New Directions in Nonlinear Observer Design (Lecture Notes in Control and Information Sciences)

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Author: Henk Nijmeijer | Thor I. Fossen

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

244

Springer London Berlin Heidelberg New York Barcelona I-Iong Kong Milan Paris Santa Clara Singapore Tokyo

H. Nijmeijer and T.I. Fossen (Eds)

New Directions in Nonlinear Observer Design

~ Springer

Series Advisory Board A. B e n s o u s s a n • M.]. G r i m b l e J.L. M a s s e y • Y.Z. T s y p k i n

• P. K o k o t o v i c

• H. K w a k e r n a a k

Editors H. Nijmeijer F a c u l t y o f M a t h e m a t i c a l S c i e n c e s , U n i v e r s i t y o f T w e n t e , P O B o x 217, 7500 A E E n s c h e d e , T h e N e t h e r l a n d s T.I. F o s s e n Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7034 Trondheim, Norway

ISBN 1-85233-134-8 Springer-Verlag L o n d o n Berlin Heidelberg British Library Cataloguing in Publication Data New directions in nonlinear observer design. - (Lecture notes in control and information sciences ; 224) 1.Observers (Control theory) 2.Nonlinear control theory 3.Feedback control systems l.Nijmeijer, Henk, 1955- II.Fossen, Thor I. 629.8'36 ISBN 1852331348 Library of Congress Cataloging-in-Publication Data New directions in nonlinear observer design / H. Nijmeijer and T.I. Fossen (eds.). p. cm. - (Lecture notes in control and information sciences ; 244) Includes bibliographical references and index. ISBN 1-85233-134-8 (alk. Paper) 1.Observers (Control theory)--Congresses. 2. Nonlinear control Theory--Congresses. I. Nijmeijer, H. (Henk), 1955- . II. Fossen, Thor I. III. Series. QA402.3.N487 1999 99-12174 629.8'312--dc21 CIP Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. © Springer-Verlag London Limited 1999 Printed in Great Britain 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 laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Typesetting: Camera ready by contributors Printed and bound at the Atheneum Press Ltd., Gateshead, Tyne & Wear 69/3830-543210 Printed on acid-free paper

vi

Acknowledgments The editors are grateful to: S t r a t e g i c U n i v e r s i t y P r o g r a m (SUP) in M a r i n e C y b e r n e t i c s at the Norwegian University of Science and Technology (NTNU), Departments of Engineering Cybernetics, Marine Hydrodynamics and Marine Strutures (Professor Dr.-Ing. Olav Egeland, Program Man-

ager). • A B B (Professor Dr.-Ing. Asgeir J. SCrensen, Technology Manager -

Business Area Marine and Turbochargers) for their financial support. The authors want to thank all the workshop contributors for contributing to this book project. Finally, Mrs. Alison Jackson at Springer-Verlag London should be thanked for editorial suggestions and for helping us with general publishing questions.

Trondheim, February 1999 Enschede, February 1999

Thor I. Fossen Henk Nijmeijer

Contributors Alcorta Garcia, E., Department of Measurement and Control, University of Duisburg, Duisburg, Germany. Ashton, S. A., School of MIS, Coventry University, U.K. Astolfi, A., Centre for Process Systems Engineering, Imperial College of Science, London, U.K. Bastin, G., Centre for Systems Engineering and Applied Mechanics, Universite Catholique de Louvain, Louvain-La Neuve, Belgium. Battilotti, S., Dipartimento di Informatica e Sistemistica, Universit~ di Roma "La Sapienza", Italy. Besan~on, G., Laboratoire d'Automatique de Grenoble, ENSIEG, SaintMartin d'H~res, France. Blanke, M., Department of Automatic Control, Aalborg University, Denmark. Canudas de Wit, C. Laboratoire d'Automatique INPG, ST. Martin d'H~res, France.

de Grenoble, ENSIEG-

Cruz, C., Department of Electronics ~ Telecom., Scientific Research and Advanced Studies Center of Ensenada (CICESE), M~xico. Deng, H., Department of Applied Mechanics and Engineering Sciences University of California at San Diego, La Jolla, USA. Egeland, O., Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway. El Bahir, L., Department of Control Engineering, Universit~ Libre de Bruxelles, Brussels, Belgium. El Yaagoubi, E. H., LCPI ENSEM

Cassablanca, Morocco.

Fossen, T. I., IDepartment of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway and 2ABB Industri AS, Marine Division, Oslo, Norway.

Vlll

Frank, P. M., Department of Measurement Duisburg, Duisburg, Germany.

and Control, University of

Glumineau, A. Institut de Recherche en Cybern~tique de Nantes, France. Hammouri, H., LAGEPT

University of Lyon, France.

Huijberts, H. J. C., Department of Mathematics and Computing Science, Eindhoven University of Technology, The Netherlands. Horowitz, R. Department of Mechanical Engineering, University of California, Berkeley, CA, U.S.A. Isidori, A., IDepartment of Systems Science and Mathematics, Washington University, St. Louis, USA and 2Dipartimento di Informatica e Sistemistica, Universit~t di Roma "La Sapienza", Italy. Izadi-Zamanabadi, R., Department of Automatic Control, Aalborg University, Denmark. Jiang, Z.-P., Department of Electrical Engineering, Polytechnic University, Brooklyn, U.S.A. Junge, L., Drittes Physikalisches Institut, Universitttt G6ttingen, Germany. Khalil, H. K., Department of Electrical and Computer Engineering, Michigan State University, USA. Kinnaert, M., Department of Control Engineering, Universit~ Libre de Bruxelles, Brussels, Belgium. Kocarev, L., Department of Electrical Engineering, St Cyril and Methodius University, Skopje, Macedonia. Kristiansen, D., Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway. Krstid, M., Department of Applied Mechanics and Engineering Sciences University of California at San Diego, La Jolla, USA. Lilge, T., Institut flit Regelungstechnik, University of Hannover, Hannover, Germany. Lohmiller, W., Nonlinear Systems Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA. Ldpez-Morales, V., Institut de Recherche en Cybern~tique de Nantes, France.

ix Loria, A., Laboratoire d'Automatique de Grenoble, ENSIEG, St. Martin d'H~res, France. Nijmeijer, H., 1Faculty of Mathematical Sciences, Dept. of Systems, Signals and Control, University of Twente and 2Faculty of Mechanical Engineering, Eindhoven University of Technology, The Netherlands. Ortega, R., Laboratoire des Signaux et Syst~mes, Ecole Sup6rieure d'Electricit6, Paris, Prance. Panteley, E., I.N.R.I.A., Rh6ne Alpes, St. Martin d'Hfires, France. Parlitz, U., Drittes Physikalisches Institut, Universitat GSttingen, Germany. Pettersen, K. Y., Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway. Praly, L., Centre Automatique et Syst~mes, t~cole des Mines de Paris, Fontainebleau, prance. Rodrigues-Cortes, H., Laboratoire des Signaux et Syst~mes, Ecole Sup~rieure d'Electricit~, Paris, prance. Schaffner, J., Institute for Systems, Informatics and Safety, European Commission Joint Research Centre, Ispra, Italy Schreier, G., Department of Measurement and Control, University of Duisburg, Duisburg, Germany. Shields, D. N., School of MIS, Coventry University, U.K. Shiriaev, A., Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway. Slotine, J. J. E., Nonlinear Systems Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA. Strand, J. P., ABB Industri AS, Marine Division, Oslo, Norway. Teel, A., Department of Electrical and Computer Engineering, University of California, Santa Barbara, USA. Tsiotras, P., Georgia Institute of Techology, School of Aerospace Eng., Atlanta, Georgia, USA. Vik, B. Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway. Zeitz, M., Institut ftir Systemdynamik of Stuttgart, Germany.

und Regelungstechnik, University

Contents Nonlinear Observer Design A Viewpoint on Observability Nonlinear Systems

and

Observer

Design

for 3

G. Besanf~on 1 2 3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Definitions a n d P r o p o s e d "Classification" . . . . . . . E x a m p l e s of N o n U n i f o r m a n d U n i f o r m O b s e r v a t i o n . . . . 3.1 N o n U n i f o r m O b s e r v a t i o n : the Case of State-Affine Systems . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 U n i f o r m O b s e r v a t i o n : the Case of U n i f o r m l y O b s e r v able Systems . . . . . . . . . . . . . . . . . . . . . . 3.3 A n E x a m p l e of U n i f o r m O b s e r v a t i o n of N o n - u n i f o r m l y observable S y s t e m s . . . . . . . . . . . . . . . . . . . Observer I n t e r c o n n e c t i o n . . . . . . . . . . . . . . . . . . . S t a t e T r a n s f o r m a t i o n s a n d Observer Design . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Model-Based

Observers

for

Tire/Road

Contact

1

3 4 5

7 8 9 11 15 20 20

Friction

Prediction C. Canudas de Wit, R. Horowitz and P. Tsiotras

2

3 4 7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . T i r e - r o a d Friction Models . . . . . . . . . . . . . . . . . . . 2.1 P s e u d o - S t e a d y S t a t e Models . . . . . . . . . . . . . 2.2 L u m p e d D y n a m i c Models . . . . . . . . . . . . . . . 2.3 D i s t r i b u t e d D y n a m i c Models . . . . . . . . . . . . . 2.4 Relation Between Distributed Dynamical Model and the Magic F o r m u l a . . . . . . . . . . . . . . . . . . . Problem Formulation . . . . . . . . . . . . . . . . . . . . . . G e n e r a l Observer Design . . . . . . . . . . . . . . . . . . . . A p p l i c a t i o n to the O n e - W h e e l Model . . . . . . . . . . . . . 5.1 Simulation Results . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 23 25 26 28 29 30 32 33 36 39 40 41

xii

Observer D e s i g n for Nonlinear D. Kristiansen and O. Egeland Introduction . . . . . . . . . . 1 Contraction Theory . . . . . 2 System Equations . . . . . . . 3 4 5

6 7

Oscillatory S y s t e m s

. . . . . . . . . . . . . . . . . . 3.1 Analysis . . . . . . . . . . . . . . O b s e r v e r Design . . . . . . . . . . . . . Simulations . . . . . . . . . . . . . . . . 5.1 E x a m p l e 1: 2 - D O F O s c i l l a t o r y Nonlinearities . . . . . . . . . . . 5.2 E x a m p l e 2: C y l i n d e r G y r o s c o p e Conclusions . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . System with . . . . . . . . ............ . . . . . . . . . . . . . . . .

Transformation to State Affine S y s t e m Design A. Glumineau and V. Ldpez-M. Introduction . . . . . . . . . . . . . . . . . . . 1 Defi n i t i o n s a n d N o t a t i o n . . . . . . . . . . . . 2 Problem Statement . . . . . . . . . . . . . . . 3

4

5 6

and

Observers

1 2

. . . . . .

43 44 46 47 49 51 51 52 56 57

Observer

. . . . . . . . . . . . . . . . . . . . . for S t a t e A t t i n e . . . . . . . ........ ........ . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 60 61 61 62 65 66 67 68 69

f o r N o n l i n e a r Discrete73

Introduction . . . . . . . . . . . . . . . . . . . . . . Differential F o r m s . . . . . . . . . . . . . . . . . . O b s e r v e r D e s i g n using O b s e r v e r F o r m s . . . . . . . . . . . . O b s e r v e r Design using E x t e n d e d O b s e r v e r F o r m s Conclusions . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

Stability Analysis and Observer Diffusion P r o c e s s e s W. Lohmiller and J.-J. E. Slotine

. . . . . . . . . . . . . . . . . . Cubic . . .

59

. . . 3.1 T h e I n p u t - O u t p u t Differential E q u a t i o n Systems ~a - . . . . . . . . . . . . . . . 3.2 S t a t e Affine T r a n s f o r m a t i o n A l g o r i t h m S y n t h es i s O b s e r v e r for S t a t e Affine S y s t e m s 4.1 Physical Example . . . . . . . . . . . . 4.2 Simulation Results . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

On E x i s t e n c e o f E x t e n d e d Time Systems H. J. C. Huijberts

43

Design

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

73 75 79 84 90 91

for N o n l i n e a r

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Contraction Analysis . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic Tools . . . . . . . . . . . . . . . . . . . . . . .

93 93 94 94

xiii

4 5 6 A

2.2 N o n l i n e a r O b s e r v e r Design using C o n t r a c t i o n T h e o r y 2.3 Weakly Contracting Systems . . . . . . . . . . . . . N o n l i n e a r Diffusion E q u a t i o n s . . . . . . . . . . . . . . . . . 3.1 C o n t r a c t i o n P r o p e r t i e s of R e a c t i o n - D i f f u s i o n P r o cesses . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Observer Design for N o n l i n e a r Diffusion Processes Spatial Discretization and Numerical Implementation . . . . Further Extensions . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . C o m p u t a t i o n of C o n t r a c t i o n R a t e s . . . . . . . . . . . . . .

96 97 99 100 103 104 105 109 109

N o n l i n e a r P a s s i v e O b s e r v e r D e s i g n for S h i p s w i t h A d a p t i v e

Wave Filtering J. P. Strand and T. L Fossen 1 2

3

4

5 6 7

113

Introduction . . . . . . . . . . . . . . . . . . . . . . . Modeling . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Kinematics . . . . . . . . . . . . . . . . . . . 2.2 Vessel D y n a m i c s . . . . . . . . . . . . . . . . . . . . 2.3 T o t a l Ship Model . . . . . . . . . . . . . . . . . . . . N o n - A d a p t i v e Observers . . . . . . . . . . . . . . . . . . . . 3.1 Observer i n the E F frame . . . . . . . . . . . . . . . 3.2 Augmented Observer . . . . . . . . . . . . . . . . . . A d a p t i v e Observer . . . . . . . . . . . . . . . . . . . 4.1 A d a p t i v e Observer E q u a t i o n s . . . . . . . . . . . . . 4.2 A d a p t i v e Observer Error D y n a m i c s . . . . . . . . . . 4.3 Stability a n d P a s s i v i t y . . . . . . . . . . . . . . . . . Experimental Results . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

Nonlinear INS

Observer

Design

for I n t e g r a t i o n

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

113 115 115 115 118 118 119 123 125 126 126 126 128 130 133

. . . .

. . . . . . . .

of DGPS

and

135

B. Vik, A. Shiriaev and T. L Fossen 1

2 3

4

5

Introduction . . . . . . . . . . . . . . . . . . . . . . 1.1 Nomenclature . . . . . . . . . . . . . . . . . 1.2 Motivation . . . . . . . . . . . . . . . . . . Review of G P S F u n d a m e n t a l s . . . . . . . . . . . . . . . . . Strapdown Equations ..................... 3.1 Local F r a m e R e p r e s e n t a t i o n . . . . . . . . . . . . . . 3.2 Earth Frame Representation ............. 3.3 A n g u l a r Velocity E q u a t i o n s . . . . . . . . . . . . . . N o n l i n e a r Observer Design . . . . . . . . . . . . . . . . . . . 4.1 A n g u l a r Velocity Observer . . . . . . . . . . . . . . . 4.2 Velocity a n d P o s i t i o n Observers . . . . . . . . . . . . Case S t u d y . . . . . . . . . . . . . . . . . . . . . .

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

. . . . .

135 135 136 138 140 140 141 142 142 143 145 157

xiv

6

Conclusions and Future Work

7

References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Variants

of Nonlinear

Normal

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

Form

Observer

158 158

161

Design

J. Schaffner and M. Zeitz Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Normal Form Observer . . . . . . . . . . . . . . . . . . . . . 2 Continuous Observer . . . . . . . . . . . . . . . . . . . . . . 3 Extended Luenberger Observer . . . . . . . . . . . . . . . . 4 Block-Triangular Observer . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1

167 171 179 179

181

Output Feedback Control Design

II

161 162 163

Separation Results for Semiglobal Stabilization Nonlinear Systems via Measurement Feedback

of 183

S. Battilotti 1

2 3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regional Stabilization via Measurement Feedback ......

183 185 186

3.1 3.2 3.3

186 191

Tools . . . . . . . . . . . . . . . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . S e m i g l o b a l S t a b i l i z a t i o n of U n c e r t a i n N o n l i n e a r S y s tems . . . . . . . . . . . . . . . . . . . . . . . . . . .

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observer-Controller

Design

for

Global

Tracking

of

N o n h o l o n o m i c Systems Z.-P. Jiang and H. Nijmeijer Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . 2 Reduced-Order Observer . . . . . . . . . . . . . . . . . . . . 3 Output-Feedback Design . . . . . . . . . . . . . . . . . . . . 4 4.1 Backstepping-Based Trackers . . . . . . . . . . . . . 4.2 A Modification . . . . . . . . . . . . . . . . . . . . . Example: A Knife-Edge . . . . . . . . . . . . . . . . . . . . Conclusions and Future Work . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

196 205

Separation Principle f o r a C l a s s o f Euler-Lagrange Systems

207 207 208 210 212 213 217 219 225 226

A

229

A. Loria and E. Panteley 1

Introduction

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

229

xix

IV

Synchronization

467

and Observers

1 Synchronization Through E x t e n d e d Kalman Filtering

469

C. Cruz and H. Nijmeijer Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 A n E x t e n d e d K a l m a n F i l t e r as Receiver . . . . . . . . . . . 2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.1 Synchronization .................... 3.2 Secure C o m m u n i c a t i o n . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Nonlinear Discrete-Time

469 472 479 479 483 487 488

Observers for Synchronization

Problems

491

T. Lilge 1 2 3 4

7 8

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . S t a t e E q u i v a l e n c e to a S y s t e m in E x t e n d e d O b s e r v e r F o r m Observer Design via E x t e n d e d Observer F o r m . . . . . . . . A l t e r n a t i v e Observer S t r u c t u r e s via E O F . . . . . . . . . . 4.1 Observer E q u a t i o n s . . . . . . . . . . . . . . . . . . 4.2 M a i n C h a r a c t e r i s t i c s of t h e Observers . . . . . . . . A n E x a m p l e in the F i e l d of C o m m u n i c a t i o n . . . . . . . . . Observer Design for the R6ssler S y s t e m . . . . . . . . . . . 6.1 Observer Design in C o n t i n u o u s - T i m e . . . . . . . . . 6.2 Observer Design i n D i s c r e t e - T i m e . . . . . . . . . . 6.3 Observer Errors for Slow E r r o r D y n a m i c s . . . . . . . 6.4 Observer Errors for Fast E r r o r D y n a m i c s . . . . . . 6.5 Concluding Remarks .................. Discussion a n d C o n c l u s i o n s . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

511

Chaos Synchronization U. Parlitz, L. Junge and L. Kocarev Introduction . . . . . . . . . . . . . . . . . . . . . 1 S y n c h r o n i z a t i o n of S p a t i a l l y E x t e n d e d S y s t e m s 2 Generalized Synchronization . . . . . . . . . . . . . . . . . . 3 Phase Synchronization ..................... 4 Conclusions . . . . . . . . . . . . . . . . . . . . . 5 References . . . . . . . . . . . . . . . . . . . . . . 6

491 494 497 499 499 500 501 503 505 506 506 507 508 509 509

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

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

511 512 515 518 522 522

XV

4 5 6 A 4

Model a n d P r o b l e m F o r m u l a t i o n . . . . . . . . . . . . . . . A Cascades A p p r o a c h to a S e p a r a t i o n P r i n c i p l e . . . . . . . 3.1 Observer Design . . . . . . . . . . . . . . . . . . . . 3.2 S t a t e Feedback C o n t r o l l e r . . . . . . . . . . . . . . . 3.3 A Separation Principle . . . . . . . . . . . . . . . . . A p p l i c a t i o n to R o b o t M a n i p u l a t o r s . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . A T h e o r e m o n U G A S for Strictly Passive S y s t e m s . . . . .

High-Gain

Observers

in Nonlinear

Feedback

232 234 235 236 237 240 243 244 246 249

Control

H. K. Khalil 1 2 3 4 5 6 7 8 9 10 11 5

Introduction . . . . . . . . . . . . . . . . . . . . . . . Motivating Example . . . . . . . . . . . . . . . . . . Separation Principle . . . . . . . . . . . . . . . . . . S t a b i l i z a t i o n a n d Semiglobal S t a b i l i z a t i o n . . . . . . . . . . Nonlinear Servomechanisms . . . . . . . . . . . . . . . . . . Adaptive Control . . . . . . . . . . . . . . . . . . . . Sliding Mode C o n t r o l . . . . . . . . . . . . . . . . . . U n m o d e l e d Fast D y n a m i c s . . . . . . . . . . . . . . . . . . Discrete-Time Implementation ................ A p p l i c a t i o n to I n d u c t i o n Motors . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

Output-Feedback tems

Control

of

Stochastic

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

. . . . . . . .

. . . .

Nonlinear

Feedback

Sys269

M. Krstid and H. Deng Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1 P r e l i m i n a r i e s on Stochastic S t a b i l i t y . . . . . . . . . . . . . 2 O u t p u t - F e e d b a c k S t a b i l i z a t i o n in P r o b a b i l i t y . . . . . . . . 3 Output-Feedback Noise-to-State Stabilization . . . . . . . . 4 Output-Feedback Adaptive Stabilization ........... 5 References . . . . . . . . . . . . . . . . . . . . . . . . . 6 P r o o f . . . . . . . . . . . . . . . . . . . . . . . . . . . . A P r o o f . . . . . . . . . . . . . . . . . . . . . . . . . . . . B P r o o f . . . . . . . . . . . . . . . . . . . . . . . . . . . . C Output

249 250 255 258 259 261 262 262 263 263 264

Control of Food-Chain

. . .

. . . .

. . . .

. . . .

Systems

R. Ortega, A. Astolfi, G. Bastin and H. Rodrigues Cortes Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 C o n t r o l l e r Design P r o c e d u r e . . . . . . . . . . . . . . . . . . 2 S t a t e - F e e d b a c k C o n t r o l of a Simple P r e y - P r e d a t o r S y s t e m 3 Output-Feedback Stabilization ................ 4 M ain Result . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 S i mulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

269 270 271 276 280 284 287 288 289 291 291 292 295 299 301 305

xvi 7 8 A 7

8

Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maple Code . . . . . . . . . . . . . . . . . . . . . . . . . . .

306 307 308

O u t p u t Feedback Tracking Control for Ships

311

K. Y. Pettersen and H. Nijmeijer Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 T h e Ship M o d e l . . . . . . . . . . . . . . . . . . . . . . . . . 2 D e s i g n of a n O u t p u t F e e d b a c k T r a c k i n g C o n t r o l L a w . . . . 3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Bias E s t i m a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . 5 S i m u l a t i o n s w i t h an E n v i r o n m e n t a l D i s t u r b a n c e . . . . . . 6 Conclusions and ~ t u r e Work . . . . . . . . . . . . . . . . . 7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

311 313 314 319 324 327 329 331

D y n a m i c U C O Controllers and Semiglobal Stabilization o f U n c e r t a i n N o n m i n i m u m P h a s e S y s t e m s by O u t p u t Feedback 335 A. Isidori, A. R. Teel and L. Praly 1 2 3

4

5 6

III

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . S t a b i l i z a t i o n of N o n m i n i m u m P h a s e S y s t e m s by O u t p u t F eed back . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 T h e R e l a t i v e D e g r e e O n e Case . . . . . . . . . . . . 3.2 T h e R e l a t i v e D e g r e e G r e a t e r t h a n O n e Case . . . . On Dynamic U C O Feedback . . . . . . . . . . . . . . . . . . 4.1 General Results . . . . . . . . . . . . . . . . . . . . . 4.2 A p p l i c a t i o n to N o n m i n i m u m P h a s e S y s t e m s . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

335 336 338 338 341 344 344 347 349 349

351

Fault Detection and Isolation

Fault D e t e c t i o n O b s e r v e r f o r a C l a s s o f N o n l i n e a r S y s t e m s 353 S. A. Ashton and D. N. Shields 1 2 3 4 5

6 7

Introduction . . . . . . . . . . . . . . . . . . . . . . . System Description . . . . . . . . . . . . . . . . . . . Observer Design . . . . . . . . . . . . . . . . . . . . General Detectability Conditions ............... Testable Detectability Conditions ............... 5.1 A S p e c ia l Class ( S t e p - F a u l t s ) . . . . . . . . . . . . . 5.2 Numerical Calculation Procedure ........... Concluding Remarks . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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

. . . . . . . .

353 354 354 363 365 368 371 372 373

xvii

N o n l i n e a r Observer for Signal and P a r a m e t e r Fault D e t e c t i o n in Ship P r o p u l s i o n Control 375 M. Blanke and R. Izadi-Zamanabadi 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Ship P r o p u l s i o n S y s t e m . . . . . . . . . . . . . . . . . . . . 2.1 Propeller Thrust and Torque ............. 2.2 Diesel E n g i n e P r i m e M o v e r . . . . . . . . . . . . . .

375 376 377 377

2.3 2.4

Hull R e s i s t a n c e . . . . . . . . . . . . . . . . . . . . . A c t u a t o r s for Fuel I n j e c t i o n a n d P r o p e l l e r P i t c h . .

378 378

2.5

Sensors

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

378

3 4

Control Hierarchy . . . . . . . . . . . . . . . . . . . . . . . Structural Analysis . . . . . . . . . . . . . . . . . . . . . . . 4.1 D e s c r i p t i o n of t h e M o d e l . . . . . . . . . . . . . . . 4.2 Formal Representation . . . . . . . . . . . . . . . . . 4.3 Sensor Fusion for R e - c o n f i g u r a t i o n . . . . . . . . . .

379 380 380 380 381

5

I s o l a t i o n of Shaft S p e e d a n d E n g i n e F a u l t s ......... 5.1 Adaptive Observer . . . . . . . . . . . . . . . . . . . 5.2 I d e n t i f i c a t i o n of P r o p e l l e r P a r a m e t e r s . . . . . . . .

384 384 386

5.3

....

388

6

Fault Isolation . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Re-configuration . . . . . . . . . . . . . . . . . . . .

389 392

7

Simulation Results

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

394

8

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .

395

9

References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

395

Nonlinear Observers for Fault D e t e c t i o n and Isolation P. M. Frank, G. Schreier and E. Alcorta Garcia Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Observer-Based Residual Generation ............. 3

399

3.1 3.2 3.3 3.4 3.5 3.6 4

Identifiability from Usual Maneuvering D a t a

399 400 401 Nonlinear Identity Observer Approach ........ 401 N o n l i n e a r U n k n o w n I n p u t O b s e r v e r A p p r o a c h . . . 403 The Disturbance Decoupling Nonlinear Observer Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 404 Adaptive Nonlinear Observer Approach ....... 406 T h e N o n l i n e a r F a u l t D e t e c t i o n F i l t e r A p p r o a c h . . . 408 O b s e r v e r for F a u l t D i a g n o s i s in B i l i n e a r S y s t e m s . . 410

Nonlinear Observer Design via Lipschitz Condition . . . . .

412

4.1 4.2 4.3

Observer Presentation . . . . . . . . . . . . . . . . . C o n t r i b u t i o n of this O b s e r v e r . . . . . . . . . . . . . Residual Generation . . . . . . . . . . . . . . . . . .

412 415 417

Conclusions . . . . . . . . . . . . . . . ............ References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

418 418

XVlll

4

A p p l i c a t i o n o f N o n l i n e a r O b s e r v e r s t o Fault D e t e c t i o n a n d Isolation 423 H. Hammouri, M. Kinnaert and E.H. El Yaagoubi 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .

423

2

R e s i d u a l G e n e r a t i o n for L i n e a r S y s t e m s

424

2.1

3

...........

Problem Statement . . . . . . . . . . . . . . . . . . .

424

2.2

Second Problem Formulation

425

2.3

P r i n c i p l e of t h e S o l u t i o n . . . . . . . . . . . . . . . .

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

R e s i d u a l G e n e r a t i o n for N o n l i n e a r S y s t e m s

426 .........

428

3.1

Introduction

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

428

3.2

Basic N o t i o n s . . . . . . . . . . . . . . . . . . . . . .

428

3.3

H i g h G a i n O b s e r v e r s for U n i f o r m l y O b s e r v a b l e Systems . . . . . . . . . . . . . . . . . . . . . . . . . .

429

3.4

T h e F h n d a m e n t a l P r o b l e m of R e s i d u a l G e n e r a t i o n for N o n l i n e a r S y s t e m s . . . . . . . . . . . . . . . . .

431

3.5

A p p l i c a t i o n of N o n l i n e a r O b s e r v e r s t o t h e F P R G

. .

Hydraulic System . . . . . . . . . . . . . . . . . . . . . . . .

434 437

4.1

M o d e l l i n g of t h e S y s t e m . . . . . . . . . . . . . . . .

437

4.2

Design of a R e s i d u a l G e n e r a t o r . . . . . . . . . . . .

438

5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .

441

6

References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

441

A

N u m e r i c a l Values used for t h e S i m u l a t i o n of t h e H y d r a u l i c System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

443

I n n o v a t i o n G e n e r a t i o n for Bilinear S y s t e m s w i t h U n k n o w n 445

Inputs

M. Kinnaert and L. El Bahir 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .

445

2

Problem Statement . . . . . . . . . . . . . . . . . . . . . . .

447

3

Design Procedure . . . . . . . . . . . . . . . . . . . . . . . .

448

4

Innovation Monitoring . . . . . . . . . . . . . . . . . . . . .

456

4.1

Introductory Remark . . . . . . . . . . . . . . . . . .

456

4.2

I n n o v a t i o n in t h e P r e s e n c e of A d d i t i v e F a u l t s . . . .

456

4.3

Generalized Likelihood Ratio Test

457

5

..........

D e s i g n a n d V a l i d a t i o n of a F D I S y s t e m for a t h r e e T a n k Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

459

5.1

Process Description

459

5.2

Design a n d V a l i d a t i o n of t h e I n n o v a t i o n G e n e r a t o r .

5.3

E v a l u a t i o n of t h e I n n o v a t i o n S e q u e n c e . . . . . . . .

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

460 462

6

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .

463

7

References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

463

A V i e w p o i n t on Observability and Observer Design for Nonlinear S y s t e m s Gildas Besan~on

ENSIEG

1

Laboratoire d ' A u t o m a t i q u e de Grenoble BP46 - 38402 Saint-Martin d'H~res, France

Introduction

Given a dynamical system, the observer aims at obtaining an estimate of the current state by only using available measurements. For linear systems, the property of observability, characterized by the K a l m a n rank condition, guarantees the possibility to indeed design an observer. In the case of nonlinear systems, observability is not enough, basically because this p r o p e r t y in general depends on the input of the system. In other words, observability of a nonlinear system does not exclude the existence of inputs for which two distinct initial states cannot be distinguished by using the knowledge of the measured output. This results in the fact t h a t in general, observer gains can be expected to depend on the applied input. Moreover, the existing observers generally tightly depend on some specific structure of the considered system. This chapter discusses such characteristics of observer design for nonlinear systems, basically following the recent work of [1]: on the basis of background definitions, the main observability requirements for observer design are first recalled in Section 2, so as to put into relief the various contexts which can be found. In particular, designs which are non uni.form w.r.t, the input are distinguished from those which are uniform, with a special attention paid to the case of uniform design for non uniformly observable systems in Section 3. Two directions of extension of available designs are then highlighted and illustrated, namely the possibility of interconnecting sub-observers to still obtain an observer in Section 4, and the issue of state transformation to make some observer design possible in Section 5. Some conclusions are finally given in Section 6.

4

2

1. A Viewpoint on Observability and Observer Design for Nonlinear Systems

Basic Definitions and Proposed "Classification"

Let us consider a nonlinear system defined by the following representation:

(~n~)

{~ = f(x,u) y

=

h(x)

x c ~ , u

9

y9

m

,

(1.1)

and let X~,(t, xo) denote its solution at time t, with initial condition x0 at time t = 0 and control u(t). Admissible inputs u(.) axe assumed to be taken in some set L / o f measurable and bounded functions. Observability can then be defined by the notion of indistinguishability [22] (see [11] for a synthesis). D e f i n i t i o n 1.1 Indistinguishability.

A pair (xo, ~o) will be said to be indistiguishable by u i.fVt > O, h(x~(t, xo)) =h(Xu( t, ~o) ). The pair is just said to be indistinguishable, i.f it is so .for any U.

From this definition, observability of (1.1) can be defined as follows: Definition 1.20bservability.

A nonlinear system (1.1) is observable if it does not have any indistinguishable pair o.f states. At this point, one can notice that from this definition, observability does not exclude the possible existence of inputs for which some states are indistinguishable. As an example, the following system: :~1

~

UX2

J:2

=

--x2

y

~

X1

(1.2)

is clearly observable, and yet u _= 0 makes every pair

indistinguishable. This means t h a t in general observability is not enough to be able to design an observer and t h a t the problem of inputs must be taken into account. A particular case of interest, is the case of inputs for which no indistinguishable pair can be found: D e f i n i t i o n 1.3 Universal inputs. An input u is universal on [0, t] i.f.for every pair of distinct states xo ~ 20, there exists T 9 [O,t] such that h(x~(7, xo)) ~ h(x~(7, 20)).

If u is universal on Kt +, it is just said to be universal.


5

From this definition, the notion of singular inputs can be derived: D e f i n i t i o n 1.4 Singular inputs. A non universal input is called singular. As one can guess from example (1.2) above, a typical class of systems which have singular inputs is the class of so-called state-affine systems of the following form:

y

= =

A(u)x Cx.

(1.3)

For such systems, one can define - as for linear time-varying systems - the following quantities: 9 the transition matrix @~(T, t) by: d(I)~('r, t) dT @~(t,t)

-

A(U(T))O~(T,t)

=

Id,

(1.4)

9 the observability Grammian F(t, T, u) by: F(t, T, u) :---

f

t+T

@T(T,t)cTV~2u(T,t)dT),

(1.5)

Jt

9 and some universality index "y(t, T, u), defined as the smallest elgenvalue of F(t, T, u). On this basis one can characterize inputs which are "universal enough" so that an observer design will be possible: D e f i n i t i o n 1.5 [10] Regularly persistent inputs. A n admissible input u is said to be regularly persistent .for system (1.3) if 3 T > O, a > O, and to > 0 such that 7 ( t , T , u ) >_ ~ for t >_ to. These remarks show that in general, the observer gain, as well as its stability property, depend on the input. In view of the above definitions, a particular case of special interest is the one of systems without singular inputs: D e f i n i t i o n 1.6 Uniformly observable systems. A system whose all inputs are universal is called uniformly observable. If, .for every t > O, all inputs are universal on [0, t], the system is locally uniformly observable. A class of "sufficiently regular" locally uniformly observable control affine systems have been characterized in [14]:

6


Theorem

1.1 [14] A n observable nonlinear system in the form:

= y

f(~)+ug(~)

=

(1.6)

with u = 0 as a universal input, and nonsingular Jacobian of ~(~) = (h(~),Lf(h(~)), ... L ~ - l ( h ( ~ ) ) ) at ~0, is locally uniformly observable at ~o i.f and only i.f the change of coordinates x = ~(~) turns it into the .following .form: Xl X2

= =

X2 + ~ I ( X l ) u X3 + ~ 2 ( X l , X 2 ) u

Xn--1

=

Xn+~n--I(Xl,...Xn--1)U

y

=

(1.7) =

+

xI []

Extensions to non-control-affine case mono and multi o u t p u t can be found in [15] and [12] respectively. For such systems, so-called high gain observers m a y exist [16], which are observers with gain and stability independent of the input. However, one can find several other cases of observers which have been proposed irrespective of the input, although the considered systems are not uniformly observable [21, 29, 13, 2]. The only possible explanation for such a phenomenon is t h a t in these cases, the difference between trajectories resulting from two distinct indistinguishable states naturally tends to zero. This is for instance what happens in system (1.2) : for this system u = 0 is singular insofar as it cannot distinguish

However, for such initial conditions and input, the error e2 on trajectories of x2 satisfies 42 = - e 2 , while t h a t of xl is identically zero, and thus, it is clear that the difference between the two trajectories asymptotically goes to zero. In this case, an observer irrespective of the input can be designed, simply as ~:2 = - x 2 Following the terminology of linear systems, this suggests to define some detectability property as follows: D e f i n i t i o n 1.7 Detectability. A nonlinear system (1.1) will be called detectable if for every couple ((2o, ~o), u(.) ) in (j~n • H:tn) • 5t such that there exists to .for which

1. A Viewpoint on Observability and Observer Design for Nonlinear Systems Vt _> to; h(x~(t, xo)) = h(x~(t, 5:0)) then IIx,,(t, xo) -

x=(t, ~0)ll

,0. t ---~oo

In the case of uncontrolled system, we only consider pairs of initial conditions. As a summary, in view of the above observability properties w.r.t, the inputs, one can distinguish the following cases in observer designs: 9 Either the system does not have singular inputs, and in t h a t case, one can hope to be able to design an observer irrespective of the input (uniform observation of uniformly observable systems), but m a y also only find an observer depending on the input (non uni.forTn observa-

tion of uniformly observable systems); 9 Or the system may have singular inputs, and in t h a t case possible observer designs will generally depend on the inputs (non uniform observation of non uniformly observable systems), except in special cases of systems which are detectable in the sense of definition 1.7

(uniform observation of non uniformly observable systems).

3 3.1

Examples of Non Uniform and Uniform Observation Non Uniform Observation: the Case of State-Affine Systems

Let us consider here a system described by the following equations:

y

= =

A(u)x + B(u), Cx

x C ~n, u C ~m y C P:tp.

(1.8)

For such a system, the observability generally depends on the input, and under appropriate excitation, an observer has been proposed with a gain indeed depending on u, as recalled below: 1.2 [10, 17] I.f u is regularly persistent for (1.8), and A(u), B(u) are uniformly bounded on the set of admissible inputs, then there exists 0o s.t. .for any 0 >_ 0o, the following system is an observer for (1.8):

Theorem

= So

= >

A(u)2 S - 1 c T ( c : ~ y) -~- B(~t) - 0 S - d ( u ) V S -- SA(u) + C T C O,

(1.9)

8


and V~ > 0, 30 > 0 : II~(t) - x(t)l I < A e x p ( - ~ t ) , .for some A > O. [] T h i s result c a n c l e a r l y b e e x t e n d e d t o t h e case of s y s t e m s in t h e f o r m ic = A ( s ) x + B(s) for a n y m e a s u r e d signal s w h i c h is r e g u l a r l y p e r s i s t e n t for k = A(s)x, for i n s t a n c e s = (u, y) [18]. W e p r e s e n t in S e c t i o n 4 a n o t h e r t y p e of e x t e n s i o n , b a s e d on i n t e r c o n n e c t i o n s of o b s e r v e r s in t h e form (1.9).

3.2

U n i f o r m O b s e r v a t i o n : the C a s e o f U n i f o r m l y O b s e r v a b l e Systems

W e consider h e r e a s y s t e m of t h e following form:

ic y

= =

A x + ~(x, u), C x e Kt

x C ~:tn, u E1R m (1.10)

0 1 ... 0 / withA=

". ". 0 andC= (1 0... 0 ). 0 ... 0 1 0 ... 0 U n d e r s t r u c t u r e c o n d i t i o n as in (1.7) - e n s u r i n g u n i f o r m o b s e r v a b i l i t y - a n d s o m e L i p s c h i t z c o n d i t i o n on ~, one c a n here d e s i g n a n o b s e r v e r w i t h a g a i n which is u n i f o r m w.r.t, u as recalled h e r e a f t e r ( w h e r e xi - resp. ~ i - d e n o t e s each c o m p o n e n t of x - resp. ~):

Theorem 1.3 [16] If: ,, ~ is globally Lipschitz w.r.t, x, uni.forTnly w.r.t, u; 9 ~xj = - - O , . f o r i = l , . . . n - l , j = i + l , . . . n . then there exists 00 such that for all 0 > 00, the .following system is an asymptotic observer .for (1.10): x 0 andV

> 0,30

= =

A 2 - S - 1 C T ( C 2 - y) + V ( } , u ) -OS- ATs - SA + cTc,

> 0 : Ilk(t) - x(t)ll _
O. []

W e will use this r e s u l t in S e c t i o n 4 t o p r o p o s e a u n i f o r m o b s e r v e r for s o m e n o n u n i f o r m l y o b s e r v a b l e s y s t e m . B u t let us first i l l u s t r a t e t h i s p h e n o m e n o n in n e x t s u b s e c t i o n .


3.3

9

A n E x a m p l e o f U n i f o r m Observation of N o n - u n i f o r m l y observable S y s t e m s

To be able to find an observer which is non uniform w.r.t, the input for a system which is not uniformly observable, as well as a uniform observer for a system which is uniformly observable, is to some extent quite consistent. More impressive are cases of uni.form observers for non uniformly observable systems. An illustrative example of the phenomenon is given by the case of systems of the following form:

y

= =

Ax + f ( x , u ) Cx

(1.12)

for which one can find matrices K, D such t h a t

T=(

C

is invertible, ( K C + D)f(x, u) = cp(Cx, u), and A22 + NA12 is stable, with

( ~-T-~T-IA21 All A22 A12) . Such systems indeed, generally admit singular input (like system (1.2) for instance, for which D = (0 1) and K = 0 satisfy the above conditions, and u = 0 is a singular input), and yet: =

w

(A22 + KA12)z + ( K A u + A21 - (A22 + KA12)K)y + ~(y, u)

=

KC+D

z (1.13)

is an observer for (1.12) irrespective of the input [2]. Such systems in fact enjoy the following structure:

x= / All A21

A22

A

r

u)

' y = Cx = (Ip, O)x (1.14)

and as design Such in the

soon as A makes A22 - AA12 asymptotically stable, then one can an observer in the form (1.13) for (1.14). a property can in particular be used to design robust observers sense that the state estimation does not require the knowledge of

F(x,u). For instance, various manufacturing s y s t e m s admit a representation in the form (1.14), and an example can be found in [28] where the t h e r m a l

10


behaviour of some machine-tool spindle-bearing system is considered. Considering temperatures of the various elements involved as state variables, a numerical realization of the systems reads as follows:

---

+ y

=

(1

-8.000.10 5

2.982.10 -6

0.0421 0

-0.0325 2.483.10 -6

1.407.10 - s 0.008O 2.495.10 .5

) 0(u,x)

0)

0.0104 -1.724.10 -4

(1.15) (1.16)

00)x

where Q(u, x) is some inaccurate model of the friction heat flow. This system is thus in the form (1.14), with the nonlinear part in the form B1

B2 ) F(x,u) and one can check that here A = B_z B1 leaves A 2 2 - AA12 stable. Hence an observer can here be designed with complete decoupling of the uncertain part Q. Some more general conditions for such a design to be possible are given in [2], but a general formulation of the idea can be expressed as follows: P r o p o s i t i o n 1.1 If a nonlinear system:

y

=

f(x,u)

=

h(x)

(1.17)

can be transformed into: Zl

z

f l ( Z 1 , Z2, U)

~2 =

f2(zl,z2,~)

y

Zl

z

(1.18)

by change of coordinates z = O(x), such that .for any couple (u, Zl) of admissible functions and any pair of initial conditions z ~ ~ ~o we have ][X~u,zl)(t, z o) _ X~u,*i)(t, ~0)]] __~ 0 when t ---* ~ , then:

z2 =

f2(y, ~2,~) (1.19)

is an observer for (1.17).


11

This illustrates how an observer can be designed on the basis of some state transformation, here in some particular conditions ensuring detectability. Other results on state transformation for observer design are given in section 5.

4

Observer Interconnection

One way to extend the class of systems for which an observer can be designed is to interconnect observers in order to design an observer for some interconnected system, when possible. If indeed a system is not under a form for which an observer is already available, but can be seen as an interconnection between several subsystems each of which would a d m i t an observer if the states of the other subsystems were known, then a candidate observer for the interconnection of these subsystems is given by interconnecting available sub-observers. Notice t h a t in general, the stability of the interconnected observer is not guaranteed by that of each sub-observer, in the same way as separate designs of observer and controller do not in general result in some stable observer-based controller for nonlinear systems (no separation principle). However, Lyapunov-based sufficient conditions can be given so t h a t the existence of sub-observers results in t h a t of an interconnected observer [7]. Consider for instance the case of systems made of two subsystems of the following form:

it1 Y

:

= =

f l ( X l , X 2 , U ) , u C U C 1Rm; fi C ~ function, i = 1,2; f 2 ( x 2 , x l , u ) , x~ E X~ C ~ , i = 1,2; ( h i ( x 1 ) , h2(x2)) T = (Yl, Y2) T, Y~ E H~n~, i = 1,2. (1.20)

Assume also t h a t u(.) E U c / : o o ( ~ + , U), and set Xi := A C ( ~ + , ~ '~') the space of absolutely continuous function from ~7~+ into ~ n , . Finally, when i E {1, 2}, let ~ denote its complementary index in {1, 2}. The system (1.20) can be seen as the interconnection of two subsystems (Ei) for i = 1, 2 given by: (E~)

2~ = f~(xi,v~,u),

y~ = h~(x~),

(v~,u) e X~ x l~.

(1.21)

Assume t h a t for each system (Ei), one can design an observer (Oi) of the following form:

(Oi)

zi = fi(zi, v~, u) + ki(gi, z i ) ( h i ( z i ) - Yi),

gi = Gi(zi, v~, u, gi), (1.22)

for smooth ki, G~ and (zi, g~) E (~:~n~ x(~i), dT~ positively invariant by (1.22). T h e idea is to look for an observer for (1.20) under the form of the following


12

interconnection:

CO)

~ gi

f~(~, ~ , u) + ki(~, ~ci)(hi(~i) - yi); i = 1, 2; G~(2i, ~ , u, ~i); i -- 1, 2

=

(1.23)

Set ei := z~ - xi, and for any u C/4, v~ E A'i consider the following s y s t e m (where k~"(t) denotes gain ki(g~, zi) defined in (1.22)) :

c(~.... )

~

=

f~(z~,v~,u)+k~

(t)(h~(z~)

h~(z~-e~))

T h e n sufficient conditions for (1.23) to be an observer for (1.20) have b e e n expressed in [7] as follows: 1.4 [7] If.for i = 1,2, any signal u C Lt, v~ E A C ( ~ + , ~ n ~ ) , and any initial value rz0 [ i ,gi0~) E ~n~ x dg~, 3Vi(t, el), Wi(ei) positive de.finite .functions such that: Theorem

(i) Vx~ E X~;Ve~ E z~'~;Vt _> 0, ~

(t, ei) + ~

(t, ei)[fi(xi + ei, v~(t), u(t) ) - fi(xi, v~(t), u(t) ) +k~(t)(h~(xi + e~) - hi(xi))] ~ - W i ( e i )

(ii) 3a~ > 0;Yx~ C X~;Vx~ C ~ ; Y e ~

E ~';Ye~

C ~';Vt

>_ 0,

O~(t, e~)[f~(x~, x~ + e~, u(t)) - f~(x~, x~, u(t))] < ~ i ~ X / / - ~ e ~ ) , (iii) a~ + a2 < 2, then (1.23) is an asymptotic observer for (1.20). [] In the weaker case of cascade interconnection, n a m e l y when f l ( X l , x2, u) ---fl (x~, u) in (1.20), a s s u m p t i o n s can be weakened in the following way: Theorem

1.5 [7] Assume that:

L System :~1 fl(Xl,U); Yl = hi(x~) admits an observer (0~) as in (1.22) (without v2), s.t. Vu ~ Lt and Vxl(t) admissible trajectory of the system associated to u: ~-

lim el(t) = 0 a n d

Ile~(t)lldt < +o~

(with e~ :=

Z 1 -- Xl) ;

t ----~0 ( 3

(1.24)

IX. ~c > 0; Vu e U; Vx~ e X2, Ilf2(x2, xl,u)-f~(x2,x~,u){I

~ cllxl-X~lII,


13

III. Vu ELt, Vvl E A C ( f t +, ITtTM ), Vz2, o g2, o 3v(t, e2), w(e2) positive de.finite .functions s.t for every trajectory of ~(u'vl 2 with z2(O) = z ~ g2(O) = gO: (i) Vx2 E X2, e2 E j~n,2, t ~ O, ov

--~(t, e2) +

(t, e2)[f2(x2 + ee, vl(t),u(t)) - f~(x2,vl(t),u(t)) +k~ 1(t)(h2(x2 + e~) - h2(x2))] < -w(e~)

(ii) re2 E / R n2, t >_ O; v(t, e2) >_ z~(e2) (iii) Ve2 E Kgn2\B(O,r),t >_ O;

~--~eo(t, --o~

0 and B(O,r) := {e2:11~211 _< ~}. Then: X1

fl (:~1, lt) Jr- k 1 (gl, :~l)(hl (Xl) - hi (Xl))

X2

f2( 1,

gl

- h (xl))

u) + k (02,

(1.25)

----

is an observer" .for (1.20) where fl (xl, x2, u) = f l (xl, u). [] In view of these conditions, and using available observers for systems in some particular forms, one might be able to design observers for further nonlinear systems. As an example, one can obtain in this way, and on the basis of observer (1.9) for system (1.8), a non uniform observer for a class of cascade block state affine systems of the following form:

~c

:bl x2

= =

Al(u,y)xl + Bl(u,y) A2(u,y, xl)x2 + B 2 ( u , y , xl)

2Cq Yl

~~-

Aq(u,y, xl,...Xq_l)Xq-]-Bq(U,y, CIZ1

yq

-~-

Cqxq

Xl,...Xq_l)

(1.26)

where xi E ~ n , : yi C zT~V~,u E ff~'~,y = ( y T ...yT)T = Cx and Ai,~i are continuous functions 9 Here the stability of the interconnected observer can only be guaranteed provided the inputs are "rich enough". Denoting by x~(t, xo) the projection of the solution onto llr~TM + ....... which takes components from 1 to nl + . . . ni of x, a~i(t, u, x0) the extended input

cx

(t, t ,x~

,

14


and B(u) the set {x c s II~(t,u,x)ll < c~, t e [ 0 , + ~ [ } of initial conditions generating bounded trajectories with u, we define this "richness" as: D e f i n i t i o n 1.8 Given E C 1l~"~, an input u will be said to be E - r e g u l a r l y persistent .for E~ if.for any compact K of E such that K A 13(u) ~ ~ there exist to > O, (~ > O, T > 0 such that: Vt>_to;VxEKAB(u), Theorem

Vi=ltoq-1,

~/(t,T,w~(t,u,x))>_a.

1.6 [~] Given E C 1Ft'~, assume that:

9 For i = 2 , . . . q, Ai, Wi are globally Lipsehitz w.r.t. ( X l , . . . x i - 1 ) uni.formly w.r.t. (u, y). 9 Input u is E-regularly persistent .for (1.26). 9 x(O) e E n u(u). then .for any ~ > O, there exist 01 > 0 , . . . ,0q > 0 and A > 0 such that the .following system: A1 (u, y)21 + ~ (u, y) - g~-IcT(c~&~ - yl) A2(u, y, ~,)~2 + ~2(u, y, &~) - S ~ c T ( c 2 2 2 - y2)

~E1

A q ( u , y, X l , . . .

Oc

~1

=

- - 0 ~ 1 -- A T ( ~ , y ) g l -02S2

~q

=

, Xq-1)Xq ~- (~:~q(U, y , : E l , . 9 . , : ~ q - 1 )

-

- glAI(~,y)

m~(u,y,2~)S2

-OqSq - AT (u,y,2l,

+ CyC~,

-

SqlCT(Cqxq

-- Yq)

g~(O) > 0

- S2A2(u,y, xl) + cTc2,

. . . ,~q-1)Sq - SqAq(u,y,~l,

g2(0)

>

0

. . . , ~ q - 1 ) + c T Cq,

gq(O) > 0 (1.27)

is an observer.for (1.26), with: ][2(t) - x(t)[] _< Ae -r [] The above design illustrates the case of cascade interconnection. As an example of "full interconnection" of observers, let us consider system of the following form: d:l X2 Yl Y2

= = = =

Alxl-b fl(Xl,U)+gl(xl,x2,u); A2(u)x2 ~ - f 2 ( x l , u) : : r ClXl C C2x2 C1R p

Xl C ]~ TM x2 E ~ n 2

(1.28)

with: (C1) A, C as in (1.10) and f ( x l , u ) Lipschitz assumptions;

satisfying uniform observability and


15

( 6 2 ) g(Xl,X2,U ) : ( 0 , . . . O, g n ( Z l , X 2 , U ) ) T and gn is globally Lipschitz w.r.t, xl (resp. x2), uniformly w.r.t. ( x 2 , u ) (resp. (Xl,U));

(C3) f2 is globally Lipschitz w.r.t, xl, uniformly w . r . t . u . For such a system, one can easily identify two subsystems for which (1.11) and (1.9) are candidate sub-observers, and on the basis of the associated Lyapunov functions [16, 17], one can check t h a t as soon as u is regularly persistent for ~ = A2(u)z, conditions of theorem 1.4 can be satisfied for 01 large enough. This gives an observer of the following form: X1 s x2 0 602

=

= -=

A13Cl q- fl(a?l, u) q- gl(~l, a:2, u) - - S l l C I T ( C l g C l Yl); A2(u):?2 + f2(:rl,U) - $21c2T(c2:c2 - y2); 01S1 -- A1Ts1 -- SIA1 + cT1c1 - 0 2 S 2 - d 2 ( u ) T & - S e A s ( u ) + C~C2; $2(0) > O.

Notice t h a t here the observer gain is non uniform due to the state affine p a r t of the system. But one could imagine a similar case where some uniform gain can be used, provided that detectability is guaranteed. As an example, consider system (1.28) again, now with 6'2 = 0 and p being some function now enjoying the following property: (C3') ~ is globally Lipschitz w.r.t. Xl, uniformly w.r.t. (u, x2) and there exists V positive definite s.t. V(~,e) 9 ~n2 , IIoy ll _< -9211ell and OV e ) + -~-(t, OV e)[~(~ + e , x(t), u ( t ) ) - ~(~, x(t), u(t))] _< -~111ell 2, for -57-(t, every admissible input function u and absolutely continuous function X. T h e n one can again check t h a t under conditions (C1), (C2), (C3'), an observer can be obtained as follows [7]: :~1 x2 0

5

= = :

A l X l q- f l ( X l , U ) q- gl(Xl,3:2, u) - S l l C 1 T ( C I : ~ I - y l ) qp(x2, :~1, u) -01S1 - ATs1 - SIA1 -/- cT1c1 .

(1.29)

S t a t e T r a n s f o r m a t i o n s and O b s e r v e r D e s i g n

One can notice t h a t observer designs presented till now are all based on a particular structure of the system. The subsequent idea is t h a t these designs also give state observers for systems which can be turned into one of these forms by change of state coordinates. We will call equivalent, two systems related by such a relationship: D e f i n i t i o n 1.9 Given xo E 1R n, a s y s t e m described by: 5c = y =

f(x,u)=fu(x)xe~n,u 9 h(x) 9 1Rp

"~ (1.30)

16


will be said to be e q u i v a l e n t at xo to the system: { ~

=

F(z,u)=F~(z)

y

=

H(z)

~f there exists a diffeomorphism z = q~(x) defined on some neighbourhood of xo such that: Vu

e/R m,

0q5 -~xf~(x) I~=o-l(z)= F~(z)

et

h o q5-1 --- H.

