

Zg + r . ~ c

~ ' a n d put

c (u) = r.u, %tO {u 9 Nv I J'Nv g(z) 9 e x p { < u , z > l d z < + c~}, K(x;x') ~ g(x'- r.x) The triple (c,~

(2.8)

K) satisfies the condition (2.2).

Proof J ' t e x p { < u , x ' > } 9 K(x;x') dx' = J'~. g(x'- r.x) 9 e x p { < u , x ' > } dx'. For any fixed x 9 i , g(x'- r 9 x) is positive if and only if x' 9 Z g + r 9 x and we can write J'~ e x p { < u , x ' > } 9 K(x;x') dx' =

J'tn(zg

+ r.x)

exp{ } 9 g(z) dz (2.9) The hypotesis Z o + r-x c :F, V x 9 ~:, implies that the set {[IM(Zg + r.x)] - r-x} does coincide with Zg, V x 9 ~; and so (2.8) entails J'~ e •

9 K(x;x') dx' = e x p { < r - u , x > } . J ' Z 9 e x p { < u , z > } - g(z) d z .

This completes the proof since the integral in the right hand side does not dcpcn on • [] The arguments contained in this Section can be resumed by the following s t a t e m e n t s 2.3 T h e o r e m Let the family of observation densities f(y[x) be of the form (2.1) and g (g: ~v_.. ~ + U { 0 } ) be such that Z g + r .~" c ~ for some r > 0. Suppose that for some 7 > 0 we can find a non null

p(x)

solution p(x) _> 0 for the equation p(x') = 7 9 J'~" ~

9 g(x'- r.x) dx

Suppose moreover we can find a (non empty) set % c ~.v such that r {u 9 ~v I J ' t p(x') 9 e x p ( < u , x ' > } dx' < ~ , d ( ~ ) -I- r. ~ c ~

J'Zg g ( x ' ) . e x p { < u , x ' > } d x ' < co} (2.10)

p(x)

F o r p ( x Ix')----- 7 " atx~./x,~.g(x'-~ J ~ ) r.x), there exists a f i l t e r - c o n j u g a t e f a m i l y g i v e n by

243 9~ = {~u(X); x E ~'}uEOd

with

p(x)-exp(} ~Ep(9).exp{}d 0 , ~o(u,y) = r.u + d(y).

~u(X) = Corollary

Consider t h e a s s u m p t i o n s of T h m . 2.3; suppose m o r e o v e r t h a t t h e "initial" density of X 1 is given by p l ( x ) cx p(x) . e x p { < u D x > } ' r

I YI . . . . . Yn) or p(x) 9 e•

pn(X

l Yl . . . .

with

u1 e ~

x E E, t h e n

{ )

Un =

rn-Z" Ul +

n-i rn_h E1 h " d(Yh)

3. C o n v o l u t i o n semigroups and solutions of Eq. (2.2) In this Section we w a n t to o b t a i n triples ( c , % , K ) satisfying t h e condition (2.2) with c : ~v _, ~ v of more general form. First we shall briefly recall, some well-known definitions and results (see e.g. [3]) a b o u t infinitely divisible d i s t r i b u t i o n s whose s u p p o r t is c o n t a i n e d in [0,+oo). This t h e o r y has indeed a f u n d a m e n t a l role in o b t a i n i n g our results (as it m a y be already clear by t a k i n g into a c c o u n t the R e m a r k presented in t h e Sect.2). Let M be a probability d i s t r i b u t i o n on t h e real line; M is infinitely divisible if for all n E N t h e r e exists a probability d i s t r i b u t i o n M n such t h a t M = Ma *...* Mn = (Mn) n* where * d e n o t e s t h e convolution operation. A d i s t r i b u t i o n M r is infinitely divisible if and only if it is the

probability

distribution

of X s + r - X s

(s>0)

for a

time-homogeneous

i n d e p e n d e n t i n c r e m e n t s {Xt}. T h e family of p r o b a b i l i t y d i s t r i b u t i o n s { M r } r > 0

process

with

associated to a

fixed t i m e - h o m o g e n e o u s stochastic process with i n d e p e n d e n t i n c r e m e n t s forms a convolution semigroup: Mt~+t2 = Mr1. Mt2. For a p r o b a b i l i t y distribution M on [0,+oa) d e n o t e by ~M its Laplace t r a n f o r m :

A

~M(s) = I 0 real f u n c t i o n

exp{- s 9 z} dM(z) ~

on

[0,+oo)

is t h e

Laplace

transform

of a p r o b a b i l i t y

distribution.

M if a n d only if ~(0) = 1 and it is completely monotone, i.e. it possesses derivatives ~(n) of all orders a n d (-1) n. ~(n)(s) > 0, s > 0. M is infinitely divisible if a n d only if, for n = 2, 3, . . . , the positive n - t h root !0n=(~M) 1/a is the Laplace t r a n s f o r m of a p r o b a b i l i t y d i s t r i b u t i o n ; this happens r

if a n d only if ~M = e x p { - r

where r has a completely m o n o t o n e d e r i v a t i v e a n d

= 0. If { M r } is a convolution semigroup a n d ~M1(S) = e x p { - r ~ M r ( s ) = e• Given r

{- ~-. r

V ,- > 0

we shall henceforth d e n o t e by M r (r

then (3.1)

t h e d i s t r i b u t i o n with Laplace t r a n s f o r m

244 exp {- r. r

In order to follow the line of the present discussion, the following remark is

now of interest Remark Let M r (r

be an infinitely divisible distribution on [0,+co)

0. T h e Eq. (3.1) can be written I

exp{- s. x ' ) m r ( r

admitting a density m r ( r

v ;,

') dx' = exp{- r. r

0

i.e. the function K(x;x') = mx(x') satisfies (2.2) with t stress t h a t M r (r

-- [0,-t-co) and c(s) = - r

We

can be seen as the distribution, at time t = r, of a process with

independent increments {Xt (~)

We recall finally that an infinitely divisible M is (strictly) slable if, in particular, M ---(Mn)n*where, V n, Mn is of the same type of M: M(z) ---- Mn( c~ )

(3.2)

for constants c n > 0. The norming constants arc necessarily of the form c n ~- n 1 / a with 0 0

and define c(u) -- - r

-u ), % -

(-co, 0],

x' K(x ; x') -- J" q(x'- z) dM• 0

(3.4)

3.1 T h e o r e m T h e triple (c,~

K) defined in (3.4) satisfies the condition (2.2) with t = [0,+co).

Proof T h e proof is now straightforward. V x > 0, K(x;x') is the density of the convolution of Q and Mx (r then, by the convolution theorem for Laplace transforms~ for s > 0 +00 4-00 +oo ]" exp {-s 9 x'} 9 K(x ; x ' ) d x ' = J" exp{-s 9 x'} 9 q(x') d x ' - f exp{-s.x'} dMx(x') o

= ~Q(s)

0

9 exp{-•

- r

r

exp {-x 9 r

0

= exp {x 9 c(-s)}.

[]

245

We remark that if r e.g. [3] pg. 295):

= r.s

Mr(r

(r>O), then { M r (r

= l[r.r ' +oo)(Z)

r>0, r > 0

K(x;x') ----q(x'- r-x); so we reobtain, for the trivial case 9 -

is a "translation semlgroup" " (see and K ( x ; x ' )

in (3.4) reduces to

[0,+oo) Z9 c [0,+oo), the result of

Lemma 2. When Mx(~b)admits a density mx (r

K(x ; x') in (3.4) can be replaced by

Xf

K(x ; x ' ) ~ f

mx(r

'-z)

dQ(z)

(3.5)

0

which shows that, in this case, we can consider also non absolutely continous measures Q. If in particular r K(x;x') = ~

= - s ~ (0 < a < 1), then by (3.3) T h m . 3.1 applies with 9 J'X'mt(e) ( x'-____s ) dQ(z) 0

(3.6)

x 1/a

Remark. We see that when we build K ( x ; x ' ) elements: the convolution semigroup {M• (r {Mx (r

by means of (3.4), we must choose two

and the measure Q. The form of the semigroup

is directly determined by the choice of the "connecting function" c(u); the choice of Q

has consequences on the form as well on the existence of the " i n v a r i a n t function" p(x). The latter aspect is clarified by means of the example presented in the next Section.

Analogously to what we did in T h m . 2.3 we can now draw the following conclusions: Let f(y[ x) be of the form (2.1) with d(y) _< 0, t -= [ 0 , + o o ) and let M r (r_>0) be a convolution semigroup of probability distributions on [0,+oo). Suppose we can find a density q on [0,+oo) such that, for some 3' > 0, the equation -~oo

p(x') = 7 " ]'o

e(x)

X'

a(x) "[~0 q(x'- z)dMx(z)] dx

as a non null solution p(x)_>0. Suppose moreover we can find so_> 0 such that

;

--}-oo p(x') 9 e x p { < - s , x ' > } d x ' < oo, o

-[-oo J" q(x') . e x p { < - s , x ' > } d x ' < oo V s > so. o

p(x) x' With respect to the transition kernel p(x [ x') = 3' 9 a(xi - p ( x ' ) J'0 q(x'- z) dMx(z) there

exists a filter conjugate family given by 9 ~ {~u(X)}uEq.t with ~ = (-co, -So] and ~u(~x) cr p(x) 9 exp{u - x}, indeed ~ satisfies the Eq. (1.2) with the expression -{-r

~0(u,y) = log [ f

exp{u 9 x'} dMl(x')] + d(y). 0

246 If 2e = (.co, X] or :E = [~, +oo) for s o m e X e ~ we can apply Thin. 3.1 in order to obtain triples (c,qs

satisfying the condition (2.2), by merely considering a linear transfor-

m a t i o n of the signal process; when X -- {x 6 ~v[ x = (xl,...,Xv) , xj > ~j, j =

1,2,...,v}

for a

fixed ~ E ~v we m a y use the theory of processes with i n d e p e n d e n t i n c r e m e n t s with s t a t e space ~ + v . Indeed let M z (~j) (7->0, j = l , . . . , v ) be t h e (absolutely continous) probability distribution at time t = v of a process {Xt (r

with i n d e p e n d e n t i n c r e m e n t s with s t a t e space ~ + v and, for

x -- (xl,...,xv), d e n o t e by K(x;x') the j o i n t p r o b a b i l i t y density function of the distribution MxI(T1)* ... * Mxv(!PV)(x'). T h e n

~+v

e x p { - < s , x ' > } 9 K(x;x') dx' = l]~ S

~+v

1-Ij exp{- r In

analogy

exp{-<s,x'>}

dMxj(r

x/ ---- e x p { - < r with

(3.5)

further

solutions

of

the

Eq.

(2.2)

can

be

obtained

by

c o n v o l u t i o n with positive measures Q on ~ + v .

4. Discussion of a particular case T h e p r o c e d u r e described in Sect. 2. and the results of Sect. 3. are now illustrated

by

discussing a p a r t i c u l a r case. Let f(Y I x) = ~

9 x 6 9 y6-i . exp {-x 9 y}

(4.1)

+oo x > 0, y > 0, 6 > 0, r(6) = ;

s 6-z - e x p {-s} ds. 0

We have

q.J = [0,+co,

E c [0, + c o ) ,

a(x) = x 6, d(y) = -y. Since d(q.]) = (-oo, 0], we need

cO. = (-oo, u0] for s o m e u 0 in order to satisfy the condition d ( ~ ) + r - eLI. c ~ with r > 0. W e put E - [0, + c o ) and consider the c o n v o l u t i o n s e m i g r o u p associated a stable process z-3/2

{Xt}t_>0 with cr = 1/2 ; the density of X t is m(1/2)(z) = ~ +co

f

0

exp {-s 9 z} 9

By (3.2) we h a v e Q(0) = 0,

Z-3/2 ,/~rr

9

exp {- ~ z } dz =

9 exp {- ~ z } :

exp {- -/2 9 s 1/2}

7-2 m r ( 1 / 2 ) ( z ) -- J 2 -1 ~ " z 3~2 exp {- ~ }. If we put in (3.5) Q(z) = 1

V z > 0

(4.2)

we o b t a i n K ( x ; x') = m x ( l / 2 ) ( x ') = J2. 7r

,

X

x,a/---5

X . ~2

9 exp {- 2-x' }

(4.3)

247 E q u a t i o n (2.4) then becomes ? ~176

p(x')oc3'.

0

4~-~r'

x (1-~) x ' ' 3 / 2 "

X2

"

exp { - 2.x-------3} dx

this a d m i t s solutions of the form p(x) oc x -6-1 which do not satisfy the condition (2.5). W e t h e n consider the measure Q such t h a t d Q ( z ) = z -1/2 dz. W i t h such a choice, (3.4) yields K(x ; x ' )

=

x , t1/ 2

. exp {- 2.x' x----~-2}" F u n c t i o n s of the form p(x) or x "6 are positive

eigenfunctions of the integral o p e r a t o r in (2.4); p(x) 0r x "6 satisfies the m e n t i o n e d condition (2.5) when 6 < 1, so we find t h a t , when f(y [ x) is given by (4.1), (0 < 6 < 1) and the t r a n s i t i o n s kernel is

X2

p(x I x') (x p ( x 'p(x) ) 9 a(x) 9 K ( x ; x ' ) = x -2"5 9 x '6-1/2 9 exp { - 2.x' }

a filter-conjugate family of densities 9 ~o(x) = (su ) 1 4

" x -6 9 exp {u 9 x};

--- {~u(X)}uEOd. exists

x > 0 . It results ~,(u,x) = -

with qJ. =

(-co,0)

and

J 2 . lu] - x. T l f f s i s a s l i g h t

extension of t h e situation considered in t h e e x a m p l e iv) of [1], where 6 -- 1 Aknowledgments I like to t h a n k M a r t i n J a c o b s e n and W o l f g a n g J. R u n g g a l d i e r for valuable discussions.

References [1] B a t h e r J.A.

Invariant Conditional Distributions. A n n . M a t h . S t a t . 36 (1965), 829-846

[2] Diaconis P. and Ylvysaker D.

Conjugate priors f o r exponential f a m i l i e s A n n .

Star. 7"

(1979), 217-226. [3] Feller W. [4] F e r r a n t e

P r o b a b i l i t y T h e o r y and its Applications (Vol. 2). Wiley (1970) M. and t~unggaldier W.J. On necessary conditions f o r the existence o f f i n i t e dim-

ensional f i l t e r in discrete time. To a p p e a r on Systems and C o n t r o l Letters. [s] Levine J. a n d Pignie G.

Exact Finite D i m e n s i o n a l Filters f o r a class o f N o n l i n e a r Dis-

c r e t e - T i m e S y s t e m s . Stochastics 18 (1986) , 97-132 [6] l~unggaldier W . J . and Spizzichino F. F i n i l e - d i m e s i o n a l i t y in discrete time non linear filtering f r o m a Bayesian Statistics viewpoint. S t o c h a s t i c Modelling and Filtering, G e r m a n i A. (Ed.),

Lecture notes in Control and I n f o r m a t i o n Sciences. 91. (1987) Springer Verlag

[7] Sawitzki G. Finite D i m e n s i o n a l Filter S y s t e m s in Discrete Time. Stochastics 5_ (1981), 107114 [8] V a n S c h u p p e n J.H. Stochastic filtering theory: a discussion on concepts, methods and resuits. S t o c h a s t i c control theory a n d s t o c h a s t i c differential systems.

K o h l m a n n M. a n d Vogel

W. (Eds.). Lecture notes in C o n t r o l and I n f o r m a t i o n Sciences 1 6 (1979), Springer-Verlag.

ON THE DECOMPOSITIONS OF OBSERVATIONS WITH NON-GAusSIAN ADDITIVE NOISE AND THEIR INNOVATIONS PROCESSES* YOSHIKI TAKEUCHI A-2361

Summary.

IIASA Laxenburg,

Austria ~

This paper is concerned with information structure of the observation with

additive non-Gaussian noise under the assumption that the noise belongs to a class of continuous martingales.

It is known that such an observation is decomposed into

a process with additive Gaussian noise and the quadratic covariation process of the additive noise.

It is also known that the innovations process is decomposed into a

standard Brcwnian motion process and the quadratic covariation process.

In this

paper, a number of sufficient conditions are obtained for the observation to have the information structure such that the information in the quadratic covariation process is not contained in the additive Gaussian part of the observation and/or the Brownian motion part of the innovation process.

I. INTRODUCTION Up to the present, many researches were devoted to nonlinear filtering and its associated innovations problems If] - [ 1 7 ]

Mainly,

they are concerned with the case

of additive Gaussian noise Ill - [12], and relatively small number of researches have been reported on the case of additive non-Gaussian noise. For the class of observations in which the additive noise is a non-Gaussian continuous martingale, we reported results on nonlinear filtering problems [13] and innovations problems[14]f" - [17]

In [13], it was shown that if the additive noise

is a non-Gaussian continuous martingale,

the observation is decomposed into two com-

ponents: one is a process with additive Gaussian noise and the other is the quadratic covariation process of the additive noise.

Also, the author showed in [15] that

the innovations process of such an observation process is a non-Gaussian martingale with the same quadratic covariation process as the additive noise.

The innovations

also decomposed into two components: one is a standard Brownian motion process and the other is the quadratic covariation process, and the latter is common in both decompositions of the observation and innovations processes. In this paper, we will give a number of sufficient conditions for having the unusual information structure such that the information in the quadratic covariation process is not contained in the additive Gaussian part of the observation and/or the Brownaian motion part of the innovations.

~

The first condition is that the family of

On leave from Dept. of Industrial Engineering, Kanazawa Institute of Technology, 7-i Ohgigaoka, Nonoichi-machi, Ishikawa 921, JAPAN

249 ~-algebras t.

generated by the observation or innovations

The second one is the stochastic

process is discontinuous

independence between

and the quadratic covariation process of the additive noise. the existence of a process which is computable

in

the Brownian motion part The third condition is

from the observation and contains a

martingale part that does not belong to the space of martingales Brownian motion part of the observation noise.

generated by the

These conditions cover a large class

of observations with non-Gaussian additive noise, and we will see that usually, information structure of such observation is different with additive Gaussian noise. understanding

We will also give a number of examples for better

of the results reported in this paper.

In this paper, mathematical denotes

symbols are used in the following way.

tile transpose of a vector or a matrix.

a nonsingular

square matrix, A -I denotes

The prime

The Euclidean norm is I.lo

the inverse matrix of A.

(~,~, P) is a complete probability space where ~ i s a s a m p l e

If A is

The triplet

space with elementary

events ~, ~ is a ~-algebra of subsets of ~, and P is a probability measure. denotes

the

from that of the observation

the expectation and E{.I~} , ~ C ~

the conditional

expectation,

E{.}

given ~, with

respect to P. ~[.} is the minimal sub-~-algebra

of ~ with respect to which the fami-

ly of >-measurable

is measuraie.

sets or random variables

{~

If ~I and ~2 are sub-

~-algebras of ~, then ~ I V ~ 2 denotes the minimal ~-algebra which contains b o t h ~ 1 and ~2"

Also, for ~ C ~ a n d

AE~,

{~t; 0 E t S T }

be a non-decreasing

{xt; 0 K t K T}

is said to be

~AA

denotes

the family { B A A ,

family of ~-algebras.

adapted to

E or

F-a~pted

all t E [0,T].

It is assumed that all random variables

~-measurable.

Unless otherwise stated, stochastic

BE~}.

Let F E

A stocastic process x E

if xt(~) is ~t-measurable

for

and stochastic processes are

properties are that with respect

to P.

2. OBSERVATION AND INNOVATIONS PROCESSES AND THEIR DECOMPOSITIONS Let y E {Yt; 0 J t iT}

be an m-dimensional

yt = O + f~hs(~)ds + n t ,

observation

process given by

t E [0,T] ,

where h E [ht; 0 ~ t i T }

is an m-dimensional

notes an m-dimensional

noise process.

(i)

signal process and n E {nt; 0 E t i T }

Most discussions

de-

on the innovations processes

have been made for the case where the noise n is a Gaussian process,

especially a

Brownian motion process.

In this paper, we are concerned with a more general case

where n is a non-Gaussian

continuous martingale with an absolutely

continuous quad-

ratic covariation process. Let (~,~, P) be a complete probability decreasing and right-continuous

space and F E { ~ t ; 0 K t K T }

family of sub-~-algebras

of 5.

We will assume that

all the stochastic processes including y, h and n are adapted to F. we will consider the case where n belongs that n has the form:

be a non-

In this paper,

to a class of continuous martingales

such

250 nt = fO Rs (~ where w - {wt; 0< t denote

the quadratic covariation process of m and ~, i.e., <m,w> is an m-dimensional Fadapted process such that {mt~ t- <m,~>t; 0 i t iT} well-known that there exists f ~ 2 ( F ) dition m ~ 4 2 ( ~ , F )

such that

is an F-martingale [I0] <m,G> t= f~ fs(~)ds.

is equivalent to <m>t ~ 0, where m E {Nt(~); 0! tiT}

nent of m orthgnnal to ~2(w,F),

i.e.,

It is

Then, the conis the compo-

255 ~t(~) ~mt(~ ) -f~f~(~)dOs,

(46)

and is the quadritic variation process of ~, i.e., is an F-adapted process such that {~t 2- <m>t; 0 i t iT}

is an F-martingale [II]

4. PROOFS OF THEOREMS 5 AND 6 In this section, we will use the following notation for simplicity. and ~2(y)

Let IM2(F)

respectively denote the space of square integrable F- and Y-martingales.

Let ~2(y) denote the set of ~m-valued Y-adapted f E[ft(~); 0~ t i T} with the property (41).

We can define ~i2(v,Y), i.eo, the space of square integrable martingales

generated by ~ similarly as (42). Now, we have the following lemma. Lemma 7.

Let E~$2(F;Y) and x be defined by

x t ~ t - ~ 0 - f ~ E { a s ( ~ ) I ~ s } d s E f~[as(~ ) -E{as(~ ) I~s}]ds+mt(~ ). Then, x E ~ 2 ( Y )

and xE$2(F;Y).

If m ~ 2 ( F )

(47)

is its martingale part, then we have

<x>tI~t = <m>tI~ t

(48)

t[%t = <m,~>tr> t.

(~9)

and

Proof: Noting (39), we see that It$'s stochastic differential formula implies xt2 = x02 + 2f~xsdxs +<m>tI~t. By (50), it is clear that <m>tI~ t is ~t-measurable~

(50) Since

x02 + 2f~xsdxs = xt2 - <m>tl~ t (51) and the left-hand side of (51) is a Y-martingale, we have (48) by the definition and the uniqueness of .

Similarly, by applying It$'s stochastic differential formula

to the F-adapted process {xtvt; 0N t ST}, we have xt~t=Xo~O+f~sdxs+S~xsd~s+<m,~>tI~t . (52) Since the sum of the first three terms in the right-hand side of (52) is a Y-martingale by

v, xE~2(Y), we have (49). This completes the proof.

L e n a 2.

If ~ is an orthogonal base of ~2(y),

then w is an orthogonal base of

~2(F;Y) ~{mE~2(F); m is a martingale part of x ~ 2 ( F ; Y ) } .

(53)

]M2(~,Y)=~(Y)

(54)

Namely, > ~2(~,F) ~ 2 ( F ; Y ) ,

where ~(Y)~

Proof:

{mE~i2(Y); m0(~) =0}.

Let x~$2(F;Y), and assume that its martingale part mE~I2(F;Y)

fies <m>tl~t# 0 and <m,w>tI~t = 0, ~(Y).

(55)

By eemma i, we have <x>tI~t#0 and <x,v>tI~t=0.

consequently, ]M2(v,Y) ~ ( Y ) .

satis-

Then, the process x defined by (47) belongs to Hence,

x~2(~,Y)

and

This completes the proof.

The following lemma is easily shown by the well-known results for the observation with additive Gaussian noise.

256 Lemma 3.

If v is not an orthogonal base of ~2(y),

i.e., if ~ 4 2 ( v , Y ) ~ ( Y ) ,

then we have (22).

Proof:

If ~t=q]t for all tE [0,T], then v is the Gaussian innovations process

for the observation 9 with additive Gaussian noise. Theorem 3.1], we have ]M2(~,Y) = ~ ( Y ) .

Hence, by (C- 3) and [i0;

This completes

the proof.

Now, we will give the proof of Theorems 5 and 6.

Proof of Theorem 5:

By Lemma 2, we have

~2(w,F) ~ 2 ( F ; Y )

> ]M2(~,Y) ~ ( Y ) .

Hence, if there exists xE$2(F;Y) ~2(w,F),

whose martingale part m does not belong to

then v is not an orthogonal base of ~ ( Y ) .

Hence, by Lemma 3, we have

(22).

Proof of Theorem 6: $2(F;V),

First, let us show (22).

Since q/tC~t, tE [0,T], if x E

Hence, S2(F;V) CS2(F;Y)

and we have (22) by Theorem 5.

then xES2(F;Y).

In order to show (23), note that from (i), (4), (14) and (16), we have (56)

v t = 0 + S~(hs (w)- h(s)}ds + S~ R~(s,~)dw s and vt =

ft ~-89 O+~ K 0 ~s,~){hs(~)-

fi(s)}ds+wt.

(57)

Let us regard v as an observation process and ~ as its component with additive Gaussian noise.

Then, we see that the innovations process and its Brownian motion

component of "observation v" are respectively v and ~ since E{ht(~)- fi(t)I~/t} = E{E{ht(~)- fi(t) I%t } 12/t} = 0.

(58)

Hence, we can apply Theorem 5 by replacing y and y by v and v, respectively. if there exists xE$2(F;V) ~2(w,F),

whose martingale part m E ~ 2 ( F ; V )

then we have (23).

Remark 5.

Namely,

does not belong to

This completes the proof.

As we can see from the proof of Theorem 6, for the observation with

feedback: yt=8+S~{gs(~)-~(s)}ds+f~ where ~(t) ~E{gt(~)i~t},

we have v = y

R~(s,~)d~s,

and the innovations informational

(59) equivalence

to the observation always holds, provided that (59) has a unique strong solution.

5. CONCLUDING REMARKS We obtained sufficient conditions for (22) and (23) and it was shown that for a large class of non-Gaussian martingales n, the information in Z is not contained in y and v.

We were not concerned with the innovations informational equivalence

to the observation in this paper.

It should be noted that the condition of innova-

tions informational equivalence is mainly related with the stochastic properties of h and {Ro89

0 ~ t~T}

whereas the condition for (22) and (23) is described

by the properties of R 0 and ~ and is almost independent of h.

257 REFERENCES [i] T. Kailath, "An innovations approach to least-squares estimation--Part I: Linear filtering in additive white noise," IEEE Trans. on Automat. Contr., vol. AC-13, pp. 646-655, 1968. [2] P.A. Frost, Estimation in Continuous Time Nonlinear Systems, P h . D . Dissertation, Stanford Univ., Stanford, Calif., 1968. [3] P.A. Frost and T. Kailath, "An innovations approach to least-squares estimation --Part I~: Nonlinear estimation in white Gaussian noise," IEEE Trans. on Automat. Contr., vol. AC-16, pp. 217-226, 1971. [4] T. Kailath, "A note on least squares estimation by the innovations method," SIAM J. Contr., vol. I0, pp. 447-486, 1972. [5] J. M. C. Clark, "Conditions for the one-to-one correspondence between an observations process and its innovations," Center for Computing and Automation, Imperial College, London, U.K., Tech. Rep. i, 1969. [6] V. E. Benes, "Extension of Clark's innovations equivalence theorem to the case of signal Z independent of noise, with fZs2dS< ~ a.s.," Mathematical Programing Study, vol. 5, pp. 2-7, 1976. [7] D. S. Cirelson, "An example of a stochastic differential equation having no strong solution," Theory Probability Appl., vol. 20, pp.427-430, 1975. [8] V. E. Bene~, "Nonexistence of strong nonanticipating solution to stochastic DEs: Implications for functional DEs, filtering and control," Stochastic Processes and their Appl., vol. 5, pp. 243-263, 1977. [9] D. F. Allinger and S. K. Mitter, "New results on the innovations problem for nonlinear filtering," Stochastics, vol. 4, pp. 339-348, 1981. [I0] M. Fujisaki, G. Kallianpur and H. Kunita, "Stochastic differential equations for the nonlinear filtering problem," Osaka J. Math., vol. 9, pp. 19-40, 1972. [ii] H. Kunita, Estimation of Stochastic Processes, Sangyo-Tosyo, Tokyo, 1976. (in Japanese) [12] R. S. Riptser and A. N. Shiryayev, Statistics of Random Processes I: General Theory, Springer-Verlag, New York, 1977. [13] Y. Takeuchi and H. Akashi, "Least-squares state estimation of systems with statedependent observation moise," IFAC J. Automatica, vol. 21, pp. 303-313, 1985. [14] Y. Takeuchi, "Generalized innovations equivalence for observations with nonGaussian noise," Proc. International Sessions 26th SICE Annual Conference, Hiroshima, Japan, July 15-17, vol. 2, pp. 1069-1072, 1987. [15] Y. Takeuchi, "On non-Gaussian innovations processes for observations with nonGaussian noise," Proc. 25th IEEE Conference on Decision and Control, Athens, Greece, Dec. 10-12, pp. 1035-1036, 1986. [16] Y. Takeuchi, "Innovations informational equivalence for observations with nonGaussian noise," Proc 26th IEEE Conference on Decision and Control, Los Angeles, California, Dec. 9-11, pp. 1163-1168, 1987. [17] Y. Takeuchi, "Innovations informational equivalence for a class of observations with independent non-Gaussian noise," submitted to IEEE Trans. on Inform. Thoery.

NONLINEAR THE

FILTERING

CASE

OF

WITH

UNBOUNDED

Patrick UF~A C N P ~

No

paper

noises

concerns

and

a one

coefficients. respect define

to

prove

paths

of

the

a formal

the

Malliavin

density

for

filter

is

the

the

formula.

unique

problem

continuity

Then,

we

filter.

Metz

by

prove

Finally,

solution

of

of

filter means

of

the

filter

With

this

and

we

~iven

of

a

the

with

aim,

we

prove

a result

that

Zakai

unbounded

the

existence

prove

correlated

with

process.

the

we

with

process

unnormalized

calculus, the

de

filtering

observation

continuous

Kallianpur-Striebel

Universit~

observation

we

the

COEFFICIENTS.

LORRAINE

a nonlinear

dimensional

First,

NOISES

FLORCHINGE~

399,

INDIA

This

DEPENDENT

a by

regular

unnormalized

equation.

INTRODUCTION

The

aim

compute given It

the

is

of

the of

known

that

minimize

that

the

nonlinear

history

well

which

in

"'at b e s t "

the

of

compute

this

obtain,

for

the

an

of

the

best

with

{yt}

is

to

the

following

process.

a

functional

to

to

any

t.

{x L }

of

xt

expectation

field

time

:

process

conditional

for

conditional

the

this

sigma

up

expectation

the

on

the

process

is

stochastic

estimation

respect

conditional t,

theory unknown

risk

observation

each

an

observation

quadratic

functional

paths

filterin~

paths

~enerated Notice

functional

distribution

by

that

to

reduces

of

xt

to

given

{ys,OSsst}. Some be

theorical

solved

9 The

are

continuity

observation 9 The state

an The of

scalar aim,

we

discuss

questions following

of

the

in

nonlinear

filterin~

which

have

to

:

filter

with

respect

to

the

paths

of

the

process.

existence

9 Gompute

case

the

the

of

uniqueness purpose

of

a nonlinear observation present the

a regular

general

main

form result this

ideas

the

paper

process of

and

for

nonlinear

for

filtering

a survey

density

of

is

the

solutions to

solve

problem

with

with the

unbounded previous

results

of

filter.

filtering to

equations

these

those

questions

dependent

works

on

[i?]

topics

[i8].

the

and

With

these and

in

noises

coefficients.

[i63,

and

equations.

a

this and

259 I STATEMENT

OF THE

We consider process

P~OBLEM. on a probability

- observation

stochastic

(xt,yt)~nx~P

differential

system

(O,~,P)

solution

the of

pair the

system

followin~

:

t=l

~=1

dyt=h(xt)dt

+ dvt

x o donn~

loi

de

space

(i.l)

mo

Yo=O where

w and

~m and Notice the

v are

two

independent

~P respectively. that since the process

system

process

and

is c o r r e l a t e d . Furthermore, assume 9 X o is a vector

the

the

field

9 XI,..,Xm,XI,..,Xp 9 h is a function

Wiener

v t appears

observation

followin~ on

are

in c ( ~ n , ~ p)

with

in

definitions

both

process,

hypothesis

~n with

bounded

processes

less

vector

such

than

X"o +

system

on the

in

of Ci.l)

coefficients

linear

fields

that

the

values

:

~rowth.

on ~n

~ hs163 h a s

less

than

t=l

linear For

~rowth

any

fin

t~s

associated

with

~o(s

order

and

any

the

system

p)

2 CONTINUITY

In this

we

OF THE

obtain

are

a paths remains

process of

of the notion

we are

function

~,

as

the

of

define

functional

= Es

t)

interested

This means that on the measures by

paths valid

i s used.

interest

by Chaleyat-Maurel Next, we study the paths That

solutions

(l~ filter

the

:

/ F~]

a.s.

FILTER.

section,

formula

observation which

(l.i)

> ~t~(y)

o f t h e f i l t e r ~t" sense errors made filter

bounded

explosive

O(Ys/OSs~t).

F~ =

hand,

avoid

>

y where

to

The

in nonlinear

in s continuity

of

in

continuity

properties

the model drives in the of the observation. On the formula when

which

an

different filterin~ the

filter

implies of

the of the continuity

been

discussed

approximation notions have with

right other

that

respect

observation p r o c e s s f o r t h e B a n a c h norm. of continuity in nonlinear filterin~

has

Co

the been

260 introduced bounded

by Clark

[8]

coefficients.

in t h e

We aim

F mappin~

Co([O,T],~P)

in ~,

norm

Co(s

such

on

That

notion

of

continuity

allows to estimate observation. The

continuity

where been

the

of

the the

studied

first

of

uncorrelated

the

continuous

that

filter

= ~t~(y) by

paths

one

has

with

by S u s s m a n n

for

by Fleming-MiLLer

[153

last results have continuity of the

been improved by Sussmann filler with respect Lo the that

Vc>O B Kc>O /

Banach

formula only

"cubic

Lwo the

such

the

which

a

unique system~

coefficients

and

h is a C 2 f u n c t i o n

h is

the

to

uncorrelaLed

unbounded

[303

when

with

with

a funcLional

a.s.

a paths

associated

of

respect

whenever

is s c a l a r

systems

existence wiLh

F(y)

~ives signal

observation

case

to prove

sensor

a polynomial

has

problem"

function.

These

[31] who proved Banach norm when

:

[Lh[

§ ~

~ X t h l 2 < c h 2 § Kc

L=I

The

of

continuity

the

(i.e.3~,i~p,s.t.

filler

X ~0)

associated

has

been

with

handled

correlated

by Davis

[9]

systems

in

the

case

t

of

a scalar

observation

with

bounded

Kolhman C123 w h e n t h e v e c t o r f i e l d s notice that Davis-Spathopoulos [113 of correlated system with unbounded Here,

we aim

to study

the case

observation

is scalar

Assume

XI,... ,Xm,X I are

that

function Denote

with

b y ~t

less the

with

than

of

the

~t ( x ) ' x § Moreover,

assume

that

a correlated

by

E11iott-

system

where

the

coefficients.

C 3b v e c t o r

of

and

X~, l ~ p commute. Moreover, have studied a particular case coefficients.

unbounded

exponential

solution

coefficients

and

fields

that

h Is

a

equation

:

Cs

~rowth.

followin~

~(~

integral

)(x)ds.

condition Tn

(H)

Vr>O,Vc>O,3K~>O

/[Xh~+

sup

[GO

L defined

denoted

a.

for

some

constant

such

that

:

~ in

[0,I[.

L,j=i

D.h (C~)

For every

i,j,iSi,jsn,

the f u n c t i o n s

D~Dih

~

I/2 and

(I + h x) are

i + hz

in C~(~n).

Thus, a c c o r d i n ~ w i t h t h e o r e m 5.2 from [4], we can f o l l o w i n g u n i q u e n e s s r e s u l t to t h e s o l u t i o n o f e q u a t i o n

THEOREM

equa*ioa

4.2

: Se~

(4.1)

~(x)

has a u n i q u e

l" b (~n))

in C ( s

: i + h 2zf x )

n

~or

solution

p

every

such

C ( [ O , T ] ; C 2,1" b (~n,Z))

x in ~n.

state (4,1)

the :

Suppose

that

~ha*

e x p ( M ' ~ M) p.

for

every

H' 0.

These algorithms obviously satisfy : qf0 D ~.d0 D q f l D ~ 1 ~ ... D Clf~tD r

D ...

Hence the respective limits, qf* and r are equal. This limit is well known as being the supremal (E,A,B) invariant subspace contained in ffrv IV 81], [O 85]. Both algorithms will be used hereafter in order to describe some precise structure of (1). A

A

A

A

When dealing with (E, A, B, C), the associated recursions (8) and (10), say e~eit and ~0it obviously satisfy (see for instance (12)): ~ 0 D q~St0= ~V'1 = ~ 1 = ~ 2 ~... A

A

A

A

Hence, for (E, A, B, C), we shall only deal with : (11) ~ , e 0 = ~ , ~ , e I t + l = 5 % c ' ~ - 1 ( h n B + E ~ f I t )

It>0.

We fwst have the following result : Lcmma 1 : The following algorithms are equivalent to (11) (see (4)-(5)) : (12) ~ , e 0 = ~ ,

c~eit+l=,~nA-l(imB+E~lfit + .~)

(13) ~ l f 0 = ~ , c ~ e I t + l = f f ( , n [ i - l ( I m B + E C ~ e I t ) r Proof :

32 ]

It20, It>0.

i) the equivalence between (11) and (12) is direct since Ker P0= . ~ . ii) for the equivalence between (12) and (13), we shall first prove that, for any c~c ~(~ :

(14) A - I [ ( I m B + E ~ ) ~ . C ~ I ] = A - I ( I m B + E ~ ) $

$2 "

For that, let x e A-l[(Im B + E~) 9 -~1 ] ' then, because of (5), there exist Xle g l @ Ker A, x2e 32, t ~ ~, b ~Im B and XlE_~ 1 such that : x = Xl~ x 2 and AXl+ Ax2 = (b+Et) $ x 1, that is : Xle A'l(Im B + Er

and x2e A'I(_.~_1 )=: Ker A 9 82 , which imply :

A-l[(Im B + ECg) ~ -~1 ] c A-l(Im B + EcC) + Ker A 9 ~2 = A-l(Im B + E~) $ 82 . On the other hand, A'l(Im B + E~g) 9 82 c A'l(Im B + Ec~) + A-I(_.~I )c A-l(Im B + ECg+_~l ), which ends the proof of (14) and therefore that of ii) * The relation between algorithms (8) and (13) is described in the following Lemma :

282 Lemma 2

:

(15) ~ 0 nr Proof :

~cr c!,eIt nqATix+l=,cf(, n A -1 [( Im B + ECV'it-1)n( Im B + E r

Ix > 1.

First note that q.e 0 n~121=: ~(~ nSes = ,5~. Next, from (8) and (13), it follows :

q e Ix n~.eIt+l=5% n [A -1 ( Im B + E q ? I x - 1 ) l n [ a -1 ( Im B + E ~?It) @ g2 ]" On the other hand, let x 9 [A -1 ( I m B + EqfIt-1)] n [A -1 ( Im B + E r s 11 e A - l ( I m B + E q

?It-1) C 81 ~ K e r A , Sl2 9 A " I ( I m B + E

$ 8 2 ], there then exist

~eIt) c 81 ~ K e r A a n d s 2 9

that : x = s 11 = s12 + s2, which implies : s2= s 11 - s12 9 (81 ~ Ker A ) n $2 =: 0 (remember (5)). Therefore, x e A -1 ( I m B + Eq2 I t - l ) n A -1 ( I m B + E ~ i t ) and (15) follows,

2 STRUCTURE ALGORITHM AND ZEROS AT INFINITY : In the proper case (rank E = n), the infinite zero structure only depends on the transfer matrix of the system (see for instance [CD 82]). In that case, this structure at infinity has been characterized algebraically with the help of the famous Silverman's Structure Algorithm [S 69] (see [SK 83]). In the case of generalized systems, there exist zeros and poles at infinity, and the infinite structure is not completely characterized by the transfer matrix ; more precisely, the invariant zeros at infinity are not only transmission zeros but also include (possibly) some input / output decoupling zeros at infinity [R 74]. The Structure Algorithm has been extended by Lewis to this class of generalized systems (see [LB 85]) and it is thus possible to define some kind of structure of zeros at infinity directly from this Structure Algorithm in a similar way to that of [SK 83]. A preliminary result in that direction has been given in [M 89]. Meanwhile, as a part of a new canonical form ILL 89], four different types of zeros at infinity have been geometrically characterized as follows. Their respective orders, say mi,ni,Pi and q i ' satisfy : I c a r d {m i = g + l } = d i m [ ( K e r E n r (16)

q c a r d {n i = I x + l } [card

It) / (Ker E n q,e Ix+l)[ VIx >_ 0

= dim[(ImB n E"~ It-1)/(imBn

{Pi = g+1}

[,.card {qi = It+l}

Eq, e It)]VIx_> 1

d i m [ ( I m B n Eq,~ It) / (Im B n E~fa9- Ix)[

Vit >_ 0

d i m [ ( K e r E n q,e Ix) / (Ker E n r

VIx _> 0

Ix)[

The number of n i equal to 1 is given by : card{n i = 1 } = dim [ Im B / ( Im B n Im E)] = dim ( ~ ). For more details, see [LL 89]. Recall however that, for the time being, there exists no dynamic interpretation of these lists mi,ni,Pi and qi in terms of transmission or input / output decoupling properties. The main objective of this paper is to provide an algebraic characterization of the four lists described in (16) with the help of the following Refined Structure Algorithm (see Corollary 1) : Refined Structure Algorithm : Initialization (k=O) :

283 (17.a) T 0 [ E A B ] = [ E1 A1B1]

with TO a maximal row compression on E

(18.a) S0[~O0 ~COc]=[ DI00~0'Cl]

w~ ~o, ma~m~ ~ow com,~,~o~ ~176 [~0 ]

(17.b) I[ E A ; ] : [ E'I AI BI] (18.b) I [ O C ] = [ O ^

^

A

C1]

^

A 0, 13(i, D 1, and C 1 are empty. Iteration k :

j

(19.a) 0 (20.a)

0

Sk [ D k

Ek+i Ak+l Bk+l] withTk amaximal row compression on

0

7 Dk+x ck+l ]

with Sk a maximal row compression on []/Dk[

LB--kj

Ak ^ A A ^ Bk E k + 1 Ak+ 1 Bk+ 1 with Tk a maximal row compression on (19.b) ,', TkL 0~ 0 = 0 Ak B__kJ 0

(20.b)

(19.c)

A [o,c,I D,,, ,+i

AAEA 1

Sk B-k Akj ^ Ek Ek+l

0

Ck+ I

0

0

[A1

^ . Dk with Sk a maximal row compression on ^

L--BkJ

] =Zk

(20'c) ]I ^= D Ak 1L D k + Note that this proposed algorithm is exactly Lewis' one, but applied simultaneously (in parallel) to both systems (2) and (6). Then, the following hold for k >_.0 and directly follows from [M 89] and [LB 87] : Fac~ l, : f Ker Ek+ I = Ker E ~ qf, k (21) Ker Dk+ l = B ' I ( E qfk) qy, kr KerCk+ I = q,ok+l

284

Ker Ek+l (22)

= K e r E n c~ek

Ker Dk+l

= B-I(E

~/'k~_0)

~l-ekn Ker C^ k + 1 = d~ek+l A

for (22), note that Ker E = Ker E and Ker P0 = -~---0" We are now in a position to establish our main result : Theorem 1 : For any k _> 1, the following hold : A ,'~.k- 1 (23) Ker E k n Ker E k + l = Ker E n (24) Ker D k n Ker [ ) k + l = B-I(E ~ k - 1 ) A k (25) qfk n Ker C k + l = The following Corollary is thus immediate and provides an algebraic characterization (through our Refined Structure Algorithm) for the different types of zeros at inf'mity of (I) recalled in (16) : Corollary_ 1 : card { m i = ~t+l } = rank (El.t+2) - rank ( Z ~ + l )

"v' ~ > 0

card { n i = ~+1 } = rank ( Dlx+l ) - rank ( Atx) card {n i = 1 } = rank B(_B0)

k/~t > 1

card { Pi = p.+l } = rank ( A~t+l ) - rank ( D g + l )

V l.t > 0

card { qi = p.+l } =rank (E~+I) - rank ( Ep.+l )

'v' ~ > 0

The proof of Theorem 1 deeply relies on the following Lemma which depicts some interesting distributive properties and the proof of which is given in Appendix : Lemma 3 : For any k > 0, the following hold : (27) Ker E n (qfk + ~ , k + l ) = Ker E n cv'k + Ker E n ~ f k+l (28) Im B n (EC~"k + E~ ek+l) = I m B n Eq7 k + I m B n E ~ k+l (29) , ~ k = clpkn~lek+l Proof of Theorem 1 : i) ii)

(23) follows from (21.a), (22.a) and (29) 0 From (21.b) and (22.b), it follows : A

Ker D k n Ker Dk+ 1

-----

1

A

B- [ Eclpk-ln (E q ek ~ .r A

A

A

On the other hand, for any x e E%ek-ln (EClf'k ~ ~,q3), there exist v ~ q ek-1, v E c f k and -~)~-~:0 A

A

such that x = Ev = Ev 9 -~0" Then : -~0 = Ev - Ev ~ Im E n ~

=: 0 (remember (4')).

Therefore : E q f k - l n (E~ ek ~ ~...~_0) = E q / ' k ' l n EC~ k , which implies : KerDk nKer Dk+l

= B - l ( E q f k - l n E~l'ek) = B - I [ E ( q f k - l n c~k)] =

B-1 E ~ k - 1

, because of (27) , because of (29),

285 which establishes (24) 0 iii)

From (22.c), it follows :

clTk nc~ekc~ Ker

Ck+l= q2k n c~k+1

^ 1= then : cv'k n Ker Ck+

~k

(since

~k

c

~k

for all k and because of (29)), which ends the proof of

Theorem 1 .

CONCLUSIQN : We have presented here a refined version of the Structure Algorithm introduced by Lewis (See [LB 85]) in the case of generalized systems. This refinement amounts to performing simultaneously Lewis'Algorithm onto the given system and on some projection of this system, where the algebraic part of the state equation is eliminated. In this way, we have been able to characterize algebraically all the four types of invariant zeros at infinity of the system, which are geometrically described in iLL 89]. Furthermore, we have extended the well-known algebraic characterization of q ek (see (21.c)) [LB 87] to ~fa9-k (see (25)) : each step of these algorithms (8) and (I0) can be computed as the Kernel of some map derived from this Refined Structure Algorithm. Connections between the geometry of the system and that of its projection have also been established (see for instance Lemma 1 and (29)). Our future objective will be to use these results in order to derive a left inverse of a given (left invertible) system with a minimal pole structure at infinity, that is with the least possible number of differentiators. This minimal pole structure at infinity is in correspondence with the tran~mis~i0n ?:eros at infinity of the system (recall [CP 84]), which have to be further characterized from a geometric point of view.

REFERENCES : [CD 82] C o m m a u l t C., Dion J.M. : "Structure at infinity of linear multivariable systems : a geometric approach", IEEE Trans. on Automat. Contr., AC-27, n~ pp. 693-696, 1982 [CLM 89] C o m m a u i t C., Lafay J.F., Malabre M. : "Structure of linear systems : geometric and transfer matrix approaches", IFAC Workshop "System Structure & Control : State Space & Polynomial Methods", September 25-27, 1989, Prague [CP 84] Conte G., Perdon A.M. : " Infinite zero module and infinite pole module", Proceedings of the 6th Int. Conference on Analysis and Optimisation of Systems, Lecture Notes in Control and Information Sciences, Springer Verlag, Vol. 62, pp. 302-315 [CPW 88] Conte G., P e r d o n A.M., W y m a n B.F. : " Fixed poles in transfer function equations", SIAM J. Control & Optimization, Vol. 26, n ~ 2, pp. 356-368, March 1988 [LB 87] Lewis F., Beauchamp G. : "Computation of Subspaces for Singular Systems", MTNS' 87, Phoenix, June 1987 iLL 89] Loiseau J.J., Lebret G. : "A new canonical form for singular systems with outputs", in this book of Proceedings. [ L O M K 89] L o i s e a u J.J., Oz~aldiran K., M a l a b r e M., K a r c a n i a s N. : "A feedback classification of singular systems",IFAC Workshop "System Structure & Control : State Space & Polynomial Methods", September 25-27, 1989, Prague [M 89] Malabre M.: "On infinite zeros for generalized systems", MTNS'89, Amsterdam, June 1989 [M 73] Morse A.S. : "Structural invariants of linear multivariable systems", SIAM J. Contr.& Opt., Vol. 11, n~ pp. 446-465, 1973 [O 85] Oz~;aldiran K.: "Control of descriptor systems", P h . D . Thesis, Georgia Institute of Technology, May 1985 [R 74] Rosenbrock H.tl. : "Structural properties of linear dynamical systems", INT J. of Control, vol. 20, n ~ 2. pp. 191-202, 1974 iS 69] Silverman L.M. : "Inversion of multivariable linear systems", IEEE Trans. on Automat. Contr., AC- 14, n~ pp. 270-276, 1969

286 [SI~ 83] ~iiverman L.M., Kitap~i A. : "System Structure at infinity", Systems & Control Letters, 3, pp. 123-131, 1983 [V 81] Verghese G.C.: "Further notes on singular descriptions", JACC, TA4, Charlottesville, 1981 [WS 85] W y m a n B.F., Sain M.K. : " On the design of pole modules for inverse systems", IEEE Trans. on Circuits and Systems, CAS-32, n ~ 10, pp. 977-988, October 1985

A P P E N D I X : P r o o f of L e m m a 3 : Consider the generalization of Morse's group [M 73] corresponding to all possible changes of bases in the domain ~ (noted V), in the codomain . ~ (W), in ~U' (G) and in ~ (H) and all possible proportional state feedbacks (F) and proportional output injections (R). With each (E,A,B,C) system is associated an orbit under this group action. All equivalent systems within the same orbit have some common structural properties which are more nicely described on the particular element of the orbit called the canonical fomL As any element of the orbit, this canonical form (which is fully described in [LL 89]) is derived from (E,A,B,C) through an element of the transformation group, say (V c, Wc,Gc,Hc,Fc,Rc), namely :

(A.1)

Fc = ~A c = |B c = [C c =

W c E Vc ; where W c and V c are isomorphisms W c (A+BFc+RcC) V c Wc B Gc 9 where G c is an i s o m o r p h i s m Hc E Vc " where H c is an i s o m o r p h i s m

We shall first prove this Lemma 3 for systems in the canonical form : L~mm~ A,1 : For any k > 0, the following hold : (A.2)

Ker E c n (flec k + ~,eck+l ) = Ker E c n fleck + Ker E c n ~ c k+l

(A.3)

Im B c n (Ecfleck + E c ~ c k+l) = Im B e n Ecflec k + Im B c n Ecq~eck+l

Proof : The proof is based on both following properties : i) First of all, it is very easy, working on the canonical form (Ec,Ac,Bc,C c) (see the description of the foml in ILL 90]), to show that all the subspaces which appear in (A.2) and (A.3) can be spanned exclusively by vectors of the canonical bases of ~(~ and . ~ . ii) Next, it is relatively easy to see that for any triple ( ~ , 3 , ~ ) of subspaces of a vector space ~ , a basis of which can be chosen as a subset of a given basis o f ~ , i.e. : ff~, = {{ei }},S = {{ej }}, and cC= {{e k }}, where {ei }, {ej }and {ek }are elements of the basis of ~ , then distributlvity always holds, that is :

5%n($+~)=SZng+~n~. Therefore (A.2) and (A.3) directly follow , Now, a series of three results, which make the connection between Lemrna A.I and Lemma 3, is established : Fact A.1 : (A.4) (A.5)

Im E c = W c Im E Im B c = W c Im B

(A.6)

KerCc=Vc-1KerC

(A.7)

Ker E c = V c- 1 Ker E

Proof :

Directly from ( A . 1 ) ,

(=Vc "1 5ec,,)

287

Fact A,2 : With ~ (a.8) then :

defined in (4), let :

" ~ 0 = Wc " ~ {~

= Im Ec * ~:..c 0

~ 0 : =

Vc-1 cv'k

(A.10)

qfc k

(A.11)

~ag-ck

=

Vc-1

(A.12)

C~fck

=

Vc-I c~fk

gO. k

Proof : Only (A. 12) will be proved here, since (A. 11) and (A.10) are similar. First note that: r 0 = Vc-1 ~ f 0 . Now, suppose that (A.12) is true for some k _> 0, then, from (A.1), Fact A.1 and Fact A.2, it follows : ~fck+ 1= Vc-1 if.c n Vc-1 (A+B Fc+RcC )- 1W c- i (WcEd~fk+Wci m B+Wc._..~) = Vc-lc~ok+ 1 , We are now able to end the proof of Lemma 3 : Proof Qf Lemm~ ~ : i) (27) follows from (A.2), (A.7) and Fact A.3 0 ii) (28) follows from (A.3), (A.5), (A. 1) and Fact A.3 0 iii)

First of all, note that (29) holds for k = 0 since r -0 = q f 0 n e~;,1 . Now, suppose that (29) holds for

some given k > 1, that is : r = qfk-I n d~ek, then, from Lemma 2, (28) and (27) : q f k n c~ek+l = , ~ n A-l[( im B + Eqfk-l) n ( i m B + E~k)] = fff, n A - l [ I m B + E % ~ k - l n ( i m B + E ~ k ) ] = ~ e t . d k

,

THE DYNAMIC BLOCK DECOUPLING PROBLEM: A MINIMAL SOLUTION BY PRECOMPENSATION J. DESCUSSE Laboratoire d'Automatique de Nantes, URA CNRS 823, ENSM ,1 Rue de la No~, 44072 Nantes C~dex 03, France

Abstract : The purpose of the present paper is to provide a new dynamic solution to the Block Decoupling Problem based upon a procedure which extends to the block case that one previously given in [14] for Morgan's Problem. This procedure is minimal in the sense that it leads to the least infinite zero structure for any decoupled system reachable from the original one. Kevwords: Block decoupling problem ; block essential orders ; infinite zero structure ; minimal dynamic compensator.

1) Introduction . During the last 25 years, a great deal of interest has been brought to the theory of input-output decoupling for linear time invariant systems. Since the early works of Morgan[12] and of Falb and Wolovich [9], many authors have tackled this famous problem. It is rather difficult to give an exhaustive bibliography on this topic, so many papers have been published ! (Many references can be found in [10], [13] as well as in [7] for very recent ones). The theory is still in progress. Till a recent past , the general block decoupling problem remained unsolved without particular assumptions made on the system or on the feedback laws (see for instance [1], [5], [11], [13]). A complete solution with static state feedback is given in [18] When it is not achievable one can use dynamic state feedback [16] or precompensation.(See [10] for details on static or dynamic state feedback as well as precompensators). The scope of the present paper is to provide a new dynamic procedure which generalizes that one of Wang [14] for Morgan's Problem. Two of its main interests can be noticed : it leads to the least infinite zero structure for the dynamically decoupled system. Secondly, it can be extended with some arrangements to the nonlinear case [8]. Preliminary results are also presented in [6]. The paper is organized as follows : next section is devoted to notation and basic concepts. In the third section, we introduce the block decoupling problem with static state feedback. In the last one, we provide a new algorithmic dynamic procedure which can be used when static state feedbacks are not available. At last , we recall the definition of the "block essential orders" [17] , [18]. We also recall that they are the least infinite zero orders reachable for decouplable systems. 2) Notation and basic concepts We shall consider, in the sequel, linear time invariant systems described by :

289 ,T-,0 I ~ = Ax + Bu

[ y = Cx where xe ,,~ = ~n, u 9 ~ = ~m, y 9 ~ = [p,p; [p, denotes the field of real numbers. The set of natural integers will be written as I]t. Associated with 7_.0we shall consider a k-partition of the output y into k nonempty subsets of components Yi, each of dimension Pi (0 < Pi, '~ie k Pi = P). This partition induces a corresponding partition of C, written as {Ci} k. We shall assume, without loss of generality, that B is monic. The same symbols will be used to denote a map or any of its matrix representations in particular bases. Let 3 the image of 1 3 , a n d K e r C t h e kernel ofC. Let B b e a m a t r i x , t h e n sp{B}will denote 3 . For any space ~.,~, dim ~,~ is its dimension. If ely c ff-.~, ff.~/cV will denote the factor space of ~ - byC~. A set of p elements will be written as {.}p and k will be used for the (not necessarily ordered) set of integers {1, 2 ..... k}. The number of elements of a given list {.} is noted as card {.}. The symbol U will denote the union of sets with repeated common elements. The rank of (C, A, B) defined as the rank of its transfer matrix will be written as p ; Pi will denote the rank of (C i, A, B). In what follows, we shall use cFi* the maximal (A,B) - invariant subspace contained in Ker C i , also written as sup I(A, B, ker Ci), fi~i~ that one contained in nj~ i Ker Cj, while $~'i" will denote the maximal controllability subspace contained in ~i*Recall that %Pi* and "-~i* are the respective limits of "I.S.A." and"C.S.A." defined by : I.S.A. C.S.A. flYi 0 = ~

..~i 0 = 0

cFiP- = Ker Cin A "1 ( s

"1 + 3)

SLip. = ~i" 'q (A ,..%ip.-1 + 3)

We shall also use "C.I.S.A." defined by: 80 = o 8p. =

3

+ A(Ker

C N ,~,p._t)

of which the limit is written 8 * = i n f 0-.t (C, A, 3 ) Now, we recall possible definitions of the infinite zero structure of (C,A,B) (See for instance[2], [7]). Define P~+I = d i m ( 3 nCVp./ 3 n q P * ) # > 0 (2.1) = dim(C~/ where

{~PZ}

and %P* are

C~p.) related

p. > 0

(2.2)

to sup I(A, B, kerC).

The infinite zero structure of (C,A,B) is the list of integers {pp. } , or equivalently {n'i} defined by n' i = card {pp. ~ i}. has Pl infinite zeros of which the orders are the n'i's. Recall that Pl = P.

290 3) The Static State Feedbaqk Block Dec0vDling Problem (SSFBDP) SSFBDP is defined as follows: "For a given k-partition of the output {Ci} k find necessary and sufficient conditions under which there exist static state feedbacks u = Fx + ~-i~k Gi vi such that v i controls Yi without affecting yj, j#i, i ek". This problem remains unsolved in its most general setting. Up to now, the only available results deal with special situation when the static state feedback is "regular", which means that G = [G 1 I...I Gk] has rank m, or when it is "restricted", in other words when u = GFx + Gv, [5],[11]. Theorem 3.1 1"13] : Let ,T-.0 and a k-partition of the output be given. Then the regular SSFBDP is solvable if and only if : = )lie k ~ c~ ~i

(3.1)

As an immediate consequence, we have Corollary :~,1 : Let T.0 and a k-partition of the output be given. Then the regular SSFBDP is solvable if and only if : (3.2) = Zi~k ~ n n j ~ i 97*J The proof is not difficult to perform and is left to the reader. Lemma ~).1 : Let E,0 and a k-partition of the output be given. Then, if P = T-ie k Pi we have : = ,T-,i~k ~ n ~i r

ely* = c~iek clYi

(3.3)

Proof : It is quite similar to that one given in [3], for right invertible systems, turning Pi into pi.E] Now suppose that (3.1) or (3.2) fails. One can search for dynamic solutions. For this purpose an auxiliary system is joined to ~0. It is defined as T.a ,

/ xa = D l x + D2x a + Ev 7.a

] [ u = F l x + F2x a + Gv

where x a e "~a = I~n~ and v e Ig m. We shall say that decoupling is achievable with dynamic compensation if it is possible to find ,T-,asuch that the composite system T.0 x ,T-,adefined on [pn+no can be decoupled using static state feedbacks. Note that this problem is always solvable, under non restrictive conditions[13] , [15].

291 Our purpose here is to provide a new solution to this problem, when p = ,T_,iek Pi. This new solution will be based upon an algorithmic procedure, which generalizes that one proposed in [14] for Morgan's Problem (k=p). From it, it will be possible to define the essential orders of the output related to the given k-partition {Ci} k. 4 I The Dvnamic Block Decoupling Problem Consider the following [14] available when k=p.

Dynamic Algorithm (D.A.), which generalizes that one of

Steo 1 " From the given data ({Ci} k, A, B) compute ~1 = -T-,iek$ n nj~ i c]yj . If ~1 = ~, Stop ! If ~1 c ~,write ~ =

~1 @ ~2- Define Gl and G2 such that ~ I = I m B G 1

and ~ 2 = I m B G 2 Steo 3 : Consider ,T-,0and write u = G1 Vl + G2 v2 = A x + BG 1 v 1 + BG2v 2 assume that dim v 2 = q, and put integrators in series with the q corresponding inputs of v 2. One obtains rewriting v 1 as w 1 :~

~x + BG 2 v 2 _--

"w1 +

0

y = C(,T_,1

=Cx

Steo 4 - Go to step 1 and resume the procedure with T.1 instead of ,T-,0 Theorem 4.1 9 Let .T_,0 and a k-partition of the output be given. Then, if p = ,T-,i~kPi, "D.A." "converges" after a finite number of "loops", say N, towards an extended ,T-,Nfor which SSFBDP is solvable. Proof - The central idea of the proof amounts to show that dim n i c k c~i* (,T_,l) < dim ni~kClYi*(,T_,l+l), where I denotes the I-th loop of "D.A.". Indeed, if this inequality is true, after a finite number of steps say N, we shall fulfil clP*(T.N) = ~i~ k_ ClPi (,T-,N), and lemma 3.1 does the rest. To show the above inequality it is sufficient to perform the ;roof for I = 0. This is done in Appendix 1. [3 We shall now pay attention to the structural properties of TN, and mainly to its infinite zero structure. For this purpose, consider the infinite zero structure of the

292 subsystem ( Ci(T.N), A(,T_.N) + B(T.N)Fi(T.N), B(T.N)Gi(.T-,N)), where Fi(T.N) is any friend of ..9~i*(.T_,N)and Gi(,T_,N) satisfies Im B(,T_,N)Gi(,T_,N)= ~(T.N)r~,-q:~,i*(T.N). We shall note it simply as ~ (Ci(,~,N), ..%i*(T.N)). It can be computed using either (2.1) or (2.2). We shall note it { 5i1~ (,T-,N)} or equivalently {13iv(,T_,N)} with 13iv(,T_,N)= card { 5il~(,T_,N) > v}. The infinite zero structure of ,T-,Nwill be noted 0 i ~k (Ci(7-N), "-'%i*(,T-,N)) and, with the above notation , is defined by {T.i~kSit~(T.N)} or equivalently , from the above counting procedure , by 0 i ~k{ 13iv(,T--N)}" Lemma 4.1: Assume that p = T.ie kp i. Let N denotes the final loop of "D.A.". Then the infinite zero structure of ,T-,Nis not different from ui ~k (Ci('7--'0), "9~i*('T--0))' Proof: since .T-,N is decouplable using regular static state feedback, it follows that [4] ,T_,~(7.N) = Oi ~k,T_,~ (Ci (T.N), ,-gLi*(,T_,N)) Assume that C i ,.9:~i*P.(,~,0)=C i (T.N) ..9~.i*#(~N)

(4.1)

From (4.1) one can deduce that SIP'+1 (,~-.0) = 5il~+1 (,T-,N)

p2.0

which states the result. The proof of (4.1) is given in Appendix 2.F] Definition 4.2117]_ . [18]: The infinite zero structure ~

(C i , --%i*(.T_.0)) is said to be the

essential structure of the output Yi, i~k. It is written as ,T_,e(Ci,A,B).

The essential

structure of (C, A , B), relatively to the given output k-partition, is defined as ui~ k (Te(Ci,A,B)). The terminology of "essential orders" comes from the following property. ProPosition 4.1 [17]. [18]: Let ~

and a k-partition of the output be given. Assume that

P = .T-,iekPi. Then if SSFBDP is solvable for some feedback u = Fx + ~iek Gi vi then .T_~o(Ci,A+BF,BGi) >_T.e(Ci,A,B )

REFeReNCES

[1] Commault C., Descusse J., Dion J.M., Lafay J.F., Malabre M. " Influence de la structure & I'infini des syst~mes lin~aires sur la solution de probl~mes de commande. APII, vol. 20, pp. 207-252, 1986

293 [ 2]

Commault C., Descusse J., Dion, J.M., Lafay J.F., Malabre M. : About new decoupling invariants : the essentiel orders. Int. Journ. of Contr., vol. 44, 3, pp. 689-700, 1986

[ 3]

Descusse J. : Sur la structure & rinfini des syst~mes lineaires d~couplables : le cas des syst~mes inversibles & droite. Outils et ModUles Math~matiques pour I'Automatique, I'Analyse des Syst~mes et le Traitement du Signal, vol. 3, Editor I.D. Landau. Ed CNRS, Paris 1983

[ 4] Descusse J., Lafay JoF., Malabre M. : On the structure at infinity of linear block decouplable systems: the general case. IEEE Trans. on Automat. Contr., AC-280 12, pp. 1115-1118, 1983 [ 5]

Descusse J., Lafay J.F., Kucera V. : Decoupling by restricted static state feedback: the general case. IEEE Trans. on Automat. Contr., AC-29, 1, pp. 79-81, 1984

[ 6]

Descusse J. : A new approach for solving the linear decoupling problems. Proc. of the 10th World Congress on Automatic Control, IFAC, July 27-31, 1987, Munich, FRG

[7]

Descusse J., Lafay J.F., Malabre M. : A survey on Morgan's Problem. Proc. of the 25th IEEE Conf. on Decision and Control, Dec 10-12, 1986, Athens, Greece ; Solution to Morgan's Problem. IEEE Trans. on Automat. Contr., AC 33, 8, Aug. , pp. 732-739, 1988

[ 8] Descusse J. : Towards a dynamic solution to nonlinear block decoupling . Preprints of the International Conference "Automatique Non Lin~aire" of C.N.R.S. , June 13-17 , Nantes France , 1988; "New Trends in Nonlinear Control Theory", LNCIS, vol. 122, Springer Verlag.

[ 9] Falb P.L., Wolovich W.A. : Decoupling in the design and synthesis of multivariable control systems. IEEE Trans. on Automat. Contr., AC-12, 6, pp.651-669, 1967 [10] Hautus M.L.J., Heymann M. : New results in linear decoupling. Proc. of the 4th INRIA Conf. on Analysis and Optimization of systems. Springer Verlag, vol 44, 1980 [11] Kamiyama S., Furuta K. : Decoupling by restricted state feedback. IEEE Trans. on Automat. Contr., AC 21, pp. 413-415, 1976 [12] Morgan B.S. :The Synthesis of linear multivariable systems by state feedback, JACC 64, pp. 468-472, 1964 [13] Morse A.S., Wonham W.M. : Status of non interacting Control. IEEE Trans. on Automat. Contr., AC-16,6, pp. 568-581, 1971 [14] Wang S.H. : Design of precompensator for decoupling problem. Electronics letters, 6, pp. 739-741, 1970 [15] Wonham W.M. : Linear multivariable control : a geometric approach, 2nd edition, Appl. of Mathematics, Springer Verlag, vol. 10, 1979

294 [16] Commault C., Dion J.M., Torres J.:lnvariant spaces of linear systems. Application to block decoupiing. Proceedings of the 27-th IEEE CDC , Austin , USA ,Dec 1988.

[17] Commault C., Descusse J., Dion J.M., Torres J.: Block decoupling invariants. Geometric and transfer matrix characterization.-IFAC Workshop on Systems Structure and Control; 25-27 september, Pague, Tch~coslovaquie.

[18] Descusse J.:Block noninteracting control with (non)regular static state feedback. A complete solution. -IFAC Workshop on Systems Structure and Control; 25-27 september, Pague, Tch~coslovaquie.

Vl>0"

"dim niek cF i (7-1+1)< dim nie k cF i (ZI) ,

It is enough to perform the proof for I = 0. To proceed we need several auxiliary lemmas. Let P 9 ,~, x ~'a ~ ~ x '~'a the projection into ~, along "~'a, where ~. x '~'a is related to T.1 , then t

*

P c[Pi (7-1) = cFi (,T-,0)

i ek

The proof of this lemma is not difficult to perform and we shall omit it. For 7.1 = (Ae, Be, Ce) above defined, we have 9 niek flPi*(Z1) = P n i e k c/Pi*(Z1) Proof 9 Start from Ae n i e k flPi*(7.1) c flYi*(,T_,l)+ ~e c nie k (Vi*(T.1) + ~e)

x[-4] :,,,n[ {[Xo] l+

,o,1,,Jll

where sp{(Xli/UlI)} =flYi*(Z1). From the respective definitions of B 1 ( $1 = 7.iek ~ *

n nj~ i clYj ) and q'Pi (7.0) = sp {(Xli)}, by lemma A1 , we have also, using a repeated action of the modular distributivity rule "

Ix4

A e n ~. (Y. 1) c n sp { iek

I

iek

} + Sa

(A. 1 )

295 Now assume that r~ie k fiYi (,7-,1) = P n i e k c~Yi*(-T-,1)-This means that there exist

onto~.generat~of niek_ctYi*(Y4)written as [~I" From (A.1) we then obtain, by projection B 2 U e hie k 1,f'i*(~O) + @1

c $ r~c~iekqYi*(.~.,0)+ $1 = $1 which is impossible by the respective definitions of ~1 and ~ 2 13

Lemma A.:~ " Assume that SSFBDP is not solvable for ,T_,0. Then, for ,7_,1 = (C e, A e, Be) above defined, we have 9 dim hie k cFi*(,T_.,1)< dim nie_k (s

PrQQf " From the preceeding lemma we know that the generators of hie k c~Yi*(,T_,l)are written asIXI 1, where 0 ~ X I e hie k cFi*(.T_.0).

LU j So

dim nie_k ~qYi*(T.1) < dim hie k cV'i*(,7_,0)

(A.2)

Assume that equality holds. Then, from (A.1), one obtains A X I + B 2 U 1e hie k cFi*(T.0) + $1 which leads to A hie k flYi*(T.0) c rhiek cFi*(T_.0) + 9 The latter implies that chic k clYi*(,T_.0) is (A,B) - invariant, and so decoupling would be achievable for TO, which is impossible by assumption. Thus, (A.2) holds as a strict inequality. [] APPENDIX 2 "PI+I "-q~'i# (~1+1) = "-%ip" (E,I)

~' I> 0,

I~> 0,

e k"

It is enough to show that this equality holds for I = 0. The rest will follow by transitivity.

i) "P .~i I~ (,7_.1)c ..,%i# (-~,0)" The proof is performed by induction. This inclusion is true for I~ = 0. Assume that it is also true for I~ - 1. Then, from CSA, "-q~'p" i (TI) = ~'i* (,7-,1) n (A e ..%i1~'1 (,7-,1) + ~e)

296 By projection on to ~. along '~'a, we have 9 P --,~ip" (7-,1) = P (~i* (7-,1) c~ (A e ,~q,iIx-1 (,~,1) + Se))

= P ~'i* (7-,1) c~ P (A e ..~i #-1 (7-,1) + ~e)

1"'3 since Ker P =/D- I [q~l

It follows, from Lemma A.1, that 9 P ~"iix (7-,1) = ~i (T0) c~ (Sp {A X I + B 2 UI} + 51)

= P ~i (7,1) c~ P (A e -~,iix -1 (7_,1 ) + ~e)

sinceKerP;t]

It follows, from Lemma A.1, that : P -..~iP" (.7_,1)= ~i* (T.0) n (Sp {A Xl + B2 UI} + 51) where

(A.3)

sp

fx,]

(A.3)

} ---b21~ ~'1 (21)

From (A.3) we have : P ..~i # (T.1) c ~'i* (-7--0) n (A -.~i p--1 (7_,0) + 5) = ..~i p- (7,0) ii) "..~i p- (,T_,0) c P ...~i I~ (7_,1 )" It is enough to show that V X I e '~iix (7,-,0), there exists U I such that

The proof is still perform by induction. The inclusion is true for Ix = 0 . A s s u m e that it is a l s o true for g-1. Then V R I a v e c t o r of ~ i I x -1 , t h e r e exist X 1 , U I, U 1 such that :

"%iIx ) RI = A X I + B2U I + B1U I e ~di* (7,0) Moreover, from lemma A.1, there always exists W I such that :

RI = Ae

+

I+

Wl

(~1)

By assumption

and so"

R--L] E ~i (~I) ('h (A e,~i -I (~'.I) +~e ) = '-'~Pi(7,1) Wi which ends the proof. 0

Minimal rational interpolation and Prony's method A. C. Antoulas Rice University & E.T.H. Ziifich and

J. C. Willems University of Groningen

Abstract. A new method is proposed for dealing with the rational interpolation problem. It is

based on the reachability of an appropriately defined pair of matrices. This method permits a complete clarification of several issues raised, but not answered, by the so-called Prony method of fitting a linear model to given data.

1. Introduction.

Given the array of scalar pairs points

(xi, Yi), i E N , xi~xj, i ~ j ,

(1)

with finite entries, we seek all rational interpolants, that is rational functions

y(x) = n(x) d(x)'

(2.1)

n, d: coprime,

(2.2)

=y,, i

(2.3)

where

which interpolate the array (1):

N_

Moreover, we wish to keep track of the complexity of interpolants, which is defined to be their McMillan degree:

deg y := max {deg n, deg d}. The following problems arise: (a) Find the admissible degrees of complexity, i.e. those positive integers ~ for which there exist interpolants y(x) with deg y = ~c (b) Given an admissible degree ~r construct all corresponding solutions.

298

The first complete answer to the above questions was provided by Antoulas and Anderson [1986] using the so-called L6wner matrix as the main tool. For further developments, see Antoulas and Anderson [1989], [1990] as well as Anderson and Antoulas [1990]. In the present paper we will discuss a novel approach to answering the above questions. This allows a complete analysis of Prony's method of modeling, which for example, although widely used in digital signal processing is not well understood. Suppose that there exists an interpolant y ( x ) = n(x)/d(x) having complexity ~; let n(x) := no + n l x + . . .

+ n ~ ~, d(x) : = d o + d l x + " ' " + d)cx ~,

with either n): or d K different from zero. The c o e f f i c i e n t v e c t o r s corresponding to n, d

are

defined as:

c~(n) := (n0

nl

...

n~),, c K ( d ) : = ( d o d I . . .

d,)'.

Conditions (2.3) imply that the coefficients ni, d i satisfy the following sets of linear equations: n(xi) - yid(xi) = O, i ~ N.