The interest of such a relationship for observer design is then motivated by the following proposition: P r o p o s i t i o n 1.2 Given two systems (El) and (E2) respectively defined

by: X(x,u) { ~ = Z(z,u) h(x) and(r2) y = H(z) and equivalent by z = C~(x), If: Z(~,u) + k ( w , g ( # ) (21)

{ ~ y

= =

(o~)

w

=

~-

=

x(:~,~)+

(v

=

F(w,u,y)

is an observer for (E2), Then: (o~)

{

y))

r(w,u,y)

(0o), ~

k, k(~,, h(:~) - y)

is an observer .for (El). This kind of remark has motivated various works on characterizing (rank observable) systems which can be turned into some "canonical form" for observer design, from the linear one up to output injection [23, 8, 24] to several forms of cascade block state afiine systems up to nonlinear injections from block to block [3, 26, 4, 5]. Using the formalism of differential forms [9] e.g. used in [19], we can indeed characterize systems equivalent to "special forms" of (1.26) (a general characterization would further require the use of explicit PDE's in its formulation). With the following notations:

9 d, L z , iz, A to respectively denote usual differentiation, Lie derivative along a vector field Z, inner product with Z and exterior product of differential forms; 9 d~v := dVl A . . . A d v , , and i x v := ( i x v l , . . . i x v ~ )

ifv = (vl,...v~);


17

9 i x ~ : = { i x w , w 9 f~}, Adf~ : = {wlA...Wd,Wi 9 Q}, df~ : = {dw,w 9 f~} if f~ is a set of differential forms; 9 f~ | O to denote the set of finite linear c o m b i n a t i o n s of elements of ft with coefficients in O; 9 i x w : = i x .... i x , . w if w is an (r + 1)-differential form, a n d X ( X 1 , . . . X ~) an r - t u p l e of vector fields;

=

9 (9(y) to d e n o t e the observability subspace of the considered s y s t e m with o u t p u t y, n a m e l y the smallest vectorial subspace of ~ which contains all o u t p u t functions, and is invariant u n d e r Lie derivation along the vector fields of the system, o b t a i n e d w h e n u describes ~ m ; let us privilege systems (E~,...~') of the following form:

A l ( u , y l ) z l + ~ l ( U , y 1) A 2 ( u ' y2, Z l ) Z 2 ~_ qp2 ( u , y 2 , ~.q

=

y

=

~'~1. . .7~q

Aq(u,

Zl )

yq, Z l , . . . Zq--1)Z q ~- r

y q , . . . Zq_l)

(1.31)

Cqzq ;q . U 9 j ~ m ~Zi 9 ~2~n.i y~ 9 H:C~, a s s u m e d to satisfy the following cascade rank observability condition at x0: for any x in some n e i g h b o u r h o o d of x0,

dimdO(yl)(x) difrtdO(y 2) A dnl O(yl )(x) d i m d O ( y q) A d TM O ( y l ) . . .

/~

dnq-lO(yq-1)(X)

~-

n 1

~.

n2

~-

nq

(1.32)

where ni E Pc'* such t h a t ~ i =q 1 ni = n. We will call those integers cascade observability indices. This c o n s t r u c t i o n means t h a t in (1.31), variables zl of each block are exclusively "observed" by o u t p u t hi, as soon as Z l , . . . zi-1 are k n o w n . T h e characterization of such systems will use the following tools: Given n .... Vm,q 9 fV*, V m - t u p l e s of functions y m = (y~n,...y~,~,,) a n d v , ~ - t u p l e s of vector fields X "~, for 1 < m < q, we define: 9 ~ ( y m ) the space such that dTl(y m) A d'""y "~ -= O, (f~x''(y~,.) = Span~{dLi.,,(y jfn, ) A d vl~~y rn, ,u E ~:~nq, ,1 1 and a x ' ( y l ) @ ~_((yl) for m = 1. ?4t

These definitions associate to each block (m) - c o r r e s p o n d i n g to a l] m tuple of o u t p u t s ym _ the set of functions of these o u t p u t s T/(y'~), a space of differential (urn + 1 ) - f o r m s x ' " , a set of functions 6) _Yn ~1 "Y "+~ g a t h e r i n g , ,nTT~, all functions of zl to zm-1 a n d a m o d u l e to characterize the state affine s t r u c t u r e of each block. We can t h e n state: T h e o r e m 1.7 A nonlinear s y s t e m (1.1) cascade observable w.r.t, outputs h = ( y l , . . . y q ) in the sense of (1.32) is equivalent at xo to a s y s t e m I11 . . . t J q (E . . . . ... . ,) descmbed by (1.31) If " a n d o n l y If 9 y i E J~ , ", ( n l , . . . n q ) are cascade observability indices of (1.1) and there exist q Um-tuples of vector fields X m = ( X ] n, X m ] 1 .< r e < q ,. such. that, . .for l < r e < q :

9

9

.

.

l]m.

/

~

1. Lxj,,(y'~') ----0 if j • k and 1 otherwise, .for 1 < j , k < p yl ..,ym, 2. d i m ( f P " ~ ' " (E)) = n,,, - u m on On, ......... I

[ dix,,,~x'"(y ') A A~'-I (A "~-'~ ~x..,ex'(y j) A d~Jyj) = 0 3.

m-1 J dix, " ~x'" ( ym ) A A~:I(A v.,

" ~ - ~ i x. ~ X~ ( yJ ) A d ~ 'y j ) rn--1

nj--uj

.

~'"a~'"(y~) Adh,A A ( A i~,a~"(YJ) Ad~JY~)| k

1=1 q

j=l n , ) --~,:j

~ A(e~"y ~A A "

x~

j=l

This s t a t e m e n t follows previous results of [20], [4] or [6], a n d can be checked by the same kind of arguments: necessity is o b t a i n e d by verifying t h a t conditions I to 4 are indeed satisfied for a s y s t e m of the f o r m (1.31) w i t h X~ ~ = a a n d sufficiency is established by inductively defining new c o o r d i n a t e s OgF" under the form dzj A dzy-1 A ... A dzl = M j ( y j, z j - 1 , . . , z l ) i x J f t Xj (yY) A dZj_l A ... A d z 1 where M can be found on the basis of condition 3 along the same lines as in [6]. T h e p r o b l e m in such a characterization is t o find a p p r o p r i a t e vector fields X i. Let us sketch a constructive procedure giving such vector fields in the case of systems equivalent to (1.31) where each block takes the following


19

form: Zil

~-- A~I (u, z i _ l , Yi)+A~2 (u, _zi-1)zi2

Zi2 Zi3

~-~--

A ~ (u, z_~_~,yi) + A~2(u, z~_~, y~)z~2 + A~a(U, Z~_l)Z~3 Ai31(u,z__i_l , Yi) +A32(u, i i (u, _zi_ 1 , Y i ) Z i 3 Zi-- 1 , Y i ) Z i 2 ~-A33 +A~4(u, z _ i _ l ) Z i 4

Zini_ 1

Ai~-I,I(u, zi-1, Y i ) + A ni~ - l , 2 ( u , z i - 1 , y i ) z i 2 + i Ai.,:-1,3( u, _zi-1, Y i ) Z i 3 Jr- An.~_l,4(u, Zi-1)Zi4

i (~, z~_l)z~, +... + An.,:_l,n.~

z~n~

i Z = An,l(u,_~_l,yi) + A~2(u,Zi_l,yi)zi2+

yi

= Ciz~ = z~l E ~:t

9

A , , , 3 ( u , - i -Z1 ,

Y i ) zi3 -~ A ni ~ 4 ( u , z i - 1 ) z i 4

~- "" ~-

A~,~,

(u Zi_l)Zin,,

(1.33) where ( Z i l , . . . , z~,~,.)T = zi and z~_ 1 = (z~, z T , . . , zT_I) T for i > 2 a n d is e m p t y otherwise 9 Notice t h a t any s y s t e m equivalent to a form (1.31) where A~ (u, z i _ l , yi) = Ai(u, zi_l) - as characterized in [4] - is equivalent to a form where each block has this triangular s t r u c t u r e (1.33). Notice also that, as in the case of state-affine equivalent s y s t e m s where A(u, y) does not d e p e n d on y, e.g. considered in [20], one can c o m p u t e sets of c o n s t a n t control sequences I~ in the form { (uiH, " " " ~tilk)' " " " (~tivil' " . U ..,k } such that: { d L f ~ ( h l ) , k = 0 . . . r l } spans dO(hi) a n d inductively,

{dLf,~(hi) , k = O. ". r~}AAi~_~dO(hl ) spansdO(hi) A Al=l i-1 dO(hi) (where hi denotes the o u t p u t function for Yi and Lfq: (hi) is the vector of c o m p o nents LI, h ... Lf,,t (hi))9 On this basis, one can inductively c o m p u t e candidates for X 1 to X q on the s a m e p a t t e r n , given hereafter for X I : 9 C o m p u t e Y (uniquely) defined by:

gy(hl) L r L L } (hi)

=

0,

=

0 if j % T1, 1 otherwise;

LyLL~(hz)

=

0

and forj=l

forj=l,..rt,

top

(1.34) (1.35)

l=2,...q.

9 B y successive Lie Brackets, c o m p u t e Y,~....... 3 := [fv~,-,,... [f-~, Y ] . . . ] a n d Y~...... , := [f,,,, Y,~,_~ .... ~] for some c o n s t a n t vi's, a n d set: Lf,,, (hi) Z := Yv,,...Vl + - 9 Check: d L z ( h l ) =- 0; L z ( h l ) ~ 0 and finally set: X 1 .--

1

Lz(hl)

Z.

20


One can check by inspection that such a construction necessarily gives a candidate for X 1. The same construction can be used to find X 2 to X n and finally, verification of conditions of Theorem 1.7 reduces to differentiations and tests of linear dependencies. In this way, the particular structure (1.33) can be fully intrinsically characterized, as this is done for several cases in [6].

6

Conclusions

The purpose of this chapter was to draw some lines of recent advances in the problem of observer design for nonlinear systems and highlight several further directions of research. In particular, the problem of the input has been underlined for the observability properties of the systems, and several aspects of observer designs based on interconnection of sub-observers as well as state transformations have been discussed. In terms of the technique used for the design, obviously further methods can be thought of, for instance including optimization [25], sliding modes [27] etc.

Acknowledgement The author would like to thank Professors Hassan Hammouri and Guy Bornard for having awoken and fed his interest in nonlinear observers.

7

REFERENCES [1] G. Besan~on. Contributions d l'Etude et i~ l'Observation des Syst~mes Non Lin~aires avec Recours au Calcul Formel. PhD thesis, Institut National Polytechnique de Grenoble, 1996. Laboratoire d'Automatique de Grenoble. [2] G. Besan~on and H. Hammouri, "On uniform observation of nonuniformly observable systems," Systems ~4 Control Letters, vol. 33, no. 1, pp. 1-11, 1996.

[3]

G. Besanqon and G. Bornard. "A condition for cascade time-varying linearization," in IFA C Proc., Nonlinear Control Systems Design Symposium, Tahoe City, CA, USA, pp. 684-689, 1995.

[4] G. Besan~on, G. Bornard, and H. Hammouri. "Observer synthesis for a class of nonlinear control systems," Europ. Journal of Control, vol. 3, no. 1, pp. 176-193, 1996.


21

[5] G. Besanqon and G. Bornard. "State equivalence based observer design for nonlinear control systems," in Proc. IFAC World Congress, San Francisco, CA, USA, pp. 287-292, 1996. [6] G. Besan~on and G. Bornard. "On characterizing classes of observer forms for nonlinear systems," in Proc. ~th European Control Conf., Br~tssels, Belgium, 1997. [7] G. Besan~on and H. Hammouri. "On observer design for interconnected systems," Journal o.f Mathematical Systems, Estimation, Control, vol. 8, no. 3, 1998. [8] D. Bestle and M. Zeitz. "Canonical form observer design for nonlinear time-variable systems," Int. Journal of Control, vol. 38, no. 2, pp. 419431, 1983. [9] W. M. Boothby. An Introduction to Differentiable Manifolds and Riemannian Geometry. Academic Press, New York, 1975. [10] G. Bornard, N. Couenne and F. Celle. "Regularly persistent observer for bilinear systems," in Proc. of the Colloque International en Automatique Non Lin~aire, Nantes, June 1988. [11] G. Bornard, F. Celle-Couenne and G. Gilles. "Observability and observers," in Nonlinear Systems - T.1, Modeling and Estimation, pp. 173 216, Chapman & Hall, London, 1995. [12] K. Busawon. Sur les Observateurs pour des Syst~mes Non Lin~aires et le Principe de S~paration. PhD thesis, Universit~ Claude Bernard, Lyon I, 1996. [13] D. Dawson, Z. Qu and J. Carroll. "On the state observation and output feedback problems for nonlinear uncertain systems," Systems Control Letters, vol. 18, pp. 217-222, 1992. [14] J.P. Gauthier and G. Bornard. "Observability for any u(t) of a class of nonlinear systems," IEEE Trans. on Automatic Control, vol. 26, no. 4, pp. 922-926, 1981. [15] J.P. Gauthier and A. Kupka. "Observability and observers for nonlinear systems," Siam Journal on Control and Optimization, vol. 32, no. 4, pp. 975-994, 1994. [16] J.P. Gauthier, H. Hammouri and S. Othman. "A simple observer for nonlinear systems - applications to bioreactors," IEEE Trans. on Automatic Control, vol. 37, no. 6, pp. 875-880, 1992. [17] H. Hammouri and J. D. L. Morales. "Observer synthesis for stateaffine systems," in Proc. 29th IEEE Conf. on Decision and Control, Honolulu, Hawaii, pp. 784-785, 1990.

22


[18] H. Hammouri and F. Celle. "Some results about nonlinear systems equivalence for the observer synthesis," in New Trends in Systems Theory, pp. 332-339, Birkh~user, 1991. [19] H. Hammouri and J.P. Gauthier. "Bilinearization up to output injection," Systems ~ Control Letters, vol. 11, pp. 139-149, 1988. [20] H. Hammouri and M. Kinnaert. "A new formulation for time-varying linearization up to output injection," Systems ~ Control Letters, vol. 28, pp. 151-157, 1996. [21] S. Hara and K. Furuta. "Minimal order state observers for bilinear systems," Int. Journal of Control, vol. 24, no. 5, pp. 705-718, 1976. [22] R. Hermann and A. Krener. "Nonlinear controllability and observability," IEEE Trans. on Automatic Control, vol. 22, no. 5, pp. 728-740, 1977. [23] A. J. Krener and A. Isidori. "Linearization by output injection and nonlinear observers," Systems ~ Control Letters, vol. 3, pp. 47-52, 1983. [24] A. J. Krener and W. Respondek. "Nonlinear observers with linearizable error dynamics," Siam Journal on Control and Optimization, vol. 23, no. 2, pp. 197 216, 1985. [25] H. Michalska and D. Mayne. "Moving horizon observers," in IFAC Proc., Nonlinear Control Systems Design Symposium, Bordeaux, France, pp. 576 581, June 1992. [26] J. Rudolph and M. Zeitz. "A block triangular nonlinear observer normal form," Systems ~ Control Letters, vol. 23, pp. 1-8, 1994. [27] J. J. E. Slotine, J. Hedrick and E. Misawa. "On sliding observers for nonlinear systems," Journal of Dynamic Systems, Measurements, and Control, vol. 109, pp. 245 252, 1987. [28] J. Tu and J. Stein. "Model error compensation and robust observer design - part 2: Bearing temperature and preload estimation," in Proc. American Control Conference, Baltimore, Mawland, USA, pp. 33083312, June 1994. [29] B. Walcott and S. Zak. "State observation of nonlinear uncertain dynainical systems," IEEE Trans. on Automatic Control, vol. 32, no. 2, pp. 166-170, 1987.

M o d e l - B a s e d Observers for T i r e / R o a d C o n t a c t Friction Prediction Carlos Canudas de Wit 1, Roberto Horowitz 2 and P. Tsiotras 3 1Laboratoire d ' A u t o m a t i q u e de Grenoble, U M R CNRS 5528 E N S I E G - I N P G , ST. Martin d'H~res, France. 2Department of Mechanical Engineering, University of California Berkeley, CA 94720-1740, U.S.A. 3Georgia Institute of Techology, School of Aerospace Eng. Atlanta, Georgia 30332-0150, U.S.A.

1

Introduction

This contribution is devoted to the problem of tire-road friction estimation. The need for such type of studies, steers from the difficulty of direct sensing of tire forces, slip, slip angles and other external factors. Observer algorithms are, in this context, a low cost alternative for sensors. Tire forces information is relevant to problems like: optimization of Anti-look brake systems (ABS), traction system, diagnostic of the road friction conditions, etc. Literature for tire/road friction estimation is numerous. Bakker et al [1] and Burckhardt [4] describe two analytical models for t i r e / r o a d behavior t h a t are intensively used by researchers in the field. In these two models the coefficient of friction, #, or more precisely, the normalized friction force, i.e. F Friction force # - F~ Normal force is mainly determined based on the wheel slip s and some other p a r a m e t e r s like speed and normal load. Fig. 1 shows two curves, obtained from H a r n e d et al [9], t h a t represent typical # versus s behavior. It is current practice to n a m e the ratio between the friction and the normal forces, #, as being the "coefficient" of friction. Under constant normal force conditions, #, is a constant if and only if the Coulomb model is used to describe friction. Nevertheless, the Coulomb model is too simplistic to suitable represent forces between the rubber tire and the road, which are dominated by the elesto-plastic force/displacement characteristics. Therefore, to consiser it as a constant is a pure idealistic view. p should thus

24

2. Model-Based Observers for Tire/Road Contact Friction Prediction Relali~ship ol ~

andi

Relallonship of ~ and i !

a~pha,'t

,

~

.

.

.

o~

.........

~

o~

. . . .

.

~M~

.......

:4o MP~

:

L ~ e g~avel .......

i

......

;

.

.

.

.

.

~o

_I o~1 i

0.1

02

03

04

05

06

LonOIttdnal sl~

07

08

09

1

-- ol

ol2

0'3

o'4

o'.5 or6 Long#u~naL slW

ol7

018

ol9

FIGURE 1. a) Variations between coefficient of road adhesion # and longitudinal slip s for different road surface conditions (left). b) Variations between coefficient of road adhesion # and longitudinal slip s for different vehicle velocities (right). be viewed more as the ratio between friction and normal forces (i.e. the normalized force), which is indeed a (static or dynamic) function of the system state variables. The expression given by Bakker et al [1], and Paceijka and Sharp [14], also known as "magic formula" is derived heuristically from experimental d a t a to produce a good fit. It provides the tire/road coefficient of friction # as a function of the slip s. The expression in Burckhardt [4] is derived with a similar methodology. The final m a p expresses # as a function of s, the vehicle velocity, v and the normal load on the tire F,~. Kiencke [10] presents a procedure for real-time estimation of #. A simplification to the analytical model by Burckhardt [4] is introduced in such a way that the relation between # and s is linear in the parameters. Kiencke [10] uses a two stages identification algorithm. In the first stage, the value of # is estimated. This estimate of # is used in the second stage to obtain the parameters for the simplified # versus s curve. The paper by Gustafsson [8] derives an scheme to identify different classes of roads. He assumes t h a t by combining the slip and the initial slope of the # versus s curve it is possible to distinguish between different road surfaces. The author tests for asphalt, wet asphalt, snow and ice and identifies the actual value of the slope with a K a h n a n filter and a least square algorithm. Ray [16] estimates # based on a different approach. Instead of using the slip information to derive a characteristic curve, Ray [16] estimates the forces on the tires with an extended K a l m a n filter. Using a tire model introduced by Szostak et al [17], that expresses the tire forces as a function of #, the author tries this model for different values of #. A Bayesian approach is used to determine the value of # that is most likely to produce the forces estimated with the extended K a l m a n filter. The works of Kiencke [10], Gustafsson [8], and Ray [16] do not consider any velocity dependence in the derivation of #, as suggested by B u r c k h a r d t

2. Model-Based Observers for Tire/Road Contact Friction Prediction

25

[4] and Harned et al [9]. An a t t e m p t to consider the velocity dependence for ABS control is presented in Liu and Sun [13]. The authors assume the tire/road characteristics to be known. Due to the limitations in the available data, the authors are not able to compare their algorithm with other methods. There are other works related to the on line identification of the t i r e / r o a d friction, as for example Lee and Tomizuka [12], and Yi and Jeong [18]. However, in these papers only the instantaneous coefficient of friction is identified. The coefficient of t i r e / r o a d friction, or coefficient of road adhesion, # is mainly a function of the longitudinal slip, the velocity of the vehicle and the normal load. The estimators proposed in the literature depends very much on the type of used models, and verification of the hypothesis used for the model derivation. As shown by the figures above, the relation of the curves # - s , depends very much on system operating conditions, such as the vehicle velocity. It is clear t h a t p a r a m e t e r s describing a curve like the one in Fig .1-(a), will not be invariant, as shown in Fig .l-(b). It is thus interesting to introduce models described by parameters t h a t are more likely to be invariant and have physical significance. Theory never exactly matches reality, but some times closely resembles it. To achieve this goal, we propose in this paper to use a dynamical t i r e / r o a d friction model, together with a nonlinear observer specifically designed for this application. This paper is organized as follows: The next section reviews some of the existing tire/road friction models, and also introduces lumped and distributed dynamic representations. In Section 3 we set-up the observation problem, using the particular case of a one-wheel system with lumped contact friction. Inspired from previous works by Canudas-deWit and Lischinsky [6] on adaptive friction estimation and compensation, Section 4 presents a general framework for the design of nonlinear observers for the on-line estimation of the road conditions. In Section 5 we apply this design to the case study case set in Section 3. Finally, Section 6 presents simulation results.

2

Tire-road Friction Models

This section reviews some friction models t h a t can be used for the study of the on-line identification of the friction force (or coefficient, if we consider normalized force). We first present the some of the pseudo steady-state models proposed in the literature, then we discuss some alternative dynamic (lumped and a distributed) models. T h e sep up for this study is the simple case of an one-wheel model with tire-road contact friction, shown schematically in Fig. 2. In this s t u d y we

26

2. Model-Based Observers for Tire/Road Contact Friction Prediction Wheel with

Wheel with

lumped friction F

distributed friction F

or

p

F I G U R E 2. One-wheel system with: lumped friction (left), distributed friction (right) will thus consider a s y s t e m of the form m~

=

F

J&

=

-rF

(2.1) + u,- - cr~w ,

(2.2)

where: m - wheel mass, J

wheel inertia,

r - wheel radius, v

linear velocity,

w - angular velocity, u~ - b r a k i n g / d r i v i n g torque, F

t i r e / r o a d friction force.

Therefore, only longitudinal m o t i o n (longitudinal slip) will be considered. 2.1

Pseudo-Steady

State

Models

This t y p e of models are currently used in the literature. T h e y are defined as one-to-one ( m e m o r y less) maps between the friction F , a n d the longitudinal slip r a t e s, defined as: s=

.... r ~r e -d v

if if

v > rw, v # 0 v 0. From the Kalman-Yakubovich-Popov lemma, we thus ensure with this choice of K that there exist P satisfying the L y a p u n o v equation with P B = C. C o n d i t i o n A4 ( P e r s i s t e n c e o f e x c i t a t i o n ) . To ensure p a r a m e t e r convergence we need to guarantee t h a t lim

t ----*oo

~(y(t), u(t), x(t))

=

lim

olry(t)

-

v(t)l z(t) r o

t --* oo

This implies that the relative velocity should not tend to zero in order for the estimated p a r a m e t e r to converge. This in turn implies t h a t the internal friction state z(t) will not asymptotically converge to zero. Finally, we have,


39

2.2 Consider the one-wheel model with lumped dynamic .friction (2.~9)- (2.51), then the .following observer:

Theorem

=

_0-0~ + (j0-0 _ 0-~)w + u~ O"1

=

0

=

~) =

(2.54)

0-1

0-01r

~(~-

v) -

0-01r g(vr) -

vl

rFn0-1~),

_

-

(2.55)

(2.56)

(2.57)

with positive nonzero k, and % ensures that all the estimated states are bounded, and that: lim ~ ----X

(2.58)

lim 5 = z.

(2.59)

t ---* ~:) t ----*oO

If in addition, the relative contact velocity does not vanishes, then we also have that lira 0 = 0 . t----*OO

5.1

Simulation Results

Simulations have been performed with the one-wheel system and the l u m p e d LuGre model. The friction parameters used in the simulations are the ones given in Table 1, with the following additional values for the wheel: r -25[cm], m = 5[Kg], J = 0 . 7 5 . m * r 2 = 0.2344[Kgm2], F,~ = 14[Kgm2/s2]. Fig. 5, shows simulation results. Fig. 5-(a) shows the time-profile of the contact friction force resulting form the application of the time torque profile u~(t) shown in Fig. 5-(e). T h e simulation has first an acceleration phase, and then a breaking phase. From Fig. 5-(a), we can see t h a t a b o u t 2 seconds are needed for the friction torque to reach its m a x i m u m value. The observation error of the X, and z is shown in Fig. 5-(d). According to the theorem these two variables should converge to their true values regardless the profile evolution of the system states. This is verified by this curve showing the exponential convergence of the II(~(t), 5)11 to zero. Since the ultimate goal of this work is to be able to on-line estimate this variation, the simulation was done under variations of the p a r a m e t e r 0, representing the road variation conditions (see Fig. 4). Fig. 5-(b) shows in bold lines the value of 0, which evolves within fourth different conditions: the first quarter of the simulation corresponds to dry asphalt conditions. The second quarter corresponds to a sudden change from dry to wet. During the third quarter, there is a smooth variation from wet to snow. T h e last

40

2. Model-Based Observers for Tire/Road Contact Friction Prediction Contact friction force

Estimation of theta 4.~

0.=

0.,

3.~

o~ 0.,

WET 2.~

o ~'-o.

12

"~ -OA

l

-o,E

O~

-o.e -1

DRY

0 ,

2

4

6

8 tO 12 Time [secondsJ

14

16

18

20

-O.5

2

4

6

Applied torque

'

1'0

1'2 2

Time [seconds]

8

10

12

Time [seconds I

14

16

18

20

1'8

2o

Observer error norm

1'4 ' '

1'6 '

1'8 '

0

~

~

;

1'0

,'2

Time [seconds]

~'4

l's

FIGURE 5. a) Contact torque friction F(t) (up left), b)Estimated parameter 0(t), and evolution of 0 (up right), c) Applied wheel torque uT(t) (low left), d)Observe error norm of ()~(t), ~) (low right). quarter is keep constant at the snow conditions. In dotted lines we can see the evolution of the estimate t~(t). As we can observe, a good parameter tracking is obtained, as long as the relative contact velocity is different from zero. During the small time-period when this velocity is small or zero, the adaptation law yields a constant 0(t).

6

Conclusions

We have presented a m e t h o d to estimate on-line the changes in road condition. To achieve this goal we have introduced dynamical friction models that, one hand provide a more accurate description of the contact friction, and one the other hand, allow us to characterize road condition variations via a single parameter. It has been shown that the distributed parameter version of these m o d e l also capture stationary shape profiles between normalized friction and slip rate that are similar to the ones obtained from experimental data (i.e. magic formula).


41

We have introduced a model-base observer that ensure asymptotic tracking of road condition, under mild conditions implying a non-vanishing evolution of the slip rate. This condition are quite natural in this context (they imply that the vehicle should operate away to the ideal pure rolling condition). Mathematically, this condition correspond to the persistently excitation condition, which is well known in the adaptive control literature. In the context of nonlinear observers, this condition appear as being the characterization of "good " inputs, which are required to recover state observability. The observer presented here has been derived in a general framework allowing to extend our study to the case where the vehicle velocity is not measurable. In particular, assumption A2 - (ii) will allows for this extension, if it can be shown that the assumption A3, also holds. This study and the introduction of other factors like: wheel vertical deformation, and suspension dynamics, are currently under study.

Acknowledgements The LuGre version of the dynamic friction model presented here, was derived during the first author visit at the Department of Aeronautics at the Georgia Institute of Technology ( C N R S / N S F collaboration project). A more complete report on this topic is in preparation. The first author would like also to thanks M. Sorin and P.A. Bliman for the interesting discussion on distributed friction models. 7

REFERENCES [1] E. Bakker, L. Nyborg and H. Pacejka. Tyre Modelling for Use in Vehicle Dynamic Studies. Society of Automotive Engineers Paper 4P 870421, 1987. [2] P. A. Bliman, T. Bonald and M. Sorine. Hysteresis Operators and tire Friction Models: Application to vehicle dynamic Simulator. Prof. of ICIAM. 95, Hamburg, Germany, 3-7 July, 1995. [3] M. Burckhardt. ABS und ASR, Sicherheitsrelevantes, RadschlupfRegel System, Lecture Scripture. University of Braunschweig, Germany, 1987. [4] M. Burckhardt. Fahrnverktechnik: Radschlupfregelsysteme. Verlag, Germany, 1993.

Vogel-

[5] C. Canudas de Wit, H. Olsson, K. J..~strOm and P. Lischinsky. A New Model for Control of Systems with Friction, IEEE TAC, Vol. 40, No. 3, pp.419-425, March 1995.

42


[6] C. Canudas de Wit and P. Lischinsky. Adaptive friction compensation with partially known dynamic friction model, International Journal o.f Adaptive Control and Signal Processing, Vol. 11, pp.65-85, 1997. [7] P. R. Dahl. Solid Frictioin Damping of Mechanical Vibrations. AIAA Journal, 14, No. 12, pp.1675-1682, 1997. [8] F. Gustafsson. Slip-based Tire-road Friction Estimation. Automatica, 33(6):1087-1099, 1997. [9] J. Harned, L. Johnston and G. Scharpf. Measurement of Tire Brake Force Characteristics as Related to Wheel Slip (Antilock) Control System Design. SAE Transactions, 78(690214):909-25, 1969. [10] U. Kiencke. Realtime Estimation of Adhesion Characteristic Between Tyres and Road. In Proceedings o.f the IFA C World Congress, volume 1, 1993. [11] U. Kiencke and A. Daiss. Estimation of Tyre Friction for Enhaced ABS-Systems. In Proceedings of the AVEG'9~, 1994. [12] H. Lee and M. Tomizuka. Adaptive Traction Control. PATH Technical Report UCB-ITS-PRR-95-32, Institute of Transportation Studies, University of California at Berkeley, 1995. [13] Y. Liu and J. Sun. Target Slip Tracking Using Gain-Scheduling for Antilock Braking Systems. In The American Control Conference, pages 1178-82, Seattle, Washington, 1995. [14] H. B. Pacejka and R. S. Sharp. Shear Force Developments by Psneumatic tires in Steady-state conditions: A review of Modeling Aspects.. Vehicle Systems Dynamics, Vol. 20, pp.121-176, 1991. [15] W. R. Pasterkamp and H. B. Pacejka. The Tire as a Sensor to Estimate Friction. Vehicle Systems Dynamics, Vol. 29,(1997) pp.409-422, 1997. [16] L. R. Ray. Nonlinear Tire Force Estimation and Road Friction Identification: Simulation and Experiments. Automatica, 33(10):1819-1833, 1997. [17] H. T. Szostak,R. W. Allen and T. J. Rosenthal. Analytical Modeling of Driver Response in Crash Avoidance Manuevering. Volume II: An Interactive Tire Model for Driver/Vehicle Simulation. Report no. DOT HS 807-271, U.S. Department of Transportation, 1988. [18] K. Yi and T. Jeong. Observer Based Estimation of Tire-road Friction for Collision Warning Algorithm Adaptation. JSME International Journal, 41(1):116-124, 1998.

Observer Design for Nonlinear Oscillatory Systems Dag Kristiansen and Olav Egeland D e p a r t m e n t of Engineering Cybernetics Norwegian University of Science and Technology Trondheim, Norway

1

Introduction

Numerous vibration phenomena which are theoretically interesting as well as practically i m p o r t a n t can only be understood on the basis of nonlinear vibrations. For instance, the wide field of self-excited, p a r a m e t r i c and auto-parametric vibration demands nonlinear t r e a t m e n t from the very beginning. The sources of the nonlinearities m a y be either geometric, inertiM, material, damping or a combination of these things. Nonlinearities bring a whole range of p h e n o m e n a that are not found in linear systems. In single-degree-of-freedom systems these p h e n o m e n a include multiple solutions, jumps, limit cycles, natural frequency shift, subharmonic and superharmonic resonances, period-multiplying bifurcations, and chaotic motions [12]. Large excitation levels are usually needed to produce periodmultiplying bifurcations and chaotic motions in single-degree-of-freedom systems. In addition to the above mentioned phenomena, the response of nonlinear multi-degree-of-freedom systems can exhibit combinations resonances and modal interactions. T h e latter m a y provide a coupling or an energy exchange between the system's modes and arises if there exists a special relationship between two or more natural frequencies of the linear modes and an excitation frequency. This means that the long-time responses of the system can contain significant contributions in m a n y modes of vibration. The presence of significant responses in more t h a n one m o d e increases the number of modal equations t h a t must be analyzed, and this generally serves to complicate the dynamics of the system. More importantly, modal interactions can lead to dangerously large responses in modes t h a t are predicted by linear analysis to have insignificant response amplitudes. T h e extent of the interaction and its conditions depend on the linear natural eigenfrequencies wi and the nonlinearities of the system. More precisely, autoparametric resonances in systems with n linear natural frequencies (wl,. 99 , w,~) and n corresponding modes (the eigenfrequencies are assumed to real and

44

3. Observer Design for Nonlinear Oscillatory Systems

nonzero) occur whenever two or more eigenfrequencies are c o m m e n s u r a b l e or n e a r l y c o m m e n s u r a b l e (see e.g. [12, 1]). If a harmonic external excitation of frequency f~ acts on a multi-degree-of-freedom system, then in addition to all primary and secondary resonances ( r f ~ ~ s w i , where r and s being integers) of a single-degree-of-freedom system, there might exist other resonant combinations of the frequencies in the form r f ~ ~ S l W l + 9 9 9 + S n W n , n where r and si are integers such that r + ~ i = 1 Isi[ = N, where N is the order of the nonlinearity plus one and n is the number of degrees of freedom. This means for multidegree-of-freedom systems with cubic nonlinearities, to the first approximation, combination resonances may occur

if a

I•

•

a

1•

or a

I•

If

quadratic nonlinearities are added, additional combination resonances m a y occur if ~t ~ [+win + wk[. Thus, a high-frequency excitation may produce large amplitude responses in low-frequency modes that are involved in the combination resonance and vice versa. Interestingly, the concept of modal interactions can also be utilized in control design, see e.g. [14, 13, 3]. In this chapter we will focus on designing full-state nonlinear observers for systems where i n t e r n a l r e s o n a n c e is present. We will assume that we do not have any measurements of the velocities and also that we cannot measure each position separately. Direct applications include e.g. cylinder gyroscopes [4]. As an analysis-tool, we shall use the concept of contraction theory [10]. A short review of this concept is given in Section 2.

2

Contraction Theory

In connection with the observer design, c o n t r a c t i o n t h e o r y will play an important role in the analysis. Here we will give a short review of the theory known as contraction theory which was proposed by [10]. The results are based on ideas from fluid mechanics and tools from differential geometry. The basic idea is to view the system differential equations as an n-dimensional "fluid-flow" described by Euler coordinates. By calculating the squared distance between two trajectories in the "flow-field" one ends up with a concept called contraction region. The interested reader is referred to [10] and the references therein for more on this subject. It is also worth mentioning that a thoroughly mathematical treatment of similar ideas can be found in [2]. Given the nonlinear, non-autonomous system • = f (x, t)

(3.1)

where x C ~n and f :~n • I1~+ ~ R n is assumed to be sufficiently smooth. This equation can be written differentially as 5/r

0 f (x, t) 6x 7xx

(3.2)


45

where 5x is a virtual displacement. The squared distance between two neighboring trajectories can be defined as 5xT6x, which means that the rate of change is given by s dt

= 25x 5

=

(3.3)

Let Area• (x, t) denote the largest eigenvalue of 89( o f + 0,, of T~/ , then

II xll < II xolleJ

....

(3.4)

If )~max (X, t) is uniformly strictly negative, (3.4) shows that [lSx[[ converges exponentially to zero. This implies by path integration that the length of any finite path converges exponentially to zero. Now consider the differential coordinate transformation

6z = o (x, t) 5x

(3.5)

where @ (x, t) is a square matrix. Then a generalization of the squared length is

5zTSz ----5 x T M (x, t) 5x

(3.6)

where M (x, t) = O T (X, t) O (X, t) represents a symmetric and continuously differentiable metric. If M (x, t) is uniformly positive definite, exponential convergence of 5z to zero implies exponential convergence of 5x to zero. We also have that d S z = FSz, where F = ((~ + O ~- 1 5 -O ~ ] , and we can state the following definition and theorem: D e f i n i t i o n 3.1 ([10]) Given the system equations • = f ( x , t ) , a region of the state space is called a contraction region with respect to a uniformly positive de.finite metric M (x, t) = O T (x, t) O (x, t) /f F is uniformly negative definite in that region. Regions where F is negative semi-definite are called semi-contracting, and regions where F is skew-symmetric are called indifferent. T h e o r e m 3.1 ([10]) Given the system equations • = f (x, t), any trajectory which starts in a ball of constant radius with respect to the metric M (x, t), centered at a given trajectory and contained at all times in a contraction region with respect to M (x, t), remains in that ball and converges exponentially to this trajectory. Furthermore global exponential convergence to the given trajectory is guaranteed of the whole state space is a contraction region with respect to the metric M (x, t).

46


R e m a r k 3.1 ([7]) Note that V = 0 T O-~x can be written in terms of Christoffel symbol of the first kind, i.e., [11]: 1 {OMm

rh k =

OMlh Ox k

\-a-7

OMhk ) Ox t

+ - -

(3.7)

Since Fhlk = Fklh [11], we have that Fmkahb k = Fklhakb h = Fmkbha k

(3.8)

where a k is the k-th component of the vector a and bh is the h-th component of the vector b. This means that.for autonomous 0 (x), we can analyze the i.e., dynamics ~ = O• in place of F = O + Ox J

O0f'~O-1,

d (SzTSz) = 25z TS~ dt

(3.9)

since 62 = 5 (O•

=

5x~ + O

0• 25zT 5i = 25xT OT (\ 0OX

5x = -~-x•

+ O~xx~X

f ' ~/I 6X =--~ d (SzrSz) q- o O'~X

(3.10)

(3.11)

A last result which will be used in the observer design is the following [10]: Consider a smooth virtual dynamics of the form

XQ 5Zl ) dt

5z2

(Fll =

F21

0 F22

)(SZl)

(3.12)

5z2

and assume that F21 is bounded. Exponential convergence of 5zl can be concluded for uniformly negative definite F n . Also, if F22 is uniformly negative definite, this implies exponential convergence of the whole system to a single trajectory since F215zl represents an exponentially decaying disturbance in the second equation. We can think of the dynamics of 5zl as the plant, and the dynamics of 5z2 as the observer. By designing the observer such that the system trajectory is contained in the "flow-field" of the observer, this means that the observer is exponential convergent if F u and F22 are uniformly negative definite, and F2a is bounded.

3 System Equations We will assume that our system is given as n nonlinearly coupled oscillators with constant mass M =diag{mi} > 0 and linear viscous damping C =diag{# d > 0, i.e. [12],


M~

=

Cl --

v

y

~qi

=

OV

-Cv

-~q + F

47

(3.13) (3.14) (3.15)

i

where F is an external forcing and V = V (q) will decide w h a t t y p e s of nonlinearities which are present in the s y s t e m (e.g. quadratic, cubic or both). More precisely, V can be w r i t t e n as V = V1 + V2 where V1 is due to the linear spring constants a n d is a s s u m e d to be positive, and V2 reflects the nonlinear coupling terms. Note t h a t our m e a s u r e m e n t y, given by (3.15), m a k e s this p r o b l e m in some sense different from e.g. robotics where one usually can m e a s u r e each position separately. We will a s s u m e t h a t t h e s y s t e m p a r a m e t e r s (mass, d a m p i n g etc.) are known.

3.1

Analysis

C o n t r a c t i o n analysis of mechanical s y s t e m s in H a m i l t o n i a n f o r m were inv e s t i g a t e d in [9, 8, 6], while s y s t e m s in L a g r a n g i a n form were considered in [7]. Here we will give an alternative analysis, which is a direct consequence of energy considerations. We will a s s u m e t h a t 1. V can be w r i t t e n as V = q T p (q) q where P is positive definite. 2. ~

can be w r i t t e n as ~ q = K ( q ) q where K is a square m a t r i x .

Now (3.13) can be w r i t t e n differentially as ( F = 0) Mdv = -Cdq-

OVdt

(3.16)

0q

Using -~q = K (q) q, (3.16) can be w r i t t e n as Mdv = -Cdq

- K (q)

qdt

(3.17)

Since P > 0, there exists a m a t r i x W (q) such t h a t W T (q) W (q) = P (q)

(3.18)

Define

0

x/2W (q)

q

a n d introduce q~ = q

(3.20)

48

3. Observer Design for Nonlinear Oscillatory Systems then dz

~Z

0

~

o de ) v/2W(q) ) ( dq

(3.21)

)

(3.22)

0

( v/-~ 0

Also from (3.19) (3.23) Then, using (3.9) dt =

( 5qTv/-M

V~5r

T (q))

v~o(~(qq/q)Sq

_-- --~qTC5 q - 5qT K (q) 5r +25r

T (q) 0 (W (q) q) 5q 0q

(3.24)

Note that OV m

0q

z

0 (qTWT (q) W (q) q) 0 ( w (q) q ) T w (q) q = K (q) q =2 cOq Oq (3.25)

i.e., KT (q) = 2~vvT (q) 0 (W0q(q) q)

(3.26)

such that d--~

$zTSz

= --SqTCSq

(3.27)

which means that the "flow-field" is semi-contracting. Bounded 5q and ~r and by assuming that K (q) is bounded, leads to bounded 5v (using MSv = - C S q - K (q) 50). Assuming bounded ~ means that 5~ is bounded since MS~ = - C 5 v - ~ 5q. This means that 5q and 5v c o n verges asymptotically to zero.


49

R e m a r k 3.2 As in [7], the above analysis can be regarded as a generaliza-

tion of the energy conservation since d ( 2 6 q T M 6 q + 5r

(q) W (q)

--SqTCSq

(3.28)

d ( l dqTMdq + dCTwT (q) w (q)

-dqTCdq

(3.29)

Multiplying with 7-iv. 1 . d ( 1 v T M v + q T w T (q) W (q) q )

h7 4

=

--vTCv

(3.30)

--vTCv

(3.31)

Observer Design

Since we do not have any measurements of the velocities, we will take advantage of the following result due to [5]: Given the system = f (x, t)

(3.32)

with measurement y = h (x)

(3.33)

and the following general observer X

:?

=

g (~, y, t)

(3.34)

h (~)

(3.35)

where g, h are assumed to be smooth functions. We can state the following result: P r o p o s i t i o n 3.1 ([5]) Given a smooth coordinate transformation of the observer dynamics ~ = N (R, ~), where .for each ~, the mapping ~ ~-~ ~ is in-

vertible, and given the n-dimensional system equations and m-dimensional measurements ~r =

f(x,t)

(3.36)

y

h(x)

(3.37)

=

50


then the observer equations x

=

g(~,y,t)

:9

=

h (R)

(3.38) (3.39)

transform to (3.40)

:~ = g (:~, y, t) + 7a y- (~' - 5')

i.f _

0R (:~ (.~, y), y) g (:~ (R, y) y, t) OR O~ (:~ (R, y ) , y ) O-~Oh(:~ (~, y)) g (Yc (~, y ) , y, t) +~yy

(3.41)

is integrated instead of (3. 38) and (3. 39). T h e proof can be found in [5]. We now propose the following observer for (3.13)-(3.15): Mv

=

s

q

-C9-

~

0~

+F

(3.42)

=

Introduce M ~ = M ~ ' -

"

(3.43)

~), t h e n ( F = O)

M ~ = - C r 1 6 2 OV

(~1) H~

(3.44)

0~

where H =

V1 "

'" "

"Yl ) " and

7n

" " "

~n

M v = M~5 +

" %

= -C9

0 V (s

00

H (~, - v)

(3.45)

We can now view (3.13), (3.14), (3.43) and (3.45) as a hierarchical combination as in (3.12). Since H is bounded, this means t h a t under the assumption t h a t K ((t) and ~O ~ V are bounded, 5(7t and 5~r converge a s y m p t o t i c a l l y to zero.


51

R e m a r k 3.3 Note that due to our measurement (3.15), i.f the observer equation (3.43) is changed to q = 9 - H ( q - q), this implies that (using the relation K T (0) = 2 w T (q) 0(W~l)0))

dt

= ( 5~Tv/-M _ & ~ T H T v / ~

4555 w

(

)

v e,

= - 5 ~ T (C + H) 6~ + ~ ) T H T (C + H) 5~ _~T

( H T K (~1) + 2WT ((?t) 0 (W0~l((?t)q) H'~] 5r

where~=~,r

~z =

o)(o)

0

v ~ W (0) ~q;

,/~W(el)

, ~

=

0

o

(3.46)

'

.

~.0

Generally, there seems to be no conclusion about the contraction behaviour o.f this observer design. However .from (3.46) we see that when the gains in the observer (7i) are in some sense small, this observer "behaves" in the same way as (3.42) and (3.43).

5

Simulations

5.1

E x a m p l e 1: 2 - D O F O s c i l l a t o r y S y s t e m w i t h C u b i c Nonlinearities

Nonlinear oscillations in multi-degree-of-freedom systems with cubic nonlinearities can be found in many physical systems such as the vibration of strings, beams, membranes, and plates for which stretching is significant, the motion of spherical, centripetal, and double pendulums, and the motion of masses connected with nonlinear springs [12]. For a 2-dof mechanical system with cubic nonlinearities, V is given by kl 2 + --~-q2 k 2 2 + alq 4 + ct2q31q2 + a3qlq2 V = --~-ql 2 2 + o~4qlq23 + 0~5q4

(3.47)

where ki > 0 are the linear spring constants and ai are constants. Note that V can be written as V

=

( ql

q2 )

~" + ~

=

(ql

q2 ) P ( q ) (

2+g

89( 2ql + ql )q2

leg

2

3q2

I, 2ql ~- 4(/2] _~a + a5q~ + 89

ql q2 (3.48)

52


Also 0V

_

// kl -t- 4alq12 + 2a3q 2

=

~ a2ql2 + 3a4q2 2 K(q) q

0q

30~2q12 ~-o~4q 2

)(

k2 + 2c~3q 2 + 4a5q 2

ql )

q2 (3.49)

T h e following p a r a m e t e r s were used in the simulations: rnl = rn2 = 1, O~1 = 1, OZ2 ~---0.9, ~3 : 0.8, OL4 = 0.6, O~5 ~---0.5, #1 = #2 : 0.001, k I ---~ 1,

is o i ivo o n 0. nceo >0 ~I~3

d

~ 4

> -

0

(3.50)

c~ 4

> -

0

(3.51)

0~20~4 OL10~5 Jr- - ~ -}- T

> --

0

(3.52)

2 0~3015

2

T h e initial conditions of the plant were: qz (0) = q2 (0) = vl (0) = v2 (0) = 0, while the initial conditions of the observer were ql (0) = 1, q2 (0) = 1, Vl (0) :

0.5, Y2 (0) :

--0.4. T h e r e s u l t s u s i n g ")/1 = "/2 :

-1

is s h o w n in

Figures 1-5

0

-9.5

10

20

30 t~me [s]

40

1

50

60

F I G U R E 1. ql (t) [solid line] and 01 (t) [dotted line].

5.2 Example 2: Cylinder Gyroscope T h e nonlinear d y n a m i c s of a cylinder gyroscope was m o d e l l e d a n d a n a l y z e d by [4]. T h e m o d e l included geometric nonlinearities, a n d it was shown t h a t V is given by


'I

53

t

10

"-0

2o

3o time [s]

4o

so

6o

FIGURE 2. vl (t) [solid line] and ~1 (t) [dotted line].

1

o.

I I I'~ Ib j~ C

-0.2

h

-0.4 ~3.6

10

FIGURE

20

ti

4o

Is]

~o

~o

3. q2 (t) [solid line] and 02 (t) [dotted line].

1

2

1

2

1

2

/ ~]gl -}- alql + ~a4q2 + ~a5q3 [ V = ( ql q2 q3 ) I lasql + lalOq3 1 1 2 1 2 \ 7a7ql + gagq2 ] la - - - la ]a _2 - - 1 a ~2 8r

T ~

I0r

~ 7 ( / 1 - I " - ~ 9~/2

)(ql)

1 1 2 1 2 ~k2 + a2q22 + ~a4ql + ~a6q3 0 q2 1 2 1 2 0 lk3 + a3q 2 + ~asql + ~a6q2 q3 where ki are positive constants and aj are constants depending on the linear axial mode shapes of the gyroscope. Straightforward calculations show

54


U 0 ili" =0.51

i t,me [sl

F I G U R E 4. v2 (t) [solid line] and ~2 (t) [dotted line].

-0.

-1

10

20

30 t,me [s]

40

SO

60

F I G U R E 5. y (t) [solid line] and ~)(t) [dotted line]. that

OV Oq

kl ~- 4alq 2 + 2a4q~ + 2a5q~ + 2asq2 asql § a l o q a a7q 2 + a9q 2 + aloq2 0 k2 + 4a2q2 + 2a4q~ + 2a6q2 + 2a9qlq3 0 3arq2+agq2+az~ k3 + 4aaq2 + 2asq~ + 2asq 2

) ( ql qa

K (q) q T h e following p a r a m e t e r s w e r e u s e d in t h e s i m u l a t i o n s : m l = m2 = m a --- 1, a l = 1, a2 = 1, a3 = 1, a4 = 0.3, a5 = 0.4, as = 0.3, a7 = 0.5, as = 0.5,

3. Observer Design for Nonlinear Oscillatory S y s t e m s a9

=

0.7, alo = 0.3, #1 = #2 = #3 = 0.001,

a n d F -be q; of ~2 in

0 0

k1 =

1, k2 = 9,

k3

55 -----

25,

. Note that with these data, V can be shown to

p o s i t i v e for q r 0. T h e initial c o n d i t i o n s of t h e plant were: q l ( 0 ) ---(0) = qa (0) = vl (0) = v2 (0) = v3 (0) ---- 0, w h i l e t h e initial c o n d i t i o n s t h e o b s e r v e r w e r e 01 (0) = 1, 02 (0) = 1, 03 (0) = - 0 . 2 , ~1 (0) = - 0 . 5 , (0) = - 0 . 4 , ~3 (0) = - 0 . 1 . T h e results u s i n g 71 = 3'2 = 73 = - 1 is s h o w n Figures 6-12.

1 O.E OE 0.4 0.,~ 0

41,2 -0.4 ~).6 ~.8 -1

10

20

30 time [s]

40

50

60

F I G U R E 6. ql (t) [solid line] and ql (t) [dotted line].

-0

-1

-1 .S

10

20

30 time Is]

40

50

60

F I G U R E 7. vl (t) [solid line] and 51 (t) [dotted line].

56


o,

o

i',,,

0

10

2o

~

4o

5O

60

Is]

ti

FIGURE S. q2 (t) [solid line] and 02 (t) [dotted line].

1A 1 0..= C -9.5 -1

iI

-1.5 -2 -25 -3

10

20

3O

4O

5O

6O

t~rne ($]

FIGURE 9. v2 (t) [solid line] and ~2 (t) [dotted line].

6

Conclusions

We have proposed an observer for nonlinear oscillatory s y s t e m s in Lan grangian form with a single m e a s u r e m e n t given by y = 7:i=1 qi. T h e analysis was mainly based on contraction theory which can be found in the papers by Lohmiller and Slotine [5]- [10]. It was s h o w n that the p r o p o s e d observer was asymptotically convergent. T h e observer was simulated first on a 2-dof s y s t e m w i t h cubic nonlinearities, and then on a m o d e l of a cylinder gyroscope. T h e simulations s h o w e d agreement with the theoretical analysis.


57

J,

0

10

20

3O time [s]

4O

5O

6O

FIGURE 10. q3 (t) [solid line] and c)3 (t) [dotted line]. 1.S

~0

-1

-1.5

10

20

30 time [s]

40

50

60

FIGURE 11. va (t) [solid line] and ~a (t) [dotted line].

7

REFERENCES [1] R. Evan-Iwanowski Resonance Oscillations in Mechanical Systems, Elsevier, New York, 1976. [2] P. Hartman. Ordinary Differential Equations, Birkhauser Verlag, Boston, 1982. [3] A. Khajepour, F. Golnaxaghi and K. A. Morris. "Modal Coupling Controller Design Using a Normal Form Method, Part 1 & 2," Journal o.f Sound and Vibration, vol. 205, pp. 657-688, 1997. [4] D. Kristiansen and O. Egeland. "Nonlinear Oscillations in Coriolis Based Gyroscopes," Accepted for publication in Nonlinear Dynamics. [5] W. Lohmiller and J. J.-E. Slotine "On Metric Observers for Nonlinear Systems," Proceedings IEEE International Conference on Control

58


O.

-o.:

-1

10

20

tirn3eO[~]

40

50

60

FIGURE 12. y (t) [solid line] and y (t) [dotted line]. Applications, Dearborn, MI, pp. 320-326, 1996.

[6] W. Lohmiller and J.-J.E. Slotine. "On Metric Controllers and Observers for Nonlinear Systems," Proceedings 35th IEEE Conference on Decision and Control, Kobe, Japan, pp. 1477-1482, 1996. [7] W. Lohmiller and J.-J.E. Slotine. "Applications of Contraction Analysis," Proceedings 36th IEEE Cor~ference on Decision and Control, San Diego, CA, pp. 1044-1050, 1997. [8] W. Lohmiller and J.-J.E. Slotine. "Applications of Contraction Analysis," Proceedings IEEE International Conference on Control Applications, Hartford, CT, pp. 699-704, 1997. [9] W. Lohmiller and J.-J.E. Slotine. "Simple Observers for Hamiltonian Systems," American Control Conference, Albuquerque, NM, 1997. [10] W. Lohmiller and J.-J.E. Slotine. "On Contraction Analysis for Nonlinear Systems," Automatica, vol. 34, pp. 683-696, 1998. [11] D. Lovelock and H. Run& Tensors, Differential Forms, and Variational Principles, Dover Publications, New York, 1989. [121 A. H. Nayfeh and D. T. Mook. Nonlinear Oscillations, Wiley, New York, 1979. [13] S. S. Oueini, A. H. Nayfeh and J. R. Pratt "A Nonlinear Vibration Absorber for Flexible Structures," Nonlinear Dynamics, vol. 15, pp. 259-282, 1998. [14] K. L. Tuer, M. F. Golnaraghi and D. Wang. "Towards a Generalized Regulation Scheme for Oscillatory Systems via Coupling Effects," IEEE Transactions on Automatic Control, vol. 40, pp. 522-530, 1995.

Transformation to State Atfine S y s t e m and Observer D e s i g n A. Glumineau and V. L6pez-M. Institut de Recherche en Cybern6tique de NANTES, IRCyN, UMR 6597 1 rue de la Noe, B.P. 92101, 44321 Nantes cedex 3. F R A N C E

1

Introduction

The observer design problem is completely solved for linear time invariant systems, whereas in the nonlinear case, there is no general theory. In order to tackle this problem, some methods have been employed: Lyapunov-like technique, linearizations, numerical differentiation, and geometric and algebraic methods (cf. [2, 10, 16, 18, 27, 6, 9, 21, 28, 31]). In order to combine the advantages and improve the shortcomings of two different approaches, structural and numerical differentiation have been sucessfully dealed with input time derivatives [25] and input and/or output time derivatives [21]. Table 1 summarizes the existing literature and shows some observer applications. Table 1 L i n e a r i z a t i o n by i n p u t - o u t p u t injection

System =

A( + ~o(y, u)

Approach Geometric: [16, 22, 31] Algebraic: [9, 10, 19]

Applications Motor: Shunt DC, Series De: [5, 24], Flexible joint: [23].

Geometric: = A~ + ~o(y,u,i~,... , u (~))

= A~ + ~(y,--. ,y(S),u,--- ,u (~))

[2s, 15] Algebraic: [26, 25]

[21]

Biological systems: [28], Numerical differentiation: [6].

In order to extend the class of linearizable systems, some results about the transformation of nonlinear systems into state affine systems have been obtained. High-gain observers are useful for state alZfine systems as shown in [3, 12, 30] and the references therein. These observers are based on optimal Kalman's observer and used in physical processes, for instance chemical reactors, distilling columns and mechanical systems [8, 30, I].

60

4. Transformation to State Affine System and Observer Design

T h e following table summarizes the main contributions on the equivalence between a nonlinear system and a bilinear or state afflne system, as well as some observer design applications. T a b l e 1. Authors Construct.

System

= A(u)~ + ~(u, y) = A(u, y)~ + ~(u, y)

[11]

~o

[141 [1]

Yes No

[20]

Yes

Applications

Synch. Generator [17] Inverse Pendulum [1] Chemical reactor [8] Distilling columns [30]

In the following our new results [20] are introduced. One of the contributions of [20] is the definition of a first algorithm to compute the transformed system functions, from the I / O differential equation. The chapter is organized as follows. Section 2 introduces some definitions and notation. Section 3 we state the problem of state affine transformation of nonlinear systems, and gives the aim of our approach we introduce by an example. We define an algorithm t h a t permits to give a NSC in order to solve this problem. Section 4 obtains the synthesis observer for the s t a t e affine system founded in Section 3. This is achieved with a well defined coordinates transformation and a Kalman-like observer. Some conclusions are given in Section 5.