In order to write the above system of equations in matrix form, the N•

constant matrices I~:

and I ~ which depend just on the data, are defined; their i th row is [I~]i := ( 1 x i . . . x~ r ), [ l ~ j i : = - y i ( 1 x i . . . x ~ ). With the notation introduced, the above system of linear equations can now be expressed as follows:

f~,c,(n) + ft,c,(d) = O,

(3)

while the coprimeness condition n, d: coprime,

(2.2)

must be fulfilled. It readily follows that every interpolant of complexity 1< must satisfy (3), (2.2) and vice versa. The former being a linear constraint it is straightforward to deal with. The difficulty lies in dealing with constraint (2.2), since as already mentioned we are interested in keeping track of interpolant complexity at the same time. A very common approach to rational interpolation is sometimes referred to as Prony's method. It is based on the assumption that N = 2rn+l data pairs are always interpolated by a rational function of complexity m, since such a rational function contains exactly 2m + 1 free parameters. This assumption however is true for g e r e r i c , i.e. randomly generated, data only. In such a case, (I~ra l~m) has size Nx(N+I) with full row rank; therefore, up to a non-zero constant, there are unique c~:(n), c~:(d) which solve equation (3). Moreover, the coprimeness constraint (2.2) is generically always satisfied. Consequently in the generic case, one interpolant (of minimal complexity m) can be computed this way. For n o n - g e n e r i c data however, trying to fit 2m + 1 data pairs with a rational function of complexity m, may or may not give a solution. Even if a solution is obtained

299

this way, questions (a) and (b) formulated above, remain unanswered and no understanding of the coprimeness constraint is achieved. In the sequel we will expand on the approach which is based on equation (3) and constraint (2.2), and show how the system-theoretic concept of reachability allows us to take care of this coprimeness constraint. 2. The main results.

elii

Consider the pair of NxN, Nx2 matrices defined in terms of the data array (1):

X2

q

F :=

G : = ( g t g2): = x/v

Y2

(4)

N

A closer examination of the matrices 1~ and 1~: defined in the previous section, shows that they are partial reachability matrices of the above pair, containing ~ 1c2, can be represented in the f o r m

where

p, q

c~:(n) = Spc~l(Oll ) + Sqc~2(012),

(17.1)

c~(a3 = Spc~1(021) + Sqc~(Oz2)

(i 7.2)

are coprime polynomials of degrees ~c-~ct, ~c-~c2 respectively, and satisfy the

cop rimeness constraint (11).

303

Remarks (i) In view of the above theorem, here is the solution procedure involving equation (3) which is the starting point for Prony's method. First, construct the full reachability matrix R N of (F, G); as already mentioned, this is the same as the matrix (I/N I1N) which enters in (3), up to column permutation (this column permutation is crucial in the sequel). Proceeding from left to right, determine the first column, say FKIg2 of R N which becomes linearly dependent on the pre-

vious ones (i.e. linearly depend

on

FJgi, j < 1'~1,i =

1, 2).

The corresponding vector, after permuta-

tion of its entries, will be [ c~'(011) ]. %(021) J Because of reachability of the pair (F, G) the next linear dependence which is needed is that of

F':2gt, where lc2 : = N - ~cl, on the previous columns of R w. The corresponding vector, after permutation of its entries, will be [%(012)

r 1" The linear dependencies of all intermediate or subsequent columns are of no interest. All solutions to the problem are now parametrized by means of (17.1,2). (ii) The kernel of (ll N I~N) has dimension N + 2. But out of all coefficient vectors in this kernel, only two are needed. Their choice is dictated by the teachability property of the associated pair (F, G). Moreover, these two coefficient vectors are the basis for parametrizing interpolants of complexity below, equal to, and above N. (iii) The matrix (Sp Sq) is very closely related to the Sylvester resultant of p, q. The difference between the two lies in the fact that the Sylvester resultant has more columns. Therefore, because of the coprimeness of p, q, (Sp Sq) in the theorem above, must have full rank. [] 5. An example. Consider the data array containing 4 pairs: (0, 0), (1, 3), (2, 4), (1/2, 2), According to (4):

i0 i [!0] 1

-3

, G:=

2

-4

1/2 It follows that the fuU teachability matrix of the above pair is

-2

304 iI R4 =

]

0 -3

0 0 1 -3

0 1

0 -3

0 1

0 -3

0 1

0 1 -3

-4

2

4

-16

8

-32

16

-64

-8

-2 112 -1 1/4 -1/2 1/8 -1/4 1/16 -1/8

Recall remark (i) of the previous section. The first linear dependence occuring in R 4 is that of the fourth column: Fg2+6Fg 1+ g2=0.

It follows that x:t = 1. Therefore according to the same remark, ~c2 = N - ~cI = 4 - 1 = 3. This means that the next linear dependence is that of the 7 th column on the previous ones. It turns out to be 2F3gl - 9F2gl + 4Fgl - 2Fgl + g2 = O. The corresponding coefficient vectors are c1(011) = (0 6)', cl(O21 ) = (1 1)',

c3(012 ) = ( 0

4 -9

2)',

c3(022 ) = (1 - 1)'.

One column reduced version of O is therefore

e(x)= [ 6x

x(x - 4)(2x - 1) 1.

x+l

-

(2.x-

1)

J

By (i) of the main theorem, since 6x and x + 1 are coprime polynomials, there is a unique minimal interpolant with McMillan degree 1, namely y~in(x) =

6x x+l "

By (i) of the same theorem, there are no interpolants of degree 2. The next family of interpolants has McMillan degree 3. It can be parametrized in terms of the second degree polynomial p

as

follows: y(x) =

x(x - 4)(2x - 1) + (.o2x2 + p l x + po)6X -

(2x - I) + (p2x2 + p ~ x +po)(X+ 1)

The coefficients of p must satisfy the constraints (11), which in this case turn out to be po + 1 s O , 2p2 + 2pi + 2 p o - 1 s O , 4P2 + 2Pl + Po - 1 s O , P2 + 2pl + 4Po ~ O.

By letting p 2 = p i = 0

and p 0 = 2

in the above family of interpolants, we obtain the Lagrange

interpolating polynomial for our data array l(x) = 3 ( 2 x 2 - 9 x + 16).

The next family o f interpolants has McMillan degree 4. It can be parametrized in terms of a third degree polynomial p and a first degree polynomial q , as follows:

305

y(x)=

(p3x3 +p2 x2 + p l x + p0)6x + (qt x + qo)x(x - 4)(2x - 1) (P3X3 +p2 x2 + pl x + po)(X + 1) - (qlx + qo)(2x - 1)

Firstly, p, q must be coprime, i.e. p3q 3 -p2q2oql + p l q o q 2 - p o q 3 ~ O.

Then, constaints (11) must also be satisfied: po + qo ~ O, 2p3 + 2p2 + 2pl + 2po - ql - qo ~ O, 8p3 + 4p2 + 2pl + po - 2ql - qo ~ O, p3 + 2p2 + 4pl + 8po ~ O.

Notice that the first constraint is a non-linear one while the latter four are linear ones, as mentioned in remark (ii) of section 2. 5. Generalizations and concluding remarks. The above results can be generalized considerably. Consider the array consisting of the data

xi, vl, ri, i ~ N_, of size l / l , rixp, rixm respectively, satisfying the constraints x i ~ x j , i~:j, and r a n k V i = r i < p.

The former constraint is the so-called distinct-point constraint. The more general multiple-point interpolation problem involves information about derivatives. As it turns out, the latter constraint can be made without loss of generality. The left tangential or left directional interpolation problem consists in finding all p x m rational matrices Y(x) satisfying viY(xi) = Yi, i ~ N_,

keeping track of their complexity at the same time. The right tangential or right directional interpolation problem, as well as the bi-tangential or bi-directional interpolation problem, can be defined similarly. The procedure presented in the preceeding sections for the scalar interpolation problem, can be used to answer questions (a) and (b) formulated in the introduction, for all of the interpolation problems mentioned above. The complete solution of these problems, using the idea of defining appropriate reachable and observable pairs of matrices, is given in Antoulas, Ball, Kang, and Willems [1990]. In conclusion, the observation that the matrices 1~K and 1~ can be regarded as partial reachability matrices of a reachable pair (F, G), allows us to obtain complete insight into the rational interpolation problem. As a consequence, it becomes clear in what respect Prony's method of data fitting is deficient, and how to modify it so that it yields the complete answer.

306

6. References. A. C. Antoulas [1981], "On canonical forms for linear constant systems", Int. J. Control, 33: 95-122. B. D. O. Anderson and A. C. Antoulas [1990], "Rational interpolation and state variable realization", Linear Algebra and Its Applications, Special Issue on Matrix Problems, in press. A. C. Antoulas and B. D. O. Anderson [1986], "On the scalar rational interpolation problem",

IMA J. of Mathematical Control and Information, 3: 61-88. A. C. Antoulas and B. D. O. Anderson [1989], "On the problem of stable rational interpolation", Linear Algebra and Its Applications, 122-124:301-329. A. C. Antoulas and B. D. O. Anderson [1990], "State-space and polynomial approaches to rational interpolation", Proc. MTNS '89, Birkh/iuser, Boston. A. C. Antoulas, J. A. Ball, L Kang, and J. C. Willems [1990], "On the solution of the minimal rational interpolation problem", Linear Algebra and Its Applications, Special Issue on Matrix Problems, in press. T. Kailath [1980], Linear systems, Prentice Hall.

Discrete Normalized Coprime Factorization Peter M.M. Bongers, Peter S.C. Heuberger Laboratory for Measurement and Control Department of Mechanical Engineering and Marine Technology Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands

ABSTP~ACT

In this paper a reliable algorithm is developed to perform a normalized coprime factorization of proper discrete time finite dimensional linear time invariant systems. Instead of using the bilinear transform the factorization is calculated directly. The system is allowed to have a singular state-space matrix. It is shown that a modified discrete time Riccati equation plays a crucial role to obtain a state-space realization for the factorization.

I. INTI~ODUCTION

IlL {,he theoretical work of Desoer et al. (1980), Vidyasagar et a1.(1982), Vidyasagar (1984) the benefits of using coprime representations in stability analysis of controlled systems are shown. In the continuous time case Nett et. al. (1984), Meyer and Franklin (1988) and Vidyasagar (1988) derived state--space representations for the normalized coprime factors. Glover and McFarlane (1988), (1989), McFarlane (1988) established the importance of normalized coprime factors in the Hoo controller design. They explicitly solved a continuous time four block Hr control problem by using a normalized coprime representation of the plant. In practical situations a continuous time plant is controlled by a discrete controller using sampling and zero order hold circuits. So in order to design accurate discrete time controllers the controller design procedure has to be done in discrete time. The first step in discrete Hoo control design with normalized coprimc factors is to establish whether or not in discrete time normalized coprime factors exists and can be represented in state--space forms. Chu (1988) gave state-space representations for discrete coprime factors with an inner numerator under the condition that the system has no poles in the origin. Poles in the origin are of major importance since in discrete time very often time--delays have to be incorporated in the system model (appendix 1). In this paper we will show the existence of a normalized coprime factorization of a discrete time plant with possibly poles in the origin.

3O8 II. PB.ELIMINAILIES

Stable multivariable linear systems can be studied by considering them as transfer function matrices having all entries belonging to the ring 7/. Moreover, in many cazes (e.g. convolution operators) the ring 7/is commutative and is an integral domain (i.e. 7/has no divisors of zero). The class of possibly unstable systems are elements of the quotient field F of 7/. Throughout this paper we let (Vidyasagar et.al. 1982, Desoer et.al. 1980): :7: := { a/b [ a e 7/, b E ~ 0 }, a quotient field of 7/ G := a (not necessarily commutative) ring with identity. 7/:= a subring of G which includes identity I := { h ~ 7/I hq e G }, the set of multiplicative units of G 3" := { h e 7/[ h-l e 7/}, the set of multiplicative units of 7/ Note that:

Jczc~c~c7

(1)

In the sequel of this paper we will study real rational finite dimensional discrete time invariant systems. The ring {~ is identified with [RL~ the space of proper real-rational functions with no poles on unit circle with norm H-f]| []J(z)Hoo= sup ~[j(eJ0)]. The subring 7/is identified with U~IIco 00

The procedure to obtain a normalized right coprime factorization for the plant G is to solve the fticcati equation (12 c,d) to obtain Q, calculate F and choose an It. The equivalent for the normalized left coprime factorization is a direct result from theorem 1 and is given in the following corollary. COItOLLAR.Y 1 Given a minimal realization (10): G(z):=D+C(zI-A)'IB:= [ ~ ]

E5r

and define:

then [1q IN] is a normalized left coprime factorization of G(z) iff: a)

K = (APCt+BDt)(I+CPCt+DDt)q

b) c)

RtR = (I+CPCt+DDt)q P - A P A L- BBt+(APCt+BDt)(I+CPCt+DDt)-I(CPAt+DBt) = 0

d) p = pt > 0 Plt00F Let G = i'r

with (IQ,I~) E 2' a nlcf of G, then G t = ~@[-t with (~lt,l'r

a nrcf of G t, so the

realization of [IQ N] follows from theorem 1. cl I~.EMMtK Note that we don't need the assumption that A is invertible, as is the case in Chu (1988). Using proposition 2.1 it is straight forward to show that this assumption is indeed superfluous. This is of major importance since in discrete time very often time-delays are incorporated in the system. In order to solve the normalized coprime factorization for the plant in discrete time by means of standard techniques, the equation (12@ Q - AtQA - CtC + (AtQB + CtD)(BtQB + DtD + I)-I(BtQA + DtC) = 0 can be written as a standard Riccati equation. Define:

A,: [A

E0IJ,

:[0

The standard Riccati equation with AI,Bt,CI,RI,Q1 is : Q I - A I t Q t A I - CttCl + AItQIBI(BltQLBI + Rl)-lGtQiA1 = 0

312 Substituting the definitions of Al etc. gives:

[77]-r-ooo1[o evaluating this equation gives (12c). Which means that using standard techniques the discrete time normalized coprime factorization problem can be solved. Note that the sufficient conditions for the existence of a positive solution of the Riccati equation are still valid.

IV. CONCLUSIONS

Theorem 1 and Corollary 1 show that with standard mathematical tools the normalized coprime factorization can be calculated, which is necessary to design discrete time controllers,that satisfy H~o robustness bounds. Since in practical applications one will in general be dealing with a discrete time problem, this is an important step towards the solution of the Hoe control problem in discrete time.

V. LITEKATUgs

Anderson, B.D.O., J.B. Moore (1979), Optimal Filtering, Prentice Hall, Inc., Englewood Cliffs, N.J. Chu, C.C. (1988), On discrete inner-outer and spectral factorizations,Proc. Amer. Control. Conf., Atlanta, Georgia Desoer, C.A., R-W. Liu, J. Murray, It. Seaks (1980), Feedback System Design: The Fractional Representation Approach to Analysis and Synthesis, IEEE Trans. Automat. Contr., AC---25, 399---412 Glover, K., D. McFarlane (1988), Robust stabilization of normalized coprime factors: an explicit Hoe solution, Prec. Amer. Control. Conf., Atlanta, Georgia Glover, K., D. McFarlane (1989), Robust stabilization of normalized coprime factor plant descriptions with Ho0---boundeduncertainty, IEEE Trans. Automat. Contr., AC-33, 821-830. Heuberger, P.S.C (1990), PhD Thesis (to appear), Delft University of Technology, The Netherlands. Huang, Q., R. Liu (1987), A necessary and sufficient condition for stability of a perturbed system, IEEE Trans. Automat. Contr., AC---32, 337-340. McFarlane, D. (1988), Robust controller design using normalized coprime factor plant descriptions, PhD. thesis, University of Cambridge. Meijer, D.G., G.F. Franklin (1988), A connection between normalized coprime factorization and linear quadratic optimal regulator theory, IEEE Trans. Automat. Contr., AC--32, 1041-1047

313

Nett, C.N., C.A. Jacobson, M.J. Balas (1984), A connection between state---space and doubly coprime fractional representations, IEEE Trans. Automat. Contr., AC--29, 831---832. Rosenbrock, H.H. (1970), State-space and mullivariable Theory, Thomas Nelson and sons Ltd. London, Great Brittain. Vidyasagar, M. (1984), The Graph Metric for Unstable Plants and Robustness Estimates for Feedback Stability, IEEE Trans. Automat. Contr., AC--29, 403--418. Vidyasagar, M. (1988), Normalized coprime factorizations for nonstrictly proper systems, IEEE Trans. Automat. Contr., AC,---33, 300-301. Vidyasagar, M., H.Schneider, B.A. Francis (1982), Algebraic and Topological Aspects oI" Feedback Stabilization, IEEE Trans. Automat. Contr., AC-27, 880---894.

APPERDIX 1 In this appendix we show the existence of a singular state--matrix due to time-delays. Let the discretized plant be given by: Xk§ ---- A x k 4- Bvk Yk = Cx k 4- Dvk

In general the discrete time controller will have one time---step delay between the measured outputs and computed plant inputs. Using discrete time control design algorithms the time-delay has to be incorporated in the plant description. Let the time-delay be described by: Zk§ = Uk Vk ---- Zk

Then the augmented plant can be written as:

[x< -Zk*lJ

[::] +

Yk = [C D] Zk Hence the augmented state matrix is singular.

A FORWARD FOR

ACCESSIBILITY

NONLINEAR

DISCRETE

ALGORITHM TIME

SYSTEMS

J.P.BARBOT LABORATOIREDES SIGNAUX ET SYSTEMES CNRS-ESE PLATEAU DE MOULON, 91190 GIF SUR YVETTE, FRANCE.

ABSTRACT In this paper we present a SAMDI algorithm (Symbolic Algebraic Manipulation for Discrete time system), which permits us to know whether a system is accessible for nonlinear discrete time. This algorithm uses the symbolic language REDUCE. In our algorithm we use a set of Gij vector field, which allows us to work with the same geometrical tools as for the continuous time system. Therefore, the SAMDI algorithm uses the same procedures of the SAM (Symbolic Algebraic Manipulation) programme, which was realized for continuous time systems.

INTRODUCTION The local controllability of a linear analytic system is based on the concept of infinitely small displacements in the neighbourhood of a state point. This displacement is generated during an infinitely short period o f time, which is impossible to realize for discrete time systems, because there the sampling period is fixed. In order to create this infinitely small displacement in the neighbourhood of a state point, M. Fliess and D. Normand-Cyrot [1] have chosen a very short input variation. This new concept allowed us to introduce the notion of weak accessibility in discrete time [2], [3], which is an equivalent relation on the state manifold and allows us to use of the orbit's theorem [4]. Thus we obtain a criterion of weak accessibility in terms of the tangent state space at a point of the Lie algebra. This is generated by a set of Gij vector field, introduced by S. Monaco and D. Normand-Cyrot [5]. We may consider this vector field as a derivation of the input of a causal flow (which is the difference equation of the discrete time system) and a non-causal flow (which is the inverse of the difference equation of the discrete time system), or vice versa. This

315

composition of functions draws all the field vectors back to the initial point. For reasons of computing and because of its more physical character [6], we use in this algorithm the concept of "forward accessibility" [7], which is a restriction to the accessibility in the causal, followed by the non-causal, displacements. From the above-mentioned works, we will introduce below a SAMDI (Symbolic Algebraic Manipulation for Discrete time system) algorithm of forward accessiblity for the discrete time system. This programme has been realized with the symbolic language REDUCE [8] and also using some procedures of SAM (Symbolic Algebraic Manipulation) [9] (for continuous time systems). In order to simplify our calculations the SAMDI algorithm has only been realized for discrete time systems with a single input. Following theory is only used to introduce the algorithms easily and the reader can turn to the references to get a more extensive account on them.

Theory

on the Gid vector

fields

Consider a linear analytic discrete time system Z: X(k+l) A__f(X(k)) +g(X(k)) U(k) =Ah(X(k),U(k)) where X e R n, U e R and f, g, h are the vectorial analytic functions of the appropriate dimension. Definition 1 is the set of inputs U, such that h(.,U) is inversive for all X. Remark In this paper we consider the set of ~ , which is not empty and where all the inputs U belong to f2. We have to remind the fact that, if in a point X, we apply a composed function with its inverse, we remain in that same point. For the analytic discrete time system S. Monaco and D. Normand-Cyrot [5] introduced the following set of vector fields:

G(U) I XO = ~ h'l("U)'h("U+e)oe I XO, e = 0

316

We can interprete these vector fields graphically in the following figure:

/

( "

7,0 /'

Figure 1

\ xs" "

Prooosition 1 The differential operator G(U) applied to function h is equal to the Lie derivative of this function h on the vector G(U).Id (see [5]). From the G(U) vector fields we deduce of new set of vector fields Gij (where i e N andj Z). Thus

Gi,01x 0 =

~G(U) ~ U i [U--0,x0

with G0,0Ix 0 = G(U)Iu=0, x 0 In the same way we find

G0,j [X 0 =

f-J h(.,u)-l.h(.,U+e).fJ [ U=0, ~0,X 0 ~ e

and Gij Ix 0 = ~i+l f-j h(.,U)-l.h(.,U+e).fj [O=0,e--0, X0 ~UiOe

Remark The fJ function is equal to the fJth function of f and f0 is equal to the identity function.

317

FORWARD ACCESSIBILITY As a result of the research on weak accessibility of discrete time systems [1], [2], and [3], and on the notion of action [11], B.Jakubczyk and E.D.Sontag [7] introduced the notion of forward accessibility. Below you will find the definition of a linear system with discrete time system 2; with a scalar input. We assume that the elements of input A are open in ~ and zero inputs in element A. We first define A+(Xo ) as a forward accessible set of X0. A+(X0)= { X 9 R n / 3 k 9 N , ~ / i 9 k , 3Ui+,Ui - 9 Asuchthat X = h(.,U0" )'l....h(.,Uk')-l.h(.,Uk+)....h(.,U0 +) Ix 0 } From A+(X0) it is possible to define the forward accessibility. Dgfinition 2 A system is said to be forward accessible if, and only if, there exists V(X0) in the neighbourhood o f X 0 such that A+(X0) n V(X0) = V(X0). From this definition and the field vectors Gij [5], B.Jakubczyk et E.D.Sontag formulated the following theorem: Theorem 1 [7] The discrete time system 2; is forward accessible in X0 if, and only if, the vector fields generated in X 0 by the smallest Lie algebra F+(~) contains all the elements of Gij where id 9 N, is the dimension of n. R~mark The forward accessibility in X0 (r+(X0)) is the dimension of the general field vector generated in X 0 by F+(Y.). We find the proof of this theorem in [7] or in [6] with the mathematical tools introduced in [2]. The set of vector fields used in the above theorem is the only difference with the one given by D. Normand-eyrot [2]. TheQr~m 2 [21 The discrete time system s is weakly accessible at X 0 if, and only if, the vector fields generated in X 0 by the smallest Lie algebra F(Y.), which contains all the elements of Gij with i e N andj 9 Z.

318

Remarks a. F+(Y.) is included in F(~) and therefore forward accessibility means weak accessiblity. b. In a recent paper of A. Mokkadem [11], he introduces the concept of quasiaccessibility (on a Zariski topology) and studies the links of these concepts of accessibility. Thus, for some systems the results of our algorithm can be applied to the quasi-accessibility. COMPUTATION OF THE Gkj.VECTOR FIELDS As shown in [5], the Gi,j, with i, j e N, are easy to compute. Thus, like before, the vector fields G0,0 can be defined as follows

G0,01 X0 =

0 FG(.,0)-I.FG(.,e) de I X0,~0

which is equal to GO,O IX0 = j(f-1) I f(X0) -g [ x 0 where j(f-l) is the Jacobian matrix of function f-1. As a result of the following equation 3f-l.f

I X0= j(f-1) I rcx0) J(0 t x0 = Id

where Id is the matrix identity, we get: IS0,01 x 0 = J(f)-I ] x0.g [ X0l where the vector fields Gi,0 are equal to:

Gi,01 x 0 =

~i[J(f)+J(~).U]-I 0U i I X0,U=0 .g Ix 0

The properties of the derivatives of the matrix give us the following equation: [Gi,0 ]X 0 = (-1) i i! ([J(f)'l.j(g)]iJ(f)-l.g) I X0]

In the same way, the vector fields Gk,1 are equal to:

Gk,l IX 0 =

0k+l f-1 FG(.,U)-I.FG(.,U+e).fl 0 Uk 0 e [U--O'~O'Xo

319 which gives Gk,1 IX 0 = [j(f)]-I

1x0.Gk,01 r(x0) ]

Finally, from their recurrent property, we obtain: [Gk,j+l ] X0 =

[J(f)]-I I x0.Gkd l r(x0) ]

INTRODUCHTON OF THE "SAMDI" ALGORITHM AND CRITERIA TO STOP As shown above, the vector fields Gi,j are products of the Jacobian matrix, which are easily computable in the symbolic language REDUCE [8]. As an example we given the following little programme: FOR I:=1 : n DO BEGIN FOR J:=l : n DO BEGIN JF(I,J):=DF(F(I),X (J)); END; END; where DF(F(I),X(J)) is the symbolic function of the derivation of the first component of the vectorial function F, compared to the variable X(j). Note that the inverse of the matrices and their product are basic functions of the REDUCE language. Moreover, the calculation of the Lie brackets and the dimension of the vectorial space generated by a certain number of vectors are given in [9]. Therefore we limit ourselves to the following introduction of some stopping criteria. In this algorithm the first step must be to calculate the dimension of the space generated by the vector fields Gi,0 and X0 with i E N, for a finite number of vector fields. Therefore, as a result of equation (2) and the Cayley-Hamilton theorem, we find the following stopping criterion: CRITERION 1 We stop the calculation of Gj,0 when rank [Gj,0....G0,0] I x 0 = rank [Gj-I.0..... G0,0] ix0. The second step consists of calculating the dimension of the vector fields generated in X0 by the smallest Lie algebra 1-'+(~), which contains all the elements of Gi,j. For this we use the property of involution, see [12] and [9].

320 CRITERION 2 Consider a set of 13vector fields. If ~ Lf a 13and "~ Lg ~ 13we have [Lf,Lg] Ix ~ EB(X) where EB(X) is the vectorial space generated by the vector fields 1] en X. From this we can conclude that the dimension of EB(X) is equal to the dimension of space vector X, by means of the smallest Lie algebra containing 13. The following step is to calculate the vectorial space of the Gk,l where k and 1 are the integers. In order to stop this calculation we use criterion 1 for exponent k and the Cayley-Hamilton theorem for exponent I. However, in equation (4), the matrices are function of the exponents. For this, we need to assume that X0 is not a unique point. Also, the rank of a Jacobian matrix is a function for which there exists an open set of 7 state points, such that x ~ 7, and that the rank is maximal. Therefore we can introduce the following criterion: CRITERION 3 We stop the calculation of Gj,p as soon as rank [Gj,p+l ...... Gj,0] Ix 0 = rank [Gj,p ...... Gj,0] I X0.

The last criterion is evident, but we should not omit to use it in a programme: CRITERIQN 4 Whatever the vectorial space Eli generated by vector fields of n dimension, the dimension of El3 is smaller or equal to n. With the help of these criteria we can realize a SAMDI algorithm, represented in Figure 2. EXAMPLES The following examples are arbitrarily chosen to illustrate some properties of forward accessibility. Examole 1 Consider the following discrete time linear system: X(k+l) = A X(k) + B U(k) where X e R n, U ~ R and A, B are the appropriate dimensions of the matrices.

321

The forward accessibility is different from the usual discrete time commandability [12], by the fact that matrix A must be regular (see [6] for a comment on this subject). If matrix A is regular, all the inversive inputs are equal to R~ The forward accessiblity rank r + is then r+= rank [B,AB ........An'IB]

Consider the following linear analytic system Y= X1 (k) 3+X2(k) X(k+l) = FG(X(k),U(k)) - ~.X ] ( k + l ) = - [X2(k+l) = X2(k)+U(k) This system is inversive on levels XI>0 and XI0

G0,t = (9 X12 (X13+X2)2) "l (-3(XI3+X2)2-1, 9 X12 (X13+X2)2) T

The system is forward accessible, but we find a set of singularities for (X13+X2) 2 equal to zero, which correspond to Xl(k+l) equal to zero. Note that all the singularities found for this system are a result of the too restrictive definition of forward accessibility. Indeed, whatever the values of X(0), with the help of U(0) = - (X13+X2)3-X2 and U(1) = (X13+X2) 3 we find that state X(2) is equal to (0, 0)T. In the same way, from (0, 0) it is possible to go to any point (XI, X2) with the help of control law U(0) = X1 and U(1) = X2-X1.

CONCLUSION The above presented algorithm of forward accessibility is a symbolic tool which is useful when we study a linear analytic discrete time system with scalar input. Indeed, if the system is not forward accessible in an open set which is not empty, it is of no use to look for control law. However, if the system is forward accessible, it is not always possible to control all the states. We also have to limit ourselves to inversive systems, but this restriction does not hold for sampling systems [ 14]. Finally also note that the extension of SAMDI algorithms to linear analytic discretetime systems with multi-input only give some computational problems (sufficient memory, time of calculation, increase of the number of vector fiels index in order to take into account eachinput .... ).

322

ACKNOWLEDGEMENT I especially wish to thank M rs Doroth~e Normand-Cyrot and M r Salvatore Monaco for their precious help and suggestions. Without their publications [1], [2], [3] and [5], this article could not have been written. The research has been made possible by a grant from the European M.R.E.S.

,

t

GT,0 C,)HPIJTI~T I OH TR I RI IGU LP,~ 1Z,~T 1 C+

X ) ' ~ . 'i.~-

,+,,'+'O_.~F:~TE~XO~

I ~ '..t.o

-.,,

,p-

[

~/~-.

vm

'v~

TIlE

,f.

-,II--C}-~;:R I T E R I

fK' ~RITEF.~ON~ #

-'-...5/

C~r

.o ~".... I L

I,

-

INIjOLUT i C,N R~-IC'l T ~ IRNGULRR Z Z ~ T I O N

"NO

~

TIIE SYSTEM IS F011NAKD ACCESSIBLE

SVSTEH ISII'T 1

FOnHanD A C C E S S [riLE

Figure 2

323

REFERENCES

(1)

M.Fliess, D.Normand.Cyrot (1981) " A Lie-theoretic approach to nonlinear discrete-time controllability via Ritt's formal differential groups" Systems and Control Letters, Vol 1, N~ pp 179-183

(2)

D.Normand.Cyrot (1983) " Throrie et pratique des syst~mes non linraires en temps discret" Thesis of Doctorat d'Etat, University of Paris Sud, Orsay

(3)

B.Jakubczyk, D.Normand.Cyrot (1983) " Orbites de pseudo groupes de difffomorphismes et commandabilit6 des syst~mes non linraires en temps discret" C.R. Acad. Sciences Pads

(4)

H.J.Sussmann (1973) " Orbits of families of vector fields and integrability of distributions "Transactions of the American Mathematical Society Vol 180 pp 172-188

(5)

S.Monaco, D.Normand.Cyrot (1984) "Developpements fonctionnels pour les syst~mes non lineaires en temps discret" Report of I.A.S.I, C.N.R Italien (in French) and "Functional expansions for nonlinear discrete time Systems" Math. System Theory 21 pp 235-254 (1989).

(6)

J.P.Barbot (1989) " MEthodes de calcul appliqudes aux syst~mes non linEaires sous 6chantillonnage" Thesis of Docteur en Science, University of Paris Sud, Orsay

(7)

B.Jakubczyk, E.D.Sontag (1988) " Controllability of nonlinear discete time systems: A Lie-algebraic approach" Proposed to S.I.A.M J. Control and Optimization

(8)

A.C.Hearn (1984) " R.E.D.U.C.E User's manual " The Rand Corporation, Santa Monica USA

(9)

R.Marino, G.Cesaro (1982) " Nonlinear control theory and Symbolic Algebraic Manipulation " Ed Springer-Verlag Proc. Beer-Sheva Israel

(10) E.D Sontag (1986) "Orbit theorems and sampling " M.Fliess and M. Hazewinkel Eds Algebraic and Geometric Methods in nonlinear control theory pp 441-483 D.Reidel Publishing Company (11) A.Mokkadem (1989) "Orbites de semi groupe de morphismes rrguliers et systrmes non linfaires en temps discret" to appear in Forum Mathrmafique. (12) A.Isidori (1989) " Nonlinear Control Systems : An introduction " Communications and control engineering series (2nd Edition) Springer-Verlag (13) R.E.Kalman (1960) " On the general theory of control systems" Proc. First International Congress on Automatic Control, Moscou. London, Butherworth 61, Vol 1 pp 481-492 (14) S.Monaco.D.Normand.Cyrot (1985) "On the sampling of a linear analytic control system" 24 th Conf. Decision and Control Ft Lauderdale pp 1457-1462.

S t a b i l i s a t i o n globale de s y s t ~ m e s n o n - l i n ~ a i r e s p a r un contr61e positif. Jean-Luc Gouz4 INRIA Sophia-Antipolis 06565 Valbonne Cedex, France A b s t r a c t : We give sufficient conditions for global stabilization in a fixed domain around an equilibrium of a class of controlled systems of non-linear ordinaa-y differential equations in n-space, that can be expressed as the difference of a quasi-monotone increasing and a monotone increasing system. These systems are usual in biological modelling. We consider the cases when the control is scalar, or scalar and positive. We apply the theory to LotkaVolterra systems. R 4 s u m 4 : Nous donnons des conditions suffisantes de stabilisation globale dans une r4gion fix4e par un contr61e autour de l'4quilibre pour une classe de syst~mes diff6rentiels nonlin6aires en dimension n qui sont diff6rence d'un syst~me quasi-monotone croissant et d'un syst~me monotone croissant, fr4quents dans la mod41isation biologique. On consid~re le cas d'un contr61e scMaire et d'un contr61e positif scMaire. On prend pour exempte les modules de Lotka-Volterra.

I

Introduction

Nous nous int4ressons dans cet article au probl~me de la stabilisation globale vers l'4quilibre, dans une r4gion fix~e born4e de l'espace, d'une classe de syst~mes diff~rentiels non-lin4aires, par un contrSle en boucle ferm4e. L'approche la plus usuelle, et sans doute la plus g4n4rale, clans ce type de probl~me, consiste k utiliser des fonctions de Liapunov (voir [8]); mais il est souvent difficile, quand le syst~me est sp4cifi4, d'exhiber une telle fonction. D'autres approches existent, qui utilisent par exempte la th6orie de la vari4t4 centrale ([1]), mais elles sont surtout locales. Nous utiliserons ici une autre approche globale, ne faisant pas appel aux fonctions de Liapunov, mais nous restreindrons la classe des syst~mes non-lin4aires 4tudi4e. Ces syst~mes soar issus, dans notre cas, de probl~mes de mod41isation biologique, par exemple en dynamique des populations. I1 est alors normal que les signes de certaines grandeurs jouent un rSle important, car ils peuvent s'interpr4ter biologiquement (voir par exemple [7] et [2]). Nous consid4rons donc la classe de systbmes non-lin4aires n-dimensionnels: ~ = f(x)

- g(x)

(1)

avec les conditions suivantes vMables pour x dans un ouvert convexe W C R~: ~Ofi , (1:) >_0 pouri # j e t

( x ) > _ O ( i , j = l . . . . ,n)

(2)

Les fonctions f e t g sont suppos4es C 1. On dit que f e s t quasi-monotone croissante et que g est monotone croissante si elles v4rifient (2)(volt [13]). S i g -- 0, o11 dit aussi clue le syst~me

325 (1) est coop~ratif (voir [4]). Des conditions de ce type i n t e r v i e n n e n t souvent dans les syst~mes difffirentiels modfilisant des ph~nombnes biologiques ([7]), car, par exemple, le signe de ~ ( x ) traduit si la variation de l'esp~ce j influence positivement ou nfigativement 13 variation de l'esp~ce i. Remarquons que cette classe de syst~mes est d~jk tr~s vaste ([3]). Un exemple i m p o r t a n t est c e h i des modNes de Lotka-Volterra (voir [12,11]) qui s'fierivent:

,i:i = xi(bi + f i cqx~)

i =

1 ....

, n

(3)

j=l

Les syst~mes coop~ratifs et comp~titifs (en inversant le temps) ont ~t~ beaucoup ~tudi~ m a t h ~ m a t i q u e m e n t (voir [5,9]). Nous avons utilis~ ces r~sultats pour obtenir u n critbre surfisant de convergence globale pour le syst~me (1) vers un ~quilibre suppos~ unique ([3]). Nous appliquons ici ce critSre k u n probl~me de stabilisation globale dans une r~gion fix~e de ce systSme par u n contr61e scalaire et par un contr61e scalaire positif (la positivit~ dn contrSle a yant souvent un sells dans les probl~mes biologiques). Plus pr~cis&nent, nous cherchons k stabiliser (en boucle ferrule ou par des contr61es constants) le syst~me : 2 = f ( x ) -- 9(x) - ud(x)

(4)

o~ u est un scalaire et d une application que nous prendrons d'abord trbs simple pour l'~tendre ensuite. Dans la suite, quand nous parlerons de stabilisation globale, cell voudr3 dire que nous savons exhiber une r~gion born4e conten~m dans W, c o n t e n a n t un seul point fixe a s y m p t o t i q u e m e n t stable dont le bassin d ' a t t r a c t i o n contient la r4gion. Nous l y o n s limit~ notre exposition au cas u scalaire, mais on peut facilement ~tendre l' dtude au cas multivariable (volt la premiere remarque de la section 6). Dans la section 2, nous citons le crit~re de convergence globale vers l'~quilibre, que nous appliquons au probl~me de stabilisation dans la section 3, avec u n contr61e positif en section 4. Nous terminons en section 5 par une application k la stabilisation du module de LotkaVolterra et quelques remarques et interpr4tations biologiques. Notations: Pour x E R ~, on ~crit x >__ 0 si xi >_ 0 i = 1 . . . . , n et x > 0 si xi > 0 i = 1 , . . . , n . On note R~_ = {x E R"; x >_ 0} (c'est l ' o r t h a n t non-n~gatif fermi) et, si z < y, [z,y] = {z ; x < z < y} (intervalle ferm~ m d i m e n s i o n n e l ) . De m~me, on notera par e ~ le vecteur ( e ~ ' , . . . , e~"), par In x le vecteur (In a'l . . . . . In x,~). Si W C R " est ouvert, h : W ~ R " est C 1, et x E W, on note par Dh(x) 13 matrice jacobienne Ohi/O.rj(x). Soil x : J --+ R ~, o~l J e s t u n intervalle rdel, on note if: la d~rivfie de x pat" rapport an temps t. Si W C R ~ est ouvert, h : IV ~ R " est C 1, pour l'~quation diff~rentielle .i: = h(x) et une condition initiale x(0) = xo dans W, nous noterons .v(t, xo) ou parfois x(t) la solution maximale issue de x0. Enfin, on note (ei), i = 1 . . . . . n la base canonique de R ". 2

Un

crit&re

de

convergence

globale

Nous citons ici le crit~re de convergence globale en renvoyant ~ [3] pour t'exposition. Soil donc le syst~me (1) v~rifia.nt d,~ns I.V ouvert COllvexe les conditions (2), avec f et g suppos~es

326

C 1. Nous construisons alors un syst~me dans R2~:

{v

= f(v)

-

g(~)

= f(~)

-

g(v)

(.~)

ddfini pour (y, z) E W x W. On a Mors le th6or~me suivant: T h 4 o r ~ m e 2.1 S'il existe (yb,zb) E IV x W tels que :

z b 2. Extension

et exemples

biologiques

On peut 6tendre les r6sultats des sections 3 et 4 au cas de plusieurs contr61es E uiei. Si l'on a k contr61es, on supposera que les (n - h) in6galit6s restantes des th6or6mes 3.1 et 4.1 sont v6rifi6es, et l'on choisira ui fonction de xi. On peut 6tendre les r6sultats de la section 5 au cas o{t d(x) est une application diagonale (Dd(z) est une matrice diagonale) n6gative pour x _< x* et positive pour a" >_ x*, et donc S~allnulant en x ' . On va prendre comme exemp!e les modNes de Lotka-Volterra (3). On suppose que le systhme a un 4quilibre unique x* > 0 (voir ([11]), et, apr~s le changement de variable = In x, on obtient:

= ~ + ce~

(10)

On pose C = F - G, off F est une matrice & 416ments en dehors de la diagonale positifs et G une matrice positive. On suppose que l'on peut exercer un contrSle sur la premi&re composante, le syst6me boucl6 s'6crit donc: = F(d - d') - a(e ~ - d') - u(~)q

(11)

et le syst6me associ4 (7) s'6crit:

~

u

,,(z)e~ )

(12)

avec A=

-G

F

et

V = e z - e ~"

(13)

Nous allons choisir u(~) = k(e ~* - e~;), k constante positive; de plus, nous supposerons d bijective (volt ([3])); le syst~me (12)s'6crit alors:

~"

U

off le d6ternfinant de Al(k) est u n polyn6me du deuxi~me degr6 en k; si les hypotheses sur les in6galit6s du th6or6me 3.1 sont remplies, on pourra donc de plus choisir k pour

330

que Al(k) soit bijectif, donc que le syst~me (14) ait (~*,~') comme unique @quilibre. On pourra donc appliquer le th@or~me. Donnons deux exemples num@riques, le premier avec un contr61e u de signe variable, le deuxi~me avec un contr61e de signe fix@ : - Le syst~me (10) avec C =

4 -5

et b =

1

peut @ire stabilisfi vers son @quilibre instable (0, 0) avec un contr61e sur la premiere composante u(4) = 3 ( d ' - d r ) dans l'intervalle (dans R 2) ] - oo, ln2]. - Le syst~me (10) avec

50 1 ) ( : : ) C =

0

-2

3

I

i

--3

et b =

(16)

admet un @quilibre instable en (0,0, 0); il peut ~tre stabilis~ globalement dans l'intervalle (darts R a) [-ln(0.9, 0.88, 0.2), ln(1 + 10 -a, 1 + 2.10 -a, 1 + 10-3)] avec un contr61e positif sur la premiere composante que l'on peut prendre nul si ~1 _< ~ et @gal ~ u(~l) = 6.104(e ~ - e ~) si ~1 _> ~[- Remarquons que ce contr61e n'est pas d@rivable en ~*, mais on v@rifie que le syst~me boucld reste C 1 partout. * Biologiquement, ~ repr~sente un taux de croissance; l'exemple des modules de LotkaVolterra nous dit que, dans certains cas, on pourra stabiliser un syst~me biologique si on salt agir sur le taux de croissance d'une esp~ce dans les deux sens; et, dans d'autres cas, si on ne salt agir que dans un sens (en la chassant ou en augmentant le taux de reproduction par exemple). R e m e r c i e m e n t s " L'auteur tient & remercier les rapporteurs pour leurs remarques judicieuses. lZdfdrences [1] D. Aeyels, Stabilization of a class of nonlinear systems by a smooth feedback control, Systems control Lett. 5 (198.5), pp. 2S9-294. [2] J.L. Gouz6, Structure des modbles matMmatiques en biologie, in: A.Bensoussan et J.L. Lions, eds., Analysis and optimization of systems, Lecture Notes in Control and Information Sciences No. 111, Springer-Verlag, Berlin (1988) [3] J.L. Gouz4, A criterion of global convergence to equilibrium for differential systems, Rapport Inria No. 894, (1988) [4] M.W. Hirsch, Systems of differential equations which are competitive or cooperative I: Limit sets, SIAM J. Math. Anal.,13 (1982),pp. 167-179.

331

[5] M.W. Hirsch, Systems of differential equations which are competitive or cooperative II: Convergence almost everywhere, SIAM J. Math. Anal., I6 (1985), pp 432-439. [6] M.W. Hirsch, The dynamical approach to differential equations, Bull. Amer. Math. Soc., ll (1984), pp 1-64. [7] R.M. May, Stability and complexity in model ecosystems, Princeton University Press, Princeton (1974) [8] N.Rouche et J.Mahwin, Equations diff~rentielles ordinaires, tomes 1 et 2, Masson, Paris (1973) [9] It.L. Smith, Systems of ordinary differential equations which generates an order preserving flow. A survey of results, SIAM Review, 30 (1988), pp. 87-113 [10] H.L. Smith, Competing subcommunities of mutualists and a generalized Kamke theorem, SIAM J. Appl. Math, 46(1986), pp 8.57-874. [11] Y. Takeuchi, N. Adachi and H. Tokumaru, Global stability of ecosystems of the generalized Volterra type, Math. Biosci., 10 (1980), pp 119-136. [12] V. Volterra, Variations and fluctuations in the numbers of coexisting animal species, (1927) in: F.M. Scudo and J.R. Ziegter, eds., The golden age of theoritical ecology: 1923-1940, Lecture notes in biomathematics No. 22, Springer-Verlag, Berlin (1978)

[13]

w. Walter, Differential and integral inequalities, Springer-V'erlag, Berlin (1970)

CONTROLLABILITY OF DELAY-DIFFERENTIAL SYSTEMS Paula Rocha Technical University of Delft P.O. Box 356 2600 AJ Delft

Jan C. Willems University of Groningen P.O. Box 800 9700 AV Groningen

A b s t r a c t : The concept of controllability is introduced and investigated for the class of AR delay-differential systems with separable AR descriptions. For this class of systems, it is shown that a system Z described by the AR equation

R(al,o'2)w=o (with a I the

differentiation- and a 2 the delay-operator) is controllable if and only if rank R(A,e-~) is constant for all A9 s

1. INTRODUCTION The aim of this paper is to investigate the concept of external controllability for delay-differential systems. The contributions to the area of delay-differential systems are vast, and can be divided into two main streams. On the one hand, the infinite dimensional state space approach of, among others, Bhat and Koivo, and Manitius Trigianni, [BK], [MT], and on the other hand the algebraic approach developed, for instance, in [L]], [M] and IS], which views delay-differential systems as systems over rings. Both these approaches are concerned with internal system properties, such as, for instance, controllability and observability of the state space. Recently, an external characterization of approximate controllability of the standard observable realizations

of delay-differential

transfer

functions has been given by Yamamoto, [Y]. However, although externally characterized, this controllability

property is defined on an

internal level,

as it concerns state

space realizations. Our purpose is to investigate for the particular case of delay-differential systems the concept of controllability introduced by Willems [W2] for general dynamical systems. In the behavioral framework of [W2], a system is viewed essentially as a set of external trajectories,

and thus controllability is defined as a property of these trajectories.

333

Here, we will consider the class of the s o - c a l l e d AR d e l a y - d i f f e r e n t i a l systems, and derive a controllability test in terms of the system AR representations. 2. CONTROLLABILITY OF DYNAMICAL SYSTEMS

We start with the basic definition of a dynamical system introduced in [W1].

D e f i n i t i o n 1. A dynamical system Z is defined as a triple Z := (T,W,~), where T c_R is the time set, W is the signal set and ~9 _c Wr := {w:T-+W} is the system behavior. Thus, a system is essentially characterized by its behavior, i.e. by the family of all trajectories in the domain W"T which are compatible with the system laws. Note that in this definition no distinction is made between inputs and outputs among the system variables. In fact, our approach is more general than the input/output framework, as input/output systems can be viewed as dynamical systems in the sense of Definition 1, if instead of inputs u and outputs y we speak of system trajectories = col(u, y).

In the sequel we will consider excIusively continuous time systems, i.e., systems with time set T=R. Intuitively, we will say that a system is controllable if it has limited range memory, i.e., if the values of the system trajectories on two intervals ( - c o , t ] and [t+d, +co) are in some sense independent provided that d > 0 is sufficiently large. Given two trajectories wl,w2eW R we define the concatenation of tox with w~ at time t e R as the trajectory w e W R given by w[(_|

and wl[t+|

We will

denote w := w 1 A w 2. D e f i n i t i o n 2. A dynamical system E = ( R , W , ~ ) is said to be controllable if the following condition holds. {Wle~ , W2e~ , teR}.~{3d>0

3w*e~B s.t. WlAUr*t+AdW2EB }

3. AR AND MA DELAY-DIFFERENTIAL SYSTEMS

The dynamical system E=(R,Rq,B) will be called an AR delay-differential system if its behavior B can be described as the solution set of a finite number of linear equations on the derivatives and delays of the trajectories, as indicated below.

334

~'*~ ~'~

d*

k=kl ~ ~=el Rke

VteR

--

dt k w(t - ~ ) = 0

with k l < k 2 positive integers, ~l

. .

ni

Bsz=block-diag

L 'sJ 4---- hi-1 -"-->

9

?J B61=0

Now, if one applies this result to systems without input but with outputs (E,A,0,C), one obtain the following result that is deduced by duality9 Split in two of the lists {qi} and {vi} under the action of ..q'R={(W,V,0,S,0,R)}:

375 Any triple (E,A,C) can be given, through the action of E'R, the following equivalent form: [ sE-A ]= block-diag [ sEi -Ai]

C = [Cij]

i=1 to 2, j=l to 6

* [ sE 1 -A 1 ]= block-diag [Lkij] with Lkii unchanged and Cil =0 'Vi * [ sE 2 -A 2 ]= block-diag [L~i] with

Ls unchanged and Ci2 =0 'v'i

* Associated to the two types of {qi} there are two types of blocks:

[ sE 3-A 3 ]= block-diag [L~i]

with L(i =

is 1 . -11, . 9 ~i

Ci3=0 Yi

L 1J* 4--- ~i-1 --~

[ sE 4-A 4]= block-diag [L~i]

with L~i =

I

s

1`

1

is $ 4--+

4--

.ol]}

014---- block-diag { [ 0

024=0

* Associated to the two types of {vi} there is two types of blocs:

[ sE 5-A 5]= block-diag [Lmi]

with

S . Lm i =

.

Ci5=0 Vi

| mi

sl -J $ is1 [sE6-A6]=block-diag[Lqi ]

.

11,

~

with Lq,= L

.

jqi-1

sl 4--

qi

C26= block-diag {[1 0 . .

4.

"-')'

0 ]}

C16=0

Let us now come back to general systems with input and output equations; The reduction procedure of ~l(s), ~2(s) and s lead to a split in two of the lists {~i} and {qi}, and a split in four of the list {vi}; the list of the finite elementary divisors is unchanged.

376 We are now able to state our main result : Theorem 2,1 9 Any (E,A,B,C) quadruple (1.1) can be given through the action of ,.q"the following equivalent block-diagonal form :

II

II

_~

j i

-,Z"

3 m

3

/

I / ~ Oj

II

I ,_S

377 One can remark the following important particular cases : oi =1 corresponds to a zero column in sEc- Ac and Cc [,i =1 corresponds to a zero row in sEc- Ac and Bc qi =1 corresponds to a zero column in sEc- Ac but a non zero column in Cc ni =1 corresponds to a zero row in sEc- Ac but a non zero row in Bc Note again than some zero columns (respectively zero rows) appear in Bc (resp. in Cc) when B is not monic (resp. if C is not epic).

11.2 INVARIANTS. CANONICAL FORM : Let us now introduce the following algorithms. (2.2.1)

q2"~

q/~+]=KerCc-~-](E~/~+lmB)

(2.2.2)

ff.,~0=,,~'C "t~#+l=KerCc'~A'i(E~'ro~-~'+lmB)

J

(2.2.3)

8~

8#+i=E'l(A(KerC~3~)+ImB)

l,U, oc$oc:,U,1 c ... ~j,~S~..."

(2.2.4)

flJ'~

"lJ'~'+l=E'l (A(Ke rCc~fl.Y~)+l m B)

J

(2.2.5)

-~'~

%~+l=E(KerCnA'l(~'-#+lmB)) l ,,~o ~q/,oD~l ... ~ p - _ ~ . . .

(2.2.6)

~o=,,~,

,,~+l=E(KerC,qA-1(,~#+lmB)) J

(2.2.7) _8~

l ~2"0Dff-~'0D"V1...Dq/'#D'V'~'...

-

-

~#+l=A(KerCc~E "18_~)+lmB

(2.2.1), (2.2.2) and (2.2.4) were introduced by 0zs [6]; (2.2.3), (2.2.5) and (2.2.7) were defined by Malabre [7]. The convergence of (2.2.6) is a consequence of the above mentioned inclusions. We shall now characterize the indices which appear in theorem 2.1 in terms of the dimensions of the steps of these algorithms. Prooosition 2.1 9 {(S-Or.i)kij} is the list of the invariant factors of (Eq/*/E ,,,,,,,,~*[IA+BF+RCII"V*/,.%*), the map induced in the, quot,ient subspa,ces,~"/,.,%* and Eq/' /F_.,~ where F and R are such (A+BF+RC)'],/' c:E'12" and ~ ='1,/" n3 9

-

.

[ W ' * n . ~ '-I ]

- card {oi > P.}=d'm tfl),,n flj,~._ll

V #>1

card {~>__p.}=dim Lfl), n8 ~-I

V p.>l

378

-

card {r~i> i~}=dim I --~+'~#-tl

9 rKe rE n ff.,.9.~-i] -card{mi=~}=omlKerEnfls I

- card {pi = #}=dim

~'lmBnE~"-'l

llmBnEff~_] j

Particular case:

v

1

V ~>1

V p.> 1

card {~i> I~}=dim I--~*+-~-' l

V I1> 1

card {hi = p.}=dim I ImBnE%~rL-2IV i imBnE1s j 1~>-2

card

= ,}=dim

IKerEn f17~-' 1

V

>_ 1

card {ni = ;}=dim ,J ImB "[ llmBnlmEJ

Proof: The characterization of the polynomials {(s-oq)kij} is due to Malabre [7]. The other relations can be obtained by direct counting of the number of vectors in bases of the considered subspaces (see for ideas Loiseau [8]). We can remark that the algorithms (2.2.1) to (2.2.6) are invariant under feedback, output injection and changes of basis. This shows us that the indices {cri}, {7i}, {~J, {~}, {mi}, {ni}, {Pi}, {qi}

and the polynomials {(s-ai)kij} constitute a complete set of invariants for quadruple (E,A,B,C) under the transformation group S. In other words, this proves the canonicity of the form described in Theorem 2.1 . It is now important to compare this form with other existing ones and particulary the canonical form of Van Der Weiden and Bosgra.

III COMPARISON WITH THE CANONICAL FORM OF VAN DER WEIDEN AND BOSGRA; Van der Weiden and Bosgra (VDW & B) introduced [1], for somes quadruples (E,A,B,C), a canonical form which looks like our form. We shall compare both canonical forms. Two facts must be pointed out : 1)In the VDW & B's process, only the dynamical part of Dg is considered as the "actual" state space. Thus the system is described by the so-called "strictly generalized state space form" where E has a particular description "

o] From our point of view, the system is in a general form (see (1.1)): the state space is .,%. This is not a restrictive constraint : one can also, in our case, consider systems in strictly generalized state space form ( this is just a matter of basis transformations.).

379 2)The authorized transformations, in both cases, are in close connextion with the underlying state space. - This is why VDW & B use the" strict pencil equivalence" : PVDW&a(S)= M P(s) N with

[ o u=~,o//-A=I

Mll M12RI]~-sI-A11"A12B1]rNll

-

0 y~2]

-A=2B2/|N21 N22

=Pvow~.a(s)

0 X2 S JL -01 -02 0 JL F1 0 R~ and F1 only act on the dynamical part of 3(5 X2 and Y2 allow combinations between the other part of ~ and the inputs or the ouputs. In our case, the authorized transformations can be formalized by :

wl, w12RllEs A1 AI , :tE v11v12i]

W021W22 R2 0 S

-A21 -A22 -C1 "C2

V21 V22 F1 F2

=

Po(S)

R21 and [ F1 F2] act on the full state space 3P.,. JR' Only internal transformations (changes of bases) are allowed on the inputs and outputs. This second point shows that the "strict pencil equivalence" and our "proportional equivalence" are really different. We shall now illustrate this difference with an example. Consider the following systems which are equivalent in the sense of VDW & B :

l-s lolo- I 9 [-sE1-A1B1

I0s

P2cs~;EsE2A2~I

111 1

~'IS~=L ci o]=Ioo ~IoI

02

L. q-5-5 o[6-_.1

Esl~ o silo

=

0 0 1 I1 100

Indeed, they have the same VDW & B canonical form:

,oo iooiI oiOOI

r,oOOoqro,_, 0

s

sll

010

o., .,Jl., 0 1 0 j l 0 0 1 ~oo ~ 1 7060 0 0 0 0

s01 = 1 0 0 1 0 =PlvDW&B(S) L1 0 0 0

loolioooj lOOI

rio oo O]ro s,o 1-1 0

L~176 I~ oo,

P1VDW&B(s)=P2vow&B(S)

loo roso, oll II oo,,1 -L~olo _-P2vow,o~s~ 00 0 / L 0 0 0 -1

0 00

Now look at their canonical form under the action of our group (W,V,G,S,F,R).

380 s01 ~1 ro '~176 =Plo(s)

LSOo'tJL? o Oo'jLOo o Oo;, ~J--I oo,o

Pl =2 "ml =1

L1 o o o

o1 Fo,oo sl0

?JL?'o'oJL~ oj=Looo,

= P2o(S)

ql =3 ;nl =1

1000

Since they do not have the same lists of invariants, these systems do not belong to the same orbit. This implies that both transformation groups are actually different: and the associated canonical form too.

IV CONCLUSION: We have introduced a new canonical form for descriptor systems with an output equation. In other words, we have characterized by nine lists of invariants the orbit of any (E,A,B,C) quadruple under the action of the proportional group. It generalizes the form given by Morse [3] in the strictly proper case. The considered systems have been taken in a general form without any restriction of regularity, regularizability, controllability. E and A are not even assumed to be square. More properties of this canonical form will be investigated in future work. AcknowledgemQn(; The authors gratefully thank Dr. M. Malabre for helpful conversations.

References: Van Der Weiden ,A.J.J. & Bosgra,O.H., "The determination of structural properties of a linear multivariable system by operation of system similarity.", Int.J.Control, 1980, Vo1.32, N~ pp 489-537. [2] Gantmacher.F.R (1959), "The theory of matrices, Vol.2, Chelsea, New York. [3] Morse,A.S. "Structural invariants of linear multivariable systems", SIAM.J.Control, Vol.11, N~ August 1973. [4] Loiseau,J.J.;C)zs & Karcanias.N, "A feedback classification of singular systems", IFAC WORKSHOP "System Structure & Control :State Space & Polynomial Methods" sept 25-27,1989 Prague. [5] Jaffe,S. & Karcanias,N. (1981), " Matrix pencil characterization of almost (A,B)-invariant s..ubspaces: A classification of geometric concepts", Int.J.ControI,Vol.33, n~ 1, pp.51-93. [6] Ozs of descriptor systems", PHD thesis Georgia institut of technology, May 1985. [7] Malabre,M., " A structural approach for linear singular systems", International Mini Symposium on Singular Systems, Atlanta, 4-6 December 1987. [8] Loiseau.J.J., "Some geometric coniderations about the Kronecker normal form", Int.J.Control, Vol42, n~ pp. 1411-1431, (1985). [1]

E v a l u a t i o n T r a n s f o r m a n d S y m b o l i c C a l c u l u s for N o n l i n e a r

Control Systems

V. Hoang Ngoc Minh ~ G. Jacob, L.I.F.L. - U.A. 369 C.N.I~.S. Universit6 Lille I, 59655 Villeneuve d'Ascq, France. A b s t r a c t : Given a nonlinear control system, one can view its output function as a signal, parametrized by the primitives of the input functions. This signal can be formally described by its Fliess' power series, that is a formal power series on noncommuting variables. The temporal behaviour of the system can be derived h'om this symbolic description by a transform, that we call "Evaluation transform" and that generalizes the inverse Laplace transform to the nonlinear area. We developpe here the basic tools of that symbolic calculus by introducing a "kernel" for our Evaluation transform. This kernel can be viewed as some "temporal memory" of the systena in the Volterra's meaning as well as in the programmation meaning.

1o I n t r o d u c t i o n

and notations

Following M. Sliess ([2]) the output Y(~) of any nonlinear control system can be symbolically described by its generating series S = ~ < SIw > w (also called Fliess' wEZ ~ power series) that is a formal power series on noncommuting variables belonging to an alphabet % -- {z0, Zl,...,zm}. This series is nothing else as the adequate generalization of the Heaviside calculus ([2], [3], [8], [9]) in the nonlinear area : the notions of transfer function and of impulsive response, coding signals produced by linear or multilinear control systems, axe generalized to generating series and Volterra series, coding signals produced by nonlinear control systems. The inverse Laplace transform allows to recover the signal symbolically described by some transfer function. In the same way, the Evaluation transform ([5], [6]) allows in return to easy derive the temporal behaviour of the system from its symbolic description by computing the Evaluation Eu(w)

of each word w, for the i n p u t a :

integral

( a z~

a z~

...

a ~m), as b e i n g tile i t e r a t e d

/0'

~uw. It follows from the Fliess' f u n d a m e n t a l f o r m u l a (also called Peano-

Balker formula) which presents the o u t p u t y(t) as the E v a l u a t i o n of the g e n e r a t i n g series S : y(t) =

~ < S l w > g~,(w)(~). So the o u t p u t y(t) can b e viewed as a signal wEZ9

d e p e n d i n g on the parameters ~(~) =

/0'

a~(~-)dT,

z E Z ([5], [6]).

We develop here, in c o n t i n u a t i o n of [5] a n d [6], the basic elements of this symbolic calculus using the notations of E v a l u a t i o n via kernel functions which plays the rule of t e m p o r a l m e m o r y of systems in the Volterra's m e a n i n g as well as in the p r o g r a m m a t i o n m e a n i n g . So the Evaluation transform becomes a f u n c t i o n a l d e p e n d i n g o n the kernel a n d on the inputs. This systematic t r e a t e m e n t has been used in [7] to write M A C S Y M A programs by use of its recursive definition a n d of the recursive i n t e r n a l r e p r e s e n t a t i o n by the b i n a r y tree of the linear c o m b i n a t i o n s of the n o n c o m m u t a t i v e r a t i o n a l fractions describ ed by (Coz j o ) * P ~ z q ( c l zj~ )*P~zi~ (c2 zj~ )'1'~... zi~_~ (ck-1 z j ~ _ ~ )*P~-~ z i k (cl, zj~ )*pk, where p o , 9 9 9 Pk a r e integers, Co, .. 9 ck are complex n u m b e r s , Zjo , . . . , z j k a n d Zio , 9 9 9 zi~ are letters in Z.

382

Recall that a f o r m a l p o w e r series o n t h e a s s o c i a t i v e v a r i a b l e s z E Z ( n o n c o m m u t i n g if card Z > 2) with coefficients in/it(([1]), is any m a p p i n g : S:Z*

~ ~( w l

~< S I w >,

and the set of ~11 formal power series over Z is denoted by Zr( >. A formal power series S will be written as a formal sum E < SIw > w, where w6Z" < SIw > is the coefficient of the word w in S. A formal power series S E ~r( > will be said q u a s i r e g u l a r if and only if its constant term < S]~ > is equal to O. The s u m of two formal power series S and T is the formal power series S + T in ~f > defined by : V w E Z*,

< S + T I w > = < S [ w > + < TIzu > .

The C a u c h y p r o d u c t , noted by ".", of two formal power series S mad T is the formal power series S.T in iK > defined by : V w E Z*,

< S.TIw > =

~ ltjvEZ*

< S[u > < T I v > . ~It l J : l o

The symbol "." will be omitted when there is no ambiguity. For aa~y quasiregular formal power series S in/T*2 >, S* represents classically the formal power series E S'~' In commutative variables, it coincides with the rational n>0

fractio~ 1/(~

-

s ) , ~nd in ~his case ,,~ have S "~ = (1/(~ - S)) ~. 2. S y m b o l i c c a l c u l u s

2.1. E v a l u a t i o n o f f o r m a l p o w e r s e r i e s Let Z = {zo, z l ; . . . , zm} be a finite alphabet. D e f i n i t i o n 2.1.1. : W e wii1 call i n p u t r e l a t i v e to Z t/~e given of a vector a = ( a z~ a z' ... a z~ ) of" p i e c e w i s e c o n t i n o u s rea~ vaJued f u n c t i o n s defined on a c o m p a c t intez~val [0, t] C ~ + . ConventionaJly t h e O_component o f a n y i n p u t is a z~ ~ I. F o l l o w i n g K . T . Chen ([2]), we will coil p a t h a s s o c i a t e d t o t h e i n p u t a --(a

z0

a"

...

a zo,

), the time depenae~t vector r = ' ( ~,o

~,,

...

~,m ) de~ned

as follows :

/"

/o"

In all the sequel, we suppose that the function f is Stieltjes integrable with respect to { ~ } ~ z defined over [0,t], (t _> 0), and f vanishes at zero. The unit step (vanishing at zero) is noted " u n " : v , E ]O,t],

~(~)

= 1,

~(0)

= O.

383

D e f i n i t i o n 2.1.2. : T h e E v a l u a t i o n o f t h e w o r d w in Z* w i t h r e s p e c t t o t h e k e r n e l f , for the i ~ p u t a = ( a ~~ a ~ . . . a ~" ) related to t h e finite a l p h a b e t Z , is defined by i n d u c t i o n on the l e n g t h o f w as follows :

&(f;w)(O

=

,

&(f;v)(r) d~,(r)

if

w = vz.

This definition is extended to g ( > in the following way : D e f n i t i o n 2.1.3. : W e will call E v a l u a t i o n o f t h e f o r m a l p o w e r series S in g~" > w i t h r e s p e c t to t h e k e r n e l f , for theinpu~ a = ( a ~~ a ~* . . . a ~=) related to the finite a l p h a b e t Z , w h e n it is defined, the functionaJ : Ca(f;S) =

E

< SIw > Sa(f;w).

wEZ ~

In particular, for f = u n , the E v a l u a t i o n o f t h e f o r m a l p o w e r series S in f f ( < < Z > > , f o r the i n p u t a = ( a ~~ a ~ . . . a == ) r e l a t e d to the finite a j P h a b e t Z , is the function ([5], [6]): & ( s ) = & ( ~ ; s).

According to the Fliess' fundamental formula, we see the Evaluation of a formal power series S can be viewed as a transform that associates to S the signal depending on the primitives {(~ },~z of the inputs functions, and the Evaluation transform is notlfing else as a generalization of inverse Laplace and Fourier transforms ([5], [6]). The two following lemmas can be obtained easily by induction : L e m m a 2.1.1. : L e t u, v be two w o r d s in Z * . T h e n : Ca(f; UV) -'- Ca(~Ta(f ; u); V).

L e m m a 2.1.2.

:

L e t z be a letter in Z . For any n ~_ 1, we h a v e :

C,(/; z'~)(t) =

fo' f ( r ) (r

7n_~ - r ~).v

d~,(r).

I~ particular, for z = z0, we h a v e : Ea(f;z'~)(t) =

~

t

(t --

f(r)

T) n-1

(n-1)!

dr.

From the lemma 2.1.2., we can see that tile Evaluation transform is an "extension on noncommutative variables" of the exponential transform of the "ordinary generating

series" S = E

c n z " , which associates to S the "exponential generating series" ga(S) =

n>0

c,, ~ n>O

well known in "combinatorics" ([4]):

384

C o r o l l a r y 2.1.1. : Let z be a Jetter in Z. T h e n for any n >_ O, S~(z'~)(t) = ~"n-~. " zn p ~ r t i c ~ , ~or z = zo, ~hen ~o~ ~ y ~ >__ o, s a ( ~ ) ( t ) = ~t,~.

T h e o r e m 2.1.1. : Let S and T be two formal p o w e r series in g~ >. Let r be a scalar. T h e n : Z~(f; S + r.T) = E~(f; S ) + r.E=(f; T).

P r o o f : The proof is immediate since the linearity of the ordinary integrals

9

T h e o r e m 2.1.2. : Let S and T be two #orrnaJ power series in ~ ( >. T h e n : E~(f; S . T ) = E~(E~(f; S); T).

P r o o f : Actually, by the definition 2.1.3., we have : &(E=(f;S);T)=

~

~

< S]u > < T l v > E ~ ( E ~ ( f ; u ) ; v )

uEZ* v6Z*

= ~

~

< Sl~ >< TI~ > eo(f;~)

(lemma 9LL)

uEZ* vEZ*

\ u E Z * vEZ*

= Eo(f; S . T )

9

The introduction of the kernel f for the Evaluation function G ([5], [6]) allows to give a notion of memory for the system. This kernel can be viewed as the temporal memory of the system in the Volterra's meaning as well as in the p r o g r a m m a t i o n meaning ([7]), that justifies our approach. So the Evaluation transform becomes a functional depending on the kernel f and on the inputs {aZ}zEz. For implement these functionals, we used the A-notation of MACSYMA and the recursive internal representation by binary trees of the linear combinations of the noncommutative rational fractions described by

(co Zio )*P~ z , (Cl z~, )*p, z,2 (c~ zj~ )*~.~.. z~ _, (c~_~ zi~ _, )*~-'z,~ (ok zj~ )*~, where p 0 , . . . , ;k are integers, Co, 9

ca are complex numbers, Z j o , . . . , zjk and Zio , . . . , zik are letters.

T h e o r e m 2.1.3. : (Convolution theorem) L e t z be a letter in Z. Let H be the formal p o w e r series on the only letter in z. T h e Evaluation with respect to the kernel f , for the input a = ( a z~ a zl ... a zm ) related to the finite aiphabet Z , of the formal p o w e r series H is : E~(f; H ) ( t ) =

/o'

h(~z(t) - ~z(~-))df(r),

385

where h(~z(t)) the EvaIuation of the formal power series H. In particular, if H is a formal power series in g( >, then :

g~(f; I t ) ( t ) =

/'

h(t - , ) d r ( r ) .

Proof. : (i) F i r s t ease : H = z", n _> 0 : For n = 0, the result is i m m e d i a t e . If n > 0 then we have : 8 ~ ( f ; zn)(t) =

f ( T ) ((~(t)--U ( n - -~-z(r))"-I 1)! d~z(r')

= /'f(~)d[ - (4z(t)-~-~('))"]~,

(leumaa 2.1.2.)

J

(integration by parts)

=

/'

h(~,(t) - ~ , ( r ) ) d f ( r )

(ii) G e n e r a l case : H = E

( / ( 0 ) = 0)).

H " z " ' we have :

n>__o

E=(f; H)(t) = ~

H , e ~ ( f ; zn)(t)

n>O

~(~))~

(~(t)

-~i

df(~) (case (~))

n>o

=

i'

h((,(t) - ~(r))df(r).

In p a r t i c u l a r , for z = zo, since ~ 0 ( t ) = t, t h e n we have result

9

2 . 1 . 2 . : Let G E f~2 > be a s series, and let H be a power series on the only letter in z. T h e Evaluation with respect to the kernel f , for the i n p u t a = ( a ~~ a ~ . . . a ~'~ ) r e l a t e d ~o t h e / ~ n i t e alphabet Z, of the formM power series C H is : Corollary

s

~ ( a H ) ( t ) = < al~ >

h(~(t)) +

/' h(~.(t) -

~(T))dEo(a)(,-),

where h((~(t)) the E v a / u a t i o n o f the f o r m a / p o w e r series H. /'ormai power series in g ( >, then :

e ~ ( c ~ ) ( t ) = < cl~ >

h(t) +

In particular, i f 2I is a

/' h(t-,)aeo(C)(,).

386 P r o o f : Actually, since G H = < Gle > H + G I H , where G1 is the quasiregular formM power series Lr.2 >. Hence by the theorem 2.1.1, the Evaluation of the formal power series G H , for the input a = ( a ~~ a z~ ... a ~ ) related to the finite alphabet Z, is < G]e > h ( ~ ( t ) ) + g ~ ( G l t I ) ( t ) . Using the Convolution theorem in the particular case where the kernel f is the Evaluation of the quasiregular formal power series G1, we have the expected result * 3. C a l c u l u s e x a m p l e s Example

3.1.

In this example, we c o m p u t e the Evaluation of the series z *'~ for any letter z and for any integer n > 0. Lemma

3.1.1. : For any integer n > 1, we have g~((c~z)*~)(t) = exp(~(~(t))g~(c~(z(t)) ,

where the g,, are polynomials and verifie the following inductive equations :

{

g,,(a(~(t)) =

/o

g~-l(a(~(t)) + a

g~-l(a((~(t) - (~(r)))d~0-)

if

n > 1.

P r o o f : Since ga((az)*)(t) exp(c~(z(t)), we can write gl((~(z(t)) = 1. ~Ve suppose that the result is true for any u, 0 _< u < n - 1. For v = n, we have ( a z ) * ' * = ( a z ) * ( a z ) *~-~. By the induction hypothesis and by the corollary 2.1.2, we obtain : =

sa((~z)*")(t)

=

exp(~(z(t))j._l(~4z(t)) + ]o exp(~((~(t) - (~(~)))g.-l(~((~(t)

= exp(c~,(t)) g , - l ( a ~ ( t ) ) hence we have the expected result

Lemma

3.1.2.

:

9

+ a

- (~0-)))dexp(~(~0-))

g , , - l ( a ( ~ ( t ) - ~('r)))d~z(T) ,

_i()

The f~ilj, j.(~(,(t)) = j----0 ~ '~ -J i (~(,)J(~) j! , f o r n >_ I, is the

unique solution o f the inductive, equations :

~,,(~(t)) =

I

j,,-l(,~(t)) +,~

/

1 if ~ = 1 t

9,,-~(~(~(t)- ~O-)))d~(T)

if ,, > i.

387

Proof:

Given a I = 1 E •(


1, we have g , ( a ~ ( t ) ) = E~(Gn)(t), with G,, E f/( > . Thus we have (see the corollary 2.1.2.) E~(G,,) = $~(G,,-1)+S~(azG~-l). This equation is true if G~ = (1 + az)G~_l. In other words : n--1

~n : ('l'JvOlZ)n--i = ~ ( ~ j=O Since $~((az)J)(t) =

j!

1}(O~Z)J.

, j >_ 0, tl~en we have for any integer n >__1 : n--1

By the l e m m a 3.1.1. and the l e m m a 3.1.2., we have the following proposition : Propositon

: For any positive integer n, for any complex number a, we

3.1.1.

.,/]ave :

e~[(c~z)*~](t) =

exp[c~z(t)]~.; n J- i [c~G(t)]Jj! if

n > O.

In particular,/'or z = z0, we have : C

,.~[(aZo)

*n

](t) =

n--1

exp(at) ~

j=o

1 (at) j

n-

if

7z = 0

if

n>O.

J

By the theorem 2.1.2. and the proposition 3.1.1, we deduce the following theorem Theorem

3.1.1. : For any positive integer n, for any complex number a, we have

f(t)

[a(4~(t) j~ ~ ( r ) ) ] j df (7)

if

n=0

if n > O .

In particulaz, for z = zo, we have :

g~[f; (azo)*"](t) =

~ j=O

n J- 1

exp[a(t - ~-)] [a(t J[ - T)] j df(-r)

if

n=O

if

n>O.

388

E x a m p l e 3.2. In this example, we compute the Evaluation of the formal power series S that verifies the following polynomial equation :

S + #lSz + fl2Sz 2 + ... fl, Sz" = G, where z is a letter of Z, and G a formal power series of ~" >. We have S K = G, 71

where K is the formal power series of Zrx"> defined by E

flkzk with/30 = 1.

k=0

Since the constant term < Kle > = fl0 = 1 does not vanish, then the formal power series/x~-1 exists and is a formal power series in the single commutative variable z. Suppose that K admits r complex distinguished roots # 1 , . . . , #r of respective multiplicity order m l , . . . , mr. One can express mfically K -1 under partial fraction decomposition form

~_~ Ern~ (_-"~-l)k ~l,k f~l,k,

where for any I e [1..r] and for any k e [1..rn,], A,,k e ~T, and

l=l k=l

f V' " Let hl,k({~(t)) be the Evaluation of Hi,k:

each Hl,k is \-fiT]

hl,k(~z(t))=exp(~l))~(k--1

So S = GK -1 can be expressed as ~ m ,E

1

(proposition 3.1.1.).

Al,k GHl,k. By the theorem 2.1.1. and by (_#l)k

l=l k=l

the corollary 2.1.2, we obtain the Evaluation of S :

Za(S)(t) =

(_#l)k < Gle > hl,k(~z(t)) -t-

hl,k(~z(t)

--

G(r))dg.(G)(r)

.

1=1 k = l

Let us indicate that in the particular case z = z0, the above calculation corresponds mutadis mutandis to the inverse Laplace transform in the study of linear control systems. As conclusion, we can say that if the Fliess' series is considered as a symbolic encoding of input/output behaviour of the nonlinear control systems, then the Evaluation transform allows in return to easy derive the temporal behaviour from this symbolic description, (see [6] for a simple obtaintion of the Taylor expansion of the Volterra kernel). Thus, we get a generalization of the notion of transfer function (generating series on one variable) and impulse response, encoding signals produced by linear or multilinear systems, and the Heaviside calculus (Fourier and Laplace transforms) as already pointed out by M. Fliess and al. ([2], [3], [8], [9]). In the symbolic calculus for linear control systems area, the integration operator is noted by ,,1,, ~ . Here, it coincides with the letter z0. And the letters z of the Fliess' encoding alphabet Z plays an analogous

389 part : they encode the "Stieltjes integration operators". And we have the following Evaluations of some usual formal power series : S

$.(f; S) f(t)

d Zn

~f

~ (n - 1) f t exp({z(t)- {~(r)) ({*(t) -{*(r))Jdf(r) ,=o J ,o J! f ' (r - ~(,-))~ an ni df(r)

z*",n > 1

E

e/o-)

E

CnZ n

n>O

Jo

n>O

then, in particular Z* ~ ~

Z ~"

' e~p(r

n_~O

/

n:>O

~--~(_1)% 2"

'(r

t

~z(,))df(~)

-

-

~(~))

e~p(r

-

r

cos(~.(t) - (~(r))df(r)

n~O

~--Lj(__ 1)- z2n+ i

t sin({,(t) -

~.(r))df(r)

n>O

In particular, for f = un, we have the following Evaluations ([5], [6], [7]):

s

&(s) 1

s

Zn ,n

z ,n>_l

exp(~z(t)) ~

C(t)

~ - ~ Cn Zn n>O

n>O

then, in particular Z* ~ ~

Zn

exp(~(t))

n>O

Z

nZ n

4~(t) exp(r

n>O

~-~j(-1)~z 2~

cos(r

n)O

sin(~(t)) n~O

j

j[

390

4. R e f e r e n c e s [1] [2] [3]

[4] [5]

[6] [7] [8] [9]

J. Berstel and C. R e u t e n a u e r , S6ries rationelles e~ leurs langages, Masson, Collection E.r~.I., 1984. M. Fliess, Fonctionnelles causales non lindaires et ind~termin~es non commutative, Bull. Soc. Math France, 109, 1981, pp. 3-40. M. Fliess, M. L a m n a b h i and F. Lamnabhi-Lagarrigue, An algebraic approad~ to nonlinear functional expzmsions, IEEE Trans. Circ. Syst., CAS-30, 1983, pp. 554-570. D. Foata, La s~rie g~n~ratrice exponentielle dans les probl~mes d'~num~ration, S4mlnaire de Math. Sup., Presses de L'Universi[5 de Montr4al, Montr6~tl, 1974. V. Hoang Ngoc Minh, ElEments d'un calcul symbolique pour les syst6mes dynamiques non lindaires, Journ6es-S6minaire "TraJtements Alg6briques ct Informatiques des S~ries Formelles Non Commutatives", Lille, December 1988. V. Hoang 1Ngoc Minh and G. Jacob, Symbolic calculus and Volterra series, IFAC Symposium "Non Linear Control Systems Design", Capri, Juin 1989. V. Hoang iNgoc Minh and G. Jacob, Transformation d'Ewlua~ion et Calcul Symbolique pour les Systbmes Non Lindaires, LIFL technical report, IT 168, 1989. M. Lamnabhi, A new symbolic calculus for the reponse of nonlinear systems, Systems ~z Control Letters, 1982, pp. 154-162. M. Lamnabhi, Functional za2aJysis of nonlineaa" circuits : a generating power series approach, IEE proceeding, Vol. 133, Pt H, N ~ 5, pp. 375-384.

IMMERSION IN INFINITE DIMENSION H. Hammouri & S. Othman Laboratoire d'Automatique et de G6nie des Proc~d~s Universit~ Claude Bernard Lyon I, 43 Bd de 11 Nov. 1918; 69622 Villeurbanne FRANCE.

Abstract

:

The input-output map is the intrinsic object of a dynamical system . The immersion problem consists of sending a dynamical system to another one via a transformation which preserves the input-output maps. Many authors have studied the immersion of nonlinear systems into a finite dimensional state affine systems ( up to output injection ). This result can be applied to the synthesis of observers of nonlinear systems. In this paper we prove that under some assumptions, any autonomous single output system can be immersed into an infinite dimensional linear system on some Banach space. Keyw0rds : Nonlinear systems, infinite w

dimensional systems, immersion.

Introduction.;

The immersion problem is introduced by Fliess and Kupka [3] . They proved that if the observation space of some nonlinear system is finite dimensional then we can immerse the nonlinear system into a state affine system. We recall the definition of the immersion. Let (.T_.I), (~2) be the two following systems : 1

(T.1) k= F(u,x) y

(T_.2) ~'=

x(t) ~ M 1 , y(t) ~ Bm , u(t) c Rp

Ht(x) F2(u'z)

z(t) r M2 , w(t) c Rrn , u(t) c B p

W = H2(Z)

Where M 1 and M2 are two manifolds. We say that (T-l) is immersible into (~,2) if there exists a continuous function : T" M 1 ~ M2 SuChthat: If x(u,xO,t) and z(u,T(xO),t) are the unique trajectories of (~1) and (]~2) which start at xOand :(x O) respectively at t=to, then y(u,xO,t)=Hl(x(u,xO,t)) coincides

392 with W(U,T(X0),t)=H2(z(u,T(X0),t)) for all x0 and all admissible controls u(.) along some interval of t. Note that to construct an observer system, many authors classify the systems which are immersible into state affine systems up to output injection. See [1], [4], [5], [6], [7], [8]. All these immersions are finite dimensional. Under some assumptions, the authors in [2] constructed an infinite dimensional unitary immersion. In this paper we give an immersion of some autonomous nonlinear system into two infinite dimensional linear systems. The generalization to the non autonomous case can be obtained in the same w a y . w 2 An infinite dimensional immersion : A)

Banach

immersion

:

Let (T_.) be the system : {~( = f(x) y = h(x)

x(t) c M

y(t) E R

Where M is an analytic manifold, f a vector field and h a real analytic function Such (T_.) is said to be analytic. Let O be the observation space ( i.e. the R-vector space generated by the family { Lkf(h), kr N }, where Lkf(h) is the kth Lie derivative and L0f(h) = h ). Remark that if this family is not free then O is finite dimensional and by Fliess-Kupka's theorem [3], we can immerse (~) into a finite dimensional linear system. Along this paper the family { Lkf(h) , kc N } is free and satisfies the following assumptions : (A1) I I Lkf(h) II oo = Pk < oo Where I I I loo is the norm of L~176 functions on M ). (A2)

sup~ o

~k+l

( the space of essentially bounded mesurable

= ;k < +oo

IJk Now set e k = Lkf(h) and consider the R-vector space : E=

ale i

s.t:

~lailPio ~

li = I(ei) .

< +oo. ~ui

b)

III II = suPi~ o

where II i II is the norm of the continuous linear form .

In this way, we identify a continuous linear form on E with

. ..

an

ordered

lail

i['[0ai ( supi~o ._~ < +o= ) and we denote the norm by II II, 9 =

~.Li

N o w let L be the shift ooerator on E defined bv L(e=~ = e=..l,

for i r N

sequence

394 Remark 2 : 1) L is well defined and continuous on E §

§

§

Since L ( ~ ale i ) = % aiei+ 1 and II i=0

i=0

,.I- ee

+~

,Y_.,aiei+ 1 II = ,Y_.,la~l#~§ < ~,% la~l#~ i=0

i=0

i=0

~li+ 1

where

;~ =supiz o - -

which is finite by (A2) .

2) L is exactly the extension of the Lie derivative Lf to the functional this m e a n s that L restricted to O coincides with Lf.

space E,

NOW let A be the adjoint operator of L, A is a linear operator of H into itself defined by : = "=i--~oai+l

A(~C[ ) i = oai Let

T " M

T(x) : E - +~,o

--, H be the valuation function defined by : 9B +ee

for every x r M

,

i

where e i = Lf(h).

aie i ----, ~ aiei(x) I-0

i=0

By claim 1 part (i) T is well defined. In this paper we consider on H two topologies : (i) Its natural Banach topology as above. (ii) T h e *-weak topology. The second topology is defined by ' A s e q u e n c e {fn}ncN of H converges to fcH if and only if 9 V zcE ;

limn-.,o~ (fn-f)(z) = 0

Definition

3 :

We say that a function g : M ----. H is "-weakly continuous if and only if g is continuous with respect to the "- weak topology. Lemma 4 : (i) (ii)

The operator A is continuous. T h e function T is "-weakly continuous.

395 Proof (i) By a s s u m p t i o n

(A2) :

supi~0

~'li+ 1

= ;~ < +oo

Pi Let ~ E H ;

~= ~

,

i=0

II ~ IIH = sup=z 0 p. I

I ~:i+~l II A~; II H =suPiz0 ~

I ~i+1 I ~i+1 _< supiz0 _ _ 'Ui+l

~i

_ _ _0

;

3 No

,

V N>_N0

;

0 < ~ lanlPn < n=N - "~

A n d then : +~

oo

NO

I (~xo) - T(x))( T. ae i ) I i=0

-
0 such that if d(x n, x ) < q ( d ( . , . ) is a metric on M which is compatible with the topology of M ) then I Lfi(h)(xn) - Lfi(h)(x) I -< d K for 0 ~; i < N O , wh e re q depends on K and ~. +oo

Set K =

~ l a n l , w e can choose n=0 This m e a n s that :

Let ~ C H ;

~=

.=

~;~ ,

N 1 E N, such that

II ~ IJH = supi~0

Pi

Vn_>.N 1 ; d(Xn,X)_< q .

396 +

eo

I ("Kx.) - ~x) X)-'. aie i ) I ~ r

[]

i=O

Now consider t h e system :

(S)=

(t)=Cg t

~tcH

; w(t) c R

Where C is the continuous linear form : C : H - -

-'[a i i=O

.B ._, a o

Because A is a linear bounded operator, it becomes an infinitesimal generator of an analytic group e tA . And V ~0r H ; :3 ~(t) a unique analytic solution of ~t = A ~;t such that ~(0) = ~0 Theorem 5 ; Under the assumptions (A1), (A2), the system (~) is immersible into (s) for the "-weal~ topology on H . Proof of theorem 5 : Let xOr M and y(t) the output of the system (T_.) with respect to the initial state x O . Let w(t) be the output of (S) with respect to T(x0) . Let us prove that : y(.) - w(.) in some interval [0,T] . By analyticity of (T.) : +=o

13

+~

rl

t (n)/n ~ t Ln h o y(t) = ~: ~y ~v, = :~ ~ f()(x) n=o n=o Since

on some interval

[O,T1], TI>O .

w(t) is analytic, 13

+co

w(t) = :~ -~w(n)(o) n=O

on some interval

ill

More generally : Since C is a continuous linear form :

[O,T2], T2>O 9

397 n

d w = CAn~(t)

,

where A n is defined as follows :

dt n

for n>l for

An~ = A(Anq~)

V~r H

A ~ is the identity operator.

n=O

This gives " w(n)(0) = CAn~(0) = CAnT(x 0) + oo

= C [ ' ~ T i + n ( X 0) i=O + ~ i+n

=Cl-[Lf

0

(h)(x)

i=O

= Lnf(h)(x0) This implies that w(t) = y(t) on [0,T] where T = inf (T1,T2). [7 In order to construct an observer for the above infinite dimensional linear system, sometimes we minimize some quadratic convex criterion, for this we need a smooth norm. For this reason we have to embed H into a Hilbertian vector space FI for which the old immersion can be extended. B)

Hilbert

Now Let

l

immersion

H be the Hilbert space :

§ a i s.t 9 +~ ~ "=

:

i=O

Jaij2 < +co / 2

P i (i+1)

2

The scalar product and the norm of H are defined by : + ,=

+ .

+ ~

akb k

N 1 ;

2

I Lfk(h)(xn) - Lfk(h)(x) J < d M

for 0< k 2n + V"2"n, we have that

~(~+,,~+~ ~+,,~k+1) ___600.

n 100 200 300 400 500 600 700 800 900 1000

Itr. 14.90 15.10 15.40 15.70 16.10 16.00 16.00 16.00 16.00 16.00

Time 0.57 3.27 10.98 25.02 47.72 83.99 129.29 197.37 274.35 372.31

Table 1: fl=0.99, p=n Ls, cond=6, deg=6, na(x')=n/2.

3.2.

Computational l~esults with Sparse Matrices

Next, we use our algorithm for solving large-scale practical engineering problems. The obstacle problems, elastic-plastic torsion problems, and journal-bearing problems can be formulated as (1.1). Using instances of tile obstacle problem, we are going to test our algorithm. T h e obstacle problems are generated in the manner explained in [11]. For these problems, Q is a special block tridiagonal matrix. Superdiagonal and subdiagonal blocks of (2 are - I E/~,~x,~, where I is an identity matrix, and diagonal blocks are also tridiagonal matrices whose diagonal elements are 4 and their superdiagonal and subdiagonal elements are -1. All blocks of Q have a same size rn by m, where rn 2 = n. Note that Q is a very sparse syrmnetric positive definite matrix. Let c=-h

2 where h =

1 m + I"

Upper bounds and lower bounds of variables are given by (sin(9.2al) sin(O.3a2)) 3 < x, __ 0 et 3' > 0 song choisis trSs grands par rapport s c, La forme (2.6) correspond ~ la minimisation des pertes actives, alors que la forme (2.7) correspond h la recherche du plan de tension le plus ~lev~ possible. 0 u t r e les infigalit~s (2.4)(2.5) et les in~galitds V " s'5crivent : Pi~ + s = vi(O, V), Pi~ = VI(0, V),

V i e G,

Vi r G,

Q,g + Ci - S, - QI~ = r Ci - Si - Qir = r

S: L-y"

p? z~2 -~ + L2 n + ~. / ( : )


__ - tl, en i ( s ) = i a n d go t o S t e p III.

8.

i = i + 1, return to point 2.

-

psiII~g s.

II: + A * ' c :

S t e p III

x ' + l = z~,g "+t = 9~.

S t e p IV

s = s + 1, return to Step II.

Let us introduce some additionM designations. Let

T ( x ) = {y E an: f ( y ) f(alX + c~2y) for all cq, a2, x, y such that al+a2=

1,

al >0,

a2>O,

xET(x~

yET(x~

We next formulate a theorem about the convergence of Algorithm 4.2. Theorem

4.2 Lct a function f: R '~ ---* 12 be strictly convex (possibly nonsmooth) on the set T(x~

be given, and let the sequence {Aj}, j = 0, 1 , . . . of positive numbers satisfy the conditions

i=0

i=0

let v > 0

438 Then there ezists 5 such that

f(x ~) - f* _ ( g , Y - z) + ( ( z , y )

for all y E X

is nonempty, where ((x, y) is uniformly small with respect to [Ix - yl] on each compact subset I( C X, i.e. for each e > 0 there exists 6 > 0 such that I1~;(=, y)ll/ll= - yll < for =, y ~ so, 11= - yll < ~.

L e m m a 6.1 Let the function f : R n --~ 1~ be convez on 1~", and the set 13 E R nx,~ be convez.

Then the

function (p(B) = f ( x - pBBT~) is weakly convez on 13 and O~( B ) = { - p ( ~ ( ~ + r

e Of(= - pnnrr

L e m m a 6.1 gives a formula for the subdifferential of tile function qo,(B). For the a d a p t a t i o n of matrix B ' , the following gradient method can be used: B~+, = B~ + %1 (~k'9 "T + 9 " ~ r ) B~,i = 0, 1 . . . . .

(16)

where ~' denotes the normalized vector g~ E Of(x~) (see (12)). Analogous to Algorithm 4.2, we write an algorithm with t h e m a t r i x modification formula (16). Algorithm Step I

6.1 Ini tialization s = 0, Bo 1 = f , i = - 1 , j = - 1 , z ~ = xinlt,g ~ E Of(z~

S t e p II

1.

B8 =

2.

i=0.

S, B.%l,

if IIB$+~g'll = o otherwise

440 3.

P,i = argmino> o f (x" - pB~B~rg').

4.

j=j+l,j(s,i)=j.

5.

x~ = x" --psiB~B~Tg ,.

6.

Compute g~ e Of(x~) such that (g~, 23~B~'g ") < O.

7.

Bt+l = B Z + ) U ( ~ g ~ r

8.

If ]]x~ - x'll :> v, then i(s) = i and go to Step III.

9.

i = i + 1 and return to point 3 of Step II.

~_

g s ~is T )B~"

Step III

x~+x = -,,~z'~ a~+l = g~.

Step IV

s = s + 1 and return to Step II.

We formulate a theorem about the convergence of Algorithm 6.1 for smooth objective functions. T h e o r e m 6.1 Let the function f: 12'~ ~ R be strictly convex and smooth, Ll be a Lipschitz constant of the function f on the set T(x~

and L2 be a Lipschitz constant for gradient V f ( x ) on the set

T~(xo) d=ef[~'x:~eT(~0)min[Ix--y]l 0 and a sequence of positive numbers {A./} satisfying co

./=o

j=o

Then for Algorithm 6.1 there exists ~ such that

IIg~ll ~ 2~L=. R e m a r k As was mentioned before the exact steepest descent (see Step II-2 of Algorithm 6.1) is not implementable. Rough approximations are used for practical purposes.

7 1. 2. 3. 4.

5. 6. 7. 8.

References EAVES, B.C., ZANGWJLL, Generalized Cutting Plane Algorithms, SIAM Journal on Control, 9, pp. 529-542, 1971. BERTSEKAS, D.P., At~o ~r S.K., A Descent Numerical Method for Optimization Problems with Non-Differentiable Cost Functionals, SIAM Journal on Control, 11, pp. 637-652, 1973. WOLFS, P., A Method of Conjugate Subgradients for Minimizing Nondifferentiable Functions, Mathematical Programming Study 3, pp. 145-173, 1975. LSMAltECIIAL,C., STrtODIAT, J.J., AND BHIAIN, A., On a Bundle Algorithm for Nonsmooth Optimization, Nonlinear Programming 4, Edited by O.L. Mangasarian, R.R. Meyer, and S.M. Robinson, Academic Press, New York, I981. MI~F'LIN, R., A Modification and an Extension of Lemareehal's Algorithm for Nonsmooth Minimization, Mathematical Programming Study 17, pp. 77-90, 1982. KIWlEL, K.C., Methods of Descent for Nondifferentiable Optimization, Springer-Verlag, Berlin, 1985. DI~M'JANOV, V.F., AND VASlL'I~V,L.V., Nondifferentiable Optimization, Springer, New York, 1985. ROBINSON, S.M., Newton's Method for a Class of Nonsmooth Functions, SIAM Journal on Numerical Analysis, to appear.

441

9. Sltolt, N.Z., On a Structure of the Algorithms for Numerical Solution of the Optimal Planning and Designing, Kiev, PhD Thesis, 1964 (in Russian). 10. DENNIS,J.N., AND Moltl~, J.J., Quasi-Newton Methods, Motivation and Theory, SIAM Review, 19, pp. 46-89, 1977. 11. SltoR, N.Z., Minimization Methods for Non-Differcntiable Functions, Springer-Verlag, 1985. 12. POLJAR,B.T., Subgradient Methods: A Survey of Soviet Research, Proceedings of a IIASA Workshop "Nonsmooth Optimization," Edited by C. Lemarechal and R. Mifflin, 1977. 13. UR.YAS'EV,S.P., Stochastic Quasigradient Algorithms with Adaptively Controlled Parameters, IIASA, Laxenburg, Austria, WP-86-32, 1986. 14. EJtMOLIEV, Yu., AND WETS, R.J.-B., EBS. Numerical Teehiques for Stochastic Optimization, Springer-Verlag, 1988. 15. U~tYAS'SV,S.P., Adaptive Variable Metric Algorithms for Nonsmooth Optimization Problems, IIASA, Laxenburg, Austria, WP-88-60, 1988. 16. ROCKA~'ELLAR,R.T., Convex Analysis, Princeton Mathematics, Vol. 28, Princeton Univ. Press, 1970. 17. PSIIENYClINYI,B.N., Necessary Conditions for an Extremum. Dekker, New York, 1971. 18. NESTEROV,YU.E., Minimization Methods for Nonsmooth Convex and Quasiconvex Functions, Economika i mat. metodi, USSR, XX, pp. 519-531, 1984 (in Russian). 19. NIJRMINSK1, E., Numerical Methods for Solving Deterministic and Stochastic Minimax Problems, Naukova Dumka, Kiev, 1979 (in Russian). 20. HOVFMAN,A., Weak Convex Functions, Mullifunclions and Optimization, 27. IWK d. TH Ilmenau, tieft 5, pp. 33-36, 1982.

COMPOSITE

OPTIMIZATION:

SECOND

VALUE FUNCTIONS

problems

=

By

~(F(x)),

not

constraints.

is

Such

the of

a

all)

paper

we

composite

we

map

Composite minimization

problems

and

~ appear

give

Formula

for

Further

F

given on

is a s m o o t h

point

convex

(in

reduction

optlmlzatlon the

It

general

Forms

for

problems

wlth

order

lower

second

apply

f(X)

Form

to

obtaln

new

a neigborhood

of a

INTRODUCTION.

is the n ~ m e

involving

for a class

functions

of

of u n c o n s t r a i n e d

the f o r m

(I)

map from a Banach

space

x ~ X) into a n o t h e r B a n a c h

space

of

is

X Y

(or f r o m and g

is a c o n v e x

function

Y. The

role

important

composite

classes

programming, some

the

optimization

In composite optimization.

r(x) = g ( F ( x ) ) where

of

a

as

of

and

of is

classes

a

class

Functions

typically

Functlons

optimization

a

important

second order conditions and sensltlvlty estlmates

i.

mean

composite

smooth

problems

practically

In

epi-derlvatlve

Involvlng

F

Function.

(If

AND SENSITYVITY

opt|mlzatlon

constraints

where

non-smooth) many

composlte

without

CONDITIONS,

Alexander Ioffe of M a t h e m a t i c s , T h e T e c h n i o n H a i f a 32000, I s r a e l

Department

ABSTRACT.

ORDER

problems

equivalently,

of

optimization

semi-infinite of

optimization

shape

As an example,

problems

programming,

optimization

in o n e or a n o t h e r

sense,

let us c o n s i d e r

determined

-

with

relaxed see

reduced

the p r o b l e m

[Ill,

by

the

constraints optimal

that

many

(mathematical

control

[I2],[RI],

to a p r o b l e m

fact

problems,

[DZ])

can

of s u c h sort.

be

443

(p)

fo: X ~ R,

where

fo(X)

minimize

G(x) E K,

subject to

Set

G: X ~ U, E: X ~ Y

Y = RxUxV,

K = {u:O(u)

~ O}

and

take a continuous

and

dist(K,u)

E(x) =0, K

is a closed convex cone in

sublinear function

s ce+(u)

for some

function of the intersection of the polar of K

~(u)

U.

such that

c > 0 (say,

the support

w i t h the unit ball in

U~ )

and consider the functions

g1(y) = g1(=,u,v)

= max {~,e(u),llvll},

L

g2(y) = max {~,O(u)} + kllv]l; + Ilvll)

g3(y) = ~ + k(~+(u) +

(where

~

:

max { ~ , 0 } ) .

Finally,

we fix an

x ~ X

and set

F(x) = (fo(X) - fo(~),G(x),E(x)), fi(x) = Ei(F(x)),

i =1,2,3.

Then the following "reduction principles" I. If

x

is a strict local minimizer

hold:

to eiteher of

f.

and an admissible

1

element for II. local

(P), then

is an isolated local solution to the problem;

If the equality constraint

in

(P)

solution

f

attains

sufficiently III.

x

to

large

(P),

then

is an

in

(P)

are

f

Q(x)

r

=

{x:

G(x)

~ G}.

If

(cf.

[Ii]),

C

if, say, x

x ~ x

s u f f i c i e n t l y close to

=

x

(i.e.

x

x

x,

x

close to

{0},

then

at

is a x

for

the cost then

x

function is

is

a local

x.

means

that

x

we have

x ~ G and

a

fIT]

sufficient is that

condition

Q'(x)

is an isolated local solution of

we have (other than

for

is onto.

[P].)

satisfying

G(x)

f (x) > f (x). 0

of a neigborhood of

x

~ r.dist(O,Q(x)),

(For a survey of regularity results see Indeed

and

minimum

(P),

at

regularity given by the theorem of Ljusternik

for any admissible

at

in

~ C

such that for any dist(A,x)

A

local

x

attains a local minimum at

3

(Regularity of the constraint there is a constant

regular

admissible element

solution to (P) if and only if

where

a

at

k;

If the constraints

Lipschitz and

2

is regular

~ K, E(x)

(P), then

= O) which

is

This means that for any

x

0

x

itself)

either the inequality above

444 holds,

IIECx)II > 0

or

or

@(GCx)) > 0 which means that

f (x) > 0 , whereas 1

f (x) = O. On the other hand, if the latter is true and I than fo(X) > fo(X) which means that x is an isolated (P). This proves Principle Let

us

minimizer

prove for

f

valid and let

x

admissible other hand,

also

I for

Principle

III.

It

is

such

that

fo

since

~ fo(U)

of

local

to

that

CP).

solution

x

Suppose

is the

x. By regularity,

(r+l)(dist(K,G(x))

Ux - ull ~

is Lipschitz,

~(x) = s

obvious

solution

be any point close enough

u

is admissible, of

g1"

if it is a local

3

x

a

local

latter

is

there is an

+ llE(x)ll). On the

we have

~ fo(X)

+ lllx - ull

fo(X) + 1(r+l)(dist(K,G(x)) + llE(x)U) ~ f3(x). Using principle (P)

from

I, we can obtain sufficient

sufficient

conditions

for

either

and Ill, we can do the same for necessary f3

([BP],

[Ii],

[12],

reduction

to

procedures

for problems

[R2]).

composite

Thus, of

composite

unconstrained

optimization

advantages; of certain this

theory.

it also

connection

we

This

offers

difficulties

a good

number

of questions

(e.g.

condition

of

of [R2],

why

by

epi-derivative

of

instrument

technical

many

point"

them

problems

describing

the standard

sometimes

that

contain

a

composite

to many and

problems,

in specific formula

function

aesthetical the nature

better a pair

branches

seen.

In

of

"twin"

that

cover

[I4] and actually clarify a

Hessian

must be solved

a

4 and 5 stated

from which

are

[K2],

numerical

optimization.

and the other sufficient, [KI],

complicated

pair of second order conditions begin

has

second

order

conditions

or why the necessary

[KI] cannot have a sufficient

what kind of technical

We

approach

3

develop

[W]. Theorems

a unified

2 and

one necessary

cannot work in sufficiently order

of

to

via composite

"observation out

Theorems

the most recent developments

[FS],

analysis

approach

and ways

mention

second order conditions,

optimization

offers

for

f.; using principles II 1 associated with f and 2 makes it possible to use

III

with constraints

optimization

of

(of any order)

conditions

Principle

in section 4 show a way to sensitivity

conditions

"twin"

in order

ete.).They to obtain

second

also

show

a workable

cases. for

which

in proofs of all the above mentioned

the is

the

results.

lower

second

principal

order

technical

445 2.

Let

LOWER SECOND ORDER E P I - D E R I V A T I V E OF A COMPOSITE FUNCTION

f

be a function

order e p i - d e r i v a t i v e

X

which

is finite

at

x

along

h E X

f

at

with respect

The

is

limit as

x" = O, we write basic

if

t 9 0

0

for

[RI]

of

the

lower

all

second

nonzero

h,

then

A = F'(~),

= {y* 9 a g ( y ) :

K(x') = {h: g ' ( y ; A h ) (we write simply

~

and

K

if

f

w(h)

A'y"

c)

is

C 2 near

then

S

be

where

if

L

at

containing

defined by

x

a

local

x.

(I) . We set

= F"(~)(h,h);

s <x',h>}.

x;

is the linear subspace of and

7 + L

the collection

converging to zero. For any #s(h,w)

attains

x" = O) and assume that

L + Im A = Y

continuos with respect to Let

that

Y.

(regularity condition)

dom g - Y,

epi-derivative

= x'},

a) g is finite and lower semicintinuos at b) F

f

linear manifold

In what follows we consider functions 7 = F(~),

order

is the following:

m i n i m u m on e v e r y f i n i t e d i m e n s i o n a l

~(x*)

).

f"(x;h).

property

>

- t<x~,h'>)

of the epigraphs of the functions (see

its role in o p t i m i z a t i o n

f'~(x;h)

to

second

this means that the e p i g r a p h of the function h ~ f " ( x , x ~ ; h )

h ~ t-a(f(x + th) - f(x) - t<x",h>)

determines

lower

is

the upper Kuratowski

If

The

[RI]:

f'i(x,x';h)- = lim inf t-2(f(x + th') - f(x) h'~ h t 9+0 Geometrically,

x.

x" 6 X ~

defined as follows

of

on

at

there

z ~ X

such

spanned b y that

g

is

y + Az

of sequences

s E S

is a

Y

and any

= tim sup inf t 2(g( 7 + tn Ah" ng~ h'gh

s =

{t } of p o s i t i v e n h e X, w E Y, we set + t n% )

- (7)

- tn < x ' , h ' ~ ) ,

numbers

446 lim sup inf = sup lim sup n 9 m h'9 h e > 0 n ~ ~ i. The following

THEOREM

- either or

-

f"_(x,x~;h)

f2(x') ~ ~

= - ~

alternative for some

In

the

last

holds:

h;

and

f"(x,x';h)

+~

case

min ~s(h,w)). sES w -> ~s(h,w) is either =

the f u n c t i o n

or it is convex

continuous

with

in case w h e n g(-)

is a continuous

of the first

convex

is continuous,

f"_(x,x';h)

part

function.

g ( F ( x + th)) = g ( y +tAh w(.)

identically

equal

to

d o m ~;(h, ") = ~(x').

We shall give a sketch of the proof

and since

inf IIh'-hll<e

In this case

+ t2w(h))

it follows

of the theorem

+ o(t 2)

that

= lim inf t-a(g(y +tAh + t2w(h')) h'-) h

-g(y)

- t<x',h'>)

t ->+0 min

=

s ~S

lim sup tn~(g(y

+tnAh"

+ t2w(h))n - g(Y)

- tn<x%h'>)

h'-~h n-~oo

=

If

min #s(h, w(h) ). s E S

f[(~,x';h)

> - m

for any

lim inf t-1(E(~

h,

then

+tAh + t2w(h))

- g(y)

- t<x~,h>)

z 0

t 4+0 and,

consequently,

~(x ~) ~ ~.

g'(y;Ah)

In a similar

z <x',h>

for any

way we conclude

that

h

which

implies

h ~ K(x ~)

if

that

f[(~,x';h)

is

finite.

3. SECOND O R D E R

It is not a difficult second

order necessary

THEOREM

is that

2.

~ ~ ~

to d e r i v e

from Theorem

1 the f o l l o w i n g

condition.

A necessary and

matter

CONDITIONS

condition

#s(h,w(h))

~ 0

for

x

for all

to be a local h ~ K,

s E S

minimum or,

of

f(.)

equivalent-

447 ly, that for any

h E K, s ~ S

there is a

A

y* E ~

- ~[(h,y*)

such

that

~0.

For the "twin" sufficient condition we need some attraction property for

the cone

K

of critical directions.

(AP) ~ ~ z

and any bounded

This is a convenient one:

{h n}

sequence

a subsequence converging to a certain We observe

THEOREM 3

(AP)

for

and all nonzero h E K

0

for all

condition).

If

provided

that

f(.),

h E K

there is a

a

h

if

has

~ ~ z.

dim X < m.

holds,

(AP)

~s(h,w(h))

or, equivalently,

y* E ~

g'(y;Ah n) ~ 0

that

h E K.

is trivially satisfied if

(sufficient

minimizer

local

g'(y;Ahn)

that

The property

such

then

x

is a strict

> 0

for

a11

s E S

that for any s E S and any nonzero

such that

- #~(h,y*)

> 0

(2)

It follows from Theorems 2 and 3 that in every specific s i t u a t i o n we must be

able

to

calculate

@s

second order conditions. case

when

the

programming any

original

say,

problem



Since

with the Lagrangian, has

If,

#g

to

non-positive

(P)

order

to

be for

obtain

is a polyhedral is a

standard then

is p r e c i s e l y

a workable

function problem

~(h,y*)

the H e s s i a n

observed

in

this

connection

> O,

V

h E K,

condition but a weaker

natural necessary counterpart

that

calculated. terms.

in finite dimensional so called piecewise of them the

g

function

on

~

for

associated

is

always

(3)

In fact,

in

[RI]

(that is both

linear quadratic

g(').

is actually

(3)

>

by

ill]).

in some specific

Rockafellar

spaces

it does not have a

(by which we mean that replacing in

is either affine or quadratic.) f(-)

= 0

h ~ 0

the f u n c t i o n

situations

considered

dim X < ~

composite and

#s can

this has been functions

dim Y < m) with

(The latter means that the domain

can be broken into finite number of polyhedral g(.)

is the

mathematical

#~(h,y*)

It is possible to mention a few more cases in w h i c h

in different

of

function

one because

does not make the condition necessary - see

done

(which

of

y* ~ ~. Therefore the standard Lagrangian c o n d i t i o n

is still a sufficient

be e x p l i c i t l y

pair

we obtain the standard second order conditions.



a

g

in

with finitely many constraints),

h E K,

It

or

pieces and on each of

It turns out

epi-differentiable

and

that

in this case

a formula

for

the

448 epi-derivative class

was

obtained.

of functions

(see

This

[C2],

hold already for a function on quadratic functions.

result

[14]) R2

hidden

in

function of tiability

[K2]

extended

too more

to a more

general:

is the m a x i m u m

general

it may

of a linear

not

and a

A formula for the e p i - d e r i v a t i v e of the m a x - f u n c t i o n

in

t (and

of

not

which

f(x) =

is

can be

but

case

x

when

f(t,x)

max

f

has

a

polynomlal-type

is finite dimensional).

general

max-functions

(involving

behavior

a

A c r i t e r i o n for epi-differenmaximization

over

arbitrary

compact spaces and also defined on infinite dimensional Banach spaces) ven in [14]. We also mention the p a p e r b y Darkowskii and Levitin similar work was done in case when m a x i m i z a t i o n

as

is gi-

[DL] where a

is a p r o b l e m of concave prog-

ramming. In all

these

cases

epi-differentiable. involved

infinite

~s

does not

actually

This, unfortunately, number

possible

condition

sequences

not

taking

s

and

into

there

account

the

on

s

and

f(.)

is

is not a typical case when problems

of constraints

is considered,

semi-lnfinite p r o g r a m m i n g or optimal control. all

depend

is

such

as

problems

of

In general we have to scan over no

variety

hope of

that,

say,

second

a

order

necessary

directional

behaviors of a convex function can be s u f f i c i e n t l y powerful.

4. SENSITIVITY ANALYSIS

We now c o n s i d e r what happens Namely,

if we begin to slightly perturb

p(u) = inf g(F(x) x (For

simplicity

infimum above fixed small We

the map

F.

we consider the "value" function

shall

we

consider

is calculated

the

case

subject

of

a

+ u). "local"

to the condition

value

function

when

llx - xll < ~

for

the some

e > O. This does not cause any real loss of generality.) formulate

second order sufficient order directional

two

theorems

with

estimates

condition is satisfied,

derivatives of

p

at

and,

in

case

exact formulas for

when

the

the first

zero. (It should be remarked at this

449 point

that

order

derivatives

classical

we

typically of

problems

need

value

with

second

order

functions

equality

in

information

extremal

constraints

standard problems of mathematical

to

calculate

problems

and

-

see

first

lIT]

[ B ] , [ G ] , [GJ],

for

iS], for

programming.)

We set

p [ ( O , u ) = lim inf t-*(p(tu) - p ( O ) ) t ~ +0

and f o r

any

s

e S,

t~1(p(tn) -

p~+C0;u) : lim sup n9 THEOREM 4. For any

s ~ S

p(0))

~

we have

p~+(O;u) ~ inf ~s(h,w(h) + u) h If the sufficient condition of Theorem 3 is satisfied, p~(O;u) = Therefore,

if

Cs

differentiable

at

does 0

depend

along every

p'(u) = To formulate

Cs(h,w(h) + u)

min min s~Sh~K

not

on

u

~s(h ) = {ym ~ ~:

s,

then

p

is

directionally

and

r

min h~K

the second theorem,

then

+ u).

we set

for

h ~ K

- r

z 0}.

THEOREM 5. We always have

p~(O;u) ~

This

becomes

Moreover,

if

an

equality

Cs



(3.7)

and their norms by : lt("(t)ll = ~ ,

lb(O}l = IlA{t)l} ( 1 - < . , iiy(oil~.

A(t)

>=)~

(a.s)

At points to of a pa.th where V(to) --- 0, one may still define v(t) and a(t) by continuity provided that { T h e fight-hand sides in (3.7) are defined (3.9) V(to) = 0 =, and have a limit when t --+ to, t 7~ to Notice that (3.9) holds automatically as soon as ~o'(x) is injective for any z in C! Now that 90(C) is equipped with C2-pathes with respect to the arc-length, we define the r a d i u s o f c u r v a t u r e p(t) of a path P = ~o([z,y]) at parameter t by : p ( 0 = lla(t)lV ~ =

Itv(t)ll=cx- < v, A(t) ilA(t)ll" ~

>=)-~, > [IV(~)ll 2 -IIA(t)II -

(3.x0)

-

Following [51, we introduce also the g l o b a l r a d i u s o f c u r v a t u r e pa(t,t') of P at ~ seen from ~', which is given by the formula 0

pa(t,t') =

Sgn(f - t ) < P ' - P , v ' > ( 1 - < v,v' >=)89 Sgn(t'-Q

ifSgn(t'-t)

2 0

>0

ifSgn(t'-*)

>0

and = 1. Geometrically, pa(l, *') is the distance of P ( t ) to the intersection of the two haK spaces normal to each end of the subpath of P located between P(t) and P(t'). It is related to the usual radius of cm'vature p(t) by limp(t,t') = p(t) (3.12) t~t I

455

We denote also by 5(t,t') the arc l e n g t h between the two points P(t) and P(t') of the path P = ~([~, v]) :

a(t,t') = / " IlV(r)lldr

(3.13)

and by O(t, t') the d e f l e c t i o n between the path directions at the two saane points : @(t,t') = c o s - l ( < v(t),v(t') >)(in radian),

(3.14)

which statisfles (cf [5]) :

O(t,t') 2 )rdr A(C,%P) (3.21)

< e,~(c,~,p), e >_ e(c,~,~)

In view of (3.20) it is reasonable to suppose that R a n d / ~ has been chosen such that as to satisfy

o

(4.4)

~(C)closed in F

(4.5)

Then : i) The NLLS problem (2.3) i~ Q-wellposed on 0 = {z 9 FId(z,~(C)) < / ~ }

(4.6)

for ~he p.~eudo-dis~ance 6(x,y) on C defined in (~.1). ii) More precisely, if zj 9 O,j = O, 1 satisfy

Iiz0 - z, lir + m ~ d(zj,~(C)) < d < ~ i=0,1

(4.7)

457

for some d, then the corresponding solution~ &j,j = O, 1 of (2.3) ~atisfy the s t a b i l i t y e s t i m a t e :

6(:~0,:~1) _< (1

d/R)-'ltzo - =,ll~

-

(4.8)

P r o o f : Hypothesis (2.1) (3.9) and (4.4) imply, using theorem 5.12 of [5], that r is strictly quasiconvex with a neighborhood v~ given by (4.6), which, together with the fact that ~(C) is closed, implies existence, uniqueness and Lipschitz stability of the projection on r all over ,9, and absence of local minima by theorems 3.6, 3.9 and 3.5 of [5]. [] Of course, if the strong property (4.2) on the derivative c2'(x)-the so-called "sensitivity Matrix" in the finite dimensional case-holds, then one obtains stability for the ]Ix - Y[IE distance on C, and c2(C ) is necessarily closed : C o r o l l a r y 4.1 Let hypothesis (2.1) and (.{.2) hold, and R >_ Ra be lower bounds ~o R(C,~,79) and RG(C,~,79) a~ defined in (3.16) (3.17) and (3.21).

If: RG > 0

(4.9)

Then T(C)

is closed in F

(4.10)

and conclusion~ i) and ii) of theorem (4.2) hold, with (4.8) replaced by o',,,ll:~o - :~,IIE

< ( I --

dlR)-'llzo

-

~,11,~

( 0, Ik,"(=)(y, y)ll~ < #llyll :v= e C, Vy e c Then we have the

(4.1~)

458 C o r o l l a r y 4.2 Let hypoihe~i~ (2.1) and (4.2) hold, and define : r: =

,~1~

,

e

=

(~l,~..)diam c

(4.17)

and : P~ =

R 0 < O 0. (This of course requires that the "variety" r itself has a "bounded curvature"). The condition Re: > 0 then can ahvays be satisfied by reducing the size of C, and is hence less critical. Let us first consider the case where E is finite dimensional, for example in parameter estimation in O.D.E.s, or in P.D.E.s after discretization has been performed. This case is very important practically, as it is the only one which one can actually attempt to solve on a computer ! One may then try to use the above geometrical theory to determine if the setting of the parameter estimation problem is satisfying, ie if the knowledge of C and ~ allows for a unique, stable determination of the parameter, and if so, which accuracy is required on the data. This requires the estimation of lower bounds R a n d / ~ to the smallest radius of curvature (3.16) and global radius of curvature (3.17), which for sure is not an easy task. When the dimension of E is not too large, one can try a numerical determination of R using (3.16) (3.10) and Ro using (3.17) (3.11), and use theorem 4.1 which given the most precise sufficient condition. This includes intensive computation (namely, along all segments [x, y] with extremities x and y located on the (relative) boundary OC of C[), which may quickly become unaffordable when the number of parameters is larger than a few units... But the reward for this computational effort is a treasurable ixfformation on the wellposcdncss of the NLLS problem and the absence of local minima in the objective function for any z in O, which is practically very useful both for the engineer who has set the

459

parameter estimation problem ("do I have enough information for recovering my parameter in a unique and stable way ?") and the numerical analyst in charge of the computations ("is my optimization routine going to be stuck in local m i n i m a 7"). W h e n this numerical approach is impossible, one may think of calculating analytically (i.e. with paper and pencil) a lower bound R using (3.16) (3.10), upper bounds @ using (3.25) and A using (3.24) (all these quantities are expressed by simple formula involving only r - x ) and ~"(z)(y - z , y - z), and then use theorem 4.2 to get information on the wellposedness of the NLLS problem. There are yet no example where this approach has been used, but the corresponding theory is just being released now in [5] and in this paper, so we hope that some application along this line will show up in the future. Let us turn now to the case where E is infinite dimensional, as for example in parameter estimation in P.D.E.s. We expect here that the generic (in an imprecise sense ...) situation is R(C,~o,'P) = 0, so that the above geometrical theory does not apply. As a support for this assertion, we refer to [2] where it was shown, for the model problem of the estimation of a diffusion coefficient function in a 1-D elliptic equation, that one can find, when the discretization is refined, a sequence of pathes on which the smallest radii of curvature tends to zero (these small radii of curvature are obtained for perturbations of the diffusion coefficient having smaller and smaller support containing one stationary point of the solution to the elliptic PDE). It would however be maybe possible to prove some wellposedness results in the somewhat academic case where the solution to the elliptic equation does not possess any stationary point. A bright singular point in this dark picture of the situation for the ilffinite dimensional case is given in [6], where tl~e estimation of the shape (i.e. a function) and the phase (a number) of a plane wane is discussed, and analysed using the above geometrical theory. To conclude on ilffinite dimensional E, let us mention that the geometric theory may reveal as a useful tool for analysing how the well-poscdness of the NLLS problem deteriorates when E is approximated by larger and larger finite dimensional spaces - multiscale aa~alysis of functions should play a crucial role here (see [2] and [71 for vet3 preliminmT results).

5

Well-posedness of the regularized N L L S problem

We investigate in this paragraph the wellposedness of the regularized problem (2.3) under the minimum set of hypotheses (2.1) (2.4). Of course, as we have not required any compact injection from s into E, this minimum set of hypothesis does not ensure in general even the existence of a solution .~, in opposition to the linear case (i.e. ~2 E s F)) where the same hypotheses ensure the existence of a unique f&. We shall be able in this paragraph to quantify the natural intuition that "a m i n i m m n amount" of regularization should be added in order to compensate for the non-linearity of ~o, mad restore a situation similar to that of the linear case. As mentioned at the end of paragraph 2, the study of the wellposedness of the regularized problem (2.3) can be made very simply by applying all results of paragraph 4 to the NLLS problem (2.3) with a proper choice for E, C, F, ~ m~d z as exptaned in (2.5). For sake of simplicity, we shall explicit this approach only for the case of corollary 4.2. Using hypothesis (2.1) and (2.4), we know that :

3&M > 0, II~,'(~).yllr -< aMII'JII~,W E C n C,V,j e C

(5.1)

3B > 0, IW'(z).(y, Y)IIF -< Bllyllg,w E c n E, vu E x

(5.2)

(notice that if E itself happens to be aa~ Hilbert space and the choice e = t3 is made, then &M coincides with aM defined in (4.2) and r coincidcs with/3 defined in (4.16)).

460 As in corollary 4.2, we shall need the size of C in the parameter space g :

m~(C,g)

= sup II= - Y l l ~

(5.3)

aa,yEC

but the position of the a-priori guess x0 with respect to C with also play a role through the "radius of C seen from x0 in g" : rad(C, xo, $) = sup

a:EC

11~ -

~01l~

(5.4)

In order to express the results in a simple form, we introduce the dimensionless quantities : (5.5) (5.6) (5.7) (5.8)

q g d(z,z') r

= = = =

rad(C, z0, s e/(/~diam(C, $))

S)

d(z,z')/(fldiam(C,s &M/(3diaan(C,$))

(position index of x0 w.r.t. C) (adimensional regularization pm-ameter) (adimensional distance in data space) (adimensionat upper bound to sensitivity)

Notice that 1/2_ ~

(5.zo)

,-ad(C, zo,~).

Using the adimensional variables (5.5 thru 8), we rewrite (5.23) (which implies/Q;.~ > 0!) as : g > 77

rsin 1/z + ((1 + ~ / ~ ) ~ / :

- 1) cos 1/g > r~

if 2/rr _< g if 1/~r < g < 2 / ~

(5.20)

and (5.22) as : ~

= ~{(~sln 1/~ + ((1 + r

- 1) cos 1/~) ~ - ,7;}'/~

(~.21)

if l/;r < ~ < 2/7r Hence we see that if g satisfies (5.24), then the regularized problem (2.3)~ is Q-wellposed on the neighborhood O of c2(C ) of adimenslonal size ~n.x given by (5.25), which is the announced result. [] \u have illustrated on figure 1 the q -+ g.fin function for various values of (, which makes clearly visible that the choice > m ~ { 2 / , ~ , ~} (5.22) ensures the Q-wellposedness of the regularized problem independently of the sensitivity index (. Figures 2 and 3 illustrate how the size dm~ of the neighborhood of ~p(C) depends on ~ (ta.kcn larger than g.~. of course !), ~7 and (. These curves can be used for example to determine, given an estimation of an upper bound dm~ of the measurement and model error, the smallest mnount of rcgularization g to be used in order to restore wellposedness of the NLLS problem and suppress local minima on a neighborhood of ~(C) large enough to contain the expected data,.

462

.p._t,o,' _~, .

I

1.0 from top to bottom : zeta = 10 zeta = 5 zeta= 1

O.B

/ /

"

/

0.7 2/.rr.

0.5 '

'

+

'

0.5

I

'

'

'

0.7

'

I

'

'

O.B

'

'

1.0

Figure 1: T h e n f i n l m u m value ~ m i n of t h e r e g u l a r i z a t i o n p a r a m e t e r as f u n c t i o n of the positio1~ index q of t h e a-priori guess and t h e sensitivity i n d e x ~ of t h e ~ m a p p i n g

I 25.0

2.2

/

20. o

/ /

from left to fight : eta = 1/'2, zeta = 0 eta = 0.57, zeta = 0 eta = 2/pi ~ eta=0"76~an zeta eta=0.88/ Y eta=l.OI

.ts. 0

10.0

-

/ r

/ / / /

/

from ]eft to fight : eta = 1/2, zeta = 0 eta = 0.57, zeta = 0 eta = 2/pi / eta = 0.76~ an zeta eta=0.88/ Y = .

1. 5

r r

ops_bor

'

/

#/// //7/// ////// ////// //////

0. 7

5.0

0.0

''l'l''J'l''''l''''l~l''

0.0

1.0

2.0

0.0 3.0

4.0

5.0

0.0

0.5

1.0

1.

.p~_bor Figure 2: T h e d.,~x f u n c t i o n giving Lhe size of t h e c y l i n d r i c a l n e i g h b o r h o o d for t h e regularized p r o b l e m , as f u n c t i o n of 7, for five values of t h e p o s i t i o n i n d e x 77 of t h e initial guess z0, a n d a zero value of t h e s e n s i t i v i t y p a r a m e t e r s ~ for t h e two curves 77 = .5 a n d q = .57 c o r r e s p o n d i n g to ,7 < 2 / 7 r . Left : general

overview

; Right

: close

up on the

[0,1.5]

interval.

463

0.40

,-~~

I

0. 3 0 eta = 1/2

0. :30

0. 2 0

eta

from left to right :

0, 2 3

zeta = 0 z e t a = 1/2 zeta = 1 zeta = 5

zeta = 10

~

0. 15

0. 1 0

0. 0 0

/

0.57

from left to right : zeta = 0 z e t a = 1l'2 zeta = 1 zeta = 5 zeta= 10

0. 0 8

I I ~ 0. 5 0

0. 55

I I 0. 60

0. 0 0

1 I 1 I I I 0. G5

[

.p,_b~

0. 50

O. 7 0

0. 55

]

0. 60 I

0. G5 0. 70 ,P,-~o' ]

Figure 3: Influence of the sensitivity parameter ( on ~/-~x for values of the position index 71 of the in]tiM guess smaller than 2/~.

References [1] C. CHAVENT,"Local Stability of the Output Least-Square Parameter Estimation Technique'", Mat. Apllc. Comp., Vol 2 no1, pp 3-22, (1983) [2] G. CHAVENT, "New Trends in Identification of Distributed Parameter Systems", Proceedings of the IFAC World Congress, Mnchen, Pergamon Press, (1988) [3] G. CHAVENT, "On the Uniqueness of Local Minima for General Abstract Non-linear LeastSquare Problems", Inverse Problems 4, pp 417-433, (1988) [4] G. CHAVENT, "Quasiconvex Sets and Size • curvature condition ; Application to non linear inversion", INR.IA Report 1017, April 89 (Submitted to JAMO) [5] G. CHAVENT, "New size x curvature conditions for strict quasiconvexity of sets", INRIA Report to appear, 1989 [6] W. W. SYMES, "The Plane Wave Detection Problem", to appear [7] G. CHAVENT, J. LIU, "Multiscale parametrization for the estimation of a diffusion coefficient in elliptic and parabolic problems", Preprint IFAC Symposium on Control of Distributed Parameter Systems, Perpignan, France, June 26-29, 1989

AN IDENTIFICATION T E C H N I Q U E FOR A D A P T I V E S Y S T E M S IN THE CASE OF POOR EXCITATION

Hong Yao Xu and Co Robert Baird Department of Electrical Engineering Technical University of Nova Scotia Halifax, Nova Scotia, Canada B3J 2X4

Abstract. An Adaptive Identification Technique (AIT) is presented primarily for adaptive systems with insufficient persistent excitation. In this technique, the P(t-1) matrix is reset reasonably back to P*(t-1), and a new forgetting factor ~.(t) is designed to track the parameters of slowly timevarying processes. In addition, a weighted filtering algorithm is proposed such that a modified least squares algorithm is presented in terms of the procedures mentioned above. Theoretical analyses and simulation results have demonstrated that the proposed AIT methodology can overcome the bursting problems of adaptive systems in the case of poor excitation. Key words:

Identification; adaptive systems; least squares algorithm; bursting; excitation;

simulation.

1

Introduction

Many identification techniques have been proposed for adaptive systems in recent years. However, the estimated parameters may burst from instability in adaptive systems, if the preconditions of these identification techniques cannot be satisfied. It is well-known that when persistent excitation is gone in adaptive systems, most existing identification algorithms will not only fail to identify the parameters correctly, but also exhibit a bursting phenomenon [1], or so-called "estimator windup" [2]. A variable forgetting factor was presented [3] to solve this problem, which can be only used in deterministic systems, so the identification problem is still unsolvable. Isermann has proposed a simple remedy so-called "switch off" when identification conditions are violated [4]. However, the method [3] is not suitable for stochastic systems, and to "switch off" the identification is not effective [5]. If there is poor persistent excitation and when the adaptive control systems run for a certain time, the bursting phenomena may occur sometimes in practical applications. The proposed AIT has a strong

468 robustness to avoid this problem and has a good convergence property. Simulation examples are given to illustrate its results.

2

Bursting

Phenomena

In

Adaptive

Systems

Consider a linear system described as follows: A(q "1 )y(t)=B(q" 1)u[t-d(t)]+C(q "1 )0~(t)

(2.1)

where, A(q'l)= a o + alq "1 + . . .

+ anq-n

B(q'l)= b o + blq-t + . . . + bnq -n , b o ~ 0 C(q-1)= 1 + c l q -1 + . . .

+cnq -n

and y(t), u(t) and ~(t) represent the output, input and noise sequence, respectively, t is the sampling instant and d(t) denotes variable time delay of the system. The noise sequence {~(t)} is a Martingale difference sequence defined on a probability space (~, E, P) and adapted to (E t, t e h0, where Et is generated by the observations up to t and Eo includes initial condition information. Considering the system (2.1), we propose: Assumption set A : (i)

(ii) (iii)

E{('~

Et-t} = 0

a.s.

(2.2)

E{ o)(t) 2 [ F.t-1} =0"2< oo

a.s.

(2.3)

N lim 1 N._~=,sup~ ~ (0(02 0

(3.5)

where Equations (3.3)_(3.5) are described similarly to Sin and Goodwin [10] in which do(t)=1 and X(t)=l. r(t-1)=r(t-2)[;L(t-1)+~[t-d

o(t)] TP(t-2)~[t-d

o(t)],

=:(-1)>0

(3.5)

If [r(t-1)13minP1(t-1)
p., V i e [ t-M+1 . . . . .

t ]

(3.24)

check the diagnosing condition (3.10) and the filtering condition (3.25) so as to decide if some proper actions need to be taken correspondingly, and keep e*t(t ) and P*(t-1) constant until (3.21) is satisfied. Step 5. The weighted filtering procedures of e(t), y(t) and u(t) are implemented through a detecting criterion such that the system can be protected to avoid such a bursting phenomenon.

(1) If the condition A

Q(t) = inf II e(k)-e*t(t-M) {I2 > E max > 0

for E: max chosen

(3.25)

ke [t-M,t] is fulfilled , we obtain the filtered parameters: k e ( t ) = ~ ) f ( k ) = e * t ( k - M ) + 1--~ ,~, {Ae(i)-e*t(t-M)} M. i=k-M+l

y(t)=yf(k)={

k ~,e'~ i=k-M+l

u(t)=uf(k)={

k ~L, e "~i u(i)}/{ i=k-M+l

y(i)}/{

k ~/_, e'F,i}, i=k-M+l

V k ~ [ t-M+1 . . . . .

> 0

k ~/_~e-~i}, i=k-M+l

> 0

ibid

ibid

t ]

(3.26)

(3.27)

(3.28)

(2) If we have A

R(t) = sup II e(k) - e*t(t-M){{ 2 < s rain

forE; min chosen

(3.29)

k e It-M+1, t ] where 0 < Emi n < E.max, k = t-M, t-M+1 . . . . .

t. Hence, no filtering procedure is needed.

A

(3) Otherwise, we only require e(t) to be filtered using Equation (3.26). Equations (3.28) and (3.29) show the weighted filtering, where e-~ i represents a weighting factor, its decreasing rate is depended on ~. M denotes the length of measurements, M=10~20, ~=0.01~0.1 are recommended here. In (3.27) and (3.28), we use a weighting factor e-~ i to emphasize old data which indicate that the system operated better in the past. Simulation results have shown that the variations of e(t), y(t) and u(t) are reduced obviously by (3.26)~(3.28).

473 It must be pointed out that in Equations (3.13), (3.14), (3.21), (3.24), (3.25) and (3.29), ^

since the same term {{0(i)-e*t(t-M){{ 2 {s calculated only once every M sampling intervals, a number of computational efforts can be saved. The MRLS algorithm has good convergence properties. Its theoretical analyses will be derived from the following lemma 2 and lemma 3. Lemma 2. For the system (2.1), the MRLS algorithm (3.1)_(3.15) is used in order to achieve:

II~(t)-ell 2

(1)

_< K l l l ~ ( 0 ) - e l l

2,

t > 1

(3.30)

where K 1 = condition number of [P(-1) " l ] - - A l . ) m a x P ( - 1 ) ' l / ' O m i n P ( - 1 ) ' l = ' o m a x and.Orni n denote the maximum and minimum eigenvalue, respectively.

lira " ~ (2) N--,o~ t=l

e(t) 2 ;L (t) +,~ [ t . d o (t)]T p (t. 2),~ [t_do ( t ) ]

40, ~(t) was very largely reduced to 0"2=0.005 which implied that the persistent excitation became poor. As a result, all the estimated A

parameters e(t) shifted to the wrong area. ~2, one of the estimated parameters, is shown in Figure 1. ^

During 500 < t < 1000, several bursting phenomena occurred and then e(t) all became false value: e(2500) = [ 0.41, -0.302, 0.005, -1.01 ]. However, in the same case, the proposed AIT can demonstrate its strong robustness, one of its results is shown in Example 2. Example 2. Consider that the system and simulation conditions are the same as those in Example I. Assumed that at t > 40 the persistent excitation o)(t) was numerously decreased to be insufficient. In this case, we selected M=15, s s 6ma• 6min =0.015 and ,u.=0.0015. The A

estimated parameters 0(2000)=[-0.428, -0.344, 1.096, 0.303] almost exactly converged to the true values 0=[-0.43, -0.34, 1.10, 0.30] for the minimum variance control of STR. This indicates

475 that the AIT can still identify the real parameters even if the input exciting suddenty becomes insufficient. One of the estimated parameters, 92, is shown in Figure 2.

5

Conclusion

Adaptive identification becomes a very difficult task of an adaptive system in the case of poor excitation. The proposed AIT can fulfill it and overcome the bursting problem. Theoretical analyses and simulation studies have demonstrated the AIT methodology that is basically established by four proposals: 1 ) The P(I-1) matrix is reset as P*(t-1) based on Equations (3.5)~(3..12); 2) Weighted filtering is given to 8(t), y(t) and u(t) by (3.25)~(3.29); 3) Remembering units are employed to store some historical data through (3.21)~(3.24) for filtering and resetting; 4 ) A forgetting factor function ~.(t) is designed according to (3.13) ~ (3.15). However, it is recognized that if system excitation is too insufficient, the AIM may not work well. Therefore, further research work needs to be done to find out the minimum persistent excitation that the AIM can tolerate.

6

References

I . Anderson, B. D. O. (1985). "Adaptive Systems, Lack of Persistency of Excitation and Bursting Phenomena". Automatica, Vol. 21, No. 3, pp.247-258. 2 . Astrom, K. J. (1983). "Theory and Applications of Adaptive Control--A Survey". Automatica, Vol.19, No.5, pp.471-486. 3. Fortescue,T. R., L. S. Kershenbaum and B. E. Ydstie (1981)."Implementation of Selfquning Regulators with Variable Forgetting Factors". Automatica, Vol. 17, No. 6, pp. 831- 835. 4 . lsermann, R. and K.H. Lachmann (1985). "Parameter-adaptive Control with Configuration Aids an~ Supervision Functions". Automatica, Vol. 21, No. 6, pp.625-638. 5. Foley, M., K. Walgama, Y. H. Jin and G. Fisher (1988). "Adaptive Control Based on a Modified Kalman Filter Predictor". Preprints of the 8th IFAC Identification and System Parameter Estimation. Beijing, PRC. Vol.1, pp.342-347. 6. Goodwin, G. C. and K. S. Sin (1984). Adaptive Filtering, Prediction and Control Prentice Hall, New Jersey. 7. Goodwin, G. C., P. J. Ramadge and P. E. Caines (1980). "Discrete-time Multivariable Adaptive Control". IEEE Trans. Automatica Control, AC-25, pp.449-456. 8. Anderson, B. D. O. and C. R. Johnson, Jr. (1982). Exponential Convergence of Adaptive Identification and Control Algorithms ". Automatica, Vo~.t8, pp. I.

476 9. Xu, H. Y. (1988). "Robust Protection-Time Delay Tracing Adaptive Control and Its Application". Preprints of the 8th IFAC Identification and System Parameter Estimation9 Beijing, PRC. Vol. 2. pp.1064-1069. 10. Sin, K. S. and G. C. Goodwin (1982). "Stochastic Adaptive Control Using a Modified Least Squares Algorithm". A u t o m a t i c a , Vo1.18, pp. 315-321.

1

t32

9 .

1.00,50.0-0.5 -1.0 -1.5

'

I

100

'

I

'

200

I

'

300

I

'

400

I

9

500

I

'

600

I

9

700

|

'

800

I

"

/

K

900 1000

Fig. 1. The bursting parameter [32 of the RLS in Example 1

0.6

132

0.4 0.20.0-0.2 -0.4

9

0

I

|

50

100

9

I

150

'

I

200

9

I

250

9

I

300

9

I

350

9

I

400

9

I

450

9

I

500

Fig. 2. The convergent parameter ~2 of the AIT in Example 2

Asymptotic

Properties

in Rational

/2-approximation

L. Baratchart, l~f. Olivi, F. |Vielonsky Institut National de Recherche en Informatique et Automatique Avenue E. Hughes, Sophia-Antipolis 06560 Valbonne (France)

Abstract: This paper is concerned with the problem of best rational approximation of given order n in the Hardy space/:/2. We show that, generically, all critical points converge to the fnnction in tI~ as n increases to infinity. This property shows in turn that local maxima can appear only for a finite range of orders. This has consequences on an algorithm to find local minima previously described by some of the authors [3].

I

Introduction

Let us recall briefly the/2-approximation problem as described, for instance, in [2]. Consider the real Hardy space H~,I~ of functions f analytic in the complement of the closed unit disk 0 , vanishing at infinity, that can be written f ( z ) = F_.k~__l.f~z -k with .fk E R. and such that the norm Ilfll 2 = g:k .t2 is finite. The assertions that the coefficients fk are real and square integrable are respectively equivalent to the facts that the function f maps the real line into itself and satisfies a growth condition at the frontier T of U: sup- 1 r>l

27~

fo ~,"]f(rei~

< c~.

As in [2], let P~ be the set of real polynomials of degree at most n, and T'~ the subset of monic polvnomials, of degree n whose roots are in U. Moreover, let ",x'l C H~,I~ consist of all rational fractions h = p/q, where p E P,,-1 and q E T'~. The problem is to minimize, for arbitrary n, the criterion F)(h) = I[/- hll ~ where h is in x-1 The relevance to system theory of this question arises from the need of describing the input-output behaviour of a given system (i.e. its transfer function) by a finite dimensional model, whose transfer function is therefore rational. For more details, and a stochastic interpretation of the 12 norm, we refer the reader to [1]. Of course, in this context, our problem can only appear as the prototype of an integral criterion, which can admit many variations and generalizations, putting for instance weight functions, aclditional constraints, a.nd going over to the multi-input multi-output case. In the sequel, however, we shall restrict ourselves to the simple formulation given above. Apart from a classical formulation using the so called normal equations [4], this problem has been mostly tackled using differentiation, namely a gradient algorithm [6]. In [:3], an algorithm is described, which converges to a local minima by integrating numerically a differential equation. In all three cases, critical points (i.e. points where the derivative is zero) are on the saddle, because they are the ones that can be computed, and the trouble comes of course from the possibility of having several local minima. 9a . ~ r L

9

478

Let us take a closet" look at ti~e criterion F}(p/q) that we want to minimize. As q is settled, the polynomial p is uniquely determined as the orthogonal projection of f onto the n-dimensionM linear subspace 1.~ of H~,it defined by lq = P~-l/q. We shall denote by L'~(q) this projection and replace tile former criterion by the following one:

~L,~(q) = f

L}(q) ~" q

At each order, we get a set of critical points for this criterion and the goal of this paper is to describe their asymptotic behavior. These critical points may be of two different kinds: either L~(q) and q are coprime, in which case the point q is sa.id to be irreducible, either they share a common factor and we get a reducible point. When q is reducible, the variational argument (due to Ruckebush) used in [1] to show that it cannot be a minimum applies as wel! to show it cannot be a maximum either unless .f is already rational. In the sequel, we shall suppose that the element f of H~.it to be approximated is not already a rational function a.nd we shMl make the mild assumption that it is analytic on a domain strictly containing the complement of the open disk U. Then we show that, generically, critical points converge in /2-norn- to the function f as the order (i.e. tile degree of the polynomials) growths to infinity. Using this fact, we deduce that criticM points which are local maxima can only appear for a finite range of orders.

2

P r o p e r t i e s o f t h e c r i t e r i o n d,i a n d its c r i t i c a l p o i n t s 9

,;'7

As in [2], we introduce a.n other Ilardy space H+lZ, which contains analytic functions g in the unit disk U, naapping tile real axis into itself and such that

sup

Ig(rJe)!=g0 < ~ .

r =

As 5 = z -~ on the unit circle T and the coefficients in the power series expansion of 5 are real, we also have I ,;" ; , I dz < f ' g > = 2 i T r J ~ .f~-:JJ(-)--. z z This scalar product verifies the two following obvious properties which will be used in the sequel: 1) for all/," E Z, the multiplication by z k is an isometry of L~.~(T), i.e. for IT' ), all .f, 5' E L2.r~t< zkf, z~g > = < f, 9 >; (1) -

-

2) for al! f , g, h in L2,rc(T) such that f9 and f h are in L=,Iz(T).

< fg, h > = < s, .fh > .

(:2)

479

Let f be the function of H~,rt to be approximated and let g = f~, its image in the space FI2,1L" + W i t h the assumption made on f, there exists a real ,\ > 1 such that g is analytic in the open disk Ua centered at 0 of radius ,\. Let q in 79~ and (7 defined as in [2] in the following way: ]

~(:) = ~-q(:). Z

T h e Weierstrass division theorem (cf [5]) or more precisely its one dimensional version (also known as the Hadamard representation, cf [7]), applied to the function cq~ of/:f~rt shows ~hat there exists a unique function v(g , q) analytic in Ua and a unique polynomial w(g, q) of degree n - t such that: 9~ = o(g, q)q + w(9, q). It follows from [2] that if you seek in the space of rational fractions Vq for the minimum Lj(q)/q of ~by, you get Lf(q) = w(g, q). The quotient v(g,q) of the former division which we shall simply denote by v when no confusion can arise, possesses the helpful property to give the value of the criterion 02 at ;he corresponding point q: p r o p o s i t i o n 1 Let q be eLpoint of 79~ and v the correspondin 9 quotient, II~ll ~ = eTfq).

Proof: Using (1) and (2), the value of the criterion at q is:

Z(q) ' ' _ , g - L(q) _ >.

L ( q ) , f _ L(q) > - - < g -

62(q)= .

This set of derivatives vanishes iff for all polynomials P in C ~-1 [z]

< f -L(q) q

'

pL(q) >=0. q2

From the definition of the scalar product, we get

or

q

q

If a is a root of order m of the polynomial q, this implies f ,_, L ( q ) , , , dz Vl ~ {L...,m},jT v t , ) - - ~ - - t ~ j i : _ a)~ - 0. Then, hy the residue formula, the following derivatives should vanish:

W 6 {O,...,n~- i}, rvZ(~)] ~''

L--~J

B y induction, o' is a zero of order m

of

vL(q).

(~) = 0.

This proves the equivalence of the two

assertions and proposition 2.

3

A s y m p t o t i c behaviour of critical points

We denote by Cn the subset of T'~ containing the critical points at order n, and we put C = U,C,. By choosing a point in C, for each n, we construct a sequence of quotients (v,~). In order to prove that the family of functions (v,,) is normal, we use the integral representation:

1 f~ g(r162

v ( z ) = 2i---~..,

q(()

de

(-

z

where it is any real such that 1 < ;L < .\. Using this expression we prove the l e m m a 1 Let /

a real number such that 1 < #' < ~, there exists on the open set Uu, an uniform bound for the set of functions (v~) which depends only on the function g.

Proof: On the unit circle T, the quotient i[/q is of modulus 1. Then by using the maximum principle over the complement of the unit disk U, we get

v~ ~ c-u, lq-~I ~(() _ N , W e :r.,, Iw.(:) - w.~(~)l < Iw.m(z)l. By Rouch~'s theorem, w, and wum will have the same number of zeros in the open set U,,, but using proposition 2, a quotient corresponding to an irreducible critical point of order n has at least n zeros in U. As the order of points in the subsequence (wp) tends to infinity, wti,,, must be equal to zero. This is a contradiction with the assumption made on the circle T~,,. By letting IF vary continuously, we get a compact circular annulus containing infinitely many zeros for wti~ and thns this limit n-lust vanish on the open disk Uu. We just showed that every convergent snbsequence of (v,a) converges to zero uniformly on every compact set of U,,. Since it is normal, this is true for the sequence (t,,) itself. By using tim proposition 1, we get the F-convergence of any sequence of irreducible critical points to tile function to be approximated as the order of points tends to infinity. In order to generalize this fact to sequences containing also reducible points, we shall have to restrict ourselves to functions with tile property that Cn is finite for each n. This can be proved to be generic in various contexts. For instance, it is shown in [1] that such functions form a set of first category in the disc algebra of U u where # > 1. We first prove the p r o p o s i t i o n 3 Let p E 79~, a critical point such that the fraction Ly(p)/p is irreducible. Let r ET)~ a n d q = p r . Then (i) LI(cl) = rLl(p) ~ff r divides v(g,p). (ii) if (i) is verified, we have the following equivalence: q is a critical point iff p is a critical point and r divides the quotient v(g,p)/p.

Proof:

Apply the above mentioned division theorem to gc7 and gl'5:

J~7 = v(j, q)q + rz(q ) and ~ = ~,(g, p)p + LAp). Multiply the second equation by F:

g~7 = v(g,p)~p +/:LAp). Let us denote v(g,p) by vp and divide vpF by r:

vpF = v(vp, r)r + L~g(r). Plugging this expression in (3), we get

s~ = v(,,,, r)~ + (Lo#(r)p + ~L:(?))

(3)

482

and the second term on d~e right-hand side is of degree strictly lower than that of q. Thus we have t,~F = v(9 , q)r + Lff.(r)

(4) In order to prove (i), suppose first that L/(q) = rL/(p) holds. Applying to the second equation of (4) the assumption that p divides Lj(q), and hence that + divides Lf(q) we get (nsing the nsual notation for division) dpL~(~). As roots of/5 lie in the complement of the unit disk, r divides L~g(r). But L~-(r) is the remainder of a division by r and then it must be zero. The previous pair of equations beconles

I zj(q) = ,'ZAp),

(5) t,pr

= v(g,q)r.

The second equation of (5) shows that r divides vp. Conversely, suppose that @% The first equation of (4) implies that rIL~7,(r ). As the degree of L~7,(r) is strictly lower than that of r, it must be zero and (4) reduces as in the former case to (5). Suppose now that the assertions of (i) are verified and let prove (ii). By proposition 2, the fact that q is a critical point means that qlu(9, q)Ll(q) i.e.

p,-1~,(7)~G(p) or

pq~,~&(,,)

which yiekls that p]vpL1(p) i.e. p is critical. This yields also that r ] ( ~ ) L f ( p ) . Moreover as (1) is verified, r ] ( ~ ) p . As LZ(P) and p are relatively prime, we deduce that r]-~. Conversely, this last relation implies that q[vp. By the second equation of (5), we get q]u(g, q)r~. As roots of ? lie in the complement of the unit disk, q divides v(9, q)r and in particular v(9, q)rLl(p). Then using the first equation of (.5), q divides v(fl, q)Lf(q) and q is critical. O If critical points are irreducible, there exists an order over which the corresponding quotients v have more than any preassigned number of zeros. To get our generalization, we prove that such an order exists even in the case of reducible points. Following proposition 3, these points are generated by adjoining to irreducible critical points q of lower order, zeros from v/q. \u show that for a fixed order, the number of such zeros is bounded from above. Let I,~ be the subset of C, containing irreducible critical points of order n and let q in I , . We denote by Z(v/q), the number or zeros of the quotient u/q in the disk U. Then Z(v/q) is finite. Indeed, with the assumption made on f , the quotient v is a.nalytic on the open set U.\ which conta.ins the compact disk C. If Z(v/q) is not finite, v vanishes in U which means that the function f to a.pproximate is already a rational fraction, but we discarded this case ill the introduction. Let us set one more notation:

R. = max{Z(,,/q),q ~ z.}, then R , is finite. It is obvious when I , itself is finite. Otherwise, let suppose that R , is not finite, then we can select a sequence of critical points (qt) in In whose corresponding

483 quotients (vl) have a n u m b e r of zeros growing to infinity. From this sequence, we can extract as before a subsequence which tends to zero. But this means that there is a. sequence of critical points of order n which converges to the function f and then f is again rational. Indeed, we have

gqt = vzql + Lf(ql).