2

Definitions and N o t a t i o n

Consider the nonlinear system:

{~ --

= y

f(x,u)

=

(4.1)

with x ~ M where M is an open and dense subset of ~n, u C ~ m and y E ~. T h e entries of f(., .) and hi. ) are meromorphic functions of their arguments. Let us define the state affine system, considered here

{ A(y(t), (t))Zcz +

(4.2)

where z(t) 9 ..~n y(t) 9 '.~,u(t) 9 ~m. W h e n one measures y(t), one can define ~ :--- (y, u) as a new input a n d as recalled in [13] if it is regularly persistent [3], thus the system

~o { "~--- A(v~). ~. + qo(~)- S - I c T ( c ~ -- y) = - O S - AT(~)S - SA(~)) + c T c

(4.3)

4. Transformation to State Affine System and Observer Design

61

is a Kalman-like observer for ~--~a" Where z(t) C ~ n S(t) E ~ + is a symmetric positive definite matrix and 0 > 0. The norm of the estimation error converges locally exponentially to the origin. From now on, ~ is supposed to be generically observable [25] and will be called observable.

3

Problem

Statement

The goal is to find a state coordinates transformation z ----O(x), such that system ~ (4.1) is locally equivalent to system ~'~.~ (4.2, in order to design the observer ff-~-o (4.3)9 The approach consists in checking t h a t the I / O differential equation associated to the system ~ has the same form than the ~ one. The uniqueness of this equation for an observable system is shown in [29].

3.1

The Input-Output Differential Equation for State Affine Systems ~-~

The I//O differential equation for }-~ verifies

P~

:= y(n) = F n ( A ~ , . . . , A,~_~)+ + F n - 1 (A1, " 9" , A n - l , t/91)+ + A 1 F n - 2 ( A 2 , . . . , A n - l , ~2) + " " +A1A2"'" A n - 2 F 1 ( A n - l , ~gn--1)+ A1A2"" A~-IFo(~,~),

(4.4)

where F~_j (0 N we do not only know the output y(k) at t i m e k, but also the past outputs y(k - 1),-. - , y(k - N). An extended observer with buffer N for E then is a dynamical system E of the form E : ~(k + 1) --- f ' ( ~ ( k ) , y ( k ) , - - . , y ( k - N ) ) , k _> N where ~ E ~ ' ~ and the m a p p i n g f ' : /R n + g + l ~ ff~'~ is smooth, with the property t h a t x(k) - ~(k) --* 0 (k ~ oo), for all x(0), ~ ( N ) E ff~n. To study the design of extended observers for a discrete-time system E of the form (5.20), we first consider a system E~ of the form

~

[

z ( k + l) 9(k)

= =

Az(k)+O(~(k),... ,~)(k-N)) Cz(k)

(5.44)

where the state z E ~'~, the output ~ E ~ , A, C are matrices of a p p r o p r i a t e dimensions, the mapping ~ : ~ g + l ~ ~,~ is smooth, and the pair (C, A) is in Brunovsky form. Note that for N = 0 the system E~ is identical to the system E in (5.18). Therefore, a system E~ of the form (5.44) will be referred to as a system in extended observer.form with buffer N. As for a system in observer form, the design of an extended observer for a s y s t e m in extended observer form is relatively easy. Namely, it is straightforwardly checked that the system

{

E(k + 1)

=

AE(k) + K(~](k) - y(k)) + O(~)(k),... ,~)(k - N ) )

=

c

(k)

(5.45) where the matrix K is such that all eigenvalues of A - K C are in the open unit disc, is an extended observer for E ~. As in the previous section, we now consider the question under which conditions a given discrete-time system can be put in extended observer form for some N E / N . The transformations we are going to use here, are more general t h a n the ones in the previous section, in the sense t h a t we also allow them to depend on the past output measurements y ( k - 1 ) , - . 9 , y ( k N). More specifically, we will be looking at p a r a m e t r i z e d transformations Z = /:~(X, ~ n , ' ' " , ~N), where z E ~ n , with the property t h a t there exists a

5. On Existence of Extended Observers for Nonlinear Discrete-Time Systems

85

m a p p i n g p - l ( . , f l , ' " 9 , iN) : ~ n __~ ~ n p a r a m e t r i z e d by (fl," " " , i N ) such t h a t for all ( f l , " " , iN) we have t h a t

P(P-l(z, fl,""

,fN),fl,''"

,iN)

:

Z

A m a p p i n g having this property will be referred to as an extended coordinate change. We will then say t h a t the system (5.20) can be put in extended observer .form with buffer N if there exists an extended coordinate change P ( ' , fl," "" , iN) : ~ n --, ~ n parametrized by (fl," "" , iN) and a diffeomorphism p : / R -~ H~ of the output space such t h a t the variable

z(k) := P ( x ( k ) , y(k - 1),--. , y(k - N ) )

(5.46)

satisfies (5.44), where ~) :-- p(y), and the pair (C, A) is in Brunovsky form. As pointed out above, one m a y then build an extended observer (5.45) for z(k) in (5.46). From this extended observer, one then obtains an e s t i m a t e ~(k) for x(k) by inverting the extended transformation P: ~(k) := P - l ( ' d ( k ) , y ( k - 1 ) , . . - , y ( k -

N)),k > N

(5.47)

The following generalization of T h e o r e m 5.4 gives conditions under which a discrete-time system (5.20) can be put in extended observer form. 5.7 Consider a nonlinear discrete-time system (5.20), and assume that f(O) = O, h(O) = O. Let N E be given. Then E can be put in observer .form with buffer invariant subset U C ~ztn containing the origin i.f and only

E of the .form

Theorem

{ 0 , . . . , n - 1}

N on an open if

(i) E is strongly obsewable on U. (ii) There exist .functions r ,r : ~ N + I --~ ~ and a diffeomorphism p on h(V), such that on V the .function fs in (5.22) satisfies n-N

p o f~(s) ---- E

r

, si)

(5.48)

i=1

P r o o f . The proof for the case where only coordinate transformations are considered (or, in other words, the case where p = idt~) m a y be found in [8]. The case where also output transformations are considered is an (almost) immediate consequence of the case where p = idt~. 9

If there exist functions p, r defines the following variables:

,

Cn-N such t h a t fs satisfies (5.48), one

z (k) E Cj(s~_j(k),..- , s l ( k ) , y ( k j=l

(i----1,...,n)

1),--. , y ( k -

N-

1 +i-j))

(5.49)

86 5. On Existence of Extended Observers for Nonlinear Discrete-Time Systems where c~i : = m i n i / t~ y + l by

~)n-N : ~ N + I

1, n - N). Further, define r

~i(~1,''" ,~N+I) : : (~I(p--I(~I),''" ,P--I(~N+I)) (i=l,-.-,n-N) It m a y t h e n be shown t h a t in these variables we o b t a i n the following ext e n d e d observer form: zl(k-4- 1)

=

z2(k)

+ q~l(y(k),... ,y(k -

N))

z,~_g(k+ 1)

=

Zn_NA-l(k)-~-~n_N(y(]'g),''"

z n - g + l ( k + 1)

=

Zn-N+2(k)

= = =

o z (k)

zn_l(k

+ 1)

,y(]~ -- g ) )

(5.50)

Note t h a t T h e o r e m 5.7 generalizes T h e o r e m 5.4. ~hlrther, f r o m T h e o r e m 5.7, we o b t a i n the following result for N = n - 1 (see also [7],[8]). C o r o l l a r y 5.1 Consider a nonlinear discrete-time s y s t e m E of the .form (5.20), and assume that f ( 0 ) = 0, h(0) = 0. Then E can be put in observer .form with buffer N on an open invariant subset U C i~:tn containing the o,~igin i.f and only i.f E is strongly observable on U. Thus, we see t h a t every strongly observable s y s t e m can be p u t in ext e n d e d observer with buffer N = n - 1. F r o m a practical point of view, e.g. when n is large, it m a y be desirable to reduce the size of the buffer. T h e r e fore, we next investigate under which conditions an e x t e n d e d observer with buffer N E { 1 , . . . , n - 2} exists. As in the previous section, these conditions are again given in terms of the one-forms wi in (5.26) a n d the vector fields ~-~ in (5.30). Using the one-forms w~ in (5.26), we define the following codistributions: ~i : = span{wk - wk-1 ] k = i + 1 , . - . , m i n ( n , i + N ) } ( i = 1 , . . . , n ) (5.51)

~i : = s p a n { w ~ , . . - , w i + g } ( i = 1 , . . . , n - - N -

1)

(5.52)

We first consider the case w i t h o u t o u t p u t t r a n s f o r m a t i o n s , i.e., the case where in (5.48) we have t h a t p = i d a . This result generalizes T h e o r e m 5.5. T h e o r e m 5.8 Consider a discrete-time s y s t e m E o.f the .form (5.20) that is strongly observable on an open invariant subset U C ~ n containing the

5. On Existence of E x t e n d e d Observers for Nonlinear Discrete-Time Systems

87

origin. A s s u m e .further that U is smoothly contractible to the origin, and that the o n e - f o r m s wl , . . . , a~n in (5.26) generate a codistribution on U. L e t N E {1,. - 9 , n - 2} be given. T h e n .for p = i d a , E can be p u t in extended observer .form with buffer N on U i f and only i f the o n e - f o r m s wi in (5.26) satisfy dwi ~- 0 m o d f ~ i ( i = 1 , - - - , n )

(5.53)

P r o o f . (necessity) Follows b y d i r e c t v e r i f i c a t i o n . (sufficiency) A s s u m e t h a t t h e o n e - f o r m s wi i n (5.26) s a t i s f y (5.53). N o t e t h a t b y t h e d e f i n i t i o n of t h e wi we h a v e t h a t f~i = s p a n { d s k [ k = i + 1 , - . . , r a i n ( n , i + N ) } ( i = 1 , - - - , n ) D e f i n i n g c~i : = r a i n ( n , i + N ) (i = 1 , - - . , n ) , (5.53) t h e n gives

0 = dwi A dSi+l A 9 .. A d s ~ . . . . . i

E j=l

E

(\ os~.Os~ o s, ))

k=c~i+l

'

A

A

A.. A

(5.54)

"

(i = 1 , . . . , n ) w h i c h is e q u i v a l e n t to

( Osksj o2f ])

= O(j, k = 1 , . . . ,n; IJ - kl > n)

(5.55)

I t is e a s i l y checked t h a t t h i s c o n d i t i o n is e q u i v a l e n t t o t h e e x i s t e n c e of f u n c t i o n s 4 ) 1 , " " , c/)~-N s u c h t h a t fs satisfies (5.48). 9 F o r t h e case t h a t in (5.48) we h a v e t h a t p ~ i d a , t h e f o l l o w i n g r e s u l t holds. Theorem 5 . 9 Consider a discrete-time s y s t e m E o.f the .form (5.20) that is strongly observable on an open i n v a r i a n t subset U C ~T~~ c o n t a i n i n g the origin. A s s u m e .further that U is smoothly contractible to the origin, and that the o n e : f o r m s Wl,. 9 9 , wn in (5.26) generate a codistribution on U. L e t N E {1, - 9 9 , n - 2} be given. T h e n the .following s t a t e m e n t s are equivalent:

(i) E can be p u t in extended observer .form with buffer N on U. (ii) There exists a .function U : ~

-~ ~zt such that

dwi - d S A wi = 0 m o d f t i ( i = 1 , - . . , n )

(5.56)

(iii) The o n e : f o r m s wl," " 9 , wn and the vector .fields ml,. 9 9 , "r,~ satisfy daJi - 0 m o d F t l + s p a n { w n } ( i = 1 , . - . , n - N - 1)

(5.57)

88

5. On Existence of Extended Observers for Nonlinear Discrete-Time Systems dwi =- 0 m o d ~ i ( i = 1 , . - . , n - N - 1)

(~.ss)

dwi -=- 0modf~i(i

(5.59)

= n-

N,...

,n -

1)

[cri, 7,]_lwn = [c~j, T n ] J w ~ ( i , j = 1 , . . , n -

N-

I,''" ,n-- N--1;j=

l,...

s

1)

(5.60)

,n--l)

(5.61)

where the vector.fields or1,... , crn_N_l are de.fined by o'i

:=

T i ~ - " "" n a T i + N ( i

----

1 , . . . , n - N - 1)

(5.62)

P r o o f . ( i ) ~ ( i i ) A s s u m e t h a t there exist functions p, r , C n - N , such t h a t p is a diffeomorphism on h ( U ) a n d fs satisfies (5.48). Define o n e - f o r m s &l,'", 9 by

o?i :=

( O(po_f,) ~ dsj(i = 1,... j=l

\

n)

Osj ]

(5.63)

T h e n it follows from T h e o r e m 5.8 t h a t d&i ~ 0modspan{ 9

- 0~k_ 1 I k =

i + 1,... ,min(n,i + N)} (5.64)

(i = 1 , . - - ,n) Note t h a t we have t h a t &~ = ( p ' o f ~ ) w i (i = 1, .. , n). Defining the f u n c t i o n S := - log IP' o fsl, this gives 1 d '[p ' 0 f s ) ) A --:--: dwi - d S A w i = d (:---k--~.) p'of8 ~ -~ [~-~-~.~ p'of., o?z. . . . . .

p'~I., d&i(i = 1,--- , n)

(5.65) Further, it follows t h a t

span{i~2k -- ~]k--1 I ]~ ~---i ~- 1,--. , m i n ( n , i + N ) }

= f~i(i = 1,--.

,n) (5.66)

O u r claim is then established by combining (5.64),(5.65),(5.66). ( i i ) ~ (i) A s s u m e t h a t there exists a function S : U ~ H~ such t h a t (5.56) holds. Note t h a t from (5.26) we have t h a t wn -- df~. Thus, (5.56) for i = n gives t h a t 0 = dwn - d S A wn = - d S A wn

(5.67)


89

By C a r t a n ' s L e m m a , this gives t h a t d S e span{dfs}. Define T : = e x p ( - S ) . T h e n we also have t h a t d T E span{df~}, a n d thus there exists a f u n c t i o n t5 : ~ --~ ~ such t h a t T = 15 o f~. Define p : = f i S ( r ) d r , a n d one-forms 9 , 9 as in (5.63). We t h e n have t h a t &i = Taxi (i = 1 , . . . ,n), a n d thus d&i = d T

Awi +

Tdwi = T(d~i - dS

Awi)

=-- 0modf~i(i = 1,--- , n) (5.68)

Together with T h e o r e m 5.8, this establishes our claim. (ii)r F r o m the fact t h a t ~,~ = dfs, it follows t h a t the f u n c t i o n S t h a t needs to exist has to satisfy d S A w n . By C a r t a n ' s L e m m a , this implies t h a t there should exist a function a such t h a t d S = o~w,~ a n d da A w~=O

(5.69)

Thus, the existence of a function S such t h a t (5.56) holds is equivalent to the existence of a function a satisfying (5.69) a n d dwi - (~Wn A wi -- 0 m o d f t i ( i = 1,. -- , n - 1)

(5.7o)

We now have t h a t a two-form w 2 satisfies w 2 _= 0mode2 for some codistrib u t i o n 12 if and only if X_I Y_I w 2 for all X, Y E f t • It is easily checked t h a t we have ~/l = span{r1,- 9 , ri_], r i + N + l , ' ' '

, r~, c~i}

(5.71) (i = 1 , - . . , n -

N-

1)

and 12~ = s p a n { r 1 , . . . , r i _ l , c l i } ( i = n -

N,-.. ,n-

1)

(5.72)

where, analogously to (5.62), we have :=

+...

+

= n - N,...

, n - 1)

(5.73)

NoB, let i E {1,--- , n -- N - 1} be given. We t h e n have 0 = %_1 re-I (da;i - a w n A w i ) = "rkJ re-I dwi = ['rk, re]-I wi (5.74) (k,(=l,...

,i-l,i+N+l,...

,n)

0 ---- Tk--I 0",_1 (dwi - aw,~ A wi) = "rk-I o'{_1 dw{ = [%, o-{]_.1 oJ{ (5.75) (k=l,.-.,i-l,i+N+l,---,n-1)

90


and 0 = cr~AT,~_I(dcvi - awn A wi) = cri--I7nA dwl - a = [cri, 7-n]-Iw~ - a (5.76) Where in (5.74),(5.75),(5.76) and (5.14). Combining (5.74) ther, (5.76) gives (5.60), while Next, let i E {n - N , - . . , n -

the last equality follows by applying (5.15) and (5.75), we obtain (5.57) and (5.58). Furcombining (5.69) and (5.76) we obtain (5.61). 1} be given. We then have

0 = Tk/Te-I (dwi - o~w,, Awi) = 7k-I r e / d w i = [Tk, Te].-Iwi (5.77)

(k,e =

1,...

,i -

1)

and 0 = 7 k l c~iI (dwi - ozwn A wi) = Tk I cril dwi = [7k, ai] I wi

(5.78) (k=l,...,i-1) Combining (5.77) and (5.78), we then obtain (5.59). m R e m a r k 5.1 Theorem 5.9 generalizes Theorem 5.6. From the .first two items of both theorems this is seen immediately. I f however, one considers the third i t e m of both theorems the generalization is .far.from obvious at .first sight. This is due to the .fact that the equivalence (ii)~==>(iii) in Theorem 5. 6 holds .for general independent one:forms W l , ' " ,Wn, while this equivalence in Theorem 5.9 only holds .for Oneaforms w l , " ' , w , of the .form (5.26).

5

Conclusions

In this chapter, we have given conditions for the existence of extended observer forms and extended observers for single-output nonlinear discretetime systems. All conditions are valid on an open invariant subset of the state space t h a t is smoothly contractible to an equilibrium point of the system and on which some regularity assumptions are satisfied. This raises the question what can be said for the case t h a t (some of) the regularity assumptions are not satisfied. This remains a topic for future research. A further topic for future research would be the question when extended observer forms and extended observers for multi-output discrete-time systems exist. As also mentioned in the Introduction, it seems t h a t the conditions given in [13] for the existence of an observer form when only coordinate transformations are allowed, seem to be incorrect. Preliminary investigations suggest that in fact the problem of coining up with correct conditions m a y be quite intractable. On the other hand however, it m a y

5. On Existence of Extended Observers for Nonlinear Discrete-Time Systems 91 be shown by using the same techniques as in [7],[8] that a strongly observable multi-output system may always be put in observer form with buffer N = n* - 1, where n* equals the maximal so called observability index of the system.

Acknowledgments Part of this research was performed while the author was visiting the Laboratoire d'Automatique de Nantes, Nantes, France, supported by a grant from the R6gion Pays de la Loire. 6

REFERENCES [1] M. Brodmann. Beobachterentwur.f .fiir nichtlineare zeitdiskrete Systeme, VDI Verlag, D~isseldorf, 1994. [2] R. L. Bryant, S. S. Chern, R.B. Gardner, H.L. Goldschmidt and P.A. Griffiths, Ezterior differential systems, Springer, New York, 1991. [3] H. Cartan. Formes diff~rentielles, Hermann, Paris, 1967. [4] Y. Choquet-Bruhat and C. DeWitt-Morette (with M. Dillard-Bleick), Anaysis, manifolds and physics, Part I: Basics, North-Holland, Amsterdam, 1991. [5] H. Flanders. Differential forms with applications to the physical sciences, Dover, New York, 1989. [6] A. Glumineau, C.H. Moog and F. Plestan. New algebro-geometric conditions .for the linearization by input-output injection, IEEE Trans. Automat. Control, 41, pp. 598-603, 1996. [7] H. J.C. Huijberts, T. Lilge and H. Nijmeijer. A control perspective on synchronization and the Takens-Aeyels-Sauer Reconstruction Theorem, to appear in Phys. Rev. E, 1999. [8] H. J. C. Hnijberts, T. Lilge and H. Nijmeijer. Synchronization and observers for nonlinear discrete time systems, submitted to European Control Conference 1999. [9] H. J. C. Huijberts, H. Nijmeijer, and A.Yu. Pogromsky. Discrete-time observers and synchronization, in G. Chen (Ed.), Controlling chaos and bifurcations in engineering systems, CRC Press, Boca Raton, Florida, 1999.

[10] A. J. Krener and A. Isidori. Linearization by output injection and nonlinear observers, Syst. Control Lett., 3, pp. 47-52, 1983.

92


[11] A. J. Krener and W. Respondek. Nonlinear observers with linearizable error dynamics, SIAM J. Control Optimiz., 23, pp. 197-216, 1985. [12] T. Lilge. On observer design for nonlinear discrete-time systems, Eur. J. Control, 4, pp. 306-319, 1998.

[13]

W. Lin and C.I. Byrnes. Remarks on linearization of discrete-time autonomous systems and nonlinear observer design, Syst. Control Lett., 25, pp. 31-40, 1995.

[14] M. Spivak. A comprehensive introduction to differential geometry, Volume I, Publish or Perish, Houston, 1979. [15] X. Xia and W. Gao. Nonlinear observer design by observer canonical forms, Int. J. Control, 47, pp. 1081-1100, 1988.

Stability Analysis and Observer Design for Nonlinear Diffusion Processes Winfried Lohmiller and Jean-Jacques E. Slotine

Nonlinear Systems Laboratory Massachusetts Institute of Technology Cambridge, Massachusetts, 02139, USA

1

Introduction

The stability of a nonlinear reaction-diffusion process and its convergence rate can be determined very simply. This allows in turn the design of simple observers for such processes. The technique is based on an extension of the recently developed tools of contraction theory. Reaction-diffusion processes are pervasive in physics. Mathematical properties such as existence and smoothness of solutions are well understood for many such processes (Evans [3]). This paper shows that analyzing global stability - are initial conditions or temporary disturbances eventually "forgotten," and if so, how fast? - and determining convergence rates given boundary conditions is very simple for such processes, and shows how this result applies naturally to the design of observers. This is achieved by taking advantage of recent results on stability theory, referred to as contraction analysis (Lohmiller and Slotine [8]), and extending them to partial differential equations describing nonlinear reaction-diffusion processes. After a brief review of contraction analysis in Section 2, Section 3 analyzes the contraction properties of the Laplace operator with general boundary conditions. The result is applied to reaction-diffusion equations of the form

c9_~r = h(t) V2r + g(t) V r + f ( r Ot

(6.1)

whose stability and convergence rates are explicitly quantified, and then to exponentially convergent observer designs for these systems. Numerical aspects are discussed in Section 4, and extensions to other classes of distributed systems in Section 5.

94

6. Stability Analysis and Observer Design for Nonlinear Diffusion Processes

2

Contraction Analysis

Differential approximation is the basis of all linearized stability analysis. W h a t is new in contraction analysis is that differential stability analysis can be made exact, and in turn yield global results on the nonlinear system. We first summarize some basic results of (Lohmiller and Slotine [8]), to which the reader is referred for more details, and then discuss how these results apply naturally to the design of nonlinear observers.

2.1

Basic Tools

We consider general deterministic systems of the form ~=f(O,t)

(6.2)

where 9 is the n • 1 state vector and f is an n • 1 nonlinear vector field, All quantities are assumed to be real and smooth, so that we can write the exact differential relation 0f 5 6 = ~--~(O, t) 5 0

(6.3)

where $0 is a virtual displacement - recall that a virtual displacement is an infinitesimal displacement at .fixed time. Note t h a t virtual displacements, pervasive in physics and in the calculus of variations, are also well-defined m a t h e m a t i c a l objects (Arnold [1] and Schwartz [10]). Consider now two neighboring trajectories in the flow field ~ = f(O, t), and the virtual displacement 50 between them (1). The squared distance between these two trajectories can be defined as ~ 0 T 6 0 , leading from (6.3) to the rate of change

virtual displacement 8 ~

velocity 8

two neighboring traiectories

~K

FIGURE 1. Virtual dynamics of two neighboring trajectories.

6. Stability Analysis and Observer Design for Nonlinear Diffusion Processes d ( ~ ~ T (~~ ) = 2 5 ff2T h Oo = 2 5r T ~--~f~ hrb < 2 /~max ~ O T ~ o

95

(6.4)

where Am~x(q~,t) denotes the largest eigenvalue of the symmetric part of of i.e., the largest eigenvalue of 7(b-~ o of ~ + 1 of T~j. Assume now the Jacobian b-~, that Am~x(q),t) is uniformly strictly negative (i.e., 3 /3 > 0, VO, Vt _> 0, Amax(gP,t) < --Z < 0. ). Then, from (6.4) any infinitesimal length [[5~][ converges exponentially to zero. By path integration, this immediately implies that the length of any finite path converges exponentially to zero. Thus, as in stable linear time-invariant systems, the initial conditions are exponentially "forgotten." We can state the following definition and basic result (Lohmiller and Slotine [8]) T h e o r e m 6.1 The system (~ = f(qh, t) is said to be contracting if -of ~ is uni.formly negative definite. All system trajectories then converge exponentially to a single trajectory, with convergence rate IAm~l, where )~max i8 Of the largest eigenvalue of the symmetric part of -5"~" The system is called semi-contracting if b-~ of is only negative semi-definite, and indifferent if of is skew-symmetric. More precise local versions of the above theorem can also be derived. In addition, departing further from classical Krasovskii-like results for autonomous systems (Krasosvkii [6], Hahn [5] and [7]), the approach can be vastly extended by allowing for a prior differential coordinate transformation, leading to a a necessary and sufficient condition for global exponential convergence (Lohmiller and Slotine [8]). Specifically, the line vector 5~ between two neighboring trajectories in Figure 1 can also be expressed using the differential coordinate transformation = o

(6.5)

where O(q~, t) is a square matrix. This leads to a generalization of our earlier definition of squared length 6~JT 6tII = 5 0 T M

5~

(6.6)

where M(O, t) = o T o represents a symmetric and continuously differentiable metric - formally, equation (6.6) defines a Riemann space (Lovelock and Rund [9]). Since (6.5) is in general not integrable, we cannot expect to find explicit new coordinates ~(O,t), but 6~ and 5 ~ T 5 ~ can always be defined. We require M to be uniformly positive definite, so that exponential convergence of 50 to 0 also implies exponential convergence of 5q~ to 0. Distance between two points P1 and P2 with respect to the metric M is defined as the shortest path length (i.e., the smallest path integral f 2 ~ [15~l] ) between these two points. Accordingly, a ball of center c and radius R is defined as the set of all points whose distance to e with respect

96


to M is strictly less than R. Computing I

d 5~ = F S ~ dt

or alternatively ~-~

where

F---(E)+o0f)o

0,)

\ ~ - ~ M + 1~I + M~-~

-1

(6.7)

5(I)

(6.8)

we can state the following definition and main result (Lohmiller and Slotine

[8]) D e f i n i t i o n 6.1 Given the system equations ~ = f(O,t), a region of the state space is called a contraction region with respect to a uniformly positive de.finite metric M(O, t) = OTO i f F in (6.7) or equivalently ~O 0 T M ~- - - M Oof 4cP -1(r is uniformly negative de.finite in that region. Regions where F is negative semi-definite are called semi-contracting, and regions where F is skew-symmetric are called indifferent. T h e o r e m 6.2 Given the system equations ~ = f(O,t), any trajectory, which starts in a ball of constant radius with respect to the metric M(O,t), centered at a given trajectory and contained at all times in a contraction region with respect to M(O,t), remains in that ball and converges exponentially to this trajectory. Furthermore global exponential convergence to the given trajectory is guaranteed if the whole state space is a contraction region with respect to the metric M(O,t). It can be shown that the existence of a uniformly positive definite metric with respect to which the whole state space is a contraction region is actually a necessary condition for global exponential convergence. In the linear time-invariant case, a system is globally contracting if and only if it is strictly stable, with F simply being a normal Jordan form of the system and O the coordinate transformation to that form. Note that the metric is unchanged by an additional (perhaps time-varying or state-dependent) orthonormal transformation, i.e., by left-multiplying O by an orthonormal matrix. 2.2

Nonlinear Observer Design using Contraction

Theory

By using a differential approach, contraction theory in a sense treats convergence analysis and limit behavior separately. Guaranteeing contraction means that after exponential transients the system's behavior will be independent of the initial conditions. In a control context, once contraction is guaranteed through feedback, specifying the final behavior reduces to the problem of shaping one particular solution, i.e. specifying an adequate


97

open-loop control input to be added to the feedback terms, a necessary step of any control method. In an nonlinear observer context, contraction theory is a rather natural tool, since once the observer contraction behavior has been shown and quantified, one needs only verify t h a t the observer equations contain the actual plant state as a particular solution to automatically guarantee convergence to t h a t state. If y((I), t) is the available m e a s u r e m e n t vector, and q) the estimated state, with ~, = y(q), t), this m a y be achieved by simply copying the system dynamics (identity observer) and adding to the right-hand side a term of the form k ( p , t) - k ( y , t) , where the vector field k is selected to guarantee or enhance the contraction behavior of the observer. W h e n the actual system is itself contracting, as will be the case for most of the nonlinear diffusion processes considered in this paper, k needs only to be selected to enhance (speed up) the natural contraction behavior of the system.

2.3

Weakly Contracting Systems

In this section we derive an exponential convergence condition for classes of semi-contracting systems. The idea is simple: whereas in Section 2 we have used only first time-derivatives of a virtual displacement to characterize a flow field, we now perform a complete Taylor series expansion to analyze a semi-contracting virtual dynamics. This section m a y be skipped in a first reading, as it will only be needed for an extension in Section 5. Consider a semi-contracting, analytic virtual dynamics in 5 ~ d The corresponding virtual length dynamics is

d (e~Te~)=--2er dt T

with positive semi-definite F~ = - ~1 ( F + F T ) . Factorizing F s as x / ~ v ~ , say with a Cholesky factorization, allows one to compute the time-derivatives of (~tI/T~ / as

o

T

dt 2 d3 =

+

2 L l v ~ T nlv/-Fs + n o v ~ sT n2x/F~)~k0

98


with the Lie derivatives L J v ~ ( x , L~

=

t) (Lovelock and R u n d [9])

~;~

-- L ~ v ~ F + ~ L J v ~

vj>_o

dt

As a result 6k~T6k~(t A- T ) along a trajectory x(t) can be written as the Taylor series expansion

(

6q2T6q~(t + T )

O

and 21}/=0.

(7.4)

P 2 The linear damping matrix D is strictly positive: D>0.

(7.5)

7. Nonlinear Passive Observer Design for Ships with Adaptive Wave Filtering

117

Linear Wave Frequency Model The W F motions are mainly generated by the 1st-order wave forces acting on the ship. Based on linear approximations of existing wave s p e c t r u m descriptions, see Fossen [5] for details, a linear W F model can be formulated as:

~

= A ~ + Ewww

(7.6a)

= C~,~

(7.6b)

where p is the order of the W F model, ~ E !Rap, w~ C ~3 is a zero-mean bounded disturbance vector and Aw, C~ and Ew are constant matrices of appropriate dimensions. The components of the W F motion is represented by the vector r/w = [xw,yw, ~bw]T. In one degree of freedom, this can for example be a 2nd-order d a m p e d oscillator (p = 2):

~{~} (s) = s2 + 2r

e~is

2 w{~i} + Woi

(i = 1, 2, 3)

(7.7)

where (.){,:} denotes the i-th vector element. Here (i is the relative d a m p i n g ratio and Woi is the natural frequency, which is related to the dominating wave frequency of the incoming waves. From a practical point of view, these are slowly varying quantities, depending on the sea state. Typically, the periods of the dominating waves are in the range of 5 to 20 seconds in the N o r t h Sea. In the case of a 2nd-order W F model the matrices in (7.6a)--(7.6b) are:

,]

-2af~

'C~:

[ 0 I ], E ~ =

[0] E~,2

where A = diag {~1, @, ~a}, = diag {Wol, wo~, Woa} , Evo2 = diag {ewl, ew2, ew3} .

(7.9) (7.10) (7.11)

Bias Modeling A frequently used bias model for marine control applications is:

= - T b l b + EbWb

(7.12)

where b E .~3, Wb C {R3 is a zero-mean bounded disturbance vector, Tb C ~axa is a diagonal m a t r i x of bias time constants and Eb a diagonal m a t r i x scaling the amplitude of Wb. T h e bias model accounts for slowly-varying forces and m o m e n t due to 2nd-order wave loads, ocean currents and wind.

118 7. Nonlinear Passive Observer Design for Ships with Adaptive Wave Filtering In addition, a bias model wilt account for errors in modeling of the constant mooring loads, actuator thrust losses and other unmodelled slowly-varying dynamics. Measurements

For conventional ships usually only position and heading measurements are available to the positioning system, whereas accurate velocity measurements are not available. Hence the measurement equation is written: y=v+~

+ vy

(7.13)

which consists of the LF and W F motions and measurement noise Vy c .~3.

2.3

Total Ship Model

When designing the observer, the following assumptions are made in the Lyapunov analysis regarding the ship model: A1

J(r ~ J(~ + r = J(r where Cy A r + r denotes the measured heading. This is a good assumption since the magnitude of the wave-induced yaw motion r will be less than 5 degrees in extreme weather situations and less than 1 degree during normal operation of the ship/rig.

A2

Position and heading sensor noise is omitted, vy = 0, since this noise is negligeable compared to the wave-induced motion.

F r o m Assumptions AI-A2 the total motion of moored and free-floating

ships is represented by the following equations: (7.14a)

= Aw~ + E w w w

(7.14b)

i] = J ( ~ y ) u = -T[-lb + Mi, = -Dr

y = ~+ ~

EbWb

- JT(r

= ~] + Cw~.

(7.14c) + JT(r

+ Tthr

(7.14d) (7.14e)

3 Non-Adaptive Observers Two different non-adaptive observers will be derived in this section. T h e first one is similar to the observer of Fossen and Strand [6] for dynamically positioned (free-floating) ships, where here also the effect of a spread mooring system is taken into account. In the second design, the observer is augmented by a new filtered state of the innovation signals. This adds


119

more flexibility to the observer design. By using feedback from the highpass filtered innovation in the W F part of the observer there will be no steady-state offsets in the W F estimates. Moreover, by using the low-pass filtered innovation in the bias estimation, these estimates will be less noisy and can thus be used directly as a feedforward t e r m in the control law. The adaptive observer proposed in Section 4 is based on the a u g m e n t e d observer. In the design we use an S P R - L y a p u n o v approach for obtaining passivity and stability of the observers. By including the synthetic wave model in the observer, wave-filtering is obtained, see Definition 1.

Wave .filtering can be defined as the reconstruction of the LF motion components .from noisy measurements o.f position and heading by means of an observer. In addition to this, noise.free estimates o.f the LF velocities should be produced. This is crucial in ship motion control systems since the W F part of the motion should n o t be compensated .for by the positioning system. If the W F part of the motion enters the .feedback loop, this will cause unnecessary tear and wear of the actuations and increase the .fuel consumption. D e f i n i t i o n 7.1 ( W a v e F i l t e r i n g )

3.1

O b s e r v e r in the E F f r a m e

The observer in this section is similar to the observer in [6], except t h a t the effect of a spread mooring system attached to the ship is included.

Observer Equations A nonlinear observer copying the ship-mooring dynamics (7.14a)-(7.14e) is: = A ~ + KI~)

;) = J ( r

+

b = --Tb-1/~ + K3y

M ~ = - D ~ - JT(fv)G~ ) + J T ( f v ) b + Tthr + JT(~y)K4~] 9 =

+

where ~ = y - ~) is the innovation vector and K1 E ~2.p• ~3• are observer gain matrices to be determined later.

(7.15a)

(7.15b) (7.15c) (7.15d) (7.15e) K2 ' K3 ' K4 C

Observer ET~ror Dynamics The estimation errors are defined as ~ = ~ - ~, ~ = ~ - ~), b = b - / ~ and = v - ~. Hence, from (7.14a)-(7.14e) and (7.15a)-(7.15e) the observer

120 7. Nonlinear Passive Observer Design for Ships with Adaptive Wave Filtering error dynamics is: :

A~

- KI~]

: fl(r

+

E~w~

(7.16a)

- K2y

(7.165)

: - T b l b - K3y + Ebwb M ~ : - O f , - JT(r

(7.16c)

+ jT(jy)~ _ jT(~y)K4~

9 = ~ + c~.

(7.16d)

(7.16e)

By defining a new output

50 A K4~I + G@ - b A Co,co

(7.17)

and the vectors

the error dynamics (7.16a)-(7.16d) can be written in compact form as:

M~, = - D e , - Jr(r

(7.19a)

)o = Ao?co + BoJ(~y)i, + EoW

(7.19b)

where

[ Aw-K1Cw Ao =

-I(l

0

-K2Cw

-K2

0

-K3Cw

-K3

--Tb 1

Co=[ K4C Bo =

I 0

, Eo:

4+C 0 0

] ,

-I], 0 Eb

.

Next the requirements on the observer gain matrices for stability and passivity of the observer error dynamics is provided.

Stability and Passivity By rewriting the observer error dynamics as (7.19a)--(7.19b) stability of the observer is provided by a SPR-Lyapunov design. The error dynamics is shown in Figure 3 where two new error terms ez and e , are defined as:

cz

A

-- flT(r

eu A J(r

(7.20)

Thus, the observer system consists of two linear blocks, interconnected through the bounded transformation matrix J(r Based on the physical properties of the ship dynamics, we can make the following statement:


121

FIGURE 3. Block diagram of the observer error dynamics.

P r o p o s i t i o n 7.1 The mapping Cz H 5 is state strictly passive and the block 7-ll in Fig. 3 is strictly passive. Proof. Let S1 ~--

I DTMD

(7.21)

be a positive definite storage .function. From (Z19a) we have: 1

: - - 2 i T ( D + DT)D + 5TE~

E'Tr >-- $1 + fl5 TE'

(7.22)

(7.23)

where fl ---- 1/~min(D q- D T) > 0 and/kmin(.) denotes the minimum eigenvalue. Thus, (7. 23) proves that cz ~-~ ~ is state strictly passive [8]. Moreover, since this mapping is strictly passive, post-multiplication with the bounded transformation matrix J(r and pre-multiplication by it's transpose will not affect the passivity properties. Hence the block Tll is strictly passive. [] Passivity and stability of the total system will be provided if the observer gain matrices K1, ..., K4 can be chosen such that the mapping c, ~-* So is passive. This is obtained if the matrices Ao, Bo, Co in (7.19a)--(7.19b) satisfies the KYP Lemma which is stated as below: L e m m a 7.1 ( K a l m a n - Y a k u b o v i c h - P o p o v ) Let Z ( s ) = C(81--J4)--1~ be a n x n transfer .function matrix, where .4 is Huvwitz, (.4,/~) is controllable, and (.4, C) is observable. Then, Z(s) is strictly positive real (SPR) if and

122 7. Nonlinear Passive Observer Design for Ships with Adaptive Wave Filtering only if there exist positive de.finite matrices P = p T and Q = Q T such that: "P*A + *ATT~ = _ Q,

13TJ i) = ft.

(7.24)

Proof: See e.g. Khalil [8]. []

Given a set of observer gains K1,...,/s the existence of the system to satisfy the KYP Lemma can be checked numerically by using the Frequency Theorem, originally formulated by Yakubovich [20], explicitly contained in Gelig, Leonov and Yakubovich [7]: T h e o r e m 7.1 ( F r e q u e n c y T h e o r e m )

Consider the system

= *Ax + 13u

(7.25a)

(7.25b)

y = CTx

where x E ,~n, u E ,~m, y E ,~m and,A, 13, C are real matrices of appropriate dimensions. Suppose the pair (*A,13) is stabilizable and det(jwI,~ - *A) ~ 0 Vw E !}~1. There exists a P : 7~T > 0 with

7) +*ATP < 0,

7~13+C = 0

(7.26)

i f the .following conditions

Re

( C T ( j w I n --

lirno w2Re (C T (jwIn -

,A) -1 13) < O, V~ E !}~1

(7.27)

*A)-I 13) < 0

(7.28)

hold. []

If the Frequency Theorem is satisfied for A = Ao, 13 = Bo, CT = - C o , the mapping c, H 4o (block ~2 in Fig. 3) is SPR and the observer error dynamics system is passive and GES as stated in the following: T h e o r e m 7.2 ( P a s s i v e O b s e r v e r ) The nonlinear observer (7.15a)-(7.15d) is passive. P r o o f . Since it is established that Tll is strictly passive and TI2 is SPR, the nonlinear observer is passive. [] T h e o r e m 7.3 (ISS a n d G E S O b s e r v e r ) The observer (7.15a)-(7.15d) with disturbance w is input-to-state stable (ISS). In addition, the observer error dynamics is rendered GES i.f we disregard the zero-mean disturbance, w~-O.

P r o o f . Consider the .following Lyapunov .function candidate: Vo =

+


123

Time differentiation of Vo along the trajectories of D and Xo yields: ~:o = - D T ( D + DT)D + Xo -T (PoAo + droPo) JCo + 2DTjT(r -- 2DT jT(r

(7.29)

+ 2~ToPoEow.

If the K Y P Lemma is satisfied .for the mapping e, ~ AoT 10o = - - Q o and BoT Po = Co, Vo can be written as:

zo, with PoAo +

17o = --DT(D + DT)D -- YcTQo&o + 2?cToPoEow.

(7.30)

From (7.30) it is seen that

~:o < O,

II~:o[I > 2

HQO1Eow[I

(7.31)

which shows that the observer is ISS. Moreover, in the disturbance free case, w =- O, the equilibrium point of the error dynamics is GES. []

Regarding the choice of observer gain matrices, the tuning procedure can be similar as for the observer for free-floating ships in Fossen and Strand [6]. Pole placement techniques can also be applied.

3.2 Augmented Observer The proposed observer in Section 3.1 can be further refined by augmenting a new state. The augmented design provides more flexibility and it is the basis for the adaptive observer in Section 4. We start by adding a new state, x i , in the observer, which is the low-pass filtered innovation ~: ~cI = - T f i x l

§ ~1= - T ? l x I

+ ~ + C~,~

(7.32)

where x I E !tP and T$ =diag{T$i,Ti2,T$3 } contains positive filter constants. High-pass filtered innovation signals can be derived from x I by:

~.f = - T f l x f + ~ = - T ? l x f -t- (] + Cw~

(7.33)

Thus, both the low-pass and high-pass filtered innovation is available for feedback. Moreover,

-

i+~I:'Y{O(S)T[: ~,{i}(s~ } , l+Tfis ~

\

(i = 1,2,3)

(7.34)

2

The cut-off frequency in the filters should be below the frequencies of the dominating waves in the WF model (7.6a)--(7.6b). Augmented Observer Equations

124 7. Nonlinear Passive Observer Design for Ships with Adaptive Wave Filtering The augmented observer is formulated as: = A~ + KlhYf

(7.35a)

= J(r163 + K2~ + K21xf -t- K2hYf

(7.35b)

= -Tb1~, + Ka9 + Ka~xl

(7.35c)

M~, = - D i - Jr(r

+ JT(r

+ Tthr (7.35d)

~_ jT(@y) (K4y ~- K41xj: -I- K4hYf)

9=~+c~

(7.35e)

where xy is the low-pass filtered innovation vector and Yl is the high-pass filtered innovation given by (7.32) and (7.33), respectively. Here Klh E ,~6x3 and K21, K2h, /(31, K41, K4h C ~3x3 are new observer gain matrices. Augmented Observer Error Dynamics

The augmented observer error dynamics can be written compactly as: (7.36a)

MP, = - D~, - J T (~py)Cj:a

(7.365)

~ca = AaS:a + B,,I(*py);' + Eaw

where

[

(7 7)

Za A_~K4~1 + K41xf + K4hYf -t- G~ -- b ~ Ca~ga,

(7.38)

and

Aa =

A~. - KlhCw - ( K 2 -~- K2h)Cw c~

-KaC~ 0 I 0 0

Ba =

Ca =

--Klh - ( K 2 -t- K2h) I -Ka Ea=

'

(K4 -t- K4h)Cw

0 0 0

Klhrf 1 K2fr) -1 - K21 -Kal

0 0 0 _Tb-1

0 0 0

Eb

(K4 -I- K4h) -F- G

- K 4 h T i I -t- K41

-i]

The signals Sf and x I are extracted from xa by Yl = ChiC. and xf = Cz2a where Cs--[C~

I-T~-'

Passivity and Stability

0],

Cz=[O

0

I

0].

(7.39)


125

Let Va = ~'TM ~ + YcTPaYca

(7.40)

be a Lyapunov function for the observer error dynamics (7..36a)--(7.36b). As before, the cross-terms of ~ and ~a in the expression for Va are cancelled by using an SPR-Lyapunov design, where it is required that: A T pa + PaAa -- - Q a ,

B T pa = Ca

(7.41)

The existence of a Pa = p T > 0, satisfying the KYP Lemma for the augmented observer error dynamics, can be tested numerically for a fixed set of observer gains by using the Frequency Theorem with 7~ = Pa, .4 = Aa, 13 = Ba, CT = - C a and u = e~. If so, the passivity and stability properties are similar as for the observer in Section 3.1 where: Va = --;'T(D + D T ) ; ' -- xTQaxa + 2y:TaPaE~w

(7.42)

and ~'a < 0,

[[Xa[I ~> 2 [[QalEaw[]

(7.43)

4 Adaptive Observer In this section we treat the problem when the parameters of Aw in the W F model (7.6a) are not known. The parameters vary with the different sea-states in which the ship is operating. Gain-scheduling techniques, using off-line frequency trackers and external sensors such as wind velocity and wave radars can be used to adjust to the W F model to varying sea states. However, this can be circumvented by using an adaptive observer design. Since the wave models are decoupled, A and ~ in A~ are diagonal matrices, and we have: A~o(0~)= [

0 _~2

T

I ] ~ [ 0 -2A~ = -diag(0~l)

I ] -diag(0w2)

(7.44)

S}~3

where 0~ = [0~1, T 0~2 ] , 0wl, 0~2 E , contains the unknown wave model parameters to be estimated. We start with the following assumption: A3

( C o n s t a n t e n v i r o n m e n t a l p a r a m e t e r s ) . It is assumed that the unknown parameters ~ and A in the W F model (7.7) are slowly varying and within the range of

0 0 (i = 1, 2, 3) are diagonal matrices of time constants, n = [n~, ny, nz] r are three gyro scale factor errors, and a = [axy, C~z, aye, ayz, azx, azy]r are six small gyro misalignment angles, bl = - b g y r o represents the biases of the gyros. The nonlinear observer of Saleudean [6] is extended to include bias and error update laws according to:

[

~

1

_gT

2

[?)I -- S(~)]

(8.32)

02il

9 1 g~ = -T~ 1~,~ + ~sgn(9) -

(8.33)

1

k = - T ~ - l k + ~diag(g) a)imu sgn(ff])

(8.34)

(~ = - T 3 1 5 + iF(g)Wimu sgn(~)

(8.35)

where 0 0 r(e) =

e2

0 g3 0

gl 0 0 0 0 g3

0 gl 0 g2 0 0

(8.36)

The error model is found by combining (8.29)-(8.35):

1[ [~I + S(g)] ]E , imu l Kl sgo( ,l bl= -Tllt)l

1

- ~gsgn(~)

,837, (8.38)

1 ~ = -T~-I~ - ~diag(g) wire. sgn(9)

(8.39)

1 5 = - T 3 1 5 -- ~r(g) Odimu sgn(~)

(8.40)

Note that the equilibrium points (~/, g, I~1, ~, 5) = (4-1,0, 0, 0, 0). T h e o r e m 8.1 ( E x p o n e n t i a l l y S t a b l e A n g u l a r V e l o c i t y O b s e r v e r ) The equilibrium points (=t=1,0,0,0,0) of the error model (8.37)-(8.40) are exponentially stable. P r o o f . Consider the following Lyapunov function candidate: 1~ T _ I ~ T _ 1 { ( ~ - 1)2 + g g ifT) > 0 V = lt)lT~)i + ~a; n + ~ c~ + -2 (~ + 1)2 + gTg i f ~ < 0

(8.41)

8. Nonlinear Observer Design for Integration of DGPS and INS

145

T h e time derivative along the trajectory of (8.37)-(8.40) is:

=

+

+

+ {

if 0

_-- _ l ~ T T 7 1 ~ 1 _ 1gl~T/sgn(~) -- ~ T T 2 1 ~ -- l ~ T d i a g ( [ ) Wimu sgn(~)

--&Tw31a -- 2&TF(g) Wimu sgn(~) + ~ 1-T [/TklCdimu-]- bl -- gl~sgn(?~)] s~n(?)) = -I~ITTI-lbl - ~TT21~--&TT31(~ -- ~1 [ T g l [

0 is a large time constant, all


147

assumed to be known. The following observer is proposed for (8.47)-(8.53): ~ l = 1~ [(I + A2)aimu + 1~2] -4- ~ l _ [2S(w~e) + S(w'el)] 9t + K2~" + ( R ~ ) T p e

(8.54)

b e ~- R~g/d- K3p s 132 = -T411~2 q-

(8.55)

(I:~Zb)Tv'

(8.56)

= -T~-I~ + diag(aimu)(l~/)Tv "/ = T ^l T~l

(8.57)

- T ~ - ~ + T (a~m.)(Rb) v

(8.58)

) = - t ~ 1] + k4f + g-

(8.59)

~- = ] + kh~

(8.60)

where T is defined in (8.92), K/ = K T > 0, (i = 2, 3) are two 3 x 3 gain matrices, and k4 and k5 are scalar gains. The error dynamics takes the form: v l = 1~ [/k2aimu + ]32] -4- E [(I + A2)aimu + b2] - [2S(w~e) + S(Jet)] 91 - K29t - (R~)T~ e --e

p = R~fi z - g 3 ~ ~ b2 = - T 4 1 l ~

-

(8.61) (8.62)

(I:[tb)Tv'

(8.63)

= - T 5 1 g - diag(aim,)(t~)T9 ' T ^l T l ~=-T~-I/3-T (airnu)(Rb) "r

(8.64)

) = - - t 7 1 / - k 4 ] - 7-

(8.66)

r = ]-

(8.67)

(8.65)

kh~

where E = R~ - l ~ . Now, (8.61)-(8.67) can be written: = f(x, t) + g(t) where

x =

;?,],

(8.68)

,

1~ [/X2aimu + 1~2] -- [2S(w~e ) + S(wlel)] ~l__ Z 2 ~ . / - (R~)T~e RT~ l - K 3 ~ e -

f(x, t) =

-T51g-

-W~l;~

(1%)Tr

diag(aimu) (I?t~)T9 l _

T

^l

T (aimu)(Rb)

T

l

kj-

f - kh~ (8.69)

148

8 Nonlinear Observer Design for Integration of DGPS and INS

and g(t) = [ E [ ( I + A 2 ) a i m u + b 2 ]

0 0 0 0 0 0] T

(8.70)

By recognizing that the rotation matrix R~, as seen from (8.24), is a nonlinear function of the quaternion q, and using the angular velocity observer above, it can be seen that the error matrix E = R~(q) - R~(~I) -~ 0 as t --* oc. Thus, it is established that g(t) --* 0 as t -~ o 0 and a continuos function c~(x) > O, well defined when [x[ > such that for any solution x(t) of the system

5: = f(x, t)

(8.71)

when [x] > { the relation dv(x(t))

< -~(x(t))

(8.72)

hold almost everywhere. Choose the number 7? > 0 such that

F = E{V(x) _< 77} D E{lxl _< ~}

(8.7a)

Then: a) F is an invariant set of (8.71). b) For any solution x(t) of the system (8.71) ~T _> 0 where x ( T ) c F.

P r o o f . See Yakubovich [11]. 9 R e m a r k 8.1 Lemma 8.1 remains true ~f the relation (8.72) becomes valid .for any solution after some time T* > 0, which may depend on initial conditions, provided that this solution o.f (8. 71) is bounded on [0, T*].


149

L e m m a 8.2 ( A s y m p t o t i c S t a b i l i t y o f F o r c e d S y s t e m s ) Suppose that .for the system ~ = f ( x , t ) , there exist a scalar Cl-smooth .function V(x) and a continuos .function ~(x) > 0, such that (I) There exist class 1C .functions ~1,~2 such that V x

~Cl(IXl) ~ V(x) ~ ~2(Ixl)

(8.74)

(II) Along any solution x(t) of the system ~r = f ( x , t ) the following relation holds

dv(x(t)) < -~(x(t))

(8.7~)

limsup I-~xx~] - 0

(8.76)

(Ill)

Ixi-~ ~(x)

Then,

a) A n y solution x(t) o.f the system = f(x, t) + g(t)

(8.77)

where g(t) -~ 0 as t --~ cxD, tends to zero as t --* cx~.

b) Ve > 0 there exist a 5 > 0 and a T** = T**(g(t),e) _> 0 such that ff Ix01 < 5, t h e n V t _>T**: [x(t, x0)] < e. P r o o f . Given g(t). Suppose that Ix(t)[ does not tend to zero as t --~ oo. Then there are two possible cases: limsup Ix(t)l --, oo or [x(t)l < R. t~o 0. T h e n 3 T > 0 such that $--* oO

Vt > T : [g(t)l < e. Due to the assumptions, the derivative of V(x) along any solution of (8.77) satisfies the inequality

_~V(x(t)) ~ - a ( x ( t ) ) + ov(~)., ~gt~ )

(8.78)

Due to the assumption 8.76), there exist a N > 0 such that

Oy(x)

J

< ~(x) A~

Vx e ~ \ B ( 0 ,

N)

(8.79)

150

8. Nonlinear Observer Design for Integration of DGPS and INS Therefore, Oy(x) c Ox 1 < - a ( x ( t ) ) ( 1 - ~) = - l a ( x ( t ) ) Vx 9 ~ n \ B ( 0 , N)

dv(x(t)) < -~(x(t)) +

(8.80) Due to Lemma 8.1 and (8.74), x(t) is bounded. The contradiction proves that Ix(t)[ _< R. 2) Let [x(t)[ be bounded, and assume that x --~ 0 as t ---* ~ . Take any small 5 >0. If x --~ 0 as t -~ oo, 3 {T ,~}n=l, Tn ---* +c~ such that Ix(T~)[ > 5 >0, :: Vn_>l

(8.81)

Due to (8.74) there exist a 51 > 0 and a 52 > 0 such that

E{]x]~_52}cE{V(x) 0. Due to (8.74) there exists a 61 :> 0, and a 52 > 0 such that

E{Ixl _< 62} C E{V(x) < 61} C E{Ixl < ~}

(8.86)

Fix any 6 > 0. Then following Step 2, there exists a T* : Vt > T*, 62 < Ix(t)] < R5 where the relation

tV(x(t))

(8.87)

is valid. Then,

V(x(t)) -- V ( x ( T * ) ) +

~'(x(r

_< V ( x ( T * ) ) - ~1( t - T*)ao where

c~ o =

(8.88)

min a(x). 52 0

(8.133)

K3 > 0

(8.134)

k4, k5 > 0

(8.135)

this results in: l/=-a(x) 0

9 ( n o n l i n e a r c o u p l i n g ) there exist a E (0, 1) and C O .functions 6 : -~ (0, 1 - a] and 71 : 1Rm ---* ~ m such that if F2(c)

def =

-R~I(c)BTpsF(C)

(1.17)

~(s)

aof =

as +

(1.18)

o

6(~))d~

one has ~(F2(c)x)

=

F2~(c)x,

i = 1,...,m

(1.19)

and

IIv(F2(e)(x - e)) - F~(c)xlb~,(~) - I l x l l ~ ( ~ ) IlF2(e)ell~,(c) + Ilell~,,~(~)

< a + 5(IHl2gm(c))

(1.20) for all x E ~(c) and e e KU such that 0 < ~(]le][p,,,(c)) 1

(1.27)

(lo~) (C)} = ~ - ~ lim .~min{P~, 1ol--~+oo

(1.28)

lim e ~ ) ( c ) lo~+oo lOI

-- +oo

(1.29)

= 0

(1.30)

lim I]R~'~ 101---,-~-oo

Using (1.26), (1.28) and (1.29), choose lOi,l(C ) for all lot E [Ioi,1(c), +oo)

( ~~2 ( c ) e ~,

,

+cx~) such that

192

1. Separation Results for Semiglobal Stabilization of Nonlinear Systems

1. P ~ ~

d~j 72(c)p(~?,)(c)

PSF(C) > 0

_

2. w}(l~ ( C) def =

- Qs

(e) > 0

and pick L(c) E J~+ such that, if

um,~,~,i

----- max {]F2i(c)x]} ~e~(c)

7?~(s~)

=

s~min{1, u m ~ # }

~(81,...,8m)

~-

COI(TI1(81),...,~lm(Srn)) 1 ifs>_O ~ ifs L(c). Note t h a t with our choices the region of attraction of the closed-loop system (1.7)-(1.22) is given by ~ e ( e )def {(Vl .02 ) E ~ n 1 ln(1 + Itvl

+ loz

2 X ~ n : tlVlllpsl.,(c) 2

,,,

_< c 2}

(1.32)

Since

lns k

lira ~ = 0 , s-~+oo 8

Yk_>l

and from (1.27), it follows that, for each pair of compact sets $, • lira

sup

~1

to~-*+oo xC,S;aE~V I loI

C / R n,

ln(l+[]x-crl[~(,:o,)(c))}=0

We

conclude that for each pair of compact sets $, ]A2 c ~n one can pick lol E ~( ~~(c) , +oe) sufficiently large in such a way that fie(c) contains

9S • l/Y. For the class of systems (1.7), by using the above arguments one can also recover the fact t h a t semiglobal stabilizability via state .feedback (in

the sense of (1.14)) plus complete uniform observability implies semiglobal stabilization via output feedback. Indeed, under these assumptions, the output can be taken as a state and, thus, one can assume C l ( u , x , t ) = 0 for all x , u and t. However, since throughout this paper we consider a nominal systems which is linear, Teel and Praly's separation result remains still more general t h a n ours as long as C1 (u, x, t) = 0 for all x , u and t and the dynamical model (1.2) is available.