(6)

T h e functions vt converge uniformly to zero on ~" and the polynomial.~s ql and (7~ are also b o u n d e d on/_7 as their degree and their coefficients are. T h e n by (6), Lf(q~) is bounded. Vve can successively extract two subsequences such that Li(qt ) and ql will converge respectively to some polynomials p and q, uniformly on U. By taking the limit, the equation (6) becomes gq = p on U and thus f is equal to/3/(~. As a conclusion, at order n + R~, quotients v corresponding to irreducible critical points as well as reducible ones which come fl'om irreducible points of order n have all a.t least n zeros. At order maxp=0

or

gi E {0 .... , n -

1},< f

L(q__i),.o~7(L(q)) . > . < f _. L(q), ziL(q) > = 0 . q q q q=

As L_~ is the orthogonal projection of f on tile space l,~, we know that q Vk E { O , . . . , n - 1 } , < f

L(q) z k --, q

q

>=0,

(7)

so the last equality reduces to Vi E { O , . . . , n - 1 } , < f

L(q) -iL(q) - -q, - q2 > = O.

(s)

484

Combining (7) and (8), we get q- - < f -

q

>

>

J~q/

~

>

and using (9), we gct

2 < 7' q

q

Oqi

> "

But

=

[Cl2Z'~-i _ 2(lqz i] =

2q~13,

so by (9) again, we obtain 05

OqiOqj < f -

_L _( q ) , f _ q

L(q) > = 2 [ < - -v,, - -vj> + < f q q q

L(q),9(tzivj > J . q ~ q3

Now, the variation of the criterion in a neighbourhood of the critical point q following a direction given by tile real vector (.\1, ..., An) in the space 7~ is

;~ As the family of polynomials (ui) is independent, we choose the numbers ,\i such that

486

The value of (})Aq(,\I,..., ,\,) becomes

[ ui u,_>+

As the order of q growths, f _ L_~ tends to zero, then the variation following the chosen II q Ii . . . . direction becomes positive which means that the critlcat pomt q ma.y not be a maxnnum. t h e o r e m 2 Let f , be a function as in theorem 1. maxima can only appear for a finite range of orders.

Then, critical points which are local

References [1]

L. Baratchart, Sur l'approximation rationnelle l a pour les syst~rnes dynamiques lin4aires, Th~se de doctorat d'dtat, Universitd de Nice, September 19S7.

[2]

L. Baratchart and M. Olivi, Index of critical points in /=-approximation, Systems & Control Letters 10 (19SS), 167-174.

[3]

L. Baratchart and M. O!ivi, New tools in rational L 2 approximation, proc. of tile 8th IFAC Syn~positun on Identification and System Parameter Estimation, Vol. 2, Beijing 198S.

[4] J. Delia Dora, Contribution ~ l'approximation de fonctions de la variable complexe au seas de Hermite Paddet de Hardy, th~se d'dta.t Univ. Sc. et Mdd. de Grenoble, 1980. [5]

R.C. Gunning and H. Rossi, Analytic functions of several complex variables, PrenticeHall, Englewood Cliffs, N J, 1965.

[6]

B. Wahlberg, On system Identification and Model Reduction, rep. LiTHISY-I-0847, LinkSping University, 1987.

[7]

J.L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, AMS colloq. Pub. XX, 1969.

A M i n - M a x T e s t i n g A p p r o a c h to Failure D e t e c t i o n and I d e n t i f i c a t i o n Elias Wahnon IRISA/Centre IINRIA de Rennes 35042 Rennes Cedex, France

Abstract The problem of failure detection and identification in stochastic linear timeinvariant systems is approached as a Min-Max Hypothesis Testing problem. A statistical test is proposed and a Min-Max testing technique is followed, leading to a linear transformation of the residual vector, which results in a new statistical decoupling of failure influences. The resulting problem is to detect a change of the mean in a gaussian vector sequence and to estimate the random time of failure occurrence.

1

Introduction

Detection and identification of actuator and sensor failures is an important problem in engineering systems which rely on sophisticated control systems to achieve high Performances. The methods to deal with the Failure Detection and Identification (FDI) problem are generally classified in two main categories : 1. Redundancy Methods : the simplest way is the hardware redundancy, i.e., several identical measurement and/or control units, such as the quadriplexed digital flight control system, where the failure decision is generally made by a majority voting scheme. However, this passive m e t h o d increases the system cost and complexity. An active way is to exploit the analytical redundancy or parity checks, i.e., temporal relationships between the actuator inputs and sensor outputs which are, for perfectly known deterministic systems, zero when the system is operating normally and deviate from zero when a failure occurs, (see [7]). 2. Filtering Methods : known also as Detection Filters and originally proposed by Beard[2], this methodology is based on the design of a full-order observer where the gMn is chosen so that the direction of the observer output error vector in the output space, allows to identify the failed component. White and Speyer[ll] reformulated this theory as an eigensystem assignment problem, developing the required linear implicit equations for the observer gain and the solvability conditions. Massoumnia

488

et al. [8], relying on geometric concepts and exploiting the dual relationship between the FDI and the Control Decoupling problems, presented a method for designing a dccoupling filter. Viswanadham et ai.[10] proposed a frequency d~,main approach to develop generalized parity space techniques based on the theory of stable factorizations and showed the equivalence of this generalized parity vector approach with the concept of detection filters. Essentially, the so far cited references concern the problem of FDI for deterministic linear-time-invariant (LTI) systems. However, as was recently pointed out by Kerr[5], the original motivation was the problem of failure detection in navigation applications which are described by stochastic LTI models. In this paper, the FDI problem for stochastic LTI systems is approached as a Min-Max Hypothesis Testing problem. A 'detection filter', i.e., a linear filter, based on the no-failure hypothesis H0, where the gain matrix is chosen so that, upon the failed system hypothesis Hi, the mean of the filter residual vector propagates along distinct but fixed directions for each failure, is implemented. A statistical test is proposed and a Min-Max hypothesis testing technique is followed (see [9], [6], where this technique was proposed for the detection of changes in the AR part of a multivariable ARMA process). This technique leads to a linear transformation of the original residual vector which results in a statistical decoupling of failure influences. The problem, finally being a problem of detecting a change of the mean in a non-white gaussian random vector sequence and the estimation of the random time of occurrence, simple well-known detectors can be used. Two methods are also discussed for the special case where the number of failures is higher than the dimension of the measurement vector. In Section 2, the FDI problem for stochastic LTI systems is formulated. The Min-Max hypothesis testing approach is developed in Section 3. Extension to the special case of more failures than measurements is presented in Section 4. Simulation results of a working example are discussed in Section 5. 2

Problem

Formulation

Consider the normally operating system described by the following state space model :

x(k+l)

= Ax(k)+Bu(k)+w(k) y(k) = Cx(k) + v(k)

where x E R" and y E R "~.

w(k)~N(O,Q) v(k) ~ N(0, R)

(1)

489 The failed system is modeled as :

x(k + 1) = Ax(k) + Bu(k) + w(k) + ~ I{k>_n}fimi(k)

i=l

(2)

y(k) = Cx(k) + v(k) where fi 6 R '~, assumed known, is the failure direction associated with the i th actuator failure, rni(k) is a general time-varying unknown scalar which may be a function of x(k) or u(k), 7/ represents the random time of the i-th failure occurrence and IA is the indicator function of the event A. The FDI problem can be stated as the problem of detecting on-line a failure and to isolate, statistically, the influence of the i-th failure from the j-th, j # i, so that an identification of the failed component should be possible. For simplicity of notation, take 7i = T, i = 1 , . . . , r. Assume now that a linear filter, based on the nominal model, Eq. (1), has been implemented where the gain matrix, K, should be later specified :

u(k) = ,.,(k)-C~o(k) ko(h + 1) = A~:o(k) + Bu(k) + IC,(k)

(3)

Let H0 be the hypothesis which states that the system is described by the nominal model, Eq. (1), and H1 the hypothesis that the system is described by the Eq. (2). Under Ho, u(k) is a zero mean non-white gaussian sequence with variance matrix, V(k), given by :

v(k) = cP(k)c T + R P(k + 1) = (A - I(C)P(k)(A - KC) T

(4)

+ KRK T + Q and assume that the steady state value of V(k) has been attained, denoted here by V. Under HI, decompose u(k) into : ~(k)

=

y(k) - c ~ 0 ( k )

=

v(k) - c ~ ( k )

+ c [ ~ , ( k ) - ~o(k)V{k>,}

~ ( k ) + ~(k)hk_>, } where ux(k) ,,, N(O, V) and/~(k) is given

by:

o~(k -I- 1) -- (A - KC)o~(k) -F f, ml(k) ,(k) = Ca(k) ; ~(k_~,)e~,(k)

where ei is the i-th m 9 1 unit vector, leads to the following equation for ~he mean variation :

o,(I~ + 1) = (A - Z(C)~(k) - K~i6,(k) The presence of the i(ei as a failure direction is a problem since K is not known. However, as pointed out by White and Speyer[11], taking gi being any direction such that e{ : Cg~, this equation can be written as Eq. (5) and #(k) can be made to lie in the plane generated by CKel and ei. Therefore, we use the actuator failure case, Eq. (1) as the general problem. The F D I problem can be formulated as the following hypothesis testing problem :

U,~de,'Ho: ~,(k) ~ iV(O, V) UnderHl : 3% k < T : V ( k ) ~ N ( O , V ) k >_ ~: . ( k ) ~ N ( , ( ~ ) , V) 3

The

Min-NIax

Testing

Approach

The statistic :

(G)

to 2 .r(k)v-l.(~) allows to test beLween H0 and Hi. Furthermore, Eo(to) =

m

El(to) = I m

m +

,~(k)v-b(k)

kT

where Ei(') is the expectation under Hypothesis Hi. Now, assume that there exists a gain matrix K such that the failures associated with the fl, i = 1 , . . . , r, in the system described by the mean Eq. (5), are dctcctablcs according to the definition given by Beard [2], i.e., #(k) maintain a fixed direction in the output space due to fi and all eigenvalues of A - K C can be arbitrarily specified, except for the constraint on the conjugate symmetry. This gain matrix ensures the decoupling of the failure set, { f l , . . - , fr}, influences in tlle output space and can bc computed, for cxamplc, by the method presented in

491

[11]. Therefore, assuming that a such gain matrix has been found, we have that for each fi,i = 1 , . . . , r :

rank[Cfi, C(A - [(C)fi, C(A - KC)2fi,...] = 1 and

assume

(7)

also :

rank[CfI,Cf2,...,Cf~] = r, r _ r) , given by :

Under the condition in Eq. (7), #(k) maintain a fixed direction, Cf~, in the output space due to {mi(1);l = k - 1, h - 2,...,7"},i = 1 , . . . , r . Therefore,

(adl,. .,Cf,mx){[k 1,] + +

i A21rnl(k - 3)

;~2~mr(k 3)

i

+ ....

2,1

(Cft,Cf2,...

Cf~)

I

3.42

and 7, Eq. (10), can be formally writen as : =

....

We look now to isolate, statistmally, the different failure influences by building, for each failure, a test which should be insensitive to the other failures. Take, for example, the first failure fl and partition the matrix )c =zx [ 3S cTl i - T t2] ,whereSll F22 is a scalar and ~c22 is a ( r - 1) * ( r - 1) matrix. We shall follow a Robust Min-Max hypothesis testing technique (see [9], [61), by considering for the failure f;, the least favorable case for the remaining failures ( f 2 , - . . , f.)- For a fixed test leve! c~, given by : PHo(to > A) < a', where, A is the threshold fixed by c~, the power of the X2-test to, Ptil(to > A) is an increasing function of the non-centrality parameter 7, Eq. (10). By the Min-Max approach we are looldng to find :

min

(~,-..,~,)

7 =

min

(~,...,~r)

[.All, (.s

, .s

[-7-11 F12 ] ] .T-r .T.22

3,41 A'42 :

3~I,

492

The minimum, obtained for : (.M2,...,.M~) r = --~-~19rT.A41, is given b y : min 3' = 3Al(Scm - ~129r~)'Su~)-A41 (~4~,...,M,)

-~175axl ] = 3/li{(1,-.7-i2.7-~i)(cfl,..., C f ~ ) r v - i } v {y-l(c

f l , . . .,

cA)(1,-f~27~)r}ax~

Tlicrefore, considering the following transformation of the original residual vector sequence, u( k ), : ~,,(k) = ( 1 , - 7 1 2 : r ~ ) ( c f ~ , . . . ,

cA)rv-l,(k)

(11)

we have that ul(k) is a non-white gaussian scalar sequence, zero mean under H0 and witll a non-zero mean under Hi (for k > r) given by :

E(vt(k))

= (1,-.7"i2.Y~i)(cfi,...,Cf,.)Tv-1/.z(k)

= (l'-212"~)(CI~'""C/~)rW-~(C/~'""C/~) =

(~,-j:,~:#)j:

J(4.

~

and, V a r ( vl ( k ) ) = .T m - .T12.T~I.T'f~2. i.e., ul(k) is only sensitive to the failure fl and is not affected by the remaining

failures (f2,.-., A)Finally, returning to the original problem stated in Eq. (2), given the failure set {f~,.-., fr} with the unknown random times of occurrence {T1,..., Tr}, we build for eacli failure, f/, the required transformation, ui(k), of the residual sequence u(k) which is essentially a statistical decoupling of failure influences. The problem being now of detecting a change of the mean and the estimation of the r a n d o m time of occurrence, simple detectors can be proposed. In particular, the well-k/iown Hinldey detector can be useful, implementing for each ui(k) two sucti detectors, to detect an increase or a decrease in the mean of ui(k). The Hinldey detector, introduced in [4] and analysed in [1] is given (for example, to detect an increase

493

in the mean of/]i(k)) by the following recursive procedure : TO

~

0.

~ffk

:

T k _ 1 "-~ Y i ( ] r

Mk

=

Alarm

5

- - ry

rain Tj

(12)

l<j

where A is the threshold and 5, fixed apriori, is the minimum jump to be detected. Remarks: 1.) The Min-Max technique leads to a linear transformation of the residual vector v(k) which hides, for each failure, the subspace spanned by the remaining failures in the output space. Consequently, the existence of a gain matrix satisfying the condition in Eq. (7) is required. Otherwise, different failures m a y span the same subspaces and no hiding should be possible. 2.) Even though the Hinkley's detector was developed to detect a change of the mean in a white gaussian sequence, the experience shows that this detector is robust with respect to the whiteness assumption. 4

More

Failures

than

Measurements

In the special case, r > m, i.e., more failures than measurements, the assumption, Eq. (8), cannot be satisfied and some modification are required for the identification of the failed components. Two methods are here discussed : 1. Partial Filters : Take, for simphcity, rn = 2 and r = 3, i.e., (fl, f2, f~). Then, if we assume that the probability of simultaneous failures is very low (i.e., zero), then by building two filters, according to Section 3, the first for f; and f2, and the sccond for fa and, for example, ft, we obtain, by adding an heuristic reasonning, a complete failure identification system. However, if simultaneous failures are assumed, we are not generally able to identify the failed components, except for the case where the mean jump size m a y be used to make the failure decision. 2. Clustering : Define the following distance measure between fi and fj (see an analogous definition in [3]) :

(Cfl)T(cfJ) d(fi, fj) = l - pii = l - { Cfi ll C f j l

(13)

and make the required clustering, i.e., replace by a single failure vector fi the failure vectors fj which are close in the sense of the distance measure, Eq. (13), such that we stay with the required number of failure vectors. Note that pij is nothing but the cosine angle between the images of fi and fj in the measurement space. Let us r c m ~ k that with the clustering method, some failed components m a y not be

494

well identified since the failed component direction in the output space has a main projection in the clustered direction but also a smalt component in other failure directions.

5

Simulation Results

Given an stochastic LTI system described by the Eq. (1), where : A=

034 123

-3

; /3 = [fl, f2] ; fl ----

0 2 5

1 0

=

11 ] -i/2 i/2

1

C=

[0101 0 0 !

; Q = G G T;

G=

1/2

t/3

0

.05

the condition in Eq. (8), rank [Cfi, C f2] = 2 = dim y, is satisfied and the gain matrix K, calculated by the method presented in [11], which ensures the condition 25/8 91/24 in Eq. (7) is given by : f ( = 5/4 53/12 . Furthermore, the closed-loop matrix 2 29/6 A - I ( C is stable and its eigenvalues are freely chosen ( here, Ii = 1/2, A2 = 1/4, and Aa = 1/6). The steady-state variance matrix, V, has been computed according to Eq. (4) and the required transformation of the residual vector is given, according to Eq. (11), b y :

where z,r(k) isolates the influence of {ml(j))~=r and u.~(k) the influence of {m:(j))~: T. Since the system matrix A is not stable a feedback control has been used : r .4 l.S9 4 . a 7 ] u(k) : - D ( x ( k ) - Xre/), where D = ] j and ~.T : (1, 1, 1). 1.27 8.52 17.33 "~ef L Fig. 1 presents the simulation results for the case where at the instant 300 the second actuator faiied (a constant bias failure of 10). For the nominal residual vector, both elements of this vector jump at the failure instant while for the transformed residuals, only u~(k) jumps while u~(k) rests insensitive to the failure f2Therefore, the transformed residual sequence performs a statistical decoupling of failure influences, where u~(k) is only sensitive to the failure fl and u~(k) to the failure f2-

495

6

Conclusion

A Min-Max hypothesis testing approach for failure detection and identification in stochastic LTI systems has bcen presented. A statistical test has been proposed and a Min-Mmn approach has been developed. This approach leads to a linear transformation of the original residual vector which results in a statistical decoupling of failure influences. Simulation results of a working example have also been presented. The problem of failure isolation by the Min-Max approach, when the optimal gain matrix is used in the filter which generates the innovation sequence, is actually investigated.

Acknowledgment I would like to thank Dr. A. Bcnveniste and Dr. M. Basseville from IRISA/INRIA, Rennes, France, for the fruitful discussions.

References [1] M. Basseville, 'Edge detecticn using sequential methods for change in level Part II : Sequential detection of change in the mean', IEEE Trans. Acoust. Speech Sig. Process., vol ASSP-29, 1981, pp. 32-50. [2] R.V. Beard, 'Failure accommodation in linear systems through self reorganization', Mass. Inst. Technol., Cambridge MA, Rep. MVT-7I-1, Feb. 1971. [3] A. Benveniste, M. Basseville and G.V. Moustalddes, 'The asymptotic local approach to change detection and model validation', IEEE Trans. Automat. Control, vol AC-32, 1987, pp. 583-592. [4] D.V, Hinkley, 'Inference about the change-point from cummulative sum tests', Biometrika, voI 58, 1971, pp. 509-523. [5] T.H. Kerr, 'A critique of several failure detection approaches for navigation systems', IEEE Trans. Automat. Control, vol AC-34, July 1989, pp. 791-792. [6] G. Le Vey, 'Analyse modale et surveillance vibratoire des machines tournantes', Thhse de l'Univ. Rennes 1, 1988. [7] X.C. Lou, A.S. Willsky and G.C. Verghesse, 'Optimally robust redundancy relations for failure detection in uncertain systems', Automatica, vol 22, 1986, pp. 333-344.

496

[8] M.A. Massoumnia, G.C. Verghesse and A.S. Willsky, 'Failure detection and identification', IEEE Trans. Automat. Control, vol AC-34, March 1989, pp. 316-321. [9] A. Roug6e, M. Basseville, A. Benveniste and G.V. Moustakides, 'Optimum robust detection of changes in the AR part of a multivariable ARMA process', IEEE Trans. Automat. Control, vol AC-32, 1987, pp. 1116-1120. [10] N. Viswanadham, J.H. Taylor and E.C. Luce, 'A frequency-domain approach to failure detection and isolation with application to GE-21 turbine engine control systems', Control Theory and Advanced Technology, vol 3, MITA Press, March 1987, pp. 45-72. [11] J.E. White and J.L. Speyer, 'Detection filter design : Spectral theory and algorithms', IEEE Trans. Automat. Control, vol AC-32, July 1987, pp. 593603.

1

0.5 0 -5 -0.5 -I0

0

200

400

i

i

15

600

-1

0

*,,(k) : N.mi.al II~idual

200

400

600

*.~(k) : 'l'r,.~h.rm.M Ih:~id..I

10 5 0

0

-5 -10

9

0

a

t

200

400

_~

600

0

l

a

200

400

600

Fig 1 Time histories of Nominal and Transformed Residuals for the case of a Constant Bias Failure of the Second Actuator.

I

ON-LINE DETECTION OF MINIMAL ORDER FOR LINEAR PIECEWISE I STATIONARY SYSTEMS

t

M. GUGLIELMI Laboratoire d'Automatique de NANTES U.R.A. 823 Ecole Nationale Sup6rieure de M6canique 1 rue de la No~ 44072 NANTES Cedex 03 Membre du GRECO Traitement du Signal et des Images Abstract: This work deals with the detection of minimal order for linear systems with piecewise constant coefficients. In this case, it is impossible to use off-line algorithms. We propose, here, a real-time method able to detect the minimal order of a linear system. The basic mathematic tool is given by a simple divisibility condition of the charateristic polynomial of the differential equation for stationary systems. This necessary and sufficient condition is used to define an algorithm able to give the reduced equation. Then, we show that this result can be directly extended to linear piecewise stationary systems for which we search a change if the order. The real-time method that we obtain, is checked in simulation on deterministic systems and on systems with measured noisy informations. I- INTRODUCTION: The knowledge of the minimal order for dynamical systems is a very important information. In control systems (where supplementary modes may be a drawback) as in the pattern recognition problem in the aim of classification (diagnostic...), it is fundamental to describe the linear observed system by a lowest order differential equation. For stationary system, it exists well-known algorithms able to get the exact minimal order. In the determistic case, the principle of these algorithms, like the HO's one, are based on the calculation of the rank of the HANKEL matrix [1] build with the input/output of the system. In the stochastic environment, the order is also obtained by using algorithms which compute the rank of matrix build with the variance/covariance informations [2]. The mimal dimension of the system can also be directly obtained by using the AKAIKE's criterion [3]. Finally, some parametric identification methods can be implemented with a on-line test to check the process order (BURG-LEVINSON [3]). The most important drawback of these methods is that they need a large sample of signal, so they are unable to work on real time. In many cases, it is very interesting to have a mean able to get the order information of a dynamical system from real time input/output information, specially if we turn our attention on systems for which the

498

order can decrease. The reducibility of a differential equation can be studied by the means of the one-parameter groups [4]. But this approach is not quite easy to use in an algorithm based on input/output informations. Our paper deals with the definition of areai time algorithm able to detect a change in the order for piecewise stationary linear systems. Its organization is: after a simple example, in the paragraph 1I, we give the necessary and sufficient condition which must be satisfied by a nth linear stationary system to be represented by a n-1 th differential equation. This condition is a relation concerning the initial values, which uses the quotient of the division of the characteristic polynomial of the differential equation by its factors. From this condition, we obviously get a generalization (paragraph III) for a preducibility greatest than one. In the IV paragraph, we propose a algorithm able to detect on-line a change in the order when the linear system has piecewise stationary coefficients. Two examples are given in paragraph V. I[: N E C E S S A R Y and S U F F I C I E N T R E D U C I B I L I T y C O N D I T I O N : II-1: A simole example: Consider the following second order linear stationary system: y(2)(t) + aly(1)(t ) + a 0 y(0)(t) = 0 diy(t) . dt I Among all the trajectories obtained in the phase plane @~ ) when the initial conditions y(0)(0) and y(1)(0) are varying, two of them are straight lines. In this case, the trajectories can be represented by a first order differential equation. For example, the figure I shows that for the system described by: y(2)(t) + 3 y(1)(t) + 2 y(0)(t) = 0 In the paper, we note: y(i)(L) =

.............................. p 09, -2.00

-1.50

-1.00

Notice that the two straight line trajectories are the solutions of the 2 nd order differential equation: y(2)(t) + a!y(D(t ) + a 0 y(0)(t) = 0

-0.50

- 1 . 0 0 -"

(only if the initial conditions verify y(1)(0) + 2 y(0)(0) = 0 case [~ or y(1)(0) + y(0)(0) = 0 case [~]). Here, we

- 2 . 0 0 "3

~3.00 -~ [rojectolce de :~ + AY + BY = 0 (A=3, B=2) en fonctlon des conditions inltio}es

FIGURE 1

propose a method which detects directly, in the general n th order case, the presence and which deletes the supplementary useless dynamics in the differential equation.

499

II-2: Reducibility of 1:

Let's have the linear homogeneous system (Y) described by the linear differential equation with constant parameters: y(n)(t)+ aly(n-1)(t)+ a2y(n'2)(t) + ......... + any(0)(t) = 0

(eq. tI-1)

With the n initial conditions y(n-1)(0), y(n-2)(0) ..... y(0)(0) We suppose that y(t) belongs to C ~176 diy(t) We note: y(i)(t) - dt i -T :~ (t) = [ xl(t ), x2(t) ......... Xn_l(t)] T = (t) = [ Xl(t), x2(t) ......... Xn_!(t), Xn(t) ] P()~) is the characteristic polynomial of the differential equation II-1 When it will be necessary and when no ambiguity will be possible, we shall use equally, in the paper,P@) like a differential operator or like a polynomial in of X.When P(X) is the fifferential operator, we note: P(X) o x(t)= x(n)(t)+ alx(n-1)(t)+ a2x(n-2)(t) + ......... + anX(0)(t) II-2-1:Theorem: The trajectory described by the equation (II-1) can be obtained by a n-1 th order linear differential equation if and only if it exists a set of n-1 coefficients {a i ) such as if the polynomial: Q(X) = ) n - l + o~ ~n-2+ c~2 )~n-3 § ....... + O~n.1 1 we have: y(n'l)(0)

+ Cr

divides

) + cc2y(n'3)(0) + ....... + c~n_ l y(0)(0) = 0

In other words:

y(n'l)(0) = -

where

y T ( 0 ) = [y(n'2)(0), y(n-3)(0)...y(0)(0)]

P()~)

(eq ii-2)

~Ty(0)

Demonstration: 1 ~ Necessary_ condition: Suppose that y(t) can be represented by the n-lth order differential equation: y(n-1)(t)+ C~l y(n'2)(t)+ (z2y(n-3)(t) + ........ Then, the characteristic polynomial:

+ Ctn_ly(0)(t) = 0

(eq. II-3)

5OO

Q(X) = xn-l+ oc1 ~n-2+ a2 )~n-3 + ....... + an_ 1 divides P(X). From the division of P(~,) by Q(X) which gives: P(X) = (7~ - ~0) Q 0 0 + K

where K is a constant.

we have: P()0 o y(t)= [(X- X0 ) Q(L) + K] o y(t) Thanks to the properties of the differential operator, we get: 0 = [ (X- %0) Q(X) ] o y(t) + K o y(t) 0 = (7~ - 7~0) o[Q(X) o y(t) ] + K o y(t) Thus:

0= 0 + K o y(t) = K y(t) Finally K=0 so

Q(X) divise P(X)

In the other hand, the condition II-2 is obviously satisfied: Q.E.D. 2 ~) Sufficient condition: Let's have the differential equation with the characteristic polynomial Q(X). By hypothesis, we suppose that: Q(~) = ~n-l+ o~1 ~n-2+ cc2 ~n-3 + ....... + ~n-1 divides P(X) If w(t) is a trajectory defined by: w(n-1)(t)* a 1 w(n-2)(t) + ........ +con_ lW(0)(t) = 0

(eq II-5)

with: w(i)(0)= y(i)(0) for i varying from 0 to n-2. We define e(t) by:

e(t) = y(t) - w(t).

From the expression: e(n)(t)+ a l e ( n - 1 ) ( t ) § a 2 e ( n - 2 ) ( t ) § .........

+ ane(O)(t)

We can easily show (Annex 1) that e(t) satisfies: i=n i=n-1 e(n)(t) + ~ ai e(n-i)(t) = - anW(0)(t ) + ~ (~ - ai)w(n-i)(t ) i=1 i=1 The second member of this equation is equal to 0 due to the fact that the characteristic polynomial of this part is:(7~ Q(X) - P(X)). Using the hypothesis, we have:

P()0 = (7~ - X0)Q(X).

Thus: (Q(X) - P(X)) ow(t) = {9~ - (9~ - 9~0) 1o [Q(k) o w(t)] = ~0 [Q(X) o w(t)] = 0 So e(t) -- 0

t , if and only if all the initial conditions are equal to zero. For

i from 1 to n-2, it is satisfied by definition of e(t).

501

To get e(n-l)(0) =0 we have: e(n-1)(0) = y(n-1)(0) - w(n-1)(0) y(n-1)(0) = w(n-1)(0 ) = . {o~1 w(n-2)(0) + ........ +t:Zn_lW(0)(0)} = - {(z1 y(n-2)(0)+ ........ +an_ly(0)(0)} = -

-T ~ Y(0)

So, the trajectory w(t) is the same that y(t) if the above condition is satisfied and we can say that: y(t) has a model described by a n-1 o r d e r d i f f e r e n t i a l e q u a t i o n Q.E.D.

II-3: Reducibility of order >1; The above result can be enlarged: 11-3-1 :Theorem: The following linear differential homogeneous system: y(n)(t)+ aly(n-1)(t)+ a2y(n-2)(t ) + ......... + any(0)(t) = 0 can be represented by a linear differential homogeneous equation of order 1 (l O, k E f_fl(to,T) such that for almost all t E [to,T] the map F(t,.) is k(t) - Lipschitzian on y(t) + f i b Ha) The function t~-.~ dist(g(t),F(t,y(t))) belongs to s )

From [2, Theorem 8.2.8 and Corollary 8.2.13] we deduce that the function t ~ dist(g(t), F(t, y(t))) is measurable whenever assumptions H1), II~) are satisfied. Theorem

=

1.2 ([15]) Let 6 > 0, M = sup~e[o,T_to ]HG(t)[I. Assume that l i t ) - II3) hold true and set

re(t):

(. Z

,(t)= re(t)(, + Z (s)ds)

If , ( T ) < fl, then for all xo E X with I[yo-xo[I O, there exist x e S[to.T](X0) and f Es T; X ) satisfying (g), (10) such that for all t E [to, T] fix(t)

-

y(t)l I
0 such that for all 71 near y(to) we have

d i s t c x r ((y,g), { ( x , f ) is a trajectory-selection pair o[(8) on [to,T] ,x(to) = ~}) _< LILT]- Y(to)]l We compare next trajectories of (8) and of the convexifted (relaxed) differential inclusion:

z'(t) e Ax(t) + ~'~ F ( t , z ( t ) )

(12)

We say that a set-valued map U : [to,T] "-~ X has an integrable selection if there e ~ s t s an integrable single-valued map [to,T] ~ t ~ u(t) E U(t). T h e o r e m 1.4 Let (y,g) be a trajectory-selection pair of the relaxed inclusion (12) on [to,T]. Assume that F and y satisfy all the assumptions of Theorem 1.2 and that F(.,y(.)) has an integrable selection. Let ~](.) be defined as in Theorem 1.2. If rl(T ) < fl then for every s > 0 there exists a mild trajectory x of (8) on [to,T] satisfying x(t0) = y(to) and Itx - Y][c O, k E s and for almost every t E [to,T], F(t,.) is k(t)-nipschitz on xo + 5B. Then for every u e eaF(to,xo) dist ( a ( h > o + hu, ~(to + h, to)xo) = o(h) whcre limh~o+ o(h)/h = O. Fix u e ~ F(to, x0) and set y(to + h) = C(h)zo + f,~+t, a ( t o + h Set r = sup,e[O,h] dist(u, ~6F(to + s, y(to + s))). Then, by continuity of F , Theorem 1.2 yield that for some C > 0 and every small h > 0, there exists ]ly(to + h) - xh(to + hll < Cr Applying Theorem 1.4 we end the proof.

Proof--

s)uds = a ( h ) x o + l~u + o(h). limb-o+ ~(h) = 0. This and co X xh E $[to,to+h](o) such that El

C o r o l l a r y 2.2 Under all assumptions of Theorem 2.1 ~6 -/;'(to, Xo) C lim inf /~(to + h, to)xo - G(h)xo h~o+ h

Consequently, if xo E Dora A, then Axo + e-O F(to,xo) C liminf R(to + h, t o ) x o - xo h~o+ h We introduce next an analog of the variational equation for differentia] inclusions. For this we need to extend the notion of derivative to set-valued maps. D e f i n i t i o n 2.3 Let ~ be a set-valued map from a Banach space X to another Y and lel y E .T(x). d e r i v a t i v e d ~ ( x , y) is the set-valued map from X to Y defined by

v e dT(x,y)(u)

~=.

hn~+_ d ( "~,g ( x + h u h )h- - Y )

= 0 for some uh -+ U

When .T is locally Lipsehitz at x then the above definition may be rewritten as v e d.T(x,y)(u)

~

lim dist h~O+

v,

= 0

The

524 We refer to [2] for properties and to [8] - [12] for the applications of set-valued derivatives in the finite dimensional context. Below dF(t; x,y) denotes the derivative of the set-valued map F(t,., .), i.e. its partial derivative with respect to the state variable. Let (y,g) be a trajectory-selection pair of the differential inclusion (8) defined on the time interval [t0,T]. We "linearize" (8) along (y,g) replacing it by the " v a r i a t i o n a l i n c l u s i o n " :

w'(t) E Aw(t) + dF(t;y(t),g(t))(w(t)), w(to) = u Consider the solution map Sc,z from X to the space C(to,T;X) • s

(15)

T ; X ) defined by

5cx(~) = { ( z , / ) i s a trajectory-selection pair of (8) on [t0,T], z(t0) = ~ } The following result was proved in [15]. T h e o r e m 2.4 ( V a r i a t i o n a l i n c l u s i o n ) If F and y verify assumptions HI) - It3), then every trajectoryselection pair (w,~r) of the linearized inclusion (15) on [t0,T] satisfies (w, ~) E dSc,L(y(to),(y, g))(u). A stronger result may be proved when the map t ~ F(t,y(t)) contains an integrable selection. Considcr the "convex" linearization of (8) along (y,g):

w'(t) E Aw(t) + d~-6 F(t;y(t),g(t))(w(t)), w(to) = u

(16)

In the theorem stated below we consider the solution map Sc(() = 3[t0.Tl(~) as the set-valued map frmn X to the space C(to, T; X). T h e o r e m 2.5 Under all assumptions of Theorem 1.2 assume that F(.,y(.)) has an integrable selection. Then every mild trajectory w of the lincarized inclusion (16) defined on [to, T] satisfies w E d Sc(Y( to), y)(u). Proof--

By Theorem 1.4 we may replace F by ~ F . Then the result follows from Theorem 2.4.

[]

The derivative of the set-valued map ~ F(t,x) has the following useful property: If F(t,.) is locally Lipsctfitz on a neighborhood of x, then for every y E F(t,x)

dF(t;x,y) + T~F(t,~)(Y) C d-~'hF(t;x,y)

(17)

where T-~f(t,~)(y ) denotes the tangent cone of convex analysis to "~F(t, x) at y. See [15] or [2] for the proof. We provide next two comparison theorems for trajectories of the control system (2) and the differential inclusion (1). Let Z be a complete separable metric space, X be a separable Baaach space and f : 1Z+ x X • Z ~ X be such that for all (x, u) E X • Z the map f(-, x, u) is measurable and for every t E 1%+, f(t,., .) is continuous. Consider a set-valued map U : It+ x X .-~ Z with closed nonempty images and define the set-valued nlap

F : IZ+ • X...* X by V ( t , x ) ~ It+

x

X, F(t,z) = I ( t , x , U ( t , z ) )

T h e o r e m 2.6 If U is measurable with respect to t and continuous with respect to x, then solutions of (1)

and (2) do coincide. Proof-It is enough to show that for every trajectory-selection pair (x,v) of the differential inclusion (1) defined on [t0,T] there exists a measurable selection u(t) E U(t,x(t)) such that for almost every t E [t0,T], v(t) = f(t,x(t),u(t)). By [2, Theorem 8.2.8] the map t ",-* U(t,x(t)) is measurable and for every u E Z the single-valued map f(.,x(.),u) is measurable. Applying [2, Theorem 8.2.9] we end the proof. [] We observe that in general the images of the map F defined above are not closed, while the calculus of differential inclusions developed in previous sections deals only with closed-valued maps. For this reason we provide one more comparison theorem. Denote by s the set of maps f : It+ ~ X such that for

all 0 O

suchthat Vhe[O,e], z+hB~(w) CK}

respectively. See [2, Chapter 4] We apply results obtained in the previous sections to derive necessary conditions for optimality for three optimal control problems. Consider a continuous, Gs differentiable function ~p : X • X ~ R. aald closed subsets K0, K1 C X. For every T > 0 set UT = {u : [0,T] ~ Z I u(t) e U(t) is measurable}.

526 3.1

Problem

with fixed end time and free end point

Consider T > 0 and the optimal control problem minimize qo(x(0), x(T))

(18)

over mild solutions of the semilinear control system

z'(t) = Am(t) + f(t,x(t),u(t)), u E LtT, z(O) E I(o

(19)

Let (z,~) be a trajectory-control pair of (19). We associate with it the linear equation

z'(t) = AZ(t) + ~(t,z(O,~(t))z(t)

(20)

and denote by S~(t; s) the solution operator of (20). That is the only strongly continuous solution of the operator equation

VO<s a.e. in [0,T] ,~eU(t)

(22)

and the transversality condition: p(O) e ff'~xl(Z(O),z(T)) + TKo(z(O))-

(23)

where Tlco(z(O))- denotes the negative polar cone to the contingent cone Ti(o(z(0)). ProofDefine the set-valued map F : [0,T] x X -,~ X by F(t,x) = f(t, x, U(t)). From [2, Theorem 8.2.8] follows that F is measurable with respect to t. By the assumption a) it is also k(t)-Lipschitz with respect to x. Consider the differential inclusion

x'(t) E Ax(t) + F(t,x(t)), z(O) E Ko

(24)

By Theorem 2.7 and continuity of ~v, the trajectory-selection pair (z(.),f(.,z(.),~(.))) is optimal for the problem (18), (24). Thus we may replace the control system (19) by the differential inclusion (24). Consider the linear control system w'(t)

=

Aw(t)+ ~

y(t)

e

Tazi(t,~(t),cr(,))(f(t,z(t),~(t)))

w(O) E TKo(z(0)) (25)

The reachable set/~L(~) of (25) by the mild trajectories from ~ at time T is given by RL(() =

{

S~(T;0)~+

/;

S~(T;s)y(s)ds l Y E s

e T~/(,M,),u(,))(f(s,z(s),g(s))) a.e.

}

From Theorem 2.5 we deduce that every mild trajectory w E C(0, T; X) of the "linear" differential inclusion

w'(t) E Aw(t) + d'd-S F(t;z(t),f(t,z(t),g(t)))(w(t))

527

on [O,T] verifies w E dSc(z(O),z)(w(O)). From the definition of the derivative for almost every t E [0,T]

V w E X , ~x(t,z(t),-ff(t))w E dF(t;z(t),f(t,z(t),~(t)))(w) I[ence, using (17) we deduce that every solution w E C(O,T;X) of the linear control system (25) verifies w E dSc(z(0), z)(w(0)). Since z is optimal, using Corollary 1.3 we deduce that for every trajectory w of (25) we have (U(p(z(0),z(T)),(w(0), w(T)) I >_ O. The last inequality yields that for every integrable selection y(t) E "d'5f(t,z(t), U(t)) and every ~ E T1r we have

Define p(.) by (21). Setting y(t) = f ( t , z ( t ) , g ( t ) ) i n the above we get p ( 0 ) - -~7(z(O),z(T)) E TK0(z(0))-. llenee the transversality condition (23) holds true. On the other hand applying (26) with ~ = 0 we obtain: for every integrable selection y(t) E ~ f(t, z(t), U(t))

fo T (3~(T; t)*p(T), y(t)) dt = < p(t), f(t, z(t),g(t)) > almost everywhere in [0, T]. This yields the ma.ximum principle (22). t::l 3.2

Problem

w i t h free e n d t i m e a n d f r e e e n d p o i n t

Consider the optimal control problem minimize { (p(x(O),x(T))iT > 0 }

(27)

over mild solutions of the semilinear control system (19). T h e o r e m 3.2 Let (z,~) be an optimal trajectory-control pair of the above problem and T denote the cor-

responding optimal time. Then the same conclusions as in Theorem 3.1 are valid. Furthermore if f(.,x, u) and U are continuous at T and z(T) E Dora A, then sup (p(T),f(T,z(T),u))

~ (p(T),-Az(T))

~eu(o P r o o f - - By the proof of Theorem 3.1, z is optimal for the problem (27), (24) and it is enough to prove the last statement. By Corollary 2.2 and optimality of z, for every u E ~5 F(T, z(T)), 0~2 (z(0), z(T))(Az(T)+ u) _> 0. tfence the result. [] 3.3

Problem

w i t h fixed e n d t i m e a n d e n d p o i n t c o n s t r a i n t s

Consider T > 0 and the optimal control problem minimize { V(x(O),x(T)) [ x(T) E Kx }

(28)

over mild solutions of tile semilinear control system (19). T h e o r e m 3.3 Let (z, ~) be an optimal trajectory-control pair and let Q C DKr(z(T)) be a convex cone with nonempty interior and P C T1~'o(z(0)) be a convex cone. Further assume that for every (t,x) E [0, T] x X

the set f ( t , x , U ( t ) ) is closed. Then there exist A > O, ~o ~ P - , ~T E Q- not vanishing simultaneously such that the map pit)=

S~T;t)* (-)~ ff--~x (z(O),z(T)) - ~T)

(29)

satisfies the maximum principle (22} and the transversality condition (p(0), -p(T)) The proof can be found in [15].

=

~V~(z(0),z(T)) + (r

~r)

(30)

528 R,eferences [1] AUBIN J.-P. ~: CELLINA A. (1984) DIFFERENTIALINCLUSIONS.Springer-Verlag, Grundlehren der Math. Wissenschaften, Vol.264 [2] AUBIN J.-P. & FI~ANKOWSKA H. (1990) SET-VALUED ANALYSIS,Birkh~,user, Systems and Control: Foundations and Applications [3] CASTAING C, & VALADIEI~ M. (1977) CONVEX ANALYSIS AND MEASUR.ABLEMULTIFUNCTIONS. Lecture Notes in Mathematics, n ~ 580, Springer Verlag, Berlin [4] CLAR,KE F. (1983) OPTIMIZATIONAND NONSMOOTII ANALYSIS.Wiley Interseience [5] DUNFOR, D N. & SCIIWAI~TZ J.T. (1967) LINEAIL OPERATORSPart I: General theory. Interscience Publishers, Inc., New York [6] FATTORINI II. ~z FR,ANKOWSKA H. (to appear) Necessary conditions for infinite dimensional control problems. Mathematics of Control, Signals, Systems [7] FILIPPOV A.F. (1967) Classical sobttions of differential equations with multivalued right hand side. SIAM J. Control & Optimization, 5, 609-621 [8] FR,ANKOWSKA H. (1987) The maximum principle for an optimal solution to a differential inclusion with end point constraints. SIAM J. Control ~z Optimization, 25, 145-157 [9] FR,ANKOWSKA II. (1987) Local controllability and infinilesimal generators of semi-groups of set-valued maps. SIAM J. Control ~: Optimization, 25,412-432 [10] FI~ANKOWSKA ]I. (1989) Set-valued analysis and some control problems. Proceedings of the International Conference 30 years of Modern Control Theory, Kingston, June 3-6, 1988, E.R,oxln Editor, Marcel Dekker [11] FI~,ANKOWSKA H. (1989) Local controllability of control systems with feedbacks, Journal of Optinfization Theory and Applications, 60, 277-296 [12] FI~ANKOWSKA tI, (1989) Contingent cones to reachable sets of control systems. SIAM J. Control &: Opt. 27, pp. 170-198 [13] FR.ANKOWSKA lI. (1989) Optimal trajcctories associated to a solution of contingent lIamillonJacobi equations. Applied Math. & Optim., 19, pp. 291-311 [14] FII.ANKOWSKA II. (to appear) Some inverse mapping theorems, Ann. Inst. IIenri Poincar~, Analyse Non Lin~aire, [15] FR,ANKOWSKA II. (to appear) A priori estimates for operational differential inclusions, J. Diff. Eqs. [16] HILLE E. & PIIILLIPS R,.S. (1957) FUNCTIONALANALYSISAND SEMI-GKOUPS.American Mathematical Society, Providence, l~hode Island [17] PAZY A. (1978) SEMI-GR,ouPs OF LINEAI~.OPER.ATOItSAND APPLICATIONSTO PARTIALDIFFEItENTIAL EQUATIONS.Springer, Berlin [18] TOLSTONOGOV A.A. (1986) DIFFERENTIALINCLUSIONSIN BANACfl SPACES. Nauka, (in Kussiaa) [19] WAZEWSKI T. (1963) On an optimal control problem. In: Differential Equations and Appfications, Proc. Conf. Prague, 1962 [20] QIJI ZIIU (to appear) On the Solution Sets of Differential Inclusions in Banach Space, J. Diff. Eqs.

Relaxed Controls for Time Delay Systems by l's

B. Vinter

(Department of Electrical Engineering, Imperial College, Exhibition Road, London SW7 2BT, England)

Despite the attention timt optimal control problems involving time delays have received over the last two decades, little attention has been paid to the important question of how such problems should be relaxed in order to assure existence of minimizers. Recently Warga has proposed a relaxation procedure for fully nonlinear problems with delays in the dependent variables and in the controls, and showed that the resulting relaxed problem has a solution. We show, through an example that this relaxation procedure can fail to give a p r o p e r e z t e n s i o n , i.e. the effect of relaxation can be to reduce the infimum cost. Sufficient conditions are given for a proper extension. A new relaxation prot:edure is provided for which the extension is proper in certain situations where the former extension is not,. Abstract:

1. Introduction. There is a substantial literature treating optimal control problems with time delays. In a number of respects our understanding of problems with time delays matches that of delay free problems. Wc note in particular that necessary conditions of optimality, akin to the Pontryagin Maximum Principle for delay free prob[enls, have also reached a high level of refinement [4] for problems with time delays. Itowcver one important strand in the delay free theory is the study of relaxation procedures and of conditions for c•

of minimizers. This is an area which, in the context of time delay problems,

has received very little attention. Our objcct here is to describe some of the difficulties involved, and to indicate how, in some respects, they can be overcome. The optimal control problem we address is Minimize g(x(1)) subject to

(e)

x(t) = f(t,x(t), u(t), u(t-01), u(t-02)),

a.e. [0,11,

x(0) = •

u(t) ~ ~,

a.e. [--02,11.

The data associated with this problem are: real numbers 0 < 0i.< 0 2 < 1, a point x0E Rn, a set f~ C Rm and functions g: Ru

R f R 1+n Win3

Ru

We have chosen this problem for simplicity of exposition, and to keep notationM complexity to a minimum.

No difficulties arise in extending the theory to follow to apply to control problems where

delay terms are present also in t.he state variable. We could also have looked at problems involving more than two delays. Our reason for clmosing precisely two delays is that cert,ain difficulties encount.ered with two delays typify those for several delays, and are not shared by problems wit.h one delav. Notice lhat we have adopted dynanfical equations where there isageneral .onlinear dependence

530 on u(t), u ( t - - 0 t )

and

u(t--02). (In the special case that the dependence is 'separable',i.e. f is

expressible as f(t,x(t), u(t), u ( t - 0 1 ) , u ( t - 0 2 )

= fl(t,x(t), u(t)) + f2(t,x(t), u(t--01) ) + f2(t,x(t), u(t--O2)

the task of finding an appropriate notion of relaxation is a simple adaptat, ion of standard delay free theory). T h e hypotheses we impose are f(-, x, u0, Ul, u2) is measurable and f(t, x,.,.,.) is continuous, f2 is compact, g is continuous, 3 a constant c and a function k(-) E L 1 such that Ill(t, x, u o, u 1,

u2)ll _< c(

t + Ilxll), for all (t, x, (u0, ux, u2) ) E [0,1] • R n x f23 , and

Hf(t, x, u 0, Ul, u2) - f ( t , y, u0, Ul, u2)l[ < k(t) l l x - y l l for all (t, (u0, ul, u2) ) E [0,1] x f23 and x , y E N n. W a r g a [5] was the first, to call attention to the speciat difficulties encountered in addressing existence questions for problems with non-separable time delays in the control. relaxation procedure ( 'weak' rela.xation, as we shall call it,).

Warga also proposed a

The contributions of this article are,

firstly to illustrate through an example that weak relaxat.ion can have the undesirable effect of reducing the m i n i m u m cost (in other words weak relaxation m a y fail to be proper), to give a simple testable criterion for properness of weak relaxation and, finally, to supply a more refined notion of rela~,:ation which is proper in certain situations where this criterion fails.

2. Delay Free Problems. For purposes of comparison we briefly review a standard procedure for relaxing delay free problems. Throughout this section then we look at Minimize g(x(1)) subject to

.~(t) = f(t,x(t), u ( t ) ) ,

a.e. [0,q,

x(0) = x0, u(t) e f~,

a.e. [0,1].

The hypotheses on the data are those of the previous section, when tile delay free problem is regarded as a special ease of (P). A measurable function u: [0,1] ~

R m satisfying u(t) E f2 a.e. is called an ordinary control. A

pair (x(.), u(-)), comprising an ordinary control u(.) and an absolutely continuous function x(-) which satisfies the differential equation is called an ordinary process. called admissible.

If x(0) = x 0 the ordinary process is

The optimization problem, posed over admissible ordinary processes, is called the

original problem (Poriginal)"

Two questions now arise. When does rite original problem have a

minimizer and. in the event that there is no minimizer, how can we supplement the class of admissible ordinary processes to ensure existence of a minimizer? 13orb have very satisfactory answers. As for the first question, a simple geometric condiLion "f(q x, .Q) is convex for all (t, x) E [0,1] x ~n,, is I~nowll to assure existence of an optimal admissible ordinary process (See, e.g. [3]). An existence

531 theory, applicable for nonconvex 'velocity sets' f(t, x, f~), involves introduction of the notion of relazed process.

A relazed control is a measurable essentially bounded function, often written ~ or t---~ ~t' taking vMues in the space of regular probability measures on ft. ('Measurable' here is understood in the sense that t - - < g , # t >

is measurable for each g E C(fl).)

We denote the set of relaxed controls by Jtl,. The

set .46 can be regarded as a subspace of the topological dual space (L[0,1; C(fi)])*; it acts on elements r in the primal space according to 1

r ~f

J" O(t,u) d#t(u ) dt. 0

We equip it with the relative weak* topology. A relazed process is a pair (x, #) comprising a relaxed control and an absolutely continous function x which is a solution to the differential equation, in the sense that :~(t) = f f(t, x(t), u) d , t ( u ). It is admissible if x(0)=x 0. (Under the hypotheses, given /~, there is a unique x such that (x,#) is an admissible relaxed process.) An ordinary control u(.) can be regarded as an element in .21,; we identify it with the point t ~ 6 ( t ) , where 6~ denotes the unit measure concentrated at the point a.We find that .At, is a non-empty, compact set. .At, = closure{ordinary controls} The mapping

u - - x(-) : .Ag ~

C([0,1];~I n) is continuous, where (x,,u) is the admissible

relaxed process associated with /~. The problem posed over admissible relaxed processes is called the relazed problem

(Prelaxed).

We

deduce immediately from the preceeding assertions:

Theorem 2.1 The relaxed problem has minimizer, and tile infimum cost for the original problem coincides with the minimum cost for the relaxed problem. In symbols Inf{ Poriginat } = Min{Prelaxed}. The fact that the infimum costs coincide is summarized by the statement "tile relaxed problem is a

proper extension of the original problem".

Properness of the extension is desirable for a number of

reasons. It means that there is a close connection between the problem of primary interest (the 'original' problem) and the problem which replaces it.

It also suggests a methodology for finding

ordinary admissible processes which come close to achieving the infimum cost in situations where existence of a minimizing ordinary admissible process is not assured: we should solve the relaxed problem and then approximate the 'relaxed' minimizer by an ordinary one.

A full account, of these

ideas is to be found in Warga's book [4].

3. Relaxation Via a Reduced Problem. We return now to problem (P). In relation to this problem we term ordiT~ary controls measurable fuuctions u(.): [ - 0 2 , 1 ] ~ n such that u(.) E f~ a.e.

Ordin~Lr~

532 processes and admissible ordinary processes have the obvious meanings (c.f. Section 2). Problem (P), posed over admissible ordinary processes, is the original problem (Poriginal). A well-known technique (see e.g. [4]) reduces a control system with commensurate time delays to a delay free problem. ('Commensurate' means that 0t/0 2 is a rational number.)

To avoid some minor

technical difficulties, in describing this technique we assume a little bit more than commensurability, namely 9

01 and 0 2 are rational.

This means that there exist integers 0 < N 1 < N 2 < N such that 01 = N1/N and 0 2 = N2/N.

We

write A = 1/N. The basic idea is to section ordinary controls and the corresponding trajectories into segments of length A, and to stack these segments to form higher dimensional

vector valued functions

on the interval [0,A]. The resulting functions satisfy a delay free differential equation. The reduced problem is expressed in terms of a partitioned state vector Y~ = col [x0,.., XN_l] and control vector fi = col [U.N 2 .... u 0 .... UN_l]. It is

Minimize ~(2(A)) subject to subject to ~,(s) = f(s,~(s), fi(s)) , (R,P)

a.e. [0,~],

fi(t) E f2, a.e. [0,A], together with the mixed endpoint conditions x0(0 ) = x 0 and xi(0 ) = Xi.l(1), for i = 1,2,..,N-1 .

The d a t a for this problem is constructed from that for (P): = ~ (N2+N), ~(.~) = g(xN_i) and the components {fi}iN-~ of f are fi(s,~.fi) = f(iA+s, xi, ui, Ui_N1 , Ui_No).. It is a straightforward matter to show that (RP) and (P) (posed over ordinary processes

are

equivalent, in the sense that there is a one to one mapping from ordinary admissible processes for (P) and ordinary processes for (RP) which satisfy the mixed boundary conditions. This mapping carries an ordinary process (x,u) for (P) into (2,fi) where ~(s) = col[x(s), x ( A + s ) , x(2/',+s),..,x([N-1]A+s] and fi(s) = col[u(-N2A+s ) .... x([N-IlA+s]. This reduction procedure, together with the theory of the preceeding section, suggests a relaxation procedure for the time delay problem.

We can relax (R.P), which has no delays, along the lines of

Section 2. It is not difficult to show that the relaxed extension of (RP) has a solution and is a proper extension (the a r g u m e n t s involved in establishing riffs latter property are a little bit different to those underlying Section 2, since we must ensure that. given an 'admissible' relaxed process for (RP), we can find a suitable ordillary process approximating it and which satisfies the mixed botmdary comlitions). The procedure of replacing (P) by the reduced problem (liP) and relaxing (R.P) we shall refer to ;us 'relaxation

via the reduced problem'. This procedure in certain respects achieve~, the desired

533 objectives. The effect of relaxtion via the reduced problem is to guarantee existence of minimizers, and the relaxed problem is a proper extension of (P). IIowever it is unsatisfactory ill certain respects. Notice first of all that the dimension of the state and control spaces in (P~P) can be very large ((Nxn) and ([N2+N]xm respectively). The fact that we have posed (P) on the time interval [0,1] is merely a normalization procedure, and the above technique for eliminating delays works on an arbitrary finite time interval; however we note that, for fixed time delays 01 and 02, t.he dimension of the spaces involved increases rapidly with the length of the underlying time interval. Apart from this, in passing to the reduced problem on the time interval [0,A] the connections with the original problem are somewhat obscured.

Finally, relaxation via the relaxed problem is only possible for commensurate

delays.

4. Weak Relaxation. We now describe another relaxation procedure for the time delay problem, due to Warga [5]. Take an ordinary process (x(-), u(.)), and define u0(t ) = u(t), ul(t ) = u ( t - 0 1 ) and u2(t ) = u ( t - 0 2 ) , for t e [0,1].

Then (x(-), (u0(-), u l ( ) , u2(.)) ) satisfies C .~(t) = r(t,x(t), u0(t), Ul(t), u2(t)),

a.e.

[0,1]

(4.1) /(u0(t.), ul(t), u2(t)) ~ f~3,

-

x(0) = x 0

together with Ul(t ) = u 0 ( t - 0 t )

,01 Min (Pw_relaxed).

(strict inequality). But problem (Ps_relaxed) is an extension of problem (Poriginal) so Inf (Poriginal) _> Min (Ps_relaxed)

> Min (Pw_relaxed).

We summarize our findings:

Proposition 7.1 The data for problem (P) can be chosen so that 0 i and 02 are rational, 02= 201_ , and (Pw-relaxed) is not a proper extension of (Poriginal). Observe that Proposition 6.1 predicts that, in this example, (Ps.relaxed)

is a proper extension of

(Poriginal)" There is one further respect in which this example is revealing. Define

11(01,02) : = {the infimum cost of the original problem for delay parameters O1 and 02}. It is of interest to know if this function is continuous, for if it is not the well-posednness of problem (17) is called into question in circumstances where the delay parameters have uncertain values,

l:;xamining

can be discoa~inuous. Let 0 be a fixed rational number, 20 and {0(I)},{0( ')} be sequeuces of rational points such that 0(1)~ 0 and 0(.2)

variants on the above example tells us that r/ 0 < 0 < 0.5.

Let

for which 0f'e)~90(. I) for i = 1,2,.. l

---

1

For each i sufficiently large, the function #t = u is a weakly

relaxed control in tile above example when we adopt

0 (.I) and 0(e) as delay parameters, and the cost is 1 1

zero. Since the cost cannot be negative, anud in view of Theorem 6.1, we must have

,1(0[12,0(e)) = 0, llowever ~/(fl, 20) > 0

as

for all i sufficiently large.

we have shown. Tiros ~l is discontinuous at (0,20).

538 REFERENCES

[1]

T. Andrews, J. Rosenblueth and R. Vinter, Relazation Procedures for Time Delay Systems, in preparation.

[2] [3]

C. Dellacherie and P. A. Meyer, Probabilit~s et Potentiel, Iterman, St,rasbourg, 1975 W. It. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, New York, 1975.

[4]

J. Warga, OpIimal Control of Differential and Functional Equations, Academic Press, New York,

[5]

J. Warga, Nonaddilively Coupled Delayed Controls, preprinL

!966.

MAXIMUM

PRINCIPLE DIFFERENCE

FOR

NONCONVEX

CONTROL

FINITE

SYSTEMS

Boris S. Mordukhovich Department

of Mathematics,

Wayne

State University

Detroit, M i c h i g a n 48202, U S A Optimal control problems are considered for a family of finite difference systems with constraints on the control and phase variables. Such systems arise in computer calculations of optimal control systems with continuous time and aiso in the simulation of discrete large-scale optimization problems. We regard discrete approximation systems as a process with decreasing time quantization period. New necessary optimality conditions for such systems are obtained in the form of ~he approximate maximum principle which is fulfilled the more precise the less is the time quantization period. We consider arbitrary nonlinear control systems without any asssumptions about convexity. Constructive methods are proved for approximating the state constraints so thai, the stability of the Pontryagin maximum principle is ensured in computer calculations of nonconvex control systems with coutinuous {~ilne. The results obtained are used for the analysis o[ qualitative aspects and the development of numerical algorithms of opt,treat control. 1. I n t r o d u c t i o n The paper deals with optimal control problems for finite difference systems of the form

z(t + h) = z(t) + hf(x(t),u(t),t), v(t) E U(t),t E T/, = {to, to + h , . . . , t l - h},

(t.l)

(1.2)

with various constraints on the state variables. Such problems arise in computer work when the time derivatives are replaced by finite differences : ~:(t) ~ [x(t + h) - x(t)]h -~, et,c. It is then natural to consider the problems ~ a constructive interpretation of the corresponding problems for continuous dynamic systems without using the abstraction of continuous time. We note also that finile difference dynamic systems can be used for modeling a variety of static problems of discrete optimization of large dimensiouality. Optimization problems for systems (1.1), (1.2), regarded as a process as h ; 0, occupy an intermediate position between optimal control problems for systems with continuous and discrete time (with fixed h) and lead to the appearance of essentially new effects some of which are analyzcd below. I~ is well-known that the Pontryagin maximum principle [P 61] gives some necessary conditions for an extremum in general optimal control problems for continuous-time systems described by ordinary differential equations. But, for its discrete analogue to hold as necessary optimality conditions, certain assumptions about convexity [B 73, CCP 70] are required, since wil,hout ~hem the

540 discrete maximum principle is invalid even in the simplest problems with a fl'ee right-hand end. The connection between convexity and maximum type conditions in the theory of extrernal problems in reflected in the classical Kuhn-Tucker theorem [R 70] and follows naturally from the general results of nonsmooth analysis [C 83, Mot 88] (the d~aracteristic form of the normal cone to the convex set). On the other hand, the unconditional satisfaction of the maximum principle in systems with continuous time is linked with the "hidden convexity" property inherent to such systems fIT 74], which can be traced back to Lyapunov's theorem on convexity of the values of a nonatomic vector measure [L 40]. It is important to emphasize that, in order to utilize the Pontryagin maximum principle correctly in computations of actual control systems with continuous time on computers operating discretely, we have to prove the stability of necessary optimality conditions supplied by the maximum principle in difference approximations of nonconvex systems with constraints on control and state variables. Our main result is to prove the necessary optimality conditions, spccifically for finite difference control systems with constraints on the trajectory, in the form of the approximate maximum principle which is valid when the perturbation of constraints is constructively matched with the step of a difference mesh without any assumptions about convexity. In control problenls where the right-hand end of trajectory is free and the left-hand end is fixed, our result is in accord wit,h the quasimaximum principle of [CK 71]. During the proof of the approximate maximum principle, an approximate analogue of the hidden convexity property is found to be inherent to finite difference systems so that we can solve the problem posed in [Mo 71] of finding the reasons why finite difference systems with a small quantization period behave like any discrete systems with a priori convexity conditions. We shall use the following standard notations: R" is the n-dimensionalEuclidean space of column vectors (they are written in rows) with the norm II "II and the scalar product < -,. >; coX, d X are the convex hull and the closure of the set X; E is tile identity matrix; o(h) and O(h) a.re vector quantities such that

o(h)/h

--, 0, IlO(~dll _< K~ as z, s 0. In addition to the usual notation for a

matrix product, we put

.l:=i

( AiAi-1 ... Aj for i > j,

fiat= J

J E 0

for i = j - 1 , for i < j - 1

2. P r o b l e m F o r m u l a t i o n and Definitions. Wc take the control problem

5c = f(x,u,t), u(t) E U(t), t E T = [to, t1],

(2.1)

~ ( z ) < O,i= 1 , . . . , m ; ~ ( z ) = O,i= m + 1 , . . . , r e + p ,

(~.2)

I = ~Oo(Z)~ min,

(2.3)

541 where z = (xo, xx) = (x(to), x(t~)) E R 2~ and T is a fixed intervM. For each positive integer we consider the quantization step hN = (q - to)/N and construct, a sequence of finite difference problems of minimizing the functional (2.3) with z = ZN = (XN(t0), zN(t~)) and the constraints

xN(t + hN) = XN(t) + hNf(xN(t),uN(t),t),

(2.4)

UN(t) ~ U(t),t ~ TN = {to, t0 + hN,... ,t, -- hU),

(2.5)

(PI(ZN) 0,i = 0,1,...,m,()~oN) 2 + . . . + (.~+p,N) 2 = 1,

(2.~3)

where z} = (x~(to),x~(t,)),ei(h) --+ 0, e(t,h) + 0, as h .L 0 uniformly with respect to t C TN, N = 1,2,.... It follows from (2.10), (2.12) and (2.13) that e(t, hN) 0 there exist No such that

~oi(x/v(to),ZN(tx)) --7i/v _ No, N e A.

(4.2)

We note that the concept of an essential constraint in a sequence of finite difference optimization problems corresponds to the concept of an active inequality-type constraiat in usual problems of mathematical programming and optimal control.

It is easy to see that, if the constraint is

not essential along the sequence A = {N}, then it is inessential along the sequence A, C A. IIence it can be assumed without loss of generality that, for the sequence of optimM trajectories {x~v(')} in problems (2.3) - (2.6), the first g constraints in (2.6) are essential and the remaining

m -g.,O < g < m, are inessential along all N = 1 , 2 , . . . . From the optimal trajectory increment A z } = (Az~v(10) , Az~v(q)) we construct the (g + 1) -dimensional column vector

e/v(Az~) .

((-~e~ . .

and wc consider the vectors b/v = Ax~

.

,(Tgz ~~176

(~.3)

g/v(b/v, O, u) = g/v(b/v, A0,.x}(t,)), and the set

C-'/V = {aN(b,,O,u) : bN e B/V,O < 0 < N -- 1,u e U(ro)},

(4.4)

where B/V = {x E /{", [[xl[ _< h/v}. We form the convex hull COGN of the set (4.4). The following lemma shows that the set coG/v can be shifted by an amount of order o(h/v) as hN ~. 0 in such a way that it does not intersect the negative orthand in Re+x:

Y = {y = ( v ~

e R TM : v' < 0,i = 0 , 1 , . . . , e } .

(4.5)

L e m m a 4.2. There exists a sequence of (g + 1)-dimensional quantities of order o(h/v), h/v J. O, such that

( c o a ~ + o(h/v)) a z = r N > No

(4.6)

Using this result, we can prove the approximate maximum principle in problems (2.3) - (2.6) in the case of any perturbations ")'iN of inequality-typed constraints.

546 T h e o r e m 4.3.

Let {x~

be optimal processes in problems (2.3)- (2.6) where the

inequalties in (2.6) arc essential along the sequence {x~

N = 1 , 2 , . . . , for i = 1 , 2 , . . . , g and are

inessential for i = g + 1 , . . . , m. Then the relations of the approximate maximum principle (2.8) (2.13) hold for these processes, and

e,(hN) = O(h,v),i = 1,2,...,g;AiN = 0, i = g + 1 , . . . , m , N

>_ No.

(4.7)

P r o o f . By Lemma 4.2, the separability theorem for convex sets and the structure of sets (4.4), (4.5), there exist vectors #N = ( # ~ . . . #~) E R ~, (#~ i rtOtPi/ o',

O~al

+ . . . + (#~)2 = 1, N > No, such that

o

ttNK~xo~ZN), b~) + (3-~-(z,-,,, ao,,x~(t,))] + o(1,,) ___0

(4.S)

for any bN e BN, IIbNII _< hN, and parameters {0(N), u(N)} of one-needle-shaped variation, where It'y >_ O,i = 1,...,g. We put Auv = #'N,i = O, 1,...,g;.~iN ----0, i = e + 1 , . . . , m , and obtain sets of Lagrange mutipliers which satisfy (2.12), (2.13), (4.7), while el(h) = O(h) follows from (4.1). Using thc standard transformation of the method of increments [GK 71] we obtain from (4.S) the relation

O~~ o O~t o (AoN-~x (zN) + . . . + ,~eN-~x (zg) -- ~b~

bN) -- hN[H(x~

(4.9)

v

r b-0+,), ,,,-0) - ~r(~o (To), r

,6(,-0), ~-0)] + o(I,.) >__o

along the trajectories (2.8), (2.9) corresponding to {x~

With bN = 0 we get (2.10) from

(4.9), and with u = u~(ro) we obtain (2.11) by reduction ad absurdum. [] 5. T h e A p p r o x i m a t e M a x i m u m P r i n c i p l e in G e n e r a l C o n t r o l S y s t e m s w i t h E q u a l i t y Type Constraints Let us study the discrete approximations (2.3)- (2.7) of problem (2.1)- (2.3) with cquality type constraints. Our example below shows that if 51N = 0 or 5iN J. 0 too rapidly, the approximate maximum principle may not be satisfied in ( 2 3 ) - (2.7). E x a m p l e 5.1 We consider the two-dimensional control problem =

v,y(0) = 0,.~ = w,s(0) = 0, x = (y,s) E R 2,

=

(,,, ~) e

(5.1)

m , t e [o, 11,

I = ~o(x(1))

= s(1)

--~ min

(5.2)

with fixed left-haad end and the constraint at the right-hand end of the trajectory ~,,(x(1)) = y(1) = 0

(5.3)

with the quantized control domain in (2.1):

V = {(0, 0), (0,-1), (1, -2), (vq, -3))}

(5.4)

547

It is easy to show that, in the corresponding finite-difference approximation (2.3) - (2.5), (2.7) of the problem (5.1) - (5.4), there exists the unique sequence of solutions u~v(t) = ( 0 , - 1 ) , x ~ ( t ) = (0,--t),t e T N , N = 1 , 2 , . . . , where r

(5.5)

- (--~lN,--MN) for the corresponding trajectories of the adjoint system (2.8), (2.9)

with Lagrange multipliers ~dmissible in (2.13). The Hamilton-Pontryagin function along {X~v('), r h ~ the form [IN(U, t) = - - & N V - - ~ o u W , and for the optimal control (5.5) we obtain H N ( u ~ ) = ~o~v. It can be shown directly that max{I-IN(u) : u e U} - HN(U~ in

(2.13),

>__ 1 for any {-'~0N,/~IN} a,dmissible

i.e. condition (2.10) does not hold.

We introduce a coordination condition between perturbations 51u and the step hN: . 5~1v lm - - = oo, i = m + l , . . . , m + p ,

(5.6)

n~oo hN

and we show that in the caze of (5.6) the relations (2.8)- (2.13) are necessary for optimality in (2.3)

-(2.7). T h e o r e m 5.2.

Let the processes {x~(-),u~(.)} be optimal in problems (2.3) - (2.7) with

any perturbations 7iN and the coordination condition (5.6) for &N- Then approximate maximum principle (2.8)- (2.13) holds in (2.3)- (2.7); moreover ei(hN) = O ( h s ) , i =

1,... ,m.

P r o o f . We can assume for simplicity that ~ol -= 0 in (2.6), since these constraints were considered in Theorem 4.3. We put ~a+N(z) = ~a~(z) - 5m, ~a~-u= -~o~(z) - 5,v, i = m + 1 . . . . , m + p, and write constraints (2.7) as

~+,u(ZN) _ 0 9 T(t)~,: The function of C(] - oo, 01, X ) defined by: (T(t)~)(s)=~(t+s)

for all.~ 0. Thence, we will study the set s = h?. M Ca In this section we will first prove results about the simplest case which happens when the dynamics are just constrained by one delay. T h e n we will see that the other case can be generalize without great efforts. 1.1

The

one

delay

case:

We suppose that p = 1 and that:

;c -- {~ e c(] - oo, 0], x )

I ~o(0) E M ( ~ ( O ) ) } , 0 < 0 is given

No other assumptions need to be made about the set valued-map M in the first result we shail obtain. But the generality of the proposition does not provide us a useful way to compute iPA:(c;). T h a t is why we wiI1 need further results.

552 Proposition

1.1 Dpc(%0) = [D(M o %0)(0,%0(0))](1)

Proof-The next proposition involves the contingent derivative of the set-valued map M o %0.3 Let v E D~c(%0). By definition, we know that it is equivalent to: 3hl ~

0+ and %01E C ( ] - o o , h i ] , X )

such that:

{ 0 0 because, in fact, it is the set that will be useful in the application of the theorem (0.2). A new proposition will help us to do so. P r o p o s i t i o n 1.5 ~D~(~o)fl I n t B x ( A ) C 2?~c,(~) C g)~(~o) Cl Bx(A) Bcforc thc proof, let's state a preliminary result.

l ~ m m ~ 1.6 ~o~(90) = Bx(A) Proof-Let v E 7?jc(~o) Cl I n t B x ( A ) , then 3hl ~

0 , vl ~

Vk=l,...,p,

v such that:

0 O,

,(.)eu

//

/~

> -C, -

> 1. -

Now, we let tl~ be the set of all controls u(.) E U, such that the corresponding solution Za(') of (3.7) satisfies constraint (3.15). Then, by the construction of Q,, we see that

(3.17)

~(.) E Ua~d, Vn >_ 1.

565 Next, let us denote (3.18)

J,,=

inf

,,(.)eu2~

J,,(u(.)).

Then, we have the following result.

Theorem 3.4. Let (H1),(H2) a~d (H3) hold. The~, (3.19)

nm J: = J(~(.)) = J. f/,----+ OO

Pro@ First of all, by (3.17), we have J,, < Jn(fi(')). Thus, it follows from Theorem 3.3 that

~l~ J,, < J".

n-..-+ (:~

Now, suppose (3.19) is not the case. Then, one can find a constant 6 > 0 and a sequence

{urn(.)} c u~, such that Jm(um('))
0,

v~(.) 9 U.

Now, we let 2,,(.) be as in Section 3 and 2 % ) be defined as in (4.2) replacing (x,(.), u(.)) by (2,~(.), a(-)). Then by Theorem 3.4, we know that as n - , oo, (4.4)

0 < a(n) = Fn(~(.)) = d(_oo,j'_,/a](J,(~(')) _I.

568

Substitute (4.1S) in to (4.11), we have (see [9]) v~T (4.20)

_> (r

~,,(T)) + r

o 0

= /oT[H,~(t, xn(t),u(t),r162

_ Hr,(t,z,~(t),un(t),r162

This can be regarded as an approximating maximum principle. Then, it is natural to expect that we can take the limits to get the final result. The rest of the proof is to achieve this. We note that (by (4.5)) (4.21)

{ Iz•(t)- ~(t)] ~ 0 ,

ir

/

_

r o 0,

uniformly in t E [0,T],

r

• r

On the other hand, from the equation (4.18) and the Gronwall's inequali~, we see that (4.22)

Ir

_< c,

Vn >_ 1,t 6 [0, T].

Thus, by the reflexivity of X, we may assume that

{

/~(.,x"(.),u"(.))*r

(4.23)

~ r

in L2(GT;X),

fo,,(.,x"(.),u"(.)) L A(.), in L2(O,T;X), r ~ r in L2(O,T;X),

for some functions r A(.) and 0(.). Next, we will derive the equation which the function r satisfies. To this end, we need the following result. Lemma 4.1. Let eAt he a compact semigroup on reflexive Banach space X. Let gu(') 6 L~176 with

(4.24)

Ig,(.)l~(o,r~x) _ 1.

Then, there exists a subsequence of ga(') (stiU denoted it by itself) and a g(.) E L ~176 (0, T; X), such that for all t 6 [0, T],

~m I

(4.25)

/:

eX(~-r)Cg,~(r) - g(r))&" l = 0.

By noting the fact that the semigroup eAt is continuous in the uniform operator topology for t > 0 ([11]), one can easily prove Lemma 4.1. Now, by the above lemm% we can assume the following (see (4.18), (4.22) and

(4.23)): (4.26)

r

o r

strongly in X,

Yt 6 [0, T].

569 Then, for t < T, we have (4.27) ir ) - ea'(r-t)~b(T) _< I~P.Ct) - r

/"

ea'(,-t)--,, ~,~ ~-, ~"(~-), ~(~), r176r

+ leA'(r--t)CCr, CT) -- r

+ ML

+ Omens(u" # a) + 01r ~ - r176 I -~ o,

.

-*

f

e~Cf-t)lr

) - ~p(T)ldr

oo.

While, by [4] (Proposition 1.12 and Lemma 5.4), we know that if (4.28)

H~"(',z"('),~('),r176

~s'('),

inL~176

then, (4.29)

;(r) eO~H(r,~(r),~(r),r176162

a.e. r E [0,T].

Thus we obtain (2.12). While, the rest conclusions follow easily.

[]

References

[1] N. U. Ahmed and K. L. Teo, Optimal Control of Distributed Parameter Systems, North Holland, New York, 1981. [2] E. hsplund, Averaged norms, Israel J. Math., 5 (1967), 227-233. [3] A.V. Balakrighnan, Applied Functional Analysls, Springer-Verlag, New York, 1976. [4] V. Barbu, Optimal Control of Variational Inequalities, Pitman, Boston, 1984. [5] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. [6] I. Ekeland, Nonconvez minimization problems, Bull Amer. Math. Soc. (New Series), 1 (1979), 443--474. [7] H. O. Fattorini, A unified theory of necessary conditionz for nonlinear nonconvez. control systems, Appl. Math & Optim., 15 (1987), 141-185. [8] X. Li and Y. Ya~, Maximum principle of distributed parameter systems with time lags, Distributed Parameter Systems, Lecture Notes in Control and Information Sciences, Springer-Verlag, New York Voi.75,410--427, 1985. [9] X. Li and J. Yong, Necessary conditions of optimal control for distributed parameter syatems, submitted [10] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, 1971. [11] A. Pazy, Semigroups od Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. [12] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mischenko, Mathematical Theory of Optimal Processes, Wiley, New York, 1962.

NE~ ALGORITHMS O~ SOLVING EXTREN~L PROBLEMS Rafail Gabasov Byelorussian State University,

220080,f/insk,USSR

Faina M.Kirillova Institute of )~thematics,

22060~,

rvinsk, USSR

Algorithms os solving mathematical programming and optimal control problems using a new approach to the solution of linear programming problems are discussed. This approach has been developing in Einsk (USSR) from the begimling of the 70th. Numerous experiments by computer carried out for linear programming problems (in three computer centers independently) have demonstrated advantages of the adaptive method (R.Gabasov, F.N.Kirillova,

O.I.Kostuykova,

1980) and

another methods regarding to the simplex method and other known methods. 2rinciples of the adaptive method were used while algorithms of solving mathematical programming and optimal control problems were constructed. The paper gives a brief account of recent results obtained by participants of the Y~nsk Seminar on constructive problems.

theory of extremal

They deals with exact finite algorithms for solving optimal

control and nonlinear programming problems tion of nonlinear functions), ciples.

(using network interpreta-

support maximum and

6 -maximum prin-

Consider the classical optimal control problem

1 .

J(u) i

(1) - ~ Cx, u , ~ : ) ,

~(~(~),-~)

~0

h (.:~ C~'))

-

~(~.)

=:r.

,

(2) (3) (4)

,-t. e T , o,

(5) , Piecewise-continuous

R"

Rt

vector-functions

R"

~(s

k

s E T , which take on the

values from the set (5) generate continuous trajectories ~(%), s ~ T, of the system (2) satisfying the phase

(3) and the terminal constraints

571 (3),(4).

They are usually called admissible.

is called optimal if the cost function Complexity

I) infinite number of nonlinear dimensionality

are chosen;

3) nonlinear nature

Lhe problem

(I)-(5)

constraints

in finding

of transformations.

can be considered

efficient

methods

to the problem

of formation

which allow to solve

and numerous

lems are known.

methods

operation gradient

~rom the general point of

of passing feature

to finite-dimensional

the limit

(or its analogue) processing

dimensional

simplicity

of the basic method

consists

in calculating

of current

information

On the other hand,

finite-dimensional methods as well.

we introduce

problems However,

are always finite and determined

are used in the mathein our methods

interval

They accumulate

of the problem

and linear-quadratic

only by optimal control differ fundamentally

optimal

(I)-(5).

optimal

intervals

2. ;~&ile solving the problem

,

d ' 6 ~ ~,

Various methods known among

6.

,

(I).-(5) in

shortening

the time

The methods are finite for linear

control problems,

(I)-(5)

i.e. their realization

6"e A"

,

linear programming

A ~ R""

of solution are known for the problem

them is the simplex method.

of primal and

the principal part is ful-

filled by a linear model which is an interval problem:

d,

They

structures~

from the tradi-

by computer needs a finite number of full integrations conjugate systems and a finite storage.

c,x,

do

to solve the extremal problem.

As a result we obtain a method of solution of the problem approximations

a finite-

of solution.

unlike with mathemati-

the sizes of auxiliary problems

not depend on the accuracy needed

tional ones.

the

and a step along it. Instead for a com-

matical programming

which successive

is

of these principles.

linear problem with an efficient algorithm

cal programming

prob-

of the suggested approach

which in many known approaches

prehensive

(I)-(5) with a given

to the object being opti-

method of realization

On the one hand we abandon

consists

is not new: both various linearization

Characteristic

defined by a particular

(I)-(6)

of this comple-

of linear finite-dimensio-

the problem

mized and a finite storage of computer. view the idea of the approach

By virtue

as one of the ultimate

accuracy using finite number of references

methods

on output .signals ( ~ ) ,

ofthe space from which elements ~ s

The main idea of our approach

nal problems

control

the optimal value.

of the extremal problem is defined by three factors:

2) infinite

xity.

An admissible

(I) reaches

(6)

(6). The most

This is a cause for many

572 scientists

to confine

their investigations

reduces to the classical ting effective constrained

algorithms~

The same conclusions

optimal control problems

zation problems when methods

reduce

(the penalty function

of solution are grounded

However,

of solving complex optimization

For example

(mathematical

and finally

there exists a nt~ber of optimal control

package

LP ASY realizing

problems

such peculiar programming

solution

the known methods

software

of the phenomenon

me-

as methods

The fact is that an increase

which caWt be solved using the most powerful

source

to finite-dimenprogramming

to a more complicated

problems

programming

minimi-

The same is said

on a reduction

problems.

of solution leads

auxiliary problems

become helpless.

are often used when

in reality such a way cannot be considered

of a given accuracy of "simplified"

"the problem

to unconstrained

methods).

sional nonlinear programming problems thods).

to the phrase

L2 problem which can be solved by the exis-

and perfect

the simplex method

[d]

is clear enough for specialists.

generated from optimal

The Linear

control problems,

have

features which are seldom occured in general.linear

problems

and they are not considered

by the simplex

me-

thod created in the middle of the 40th. In this connection the problem method

a new method was created in ~ n s k

(6). Its characteristic

called a support preserves ons: The support unlike

are as follows:

the feasibility

is similar to the basis

of vectors

result

of decreasing

all the variables varies;

iteration

at iterati-

solution and

element;

a suboptimality

which do not satisfy

rion vary at the iteration. variable

~

of the simplex method but

in the method as an independent

is based on a principle

3) an iteration estimate.

the optimality

solution is improved at an

but the support is improved

5) the solution can be stopped using the suboptimality s -optimal

As a

crite-

As for the simplex method there only one

4) not only a feasible

(as in the simplex method),

on fo~ any

I) the

~ ; 2) a special construction

the latter itTis separated from the feasible

participates

well;

features

starts with any initial vector

to solve

feasible

solution with a preassigned

9 The method called "adaptive"

as

criteriaccuracy

~ 0

experiments

programmed

both on general problems

random-number

generators

mal control problems. published [ L , 3 ] . 3. The continual tion elements

~ny

and

and on special problems results

by different

obtained from opti-

of the experiments

constraint-inequality

of extremal problems.

~ested in numerous

(6) constructed

have been

is one of the complica-

Such a constraint

can be taken

573 into account

using the semi-infinite

;

d. -/mz,

(18)

the maximum condition

~'(r

=

m~

~'ce) 6 u

,teT

,

(19)

is true. The prototype Lagrange

of such optimality

multipliers

of variational

calculus

carried over to nonlinear programming method and the adaptive methods ther form.

conditions

which has been later

problems.

the optimality

They come not from the multipliers

is the rule of In the simplex

conditions

have ano-

but from the basis and

the support. The Lagrange multipliers (called simplex multipliers, potentials) are obtained from the basis and the support. The support

of the problem

(17) is closely connected with

579 locally controllability of the system. The support $,up = ~ s ~ p , ~ p ~ r e p r e s e n t s such a set of moments from the i n t e r v a l T and a set of natural numbers assigned to each moment that the system (17) is locally controllable. The Support ~aximum Principle: for each normal optimal control there exists such a support that along the accompanying trajectory of the conjugate system (1~)the maximum condition (19) is true. The statement extends the Pontryagin maximum principle: in the Support ~iaximum principle only special multipliers generated by the support are used. The situation is the same as in classical result of the simplex method, i.e. among optimal feasible solutions there are always basic feasible solutions. By virtue of this fact we can speak of the Lagrange basic vectors. Algorit]~ms of solving nonlinear optimal control problems are given in [ 5].

~EFEREUC ]~S [I] Gabasov R., liirillova F . M . 1984). Consideration of Optimal Control Problem Specifity of Generalizing 1~lathematical Programming.- In: Preprints of IFAC 9th Triennial World Congress, v.5, Budapest, 264-269. 2 ] Oabasov R. Xirillova F.~I. (19~I). Constructive ~iethods of Parametric and Functional Optimization.- In: Preprints of IFAC 8th Triennial World Congress, v.4, Kyoto, 111.-116. [ 31 Gabasov I{., Kirillova F. ~,{., Kostyukova 0.I., Pokatayev A.V. (1987). Optimal Program Controls and Flexible Feedback.- In: Preprints of IPAC Xth Trieilnial World Congress, v.8, Munich, 119-124. 4] Abstracts of International Soviet-Polish Seminar on ~hthematical J r/lethods of Optimal Control and their Applications (1989)~ Institute of ~athematics, f~insk, USSR. [ 5 ] Oabasov R., Kirillova F.~L (co-authors Thatyusl~in A.I., Kostyukova 0.I., Raketski V.M., Pokataev A.V.)(1984.-1990). Constructive ~ethods of Optimization, parts I-5. Linear Problems. Control Problems. Network Problems. Convex Problems. Nonlinear Problems. University Press, ~Insk,USSR. [6] Gabasov i~., Kirillova F.M., Pokataev A.V., Kalinin A.I. (1990). General Approach to Construction of Algorithms of Optimization of Nonlinear Control Systems with Nonsmooth Characteristics. In: Preprints of IFAC Xlth Triennial V~orld Congress, Tallinn, USSR (to appear).

CONSTRAINED

CONTROLS

IN L I N E A R

OSCILLATING

SYSTEMS

F.L.Chernousko Institute

f o r P r o b l e m s i n M e c h a n i c s , USSR Academy o f p r . V e r n a d s k o g o l O 1 , M o s c o w , 1 1 7 5 2 6 , USSR

~_U_~.@_EX.- We c o n s i d e r linear state

controlled by

The

means

of

Nell-known

control

as

a

linear

obtained is

Some

to

a

open-loop

the

system,

example

are

control

obtained;

also

of

arbitrary by

on

as

a

well

depend

on

initial

in

of

the under

which

constraints. number

a

absence

of

This linear

scalar

constrained

as

time

the

of

eigenfrequencies

and

which

obtaining

in

given

of

law

considered

are

imposed

a

constraints.

motions

applied

of

to a t e r m i n a l

for

natural is

the

they

and

applied

conditions

system

transfer

state

geometrical

is

controlled

on c o n s t r a i n t s

is

to

technique

satisfies

control.

An

initial

subject

sufficient

(pendulums)

process

point-to-point

combination

control applied

a

technique

this

oscillators

control

of

a given

control

system;

of c o n s t r a i n t s .

approach

a

from

Kalman's

non-controlled

the

a problem

system

Sciences

terminal

the

states.

system

has

the of An

zero

eigenvalues. INTRODUCTION Property systems paper

of

controllability

without [i]

proposed

Kalman to

consider

on

control.

satisfied. number control.

of

a

We

use

conditions These linear

with

Kalman's under

results

are

oscillators

on

as

solutions

systems

usually

considered

control.In

conditions

control

(natural)

linear

is

imposed

obtained

find

non-controlled we

sufficient

constraints

of linear

of

the

method

for the

applied controlled

combination

system.

control

a by

In

this

constraints

imposed to

linear

well-known

controllability

a

geometrical

which

for

his

and

scalar

give

of

of paper

imposed

constraints

system a

and

some are

arbitrary

constrained

581 GENERAL RESULT Ne c o n s i d e r = A(t)x lu(t)l Here

x

The

functions

problem

(i)

x(t

(2)

to all

satisfies

0

x~

fixed

or

x~

x(T)

are

control (1) (2)

n

vector, x m

u

is

matrices

a control, which

are

A and

find t

a

control

u(t)

satisfying

c

[t o , T]

and

such

that

and

terminal

conditions

the

= X1

given

the

solution

(3)

vectors,

t

is

fixed,

T may

be

either

o

free

(T

> t

). Me

denote

by

~t)

the

n~

matrix

defined

o

by

= A(t)m, where

E

~t 0) is

a

= E

(4)

unity

n

x

n

matrix.

The

solution

of

(i)

n

satisfying

x(t)

the

initial

= r176

condition

(3)

is

(5)

+ ~ ~-l(s)S(s)u(s)ds~ t o

Substituting

(5)

into

the

terminal

condition

(3)

we

obtain

T

~-1(t)B(t)u(t)dt

(6)

= X*

t o

where X)K :

~-I(T)x%

The c o n t r o l ( see

obtain

Xo

(7)

must satisfy

(6)

and (2).

Ne s e e k

it

as

[1] )

u(t) where

u(t)

-

= OT(t)c, c

O(t)

is a c o n s t a n t

B

piecewise

t

initial

) = x O,

Here

n,

of

is

a constrained

u c Rm),

a state

x

for

with

> 0

is n

system

c Rn ,

a = const

is time,

constraint of

(x

respectively

continuous

dynamic

+ B(t)u,

~ a,

t

are

a linear

(8)

= ~-1(t)B(t) n-vector.

Substituting

(8)

into

(6)

we

582

(9)

R( T )c = x * where

R(T)

is

a

symmetric

n

x n

matrix

T

R(T)

(lO)

Q( t )Or( t ) d t

= t o

Consider

a quadratic

f o r m (v

is a c o n s t a n t

n-vector)

T

(R(T)v,v)

= ~

IQT(t)vI z dt

(11)

> 0

t o

It f o l l o w s

from

the

(i)

system

v = O, the

see

R(T)

is c o n t r o l l a b l e

[2],

linear

and

R(T)

is a n o n - n e g a t i v e then

matrix.

is p o s i t i v e

definite.

In

If

for

all

this

case

solution

x

%.bAgr__e_m_. Let (i)

(ii)

is p o s i t i v e

s y s t e m (9) has a u n i q u e

c = R 1 (T)

system

(II) that

(i2)

the is

matrix

R(T)

be

controllable)

and

positive for

definite

(the

n-vector

the

any

inequalities T

IQ

(t)

K(T)

IR(T) K(T) hold

for

vl

vl

g ~(T)Ivl,

V t,

t

e [t

o

, T]

(13) (14)

Z %(T)Ivl

some

T > t

~T)

and

. Here

K(T)

is

n x n

matrix

(det

K

o

0), arbitrary

Ix*l then

~-I(T)

control

u(t)

The c o n t r o l

According lu(t)l =

to =

lOT(t)

(8),

are

positive

scalars,

v

is

an

If

< a %(T)

the

Ey_o.~.

n-vector.

%(T)

(8),

(15)

given

by ( 8 ) ,

(12)

satisfies

(12)

IQT(t)R-1(T) K(T) K-I(T)

x*l

=

R-I(T)

x*l.

(12) (2)

satisfies by i t s

(2),(3). definition.

583 Using

(13)

lu(t)l

we o b t a i n

~

/ZT)IK-I(T)

Now we s u b s t i t u t e lu(t)l

R-~(T)

x

= R(T)

~ ~T)~-I(T)I

The p r o o f

is

over.

R#_m#.!K.

The

matrix

matrix:

K(T)

= E

K(T)

R(T)

v

K(T)

K(T) In

x*l.

in

this

and use

vl

= ~T)

(13),

case

(14)

and (15)

~1 ( r ) l x

(14) MT)

is

can

be

a

lower

I*~ a

equal

to

bound

unity

for

the

n

minimal

eigenvalue

of

the

matrix

SYSTEM Ne

consider

a system

+ z

= u

I

Here

i

~i

are

describe

by

means

constraint

(2)

oscillators

..,

is

of

pendulums

of

springs.

a scalar

control

(16)

constants

a control

boundary

with

n

,

u

and

= ~l'o

~l ( 0 )

'"

coot dinates

a system

body

OSCILLATORS

linear

i = 1 '

eigenfrequencies,

a

of

OF

R(T).

or

e

subject a

to

system

> 0 (2).

of

u(t)

Control

are Equations

masses must

(16)

attached satisfy

to the

conditions

%i(o ) = ~l' o

i

= 1, ....

n, (17)

1

~(T)

. ~,,

We a s s u m e

&l(T)

that

.

all

n~,~

i

are

positive

= 1 .....

n.

and

different.

Denote

I

e

=

0

o

and

Q =

min

(~

1§

- ~ ) > 0 1

(18) 0 = ~

The of

0

system change

< e

1

0, U.~i~< U~ < Um~ and j = 1 , 2 , . . . , N - 1

i=0,1,.--,N-1

Assuming that the second-order Kuhn-Tucker and constraints - qualification conditions hold for the constrained optimization, a local saddle point of the Lagrangian may be found if ~ large positive value of c is chosen.

6

Algorithm

and

Simulation

The above method was applied for the optimal path planning of the first three finks of robot PAMIIL In the joint space defined by Q3 = {-~r > 0, 11Y(t) II IlY(t)ll"0

-l IlY(t)ll"0

-l 0 there is a partition s of [T - r,T] such that if {tp } refines s then NH( 2 n

p=0

L(, tp)[Htgt](p))

dt

Notice that ~--{f o sin(t - s) du(s) = f 0 cos(t - s) du(s), i.e., if f =

L;J

is in Ag(I - P0)

G H then fz = fl". The projection 1-I of (I - Po)GH onto clAg(I - P0)GH can be represented as follows:

,,-, where ~ =

,tl..,-:,:,_ -1 cosh T

,..,.:

de

639

If f =

is not Ap(I - Po)GH, i.e., f2 :~ f ; then f - FI f is orthogonal to Ap(I - Po)GH.

Clearly, for any 0 < r < T, (I - PT-r)GH is not contained in clAp(I - Po)GH. On the other hand, the first condition for weak controllability does hold. Since rank [13, oc 13] = rank [ 0

1] = 2 the first condition must hold for 0 < r < T.

Still computing L and checking the column independence condition for L ( , T - r) is instructive. If 0 < s < t _ 0

(1-p)liutII2+ (29)

small.

(1-e-el-e2)ilUxxll2

inequality +

1 ~ . we h a v e

K(/],e)llull 2 From ( 2 6 )

and

(2S)

we h a v e

+ (b+p(a-1)-K(~,e)-C2(~.e2))tlull2

+ (6 -CI(K'el)-C2(~'e2))

u6dx

~ G(t)

=

G(t;p,a,b,c.d)

9

(l+p)llutll2+(l+e+el+e2)llUxxll2+(b+p(a+l)+K(p,e)+C2(~,e2)) + ( ~d ' + C 1 ( K . e l )

Ilull2

+ C2(~,e2))lu6dx

fl From ( 2 7 ) N(t) (30)

and

(28),

it

s

that

= N(t;p,a,b.c,d)

~ (a-p

)llutll2

+ (p-pc

- ~3 p e l - P e 2 - e 3 ) l l U x x

+ (pb-pcCs-PC2(~,e2)-PK(~,e))llull2 + ( (l-p) c-C3(~, e3) )Zu~dx

9 + (pd - ~1 p3 C

Now we r e q u i r e

(K,el)-PC2(~,e2)-Cd(~,e3)-pcCS)~U

that 1 > p > 0 1

-

e

-

e I

a

-

p

> 0

-

,

e 2

>

0

6dx

-

II2

649 3 - ~el-e2)

p(1-e

-e 3 > 0

,

b+p(a-1)-K(p.e)-C2(~,e2)

(31)

b -

cC 5 - C 2 ( ~ . e 2 )

(l-p)c d

- Cl(~,el)

choose

parameters

requirements. e.el,e2.e and

e3

First

3

small 3 32

-

-

K(~,e)

in

take

C2(~.e2)

enough

' so

order 1

p

= ~

and

fix

-

-

> 0

,

> 0 ,

- ~3 p C 1 ( ~ , e l ) _ P C 2 ( ~ ,

pd

We c a n

- C3(~,e3)

-

> 0 ,

> 0

,

e2)-C4(~,e3)-pcC

5

as

follows

the

so

that

the

to 6




0

and 3 - ~

p(l-e Then

take

a

=

1

1 > p

= ~

eI -

.

Choose c =

Next

choose

b

> 0

such

b = max(K(~.e) with

c

chosen

and

e2)

-

e3

c

> 0

2C3(~,e3)

+

=

1/d such

i

> 0 that

.

that

+ C2(~,e2).cC5 fixed

as

above.

+ C2(~,e2) At

last

+ K(p,e)}

we c h o o s e

+ 1 d > 0

satisfying 3

d=max{6Cl(~,el)+6C2(~,e2),~Cl(~,el)+C2(~,e2)+cCs+2C4(~,e3)}+3 With obtain

such

from

a

(29)

(non-unique)

choice

and

following

(30)

gI { l l u t l l 2 (32)

the

of

+ IlUxx 112 + llu[I 2 +

~ max(3/2,2b,d/3){llutll2 ~ (2b+d){llutll2

all

the

involved

parameters,

relations,

fuedx) ~ G ( t ) + IlUxxll2 + Ilul[ 2 + f u 6 d x }

+ IlUxxll2 + Ilull 2 + f u 6 d x }

,

n and

(33)

1,ut,2 + 1,Uxx,2 + 89 ~1{ n u t l t 2

+

llUxx

]12 + llull2

i; q d x § 3;uedx + fu~dx + fuedx) +

[2

9

we

650 Substitute

the

relations

(32 / a n d

(33)

into

the

identity

(19).

we

obtain I

d

It follows

1

dt G(t)

2

+ 4(2b+d~

G(t) also

Define

d

~ 2

dt G(t)

shows

that

an equivalent

(34)

and

where

(32)

C(Uo,Vo)

u t = v~

, and

By

G(t)

is a L y a p u n o v - l i k e

norm of

yields

= G(O)

H

functional

of s o l u t i o n s .

by for

(~)

6 H

that

is d e f i n e d

the p a r a m e t e r s

the c o n t i n u a t i o n

of h y p e r b o l i c

as mild

solutions

remain

strong

these

solutions

solutions

of

by

(u,v)

(17)

as

chosen.

exist

[5],

uniquely

(Uo.Vo)

H

at a u n i f o r m

the [5, Thm.

(u,~/

1.1]

of

with

semilinear

since

in

and

of

by

equations

(11),

decay

fixed

theory

evolution

solutions

Finally.

^

.

t II(: )II~ ~ 2G(t I ~ 2G ( U o , V o l e x p (- 2 ( 2 g + d ) ) t o

(35)

and

= 0

t 2(2b+d))

~ G(O)exp(-

H(~)JI~ = ,,UxxH2 + ,,uH 2 + Hvl, 2 Then

+ N(t)

that

(34) (32)

1

G(t)

and

(11)

evolution

(35) for

6 E

t = O, u = u ~

.

implies

with

that

t 6 [0.~) Moreover.

exponential

its

equations

(u.ut)

and (35)

that

they

shows

that

rate.

references,

initial

[g]

for

two

states

(Uo.Vo)

and

^

(Uo,Vo)

respectively,

there

is a relation t

(361

H (vu((ttl)) -

for

t

in

continuity and

( ~^ I ) tH) H t)

( ^u~ v~

the

common i n t e r v a l

of

g : H~(O) ~ L 2 ( ~ )

the G r o n w a l l

there

~ I I (uv ~ o

exists

corresponding

a

inequality,

sequence

of

solutions sup

+ f "g(u(s))-g(:(s)) 0

of existence. , the

From t h e

denseness

it f o l l o w s

that

points

of

~)

local

in

(Uo.Vo)

s H

such

that

the

of

the

satisfy

If(u(t)v(t)) - (vn(t)un(t))llH ~ 0

as

n ~

te[u.T] for

each

compact

interval

[O,r]

in

the

existence

interval

ds

Lipschitz

E = ~(A)

for any I c E

ILL2(

H ,

I

651 solution

(u,v)

continuity

os

globally initial

starting Go(-,.

)

in

C([O,m);H)

data

(Uo,Vo)

from

(Uo,Vo)

implies

that

Besides in

H o

.

the

Thus

(35)

solution

(35)

holds

The p r o o f

for

is

together

(u,v) all

with

exists solutions

with

completed.

REFERENCES [13

O.P. Bhutani, P. Mital, and G. Chandrasekaran, On invariant solutions of the generalized Boussinesq equaitons, Intern. J. Engr. Sci., 26 (1988), pp. 307-310.

[2]

P. C o n s t a n t l n , C. F o i a s , B. N i c o l a e n k o , a n d R. Temam, Integral and Inertial Manifolds for Dissipative Partial Differential Equations, Sprlnger-Verlag, New Y o r k , 1 9 8 8 .

[3]

J.M. Chidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations, J. Math. Pures Appl., 66 (1987), pp. 273-319.

[4]

J. Hale, Asymptotic behavior of Dissipative systems, Math S u r v e y s & M o n o g r a p h s , V o l . 2 5 , AMS. P r o v i d e n c e , 1988.

[~]

A. H a r a u x a n d E. Z u a z u a , D e c a y e s t i m a t e s f o r some s e m i l i n e a r damped hyperbolic problems, ARch. R a t i o n a l Mech. Anal., 100 (2) (1988). pp. 191-206.

[G]

R. H i r o t a . Exact N-soliton solution long waves in shallow-water and in Math Phys., 17 ( 1 9 7 3 ) , p . 8 1 0 - 8 1 4 .

sol the nonlinear

wave equation lattices, J.

[7]

J.L. Lions, Control of Distributed Grauthier-Villars, Paris, 1985.

Singular

Systems,

[s]

H.P.

Appl.

McKean, Math.,

Boussinesq's equation on 34 ( 1 9 8 1 ) , p p . 5 9 9 - 6 9 1 .

the

circle,

Comm.

A. P a z y . S e m i g r o u p s partial differential

[10]

J. Sprekels. One-dimensional thermomechanical phase transitions with non-convex potentials of Ginzburg-Landau t y p e , IMA P r e p r i n t , ~505 (1989).

[11]

F. U r s e l l . The l o n g - w a v e p a r a d o x i n t h e waves, Proc. Cambridge Philos, Soc., 49

[12]

Y. You, T e c h n i c a l University. 1989.

of linear operators and applications equations, Springer-Verlag, 1983.

~102,

Center

of

Pure

[9]

Report

to

theory of gravity (1953), p. 685-694.

for

Appl.

the

Math.,

Purdue

DIRECT ADAPTIVE STATE-SPACE

CONTROL IN A SETTING*

Edward W. Kamen Department of Electrical Engineering University of Pittsburgh Pittsburgh, PA 15261 (USA)

SUMMARY Direct invarlant

adaptive

n-dimensional

2n-dimensional difference

adaptive

discrete-time

state representation

equation.

simple procedure

minimal

control of a single-input

The central

system is approached

constructed

result

The only assumption

is not required to be minimum phase.

(4n-dimensional) relationship

regression

estimation

this paper Elliott

schemes.

The estimation

exciting

system i3 exponentially desired poles

The approach

parameters

estimation

is that the

the given system

taken in the paper is based of an expanded

vector of the linear

parameters.

In this way the

is reduced to a

The particular

parametrization

of the parametrization

of controller

parameters

If the expanded

(as opposed to persistently

large values

constructed

in

derived by

i3 based on the standard

regression

exciting),

stable with the closed-loop

for sufficiently

of a

in

problem which can be solved using existing

version

least squares algorithm.

sufficiently

in particular,

vector with the coefficient

is a "state-space"

[I].

recursive

2.

on the system model

the desired controller

of the controller

linear-in-the-parameters parameter

parameters

of the system input as a linear function

containing

determination

in terms of a

from the input/output

the controller

order of the system is known a priori;

on the expression

directly

linear time-

of the paper is the derivation

for directly determining

pole placement.

single-output

poles

vector is

the closed-loop

located close to the

of the time index.

Preliminaries Consider

the single-input

single-output

time system defined by the input/output

y(k)

=

n Z aiY(k-i) i=l

In (i), k is the integer-valued output,

and u(k)

is

the

+

n Z b.u(k-i)~ i=0 discrete-time

real-valued

linear time-invariant

difference

control

equation

o index, input.

discrete-

(ARMA) model

(I) y(k)

is the real-valued

Note that

if

the

*This work was supported in part by Eglin Air Force Base under Contract F08635-90-K-0210.

No.

656 coefficient

b 0 is n o t

zero,

there

is a d i r e c t

feed between

the input

and

output. The coefficients assumptions

on the

is t h e m i n i m a l transfer

a. a n d b.

in

system model

order

of the

(i) a r e

unknown

(1) a r e t h a t

system.

the

fixed

order

Here minimal

constants.

n is k n o w n

order means

The only

and that

that

the

function n Z b,z -i 1 a(z)

n -i 7. a.z i=l

i-

does

not have Now

let #(k-l)

#T(k-l) where

any common

-

denote

[y{k-1)

"T" d e n o t e s

system

at t i m e

poles

the

a n d zeros.

the 2 n - e l e m e n t

y(k-2)

...

transpose

the

y(k-n)

vector

u{k-l)

operation.

u(k-2)

Defining

defined

by

... u(k-n)]

the

state

representation

(see

x(k)

of t h e

k to b e x(k)

we h a v e

regression

following

x(k+l)

= ~(k-l)

2n-dimensional

= Ax(k)

,

state

[2, p. 75])

+ bu(k) (2)

y(k)

~ cx(k)

+ b0u(k)

,

where

A=

b -

bI

b2

9 . .

0

0

0

0

~

0

0

...

1

0

0

0

.,.

0

0

0

...

0

0

0

0

9 . .

0

0

...

0

0

1

0

.,,

0

0 0

:

:

a2

...

1

0

...

0

0

0 0

an_ 1

b n 84

an

"aI

0

0

...

0

0

0

0

0

0

...

0

0

0

0

[b 0

0

...

0

1

0 th

...

0

bn_

...

1

0

0

1

0

th +(n+l) position

0] T

(n+l) position

c = Now

[a I

consider

a2

9 -.

an_ 1

the control

u(k)

an

bI

b2

generated

... b n _ 1 by

b n]

n

657 n

n

u(k) = - Z diu(k-i) i=l

-

Z ciY(k-i) i=l

,

(3)

where the c. and d. are the real-valued controller parameters. Note that in l l (3) we are not allowing the input u(k) at time k to depend on y(k) at time k since there may be a direct Let 8

c

feed from u(k)

denote the 2n-element

to y(k)

in the given system

column vector of controller

(i).

parameters

defined by 8~ = [c I

c2

... Cnd I

d2

Then the feedback

control

where x(k)=~(k-l)

is the state defined

u(k) = -e~x(k)

feedback

control

... d n]

(3) can be written

in the form

(4)

, in the previous

(4) into the state representation

state description

of the resulting

x(k+l)

closed-loop

section.

Inserting

the

(2), we have the following

system:

= (A-b0~)x(k)

(5) y(k) = cx(k)

+ b0u(k)

The 2n poles of the closed-loop of the closed-loop following

system are of course equal to the zeros

polynomial

l:

If n is the minimal

order of the system

all 2n poles of the closed-loop state feedback of the form

(ii)

det(zI-A+bsT). c

We then have the

result.

Proposition (i)

characteristic

the state representation teachability U = [b

system

(I), then

(5) are assignable

by using

(4);

(2) is reachable;

that is the 2nx2n

matrix Ab

...

A2n-lb]

is invertible. Proof:

From known results,

control

of the form

system

(I).

all 2n closed-loop

Thus if n is the minimal

assignable

by using state feedback

Since pole assignability

implies

order,

in [2, pp. 75-77];

manipulations

teachability,

however,

we also have

(2) is pole which proves (ii).

(i).

9

of the state representation

(2) is

the above proof avoids the matrix

{Pi:

i=I,2,...,2n}

within the unit disk of the complex plane, there i8 a unique controller det(zI-A+bST)c In addition,

by using a

given in [2].

Now given a desired set

reachable,

the state model

of the form u(k)=-8~x(k),

It should be noted that teachability proved

poles are assignable

(3) if and only if n i8 the minimal order of the given

=

(z-Pl) (z-P2)

using Ackermann's

formula

of closed-loop

if the state representation

parameter ... (see

poles

vector 8

c

located (2) is

satisfying

(z-P2n) [3]), we can express

(6) 8

c

in the

658 following

form.

~(z)

First,

=

writing

...

(z-pl) (z-p2)

(z-P2n)

2n-i Z ~i zi i=0

= z 2n +

define ~(A)

2n-1 ~ ~'iA i i=0

= A 2n +

w h e r e A is the s y s t e m m a t r i x of the s t a t e solution

8

satisfying

c

is the i n v e r s e

In the next be e 3 t i m a t e d

3.

Direct Again

0

...

0

(7)

i]

(8)

matrix

defined

we show that the c o n t r o l l e r

from measurements

Determ/nation consider

[0

of the t e a c h a b i l i t y

section

directly

Then the u n i q u e

(2).

8 T = ~U-I~(A) , c row v e c t o r g i v e n b y =

and U

representation

(6) is g i v e n b y

w h e r e ~ is the 2 n - e l e m e n t -i

,

of the

of the C o n t r o l l e r

the s y s t e m

system input

Parameter

(i) with t h e s t a t e

x(k+l)

= Ax(k)

+ bu(k)

y(k)

= cx(k)

+ b0u(k)

in P r o p o s i t i o n

parameter

vector 8

i. can

e

a n d output.

Vector

representation

(9) w h e r e A, b, c are as d e f i n e d controller Theorem

i:

parameter Suppose

in S e c t i o n

vector 8

c that n is the m i n i m a l

state m o d e l

(9) is r e a c h a b l e

closed-loop

characteristic

denote

Then the s y s t e m

u(k) where

8

= z

= x(k)

input u(k)

is the c o n t r o l l e r c d e f i n e d b y (8).

Proof:

With ~ defined by ~U-iAib-

the s t a t e

the

k e y result.

of the s y s t e m

vector

defined

2n-i + • ~ix(k+i-2n) i=0

Given

(I) so that the

a desired

parameter

vector

(9) y i e l d s

.

(10)

,

given by

(ii) (7) a n d ~ i3 the row

of U we have

i=0,1,2,...,2n-2

= 1 .

by

in the f o r m

- 8Tx(k)

(8), b y d e f i n i t i o n

equation

to d e t e r m i n i n g

2n-I i Z ~i z , i=0

can be e x p r e s s e d

0 ,

aU-iA2n-lb Solving

+

= ~u-l~(k+2n)

vector

order

(i.e. U i3 invertible) .

the 2'n-element column

~(k)

Our approach

polynomial 2n

~(z) let ~(k)

2.

is b a s e d o n t h e f o l l o w i n g

(12) (13)

659 .

x(k+i)

Multiplying

i-l.,

= Alx(k)

+

b o t h sides of

E A~-3-1bu(k+j) j=0 (14)

,

i=l,2,...

on the left b y ~U

-i

(14)

and using

(12) a n d

(13), we

obtain ~U-ix(k+i)

= ~U-IAix(k)

~U-ix(k+2n) Multiplying

,

= ~U-IA2nx(k)

i=0,i,2,...,2n-I

(16)

b o t h sides of the ith e q u a t i o n

in

~U -I ~i x (k+i) = ~U-iZiAix(k) Adding

(16)

and

Ix

~U -I

(k+2n)

+

(15)

+ u(k) (15) b y Zi yields

(17)

i=0,1,2, ...,2n-I

(17), we have

n

7. (k+i) i=0Zi x

]

~U-I

2nl

2n

=

]

7. ziA~ x(k)| i=0 ]

+

+ u(k)

.

Then s e t t i n g

~(k)

= x(k)

~(A)

and rearranging Throughout order

terms,

+

= A 2n +

2n-1 7. ~i Ai i=0

8T . ~U-I~(A) c we o b t a i n (ii).

the r e m a i n d e r

of this

so that the state m o d e l N o w let w(k)

2n-i 7. ~ i x (k+i-2n) i=0

denote

,

9 section

(9) is r e a c h a b l e

the e x p a n d e d

we a s s u m e

that n is the m i n i m a l

and the f o r m

4n-element

regression

(II) exists. vector defined by

~(k) ] w(k)

=

Lx(k-2n)] a n d let 8 d e n o t e

the 4 n - e l e m e n t

parameter

vector defined

by

= [~u-I _8T]

eT

c Then

(ii)

can be r e w r i t t e n

in the f o r m

u(k)

= 8Tw(k+2n)

,

or u(k - 2n) From

(18)

we see that 8 can be e s t i m a t e d

s y s t e m inputs direct

and outputs,

procedure

parametrization Elliott's

and since

for e s t i m a t i n g (ii) or

parametrization

The e s t i m a t i o n parameter

= 8Tw(k)

estimation

(18)

8

directly

from measurements

is c o n t a i n e d

o the c o n t r o l l e r

turns

(18) of the

in 8, we t h e r e f o r e

parameter

vector

out to be a " s t a t e - s p a c e "

8 . c version

have a

The of

[I].

of 8 b a s e d on schemes

(18)

c a n be c a r r i e d

such as g r a d i e n t - t y p e

out u s i n g e x i s t i n g

algorithms

or r e c u r s i v e

660 least squares

(RLS).

We shall use the RLS a l g o r i t h m

which is defined as

follows. Given a 4nx4n positive sequence

definite matrix Q, let P0,PI,P2,...

of 4nx4n symmetric matrices

denote the

defined by

P0 = Q

PkW(k+l)wT(k+l)Pk Pk+l " Pk -

,

k~0

(19)

.

1 + wT(k+l)PkW(k+l) Then the RLS estimate

~(k)

of 8 is given by

~(0) = e

o

= initial estimate

PkW(k+l) ~(k+l)

= ~(k)

+

[u(k-2n+l)

-wT(k+l)~(k)]

, k>0

. (20)

l+wT(k+l)PkW(k+l) Defining ~(k-2n) we can interpret time k-2n.

~(k-2n)

= wT(k)~(k-l)

as the estimate

,

at tim~ k of the system input u a t

We can then define the "input estimation e(k) = u(k-2n)

Rewriting

(20) in terms of e(k),

- ~(k-2n)

error"

.

we have

PkW (k+l) ~(k+l)

= ~(k)

+

e(k+l)

,

k>0

(21)

l+w T (k+l) Pk w (k+l) From known properties Proposition

~:

of the RLS ~igorithm,

The estimate

k*~, where 8* is an element

~(k) defined by of 4n-dimensional

By the above proposition, e*.

Unfortunately,

we have the following

(19) and

(20) converges

Euclidean

the RLS estimate

~(k)

result.

to 8* a3

space.

always

converges

to some

e* is not equal to 8 in general.

Thus, letting 8* denote c to 8 , we have that 8* is not equal to 8 in c c c if the expanded regression vector w(k) is sufficiently

the part of 8* corresponding general. exciting

However,

(SE), we can get 8* to be as close to 8 as desired. c c

First,

w(k)

SE if for some k~4n, k Z w(i)wT(i) i-0 Then applying

is positive

known results

(e.g. see

that w(k)

definite. [4]), we have the following

Theorem 2:

Suppose

is SE and that P0=Q=~I,

algorithm.

Then given any C>0, if ~ is sufficiently

lle*-ell< where

l J l J is the Euclidean

norm.

~>0, large,

result.

in the RLS

is

661 in T h e o r e m 2 i m p l i e s

The result

that we can get

as desired by taking the initialization that w(k)

is SE).

[4] to guarantee of ~(k)

corresponding

The Adaptive

to be

large

small

as

(assuming

One could also use the modified RLS algorithm developed that ~(k)~e as k~m and thus ~

in

(where ~ (k) is the part c c c that w(k) is SE. Thus it is possible to

to 8 ) assuming c parameter vector

identify the controller

4.

Ile~-el I

P0 to be sufficiently

8

c

(k)~8

directly

from input/output

data.

Controller

Again let ~

(k) denote the part of the ~(k) corresponding to the C controller parameter vector 8 . Then the adaptive controller is simply c realized by setting u(k) = _~T(k)x(k) . c closed-loop system is given by the state equation

The resulting

x(k+l) which can be rewritten x(k+l)

= Ax(k)

- b~T(k)x(k)

,

(22)

in the form = (A-becT)X(k)

Since the desired pole locations

+ b[sT-~T(k)c c ]x(k)

were chosen

to be within the unit disk,

the

system x(k+l) is exponentially

stable.

=

(A-bS~)x(k)

Then if

11%is sufficiently

small for k large,

II it follows

(22) is exponentially

from standard

closed-loop

system

closed-loop

poles will be "close to" the desired poles.

then have the following Theorem 3: (19),

Suppose

(20).

and in the limit as k ~ ,

the

Using Theorem 2, we

result.

that w(k)

is SE and P0=Q=~I,

Then if ~ is selected

(22) is exponentially

stable

results that the

sufficiently

~>0 in the RLS algorithm large,

stable with the closed-loop

the closed-loop

system

poles located close to the

desired poles in the limit. As noted in the previous algorithm

in

closed-loop

5.

section,

if w(k)

i3 SE and if the modified

RLS

[4] is used, ~ (k) will converge to e . In this case the c c poles will be assigned to the desired locations in the limit as

A n Example

Consider difference

the discrete-time

system given by the second-order

equation y(k) = y(k-l)

- y(k-2)

+ u(k-1)

+ 2u(k-2)

input/output

662

The poles of the system are located at 0.5+j.866 circle,

and thus the system is unstable.

which is outside the unit

The system is also not m i n i m u m phase

since there is a zero at z=-2. Letting x(k)

the coefficients

A =

denote the state vector d e f i n e d by T x (k) = [y(k-1) y(k-2) u(k-l) u(k-2)] of the state model

,

(2) are given by

1 0 0

c

-

-i

[i

1

2]

The teachability matrix is

U = [b

Ab

A2b

The objective

A3b] = 0

0

1

0

is to design the adaptive

closed-loop poles are at z=0,0. parameter vector 8

which accomplishes

c

8T = ~U-IA4

c

=

controller so that the

By Ackermann's

the controller

this is given by

[ 2

1

-7

-7

AS discussed in the previous section,

formula,

1

2

(23)

7]

the adaptive

controller

is realized by

setting u(k) = -~T(k) x(k) , (24) c where ~ (k) is the estimate of 8 generated from the RLS algorithm (19) and c c (20). Since w(k+l) depends on y(k-5) and u(k-5), to evaluate (19) and (20) for k>0 we need initial values of y(k)

for k=-5,-4 and initial values of u(k)

for k=-5,-4,-3,-2,-i.

(20) and the control law

Equations

(19),

(24) were

p r o g r a m m e d with the initial conditions y(-5) u(-5)

- 0, u(-4)

= S, y(-4)

= 0, u(-3) P0 = ~(0)

It should be n o t e d that u(-3) (Since ~(0)=0,

if u(-3)=0 by

The computer Figures

8 . c

- 0, u(-l)

= 0

(104) I =

o

was set equal to 1 to insure that ~(i)~0. (19) we see that ~(k)=0

implementation

1 and 2, respectively.

converge to zero.

= 0

- I, u(-2)

for all k>0.)

resulted in the control and output shown in Note that both the control and ~he output

Shown in Figure 3 is the estimate of the third element

This estimate does converge to the correct

value

(see

(23)).

The

of

663 e3timates

of t h e other e n t r i e s

Thus the c l o s e d - l o o p desired

poles

of e also c o n v e r g e to the correct c are p l a c e d (after an initial t r a n s i e n t

values. at the

locations. ACKNOWLEDGEMENT

The a u t h o r

wishes

tO the work p r e s e n t e d relevance

to t h a n k John C h i a s s o n in this paper,

of A c k e r m a n n ' s

for s e v e r a l

a n d in p a r t i c u l a r

discusslons

for p o i n t i n g

relating

out the

formula.

REFERENCES [i]

H. Elliott, "Direct A d a p t i v e P o l e P l a c e m e n t with A p p l i c a t i o n to N o n m i n i m u m Phase Systems," IEEE Trans. A u t o m a t i c Control, Vol. AC-27, 720-722, 1982.

[2]

G.C. G o o d w i n and K.S. Sin, A d a p t i v e Filtering, P r e n t i c e Hall, E n g l e w o o d Cliffs, NJ, 1984.

[3]

T.K.

[4]

E.W. Kamen, Convergence Processing,

Kailath,

Linear Systems,

Prentice-Hall,

Prediction

Englewood

and Control,

Cliffs,

NJ,

1980.

"A R e c u r s i v e P a r a m e t e r E s t i m a t o r Y i e l d i n g E x p o n e n t i a l u n d e r S u f f i c i e n t E x c i t a t i o n , " Circuits, Systems a n d S i g n a l Vol. 8, No. 2, pp. 207-228, 1989.

y (k)

8

u (k) '

1.2

6 9

~

4

0.4

2

. . . . . . . . . . . . e~--~ k 9 o9 -0.4 ) -0.0 "o -i. 2 9

Figure i. The control signal.

:=::: -2

......... 6 ......

9

Figure 2. The output response.

1.0~ 0.8 0.6 0.4 0.2'

.99 "::-.-e.

00,0,00

00000 4 ...................

Figure 3. Estimate of third entry of 8c.

pp.

k

Pole Placement via Generalized Predictive Control Youbin PENG"and Raymond HANUS Laboratoire d'Automatique, C.P.165,'Universit~ Libre de Bruxelles 50, Avenue F.D. Roosevelt, 1050-Brussels, Belgium Abstract: T h i s paper presents a tuning strategy for generalized predictive control(GPC). I t is shown that the GPCdesign can be made to be equivalent to the usual pole-placement technique by the introduction of suitable f i l t e r s and the selection of particular horizons. Hence, the desired closed-loop behaviour can be obtained and the s t a b i l i t y of the closed-loop system is guaranteed. The resulting GPC proves to be robust even i f overparameterization or near pole-zero cancellation is encountered. Two simulation examples i l l u s t r a t e the good performances of the proposed algorithm.

I. Introduction The basic idea behind long-range predictive controllers, is to build a predictive model of the plant over a range which is larger than the plant delay and to use this model in the control law design. Several long-range predictive control designs have been developed by many authors, for instance, Mosca et al.(1984),De Keyser et ai.(1985) and Clarke et ai.(1985). In particular, the generalized predictive control(GPC) approach derived in Clarke et ai.(1985) is able to handle control problems such as unknown and possibly variable time-delay, non-minimum phase behaviour and plant-model mismatch. GPC is essentially an algorithm in which the output y(t) is predicted over f i n i t e horizon and a sequence of control signals are calculated to minimize a quadratic cost function subject to an assumption made about the future control increments. I t offers several possible tuning knobs such as costing horizons and control horizon. However, there is no straightforward relationship between the cost-function used in GPC and the characteristic polynomial of the resulting closed-loop system. Usually, the desired dynamic behaviour can be approached by introducing suitable design f i l t e r s or polynomials. In particular, Gorez et ai.(1987) imposes the desired closed-loop poles by f i l t e r i n g the input, output and reference signals and using the f i l t e r e d signals in the cost function. The f i l t e r poles coincide with the desired closed-loop poles. However, there is s t i l l one part of the poles of the closed-loop system which depend on the horizons of GPC. These poles are not necessarily inside the unit circle, and the s t a b i l i t y of the closed-loop system is not always guaranteed. Here, we show that these poles can be assigned at the origin of the q-plane by the selection of particular Un leave from the Department of Automatic Control and Computer Engineering, Huazhong University of Sciences and Technology, Wuhan, China

665 horizons of GPC without influencing the poles assigned with the f i l t e r i n g process. In this case, the GPC design is equivalent to the usual pole placement technique. The major theoretical and algorithmic problem of the pole placement technique lies in the solution of the Diophantine equation. For instance, i f the system model equation is overparameterized, then a common factor between the numerator and denominator polynomials of the system transfer function w i l l certainly occur and the Sylvester resultant determinant w i l l be n u l l . In this case, the algorithm is ill-conditioned. When GPC is made to be equivalent to the pole placement, the same ill-conditioning problem w i l l occur i f overparametrization is encountered and there is no precaution. However, i f we use the recursive matrix inversion algorithm proposed by Favier(1987) in GPC, such drawback can be avoided by simply tuning the control horizon. Clarke et al.(1987a,b) and Mclntosh et ai.(1989) also derived conditions on the horizons of GPC so as to assigne the poles of closed-loop system at the origin of the q-plane. However, their conditions are less general. Furthermore, overparameterization can not be tolerated when a deadbeat control is achieved. This paper is organized as follows. Section I I defines the modelling assumptions and predictive model. Section I l l recalls the generalized predictive controller. Section IV investigates the closed-loop performances. Section V presents a new tuning strategy of GPC. Section VI shows the simulation results, and conclusions are given in section VII.

II. Modelling Assumptions and Predictive Model Consider a single-input single-output(SISO) discrete linear time-invariant system described by the following CARIMAmodel:

A(q~)y(t) =B(q") u (t)+C (q") ~ ( t ) / &

2. I

where { y ( t ) } and { u ( t ) } are sequences of outputs and inputs respectively. { ~ ( t ) } is a sequence of random variable with E{~(t)/F,.~)=O, E{~(t)2/F,.~):o ~ and F,., is the sigma algebra generated by {y(t-1) . . . . . y(O)} together with i n i t i a l conditions. A:l-q" is the differencing operator. A(q"), B(q") and C(q") are polynomials of degrees n,, nb and no in the backward s h i f t operator q": A(q"):l+a,q% ... +ao.q''~ B(q") = b,q"+ . . . +b,bq~b C(q'1)=1+c,q"+ . . . +c=q~ The optimal j-step-ahead predictor is given as follows. See Goodwin and Sin(1984). Theorem 1: For the system described by (2.1), given measured input and output data up to time t and any given u(t+j) for j>O, the optimal j-step-ahead predictor y ' ( t + j / t ) satisfies

C(q")y" ( t + j / t ) =Fj(q")y (t) +Ei(q")B(q") Au(t+j)

2.2

where

y" (t+j/t)=ECY(t+j)/F,}=y (t+j) - Ej{q") ~(t+j)

2.3

and

C(q") :Ej(q") A(q")+q JFj(q' )

2.4

666

where A(q"):AA(q"):I+A,q"+...+A~q ~, ni:n,+l, and Ej(q"):eo+e,q"+...+eF~qJ+', Fj(q")=~+~.,q"+...+~q', n,=n,, also,

I-~

E{ [y(t+j) -y* (t+j/t) ]2)=E{E[Ej(q")~ (t+j) ]~F,)= Z e,2o2

2.5

VVV

I-0

For the predictor (2.2) to be optimal for all t, i t is necessary that the initial conditions for y ~ be appropriately chosen. However, since C(q") is asymptotically stable, the effect of arbitrary i n i t i a l conditions will diminish exponentially. Thus we can simply define a prediction 9(t+j) with any i n i t i a l conditions as C(q-1))(t+j)=Fj(q~)y(t)+Ej(q1)B(q-')Au(t+j)

2.6

In the sequel, we will replace y * ( t + j / j ) by ) ( t + j ) . For j~k(k is the system time-delay), the prediction 9(t+j) depends entirely on available data, but for j>k, assumptions need to be made about the future control actions. It is generally assumed that the controls are to be performed in open-loop by ignoring the future noise sequence {~(t+j)}. III. G e n e r a l i z e d P r e d i c t i v e C o n t r o l

The generalized predictive controller(GPC) minimizes a multi-stage cost function of the form: N2 N2 J=E{Z [y(t+j)-w(t+j)]%>,Y-~u(t+j-l) 2} 3. I l-N1

1-1

where N, i s the minimum c o s t i n g horizon, N2 i s the maximum c o s t i n g horizon and ;~ i s a

nonnegative weighting factor, {w(t)} is the sequence of the reference signal. The expectation in (3.1) is conditioned on data up to time t assuming no future measurements are available. One can then ignore the future noise signals and use the prediction ~(t+j) to generate the control law. The prediction .~(t+j) can be separated in two parts as follows: Partioning Ej(q")B(q") according to Ej(q") B(q") =q"C(q") Gj(q-')+q+'Qj(q~)

3.2

where Gj(q")=go+g,q"+...+gj.,q-j§ Qj(q~)=qj.o+qj.lq"+...+qj.,qq"~, nq=nb-2 Substituting (3.2) in (2.6) leads to ~ (t+j)-Gj(q") Au(t+j- l)+~(t+j/t)

3.3

where ~(t+j/t) is the prediction of the output assuming there are no future changes in Au(t), i.e. C(q")) (t+j/t) :F~(q")y (t)+Qj(q")Au (t- I)

3.4

In the GPC, a control horizon Nu is introduced. That is, after an interval NuNu. Taking this hypothesis into account, the minimization of (3.1) with respect to the future auxiliary control sequence {Au(t) . . . . . Au(t+Nu-1)} yields

Or=[G~TG,+~I]"G,T(~-? (/t))

3.5

667 where

~r=[9(t+N,) . . . . . ~(t+N2) ] OT:[Au(t) . . . . . Au(t+Nu-I) ]

W%[w(t+N,). . . . . w(t+N2) ] ~(/t)r= [~(t+N,/t) . . . . . 9(t+NJt)]

3.6

and a (N2-N,+I)xN~ matrix

I gN,-, "'" gm-Nu gm 9gNI-Nu+I GI = 9 gN2-, " g:Nu

3.7

in which gj=O i f j,I]"G,r as KT=[KN,. . . . . KN2] and using (3.3) and (3.4) results in R(q") u(t)=T(q" )w(t )-S (q')y (t) N~

4. I N2

where R(q")=[C(q")+~ KjqIQ~(q~)]D(q"), S(qI) = Z K~Fi(q"), T(q")= ~.=KjC(q") I'N1

J-N1

4.2

I-NI

Taking (2.4) and (3.2) into account, the closed-loop polynomial can be written as

A'(qi):C(q ') [A(q")+q" N~ Kjqj(B' (q") -A(q")Gj(q") ]

4.3

J-N|

Hence, the closed-loop system has the poles corresponding to the roots of C(q") and that of Am(q")=A(q")+q' ~2KJqj(B' (q")-A(q")Gj(q"))

4.4

l-N1

By modelling assumption, C(q") is asymptotically stable. However, Am(q") is not always stable and more analysis must be undertaken. We note that Am(q") is depending on the choice of N~, N2, Nu and ~. In fact, i t is not a simple task to adjust the weighting factor X in such a way that Am(q") is stable. Here, we only discuss the case when ~=0. I f ~ is increased, the controller is progressively detuned. However the

668 closed-loop behaviour caused by non null I ,

if

it

is good and known, can be

alternatively obtained by introducing suitable design f i l t e r . This w i l l be discussed at the end of this subsection. As the calculation of K involves the inverse of matrix G~TG~, the rank of G, should be equal to N,. We now show when this condition is satisfied. The proof can be found in Peng(1990). Theorem 2: Two polynomials ~(q") and B(q") of order na and nb are r e l a t i v e l y prime i f and only i f rank(G,)=N, with either of a)N,>%, N2>_n~+N,-I, N.:nl; b)N,=%, N2>Nu+nb-1, Nu>nA. VVV Now, l e t us show when A'(q ~) is reduced to C(q"). Theorem 3: For the GPC approach described above, i f A(q") and B(q") are r e l a t i v e l y prime and i f N,>nb, N2>_%+N,-I, No=%, >.=0, or N,=%, N2>_Nu+n:-l, N,>%, >.:0, the closedloop c h a r a c t e r i s t i c equation (4.3) is reduced to A'(q")=C(q"). Proof: From theorem 2, G,TG, is i n v e r s i b l e . Let

~ Kjq~B' (q"):B~

"~. KiqJA(q-')G1(q"):A.(q~)+a,(q)

|-NI

4.5

i-~1

where B.(q") and A.(q") are polynomials without positive powers of q, Bp(q) and Ap(q) are polynomials with positive powers of q only. From the fact that the f i r s t j terms of Gj(q~) are the f i r s t j terms of the impulse response of B'(q;')/A(q'~), we deduce that Ap(q)=Bp(q). Hence, (4.4) can be rewritten as Am(q')=A (q") - q"A. (q") +q~B. (q '' )

4.6

A.(q") can be derived from (4.5) as N2

A.(q")= Z KjqJA(q")G~(q")-A.(q) J-N1 N2

= ~. Kj[(gF,~,+...+g~.~)+.. "+ (gF,a.A-,+gF2aJ - q~.2+ gH~.Aq-.~+,]

4.7

I'NI

Noting that KTG,=[I N2

0.....

0], we have

N2

Kjgj.,=1,

~ Kjg~=O, i=2 . . . . ,N.

4.8

I f Nu>nl, we can substitute the f i r s t nA equations of (4.8) into (4.7). This yields A.(q')=q(A(q")-l). Moreover, i f N,_>%, we have B~

Hence, for the values of N,, N2

and N, mentioned in the theorem statement, A~(q")=1 and A'(q")=C(q~).

W/V

I f C(q") is taken as unit, which is often the case of adaptive control, the selection of particular horizons described in theorem 3 w i l l lead to a deadbeat control law(i.e. A'(q')=1). In this case, the system output has generally a large over-shoot to reference and disturbance changes. In order to penalize this over-shoot, a desired characteristic polynomial can be assigned. This is the case of the pole placement control I er. The GPC algorithm can be easily adapted to provide a standard pole placement controller by considering the augmented system: A(q")y,(t) =B(q")Au,(t) +P(q") C(q~)~(t)

4 .g

where y,(t)=P(q")y(t), u,(t}:P(q")u(t) and P(q~) is a suitable polynomial.

669

We can derive the control law by minimizing the cost function which uses the f i l t e r e d signals: N2

N2

J=E{~ (y,(t+j) -w,(t+j) )2+lT_~u,(t+j- I )2) J-N1

4.10

l-I

Since the role of P(q")C(q") in the augmented system is similar to that of C(q") in original system, all the subsequent results of the original GPC s t i l l hold only with C(q") replaced by P(q")C(q") and y ( t ) , u(t) and w(t) replaced by y,(t), u,(t) and w,(t). However, the control law is s t i l l valid for y ( t ) , u(t) and w(t) since the common factor P(q") may be dropped in both sides of (4.1). Hence, the closed-loop characteristic equation is A" (q")=P (q'~)C (q") Am(q ~)

4.11

where Am(q" ) is the same polynomial as t h a t defined by (4.4). We can choose P(q") so as to assign part of the closed-loop poles to a desired l o c a t i o n . The closed-loop system has also the poles of Am(q" ) which is independent of P(q')C(q"). However, i f we take N,>nb, N~>n~+N,-I, No:nA, I=0, or N~:%, N=>N,+%-I, N,>nA, X=O(i.e. Am(q':)=1), the closed-loop system has only the poles of P(q')C(q") and poles at the o r i g i n of the q-plane. In t h i s case, the GPC design is equivalent to a standard pole placement c o n t r o l l e r where C(q") is the optimal observer polynomial. I t is c l e a r t h a t more general alternatively

Am(q") caused by non n u l l

X, i f

it

is good and known, can be

obtained by introducing i t to P(q").

V. Tuning Strategy of GPC The condition required when solving the Diophantine equation for a pole placement controller, is that A(q') and B(q~) are relatively prime. Here, we see from theorem 2 that this condition is also necessary for assuring the i n v e r t i b i l i t y of G,TGI. However, we can avoid the singularity of GITG~by simply ruducing the control horizon N. until G,TG, is regular. To this end, we use the recursive algorithm for inverting G,TG~with respect to N~. This recursive algorithm was f i r s t proposed by Favier(1987) in order to decrease the computation time of GPC. Here, i t is also used for avoiding the singularity problem of the matrix inversion. We summarize the idea as follows: The matrix M=G,TG, has the following particular form: N2-1

N2-1

I-NI-I

i-NI-I

7. g Z

N2-1

Z g,.,g,

M =

T. g,g,.,

N2-1

...

N2-1

Z g,.2

X g~g~-.u+,

I-NI*I N2-1

...

~' g~-Ig~N.*,

I-NI-1

I-N1-1

I-N1-1

N2-1

N2-1

N2-1

Z g~N.+,g, i=Nl-I

or in a concise form

2 g~.u.,g~, --i-N1-1

~ g~,=.2 I-N1-1

5.1

670

i M, : 9 : M :

with

5.2

MN~ 1

.2-1 F MH M, =,-NI-17" g2, MN:M and M'=|'NL"L

N2-1

with

M2

.........

N2-I

N,=[ -~a igH+'g} . . . . . j i-

N2-1

~" gH+'gH+2 ] ' LI=N'T and ~,= T. gj_i+,2

I-NI-I

where Mr, is a ( i - 1 ) x ( i - 1 ) matrix and e, is a scalar.

J-NH

matrix,

L, is a ( i - 1 ) x l

column matrix, N, is a I x ( i - l )

By using the block matrix inversion f o r m u l a [ K a i l a t h , algorithm f o r i n v e r t i n g M.

row

1980], we get the recursive

N2-1

I . Calculate MI'= [ 7. g2 ].i; I-NI-I

2. For i : 2 1 . . . , N , ,

calculate M,I by using the formula M,.,"+M,.II L~N~M,.I-,/#' _M,.I-,LJ#,"

M,I :

5.3 -NjM,.,"/#,

1/#,

where #h=e,-N,M,.I~L~ is known as the Schur complement of Mrl. The above recursive algorithm is v a l i d i f and only i f M,I e x i s t s f o r i : l , . . . , N

u.

In f a c t , M,~ e x i s t s i f and only i f MHI e x i s t s and #,~0. Hence, we can f i n d the maximum achievable value of Nu by examining when #N.+, is n u l l . We now present a tuning strategy of GPC: I . Choose P(q17 as the desired c h a r a c t e r i s t i c polynomial and use the augmented system (4.97 to derive the control law by minimizing the cost f u n c t i o n (4.107. 2. Assign the f o l l o w i n g values to the design parameters: NI>%,

N2>%+N,-I,

3.N, is set as f o l l o w s : replaced by N,+I.

~=0

5.4

Select N, so that #N. is non n u l l but ceases to be i f N, is

For p r a c t i c a l implementation, we require R(q17, S(q I ) to have bounded c o e f f i c i e n t s and hence near s i n g u l a r i t y of M must also be avoided. This is necessary f o r numerical

stability. To avoid near singularity of M, we proposed to stop step 3 when #,.+, is smaller than a lower bound EMI, where ~ is a small positive number for numerical robustness. Remark: From theorem 2, we have some interesting results when N,>%, N2_>%+NI-I and ~,=0: i) I f A(q" 7 and B(q") are relatively prime, M,~I exists (i.e. GI has full rank) i f and only i f N. (Feasible control set) Given set XcR n and a time stage k, a set of feasible controls at the k-th time stage for given X is defined as

uok(x) ~ {~cuk: x~xok(,~,~+ 1 . . . .

,%_~), ~+~uk+ 1 . . . .

If a given control law u(.) is feasible, that is u ( - ) C ~ f UDk(~)

(~9)

u~_lcuN_ 1] , then U k ( % )

g

for every q s Q. We shall also introduce a set of estimation type of notions

we have introduced so far by defining prediction ~ + 1 ( % , U k )

of measurement ~ + I

the k+1-th time stage which is consistent with information available at the k-th stage in a form of (%,Uk).

at

680 Def.6

(Measurement prediction)

zk+l(%, ~ ) ~= (Zk+ 1 a s 9 ( satisfy

the state

qeQ) (Xo,Wo,W1 , . ,-

wk ,

Vl,V 2 . . . .

Vk+l,~,UkJ

(20)

a n d measurement e q u a t i o n s f o r i r O,k .

We a r e now i n a p o s i t i o n

to f o r m u l a t e t h e main r e s u l t .

Theorem 1 An o p t i m a l v a l u e F o f t h e s y s t e m p e r f o r m a n c e i n d e x (10) i s g i v e n by :

= min

max

Uo~Uo

[g ( z . , u )

ZIBZl(Uo).

o

I

The f u n c t i o n H l ( Z l , U o) i s g i v e n by t h e l a s t

~(~) subject

where

to

= rain

step of the recursive

max [gk(Zk+l,Uk) + ~ + l ( ~ k , Z k + l , U k ) ] ~+1

uk r

Un_ 1

max Zn,X n

, for k ~ 1,N-2

(22)

(23)

[gN(ZN,UN_l) + f(xN) ]

subject to UN_I~UDN_I(X(~N_I)) , ~eZN(~N_I,UN_I),

If

procedure

),

, and Z k + l r

~_1(~N_1) = rain

XNr

(21)

+ Hl(Zl,Uo)]

o

: XN = fN-1 'UN-1 'WN-1)C

and (24)

~-I~X(~N-1)'WN-lCWN-1 ] "

uA(") is the CDP solution, then the following hold uAo = Arg

for every

min[ UoeUo

max

(25)

[go(Zl,Uo) + Hl(Zl,Uo) ]] ;

ZleZl(Uo )

%, Uk(~k) = Arg

subject to (22), for kcl,N-2

minimax

J

(26)

+ f(~)] ]

(27)

[gk(Zk+l,Uk) + ~ + l ( ~ , Z k + l , U k ) ]

,

and for every ~N-I' ~-1(~N-I ) = Arg

rain [ max Un_ 1 ~ ' ~

[gN(~,~_l)

subject to (24) A proof of the theorem is given in [4 ] A structure of the set Q (see Eq.4), Def.2 and Def.6 imply that

~ + I ( ~ % ) = [zk+1~s: ( ~ x ( ~ )

A ( 3wk~Wk) ^

( ~ Vk+leVk)Zk+ 1 = ~ + l ( X k + l ,Vk+ 1 ), where Xk+ 1 = fk+l (Xk'Uk'~k)] Therefore, Eqn.(20) can be w r i t t e n in the f o l l o w i n g form :

(28)

681

z.,k+~(~k,,.,k) = Zk+l(X(~),~)

(29)

Let us notice now that due to (29) the equation and inclusions (21) .. (27) do not depend explicitly on ~k; the information,vector ~k affects them only through that state estimation set X ( ~ ) .

We shall formulate this fact in terms of a sufficient

information function [2]. Corollary I The mapping ~

~ X(~)

is the sufficiently informative function for the CDP.

The Theorem and Corollary I together constitute the separation principle. The principle says that at any time stage k the closed-loop minimax control signal can be generated in the two following steps : first, the state estimate X ( ~ )

is pro-

duced and then based on this estimate a deterministic minimax control problem is solved and the control signal ~ ( ~ )

is obtained.

It is not difficult to show that the separation principle doesn't hold if the k-th time stage cost function depends explicitly on xk. Therefore, it is not applicable if

N-I

F (z,~,u)

= F(x 1 . . . .

xN,u 1 . . . .

UN_1) = ~ g k ( x k , ~ ) k=0

+ f ( x N)

It is well known that for such strucfures of the performance indices the separation principle is applicable in the case of stochastic systems. This is an important difference between stochastic and set-bounded modelling of uncertainty in optimal control of dynamic uncertain systems. An applicability of a separation principle can be achieved by introducing a new state variable Xn+ I in a standard way [2] Xn+1,k+ 1 = Xn+l, k + gk(Xk,Uk), with initial condition Xn+l, ~ = 0 The performance index then becomes equal to Xn+1, N and the Corollary 1 and consequently the separation principle is applicable to the extended system. Another interesting problem arises where some of the components of the vector Wk, k ~ O,N-I are constant but unknown parameters. Let us denote the corresponding vector of parameters by' a. Clearly, there is now a time structure in uncertainty associated with the state equations ~id the previous model is no longer valid. Again, by introducing new state variables in a form of the parameters, the applicability of the theory presented in the paper is achieved where the additional state equations are as follows : ak+ I = ak with an initial condition a ~ = a.

The exact solution of

the problem in this case and some suboptimal schemes in a form of adaptive algorithms will be reported elsewhere. 5.

Perfect State Information. If a state vector is available we have a qualitatively simpler case. We do not

now need to define a safe tube through the family [ ~ k ]. As previously, we define

682 recursively the following sets: XD k and

= xz ~

XD N = ~

[~n

(3~Uk)fk(~,~,Wk)C_XDk+1 ] for k ~ 0,N-I

(3O)

.

Clearly, if the state vector belongs to the set XD k then there exists at the k-th time stage control

uk

which independently on uncertainty realization guarantees that

X k + 1 ~ k + I and the same applies for k+1~ ..., N-I. The feasible control set at the k-th stage has the form :

~k(x) : [ ~ U k : fk(x,~,Wk)c_~k+1~ and the set is non-empty if

xgXD k .

The formulae (21) .. (24) are now simplified to where the operation max is taken with respect to the variable wk subject to WkeW k and where the state estimates X({ k) do not occur. For example, ~ ( x k) :

subject to

min uk

Ukr

and t h e r e s u l t i n g 6.

max wk

the formulae (22) and (23) are now as follows :

[gk(xk+q,Uk) + Gk+1(fk(Xk,Uk,Wk)) ]

k) and Wkr k , A A o p t i m a l c o n t r o l law i s a f u n c t i o n o n l y o f Xk, t h a t i s uk = Uk(X~).

Conclusions A problem of design of closed-loop minimax control strategies for state-

constrained dynamical discrete-time systems, with uncertain initial conditions, parameters (possibly time-varying) and exogenous input signals, and with noisy measurements has been considered. The uncertainty has been assumed to be modelled through bounds which define fixed sets within which lie unknown constants (certain of the parameters and initial conditions) and tubes for time-varying quantities (other parameters, plant disturbances and measurement noise).

A DP type recursive procedure has

been derived to design the control rules. It has been shown that the separation principle holds for .the class of problems considered in the paper. Based on applicability conditions of the separation principle and the conditions for the separation principle derived for stochastic problems, an important difference between deterministic set-bounded and stochastic modelling of uncertainty has been pointed out. 7. [1]

References D.P.Bertsekas and l.B.Rhodes (1971). On the minimax reachability of target sets and target tubes. Automatica, Vol.7, pp 233-247.

[2]

D.P.Bertsekas and I.B.Rhodes (1973). Sufficiently informative functions and the minimax feedback control of uncertain dynamic systems. AC-18, No.2, pp 117-124.

IP;~;~;Trans.on Aut.Gontr.,

683 ~]

M.A.Brdys and S.Nowacki(1982). Design of safe controls and feasible control rules for uncertain dynamical systems under state constraints with applications to water systems. Technical Report, Institute of Automatic Control, Warsaw University of Technology (in Polish).

[4]

M.A.Brdys and B.Ulanicki (1989). Separation principle in optimizing control of state-constrained dynamical systems under bounded uncertainty. Research Memorandum No.20, School of Electronic & Electrical Engineering, University of Birmingham.

[5]

J.D.Glover and F.C.Schweppe (1971). Control of linear dynamic systems with set constrained disturbances. IEEE Trans.on Aut.Contr., AC-16, No.5, pp 411-423.

[6]

W.Findeisen (1982). Some definitions and notions of feasible control. Unpublished notes. Institute of Automatic Control, Warsaw University of Technology.

[7]

A.B.Kurshanski and I.Valyi (1988). Set values solutions to control problems and their applications. In 'Lecture Notes in Control and Information Sciences INRIA'. A.Bensoussan and J.L.Lions (Editors). Analysis and Optimisation of Systems. Springer-Verlag, pp 987-988.

[8]

A.B.Kurzhanski (1988). Identification - A Theory of Guaranteed Estimates. Working Paper. International Institute for Applied System Analysis. Laxenburg, Austria.

[9]

J.P.Norton

(1987). Identification and application of bounded parameter models.

Automatica, 23, pp 497-507. [10]

F.C.Schweppe (1968). Recursive state estimation : Unknown but bounded

errors

and system inputs. I~;~;~iTrans.Automat.Contr., Vo.AC-13, pp 22-28. [11]

A.J-Subbotin and A.G.Chencow (1981)- Optimization of Guarantee in Control Problems. Nauka (in Russian).

UNI~?CATION OF

5ONE

HI~*IO ADAPTIVE

CONTROL

ALt~ORI~I~ISAND GLOBAL CONVERGENCE ANALYSIS

T i a n You C h a i Department of Automatic Control, Northeast University o f T e c h n o l o g y , S h e n y m l g 1 1 0 0 0 6 , L i a o n i n g , P. R. C h i n a

Su~taary This

paper

schastic some

presents

multivariable

HIMO a d a p t i v e

cases.

the general

control

structures

wide variety

of adaptive

viewpoints.

tile

adaptive

adaptive

control.

~daptive

For t h i s

scheme

approaches

arc

special

schcxae u s i n g a m o d i f i e d

control

theorist

which includes

(1980),

Allidina

and

has been significant Dugard

control,

and

arbitrary

peper,

other

progress

Dion

schemes and extensions

this

control

effort

least

adaptive

a

in extending

(1985) pointed open,

out

form

many

one of challenges

a unifying

view

of

has been devoted to determining

adaptive

schc~es and presenting

and self-tuning

SISO d e s i g n s that

in particular,

in

results

albeit

in

this

the fact

t o t h e Mlr~} c a s e .

multivariable

global

a

schemes in Egardit

Ilewcver,

systems have not been given yet,

some p r o b l e m s a r c s t i l l

In

out that

Hughes ( 1 9 8 3 ) a n d Chai ( i 9 f f 7 ) .

for multivarlable

addition,

have been developed

i s Cite p r o b l e m o f p r o v i d i n g

between the seemingly disparate

algorithm

direction

algorithms

Ljung and Anderson (1984) pointed

general

certain

by s p e c i f i c

.

linear

Introduction

control

F o r 5150 a d a p t i v e

tile relationship

tbcrc

matrix

control

algoriClu~s is also given.

different to

scheme which can

interactor

dcrived

convergence of this

1. A

adaptive

systems with arbitrary

A proof of global

squares

direct

stability

that In

adaptive analysis

of

o f known s c h e m e s c a n a n d m u s t b e c o n s i d e r e d .

general

adaptive

s c h c m e s f o r Nlr~0

systems

interactor matrix is presented and the relationship bctwocn

having

the

the

scheme

proposed and other scheraes developed by specific approches is discussed. The global convergenoa analysis for the sche~ao will be established.

685 2 , General Adaptive Scheme Consider the ~It~ system described by the linemr vector difference equation A(z-' )y(t)=B(z" ' )u(t)+B, (z")v(t)+C(z- ' )w(t) where

u(t),

(2.1)

y(t) and v(t) arc n-vectors defining the system input output

and measurable disturbance vector respectively,

and A,

matrices in unit delay operator z-' such that A(O)=I, (I

is the (nxn) identity matrix).

B, ,

B(O)=O,

C are

polynomial

B, (0)=0, and C(O)=I

The nxl disturbance sequence w(t) is assumed to

be a real stochastic p r o c e s s defined on a probability the

B,

vectors

D-- algebra generated by the observations

space (l),~

,

F). Ft denotes

up to and including time t.

It

is

a s s u m e d that E(w(t)/Ft - ,)=0

a.s.

E(w(t ) w ( t ) ' / F t - , ) = ~ , with trace ~ = T

Under quadratic

of

+

these

llm n~ ~



minimize

J(u)

T

V x ~ H.