193

3.2.2 Input Saturations Let us consider the system ~(t)

=

A x ( t ) + B2u(t) + B l ~ ( u ( t ) , x ( t ) , t )

y(t)

=

C2x(t) + C l ( u ( t ) , t ) ~ ( u ( t ) , x ( t ) , t )

(1.33)

with B 1 C T ( u , t ) = 0 for all u E /R m and t E /R (the case B 1 C T ( u , t ) 0 for some u and t can be studied as well). We will make the following assumptions ( H 1 ) the pairs (A, B1) and (A, B2) are stabilizable and the eigenvalues of A have nonpositive real part ( H 2 ) the pair (C2,A) is detectable " ( ) : ( 0 , 1 ] ~ 8 7 ~ n and % A E ( H 3 ) there exist continuous j'f u n c tw ns ~"~(') S F , I~f~F Kt + such that

/. II~(u,x,t)ll 2 _< ~llE2ull 2 for some E2 C ~ •

for aU t C ~ ,

x C 1~n and u C 1Ftm such that ]lulI _< A 2. R1 de_fETE2 is invertible and there exists R2 E 8 P p such that

R2 > cl(u, t)C~(u, t) .for all u E ff~m such that Iiul] _< A and for all t E 2R; 3. the .following Riccati equation is satisfied ,~4(lsf) ' ~SF

__ def

~---

( I s F ) "~ ,d A T p ( ~ F) -b .I DSF

P(lSF)fBIBT ~ SF

k

V

B D--1BT~p(ISS) 2x%1 2 } SF

_ r '~r ) q S~F" )

(1.34)

.for all lsF 6 (0, 1] 4-

lim ] ] Q ( ~ ' ) i l = O a n d

l,q F----*O

lim

l~,;~ ---*0

[pq~')l]--O

The interest in the class of systems (1.33) relies on the possibility of taking into account input saturations. As an example, consider ~(t)

=

Ax(t) + B a l ( u ( t ) )

~(t)

=

C2x(t) + Ca2(u(t))

(1.35)

where cyl, a2 are locally Lipschitz continuous, uniformly with respect to t, and such that

C(g2(u, t) - u) ~- C l ( u , t)(ol(U, t) - u) -[- (y3(u, t)

194


for some continuous 6"1(',-) and c,3(.,-), with cra(0, t) = 0 for all t e tg. Clearly, (1.35) can be rewritten as (1.33) with ~(u(t), x(t), t) = C*l(u(t), t ) u(t), B1 = B2 -- B and y(t) --- ~(t) - Cu(t) - rr3(u(t), t). In (1.35) the term c*2(u) may also capture any (unknown) erro affecting C2x, due to torque disturbances, etc. etc. Assumptions (H1)-(H3) are exactly the same invoked in [16], Lamina 3.1 (see also ([8]). By (H2), there exist Qol, P o l c ST ~n such that the following Riccati inequality is satisfied ~k~OI

clef ATpox + P o I A + P o z B 1 B T p o I - CTR~IC2 89 for all l E (0, 1]). From (H2) it follows that, whatever the C o function R~ ) : (0, 1] --, ~ + is, one has (1.42)

IiC2xi]2 + ]IuiI21"

II~(u, ~, t)ll 2
. . . . . h p (SF, 0 j

(1.78)

3j2r~(1) 'qcOI,j - Qq) SF, j

> --

(1.79)

hQ SF, (0 j

200

1. Separation Results for Semiglobal Stabilization of Nonlinear Systems .for some h E IR + and, in addition, =

!imII.~[j[ I < b--,13

o

(1.81)

+oc

- - -

.for j = 1, . . . , n;

(A3) for all l 9 (0, 1]

1 ~ IINAZJ+~,ZJ+I)r ~[ ~Y3 j+lto

(2.15)

In view of (2.11), there exist two positive constants 5 and T such that, for all t > T > to,

'[

t -- to

luld(T)ldr -- 7 sup tul(r) -- Uld(r)I >_ 5 )

(2.16)

T__~tl)

From (2.15) and (2.16), we have

V ( t , g ( t ) ) _ T

which completes the proof.

(2.17) 9

R e m a r k 2.1 If the left-hand side of (2.11) is positive, this "persistent excitation" condition can be met via two different ways: (i) design a s a t u r a t e d single-input Ul with the saturation level as small as possible; (ii) design a (not necessarily saturated) control ul so t h a t U1(t) -- Uld(t ) tends to 0 as t ~ oo. In b o t h cases, ~ = ")'(to) can be made large for some to > 0. We develop this idea in the next section.

4

Output-Feedback Design

Using the observer (2.7), we will design a dynamic output-feedback law (2.4) to drive the tracking error x(t) - xa(t) to zero. To this end, we notice that x, =

,~\_, + g,~_, + k,,_,(t)x,~

v2 < i < n - 1

(2.18)

2. Observer-Controller Design for Nonholonomic Systems

213

Introduce the new variables ~1

~

X n -- X n d A

i~ =

~-1 - (x(n-~+~)d-- k~-~Xnd)

~n

X l -- X l d

~-

V2 < i < n -

1

(2.19)

Let 9 = ( x 2 , . . . ,xn), ~d -= (X2d,... ,Xnd) and ~---- (~1,... ,I,~-1). If all signals are bounded and the conditions of Proposition 2.1 hold, then (2.19) together with Proposition 2.1 implies that ~(t) --'2d(t) converges to 0 if and only if ~(t) converges to 0. Notice that, when ki is selected as in Section 3, ]~ = 0 a.e. for all 1 < i ~ n - 2. Differentiating the variable ~ along the solutions of (2.1)-(2.3) yields

~1 :

(~2 + ]~l~l)Uld -~- ~1Ul ~l- (~2 "~- X(n--1)d -~- ]gl;1)(Ul -- Uld)

(~i+1 + k i l l -- ki-1(~2 + k l ~ l ) ) U l d -~- [~i+1 -t- ki~l - k i - l ( ~ 2 ~- k1~1) -+- X ( n - i ) d -- k i - l X ( n - 1 ) d ]

~n--1 ~n

it2 -- kn--2(~l ~- ]r

--

(Ul -- U l d )

(U2d -- k n - 2 X ( n - 1 ) d l t l d )

Ul -- Uld

(2.20) In the sequel, we will apply the backstepping approach to the transformed system (2.20) in order to design two desired output-feedback control laws u subject to (29 The first one is based on a combined application of backstepping and Jurdjevic-Quinn methods. The second one is a mixture of backstepping and cascade designs.

4.1

Backstepping-Based

Trackers

Noticing the lower-triangular structure of the (~1,... , ~n_l)-subsystem of (2.20) with u2 as the input, the backstepping technique will be first applied to design the control u2. Then, the design of the single-input ul is carried out via the Jurdjevic-Quinn m e t h o d [12]9

Step 1 : Begin with the ~l-subsystem of (2.20) with @ viewed as the virtual control. Let zl = ~1 and write the ~l-subsystem in more compact form Zl = (~2 -~ ~21(~1)) Uld -t- ~lUl -~- r (t, ~1, ~2)(ul - Uld)

(2.21)

Consider the quadratic function

yl =

(2.22)

214


We have

+ zl~lUm + Zlr

1/i = zl (r + r

~1, ~2)(ul - uld)

(2.23)

P e r f o r m the change of variable z2 = ~2 - al(Uld 2t-1 , z l ) where O~1 = - - C l ~ t l2t-1 d Zl - r

(2.24)

with Cl > 0 a design p a r a m e t e r and l > n - 2 an integer. Then, (2.23) implies Yl = --~lUldZl ~ 2l 2 + zlz2Uld + z1~1u1 + Zlr

~1, @)(ul -- Uld)

(2.25)

For later reference, the z2-dynamics satisfy "~2

~"

(~3 -r-W2\ -- .I. [U2l--1 "~ /" ~ U ld"t-~2~, -I t )~1 ~y U1 ld ' U2l--3~ ld ld, t~l,~2}) -[-r ( t, ~I, ~2, ~3)(Ul -- Uld)

(2.26)

where qo2(t) is d e p e n d e n t on Uld(t) and kl.

Step i (2 < i < n - 2) : Assume that, at Step i - 1, we have designed i - 1 virtual control functions aj (1 < j < i - 1) and o b t a i n e d new variables , 2/--1

2(/--j)+1

zj+l =~j+l --aj(Uld , . . . ,Uld assumed that, for all 1 _< j _< i, 2:j

:

(~j

--,

, 2/--1

-t-1-i-~ji, Uld

(j--l)

Uld

2t-a-

,Uld

Uld,...

,~l,.-.,~j).Furthermore, (j-l)

2(l-j)+l ,t~ld

t~ld

f

,~1,...

,

itis

Cj))

Uld

/

%

+~j(t)~lUl + Cj(t, ~1,--. , ~j+l)(Ul - Uld)

(2.27)

W i t h respect to the solutions of (2.27), the time derivative of V/_I ---- l z l 2 ~- . . . L

-1- l z 2

2

-

1

(2.28)

satisfies i--1

~-1

-- E

i--1 CjUldZj ~ Zi--1Zi~ld

j=l

j=l

i-1 "~ E

ZjCj@, ffl,.--,

f f j + l ) ( U l -- Uld)

(2.29)

j=l

where ~ol(t) = 1. We wish to prove t h a t the above properties hold for the (fix,... ,ff~)subsystem with ffi+l considered as the virtual control. To this end, consider the quadratic function t~ = V i _ l ( Z l , . . . , z~-l) + l z ~

(2.30)


215

In v i e w of (2.27) and (2.29), differentiating V~ w i t h respect to t i m e yields i--1

v,-

i--1

CSUldZj + E 5=1

i--1

ZJ~J(t)~lUl Ar E

j=l

+Zi (~iq-1 -~- r

Uld i

~- Zi~gi~lUl ~- Zir

2/--1

Letting Zi+l -- ~i+1 - aikUld 2/-1

a i = --C~Uld

ZJCj(Ul -- u l d ) + Zi--lZiUld

5=1

~

,.."

,"

-- Uld)

2(l-i)-F1 (i--1) , Uld Uld

2/--1

zi -- z i - 1 -- W~tUld

2(/--i)-bl

,...

,Uld

, ~1,

(2.31)

9 - 9 , ~i)

where

( i - 1 ) .~

,ql,...

Uld

,~i)

(2.32)

w i t h ci > 0, it follows from (2.31) that i

.o

=

i

V'

_ z__ c s u l d21 z5

2

+

ZiZi+lUl d

+

5=1

i

E ZJ~J~Ul "~ E ZJr j=l 5=1

Uld)

(2.33) S t e p n - 1 : At this step, w e c o n c e n t r a t e on the design of the true control u2. Consider the L y a p u n o v function candidate

1 2

Vn-1 = V n - 2 ( Z l , . . . , zn-2) -[- ~Zn_ 1

(2.34)

F r o m Step n - 2 and (2.20), it holds n--2

yn_l

=

n--1

-- ~ CjU211dZy-~ Zn_2Zn_lUld -~5=1 5=1 n--1

+

zsr

-

ld) + Z _l (u2 +

zs~#~lu1 n-1)

(2.35)

j=l

w h e r e #J~-I is a function d e p e n d e n t on (t, u l , ( 1 , . 9 9 , ~,~-l), r is a function of (t, ~ 1 , . . . , ~n-1) and ~ n - 1 is a function of U l d ( t ) and its derivatives up to order n - 2. B y choosing the control law u2 as

U2 = --Cn--1U211dZn--1 -- Zn--2Uld -- C n - - l ( t , U l , ~ l , . . - , ~ n - - 1 )

(2.36)

w e obtain n--1

Yn-1 : - E 5=1

n--1

Cjlt2ldZ2 -~ E

n--1

ZJ~PJ~Ul AF E

j=l

ZJCJ(Ul - uld)

(2.37)

5=1

S t e p n : In order to design a control law for u l , let us consider t h e L y a p u n o v function

Vn = V n - l ( Z l , . . .

1 2 , Zn-1) -}- "~n

(2.38)

216


Along the solutions of (2.20), with (2.37), the time derivative of Vn satisfies -

=

--

CjUldZ j -Ij=l

ZjqPj~lUl -Jr- ~n -["=

Zjq~j

(U 1 - -

Uld ) (2.39)

j=l

This leads us to choose the control law Ul as

U 1 = Uxd -- O" ~n -[-

Zjr

(2.40)

j=l

where a in C 1 is a saturation function such t h a t or(r) = r for small signals r, rot(r) > 0 for all r e IR \ {0} and s u p r e ~ Icr(r)l = (~m < OO. The saturation level crm is selected to meet the input constraint (2.2) with U l m a x > supt_>O {Uld(t)[ : = U l d , m a x . Under this choice (2.40), (2.39) gives

2 nl ~'~n ~- -- E j=l

CJUldZj "~ E

( Zj~Dj

--

nl ~n ~- E

j=l

)( Zj(/)j

nl (7

~n 71- E

j=l

) Zj~)j

j=l

(2.41) Finally, we are in a position to formulate the following tracking result. P r o p o s i t i o n 2.2 A s s u m e that the reference trajectories Xid(t) (2 _< i _~ n) and reference input ud(t) are bounded. It is .further assumed that the derivatives of uld(t) up to order n - 2 are bounded on [0, cx~). If there exists a constant eu > 0 such that liminf lUld(t)[ > t--*~

(2.42)

~u ,

then, cym can be tuned towards any level of size ul m~x - Uld,max SO that, .for any initial tracking error x(O) - xd(O) C IR n and any initial condition ~(0) E IR '~-1, the trajectory ( x ( t ) - - x d ( t ) , ~(t)) of system (2.1), (2. 7), (2.36) and (2.r is bounded with the .following properties lim Ix~(t) -~,~_~(t) - k~_i(t)x,~(t)l

=

0

lim Ix(t) - xd(t)[

=

0

t----*oO

t---*oO

Furthermore, the convergence rate in (2.r

V2 < i < n -

1 (2.43) (2.44)

and (2.r162 is exponential.

P r o o f . We first prove the boundedness property. By assumptions, we can choose a sufficiently small constant am such t h a t the conditions of Proposition 2.1 hold. As a consequence, the observation error ~(t) exponentially


217

converges to 0 and thus the property (2.43) is satisfied. We can rewrite the ~-system (2.20) in more compact form

4

=

f(t,-~)

~,

=

-c~

(2.45) 4- + ~ z 5 r

(2.46)

j----1 It is directly checked that f is linear in ~ for any fixed t. Therefore, the closed-loop solutions ~(t) = (~(t),4~(t)) and x(t) - xd(t) do not exhibit finite escape. Given a positive constant e, by means of the Schwartz inequality, (2.41) gives

%

1

< -

,v'rc5

j=l n--1

(n l )

-

-

+ Z

j=l

zs,

n--1

/

o- 4n + y-s zsr j

j=l (2.47)

j=l From (2.42), there exist a time instant to > 0 and design parameters cj's such that

cju~(t) - s > 0 Vt >_ to

(2.48)

Then, it follows from (2.47) that n--

1

Vn(z(t)) < V,~(z(to))+j~lftoll~3gluil2dT.=

t

(2.49)

With the help of Proposition 2.1, (2.49) completes the proof of the first statement since the ~j's and ul are bounded. Back to the inequality (2.47), using the fact that ai = 0 if 41 . . . . . 4i = 0, a direct application of Barb~lat lemma [15] and Proposition 2.1 yields the last statement (2.44). 9

4.2

A Modification

In the first step, the Jurdjevic-Quinn method was used to design a controller ul to diminish the effect of the (Ul - Uld)-related terms on the (~1,... , 4n_l)-subsystem of (2.20). In this subsection, we pursue the line of a cascade design. T h a t is, we design ul in such a way that xl(t) - Xld(t) converges to zero, regardless of the (41,... ,4,~-l)-Subsystem design. For

218


instance, looking at the ~ - s u b s y s t e m of (2.20), we can simply choose the controller Ul as U l = U l d -- O ' ( ~ n )

(2.50)

where (7 is a saturation function as defined above. The global output-feedback tracking result is stated below. P r o p o s i t i o n 2.3 Assume that the re.ference trajectories Xid(t) (2 _~ i _~ n) and re.ference input ud(t) are bounded. It is .further assumed that the derivatives o,f u d(t) up to order n -

2

bounded on

[0,

If

there exist

an integer I >_ n - 2 and a constant ~ > 0 such that 1 fto+t litra~inf ~ Jto lUId(T)IdT

>

0 Vto _> ~

liminf i f t---~oo

t~

lUld(V)12ldT

>

(2.51)

0

t Jto

then, .for any initial tracking error x(O) - X d ( O ) E ]R n and any initial condition ~(0) C IR ~-1, the trajectory (x(t) - x d ( t ) , ~ ( t ) ) of system (2.1), (2.7), (2.36) and (2.50) is bounded. Furthermore, (2.43) and (2.4~) hold with exponential convergence. P r o o f . As in the proof of Proposition 2.2, we can prove that the closed-loop trajectories do not exhibit finite escape. Thanks to the choice (2.50), the closed-loop solution ~ ( t ) satisfies ~,~ = -c~(~,~)

(2.52)

and converges to zero when t --~ cx). Moreover, there exist a finite time instant t ~ (probably dependent on the initial condition ~n(0)), two positive constants Pl (dependent on the initial condition ~,~(0)) and P2 (independent of the initial condition ~n(0)) such that I~n(t)l _~ Pl e x p ( - p 2 t )

Vt > t~

(2.53)

Notice that t ~ -- 0, pl --- [~n(0)l and P2 = c~0 if (r(r) = (~0r for c~0 > 0. Without loss of generality, we may assume that Pl e x p ( - p 2 t ) is so small for t > t ~ that c~(~n(t)) = ~n(t) and lUl(t) -- Uld(t)l _ t ~

(2.54)

With the aid of (2.54), pick a sufficiently large to >_ t ~ so as to check (2.11) and (2.51). As a result, by Proposition 2.1, I~(t)l _< ql e x p ( - q 2 t )

Vt _> to

(2.55)


219

where ql > 0 is a constant which depends on ~(0) and q2 > 0 is a constant which does not depend on g(0). Let us now look at eq. (2.37). By virtue of (2.54) and (2.55), noticing the fact that every Cj is overbounded by a~l + ai21zl with (hi1, a~2) a pair of positive constants, there exist positive constants c, al, a2, bl and b2 such that 1

V,~-I 0 ,

(2.61)

which corresponds to the center of mass of the knife-edge moving along the circle centered at the origin of unit radius with uniform angular rate. In the transformed x-coordinates, the desired trajectory is: Xld(t )

:

t,

X2d(t) = 0,

X4d(t)

=

Utd(t) = 1 ,

X3d(t) = 1 ,

X5d(t) =U2d(t) = 0 .

(2.62) (2.63)

For this particular reference trajectory (which is a straight line in the new xcoordinates), a global state-feedback tracking control law has been derived in [9] via a recursive approach. For simulation purposes, we recall that the coordinate and feedback transformation leading to (2.60) is Xl

=

r

X3

=

x~sinr162

X5

=

k, cos r + $c sin r + r

x4= sin r + y~ cos r

(2.64)

T2

Vl V2

x2=x~cosr162

m

=

7-1

-- + m

T2

. ( - x c sm r + Yc cos r

q~2(xccos r + yc sin r

For the new system (2.60), we consider y = (Xl, x3) as the o u t p u t and assume that the other states (x2, x4, xs) are unavailable to the designer.


221

We first introduce an observer to reconstruct the unmeasured state x4. Introduce a new variable ~a = x4 - k3xl with ]r > 0 a design parameter, which satisfies

~a=-ka(a- k~xl+vl

(2.65)

Then, the reduced-order observer is introduced .2.

~3 = -ka~'3 - k 2 x l + Vl

(2.66) ,&

which leads to an exponentially stable linear dynamics (3 = -k3~3 where

5=

3-g3.

Consequently, the unmeasured state x4 = ~a + k a x l + ~'a can be exponentially recovered via the observer (2.66). Next, we turn to the observer design for the unmeasured states (x2, xs). Guided by the development in Section 3, introduce the new variables (l = x2 - klx3

,

(2 = x5 - k2x3

(2.67)

where K = (kl, k2) is a vector of design parameters, which are constant here because the sign of Uld = X4d : 1 does not change. Direct computation yields ~1

=

(2+k2xa-kl((l+klx3)x4

42

=

V2 -- k 2 ( ~ l - + - k l x 3 ) x 4

(2.68)

Since x4 is not measured in the present case of e x t e n d e d chained form (2.60), in contrast to the observer (2.7) for the chained form case, the following observer is introduced in which the estimate x4 := ~'3 + k 3 x l of x4 is used in place of x4: ~1

~-

~ 2 - I - k 2 x 3 -- ~ l ( ~ ' l - } - k l X 3 ) ~4

~2

=

v2 - k 2 ( g l + k l x 3 ) ~ 4

(2.69)

Letting ~'1 = ~1 - ~'1 and ~-2 -= ~2 - ~ , (2.68) and (2.69) imply

(kl

--,-k20

-

k2

(~1 (x4 - 1 ) + (~'1 + klXa)g3)

(2.70) Clearly, we can pick two parameters kl and k2 such that A is an asymptotically stable matrix. For simulation use, take kl = 2 and k2 = 1. T h e second term in the last brackets of (2.70) is new comparing with the chained form

222


case, a special class of the so-called Chaplygin (kinematic) form. Bare in mind that the first term depends upon x4 which is a state component and thus, unlike in the case of chained form, is not free to choose. These terms together prevent us from applying Proposition 2.1 in order to conclude the exponential convergence of the observation error (~1, ~2). Nevertheless, as we will show below, we can still design an output-feedback control law to achieve the global tracking task. Before designing such a controller, we introduce a system of coordinates under which our synthesis is developed N

~1 ~-- X 3 -- X 3 d ,

~3 :

r

~2 ---- ~1 -- ( X 2 d - -

~2 -- ( X 5 d -- k 2 X 3 d )

= r

,

klX3d)

~4 --~ Xl -- X l d

(2.71)

-- (~4d -- k 3 ~ )

Then,

/

r

=

(r + k1r

+ ~1x4 + (r + k1r

+ (r + k1r162

~2

:-

~3 "[- ]g2~1 -- kl(~2 -[- k l ~ l ) ( 1 at- ~5 at. k3~4)

~3

=

v2 - k2(r

+ k3r

+ k1~1)(1 + r + k3~4)

(2.72) where x4 ---- 1 + r + k3r + ~'3

(2.73)

Notice that the states of system (2.72) are measured and available for feedback design. If ~'(t) - ~(t) goes to 0 as t --~ oo, then ~(t) converges to 0 if and only if x(t) - xd(t) does. In other words, we have converted the global output-feedback tracking problem into a global state-feedback regulation issue. In the sequel, the design of our desired dynamic output-feedback controllers vl and v2 will be developed according to the second m e t h o d proposed in Section 4.2. First, we observe that the (~4, ~s)-subsystem of (2.72) can be easily made GES (globally exponentially stable) at the origin. Indeed, a direct application of integrator backstepping generates a Lyapunov function candidate

1 2 1+

W1 -- ~

where c4 > 0 is a design parameter.

(@ + k3~4 + c4~4) 2

(2.74)


223

To render IPdl nonpositive when ~3 = 0, we are led to choose the control law vl = -c5(r

+ k3r + c4~4) - r - c4(r

+ k3r

c5 > 0

(2.75)

which gives

r162 = -c4r

- c5(r + k 3 6 + c4r 2 + [r + (kz + c4)(r + k3r + ~4r

~-3

Now, consider the quadratic Lyapunov function 1~

(2.76)

Then, we have

W2

=

--C4r

-- C5(r ~- k3r 4 -[- c4r

2 - k3~"2

4-r

Jr- (k3 -[- c4)(r

~- c4r

-~- k3r

(2.77)

Hence, 1)r is negative-definite if c4 > 0.5e -1, c5 > 0.5e-l(k3 + c4) 2 and k3 > c, with e > 0 being arbitrary. In addition, given any c > 0, we can select the design parameters c4, c5 and k3 appropriately such t h a t

~/i/'2(r

r

~'3) < -cW2(r

r

~"3)

(2.78)

From (2.78) and (2.76), it follows that there exist two positive constants 51 and 52 such t h a t

1(r

I < 511(r

)

(2.79)

Next, it remains to design a suitable control law for the input v2. As above, we approach this goal by an application of backstepping to the (r r r subsystem of (2.72). Without going into details, a direct application of backstepping generates a Lyapunov function

where zl = r

z2 = @ - a ~ l ( r

Ot1

=

--(C 1 + kl)r

0~2

=

--C2Z2

--

Zl

z3 = r

, --

r

- 0t2(r

r

r

and

C1 > 0

k2r

- c1(r

(2.81) -[- k1r

q- r A- k3r )

(2.82)

If we choose the control law V2 :

Oa2 ~t -

--C3Z3-- Z2 -4- (k2 -4- ~ 1 }t,~2 A- klr

+0(~2,. "HT-~((,3 +/;2r ~,2 . OOL2 - *

+ ' ~ ' 4 ((,5 + k3r

-- k1(r 00~2

+ k1r

q- r -4- k3r A- r + k3r 2

+ -~--. (vl - k3r - k3r

(2.83)

224


with c3 > 0 a design parameter, it follows t h a t

~0~2 ~--

- - e l z2 -- C2z2 -- C3 z2 -{- Zl(~2 -~- ]g1~1)(~5 -~- k3~4) -- z 3 " ~ 4 ~3

C90~2

+(zl + (cl + kl)z2 - ~ - 1 z3)(~lx4 + (r + k1r

(2.84)

Since the matrix A in (2.70) is asymptotically stable, there exists a unique solution P = p T > 0 to the Lyapunov m a t r i x equation

P A + A T p = --I2

(2.85)

w h e r e / 2 is the 2 x 2 unit matrix. Consider now the quadratic function

V(zl,z2,z3,~l,~2)

-~- Y 3 ( z l , z 2 , z 3 )

7t- (~1, ~2)P(~l,

~2) T

(2.86)

In view of (2.71), (2.73) and (2.84), from (2.79), it follows the existence of a positive constant ~ > 0 and two exponentially converging signals a(t) > O, b(t) >_ 0 such t h a t

f/ < _ - ( ~ - a ( t ) ) V + b ( t ) ,

Vt>O

(2.87)

From (2.87), like in the proof of Proposition 2.1, we can invoke the variation of constants formula and Gronwall-Bellman inequality to conclude the exponential convergence of V(zl (t), z2 (t), z3 (t), ~l(t), ~2 (t)) and, therefore of the tracking error x(t) - xd(t), to 0 as t goes to ~ . The simulations in Figure 1 were obtained with the following values of design parameters and initial conditions

kl = 2 ,

k2 ----k3 ~- 1 ,

r

Cl ----c2 = c3 = c 4 = 1, r

1(0) =

2(0) =

c5 = 3,

(2.88)

3(0) = 1

The responses indicate t h a t the tracking error exponentially converges to 0 under mild control effort.

2. Observer-Controller Design for Nonholonomic Systems (xc(t}-sin(t),

1.

y c(I)+cos(t),

~,(t)-t)

",

x (t)-~n(t) y~(tI+cos(t)

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

-0 5

9 0

2

4

8

t

8

J

10

225

ll2

1'4

12

14

J

16

i

18

secs

2

~o

i

f....

......

"

9...........

I .................. % 6

8

10

16

18

sees

FIGURE 1. Global output-feedback tracking of the knife-edge (2.59).

6

Conclusions and Future Work

The problem of global output-feedback tracking was addressed for a class of nonholonomic systems in this paper. T h e presented design methodology is a natural extension of our recent state-feedback tracking algorithms proposed in [8, 9, 10]. More specifically, when considering a flat o u t p u t of a chained-form system in this class as the only accessible measurements, we first design a Luenberger-like reduced-order time-varying observer in order to recover the remaining unmeasured states. Under a condition of persistant excitation on the reference input Uld(t), the observation error was shown to converge to zero at an exponential rate if ul is chosen appropriately. Then, based on this observer and using the chained form structure, two constructive methods involving the backstepping technique have been proposed to design desired output-feedback tracking controllers. An extension to the simplified dynamic model was discussed via a simple b e n c h m a r k nonholonomic knife-edge system. It is of interest to mention t h a t an arbitrary s a t u r a t i o n level can be imposed on the control input ul. However, we are unable to extend our approaches to cover the case when the other control input u2 is saturated. T h e chained form represents a good model for m a n y nonholonomic mechanical systems in the ideal case, t h a t is, when the disturbances are ignored. However, almost all physical systems are subject to some kind of disturbance. It turns out to be necessary to examine the robustness of the global tracking p r o p e r t y which was guaranteed by our current trackers. In short, the following problems are meaningful from a practical point of view and deserve our further investigation:

226


1. What happens if all control inputs of a nonholonomic system in chained-form are subject to some L~-type constraints? In relation to the knife-edge example, further difficulties arise here in the boundedness of the controller for the dynamic extension of the considered chained models. 2. In case when uncertainties occur in nonholonomic mechanical systems, how do we give a good mathematical description of these uncertainties? If the nominal system is transformable into a chained form, how will these uncertainties affect the coordinates and feedback transformation and the stability obtained from the undisturbed chained form? We are also interested to know how to modify our proposed tracking controllers in [8, 9, 10] and in this paper in order to maintain stability properties in the presence of uncertainties. 3. Experimental work ought to be done on some laboratory-type robots so as to test the effectiveness of our proposed tracking approaches. 7

REFERENCES [1] R. W. Brockett, Asymptotic stability and feedback stabilization, in: R.W. Brockett, R.S. Millman and H.J. Sussmann, eds., Differential Geometric Control Theory, pp. 181-191, 1983. [2] C. Canudas de Wit, B. Siciliano and G. Bastin (Eds), Theory of Robot Control. London: Springer-Verlag, 1996. [3] M.-S. Chen, Control of linear time-varying systems by the gradient algorithm, Proc. 36th IEEE Conf. Dec. Control, pp. 4549-4553, San Diego, 1997. [4] J.-M. Coron, Stabilizing time-varying feedback, NOLCOS'95, Tahoe City, CA, pp. 176-183, 1995. [5] G. Escobar, R. Ortega and M. Reyhanoglu, Regulation and tracking of the nonholonomic double integrator: A field-oriented control approach, Automatica, 34, pp. 125-131, 1998. [6] M. Fliess, J. Levine, P. Martin and P. Rouchon, Flatness and defect of non-linear systems: introductory theory and examples. Int. J. Control, 61, pp. 1327-1361, 1995. [7] Z. P. Jiang, Iterative design of time-varying stabilizers for multi-input systems in chained form, Syst. Contr. Letters, 28, pp. 255-262, 1996. [8] Z. P. Jiang and H. Nijmeijer, Tracking control of mobile robots: a case study in backstepping, Automatica, 33, pp. 1393-1399, 1997.


227

[9] Z. P. Jiang and H. Nijmeijer, A recursive technique for tracking control of nonholonomic systems in chained form, to appear in: IEEE Trans. Automat. Control, Feb. 1999. [10] Z. P. Jiang and H. Nijmeijer, Backstepping-based tracking control of nonhotonomic chained systems, Proc. European Control Conference, 1-4 July, 1997, Brussels. [11] Z. P. Jiang and J.-B. Pomet, Combining backstepping and timevarying techniques for a new set of adaptive controllers, Proc. 33rd IEEE Conf. Dec. Control, pp. 2207-2212, Florida, I994; also in: Int. J. Adaptive Contr. Signal Processing, vol. 10, pp. 47-59, 1996. [12] V. Jurdjevic and J.P. Quinn, Controllability and stability, J. Diff. Eqs., 28, pp. 381-389, 1979. [13] Y. Kanayama, Y. Kimura, F. Miyazaki and T. Noguchi, A stable tracking control scheme for an autonomous mobile robot, Proc. IEEE 1990 Int. Conf. on Robotics and Automation, pp. 384-389, 1990. [14] W. Kang and A. J. Krener, Nonlinear observer design, a backstepping approach, preprint, 1998. [15] H. K. Khalil, Nonlinear Systems. Prentice Hall, Upper Saddle River, N J, 2nd edition, 1996. [16] I. Kolmanovsky and N. H. McClamroch, Developments in nonholonomic control problems, IEEE Control Systems Magazine, Vol. 15, No. 6, pp. 20-36, 1995. [17] I. Kolmanovsky and N. H. McClamroch, Hybrid feedback laws for a class of cascaded nonlinear control systems, IEEE Trans. Automat. Control, 41, pp. 1271-1282, 1996. [18] M. Krstid, I. Kanellakopoulos and P. V. Kokotovi6, Nonlinear and Adaptive Control Design. New York: John Wiley & Sons, 1995. [19] E. Lefeber, A. Robertsson and H. Nijmeijer, Output feedback tracking of nonholonomic systems in chained form, preprint, October 1998. [20] R. M. Murray and S. Sastry, Nonholonomic motion planning: steering using sinusoids, IEEE Trans. Automat. Contr., 38, pp. 700-716, 1993. [21] E. Panteley and A. Loria, On global uniform asymptotic stability of nonlinear time-varying systems in cascade, Systems 8z Control Letters, 33, pp. 131-138, 1998. [22] C. Samson and K. Ait-Abderrahim, Feedback control of a nonholonomic wheeled cart in Cartesian space, Proc. of the 1991 IEEE Int. Conf. Robotics and Automation, Sacramento, pp. 1136-1141, 1991.

228


[23] G. Walsh, D. Tilbury, S. Sastry, R. Murray and J. P. Laumond, Stabilization of trajectories for systems with nonholonomic constraints, IEEE Trans. Automat. Contr., 39, pp. 216-222, 1994.

A Separation Principle for a Class of Euler-Lagrange Systems Antonio Lorfa v and Elena Panteley t ~C.N.R.S., UMR 5228, Laboratoire d'Automatique de Grenoble, ENSIEG, St. Martin d'H~res, France. *I.N.R.I.A., Rh6ne Alpes, Projet BIP, St. Martin d'H~res, France.

1

Introduction

The solution to the state feedback tracking control problem of fully damped Euler-Lagrange systems (in particular, rigid-robot manipulators) has been known from many years now - for a literature review, see e.g. [22, 27] -. Nevertheless, a drawback of many of the available results in the literature is that they require the measurement of joint velocities which may be contaminated by noise. An ad hoc solution, often taken in practice, is to numerically differentiate the joint positions. However, it has been shown experimentally [2] that this method is inefficient for high and slow velocities. This has motivated researchers in the robotics community to solve the global output feedback control problem of robot manipulators. This problem has been open for many years now. As in the regulation control problem, an approach alternative to numerical differentiation, that has been widely considered in the literature, is to design an observer that makes use of position information to reconstruct the velocity signal. Then, the controller is implemented replacing the velocity measurement by its estimate. Even though the certainty equivalence does not apply for general nonlinear systems, the rationale behind this approach is precisely that the estimate will converge to the true signal, and this should in turn entail stability of the closed loop. As far as we know, some of the earliest works on state estimation for robot manipulators are [20, 17] and some of the references therein. See also [21] for some interesting experimental results. In [17] the authors used a nonlinear observer that reproduces the robot dynamics, in a P D plus gravity compensation scheme. The authors prove the equilibrium is locally asymptotically stable provided the observer gain satisfies some lower bound determined by the robot parameters and the trajectories error norms. See also [7] where a sliding mode approach is taken.

230

3. A Separation Principle for a Class of Euler-Lagrange Systems

The authors of [4] proposed a linear observer-computed torque scheme which exploits the feedback linearizing property of the computed torque scheme providing an efficient tuning technique. Later, using the same tuning idea [3] presented a systematic procedure that exploits the passivity properties of robot manipulators into the design of controller-observer systems to solve both the position and tracking control problems. Local asymptotic stability was proved for sufficiently high gains. Later in [16], based on a computed torque plus PD-like controller first appeared in [27], we added an n-th order "approximative differentiation filter", to eliminate the necessity of velocity measurements. In t h a t paper we proved semiglobal asymptotic stability of the closed loop system hence showing that the domain of attraction can be arbitrarily enlarged by increasing the filter gain. Some more recent results addressing the same problem are for instance: [11, 18], and [19]. The authors of [11] proposed the first adaptive controller for flexible joint robots by using only position measurements. Simultaneously, [18] proposed a globally asymptotically stable observer-based controller needing only link (position and velocity) measurements and later in [19] they extended this result to link position feedback. The approaches mentioned above, rely on a Lyapunov design, that is, the principal aim is to design an observer and a controller such that, the total time derivative along the closed loop trajectories, be negative definite. A common drawback however, is the appearance of high order terms in the derivative of the Lyapunov function, and which can be dominated only for small states. In the best case, one can prove that the region of attraction can be enlarged for large control gains. As an a t t e m p t to bound the cubic terms in the time derivative of the storage function we presented in [13], as far as we know, the first smooth controller which renders the one-degree-of-freedom (dof) EL system. Our approach relies on a computed torque plus P D structure and a nonlinear dynamic extension based on the linear approximate differentiation filter. The main innovation in our controller, which allows us to give explicit lower bounds for the controller gains, in order to ensure GUAS, is the use of hyperbolic trigonometric functions in a Lyapunov function with cross terms. Global uniform asymptotic stability is ensured provided the controller and filter gains satisfy some lower bound depending on the system parameters and the reference trajectory norm. Unfortunately, the performance of our approach can be ensured only for one dof systems and nothing can be claimed for the general multivariable case. Independently, in [6] Burkov showed by using singular perturbation techniques, that a computed torque like controller plus a linear observer is capable of making a rigid joint robot track a trajectory starting from any initial conditions. The main drawback of this result is that no explicit bounds for the observer and control gains can be given. Thus, the author proves in an elegant way, the existence of an output feedback tracking controller t h a t


231

ensures GUAS. Later, A.A.J. Lefeber proposed in [10] an approach which consists on applying a global output feedback set point control law (for instance an EL controller) from the initial time to until some "switching time" t~, at which it is supposed t h a t the trajectories are contained in some pre-specified bounded set. At time t~ one switches to a local o u t p u t feedback tracking control law (such as any among those mentioned above). The obvious drawback of this idea is that the controller is no longer smooth, furthermore, the switching time may depend on bounds on the unmeasured variables. The results contained in I10] concern the existence of the time instant ts such that the closed loop system is GUAS. Most recently, based on the controller of [13] the authors of [28] proposed a dynamic output feedback controller for the multivariable case. The Lyapunov stability proof for the closed loop system is carried out relying on a nonlinear change of coordinates (See Eqs. 35 and 39 of that reference). This change of coordinates is not invertible, and therefore the controller the authors propose in [28] is not implementable without velocity measurements, for any intial conditions of the dynamic extension. In [5] an elegant alternative result for one-degree-of-freedom systems was reported. The controller proposed in [5] is based upon a global nonlinear change of coordinates which makes the system affine in the unmeasured velocities. This is crucial to define a very simple controller which has at most linear growth in the state variables, as a matter of fact the proposed controller is of a P D + type. This must be contrasted with the exponential growth of the control law proposed in [13], due to the use of hyperbolic trigonometric functions. Hence, from a practical point of view, the controller of [5] supersedes by far that of [13]. The work of [5] suggests that more attention should be payed to the modelling stage of the control design. As far as we know, the position tracking control problem stated at the beginning of this section for any initial conditions and for n-degrees-offreedom EL systems still remains openr In this chapter we will present a solution to this problem, for a class of n-degrees of freedom EL systems (including robot manipulators). The systems belonging to this class, allow a factorisation which does not exhibit the Coriolis effects in the dynamic model. Inspired by [5, 9, 24], we consider a kinematics model which in other words, provides a global change of coordinates. As it will become clear later, the model considered here covers a fairly wide class of EL systems, however, in general, it is very difficult to find such factorisation. Our main result is to prove that, for this class of EL systems, it is possible to design a state observer and a state feedback controller independently of each other. T h a t is, we will establish a separation principle for a class of EL systems. Our results are an extension of [14] to the tracking problem, i.e., to the time-varying case. In a more general context, some work on separation principles for nonlinear systems has been done recently for local stabilisation of input-output linearizable systems [1] and for the case of

232


nonaffine systems in [12]. Our results differ from those of in the latter references in that, neither high gains nor bounded feedbacks are required. Moreover, we consider here time-varying systems. This chapter is organised as follows. In next section we present the model we consider here. In Section 3.1 we construct a state estimator and prove global exponential stability in closed loop with the plant dynamics and kinematics. In Section 3.2 we construct a state feedback controller and prove global exponential stability. In Section 3.3 we establish our separation principle, i.e., we prove that, if the state-feedback control law is implemented using the state estimates, the overall closed loop system is uniformly globally asymptotically stable (UGAS). Finally, in Section 4 we discuss our results, when applied to robot manipulators. N o t a t i o n s . In this chapter, we use 1[-[I to denote the Euclidean norm of vectors and induced norm of matrices. The symbols km and kM are used for lower and upper bounds on [[K[[. The symbols := and =: mean "equal by definition". A continuous function/3 : N_>0 --~ 11~>0 is said to be of class /~ (/3 E )U), if j3(s) is strictly increasing and/3(0) = 0

2

Model and Problem Formulation

We consider in this chapter, fully actuated Euler-Lagrange systems with generalised coordinates q E N ~, and control inputs u C 11('~, i.e., --

dt

Oq

--

u

(3.1)

where the Lagrangian s ~)) := T(q, O) - ~;(q). It is assumed that the kinetic energy function is of the quadratic form,

:r(r O) = 10TD(q)q where the inertia matrix D(q) C N nx'~ is positive definite and uniformly bounded. The potential energy function, ~;(q), is assumed to be uniformly bounded from below, i.e., we assume that there exists a real number c, such that l;(q) > c for all q E ll~'~. As it is well known, using the Christoffel symbols of the first kind [26, 24], the system (3.1) can be rewritten in the form

D(q)~j + C(q, dt)dl + g(q) = u

(3.2)

where, in our notation, the matrix C(q, O) contains the terms corresponding to centrifugal and Coriolis effects, and the vector g(q) :-- ~ Oq As it is discussed in [13], a common drawback of o u t p u t feedback controllers relying on Lyapunov design, is that certain 3rd order terms t h a t


233

appear in the Lyapunov function derivative, cannot be dominated. These high order terms arise since the Coriolis and centrifugal forces vector in (3.2), has a quadratic growth in the generalised velocities, which are not measured. The global change of coordinates introduced in [5] for one degree-offreedom systems overcomes this problem by rewriting the dynamics with functions which are linear in the unmeasured velocities. A physical meaning for this "change of coordinates", which makes best sense when considering mechanical systems, is that this can be regarded as a kinematic model. The result of [9] for output feedback control of boats in slow motion tasks, combined with the underlying ideas in [5] suggest that, if we could rewrite the model (3.2) in a way which exhibited these kinematic relations and a dynamic model, linear in the unmeasured states, the problem of o u t p u t feedback tracking should be considerably simplified. For setpoint control, a first step has been undertaken in [14] where a separation principle for dynamic positioning of ships was already proven. Thus, inspired by the results of [9, 5] and motivated by those in [14], in this chapter, we extend the latter to the tracking problem. For this, we will assume that there exists a function J : R n --~ It('~xn, with the following properties P 1 J(q) is invertible for all q E IR'~ and satisfies 0 < kj,~ _< [[J(q)[[ 0 for all q 9 IR'L While this property is true in general for positive definite matrices, it is usually very hard to find such factorisation for n degrees of freedom systems. In [8, 9, 14] the model (3.5,3.6) represents the dynamics an kinematics of a surface vessel, for small motion applications. In [5], the author proposed a "change of coordinates" similar to (3.6) for one-degree-of-freedom systems. The control problem we solve in this chapter is the following. D e f i n i t i o n 3.1 ( G l o b a l o u t p u t f e e d b a c k t r a c k i n g ) Let qd : I~>_0 ~n be twice continuously differentiable and assume there exists /3d > 0 such that, max{llqd(t)ll, IIqd(t)l]} < /3d, uniformly in t. Assume that only q is available for measurement. Under these conditions, .find a dynamic controller T( t, q, ~), ~ ----r ~, q), such that, for any initial conditions (to, q(to), O(to), ~(to)) 9 IR>0 x ~n x R n x ~m, the system (3.~) in closed loop with 7(t, q, ~), be uniformly globally asymptotically stable (UGAS).

3

A Cascades Approach to a Separation Principle

Our main result in this chapter is a separation principle for EL systems under the assumptions made in the previous section. Our control design relies on defining an observer and a control law, with the aim at having a cascaded closed loop system, i.e., we seek for an error dynamics of the form :

r2

:

Xl = fl(t, Xl) q-g(t,X)X2

= Y2(t, X2)

(3.7) (3.S)

w h e r e x 1 9 ]I~2n, X 2 9 lt~2n, t h e f u n c t i o n s f l ( t , Xl), f l ( t , Xl), a n d g ( t , x ) are continuously differentiable and, both subsystems, E2 and

}-]0 : Xl = f l ( t ,

Xl)

(3.9)

are UGAS. Our motivation for considering this class of systems, is that the sufficient conditions for UGAS of cascades, are often easier to verify than to find a Lyapunov function for the system (El, E2), with a negative definite time derivative. In particular, in this chapter we will use Theorem 2 from [23]. In classic Lyapunov control design, one aims at designing a control law which yields a Lyapunov function with a negative definite derivative. In our control design for EL systems in the form (3.5,3.6), the system E2 will be the estimation error dynamics, hence, our first goal is to construct


235

an exponentially convergent observer. The system E ~ will correspond to the plant in closed loop with a state feedback controller. Then, E1 will correspond to the system (3.5, 3.6), in closed loop with the output feedback controller. In other words, g(t, x)x2 will correspond to nonlinearities of the system that result from implementing the state feedback control law, using the state estimates, instead of the true values. Then, to analyse the stability of the overall system, we will invoke [23, Theorem 2]. Hence, our design is made with aim at verifying the conditions of that theorem.

3.1

Observer Design

The observer design is based on [9]. With respect to the result in the last reference, we relax the assumption that the dynamic model (3.5) is internally damped. Consider the observer

M~, + v(q) = T + MKo2(q)~ = j ( q ) i + Ko,

.

(3.10)

(3.11)

where Ko: 9 ~n• and Ko2(q) 9 I~nxn are to be defined later and we denote the estimation error q = q - 0, correspondingly for the other variables. The estimation error dynamics (3.5, 3.6), (3.10, 3.11) is

= -Ko2(q)~

(3.12)

= J(q)D - Ko, q.

(3.13)

P r o p o s i t i o n 3.1 ( E x p o n e n t i a l l y c o n v e r g e n t o b s e r v e r ) Let P1, P2 be positive definite, Ko: be such that P1Ko: + K ~ P1 is positive definite, and let Ko~(q) := P21J(q)TP1. Then, the origin (~,~) = (0,0) of the system (3.13,3.12) is uniformly globally stable (UGS). Furthermore, assume that the trajectories q(t) and u(t), starting at (to, qo, vo) are globally uni.formly bounded, i.e., there exist c > 0 and ~ 9 3: such that II[q(t); u(t)]ll ~(llqo; Poll) + c .for all t > to > 0 and all (qo, vo) 9 ~'~• Then, the origin is UGAS. R e m a r k 3.1 The assumption on the uni.form boundedness o.f the plant trajectories is needed here to establish UGAS .for (3.12,3.13), however, this condition will be relaxed later when considering the overall closed loop system. That is, when introducing the output .feedback controller. P r o o f of P r o p o s i t i o n 3.1. Consider the control Lyapunov function candidate Vo(~, ~) = :1

(qTplq-f- ~Tp2~)

(3.14)

236


where Pi E N '~x'~ and/~ e IE~x" are positive definite matrices. The time derivative of Vo(q, F,) along the trajectories of (3.13,3.12) yields

Vo(4,~') = --~ql-T (PIKol + K ~ P 1 ) q + 4 T p I J ( q ) Y ' - qTKo2(q)Tp2~ (3.15) hence, using the definition Ko2(q) = P~-ij(q)Tp1, we obtain that

vo(4, r,) = - - i1-T q (P1Ko~ + K x P1)4 .

(3.16)

Since by assumption, P1K m + K ~ P 1 is positive (semi-)definite, the time derivative Vo(4, ~) is negative semidefinite. We conclude that the origin of the system is uniformly globally stable. To prove global exponential stability we rely on the a theorem, from [15], which is repeated in the Appendix A for the sake of completeness. To t

apply Theorem 3.3, let (i := q, (2 := ~, W ( t , ( i ) := ~ 4 T p i 4 , G(t,() :=

J(~l + O(t)) = J(q), P := Ps and h(t, (1) := -Ko~ 4. With these definitions, it is clear that the system (3.13,3.12) is of the form (3.51,3.52). Hence, we simply have to verify that conditions A1 - A 2 hold. The bound (3.53) is clearly satisfied with Pi (') = rnax{ko:M ,PIM } - T h e bounds (3.54,3.56) hold due to the property P l . Also, the inequality (3.55) is satisfied since, using J(q, 0) = 0 and (3.11,3.13), we can compute J(q,q)

l[ [0i(q, 0 ) - J(q, ~)] z~

0, independent of e, such t h a t any trajectory starting in ~-~bwill remain in ~ for all t 9 [0, T1]. Then, using the fact t h a t the fast variables ~/decay faster t h a n an exponential mode of the form ( 1 / e ) e -at~E, we can show that the trajectory enters the set [ ~ • E within the time interval [0, T(e)] where lim~-~0 T ( e ) = 0. Thus, by choosing e small enough we can ensure t h a t T(c) < T1. Figure 1 gives a sketch t h a t illustrates this behavior. The full-order observer (4.3) provides estimates (21, ~2) of the full s t a t e vector which are then used to replace (x], x2) in the feedback control law. We can use the fact that y -- xl is measured in two different ways. On one hand, we can use only 22 to replace x2 in the control law, while using the measured Xl. This approach does not change the analysis of the closed-loop system and we obtain the same results as before. On the other hand, we can use a reduced-order observer t h a t estimates only ~2. Such an observer is given by (v = Yc2 =

- h ( w + hy) + r w + hy

(4.9)

where h = c~/c for some positive constants a and e with e l - - E ,

Yt _> 0, Vx0 e IRn\{0}.

(5.8)

T h e o r e m 5.2 ([16]) Consider system (5.1) and suppose there exists a C2 .function V(x) and class ]Coo .functions ~1 and c~2, such that al(Ixl) 0 is the smallest eigenvalue of P. The second equality comes from substituting ~ = z~ + a~-l, and the inequality comes from Young's inequalities in Appendix A. At this point, we can see that all the terms can be cancelled by u and (~i. If we choose El, C2 and ~i to satisfy

bA - 3bnx/-~e2[PI 4 - -~ i=2 ~/~

and

~i and

u as

4e~ - p > O,

(5.2s)

276

5. Output-Feedback Control of Stochastic Nonlinear Systems

3

3.-4.

T

o~1 = - c l y - ~)I(Y) r

3- 4

3 ~

"~(~Y - -~e~Y - ~ E ~? (~)I(y)T ~)I(y))2Y i=2

3b~v/-ff lr

14y

(5.29)

i--100~i_l

00~i--1

~i = - e ~ z , - k , ~ l + ? _ _ . - S g - ( ~ § 3 "

-~5~z,

t=2

t

1

3

4

y

(00~i_1~ ~3

2 3

46~_1z'-~v3\ Oy ] z ~ - ~ \ n--l OO~n_l

u = -cnz~-k.~ +~--g~ l-~2

+

020~i--1

1 ~

^

e

~W

/

(00~i--1~ 4

Oy ] z~

(5.30)

~

(X~+l+kZ~)

l

Oc~n-l Fc2_+ 1 02C~n_1qol(y)Wqol(y) Oy 2 Oy2 1

3

~3 (OOLn--l~ ~

3

fOOLn_l~ 4

-45~_-----~z~-~\ 0y ] z~-4--~ \ 0y ] ~ '

(5.31)

where ci > 0, then the infinitesimal generator of the closed-loop system (5.16), (5.24), (5.25) and (5.31)is negative definite:

; v < - ~ c~z2 - pl~f4

(5.32)

i=1

With (5.32), we have the following stability result. T h e o r e m 5.4 The equilibrium at the origin o.f the closed-loop stochastic

system (5.17), (5.31) is globally asymptotically stable in probability.

4

\

~lty)~lty)

Output-Feedback Noise-to-State Stabilization

In this section we deal with nonlinear output:feedback systems driven by white noise with bounded but unknown covariance. This class of systems is given by the following nonlinear stochastic differential equations:

dxi dxn y

= xi+ldt + ~i(y)T dw, = udt +pn(y)Wdw ~

Xl~

i=1,...

,n-1

(5.33)


277

where T~ (y) are r-vector-valued smooth functions, and w is an r-dimensional Wiener process with incremental covariance E { dwdwT } =E(t)E(t)W dt. The observer is designed as in (5.15), and the entire system can be expressed as (5.17). The error variables z~ are defined as in (5.19), (5.20). With It6's differentiation rule, we have

dzi

=-

dzi

=

(22 + 22) dt

+~l(y)Tdw i-i Oa~_l

2~+i + ki21 - E

/=2

02t

(5.34)

1 102C~i_l

2 \ 0y 2 ) ~I(y)T~Tcfll(Y)] at i

=

~OLi--1

(2/+1 "~-klXx)

Oy (22 q- 22)

0 ~----1 ~91(y)Wdw (5.35) oy

2 , . . . ,n.