T,n

we

consider

the

followin~

linear-

problem:

~

*

lu(t)!

}dr

*

2 over

all

Vn { N ,

~ P T,n*1

u s N

(O,T;U), w

subject

to

the

state

equation

(T),y(T)>

696 (1.2)

Here

y(t)

w =

=

I

(w

probabillty

+

wm)

... space

expectation. ~s H and

S(t)~

(~,~,p),

Equation

(1.3)

E O such

c(~)

condition ~ T;

~

/

C&(CO,T);L(H,D(A*~))),

that

(T-t) K (2.2),

and

0 in

the

! t sense

< T; that

P(t)

~ PT

there

698 jT i[i)

'2

Vx 6 H,

IG*A~-P(t)S(t-s)xl

ds

is finir.e

0 iv)

PC.)

satisfies

(2.8)

P(t)




n

(t)]x,x> conver~es

uniformly

continuous

and

v)

convergence

of

of

of

1 hold

the

part

to P(-)x.

This

also

t=T.

Theorem

result

property

at

true.

previous

ill),

we

proposition,

alon~

readily

obtain

part

convergence

stated

in

(2.20).

Nhen Corollary of

(2.20),

iv)

Integrabillty

from

proof

the

P~(.)x

is s t r o n g l y

Parts

Usin~

the

that

P(.)

6.

=

n

follows

COROLLARY

that

n

n

that



of the continuous

this

n

finally

]S(s-t.)ds-

1/2

(t)]xl IP(t)-P

implies

J

P(s)D

inequality

n

if

have

function

2

IEP(t)-P

J

inequality.

every

fixed

D:t

j=1

monotone

a theorem From

from

for

pointwise

By

s

S(~-t):=[M

t

follows

steps

.

and

jT

*

A -t

is c o m p a c t ,

3.

Usin~

Corollary as

this 4 to

claimed

is ~ t a n d a r d

3.

SOLUTION OF THE L - Q - R the

obtain

hypotheses

the

convergence

in p a r t

to

have

obtain

It

Nith

we

v)

the

an of

stron~ one

can

identity the

(more

Theorem.

differential

easily)

in p l a c e

of

Finally,

RIccatl

repeat

the

from this

equation

the

inequality

(2.1).

PROBLEN and

notations

introduced

in s e c t i o n

1.2,

we

have: THEOREN J(u)

7.

over

(3.1) where

Given

u*(t) P

=

Is ~ i v e n

correspondln~ (3.2)

to

~tep

d

(u) n

1.

= E

and

there u*

u~.

Let

exists

a unique

is c h a r a c t e r i z e d ,

-G:=A~P(t)y*(t)

by Theorem



P~oo~.

x E H,

N~(O,T;U).,

1, a n d

control

by

the

u :t m i n i m i z i n g

feedback

formula

t ~ [O,T),

y:r i s t h e

solutlon

of

(I.2)

Moreover, =

J(u*),

J~(u)

be

the

<My(t),y(t)>

cost

*

functional

lu(t)l

)dr

defined

+

E. T n

in

identity

702 Equat, i o n of

(2-3)

is t h ~

min[m[z[n~

gqua~lon Since

Jn(U)

(1.2).

J(u)

Riccati

over

Thus

equation

all



< J(u),

we

correspondinK

u 6 ~(O,T;U),


the

[bmtn~bmax

=

the

the

states

Obviously,

Function

reach

curve

bmax.

D

a selection

SELECTION.

nature

B value.

Fig. oF

2).

the

720 The p r e s e n t e d

instances

consecutively

chosen

will

steps

be u s e d

for

in a d a p t i v e

an o p e r a t i o n

o~

system

this

synthesis

as t h e

system.

ill

T

X

Fill. 2 l EXAMPLE 4 . 2 .

Assumptions:

w h i c h has a d e n s i t y ~orm N = [ b n , b 9

the set

mQx

b is

a random v a r i a b l e

E u n c t i o n ~; ]

C [O,K -

the support of

~],

"

I

> O,

d > O.

- c - (B -

simulations

[8]

I

and i s

(B)

L follows

+ b m~x

E

if

{-c.(B

(14)

o~ t h e

>: ~ i n t ( N ) ;

)/2.

b d O,

i~ B -

b ~ O,

from p r a c t i c a l Taking

-

into

bm ~ x ) , d . ( B

together

be g i v e n

1 [bmtn'bm~x

c

B -

in

L(B,b) of

point

(B)

with

(7b)

b y B = bm~ n , = I

in

~J

(b

mt

--

n

description.

numerical of

view.

account the form o f

the mini-max loss Punction w i l l

inf B

if

f r o m some r e s u l t s

strate~.

= sup

c < d t h e n Prom

Similarly~

b)

recommended

function

mini-max d e c i s i o n w i l l

mLn

> 0 if

(13>

nO Appl~cat~o~ o / M[ns density

b)

Any o t h e r f o r m u l a can be used

The a b o v e s i m p l e f o r m o f

(b

m~x

~(x)

oG

= d-(B -

If

mtn

;

Eunction is

we d e ~ i n e by:

L i B , b)

support

< b

distribution

o~ d e c i s i o n s D = [bmkn ~bm~x ] "

The l o s s f u n c t i o n

where c

b

the

density

bm~n)}. it

(14)

follows

that

the

because (15)

).

]

> d t h e n B = bm ~ x .

Instead,

the

be g i v e n by:

by c = d,

it

~ollows B =

721 bD A p p L ~ c u ~ o ~ o~ Ba~es struLe~y. T h e by

(Sa)

will

take

here

the

function

o~ the

Bayes

loss

given

~orm: b

B

ib(B)

=

m~•

I

d- ( B - b ) ~

from

(16)

b)db

- I m

b After

calculating

i

(B)

b

=

0

c- ( B - b ) ~ ( b I d b .

we

(16)

get:

B

Using value

(171

o~

B.

I

~(b)db

mL

n

we c a n

= c/(c

realize

Equation

(17)

+

the can

d).

(171

Bayes

be

strategy,

replaced

by

selecting

the

respective

equivalent

~ormula:

b ~ x

I

B

~(b)db /

I

B

B(b

~

~(b)db = dlc.

9

(18)

b

B:b

F;~.3 EXAMPLE

4.3.

Assumptions:

Let

the

distribution

uniGorm i n N = [bm~n'bmax] ~ [O,K. be g i v e n t h e r e f o r e x(b)

=

the

bmtn
1 ~ I i = T i . lim x i ( t ) e" r i t t-~oo

=

lim t-~

q>i ( t ) e" r i t

(7)

(8)

(g) (i0) (11) = O.

(12)

737

9 The

three policies

liable to

be used along the equilibrium path

are determined by relations (9)-(11) : ( i ) Policy 1 = no investment policy : (I i = O) : firms i does not invest ~ i t s capital stock is kept constant since no capital depreciation is allowed ; this policy has to be implemented along the equilibrium path when x i < 1. ( i i ) Policy2 : steady state policy : (x i = 1) : The capital stock is determined by the condition that its marginal value (measured by adjoint variable x i) equals the inves-

Policy3 : maximum investment policy : firm i invests at

tment cost. ( i i i )

the maximum level ~i ; capital stock increases ; i t has to be implemented along the equilibrium path only when xi > 1. A sequence of policies over the horizon [0,+~] w i l l be called a strategy 9 A duopoly regime is defined as a combination of policies ; regime ( k - l ) stands for firm i using policy k and firm 2 policy I. Clearly all the regimes encompassing policy

1 or 3 are determined by the open loop condi-

tions (obtained by dropping partial derivatives of investment rates in relations (7) and ( 8 ) ) , since the upper and lower bounds o f 11 and 12 are independent on Kz and Kz. Regime (2-2) d e f i n e s the steady s t a t e o f the game which n e c e s s a r i l y holds w i t h I I = 12 = O.Consequently, open loop and closed

loop

s o l u t i o n s coincide.

Accordingly,

the c h a r a c t e r i s t i c s

of the p o l i c i e s

are summarized in t a b l e i :

Policy

ii

I

0

feasibility conditions

equilibrium conditions

none

xi < I

arl i _

r i

Xi = I

aK i

3

Ti

none

xi > I

Table 1 : characteristics of the policies. - Regime (2-2) defines the Nash long run equilibrium point (KI(N),K2(N)) of the duopoly where KI(N) and K2 (N) are solutions of the system :

anl - r I ' aR2 - r 2 " ~K1

aK2

(13)

738

These relations define the classical reaction functions K2 = RI(K1) and K2 = R2(KI) associated with each firm. A perfect equilibrium path is determined by a sequence of duopoly regimes along which equilibrium conditions given in table I, transversality conditions (12) are satisfied. I t is quite clear that adjoint variables m~and~ have to be continuous along the equilibrium because of the regularity conditions of the problem and the absence of the

constraints on the state.

All these conditions ensure the unicity and

s t a b i l i t y of the equilibrium path as i t will be seen now. Let C be the

upper envelope of RI and R2. I t is quite clear that any equilibrium path starting above C will necessarily remain at the i n i t i a l point. Since there is no depreciation, steady state (KI(N),K2(N))is only on the equilibrium path when the i n i t i a l capital stocks are on line ZIN. I t remains then to characterize the equilibrium path for other i n i t i a l conditions. This will be done in determining the equilibrium path ending at some point (R1,~2) of C. The path is univocally defined by Proposition i and 2.

Proposition 1 : Let capital stocks ~l and kz such that ~(i >I KI(N) and 7{2 = R2(TCI). (i) There exists an unique perfect Nash equilibrium path of the game ending at point (~1,? 6.

753 We notice that ~,(Y) is equalizing : against it, any 7Us strategy gives the same value of ~ ( Y ) . More generaly this computation proves that all strategies satisfaying Ci.~ are equalizing. Computation

of V~_1(Y, z)

V~_t(Y, y - r) = r

M~,_,. ~,_~(Y, y - r) 0

( LIZ[)[y ] = r

0

P'(~):-

+

a,

i -- (2r -- i)# #

4:),it]

=

1 P~ i

1 ,r,-- + r m

=

P, (1G, + p,(1-~,_l(Y)(-r))

=

(--or, + p, + ~,_i(Y)(-r)(l -- p,)

=

p,+, + ~ , + ~ , _ ~ ( Y ) ( - ~ ) ,

+ ~,_l(Y)(-r)

m

and similarly :

Vt-l(Y,y+u) Vt_t(Y,y+i)

= p,+, +cr,+iqot-,(Y)(u) =

p,+,

for u E U fori=r+l

. . . . . 3r.

The proposition is therefore proved until time l = 1, that remains to deal with now. 3.2.3

timet=l=T-(T-1)

Let's notice that there is not flying bulle*, at this time. Game m a t r i x : M = pT_2(ll)~xm + O'T-2N2[O]. O p t i m a l i t y of Pl 1 M , l ( 0 ) = (pr-2 + --~T-2)(1)n ~Z hence (Ih(O) is an argument of :

mineez- P ' M ~ i ( O ) .

O p t i m a l i t y of r r

=

r

p~--~(~).x,,. + ~.r-~ N~[o] ) i

=

pr-,(~)",

m

hence qi(O) is an argument of :

maxoez~ ~i(O)'MQ

eayo~

and furthermore :

= ~(O)'M,~(o) =

PT-1

9

754

4

Conclusions

1- F e a s i b i l i t y of (A2.1) : The following simple strategy satisfies (A2.~) :

r

= (1/m,...,1/,,), rn

r

1 0

ifi=j otherwise.

foreachi,jE

{-2r,...,+2r}.

Observe that we have pure controls as soon as t _> 2. The fact that 7-{ used mixed control at t = 1 is enough to avoid giving more information to ~ . 2- F e a s i b i l i t y of (A2.2) : we have exhibited a ~ ' s strategy that satisfies (A2.2) for each r > 1, and whatever the duration of the game is. In the case r = 1, such a strategy exists only for a game duration, T, not superior to 5. 3- F e a s i b i l i t y of (A~) : We cannot raise a general method to study the existence of strategy satisfying (At). We have therefore showed that in case where r = 1, it is impossible to compute such a strategy as soon as T > 5 4- L o w e r b o u n d o f g a m e payoff: The impossibility to find (or nevertheIess to prove existence of), ~ ' s strategy satisfying (A1), doesn't make the above developpement useless. As a matter of fact, at each step of time, the ~ ' s chosen control being equMizing, the found value of the payoff exhibited is a lower bound when A1 cannot be satisfied. Futhermore we can see that this lower bound is swiftly increasing toward 1, and so the 7~'s strategy as not so bad.

Bibliography - [1]- T.Ba~ar, "On a class of zero-sum discrete games with delayed information", Lecture notes in control and information sciences 119. pp9-34, Springer, Berlin, t989. - [2]- P.Bernhard, "information and strategies in dynamic games 982, 1989.

", rapport

de recherche I.N.R.I.A.

- [3]- P.Bernhard et A.L.Colomb, "Saddle point conditions for class of stochastic dynamical games with imperfect information", IEEE trans, on Automatic control, AC 10-23, pp. 98-101, i98. - [4]- P.Bernhard, A.L.Colomb et G.Papavassilopoulos, "Rabbit and Hunter game : two discrete stochastic formulation", Comput. Math. Applic. 13, No 1-3, pp. 205-225, 1987. - [5]- A.L.Colomb, "l~tude de jeux k deux joueurs en information incomplete ", thb.se, Universi~5 de Provence, Marseille, France, 1986.

AN INTERACTIVE MULTIPLE CRITERIA DECISION SUPPORTING TOOL WITH APPLICATION TO A SIMPLIFIED REGIONAL DEVELOPMENT PROBLEM CAO DONG and M. INSTALLE Laboratoire d'Automatique, de Dynamique et d'Analyse des Systemes Bg.timent Maxwell, Place du Levant, 3 B-1348 Louvain-la-Neuve - BELGIUM

ABSTRACT In this paper, an interactive multiple criteria decision supporting algorithm is proposed which combines a Pareto solution determination method based on the reference point technique of Wierzbicki (1982) with the use of an interactively updated explicit utility function. This function which is recursively updated by the decision makers of the problem is used to represent their preferences among the various objectives of the problem to be solved. The maximization of the non linear utility function is done through an original procedure which works in a progressively enlarged optimization space of small dimension. This procedure makes the algorithm efficient even for large scale probiems. Finally, an application of the algorithm to a simplified regional development problem is presented.

INTRODUCTION The solution of regional developement problems asks usually for the consideration of many objectives among which some may be in conflict. Those objectives represent the interests of the actors or decision makers involved in the development of the region. In this paper, a decision supporting tool is proposed that implies - through an interactive procedure - those actors in a multiple criteria decision making (MCDM) procedure. In the past, interactive MCDM methods have been proposed which allow for the intervention of the DM in the course of the planning procedure in order to progressively identify their preference structure and find eventually a satisfycing compromise solution. Among those algorithms , the one developped by Rosenthal (1985) combines a Pareto solution determination method with the use of an explicit function which was developped by Keeney and Raiffa (1976) from global utility theory. In this algorithm, the explicit utility function is interactively and iteratively updated by the DM on the basis of the sequentially generated Pareto solutions and the associated trade-off informations. Hence, the practical problem of the "a priori" determination of the utility function representing the preference structure of the DM is avoided.

756 However, this algorithm suffers from an important drawback : since each time that the utility function has been updated, the corresponding Pareto set of decisions optimizing this utility function has to be determined, one has to solve a great number of non-linear constrained optimization problems. Hence, for practical dynamical problems like regional development ones with a constraints set of great order, the search for a satisficing compromise solution may be very time-consuming. In the second part of this paper, a new interactive algorithm is described which can be used for large scale MCDM problems with linear constraints and linear objective functions. At each iteration of this algorithm, an explicit utility function is first interactively updated through information given by the DM. This information consists in a minimum acceptable value and in a full satisficing value for each objective of the problem. Then, the maximization of this updated utility function provides a feasible solution vector in the criterion space. Finally, this solution vector is projected on the efficient surface through the reference point method developped by Wierzbicki (1982). The maximization of the utility function is done through an original optimization procedure. In the third part of this paper, the algorithm is applied to a simplified regional development planning problem. DESCRIPTION OF THE ALGORITHM A linear MCDM problem is represented as : max Z(x) = [ z 1(x), z2(x) ..... Zp(x)]T = [C1x,...CpX]T s.t x~X = { x / A x < b , x > _ o } [P.1] where x is an n-dimensional vector of decision variables, zi(x) is the ith objective function, A is a given m x n matrix and X is the feasible set of the constrained decisions. Generally, for this vectorial optimization problem, one cannot obtain a solution which maximizes simultaneously the p objectives because of the presence of conflicts. However, a often used solution to this problem is a Pareto or efficient solution. In the next sections, it will be shown how : 1) the DM interact with the optimization procedure through the recursive updating of a utility function representing their preference structure; 2) a reference solution Z is determined by maximizing the corresponding updated utility function; 3) a Pareto solution Z corresponding to the reference solution Z may be found; 4) the complete algorithm is organized.

757 Interactive updating of the utility function A well-known additive utility function is chosen in order to represent the preference structure of the DM "

u(z, ..zp)=

1-e I]~(zr-11)

=

u,

where ~i is the minimum acceptable value of the ith objective, 5 i is the full satisficing value of the ith objective and ~i < 0 is the preference coefficient parameter. ui is the "attribute" of the ith objective and takes values in [0, 1] for ~'i < zi -< 8i. This utility function is relevant with the general decision making principle: The more, the better for each objective and the choice of the exponential says that the smaller the objective value, the greater the increasing tendency of U(zl ..... Zp). For the interactive determination of the parameters describing the preference structure U(zl .... Zp), the following information was used : at the current Pareto solution, the DM are asked to give their "margin of allowance", that is, the ~ and 8i for each objective. The J3i are then automatically updated such that for the middle value of the allowance interval, ui has a value equal to (~ where 0.5 < ~ < 0.9.

Maximization of the utilitv function As stated before, the reference solution is obtained by maximizing the function U(Zl...Zp). To overcome the computation inefficiency caused by a constraint set (Ax ~ b) of great order with a given non-linear preference structure U(zl...Zp), a reference solution is first determined in the approximated criterion subspace formed by the convex combination of the sequentially generated Pareto solutions [z i, i = 1 ,...k and the extreme points (zJ, j = 1 ...p) obtained from the individual optimization of each objective. Hence, the problem is : max U(z 1...zp) X,z

s.t. Z= ~, ~7'+ i=l...k

and

~

~.i =1

[Umaxl] ~_, ~

:~i

j=l...p

X.~>0

i=l...k+p

where k is the iteration number. We call the solution of [Umaxl] : [z~. If this solution is identical to one of the former Pareto solutions, the dual information is then used in order to extend the criterion subspace through the cutting planes defined below. Hence, the new problem is 9

758

max U(z, ..... zp) z

p

p

i=1

i=1

[Umax2]

where PJi, ZJi are respectively the dual information and the Pareto solution obtained at the jth iteration (j = 1 ..... k). W e call the solution of [Umax2] : ~2.

Generation of the Pareto solutions AS written before, the reference point method is used to obtain the Pareto optimal solution corresponding to the solution

Z maximizing U(Zl,..., Zp).

This method is based on the

minimization of a generalized Tchebychev distance from a given reference point Z. : min Jr max 1:i(zi-z,) ] + (g,...,g)(z- z ) } z ~.Li=l...p s.t. Z--- [ClX ..... CpX]T xeX

[P.2]

or its equivalent problem " min {(z + (8,...,E) (Z - Z)} z.cx s.t. cc>'~i(Zi-Z i) (i=l,...p), z i - c i x = 0

[P.3] 0 = 1 .... p), A x ~ b

where cc is any real number, the weighting coefficient "q > o (i = 1,...P) and E is a very small positive value. The supplementary term (E.... ~) (~. [z) guarantees the efficiency of the solution found by [P.3]. Two theorems based on the duality theory are easily derived " Theorem l "The optimization gf [p.3J gives a Pareto solution Theorem 2 "The ootimal solution of [P,3] ootimizes at the same time the followina dual oroblem 9 P max - ,~.pizi s.t. z i = cix and Ax < b i-1

and where the variables #i, i = 1..... p are the dual variables corresponding to the objectives zi = cix, i = 1 ..... p. The geometrical interpretation of [P.3] is i~ustrated at the next figure for a problem with 2 objectives -

759 Z2 Cutting plane

Criterion set

Zl p

p

The equation - ~_,lAz~ = - ~ . ~ z i i-1 ~1 generates a cutting plane of the efficient subspace which pass through the efficient point [z.

The comolete algorithm The organization of the algorithm is as follows : steel : For i = 1 ..... p, each objective is optimized independently and a pay-off table with ~i is obtained. steo 2 : Given this pay-off information, the DM are asked to specify their margin of allowance "yi and 8i for each objective, k is set equal to zero. steo 3 : The problem [Umaxl] is solved and Zl is obtained.

This point is projected on the

efficient region by [P.3] in order to have an initial Pareto solution [z. steo 4 : k = k+l. The DM decide whether the last Pareto solution ~,k is satisficing. If yes, the problem is terminated. If no, the step 5 is executed. step 5 : The DM modify their "margin of allowance" % 8i in order to have a newly updated U (zl ..... Zp).

steo 6 : The problem.[Umaxl] is solved. If ~1 is different of ~k, Z.1 is projected on the efficient region by [P.3] in order to have a new Pareto solution ~k and the algorithm goes back to step 4. If ~lis equal to ~Tk,step 7 is executed. steo 7 : The problem [Umax2] is solved. If ~2 is equal to ~1, the algorithm goes back to step 5. IfZ 2 is different of Z1, the step 8 is executed. steo 8 : ~2 is projected on the feasible region through [P.3]. The efficient solution ~k is used to extend the criterion subspace. The algorithm goes back to step 6. The next figure illustrates the various steps of the algorithm 9

760 Z2

~

Z~

The convergence of the algorithm towards a solution maximizing a given U(zl ..... Zp) is guaranteed by the fact that the combination of [Umax2] with [P.31 is essentially a cutting plane method applied to a non linear optimization problem where the following requirements are satisfied : 1) the real polyhedral objective region {Z(x) s.t.xe X} is bounded; 2) the function U(zl,..., Zp) is quasiconcave (Luenbergher, 1984).

ASIMPLIFIED REGIONAL DEVELOPMENT PROBLEM A simpliefied regional development problem within a chinese context with a planning horizon of 6 years is considered where the production system is composed of 2 sectors : the agricultural sector and the rural industry sector which receives the raw materials from the agricultural sector and does a value-adding transformation. In the agricultural sector, two types of land are considered : type 1, irrigated flatland and type 2, dry flatland. By building the appropriate hydraulic installations, lands of type 2 can be transformed into lands of type 1. Five production activities can be performed as follows : Activity/land :

type 1

type 2

Soy beans

X

X

Rice

X

Fruits

X

X

where "X" indicates that on the land of type i the production activity j is possible. Finally it is supposed that the rural industry sector of the studied region absorbs all the harvested soybeans in order to transform them into a value-added, commercialized product.

Descriotion of the model Let Xij(k) be a state variable which represents the area of land of type i devoted to the jth activity during year k. Dynamic equations relating the state variable Xij(k) to the land allocation decisions are the following ones :

761 Xij(k+l) = Xij(k) + U~ij(k) - U~ij(k)

(E.1)

where the decision variables U+ij(k), U-ij(k) are respectively the area of land of type i added, substracted, at the kth year for the activity j. The transformation of the land of type 2 into that of type 1 is described by the following equations : X l ( k + l ) = X](k) + TR(k) X2(k+l) = X2(k) - TR(k)

(E.2)

where Xl(k), X2(k) are respectively the total area of land of type 1 and 2 and TR(k) is the area of land of type 2 transformed into land of type 1 at year k. The static constraints associated with the land allocations are the following ones :

EJ Xij(k) < Xi(k)

(i = 1.2)

(CA)

For the industrial sector, the assets of the rural industry are imbedded into a state variable Xr(k) with which the production capacity is associated. The dynamic equation describing the evolution of those assets is as follows : X,(k+l) = o~X,(k) + U,(k)

(E.3)

where Ur(k) is the investement level in the industry at year k and CCris a redemption factor. Directly associated with this equation is the limit of fabrication capacity of the industry : ~11X11(k) + J321X21(k) -< PrXr(k)

(C.2)

where ~r is the fabrication yield (Ton/kS) and ~}11, ~21 are agriculture yields(Ton/Ha). Finally, the dynamic equation for the regional cash-flow is the following : Xa(k+l ) = asXB(k) + ,7_.~oijXij(k)- 7~rTR(k) - Ur(k) - Up(k)

(E.4)

where Xa(k) is the available cash-flow at year k, ZPijXij(k) is the net profit obtained from the production activities Xij(k), ~rTR(k) is the cost for the land improvement, Ur(k) is the investment in the industry, Up(k) is the budget for public welfare (the income budget for the workers is included in this term) and cq3 is a money erosion factor. Other static constraints are the following ones : non-negativity of the state and decision variables. - minimum quantity of produced rice : -

~12X12(k) > 4Q(k)

(C.3) (C.4)

762 where Q(k) is the quantity of rice necessary for the population at year k (prediction). - necessity of preinvestment in production activities : ~,'~jXij(k) + 7trTR(k) + Ur(k) < XB(k) (C.5) where "~1jXij(k)are costs of production. limit on available man-power for production : 7_,~ijXij(k) + ~rTR(k) ~ MP(k) (C.6) where =r,/qj " man-hours/Ha and MP(k) is the predicted available man-power on year k. limit on available water for irrigated crops : -

-

3

(C.7)

~__,~Xij(k) -<Wa(k) j=l

where % is the water demand coefficient for culture j (M3/Ha) and Wa(k) is the prediction of available water resource at year k. constraint that guarantees the continuity of the production at the end of the planning period of 6 years: XB(6) -> XB(5) (C.8) -

Definition of the ob!ective~ In this regional development problem, the planning horizon is divided into 2 development stages of 3 years each. At each development stage, 3 objectives are considered which represent the different interests of involved parties in this production system : objective 1 maximize the total profit of production; objective 2 - maximize the budget for public welfare and objective 3 - minimize the erosion caused by the production activities. At each development stage, these objectives are measured by their average over the years covered by the stage. Hence, the problem contains 6 objectives given by the following expressions : 3

6

objl" max k__~l{ ~,RjXij(k)t/3 ij ~

obj4 9max ~ , t ~__,PijXij(k)l/3 k=4~ ij J

2

5

obj2" max~__,{Up(k)}/3 k=0

obj5 : max ~_.~{Up(k)}/3

3

6

obj3 " min ~__~f~/___,eijXii(k)l/3 k=lL

ij

J

obj6 " min ~/_.,I ~_~eijXfi(k)t/3 I:~--4L ij

J

In these expressions, the eij, i = 1,2; j = 1, 2, 3, are erosion coefficients which may be given, for example, in M 3 of soil removed per hectarea of land of type i with an activity j.

763 Solution of the oroblern The above problem was resolved on a PC/AT computer with 640 KB memory. First, a pay-off table was computed :

Z1

Z2

Z3

Z4

Z5

Z6

max Zl

18700

8580

-5550

5910

9110

-1770

max Z2

12800

14800

-3830

5910

2840

-1770

max.Z3

5380

7430

-1610

5930

2880

-1780

max.Z4

18700

5400

-5540

26400

8970

-7470

max.Z5

17400

5400

-5220

16600

24000

-5010

max.Z6

16300

5400

-4910

5750

16100

-1730

The diagonal elements of the above table are the maximum values of each objective and were chosen as the initial full satisficing values 5i for each objective. Furthermore, the initial minimum acceptable value of each objective were chosen as follows : Y1 = 10000; )'2=7000; )'3 = - 5500; )'4= 11500; )'5 = 10000; )'6= -6000. Then, three iterations were made interactively in order to obtain a psychologically satisficing Pareto solution. The table at the next page gives, for each iteration, the margins of allowance that have been modified by the DM as well as the reference and Pareto solutions. The underlying course of actions giving rise to the last, satisficing solution is the following one: at the first stage, because of the budget limitations, the main production activity consists in fruit production which doesn't need great investment. Gradually, the investment is put in the transformation of land of type 2 into that of type 1. On the transformed land, the most profitable production activity, that is, soybeans is taken.

CONCLUSION In this paper, an interactive multiple criteria decision supporting algorithm and its application to a simplified regional development problem have been presented. The reduced dimension of the non-linear problems [Umaxl] and [Umax2] makes it possible to work in an interactive mode even for a large scale linear problem. This interactive mode was implemented through the use of an adjustable utility function which represents progressively an approximation of the preference structure of the decision makers.

764 obj.1

obj.2

obj.3

obj.4

obj.5

obj.6

51

18700

24000

-1730

10000 16400 16800

-1610 -5500

26400

Y1 z1 zl

14800 7000

11500

10000

-6000

8070 832O

-4910 -4810

16300 16800

13800 16200

-4770 -4620

i

52

t e r 2

72 z2 z2

id. id. 17200 17200

id. id. 7710 7720

id. id. -4960 -4960

id. 18000 18800 18900

20000 id 14700 16300

id. id. -5210 -5210

83 "t'3 z3 z3

id. id. 17500 17500

id. id.

id. id.

id. 20000

id. id.

id. id.

7210 7210

-5080 -508O

20500 20500

14700 16400

-5700 -5700

~t e r 1

t e Ir 3

REFERENCES Dyer,J.S. (1973) A time-sharing computer program for the solution of multiple criteria problems. Management Sci., 19, 1369-82. Keeney, R.L. and H. Raiffa (1976) Decision with multiple objective : preference and value trade-off. John Wiley, New York. Luenbergher, D.G., (1'984) Linear and non linear programming, Addison-Wesley, Reading, M.A. Rosenthal (1985) Concepts, theory and techniques : principles of multiobjective optimization. Decision Sciences 16, 133-152. Wierzbicki, A.Po (1982) A mathematical basis for satisficing decision. Mathematical Modeling, 3, 391-405.

CRONE CONTROL : PRINCIPLE, SYNTtlESIS, PERFORMANCES WITIt NONLINEARITIES AND ROBUSTNESS-INPUT IMMUNITY DILEMMA A. OUSTALOUP Equipe Systrmes et Commande d'Ordre Non Entier L.A.R.F.R.A. - ENSERB - Universitd de Bordeaux I 351, cours de la Librration - 33405 Talence Cedex Abstract The area of this paper concerns the robustness of stability degree, and more particularly the robustness of the damping of the control versus the parmneters of the plant. The approach of the CRONE Control is presented as resulting from the non integer order differential equation which represents the dynamic model directing a natural robust relaxation. A frequency illustration of robustness is given in the Nickol-Black plane through an open loop frequency template. Such a template is synthesized by means of a CRONE variable phase regulator from the phase diagram of an agricultural mobile robot using a highly non linear error detector. The corresponding robustness performances are presented. At last, a dilemma is established by considering two templates of different lenghts. The imput ot the plant is given for step responses to the reference input. It shows that the greater the robustness, file lower the input immunity. 1 - Introduction

Robustness is a very wide concept, even in a same domain such as the automatic control one. In fact, robustness is a notion which always translates the same idea, namely insensitivity. In automatic control, it is frequent to consider the robustness of stability (case of Hod approach). In the CRONE Control, a French abbrevation of "Commande Robuste d'Ordre Non Entier", namely "Non Integer Order Robust Control", the robustness considered is much stricter, that is to say the robustness of stability degree. More precisely, the robustness which is at stake translates the insensitivity of the damping factor or the stability degree of the control to the parameters of the plant. That amounts to saying that the robustness considered is the robustness of the control damping versus the plant parameters. A frequency illustration of damping robustness is deduced from the non integer order differential equation which represents the dynamic model governing a natural robust relaxation, that of water on a porous dyke. Particularly, in the Nichol-Black's plane, robustness is illustrated by an open loop frequency response locus which is reduced to a vertical straight line segment of abscissa between - x / 2 and - x around the axis OdB. This segment, defined for the nominal parametric state of the plant, is called open loop frequency template (or more simply template ). In this paper, the template is synthesized by means of a CRONE variable phase regulator from the phase diagram of a mobile robot using a highly non linear error detector. The performances obtained through a numerical simulation show that the control is robust, not only versus the open loop gain, but also versus the non-linearity of the error detector, whether the non-linearity is intrepreted as an increasing gain or a decreasing gain. In order to study the price of robustness, the last part of the paper deals with the robustness-input immunity dilemma. For two different lengths of the template, the noted performances reveal that the input of the plant is higher in the case when the template is longer and consequently when the robustness is higher. 2- From strategy

t h e r o b u s t n e s s o f s t a b i l i t y d e g r e e in n a t u r e : The non integer approach of the C R O N E Control.

2.1- O b s e r v a t i o n

to a n e w r o b u s t

control

of a natural robust relaxation

Our approach, the aim of which is the conception and the application of a new robust control strategy, makes use of the observation of a natural relaxation, that of water on a porous dyke. Already in the 17th century, the constructors of dykes had noted the damping properties of the very disturbed dykes and particularly those forming air pockets which can be compressed by the advance of water. Otherwise, an attentive observation of the relaxation of water on fluvial or coastal dykes, reveals that in the case of very damping (or absorbing) dykes through a porous volurnic structure and a rough surfacic structure : - the natural frequency of the relaxation is different whether the dyke isfluvial or coo.~tal ; - the damping of the relaxation seems to be independent of the dyke, whether it isfluvial or coastal.

768 Given that the fluvial and coastal tests can be distinguished by very different carried water masses, the observation seems to show that the relaxation is characterized by a natural frequency which depends on the motion water mass and by a damping which is independent of it. Although it should be paradoxical in the integer approach of mechanics where any relaxation presents a damping linked to the carried mass, this result reveals the insensitivity of the damping factor to a parameter, in this case the motion water mass. So, it translates the robustness of the relaxation phenomenon as for stability degree. 2.2- Non integer o r d e r differential equation as a dynamic model governing the relaxation After trying to determine the mathematical origin of the robustness of such a phenomenon, it appears that it resides in non integer derivation. Indeed, by taking into account the fractality of porosity and the corresponding recursivity, we show [5] that the relaxation satisfies a linear differential equation of non integer order n' between 1 and 2, namely : zn'(d~tt)n'p(t) + P(t) = 0 ;

(1)

P(t) designates the dynamic pressure at the water-dyke interface ; x is a transitional time constant which is a function of the water mass. 2.3- F r e q u e n c y t e m p l a t e The differential equation (1) leads to a symbolic equation of the form : ('r

+ P(S) = O,

(2)

from which one draws :

(3) This operational equation is translated by the fonctional diagram shown in figure 1. Because of a unit feedback, the direct chain determines an open loop transmittance of the form : ~(s) = ( 1 ) n' ,

(4)

which is the transmittance of a non integer integrator and which defines an open loop frequency response of the form :

in which mu = 1/'c designates the unit gain frequency. Given that arg ~3(jm) = -n' ~/2 with 1< n'< 2, the black locus of [3(jr~) is a vertical straight line of abscissa between - ~/2 and -~.

Figure 1 - Functional diagram defining an open loop transfer Physically, a water-dyke interface cannot be characterized by a non integer derivation in all frequency domain [5]. In fact such a derivation is limited to a range of medium frequencies. So, the vertical straight line so defined is reduced to a vertical straight line segment (fig. 2). This segment is called open loop frequency template (or more simply template). When the water mass M changes, the frequency cou is modified in conformity with the relation C0u = (K / M) 1/n', (6)

769

where K is a constant depending on the water-dyke interface. That amounts to saying that the template so defined slides on itself at the time of a variation of the water mass. Such a vertical displacement of the template insures the constance of the phase margin and, consequently, the invariance of the corresponding damping factor in time domain, so translating the robustness of the damping.The longer the template, the greater the robustness.

A

-7~

I I

I ~(j~)l dB

I I

0~u I

0 dB

-n'rd2 -r~ / 2

0

arg ~(jf.0)

I

-%-

t I

Figure 2 - Illustration of robustness in the Black plane: theform aM the vertical sliding of the template AB insure the robustness of tt,e damping through a constant phase margin ~)rn 2.4- The template in automatic control In automatic control, the aim is to obtain such a behavior, namely : - an open loop frequency response Black locus which forms the template so defined for the nominal parametric state of the plant ; - a vertical sliding of the template at the ume of a reparametration of the plant. Trying to find the synthesis of such a template defines the approach (said non integer) that the CRONE control uses.

3- Synthesis

of

the template

3.1- I n t r o d u c t i o n Although several synthesis methods of the template exist, here, we simply present the method which corresponds to the application considered in this paper, namely the wire guidance of a mobile robot. Let 13(jr and G(j~) be the open loop frequency response of the control and the frequency response of the plant. We assume that the phase diagram of G(jo) (fig. 3) is invariant, which is the case when only the variations of gain are considered (case of our application).

0i

argGfjr

argl3(j(o)

I .n t~/"2 -

I .

/D

.

~

Figure 3 - Synthesis of the template from the phase diagram of the plant.

770 To pass from the argument of G(jc0) to the argument of 13(jco) for coA < co < C~ the observation of figure 3 shows that the regulator CN(Jco) placed in cascade with the plant must provide both a phase delay and a phase advance which should be functions of frequency. 3.2- Idea of the synthesis : smoothing of crenels with variable cyclical ratio One considers that the phase diagram of the regulator results from a smoothing of crenels constituting a phase asymptotic diagram. So, one must introduce the notion of phase smoothing curve which replaces thephase smoothing straight line one usually used in the non integer approach. In order to obtain a smoothing whose value should be a function of frequency, the idea is to vary locally the cyclical ratio of the crenels. This is translated by the consideration of crenels whose cyclical ratio is a function of their ranks (fig. 4). The distribution of the zeros and the poles is defined by the relations:

ai = coi/co'i and with

rli = co'i+I/coi,

cq rli = 0~'i+l/co'i = cte > I

(7)

(8)

k/i,

where a i andrli are called recurrent factors. phase smoothing curve

OJl

r

0,~

0)2

0)2

0)'i

0)i

0)5+1

_L__2_L2__0)N 0)'N-t

CON-I CON fOB

Figure 4 - Obtaining of a phase smoothing curve through a modulation of the cyclical ratio of the crenels of a phase asymptotic diagram. 3.3

-

Synthesis

process

The synthesis process successively consists : 1 - in fixing a recursive distribution.of the zeros co'i, namely co'i+l/co'i = oti Tli = cte V i ;

(9)

2 - in considering as data, the values of the phase that the regulator must provide at the frequencies co'i, namely r = arg CN(.jco'i)= (Pd - arg G(jr

(10)

where (Pd = -n' n/2 is the open loop desired phase that the phase locking defines around cou ; 3 - in determining the distribution of the poles, that is to say that of the recurrent factors a i given that ai = coi/co'i. 3.4 - Exact algebraic method The algorithm which results from this exact method is that used by the CRONE software. A regulator frequency response in conformity with the phase asymptotic diagram shown in figure 4, is defined by an expression of the form : (1 + jco / co'l).-.(1 + jco / co'i)...(1 + jco / co'N)

CNOco) = Co

,

(I I)

(1 +ji0/col)...(1 + jco/coi)...(1 +JCO/CON) the corresponding argument satisfying the relation : N

N

arg CN(jco) = ]L tan -1 co/ co'i - ~] tan'l co/ coi. i=l i=l

(12)

771 For the frequency ~ = t0'i, which suggests to replace the index i by the new index j, the phase of the regulator becomes : N N arg CN(Jm'i)= ]~ tan "1 o ' i / 0 ; j - ~ tan "I to'i/o~j, j=l j=l

(13)

from where one draws, taking into account (10) : N

t a n ' l to'i/c0j = A(o'i),

(14)

j=l in putting : N

(15)

A(to'i) = ~ tan -1 ~o'i/o'j - q~d + arg G(j0~'i). j=l

So, one obtains the system of equations : tan -1 o' / 0h + tan "l o' 1 / co2 + ... + tan "I to'l / ~ tan -I 0;2/ro I + tan-1

o'2/a,, 2 + ... +

= A(to'l)

tan-I t 0 ' 2 / o N

= A(to'2)

tan- 1 ~'N / Ol + tan" 1 o, N / c02 + "-- + tan" 1 to,N / m N = A(to'N),

(16)

which constitutes a non linear system of N equations, the N unknowns of which are the toi. One puts ct0ai = C and xi=l/c q. By taking the tangente of each m e m b e r of each equation, the line of rank i admits the reduced writing : N

tan (]~ tan -I aij) = tan A(m'i) = A i with aij = Ci-Jxj,

(17)

j=l

of which one tries to determine the expression of the first member, namely: N PN q~ tan ( • tan -1 aij) = Z (-1)P s2p+l(i) / Z (-1)q S2q(i), j=l p=0 q=o

(i8)

expression in which the different magnitudes are defined as follows : PN = IN - 1 / 2], integer part of N - 1 / 2 ; qN = [N / 2], integer part of N / 2 ; S0(i) = 1 ; Sl(i) = ~ aijl;

Jt I~jl-AIB1): template (2); along AIB 1 or A2B2 , [ flO'co)[ dB= (cou/co)n' dB; along C1DI Icu6oo)l aB +lcoo )l dB. or C2D2, I oco)l

5.2. Performances obtained through a numerical simulation A simulation has been carried out with the regulators defined by relations (28) and (29), without considering the non-linearities of the error detector. Figure 9 gives the open loop Bode diagrams corresponding to the two regulators. It shows that the template obtained with the regulator CN2(S) is longer than that obtained with the regulator CNI(S). Figure 10 gives the variations of the input of the plant corresponding to the step responses of the control successively obtained with the regulators C N 1(s) and CN2(S). It appears that the input is greater for the regulator CN2(S). I1

lit

1era

et

Ii

I1

d II

,Q--~

-1~ d B

-ZT~

-2~

x \,b

i

111-'t

-1111

i

Illlll

I

1~ - Z

I

tlllll

I

llI -I

I

t[llll

I

I0 R

I

--368

Iltlltl

1D I

1B Z

11t]

Frcquencg [112:]

Figure 9 - Bode diagrams of the plant and of the open loop control ." gain; ..... phase ;(a) plant ; (b) plant corrected by the regulator CNI(S) ; (c) plant corrected by the regulator CN2(S)

777

0 r.s

0 1.Sf

t ,tSB

1.21

m ~8~

.91

t

.6~ .31

1~

81 -li]~

,

1

-..J~

I

.1 Ti~e(~)

II

.81

.82

Tiae(~)

Figure 10 - Variations of the input of the plant corresponding to the unit step responses of the control obtained with the regulator CN1(s)(a) and the regulator CN2(s)(b ) : the initial value is much greater with CN2(S)

6. C o n c l u s i o n The approach of the CRONE control arises from the non integer order differential equation which represents the dynamic model directing a natural robust relaxation, that of water on a porous dyke. A frequency illustration of robustness is deduced from such an equation. Particularly, in the Nichol-Black's plane, robustness is illustrated by an opeu loop frequency response locus which is reduced to a vertical straight line segment of abscissa between - re/2 and - ~ around the axis OdB. The template that this segment defines, slides on itself when the parametric state of the plant changes. In this paper, the template is synthesized by means of a CRONE variable phase regulator from the phase diagram of a mobile robot using a highly non linear error detector. The performances obtained through a numerical simulation show that the control is robust, not only versus the open loop gain, but also versus the non-linearities. A new dilemma, the robustness-input immunity dilemma is established by taking into account two templates of different lengths. The input of the plant is given for step responses to the reference input. It indeed reveals that the greater the robustness, the lower the input immunity, since the initial value of the input varies by an important factor when the template lengtht increases towards the high frequencies. References [I] - I. Horowitz and M. Sidi - Synthesis of feedback systems with large plant ignorance for prescribed time domain tolerances. Int. I. Control, vol 16, n~ 287-309, 1972 [2] - A. Oustaloup - Linear feedback control systems of fractional order between 1 and 2.IEEE Int. Symposium on Circuits and Systems, Chicago (USA), April 27-29, 1981 [3] - A. Oustaloup - Syst~mes asservis linraires d'ordre fractionnaire : Th6orie et Pratique - Ed. Masson, Paris, 1983 [ 4 ] - A. O u s t a l o u p and B. B e r g e o n - Frequency space synthesis of a robust dynamic command.IFAC'87, 10th World Congress on Automatic Control, Munich (FRG), July 27-31, 1987 [5] - A. O u s t a l o u p - From fractality to non integer derivation: a fundamental idea for a new process control strategy.8th Int. Conf. "Analysis and optimization of systems".INRIA, Antibes (FRANCE), June 810, 1988 [6] - A. Oustaloup - From fractality to non integer derivation through recursivity, a property common to these two concepts.Survey, session "Fractality and non integer derivation", 12th IMACS World Congress on Scientific Computation, Paris, July 18-22, 1988 [7] - A. Oustaloup - From the robustness of stability degree in nature to the control of highly non linear manipulators.Proceedings edited by Springer Verlag, Colloque Int. CNRS "Automatique Non Linraire", Nantes (France), 13-17 Juin, 1988 [8] - P. Baylou - L'automatisation de la cueillette du safran.Production des Plantes Aromatiques et Mrdicinales. Nions (France),Octobre 1988, 1988 [9] - A. Oustaloup, A. El Yagoubi, P. Baylou, J.F. Samson and P. Melchior - An algorithm of the CRONE software: Application to the guidance of a mobile robot. Eighth IFAC Workshop "Control Applications of Non linear Programming and Optimization", Paris, France, June 7-9, 1989. Proceedings Pergamon Press

V e r s u n e s t a b i l i s a t i o n non l i n 6 a i r e d i s c o n t i n u e Michel FLIESS I, Franqois M E S S A G E R

1z 3

Laboratoire des Signaux & Syst~mes. CNRS-ESE, Plateau de Moulon 91192 Gif-sur-Yvette Cedex (France). TEl : (1) 69.41.80.40 - Fax : (1) 69.41.30.60 2

Applications MathEmatiques et Logiciel. 6, rue AmEdde BollEe 92500 Rueil-Malmaison (France). TEl : (1) 47.49.14.00 - Fax : (1) 47.51.10.89

ABSTRACT : Since Sussmann [18], it is now well-known that, in opposition to the timeinvariant linear case, controllability of a nonlinear system does not necessarily imply the possibility of stabilizing it by a smooth feedback. This paper approaches the stabilization from another point of view, the discontinuous control, and therefore abandons the constraint of smooth feedbacks. It is interesting to see that this technique has already existed for a long time for relay systems (Cypkin [6]), sliding modes (Utidn [19]) and, of course, optimal control. Our method is based on a generalized controller canonicaI form [9, I0] which has recently been derived by using the analogue of the theorem of the primitive element in differential algebra (cf. Kolchin [14]). This canonical form is used to obtain an equation which defines a linearizing state feedback. By choosing the right parameters we are able to express a discontinuous stabilizing state feedback. The introduction of switching curves permits to take the structure of the system into consideration. We illustrate this technique by applying it to an example taken from Aeyels [I], which was shown not to be smoothly stabilizable.

1. I N T R O D U C T I O N I1 est aujourd'hui connu (cf. Sussmann [18]) que, contrairement au cas linEaire stationnaire, la commandabilit6 d'un syst~me non linEaire n'implique pas nEcessairement la possibilit6 de le stabiliser par un bouclage d'Etat rEgulier ou lisse, c'est9h-dire indEfiniment diffErentiabte. De nombreux travaux, souvent illustrEs par des exemples fort instructifs, ont depuis pennis de mieux cerner ce probl~me ( Brockett [3], Aeyels [1], Kawsky [13], Sontag [17], Dayawansa et Martin [7] .... ). Parall~lement, Byrnes et Isidori [4,5] ont entrepris une recherche de grande ampleur pour, gr~ce ~t la dynamique des zEros,.obtenir des conditions de stabilisation par bouclage rEgulier. Cette communication aborde la stabilisation par un autre point de vue, h savoir la commande discontinue et, donc, abandonne l'exigence de boucle rdguli~re. Quoique Sussmann [18] ait ddjh dcrit, il y a plus de dix ans, que "the introduction of disconfinous vector fields is unavoidable", en dehors, bien entendu, de la commande optimale bangbang, une tetle approche n'a re~u chez les thEoriciens qu'une attention fort limitde. Elle est cependant fort ancienne comme en t6moigne le livre de Cypkin [6], et se perpdtue, aujourd'hui, surtout par Ies modes glissants (cf. Utkin [19]), bien que certains travaux de 3

Travail de th5sc effectualdans lc cadre d'unc convcntion CIFRE entre lc CNRS ct AML.

779

nature plus th6orique paraissent de loin en loin (cf. Arstein [2], Hermes [12], Sontag [ 16], et aussi [ 11 ]). Rappelons encore que, sous certaines conditions de commandabilit6, Sussmann [18] a d6montr6 que la stabilisation est possible avec une commande analytique par morceaux. Nous proposons une approche de la stabilisation par bouclage discontinu en utilisant une forme canonique de commande [9,10], g6n6ralisant celle bien connue pour les syst6mes linEaires stationnaires monovariables, obtenue par des m6thodes d'alg~bre diffErentielle que l'un des auteurs a dEveloppEes [8]. Nous illustrons notre approche par un exemple qui, d'apr~s Aeyels [1], ne peut 6tre stabilis6 par bouclage r6gulier. II. R A P P E L S D ' A L G E B R E D I F F E R E N T I E L L E

(voir Ritt [15] et Kolchin [14])

On suppose le lecteur familier avec les propri6tds de base des corps commutatifs. Pour simplifier, les corps consid6r6s sont de caract6ristique nulle. Undcorps diff6rentiel est un corps commutatif K muni d'une d6rivation unique notee ~ - = , qm sausfait les propnetes usue les : d V a , b e K, ~ - ( a + b ) = a + b ?

IIIM

.

.

.

.

I

p

1

d (ab) = ~ + ab dt Une constante clans K est un 616ment c e K tel que c = 0. L'enscmb~e dc~; constantes de K est un sous-corps de K appeltd te corps de constantes. Une extension diffdrentielle L / K est donn6e par deux corps diff&entiels K, L avec K C L. Un 616ment de L e s t dit diffdrentiellement algdbrique sur K si, et seulement si, il satisfait une 6quation diff6rentielle algdbrique ~ coefficients dans K. L'extension L / K est dite diff&eutiellement alg6brique si, et seulement si, tousles 616ments de L sout diff~rentiellement alg6briques sur K. Dans le cas o~ L / K est liniment engendr~e, on a l e r~sultat fondamental suivant: T H E O R E M E : Pour une extension L / K finiment engendrde, les deux conditions suivantes sont dquival~ntes : (i) L/K est diffdrentiellement algdbrique; (ii) le degrd de transcendance de L/K est fini. Rappelons que ce degr6 de transcendance peut s'interpr6ter comme le nombre minimal de conditions initiales n6cessaires pour intdgrer le syst~me diff6rentiel repr6sent6 par L/K. I I I . I N T R O D U C T I O N DE L A F O R M E

CANONIQUE GE.NERALISEE

Soit k un corps diff6rentiel de base. On note k < u > k le corps diff6rentiel engendr6 par k et les composantes de l'entrEe u = (u 1. . . . . urn). Une dynamique est une extension diff6rentiellement alg6brique K/k finiment engendrEe. Soit x = (x 1. . . . . x n) une base de transcendance de K / k < u > , qui, comme on l'a vu, est de degr6 de transcendance t2ni. Les d6rivdes x sont k-alg6briquement d6pendantes de x. I1 en ddcoule (D)

Ai ~x ," i . -~,,u . . . . . . . . . . .,,(m)~=O J

( i = l ......n),

780

oh les A i sont des polyn6mes ~tcoefficients dans k. En invoquant des spEcialisations dans les reels ou les complexes, on voit que l'dcriture explicite (D')

x i = ai(x,u,~l ..... u (m)) = 0

(i = 1..... n),

a seulement une valeur "locale". (D) et (D') redonnent une image plus famiti~re de Ia notion de dynamique. Une autre base de transcendance x = (xx ..... xn) de K / k < u > conduit, 6videmment, ~ des expressions semblables. On sait que les composantes de (resp. x) sont k-algdbdquement dEpendantes de celles de x (resp. x). Ce sont les formules de passage d'un 6tat ~ l'autre qui font intervenir l'entrde et ses dErivEes. Pour des extensions algEbriques non diffErentielles usueUes, qui sont finiment engendrEes, la notion d'gldment primitif est bien connue. I1 en existe un analogue diffErentiel (cf. Kolchin [14]) qui affirme l'existence d'un ElEment ~ 9 K, dit dldment primitifdiffdrentiel, tel que K = k. C'est dire que K est diffErentiellement engendrd par k et ~. Le degrE de transcendance n de K/k est le olus petit entier n tel que ~(n) soit k-algEbriquement dependant de ~,~ .... ~(n-1). Posons ql = ~ ..... :'(n-l) ' . . q~ = ~ . Alors, q = (ql ..... qn) est une base de transcendance de K / k < u > , qm fouralt une gdadralisation implicite de la forint canonique de conmamlde :

I]

ql = q2

qn-1

=

qn

k.C(Ctn,q,u,u ..... u(n)) = 0 06 C est un polynEme ~ coefficients dans k. On en ddduit donc une forme explicite "locale":

!l = q2

I

qn-1

=

qn

ctn = c(q,u,u ..... u(n))

IV.

METHODE

DE

STABILISATION

PAR

RETOUR

D'ETAT

DISCONTINU Considdrons la dynamique, de dimension n : (Z)

{~~ = f l ( ~ ' u )

fn(~,u) 06 ~ = (~I ..... ~n) et u = (u 1..... urn). On suppose, pour simplifier, que fl ..... fn sont des fonctions polynomiales des arguments h coefficients reels. I1 est facile de ramener l'dtude de (E) ",i celle des corps diffdrcntie!s ~ S c e ~ !'idda! diffdrentiel premier [14], qui lui

781

correspond. L'analyse du paragraphe prdcddent montre l'existence d'un dl6ment primitif x I qui conduit &la forme canonique suivante : Ix~ = x2

(Y'g)

/Xn_ 1 = Xn k.G(xn,X,U,U ..... u(m)) = 0

o ~ a x = ( x 1..... xn). Parun argument du type "thdor~me des fonctions implicites", il vient :

I

xI = x2

v

(Eg)

IX--1 =

Xn

1

k.x.n = g ( x , u , u ..... u(m))

C'est la forme canonique explicite, qui, en gdndral, n'est valable que "[ocalement". Dbs lors, il est possible d'expfimer le bouclagc lindarisant, cn 6cfivmlt : (z)

n

p

g(x,u,u ..... u(m)) = i~t aixi + i~l bivi

off les a i e t b i sont des constantes rdelles et v = (v t ..... vp) est la nouvelle entrde. I1 s'agit alors de rdsoudre rdquation (E) en prenant l'entr6e u c o m m e inconnue. On obtient la stabilisation en injectant cette commande dans le syst6me (Zg') durant un petit intervalle de temps At et on rdsoud & nouveau (E) & l'instant t + At. V. E X E M P L E

DE S T A B I L I S A T I O N

DE SYSTEME

Le systSme ?~ entr6e monodimensionnelle v : x=x+ (2)

3t

y3

f(x,y) + v

off f(0,0) = 0, n'est pas, d'apr~s Aeyels [1], stabilisable par retour d'dtat r6gulier. V.1. E t u d e p r d l i m i n a i r e (Y.) peut, par un bouclage r6gulier de la forme u = fix,y} + v, 6tre mis sous la forme :

(]~')

{i = xy+3 u

11 faut distinguer quatre cas selon le signe de x et de x, donc selon la position du couple (x,y) par rapport ~ la courbe x + y3 = O.

782

Y

B' A

x A A'


0 et x + y3 > 0} {(x,y) ~ I R 2 t e l q u e x > 0 e t x + y 3 < 0} {(x,y) ~ ] R 2 t e l q u e x < 0 e t x + y 3 < 0} {(x,y) e ] R 2 t e l q u e x < 0 e t x + y 3 > 0}

Remarque : Les fl~ches repr6sentent le sens de d6placemcnt du couple (x,y) dans le plan avec une entrde u nulle. Si (x,y) e B U B', x et x ont des signes opposds. Aeyels [1] d6montre l'existence d'une commande u guidant l'6tat (x,y) de (T,') vers l'origine. Si (x,y) e A U A', x et x ont marne signe; le syst~me ne peut 6tre stabilis6 en restant darts A U A'. Par un argument analogue ~ celui du paragraphe pr6c6dent, il existe une commande qui permettra au couple (x,y) de passer de la r6gion A (resp. A') ~t la rdgion B (resp. B'). V.2. Calcul et choix des c o m m a n d e s

a) R6gions A et B a.1) (x,y) E A On peut v6rifier tr~s simplement que (x - u) est un 61dment primitif diff6rentiel, ce qui permet de mettre ( Z ' ) sous la forme ddcrite prdc6demment, en effectuant le changement de variable suivant : (F~)

XI=

X - U

x2

x1

x + y3 _ u

d' off

(Zgl)

r~xl = x2

Lx2

-u + (x2+u) §

(xx+u))2/3

Exprimons ensuite le bouclage lin6~arisant en dcrivant : (El)

-'6 + (x2+h) + 3u(x2+h - (xl+u))2/3 = axx I + a2x 2 + bw

off a l, a 2 et b sont des constantes rdeUes et w une nouvelle entr6e, qui sera supposde nul!e. Pour des raisons ~videntes de stabilisation, a 1 e t a 2 seront pris r6els et n6gatifs.

783

On consid~re (El) comme une Equation en u, u et "u que l'on rEsoud aux instants d'dchantillonnage : U=-X tl=-

1 X2

"11= - a l x 1 - a 2 x 2

Connaissant les conditions initiales de u ~ l'instant d'Echantillonnage t=nAt, nous calculons les param~tres A, B e t C de la fonction u(t)=At2+Bt+C, pour nAt0

(1-3)

and

BI(x)(Y) = ~ DX ( Y ) = ~

(1-4)

1 bk D X k ( y ) ~.~ (bk, Bernoulli numbers )

E 1 and B 1 are Lie series with only one occurrence of the variable Y. In (1-3) the fraction means the simplification of the series numerator by the denominator, this represents a compact expression which will be extensively used lateron. In this paper, the formal equality (1-3) is first extended to higher order derivatives with respect to t and then to the exponential term:

789

X

+Z

tk ~T".rYk

(1-5)

k>l

instead o f X + tY, where the Yk, k > 1 denotes a family of formal variables. For, we define repeatedly a family of Lie series denoted: Ei(x,Yjl ..... YJi-1)(YJi), Ji > 1 and we show that each derivative (1-5) with respect to t, of order p, satisfies a homogeneous, of degree p, polynomial in terms of these elements E i. These combinatorial equalities which are variant and extensions of the well-known BakerCampbell-Hausdorff formula are directly applied in this paper to the sampling problem and to explain relations between nonlinear differential equations and their discretized analogues. The present work goes further than the results proposed in [7] and leads to compact solutions which are much more appropriate for digital computing making use of formal languages. Moreover, we can show that the same algebraic manipulations can be used to analyze functional expansions associated to nonlinear discrete time equations, this gives strong analogies between nonlinear continuous, discrete and discretized systems which are extensively studied in [10, 11] The use of formal algebraic methods in nonlinear control theory were initialized in the continuous time case [4] and further developed in both mathematics and control domains [2, 5]. The paper is organized as follows: some definitions and notations are recalled in section 2, section 3 presents a combinatorial study of parametrized exponential series. The results are applied in section 4 to the sampling problem of a nonlinear differential equation.

2 - P R E L I M I N A R I E S AND F O R M A L N O T A T I O N S Consider a family (or an alphabet) denoted {Xl .... Xk} of elements of an associative algebra. Some definitions and notations from the theory of formal languages are recalled. We denote: Xil...Xip; ij E {1 . . . . . k}, p > 1 any word of length p in terms of the indeterminates {X 1,...,Xk }; 1 is the word of length zero or neutral element and: Xik = Xi... Xi (k times) denotes the word composed by the concatenation k times of the same letter Xi. According to these notations the following exponential type and logarithmic type series are easily defined. We note by: e Xi = I + Xi + (1/2!)XiXi + ... + (1/k!)Xi k + ...

(2-1)

the exponential series and in the same way: Log (I + Xi) = Xi - (1/2[)XiXi+ ... +((-1)k/k!)Xi k + ...

(2-2)

the logarithmic series. Stating by convention that any element Xi is said of degree i, any homogeneous of degree p polynomial with real coefficients is written as:

P=~ m>l

Z

ail"'aimXil'"Xim

(2-3)

il,...,im>l m

with: all.. aim • R and ~

ij = p.

j=1

We note DXi = adXi the formal derivative operator which acts on any word Xi 1..... Xij according to the rule: DXi(Xil ..... Xij) = XiXil ..... Xij - Xil ..... XijXi (2-4)

790 The shuffle product [12] is defined in a recursive way as follows: lmXi = Xi ml = Xi XiO)Xj = Xj(.oXi = XiXj + XjXi

(2-5)

Xjl .... Xjpf-OXil .... Xij = Xjl(Xj2 .... Xjpr-OXit .... Xij) + Xit(Xjl .... Xjp(OXi2 .... Xij)

3 - SOME COMBINATORIAL FORMULAE Hereafter, we will successively denote as {X,Y} and {X,Y i, i > 1} formal variables and t any element of R. The aim of this section is to generalize the formal equalities (1-1) et (I-3) to higher order derivatives with respect to t and then to the exponential term referred as the multivariable case:

tk X + kaY-'~1 ' " Yk The monovariable case: Considering equality (1-3), or equivalently at t r 0 the equality:

EI(x+tY) =

1 - e-D(X+tY) (- 1)k n r X + D(X + tY) =k>~_o(k+l)! ~" ty)k

(3-1)

we define: (3-2)

E2(X,Y) = (d/d0/t = 0 EI(x+tY)' and iteratively for i > 3: Ei(x,Y,...,Y) = (d/d0/t = 0 Ei-I(X + tY, Y,...,Y)

(3-3)

In this way, we define a family of Lie series denoted by Ei(x,Y,...,Y)(Y). Since each series Ei(X,Y,...,Y)(Y) contains i occuring of the variable Y, it will be said of "degree i". For the First elements, we obtain the compact expressions generalizing (1-3):

el(x) =

l-e -DX

D------g---= ~

(- 1)k ~ v k

~"'~

k~O

E (X,Y) =

,~

E3(X,Y,Y) =

r~-z2

20

(~ ~) = 2 , ( ~

DXk(mDY)

(3 -4)

I-DX+(DX2/2!) -e'DX . . . . . 2 x-', (-1) k+2 " ~ tc0t)x) = L ~ DXk(mDY)2 k_>0

and finally for i > 1: Ei(x,Y,...,Y) =

1-DX+(DX2/2!)+...+((-1)i'I/(i'I)!)DXi'I-e'DX.(mDY)i-I DX i

(3-5)

With these notations, it becomes possible to extend (I-1) and to express any derivative as a homogeneous polynomial in terms of these Lie series. We state:

791 Theorem 1: The following formed equality of series is verified:

e - x e X + tY = 1 + ~

( t k / k ! ) a k ( X , Y ..... Y)(Y)

(3-6)

k_>l

where:

ak(X,Y,...,Y)(Y) = (d/dtk)/t = 0 e -X e X + tY

can be decomposed as a homogeneous polynomial of degree k in the formal indeterminates of "degree i": E i ( x , Y ..... Y)(Y). Several techniques to prove this theorem are possible. W e will choose the more intuitive one which consists in the successive derivation with respect to t, expressed at t = 0, of the equality (1-1). In fact, according to (1-3) and (3-I), we can rewrite equality (1-1) at t ;~ 0 as: (d/dt)/t * 0 (e X + tY) = e X + t Y E l ( X + t y ) ( y ) = e X + t Y a l ( X + t Y ) ( y )

(3-7)

Taking into account the non-commutativity of the formal variables X and Y and applying the usual rules of derivation, with respect to t, of products of formal objects, we alternatively compute the polynomials ai (X,Y,...,Y)(Y) as follows: By definition and because of (3-7): (d/dt2)/t = 0 eX + tY = e X a 2 ( X , y ) ( y ) = (d/d01 t = 0 ( e x + tYEI(X+:Y)(Y)) = (d/dt)/t = 0 (e X + tY)EI(X)(y) + e X ( d / d 0 / t = 0 (EI(X+tY)(Y)) = e X E I ( X ) ( y ) E I ( X ) ( Y ) + eXE2(X,Y)(Y) in other words: a2(X,Y)(Y) = {E 1(X)(Y) }2 + E 2 ( X , y ) ( y )

(3-8)

from the definition: E2(X'Y) (Y) = (d/dt)/t = 0 E l (X+tY)(Y) Proceeding this way we can d e c o m p o s e successively the elements ak(X,Y ..... Y)(Y) as a homogeneous polynomial. For k = 3, we obtain: (d/dt3)/t = 0 eX + tY= e X a 3 ( X , y , y ) ( y ) = ( d / d 0 / t = 0 (eX + tYa2(X+tY'Y)(Y)) = (d/dt)/t = 0 (eX + t Y ) ( E I ( X ) ( y ) E t ( X ) ( Y ) + E2(X'Y)(Y)) + eX(d/dt)/t = 0 (EI(X+tY)CY))(EI(X)(Y)) + eXEX(X)(Y)(~Vdt)/t = 0 (EI(X+tY)(Y)) +

e X(a/dt)/t = 0 (E2(X+tY'Y)(Y)) which is: a3(X,Y,Y)(Y) = {EI(X)(Y)} 3 + 2 E I ( X ) ( y ) E 2 ( X , Y ) ( Y ) + E2(X)(y)EI(X,Y)(Y) + E3(X,Y,Y)(Y) (3-9) under the definition: E3(X,Y,Y)(Y) = (d/d0 / t = 0 E2(X+tY'Y)(Y)

792

For the higher order derivatives, we obtain the recurrent formula: (d/dtk)/t = 0 eX + tY = eXak(X,y,...,y)(y) = (d/dt)/t = 0 (eX+tYak'l(X+tY'Y"'"Y)(Y)) =

eXEI(X)(Y)ak_I(X,Y,...,Y)(Y) + eX(d/dt)/t= 0 ak.l(X ?- tY,Y,...,Y)(Y)

(3-10)

Denoting with L the Lie algebra generated with X and Y, L0 the Lie ideal of L generated by Y and successively with Lop , for p > 1, the decreasing series of ideals defined as: Lo 1= Lo, Lop= [Lo,L0P-1], p > 2. We verify: Corollary_ 1: Each element Ei(x,Y,...,Y)(Y), i _> 1 is a Lie element of degree i, belonging to L0 i, and including i occuring in the variable Y. Theorem 2: Using the compact notation: Ei(x,Y,...,Y)(Y) = Ei(DX), we verify the following equality of series: e-XeX+tY=l+ ~ (tk/k!) ~ k>l

~

c01 ..... im)Eil(DX)-..Eim(DX)

(3-11)

k>l il,...,im21

= 1 + T(Eil ..... Eim) = S(Eil ..... Eim), with ~. ij = k. Moreover, the coefficients c(il,...,im) satisfy the recurrent formulae of a shuffle product, j=l

which is f o r k > 1: C(il)C(i2) = c(il,i2) + c(i2,il) = C(ilf.Oi2) C(il)C(i2,i3) = C(il,i2,i3) + c(i2,il,i3) + c(i2,i3,il) = C(ilO~i2i3)

The following corollaries are easily deduced from Friedrichs theorem [12] or related ones. Corollary 2: The series Log( 1 + T(Eil ..... Eim)) is a Lie element in the variables Eil(DX) ..... Eim(DX) if, and only if, the coefficients c(i 1..... im) satisfy the recurrent formulae of a shuffle product. This Lie element corresponds to the Baker-CampbeU-Hausdorff exponent of the exponential product: e-XeX + tY. Log (1 + T(Eil ..... Elm) )= BCH(-X, X + tY)

(3-12)

1 + T = e BCH(-X, X + tY)

(3-13)

that is:

Corollary 3: Denoting with E i et F i two families of indeterminates, we verify: T(Eil+Fil,..,Eim+Fim) = T(Eil ..... Eim)+T(Fil,..,Fim)+T(Eil,..,Eim)coT(Fil,..,Fim) (3-14) and in a similar way: S(Eil+Fil ..... Eim+Fim) = S(Eil ..... Eim)(oS(Fil ..... Fim).

(3-15)

793

The multivariabl~ case: Let us now generalize these results to the exponentia! expansion associated with the multivariable series which is nonlinear in t : X + ~ (tk/k!)Yk. k>l

We will exactly follow the same procedures as in the previous case, in terms of a new family of formal Lie series, defined with degree i and denoted in a compact form by EI(DX) for i > 1. Because of the non linearity with respect to t, each EJ(DX), forj > 1, is now expressed as a J linear combination of Lie series of degree i = ~ ip, denoted by EJ (X,Yi 1..... Yij. 1)(Y ij), which are p=l

straightforward extensions of the elements defined in (3-5). In fact, for ij > 1, we easily define the elements: EJ(x,Yil ..... Yij.1)(Yij) (of degree ~ij), by the compact expression: 1 - DX + (DX2/2!) +...+ ((-1)j'I/(j-1)I)DXJ -1 - e -DX DX i coDYi1...mDYij_1(Yij)

(3-16)

In the first terms, we compute : El(X)(Yi) =

1 - e -DX DX (Yi) said of degree i

E2(X,Yi)(Yj) =

1 - DX-e -DX DX 2 (mDYi)(Yj) said of degree i + j

(3-17)

Besides these elements, we define by successive derivatives with respect to t the series Ek(DX) of degree k. According to (3-1), we can write at t = 0:

E1(DX) = EI(X)(Y1) =

1 - e -DX

DX

d X + ~ (ti/iI)Yi) (Y1) =di"/t = 0 (e-x e i_>1

and equivalently at t # 0, 1 - e-D(X+~

E t ( x + ~ (ti/i!)Yi)( ~ (ti/i!)Yi+l) = i>1

(ti/i!)Yi)

i_>l D ( X + E (ti/i!)Yi) i_>l

i~0

( ~ (ti/i!)Yi+l) (3-18) leO

For higher order terms, proceeding as previously, we compute the successive derivatives with respect to t and we define for example for k = 2: E2(DX)=(d/dt)/t = 0 E I ( X + ~ k_>l

(ti/iI)Yi)(~

(ti/i!)Yi+1)

i>0

= E2(X,Y1)(Y1) + EI(X)(Y2)

(3-I9)

ProPosition: Any element Ek(DX), k _> 1, defined of degree k, is a linear combination of elements of degree k of the form EJ(x,Yil ..... Yij_l)(Yij). We obtain: k

Ek(DX) = ~ j=l

~

c~(il...ij) EJ(x,Yil ..... Yij_l)(Yij),

il,...,ij> l

(3-20)

794

with ct(il...ij ) e R, Y~ij = k, EI(DX) = EI(X)(YI) = al(X)(Y1) and according to the recurrent law : k

Ek+I(DX) = EJ(x,Yil+I ..... Yij-1)(Yij) +"'+ Z

Z

j=l

a(il...ij)

i 1 ..... ij_>l

k

EJ(x,Yil ..... Yij_l)(Yij+l) + ~ j=l

~

a(il...ij) Ej+I( X, YI, Yil ..... Yij-1)(Yij)

(3-21)

il,...,ij_> 1

For the t-u-stterms, we compute: E2(DX) = E2(X,Y1)(Y1) + EI(X)(Y2) (3-23) E3(DX) = E34X,Y1,Y1)(Y1) + 2 E2(X,Y1)(Y2) + E2(X,Y2)(Y1) + EI(X)(Y3), E4(DX) = 2E3(X,Y2,Y1)(Y1)+2E3(X,Y1,Y2)(YI)+3E3(X,Y1,Y1)(Y2)+3E2(X,Y2)(Y2) +2E2(X,Y 1)(Y3)+E2(X,Y3)(Y1)+E1(X)(Y4)+E4(X,Y 1,Y 1,Y1)(Y1)+E2(X,Y1)(Y3). (3-24) Remark: When only one indeterminate is involved Y1, we recover the equality: Ek(DX) = Ek(x,Y1 ..... Y1)(Y1). According to these notations, the multivariable analogues of results stated previously can be written with respect to the generalized elements Ek(DX) for k > 1: Theqrem 3 : The following equality of series is verified: e-X

e (X + ~" :"""2.ttVx!)Yi)= 1 + ~

(tk/k[)Ak(X,Y1 ..... Yk)

(3-25)

k>l

with : Ak(X,Y1 ..... Yk)= (dk/dtkt/t = 0 e-X e(X +E (ti/i!)Yi) which can be rewritten as an homogeneous polynomial of degree k in the formal indeterminates: Ei(DX), i>l. Generalizing (3-8) and (3-9) according to the notations (3-23) and (3-24), we directly obtain: A 1(X,Y1)=EI (DX)=E1 (X)(Y 1) A2(X,Y 1,Y2)= {E 1(DX) }2 + E2(DX) A3(X,YI,Y2,Y3)= {EI (DX) }3+2E I (-DX)E2(DX)+E2(DX)E I(DX)+E3(DX)

(3-26) (3-27) (3-28)

Denoting with L the Lie algebra generated by X and Yi. i >1, with iLo the Lie ideal of L generated by Yi and successively denoting by Lo p, p>l, the decreasing series of ideals of "degree p" defined as: Lo 1= 1Lo, Lo 2= [1Lo, lLo]+2Lo, Lo 3= [[1Lo,[1Lo,ILo]]+[2Lo, ILo]+3Lo =[Lo2,1Lo]+3Lo, (3-29) Lo 4= [1Lo,[ iLo,[ ILo,1Lo]]]+[1Lo, {2Lo, ILo]]+[ 1Lo,3Lo]+[2Lo,2Lo]+4Lo = [Lo3,1Lo]+[Lo2,2Lo]+4Lo. We obtain: Corollary 5: Each element E'(DX), for i >_1 is a Lie element of "degree i" that is belonging to Loi.

795 Theorem 4: The following equality of formal series is verified: e-XeX+Y'(ti/it)Yi= 1+ ~

(tk/k!) ~

k_>l

m>l

~

c01 ..... im)Eil(DX) ... Eim(DX)

(3-30)

il,...,im>_l

= 1 + T(Ei! ..... Eim) = S(Eil ..... Eim), with ij > 1, ]~ ij = k. Moreover, the coefficients c(i 1..... ira) satisfy the recurrent formulae of the shuffle product.