As in Section 3, we employ a quartic Lyapunov function

1 4+ ~ 1 ~~ z2 + ~ b (2 wP2)2 V(z, ~) = ~y i~2

(5.36)

Now we start the process of selecting the functions a i ( 2 i , y ) to make s in the form (5.37)

~ v ~ -p(2, y, 2) + ~(l~l)

where p is positive definite, radially unbounded, and 7 is a class/C function. Since ~(y) is a smooth function, according to mean value theorem, we can write it as

~(y) = ~(0) + y~b(y)

(5.38)

where ~b(y) is a smooth function. Along the solutions of (5.16), (5.34) and (5.35), we have ,~

s

n

[

i--I

i=2

t

z=2 O:~z

= (kz+i + kt2i)

3~

ba~-i

]

1 (02c~i_1~

2 (o~_1~ 2

i=2 -b2Tp212[ 2 + bTr { ~ ( y ) E w ( 2 p ~ 2 T P

+

2Tp2P)

E~(y) T }

(5.39)

278


Therefore EV

O, ~i

a n d a i a n d u as

(~1

=

--Cly

3n41 --

3bnv/-~LP ]2e2

2

Jr

i--1 O~i

+

=

-- ~3 y

(r

4

3 4_ -- ~c~y

3 ^

~yo

-

(5.61)

0c~_1 (~:~+~+ k15:1) +

OOLi_1 ^ x2

/=2

4~LlZ~-~"~\ oy } 00~i--'------~1 ~[Z3COj-~ ~ j=2

j=2

O~n

~

Z 300gj--1 02"

(5.62) (5.63)

284


where c~ > 0 and 5n = 0, then the infinitesimal generator of the closed-loop system (5.16), (5.50), (5.51) and (5.63) satisfies: n

s

O,

Vxe~_,

capture the internal interconnections and the natural damping of the system, respectively, while g(x) defines the interconnection of the system with its environment. We assume measurable the q-dimensionai o u t p u t vector function y = h(x). This output should not be confused with the natural outputs associated to the port-controlled Hamiltonian system E defined as

gT (x ~OH(x~ The control objective is to stabilize, via output-feedback, an equilibrium c . ~ preserving in closed-loop the Hamiltonian structure. T h e latter property allows us to provide an energy interpretation of the control action.

6. Output Feedback Control of Food-Chain Systems

293

We will consider only static controllers, but as shown in [7] the procedure can be easily modified to incorporate controller dynamics. Following the principles of passivity-based control [S], [10], we will achieve the stabilization objective by the standard energy-shaping plus damping injection stages. T h a t is: 1. Assigning to the closed-loop an energy function Hd(X), which should have a strict local minimum at ~. (That is, there exists an open neighbourhood B of 9 such that Hal(x) > Hd(2) for all x E B.) We will define

Hd(x)A=H(x) + H~(x) where

Ha(x)

(6.2)

is a function to be defined.

2. Injecting some additional damping

Ra(x)

Rd(X)A=R(x) + Ra(x)

to get

>_ 0, V x E ~ _

T h a t is, we look for an output-feedback control

OH

[J(x) - R ( x ) ] - ~ x (x) +

g(x)u(h(x))

u(h(x))

= [J(x) -

(6.3) such that

OHd Rd(X)]~ (x)

holds V x E ~ _ , with Hd(x), Rd(X) defined by (6.2) and (6.3), respectively. In this way, the closed-loop dynamics will be defined as = [J(x) -

Rd(x)l-~(x),

(6.4)

and along the trajectories of (6.4) we will have

dH

[0gd(x)l T

OHd( )

(6.5)

Thus, 2 will be a stable equilibrium. For ease of presentation we will assume throughout the following: Assumption

A

[J(x) -

Rd(x)] is

invertible for every x E ~ .

It is important to remark that this does not imply that the closed-loop system is fully damped. T h a t is, we do not require Rd(x) > 0, Vx E ~ _ . Actually, it is shown in [7] that Assumption A is not needed for the proof of the proposition below. We have the following basic result.

294


P r o p o s i t i o n 6.1 [7] Given J(x), R(x),H(x),g(x). Assume we can find and output-feedback control u(h(x)) and a matrix Ra(x) such that R(x) § Ra(x) >_O, Assumption A hold, and the vector function g ( x ) , defined as,

K(x)~[J(x) - (R(x) + R~(x))]-l[R~(X)~x(X ) + g(x)u(h(x))]

(6.6)

satisfies 9 (Integrability) K(x) is the gradient of a scalar .function. That is, 0x (x) =

(6.~)

9 (Equilibrium assignment) K(x), at 2, vemfies

OH Ox (2)

K(~:) -

(6.8)

9 (Lyapunov stability) The Jacobian of K(x), at ~, satisfies the bound

OK 02 H Ox (~) > - -5~-x 2(2)

(6.9)

Then, 9 will be a locally stable equilibrium of the closed-loop. It will be asymptotically stable if, furthermore, the largest invariant set under the closed-loop dynamics contained in x e ~

N B { --~--x( )

-~d( )-O-'~--x i, ) = 0

(6.10)

equals {~}, where Hd(X) is given by (6.2). The latter condition will be automatically satisfied if we can achieve full damping, that is, if R~(x) > 0/07- eve~ x e ~?~. Proof First, notice that, using (6.2), (6.3) and Assumption A, the identity (6.4) may be equivalently written as

Ox ( z ) = [ J ( x ) - Rd(x)]-I[R~(x)

(x) + g(x)u(h(z))]

(6.11)

For every given u(h(x)), R~(x), this is a linear PDE. A necessary and sufficient condition for the solvability of this PDE (on every contractible neighbourhood of Nr~ + ) is that the gradient of the right hand side of (6.11) is a symmetric matrix. From (6.3), (6.6) and (6.11) we see that

og~

K(x) = --~-x (x)

(6.12)


295

Henceforth, the matrix mentioned above will be symmetric iff the integrability condition (6.7) of the proposition is satisfied. The stability proof is concluded invoking standard Lyapunov stability arguments [4]. Namely, from i6.5), we conclude that, under the standing assumptions, Hd(x) qualifies as a Lyapunov function. Asymptotic stability follows from a direct application of La Salle's invariance principle and i6.10). [:]DD R e m a r k 6.1 Notice that the construction above does not require the ex-

plicit derivation of the Lyapunov function Hdix). This can be obtained, though, as a by-product integrating K i x ) OH---~ix ) R e m a r k 6.2 Port-controlled Hamiltonian models (6.1) encompass a very

large class of physical nonlinear systems, strictly containing the class of Euler-Lagrange models considered, for instance, in [8]. They result from the network modeling of energy-conserving lumped-parameter physical systems with independent storage elements, and have been advocated in a series of recent papers [6], [11] as an alternative to more classical Euler-Lagrange (or standard Hamiltonian) models.

3

State-Feedback Control of a Simple Prey-Predator System

As pointed out in the introduction, to motivate our output-fedback control (which is given in the next section) we present first a state-feedback stabilizer for a simple second order food-chain system. The controller is obtained from a verbatim application of the m e t h o d described above. This is a systematic technique that can be efficiently combined with symbolic computation. See, for instance, the simple Maple code given in Appendix A.

S y s t e m Model We consider the normalized second order prey-predator system isee e.g.

[3]) =

f(x)-zl

=

-f(x)

-

+ u

(6.13)

The state variables xl, x2 represent the amount of mass of the two species (preys and predators) involved in the system. The function f i x) describes the predation mechanism, we consider here the classical Lotka-Volterra mechanism f(x) = XlX2. The terms - x l , - x 2 i n (6.13) represent the natural mortality of the species, while the control action u is a feeding inflow

296


rate of preys. For the output feedback case, we will consider that the variable available for measurement is the last one in the chain, in this case,

X2. The evolution of the system is clearly restricted to the positive orthant with u > 0. T h a t is, x i ( 0 ) > 0 , a n d u ( t ) >_0, Vt > O ~

xi(t) >_ O, Vt >_ O

It is possible to show that any equilibrium of the open-loop system with a lit constant input fi _ 0 is globally asymptotically stable. The control objective is, then, to asymptotically stabilize a given non-zero equilibrium 5: E ~R~_ with a positive control. The achievable equilibria axe ~" ----[:~1,X2]T ~-- [X~,1] T, with x~ > 0 the reference for xl. If we define the total mass

H(x)

=

x 1 -J- x 2

the system (6.13) may be written in the form (6.1) with J(x)=

[

0 --XlX2

xlx2 0

]

[

R(x)=

'

Xl 0

0 X2

]

g(X)..~_g =

[0] 1

The skew-symmetry of J(x) captures the mass-conservative feature of the system without inflows and outflows.

State-Feedback Stabilization Since the system is already fully damped, i.e., R(x) > O, Vx ~ O, x E ,~_, it seems reasonable (as our first try) to set R~(x) = 0. T h a t is, we will not inject additional damping, but rely instead on the natural damping of the system to ensure the attractivity. In this case, the vector function (6.6) reduces to

K ( x ) = [ kl(X)

k2(x) I

__

--U(Z)

I-'~-XiZ2 [ 1 1

From which we immediately conclude that

x2k2 (x) = kl (x)

(6.14)

The integrability condition (6.7) in this two-dimensional case reduces to

Ok1

ox2 (x) =

Ok2 (x),

which, combined with (6.14), yields the linear P D E 0kl

Oxl

0kl clx2

( z ) - x2 ~ - - - ( x )

= 0

(6.15)


297

A family of solutions of this P D E is easily obtained as

k (x)

=

~(x)

=

Xl+lOgx:,

for all differentiable functions O(.). From (6.14) we also obtain k2(x)

=

The equilibrium condition (6.8) imposes k1(2)

1

(6.16)

Hence, 0(.) must be such t h a t 0(C(2)) = - 1 , where C(2) = 21 + l o g 22 ---x~. It is clear then t h a t we cannot take O(4) = 4. We propose the function

9 (r

= cl e•

with cl, c2 constants to be defined. (Although this choice of function might seem a bit contrived, we should note that this is the function t h a t results if we directly apply the m e t h o d of undetermined coefficients to the P D E (6.15). See Appendix A). The equilibrium condition ~(~(2)) --- - 1 fixes the first constant as Cl = - exp -c~x~ We will now verify the Hessian condition (6.9). Some simple calculations yield OX (X) = ClC 2 expC2r

~'21 x-'~'• ~'2[1 x2_ c')1 ~2

-- 0X 2 (X)

,

which evaluated in the equilibrium point gives

T h e determinant of this m a t r i x is 1, hence it is positive definite iff c2 < 0. We will investigate now the asymptotic stability properties. To this end, we see t h a t the a~-limit set (6.10) is defined as {x e N~_ n B I - x,(1 + kl(X)) 2 - x~(1 + k2(x)) 2 = 0 } , which consists only of the points x = 0 and x = ~:. But, it can be easily shown, t h a t x = 0 is an unstable equilibrium of the closed loop dynamics. We have established the following result.

298


P r o p o s i t i o n 6.2 Consider the system (6.13), with f ( x ) -- XlX2, in closedloop with the positive control

u(x) = (1 + XlX2)X ~ exp c(xl-x~)

(6.17)

with x~ > 0 the reference for Xl, and c < O. Then, all trajectories starting in x(O) E ~2+, will converge asymptotically to the desired equilibrium point

1) DDD Let us summarize the calculations carried out above: 1. Fix the added damping Ra(x) - to 0 in this case, since the open-loop system is fully damped -; 2. Define the vector K ( x ) , (6.6), as a function of u(x); 3. Use the integrability conditions (6.7) to eliminate the control and obtain a linear P D E (6.15) to be solved for K ( x ) ; 4. Find a solution of this P D E that satisfies the equilibrium (6.8) and Lyapunov stability conditions (6.9); 5. Derive the control law (6.17) from the definition of K ( x ) . R e m a r k 6.3 As pointed out in Remark 2 as a by-product of our analysis we can get a Lyapunov function, which in our case is

Hd(X)

:

Xl "~ X2 -H(x)

1 k

";-X 2

k~zl-x*~ 1 exp ~ '~ + 7 - (1 + x~) Y H,,(x)

where the third and .fourth right hand constant terms are added to enforce Hd(~2) ---- O. It is worth noting that Hd(x) above is the classical Lyapunov function for the stability analysis of Lotka-Volterra ecologies (see e.g. [3] and [9] among many other references). The design procedure of this paper allows to rediscover this Lyapunov function in a very natural way. R e m a r k 6.4 There is an easier way to derive the structural constraint (6.14) that does not require the inversion of the matrix J ( z ) - Rd(x). To this end, rewrite (6. 6) as

OH [J(x) - (R(x) + Ra(x))]K(x) = [Ra(x)--~xx (X ) + g(x)u(x)] The first equation of (6.18) for this example yields

--Xlkl(X ) "~- X l X 2 k 2 ( x

) ~- 0

(6.18)


299

which, upon division by x l , is precisely (6.1~). The second equation simply de.fines the control law, in terms of K ( x ) , as

(6.19)

u(x) = -x2k2(x) - xlx2kl(x)

It is precisely this observation that will motivate the modification, introduced in the next section, that yields an output-feedback stabilizer.

4

Output-Feedback

Stabilization

There are two i m p o r t a n t drawbacks of the solution proposed in the previous section. First, it requires m e a s u r e m e n t of all the state 9 Second, it can not be extended to treat the general food-chain system model, which is of the form 51

~

XlX 2 -- X 1

5 2

~

X2X 3 -- XlX 2 -- X 2

5 3

~

X3X 4 -- X2X 3 -- X 3

5n

z

--X(n_I)X

=

x~

y

n - - X n -]- U

(6.20)

To prove the second statement, let us write the model in the form (6.1) with H ( x ) = E ~ l x i and 0 --XlX

J(x)

XlX2 2

0

" ""

0

0

X2X 3

" 9"

0

0

0

...

0

=

0

xl 0

0 x2

... ...

0 0 = RT(x)

R(x) =

0

0

...

= --JT(x)

> O, g ( x ) = g =

x~

Then, notice t h a t the distribution spanned by the vector fields defined by the column vectors obtained from the first n - 1 rows of J ( x ) - R ( x ) is not involutive. Consequently, the key P D E [J(x) - R(x)] ~ x a (x) = g u ( x )

300


can not

be solved. In this section we show how, for our second order example (6.13), these limitations can be overcome modifying the damping of the closed-loop. In the next section we extend this result to the general n - t h order model (6.20). Towards this end, let us remove the damping from the first coordinate. T h a t is, define Ra(x) like

Ra(x)= [ -xlO 00] Notice the negative sign. With this choice, the vector function (6.6) becomes now

g(x)=

kl(X)

1

k2(z)

_

-

~

Choosing the control law as the simple o u t p u t - f e e d b a c k u(x2) =

cx2 + 1,

with c some constant to be defined, yields

K(x)

=

~1

(6.21)

which is clearly the gradient of a scalar function. Hence, the integrability condition (6.7) is satisfied. We will now verify if we can find a constant c such that the remaining stability conditions of Proposition 6.1 are also satisfied. The equilibrium condition (6.16) imposes c ----x~. For the Hessian condition (6.9) we first observe from (6.21) and ~ H ( x ) ----0, t h a t

~x~X~=-~x

~=

o

Evaluated in the equilibrium point gives

0ox22 H(~) d= O-~xK(~)[ = 71, o ] 0

1 '

which will be positive definite for any x~ > 0. Finally, a s y m p t o t i c stability is ensured because the w-limit set (6.10) is now defined as

{ which consists only of the point x = ~.

-

1-o

},


301

T h e new L y a p u n o v function is

Hd(x) ~- Xl ~- x2 --x~ ln(xl) -- ln(x2) --(x~ -F 1 -- x~ ln(x~)), H(x)

H~(x)

where the third right h a n d c o n s t a n t t e r m is, again, a d d e d to enforce Hd(2) = O. We have established the following result. P r o p o s i t i o n 6.3 Consider the system (6.13), with f ( x ) = XlX2, in closedloop with the positive output-feedback control u(x2) --- 1 + x~x2

(6.22)

with x~ > 0 the reference for x l . Then, all trajectories starting in x(O) E 7~2+, will converge asymptotically to the desired equilibrium point (x~, 1). DDD R e m a r k 6.5 To increase the speed of convergence it is possible to inject some additional damping on the actuated coordinate x2. To this end, we choose

olxl=[Xl 0] 0

(r - 1)x2

'

with the desired damping a constant 1 < r < 1 + x~. Going through the calculations we get the control law u(x2) = r + (x~ - r + 1)x2

(6.23)

It can be shown that this control law is also globally asymptotically stabilizing. Notice that with r = 1 we recover the controller (6.22).

5

Main Result

In this section we present the generalization of the previous result to t h e n - t h order case. T h e o r e m 6.1 Consider the general food chain system (6.20) in closed-loop with the output:feedback positive control u ( z n ) = m x n + m + Xx,

?n--

n-1 2

.for n odd, and n

u(x,~) ---- ( m + x~)xn +-~,

n

m---- -~ - I

302

6. Output Feedback Control of F o o d - C h a i n Systems

.for n even, with x~ > 0 the reference f o r Xl. Then, all trajectories starting in x(O) E ~ _ will converge asymptotically to the desired equilibrium point $

=

-

[X 1 , / 2 ,

"', Xn]"

DDD Proof M o t i v a t e d b y t h e d e v e l o p m e n t s of t h e s e c o n d o r d e r case a b o v e we p r o p o s e to r e m o v e t h e d a m p i n g from all n o n - a c t u a t e d c o o r d i n a t e s . T h a t is, we choose

Ra(x)=

-xl 0

0 -x2

0

0

"'"

O

9 9 9

0

9 " 9

O

W e will n o w verify t h e t h r e e c o n d i t i o n s of P r o p o s i t i o n 6.1.

9

Integrability

T h e key e q u a t i o n (6.11) b e c o m e s t h e n

0 --XlX2

0 0

xlx2

0

0

X2X3

0 0

0 0

9 .. 9 ..

0 0

"'"

0 0

0

. . . .

Xn_lX

Xn--lXn n

k,(x) k2(x)

n-l(X)

Xn

--X 1 --X 2

--Xn_ 1

w h i c h c a n b e c o m p a c t l y w r i t t e n as f l ' ( x ) K ( x ) = ~(x). Now, , ~ ( x ) a d m i t s a f a c t o r i z a t i o n of t h e f o r m 0 -1

1 0

0 1

.-. .-.

0 0

0 0

J(x) = diag{x,}

diag{xi} 0 0

0 0

0 0

... ....

0 1

1 ___1


303

This leads to

0

1

0

"""

-1

0

1

.-.

0 0

Xl~I(X) X2k2(X)

-1 -1

Xn--l~n--l(X) X~kn(X)

-1

0 0

:

0 0

0 0

0 0

0 -1

1 _1

Xn

XT~

From which we obtain a system of equations of the form x2k2(x)

=

-1

-Xlkl(X)+X3k3(x)

=

-1

-x2k2(x)+x4k4(x)

=

-1

)

=

-1

k,~(x)

-

u(x)

-Xn_2kn_2(x

) ~- x n k n ( x

-xn-lkn-l(X)-

(6.24)

Xn

Notice that from the first equation of (6.24) we have 1

k2(~) -

X2

Subsequently, the functions k i ( x ) , for i even, have a unique solution, which is furthermore of the form k i ( x ) = ki(xi). Now, choosing C

kl(z) -

Xl

we can also obtain a unique solution k i ( x i ) , K ( x ) is finally given by P

K(x) = [

1

1 x2

Xl

"""

m xn-- 1

for i odd. The vector function

_rat

c

Xn

IT J

,

IYt - -

I1 - - 1 2 '

for n odd, and

K ( )x

-=

c

1

Xl

x2

_ m-t-c 9. .

x~_ 1

,~ -~-

] T

x,L

'

. ~ = ~ -n1 ,

for n even. It is clear that, in both cases, the integrability conditions are satisfied. Also, from the last equation of (6.24) we compute the control law U(X)

---- - - X n [ X n _ l

kn_l

( X ) "~- k n ( X ) ]

304

9

6. O u t p u t Feedback Control of F o o d - C h a i n Systems

Equilibrium Assignment

T h e e q u i l i b r i u m c o n d i t i o n is

K(2)-

OH ax

=

-

[1

=

..

Q

'

x~_l,

x~

w h i c h is satisfied w i t h c = x~.

9

L y a p u n o v Stability

W e will n o w verify t h e Hessian c o n d i t i o n . S o m e s i m p l e c a l c u l a t i o n s y i e l d

0

OK

ai(x)

=

0

0

0

...

0

0

~

0

0

.-.

0

0

.--

0

0

0

0

0

0

0

~

0 0

0

(6.25)

..-

0

0

-..

0

.--

0

0

T h i s m a t r i x will b e p o s i t i v e definite for a n y x E ~ _ a n d a n y x~ > 0. F i n a l l y , t h e w l i m i t set for n o d d is defined as {x C , ~ N 13 [x,,-(m+x;) ~- 0} a n d ~n

{x C !}~_ N 13 [ x , ,x,, - ( ~ ) __- 0} for n even. I n b o t h cases t h e w l i m i t set consists o n l y of t h e p o i n t xn = 2n. This, t o g e t h e r w i t h u n i q u e n e s s of t h e e q u i l i b r i u m , c o m p l e t e s t h e p r o o f of a s y m p t o t i c stability. 6 . 6 The proposed control design can be easily applied to the more general class of Lotka-Volterra ecologies defined as .follows:

Remark

xi

=

x~(-k~+Zaijxj)

3:n

~- Xn(-kn q- E

i=l,...,n-1

anjXj) -k u

with k~ > 0 the natural mortality rates, aij ~- - a i j , V i ~ j, the predation coefficients and u the .feeding rate of species xn, with u(t) >_0 Vt. The procedure yields the classical Lyapunov function .for Lotka-Volterra ecologies

~

xi - g% l n ( x i ) ,

i=l

and we obtain the .following output .feedback control law u(xn)

=

+

-

with ~t the constant control that assigns the desired equilibrium, and 0 < A < ~X,e an arbitrary design parameter.

,


6

305

Simulations

Numerical simulations of the second order model (6.13) were carried out in order to show the performance of the proposed controllers. T h e p a r a m e t e r s used in the simulations were, c -- - 0 . 2 for the state feedback controller (6.17), and r -- 1,2.1, for the o u t p u t feedback controller (6.23). T h e desired equilibrium of the system is 9 --- [1.2, 1] T. The initial conditions in all the simulations are xl (0) = 2 and x2(0) ----2.

2.,~ 2 1.8 1.6 14

1.2

0.8 0.6 0.5

1=,5

2

21.5

3

xl

FIGURE 1. Open-loop trajectory

For the sake of comparison, in Fig. 1 we present the behaviour of the open loop trajectory in the state space with a constant input fi -- 2.2, while Fig. 2 depicts the behaviour of the state and ouput feedback controllers. Finally, the control signals are shown in Fig. 3. As seem from the Figs. 2, 3 the addition of damping effectively increases the convergence rate with the additional advantage of reducing the control effort.

306


~sta

tefeedback

1.z

O. output feedback ~

0.~ 0.5

1

1.5

xl

2

2.5

FIGURE 2. State space of the closed-loop trajectory

statefeedback

damping output feedback eedback +

/

1~0

time

[sec]ll5

;~0

25

FIGURE 3. Control signals

7

Concluding Remarks

We have illustrated in this chapter how the application of the p a s s i v i t y based controller design technique of [7] allows us to solve o u t p u t - f e e d b a c k stabilization problems for a class of m a s s - b a l a n c e systems. T h e procedure is illustrated in detail with an n - t h order food-chain model. It can, mutataemutandi, be applied also to other m a s s - b a l a n c e models studied in [1], [3],


307

[9]. For instance, it can be shown that for the compartmental model of Section 4 in [1] the technique yields also asymptotically stabilizing controllers. However, we require in this case the knowledge of the full state. We have not stressed here the advantages of taking a physically-based approach for controller design, see e.g. [8], [7], [10] for a detailed discussion. We should underscore, however, that the preservation of a physical interpretation to the control action (in terms of damping injection) was instrumental for our result. Finally, we bring to the readers attention the simplicity of the resulting control law. This important feature is a characteristic of passivity-based controllers. As shown in this chapter the approach of [7] provides a flexible methodology to design controllers for physical systems. As discussed in t h a t paper, we can also aim at modifying the internal interconnection structure J ( x ) . In this way, we recover some of the results obtained with the technique of controlled Lagrangians, reported in [2]. Current research is under way to explore this interesting possibility for mass-balance systems.

Acknowledgements The first author would like to express his deep gratitude to Bernhard Maschke and Arian van der Schaft, with whom the basic principles underlying the developments reported here were obtained.

8

REFERENCES

[1]

G. Bastin and L. Praly. Feedback stabilization with positive control of a class of dissipative mass balance systems, accepted IFA C World Congress, Beijing, 1999.

[2]

M. Bloch, E. Leonhard and J. Marsden. Controlled Lagrangians and the Stabilization of Mechanical Systems I: The First Matching Theorem. IEEE Conf. Decision and Control, Tampa, FL, 1998.

[3] J. Hofbaner and K. Sigmund. Evolutionary Games and Population Dynamics. Cambridge University Press, 1998. [4] H. Khalil. Nonlinear systems. Prentice-Hall, 2nd edition, 1996. ISBN 0-13-22824-8.

[5]

B. Maschke, R. Ortega and A. van der Schaft. Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation. IEEE Conf. Dec. and Control, Tampa, FL, 1998.


308

[6]

B.M. Maschke and A.J. van der Schaft. Port controlled Hamiltonian systems: modeling origins and system theoretic properties. Proc. 2nd IFAC Syrup. on Nonlinear Control Systems design, NOLCOS'92, pp.282-288, Bordeaux, 1992.

[7] R. Ortega, B. Maschke, A. van der Schaft and G. Escobar. Passivity-based control of port-controlled Hamiltonian systems, LSS-SUPELEC, Prance, Int. Rep., 1998. [8] R. Ortega and A. Loria, P. J. Nicklasson and H. Sira-Ramirez. Passivity-Based Control o.f Euler-Lagrange Systems. SpringerVerlag, Berlin, Communications and Control Engineering, 1998. [9] F. Sendo and J. Ziegler. The Golden Age of theoretical Ecology. Lecture notes in B i o m a t h e m a t i c s , Springer Verlag, 1978. [10] A. van der Schaft. L2-Gain and Passivity Techniques in Nonlinear Control. Lect. Notes in Contr. and Inf. Sc., Vol. 218, SpringerVerlag, Berlin, 1996. [11] A. van der Schaft and B. Maschke. The Hamiltonian formulation of energy-conserving physical systems with external ports. Archiv fiir Elektronik und Ubertragungstechnik, 49, pp. 362-371, 1995.

Appendix A: Maple Code In this appendix we present a Maple code that guides us in the solution of the example of Section 2. The calculations proceed as follows 1. Definition of the system (with JmRZX=J(x) - R(x) and gu~gu(x)) : > with(linalg) : >

JmR := matrix(2,2,[-xl,xl*x2,-x1*x2,-x2S);

JmR:=

> gu

[ -xl [ [ - x l x2

x l x2] ] -x2 ]

:= v e c t o r ( [ 0 , u ( x l , x 2 ) ] ) ;

gu := [0, u ( x l ,

x2)]

2. Computation of K(x) and its Jacobian > K := multiply(inverse(JmR) [

,gu) ; u(xl,

x2)

K := [, [ 1 + x l x2

u(xl,

x2)

I

] x2 (1 + x l x2)]

6. O u t p u t Feedback Control of F o o d - C h a i n S y s t e m s > Jac

:= j a c o b i a n ( K , [ x l , x 2 ] ) ;

Jac

:=

[ [ [u(xl, .

.

.

.

d --- u(xl, dxl

x2) x2 .

.

.

.

.

.

.

[

.

.

.

.

.

.

2

.

.

.

.

.

[ [ [ u(xl, .

.

.

.

u(xl, .

.

.

.

~

.

.

.

.

.

d --- u(xl, dxl .

.

.

.

.

.

.

.

.

x2

.

x2) x l .

.

.

.

.

.

.

.

.

.

.

.

2

.

.

.

.

.

.

.

.

.

1 + x l x2

.

.

.

.

.

.

.

.

.

] ]

I

.

.

.

.

d --- u(xl, dx2

x2) x l .

.

.

.

.

.

.

.

2

.

.

.

2 (i + xl x2)

]

x2)

u(xl, +

.

(i + xl x2)

x2)

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

x2

.

] x2)] ]

(i + xl x2)

[ 2 [(i + xl x2)

u(xl,

.

1 + x l x2

x2) .

d --- u(xl, dx2

x2)

[(i + xl x2)

.

309

x2

.

.

.

.

.

.

.

.

.

] x2) ] ] .

.

.

.

.

]

x2 (i + xi x 2 ) ]

(i + xl x2)

]

3. D e f i n i t i o n of t h e t e r m

) _ 0kl

> eql2

:= Jac[2,1]-Jac[1,2];

u(xl, eql2

d --dxl

x2)

u(xl,

x2) u(xl,

x2) xl

:= 2 (I + xl x2) d --- u(xl, dx2

x2

(1 + x l

x2)

2 (1 + xl x2)

x2)

+ 1 + xl x 2 4. D e t e r m i n a t i o n of t h e c o n t r o l u(x) w h i c h s o l v e s e q l 2 = 0, i.e., w h i c h ensures the integrability condition.

310

6. Output Feedback Control of Food-Chain Systems > u_star:=rhs(pdesolve(eql2=O,u(xl,x2))); x2 u_star

) (i + xl x2)

:= _ F I ( exp(-xl)

Notice that in the line above _ F I ( - ) is any differentiable function. 5. Evaluation of the Hessian for the given control expression. > subs(u(xl,x2)=u_star,evalm(Jac)); [ x2 d [_FI( ........ ) x2 --- Z1 [ exp(-xl) dxl [. . . . . . . . . . . . . . . . . . . . . . . . . . [

i + xl x2

x2 d _FI( . . . . . . . . ) xl --- ZI exp(-xi) dx2 , ..........................

i + xl x2

[ x2 [_FI( ........ ) [ exp(-xl)

i + xl x2

] ] ] ]

i + xl x2]

d --- Z1 dxl

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

[ [

i + xl x2

x2

x2 _FI( ........ ) exp(-xl) . . . . . . . . . . . . .

2

(I + xl x2)

x2 _FI( ........ ) xl exp(-xl) +

d --- ZI dx2

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

x2 (I + xl x2)

x2

x2

] ] ] ]

(I + xl x2)] ]

x2 ~.1 := _FI( ........ ) (I + xl x2) exp(-xl)

6. The design can be concluded selecting a function F I ( - ) that satisties the equilibrium assignment and Lyapunov stability conditions of Proposition 6.1. In Section 2 we have chosen _ F I ( ~ ) -- ~k.

Output Feedback Tracking Control for Ships K. Y. Pettersen I and H. Nijmeijer 2 1Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway 2 Faculty of Mathematical Sciences, University of Twente, Enschede, The Netherlands 2 Faculty of Mechanical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands

1

Introduction

For most ships, measurements of the ship velocities are not available. For feedback control of the ship, estimates of the velocities must therefore be computed from the position and heading measurements. The ship position is typically measured using the Navstar differential global positioning systern (DGPS), while the heading is usually measured by a gyro compass. As the position measurements are quite corrupted by noise, numerical position differentiation is not desirable. Instead, an observer should be used to obtain velocity estimates from the position measurements. In conventional ship control systems, the estimation problem is solved using a linear Kalman filter. A linearized ship model is then used. T h e kinematic equations of motion are typically linearized about a set of 36 constant yaw angles (separated by 10 deg in order to cover the whole operating area of 360 deg). For each of these linearized models Kalman filters and feedback control gains have to be computed. The control and filter gains are then modified on-line using gain-scheduling techniques. The drawback of this approach is the considerable amount of tuning work, and the ad hoc nature of the approach which does not guarantee the desired stability and convergence properties. In [8] a nonlinear observer is developed and is proven to be globally exponentially stable. Hence, only one set of observer gains is needed to cover the whole state space. The observer is developed independently of the ship control scheme. In [1] a feedback control law for dynamic positioning of ships is developed based on the estimates from the observer in [8], giving a globally exponentially stable closed-loop system. In both these works, the dynamic positioning (DP) problem for ships is considered, and the observer and the control law are thus developed based on a ship model not

312

7. Output Feedback Tracking Control for Ships

including Coriolis and centripetal forces and moments. For the DP-problem it is a valid assumption to disregard the Coriolis and centripetal forces and moments acting on the ship, while for tracking control where the velocities of the ship cannot be assumed to be close to zero, these forces and moments must be considered in both the observer and the controller design. In this work we consider the output feedback tracking control problem for ships. The Coriolis and centripetal forces and moments must thus be included in the ship model, leading to quadratic velocity terms in the dynamics. Moreover, instead of designing an open-loop observer, we seek to combine the observer and controller design such that the controller exploits the underlying observer structure and vice versa, in order to find a computationally simple control law and observer. The observer-controller scheme is designed using a passivity-based approach. The idea of passivity-based control methods is to reshape the system's natural energy via state feedback, in order to achieve the control objective. In this way the passivity property of the system is preserved in the closed loop, and therefore the approach has been named the passivity-based approach. This approach has gained much attention, and based on this approach [17] proposed a solution to the robot position control problem, and [13] solved the problem of robot motion control. Also, for adaptive robot control the passivity-based approach has been studied extensively, see for instance [15],[9] and [12]. The output feedback control problem for robots has been considered by several authors, for instance [4] and [16] where observers based on the sliding mode concept were proposed, and in [10] where a linear high-gain strategy was proposed. The observers proposed were developed independently of the robot control scheme. In [5] a modified version of the computed-torque controller was proposed, and local exponential stability of the closed-loop system was proven. In [11] some known state-feedback controllers were considered, using velocity feedback from a nonlinear observer, and local asymptotic stability of the closed-loop systems were proven. In [2],[3] the output feedback control problem for robot manipulators was solved using the passivity-based approach. A key point in [3] was the fine tuning of the controller and observer structure to each other, providing solutions of the output feedback control problem that were conceptually simple and easily implementable in industrial applications. The output feedback tracking control problem for ships has been addressed in [14] and [18]. Both these works consider a 1 degree of freedom (DOF) nonlinear model and address the yaw angle tracking control problem (autopilot design). In [14] a passive control law without velocity feedback is proposed and proved to asymptotically stabilize the desired yaw angle. In [18], based on the ideas presented in [3] an observer-controller structure is proposed and proved to semi-globally exponentially stabilize the desired yaw angle. In this chapter we use the same ideas as presented in [3] to design a 3 DOF output feedback tracking controller for ships. However, in


313

this chapter we use the same ideas as presented in [3] for the controllerobserver design for ships. However, the control law proposed in [3] uses feedback from the position measurements together with the estimated velocities. For the ship, the gyro compass measurement noise will typically be less than 0.1 deg. However, the position measurements are quite corrupted by noise, as the DGPS ineasurement noise will be in the range of 1-3 m. Therefore, filtering of the position measurements is necessary, and we seek to find a tracking control law that uses feedback from the filtered position variables. In Section 2 the ship model is presented. In Section 3 we develop a controller-observer combination for output feedback tracking control of ships, and prove that the closed-loop system is semi-globally exponentially stable. If the Coriolis and centripetal forces and moments are negligible, as for the special case where the desired trajectory is a constant position and orientation, the system is globally exponentially stable. In Section 4 simulation results for this o u t p u t feedback tracking control scheme are presented. Then the problem of bias estimation is addressed in Section 5, and simulations for the output feedback tracking control scheme including bias estimation are presented in Section 6. Finally, conclusions are given in Section 7.

2

The Ship Model

The ship model is based on [7, 6]. We use the earth-fixed vector representation M(~b)/~ + C(~b, @)//+ D ( r

~-~

(7.1)

where r~ = [x, y, r The variables x and y are the position variables, while ~/~ is the yaw angle. The vector ~-~ E ~3 is the control vector. The model has the following properties: The matrix of inertia including hydrodynamic added inertia effects, M ( r is symmetric and positive definite. The symmetry property is based on the ship having starboard and port symmetries together with the assumption of low speed, as opposed to high-speed applications, as we assume that the ships considered are conventional ships, not high-speed crafts. The matrix is bounded with respect to r 0 < Mm < [IM(~b)]l < MM

V~ C S 1

(7.2)

The Coriolis and centripetal matrix, also including added inertia effects, C(~b,//), satisfies the properties

6(r

C(r

=

C(~,,w)v

V~l, E S 1, Vv, w e '~3(7.3)

o~v -t- •w)

:

ozC(~), v) -t- Z C ( ~ , w)

V~) c S 1 , Vv, w E ~3 (7.4)

314


~CM > 0

< CMitVll

lic(r

V~ 9 S 1, Vv 9 #

Fhrthermore, the m a t r i x / I f - 2C, where {/1)/}0 = metric

8T(f/[(~)) -- 2C(r

?)))8 = 0

0r

(7.5)

~o, is skew sym-

Vr 9 S 1, V?), 8 9 ~}~3

(7.6)

The damping is assumed to be linear in/1, which is a good assumption for low-speed applications and for cruising at a constant speed. The hydrodynamic damping matrix, D ( r is in general non-symmetric. T h e hydrodynamic damping is due to wave drift damping and laminar skin friction, and it is dissipative

sTD(r

>0

V r C S 1, V8 9 ~3\{0}

(7.7)

Moreover, the damping matrix is bounded with respect to

0 < Dm < IID(~')ll

2CMVM A2,M + ~

(7.24)

11

>

1 M 2 r* 2--M---~k,~2,M + ld)

(7.25)

316


where A2,M is the maximum eigenvalue of the matrix A2, and V M is the maximum of the reference velocity//d. The time-derivative of the Lyapunov function candidate is then

--8T1(1113• -- 6 ( ~2, 82))81

-]- sT(M(O)A2 +

ldM ( r ) )s2

--&TATAI& -- z)TATA2~

(7.26)

Using the matrix properties given in Section 2 we find that this is upper bounded by

?

0. We cannot use the analysis of [8] to prove stability of the system, as the analysis is based on an open-loop model of the ship, and also the ship model does not include Coriolis and centripetal forces and moments as it is modeled for dynamic positioning purposes, not for tracking control. To prove that our closed-loop system is semi-globally exponentially stable, we use the Lyapunov function candidate V(Sl,~, s2,~, b)

IT

=

I T AI~ § l s T M ( r

~s 1 M ( r

§ ~

§

§ l"ybTb

(7.65)

where b --- b - b, and where -y is a positive constant. The time-derivative of V is then

--8T1(1113•

-

-

-s~(~dM(r

C(~,D,82))s I §

sT(M(~b)A2 +

+ C ( r ~ - s2))s2 - 4 D ( r

IdM(r - s~

--~TAITA18 -- ~)TATA2O -- 7bTgbb - .,/~TT-I~)

(7.66)

Using the matrix properties we find that this is upper bounded by ~r

__~ --(/1 -- CMI[S2I[)I[Sl[i 2 + MM(Id + A2,M)I[Sl[[ [IS2[[

(7.67)

- ( l d M m - CMIlil -- s21I)lls2112 -- Dmlls2112 + 11]s2111]')'b[I "7 1 11"7~,1: -[IAl~ll 2 - Ilh2OlJ 2 + KbMA~-llbblJ []A27)1[ - ~---~M I where KbM and TM are the maximal eigenvalues of the matrices Kb and T respectively. By completing the squares, we find that

l~r ~

S (Zd + A2,M))lIsllI 2 2x M Mm

--(Zl --CMIlS21[

(7.68)

- ( l ldMm - CMlli7 -- s2l[ -- 1MmA2,M - 2 TM )llsill 2 "7 1 1 7 _ 2Kb2MA22)ll'7/~[]2 -Dmlls2112 -IIAl~ll 2 - ~llA27)ll 2 - ( 2TM" We see then that if the following conditions are satisfied

A2m

KbM

A2,M § ~

11

>

1 M 2 (A2 M + ld) 2 Mm '

(7.69)

2 X/"TTM 2CMVM

4TM

§ Mm----~

(7.70) (7.71)

326


for some 7 > 0, then there exists a region f~ in which for some a > 0

? _< -allyll 2

(7.72)

Vy Cgt

where y = [Sl,AI~,S2,A2~,"ff)]T. By Lyapunov theory we thus have t h a t y = 0 is an exponentially stable equilibrium of the system (7.60-7.64). For tuning purposes, we can interpret the condition in (7.69) as an upper b o u n d on the inverse of the integral time constant of the estimation error ~, el. (7.63-7.64). Furthermore, the condition (7.70) can be interpreted as a l o w e r bound on the derivative gain of the estimation error ~), and (7.71) as a lower bound on the derivative gain of the error ~. We can find an estimate of the region of attraction along the same lines as in Section 3, and we then have the following proposition

Proposition

7.2 Consider the ship (7. 60) and the observer-controller with

bias estimator d

^

M(r

=

- llSl - Ale + M(r

-C(r

=

1

+ ldI3x3)A2~l

+M(r d^ --b dt

(7.73)

z + (As + l d h x 3 ) ~

(7.74)

- T - 1 D + Kbr]

(7.75)

--D -t- C(~), ?)2)(?) 2 - 81) Jr- D ( ~ ) ) ~ 2 -(As

llSl

- M(~b)A22)~

-

Ale

nt- M ( r

1

(7.76)

where A1 =diag{All,A12,A13} > 0 and A2 =diag{Am,A22,A2a} > O. Under the conditions A2m

KbM

A2,M-t--

(7.78)

ll

>

2 Mm

1 M• (A2,M + la)

(7.79)

(7.77)

2x/~TM 2 CM VM 4TM + - Mm Mm 7

where KbM is the maximum eigenvalue of the symmetric positive matrix Kb, VM is the maximum value of the reference velocity ild , TM is the maximum of the bias time constants, A2,M = max{)~21,/~22,)~23}, A2,m = min{Am,A22,A23}, where CM,MM, Mm are defined in (7.2-7.5) and ~/ is some positive constant, the closed-loop system is locally exponentially stable.


327

A region of attraction is given by

A - - {y E ~121 []y][

2 Mm

1 M ~ (A2 M + ld)

(7.84)

'

Moreover, .for dynamic positioning purposes the Coriolis and centripetal .forces and moments can be assumed to be zero. Then, from (7.68) it is seen that under the conditions (7. 82-7. 84) the closed-loop system is globally exponentially stable.

6

Simulations with an Environmental

Disturbance

The simulations were performed with a disturbance bias that initially was

b0': [ ]0

(7.85)

which was of size ~oo of the control force magnitude saturation. The matrix of bias time constants was T =

1000 0 0

0 1000 0

0 ] 0 1000

(7.86)

328


2..= 2 1.~ 1 9-~ O.E

-0.5 -1 -1.5 -t

0

1

2

3

4

y-position

5

6

FIGURE 6. The ship trajectory (-) and the desired trajectory (- -) in the xy-plane.

[0 0 0]

The bias estimator gain matrix was chosen as Kb :

0

0

0.2 0

0 0.2

(7.87)

Furthermore the parameter 7 -- TM = 1000. The other observer-controller parameters, and also the noise conditions and the reference trajectory, were chosen equal to those given in Section 4. Note that the conditions given in (7.69-7.71) are conservative. The parameter Kb chosen for the simulations is chosen above the bound of (7.69) in order to obtain faster convergence of the bias estimate. Furthermore, as in Section 4 the parameters l d and ll are chosen below the bounds of (7.70-7.71) in order to reduce the thruster forces commanded by the controller. The simulation results are shown in Figures 6-11. We see in Figure 11 that the natural logarithm of the norm of [?~T,~T, bIT was upper bounded by a decreasing straight line, showing that the convergence was exponential. From Figures 7-9 we see how the position estimates follow the position variables x, y and r despite the measurement noise. The velocity estimates are quite noise-corrupted, giving control inputs that are quite influenced by the noise as seen in Figure 10, but despite this the controller performance is good. In particular, the position and yaw angle tracking control is good, despite both measurement noise and the environmental disturbance. In Figure 11 we see that the bias estimates converge quite slowly, but the convergence of the vector loT, aT, ~]T is still exponential. Simulations with a longer time-scale show that the bias error


3

actual a n d d e s i r e d x d o t [m/s]

actual and d e s i r e d x [m] 9 , , 9

actual a n d estimated x [m]

!o

.

o.1

.

.

.

.

.

.

.

.

.

.

.

.

-0

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

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

0

0

0.6

Lo i oo

50 100 150 200 actual a n d e s t i m a t e d x d o t [m/s] : : :

0

-0 0

3

329

i

50

100

i ........

150

200

m e a s u r e d a n d e s t i m a t e d x [m] . . .

F I G U R E 7. The position variable x (-) and the desired position Xd (- -) [m], the velocity variable x (-) and the desired velocity ~?d(- -)[m/s], the position variable x (-) and its estimate ~ (- -), the velocity ~? (-) and its estimate ~dx^ (- -), the measurement of x ( ) and the estimate ~ (- -). l) converges to zero.

7

Conclusions and Puture Work

In this work a nonlinear observer and feedback control law was p r o p o s e d for o u t p u t feedback tracking control of ships. As the m e a s u r e m e n t s were quite c o r r u p t e d with noise, the control law used the filtered m e a s u r e m e n t s together with the estimated velocities for feedback. T h e observer a n d controller design were combined in order to utilize the observer s t r u c t u r e in the controller design a n d vice versa, in order to develope a c o m p u t a t i o n ally simple observer a n d control law. T h e resulting s y s t e m was p r o v e d to be semi-globally exponentially stable. If the Coriolis a n d centripetal forces and m o m e n t s were negligible, as for the special case where the desired traj e c t o r y was a c o n s t a n t position a n d orientation, the s y s t e m was globally exponentially stable. Furthermore, bias e s t i m a t i o n was i n t r o d u c e d in order to c o m p e n s a t e for the bias of environmental forces, a n d the o u t p u t feedback

330


actual

6

.

and desired y [m] . .

-

50

1O0

150

: :

0 -0

~ 2 0 0 actual estimated 6 : and T : y [m] :4 .......... !.......... i ..................

-20

actual

0.1 0.0 0 1 ~

0

O.

200

-0~

.

and desiredydot [m/s] : : 9 : .... 0 ~

,

,

,

.

9

9

50 1O0 150 200 actual and estimatedydot [m/s]

0

0

m e a s u r e d and estimated y [m] 6

.

.

i

0

50

1O0

:

150

200

FIGURE 8. The position variable y (-) and the desired position ya (- -) [m], the velocity variable ~) (-) and the desired velocity ~)d(- -)[m/s], the position variable y (-) and its estimate ?) (- -), the velocity ~) (-) and its estimate ~ y (- -), the measurement of y (-) and the estimate 7) (- -).

d^

tracking control scheme including the bias estimator was proved to give a semi-globally exponentially stable system. The results were illustrated by simulations. The position and heading measurements of the ship will include the oscillatory wave motion. It is not desirable that the controller reacts to this wave motion, because this gives increased wear and tear on the actuators and increased fuel consumption. Therefore wave filtering should be included in future work.

Acknowledgments The authors would like to thank T. I. Fossen and A. A. J. Lefeber for the interesting discussions regarding the topic.

7. Output Feedback Tracking Control for Ships actual and desired psi [rad] 4

1

9

,

,

. . . . . . . . .

i

. . . . . .

.....

O0 4

.=i...

9 . . . . .

4

0

. . . . . . .

:

1O0

........

::

,!,

i ......

i ........

150

~ . . . . . . . . . . . . . . . . . . . . . . . . . . .

0

2

:

50 1O0 150 200 measured and estimated psi [rad] : : :

50

....

.

50 1O0 150 200 actual and estimated psi [rad] 9

0

:

actual and desired psidot [rad/s] : : :

-o.!l~s O~]r

331

0

50 1O0 150 200 actual and estimated psidot [rad/s] : : :

50

1O0

150

200

200

FIGURE 9. The yaw angle r (-) and the desired yaw angle ~Pd (- -) [rad], the yaw rate ~ ( - ) a n d the desired yaw rate Cd(- -)[rad/s], the yaw angle r (-) and its estimate r (- -), the yaw rate ~ (-) and its estimate ~~b ^ (- -), the measurement of ~p (-) and the estimate r (- -).

8

REFERENCES [1] M. F. Aarset, J. P. Strand and T. I. Fossen, Nonlinear Vectorial Observer Backstepping with Integral Action and Wave Filtering for Ships, Proceedings of the IFAC Conference on Control Applications in Mafine Systems (CAMS), Fukuoka, Japan, October, 1998. [2] H. Berghuis, Model-based Robot Control: from Theory to Practice, Ph.D. dissertation, Univ. Twente, Enschede, The Netherlands, 1993. [3] H. Berghuis and H. Nijmeijer, A Passivity Approach to ControllerObserver Design for Robots, IEEE Transactions on Robotics and Automation, Vol. 9, No. 6, pp. 740-754, 1993. [4] C. Canudas de Wit and J.J.-E. Slotine, Sliding Observers for Robot Manipulators, Automatica, Vol. 27, pp. 859-864, 1991. [5] C. Canudas de Wit, N. Fixot and K. J. Astrom, Trajectory Tracking in Robot Manipulators via Nonlinear Estimated State Feedback, IEEE

332

7. Output Feedback Tracking Control for Ships taul [N]

10

tau2 [N]

C -5 -100

50

100

150

200

150

200

,0

tau3[Nm]

0

-10

0

50

100

FIGURE 10. The surge control force T1, the sway control force ~'2 [N] and the yaw control torque ~'3 [Nm].

Trans. Robotics Automat., Vol. 8, pp. 138-144, 1992. [6] T. I. Fossen, Guidance and Control of Ocean Vehicles, John Wiley &: Sons Ltd., Chichester, 1994. [7] T. I. Fossen and O.-E. Fjellstad, Nonlinear Modelhng of Marine Vehicles in 6 Degrees of Freedom, International Journal of Mathematical Modelling of Systems, Vol. 1, No. 1, pp. 17-27, 1995. [8] T. I. Fossen and J. P. Strand, Passive Nonhnear Observer Design for Ships Using Lyapunov Methods: Full-Scale Experiments with a Supply Vessel, Automatica, Vol. 35 No. 1, 1999. [9] I. D. Landau and R. Horowitz, Applications of the Passive Systems Approach to the Stability Analysis of Adaptive Controllers for Robot Manipulators, Int. J. Adaptive Control and Signal Processing, Vol. 3, pp. 23-38, 1989. [10] S. Nicosia, A. Tornamb~ and P. Valigi, Experimental Results in State Estimation of Industrial Robots, in Proc. Conf. Decision and Control, Honolulu, HI, Dec. 1990, pp. 360-365.


333

bias} and its estimate [N]

biasx and its estimate[N] 0.02-

0.1. =

o.Oiol. ........ ~~-.!

0.1 0.0.= ....... 0

!

:. . . . . . . . .

: . . . .

i ........

! ......

ij~J

i .......

-0.0, I

-0.05

.... /

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

.... ~ ..........

-0.1

-0.15

......

~0

-0-20

150

-0.o~i /

: .....

1~0

200

biasz and its estimate [Nm]

o.o4(

......

io

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

,,o

,oo

The natural logarithm of the norm of [etatilde,ehat,b

0.0. =

0.51 . . . . . . . . . . :. . . . . . . . . . . . . . . . . . . .

0

:. . . . . . . . .

0 -0.05

-0.5

/

9i:H

-0.1

-1

i ........................

-1.5 -0.15

ii -0.20

-2

i 50

1 O0

150

200

-2.5

50

1 O0

150

200

FIGURE 11. The disturbance bias (-) and its estimate (- -) in the x - direction and the y - direction [N], the disturbance bias about the z - axis (-) and its estimate (- -) [Nm] and the natural logarithm of the norm of [~T ~T,~)]T. [11] S. Nicosia and P. Tomei, Robot Control by Using only Joint Position Measurements, IEEE Trans. Automat. Contr., Vol. 35, pp. 1058-1061, 1990. [121 R. Ortega and M. W. Spong, Adaptive Motion Control of Rigid Robots: A Tutorial, Automatica, Vol. 25, pp. 877-888, 1989. [13] B. Paden and R. Panja, Globally asymptotically stable ' P D + ' controller for robot manipulators, Int. J. Control, Vol. 47, pp. 1697-1712, 1988. [14] M. Faulsen, O. Egeland and T. I. Fossen, A Passive Feedback Controller With Wave Filter for Marine Vehicles, International Journal of Robust and Nonlinear Control, vol.8,no.15, pp.1239-1253,1998. [15] J.J.-E. Slotine and W. Li, On the Adaptive Control of Robot Manipulators, Int. Y. Robotics Res., Vol. 6, pp. 49-59, 1987. [16] J.J.-E. Slotine, J. K. Hedrick and E. A. Misawa, Sliding Observers for Nonlinear Systems, ASME J. Dynam. Syst., Measurement, Control,

334

7. Output Feedback Tracking Control for Ships Vol. 109, pp. 245-252, 1987.

[17] M. Takegaki and S. Arimoto, A New Feedback Method for Dynamic Control of Manipulators, ASME J. Dynam. Syst., Measurement, Control, Vol. 102, pp. 119-125, 1981. [18] B. Vik and T. I. Fossen, Semi-Global Exponential Output Feedback Control of Ships, IEEE Transactions on Control Systems Technology, TCST-5(3):360-370, 1997.

D y n a m i c U C O Controllers and Semiglobal Stabilization of Uncertain N o n m i n i m u m Phase Systems by Output Feedback A. Isidori 1, A. R. Teel 2 and L. Praly 3 1Department of Systems Science and Mathematics, Washington University, St. Louis, MO 63130 and Dipartimento di Informatica e Sistemistica, UniversitA di Roma "La Sapienza", 00184 Rome, ITALY. 2Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106, USA. aCentre Automatique et Syst~mes, t~cole des Mines de Paris, 35 rue St. Honor6, 77305 Fontainebleau c6dex, FRANCE.

1

Introduction

One of the most active research issues in nonlinear feedback theory is the synthesis of feedback laws which robustly stabilize an uncertain system with limited measurement information. In the case of output feedback without uncertainty, one of the major achievements in this area of research has been the "nonlinear separation principle" proved in [6], where it is shown that (semi)global stabilizability via state feedback and a property of uniform observability imply the possibility of semiglobal stabilization via o u t p u t feedback. To cope with the restricted information structure, the stabilization of [6] includes an approximate state observer (whose role is actually that of producing approximate estimates of a number of "higher order" derivatives of the output) earlier developed in [3] to cope with a similar (though more restricted) stabilization problem. A "robust" version of this stabilization result was given in [5], where it was shown that, in the presence of parameter uncertainties, semiglobal stabilization via output feedback is still possible if a state feedback law is known which robustly globally stabilizes the system and its value, at any time, can be expressed as a (fixed) function of the values, at this time, of a fixed number of derivatives of input and o u t p u t (a uniforTnly completely observable (UCO) state feedback, in the terminology of [5]). The design tools introduced in [3] and [5] have been recently used in [2], where a new (iterative) procedure has been proposed for the robust stabilization of certain classes of nonlinear systems. This procedure is not based

336

8. Dynamic UCO Controllers and Stabilization by Output Feedback

on the idea of solving separately a problem of state feedback stabilization and a problem of a s y m p t o t i c state reconstruction. Rather, it is based on the recursive u p d a t e of a sequence of "dynamic" output feedback stabilizers: specifically, the basic result of [2] is t h a t if a suitable subsystem of lower dimension is robustly stabilizable by dynamic o u t p u t feedback, so is the entire system. From the point of view of the approach of [5], the condition on which the result of [2] relies (that happens to be necessary in the case of linear systems) can be viewed as a condition for the existence of a dynamic feedback driven by functions t h a t are expressible in terms of the o u t p u t and its derivatives, i.e., driven by U C O functions. In this chapter we review and extend the result of [2] and we show how this result can also be obtained as a special case of a general stabilization result based on the existence of a dynamic feedback driven by U C O functions. More specifically, after some preliminary definitions in Section 2 including our definition of uniform semiglobal practical a s y m p t o t i c stability, we discuss stabilization of n o n m i n i m u m phase nonlinear systems by output feedback in Section 3. This discussion is split into two parts: the relative degree one case in Section 3.1, and the higher relative degree case in Section 3.2. The main results of these sections are t h a t if a reduced order, auxiliary system can be stabilized by dynamic output feedback then the original n o n m i n i m u m phase system can be stabilized by dynamic o u t p u t feedback. In Section 4 we show how the results of Section 3 can be viewed as special cases of a general result on semiglobal practical a s y m p t o t i c stabilization by output feedback. In Section 4.1 we present some additional definitions, including the notions of uni.formly completely observable (UCO) functions and uniform semiglobal practical a s y m p t o t i c stabilizability by dynamic UCO feedback, and a general output feedback stabilization result which expands on the ideas in [5]. This result is specialized to the case of n o n m i n i m u m phase nonlinear systems in Section 4.2. In this section, we compare and contrast the controllers developed in Section 3 explicitly for the n o n m i n i m u m phase nonlinear system case to the controllers t h a t result from following the synthesis steps presented in [5].

2

Preliminaries For simplicity all nonlinear functions in this chapter will be assumed to be sufficiently smooth so t h a t all needed derivatives exist and are continuous, all differential equations have solutions, etc. 9 We will use ~ n ( r ) , with r > 0, to denote a closed ball of radius r in ~n. 9 Unless otherwise noted, #(t) is a measurable function taking values in a compact set P C ~ P . The set of such functions is denoted A/I~,.


337

9 The origin of a nonlinear dynamical system

= f(x,#(t), k) ,

(8.1)

with x E ~ n and k E Lr~c, is said to be uniformly semiglobally practically asymptotically stable in the parameter k if for each pair of strictly positive real numbers 0 < r < R < cxD there exist k E ~ c an open set O D Bn(R), a function V : O --~ ~ > 0 that is proper on O and strictly positive real numbers 0 < q < Q < cx~ such that i.)

~n(R)

ii.)

Bn(r) D {~ E O : Y(~) < q},

c

{~ E O : V(~) < Q},

iii.) and

OV Oxf(X'#'k) 0 there exists 5 > 0 such that all trajectories starting in a 5-neighborhood of .4 remain in an E-neighborhood of .4 for all time, and 9 for each c > 0 and each compact subset of ~ there exists T > 0 such that all trajectories starting in the compact subset enter within T seconds and remain thereafter in an c-neighborhood of ,4. In fact, due to recent converse Lyapunov function results (see [4], [1], [7]), these latter properties are equivalent characterizations of uniform semiglobal practical asymptotic stability. However, we are using the Lyapunov formulation here so that we can more directly appeal to the results on semiglobal practical asymptotic stabilization like [5, Proposition 3.1] where a Lyapunov formulation was used.

338

3


Stabilization of N o n m i n i m u m Phase S y s t e m s by

Output Feedback 3.1

The Relative Degree One Case

Most methods for robust stabilization of a nonlinear system by relative degree one output feedback rely on the hypothesis that the system has an asymptotically stable zero dynamics. The main reason why this hypothesis is assumed is that most of the methods in question use "high-gain" feedback in order to keep the output small, thereby enforcing a behavior whose asymptotic properties are essentially determined by the asymptotic properties of the zero dynamics. In particular, asymptotic stabilization occurs only if the latter is asymptotically stable, i.e., if the system is minimum phase. Consider robust (with respect to disturbances #(t)) stabilization of the origin for the system

9

= f0(z,y,,(t)) =

(8.2)

q(z,y,~(t))+b(y)u

where z E ~ n - 1 , Y C ~ , u C ~ , #(.) G AJp and b(y) ~ 0 for all y. In the case of uniformly globally asymptotically stable zero dynamics, i.e. (see [4]) when there exists a smooth, positive definite and proper function V(z) such that

OV Ozf~

Vz%O,

V#~P ,

the control law

u-

1

b(y)

ky,

where k is a sufficiently large number, solves the problem of semiglobal practical asymptotic stabilization of the origin. This follows from the fact that, given a compact set in (z, y) not containing the origin, for large enough ov f0(z, 0, #) - ky 2 in the derivative of the k the negative definite term -5~-~ composite Lyapunov function

U(z, y) = V(z) + y2 , i.e., in

OV

Oz fo(z, ~ , , ) + 2y[q(z, y, ~) - ky] ,

is able to dominate all nonnegative terms on the given compact set. In the case where the original output does not yield an asymptotically stable zero dynamics, one approach is to look for a new o u t p u t function, of


339

the form y - y* (z), for which the resulting system is uniformly minimum phase. Then, by following the reasoning above, the control 1 b(y) k ( y - y * ( z ) )

u-

may be used to achieve robust semiglobal practical stabilization of the origin. The potential drawback to this approach is that it requires the measurement, or at least the robust observability via the actual measured output y and the input u, of the term y*(z). Looking at the structure of the system (8.2), we see that the main information about the z subsystem that is robustly observable through the measurement y and the input u is the term q(z, y, it(t)) and perhaps its derivatives. The discussion that follows, in this and the next subsection, describes one very efficient way, suggested in [2], to use the information contained in q(z, y, it(t)) to design a stabilizing feedback law without actually requiring a measurement of q(z, y, p(t)). We will suppose A s s u m p t i o n 8.1 For the auxiliary system

= =

fo(z,~,it(t)) q(z,~,it(t)) ,

(8.3)

the controller =

N(~),

(8.4)

with N(O) -- O, is such that the origin o.f the system (8.3),(8.~) is uniformly globally asymptotically stable. Under this assumption, we can state the following result for the system (8.2) under the action of the controller

=

n(~) + M k [ y - g ( ~ ) ] 1 g 9N

-

b(y) [ - 0 T

+ Mk[y

1

-

(8.5)

- k[y -

Note that this is simply a dynamic feedback of the original (nonminimum phase) output y. T h e o r e m 8.2 Under Assumption 8.1, the origin of the system (8.2), (8.5)

is uniformly semiglobally practically asymptotically stable in the control parameter k. P r o o f . The result is established by noting, with the help of the input transformation

l

[ ONu~

u~ON

u = b - ~ --5"L(~) + (1 - - ~ - M ) v

]

,

(8.6)

340


that the system ~? = =

~1 --

fo(z,y,,(t)) n(~)

-

Mv

ON ( ON q(z,y,p(t)) +-~-~n(7~) + 1 -

]

(8.7)

v

with output 0 = y - N(W) has relative degree one with high-frequency gain identically equal to one, is minimum phase and can be written, globally, in a form that matches (8.2). Specifically, in the coordinates (z, ~, 0) where := ~ + MO, we have:

=

f 0 ( z , N ( ~ - MO) + O,#(t)) L(~ - MO) + Mq(z, N(~ - MO) + 0, #(t))

=

q(z,N(~ - MO) + O,l~(t)) + v .

=

0

(8.8)

By Assumption 8.1, when 0 is set to zero, the origin of the (z, ~) dynamics is uniformly globally asymptotically stable. It follows from the discussion above that the choice v = -kO is semiglobally practically stabilizing for the origin of (8.8). And, since N(0) = 0, the origin of (8.8) corresponds to the origin of (8.2),(8.5). Moreover, with this choice for v we see from (8.6) and the ~b equation in (8.7) that we recover the control law (8.5). A R e m a r k 8.1 If a controller of a form more general than (8.4) like 2~ =

=

exists (in the case where ~ depends on g we would need an assumption that guarantees a solution ~ to the second equation), a controller of the form (8.4) can be obtained by dynamic extension as =

~v+l

=

--frt(~v+l

-- Y)

with m a positive number. Instead of achieving uniform global asymptotic stability for the auxiliary system, this controller would, in general, achieve uniform semiglobal practical asymptotic stability in the parameter rn, at least in the case where the functions #(t) are restricted to have uniformly bounded derivatives. While this would complicate the above discussion, the conclusion of the theorem would still be the same. z~ R e m a r k 8.2 As discussed in [51, various local conditions can be imposed on the system (8.8) to guarantee uniform semiglobal asymptotic stability, as opposed to only uniform semiglobal practical asymptotic stability.


3.2

341

T h e R e l a t i v e Degree G r e a t e r t h a n O n e Case

T h e result of the previous section, on stabilization by dynamic o u t p u t feedback, can be extended to the case of outputs with relative degree greater t h a n one. Consider a nonlinear system modeled by equations of the form

~,

=

f(z, 41,... ,~-,#(t))

(8.9)

~,

=

q(z,41,... ,4~,#(t)) + b(4)u

Y

=

41

in which z E ~ n - r , #(.) C A/Ip and b(4) r 0 for all 4- This normal form m a y result from applying a globally defined, perhaps # dependent, coordinate transformation to a nonlinear system given in some other form. T h e only measurement t h a t we will assume is available is the o u t p u t y. W h a t we will show is that if a particular reduced system can be stabilized with measurements of 4 and q(z, 41,.. 9 , 4r, #(t)) then the system (8.9) can be stabilized with measurement of y only. W i t h the system (8.9), we associate an auxiliary system

5:a =

f~(xa, u~,#(t))

(8.10)

= h (xa,

in which z

"~

Xa'l /

41

X& ~

k x~,2 / j

and

f(z,

= (.

41, 9 9 . , 4r--1, Ua, ~(t))

fa'l(Xa'Ua) ) 42 ~

4r-- 1

?da

342


and

hgxa,

:= q(z,

r-1, ua,

A b o u t this system, we a s s u m e the following: Assumption

8.3

The controller ~b = u~ =

L(~,x~,2) + M y a N(~,x~,2),

(8.11)

with N ( 0 , 0 ) = 0, is such that the origin of the system (8.10),(8.11) is uniformly globally asymptotically stable. Under this assumption, we can s t a t e the following result for the s y s t e m (8.9) under the action of the controller ~b =

L(~,x~,2) +

u

b(~)

--

Mk[~r - N(~,x~,2)]

Mk[i~ - N ( ~ , x ~ , 2 ) ] ] + q ON f~,2(x~,2, ~ ) - k [ ~ - g ( ~ , Xa,2)]J Ox~,2 [L(~,xa,2) +

(8.12) Q

Note t h a t this is a d y n a m i c feedback of the o u t p u t y a n d its first r - 1 derivatives. 8.1 Under Assumption 8.3, the origin o.f the system (8.9), (8.12) is uniformly semiglobaUy practically asymptotically stable in the control parameter k.

Lemma

P r o o f . T h e p r o o f is the s a m e as the p r o o f of T h e o r e m 8.2. W i t h the i n p u t transformation

U = b - 1~ [ - ~ON - ~ L ( p , x ~ , 2 ) + ~ON f ~ , a ( x ~ , 2 , ~ ) + ( 1 - - -ONM~ ~ )

v]

(8.13)

we get the s y s t e m

Mv ~r ~--- ha(xa, ~r, #(t))-~ ~b =

L(~,xa,2) -

L(qo, Xa,2) +

(8.14) O--~a,2fa,2(Xa,2,Cr) +

1-

v

that, with o u t p u t 0 = ~ - N ( ~ , x,,,2), has relative degree one w i t h highfrequency gain identically equal to one, is m i n i m u m p h a s e a n d c a n be written, globally, in a form t h a t m a t c h e s (8.2). Specifically, in t h e c o o r d i n a t e s


343

(xa, ~, O) where ~ := qo + MO, we have:

k.. = I~(xo, N(~-MO, x~.,~)+O,.(t)) =

L(~ - MO, x~,2) + M h ~ (x~,N(~ - MO, x~,2) + 0 , # ( t ) )

=

h~(x~,N(~-MO,

(8.15)

x~,2)+O,#(t))+v.

By Assumption 8.3, when 0 is set to zero, the origin of the (x~, ~) dynamics is uniformly globally asymptotically stable. It follows, as before, that the choice v = - k O is semiglobally practically stabilizing for the origin of (8.15). And since N(0,0) -- 0, the origin of (8.15) corresponds to the origin of (8.9),(8.12). Moreover, with this choice for v we see from (8.13) and the equation in (8.14) that we recover the control law (8.12). A The dynamic controller (8.12) uses the state variables ( 1 , . . - , 4 ~ , i.e., the derivatives up to order r - 1 of the output y of system (8.9), as input. Thus, in order to find an output feedback controller, these variables must be replaced by appropriate estimates, which can be provided by a dynamical system of the form il = P~l + Qy

(8.16)

/ 110 0/

in which the matrices Q and P have the form

P .

-g2c~_2 0 1 ... . . . . . . -g"-lCl 0 0 ... --grco 0 0 ...

0

,

Q =

1 0

1 /

-g2c~_2 -gr-lcl --grc o

(8.17)

As shown in [3], it is convenient to saturate the resulting control law, at least where the estimates of ~ appear, so as to avoid the occurrence of finite escape times for large values of g. For example, we can replace the controller (8.12), which for ease of notation we now write as =

(8.18) =

with the controller --

(8.19) where (re(-) is a (by abuse of notation both a scalar and vector) saturation function (re(v) = v . m i n

1,~ T .

344


A controller of this type is able to robustly semiglobally practically asymptotically stabilize the plant (8.9). In fact, using the methods of [5] for example, it is possible prove the following result. T h e o r e m 8.4 (See also [2]) Under Assumption 8.3, the origin of the sys-

tem (8.9), (8.16), (8.19) [with C(., .) and g(., .) de.fined by the identi.fication between (8.12) and (8.18)] is uniformly semiglobally practically stable in the control parameters (k, g, e).

4

On Dynamic UCO Feedback

The basic observation of [2], summarized in Section 3.2 and on which the result of Lemma 8.1 rests, is that the term q(z, ~ 1 , . . . , ~r-1, ~r, #(t)) in the system (8.9) can be (and, in a nonminimum phase system, has to be) "isolated" from the rest of the system, using measurements only of the output and its first r - 1 derivatives, and treated as a separate source of information for feedback. Then, having a dynamic controller driven by the output its first r - 1 derivatives, as in Lemma 8.1, it is straightforward using ideas initially developed in [3] to find a dynamic output feedback controller that induces the desired properties, as in Theorem 8.4. From this point of view, the contribution in [2] is the identification of a natural (in fact, for linear systems it can be shown to be necessary) condition (Assumption 8.3) that guarantees the existence of a dynamic feedback that is expressible in terms of the output and its derivatives. Then Theorem 8.4 can be viewed as a special case of a more general result that is essentially contained in [5] (see [5, Proposition 3.1 and footnote 5]), namely that semiglobal practical stabilization by dynamic uniformly completely observable (UCO) feedback implies semiglobal practical stabilization by dynamic output feedback. We make this result explicit below.

4.1

General Results

Consider multi-input, multi-output nonlinear control systems

ic = y =

f(x,u, tt(t)) h(x,u, it(t))

(8.20)

with #(.) E Ad~,. The definition of uniformly completely observable (UCO) dynamic feedback, given next, at times implicitly constrains #(t) to be sufficiently smooth, where sufficiently smooth has to do with the number of times the output needs to be differentiated to reconstruct the UCO function.


345

D e f i n i t i o n 8.1 A .function ~(x~ u, ~) is said to be uniformly completely observable (UCO) with respect to the system (8.20) if it can be expressed as a function of a .finite number o.f derivatives of the output y and the input u, i.e., if there exist two integers ny and nu and a function q2 such that, .for each solution o.f U (nu+l)

y

=

f(x,u,#(t))

~

V

=

h(x,u,#(t))

(8.21)

we have, .for all t where the solution makes sense, ~ ( x ( t ) , u ( t ) , # ( t ) ) = r ( y ( t ) , . . . , y ( ' ~ , ) ( t ) , u ( t ) , . . . ,u('~"')(t)))

(8.22)

where y(i) denotes the ith time derivative o.f y at time t (and similarly .for u(~)).

R e m a r k 8.3 As in [5, Footnote 6], note the strong requirement that 9 is independent of #(t). On the other hand, note that the functions ~i ,

q(~l,... , ( r , # )

for the system (8.9) are UCO since we can write

~ = y(~-l),

q(~l,...,

~T, ~(t)) = y(r) _ b(y)u A

Our next definitions, on uniform semiglobal practical asymptotic stabilizability by dynamic UCO or output feedback, are closely related to our definition of uniform semiglobal practical asymptotic stability. However, as was the case in [5], we don't insist that the states of the dynamic compensator eventually become small in the closed-loop. We formulate the definition in Lyapunov function terms but, again, the definition could be formulated in terms of trajectories. D e f i n i t i o n 8.2 The origin of (8.20) is said to be uniformly semiglobally practically asymptotically stabilizable by dynamic UCO feedback if .for each pair o.f strictly positive real numbers 0 < r < R < o~ there exist: 9 a UCO .function ~(x, u, #) 9 .functions 0 and n, 9 compact sets Cn~ and Cnz , with Cn~ a subset of the interior of Cvl, 9 an open set (9 D B n ( R ) x Cnl , 9 a .function V : (9 -~ ~ > o that is proper on (9, and

346


9 strictly positive real numbers 0 < q < Q < c~ such that

i.)

( ~ ( R ) • c,,) c {~ 9 o : v(r _< Q},

ii.)

(-B~(r) • c,s) ~ {~ 9 o : v(r

< q},

iii.) and OV o x F ( x , ~) < o

V#E7 ),

VXE {~EO:q 1).

D e f i n i t i o n 1.1 The system (3.1)-(3.2) is a robust fault detection observer

(RFDO) with respect to the system (2.1a)-(2.1b) for a class of faults, CI, i.f 1. for all bounded u(t), dit), and z(O ) (t >_ 0), and with f(t) ---- O, the error dynamic .for e(t) is assymptotically stable, so that lim e(t) = O,

t----~OO

lim e(t) = O,

t----~OO

where e(t) = z(t) - Tx(t) and e(t) is given by (3.2). 2. for all bounded u(t), d(t) and y(t) (t >_ O) there exists at least one .fault vector f(t) # O, f i t) E Cf, such that

# o, (t >_ to) with e(to) = O. 1.1 Assumption 1.1 and condition Definition 1.1(I) ensure that z(t) exists and is bounded for zero faults and bounded ziO ). (Note Cf is not restricted here.) Condition Definition 1.1(1) also ensures that both the observer error and fault signal converge to zero for any disturbance when no .faults are present. Condition Definition 1.1(2) ensures that at least one .fault exists, which can be detected for all bounded nit), dit ) and Yit), given that at a specific time the error signal was zero. Remark

D e f i n i t i o n 1.2 The RFDO (3.1)-(3.2) is called a strict R F D O (SRFDO)

if.for a class f(t) E Cf, with f(t) • O, and any bounded set {u(t), d(t), y(t)}, there exists some to >_ 0 such that, given e(to) = O,

c(t) # o, (t > to).

356

1. Fault Detection Observer for a Class of Nonlinear Systems

R e m a r k 1.2 An SRFDO ensures that all f(t) E Cf can be detected, al-

though not necessarily distinguishable. Design Problem: Find matrices F, J, H, T, H ~ ( i , . ,. m),. H ~.i( z , 9 - - , p ) , H~x~(z, i 9 99- ,p), L1 and L2 and parameters d and do so that Definition 1.1 is satisfied. First condition Definition 1.1 (1) is considered. Condition Definition 1.1(2) is addressed in Section 4. Consider the observer error

e(t) = z(t) - Tx(t).

(3.3)

Using (2.1a)-(2.1b), (3.1) and (3.3) there obtains

~(t) = n I (t) + B I (t) + QU (t) + CU (t) ,

(3.4)

where

L I (t) = Fz(t) + Ju(t) + Hy(t) - T lAx(t) + Bu(t) + Ead(t) + Kaf(t)], m

(3.5a)

m

u (t)H~y(t) - T E u (t)A~x(t), i=1

(3.5b)

i=1

P

QU (t) = E yi(t)H~y(t) i=1 n

- T E xi(t) [A~x(t) + Eid(t) + g~ f(t)] ,

(3.5c)

i=1 P

P

CU (t)= E

E yi(t)yj(t)H~xy(t) j=l

i=1

n

- TE i=l

n

E xi(t)xJ(t) [A~xx(t) + EiJd(t) + g i J f ( t ) ] , (3.5d) j=l

where (3.5a)-(3.5d) correspond to linear, bilinear, quadratic and cubic terms, respectively, in x(t). Using (3.4) and (3.2) the following proposition holds true. 1.1 If T (T ~ 0), J, H, F, L1 (L1 ~ 0), L2, Hil~x(i = 1,... ,m), H~x(i -- 1,... ,p) and H~x~(i --= 1,... ,p) can be .found such

Proposition


357

that the following conditions are satisfied (for some do, d _> 1)

0 > ~ e ( A i ( F ) ) ;i = 1,-.. ,d

(3.6a)

H = [TA - F T ]

(3.65)

J = TB

(3.6c) (3.6d)

L2 = - L 1 T ( ~ Od,(n--p-kq)

= [TA - F T ] D - TEor

(3.6e)

(3.6f)

Od,(,~-v+q) = L1Tf} H~x Od,(n--p-kq)

= T A ~ ,i i

.

i

=TA~;z

= 1,... ,m

(3.6g)

= 1,-.- , m

(3.6h)

2H~ =TB;~'~;i= Od,(n--p-.bq) Od,(n--p)

(3.6i)

1 , . . . ,p

E I F ) ; i = 1,'-. ,p

=T(B~-

(3.6j)

= TBix+Pq2; i = 1 , . . . , n - p

Od,v = T K i + P ; i - =

l,...

(3.6k)

,n-p

(3.61)

Od,q-=TE~+P;i = 1,--- , n - p

(3.6m)

Ou,q = T [K1, . . . , K p] [Ip | Od,q = T [El,

6 H ~0

(3.6n)

= 1,... ,v

- "" , E p ] DE.,,Z i . 9 = 1 , " " ,q

--TB~,0

(3.6o)

9 i , j = 1,. .. ,p

Od,(=-p+q) = T ( B ~ J ~

(3.6p)

- [Eij + EJi]F) ; i , j = 1 , . . . ,p

d,n ~ 1 1,7,DiTpj+p. 9xxx ~i ~a4 ~

1,

. . ., n.

(3.6q) (3.6r)

P

Od,q = T [ E ij+p + EJ+Pl];i = 1,.-. , n ; j ----1,-. 9 , n - p

(3.6s)

Od,q = T[KiX, ' ' ' , KiPl[Ip | l~(j)]E~; i, j = 1 , . . . , v

(3.6t)

Od,~ = T [ K O+p + KJ+Pi];i = 1,--- , n ; j = 1,--- , n - p

(3.6u)

Od,q = T [ E ~1 + E l i , . . . , E ip + ElP]DJE~; i , j = 1 , . . . ,q, (3.6v) where l~(i) = [0,... ,0, 1, 0,.-- ,0]' E ]R'~ is the i-th unit vector, | K r o n e c k e r product and B i x x _- A x ~i

+ [ A I ~ , " " " , A ~n]

B ~i jx = ( A x x xij+ A ~ x x +j i[ [ A x ~ + A ~l j]

+ [[A~x + A ~ ]

is the

[In | jl

, ... , [ An~j + A j~n ] ]

[In |

, - - . , [ A ~ + A ~ x ] ] [I~ | l~(j)]),

D ~ = [Ip | lq(i)] E~ + [[Ip |

...

lip | lq(q)]] [Iq | Eflq(i)]

(3.7)

358


and where

o= r

O(,~_p)•

=

[~,

I,~_p

[ O(n--p)Xq

In--p

oq,(~_p)],

'

(3.8)

then e(t) and e(t) are implicitly decoupled from d(t) and satisfy, respectively, ~(t) = Fe(t) + W(t)f(t),

(3.9)

where W(t) is of the form W(t) = LI* + BI*(t) + QU*(t) + CU*(t), where LI* = [HKs - TKa] ,

(3.10a)

m

i Be* (t) = E u i (t)H~xK~,

(3.10b)

i=1

P

QU* (t)= E

(yi(t) - Ip(i)'Ksf(t)) [2H~xK~ - T K i]

i=l P

+ E tp(i)'K~f(t)H;~K~,

(3.10c)

i=l P

P

c u * (t) = E E l(i)'Ksf(t)l(j)'Ksf(t) [H~xK~ - T K ij] i=1 j=l P

P

+ E E Yi(t)l(j)'K~f(t) [T[Kij +Kji] - 3 H ij~ K ~ ] i=1 j=l P

P

- T K ij] + ~ , ~-~yi(t)y~(t) [3H)~K~ ij

(3.10d)

i=1 j=l

and e(t) = L1 [e(t) - TOKsf(t)] .

(3.11)

P r o o f . Firstly, consider the fault detection signal, e(t), given by (3.2). Using the partition T -- [T1,T2], where T1 E ~ d • and 7"2 E ]Rd• (2.1b) and (3.3) the signal e(t) can be expressed as

e(t) = Lie(t) + L2Ksf(t) + L2Esd(t) + [L1T1 + L2] x~(t) + r~T2x2(t).


359

Now, consider L I (t) given in (3.5a). Using the partitions given in (2.1b) and (2.2), (3.5a) expands to L I (t) = Fe(t) + [FT1 - TA1 + H] xl(t) + [FT2 - TA2] x2(t) + [J - TB] u(t) + [HKs - TK] f ( t ) + [HEs - TEa] d(t). The sufficient conditions for LI(t) and c(t) to be independent of x(t), d(t) and u(t) are then given in (3.6b)-(3.6f). When these conditions hold true LI(t) and e(t) become respectively L I (t) = Fe(t) + LI* f ( t ) , where LI* is given in (3.10a) and e(t) = Lie(t) + L 2 K s f ( t ) . Now consider the bilinear terms in (3.5b). Using (2.1b), (3.3) and the partition for A,x i , in (2.2), there obtains m

m

B I (t) = E

u~(t) [H~,x - T A Ii ~ ] xl(t) + E

u i (t)H~xE~d(t)

i=l

i=1 m

i u i (t)A2~x2(t) + BI* (t) f(t),

__T E i=1

where BI* (t) is given in (3.10b). The sufficient conditions for BI*(t) to be independent of x(t) and d(t) are given in (3.6g)-(a.6h). When these conditions hold true B I ( t ) reduces to B I * ( O f ( t ) . To obtain the most general conditions for the observer to exist, given in Proposition 1.1, the non-unique structure of the polynomial forms used in (2.1a) and (3.1) must be considered. It is assumed, without loss, that Hix and H~J~ have unique forms which satisfy the conditions HIx = [ H ~ , . . . ,HPxl[I | lp( i)]; i = 1 , . . . ,p, ij ji li H**~ = H~x x = [H~x~, 9.. lj

= [H . . . . . .[Hg~, .j l .

.

pj

, g. ~ ] [. I

pi

N/p(j)] il

| lp(i)] .

.- , H ;ip~ ] [ I | lp(j)]

jp ,HJxx~][IQlp(i)];i,j = 1 , . . . ,p.

Next, consider (3.5c) which can be expanded partially in terms of x l ( t ) , x2(t), d(t) and f ( t ) using (2.1b), (2.2) and p

x i (t)A~x= , - E i=1

n--p

E i=1

i=1

n--p

x l' ( t ) A x' x -t- E x 2 ( it ) A ~ ,+p x , i=l

p

x 2' ( t ) A'+P l x x x l ( t )= - E i=1

x li ( t ) [ A l+p l x ~ , . . . , A ~ ] [ I | Ip(i)]x2(t),

360


where xl(t) and x2(t) are independent. Hence, (3.5c) can be written as

QU (t) = QU~ (t) + QUid (t) + QUxdf (t) + QU* (t) f(t), where QU* (t) is given in (3.10c) and where p (t)

n--p

xi(t)[H~ x

=

TA~xxlxl(t )

-

x2(t)A2~x2(t ) i+p

TEi

-

i=1

i=1 P

-

~(t)[A2~x ~ -~- r41+p [''lxx,''"

TEx

,

AI~][In-p | lp(i)]lx2(t),

i=l p

n--p

QUxd (t) = E

xil(t) [2H~E8 - T E i] d(t) - T E

i=1

x~ (t)Si+Pd(t)

i=1

P

+E

lp(i)'E~d(t)H~Esd(t),

i=l p

n--p

QUxdf (t) -= T E

lp(i)'E*d(t)Ki f(t) - T E

i=1

x~(t)Ki+P f(t)"

i=1

Sufficient conditions for QV(t) to be independent of x(t) and d(t) are thus

2H~x-- TBxxO;~ -- 1,... ,p l

Od,(n_p)

~--

*

TB~+Pq2; i --- 1,.. 9 , n - p

Od,(~-p+q) = TB~q2; i ---- 1,..- , p Od,v =TK~+P;i = 1,... ,n - p 2 H ~ E 8 = T E ~ ; i = 1,... ,p Od,q=TEi+p;i

=

1,... , n - p

(3.12a) (3.125) (3.12c) (3.12d)

(3.12e) (3.12f)

Od,q=T [K1, 9" , K p] [IpQlv(i)]Es;i = 1,... ,v

(3.12g)

1 " , H ~p ] lip @ Es]D iEs,"i = 1,.. " ,q, Od,q= [H~x,"

(3.12h)

where (I) and ko are given in (3.8) and D iE8 is given in (3.7). In obtaining the conditions (3.12a), (3.12b) and (3.12h) the following result is used: n

O=-Ex~(t)A~x(t),

iff

0 = A i + [A1, ... ,A~][In |

(3.13)

i=l

Using (3.12a), (3.12e) can be combined with (3.12c) to give (3.6j). Substituting (3.12e)into (3.12h) gives (3.6o). Hence, sufficient conditions (3.12a)(3.12h) reduce to (3.6i)-(3.6o) which, if true, imply that QU(t) reduces to QU*(t)f(t).

1. Fault Detection Observer for a Class of Nonlinear Systems Next, consider the expansion of (3.5d) in terms of xl(t), by using (2.1b), (2.2) and the equivalences

d(t)

p n--p

i=lj=l P =--

P

EE

x~ (t)mlj ( t ) [ m, i~~I,+` .P.

AI.~]~,~[I |

Ip(j)]x2(t),"

i=1 j=l P

P

Z~

x~(t)lp(j)'E.d(t)HiJ.~E~d(t)

i=l j•l P

q

~1

9 H ; = ] IS | EJ~(j)]

act),

i=1j=-1 rt

Tt

EE

xi(t)xJ(t)A~

i~l j=l p

p

p n--p

~(t)~(t)A~ i=Ij=l n--p

i~1~1

p

n--pn--p

x2(t)xl(t)A~.__ __

+

i=1 j=l

where

xl(t)

x2(t)

and

~

j

i+py+p

i=1 j=l

are independent. There obtains

CU (t) = CU~ (t) + CU~d (t) + CU~ei (t) + CU* (t) f(t), where

CU* (t) is

given in (3.10d) and where

P

P

i~I

j=l

(t)[H;~ -

TAlx~]xl(t )

n--p n--p

- T E E x2(t)x2(t)d2~x~'J

i+m+Px2(t)

i=1 j=l n--p n--p

i=I

j=l

p

p

i=1 j-~l

where

F;j ~-

[A i+pjTp [''lxxzc

[[AljTP 4- A j+pll 2xxx],

+ itx~'2mxx - -

f~nj+p aj+pnll[In |

" " " , L'eX2xxx + "~2xxx JJ

361 and

f(t)

362


and, also, where P

P

CU~d (t)= E E x~ (t)x~ (t) [3H~xEs

-

T E ij] d(t)

i=l j=l

p

n--p

- T E E x~ (t)x~(t)[E iJ+p + E j+pi] d(t) i=1 j=l

n--p n--p -

T E E x~(t)x~(t)E~+PJ+~d(t) i=1 j=l P

+ E

P

[3xl (t) + l(i)'E~d(t)] l(j)'E~d(t)HiJ~Ssd(t),

E

i = 1 j----1 P

P

cu~es (t) = T ~ ~ [2x~ (t) + l(i)'E~d(t)] l(j)%d(t)K~J f(t) i=l j=l p

n--p

- T E E x~(t)x~(t) [K ij+p + K j+pi] f(t) i=l j=l

n--pn--p -

T E E xi2(t)xJ~(t)Ki+PJ+Pf(t)" i=l

j=l

A set of sufficient conditions for CU(t) to be independent of x(t) and d(t) is then ~j

6H~ x = TB~;i,j = 1,... ,p --TY#+PJ+PffI"i,j 1, . . .,n. Odin - - "L a J x x x ~1 =

(3.14a)

p

(3.14b)

Od,(n-p+q) = T B ~ q 2 ; i,j = 1,.-. ,p -- rrl~i+PJTPm. Z d,n--~Z)xx x "~,~,j ~ 1,''"

(3.14c) (3.]4d)

,n --p

6H~iE~ = T[E ~i + E3~];i,j = 1,... ,p

(3.14e)

Od,q

--TIE ij+p + EJ+Pi];i = 1,--- ,p;j = 1,.-- , n - p (3.14f) = T[E i+pj+p + EJ+Pi+P];i,j = 1,... , n - p (3.14g)

Od,q

= T [ K i l , -.. ,KiP][Ip|

Od,q

= 1,... ,v

(3.14h)

Od,(v)

= T [ K ij+p + KJ+Pi];i = 1,..- , p ; j = 1 , . . . , n - p

(3.14i)

Od,(v)

= T[Ki+pJ+B § KJ+pi+p];i,j = 1,.. . , n - p

(3.14j)

= [g~l, .-- , H~P][I~ | E~]DiE~;i,j = 1,..- , q,

(3.14k)

Od,q


363

where 9 and ~ are given in (3.8) and D ~ Es is given in (3.7). In obtaining (3.14c)-(3.14e), (3.14g) and (3.14j) the following result is used:

O=

x~(t)x~(t)AiJx2(t),

iff

0 ----A ~j + A ji,

i=l j~l

where xl(t) and x2(t) are independent. Also, for deriving (3.14a) and (3.14b) the following is used n

n

0=EExi(t)xJ(t)AiJx(t),

iff

i=l j=l

0 = A ij + A j~ + [[mil + A l i ] , ... , [Ain + A~']][In | In(j)] + [[Aj ' + AlJ], ..- , [Ajn + AnJ]][In | l~(i)]

(3.15)

and (3.14k) is obtained by using (3.13). Using (3.14a), (3.14e) can be combined with (3.14c) to give (3.6q). The conditions (3.145) and (3.14d), (3.14f) and (3.14g), and (3.14i) and (3.14j) can be combined to give the conditions (3.6r), (3.6s) and (3.6u), respectively. Finally, using (3.14e), (3.14k) can be written as (3.6v). Thus, sufficient conditions (3.14a)-(3.14k) reduce to (3.6p)-(3.6v), which, if true, imply that CU(t) reduces to CU*(t)f(t).V7 R e m a r k 1.3 If (3.6a)-(3.6v) hold, then ~(t) by (3.~), is independent of

d(t), and x(t). A subset of these conditions have been used in [10] [3] [4] .for bilinear systems and in [1].for quadratic systems. The full set is compact, much more general and the non-uniqueness of polynomial forms has been addressed.

4

General Detectability Conditions

A general set of sufficient conditions will be given for a REDO and a SRFDO to exist for f(t) C C I, where C I is defined as the restricted class

Cf = {f(t); f(t) = a.q.q(t)}, (to _< t ~ to + h),

(4.1)

where g(t) ~ 0 is a scalar function and _.a E IRv, _.a r 0. P r o p o s i t i o n 1.2 Assume (3.6a)-(3.6v) are satisfied. Then for class C I (h =

1. (3.1)-(3.2) is a REDO i.f for some f(t) E C I (a) Hl(s,a_.)X(s) ~ 0 for all s and (b) rank(Hl(s,a)X(s), H2(s, a)) r rank(H2(s,a)) for all s,

364


where ~ is the s and

operator with respect to time T, T = t -- to,

H1 (s, ~) -- L1 [(sI - F ) - I [jg, Jgg, Jggg] - T O K ~ [ I , O, 0]], (4.2a) X(s) = s

g2(t), g3(t)]', (t = 7- + to)

(4.2b)

H2(s,_a) = L l ( s I - F ) - l [ J u , Jy, Jyv, Jy]

(4.2c)

and where Jg = [HK, - TKa]a_,

(4.3a)

P

Jgg = E lp(i)' K , ~ [TK' - H' Ks] a_.,

(4.3b)

i=l P P

J99g= E

E

lp(i)'Ksalp(j)'Ksa_[HiJKs - TKiJ] a_,

(4.3e)

i=1 j=l

J~ = [H~x, 9- - , g m] [Ira | Ks_~],

(4.3d)

Jy = ( 2 [ H L , - 9 9 g L ] [ z p | K~] -

T[K1, ... , KP]) lip | hi,

J y y = [3[g~xx, 11

...

(4.3e)

l p + g~xxx,... pl ,H~cxx ,gppxx][X~(p+2) |

T [ K l l ... , K ip + KP~,- 9 9 KPP]] [I~ (p+2) | P

JY = E lp(j)'Ksa_ ( T [[K lj + Kill, ... , [K pj +

(4.3f)

K3Pl]

j=l

. 3[H~x~, . . lj.

pj

, H~x]

[Ip

| K~]) [Ip | _a].

(4.3g)

2. A SRFDO exists if (la) and (lb) hold for any f(t) C C/. P r o o f . Considering the class Cf and taking the Laplace transform of the residual in (3.11) there obtains (t = T + to) ~(S) = L1 [~(s) - T~Ksa_g(s)],

(4.4)

where ~(s) = 1:e(7) and g(s) = Ce(T). From (3.9),

~(s) = (sI -- F ) - l s Expanding the summations in W(t), (4.4) can be written in the m a t i x form

f(s) = Hi (s, ~ ) X ( s ) + H2 (s, a_)Y(s), where Hl(S,_a), X(s) and H2(s,_~) are given in (4.2a)-(4.2e) and where

Y(s) = s

y'(t)g(t), yy(t)g(t), y'(t)g2(t)] '


365

and where

yy(t) = [(yl (t))2,... , yl (t)yP(t), (y2(t))2,.. " , y2(t)yP(t),... , (yP(t))2]. [] R e m a r k 1.4 Condition Proposition 1.2(la) is only testable if g(t) is known

which is usually not the case (only ~ is known).

5

Testable

Detectability

Conditions

A set of testable (numerically tractable) sufficient conditions will be given in this section for C I defined in (4.1) for fixed h _> 0. P r o p o s i t i o n 1.3 Assume (3.6a)-(3.6v) are satisfied. Then

1. system (3.1)-(3.2) is a RFDO if there exists at least one f(t) E Cf such that (a) L~TOK~_ r 0 or

(b)

i. J~a* # O and i~. Rank(J;a*, J~) r Rank(J~), where J~ E N d~215 and J~ E ]~do• are constant and G* E N ad+l depends only on g(t), where J~ = L1 [-TOKsc~, F* ([Id | Jg], lid | Jgg] , [Id | Jggg])] , (5.1a) G* = [g(to + h), Gg, Ggg, Ggggl' ,

(5.1b)

J~ = L1F* [[Ia | J~], [Ig | Ju], [Id | ]y], [Id | Jyy]],

(5.1c)

N=d(m

+-~P(p + 5 ) ) ,

(5.1d)

and where Jg, Jgg, Jggg, Ju, Jy, Jy and Jvy are given in (J.3a)(4.3g) and

F* = e Fh [F~ Cg . . [G . O .,

, Fd-1], , G gd - - l]' ,

. . . . Ggg [G~

G.~. = [a~

...

(5.2a) (5.2b)

(~.d-1] t ,_gg j ,

(5.2c)

Cd-ll '

(5.2d)

~~ g g g J

366


and where Gkg = ~0 h ak(T)g(7 + to)dT,

(5.3a)

h

Gkgg= / Ggggk :

ak(T)g2(7 + to)dT,

(5.3b)

ak(T)g3(T +

(5.3c)

to)dT.

2. system (3.1)-(3.2) is a SnFDO for the class Cf if (la) or (lb) holds true .for any f(t) E Of. P r o o f . By assumption, for is

f(t) E CI, the residual from (3.11), at t = to+h,

e(t0 + h) = L1 [e(t0

+ h) - TOK~_.g(to + h)].

(5.4)

Solving (3.9) with e(to) = 0 and using the Cayley-Hamilton t h e o r e m [6], ak(7) (k = 1 , - . - , d 1) exist such t h a t

d-1 e -F~r ~_ E a k ( T ) F k ,

k=0 and then

e(to + h) can be written as hd-1

e ( t o + h l = e F h fo EakO-)FkW(T+to)a_g(w+to)dT.

(5.5/

k=O T h e error, (5.5), can be written as

d--1

(

k=0 ~'rt

+E

p

- ~+E ~txKs~V;

i=1 P P

+EE

i=1 j=l

[2Hx~xKs- TK ~] -~Y;~

i=1

[3H:~Ks - T K ij] a_Yigjk '~

+ E E I ( j ) ' K ~ a _ ( T [ K i J + K ji] - 3H~K~)a__Y~ i=l j=l

)

,(5.6)


367

whe,~ a~, a ~ ~nd a G ~e g~ve~ i~ (5.3a)-(5.3c), J~, J~9 and J ~ given in

(4.3a)-(4.3c), G k=

~e

and where

/o ak(T)r

+ to)g(r + to)dr,

(5.7a)

r~;k= ~ ~kO_)y~(r+ to)g0 + to)dr,

(5.7b)

/,

h

h

r[;Jk = fo ak(r)Y~O- + t~ I"

+ to)g(r + to)dr,

(5.7c)

+ to)d~.

(S.7d)

h

Y;'2 = Jo ~(r)y'(~ + t~

The summation signs can be eliminated in (5.6) giving

e(to + h) = J;G" + J;Y*,

(59

where J~', G* and J~ are given in (5.1a)-(59 and Y* E IRu depends upon {u(t),y(t),g(t)}, where Y is given in (5.1d) and where

r* = [G, r~, ro~, ~9]'

(5.9)

and where ,

,vg

,

..

[]-l(d--1)

...

Ug(d-1)]

t

(5.30a)

(59 9 , yl(d--1) ~gg

rg_ [y ,o 9

yip0 + y;10

,''"

, Y;g(

,v;p0 9

d--l)] t ,

(5.1oc)

v l(d-1) (5.30d)

where U~k, ygk, ygjk and yg~gkare given in (5.7a)-(5.7d) above9 [] C o r o l l a r y 1.1 Let the assumptions of Proposition 1.3 hold so that (5.8)

holds true. 1. If J~ =_ O, for all a.q., then system (39 class Cf.

is not a RFDO for the

368


2. J{ -- 0 .for any a_ ~ 0 iff all the following hold true: O-=LIT~Ks,

(5.11a)

0 = L~ [HK~ - TKa],

(5.11b)

0 = LI* (T[KI, ..

. , K p] _

[Ip | K~]DJKs

0 = L~/~/( [Iv~ |

+

[Hx~x, I ... , HPx][/p | K~]) D iK~' (5.1 lc)

|

|

DL ,[Ip,~Qlv(v)]] [G~ + G ~ , ] ) ,

+ [[Ip~ |

(5.11d)

where (5.11a)-(5.11d) hold for k -- 1,... ,d - 1, where L~ = L l e F h F k, I(=

(5.12)

[Hl~x,... , H i p s , . .

-T[Kll, .-.

,Kip, ..

, g ~ l x , . . 9 ,HPP~] lip2 @Ks] ,Kpl, ... , K pp]

and where D Ks ~ = [I~ | lv(i)] K~, + [[Ip | lv(1)],--.

[Ip |

[I|

G~, = [Ivp | Kflv(i)] [Iv | Kflv(j)] ,

(5.13a) (5.135)

where i , j = 1,... ,v. P r o o f . From (5.8) if Y* = 0 then e(t0 + h) = J~G* and hence, Corollary 1.1(1) holds true. Now, also, J~ _~ 0 for any _~ ~ 0 iff

0 =- LITOK~_, 0 =- n~ [Jg, Jgg, Jggg], for k = 1,-.- , d - 1, where L~, Jg, Jgg and Jggg are given in (5.12), (4.3a)(4.3c). Using the equivalences given in (3.13) and (3.15) these equations can be written as (5.11a)-(5.11d) and hence, Corollary 1.1(2) holds true. []

5.1

A Special Class ( S t e p - F a u l t s )

Consider the class of faults Cf described in (4.1) where g(t) = 1 (to < t < to + h). Thus, step-type faults are considered along direction _a. P r o p o s i t i o n 1.4 Assume (3.6a)-(3.6v) are satisfied. When g(t) = 1 Proposition 1.3 can be simplified to

1. system (3.1)-(3.2) is a RFDO if there exists at least one f(t) C Cf such that


(a) LITr

369

#0

or

(b)

i. 3 ; # 0

and ii. Rank(Jr, J~) # Rank(3~), where J~ E IRd~ and 3~ E l~d~ J~ = L*I [Jg + Jg~ + ']ggg]

-

where -

L1Tg2Ks~_,

(5.14a)

J~ = L1F* [lid | J~], [Id | [Jy + Jy]], [Id | Jyy]] , (5.14b) P M=d (m+ 5(p+3)) , (5.14c) and where L~ = L~F -1 [e ~ - ~ ]

(5.15)

and Jg, Jgg, Jggg, Ju, Jy, Jy and Jyy are given in (~.3a)-(~.3g). 2. system (3.1)-(3.2) is a SRgDO.for the class CI, i.f.for any f(t) 6 Cf , (la) o~ (Ib) holds t~.e. P r o o f . From proof of Proposition 1.3, when g(t) = 1, the residual in (5.4) can be reduced to

e(to + h) =L1 [e(to + h) - T O K ~ ] ,

(5.16)

where

e(to + h) = e Fh fo

ak(~')FkW (T + to)a_dT.

(5.17)

From Proposition 1.3, when g(t) = 1, there follows h

G~ = Gkgg -= akggg = fo ak(7)dT and this leads to d-1

h

h

E ffO ak(7)Fkd~'= fo e-F~d~k=O

_-F-1 [I_e-Fh].

(5.18)

Also, note that when g(t) = 1 (5.19)

370


Using the equivalences in (5.18) and (5.19), (5.17) becomes e(to + h) = F -1 [eFh - Id] [Jg + Jgg + Jggg]

k=O

i=1

+ E

i=1

I(j)'Ksc~ (T[K ij + K ji] - 3 H ~ x K s ) a_.Yik

j=l "~- ~

[3HixJxKs-TKiJ] o~yijk)

where h

U ik =

ak(T)U~(T + to)dT,

(5.20a)

y~k = In ak('r)Y~(T + t0)dT,

(5.205)

Ph yijk --_

/0

ak(w)yi(7 + to)yJ(w + to)dv

(5.20c)

and Jg, Jgg and Jggg are given in (4.3a)-(4.3c). These results can then be used to write (5.16) in the form e(to + h) = 3; + 3~1>*,

(5.21)

where 3~ and 3~ are given in (5.1a) and (5.1c) and 1~* C ~M is defined as

where U =

[

y_

[y10,...

ul~

... ,

]p=[ynO,..

umO, "'" , u l ( d - 1 ) , ypO,...

,yl(d-l),...

"'" , U rn(d-1)

]'

(5.22a)

,

,rP(d-1)]',

,YlP~ + ypl~ ... ,YPP~

yll(d-1), .. , y l p ( d - 1 ) + Y P l ( d - 1 ) " "

(5.22b)

, ,YPP(d-I)

]' ,

(5.22c)

where U ik, Y i k and yijk are given in (5.20a)-(5.20c) above. [] C o r o l l a r y 1.2 Let the assumptions of Proposition 1.4 hold so that (8.21) holds true.


371

1. If 3~ -- O, all a_., then system (3.1)-(3.2) is not a RFDO for class, Cf of .faults. 2. 3~ =- 0 .for all ~ ~ 0 iff (5.11c), (5.11d) hold true, with L~ replaced by L~ in both, and

(5.23)

0 = L~ [HKs - TKa] - L I T ~ K s holds true, where L~ is given in (5.15).

P r o o f . Corollary 1.2(1) holds by letting Y* = 0 in (5.21), for which e(to + h) = J{'. If 3{" - 0 for all _~ r 0 iff ^~

0 =--L1Jg - L1TOK~_, 0 -- L~ [J~,, J,,9],

where Jg, Jg9 and Jggg are given in (4.3a)-(4.3c). >From the proof of Corollary 1.1 these equations can be written as (5.11c) and (5.11d) (replacing L~ by L~ in both) and (5.23), thus proving Corollary 1.2(2). []

5.2

N u m e r i c a l Calculation Procedure

The gain matrices in the design of (3.6a)-(3.6v) can be calculated efficiently. The equations given in (3.6h), (3.6j)-(3.6o) and (3.6q)-(3.6v) can be arranged to give the form (5.24)

Od,A; = T Z ,

where the order of the contribution to the equation in (5.24) is not important and where N ' - n - P (2m + 3v + 3q + (p + n)(q + v + 1) + p ( p -

n) + n(2 + n))

+ 2 ( 2 m + p ( 3 + p ) + 2v(v + 1) + 2q(q + 1)). Equation (3.6e) and (5.24) are combined to give the form 0d,(5q+Tn-6p = [[TA - FT] ~ - TEa [Iq, 0q,(,~_p)] ,TZ] .

(5.25)

By splitting the two terms on the right hand side of (5.25) there follows two equations in F and T which are equivalent to (5.25) Od,(5q+Tn-6p) = F T X 1 + T X 2 or Od,(5q+7n_6p ) ~- [FT, T] X ,

372


where X 1 = [~, On,(4q-1-6n--5p)] , [Eo

-

Aa, Z],

X = [Xl, X2]'. Using the algorithms developed in [10] [11, F, T, L1 and L2 (and the other gains) can now be calculated using SVD decompositions. Then conditions (la) and (lb) of Proposition 1.4 can be tested (or similar conditions in Proposition 1.3 if g(t) is known). A recursive algorithm for the complete design of a RFDO (or SRFDO) can be given along the lines developed in [10] [11 .

6 Concluding Remarks A nonlinear fault detection observer has been proposed in this chapter for a nonlinear system involving polynomial nonlinearities of bilinear, quadratic and cubic forms. Proposition 1.1 gives sufficient conditions for the error dynamics and fault detection signal to be robust with respect to a disturbance and Propositions 1.2-1.4 give conditions for a fault to be detectable (RFDO and SRFDO). Fault isolation can be performed by using a bank of RFDO's [7]. The design procedure here involves only efficient linear matrix calculations, thus ensuring easy assessment of fault detectability.


7

373

REFERENCES [1] S. A. Ashton, D. N. Shields and S. Daley. Application of a Fault Detection Method for Pipelines, System Science, Vol. 23, No. 2, pp. 97-109, 1997. [2] P. M. Frank. On-line Fault Detection in Uncertain Nonlinear Systems Using Diagnostic Observers : A Survey, Int.J.Systems Sci, Vol. 25, No 12, pp. 2129-2154, 1994. [3] A. Hac, Design of Disturbance Decoupled Observer for Bilinear Systems, ASME, J. Dynamic Syst. Measure. Control, Vol. 114, NO. 12, pp. 556-562, 1992. [4] M. Kinnaert, Y. Peng and H. Hammouri. The Rmdamental Problem of Residual Generation for Bilinear Systems up to Input Injection, Proc. IFAC con.f. ECC'95, Rome, Italy, pp. 3777-3782, 1995. [5] A. J. Krener and A. Isidori. Linearization by Output Injection and Nonlinear Observers, Systems and Control Letters, Vol. 3, pp. 47-52, 1983.

[6] H. Nijmeijer and A. Van der Schaft. Nonlinear Dynamical Control Systems, Springer Verlag, 1990. [7] R. Patton, P. Frank and R. Clark., Fault Diagnosis in Dynamic Systems, Theory and Applications, Prentice Hall, 1989. [8] R. Seliger and P. M. Frank. Robust Component Fault Detection and Isolation in Nonlinear Dynamic Systems using Nonlinear Unknown Input Observers, Preprints of SAFEPROCESS '91, Sept. 10-13, BadenBaden, FRG. Vol. 1, pp. 313-318, 1991. [9] X. H. Xia and W. B. Gao. Nonlinear Observer Design by Observer Error Linearization, SIAM J. of Control and Optimization, Vol. 27, pp. 199-216, 1989. [10] D. Yu and D. N. Shields. Bilinear Fault Detection Observer and its Application to a Hydraulic System, Int. Jnl. of Control, Vol. 64, No. 6, pp. 1023-1047, 1996. [11] A. N. Zhirabok. Fault Diagnosis in Nonlinear Systems with Uncertainies, Proc.of IFAC Syrup., Sa.feprocess '97, Hull University, Vol. 1, pp. 528-533, 1994.

Nonlinear Observer for Signal and P a r a m e t e r Fault D e t e c t i o n in Ship Propulsion Control Mogens Blanke and Roozbeh Izadi-Zamanabadi Department of Control Engineering Aalborg University Fredrik Bajers vej 7C DK-9220 Aalborg, Denmark

1

Introduction

Faults in ship propulsion and their associated automation systems can cause dramatic reduction on ships' ability to propel and maneuver, and effective means are needed to prevent that faults develop into failure. The chapter analyses the control system for a propulsion plant on a ferry. It is shown how fault detection, isolation and subsequent reconfiguration can cope with many faults that would otherwise have serious consequences. T h e chapter emphasize analysis of re-configuration possibilities as a necessary tool to obtain fault tolerance, showing how sensor fusion and control system reconfiguration can be systematically approached. Detector design is also treated and parameter adaptation within fault detectors is shown to be needed to locate non-additive propulsion machinery fault. An adaptive observer is suggested for this purpose, est trials with a ferry are used to validate the principles. Propulsion system availability is crucial for a ship's ability to maneuver. Nevertheless, control systems associated with propulsion required to be fail-operational or fault-tolerant. Instead, local safety systems protect machinery. They prevent continued operation or start-up if sensors inform that local shut-down. While fail-safe for each piece of machinery, the local safety approach is not globally fail-safe for the ship. The consequence has been many events where consequences vary from irregularity to major economic loss and causalities. Several events could have been prevented if automation systems had been designed to be tolerant to faults, with overall availability in mind. Fault-tolerant control (FTC) is a methodology where analytical redundancy is employed using software that monitors the behavior of components

376

2. Nonlinear Observer for Signal and Parameter Fault Detection for Ships

and function blocks. Without hardware redundancy, some faults m a y inevitably cause a plant shut-down, but the F T C strategy is t h a t the m a j o r i t y of faults, and in particular the ones with severe consequences, are accommodated. T h e objective is to keep plant availability but accept reduced performance as a trade-off. The first step to achieve fault tolerance is efficient detection and isolation of faults. This is a particular challenge when a system is non-linear. In this chapter, an active solution to the F T C problem is employed where on-line fault detection and isolation can trigger a discrete event signal to a supervisor-agent when a fault is detected. The supervisor-agent will activate remedial actions. Re-configuration possibilities are analyzed for a ship propulsion system consisting of a main engine with a controllable pitch propeller. It is shown that combined p a r a m e t e r and o u t p u t estimation is required and an adaptive observer is proposed for fault detection. A continous-time non-linear observer is shown to possess very useful features and can be used during b o t h detection and re-configuration. Simulations on a model of a ferry [12] illustrate performance for a selected fault scenario.

2

Ship Propulsion System

This section introduces m a t h e m a t i c a l models for ship speed, propeller and prime mover, the essential propulsion system components. T h e purpose of the modeling is to obtain information to design fault detection and isolation (FDI) modules for essential faults and to give the prerequisites for design of re-configuration when faults occur. The block diagram in Fig. 1 illustrates the structure of the propulsion system.

Shaft

YP!

~

~

mech~sm mc

J ship- .d L . u I

.....

I

T

I

.....

I

I Q~(~Vam) Ship sI~d

FIGURE 1. Structure of dynamic relations for CP propeller, shaft and diesel engine.


377

I

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

r.n. ".r

_i~. ~ n m .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

0".,-~.~3 u.

N ~ ~ ,im ) .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

FIGURE 2. Hierarchy of controllers for propulsion system. The handle gives input to a combinator, efficiency optimizer, and ship speed control. Lower level controls are shaft speed (governor), propeller pitch and diesel overload blocks.

2.1

Propeller Thrust and Torque

Controllable pitch (CP) propellers have blade angle (pitch) controlled by a hydraulic servo system. Developed thrust and torque are functions of pitch, shaft speed and flow velocity through the propeller Tprop Qprop

=

:

fT_prop(O, n, Va) fQ_prop(O,n, lra)

(2.1)

These can be shown to approximately follow quadratic relations, for thrust

Tprop -~ ~nln~'~ Inl n + T,~vnYa

(2.2)

Qp~op = Q0 bl n + Qn~,o, IOl in[ n + QnvoO [hi V~

(2.3)

and for torque

These relations give a quite good approximation in the steady state cases whereas they are less applicable during large transients. The term Q0 [nl n accounts for the torque at zero pitch.

2.2

Diesel Engine Prime Mover

Elaborate details of the dynamics [3] are not important in this context, but would be for detailed design of FDI for the engine. Here, diesel torque can

378


be considered linearly related to the fuel index, without dynamics involved, Qeng = K y Y

(2.4)

The dynamics of propeller and shaft is merely that of rotating inertias subjected to torque balance between prime mover torque and load torques, d r l r n2 ~ ~ t ) ~-n(Qeng-Qprop-Qf)

(z.5)

The dynamics of the prime mover and its control system is tightly coupled to the speed dynamics of the ship through the propeller (2.3). The structure of prime mover control was also shown in Fig. 1. The measured shaft speed is compared with a reference speed and the governor (speed controller) regulates the fuel injection to the engine to obtain the desired speed. Limit curves are incorporated for shaft speed dependent torque and air pressure.