4 - NONLINEAR SYSTEMS UNDER SAMPLING When we consider a nonlinear differential equation driven by controls, which are constant over time intervals of amplitude 5, the question of finding a nonlinear difference equation, so that its state evolution corresponds, at sampling times, to the solution of the nonlinear differential equation, when the same initial states are assumed, is referred to as the sampling problem. With respect to usual numerical integration methods, the solution of this problem leads to an implicit sampled representation described by a nonlinear difference equation. On the basis of these equations we can design various digital control schemes and study the preservation of special properties under sampling [8, 9]. This problem, solved in the linear analytical case (equation 1-1) [7] is posed here in a strictly formal way. This approach enables the extension to differential equations including non-linearities with respect to the controls; the solution appears as a direct consequence combinatorial formulae proposed in section 3. It has to be noted that the recursive properties of the solutions are very adequate to the use of symbolic programming language as REDUCE language for example already tested in [1]. Consider a nonlinear differential equation:

x(t) = f(x(t) + ~

(ui(k)/i!)gi (x(t)

(4-1)

i>1

defined on R n by the vector fields f and gi : Rn .... R n, assuming the controls constant over time intervals of amplitude 5 : u(t)=u(k), pour k5 < t 0, the question which is posed is to find a nonlinear difference equation of the form:

x(k+l) = F(x(k)) + ~

(ui(k)/i!)Gi(x(k)

(4-2)

i~l

such that the state evolutions of (4-I) and (4-2) coincide, at sampling times, when x(t -- 0) = x(0); that is x(t) = x(k), for t = k& k >- I. Denoting by Lf and Lgi, the formal directional derivatives associated respectively to the vector fields f and gi, we define an alphabet of formal variables on the basis of which computing rules established in section 2 and 3 can be applied. More precisely, elements Lf and Lg i are called of degree 1 and i respectively; the operation of concatenation of two letters corresponds to the composition of differential first-order operators and consequently exponential and logarithmic expansions, Lie and shuffle products are defined in a very natural way with respect to this composition.

796 Integrating during time interval [kS,(k+l)5[, the solution, at time t = (k+l)8, of the equation (4-1) can be expressed as the flow associated to the constant vector field : f(x(k)) + ~ (ui(k)/il)g i (x(k)). i>l We obtain the formal solution : x(k+l) = eS(L~Y, (ui(k)/i!)Lgi)(Id)/x(k) i_>l

(4-3)

The results proposed in section 3, in particular th e application of theorem 4 to (4-3), directly lead to a compact solution of the sampling problem. Comparing it with (4-2), we can write: F(x) = eSLf(Id)/x Gk(X) = (d k/duk)/u= O jS(Lf+X(ui(k)/i!)Lgi )(Id)/x, k > 1 = e~SLf ~ ~ c(il ..... im)Eil(DSLf) ... Eim(D~SLf)(Id)/x, m_>l il,....im>l with ij >- 1 and Y. ij = k

(4-4)

For the first terms, we compute: G 1(x) = erLf E 1(D~SLf)(Id)/x G2(x) = eSLf{ {El (DSLf) }2 + E2(DrLf) ] (id)/x G3(x) = eSLf{ {E l(DSLf) }3 + 2E 1(D~SLf)E2(DSI_,f) + E2(D~iLf)E I (DSLf) + E3(DSLf) }(Id)/x

(4-5) (4-6)

with: E 1(D~Lf) = E 1(SLf)(SLg I) E2(DfiLf) = E2(SLf, SLg 1) (SLg 1) + E 1(SLf) (SLg 2) E3(DSLf) = E3(SLf, SLgl,SLgl)(SLgl) + 2E2(SLf, SLgl)(SLg2) + + E2(SLf, SLg2)(SLgl)+ El(SLf)(SLg3),

(4-7) (4-8) (4-9)

The elements Ei(SLf, SLgil ..... 5Lgij.1)(fLgij ) are defined by the expressions (3-4) and (3-16) : For the first terms, we compute : El(SLf)(SLgi) = l'e'DSLf = ~ ~(-1) D S kL t k ( S L g l ) , DSLf ~o

k _>0

(4-10)

-- 5Lg i - (52/2!)adLf(Lgi) +(~3/3 !)ad2Lf (Lgi) + 0(53) E2(SLf, SLgi)(SLgj) =

1-DfLf-e-DSLf DSLf 2 (c~

1)k+ 1 DSLfk(c0DSLgi)(SLgj) =~z~ (-(k+2)! k>O = _ (52/2[)adLgi(Lgj) + (53/3 0adLfadLgi(Lg j) + (53/3 !)adLgiadLf(Lgj) + 0(53)

(4-11)

797

E3 (SLf, SLgi,SLgj)(SLgk) =,

I - D 8 L f - + (DSLf2/2!) - e - D S L f (coD8Lgi)(coDSLgj)(8Lgk) DSLf3

= ~ ~ ( - l ) k + 2 DSLfk(o~DSLgi)(coDSLgj)(8Lgk) k>0 = (5313!)adLgiadLgj(Lgk) + O(83)

0-12)

Remarks; l~i)Defining, according to (3-29), the successive Lie ideals associated to the elements 8Lf and 8Lg i ,for i > 1, it follows that Ei(D8Lf) belongs to L01. (it) Considering now the reverse problem which consists of associating to any difference equation of the form (4-2), a differential equation of the form (4-1), in such a way that the solutions coincide at sampling time, some remarks can be deduced from the previous analysis. Let us first assume the sampling period equal to 1 and the existence of a vector field f, such that eLf (Id), is equal to F. According to the definition (1-4) of the reverse series, we see that the vector fields gi can be successively computed as follows: gl(x) = B l(Lf)(G1)(Id)/x g2(x) = B 1(Lf)(G2_G1G I_E2(Lf,Lgl)(Lg 1))(Id)/x g3(x) = BI(Lf)(G3+2G1GIGt-2G1G2-GIG2-2E2(Lf,Lgl)(Lg2)-E2(Lf,Lg2)(Lgl))(Id)/x

(iii) For practical purpose, approximated solutions only are computed, this corresponds to the truncation of the series solutions, with respect to the sampling period, at a fixed order which define the order of approximation of the sampled system. REFERENCES

[1] J.P. BARBOT, A computer aided design for sampling a nonlinear system in "Applied Algebra, Algebraic Algorithms and Error Correcting Codes, Lect. Notes in Comp. Sc. 357, T. Moraed. pp. 74-88, 1989. [2] P. CROUCH, F. LAMNABHI-LAGARRIGUE, Algebraic and multiple integral identities, Acta Applicanda Mathematicae ~ pp. 235-274, 1989. [3] R.GOODMAN, Lifting vector fields to nilpotent Lie groups, J. Math. Pures & Appli., 57, pp.7786, (1978). [4] M. FLIESS, Fonctionnelles causales non lindaires et ind6terminres non commutatives, Bull. Soc. Math. France, 109., pp. 3-40, 1981. [5] M. FLIESS and D. NORMAND-CYROT, Algrbres de Lie nilpotentes, formule de Baker-CampbellHaussdorff et int6grales itErEes de K.T. Chert, s~minaire de probabilit6s, 1980-1981, 16, Lect. Notes in Math., J'. Azema and M. Yor eds., pp. 257-267, Springer-Verlag, 1982. [6] P. LEROUX and X.G. VIENNOT, A combinatorial approach to nonlinear functional expansions: an introduction with an example, Proc 27-th IEEE-CDC, Austin Texas,USA, 1988. [7] S. MONACO and D. NORMAND-CYROT, On the sampling of a linear control system, Rap. 6-84, DIS, Rome, Proc. 24th IEEE CDC, Fort Lauderdale, USA, pp. 90-95, 1985. [8] S. MONACO and D. NORMAND-CYROT, Invariant distributions under sampling, Theory and Applications of Nonlinear Control Systems, ( C.I. Byrnes and A. Linquist eds.), North Holland, pp. 215-221, 1986. [9] S. MONACO and D. NORMAND-CYROT, Zero dynamics of sampled nonlinear systems, Syst. and Contr. Letters, 1._LL1pp. 229-234, 1989. [10] S. MONACO and D. NORMAND-CYROT, Functional expansions for nonlinear discrete time systems, Mathematical Systems Theory, 21, pp. 235-254 1989. [11] S. MONACO and D. NORMAND-CYROT, Combinatorial relations between discrete and continuous time nonlinear systems, submitted for publication, 1990. [12] R.REE, Lie elements and an algebra of shuffles, Ann. of Math., 68, pp.210-220, 1958.

N O N L I N E A R MRAS IN ROBOTS M O T I O N C O N T R O L Bernard BROGLIATO Laboratoire d'Automatique de Grenoble - ENSIEG-INPG-CNRS BP 46 38402 Saint-Martin d'H6res, France

abstract : Inttds paper, we provide a brief overview o f the principal adaptive control laws that have been developped in the field of rigid manipulators during the last ten years, that can be turned into a common MRAS framework and analysed using passivity arguments.

1) I N T R O D U C T I O N Robot manipulators control has been a major topic for an increasing number of researchers during the last decade [1]-[15].Lndeed, besides the fact that new schemes had to be developped in order to improve the performances of new manipulators (particularly those employing direct drive arms), making it mandatory for designers to take the inertial effects into account in their totality (such as Coriotis and centrifugal torques), it appeared that those systems belong to a very particular class of non-linear systems, and even of finite-dimensionnal physical systems (ie systems described by a finite set of generalized coordinates ).Many control schemes have emerged, a large proportion of which are explicitly or implicitly inspired from the Model Reference Adaptive Systems (MRAS) approach studied in the linear case (see eg [16]).Two main phases can be distinguished concerning robots control through a model-reference approach : i) the first one consists in adaptive control laws designed as parallel Adaptive -Model -Following -Control (AMFC) schemes.(figure 1) ii)the second one consists in adaptive control laws derived from the computed torque method, based on the linearization- wit- the- unknown- parameters property of manipulators, and using some physical properties of robots.They can be interpreted as series-parallel AMFC .(figure 2) The general problem of controlling rigid frictionless manipulators can be stated as follows : Find a control law that asymptotically (in the adaptive case) linearizes that non-linear plant, ie makes it asymptotically follow a prespecified linear decoupled reference model.It is in fact essentially this model which will make laws of type i) differ from laws of type li) above. The paper will be organised as follows: in section 2, we provide a brief recall of linear AMFC.In section 3, we present an overview of the "first generation" schemes studied in [1]-[7].In section 4

799

we present the second type of adaptive laws studied in [8]-[15].Section 5 is devoted to unify stability analysis of those schemes. Conclusions are given in section 6.

~_~

reference ] ym model u

1u

riga : parallel MRAS (trajectorytracking : r ~ 0 )

fig. 2 : series-parallelMRAS (state regulation : r = 0 )

2) L I N E A R A M F C 21) Generalities Model reference adaptive systems have been studied extensively in the literature .As noted in [17], designing MRAS requires, from a general point of view, two steps: i) an algebraic part : One has to find a class of reference models such that a bounded control signal u and a controiler C(0*) exist such that lim t~+,,o ]] Xp - Xm[I =0 ,where x m and Xp are the model and plant state (or output) vectors respectively, O* being the true unknown parameters of the plant. ii) an analytic part : it consists in determining the adaptive law for adjusting the parameters estimates 0, such that the overall system will be stable. Point i) obviously requires prior knowledge about the plant that has to be controlled. It is strongly related to minimum-phase properties of the plant.It is called the perfect model matching problem .It can be considered either as a transfert matrix matching, or by establishing algebraic conditions on the error equation.It leads in that case to the well-known Erzberger's conditions on the plant and the model [16] .Note that these are sufficient only and may or may not be verified[27].AMFC problem can then be seen as an MR.AS problem for which a solution of asymptotical model matching exists. 22) A s s u m p t i o n s on the model and the plant W e give here a list of the properties that are generally required in the parallel AMFC problem for SISO systems [17][18][19] and that allow one to solve points i) and ii) above.The plant and the

800

model are described as follows: (P) (RM)

:~p=Apxp+Bpu xm=Amxm+Bm r

; yp=Cpxp ; ym=Cmxm

where r is a bounded piecewise continuous signal. i) the plant is minimum-phase and stabilizable ii)an upper-bound on the plant order is known iii)the relative degree p of the plant is known, and p < Pm (Pm is the relative degree of the model) iv)the sign of the high-frequency gain (transfert function approach) is known. Note that i) is necessary for (P + controller) to be detectable[17,p.375], as the eventuals poles-zeros cancellation that may appear between the plant and the controller (and lead to unobservable modes) are stable.It is shown in [18] that iii) is the only necessary assumption.If the plant zeros are exactly known (which may not be the case in practical applications), then i) is not necessary, but the model has to contain the plant unstable zeros, in order to avoid unstable poles-zeros cancellations.One could ask here whether this still represents a reference-model problem, or simply pole-placement.ii) can be rendered unnecessary by relaxing the objectives on the error [18]. iv) is not a necessary condition : controllers can be found that achieve the MRAC problem although it is not verified.The most general approach for which this condition is unnecessary can be found in [28]. Several remarks arise : i) All that has been done is concerning SISO systems.It can be translated to the MIMO case [17] [19], taking into account the particular structure of multivariable systems, especially for those systems that admit a Hermite normal form .But this is far from trivial, particularly in the adaptive case, and is still being under investigations. ii) The case usually studied in the literature is the parallel MRAC problem, with an input / output (transfert function) approach.The approach which seems to be the most widely used [20]- [23] is based on an augmented state vector xT= [ XpT, wiT, w2T], where w I and w 2 are n-dimensionnal vectors obtained by filtering the input and the output of the plant.It is shown [20] that a non-minimal representation of the reference model can be constructed in terms of the plant true parameters, that allows to derive an appropriate error equation, and then conclude on the convergence of the state error vector. It presents the great advantage of using the plant output instead of the state vector in the control law. Schemes designed from a state space approach have been seldom studied.An approach proposed in [16, p. 218] is based on hyperstability's arguments for the stability analysis.The control law

801

consists in output and state feedbacks, the gains of which are adaptively adjusted by the so-called integral+proportionnal adaptation mechanism .The main drawbacks of this approach are that the input matrix Bp has to be known (or at least its structure) for stability conditions,and that the plant state vector has to be measurable.Though these assumptions could appear at first sight quite stringent, there exist particular cases (see next section) for which they are verified. However, it presents the advantage of being available for MIMO systems, as no assumption is done on the output and input dimensions. remarks a) In this approach, the relative degree problem related to the SPR condition is avoided, as one can consider the output to be such that Cp Bp # 0, ie the relative degree of that "equivalent" system is one. b) MR.AS are often presented as an example of direct control.As noted in [17], it is not a necessary condition, and MRAS can also be designed through an indirect approach .In the litterature, MRAS are almost always considered through a direct approach, as it can be shown that for minimum-phase systems, a reparametrization of the process is always possible so that the estimated parameters turn out to be those of the controller .The most commonly used algorithms are gradient type, as the proportionnal + integral adaptation law [16].This is due to some nice properties of such algofithrr~ such as passivity, which is not met in least-squares algofithrr~ for instance. 23) Robustness issues Adaptive laws lead in the "ideal" case to boundedness of all the signals in the closed-loop system.Actual implementations of theses schemes make it necessary to take the "non-ideal" (ie perturbed) case into account.Several types of unmodeUed dynamics can be considered : i) internal and external bounded disturbances on the state or the output of the plant. ii)Plant parameters variations in those schemes where the process to be controlled is tinae-varying. iii)Plant dynamic uncertainties. A general way to design adaptive schemes with unmodeIled dynamics is to relax performances objectives [ 17][24], ie such algorithms should at least insure BIBO stability, which constitutes in fact the main theoritical problem.A fast approach is to take "small" disturbances only into account [t7][40,p.16] : ~is is based on the total stability concept and is related to the uniform asymptotic stability of the unperturbed system.Considering general unmodelled dynamics, two approaches are available: either modifying the adaptive law (e 1 modification[22],cr modification[21], bounds on

110"11124],dead-zone[25]), or increasing the "richness" of the reference input [26].Note also

that when no disturbance is present, those algorithrr~should at least lead to finite parameters and output errors, and verify exact or asymptotical model matching conditions if possible.Application

802

of the first type of modifications to the series-parallel case should not cause any difficulty.However, persistent excitation arguments on the reference model output are to be reconsidered, as Ym depends on yp and therefore cannot be assumed bounded (which is a necessary condition of excitation). 3) NONLINEAR AMFC 31)Generalities The problem of compensating a nonlinear system in order to make it match a linear reference model has been invastigated (see eg [29],[19, chp 7] for references, [27]).Techniques described therein may not always be very useful for the manipulator's case which represents, as noted in the introduction, a narrow class of nonlinear plants, for which global results can be derived. (Indeed, the general techniques applied to smooth affine in the input nonlinear plants of the form ~ = f(x) + g(x) u often lead to local results only).[30] provides a comparison work on several dynamic linearizing feedbacks studied for rigid robots, in the non-adaptive case.Note that all those linearizing tools present the major drawback that if there is any uncertainty on f and g, then the linearizing properties vanish. Adaptive control has recently been studied.J31] solves the nonlinear AMFC problem for a particular class of time-varying perturbed plants, employing alternatively a discontinuous VSS approach, and a continuous control law derived from a work presented in [16] for the linear case. A more general approach can be found in [19], for continuous SISO and MIMO square systems, linearizable and decouplable by static state feedback, which can be seen as an extension of the work done in [32] for robots controLIt represents an attractive pioneer work, but its major drawback is that estimated terms appear on the denominator of the control law, and must therefore absolutly be kept away from zero.This problem is far from obvious, and has been recently solved for the linear case in [28]. Due to the difficulties encountered, the nonlinear AMFC problem has been much less studied than its linear counterpart.However, it seems that similar problems arise for the design of suitable schemes, especially concerning assumptions i) and iii) in paragraph 22) (SISO plants).The main distinction lies in the definitions of those properties, particularly for the MIMO case, where several definitions of the zero-dynamics can be given [ 19, p.305].Fortunatly, for systems decouplable by static state feedback (as rigid robots) they are equivalent.

32) manipulators dynamics Let q, ~t, and ~ denote the robot's joints position, velocity and acceleration n-vectors.Rigid frictionless manipulators dynamics are then given by the set of nonlinear differential equations : M(q) ~t'+ C(q, ~) ~ = u- g(q)

(1)

803

where :

M(q) is the nxn inertia matrix, C(q, ~t) (} contains the centripets and Coriolis torques,

g(q) stands for gravity torques, u is the input vector torque. The main properties of equation (I) are : i) M(q) is a symmetric strictly positive definite matrix, bounded in q, composed of smooth functions ofq. ii) M(q) - 2 C(q, ~t) is a skewsymmetric matrix. iii) M(q), C(q, ~1) and g(q) can be expressed as Y0, where Y is composed of known functions of q and q, while 0 stands for unknown physical parameters. iv) (1) is static state feedback linearizable and decouplable by the so-called "computed-torque" control law :

u = C(q, ~t) q + g(q) + M(q) a

, a being any output signal.

v) q and q are both physically accessible for measure. i) and ii) are fundamental properties used to derive stability proofs of many schemes .i) insures among other things that global solutions of (1) exist [17, p.55], using the smoothness property of M(q).ii) shows that any system with input v and ouput M(q)~ + C(q, ~t)v, v a n-dimensional vector, represents a passive system [33]. iii) and iv) have been extensively used in the literature, at least for a certain class of schemes [8]-[15]. v) means that the state vector xT = (qT, ~1T) is entirely measurable.This is worthnoting, as minimum-phase properties of systems (linear or not) are obviously related to their output.In the finear case, transfert zeros'dynamics are given by forced (u ;e 0) state trajectories that nullify the output. In the nonlinear case, techniques have been developped to caracterize t h e so-called zero-dynamics [19, chp 7].They are also related to the definition of the output y = h(x).So if the state is known, the designer can choose among several possible definitions of h.In the case of robots, it has been shown that choosing y = q leads to a minimum-phase plant [ 19, p.322].This fact has certainly played a key role in the design of robots AMFC in joint space coordinates.Note that as long as the jacobian matrix J(q) remains nonsingular, one can easily translate joint space design into task space design [10]. The next paragraphs are devoted to a tentative classification of several schemes studied in the field of MRAS applied to robots.We will not consider in the following pioneer works (see eg [34]) that were based on "large" approximations of equation (1), due to their lack of global stability. 33) Parallel A M F C 331)Schemes derived from the linear case Chosing xT = (qT, ~tT) as a state vector, (1) can be rewritten :

~=

dl - M-l(q) [C(q, c].) ~ - g(q)]

+

u

M-l(q)

(2)

804 or :

= A(x) + B(x) u

(3)

It is then attractive to consider the manipulator as a 2n-dimensional plant, and try to make it match a linear decoupled model : im = Am Xm + Bm r

Ama 2n x 2n

Hurwitz matrix

Such is the purpose of papers [ 1]-[5].These control laws were inspired by the work done in [16,p.218].The main distinction lies in the adaptation laws, that include specific terms to compensate for the nonlinearity of (3). Several solutions are proposed, some of which are based on discontinuous (or "unit vector") control laws [1][2][5 part 1], or on continuous control laws [3][4][5 part 2].Properties i) and v) have been used.Note that property iii) has been used in [3] to rewrite the error equation derived from (3) (see [3] for details). As noted in [1], these algorithms require less computations than computed torque techniques. However, they also lead to less precise results. A major drawback of the discontinuous (sliding modes) approach is the so-called chattering phenomenon. No robustness analysis is available in the above references.It has been shown that including a dead- zone in the update law can reduce these negative effects [35].

332) Adaptive computed torque A second type of parallel schemes has been developped in [6]-[7].They appear as the first application of computed torque techniques in an adaptive context. Unfortunatly, on the contrary of schemes [8]-[ 15], no use is done of properties ii) and iii). Unknown time-varying terms are then updated using adaptive algorithms similar to those proposed in [16], and the closed-loop stability is insured by adding a large enough feedback PD term in the control law (see references for details). The reference model is then given as in the exact linearization case by a n th order decoupled double integrator that the plant + controller will asymptotically track. The main drawback of this approach is that the stability proof (as in [4]) hinges on the assumption that M(q) and several other time-varying terms in (1) remain constant during the adaptation process. 4) SERIES-PARALLEL AMFC Another way of ~onsidering equation (1) is to simply write it as : ~1"= M-l(q) [ C(q, q) q + g(q) ] + M'l(q) u

(4)

Chosing the "state" vector as x = q, (4) reads as : x = f(x) + g(x) u

f and g being this time n-dimensional vectors.

(5)

8O5 A whole class of adaptive schemes [8]-[15] can be interpreted as laws achieving asymptotical tracking between (5) and a series-parallel reference model given by : xr = - A x + r , where r = A Cla +/:ia

(6)

One key point in the stability analysis is that the error vector is given by : v = x - x r = H(s)- 1r q , with H(s) = ( sI + A) -1 and qtw = q - qd (see [36] for details) These algorithms can be considered as the most accomplished ones to date. Properties i), through v) are used. Various control laws have been proposed, from the pioneer works on the subject, derived almost simultaneously in [8] and [9], which can be seen as a continuation of schemes developped in [6]-[7], to more elaborate schemes [10]-[15]. The basic idea remains the same from one to another. Modifications essentially concerns update laws, with the so-called composite [11] or repetitive [14] controllers, or update laws using only desired trajectories (instead of measured signals) in the regressor matrix [12]. Exponentially stable schemes have also been studied [13], [15]. 5) S T A B I L I T Y ANALYSIS An interesting point is that all those schemes [1]-[15] can be interpreted as a feedback connection of a strictly-passive block with a passive one (figure 3). Some of them have explicitly been designed using that tool [I]-[3],[5],[6],[7]. The others [4], [8]-[15] have been studied employing a Lyapunov approach. It can be shown [37] that they can be turned into such a passive interpretation.

~ L~

str'~y-passive

passive

1

figure 3

All those convergence proofs have been lead in the ideal case.Robustness analysis similar to those already done in the linear case have been rarely published.Pioneer works on that subject can be found in [38] and [39].[38] is devoted to the robustness analysis of the scheme developped in [32].The main drawback of the approach is that the size of the instability domain depends on M-l(q), where M(q) is the estimate of the inertia matrix. Therefore, M(q) must be kept nonsingular.In [39], the authors have shown that the ~-modification can be applied to the laws

806 [8] and [9], considering almost all the possible unmodeUed dynamics i) to iii) (section 23)).At this stage several renmrks arise : i) Parasitic noises on the output (especially on the velocity q measures) have not been considered. ii) It would be interesting to see if the other techniques developped in the linear case can also be applied to the robots case. iii) Those robustness tools have been developped for parallel schemes. Results obtained in [39 ] confirm the remark done at the end of section 2). 6) C O N C L U S I O N In this paper, we have provided a tentative summary of the most significant adaptive control laws for manipulators studied in the last ten years. The most important fact is that all those controlled systems can be interpreted through a common framework (ie MRAS ), and analysed using passivity tools, either they were explicitly designed like that, or through a Lyapunov approach. references [ 1] Nicosia S.,Tomei P.:"MRAC for industrial robots";Automatica,vol.20,n~ 1984 [2] Balestrino A.,DeMaria G.,Sciavicco L.:"An AMFC for robotic manipulators";ASME JDSM&C,vol. 105; 1983 [3] Nicolo F.,Katende J.:"A robust MRAC for industrial robots";2nd IASTED Symposium on Robotics and Automation, Lugano; 1983 [4] Lim YK.,Eslami M.:"Adaptive controllers designs for robots manipulators systems yielding reduced cartesian errors";IEEE TAC, vol.AC-32,n"2; 1987 [5] Balestrino A.,DeMaria G.,Sciavicco L.:"Hyperstable AMFC of nonlinear plants";Systems & Control Letters,vol.l,n~ 1982 [6] Tomizuka M.,Horowitz R.,Landau ID.:"On the use of MRAC techniques for mechanical manipulators";2nd IASTED Symposium Modelling Identification Control and Robotics,Davos,March;1982 [7] Horowitz R.,Tomizuka M.:'An adaptive control scheme for mechanical manipulators;compensation of nonlinearity and decoupling contror';ASME JDSM&C,vol. 108,n~ [8] Sadegh N.,Horowitz R.:"Stability analysis of an adaptive controller for robotic manipulators";IEEE International Conference on Robotics and Automation, Raleigh; 1987 [9] Slotine JJE.,Li W.:"Adaptive manipulator controha case study";IEEE TAC,vol.33,n~ 1; 1988 [10] Kelly R.,Carelli R.:"Unified approach to adaptive control of robotic manipulators";27th IEEE Conference on Decision and Control,Austin,Texas; 1988 [11] Slotine JJE.,Li W.:"Composite adaptive control of robot manipulator";Automatica,vol.25,n~ [ 13] Sadegh N.,Horowitz R.:"An exponentially stable adaptive control law for robot manipulators";submitted for publication [ 12] Sadegh N.,Horowitz R.:"Stability and robustness analysis of a class of adaptive controllers for robotis manipulators";To be published in the International Journal of Robotics research,1987 [14] Sadegh N.,Horowitz R.,Kao WW.,Tomizuka M.:"A unified approach to design of adaptive and repetitive controllers for robotic manipulators";presented at the USA-Japan symposium on flexible automation, Minneapolis ; 1988 [15] Bayard DS.,Wcn JT.:"New class of control laws for robotic manipulators: part 2: adaptive cam"; International Journal of Control,vol.47,n~ 1988 [16] Landau ID.:"Adaptive control: the model reference approach";Dekker,New-York;1979 [17] Narendra SK.,Annaswamy MA.:"Stable adaptive systems";Prentice Ha11;1989 [18] Miller DE.,Davison EJ.:"On necessary assumptions in continuous time MRAC";28th IEEE Conference on Decision and Control,Tampa,Florida,December;I989 [ 19] Sastry S.,Bodson M.:"Adaptive control: stability, convergence and robustness";Prentice Hall; 1989

807 [20] Narendra KS.,Valavani LS.:"Stable adaptive controller design-Direct control";IEEE TAC,vol.AC-23,n~ [21] loannou PA.,Tsakalis K.:"A robust direct adaptive controUer";IEEE TAC,vol AC-3 l,pp.1133-1143;1986 [22] Narendra SK.,Annaswamy MA.:"A new adaptive law for robust adaptation without persistent excitation";IEEE TAC,voI.AC-32,n~ 1987 [23] Sastry S.:'MRAC :S tability,parameter convergence and robustness";IMA Journal of Mathematical Control & In formation,vol. 1,pp.27-66;1984 [24] Kreisselmeir G.,Narendra KS.:"Stable adaptive MRAC in the presence of bounded disturbances";IEEE TAC,voI.AC-27,n~ 1982 [25] Peterson BB.,Narendra KS.:"Bounded error adaptive control";IEEE TAC,voI.AC-27,n~ [26] Narendm SK.,Armaswamy MA.:"Robust adaptive control in the presence of bounded disturbances";IEEE TAC,vol. AC-31,n~ 1986 [27] Marino R.,Nicosia S.:"Linear model following control and feedback equivalence to linear controllable systems";International Journal of Control,vol.39,n~ 1984 [28] Lozano-Leal R.,Collado S.,Mondie S.:'Model reference robust adaptive control without a-priori knowledge of the high frequency gain~;IEEE TAC, to appear;1990 [29] Di Benedetto M.:"A condition for the solvability of the model matching problem";Conferenee on nonlinear systems analysis,Nantes, France; 1988 [30] Kreutz K.:"On manipulator control by exact linearization";IEEE TAC,vol.34,n~ [31] Balestrino A.,DeMaria G.,Zinober ASl.:"Nonlinear AMFC";Automatica,vol.20,n~ [32] Craig JL:"Adaptive control of mechanical manipulators";Addison Wesley Publishing Company;1988 [33] Landau ID.,Horowitz R.:"Synthesis of adaptive controllers for robots manipulators usinf a passive feedback systems approach";International Journal of adaptive Control and Signal Precessing,vol.3,n~ [34] Dubowsky S.,DesForges DT.:"The application of MRAC to robotic manipulators";ASME JDS M&C,vol. 10 l,September; 1979 [35] Slofine JJE.:"On modelling and adaptation in robot control";IEEE Conference on Robotics and Automation,San Fransisco,CA; 1986 [36] Ortega R.,Spong MW.:"Adaptive motion control of rigid robots: a tutorial";27th IEEE Conference on Decision and Control, Austin,Texas;1988 [37] Brogliato B.,Landau ID.,Lozano-Leal R.:"Adaptive motion control of robot manipulators:a unified approach based on passivity"; American Control Conference, San Diego,CA; 1990 [38] Kang H.,Dawson D.,Lewis FL.:"A robust adaptive controller for rigid robots";28th IEEE Conference on Decision and Control,Tampa,Florida;1989 [39] Reed JS.,Ioannou PA.:"Instability analysis and robust adaptive control of robotic manipulators"; 27th IEEE Conference on Decision and Control,Austin, Texas; 1988 [40] Anderson BDO.,Bitmead RR.,Johnson CR.,Kokotovic PV.,Kosut RL.,Mareels IMY.,Praly L.,Riedle BD.:"Stability of adaptive systems : passivity and averaging analysis";MIT Press; 1986

Adaptive Control of Feedback Equivalent Systems J.-B. Pomet*

A b s t r a c t We address the problem of stabilizing a nonlinear system depending on some unknown parameters in suck a way that M1 the systems obtained by varying these parameters are equivalent to one system, supposed to be state-feedback stabilizable. The description of the adaptation iaws make use of passivity. We consider both the "general" case, and the case where some "matching assumptions" hold.

and

L. P r a l y t

on the p a r a m e t e r p; g is the matrix field defined by (3). One particular p* in Ill will be called the "true value of the p a r a m e t e r p", and our problem is to stabilize the system gp,, p* being unknown. This will be done by means of a dynamic controller, i.e. of a system with a certain state, to be determined, input x, and output u : State: x U ~

1

~.~p.

X

Introduction

We consider a family of nonlinear affine-inthe-control systems, indexed by a parameter vector p: P =

(Pl

...

Pl) T

e

RI

.

(1)

State: to determine

u

Adaptive Controlier

Ix

Fig. 1 : Closed-loop system

T h e state of the overall closed-loop system The system Sp corresponding to a given is composed of x and the dynamic variables value of p is described by : (or the state) of the controller. By "stabilize Sv." we then mean that both x and Sp : ~ = f ( p , x ) + g(p,x) u (2) the dynamic variables of the adaptive conm troller must be bounded, and x must tend f(p, x) + x (3) k=l to a certain point 0 of M " , for all the solutions of the closed loop system (global where the state x lives in an n-dimensioned properties), or only for some (local propC ~ manifold M " and is completely meaerties). By "p* being unknown", we m e a n sured, that the adaptive controller must not deu = ( u , , . . . , u,~) (4) pend on, or use the value of, p*. is in tZ"~, and f and the gk's are k n o w n This general problem, as well as the smooth vector fields smoothly depending distinction between global and local results, is extensively discussed in [10], [8J, *Univ. of Toronto, Department of Mathematics, or [11], where a complete bibliography m a y Toronto, Ontario, M5S IA1 Canada tEcole des Mines, CAI Section d'Automatique, be found. T h e present paper is specifically 35 rue St Honor4, 77305 Fontainebleau, France devoted to the case when all the systems

809

@ are equivalent to one another by state feedabck and diffeomorphism. In section 2, we state precisely our assumptions about the systems Sp. In section 3, we describe some general adaptation schemes. Section 4 presents some adaptive controllers designed for the general case where no more assumption is satisfied. Section 5 is devoted to the case where some "matching assumption" is satisfied and presents some modified controllers. Finally, in section 6 we give some geometric conditions allowing to construct a diffeomorphism meeting this "matching assumption".

is such that

rankg(p, x)Z(p, x) = r = k g ( p , x) (7) 2. There is a system

= i(=) +

independant of p such that~ for any p,

u

=

(9)

=

c~(p,x) q- fl(p,x) w (10)

transforms (2) into =

2

(s)

f(~) + O(~)w

(11)

Assumptions

Linear Parametrisation (LP) assumption: The fields f and g in (2) depend linearly in the parameter p : I

f(p,x)

-= a~

+ ~--~piai(x)

(5)

i=l 1

k(p, x) =

+

b (x) , (6)

Local Stabilizability of the Transformed System (STS(~2) ) assumpt i o n : There ezist k n o w n fltnctions Vnom and U, of class (at least) C I and C 2 respectively, from a neighborhood ~ of 0 in M ~ to R "~ and to l:t respectively, such that: 1. U(~) is n o n n v a t i v e , and zero if and only if ~ is zero, and for any I ( > O,

i=1

{ ~ / Y(~) < I ; } where the ai's and the b~ 's are smooth vector fields 5ndependant of p). Numerous practical examples, as position control of a DC-motor [6] or control of biochemical processes [4] satisfy this linear dependance on the parameters. All the existing schemes for adaptive nonlinear control require linear parametrization. Feedback and Diffeomorphism Equiv a l e n c e ( F D E ) a s s u m p t i o n : There exists three smooth maps, o~ from R z x M ~ to I t ' , t3 f r o m R ' • M ~ to Mm• and ~o from R l x M ~ to M ~ such that

(12)

is a bounded subset of M ' L 2. For all ~ in M ~, we have:

U~=~(() : L~U(() , x) with A a Hurwitz matrix). Then, the adaptive controller (26).(60)-(28)-(29) for feed[a'(x) + ~__lukb~(x)t40) back linearisable systems is similar to those described in [3], though the point of view Notice that q is the state of the small sysis different there. We have the following tem necessary to realize (27) using only results for these controllers (see [3] or [8]): available signals.

If FDE and LP are satisfied, and S T S is satisfied locally, i.e. STS(~) is satisfied for T h e o r e m 1 ([11]) some ~, then any solution of the closed- If assumptions LP and FDE hold, and asloop system (figure I) such that 4 0 ) and sumption STS holds locally, there exists an Ko) are closed enough to zero and p(O) is open neighborhood of (p*,O,O) such that

813 any solution (~,x,~) with initial condition (i5(0), x(0), 7/(0)) in this neighborhood, exists on [0, oo), remains in a compact set and its (x,~)-component tends to zero. In addition, the point (p*,O,O) is a (non asymptotically) stable equilibrium point. T h e o r e m 2 ( [ i i ] ) I[ assumptions LP and FDE hold, and assumption S T S holds globally, and if in addition there exists a C o function d on II such that for all (p, x) in II x M , with Z defined in (39),

IIZ(p,=)ll ~ ( p , x ) < d(p) (1 + V(p, x) ~) ,

meeting, Ior any (p, q,b, ~), x)v2(p,q,[o,x)+-fi-j(p,x)[o = 0

(43) We will use here/~ to compute ul with (see (15)) : Ul = Unom(P, X) ,

(44)

and the estimate ~ of q* to compute u2 by :

u~ = v2(p,~,/~,~) .

(45)

(41) We then have =

,~(~) -]- A ( p , x, u . . . . (]), x)) (p* -- p)

then all the solutions are defined on [0, (x)), + A2(15, x, v2(i5, ~,p, x)) (p* - ~X46) remain in a compact set and their (x,q)with A2 the row vector defined by component goes to zero. This algorithm therefore either gives only local results or requires the bound (41) on the growth of the different vector fields. This was also the case for the first algorithm (h = some coordinates), but, as stressed in [11] or [10], if the systems are such that $ is not Lipschitz in the coordinates h, we cannot guarantee global stability in the previous algorithm whereas we can here.

~ ( p , x) [g(ql, x) - g(q~, x)] ~ = A~(p, ~, u~) (q~ - q~) (47) We will again use the a d a p t a t i o n schemes proposed in section 3 to obtain (#, ~). We propose three different choices :

1. First, as in section 4, we m a y chose for h some coordinates on M " , using the filter (27) with r some positive constant. AgMn, we only mention the results obtalned if the systems are fully feedback linearizable and the coordinates h are these 5 Adaptive control with in which ~ is linear. Thee adaptive controller is given by (17)-(44)-(45)-(26)-(28)matching assumptions (29). The same controllers are described in We call matching assumptions the fact that [3], but without the reparametrisation (and F D E holds with the following restriction on consequently with # in place of ~), and unthe dependance on p of ~ : der an assumption different from FDEM2, see remark 2 below9 We have the following A s s u m p t i o n F D E M 2 : Assumption F D E result : is satisfied, and ~o has the property that If FDEM2 and LP are satisfied, and S T S there exists a smooth map v2 : is satisfied globally (resp. STS(~2) is satisfied for some ~), then any solution of (p,q,~,x) ~ v2(p,q,p,x) E R ~ (42) the closed-loop system (resp. any such that

814

z(0) a~d 7(0) a~e closed enoueh to ~e~o and /5(0) is close enough to p*) is bounded and such that x(t) goes to zero. Notice that, unlike in the situation of the preceeding section, we need no additionnal assumption like global Lipschitzness to get global stability. This would not be the case if the vector field h was not linear in the coordinates h (see [8]). This is one of the reasons to prefer the following choice of h.

(p*,p*, O, O) is a (non assymptoticaIly) stable equilibrium point of this dynamical system.

This vields the following controller : A d a p t i v e C o n t r o l l e r .AC~(U) :

We then define the following modified algorithm which is defined only for U(~) smaller than [To : A d a p t i v e C o n t r o l l e r .AC~'(U, [70) : The same as .AC'I(U), but U is replaced by

Here, we get a global result as soon as assumption STS is global, without a restriction like (41) (but under assumption FDEM2, which is stronger than FDE). Yet, STS being globally satisfied is very restrictive. If it is satisfied only locally, AC'I(U ) gives a good beheviour for only some so2. Another choice for h is h = U, the filter lutions : f has to start close to 0, which being given by (27) with r equal to 1. The is natural, but (/5, q) also has to start close expression of Z is then to (p*, q*), which is more unconvenient. We will modify ACI(U ) to get a better local reZ(/5, q, b, x) = ( Z I Y ) , (48) sult. Suppose that STS(f~) is satisfied for a certain f/, and Uo is a positive number with, A2 being defined by (47), such that Y(/5, x, v2(/5,p, q, x)) (49) 0 ~ , ^, u ( ~ ) < Uo ~ ~ ~ a . (55) = -~ztp x)A2(/5, z,v2@,/5,~,z)).

p

=

U.om(/5,~) + ,~(/5,/5,0,~)(50)

=

-- (~ Z(/5, X, Unoitl(/5 , X))

(51)

- eY(/5,~,,:(/5,b,O,~))"

(52)

OU~ ).~( s~ ) + -~;(

(53)

q = =

-~

uq

(s6)

in (53) and in the definition of Z and Y. We have the following result :

where e = U - U(~),

UoU uo - u

(54)

and Z and Y are given by (39)-(40) and (49)-(47). and the following result :

T h e o r e m 4 If FDEM2, LP and STS(~) hold, and Uo is chosen according to (55), the solutions @(t), ~(t),~(t), =(t)) of the

closed.loop

system S,.-.aCi(U, Uo) such

T h e o r e m 3 If FDEM2, LP and S T S that U(~(0)) < Uo are defined on [0, oz), (resp. STS(f~) for a certain neighborhood remain in a compact 3et and their (x,7])ft of(O, O) in R zx M '~) hold, then all the so- component goes to zero, U(~) remaining lutions (/5(t), ~(t), ~(t), x(t)) of the closed. smaller than Uo. In addition, (p*,p*, O, O) lOOp system Sp.-~Ctl(U ) (resp. the solu- is a (non assymptotically) stable equilibtions with initial conditions in a certain rium point of this dynamical system. neighborhood o f ( / , / , O, 0)) are defined on [0, oe), remain in a compact set and their S k e t c h o f p r o o f : We only state the proof (x, U)-component goes to zero9 In addition, of theorem 3; the proof of theorem 4 goes

815 the same way, replacing U by (56) and considering that (56) is infinite when U is Uo. Consider a solution of the closed-loop system. Lemma 1 (30) implies that i5 and e = 9 - U are bounded on [0, T), the right maximal interval of definition of the solution, and that the time-ruction e(t) is in L2([0, T)). In addition, from (53), (13) and (54), we have :

This controller is the same as these described in [5], except that the double parametrization is not used there (replace by/3 and (62)-(63) by/~ = - Z r - y r . The approach to the synthesi presented here is however different : the authors design the adaptation, and the control u2, to make the positive function - Oil + 1-Iq* =

y(~) + 89

i1 < - c q

+ (l+e) e ,

(64)

(57) decrease

ong t h e s o l u t i o n s

(in fact, U(4)

is a quadratic function of { and the third term is absent there). The authors encounter the problem about implicit definition of i3 which we mention in remark 2. We can, as for AC'I, modify it AC~ into iA d a p t i v e C o n t r o l l e r AC,,i.rr : ~ , Uo). : t The same as .AC:(U), but U is replaced 3. A third choice for h in the algorithm of by section 3 is UoU which, together with e being L 2, implies that q is bounded too. This implies that the solution itself is bounded on [0, T) and therefore that T = +oo. Then (57) and (30) applied for T = +oo give e and 7, and therefore U, going to zero. []

i

-


0

the f e e d b a c k law (which

c o i n c i d e s w i t h the law d e r i v e d in [7] w h e n k-l) u I - w I w 2- k ( W l - W 3) u 2 - -2w I w 2 w 3 - k ( w 2 + w 3 2) locally asymptotically

Example

2,4

stabilizes

Zero d y n a m i c s

and d i s t u r b a n c e

(2.10).

also p l a y a f u n d a m e n t a l

rejection.

- s(w)

the s y s t e m

role in p r o b l e m s

C o n s i d e r an e x o g e n e o u s

of a s y m p t o t i c

tracking

system

w E I R 2, s(0) - 0

producing a reference

trajectory

(2.11a)

to be a s y m p t o t i c a l l y

tracked

YR - r(w)

(2.11b)

and a d i s t u r b a n c e

signal d(w)

x - f(x) + g ( x ) u + p(x)

to be a s y m p t o t i c a l l y

d(w)

(2.11c)

Two m a i n p r o b l e m s are to d e s i g n a f e e d b a c k scheme, only error m e a s u r e m e n t , achieving asymptotic e(t) - h(x(t))

w h i c h stabilizes

tracking;

Under mild assumptions,

x

-

~(w)

u

w h i c h are s o l u t i o n s f(~(w))

-

(2.11a)

u s i n g e i t h e r full state m e a s u r e m e n t

- (2.11c) w h i l e at the same

as t ~

(2.12)

n e c e s s a r y and s u f f i c i e n t

in [2]-[3]

conditions

in terms of the e x i s t e n c e

for the s o l u t i o n of these

of f u n c t i o n s

c(w)

to the p a r t i a l d i f f e r e n t i a l

+ g(=(w))c(w)

or

time

i.e.

- r(w(t))~ 0

problems were derived

rejected

+ p(=(w))d(w)

- a~s(w) @w

equation (2.13a)

825 There is a good geometric existence c(w).

Moreover,

(2.13b)

theory for ~(w) in (2.13a) given any choice of

under certain regularity hypotheses

is automatically

satisfied.

Indeed,

c(w) may be chosen a priori so that

viewing e, as defined

in (2.12),

as the

output of the augmented system (2.11co)-(2.11a)

one sees that if w o- 0 then, for any Xo,

to say y(t) - 0 is achieved with a control u(t)

is to say e(t) - O.

dynamics

That is, if the zero

of both the augmented system and the system to be controlled

the zero d y n a m i c s of the a u g m e n t e d s y s t e m c o n t a i n s

sets V*a and V ~ are smooth,

in case both zero constraint

system.

Moreover,

(2.13)

is e q u i v a l e n t

(2.11c)

to the e x i s t e n c e of a smooch,

exist,

then

the zero d y n a m i c s of the control s o l v a b i l i t y of

to V ~ in V a~ (see

invariant complement

[2]-[3]).

In the discussion of feedback stabilization or smoothness

and output regulation,

certain regularity

e.g. we asked that relative degree * such as (2.2) were satisfied or that V itself be a smooth submanifold.

conditions

surprisingly, and Marino

conditions were assumed to hold,

such conditions

are not necessary

as the following example,

[8], shows.

Example 2,5

[8]

Consider the system

-x 2 (-x+2y 3)

x -

(2.14a)

y - u

(2.14b)

The feedback law hand,

u-x-y 3 renders the origin locally asymptotically

if a smooth real-valued

function h(x,y)

is any feedback law maintaining constraint theorem,

Not

due to Boothby

set h(x,y)-0

stable.

On the other

is such that (2.2) is satisfied and u(x,y)

the constraint h(x,y) - 0

which may be computed

then the flow induced on the

from (2.14a) and the implicit function

is unstable.

Remark 2,6

In particular,

for the system (2.14) one cannot find an output function

for which both the regularity condition

(2.2) and the asymptotic

resulting zero dynamics hold.

one does not need relative degree hypotheses for w to a smooth V , but in this case we can show that

stability of the

Of course,

the zero dynamics

algorithm to converge

the zero dynamics

algorithm cannot converge while yielding asymptotically

stable dynamics.

826 Section 3

V~abi~tu

Ke~els

add Existence Theorems for Zero Dynamics

There are two regularity issues in the construction of the zero dynamics:

the

existence of a smooth V* and the existence of extra piecewise smooth feedback laws making V

invariant and inducing a system evolving on V .

The next calculation, due to Cannarsa

and Frankowska [5], shows that both may fail in an interesting class of problems for which, nonetheless, computing both V* and the associated feedback laws is extremely important.

We begin with some preliminary general remarks about V

in the context of a

general approach to geometric nonlinear control.

Fix a time-interval (0, T] with F C IR n

T S ~

and a constraint set

is controlled invariant for (2.1) provided for each

U C IR m.

XoE F

admissible piecewise continuous control u(t), such that the solution for tE [O, T] and is contained in F.

Given the closed subset

A subset

there exists an x t of (2.1) exists

K - h'l(0) for a continuous

function h, the maximal closed controlled invariant subset of K, provided it exists, is denoted by V (K).

Example 3.1

[5]

Consider the optimal control problem for (2.1) in Meyer form over a

finite time interval [O,T), with T fixed.

That is, for some constraint set U c IR m we

seek to minimize the end point functional

g(x(T)) over all solutions to (2.1) with

-

fo

and

u(t)

a piecewise continuous admissible control.

x(o)

It is well-known that the

value function

V(to,X o) - inf (g(x(t)): x(to) - x, to ~ T)

is nondecreasing along trajectories of (2.1) and is constant along optimal trajectories. Therefore, solving the optimal control problem is equivalent to computing V*(K)

control laws renderinE V*(K)

x -

f(x)

and the

invariant for the following system and choice of K-h'l(0):

+ g(x)u

w - 0

x(0)

-

fo

w(0)

- v(0,f)

r - i h(x,w,r)

- w-v(r,x)

In general, the existence of V*(K) and a description of those controls which render V (K) invariant can be obtained using methods from set-valued analysis. we express (2.1) as a differential inclusion

More explicitly,

827 x E F(x),

F(x) - (f(x)+g(x)u: u E U)

(3.1a)

and, without loss of generality, we set

K - h'l(0)

(3.1b)

By a solution of (3.1a) with initial condition x ~ we mean some absolutely continuous trajectory x(t), tz0 satisfying (3.1a) and x(0)-x o.

In the language of nonsmooth

analysis, a closed subset K has the property local viability if for any x ~ E K, there exists a T>o

such that some solution x(t) to (3.12) exists and belongs to K for tE[0,T].

If, moreover,

T-~, then K enjoys the global viability property.

controlled invariant subset (for infinite time).

Denoting by

In particular, K is a

TK(X )

the contingent cone

of K at x (see e.g. [9], [I0]), Haddad has shown that K has the local viability property if and only if K is a viability domain, in the sense that

Vxek

F(x)

Tk(X)~ ~

(3.2)

Furthermore, if F is bounded on K then, if K is a viability domain, K enjoys the global viability property.

In this language, then, if it exists,

V*(K) is the maximal closed viability domain

contained in K, referred to as the viability kernel of K (see [9], [i0]).

Moreover, it is

known that V*(K) exists (although perhaps empty) provided that F is a Peano map; i.e.

(i)

Graph (F) is closed;

(ii)

F(x)~@ and convex;

(iii)

F(x) - c (i + II x If)B, for some c>0 and some ball B.

From this "viability kernel" theorem, we conclude an existence result for zero dynamics. For (2.1) condition (ii) is always satisfied if U is convex.

Theorem 3,2

Suppose (2.1)-(3.1) satisfies the Peano conditions (i)-(iii).

Then

(2.1) has a zero dynamics evolving on the maximal closed, controlled invariant subset Z -V (K).

Moreover the zero dynamics is the system defined by the differential inclusion

zeZ (z), where

zeZ*

Z(z) - F(z)

T ,(z) '

Furthermore, if 0E U

and f(x) has a nontrivial

Z invariant set in K, e.g. an equilibrium,

Remark 3,3

then Z ~@.

The system (3.3) is a classical autonomous dynamical system only if Z(z)

is a singleton, for each zEZ ~

This situation occurred in the earliest works

feedback stabilization of invertible, nonlinear minimum phase systems.

[4] on

Of course this is

828 not always the case as a differential controls,

evolving on Z .

minimum phase

z(t)EW

W

of

0

if f(0)-0,

then we say that (2.1)

and any control u(t) containing

z~

suc~ that for

on the

(or(3.1))

is

t~0, z(t)EZ*,

there

such

(3.4a) as t ~ .

Remark 3,4 autonomous.

(3.4b)

Even when Z(z)

is single-valued,

tl.e zero dynamics

(3.3) may not be

This will be the case only if a selection for the set-valued map

exists.

are, of course, (see e.g.

For example,

constraint

Z

z_ 1 (see B y m e s and Isidori [3] for a more general definition of the nonlinear minimumphase concept).

854 Without entering into too many details, we can see that the fact of being minimum-phase or non minimum-phase for (2) defines a partition o f the space (u, y). Once again, taking discontinuous values for u and/or some of its derivatives, while y remains smooth, might lead our system from a non minimum-phase domain to a minimum-phase one.

4. A P P L I C A T I O N S In all the SISO applications which we present here: - The control law is obtained by inversion. -

The set point trajectory c(t) is a step function joining two constant operating points of the system 9

For the two academic examples, the reference trajectory yr(t) is defined via a linear fast order differential equation

~+'~=0 where "~is the time constant and e = Yr c. For the last heat exchanger example, the reference -

trajectory is defined via a linear third order differential equation

E + 3"~ + 3~2e + 3"~3"e" = 0. 4.1

Academic examples 9

,

S I N G U L A R I T Y : Take the input-output system y + "g'y = u + # ' u e u , where 'g' and z" are positive constants. There is an obvious singularity at u = 0. We want to rally the previously defined set point trajectory from a constant operating point where u is negative, to another one where u is positive. The usual continuous control leads to the singularity at u = 0, where a conventional computer integration stops (see Fig. 2a). If we impose a discontinuity on u to avoid this particular zero value of u, the integration works and the possibly difficult problem of analyzing the singularity disappears (see Fig. 2b).

855 UNSTABILITY: Now consider the following system: y + "~'~,= u + "c"u u, where "c' and z" are positive constants. The tangent linearized system around the constant operating point (u, Y) Ay + x'A~' = Au + z"uAu is non minimum-phase for u < 0. So, if we want to rally the same set point trajectory, the usual continuous control is divergent (see Fig. 3a). On the contrary, the introduction of a discontinuity, which yields u positive, brings back to stability (see Fig. 3b). Notice that we have also avoided the singularity at u = 0. 4.2

Heat exchanger

A brief physical description of this process is given here. For more details about this description, for the complete modelization and for nonlinear continuous control applications of this process, see [4, 5]. This system, used for energy recovery in the petrol industry, is composed o f two countercurrent liquid-liquid parallel heat exchangers (see Fig. 4). The objective is to control the output temperature y, by means of the flow repartition rate u. The other variables are here supposed to be constant. The input-output behaviour is very nonhnear in the operating domain: - The static characteristic (Fig. 5a) presents a hill top, i.e., the sign of the static gain changes.

- The dynamics (Fig. 5b) changes from a well damped behaviour to an oscillatory one, with minimum-phase or not. The non minimum-phase domain of the linearized model can be located on the static characteristic. Physical laws give a simplified state variable representation of dimension three:

Xl = a l X l + b l u + c l

x2 = a2x2 + b2u + c2 = a3y + uxl + (1 - u)x2 where the state is composed of the intermediate temperatures xl, x2 and the output temperature y. The coefficients al, a2, a3, bb b2, Cl, c2 are constants.

856

It is possible to eliminate state variables xl, x2 for such a bilinear system [8, 11, 15]. It yields the following input-output equation: [ ( a 2 - al)(U - 1)u + u ] [ - " y + a3"y+ ( a l b l - a2b2)u 2 + 3(bl - b2)ux + + (alcl - a2c2 + a2b2)u + (2(cl - c2) + b2)u + a2c2] + + [al2u + 2alu + u][(1 - u ) y - ((a2 + a3)(1 - u) - L1)~r + + a3(a2(1 - u) - u)y - (1 - u)(bl

-

b2)u 2 + (b2+ C l - c2)u + c2)] +

+ [aE2(U - 1) + 2a2u + l J ] [ u y - ((al+ a3)u + u)y + + a3(alu + h)y -- u((bl - b2)u 2 + (b2+ Cl - c2)u + c2)] = 0 Singularities arise when the coefficient of u vanishes:

)' + ((al - a2)u - a2 - a3)y + a3((al - a2)u + a2y) - ((bl - b2)u 2 + (b2 + cl - c2)u + c2) = 0 W i t h o u t m a k i n g a complete analysis o f this last equation, we notice that a particular singularity is obtained from its static solution (), = y = 0).This static singularity is located in the non minimum-phase domain (see Fig. 5a). W e want to rally the previously defined set point trajectory, from a constant operating point which lives outside the non m i n i m u m - p h a s e domain, through this domain. The usual continuous control stops near a singularity which is located in the non m i n i m u m - p h a s e domain (see Fig. 6a). The introduction o f a discontinuity on u leads the system outside the non minimum-phase domain, and therefore stabilizes it (see Fig. 6b). In the same time, the problem o f the singularities is avoided. 5. C O N C L U S I O N W e d e m o n s t r a t e d that the introduction o f control variables discontinuities p e r m i t s a straightforward solution o f classic control problems, which seem to be very difficult in the setting o f smooth controls. Note that our analysis does not apply if the Jacobian ~

3F

related to (1) does not contain u

nor any o f its derivatives. This is also the case when the inverse system has no dynamics, that is ~x = 0 in equation (1):

857 ~

F(u, y, y ..... y(B)) = 0. In order to bypass these important particular cases, which are often encountered in practice, we can possibly reverse the preceding analysis by now introducing other types of discontinuities. The problem will be dealt with in future publications. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

[12] [13]

[14] [15] [16]

S. Abu el Ata and M. Fliess: Nonlinear predictive control by inversion, Proc. IFAC Symposium on Nonlinear Control Systems Design, A. Isidori ed., Capri, Italy, June 1989. S. Abu el Ata, A. Co'fc and S. Chavy: Commande pr6dictive non lin6aire par inversion. Application hun probl~me de guidage d'avion. Actes Colloque SMAI "L'Automatique pour l'A6ronautique et l'Espace", Paris, Mars 1989. C.I. Byrnes and A. Isidori: Local stabilization of minimum-phase nonlinear systems, Systems Control Lett., 11, pp. 9-17, 1988. P. Chantre: Commande et optimisation de processus industriels non lin6aires. Mise en oeuvre sur un train d'6changeurs thermiques en parall61e, Th6se Doct. Ing., Universit6 Paris VI, Paris 1983. P. Chantre: Commande pr6dictive non lin6aire h l'aide d'un mod61e de Volterra, IASTED International Symposium, Grindelwald, Switzerland, February 1989. D.W. Clarke, C. Mohtadi and P.S. Tufts: Generalized Predictive Control - Parts I and II, Automatica, 23, pp. 137-148 and pp. 149-160, 1987. J.Z. Cypkin: Thforie des asservissements par plus-ou-moins (translated from Russian), Dunod, Paris, 1962. S. Diop: Elimination in control theory, Math. Control Signals Systems, to appear. M. Fliess: Automatique et corps diff6rentiels, Forum Math., 1_, pp. 227-238, 1989. M. Fliess and F. Messager: Vers une stabilisation non lin6aire discontinue, These Proceedings. M. Fliess and C. Reutenauer: Une application de l'alg6bre diffdrentielle aux syst6mes r6guliers (ou bilin6aires). In "Analysis and Optimization of Systems", A. Bensoussan and J.L. Lions eds., Lect. Notes Control Inform. Sci 44, pp. 99-107, Springer, Berlin, 1982. R.M. Hirschorn and J. Davis: Output tracking f0r nonlinear systems with singular points, SIAM J. Control Optimiz., 25, pp. 547-557, 1987. F. Lamnabhi-Lagarrigue, P.E. Crouch and I. Ighneiwa: Tracking through singularities. In "New Trends in Nonlinear Control Theory", J. Descusse, M. Fliess, A. Isidori and D. Leborgne eds., Lecture Notes Control Inform. Sci. 122, pp. 44-53, Springer, Berlin, 1989.' J. Richalet, S. Abu el Ata, C. Arber, H.B. Kuntze, A. Jacubash and W. Schill: Predictive Functional Control, Application to fas~ and accurate robots. Proc. 10th IFAC Congress, Munich, 1987. E.D. Sontag: Bilinear realizability is equivalent to existence of a singular affine differential i/o equation, System Control Lett., 11, pp. 181-187, 1988. V.I. Utkin: Sliding Modes and their Application in Variable Structure Systems (translated from Russian), Mir, Moscow, 1977.

858

Prt~s5

Pa~

out:put:

~h

J x2

2

ex_~.anger "

U Q, QI, Q2 T, TI, T2 y, xl, x2

~ ~ %

secondary fluid 2 Q2 T2

: f l o w r e p a r t i t i o n r a t e of the v a l v e (0 0 in a neighborhood of e = 0

S y s t e m s in which this situation occurs are said to have a singularly perturbed zero dynamics. The integer r is the r e l a t i v e degree of the perturbed systems, and the integer r + d is the relative degree of the unperturbed system. By definition, the change from r(e) = r to r(0) = r + d implies that: Lgh(x,e) . . . . . LgLfrqh(x~

LgLfr'Zh(x,e) = 0 # 0

LgLfrqh(x,O) . . . . . LgLfr+d'lh(x~

for all x near x ~ and all e

for all e > 0 LgLfr+d-2h(x,O) = 0

(2.2a) (2.2b)

for all x near x ~

# O.

(2.2c) (2.2d)

The f o l l o w i n g d e v e l o p m e n t s are based on an assumption about the e - f a m i l y of systems (2.1) which, as we shall see later on, allows the singularly perturbed zero d y n a m i c s to possess only two time scales. First of all we assume, in view of (2.2b) and the first on o f (2.2c), that: LgLfr'lh(x,e) = ema0(x,a) where a 0 ( x , e ) is a smooth function, nonzero at (x~

(2.3) and m is an integer. Without

loss o f generality, m can be set equal to d. Otherwise e can be replaced by e d/re. Then, our two-time-scale assumption is that, for all 0 < k < d,

863

LgLfr'l+kh(x,e) = ed'kak(X,e)

(2.4)

where each ak(x,e ) is a smooth function in a neighborhood of (x~ Note that for k = d in (2.4), the condition (2.2d) implies ad(X~ r 0. In particular, this assumption is e.g. satisfied when g(x,E) has the form: g(x,e) = go(x) + egl(x) and f(x,e), h(x,e) are independent of e (see [5]). 3.

The

analysis

of a

singularly

perturbed

zero

dynamics.

We proceed now to identify the zero dynamics of the peturbed system and to show how they are related to the zero dynamics of the unperturbed system. In doing so, we take the advantage of the fact that the zero dynamics of a system are not altered by addition of integrators to the input channel (see [8, page 389]). Because of this property, we can identify the zero dynamics of any system of the family (2.1) with those of the following family of e x t e n d e d systems: = f(x,e) + g(x,e)z 1 EZ 1 = Z 2

ez 2 = z 3

~Zd =

(3.1)

v

y = h(x,e). In order to identify the zero dynamics of (3.1), we set to zero the output y and its derivatives, until we arrive at a relation that can be solved for some v = v ( x , z , e ) . Then, we set v = v(x,z,e) on the system (3.1) and we consider the restriction of the systems thus obtained on the submanifold (of the state space) on which the output and its derivatives are vanishing. We first obtain: y = h(x,E) y(1) = Lfh(x,e) . . ,

y(r-1) = Lfr-lh(x,e) y(r) = Lrrh(x,e) + a0(x,e)eazl Because of the special structure of (3.1), the (r+l)-th time derivative of y has the form: y(r+l) = Lfr+lh(x,E) + al(X,E)Ed-lz1 + (~o(X,E)ed-lz2 +

864

+ (Lfao(X,s

+ (LgaO(X,s

2.

In a s i m i l a r way, we obtain: y(r+k) = Lfr+kh(x,E) + t~k(X,s

+ ... + Cto(X,e)Ed-kzk+1 +

+ ed'k+lpk(X,Z 1. . . . . Zk,s where Pk(X,Zl . . . . . Zk,e) is a p o l y n o m i a l in z 1. . . . . f u n c t i o n s of x and e. T h e last two iterations give:

z k, whose coefficients are smooth

y(r+d-1) = Lfr+d-lh(x,e) + ad.1(x,e)eZl + ... + ao(X,e)eZd + + e2pd.l(x,z 1. . . . .

Zd.l,s

y(r+d) = Lfr+dh(x,e) + CZd(X,e)Zl + ... + Oh(x,e)Zd+ C~0(x,e)v +

+ ePa(x,zt . . . . . Zd,e). R e c a l l that, by a s s u m p t i o n , o:0(x~

~ 0. Thus, in a n e i g h b o r h o o d of x = x ~ z 1. . . . .

z d = 0, e = 0, the e q u a t i o n y(r+d) = 0 can be solved for v, as a s m o o t h x,z 1. . . . . Zk,e. This function is a feedback law of the form: d v(x,z,e) = ~ ai(x,~)z i + b(x, e) + eQd(X,Z,e) i=l

f u n c t i o n of

(3.2)

where z = (z I . . . . . zk) and al(x~ = (cq(x~176 # 0. E q u a t i n g to zero y a n d its first r + d - i d e r i v a t i v e s p r o v i d e s a set o f r+d c o n s t r a i n t s d e l i n e a t i n g the s u b s e t of the state space on w h i c h the zero d y n a m i c s of (3.1) are d e f i n e d . These c o n s t r a i n t s are c o n v e n i e n t l y e x p r e s s e d in the form: 0 = H(x,e) + eR(x,z,e)

(3.3)

where: H(x,e) = col(h(x,e), Lfh(x,e) . . . . .

Lfr+d'lh(x,e))

(3.4)

a n d R(x,z,e) is an ( r + d ) - d i m e n s i o n a l v e c t o r of s m o o t h f u n c t i o n s satisfying R(x,0,e) = 0. O b s e r v e n o w that, at e = 0, the r i g h t - h a n d - s i d e of (3.3) reduces to H(x,0) and that, at x = x ~ the d i f f e r e n t i a l s of the r + d e n t r i e s o f H ( x , 0 ) are l i n e a r l y i n d e p e n d e n t b e c a u s e the s y s t e m (2.1) has r e l a t i v e degree e x a c t l y equal to r + d at

865 (x,a) = (x~ (see [8, page 148]). As a consequence, at (x,z,a) = (x~ matrix of (3.3):

the jacobian

[ ~-~(H(x,e) + eR(x,z,e)) e~ R(x,z,e)] has rank r+d. It follows that - for each e near 0 - the set of points (x,z) which s a t i s f y (3.3) n e a r (x~ is a s m o o t h ( r + d ) - d i m e n s i o n a l s u b m a n i f o l d . This s u b m a n i f o l d , noted Z*, is rendered i n v a r i a n t by the f e e d b a c k law (3.2) and the restriction o f the flow of the c l o s e d - l o o p system (3.1)-(3.2) to Z* describes the zero dynamics of (3.1), i.e. that of the family (2.1). In order to represent these zero d y n a m i c s e x p l i c i t l y , we note that since the differentials of the entries of (3.4) are linearly i n d e p e n d e n t at x ~ for each small e, the f u n c t i o n s : ~i(X,E) = L f i ' l h ( x , e )

1 _< i < r + d

can be used as a (partial) set of new local c o o r d i n a t e s for (2.1). The partial t r a n s f o r m a t i o n thus defined is smooth in x and e. If r+d is strictly less than n, then the c h o i c e of new coordinates m u s t be c o m p l e t e d by a set of additional smooth functions rli(x,e), 1 _< i ~ n-r-d. Since, for all (x,e) near (x~ we have: (span {g(x,e)})lq (span {dh(x,e) . . . . .

dLfr+d-lh(x,e) }).1. = {0}

we can always choose rl(x,e) (see [8, page 150]) in such a way that Lgrl(x,e) = 0 for all (x,e) near (x~ Without loss of generality, it is possible to choose rl(X,e ) so that rl(X~ = 0. In the new coordinates: = (~1 . . . . .

~r+d)

= (rll . . . . . "qn-r-d) the system (2.1) assumes the form:

= c(~,n,e) + d(~,n,e)u

=q(~,q,e). Recall that, by definition, the system: = q(0,n,0) r e p r e s e n t s the zero dynamics of the unperturbed system (2.1). In the new c o o r d i n a t e s , the e x t e n d e d s y s t e m (3.1) with f e e d b a c k (3.2) assumes the form:

control

866 = c(%, 11, e) + d(~, rl, e)Zl fi = q(~, 11, e) E7.1 = Z 2

(3.5)

ET.2 = Z 3

d I~Zd = E ~i(~,rl,E)zi + ~(~,TI,E) + E(~d(~'iq'Z'E)" i=l Note that, in the new coordinates (~, rl), the constraint (3.3) identifying the zero dynamics manifold becomes:

0 = ~ + el~(~,n,z,E). Solving this relation for

~

yields:

= eF(rl, z, e) where F0q, z, e) is a smooth function. We see, then, that in the (~,rl,z) the zero dynamics manifold Z* of the perturbed system (3.1) can be the graph of a smooth mapping; in particular, on this manifold the ~ are of order e. In order to render the zero d y n a m i c s of (3.1) explicit, is substitute (3.6) into (3.5) to obtain:

(3.6) coordinates, expressed as coordinates suffices

to

= q(eF(rl, z, e), rl, e) F~ 1 = Z 2 s

(3.7)

= Z3

d ezd = E ~i(eF(q, z, e),rl,e)z i + ~(FA::(rl, z, e),rl,e) + i=l + e(~d(eF(rl, z, e),rl, z, e) T h e s e equations describe the zero dynamics of the perturbed systems for each small nonzero e. They appear in a singularly perturbed form:

ri = f(n, z, e) e~. = g(B, z, e) with jacobian

matrix:

(3.8)

867

0

1

0

...

0

0

0

1

...

0

az(n,z,O) ) =

o.,

0z

0

0

0

(3.9)

...

1

...

~d(0,1],0)

which is n o n s i n g u l a r for all aq near 0, because

~1(0,0,0)

~ 1(0,1],0)

is nonzero

(recall that

A

~(x~ = 0, ~q(x~ = 0, and that ai({,rl,a) derives from ai(x,e) with x expressed as a function of (~,rl,e)). Hence, (3.8) is in the standard two-time-scale form [7]. F r o m the above analysis we see that, under the assumptions (2.2), (2.3) and (2.4) the zero dynamics of the family of perturbed systems (3.1) exhibit the twot i m e - s c a l e b e h a v i o r of a s i n g u l a r l y p e r t u r b e d s y s t e m . Its " r e d u c e d system", defind by setting e = 0 and eliminating the variable z, is exactly the system: fi

=

q(O,~,O)

(3.10)

which d e s c r i b e s the zero d y n a m i c s o f the u n p e r t u r b e d system. Its " b o u n d a r y l a y e r system", obtained by rescaling time as x = t/e and then setting e = 0, is that of a family of linear systems p a r a m e t r i z e d by a constant vector rl, n a m e l y : 0

1

0

...

0

0

0

1

...

0

0

0

0

...

1

dz

Z+

d~

~(O,n,o)

4.

... ~d(O,n,O)

0

(3.11)

(ore,o)

Remarks

Remark

1. There are cases p o l y n o m i a l o f (3.11) vanish, instance - when the system is d i s c u s s e d in [5]. In each of system are the roots of: xa _ ~(0,n,0)

in which all the c o e f f i c i e n t s of the c h a r a c t e r i s t i c e x c e p t for ~ l ( 0 , r l , 0 ) . This situation occurs for linear and g(x,e) = b0 + b l e or h(x,e) = co + cle, as these cases the e i g e n v a l u e s of the b o u n d a r y layer

= 0

and therefore the boundary layer system is unstable if d > 2.

868 R e m a r k 2. A more detailed inspection of the constraint (3.3) which identifies the

zero dynamics manifold Z* of the unperturbed R(x,z,e) can be rewritten as:

system shows

that the vector

R(x,z,e) = [ 0 .1 [S(x,z,e)] with: S(x,z,e) = A(x,e)z + eQ(x,z,r where: 9

a0(x,e)e

d-1

A(x,z) =

0 ao(X,e.)e,d'2

9 ..

9~

0

0

~

...

ao(X,e)

and Q(x,z,E) is a function which vanishes at z = 0 together with its first order derivatives with respect to z. Therefore, choosing ~' and ~" to denote:

~" = colG+x . . . . . ~r+d) the constraint (3.3), expressed in the (~,~,z) coordinates, has actually the form:

0=~' (4.1)

0 = ~" + ~(~',~",n,E)z + E2(~(~',~",n,z,~) A

where A(0,0,0,e) is nonsingular for small nonzero e. In the previous analysis, we have solved (4.1) for ~' and ~" in order to express Z* as a graph of a mapping. In A

view of the nonsingularity of A(0,0,0,e), one could have solved these equations for ~' and z, thus obtaining Z* in the form of a graph of a mapping:

~'=0 z = G(~",~,e) and express the zero dynamics obtained is not defined at e = 0.

accordingly.

However,

the transformation

thus

869 Remark 3. If the basic assumption (2.4) fails to hold, i.e. if any of the functions ai(x,e), 1 < i < d-1 has a singularity at e = 0, most of the previous analysis is still valid, and in particular the expression (3.7) of the zero dynamics of the perturbed system. However, the equations in question are not anymore in the standard twotime-scale form, and the singularly perturbed zero dynamics may exhibit more than two time scales.

Remark 4. If a system is modeled by an input-output differential relation of the form: F(y,y 0) .....y(n),u,uO) .....U (n-r)) = 0

then its zero dynamics, those induced by the constraint y(t) = 0, can be clearly identified as those of the autonomous system: F(O,O ..... O,u,u (1)..... u (nr)) = 0 This point of view suggests a simple interpretation of the calculations illustrated in the previous section and, in particular, of the e-scaling introduced in the extended system (3.1). To this end, note that the constraint y(t) = 0 necessarily induces, on the input u to the system (2.1), a constraint of the form: Lrrh(x,e) + a0(x,e)edu = 0 Differentiating this d times with respect to time, after simple manipulations one obtains a differential equation of the form: CCo(X,E)Edu(d) + ... + OCd_I(X,E)EU(1) + OCd(X,E)U+ eP(x,u,eu (1)..... 8d'lu(d'l),s + +Lfr+dh(x,e) = 0 where P(x,v 1. . . . . v d ,e) is a polynomial in v 1 ,Vd, with coefficients which are smooth functions of x,e. Replacing x by its expression as a function of the (~,~q) coordinates, adding the differential equation . . . .

= q(~, T1, e) and finally eliminating ~ (which can be expressed as a smooth function of rl,u,eu (D ..... ed'lu(d'D,e) yields an alternative description of the zero dynamics of the system (2.1). Clearly, the equations thus obtained can be given the form (3.8) derived in the previous section, after having set: z i = Ci'lu (i'D

1 ~ i ~ d.

870 5.

References

[1] W.A.Porter, Diagonalization and inverses for nonlinear system, Int. J. Control, 21 (1970). [2] A.J.Krener and A.Isidori, Nonlinear zero distributions, 19th IEEE Conf. Decision and Control, Albuquerque (1980).

[3] C.I.Byrnes and A.Isidori, A frequency domain philosophy for nonlinear systems with applications to stabilization and adaptive control, 23rd IEEE Conf. Decision and Control, Las Vegas (1984). [4] A.Isidori and C.H.Moog, On the nonlinear equivalent of the notion of transmission zeros, Modeling and Adaptive Control, Springer Verlag L.N. in Contr. and Info. Scie 105 (1988). [5] S.S.Sastry, J.Hauser and P.V.Kokotovic, Zero dynamics of regularly perturbed systems may be singularly perturbed, Syst. Contr. Lett., 13 (1989). [6] J. Hauser, S.S.Sastry and G.Meyer, Nonlinear controller design for flight control systems, IFAC Symposium on Nonlinear Control Systems Design, Capri (1989). [7] P.V.Kokotovic, H.K.Khalil, J.O'Reilly, Singular Perturbations in Control: Analysis and Design, Academic Press (1986).

[8] A.Isidori, Nonlinear Control Systems (2nd Ed.), Springer Verlag (1989).

glargissement des ohjectifs de robustesse des syst~mes de co,-n=nde

David Bensoussan D~partement de g~nie ~lectrique Ecole de technologie sup~rieure Universit~ du Quebec 4750. avenue Henri-Julien Montreal, Quebec H2T 2C8

Abstract

:

In this paper, we will review the theoretical background which led to the emergence of the Hoo theory. The advantages offered by the Hoo theory, particularly in the domain of robustness, are weighted against the Hoo theory, limitations which are the conservativeness of the results. It is proposed to use the Hoo theoretical framework, a) to reconsider the role of disturbances so as to use them beneficially through the feedforwarding of their estimates, b) to study variable structure systems, the variable structure being used as an extra degree of freedom. The combination of the variable structure with the feedforwarding of disturbance estimates improves performance objectives while aiming to achieve robustness objectives.