2.3

HuU Resistance

Ship's resistance to motion through the water can be described to the first order by a resistance curve, which is a third to fifth order polynomial in u. The order of the polynomial is higher the closer the ship operates into the wave making region. The resistance curve is known a priory but with some uncertainty. The first order equation m(] = R(U) + (1 - t)Tprop + Text is a sufficient approximation in this context.

2.4

Actuators for Fuel Injection and Propeller Pitch

The actuators can both be modeled as first order dynamic systems with limits in rate of change and in output. The electro-hydraulic pitch control system is described by the following equations: u~ = kt (0rer - G ~ )

z maX(0min, min(uo, Om~x)) = max(0min, min(0, Om~,))

(2.6)

The diesel actuator is equivalent to this with command Yc from the governor, rate limits Y C [Yd-, Yd+] and o u t p u t Y E [0, 1].

2.5

Sensors

Sensors for propeller pitch and fuel index are conventional angle transmitters. Shaft speed is usually measured by a set of pulse pickups. A maximum

2. Nonlinear Observer for Signal and Parameter Fault Detection for Ships F u n c t i o n blocks R combi:

C optim: -

nd Vgd

~)d

=C

Service

= RCB(ha)

--

hd v~m nm

OP

Ym um

379

normal (fault)

(n, ~) demand { freeze (input fault) } best efficiency use estimate (inp.fault) } { roll-back (ref. fault) } alter limits (diesel fault) }

nd

Vgd

C over:

~

= C_OL

nm

~m

constant U

u~

{ freeze ha(Urnfault) }

C speed: hd • C_SS I" Um I

I.

C shaft: Y c = C

" " "

avoid overload {freeze (fault)} { use estimate (fault) }

.

SP (nT,nm)

estimate Um(Umfault) } { roll-back(Ur fault) } shaft control { estimate n(nmfault) } { } r o l l - b a c(nT k fault)

TABLE 2.1. Function blocks treated as virtual components. logic selects the higher of the two signals. This protects against drop out of one of the pick ups but not against a "high signal" fault or failure in a common processor/rate counter servicing both channels. The ship speed is measured by magnetic log, Pitot tube or Doppler log. The two former measure water speed close to the hull and are quite prone to fluctuations from the turbulence and cross flow.

3

Control Hierarchy

The control hierarchy includes controllers for: shaft speed; propeller pitch; diesel overload control; combinator curves from handle position to generate reference values of n and ~; efficiency optimization using n and v~; constant ship speed control. The signal flow between these function blocks is shown in Fig. 2. The interested reader can find details about the control functions in [12]. The input-output of each block is listed in Table 2.1. The table lists the service of the function block in normal operation and the desired function in case of specific faults. The listing of desired remedial actions is a result of a combined fault-propagation and structural analysis of the propulsion system, including the possibilities for re-configuration after serious faults

380

2. Nonlinear Observer for Signal and Parameter Fault Detection for Ships Constraint

Description

: fnm = n f~ : Orn = ~

f

f~ : U m :

U

f~ : Ym = Y f~ : Ky ----Kyc f~ : Qeng = K y Y

re: -Qeng = Qprop + Q I f~ : Qprop = fQprop ] ~ : Tprop = fTprop

f~o: R(U) = fRu(U) f~l: R(U) = - Text - (1 - t) Tprop

I]

sensor_n sensor_t9 sensor_U sensor_Y engine gain engine torque shaft balance propeller torque propeller thrust hull resistance. ship speed

TABLE 2.2. Static constraints for shaft [4]. The table list their input and output, faults considered, and re-configuration possibilities. An example of this analysis is provided in the next section.

4

Structural Analysis

Structural analysis [7, 10, 17] is the study of properties which are independent of the actual values of the parameters.Constraints, here used as a synonym for relations, between variables and p a r a m e t e r s from the operating model are used in the analysis. The links are represented by a g r a p h or a table, on which the structural analysis is made.

4.1

D e s c r i p t i o n o f the M o d e l

The model of the system is considered as a set of constraints, 5c = { f f , f~, ... ,- 9- , f ~ } that are applied to a set of variables Z = X U )(. X denotes the set of unknown variables while ) ( is the set of known variables: sensor measurements, control variables, constants, and parameters, and reference variables. The constraints are the relations imposed between values of the variables, as given by the relevant physical laws. The constraints for the propulsion system are listed in Table 2.2.

4.2

Formal Representation

T h e structure of the system is described by the following binary relation: S:$-x

Z ~ {0,1}

c z S(][, zj) = 1 (f~' J) --* S ( f [ , z j ) = 0

iff f [ applies to zj, otherwise.


381

These relations can be represented by an incidence table or the equivalent digraph. Fig. 3 a) shows the structural table for the propulsion system. Some constraints may be expressed through non-isomorphic mappings for certain variables. Such variables can not be re-constructed through an inverse mapping from knowledge about remaining variables. Elements with this property are marked by M's (for multiple), replacing the l'es in the incidence table and unidirectional arcs in the corresponding digraph. An example of such a constraint is f~: it is always possible to compute the value of Qprop from f~ when ~, n, and V~ are known. However, knowing the values of Qprop, n, and V~ does not enable calculation of a unique r in all cases. This fact is not apparent from the equations in this chapter but is apparent when looking at the underlying propeller characteristics. The non-isomorphic problem for the Qprop relation is only present in a narrow range of transient conditions (during crash stop).

~.3

Sensor Fusion for Re-configuration

In control systems, re-configuration can be obtained either by means of hardware redundancy or the use of software redundancy. In the case where hardware redundancy exists, the scope of design is FDI algorithms and hardware switching. When analytic redundancy is available, fault tolerance is obtained by means of sensor fusion: the value of the signal which is lost or corrupted due to faults, is reconstructed using known values of other signals. The structural analysis approach is usually employed to obtain analytical redundancy relations for FDI [9]. It can, however, be used without difficulties for sensor fusion as well, since a constraint relation can be used to re-construct a signal from the other measured variables. An example for the propulsion system is a critical fault in the shaft speed measurement which can be accommodated by estimating shaft speed from other available measurements.

Fault in the Shaft Speed Measurement A critical fault in the propulsion system is a failure in the measurement of shaft speed. The constraint f~ represents this device in Table 2.2. A fault occurrence means that the constraint f~ does not hold, e.g. the values of the variable nm are not correctly related to the values of the variable n. Figure 3 b) shows that variable n is involved in 3 relations which are specified by the constraints f~, f~, and f~. Since the constraint f~ is not valid, there are two other possible ways of calculating the values of the variable n, namely through constraints f~ and f~. As it is shown in Figures 4 a) and 4 b), the ship speed can be described as a function of the other

382

2. Nonlinear Observer for Signal and Parameter Fault Detection for Ships v r

n

r~ e.~

I

I;

1

I

1

I:

1

1 1

I: I;

1

1

1 1

M

1

M

1

[,

fu ~

1 1 1 1 1

1

a)

"

../

\\

."

r_

r

b)

~r

FIGURE 3. a) The structural representation of the model by a (binary) table. l's are replaced by x's to indicate causality (calculability) between variables, b) Corresponding digraph representation. known variables as:

% = L(om, Kin, K~c, Urn) ~ = f~(O~, U~)

(4.7) (4.S)

The process to apply the sensor fusion based on this approach is the following: For the interested variable (for instance n) identify the set of related constraints (f~ and f~) and 9 choose one of the available constraints 9 check the causality for the constraint in order to find out t h a t the variable can be c o m p u t e d through this constraint.


e @~__..~.___~

383

T ~

.....

e

b)

~,r

FIGURE 4. Sensor fusion methods based on structural representation: shaft speed calculation through a) propeller thrust equations f~ and b) propeller torque equation f~. 9 for all the variable connected to the chosen constraint search backward until all end variables are known variables. R e m a r k 2.1 The described procedure shall find all the existing paths from the unknown variable to sets of known variables. Some of these paths may include loops, which are related to the existing control or natural loops in a system. By examining all the constraints, the set of equations/relation by which the variable can be calculated is identified and can be used for re-configuration purposes. For the shaft speed failure, the m e t h o d is illustrated graphically in Figures 4 a) and 4 b). Grey dashed arrows show the calculation paths to the known variables. Using quadratic representation of the propeller torque, the variable n

384


can be estimated from the constraint f~, but estimation of n based on static relations is obviously too primitive. A non-linear observer is employed instead.

5

Isolation of Shaft Speed and Engine Faults

This section deals with the problem of detecting whether a shaft speed fault or engine fault has occurred. The relevant dynamics to be considered was described above, leading to the constraints f7 to f11. T h e task at h a n d is to estimate a signal fault in n,~ and a p a r a m e t e r fault in K u. T h e dynamic equations directly determining shaft speed are

Itn = Q~ng - Qp~op -

QI

(5.9)

Q~,~g = K~Ym Taking ship speed U as a measured variable - a valid a s s u m p t i o n when Um is non-faulty, Qp,-op = Qo In[ n + Q~,,~ Ivgl Inl n § Q~,~y, v9 In[ (1 - w o ) U m

(5.10)

In the sequel, we use Qe,~u - Q~,~v,~ (1 - w0) for brevity. Shaft speed is positive in a controllable pitch installation, so

~ = ~ 1 ( K ~ Y m - Q f - Q o n 2 - Qonn [vg[n 2 -- Q o n u v q n U m )

(5.11)

Following the benchmark definition in [13], we need to consider faults in either shaft speed measurement or in the diesel torque coefficient, n m -= n + n f

Ku = Kyc - Ky I

(5.12)

and The detection task is hence increased from a single fault shaft speed sensor fault detection to a more complex one of simultaneous additive and non-additive faults. An adaptive observer providing simultaneous state and p a r a m e t e r estimates is a natural choice as a candidate for detection of the two particular faults.

5.1

Adaptive Observer

The dynamic relation (5.11) can be written in a form which is linear in the unknown p a r a m e t e r -- (I)(x, u2, u3) + Oul y=x

(5.13)


385

using 1

= ~ ( - Q o ~ 101 n 2 - Qo,~uOnUm - Qon 2 - Q f)

x=n,

(5.14)

U l = Y m , u 2 = U m , u 3 = O m , O-- Ky It

An adaptive observer can then be build by using the measured inputs: Ym, Urn, Z~m and the measured state nm. It is noted that the more general case was treated in [8]. However, the detailed assessment of the Lipshitz conditions, that determine the gains in the adaptive observer, are easily made too conservative to get useful results. A few comments are thus considered appropriate. This leads to the following theorem. T h e o r e m 2.1 An adaptive observer .for the problem

1 (KyYm - Qe,,~ IO[ n 2 - QonuOnUm - Qon 2 - Q f )

i~ = Tt

(5.15)

is the state estimator ~=~"t

1 (_Qo~nOm~2 _ QonuOm~Um - Q0~ 2 - Q f) + Y m ~ + L ( n m - ~) (5.16)

with parameter updating O = PYre (nm - ~)

(5.17)

The adaptive observer is semi-globally asymptotically stable with Ym > 0,

P > 0,

Qeng'max (o~nmax ~ - ~ ) L > It nmax \ nmin

(5.18)

where It, Qeng,ma~, nmax, nmin, O~ and ~ are plant specific parameters.

[]

The nonlinear torque function 1

O(n,u) = y~ ( - Q o ~ I ~ 1 n2 - Qo~u vgmnUm - Qo n2 - Qf ) is Lipshitz II~(n, u) - ~(~,u)ll < ~11n -

~ll

since 9 (n, u) - ~ ( ~ , u) --It 1

y~ ( - ((Q0 + Q~n~ 10ml) n + Qo~u~mUm) - (Qo + Q ~

[~m[) fi) (n - ~)

386


Practical diesel torque constraints and ship speed being dynamically related to torque lead to -

((Qo + Qonnl

ml) n +

QonumUm)
0 is a positive constant such that there exists a s y m m e t r i c and positive definite matrix P t h a t the modified Riccati equation

410

3. Nonlinear Observers for Fault Detection and Isolation

0

=

A H P + pATH + P[I~/2 - 1 C T C F ] P + I + Ie

(3.26)

is satisfied, a/ are constants such that Fi can be represented by the sumn mation ~ j = l ~J~'J, CF = [I -- CF[CF]t]C, and the superscript ~ means the pseudo-inverse of a matrix. Note that the solutions of (3.25) and (3.26) can only be obtained by an iterative algorithm. Note that (3.26) given in [20] has a misprint. R e m a r k 3.3 The Thau-type observer (3.24) .for sensor fault detection and

isolation was presented in [29]. In this ease the selection of the matrix H is simplified because no directionality is required. A different approach, including a more general class of systems, was considered in [42]. This approach is based on a generalization of the notion of (h,f)-inva riance using the so-called "algebra of functions" [41]. T h e same idea, but utilizing the disturbance decoupling approach, was proposed in [38]. Based on the invariance principle (or on the disturbance decoupling approach), a set of state transformations for the system (2.1) is defined. Each transformation maps the state of (2.1) (or the system (3.11)) into a subsystem that depends only on one fault or on a set of faults, and is robust to the rest of them. At this point, nonlinear identity observers are used to build a bank of residuals that have the desired fault directionality.

3.6

O b s e r v e r f o r F a u l t D i a g n o s i s in B i l i n e a r S y s t e m s

Fault-diagnostic methods for bilinear systems have been studied only in recent years, maybe because sometimes the linear approaches (such as the UIO) can be extended to bilinear systems, or because bilinear systems are a special case of more general nonlinear systems. The study of bilinear systems is, however, important, because a set of physical systems (nuclear reactor systems, suspension systems, fermentation processes, hydraulic drives, heat exchange systems, etc.) c o u l d b e m o d e l e d by bilinear equations [32]. Further, it is possible to take advantage of the special model structure in order to improve the design of the residuals. Different approaches have been proposed [25, 40, 48, 49]. In [40, 48, 49] the unknown input fault detection observer approach (in different versions) is extended to bilinear systems. In [25] a more general (and maybe a more realistic) class of bilinear systems was considered. The approach includes systems represented by


p

ic

=

2

A j u j x + ~ ( u , y ) + E ( E ~ x + F~)fi

Aox + E j=l

y

=

411

i=1

(3.27)

Cx.

For sake of simplicity, like in [25], only two possible faults are considered here. The approach given in [25] is reviewed. A fault-detection filter for the system (3.27) is given by P

=

Aoz + E [ A j u j z

+ Bjujy] + Pq2(u,y) + boy + R - 1 L ~ [ L l y - L2z]

j=l

r

=

L l y - L2z

R

=

-OR - A~R - Rs

+ L~L2

P j=i

if the conditions A ~ P - PA~ + [?iC

=

0

L1C - L 2 P

=

0

=

o

P [

F2 ]

with L1, L2 and P non-zero, and u(t) a 0-strictly persistent input [25] for the system P

j=l

q

=

(3.28)

L2rl,

are satisfied. Here u(t) is said to be a 0-strictly persistent input for the system (3.28) if: 3t0 > 0, 3a > 0, such that for any t > to

fo t

e--O(t--s) ~)u ( ~ - 8) T c T C~)u(f;

where I is the mxn identity matrix, r and O must be positive real.

-

s)ds

>_ o~ TI

is the transition matrix of (3.28)

412

4


Nonlinear Observer Design via Lipschitz Condition

Consider a class of nonlinear systems described by E2

=

Ax + Bu + /(x,u)

(4.29)

y

=

Cx

(4.30)

where x E ]R n, y E IRp, u E IR m and f ( x , u ) E IR q and the matrices A, B, C and E have appropiate dimensions. The matrice A, B, C and E are known. In this section, each component of the nonlinearity carl be a nonlinear function on the state and the input too. First, an observer is designed for the system (4.29), (4.30). Then its stability is discussed. It is shown in which cases this observer can be applicable. Finally the residual generation for fault detection and isolation is presented. A nonlinear observer for a class of nonlinear regular systems was presented by [24]. [36] discussed the same observer as [24], but better results of the upper bound of the Lipschitz constant were obtained by [36]. Generalizing the Lyapunov-like equation [37], the upper bound of the Lipschitz constant can be augmented/greater. The observer design was discussed also for nonlinear singular systems. [10] presented a reduced order observer for nonlinear system, which is independant on the control variable. A m e t h o d to reconstruct the whole state of a class of nonlinear singular systems is given in [19]. In this section, the gain matrix of the observer presented by [19] will be obtained with a more general Lyapunov-like equation. So a better solution for the upper bound of the Lipschitz constant can be found. 4.1

Observer Presentation

So that the observer can be designed, the following three hypotheses must be satisfied. 9 The row vectors of the matrices C and E must be a basis of the n-dimensional vector space:

rank

C

= n

9 The linear part of the system has to be observable:

rank

sE - A) C = n

(H1)


413

9 The nonlinearity f ( x , u) satisfies a Lipschitz condition, which requires that there exists a positive constant, e, such that IIf(3c, u) -

f(x,u)ll

< ell~ - yl[

(H3)

Moreover, if the measurement matrix has full row rank, i.e.: rank(C) = p

(4.31)

is satisfied, the matrix computation is much easier. Under these hypotheses the following procedure can be used to design an observer for FDI. P r o p o s i t i o n 3.1 The parameterized system equation (~,~) =

Nz+Ly+Gu+Rf(3c,

u)-P-1CT(~)-y)

(4.32)

5c =

z + Ky

(4.33)

~) =

C~

(4.34)

where k, z E IR n, ~) E IRp and f(&, u) C IR q and the matrices N, L, G, R, K and P have appropriate dimensions, is a stable observer of the nonlinear system (~.29), (4.30), where the Lipschitz constant E must hold the following inequality: < e0(a, ~)

(4.35)

with Eo(a, ,~) = Ami,~((2 - a ) c T c + ~P)

2~max(Pn)

(4.36)

and the matrices satisfy the .following conditions: N - p-1cTc G-RB NRE+LC-RA RE + KC

stable

(4.37)

=

0

(4.38)

=

0

(4.39)

=

In

(4.40)

and the matrix P, depending on the parameters a and ~, is the solution of the Lyapunov equation /u

_{_P N - o~cTc -}- ~ P = 0

(4.41)

Note that the parameters ((~,~) have to be ckosen so that the matrix P is positive definite and the condition (4.35) is satisfied.[:]

414


P r o o f . The estimation error is defined as:

e =

2 -

x

(4.42)

With the estimated state (4.33) and the condition (4.40), the estimation error becomes

(4.43)

e = z - REx

Taking into account (4.32) and (4.29), the time derivative of the estimation error becomes

=

Ne + (LC-

RA-

NRE)x

+ (G-

RB)u

+ R ( f ( Y : , u ) - f ( x , u ) ) - P - ~ c T ( ~ ) -- y)

(4.44)

Using the matrix conditions (4.38) and (4.39), the error dynamics are governed by the following equation

= (N - P-1cTC)e

+ R ( f ( 2 , u) - f ( x , u))

(4.45)

To discuss the stability, the direct method of Lyapunov is applied. Consider the following Lyapunov function (4.46)

V = eTpe

This function V is positive definite if and only if the time constant m a t r i x P is positive definite, i.e., if the eigenvalues of P are positive. So the second step is the discussion of the negative definiteness of the time derivative. Taking into account the error dynamic (4.45) and the Lyapunov equation (4.41), the time derivative becomes

II = --eT((2 -- a ) C C + ~ P ) e + 2 e T p R ( f ( : ~ , u) -- f ( x , u)) The of P R , if E is by the

(4.47)

second term must be overestimated by the greatest singular value because R can only be a square matrix if the system is regular, i.e., a square matrix with full rank. The first term can be overestimated smallest eigenvalue of the matrix:

T2

=

--

§

)CC +

,

P)llelr:

u) - f(x, u)l I

(4.48)


415

Applying the Lipschitz condition (H3), the time derivative of the Lyapunov function can be overestimated as follows:

9 = (--Am,~((2 -- a ) C C + ~P) § 2e~maxPR)Ne[I 2

(4.49)

Now it can be concluded t h a t the time derivative of the L y a p u n o v function is positive definite, if the condition (4.36) is satisfied. [] An observer for the above nonlinear systems can be designed, if the nonlinearity satisfies locally the Lipschitz condition. It has been proved above t h a t the a s y m p t o t i c stability holds if the Lipschitz condition (H3) is satisfied. There are two p a r a m e t e r s to design the observer. T h e main problem is the stability, but if there are different pairs of the p a r a m e t e r s a and ~ which satisfy the stability condition, the dynamics of the observer can be a second criterion to choose the parameters. A further interesting point will be the conditions of the p a r a m e t e r s a and ~ so that the positive definiteness of the L y a p u n o v m a t r i x P can be guaranteed. Note that the matrix c o m p u t a t i o n of the proposed observer satisfying the conditions (4.37)-(4.40) is presented in [19]. Normally, a system is not singular. But it can be said t h a t a model of a system is not fully known. In the next section details the kind of systems to which the proposed observer can be applied.

4.2

C o n t r i b u t i o n of this O b s e r v e r

A nonlinear observer for nonlinear, singular systems was presented in section 4.1. This observer is a generalization of the observer presented by [19, 24, 36, 37]. T h e proposed observer can be applied for failure diagnosis of nonlinear, singular systems. It can be shown t h a t this observer is also applicable for nonlinear systems with unknown inputs, which can be described by the following equations:

ic =

Akx + Bku + fk(x,u) + Dkd

(4.50)

y

cx

(4.51)

=

where d C lR n-q is the vector of unknown inputs and q has to be smaller t h a n n. D is a known matrix, whose rank is equal to ( n - q ) . If there exists a m a t r i x Ek, so t h a t EkDk = 0 and the hypotheses of the presented observer for the nonlinear singular system are all satisfied, the observer can also be used to reconstruct the state of the nonlinear system under unknown inputs.

416


The analytical equation of the nonlinear system with unknown inputs can be transformed into the equations of the nonlinear singular system by multiplying (4.50) from the left by the matrix Ek

Ekic = E k A k x + E k B k u + E k f k ( x , u) + E k D k d

(4.52)

Taking into account that E k D k = 0, (4.52) can be compared with the differential equation of the singular system (4.29). The following relations can be obtained

E

=

Ek

(4.53)

A

=

EkAk

(4.54)

B

=

EkBk

(4.55)

f(x,u)

=

Ekfk(x,u)

(4.56)

If the condition E k D k =- 0 is satisfied, it can be concluded, that the proposed observer can be well applied for a class of nonlinear regular systems, nonlinear singular systems and nonlinear systems with unknown inputs. Now it will be shown that the state can be well reconstructed applying the proposed observer for a class of nonlinear uncertain systems. Consider the nonlinear uncertain system described by the following equations:

5c =

(A~ + A A u ) x + (B~ + AB~,)u + f ~ ( x , u )

(4.57)

y

Cx

(4.58)

=

Under the constraint that the uncertain matrices AA and A B are parameterized as in [35]:

AA~, = D~,VFI

(4.59)

ABe, -- D~,VF2

(4.60)

Taking into account the parameterization of the uncertain matrices (4.59) and (4.60), the differential equation of the nonlinear uncertain system can be written in the form:

ic

=

A~,x + B~u + f~(x, u) + D ~ , ( V F l x + VF2u)

(4.61)

Comparing this differential equation with the differential equation of the nonlinear system with unknown inputs (4.50), it can be concluded that these two equations are equal if the unknown inputs are defined as follows:


d -- ( V F l x + VF2u)

417

(4.62)

This shows t h a t the proposed observer can be applied for the above class of nonlinear systems, even if the system is uncertain or if it has unknown inputs.

4.3

Residual

Generation

In the last paragraph, an observer for a class of nonlinear singular systems was proposed and different cases were presented of systems to which this observer can be applied. Now the aim is to give an approach for generating the residuals for nonlinear singular systems with faults a n d / o r parameter uncertainties. Consider a nonlinear singular system with faults 8f and uncertainties Od of the form:

Eic

=

A x + B u + f ( x , u ) + 62(x, U)Od + Of

(4.63)

y

=

Cx

(4.64)

where the matrices A, B, C and the nonlinear m a t r i x ~ ( x , u ) are well known. T h e residual of this observer can be defined as follows:

r

--

C(2-x)

(4.65)

where ~ - x is the state estimation error e, and the residual is equal to the estimation error of the output. T h e dynamic of the state estimation error becomes:

--- ~ - R E ~

(4.66)

Taking into account the system dynamics with faults (4.63), the observer dynamics (4.32) and the matrices conditions (4.37)-(4.40), the state estimation error becomes:

= (N - P-1cTC)e

+ R ( f ( ~ , u ) - f ( x , u ) ) - Rq2(x,u)Od -- ROf (4.67)

Determining this proposed residual, faults a n d / o r uncertainties can be detected.

418

5


Conclusions

In this chapter the different approaches to the design of nonlinear observers for residual generation for FDI in nonlinear systems have been briefly reviewed. The survey also incorperates some recent results obtained with a nonlinear observer that has been designed for a class of nonlinear singular systems. This observer is designed with a Lyapunov-like equation with two degrees of freedom. This allows the determination of an upper bound of the Lipschitz constant better than in [19]. The whole state of the system can be reconstructed if the three hypotheses given in the chapter are satisfied and if only part of the process is modeled. This observer design can also be applied to FDI of nonlinear systems with unknown inputs or for a class of nonlinear systems with uncertainties. Note however, that the relationship of of the fault sensitivity with the degrees of freedom is still an open problem. As can be seen, the fault detection problem for nonlinear systems is still neither generally nor completely solved. This chapter presents ideas for residual generation with nonlinear observers under the restriction to certain classes of nonlinear systems. Also, the fault isolation problem is a further interesting issue, in which there are still many open questions, because of the well known difficulties associated with the design of nonlinear observers not only for feedback control but also for fault diagnosis. 6

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25-29, 1996. [3] E. Alcorta Garcia and P. M. Frank. Deterministic Nonlinear Observerbased Approaches to Fault Diagnosis: A Survey. Control Engineering

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Automatica, Vol. 24, pp. 309-326, 1988.

[5]

G. Bastin and M. R. Gevers. Stable Adaptive Observers for Nonlinear Time-varying Systems. IEEE Transactions on Automatic Control, Vol.

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419

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[23] J. Gertler. Disgnosing Parametric Faults: Form Parameter Estimation to Parity Space. American Control Conference, Seatle, Washington, USA, pp. 1615-1620, 1995. [24] M. A. Hammami. Stabilization of a Class of Nonlinear Systems Using an Observer Design. 32 nd Conference on Decicion and Control, pp. 1954-1959, 1993. [25] H. Hammouri, M. Kinnaert and E. H. E1 Yaagoubi. Fault Detection and Isolation for State Affine Systems. European Journal of Control, Vol.4, pp. 2-16, 1998. [26] D. Hengy and P. M. Frank. Component Failure Detection Using Local Second-Order Observers. IFAC Workshop, Kyoto, Japan, 1986. [27] R. Isermann. Process Fault Detection Based on Modeling and Estimation Methods A Survey. Automatica, Vol. 20, pp. 387-404, 1984. tection and Isolation. European Control Conference, pp. 1970-1974, 1993. [28] M. Kinnaert, Y. Peng and H. Hammouri. The Fundamental Problem of Residual Generation for Bilinear Systems up to Output Injection. European Control Conference, Italy, pp. 3777-3782, 1995. [29] V. Krishnaswami and G. Rizzoni. A Survey of Observer-Based Residual generation for FDI. IFAC Safeprocess, Finland, pp. 34-39, 1994. [30] J.-F. Magni. On Continuous Time Parameter Identification by using Observers. IEEE Transactions on Automatic control, Vol. 40, pp. 1789-1792, 199s [31] L. A. Mironovskii. Functional Diagnosis of Dynamic Systems. Automation and Remote Control, pp. 1122-1143, 1980.


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[47] J. Wfinnenberg. Observer-Based Fault Detection in Dynamic Systems. VDI-Fortschrittsbericht, VDI-Verlag, Reihe 8, Nr. 222, Germany. [48] H. Yang and M. Saif. Nonlinear Adaptive Observer Design for Fault Detection. American Control Conference, Seattle, USA, pp. 11361139, 1995. [49] D. Yu and D. N. Shields. Fault Diagnosis in Bilinear Systems - A Survey. European Control Conference, Italy, pp. 360-365, 1995. Systems Research. 12th IFAC Congress, Sydney, Australia, Vol. 3, pp. ~85-~88, 1993. [50] Z. Zhuang and P. M. Frank. Qualitative Observer and its Application to Fault Detection and Isolation Systems. Journal of Systems and Control Engineering, I MECH E, Vol. 211, Part I, pp. 253-262, 1997. [51] Z. Zhuang, G. Schreier and P. M. Frank. A Qualitative-Observer Approach to Generating and Evaluating Residuals. 37~h Conference on Decision and Control, Florida, USA, pp. 102-107, 1998.

A p p l i c a t i o n of N o n l i n e a r O b s e r v e r s to Fault D e t e c t i o n and I s o l a t i o n H. H a m m o u r i 1, M. Kinnaert 2 and E.H. E1 Yaagoubi 3 1LAGEP, University of Lyon 1, Lyon, France 2Department of Control Engineering, Universit~ Libre de Bruxelles, Brussels, Belgium 3LCPI, ENSEM, Casablanca, Morocco

1

Introduction

Fault detection and isolation (FDI) systems differ from classical a l a r m systems by the fact t h a t they give early warning of faults. Alarm systems essentially process measured signals separately by comparing t h e m to thresholds or by computing their trend. F D I systems take into account the correlation existing between those signals thanks to the use of a m a t h e m a t i c a l model of the supervised process. A typical FDI system is made of two parts, a residual generator and a decision module. The residual generator is a filter with the a c t u a t o r comm a n d s and the measured plant outputs as inputs, which generates a set of signals called residuals. The latter have zero mean in the absence of fault (after the filter transient has vanished), and the m e a n of some of t h e m becomes distinguishably different from zero upon occurrence of specific faults. The decision module processes the residuals in order to decide whether some of t h e m have a mean significantly different from zero (fault detection). Then, by analysing the p a t t e r n of non-zero m e a n residuals, it decides what is(are) the most likely faulty component(s) . This operation is called fault isolation. In this text, only the design of residual generators is considered. T h e r e is a vast literature on this topic, and our aim is not to provide a survey but rather to stress the basic principle behind one approach to residual generation, namely one type of observer based methods. For linear systems, observer based residual generation dates back to the work of Beard [2] and Jones [11]. In their approach, the residuals are the o u t p u t error of a Luenberger observer of which the gain is tuned in a very specific way. Indeed, the particular choice of the observer gain ensures t h a t the residuals take a fixed direction or lie in a specific plane upon occurrence of a given

424

4. Application of Nonlinear Observers to Fault Detection and Isolation

fault. This problem can be seen as a simultaneous assignment of eigenvalues and eigenvectors [21]. Eigensystem assignment has also been used to tackle robustness issues in detection filters [18]. Massoumnia [14] has considered the same problem in a geometric framework. Another approach to the synthesis of observer-based residual generators was developed in [15], [22], [4]. The basic idea on which it relies in order to design a residual which is only sensitive to a given fault is the following. One has to determine from the initial model of the plant a detectable s u b s y s t e m of which the state is not affected by unknown inputs or by faults except for the specific fault to be detected. Next, a Luenberger observer can be designed for this particular subsystem and the o u t p u t error of the observer is a suitable residual. The latter approach is reviewed here for linear systems. Next its extension to nonlinear systems is considered. In particular, the application of high gain nonlinear observers for residual generation is investigated. T h e theory is illustrated by a simulation study on a hydraulic process.

2 2.1

Residual Generation for Linear Systems Problem Statement

We consider the class of continuous time-invariant linear systems described by the following state space model :

x (t)

=

Ax(t) + Bu(t) +

y(t)

=

Cx(t)

_Flvl(t) q- F2v2(t)

where x(t) C X C R n , u ( t ) E L t C R m , y ( t )

(2.1) (2.2)

E Y C R p, v~(t) C ~

C

Rn"~, i = 1, 2. X, L/, y , ?i, i = 1, 2 denote linear vector spaces. In (2.1),(2.2), x(t) denotes the state of the system, u(t), the known input signals, y(t), the measured output signals, vl(t) and v2(t) are unknown functions of time which we call failure modes. In the ith failure mode, the following relations h o l d : vi(t) 7~ O, t > to and vj(.) -- 0 , j ~ i. A, B, F1, F2 and C are known matrices, and we assume without loss of generality t h a t F1 and F2 have full column rank. Different types of faults can be modelled in the framework of (2.1),(2.2). If the dynamics of the actuators are negligible with respect to the process time constants, an actuator failure such as a valve sticking can be described as follows. The j a m m i n g of the first actuator can be modelled with F1 = B.,1, vl(t) = ~ - u ~ ( t ) where B.,ldenotes the first column of matrix B, u~(t), the first component of vector u(t), and ~ is the value at which the control signal is stuck.


425

A leak in an hydraulic system can also be modelled by an additive signal, namely the flow of the leaking fluid. Even a change in the dynamics of the plant could be considered as a fault of the type indicated in (2.1), (2.2), by choosing adequately Fi and vi(t). The simplest problem of residual generator design, called the fundamental problem of residual generation (FPRG) can be stated as follows, for system (2.1), (2.2): (FPRG) Determine a linear time-invariant system with inputs u(t) and y(t), and output r(t) E R q such that : 1) In the absence of fault (i.e. when vi(t) = 0, i = 1, 2),r(t) asymptotically decays to zero. 2)In the second failure mode (i.e. when v2(t) ~ O,t >_ to,to being the fault occurrence time), r(t) asymptotically decays to zero. 3) In the first failure mode (i.e. when vl (t) ~ 0, t _> to, to being the fault occurrence time), r(t) does not asymptotically decay to zero. A restatement of this problem using the terminology of linear system theory is instrumental in the determination of a solution, especially in the framework of geometric system theory. This is the object of the next subsection.

2.2

Second Problem F o r m u l a t i o n

The most general form of linear time-invariant (LTI) system with inputs u(t) and y(t) and output r(t) is:

w (t)

=

Arw(t) + B~u(t) + M~y(t)

(2.3)

r(t)

=

C~w(t) + D~u(t) + Nry(t)

(2.4)

where w(t) E ]/Y. Subsequently, nr and q denote the dimension of w(t) and r(t) respectively. Combining (2.1),(2.2) and (2.3), (2.4) yields:

[;(:/)]

=

o

+ EB F2

F1 (2.5)

v~(t) ] (2.6) Introducing the extended state xe(t) = [xT(t), wT(t)] T, which belongs to 2(e = X| and the extended control signal ue(t) = [uT(t), vT(t)] T, which

426


belongs to/~e = L/~) ]22, (2.5), (2.6) can be w r i t t e n :

xe (t)

=

Aexc(t) + Bcu~(t) + F~vl(t)

(2.7)

r(t)

=

Cexe(t) + Deue(t)

(2.8)

T h e definition of the different matrices is obvious from (2.5), (2.6). We now restate the F P R G as a set of conditions to be fulfilled by (2.7), (2.8). Clearly 1) and 2) in the definition of the F P R G are equivalent to : 1') the map u~(t) --~ r(t) is zero 2') the observable modes of the pair (Co, A~) are asymptotically stable. Several criteria can be considered for condition 3) in the F P R G , as discussed in [15I. As in the latter reference, the requirement that the system relating Vl (t) to r(t) be input observable is imposed here. Remember that the map Vl (t) --* r(t) is input observable if the magnitude vl, of a step like fault vx(t) can be determined uniquely from r(t), t >_ 0 when x~(0) = 0. Subsequently, we consider that, input observability of the map vl(t) r(t) is sufficient to guarantee condition (3) of the F P R G in practice. This yields a new statement for the F P R G : (FPRG1) Determine a system of the form (2.3), (2.4) such t h a t : 1') the m a p u ~ ,

~riszero,

2') the observable modes of the pair (C~, A~) are asymptotically stable, 3') the map f ,

2.3

, r is input observable.

Principle o f the S o l u t i o n

As already announced in the introduction, the solution relies on the determination of a detectable system with Vl as only unknown input, from the original state space model of the plant,(2.1),(2.2). To this end, an output injection map L : y ~ X, and an o u t p u t mixing map H : y ~ y are introduced in order to define the following system class :

x (t)

=

(A + LC) x(t) - Ly(t) + Bu(t) + Flvl(t) + Fuv2(t) (2.9)

z(t)

=

HCx(t)

The major part of the design consists in determining the matrices L and H so that the pair (HC, A + LC) is unobservable, and ImF2 is included in the unobservable subspace of the pair (HC, A + LC). Let h denote the dimension of this subspace. Once such matrices are obtained, there exists a


427

linear change of coordinates x = T 2 such t h a t s y s t e m (2.9) can be w r i t t e n in the s t a n d a r d form for a nonobservable s y s t e m [12] :

~ (t)

--

A l l T l ( t ) - Lly(t) + BlU(t) + FllVl(t)

72 (t)

=

~2171(t) + ~2272(t) -Z2v(t) + ~2~(t) + T21vl(t) + T22~2(t)

z(t)

z

~171(t )

(2.10)

(2.11)

(110 !

where 71 E R (n-h), 72 C ~ h , a n d

T - 1F2 ~_

~22

; T - 1L -

--

; H C T = CO1

O)

T h e first (n - h) rows of T-1F2 are null since ImF2 lies in the unobservable subspace of the pair (HC, A + LC). Moreover, the pair ( ~ 1 , ~ 1 1 ) is observable by construction. This last r e m a r k implies t h a t we can build a linear observer for e s t i m a t i n g 71(t) from (2.10),(2.11) when vl(t) ----O,t >_0 : w (t) = ~ l l W ( t ) - -ily(t) +-Blu(t)

_ +K(z(t) - Clw(t)),

(2.12)

We claim t h a t the o u t p u t r e c o n s t r u c t i o n error :

r(t) -- z(t) - C l W ( t ) : ~ l ~ ( t )

(2.13)

is a suitable residual provided some additional condition are fulfilled to ensure 3') in F P R G 1 . Indeed, notice t h a t E(t) is governed by: (t) = ("All - K ~ I ) c(t) + ~ l l V l ( t )

(2.14)

Hence r(t) a s y m p t o t i c a l l y decays to zero w h e n vl ----0. To fulfil F P R G 1 , the m a p vl --* r m u s t be input observable. It can be shown t h a t this condition is verified provided,

ImF1 A S(HC, A + LCIImF2) = 0 where $ ( H C , A+LCIImF2) denotes the unobservable subspace of ( H C , A+ LC) (containing ImF2). Notice t h a t $(HC, A + LCIImF2 ) is a (C, A) unobservability subspace (u.o.s.) containing ImF2. Indeed, a subspace 7r is a (C, A) u.o.s, if it is the unobservable subspace of a pair (GC, A + MC) for some p • p a n d n • p matrices G a n d M[14]. It can be shown t h a t the set of u.o.s, containing IrnF2 has an infimal element, S*. It t u r n s o u t t h a t necessary a n d sufficient conditions for F P R G 1 to have a solution can be expressed in t e r m s of this subspace, n a m e l y :

428


Theorem

4.1 [15] FPRG1 has a solution if and only if

S* (7 ImF1 = 0

(2.15)

where 8" := in f S(C, A; ImF2) is the smallest (C, A)-unobservability subspace containing ImF2. Moreover, if (2.15) holds the dynamics of the residual generator, i.e. the eigenvalues o.f All - K C 1 in (2.12), can be assigned arbitrarily.

3 3.1

Residual Generation for Nonlinear Systems Introduction

T h e basic idea behind the design of residual generators for linear systems can be extended to nonlinear systems, provided the a p p r o p r i a t e nonlinear notions are used. One of the problems that arises in the extension is the design of an asymptotic observer for the nonlinear system from which the residual is deduced as the output reconstruction error. One has to restrict the considered systems to a specific class to ensure the existence of an asymptotic observer. Here observers for uniformly observable nonlinear systems will be used. Observers with linear error dynamics have been used by Seliger and Frank [19]. Other classes of nonlinear systems have been considered elsewhere, such as bilinear systems [23], [13], and state affine systems [7]. The remaining part of this section is organised as follows. A review of some basic notions from observability theory for nonlinear systems is presented. Next, nonlinear observers for uniformly observable systems are described. Finally those preriquisites are applied to design nonlinear residual generators, and the theory is illustrated by a simulation on a hydraulic process.

3.2

Basic Notions

For the sake of simplicity, we only consider control a n n e nonlinear systems :

=

f(x) +

gi(x)u~

(3.16)

i=1

y

=

h(x)

=

(hl(x),...,hp(x))

where x(t) E ll~n,u(t) -- ( u l ( t ) , . . . ,urn(t)) r e /7 a mesurable subset of 1Rm, y(t) E ~ P are respectively the state, the input and the o u t p u t of the dynamical system (3.16). System (3.16) is said to be observable if and only if, for every pair of initial states, (x, 2), x r ~, there exist an admissible control u : [0, T]


429

/~ and a t i m e instant t 9 [0, T] such t h a t y ( x , u , t ) ~ y ( 2 , u , t ) , w h e r e y(x, u, t) = h(xu(t)), and xu(t) is the unique t r a j e c t o r y of (3.16) such t h a t xu(0) = x. If such an input u exists, we say t h a t u distinguishes (x, 2). A n input u : [0, T] ~ /~/, which distinguishes every ( x, 2), x ~ 2 is said to be universal on [0, T]. S y s t e m (3.16) is said to be uniformly o b s e r v a b l e if, for every T > 0, every admissible control u : [0, T] --* U is a universal i n p u t on [0, T]. T h e observation space O(h) of s y s t e m (3.16) is defined as t h e smallest vector space containing hi,. 9 , hp and closed under the Lie derivative L x , where X stands for the vector fields f, g l , . . . ,gin. This space allows to define a geometric notion of observability, n a m e l y the r a n k observability condition. S y s t e m (3.16) is observable in the sense of r a n k at a fixed x 9 ~ n if d i m dO(h)(x) = n where dO(h)(x) = {dT(x);T 9 O ( h ) } ( d is the classical differential operator). This notion extends to nonlinear s y s t e m s the K a l m a n r a n k observability condition for linear systems. For m o r e details on this topic see [9].

3.3

High Gain Observers for Uniformly Observable Systems

In [3] (for a short proof see [5]), single-output nonlinear s y s t e m s which are uniformly observable are characterized. To describe this result, let L I (h)(x) denote the Lie derivative of a scalar function h w.r.t, the vector field f , as already m e n t i o n e d above, and let L } ( h ) ( x ) = LI(Li] - l ( h ) ( x ) ) . If 9 : x --* (h(x), L f ( h ) ( x ) , . . . , L ~ - l ( h ) ( x ) ) T = z is a local diffeomorphism, and if s y s t e m (3.16), with y E /R, is uniformly observable, t h e n t r a n s f o r m s locally s y s t e m (3.16) into the following canonical form :

I

A z ~- ~(z) ~- E i : l ~ti~l[i(Z)

=

Cz

(3.17)

(010 0) y

where

m

~-

A =

"..'"

". ".

o

0

, C=(1,0,...

,0), ~ ' ( z ) =

0

1

...

o

"yn(Z)

(3.18) and ~ i ( z ) = [~/il(Zl), ~I/i2(zl, z2), 9 9- , ~in(Z)] T (i.e. q2ij(z) = q 2 i j ( z l , . " , zj)). Under the hypothesis t h a t ~, a n d the ~ i ' s are global Lipschitz, an observer for (3.17) can take the form :

m =

+

+

-

i=1

-

y)

(3.19)

430


where Se is the unique solution of the algebraic Lyapunov algebraic equation : OSe -4- A T se + S e A = c T c

(3.20)

More generally, consider the triangular form : { ~ y

--=

(3.21)

Az+~(t,u,z) Cz

where A and C are defined by (3.18), and the i th component q2~(t,u, z) of ko(t, u, z) is such that ff2i(t, u, z) = ~ i ( t , u, Z l , . . . , zi). Moreover, assume that ~ fulfils hypothesis H1) below: H1) ko is global Lipschitz w.r.t, z, locally w.r.t, u and globally w.r.t, t, i.e. Va > 0; 37 > 0; Vz, z' E /Rn; Vt > 0; Vu E /R m, Null < a, the following inequality holds: I[uo(t, u, z) - ~(t, u, z')II < 711z - z'll Then, an observer for (3.21) can take the form : = A ~ + @(t, u, 2,) - S [ ~ c T ( c 2 ,

- y)

(3.22)

where Se is given by (3.20). More precisely, the following result, which is a slight extension of the work reported in [5], holds : T h e o r e m 4.2 Va > 0; 300 > 0; V0 _> 00; 3Ae > 0; 3#0 > Os.t. Jt2,(t) - z(t)ll _
0; there exist T E]0, 7], e > 0 such that : r : B ~ ( v l , e) --~ L~([0, T ] ) : +1 ~ r(x(O), z(0), ~t, Vl, v2, 0) is Frechet differentiable at Vl. We let (DFr)(Vl) denote the Frechet derivative of r at Vl. In a similar way, y(-) = y(x(0), u, vl, v2, .) is Frechet differentiable w.r.t. vl, and we let ( D f y ) ( v l ) denote its Frechet derivative at Vl. D e f i n i t i o n 4.1 (3.26) is a residual generator.for the detection and isolation of .failure vl in system (3.25) if there exists Lt C Llo~176the space of locally bounded measurable .functions such that : 1) Vu 9 U;Vx(0) 9 ~ n ; V z ( 0 ) e ~ ;Vv2 9 Lzo C : r

(DO

r(x(O),z(O),u,O, v2)(t) ---* 0 2) 3u 9 U; ~x(0) 9 t~n; ~z(0) 9 that ( D F r ) ( v l ) # 0

as

t ---* +oo

3T > 0;3vl,ve 9 L~([0,TI) such

The determination of a residual signal which fulfils 1) and 2) above is called tile fundamental problem of residual generation ( F P R G ) for system (3.25). Condition 1 in definition 4.1 is equivalent to 1) and 2) in F P R G 1 , for the linear case. R e m a r k 4.1 In the linear case, if we denote (C, A, B) a realization of the map vl ~ r, the above Frechet derivative is given by : (DFr)(vl) : ~ ~ C

/o

e('-~)AB~(s)ds

It is independent o f t , x(O), z(O), v~ and v2. Clearly, in this case, condition 2 of definition ~. 1 is equivalent to the existence of a left inverse for the transfer .function between vl and r. The latter requirement also appears in the statement o.f the F P R G for linear systems when only scalar failure modes are considered, as is the case here (see Section 2.2). Before giving sufficient conditions for the existence of a residual generator for (3.25), we introduce a few known notions [17]. O(h) denotes the observation space of system (3.25), with vl = v2 = O. dO(h) defines a codistribution on ~ denoted by the same letters.


433

K e r d O ( h ) is the distribution spanned by all vector fields X such t h a t Lx(9-) = 0 for every ~- 9 O(h). It is invariant under f, g l , . . . ,gin (i.e. for each Y 9 K e r d O ( h ) and each X 9 {f, g i , . . . ,gm}, the Lie bracket [X,Y] belongs to Kerd(D(h)). Moreover, it is obviously involutive (i.e. VX, Y 9 K e r d O ( h ) , [X, Y] 9 K e r d O ( h ) ) . Finally, notice t h a t K e r d O ( h ) can be seen as the largest distribution invariant under f, g l , . . . , g m and contained in Kerdh. Now assume t h a t dim dO(h)(x) = dim dO(h)(xo) = k < n for every x in some neighbourhood of x0. T h e n it is well known that ENL, restricted to some neighbourhood Vx0 of x0, can be transformed by a change of coordinates into the following form:

~'1

=

~'2

=

m ]1(~1) ~_ Egli(~ )Ui ~_ ~li(~)Vi Jr- ~i2(~)V2 i=i m f2(~) ~- E g 2 i ( ~ ) U i ~- e21(~)V1 "~- e22(~)v2 i=i -

i

y = ~(~1) where ~1 E ~ k , ~ 2 E ~ n - k , ~

U =

= (~a T

(3.27)

(3.28) (3.29)

~2T)T, ~1 = ( ~ a , . . . , ~ k ) T and

{n)T,

( d { 1 , . . . , d{k) spans dO(h) and (O/O~k+l,.. , O/O~n) spans K e r d O ( h ) . In the sequel, the same type of diffeomorphism will be used on a different system. For the sake of simplicity, we shall only consider the case where the diffeomorphism x --~ ~(x) is a global one (When this situation does not hold, only initial conditions and inputs for which the associated trajectories lie into Vxo should be considered). Under this assumption we s t a t e the following proposition. P r o p o s i t i o n 4.1 The F P R G .for ENL has a solution i.f there exists an output map qJ = ~ o h, with : IR p -~ ~'~p' (p' < p) such that :

1) e2 9 KerdO(q2)

(3.30)

2) There exists a set U C LloC~c such that : 9 2.i)The system (3.27) with output map ~({1) = qy(x) admits an asymptotic observer for vl = v2 = 0 and u 9 U, of the classical .form : 2 ~1

:

^I U f l ( ~ l ) ~_ Egli(~ ) i ~-P(~I, G)((~(~I) -- ~(~1)) i~l

=

K(u, C, 1) (3.a)

434

4. Application of Nonlinear Observers to Fault Detection and Isolation w h e r e ~1 ~ j ~ k , ~ = dim d O ( ~ ) , p and K are smooth .functions

9 2-ii) 3u E U; ~x(0) ~ 1R'~; ST > 0; Sv~, v2 ~ L~176 T]) s.t. DFC~(~I)(vl) ~ O. 4.2 : In the case where U is such that U][0,T] = L~([0, T],L/) for some T > O, el ~ kerdO(ff2) implies condition 2-ii) (see proposition 4.1~

Remark

[17]).

P r o o f . Under hypotheses 1) and 2), there exists a residual generator of the form (3.31), with output r = ~(~1) _ ~(~1) t h a t solves the F P R G . Indeed, we show t h a t conditions 1), 2) of definition 1 are satisfied for the output r. First notice t h a t (3.30) implies ~12(~) = 0 in (3.27). Hence, by assumption 2, r(t) --~ 0 as t --* cx~ for every v2 E Llo ~ and vl = 0, and thus conditiion 1 of definition 4.1 holds. It remains to show condition 2) of the same definition. Assume t h a t Vu C L/,Vvl E Llo~ C Llo~ C ~ n , v ~ l E J~k,DFr(~(O),~l(O),u,., V2)(Vl) = 0. This means t h a t r does not depend on Vl. Hence the controlled dynamical system (3.31) does not depend of vl, and neither does t~(~l). Thus t~(~l) does not depend of vl or, equivalently, D F ~ ( ~ I ) ( v l ) = 0. This is in contradiction with 2. ii). 9

3.5

A p p l i c a t i o n o f N o n l i n e a r O b s e r v e r s to the F P R G

We now show how the observer (3.19) can be applied for fault detection and isolation. Consider again system (3.25), and suppose t h a t there exists a C ~ function ~ : ~ P --*/R satisfying the following assumptions: A1) There exists an integer k > 1 such t h a t the Jacobian of [~ o h , . . . , L ~ - l ( ~ o h)] T is of rank k at each x C V, where Y denotes some open set o f / R n. A2) dL~(~ o h) A d L ~ - l ( p o h) A ' .. A d(p o h) = 0

dLa~LJf(p o h) A dLJf(~ o h) A . - . A d(~ o h) = 0 for i = 1 , - - . , m , j =0,..-,k-l, where A means the exterior product of differential forms, and L~ (~ o h) = ~ o h . 4.2 Under the assumptions A1) and A2), consider system (3.25), with output ~ = (~ o h)(x). There exists an infinite choice of local systems of coordinates (~1,"" , ~n) in which system (3.25) takes the form (3.27),(3.28), (3.29) where

Proposition

(3.32)


~/1i(~1) = [ g1,(r -1 1

with

-j

gli(~

1

.'

~1~,(~ 1) ]:"

435

(3.33)

-j

)---- ~1,(~1, "" 9 ' ~)

and h(~1)~-~1

(3.34)

P r o o f . Using assumption A1), we can construct a diffeomorphism (I) = [ ~ 1 , " " ,O,~]T from an open subset W C V such that r = L~(T o h) for j = 0 , . - - , k 1. Now set ( ~ l , " ' , ~ n ) = (Ol(X),--. ,On(X)), ~1 = (~1," ~k) T and ~2 = (~k+l," 9 9 , ~)r. Taking the derivative of ~j along trajectories of system (3.25), for j = 1,... , k - 1, yields :

~n ~j(t) = ~j+l(t) + Eui(t)Lg~(Oj)(x(t)) + viLe, (Oj)(x(t)) + v2L~.~(Oj)(x(t)) i=1 (3.351 Assumption A2) implies that dLg,(~j) A d~j A . - - A d~l = 0, which means that Lg~(~j) = g{i(~l,"" ,~j) for i = 1,.-. ,m. Hence, (3.35) becomes: m

~ j ( t ) : ~j+l(t) + E u i ( t ) O ~ i ( ~ l , ' '

" ~j) + Vle~l(~) + v2~J2(~)

(3.36)

i=1 for j = 1,-.. , k - 1 . For the k th component, ~k, we obtain :

~k ---- Lf(42k)(x(t)) m

+

Eu~(t)La,(q)k)(x(t)) + viLe, ((I)k)(x(t)) + v2Le2 ((I)k)(x(t)) i----1

Using again assumption A2), we get : { dLf(~k) Ad~k A - . . A d~l

dLa, (~k) A d~k A... A d~l

-=

0 0

Hence,

{ Lf(~k) L.,(r

~--- ~(~1,''" ,~k) = ?~k(~ 1)

= ~d~l,"" ,r

=~(r

(3.37)

436


O t h e r choices of coordinates which bring s y s t e m (3.25) into the f o r m (3.27), (3.28), (3.29) with the particular s t r u c t u r e (3.32),(3.33), (3.34) can be obtained as follows : ~[ = ~1, ~ = ~i -~- #i(~1,-. 9 , ~i--1) for i = 2 , . . . , k, where #i : f~ i-1 ---+ /R are a r b i t r a r y C ~ functions. 9 Now consider the reduced controlled s y s t e m :

m y1(~1) _[_ E g l i ( ~ i=1 _

~'1

:

1

(3.38)

)Ui

= 5(1)

where f l , ~01i, a n d / t are given respectively by (3.32), (3.33), a n d (3.34). Let W1 denote the open set :

{~ ~ ~k;3~2 c ~n-ks.t.

~2

~(W)}

W C V, where W and V are given above. W i t h o u t loss of generality, we a s s u m e t h a t we are only concerned with a set of initial states a n d a class U of b o u n d e d admissible controls u such t h a t rlulloo _< M (M is a given constant) for which the trajectories of (3.38) lie into a d o m a i n W~ C W1, and such t h a t ~Tj, g l i , j =- 1,. 9 9 , k, i --- 1, 9 9 9 , rn can be e x t e n d e d to global Lipschitz functions ~]j,~]li,j : 1 , ' ' ' , k , i = 1 , . . . , m on Z@ (i.e. ~ j l w ; = ~ T j , g l i l w ( = g l i a n d ~,t)li are global Lipschitz functions on F~k). Under this a s s u m p t i o n , an observer of the f o r m (3.22) can be c o n s t r u c t e d in order to e s t i m a t e exponentially the c o n c e r n e d u n k n o w n trajectories of (3.38). More precisely, this observer can be w r i t t e n as follows (see Section 2):

;,1

rn ~ ^1 "t/, = / 1 ( ~ 1) -1- ~--~oOli(~ ) i -- S o l C T C ( ~ i=1

where f1(~1) = Adl +

~k--1(~1,...

,~k--1)

1)

)

1 - E 1)

(3.39)

, with A, C as in (3.18), a n d

S o given by (3.20).