Introduction

Les deux classique bas4 descriptions par

Zames

traditions

connues

sur les

r4ponses en

de variable d'~tat. [i]

a

permis

de

de

la

th~orie

L'approche reformuler

gain en relation avec les non-lin~arit~s

pourraient Feintuch

illustrer

[6].

Feintuch [8]

un

niveau

sont

le contr61e

presenter le

sectorielles

plus

entree-sortie,

introduite

abstrait

th~or~me du petit

[2,3] puis ~ reformuler nombre d'alg~bre

d'op~rateurs

et que

[5], et de Saeks et

initi~es par Zames [7] et

resum~es dans

ont permis la d~finition plus compl~te du problAme de contr61e dans le

a ~t~

pour presenter d'importants

rer ces

servi ~

Les ~tudes de r~alisabilit4,

syst~mes entr~es-sorties

traduits dans

d~veloppements

de systemes

les travaux de Saeks [4], Tung et DeSantis

contexte d'une alg~bre d'op~rateurs

ont ~t4

contr61e

le probl~me du contr61e dans un contexte

d'alg~bre norm~e; cette approche a d'abord

probl~mes de contr61e ~

du

fr~quences et le contr61e moderne bas~ sur les

causaux ou

reprise par

r~sultats de stabilit~ et de sensibilit~

les espaces

de Hardy

dans le domaine du contr61e.

deux theories

et de reconsid~rer de

strictement causaux.

La th4orie des

Zames dans une publication c~l~bre [9] : ces r~sultats

[10], ouvrant la porte ~ des nouveaux Ii est bon ~ ce

stade-ci de reconsid4-

en mettant en valeur leurs points forts et leurs limitations nouvelles

questions

quant

aux

directions

d'une recherche

future dans le but d'~largir le domaine d'application de ces theories.

874 I. Commande moderne et commande H~

La th~orie

de contr61e

moderne bas4e sur les m4thodes d'analyse des variables

d'~tat a pris un essor extraordinaire

depuis les ann~es soixante.

Malgr~ les succ~s

ind~niables des theories de Wiener Hopf Kalman (WHI 0, ]/H(s) = 0(s') as Isl tends to i n f i n i t y . Therefore, in order to make sure that IT(jw) / H(j~)I (which is the gain of the transfer function from disturbance, measurement noise,

and command input

to

plant

input) remains bounded as ~ -->m,

one should choose W(s) = 0(s') as (s( --> ~ [7]. From these considerations, we deduce the following possible structure for W(s)

W(s)

:

(T.s+2)

'~w

(8)

where I/T, is the frequency from which the modeling uncertainty starts increasing significantly, and w is an integer ~ 0 which characterizes the rate of increase of the modeling uncertainty.

883 We can now define a robustness indicator with respect to m u l t i p l i c a t i v e plant uncertainties, IM, as the maximum static gain that can be tolerated For a bounding function F of the form K(T,s+I) (+" (where K is a constant), i.e. the maximum value of K such that all the plants of the class M(H,K(T,s+I) ~'') can be stabilized. Obviously, From (7) and theorem i , IM = I/~ T , where AT is computed with the weighting function (8). 4. SOLUTION OF THE MINIMAX PROBLEM

For the sake of generality, we shall consider a weighting function W(s) of the form

W(s) = n.(s) / dw(s) where nw(s) and dw(s) are real

(9) polynomials with degrees dnW and dd~ respectively.

Moreover, nw(s) and d~(s) are coprime, t h e i r roots are located in the open l e f t - h a l f plane, and dnW> dd~ + E where c is the pole excess of the plant. The computation of the minimax weighted complementary s e n s i t i v i t y Function is now studied. Plants with imaginary axis poles and/or zeros require a special treatment, and they w i l l be dealt with in a separate section. 4.1. Plant without pole or zero on the imaqinary axis (no(s)=do(s)=1) This is a particular case of the minimax mixed s e n s i t i v i t y problem presented in ~7]. The results are dual to those derived in section V of that paper. Theorem 2 ==========

Consider the factorization (4), define W(s)

(5) of the plant transfer function H(s) and l e t (9)

a) I f d+(s) = constant, AT • i n f (I W(j~) T(j~) If- = II ~ ( J ~ ) I f . = 0 GEZ(H)

b) I f d~(s) ~ constant, AT is the largest (in absolute value) real value of ~ such that the solution of the polynomial equation I - - - n~ n'+ q'T : d+ PT + dw n~ qT (I0) consists of two real polynomials, PT and qT, whose degrees are dn~ + dnW - I and dd§ -I respectively. Moreover the optimal value AT is reached with the proper c o n t r o l l e r d. d, q~ G~ . . . . . . . . . ~ - - - where qT and P~Tare the solution of (10) corresponding to ~T, n. PT and the resulting closed-loop system has a l l i t s poles in the open l e f t h a l f plane. Part (a) corresponds to the obvious fact that, i f the plant has no RHP pole, one can make IT(j~)I uniformly a r b i t r a r i l y close to zero by making the controller gain small enough. The solution AT, p~, ~T of (10) can be computed by solving a generalized eigenvalue problem, following the same p r i n c i p l e as [7]. 4.2. Plant with poles and/or zeros on the imaqinary axis For a plant with poles on the imaginary axis, the closed-loop system obtained with the c o n t r o l l e r GT computed using the theory of [7] is not asymptotically stable.

884 Moreover, i f the plant has zeros on the imaginary axis, these zeros should be introduced in dw(s) to assure s t a b i l i t y of the closed-loop system [7]. However, i f we perform this modification of dw(s), we shall loose the l i n k between W(s) and the uncertainty bound F(s). Thus another approach is needed for systems with poles and /or zeros on the imaginary axis. [g] gives such an alternative approach. The results presented here for the complementary sensitivity function are dual to those developed in [9] for the s e n s i t i v i t y function. Their proofs follow the same lines. Theorem 3 ==========

Consider the following factorization of H(s) H(s) = H2(s) Ho(s) (11) where Ho(s) has poles and zeros only on the imaginary axis, and H2(s) has poles and zeros only off i t . Let ~'denote the optimal weighted s e n s i t i v i t y function for the factor H2(s). Under the assumption that IW(J~)I ~ II Y" I I - at each imaginary axis pole of H,

(12)

i t is possible to build a sequence {Y'w}in RH~ such that lim k->|

IIY',II.

=

II Y" II.

(13)

and such that Y, satisfies the interpolation constraints needed to ensure s t a b i l i t y of the closed-loop system. Thus imaginary axis poles and zeros do not affect the optimal weighted complementary s e n s i t i v i t y subject to assumption (12). 5 . EXPLICIT FORMULA In this section, we derive an e x p l i c i t formula for the value of %Tcorresponding to a plant with a single RHP pole. Theorem 4 ========~=

We consider a proper plant H(s) with one RHP-pole, a, factorized according to (5). Moreover we assume that the weighting function W(s) is of the form (9). Then %T= i n f II T G(~(H)

Wil, =

IB,(a)" W(a)]

(14)

Using the weighting function (8), (14) can be written : ~T = I/IM =

]

(I + 7.a)"*

B,(a)" I

(15)

For a minimum phase system, B,(a) = I. In this particular case the study of the evolution of IM, the maximum admissible static gain uncertainty as a function of T,a yields the following conclusions.

If

the

gain of

the

modeling uncertainties

increases at

885 a rate of 40 dB/decade, or more at high frequencies (( + w Z 2), IM decreases at least quadratically when rwa increases. In this case, for a fixed value of a, a value of I/T, smaller than a yields an admissible static gain uncertainty smaller than 25 %. I f the modeling uncertainty only increases as j~ at high frequencies (E + w = 1), the previous number becomes 50 % (see table). Thus, the gain of the modeling uncertainties should not increase significantly in the frequency range [O,a] in order to be able to accommodate sufficient static gain uncertainties. This limitation is strengthened i f the plant has nonminimumphase zeros, because JB,(a)'l > I in this case. For a system with a single RHP pole and a single RHP zero, the situation where the RHP pole lies to the l e f t of the RHP zero in the s-plane is more favourable than the converse situation. This can be checked by applying (15) to two transfer functions with a single RHP pole and a single RHP zero, in which the location of the RHP pole and the RHP zero is interchanged. 6. NUMERICAL EXAMPLES i . H(s) : e (s-2) /

((s-5)

(s+6))

,

~ ( R

Fig. 2 gives the evolution of IM as a function of 7W. The type of results that can be derived from f i g . 2 is the following. I f the multiplicative plant uncertainties can be described by (I),

(2) with F(j~) = F,(j~) = 0.1 (0.1 s +I) there exists a linear

controller stabilizing the class of plant M(H,F,). However, i f the bounding function is F2(j~) = 0.3 (0.1 s+i), there is no linear controller stabilizing M(H,F2). Indeed, the largest static gain that can be accommodatedwithout loss of s t a b i l i t y for a bounding function with a zero at s = 10 is IM = 0.286. 2. Inverted pendulum We consider the inverted pendulummounted on a cart sketched in Fig.3. The dynamic equations of this system are derived in [11] and they are linearized around the state (Y,Y,8,8) :

(Y,O,O,O) f o r u : O. The f o l l o w i n g t r a n s f e r f u n c t i o n between the force u

a c t i n g on the c a r t and the p o s i t i o n o f the c a r t y w i t h respect to Y has been obtained

H(s) : K(s-b) (s+b) / (s2 (s-a) (s+a)) where K = the

I/M , b = / g / l

,

pendulum concentrated at

a: ]

(16)

(M+m)g / MI, w i t h M, mass o f the c a r t ,

the t i p ,

I,

length

of

the

pendulum,

m, mass o f

and g,

gravity

acceleration. For 1 = ] meter, M : m : 0.5 kg, g=10 ms2, we obtain a = vr-2-O-s' and b = ~ s " . In the sequel, those numerical values are always used in model (16). Since the system has poles a t the o r i g i n , we use the r e s u l t s o f section 4.2. where

H2(s) = K (s-b)(s+b)/((s-a) (s+a))

(17)

To be able to use theorem 3, we have to check condition (12).For the class of weighting functions (8), we obtain IW(o)J=1. Applying (15) to system (17) we obtain

886 ~T : II T II- ~ l(a+b)/(a-b)I which is larger than I. Hence a sequence such as defined in theorem 3 always exists in our case. In model (16), the transfer function of the device generating the force u is assumed to be unity.

Let us assume that this transfer function can actually be modelled by a

f i r s t order system whose time constant is approximately 10 times smaller than I/a. Thus a more accurate model for the linearized plant would be : HT : H(s) / (0.03 s + 1)

(18)

Since the nominal model H(s) has relative degree 2, and since the modelling uncertainty w i l l increase significantly for frequencies larger than 33.333 rad/s, we choose : W(s) : ( 0.03 s + I) 2 The corresponding value of IM, given by (15),

(19) is 0.133.

I t can be checked that

belongs to the class (I),(2) with F(j~) = 0.133 ( 0.03 j~ + I )2 .

HT

(20)

We can now analyse the achievable robustness with respect to parameter variations. Assume that the mass M varies in the range [0.4

0.6]. The transfer function which is

equivalent to (18), for M=O.4, is : H' T = 2.25 (s2 - 10) / [ s2 (s2 - 22.5) (0.03 s + I) ]

(21)

Fig. 4 gives the Bode plots of H(s), H'T(s) as well as the bounds corresponding to the class ( I ) , (2) for H(s) and F(s) given by (16) and (20) respectively.

Notice that the

factor I/s 2 has not been included in the Bode plots, in order to obtain a better i l l u s t r a t i o n of the differences between the curves. following conclusion.

From Fig. 4, we can draw the

There exists no controller based on the nominal model (16) and

on the unstructured uncertainty model ( I ) , ( 2 ) , with F(s) = K (0.03 s + I )2, that can stabilize the actual plant subject to a 20 % variation of the mass M. Besides, no satisfactory controller was obtained for system (16) in [11]. To obtain adequate performances for the control of the inverted pendulum, an additional measurement was performed and a cascade control structure was used. In this way, the transfer functions appearing in the modified control problem do not have any nonminimumphase zero [11]. However, no unstructured modeling uncertainties were introduced in this work. 7. CONCLUSIONS

We have presented an index which gives an upper bound on achievable robustness with respect to a class of multiplicative uncertainties, in LTI controller design, for systems with RHP poles. This index t e l l s the designer whether the required closed-loop system properties can possibly be reached so that tedious t r i a l and error procedures can be avoided. Its computation is particularly easy. I t amounts to an eigenvalue problem. Moreover, an e x p l i c i t formula exists for systems with a single RHP pole. A similar index has been derived to characterize robustness with respect to additive unstructured uncertainties in [4]. Similar indices can be defined for multivariable systems. However, their computation will

be more involved. I t does not amount to eigenvalue problems. An interesting

computational approach is given in {12].

887 Acknowledqement This research topic was suggested to the author by Professor GrahamC. Goodwin. Some helpful discussion with him and with Doctor Rick H. Middleton at an early stage of this project are gratefully acknowledged REFERENCES

[ I ] J. S. Freudenberg and D.P. Looze (1985). Right Half Plane Poles and Zeros and Design Tradeoffs in Feedback Systems. IEEE Trans. Automat. Control, AC-30,6, pp. 555-565. [2] K. Glover and D. McFarlane (1988). Robust Stabilization of Normalized Coprime Factors: an Explicit H. Solution. Proceedings of the 1988 American Control Conference, Atlanta, Georgia. [3] D. McFarlane and K. Glover (1988). An H. Design Procedure Using Robust Stabilization of Normalized Coprime Factors Proceedings of the 27th. IEEE Conference on Decision and Control , Austin, Texas, pp. 1343-1348. {4] M. Kinnaert (1989) Robustness LimitationsDue to Right Half Plane Poles and Zeros in LTI Controller Design. Internal Report. Laboratoire d'Automatique. Free University of Brussels. [5] J. C. Doyle and G. Stein (1981). Multivariable Feedback Design : Concepts for a Classical/Modern Synthesis. IEEE Trans. Automat. Control, AC-26, I, pp. 4-16. [6] M. Vidyasagar and H. Kimura (1986). Robust Controllers for Uncertain Linear Multivariable Systems. Automatica, 22, pp. 85-94. [7] H. Kwakernaak (1985). Minimax Frequency Domain Performance and Robustness Optimization of Linear Feedback Systems. IEEE Trans. Automat. Control, AC-30, 10, pp. 994-1004. [8] R.L. Kosut (1986). Adaptive Calibration : An Approach to Uncertainty Modeling and On-Line Robust Control Design. Proceedings of the 25th IEEE Conference on Decision and Control, Athens, pp. 455-461. [9] B.A. Francis and G. Zames (1984). On H" Optimal Sensitivity Theory for SISO Feedback Systems. IEEE Trans. Automat. Control, AC-29, i , pp. 9-16. [IO]H. Kwakernaak (1986). A Polynomial Approach to Minimax Frequency Domain Optimization of Multivariable Feedback Systems. Int. J. Control 44, pp.117-156. [11]R.H. Middleton and G.C. Goodwin. Digital Estimation and Control : A Unified Approach. Book to be released in 1990. [12]E.A. Jonckheere and J.C. Juang (1987). Fast Computation of Achievable Feedback Performance in Mixed Sensitivity H" Design, IEEE Trans. Automat. Control, AC-32, pp. 896-906.

888 TABLE T.a

IM

IM

E+w=l

E+w=2

0.5 0.4 0.33 0.286 0.25

I 1.5 2 2.6 3

0.25 0.16 0.11 0.081 0.063

Table 1 : Evolution of index IM as

IM (+w=3 0.125 0.064 0.037 0.023 0.016 function of r.a

FIGURES

Gain Nominal model t

Envelope

~ Locj ~,

IF(j~II

Log

Fig. 1 : Typical behavior of multiplicative

uncertainties

889 0.45

0.4 0.35 0.3 0.286 O,2fi 0.2 0.[5 0,1 0.05 ..................................

0 0.5

0 0 1

Fig. 2

:

l

1.5

I m h l mR

2.5

2

Plot of IM(rw) for H(s)=a(s-2)/((s-5)(s+6)) w=0 , --- w=l , ..~ w=2

,/?

m

Fig.3

: Inverted pendulum mounted on a cart

u (force)

5

///!

4.5 4 3.5 /; /,

3 // 2.5 _ =:~.v21

.....

2 1.5 1

0.5

-1

-0.5

0

{),5

1

1.5 log w

Fiq. 4 : Inverted

pendulum

__

IH(jm) l

....

IH'T(j~)I

....

IBoundaries

example

for the class of plants

)with H(s) and F'

(i),(2)

given by (16) and (20)

STABILIZING CONTROL OF A SINGULARLY P E R T U R B E D SYSTEM DRIVEN BY WIDE-BAND NOISES

Mohamed E1-Ansary Department of Mathematics California State University, Bakersfield 9001 Stockdale Highway, Bakersfield California 93311

ABSTRACT

A nonlinear singularly perturbed system is considered where the state equations as well as the output equations are corrupted by wide-band noises. The purpose of this paper is to spell out sufficient conditions under which an output feedback stabilizing control designed for a reduced-order model, via the use of stochastic observer, will stabilize the full-order system. Our results, which were established earlier on orderreduction and stability are applied to justify the use of the reduced-order models and the stability of the closed-loop systems which result from applying output feedback to the open-loop systems.

1. Introduction

Until recently, singular perturbation techniques [3] have primarily focused on state and output feedback design of linear systems. Advantages of these techniques, such as order reduction and seperation of time scales, are expected to have a more dramatic effect on feedback design of nonlinear systems. Stabilizing deterministic nonlinear singularly perturbed systems have been considered, for example, in [4-6]. Stabilizing nonlinear stochastic systems via the use of an asymptotically stable stochastic observer has been considered in [7], where the work has been a generalization of the Kalman filter structure [8]. In this paper, motivated by the work in [7], in which a stabilizing feedback control for a system represented by an It5 equation (a Markov model) has been designed using an observer, we designed an output feedback stabilizing control for a nonlinear full order singularly perturbed system driven by wide-band noise by designing a stabilizing control for the corresponding reduced-order open-loop Markov model. To assure the readability of this paper we will state in this section the basic ~ sumptions and the main theoren~ tha~ were developed in [11 and [2t. Section 2 consists of the main result of this paper, an output feedback control has been designed.

891 Reduced-Order Model and Convergence Theorem

In [1] and [2], we studied the nonlinear singularly perturbed system: = a l ( x ) + A z 2 ( x ) y + B l ( X ) V ' , x(O) = x o (1.1) /~) ----a21(x) § A2y § B2(x)v e, y(0) : Yo (1.2) where xeR", yeR m, v~eR r. v ~ is the wide-band noise and was defined as v~(t) = ~ v ( t / e ) . It is wide- band in the sense that thepower spectral density matrix S~(w) = S(w/e) will have a frequency band Wo/e when S(w), the spectral matrix of a stochastic process v, has a frequency band w0. # is a small positive parameter representing parasitic elements. The process v(t) satisfies: (A1) v(t) is a stationary, zero mean, right continuous, uniformly bounded on [0, co). The a-algebras induced by v(t) are assumed to have a mixing property with an exponential mixing rate [9], 81tpA,,t IP(A2/Az) - P(A2)I ~ g e - a t (1.3) for some a > 0, where Alea{v(a), a < t} and A2ecr{v(8), s > t + r}. The exponential mixing rate assumption was taken for convenience but can be replaced by a more general mixing rate as in [10-12]. The following conditions were also imposed: (A2) The coefficients az, a21, A12, B1 and B2 are continuous in x and have continuous partial derivatives up to the second order which are bounded uniformly in x. (A3) The constant matrix A2 is Hurwitz, i.e. ReA(A2) < O. (A4) The positive parameters e and/z satisfy e >/Z'~o where To > 0 is arbitrary but fixed. The reduced-order model can be represented by a diffusion process ~(t) and its operator L ~ , where ~(t) is the solution of the It8 equation: d~(t) = b(~c(t))dt § a(~(t))dw(t), ~.(0) = Xo (1.4) where b and a are given by (1.6) and (1.10) respectively, and the operator L ~, whose form was obtained from the proof of the convergence theorem, is given by:

L'~f(x) :

~bi(x)i:, (x)+ 2 i , i : , (1.5)

where

b(x) = ao(x) + hi(x) - A 1 2 ( x ) A 2 Z h 2 ( x ) + h3(x), A(x) = Bo(x)S(O)B~(x) = [aq(x)] 1, ao(x) = al(x) - A~2(x)A-~'a21(x), Bo(x) = B , ( x ) - A12(x)A-~lB2(x),

= S(to) is the spectral matrix of v,

1(') denotes transposition.

(1.6)

(1.7) (1.s) (1.o)

(1.1o)

892 ! hi, = tr[D~BoW' + D,Ax2A z- - 1 El, $ l ! h2i = tr[EiBoW + E i A 1 2 A 2 - 1 E ], h3i = tr[-F[BoW'Bt~(At2) -1 -- F[BoE'(At2) -1 + F[Ax2A;1P],

(i.Ii)

Di = [Vzr162162

(1.14)

Zi = [Vzr/ii!Vzr/i2i...iVzrlir]nxr; Fi = [V,-~iliVz~i2i...iVz~/m],,xm;

BI = [r

B2 = [rli.i]mxr, Ai2 = [~ii]r*xm,

(1.12) (1.13)

(1.15) (1.16)

oo

w = f R(~)a~,

(11z)

0

R is the correlation matrix of v, oo

r. -_ f

eA,'t,B2R'(r)dr, for some "7e["7o,oo), "70 > 0

(1.18)

0 oo

P = f eA2A(B2a' + aB~)ea'2~dA.

(1.19)

0

The following assumption was also required: (A5) b(x) and Bo(x) satisfy the growth and Lipschitz conditions:

[b(#l + IB00:)l __ K(1 + I~:l), Vx~n"

IbCx) - b(z)l + ] B o ( z ) - B o ( z ) l < K l z - z l , Vz, zeR" The probability space in which v(.) is defined and the notion of weak convergence were introduced in [1] and the following convergence theorem was proved: Theorem 1 Under the assumptions (A1)-(A5), the process x~m(.), defined by (1.1) and (1.2), converges weakly to the diffusion process ~(.), defined by (1.4)-(1.19), as e -+ 0, # --+ 0, and r --+ "7. The explicit form of the limiting diffusion model shows the interaction between two asymptotic phenomena through the dependence of the matix ~, defined by (1.18), on "7 = lira * (The implication of such dependence on the engineering practice of neglect9 ~/~--*0 ~ "

ing parasitic elements was discussed in details in [1]). The first phenomenon arises in singular pertubation analysis due to the existence of parasitic elements (/z --, 0) while the second phenomenon arises in asymptotic stochastic analysis of systems driven by wideband noise (r -+ 0). The Stability Result Most of the work that was done in the area of stochastic stability, was dealing with systems represented by R6 equations (Markov models). The stochastic Lyapunov method, that is analogous to the deterministic Lyapunov method, was effectively used [cf. 13, 14]. One of the first attempts to study stability properties of dynamical systems which are driven by wide-band noise, has been made in [12]. We utilized some of the techniques of [12] and extended them to the singularly perturbed system given by (1.1)

and (1.2).

893

In [2], we proved a Lyapunov like stability theorem about the slow variables which was based only on conditions uppon the approximating diffusion process. A Lyapunov function V(x), xeR ~ was considered with the following properties: (a) V(z) is real valued, positive definite, V(x) = 0 =:a x = O, V(=) -~ co as [x[--* co, and has continuous partial derivatives up to the third order. (b) For any vector or matrix valued function g(x,t) =- O(x) for te[O,T] and xeR '~, we have: ]V~(x)g(x,t)l < K V ( x ) , (1.20)

(v-c~)g(~,,)),~(~, t) l _< gv(~), ~OZIOZjOXk glgZ.g3 < KV(=~ 9

3

lc

--

x

(L21) (1.22)

~"

where i,j, k = 1, 2, , n and a.1. #2., and g,Z are components of vectors or matrices which are O(x). The constant K in (1.20)-(1.22) may not be the same and it is independent of T. Also the following assumption was made: (A6) The coefficients a~(z), a~2(z) and B~(z) vanish at x = 0 and for some M > 0 and 9 .,

,mr7

J

VxeR ~, [al(x) I + ]A12(=)] + ]BI(Z)] _~ M Ix[. Now we state the main theorem which is recalled from [2]. Theorem 2 Suppose that there exists a Lyapunov function V(x) defined on R ~ satisfying (a) and (b) and for some A > O,

L'W(x) ~h for "h > 0 arbitrary but fixed. Now we derive the diffusion operator corresponding to X with the aid of (1.5)-(1.19), where the assumptions that will be listed later, will validate this derivation. As in (1.6), the drift coefficient is b(X) = rio(X) + t~l(X) - ,~2(X)A;XI~2(X) + I;3(X) (2.17) The diffusion coeffiecient is A(X) = 5(X)g'(X) (2.18) where =

( ~ll (~12)

, \521 b~2 rio(X) = ri, (X) - i , ~ (X)A~Xah (X) = a0C~) + a0(~)~(~)

(2.19)

/x(~) + [F,(e) - KF~(~) - K~A~Xa~(~)]~(e) + K[~(~) - f~(~) - ~ : A ; ~ ( ~ ) ]

)

(2.20)

where

(2.21)

Go(x) = Gl(x) - A12(x)ATtGz(x),

go(X)=( B0()

0 ),

-KczA21B2(~,)

(2.22)

KB3(~.) (2.23)

5=

o)

(2.24)

where

(2.25) 0

From(2.15) and similar to (1.14) we have: ( f o r X =

o x,(:

oX,)o

(:))

fori----1,2,...,n

(2.26)

for i = n + 1 , . . . , 2 n

(2.27)

896 where /S,(x) = [V,,ail(x)i ... iVzaio(z)],xo ; KBs(x) = [a~j(x)],xo for i = n + 1,...,2n From (2.14), (2.16) and similar to (1.16) and (1.15) we have

~,(x) = P,(x) =

(?')

0~,•

,

i=

1,2,...,~

(2.29)

i = ~+ 1,...,2~

(2.30)

E,(X) = ( ( E , ) , • :) i = 1,2,...,m 0 Then similar to (1.11) and from (2.22)-(2.24), (2.26) and (2.27) we have: hxi = tr[Dibol4;" +/9~AI2A~'IE] = tr[D~BoW + D~At2ATXE] = hli for i = 1 , 2 , . . . , n From (2.27) we get: hil = 0 for i = n + 1,...,2n

(h,:,)

Hence

(2.28)

(2.31)

(2.32) (2.33) (2.34)

Similar to (1.12), we have: = h2i for i = 1,2,...,m where h2i is defined as in (1.12), and similar to (1.13) we have:

hs, = t r [ - ~ " D o ~ ' g ~ A ; = hzi

for

1 - F'[3o~'(A;1) ' +

(2.35)

~'L2A;'~I

i = 1,2,..., n

(2.36)

i=n+l,...,2n

(2.37)

and hsl = 0 for Hence ~sCX) =

(2.38)

where h3i is defined as in (1.13). Notice that, it can be verified that/5 = p. Now, since ,iCX) = &CX)SC0)B~(X) where "~(0)= ( S l i 0 )

(2.39)

$20)(0)

Then from (2.18), (2.19), (2.22), and (2.39), we get:

a(X) = ( B~ \-gc2A;1B2(x)~

0 ] KBs(x)~-C~J

(2.40)

Then the reduced-order closed-loop model corresponding to (2.10) and (2.11) takes the form:

d.~ = [b(2) + Go(.~)gC~:)ld,t + B o C x ) v / ~ e w x

(2.41)

897

d~ = [fl (x,) + (F, (k,) - KF~ ($z) - Kc2 - Kc2A~IG2(~z))g(~) + g { o ( ~ ) - f2(~) - c2A;la21(~) - c2A;lh2(2,)}]dt - K c 2 A ; 1 B 2 (~) S~/~)dwl + KB3(~-) V / ~ ( ~ d w 2

(2.42) Now we impose a consistency condition which is stated as follows: the reduced-order closed-loop model (2.41) and (2.42) has to be the same as the closed loop reduced-order model which is given by (2.7) and (2.8). Hence by comparing the coefficients in the two systems and insisting that they must be equal for all K and for all functions g(x), we have: fl i x) -= b(x) (2.43) F1 (x) = Go(x) (2.44) 0.1 (x) = Bo(x) ~ (2.45) f2 (x) = el i x) - c2A 2' a21 (x) - e2 A 2' h2 (x) (2.46) F2 (x) = - e 2 A ; 1G~ (x) (2.47) Hence, the open-loop reduced-order model (2.4) and (2.5) can be written in the form: d.~ = [b(~) + Go(~)u]dt + 0.1(~,)dwl i2.48) d2 -- leo (x) -t- Fo (x) u]dt + 0.2 (x,)dwl + a3 (2,)dw2 (2.49) where b(x) and Go(z) are defined by (1.6) and (2.21) respectively, and

C0(X) ---~C1(x)

-

-

c2A21a21C x)

-

-

c2A21h2Cx)

(2.50)

Fo(x) = -c2A~lG2(x)

(2.51) (2.52) (2.53)

0"1 (.T) = B 0 (x) ~

0.2 (x) = -c2A~ 1B2 (x) ~ o3 = B3( )

(2.54)

and the proposed observer (2.6) takes the form: d~ -- [b(}) + Go(~,)g(~,) + g{c0C~) - c0(}) + (F0(~) - Fo(k,))g(~)}]dt (2.55) +K0.2 (~,)dwl + K0.3 (~)dw2 The design problem, is to choose a function g(x) which is smooth enough and a constant matrix K such that the state x and the error e = ~ - $ will be stochastically asymptotically stable. Let us write the the It6 equations (2.48) and (2.49) in the form

,using (:), (2.56)

d)( = b()Odt + 5(2)dw where

b(X) = (

bi~') + G~

)

(2.57)

b(~,) + Go(~,)g(~) + K{eoi~) - coi~) + [F0(~) - F0(~:)]g(~)} and

0 )

(2.58)

K0.2 (5;) K0.3 Suppose we have found u = g(z) and the gain matrix K which stabilize (2.56), then the next step is to apply the same control law to the open-loop full-order system (2.1)(2.3) where u = g(~) and ~ is the reconstructed states and satisfies the equation of the following observer: ~ = b(~:) + Go(~:)g(~) + g[cl(x) - co(:~) - Fo(~)g(~)] + gc2y + gB3(~)v~ 2

(2.59)

898 Now we are ready to spell out all the conditions, under which the stability of (2.56) will imply that of (2.1), (2.2), and (2.59), when u = g(~) is applied to (2.1) and (2.2). This will be done with the aid of the stability result (Theorem 2). We require the following assumptions: (A) al,b, Bo,B3,A12, and cl to vanish at their respective origins. (B) b and 5 are required to satisfy: i

~(x) -~(Y)[ +]6(X)-5(Y)[ _ 0 g i v e n .

that

A1

in

CB. 219

stable.

CB. I S g , C B . QD-CB. II>

it

follows

that

fop

=EI m,

910

C B. 2~-D

--[o from

CB. IO9,

3

X4

where,

AN=

~

Define

and

Kl=-kI

=pT>o,

P a

CB. P-S)

P

a

m

, k>O e

A 4 = A4

C B. 243

+ K1

and

~n-mxn-m

such

that

a

pa XI + -T Pa =: -Qa O,

CB. ?.79

[o and

so,

APPENDIX

CB. 289

the

sufficiency

C

Proof

-

N e c e s s s ty: CA. 14)

and

CB. 3),

CB.:9

CA. 15).

PN CAN + TC~: N)

proved.

of T h e o r e m

Consider

and

is

the Then,

so (2.43

+ CAN

3.2

i i near from is

I

tr a n s f or m a t i on

the

proof

equivalent

+ TC~ZNDTP N

=

of to

theorem

T

in 3.1

C A. g) , we

have

]

911

=

[ ]

-

Now,

%

%

0_~

Q4

from

+

-Q

Sufficiency:

and

=

-~aP~ any


0) on W zd/M which

caricatures a circuit switched network with d y n a m i c r o u t i n g (the s t a t e m e n t s below are true for any d, but the situations d = 1, and d = 2 are likely to be of most interest). We use ~/ to denote a generic element of W zdlM, and call r/(z) the value at site x. Let M* denote ((2M+l)~-l) 2

The Markov process is described by the t r a n s i t i o n s

~(.)

, .(~)

-

1

at r a t e ~ ( ~ ) ,

T/(z) -----+r/(x)+ 1 at rate u if ~(x) # C ,

(~(z), ~(y), ,7(z))

, (~(x), q(y) + 1, ~(z) + 1) at rate . / M " if z , y , z are distinct sites with

~7(x) = C,~(y) < C,~(z) < C and y , z E x § [ - 1 , 1 ] ~ . There is no difficulty constructing such a Markov process even though the n u m b e r of sites is infinite. See Liggett, [9], C h a p t e r 1, Section 3, for details; T h e o r e m 3.9 of that section applies directly. We t h i n k of each site in the 'lattice as representing a link in our network, which consists of C circuits. We think of the value at a site as giving the n u m b e r of occupied circuits in the corresponding link. Occupied circuits become free at rate 1. At each link there is a Poisson process of calls with rate v. Each call occupies one circuit on its link if available; if the link is saturated the call r a n d o m l y picks two other links which are

942 in its [--1, 1]d n e i g h b o u r h o o d , and uses one circuit from each of these links if possible. Otherwise the call is blocked and rejected from the system. Note that because we have a compressed lattice, the i n t e r a c t i o n actually has range M on the scale of links. For x E Z a / M , let u M ( t , x , k ) denote P ( q M ( x ) = k), 0 - o o }

(ii)

963

A is also equal to the minimal mean weight of all the circuit c of G(A): )~ = rain w'(c)

o t'(~)

/

Going back to the initial event graph, this gives a proof of formula/il). The following states the connection between eigenvalues and asymptotic throughput: T h e o r e m 3 ( C o h e n e~ al.[1]) Le~ 2d be the least solution of the autonomous recurrenl

syslem (10), and ~ be ~he eigenvalue of *he irreducible matrix .4. Then, there exists E N* (~he period), and tt C N (*he length of~he transienl} such ~hal:

x(t + e) = ~ex(t) for t > t~ If X(t) represents the quantities produced, then )~ has the obvious meaning of rate of production, or periodic throughpul. In the conventional algebra: X(t + 8) = 0)~ + X(t) i.e. the vector quantity OA is produced during ~ units of time. 6.2 G e n e r a l case If the matrix A is not irreducible, we partition the graph of G(A) into k strongly connected components. By renumbering the nodes of G(A), we may assume that A has the block-triangular form A1,1 A2,1

9 A2, 2

. .. ~

An,1

A~,2

...

~" t

A= Ak,k

where A t , t , . . . , Ak,k are strongly connected. Let AA~,~ be the eigenvalue of the component Ai,~. Then, the spectrum ~r(A) (the set of the eigenvalues of A) is a subset of {AA~,~. . . . . AAk,~}. AA~,~is an eigenvalue ofA iff the upstream eigenvalues AA.... . . - , AAk_,.k_~ are greater than AA~,,. We shall only consider the smallest eigenvalue ~- = rain ~(A) = min{AA, . . . . . . AAk,k} which corresponds to the slowest part of the system. Then, ~ is also characterized by (11).

7

Symbolic

computation

of the maximal

eigenvalue

When A is a constant matrix, a well known algorithm (Karp [9]) allows computing A. In our case, A (more precisely A(q)) is a matrix function of the undeterminates q l , - . . , qk associated with the resources (cf. equation (7)). The classical analogous for (#-lA(q))* would be ( I - #-tA(q))-l. A symbolic computation of this rational matrix function of q and # would obviously be out of range. In the dioid case, we shall show that absorption properties make this computation simpler. We now recall the Gauss-Jordan algorithm for computing the * of a given matrix. 7.1

The Gauss-Jordan algorithm

Let A be a n • n matrix with entries ai,j in a complete dioid, and b a vector. ~" = A*b is the least solution of x = Ax @ b: xi = a l , l Z l (D a l , 2 X 2 . . . ~ a i , n x a ~ bl x2

=

a 2 , 1 x l $ a 2 , : x 2 . . . $ a 2 , n X n (~ b2

(12)

964

We can eliminate xl by substituting zl = a * l , l ( a l , : x 2 . . . 9 al,,~z~ (9 bl) in the n - 1 last equations. This means that the usual gaussian elimination algorithm allows the computation of A*b. There is a simple way of writing the algorithm by considering the matrix equation X = A X ~ A. This is equivalent to solving n systems of type (12) where b is replaced by a column of A. The least solution of X = A X (~ A is A * A . We have A * A = A A * , and A* = AA* (9 I d (this simply is the definition of A*). So, we can recover A* from A* A. It should now be clear that A* A can be computed by the following Gauss-Jordan algorithm (cf. Gondran & Minoux[6]): start with the matrix A (~ -- A define by induction A ('n+l) = ra (m+l)~/ by: al,(re+t) (,~) 71"mare, (ra)~ ~. l,t t = al,(m) t ~ tgd al~rn with 7rm = (a('~,))* Then, we get A (n) = A * A . It is essential to notice that A* is finite iff A * A is finite which is equivalent to all the ~'m being finite. 7.2

Rational closure of the dioid of polynomials

We need to perform the Gauss-Jordan algorithm with the initial matrix A (~ = l ~ - l A ( q ) , the entries of which are monomials in # - 1 , ql . . . . . qk. Our goal is to compute the set of constraints in problem (3). The aftlne constraint ,-~-5ql + . . + ,-~q~ +

>__~"

can be written as a monomial with rational exponents using the dioid algebra:

Therefore, we introduce: D e f i n i t i o n 4 A generalized polynomial f u n c t i o n o f q t , . . . , q ~ and t~- t is a finite sum: f ( q l . . . . . q k , # - l ) = ( ~ a a q ~ X . . . q ~ k # - - ~ k + l , with (~ = (cq . . . . ,~k+l) E (Q+)k+t and a,~ E R.

We denote by R < q t , - - . , qk, ~ - l > the dioid of polynomial functions with indeterminates and /~-1, endowed with the natural sum and product. We keep the usual

ql,...,qk

11

1

algebra for the exponents (e.g. 2q 3 | Oq~ = 2q~-). For instance, 0q~ is a monomial, which is the dioid version of "0 + ~-" a in the conventional algebra. We now study the * properties of polynomials. In a commutative dioid, we have

(o): (aeb)* =a'b* Then, we only have to consider the * of a monomial. We have already pointed out that in the scalar case, a* is 0 or - c o . This leads to defining the indicator function X by: X,(t)

f 0 L -co

if t < s if $ > s

965 Let P E R- < q l , . . . , qk, #-1 ~>, and Q, R E R < q t , . . . , qk >. The main algebraic features of indicator functions are:

(0: 9 ..q~ ) = X ~

~(~)

ql p ...qk p

(.i): Xq(t) r X~(t) = Xqe~(t) (i~): (,): [)/~(~)]* = X Q ( 0 (~): [P X,~(,~) X~(o)]* = P* X,~(,~) XR(o) We denote by ~" tile set of all the functions f of the form:

f(q,l ~) = P(q,#) Xq(q)(#) XR(q)(0)

(13)

for some P E R < ql . . . . , q k , # - i > and Q , R E R ' < q t , . . . , q k >. Then: 1 The dioid F is stable under the * operation.

Proposition

.T" is the rational closure of R < q l , . . . , q k , # -1 >, i.e., the least extension of R- < q t , . . - , q k , # -1 > which is stable under @,| and *. Thus, the entries of the matrix A ('~) defined in the Gauss-Jordan algorithm will have the form (13). As the entries of the matrix A (~ only have powers of #, we never use the (i) property. Moreover, formulas (o), (ii) and (iv) show that the* of a polynomial is an indicator function. Then, we have the stronger result: P r o p o s i t i o n 2 At the end of the Gauss-Jordan algorithm, we get: ~r,,(q,#) = for some P E R < q t , . . . , q k >

Xp(,r

T h e eigenvalue A is characterized by (11), thus it is the maximal solution of X p ( # ) > - c ~ , which gives A = P. Let us explicitly write the polynomial P: P ~- ( ~ , a ~ q ~ . 99qkak 9 Going back to the conventional algebra, we get: A = P(q) = min,~(a~ + atq~ ... + akq~) This is the symbolic expression for A(q) which we need for solving problem (2). 7.3

Complexity

estimation

This algorithm is useful when (i) the size n of the system (of the matrix A) is large, and (ii) there is only a few indeterminates (k o , which is called the faulty mode. The probability of switching from a standard to a faulty mode is known and given by 0.

For simplicity, this

paper will assume that o 1=0, corresponding to a perfectly reliable server and let the switching probability be constant, independent of the number of units produced. The switching process is therefore as that

969 shown in Figure 2. In such situations, unlike queueing models with breakdowns, the process is neither stopped, nor is it taking a vacation, but operates continuously, albeit producing, with some probability, a faulty output. In this sense, the manufacturing technology of the production process is represented by the parameters [o 1'02'0]" The smaller these parameters, the more effective (in a quality production sense) the manufacturing system. For quality conformance a CSP-1 Continuous Sampling method is applied (Dodge 1943, Duncan 1974). For such quality control, we can compute the various probabilities of producing defective units, detecting such units and of course the probability distributions of basic quality states of the queue-like production process. We summarise essential results. A production cycle is shown in Figure 3, during which N jobs are processed and the following is shown. Proposition 1: Let i be the number of defective units produced after the production system has switched to a defective mode and let k be the total number of units produced in the defective mode. Define by Pi(k) the probability that the ith defective unit is detected, then these probabilities are given by the following relationships, (1) Pi(k)= b(1-b)i'~NBD(i,k), k=i, i + l , i + 2 ..... i=1,2,3.. while the probability of producing i defectives in a production run is given by, oo

(2)

qi = ~ Pi (j) j=i

where NBD(i ,k) is given by the negative binomial distribution, (3) NBD(i,k)= ( k-l i-I(l'v ~ k-b '2i k = i , i + l , i + 2 , i + 3 . . . . . . Proof: (See Tapiero and Tsiotras 1990) The expected number of defectives production cycle is,

and

their

variance

in

a

oo

(4)

i = ~ iqt , i~i

i

i i-I

By the same token, the expected number of defective units which leaves A the production system undetected is y =E(i-1), while its variance is a 2y ' or ^

(5)

co

r

y = T. (i-1)qi; a 2= ~. (i-l-y)2qi i-I

Y i=l

If b/ is the total expected number of units produced in a cycle, then of course, the expected number of non-defective units produced in the cycle

970 ^

is

given by,

^

N-y.

In the values to these parameters.

following

proposition,

we

provide

explicit

Proposition 2: Let N be the number of units produced in a cycle. expected production in a cycle and its variance are given by, A (6) N = ( 1 - 0 ) ~ + 1/0 where Oo

=

P

~.

(8)

v a r ( N i)- 1 = k=i (1-0)~

X

kPi(k)=(l/ 02

b)o 2 A

+ P2 + 2nP]

+

(1-0)/02+(1-0)2nP 2

OO

GO

P2 =

the

O0

(7)

(9)

Then,

}:

~:

i--I

k~Pi(k)=(l+~176176176

k=i

Proof:

see (Tapiero and Tsiotras 1990)

Proposition 3: In the long run,^the probability of inspecting a unit is given by, (10) n = b + (1-b)nl/N while the probability of detecting

a

defective unit

and

shutting

down

the system is given by, A (11) J = I / N On

the

basis

of

these

probabilities,

the

production

system's

performance can be assessed. Consider first the amount of time required to perform a job. Let r be the amount of time required to produce a unit standartly, r I is the time required to inspect the unit and z2 the time to shut down the system and attend to repairs, then the production time of a job is given by the mixed distribution T below, r + z (12) The

T

w.p.

=

z + z I + z 2 w.p. r w.p. 1 - ~ 5 - ~ subsequent analysis will integrate

considerations

of

quality

and

quantity issues in the manufacturing process.

The Two Stations With Sampling And Limited Buffer

Here blocking of the first station occurs when upon completion of service in the first station attempts to move to station 2 when it is full are made.

In this case,

station 2 occurs. general and

mixture

repair.

blocking occurs until

a departure

from

In this system, the service time in station 2 is a distribution

Attendance

of

the

service

times,

can

be

time

computed

including without

inspection altering

the

971

basic structure of the queueing model.

Namely,

the mean service time

and its second~, moment are,, given by, (13) (14)

l//.t = z + (~r+~)z t + ~z"2 ET 2 = ( 1 - ~ - r 0 E r 2 + n E ( z + z l ) 2 + J E ( r

+rl+z2) 2

Since the manufacturing system has a limited buffer, under an assumption of heavy loading at the first station, the output rate of the first queue is /~ , (which equals the service rate of result, the second station behaves as if it were an Poisson server

arrival acts

as

rate

/t i

an

since

additional

during

space

the

that

this

station). As a MIGIIIR+2 queue with

blocking

belongs

to

period the

the

second

first queue.

Furthermore, due to the blocking period, the server in the first station does not process unit and remains therefore idle. That is, no more arrivals

occur

at

the

second

station.

Due

to

the

exponential

assumption, this is equivalent to stating that arrivals do occur at the second station at a rate of /~i but they are lost. In view of this observation, the second station becomes an MIGIIIR+2 queue. Such queues have been studied previously (e.g. see Makino 1964, Keilson 1966, Lavenberg 1975) and their operating characteristics are known. As result, using the adjusted service rate /t and E(T 2) we can obtain

a a

direct relationship between the buffer size R, the manufacturing process unreliabilities and the manufacturing system operating performance. These effects are two fold, first on the manufacturing capacity and second on the blocking probabilities for prior queues (causing thereby inventory accumulations and forced idleness). Let fl be the probability that the system is full, blocking thereby the first station. Then, for a buffer size R, (15)

fl = 1 - I / ~ + P

+2], with

p=l~lll.t,

R+I

PR+2= l / l ~

flj]

j=0

with (15)

1 ifk ~k

=0

all(I-at)

"~

if

{Efljak+l. j +

k--1

ak}l[l'a 1]

if k > 2

and (16) a k = f0{[1-FT(t)]e'/Zlt /Zk1 tk-l/(k_l)!} dt If we denote by F~(s) the Laplace Transform of Fr(t),

then using

the

well known transform relationship, (17) LT(tk" IFT(t)) = ('l)k" IF'T (k- 1) (S) where

F~(k'l)(S)

easily shown that,

is

the

(k-1)st

derivative

of

the

transform,

it

is

972

o~ = ~/(k-1)l][F(k)/,u~- (-1) k-I F TO ( k - 1 )(~ l ) / ~ l ] with F;(S) given by, (19) F ; ( s ) = F;(s){F;i(s)[~+SF;2(s)]+I-5-~} (18)

The expected time a job will spend in the second station is thus, R+I

(20)

W = (R+2)//,t - E JPj//t 1 j-I while the variance of the waiting time is, R+I

(21)

var(W) =(R+2)[ET2-1//~ 2] +

E

R'4-1

[j ( j + I ) P j ] / / ~

j=l

- [ T. jPj//~I] 2 j=1

( R + 2 ) (R+ 1)PR+2//~I/~ A Similarly, the expected number of jobs in the second station is L=/~IW, while the expected number in qeue, Em q is given by, L q = Emq= /Jl(1-fl) Wq - ft. Finally, the throughput of the system (at station 2) is d R, and given by, ~R = ]gl/[jO+PR+2] , p=/~t//~ and thcrcforc the loss of throughput ,4 due to limited buffcr in the station is given by, d = 5R(X+~).

These terms, combined with the production system's quality performance characteristics, can bc used to assess the overall performance of the production system. Throughout our applications, define the following cost parameters, C a = the cost per unit of blocking the first station, measuring the forced idle time imposed on the first queue. C = the inventory cost in the second station, per unit per unit time and expressing the cost of WIP in the second station Ct = the inspection cost per unit C d = the post process cost of a dcfcctuous unit, expressing the penalties associated to jobs processed in a faulty manner and that have left, undetected, the system. C = the cost due to the loss of output because of the inspection T

and defective units. C = the cost of adjusting the process, once a defective unit is F detected. These include f'Lxed stoppage, re-start costs as well as the repair or maintenance costs associated to the manufacturing system. Since our process is a renewal-reward process, the long run average cost is equal to the average cycle cost (Cinlar 1975) and therefore this cost, denote by 4 , is given by, (22) 9 = [ C

+Ci[bN+(1-b)nl]+

CdQ]/[~q/~]+

CILq+ CTZ1 + CBfl

973

where N//a is the expected cycle time, C fl is the cost of blocking per unit time of the second station, or the probability of the system being full. A A [bN+(1-b)nl] is the expected number of units inspected in a cycle which was computed earlier, CILq is the inventory cost per unit time since, L q is the average number of units in the queue, CTA is the average loss of output due to inspection and repairs/maintenance, stoppage and restart. Finally, CaQ is the expected cost of non-detected defective units which have exited the system. Since in a cycle, we produce qi defectives on the average, one of which is detected, the expected number of undetected faulty units in the cycle is given by, O0

(23)Q = ~ (i-1)qs with

qi

defined earlier

Numerical analyses were used and the following conclusion were reached. 1. When the buffer space is reduced, the production variability increases. Since such variability has negative effects on the smoothness of the production flow, it is possible to compensate such variability through an increase in automation (which reduces service time variability). We might conjecture therefore that JIT provides an incentive for automation. When we consider the quantity produced in a cycle, we note that while with increases in inspection the expected production cycle becomes smaller, its coefficient of variation increases. 2. When the buffer space R is reduced, the cost of process unreliability decreases. Further we note that A~/AO is largest at R = I , which means that it is most advantageous to invest in process quality when the buffer space equals one. When buffer space increases, the relative advantage in improving quality is reduced. In fact, when b=0.05, we see from that A ~ / A 8 is largest when R=0, at which time investment in process reliability improvement reduces costs the most. Close to full inspection (b=.8), an investment in process reliability improvement can at best be used to reduce inventory costs (explaining thereby the growth with respect to R). 3. In J r r systems, flexibility reduces the blocking probability. Thus the value of flexibility at a given station is given by the cost reduction effects such reduction in blocking has. Of course, the

974

probability of blocking can be reduced by increasing inventory, but this incurs at the same time several other costs. 4. The marginal cost of inspection

are

greater,

the

greater

R.

5. The probability of blocking decreases when the buffer space increases, but it declines faster when the buffer space is smaller. At the same time, while the loss of throughput is smaller when R increases, this loss occurs also at a declining rate. This ~improvement ~ in operating performance involves a cost reflected in the increased average buffer space (inventory cost). These costs increase when we increase inspection. Thus, if the process reliability is not increased, quality can be improved only with inspection which induce detrimental operation of the manufacturing process. For in- versus out of station repair we saw that the greater the cost of production due to process unreliability,

the

more

beneficial

it

will

be

to

inspect

in-station

rather than out. These conclusions, having a strategic implication to the design and the management of unreliable Just in Time Manufacturing Systems are in part

accepted

as

"common

knowledge".

This

paper

has

provided

a

justification based on the modeling of JIT as tandem queue like manufacturing systems, reinforcing both current experience and quantifying it to value JIT, quality control and the process of quality improvement REFERENCES Cinlar, E., Introduction to Stochastic Processes, Prentice Hail, Englewood Cliffs, N.J., 1975. Couway R, W.Maxwell, J.O. McClain and L.J. Thomas, The role of Work in Process Inventory in Serial Production Lines, Operations Research, 36, 1988, 229-241 Dodge H.F., A Sampling Inspection Plan for Continuous Production,

Annals of Mathematical Statistics, vol.14, 1943, 264-279 Duncan A.C., Irwin, Ill. 1974

Quality

Control and Industrial Statistics,

4th.ed.,

Hall R.W., Zero Inventories, Homewood, Ill. Dow Jones-Irwin, 1983 Hsu L.F. and C.S. Tapiero, A Bayes Approach to Quality Control of an M/GI1 Queue, Naval Research Logistics Quarterly, 35, 1988, 327-343 Hsu. L.F. and C. S. Tapiero, Quality Control of the M/G/1 Queue,

Euro. J. of Operations Research, Forth., 1989

975 Hsu L.F. and C. S. Tapiero, Quality Control of an Unreliable Flexible Manufacturing System: with Scrapping and Inf'mite Buffer Capacity, Int. J. of Flexible Manufacturing Systems, Forth. 1989 Keilson J., The er.godic queue length distribution for queueing systems with finite capacity, J. Royal Stat. Soc., Series B, 28, 1966, 190-201 Lavenberg S.S., The steady state queueing time distribution for the M/G/1 finite capacity queue, Management Science, 21, 1975, 501-506. Makino T., On the Mean Passage Time Concerning some Queueing Problems of the Tandem Type, Journal of the Oper. Res. Soc. Japan, 7, 1964, 17-47 McClain J.O. and L.J. Thomas, Operations Edition, Englewood Cliffs, N.J. Prentice Hall, 1985

Management,

Second

Monden Yashiro, (Ed.), Applying Just in Time, The American~Japanese Experience, Industrial Engineering and Management Press, Atlanta, 1986 Schonberger R.J., Japanese Manufacturing Techniques: Lessons in Simplicity, New York, Free Press, 1982 Tapiero C.S., Production Transactions, 19, 1987, 362-370

Learning

and

Quality

Nine Hidden Control,

liE

Tapiero C.S. and L.F. Hsu, Quality Control of an Unreliable Random FMS: with Bernoulli and CSP Sampling, Int. J. of Prod. Res., 26, 1988, 1125-1135 Tapiero C.S. and G. Tsiotras, JIT, and the Quality Improvement Process in Queue Like Production Systems, Working Paper, 1990

976 Figure 1: The Production Model

Limited Butler [][]

Saturated o ....

Re.~=esta.o.

R

M/G/1

~

~R~7;k

Unrell,lble Station

Figure 2: Production System Switching %Defectlvell Produced Failure Mode Probability of mwitch

:StandardMode # Produced

Figure 3: The Production Cycle Start I

b Probabllletlr Failure n

~Sample ~

D

~

Repair ProductionCycle

Re,tart

MODULOIDS AND PSEUDOMODULES 3. T h e L a t t i c e S t r u c t u r e P r o b l e m E. Wagneur* Abstract We determine here sufficient conditions for finite dimensioned modulo~'ds and pseudomodules to be lattices. In particular, we show that completeness of the dioM D of scalars is such a condition. The simplicity conditions for pseudomodules, which make the classification problem tractable, are also shown to be sufficient for the lattice structure. Since these conditions are clearly unrelated, both results show that neither one is necessary. A concluding example illustrates this remark.

1.

Introduction

In traditional dynamical systems discribed by differential or difference equations, the underlying time set is independent of the evolution of the system, and sudden changes in the state are associated to singularities or bifurcations. In contrast, the dynamics of deterministic discrete event systems (DEDS) is characterized by the occurrence of sudden changes in the state, appearing at instants which are determined by the dynamics of the system. For example the message flow in communication networks, or the material flow in automated production lines may be modelled as DEDS. The mathematical modelling of DEDS aims at : 1. Providing the analogue of the powerfull classical models for the analysis of the dynamics involved. 2. Giving a tool for the analysis of the relation design/performance of the underlying physical system. 3. Developing an instrument for observation and control of the system described. Although various models have been proposed in relation with one or more of these goals (e.g. 1, 3 in [7] and [11]), we will concentrate here on the model introduced in [3],[4], which has proven to be relatively efficient in connexion with objectives 1 and 2. In this approach, a DEDS is modelled as an event graph (a particular class of Petri nets), and its evolution discribed by a set of equations of the type : ;gi(~ ~- 1) = Max{ Max {aij + xj(k)}, bi + u(k)}, l<j_ ~o(x) _< ~o(y)). Examples 2.3 E l . (IN U { - o o } , max, + ) , is a dioid. E2. (IR U { - o o } , max, + ) , is a pseudoring. E3. D ~ is a moduloi'd (resp. pseudomodule), whenever D is dioid (resp. pseudoring). Since D is completely ordered, D " is a distributive lattice. E4. For D an arbitrary dioi'd or pseudoring, let xl = (1, 0, 0), x2 = (1, 1, 0), xa = (0, 1, 1) E D a 9 3

The pseudomodule M = {y~ Aixi I Ai E {0,1}} = {0, z l , x2, z3, xl + za} generated by the

xi's

i=l

over the pseudoring {0, 1} is a nonmodular lattice.

Eb. For D arbitrary, let xl = (,~, 1), x2 = (1, # ) , with 1 < # < A. The set M = {~lxl+2~2x2 I )~1, A2 E D} is a proper submoduloYd (subpseudomodule) of D 2 .

980 Although Xl and x2 are not comparable, we have xl < ,~x2, as well as z2 < ~xl. Note also that the semilattice X + = {xl, x2, x l + x 2 } is isomorphic to the semilattice ((1, 0), (0, 1), (1, 1)}. E6. Let O = ( q u { - o o } , m ~ , + ) c I3 = ( r t u { - o o } , m ~ , + ) , a n d ~ = 0 , 1 , 0 ) , ~ = (0,1,1), ~ = (0,-v~,0) E D ~ 9 Then X = {~1,~,~} generates a Z ) - p , e u d o m o a u l e M , which is not a lattice, since the set (x E M [ z _< z i , x _< x2} has no 1.u.b. in M. Again, as a poser, X is isomorphic to Y = {(0,1,0), (1,1,0), (0,1,1)} C D 3 , and Z + and Y + are isomorphic semilattices.

Let xi E M , i E I , where I is an arbitrary index set. Definition 2.4 We say that X = (zi)iE I generates M iffVx E M may be written as a finite linear combination 9, = ~ ~ i X i , with xi E X, ,ki E D, Ai = 0 except for a finite number of values i E I. iE1 Let Xz stand for the span of X , and consider the following properties, where I1,12 C I. P1.

VI1,I2, XI, f] XI2 = XltnI2.

P2. VI1,/2, /'1 N/2 = O ~

XI1 N XI~ = O.

P3. V{ E I, VIx, { f! 11 => X{i} f] Xx, = 0. P4. Vi E I, VI1, ,i ~ Ix :=> xi ~ X q . Clearly these properties are listed here from the strongest to the weakest. They are equivalent in a vector space, but not in a moduloi'd as simple examples show. W h e n M is a pseudomodule, then P4 r P3. In.[12], we show that P4 is equivalent to the condition that X is the set of irreducible (x = y + z => x = y or x = z) elements of the semilattice M . Also the existence theorems for bases of [12] imply that if M is finitely generated then it has a unique system of generators satisfying P4. T h u s weak independence is defined by P4, and independence by P3. A weakly independent system of generators is then called a weak basis and its cardinality defines the weak dimension of M . In case M is a pseudomodule, a system of independent generators is called a basis. It is unique up to a rescating map of the form zi ~ ;bz~, (LJ e I). Examples 2.5 E l . Let xl,a:2 as in E5 above, then X = {2.1,22} is independent and generates a twodimensional moduloYd, which is not isomorphic to D 2 , since morphisms are isotone. E2. For z l , z2 as above, let Yl = A - l a x , Y2 = x2. T h e n Y = {Yl,y2} is independent. Now Yx < Y2, hence, even as a poser, Y is not isomorphic to X . However, as a submoduloid of D 2 (i.e. with the relation Y2 < A#yl), the moduloid generated by Y coincides with that generated by X . E1 above suggests that the order structure on an independent system of generators is not sutficient for the characterization of the moduloid it generates. Nor is it necessary, from E2. It follows that the classification problem in the general case is a m a t t e r of non trivial combinatorial complexity.

981 Given z , y E M let L ( x , y ) = {k E D I x _< A y } . Also, any basis X generates a semilattice X + over the pseudoring {0,1}. Whenever these operations are defined, we will write AM (resp. A+) for the meet in M (resp in X + ) . Definition 2.6 We say a pseudomodule M is simple iff it has a basis X = (xi)iex such that : $1.

Vwi, x j E X , O ~

XiAMX, j = x E M

s2. vx, y E x + , Z(x, y) r

r

xiA+xj

=xEX

+ ,

~ 1 e L(x, y).

Remark 2.7 If M has a finite basis X , then X + = X + U {0} is a finite semilattice with universal lower bound, hence is a lattice. Examples 2.8 E l . D'* is a simple pseudomodule, with basis X . i = 1,...,n,.

(el)'], . . where . .ei

.

(~,

~n),i

E2. Let xl = (A, 0 , 1 , 0 ) , x2 = ( 0 , # , 0 1 ) , xa = (0, 1, A, 0,), x4 = ( 1 , 0 , 0 , # ) , with 1 < A < # . T h e n X = (xi) 4 g e n e r a t e s a s u b m o d u l o ~ d o f D 4. We l e a v e i t to the reader to check that it is not simple. Note that not all bases of a simple pseudomodule satisfy the simplicity conditions $1, $2 above. Hence when it does, such a basis is called a canonical basis. In [131 the following results are proved. Proposition 2.9 If X -- ( x l ) , e i , and Y = (Yj)jEJ are two canonical bases for some simple finite dimensional pseudomodule M ~ then there is a semilattice isomorphism ~0 : X + ---* Y + . Proposition 2.10 T h e semilattice X + generated by a canonical basis X of a simple finite dimensional pseudomodule M characterizes M up to a pseudomodule isomorphism. 3. S a t u r a t e d

linear combinations

The following concept of saturated combination will be used. The definition just requires the existence of a set of generators. Let M be a pseudomodule (or a modulo~d) generated by a set X = (zdiel. Definition 3.1 We say that x = ~ Ajxj (where J C t ) is a saturated combination of x, if the following two

jEJ

conditions hold : 1: Vj E J VA > Aj, x < x+ Axj,

2. Vi q~ J , V A E D*, x < x + Axi.

982 Proposition 3.2 In a finite dimensional modulo'id (pseudomodule) M over a complete dioid (pseudoring) D , saturated combinations always exist and are unique. Proof Let x E M , V x k E X , l e t F-k = {A E D I Axk ~ x } . We h a v e 0 E --k, h e n c e E k # (3. Since Ek is bounded and D is complete, 3Ak = Sup {~k} E --k. Let I ( = {i E I [ ;~i # {0}}. Then x = y~ Akxk is a saturated combination for x. Uniqueness is straightforward. kEK

When I is finite (say I = { 1 , . . . n}), a saturated combination is easily completed to a maximal combination involving all the x i ' s i = 1, . . . , n by setting Ai = 0 for i 6 K . 4. ModuloYds and t h e lattice s t r u c t u r e

Let M be a finite dimensional D-moduloid (rasp. a D-pseudomodule). Theorem 1 If saturated linear combinations exist and are unique, then M is a lattice. Proof Let x, y E M .

Just complete the saturated combinations for x, y given by Proposition 3.2

above into maximal combinations x = ~ Aixi, y = ~ i=1

Then z = ~

YiZi

=

for if

X A y,

u < x,

#ixi,

and let

Vi ~--- /~i

A/-zi ,i = 1 , . . . , n .

i=l

and u _< y for some u E

M,

then the (completed)

i=1

saturated combination u = ~

T]ix i

necessarily verifies

qi 0 B(r(i,j), j} = t(i, j) pij = 1 B ( r ( i , j ) , m+i) = t(i, ji , qlj = 1 c(j, r ( i , j ) ) = 0 , ~ij = 0 C(m+i, r(i,j)) = 0 , ~ij : 0

IV.

DYNAMIC

BEHAVIOR

OF

CLOSED-LOOP

(24 (25

= 0

(26 , (27 (28 (29 (30} (31)

SYSTEM

F o r a n x m FMS, if t i m e - c o n s u m i n g m a t r i x T, column sequence m a t r i x P, r o w sequence matrix Q and state variables distribution matrix K are given, then column directional matrix ~, row directional matrix Q and row permutation m a t r i x S c a n be d e r i v e d . The psuedo-linear system model can be established by using rules (24)-(31}, and its dynamic behavior can be determined from the eigenvalue a n a l y s i s of c l o s e d - l o o p system. L i t e r a t u r e s [ 2-s] s h o w that the eigenvalue is j u s t t h e s t e a d y - s t a t e running rhythm of closed-loop system, so w e p u t e m p h a s i s u p o n e i g e n v a l u e analysis of s y s t e m . N o w s o m e r e l a t i v e t h e o r e m s a r e g i v e n . Definition 6 [z] F o r a s q u a r e m a t r i x A, if t h e r e e x i s t a r e a l n u m b e r A a n d a v e c t o r V w h i c h e n t r i e s a r e n o t a l l e q u a l to 8, s u c h t h a t A V = A V, t h e n A is c a l l e d e i g e n v a l u e a n d V is c a l l e d eigenvector.

989 Theorem 1 [z] The eigenvalue of a n i r r e d u c i b l e square matrix j u s t is t h e a v e r a g e w e i g h t of c r i t i c a l circuit in g r a p h G ( A ) . The eigcnvalue of a n x n s q u a r e m a t r i x A in m a x - a l g e b r a sense c a n be c a l c u l a t e d as f o l l o w ; A = v-~ {32)

A

i=i

where

~' = ~ ~ ~z ~ ... 8 ~" 9 Definition 7 [2] . A s y s t e m is s t a b l c i f f t h e r e number A , such that for all initial conditions and i, it h o l d s t h a t l i m [XI (k)] *''k = A 9

(33) exists a real all subscript

Theorem 2 [=] . A s y s t e m is s t a b l e i f f t h e r e e x i s t a real n u m b e r A, i n t e g e r d a n d k0, s u c h t h a t f o r a l l i n i t i a l c o n d i t i o n s , it h o l d s t h a t X ( k + d ) = A d X(k) , ~ k ) k0 . The above definitions and theorems suggest a relation between the eigenvalue and the stead-state running rhythm of system. N o w w e t a k e s o m e s i m p l e F M S as t h e i l l u s t r a t i o n s . Example i. In a f o l l o w - s h o p FMS with three jobs processing on two machines, its p a r a m e t e r matrixs a r e g i v e n as f o l l o w :

T=

P= 6

5

O=

4

2

2

2

1

Assuming that the capacity of b u f f in m a c h i n e t h e n t h e m a t r i x Ao , Al., B, C, Ao* a n d M in 12) determined as f c l l o w b y u s i n g (16)-(23}

A0

=

~ 3 . . 0 . . . . . . 9 5 . 5 . 4.4

. . . . .~ ] ~q B = . . 3 . _|

.

.

At

=

.

.

!!!iliI

.

.

A0*=

.

0 .... 0 C . . . . . . 0 0 . . . . . 0

M

2

3

Mz is b~ a n d (14}

=

0, are

..... 0

12 5 0 5 0 6 . 0 125 . 5 0 1 6 9 4 9 4 0

2 2 . 0 9 9 ~ 7 14 14 12 12

=

8

8

14 18

14 18

6 12 16

where

12 16

t h e d o t in m a t r i x d e n o t e s 6 . L i t e r a t u r e [~] s h o w t h a t t h e e i g c n v a l u e of c l o s e d - l o o p system with infinite b u f f e r is A = 15. In this example, the eigenvalue of c l o s e d - l o o p system (14) is A = 16, which shows that blocking phenomenon leads to the i n c r e a s e in the steady-state running rhythm of system and the system model with finite buffer can given more suitable description for maunfactuing system 9 Example 2. In t h i s e x a m p l e , we consider a c a s e of 3 o b s h o p manufacturing s y s t e m . Its p a r a m e t e r m a t r i x T, P, Q, E, ~, Q a n d S a r e g i v e n as f o l l o w ;

T

=

p = 6

5

4

Q = 2

2

!

3

2

1

990

[: .o.-3] [2 2 o] R=

B'= 4

5

[2 3 o]

[~. 2 ~]

~=

6

0

0

s =

1

0

1

2

3

2

!

Assuming that the capacities of b u f f e r in m a c h i n e M~ a n d M~ bl = bz = 2, t h e n m a t r i x Ao, Ai , B, C, Ao* a n d M in ( 1 2 ) - ( 1 4 ) determined as f o l l o w b y u s i n g r u l e s ( 2 4 ) - ( 3 1 ) .

arc are

~

~

~0

~

.

r~

~

AI

Ao*

=

|i0

3

1

5

[:!,:

0

-' . 0

5 0

0

B

=

.

.

.

2 2 9 9 7 12 12 I0 20 20 18 14 14 12 4 4

0

2..2 7

. . . . .

C =

.

9

0

.

M

0

=

0 0

]

.

The

cigcnvalue of c l o s e d - l o o p s y s t e m is A = 20. w h e n bt < 2 (i = 1,2), Ao is n o t a n i l p o t e n t r e s u l t s in d e a d l o c k p h e n o m e n o n 9 E x a m p l e 3. In this example, m a t r i x Q, ~ a n d e x a m p l e is c h a n g e d i n t o

Q =

~_ : 3

2

=

0

1

=

3 in m a c h i n e C, A0, a n d (24)-(31)

Ao

2

1

Mi a n d M~ arc M in ( 1 2 ) - ( 1 4 )

* =

35 o 3 6 11 o ; : : 5

C

=

.

.

o 5

.

.

. :2 9 12

o..] . . . .

The

above

the

01j iii?ii I 50

~: :::~| A,

6;.. 6; "o''"

2.~:~ :] =

S in

2

.

B

which

s =

1

Assuming t h a t t h e c a p a c i t i e s of b u f f e r bl = bz = I, t h e n t h e m a t r i x Ao , A~ , B, arc dctermincd as f o l l o w b y u s i n g r u l e s

A0

matrix,

0

.]

o

0

.

0

. 0

M

=

12 7 12 . 18 15 18 12 9 12 4 4

.

eigenvalue of c l o s e d - l o o p s y s t e m is ~ = 15. W h e n b~ < 1 (i = 1,2), Ao is n o t a n i l p o t c n t m a t r i x , which results in deadlock phenomenon. In job-shop manufacturing system, cloumn sequence matrix P usually is g i v e n a c c o r d i n g to t h e t e c h n o l o g i c a l requirements of jobs, but w e c a n c h o s e the s e q u e n c e of j o b s o n e a c h m a c h i n e to reduce the steady-state running period. The difference of d y n a m i c

991 behavior of closed-loop system between different routings has b e e n s h o w n in the l a s t two e x a m p l e s 9 E x a m p l e 4. In t h i s e x a m p l e , t h e m o r e g e n e r a l c a s e of j o b s h o p is c o n s i d e d . Its p a r a m e t e r m a t r i x T, P, Q, K, P, Q a n d S a r e g i v e n as f o l l o w :

T=

3 [

4

a =

P--

22 3-1

3-1

4 7 -

~ =

3 0

3 0 0 -I

o _ - - 1 2 3 1 2-1

~ :

2 2

3 0 0 -i

s :

i 1

Assuming t h a t the c a p a c i t i e s of b u f f e r in m a c h i n e a r e bz = b3 = 0, t h e n t h e m a t r i x A0 , AI , B, C, A0* a n d (14) a r e d e t e r m i n e d as f o l l o w b y u s i n g r u l e s ( 2 4 ) - ( 3 1 )

AG

=

9

,

5

. . 0

2

. 2

9

.

9

1

9

o

0

.

.

--

.

.

.

.

5

.

.

.

.

.

.

I

9

1

.

.

.

.

.

.

.

.

.

.

3

.

.

.

c

4

0

83030 4. . 0 .J

5

.~

o

.

]

Ao*=

At

3

.

.

~

.

Nz a n d M3 M in (12)-

00

]

~ . 0

3 B

.

. . .

"2 2 -

. .

.

.

.

.

.

0

.

.

.

9

0

.

.

.

M

.

0 .

.

.

0

=

6

.

.

5

3

-

3 3 0 q I0 i0 9 9 . . Ii II 7 7 i ; i [ I0 I0 .

0

.

.

.

.

. 0

.

.

7 3 . 3 0

.

.

.

The eigenvalue of the c l o s e d - l o o p system is A = Ii and the period-order is d = I. Now, we consider the influence of the c a p a c i t y of b u f f e r o n t h e d y n a m i c b e h a v i o r of c l o s e d - l o o p s y s t e m . In t h i s e x a m p l e , w h e n the c a p a c i t y of b u f f e r in m a c h i n e M2 is i n c r e a s e d to bz = I, the eigenvalue of c l o s e d - l o o p s y s t e m is A = I0. A n d w h e n t h e c a p a c i t y of b u f f e r in m a c h i n e M3 is a l s o i n c r e a s e d to b3 = bz = i, then the eigenvalue a n d the p e r i o d - o r d e r of c l o s e d - l o o p system are c h a n g e d i n t o A = 9.5, d = 2, w h i c h is s a m e to the c a s e of s y s t e m w i t h i n f i n i t e b u f f e r s h o w n in l i t e r a t u r e [7]. T h u s , d e s i g n of t h e capacities of b u f f e r in e a c h m a c h i n e is a p r o b l e m of c o n s i d e r a b l e i n t e r e s t in f l e x i b l e m a n u f a c t u r i n g system.

V.

CONCLUSIONS

In this paper, Modelling o n n x m F M S is a n a l y s e d by using p a t h a l g e b r a 9 B a s e d o n the p s u e d o - l i n e a r system model with finite buffer, the d y n a m i c b e h a v i o r of c l o s e d - l o o p s y s t e m is analysed from the eigcnvalue of s y s t e m 9 The illustrative examples show that this new model can give a more suitable description for manufacturing system 9

992 REFERENCES

[I]

Ho, Y.C. and Cassandras, D i s c r e t e - E v e n t Systems, on D e c i s i o n and Control,

[2]

Cohen, G., Dubois, D., Quadrat, J.P. and Viot, M., A LinearS y s t e m - T h e o r e t i c View of D i s c r e t e - E v e n t Processes, Proc. of 22nd IEEE C o n f e r e n c e on D e c i s i o n and Control, San Antonio, Texas, 1983, pp. I039-I044.

[3]

Cohen, G., Moller, P., Quadrat, J.P. and Viot, M., Linear System T h e o r y for Discrete Event Systems, Proceedings of 23rd IEEE Conference on D e c i s i o n and Control, Las Vegas, Nevada, 1984, pp.539-544.

[4]

Cohcn, G., Dubois, D., Quadrat, J.P. and Viot, M., L i n e a r System-Theoretic V i e w of D i s c r e t e - E v e n t P r o c e s s e s and Its Use for P e r f o r m a n c e E v a l u a t i o n in Manufacturing, IEEE Trans. on A u t o m a t i c Control, AC-30, 1985, p p . 2 1 0 - 2 2 0

[5]

Xu Xinhe, Decision,

[5]

Xu Xinhe, Modelling and D y n a m i c Analysis for A class Discrete-Event Systems, J. of Northeast University Technology, No.3, 1987.

[7]

Dubois, D. and Stecke, K., U s i n g Petri Nets to R e p r e s e n t P r o d u c t i o n processes, P r o c e e d i n g s of 22nd IEEE C o n f e r e n c e on Decision and Control, San Antonio, Texas, 1983, p p . i 0 6 2 - ! 0 6 7

C., C o m p u t i n g C a s t a t e Variables for P r o c e e d i n g s of 19th IEEE C o n f e r e n c e Albuquerque, 1980, pp.697-700.

Linear Discrete Event Systems, No.3, No.4, 1987.

J. of Control and

of of