In order to design a residual generator, the following corollary to p r o p o sition 4.1 can be used. C o r o l l a r y 4.1 C o n s i d e r s y s t e m ( 3 . 2 5 ) , a n d s u p p o s e t h e r e e x i s t s a Coo . f u n c t i o n ~ : t ~ p --* ~ s a t i s f y i n g a s s u m p t i o n s A 1 ) a n d A 2 ) . L e t U be as above, a n d s u c h t h a t U ][0,T]= Loo([0, T ] , b / ) . f o r s o m e T > O. M o r e o v e r , assume that :


a) L~2L}(q2 ) : 0 for i = 0 , . . . , k b) 3i E { 0 , . . . , k - l }

437

1.

s.t. L ~ I L } ( 9 ) # 0 .

where ~ : ~a o h. Then system (3.39) with output r(t) = ~l(t) - 9(t) (~(t) : 9 ( x ( t ) ) ) is a residual generator which detects and isolates vl. P r o o f . It suffices to check conditions 1, 2-i) and 2-ii) of proposition 4.1. 2-i) is satisfied since (3.39) is an observer which converges for every u C U. Assumption a) is nothing but condition 1). Finally, assumption b) implies el ~ kerdO(q2), and by remark 4.2, U satisfies condition 2-ii). 9

4 Hydraulic System The considered system consists of a spool valve and a single rod piston acting on an inertial load (see figure 1). The external force Fe controls the flow entering the head side chamber of the piston from a pressure supply Pa. The rod side chamber is always connected to the r e t u r n pressure Pr.

Pr

FIGURE 1. Hydraulic system. Our aim is to detect and isolate two faults in this system : a drop of the spool control force F~, and an increase of the internal leakage of the piston (which is normally assumed to be negligible).

4.1

M o d e l l i n g o f the S y s t e m

The following notations will be used:x 1,displacement of the spool, x2,velocity of the spool, xa, displacement of the piston, xa, velocity of the piston, x5, pressure in the head side chamber, Vl, failure mode corresponding to the

438


control force, v2, failure mode corresponding to the internal leakage of the piston, Ap, area of the piston, D, diameter of the spool, B, bulk modulus, Cd, discharge coefficient, p, density of the fluid, K s and Rs, respectively spring and damping coefficients associated to the spool, K p and Rp, respectively spring and damping coefficients associated to the load, M s a n d MR, respectively mass of the spool and mass of the piston together with the load. The model of the process can now be derived : 21 = x2

(4.40)

22 = - (Ksxl + Rsx2) ~Ms + (F~ - FF -- vl) ~Ms

(4.41)

23 = x4 2 4 ~-~ ( - K p x

3 - Rpx4 +

(4.42)

Apx5)/Mp

B 12 :i:5 - Apx3CdIIDxl (Pa - x5) - BX4x3

(4.43)

x5x4Bv2 x3 Ap x3

(4.44)

where FF = 2CdIIDxl ( P a - x5) represents the resultant flow force acting on the spool p. We assume that the available measurements are y =

Ix1 x3 IT.

[y,

All the state variables x i , i = 1 , . . . , 5 take values in closed intervals [ai, bi], i = 1 , . . . , 5. The position measurements are calibrated so t h a t the lower bound of the interval is positive, and thus the division by xa in (4.44) does not cause any problem.

3.2

Design o f a Residual G e n e r a t o r

First a residual generator to detect and isolate v2 is obtained by noting t h a t proposition 4.2 can be used with (~ o h) (x) = x3 = Y2, if we consider xl = Yl as a known input in (4.44). Indeed, it is easily checked t h a t conditions A1) and A2) are fulfilled with k = 3. Besides, the change of coordinates :

[ ~1 ~2 ~3 ~4 ~5 ]T

[ X3 X4 n p x 5 / M p

Xl

X2 ] T

transforms equations (4.42), (4.43) and (4.44) i n t o :

~2

~3 -- Kp~I/Mp - RP~2/MP

q-~3) ff-'~ppt~dIID ~Yl~'IV~ . .

~1 t" Mp

-

M~

-

.

MP~I

(4.45)


439

which is in the form (3.27), with ?1 (~1) and gll (~1) respectively given by (3.32) and (3.33). Besides (~1) = ~1 = (~ o h) (x) = Y2

(4.46)

T h e functions ~72(~) = - K p ~ I / M p - R p ~ 2 / M p , ~ 3 ( ~ ) = ~-IA~B~ Mp +~3), and / /2 B dHD ~1 V - ~ _~Pa -- M__~ Ap ~3/ have bounded derivatives for any ~13(~) -~pt~ in

W~ = {~8.t.~l 9 [a3,b3],~2 9 [a4,b4],~3 e [Mpah, - ~Pb 5 -- e]}. T h e arbitrarily small number e appearing in the upper b o u n d for ~3 is introduced to take care of the fact t h a t ~3 = ~MP55 zeros the argument of the square root in g13(~) (indeed b5 = P~). This phenomenon cannot occur on the actual system, as there will always be a small leak around the piston which will prevent ~3 from reaching this extreme value. It is now straighforward to extend ~2(~), ~/3(~) and g13(~) to global Lipschitz functions on ~ 3 . Conditions a) and b) of corollary 4.1 ( in which Vl and v2 are interchanged as our aim is to detect v2 here ) are fulfilled for any input signal t h a t keeps the state trajectory of (4.45) into W~. Hence a residual generator of the form (3.39) with output rl = ~'1 - Y2 can be designed on the basis of model (4.45). This was verified by simulating the process described by equations (4.40) to (4.44) together with the residual generator. The numerical values were taken as described in Appendix A. vl (t) was chosen as a step-like failure signal between 10 s and 20 s, while v2(t) was non zero between 30 s and 40 s. Figure 2 shows that the resulting residual is indeed unaffected by vl and t h a t it allows to detect and isolate v2. We now illustrate the performance of an observer of the form (3.24) by considering the system m a d e of equations (4.40),(4.41),(4.43), (4.44), with output Yl = Xl. It turns out t h a t the output reconstruction error for this observer, r2, is a signal which is sensitive to vl and v2. Hence the combined monitoring of rl and r2 allows one to detect and isolate b o t h failure modes provided they do not occur simultaneously. The equations mentioned in the previous p a r a g r a p h can be put in the form of model (3.23) thanks to the following obvious change of coordinates: [ Z 1 Z2 Z3 Z4 Z5 I T = [ X l

X2 X5 x4(1 + Xh/B)

X3 ]T

which is indeed a diffeomorphism for any x in W = {x 9 /R 5 : x~ 9 [ai,b~],i = 1 , . . . ,5}. W h e n vl = v2 = 0, A(t) and ~ ( t , F~, z) can be written:

A=

0

1

0

0

0 o 0

0 o 0

pMs Y ~ J o 0

- y - -B~

0

0

440

4. Application of Nonlinear Observers to Fault Detection and Isolation o

,~

o

o

mq

N

o

I

/

E/J

of

';

a

7

;

g3~ 9~-~

N1

d-

.2H

o

c3 I

I0

20

T

30

40

5O

F I G U R E 2. Residual rl and failure modes vl and v2.

a n d ~ ( t , F~, z) =

0 -- 2CdIIDpMsYlP,a -

(-Kpy2+Apz3)(l+

(/(SYl

+ Rsz2)/Ms

@ Fe/Ms

z_~A1..2_ _ R r z ~ + z 4 f _ B_B~_ , BJMp Mp ~+z.3/B~ y ~ ~ + f ~ ( y l , y 2 , z3))

with f ~ (yl, Y2, z3) = A---~.CdHDyl J ~ (Pa - z3)) Py2 V ~ Notice t h a t Yl (t) and y2(t) belong to closed intervals with a positive lower bound, and the c o m p o n e n t s of 9 have b o u n d e d derivatives with respect to z on the d o m a i n W ' = { z E 1~ 5 : zl E [ a l , b l ] , z 2 C [a2,b2],z3 C [as, b5 - e],z4 9 [a4(1 + a s / B ) , b 4 ( 1 + b 5 / B ) ] , z 5 9 [a3, b3]}, where e is an arbitrarily small real introduced for the s a m e reason as above. Hence h y p o t h e s e s H1) a n d H2) can be fulfilled by e x t e n d i n g ~ ( t , Fe, z) to a global Lipschitz function w.r.t, z and t. An observer of the form (3.24) can thus be designed, a n d the second residual is defined by r2(t) -- 21 - y l . Figure 3 shows the evolution of r2(t) for faults Vl (t) a n d v2 (t) with relatively realistic values. T h e t i m i n g of the faults, and the numerical values used for the s i m u l a t i o n are the s a m e as for figure 2. T h e influence of v2 on r2 does not a p p e a r on this plot, because it is low in c o m p a r i s o n with the effect of vl.

4. Application of Nonlinear Observers to Fault Detection and Isolation o u~Q

441

o

,J

N u-~ u3

f

aJ

~ct

,L__

o tO

30

20

40

S0

T

FIGURE 3. Residual r2 and failure modes vl and v2.

5

Conclusions

T h e fundamental problem of residual generation for linear systems has been reviewed, and the principle behind its solution has been described. Next, a formulation of the the same problem for nonlinear systems which are affine in the control signals and the failure modes was derived. Sufficient conditions for the existence of a solution have been presented. T h e possibility to use high gain nonlinear observers in the design of a residual generator has been investigated. The theoretical results have been illustrated by simulations on a hydraulic process. T h e considered observers are based on a single measurement. Extension of the results to the multi-output situation is a topic for further research.

6

REFERENCES

[1] P. Alexandre and M. Kinnaert. Numerically reliable algorithm for the synthesis of linear fault detection and isolation filters, based on the geometric approach. Proc. of the 1993 I E E E Conference on Systems, Man and Cybernetics, vol.5, pp 359-364, 1993. [2] R. V. Beard. Failure accommodation in linear systems t h r o u g h selfreorganization. Ph.D. Dissertation. Dep. Aeronautics and Astronautics. Mass. Inst. Technol., Cambridge, MA, 1971. [3] J. P. Gauthier and G. Bornard. Observability for any u(t) of a class of bilinear systems. I E E E Trans. Automatic Control. 26, pp 922-926,

442

4. Application of Nonlinear Observers to Fault Detection and Isolation 1981.

[4] P. M. Frank. Fault diagnosis in dynamic systems using analytical and knowledge based redundancy. A survey and some new results. Automatica, 26(3), pp 459-474, 1990. [5] J. P. Gauthier, H. Hammouri and S. Othman. A simple observer for nonlinear systems, application to bioreactors. IEEE Trans. Automatic Control, 37 (6), pp 875-880, 1992. [6] J. P. Gauthier and I. Kupka. Observability and observers for nonlinear systems. SIAM J. Control and Optimization, 32(4), pp 975-994, 1994. [7] H. Hammouri, M. Kinnaert and E.H. E1 Yaagoubi. Fault detection and isolation for state affine systems. European Journal of Control, 4, pp 2-16, 1998. [8] H. Hammouri, M. Kinnaert and E.H. E1 Yaagoubi. Observer based approach to fault detection and isolation for nonlinear systems. IEEE Trans. Automatic Control, 1998. [9] R. Hermann and A.J. Krener. Nonlinear controllability and observability. IEEE Trans. on Automatic Control, 22(5), pp 728-740, 1977. [10] M. Hou and P.C. Muller. Fault detection and isolation observers. Int. J. Control, vol.60 (5), pp 827-846, 1994. [11] H. L. Jones. Failure detection in linear systems. Ph.D. Dissertation. Dep. Aeronautics and Astronautics, Mass. Inst. Technol., Cambridge, MA, 1973. [12] T. Kailath. Linear Systems. Prentice-Hall, Englewood Cliffs, N.J, 1980. [13] M. Kinnaert. Innovation generation for bilinear systems : application to robust fault detection. Proc. of the 1998 American Control Conference, Philadelphia, pp 1595-1599, 1998. [14] M. Massoumnia. Geometric approach to the synthesis of failure detection filters. IEEE Trans. Automatic Control, AC-31, pp 839-846, 1986. [15] M. Massoumnia, G.C. Verghese and A.S. Willsky. Failure detection and identification. IEEE Trans. Automatic Control, AC-34, pp 316-321, 1989. [16] R. Nikoukhah. Innovation generation in the presence of unknown inputs:application to robust failure detection. Automatica, 30(12), pp 1851-1867, 1994.


443

[17] H. Nijmeijer and A.J. van der Schaft. Nonlinear Dynamical Control Systems. Springer-Verlag, 1990. [18] R. J. Patton and J. Chen. Robust fault detection using eigenstructure assignment: a tutorial consideration and some new results. Proc. of the 30-th IEEE Conference on Decision and Control, pp 2242-2246, 1991. [19] R. Seliger and P.M. Frank. Robust component fault detection and isolation in nonlinear dynamic system using nonlinear unknown input observers. Proc. of the IFAC/IMACS Symposium SAFEPROCESS, pp 313-318, 1991. [20] P. M. Van Dooren. The generalized eigenstructure problem in linear system theory. IEEE Trans. Automat. Control, vol. AC-26, pp 111-129, 1981. [21] E. White and J.L. Speyer. Detection filter design: spectral theory and algorithms. IEEE Trans. Automatic Control, AC-32, 7, pp 593-603. [22] J. Wunnenberg. Observer based fault detection in dynamic systems. VDI-Fortschrittsberichte. Reihe 8, Nr. 222, 1990. [23] D. Yu and D.N. Shields. A bilinear fault detection observer. Automatica, 32(11), pp 1597-1602, 1996.

Appendix A: Numerical Values used for the Simulation of the Hydraulic System Ms = 0.1 kg, R s = 2.10 Ns/m, Ks -= 103 N/m, D p = 0.2 m (Diameter of the piston), D -- 0.01 m, p -- 840 k g / m 3, B -- 109 N / m 2, Cd = 0.7 k g / m 3, Pa = 220 105N/m 2, M p --- 5 103kg, Rp = 104 Ns/m, K p = 5 105

N/m

Innovation Generation for Bilinear S y s t e m s with U n k n o w n Inputs M. Kinnaert and L. E1 Bahir D e p a r t m e n t of Control Engineering Universit~ Libre de Bruxelles Brussels, Belgium

1

Introduction

Fault detection systems are typically made of two parts : a residual generator and a decision system. The first module generates sequences, called residuals, from the sampled input and output signals of the supervised process. These sequences have nominally zero mean in the absence of fault (after a possible transient has vanished), and their m e a n becomes distinguishably different from zero upon occurrence of a fault. T h e decision system evaluates the residuals in order to determine whether their m e a n differs significantly from zero. This task can be efficiently performed by a p p r o p r i a t e statistical tests [2]. However, such tests are often based on s t a n d a r d hypotheses, such as whiteness of the evaluated sequence. To be able to know the properties of the distribution of the residual, the problem of residual generation has to be stated in a stochastic setting. For linear models with additive faults, the innovation sequence (associated with a K a l m a n filter) has been used as a residual. Indeed, it is a sufficient statistics, which means t h a t the information a b o u t the faults contained in the measurements is retained in the innovation. Moreover, it is zero-mean in the absence of fault, and it is white with known covariance. Hence, it can be evaluated, for instance, by cumulative sum (CUSUM) or generalized likelihood ratio (GLR) tests, depending on whether the fault magnitude is known or not [2]. Yet such a signal can only be obtained from a standard linear model without unknown inputs. T h e latter often arise due to the presence of unmeasured signals or unknown p a r a m e t e r s which cannot be modelled as r a n d o m processes with known statistics. To handle this problem, Nikhoukhah [16] has developed, for linear time invariant (LTI) systems, a m e t h o d to design a filter which, from the observed quantities (measured plant inputs and outputs), generates a signal with the following properties. It is zero m e a n and white in the absence of fault; it is decoupled

446

5. Innovation Generation for Bilinear Systems with Unknown Inputs

from the unknown inputs, and it preserves, as much as possible, information on the faults. The resulting residual is called innovation because, in the absence of unknown inputs, it coincides with the innovation of a Kalman filter designed for the plant model. Our aim in this chapter is to extend the known results for LTI systems to the class of bilinear systems. There are several reasons for the choice of this particular class of systems. First, when modelling systems using physical laws, one often obtains bilinear models. This is for instance the case for processes involving heat transfer for which energy balance equations yield products of temperature (state) and flow (input). Secondly, bilinear models can give a significantly better approximation of the behaviour of a nonlinear system over its working range, than what can be achieved using a linearized model around the nominal set point. This will be illustrated by an example below. Finally, observer theory, which is a fundamental ingredient for the design of residual generators, is well developed for bilinear systems [11], [5] [4]. The latter can actually be seen as a particular class of linear timevarying systems. Observer-based residual generators for bilinear systems have been developed and studied in a deterministic setting in [20], [21] and [15]. In those papers the state equation describing the residual generator is linear and time-invariant up to an output injection. This choice of structure restricts the set of systems for which the problem admits a solution [13]. A more general structure is considered in [19]. However the authors do not deal with robustness issues in that paper, since there are no unknown inputs beside the faults in the problem they consider. Moreover, they obtain an estimate of the fault signals, which is a nice feature, but requires stronger hypotheses on the class of monitored systems than what is needed in our work. Our formulation of the problem of innovation generation for bilinear systems is directly inspired by Nikoukhah's work [16]. However, the proposed solution does not resort to transfer function manipulation, like in [16], since this approach cannot be extended to time-varying systems. First a uniformly completely observable subsystem without unknown inputs is extracted from the original system. Next a Kalman filter is designed for the bilinear system resulting from the first step. The work reported here is built on previous results obtained in [9] and [14]. The first paper describes a geometric approach to the synthesis of residual generators for deterministic bilinear systems. No constructive design procedure is obtained in that work. In the second paper, an algebraic approach is considered to solve the same problem. The algorithm derived in that paper is, however, limited to a smaller class of bilinear systems t h a n the results presented here, if we translate them for deterministic systems. The chapter is organized as follows. The problem of innovation generator design for bilinear systems is stated in Section 2. A solution is presented in section 3. Next the issue of innovation monitoring by G L R test is addressed


447

in section 4. Finally the theory is applied to design and validate a fault detection and isolation system for a three-tank process.

2

P r o b l e m Statement

T h e following class of discrete-time bilinear systems is considered :

x(k + 1) = Aox(k) + E i % l u i ( k ) A i x ( k ) + ~p"_l(Edx(k) + Fd)d~(k)

-~Bu(]~) Jr- Ei=lnf ( E [ x ( k ) + F / ) fi(k) + G w ( k ) y(k) = C x ( k ) + Du(k) + g w ( k ) (2.1) where x(k) e R '~, u(k) e R m, y(k) e R p, d(k) 9 R '~, f ( k ) 9 R "~s, w(k) 9 R n'" are respectively the state, the known inputs, the measured outputs, the unknown inputs, the failure modes, and a zero m e a n Gaussian white noise sequence with covariance matrix E ( w ( k )w(T) T) ----In,,, 6k~ (where In,o denotes the n~-by-nw identity matrix and 5k~- = 1 if k = v and it is null otherwise), ui(k) denotes the i th component of u(k), and similar notations are used for the other variables. W i t h f ( k ) set to zero, (2.1) describes the normal (fault free) behaviour of the monitored system. We assume t h a t H has full row rank. Moreover, we let L/ denote the set of admissible input sequences. Consider the following class of systems :

{

~(k + 1) = A(k)5(k) + [~y(k)y(k) + [~u(k)u(k) r(k) = 5s(k) + D~y(k) + D~u(k)

(2.2)

where A(k),/~y(k) a n d / ~ u ( k ) are time-varying matrices of a p p r o p r i a t e dimensions, while C, b y and D~ have constant entries. D e f i n i t i o n 5.1 A system of the .form (2.2) is called an innovation filter

(or an innovation generator) for system (2.1) if there exists a set Ll in Lt such that, in the absence of fault, for all u E ~ , (2.2) i s uniformly asymptotically stable, and the output r is a zero mean white noise sequence which is invariant under u and d, once the transient due to initial conditions has vanished. Notice that, in the linear case, an e x t r a condition is added in the definition in order to ensure t h a t no useful information on f , contained in y, is lost. This notion is difficult to translate in a nonlinear framework. However the innovation filter resulting from our algorithm will be seen to offer certain guarantees on this point.

448

3


Design Procedure

The algorithm is based on the following two theorems of which the proofs yield a constructive design procedure. T h e o r e m 5.1 There exists an innovation filter of the form (2.2) for system (2.1) if the .following two statements hold true : A

A

1. There exist constant matrices P, Ai, Bi, i = 0, ..., m, L1 and L2, with P, L1 and L2 different from zero, such that : PA~

-

A~P = B i G

p [EdiF d] = 0

i = O, ..., m

(3.3)

i = l,...,nd

(3.4)

L , C - L2P = 0

(3.5)

2. There exists a set l]t in LI such that, .for all u C l~ the following system is unifo~nly completely observable and uniformly completely controllable : { ~(k + 1) = A(k)~(k) + G(k)w(k) q(k) = L2S(k)

(3.6)

where fit(k) = ,4(u(k)) - S ( k ) R - 1 L 2 , ft(u(k)) = rio + ~-~=1 ui(k)fii, R = L 1 H H T L T, S(_k) = ( P G - B ( u ( k ) ) H ) g T n T, B ( u ( k ) ) = Bo + ~-'~im=lui(k)Bi, and G(k) = P G - l~(u(k))H - S ( k ) R - 1 L I H . P r o o f . As there is no need to consider the term in f ( k ) for this proof, we let f ( k ) = 0 in (2.1). Define z(k) = Px(k). Then, from (2.1) and (3.4) m

z(k + 1) = P A o x ( k ) + ~ u i ( k ) P A i x ( k )

+ PBu(k) + PGw(k)

(3.7)

i=1

Now substituting (3.3) for PA~,i = 0 , . . . , m in (3.7), and taking the output equation of model (2.1) into account, (3.7) yields :

z(k + 1) = A ( u ( k ) ) z ( k ) + B ( u ( k ) ) ( y ( k ) - D u ( k ) ) + P B u ( k ) +(PG - B(u(k))H)w(k)

(3.8)

where A(u(k)) = Ao + ~m=l ui(k)Ai and B ( u ( k ) ) = Bo + }-'~-1 ui(k)Bi. (3.8) is a bilinear system up to output injection which is only affected by d via y. In order to build an innovation filter, we shall design a Kalman


449

filter for estimating z. To this end, we have to determine that part of the measurement y which depends on z, u and w only. This is achieved by defining the signal q as :

q(k) = Lly(k) = L1Vx(k) + L1Du(k) + L 1 g w ( k )

(3.9)

and, from (3.5),

q(k) -- n2z(k) + nlDu(k) + L1Hw(k)

(3.10)

The noise terms are correlated in (3.8) and (3.10). Yet one can easily transform (3.8), (3.10) into a system for which the state excitation noise and the observation noise are not correlated, by adding and subtracting S(k)R-lq(k) in (3.8), with S(k) and R as defined in the theorem statement [8]. This yields :

z(k + 1) = ~(k)z(k) + ~y(k)y(k) + ~(k)u(k) + ~(k)w(k)

(3.11)

where [~y(k) = B(u(k))+ S(k)R-1L1, B~(k) = P B - B ( u ( k ) ) D - S(k)R -1 L1D, and the other notations are as in the theorem statement. The Kalman filter for (3.11), (3.10) is: ~(k + 1) = A(k)~(k) + By(k)y(k) + [l~(k)u(k)

+r(k)(Lly(k) - L2~(k) - L1Du(k))

(3.12)

where the gain is obtained from F(k) = ft(k)II(k)L~(L2H(k)L T + R) -1

(3.13)

with II(k) given by H(k + 1) --- -fi(k)II(k)LT(n2H(k)n T + R)-ln2H(k)ffi(k) T +A(k)H(k)fi(k) T + Q(k) with II(0) = H0

(3.14)

where II0 is a positive definite matrix and Q(k) = G(k)G(k) T. The filter (3.12) is of the form of the first equation in (2.2) with: fi~(k) = f t ( k ) - r ( k ) L 2 , By(k) = By(k)+r(k)L1, and B~(k) = B~,(k)-F(k)LID. By the second condition in the theorem statement, the system (3.12), (3.13), (3.14) is uniformly asymptotically stable for all u C/2 [12]. Now set

r(k) = q(k) - L2~(k) - L1Du(k) = - L 2 ~ ( k ) + Lly(k) - L1Du(k) (3.15) (3.15) is of the form of the second equation in (2.2), and r(.) is a white noise sequence once the transient due to initial conditions has vanished, since it is the innovation of a Kalman filter. 9 The next theorem gives necessary and sufficient conditions for the existence of a solution to the equations (3.3), (3.4), (3.5). The proof of the

450


result is constructive and directly yields a solution for P, L1, L2, Ai, Bi, i ----

0,''" ,m. For the clarity of the developments, before stating the theorem, we rewrite (3.4) as follows : PK = 0

(3.16)

where K -- [E 1d ...E,~ /~d1 " " F rtdJ" d] d The singular value decomposition (SVD) of several matrices will be used. For a matrix M r (the exponent indicates the iteration number in the recursive method resulting from the proof below), with rank rM~, the different factors of this decomposition are denoted in the following way:

M r = ~M~MVM -VTqr'

17"~T

_-- [U~41 U)~/2]

y~/~2

where E ~ is a r M ~ - b y - r M , diagonal matrix containing the singular values of M r, and U~4 , V~/ are orthogonal matrices. T h e o r e m 5.2 : Equations (3.3)-(3.5) and (3.16) have a solution such that P ~ O, L1 ~ 0 and both matrices have full row rank if and only if there exists an integer o~ > 0 such that K s = 0 or Q(~-I) has full column rank, and none of the matrices K ~ K 1 , . . . , K (a-1), Q0, Q 1 , . . . , Q(a-1) has .full row rank. The matrices K j and QJ, j -- O, ... , ~ are given by the .following recursive formulas : 9 WhenQJ ~0:

n(J+l) i

rTjTAj{ITj j j j z V K 2 , , i k ~ , K 2 -- U K I V ~ I ( E Q )

-1

jT j j U~IC UK2 )

i ~- 0 , . . .

,m

(3.18) C (j+l) = u ~ T c J u ~ 2 K(J+I) = UJKT[AJoUJK1V~2 A{UJK1V~2 . . .AJmUJK1V~21 Q(i+l) = C ( j + I ) U ~ + ' )

(3.19) (3.20) (3.21)

W h e n QJ -- O: A~ j+l) = uJKT A{uJK2

i ----0 , . . . , m

(3.22)

C (j+l) = CJUJK2

(3.23)

K(J+ 1) --- TTJ TF AjTTj J J J J vg2L "'0 ~ K1 AIU~:I 99. AmUJgl]

(3.24)

Q(j-{-1) __--c(J+l)U~I+1)

(3.25)


451

where A ~ = A~, i = 0 , . . . , m , K ~ = K , C O = C, QO = C O u O 1 = C U K 1 "

Proof. If part The sufficiency part of the proof is constructive and it can be divided into two sections : first the computation of P, Ai, Bi, i = 0 , . . . , rn t h a t fulfil (3.3) and (3.4), next the computation of L1 and L2 that fulfil (3.5) for the matrix P obtained in the first step. C o m p u t a t i o n o f P, Ai, B~, i = 0 , . . . , m Set p0 _ p and B i - Bi. Then (3.3) and (3.16) can be written as follows, given the notations introduced for the initialization of the recursive formulas : A

A

A

P 0 Ai0 - A l P ~ = B^o iC o

i = 0,...,m

P~176= 0

(3.26)

(3.27)

Equation (3.27) is fulfilled if and only if (3.28)

pO = p 1 u O T

for some appropriate matrix p1. (3.28) can only be written when r a n k K ~ < n (and hence K ~ has not full row rank), which holds by hypothe-

sis. Substituting (3.28) into (3.26), and multiplying the resulting expression on the right by U ~ ---[U~ U~ yields:

p l tZK2~itzK1 t r OnOtrO T plu~

~Ot-',OtrO

(3.29)

~ J~'i~ U'K1

A ~ 1 7 62 - A i P1 ~ B ? C ~ 1 7 62

(3.30)

for i = 0 , . . . , ra. Define Q0 as in the theorem statement, and introduce its SVD in (3.29). Three situations are considered successively: 9 Q0 # 0 and Q0 has not full column rank Equation (3.29) yields, after multiplication on the right by [V~I V~21: p1TTOTAOTTO l]'O ~0rr0 x-0 ~'~K2":~i ~K1 vQ1 ~ ~'~it~QlZ~Q

i=0,...

(3.31)

,m

p I uOT2A oUO 1V~ 2 = 0

(3.32)

For a fixed p1, any/~0 that solves (3.31) has the form : ~0 : p1uOTAOUOlV21(~)-Iu~T

+ ~Iu~T

(3.33)

452

5. Innovation Generation for Bilinear Systems with Unknown Inputs for some m a t r i x / ~ of appropriate dimension, and for i = 0 , . . . , m. Indeed, U~2 exists since Q0 has not full row rank by hypothesis. Substituting (3.33) for ~o into (3.30) yields: 0

plrrOTAOfTTO ~K2~i~vg2 =

0

0

-- U ~ I V ~ I ( E Q )

--1

OT

U~lC

~ lrr0T~0rr0 "t-~i ~ Q 2 ~ J ~ K 2

0

0

U~2) - ~ i p 1 i = O,.. . , m,

(3.34)

which can be written as follows using the recursive formulas (3.18), (3.19) : p1Ail

--

~ i p 1 = B~C 1

i = 0,...,m.

(3.35)

Moreover, the set of equations (3.32) can be written : p1K1

= 0

(3.36)

thanks to (3.20). Notice that (3.35), (3.36) have the same form as (3.26), (3.27). If K 1 = 0, one directly finds a solution for (3.35), for instance p1 = I,.~i = A~,~/1 = 0, i -- 0 , . . . ,m. If K 1 ~ 0, one repeats the above procedure, starting from (3.28), with adequate changes of exponents. 9

Qo~_o

(3.29) yields: p1TTOTAOTTO ~ K 2 .r~i t.J K 1

---- 0

(3.37)

Hence defining K 1, A 1 and C 1 respectively as in (3.24), (3.22) and (3.23), and setting/~} = ~ 0 , i = 0 , . . . , m , (3.30) and (3.37) take the form of (3.35) and (3.36). The iterative procedure continues if K 1 ~ 0; otherwise it stops since the solution for p l , . ~ , / ~ , i = 0 , . . . m can be obtained as above . Q0 has full column rank In this case, V~ = V~I and (3.33) is obtained from (3.31) as a solution for/~o. Substituting this expression in (3.30) yields (3.35). T h e difference with the first case (Q0 ~ 0 and Q0 has not full column rank) is that no equation like (3.32) arises. Hence one can directly solve (3.35) (p1 = I, A~ = A ~ , / ~ = 0, i = 0 , . . . , m). The combination of the above three cases defines a recursive procedure which stops either when K ~ = 0 or Q(~-D has full column rank, for some integer c~ > 0. At that stage, one is left with the equations : P"A?

- AiP~

= B~C"

i : O, ..., m

(3.38)


453

for which P ~ ---- I, Ai = A~ a n d / ~ -- 0, i -- 0 , . . . , m is a solution. For P ~ = I, the m a t r i x P resulting from the recurrence based on (3.28), say P*, is easily seen to be: P* = U ( 2 - 1 ) T . . . u I T UOKT

(3.39)

Moreover, /~i, i ---- 0 , . . . , m can be c o m p u t e d by a p p l y i n g backward, from j -- c~ - 1 to j = 0 the following recursive formulas d e d u c e d from (3.33) :

{ BJi ---- r ( J T 1 ) r r J T z i J r r J l'rj (v'J ~ - l r r J T ~(J+l)rlJT ~j ./~J+li '-'K2~'~i"K1 "Qlk~-"Q] VQ1 -}- --i "-'Q2

QJ whenWhen QJ =7~O0 (3.40)

with B~ = 0, i = 0 , . . . , m ,

a n d by r e m e m b e r i n g t h a t /~o = /~i,i =

0,... ~Trt. C o m p u t a t i o n of L1 a n d L2 It now remains to solve (3.5). A particular solution c o r r e s p o n d i n g to P* is : L~ : U ~ - I ) T

. . . U ~ T U ~ T2

(3.41)

L~ = U ~ - 1 ) T . . . ugoT2cuO2 . . . U(K2-1)

(3.42)

One can check t h a t (3.5) is fulfilled as follows. Let us c o m p u t e L I P * , with P* given by (3.39) : L~P* = U;2 -1)T . ..u~Tcu~ Noticing t h a t rri

rriT = I -- rri

'JK2"~K2

2 ,..U(K~-I)U(~-I)T...U rriT i = 0,

' ~ K I ~ K I ,

. . .

,

~

(3.43)

c~ -- 1, a n d s u b s t i t u t i n g

successivelyU(K2-1)U~2- I)T, U(2-2)U(a2-2)T..., uO2uOT in terms of these expressions in (3.43), one gets :

Lip*

.~_ U ~ - I )T . . . u ~ T c - V ~ 2 - 1 ) r . . . u ~ T C U O l UOT1 - Ec~--: U ~ - - I ) T . . . U ~ T 2 C U 0 2 . . . U ~ 2 U ~ f + l l ) U ~ 2 + l l ) T u ~ f T . . . N

~

(3.44) T h e first t e r m on the right h a n d side of (3.44) is n o t h i n g b u t L ~ C . Hence, to conclude this p a r t of the proof, it suffices to show t h a t all the other terms are equal to zero. Let us consider an a r b i t r a r y t e r m of the s u m (the same reasoning also applies to the second t e r m of the right h a n d side of (3.44)). Using the recursive formula (3.19), one easily deduces :

U ~ - I )T . . . s ~ T c s ~ : U~-I)T...

. . . U~72U(I~Wl l ) U}~Wl l )T u~TT . . . S Or U ~ ; 1 ) T c ( i + I ) U ~ + l l ) U ~ + 1 1)Tu~(T2... GOT

=U~-I)T...U~+I)TQ(i+i)u(~+I)Tu~T...uOT

(3.45)

454


where the definition of Q(~+I), (3.21), was used to obtain the last expression. "r i Finally notice that, in the right hand side of (3.45), U Q2 (i+l)Trl(~+l) ~ ( ~ ( i + I ) T . . ( i + 1 ) ~-~(/q-1). , ( i + I ) T

,~

.

2 %1 Z~Q v~}1 ----u, a n a hence any t e r m in the s u m is null in 3.44). This also holds for the second t e r m of the right hand side of (3.44) as already mentioned. In the expressions for L~ and L~, one should set U~2 = I when QJ = O, which corresponds to what is done in the definition of C (j+l) (compare (3.23) and (3.19)). It remains to verify t h a t P* and L~ are non-zero. As all the matrices U~T, i = 0 , . . . , ( a - 1) have full row rank, one easily checks t h a t r a n k P * = .rr((x--1)T ran~uR2 . The latter is non-zero as K (~-1) has not full row rank by hypothesis. Hence P* r 0, and P* has full row rank. On the other hand, as Q i , i = 0, .. . , a -- 1 have not full row rank, U~2 iT ,i = 0 , . . . a 1 exist. B y a similar argument as for P* one concludes t h a t L I has full row rank and is different from zero. O n l y if p a r t Assume t h a t equations (3.3)-(3.5) and (3.16) have a solution with P ~ 0 and L1 # 0, b o t h matrices having full row rank. Let the dimensions of P and L1 be denoted e x n and s • respectively. From (3.16), K cannot have full row rank. Hence P = p 0 has the form (3.28) for some m a t r i x p1 t h a t fulfils (3.29), (3.30). Equivalently, p1 must fulfil (3.35) for some matrices / ~ , i = 0 , . . . , m, linked t o / ~ via (3.40) and it must also fulfil (3.36) when Q0 has not full column rank. p1 has dimensions e x (n - r K ) where r g denotes the rank of matrix K . Now, two situations can be distinguished : 1. if Q0 has full column rank or K 1 = 0, then a = 1, and p1, -4i, i = 0 , . . . , m must be a solution of (3.35) for s o m e / ~ , i = 0 , . . . , m. 2. if none of the conditions in 1. hold, K 1 cannot have full row rank by (3.36), and p1 must be of the form p1 = p 2 u I T where p 2 is an • (n - r a n k K 1 - r a n k K ) matrix. P2 must fulfil equations of the form (3.29), (3.30) with all the exponents increased by 1. One can repeat for p2 the same procedure as for p1, and so on. This corresponds to an iterative procedure. One realizes t h a t there must exist a finite integer a, for which either Q(~-I) has full column rank or K s = 0. Indeed, as the number of columns of P J , j = 0, 1 , . . . keeps decreasing when the number of iterations increases, this iterative procedure must stop, otherwise P cannot have full row rank. It remains to show t h a t the existence of a solution implies t h a t Q0, Q1, ... , Q(~-I) cannot have full row rank. To this end, let us consider equation (3.5). Introduce (3.28) into (3.5), and multiply the resulting equation on the right by [U~ U~ This yields: LICU~

- L 2 - p l ---- 0

(3.46)

5.

Innovation

Generation

for Bilinear

Systems

with Unknown

Inputs

455

(3.47)

L 1 C U ~ = LI Q ~ = 0

(3.47) implies t h a t Q0 cannot have full row rank, as L1 ~ 0. Now, substituting p1 for its value in terms of p2 in (3.46), and multiplying the resulting expression on the right by [ U l l U~2], one d e d u c e s : L1CU~

- L2P 2 = 0

(3.48)

-- 0

(3.49)

L1CU~ T h e latter equation can be written : 0

0T

0

1

0

0T

0

1

L,U~21U~21CU~2Uk1 + L , U ~ 2 U ~ 2 C U ~ 2 U k l ----0

(3.50)

By (3.47), L1 is necessarily of the form L1 ~--- ~lVQ2Flrr0T for some non zero • (p - r a n k Q ~ matrix L~. Substituting this value for L1 in (3.50), and taking (3.19) and (3.21) into account, one obtains : L~Q 1 -- O. Hence Q1 cannot have full row rank. Proceeding in the same way as above, one gets L{QJ -- 0, 0 < < ~ - 1, where L ~ ~ L1, L{ -1 ~ rJH(J--1)T and _ j __ ~lVQ2 L~ ~ 0. Hence QJ, 0 < j _< ~ - 1 have not full row rank. 9 T h e matrices P* and L~ resulting from T h e o r e m 2 have full row rank, and moreover P* and L~ have the largest possible rank among the set of solutions to (3.3)-(3.5) . Indeed, from the necessity part of the proof of T h e o r e m 1, one notices that any pair of matrices P and LI t h a t make a solution of (3.3)-(3.5) (together with adequate matrices L2, A~, B~,i --or 1 , . . . , m ) must be of the form P = P ~ P * , L 1 = L1L1 for some matrices P~, L~' of appropriate dimensions. Hence z = P*x and q = L~y have the largest possible dimension, which is a normal requirement for avoiding loss of information on f . The design method resulting from Theorems 1 and 2 can be s u m m a r i z e d as follows: A

*

1. determine a solution to (3.3),(3.4) by applying the recursive formulas (3.18)-(3.21) or (3.22)-(3.25) for j = 0 , . . . , a . If one of the matrices K 0 , K 1 , . . . , K ( ~ - I ) Q 0 , Q 1 , . . . , Q ( ~ - I ) has not full row rank, the procedure stops : the algorithm does not give a solution. Otherwise, compute P* according to (3.39); set Ai = A{, i -- 0 , . . . , m and comp u t e / 3 i , i = 0 , . . . , m from (3.40). 2. Solve (3.5), with P = P*. A solution is given by (3.41), (3.42). 3. I m p l e m e n t the innovation filter (3.12)-(3.14). To be able to use the innovation filter described above for fault detection, one should monitor on-line its output, or ~/function of its o u t p u t , b y adequate statistical tests. This issue is discussed in the next section.

456

4


Innovation Monitoring

4.1

Introductory Remark

Two situations must be distinguished depending on whether E[ -- 0, i = 1 , . . . , n f or not in (2.1). In the first case, the faults are additive, n a m e l y they only change the m e a n of the innovation. Then, the latter is known to be a sufficient statistic for the faults f , and it can be monitored by the generalized likelihood ratio (GLR) test, for instance [18], [2]. If some of the matrices E[,i = 1 , . . . , n f are non zero, the faults are not additive, and the innovation is not a sufficient statistic for f anymore [2]. Monitoring the innovation could still allow one to detect the faults, but it is not the best solution. One potential approach could be to work with the leastsquares-score associated to the innovation filter [1]. However, some issues still have to be clarified for that method. Therefore, only additive faults will be considered in the remaining p a r t of Section 4. T h e distribution of the residual in the absence and in the presence of faults is first determined, before presenting a review of the G L R test.

4.2

Innovation in the Presence of Additive Faults

To be able to apply the G L R test, step-like faults will be considered, namely f(k) = #l{k_>to}, where # is a constant vector, and l{k_>to} is equal to 1 when k _> to and it is null otherwise. It is straightforward to c o m p u t e the signature of the fault on the innovation (also called the dynamic profile of the fault). Indeed, with f(k) non zero, (3.11) can be written :

z(k + 1) = fi.(k)z(k) + B~(k)y(k) + [~(k)u(k) + P F f #l{k>_to} (4.51) By subtracting (3.12) from (4.51), and by substituting (3.10) for (3.15), one deduces : ez(k + 1) =

(fi(k) - r(k)L2)Ez(k) + PFS~l(k>to} +(G(k) - F(k)LiH)w(k) r(k) = L2ez(k) + L1Hw(k)

q(k) in (4.52)

(4.53)

where ez(k) = z(k) - ~(k). Hence the innovation can be written :

r(k) = ro(k) + p(k, to)#

(4.54)

where to(k) is the innovation for the fault free system, and p(k, to)# is the signature of the fault. T h e latter is null for k < to and it can be obtained


457

by simulating (4.52),(4.53) with w ~- 0 and ez(t0) = 0 for k _> to. Since the noise w(k) is assumed to have a normal distribution (see model (2.1)), r(k) is also Gaussian. More precisely,

= N(0,

when no fault has occurred

(4.55)

after occurrence of a fault

(4.56)

= •(p(k, to)., r ( k ) )

where N'(~, O) denotes the normal distribution with mean ~ and variance O, E(k) = L2II(k)L T + L1HHTL T, and II(k) is given by (3.14).

4.3

Generalized Likelihood Ratio Test

For the sake of simplicity, # is assumed to be a scalar (see [18] for the nonscalar case). The GLR test is aimed at choosing between two hypotheses: 9 H0 : no fault has occurred : a fault of unknown magnitude, #, has occurred at an unknown time instant, to _< k, where k denotes the present time instant.

9 H1

To explain the idea behind this test, let us first assume that # is known. Classical tests between both hypotheses rely on the log-likelihood ratio of H1 versus H0 for the residual sequence, namely:

s(k)

= in

p•(r(k))

(4.57)

where p , (7"(k)) (P0 (r (k))) is the probability density of r (k) under hypothesis H1 (H0). This quantity has the following fundamental property:

E.(s(k)) > 0

E0(s(k)) < 0

where E~ and E0 denote expectation under the distributions associated to p , (.) and Po (') respectively. Therefore, the cumulative sum of log-likelihood ratios, Sk = ~ i =k l s(i) has a negative drift in the absence of fault, and a positive drift when a fault has occurred. Hence the maximum likelihood estimate of the fault occurrence time, t0, can be computed by maximizing, w.r.t, to, the log-likelihood ratio of H1 versus H0 for the residual samples from time to to k, namely :

k

p,(r(i))

(4.58)

i=to

This yields the cumulative sum (CUSUM) test which amounts to computing the function: gCUSUM(k)---- max S~o

l 0 and for all k > N, where O(k2, kl) = Fz ( k 2 - 1) Fz (k2 - 2)-.. Fz (kl). Similarly, the controllability Gramian of [F~, Q] along a trajectory {z (k)} of (2.22) as k-1

C(k,N)=

E

O(k'i+l)QOT(k'i+l)"

i=k- N

A system is said to be controllable (observable) along a trajectory {z (k)} if there exists N such that for all Rx > 0 there exist 0 < er < Rx, ai (Rx,er,N) and bi (Rx,e~,N), i = 1,2, such that for some arbitrary sequence {~(k)}, ]l~ (k)ll < R~, and for all {r such that I1r (k)[] < ~

alI >_C (k, N) > a2I,

0 < a2 ~_ al < 00,

(2.23)

b l I < O ( k , N ) n - 1 are given by

/ 0 ek+l =

0 r y , k + l -- )~nry,k

+

"

".

"

0

...

)~n

ek.

(4.21)

e n _ 1 = 0 and k > n - 1, ek = ( 0 ... 0 r y , k )T is a solution of this difference equation. This also holds for observer 2. Therefore, using observer 2 or 4, noise is not filtered regardless of the chosen eigenvalues whereas observers 1 and 3 are able to filter noise. This especially holds for observer 1, which, designed for a linear system For

2. Nonlinear Discrete-Time Observers for Synchronization Problems

xk+l = A xk,

Yk ~- c T Xk

501

(4.22)

with A a matrix and c a vector of appropriate dimensions, leads to the well known linear observer

(4.23)

2ck+1 = A 2 k + g ( c T x k - - Y k )

with the gain vector g. Choosing all eigenvalues of the observer error near the system eigenvalues (if possible) the gain and therefore the influence of noise at the output measurements on the observer is small. It even vanishes if system and observer eigenvalues are identical. Assuming small noise and system states near the operating point, the results from the linear case can be adopted to nonlinear systems. Therefore, good robustness of observer 1 to noise at the o u t p u t measurements designed for a nonlinear systems is achieved by eigenvalues in E O F near to those of the linearized system. To summarize, table 2.1 shows the main characteristics of the observers in E O F with hz(zn,k) ---- z,~,k. Supposing a smooth t r a n s f o r m a t i o n m a p between E O F and x-coordinates, it is probable t h a t the characteristics of the observers are also found in x-coordinates. TABLE 2.1. Characteristics of the observers in EOF (hz(z,~,k) = z,~,k). Observer Characteristic Filtering of measurement noise Robustness to model uncertainties Transient behavior C o m p u t a t i o n for step k + 1 at step

5

1

2

_

k

3

4

+

-

+

_

+

k+l

k

+

k+l

An Example in the Field of Communication

As an example for reconstructing a desired information from the transmitted signal, the presented observer design is applied to the second order system

xk+l Yk

= ~

(1 - e)#Xl,k(1 - Xl,k) + ~X2,k "~ (1 e)#x2,k(1 x2,k) +wkx2,k J ' Xl,k

(5.24)

502


where # a n d e are constants, Yk = Xl,k is the t r a n s m i t t e d signal a n d wk contains the desired information. T h e signal wk is a discrete-time signal with 0.06 _< wk _< 0.12 and a step w i d t h of 5, i.e. w takes a new value for k = 0, 5, 10, 15, ... a n d remains c o n s t a n t for other values of k. For w = e, s y s t e m (5.24) is identical to the one presented in [1]. A n observer design via E O F for w = e was already considered in [5]. T h e observer design via E O F for s y s t e m (5.24) is based on the t h i r d order model

x2,k+l

=

(1 - E)#x2,k(1 -- x2,k) + W k X l , k

Wk+l

,

(5.25)

Wk Yk

=

Xl,k.

T h e signal w is assumed to be constant. Since this only holds for t h e d u r a t i o n of five steps, this a s s u m p t i o n requires an observer t h a t converges fast enough. For xl,k > 0, the representation of s y s t e m (5.25) in observability form exists and an observer design via E O F is possible. T h e observer equations are o m i t t e d for reasons of space. 1

I

I

I

60

80

0.9 "~

0.8

~

0.7

.~

0.6

L

=~ 0.5 0.4 0.3 0

20

40

100

F I G U R E 1. Transmitted signal yk ----xl,k Figure 1 shows the t r a n s m i t t e d signal yk --- xl,k for initial conditions xl,0 = 0.4, x2,0 --- 0.2 and w k as presented in Figure 2, which also shows the r e c o n s t r u c t e d signal w~,k using observers 1, 3 and 4 via E O F w i t h 21,0 = 0.5, 52,0 = 0.5 and w~,0 ---- 0. T h e eigenvalues of the observer error d y n a m i c s in E O F were chosen to A1 = A2 = ,k3 -- 0 . 1 . T h e reconstruction of w k using observer 1 has a delay of 3 steps a n d reaches satisfactory a c c u r a c y just before the observer starts to converge to the next value of wk (A d e a d - b e a t design with "~1 : )~2 : "~3 : 0 considerably improves the behaviour). Since observer 3 shows b e t t e r transient behaviour t h a n observer 1 (see Section 4.2), the r e c o n s t r u c t i o n of wk is quite

2. Nonlinear D i s c r e t e - T i m e Observers for Synchronization P r o b l e m s 0.121-

o. 9I

0.08 I- I 0.07 F 0.06 i 0

0.11 0.1 0.09 0.08 0.07 0.06

0

'

'

'

'

'

503

' ~

~1

i 20

i 40

i 60

80

1O0

' 20

' 40

' 60

' 80

1O0

I 60

, 80

100

' 60

' 80

100

I

k t.,r

0.11

0

0.08 0.07 0.06 0

I 20

I 40 k

"~" 0.11 .~ 0.08 0.07 0.06

0

20

40 k

F I G U R E 2. Information signal wk and the e s t i m a t e d signal w~,k using observers 1, 3 and 4 with A1 = )~2 = A3 = 0.1

g o o d a l t h o u g h it also h a s a d e l a y of 3 s t e p s . B e s t r e s u l t s c a n b e a c h i e v e d u s i n g o b s e r v e r 4. F o r k > 6 t h e r e c o n s t r u c t i o n of wk is e x a c t e x c e p t for a d e l a y of 2 s t e p s , i.e. we,k = w k - 2 . T h i s o b s e r v e r e x p l o i t s t h e a c t u a l m e a s u r e m e n t Yk for e s t i m a t i n g wk w h i c h o b v i o u s l y l e a d s t o t h i s g o o d t r a n s i e n t b e h a v i o u r w h e n wk t a k e s a n e w v a l u e .

6

Observer Design for the R6ssler S y s t e m

T h i s s e c t i o n f o c u s e s o n t h e d e s i g n a n d r e a l i s a t i o n of a n o b s e r v e r for t h e continuous-time R6ssler system which has the form

504


5:(t)

--x2(t) -- x3(t) x l ( t ) -t- ax2(t)

=-

)

,

y(t) = x3(t)

(6.26)

c + x 3 ( t ) ( x 1 ( t ) -- b) with the coefficients a, b, c > 0. As shown in [13] the initial condition x3(0) > 0 leads to y(t) = x3(t) > 0 for all t _> 0 a n d the observer p r o b l e m is well posed. 15

i

i

i

i

,

20

25

30

20

25

30

i

i

5 0

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

-5 -10 -15

0

5

10

15 t

10 t",l

0 -5 -10 -15

.

0

5

.

.

10

.

.

15 t

.-.

25 20 15 10 5 0 -5

i

0

1

,

I

I

I

I

I

5

10

15

20

25

t

/

d 30

FIGURE 3. State variables of the R~ssler system Figure 3 shows the trajectories of the s y s t e m states for initial conditions x(0) = ( 5 5 5 ) T and coefficients a = c ----0.2 a n d b = 7.5. T h e r e exists two approaches for the realisation of an observer for a continuous-time system: 1. T h e observer is designed in continuous-time. T h e c o m p u t a t i o n of this observer in a real-time p r o g r a m with c o n s t a n t sampling time requires the discretization of the observer equations. Since the o u t p u t signal y between the samplings is u n k n o w n between two samplings, t h e discretization can not be exact even in the linear case. In addition, t h e sampling time has to be chosen w i t h respect to the s y s t e m s a n d t h e observer dynamics.


505

2. The system can be, at least approximately, discretized which allows an observer design for the discrete-time model. The obtained observer can directly be implemented in a real-time program. Whereas a linear system can always be exactly discretized, the discretization of a nonlinear system is in general an approximation. In what follows, a discretized continuous-time observer and the discretetime observer 4 via E O F are designed for the R0ssler system and implemented in a real-time program which reads all T -- ~ s s the o u t p u t y of the simulated R0ssler system. The two observers are compared for initial conditions ~(0) = ( 2 2 2 )T and different error dynamics.

6.1

Observer

Design in Continuous-

Time

The transformation

z2(t)

=

z3(t)

x2(t)

(6.27)

In(x3(t))

leads to a system representation in new coordinates (cf. [13])

~(t) y(t)

=

=

0 1 1 e z:~(t)

lO) (ez) a

0

0

0

z(t) +

0

,

ce -z:~(t) - b

(6.28)

which allows to design an observer with linearizable error dynamics. T h e resulting observer has the form

Q

~(t)

=

--:~2(t) -- y ( t ) ~- Ol(/~t(:~3(t)) -- l r t ( y ( t ) ) ) 5:l(t) + a~:2(t) + 0 2 ( l n ( 5 ~ 3 ( t ) ) -- l n ( y ( t ) ) ) :~3(t) (:~1 (t) ~- y--~t) -- b ~- 03 (/n(:~3(t)) - / n ( y ( t ) ) ) )

) (6.29)

with

03 = --(a ~- q2),

O1 = 1 Jr- oaa -- ql,

02 = qo + 03 -- o l a

(6.30)

and q0, ql, q2 the coefficients of the desired characteristic polynomial of the observer error dynamics. For the realisation in a real-time program, the differential equations are numerically c o m p u t e d using the R u n g e - K u t t a algorithm.

506

6.2

2.

N o n l i n e a r D i s c r e t e - T i m e O b s e r v e r s for S y n c h r o n i z a t i o n P r o b l e m s

Observer Design in D i s c r e t e - T i m e

A continuous-time system of the form

2(t) = f(x(t)),

y(t) = h(x(t))

(6.31)

can be discretized by a Taylor-Series expansion

9 (t + T)

+

T 2 ..

=

z(t) + T

x(t) + . . .

=

x(t) + T f(x(t)) + - T ~ f ( x ( t ) )

T 2 Of x t

+...

(6.32)

with sampling time T. Setting x(t) = x(kT) =: xk, x(t + T) = x((k + 1)T) =: xk+l leads to the discrete-time system representation

xk+l

=

xk + T f(xk) + - ~ - ox~: J~ ~J

yk

=

h(xk)

"'"

(6.33)

In practice it is necessary to neglect higher order terms of the Taylorseries expansion. The higher the order of the first neglected term, the more accurate is the discretization. However, the complexity of the obtained system representation also increases with the number of considered terms. For the R6ssler system, all terms with order higher t h a n one must be neglected. Otherwise, the inverse of the observability m a p is not found. Therefore, the observer design is based on the discrete-time system model

Xk+l

Yk

z

:

Xl,k - T(x2,k + za,k) ) x2,k + T(xl,k + ax2,k) x3,k + T(c + x3,k(xl,k -- b))

(6.34)

X3,k.

For this system, observer 4 via E O F with linearizable error dynamics was designed and compared to the continuous-time observer.

6.3

Observer Errors f o r S l o w E r r o r D y n a m i c s

First the eigenvalues of the error dynamics were chosen to Ai,c = - 1 0 (continuous-time) and Ai,d = e T~'c ~ 0.925 (discrete-time) with i = 1,2, 3. Figure 4 shows the observer errors ei,k = ~i,k - xi,k for observer 4 via E O F . During the peaks of x3 (t), there are considerable observer errors. Especially ]e2,kl reaches values up to 1 whereas the observer errors of the continuoustime observer are always smaller t h a n 0.003 for t > 1.5s which can be seen in Figure 5.

2. Nonlinear Discrete-Time Observers for Synchronization Problems |

9

507

I

0.8 0.6 0.4 0.2

el - e2 --e3 ---

i ~

0

r,

i!

,1

,Ib

-0.4 -0.2 -0.6 -0.8

i

-1

0

,

,-

ij

f J

I

I

5

10

15

20

25

30

t

FIGURE 4. Observer error using discrete-time observer 4, A~,a = e -l~

0.003

el - e2 ...... e3 .......

0.002 0.001 o

?

-0.001 -0.002 -0.003 0

5

10

I

I

I

15

20

25

30

t

F I G U R E 5. Observer error using discretized continuous-time observer, )%c = - 1 0

6.#

Observer Errors for Fast Error Dynamics

W h e r e a s the eigenvalues of a discrete-time observer can be chosen to Ai,d = 0 (dead-beat), the error d y n a m i c s of the continuous-time observer can not be chosen arbitrarily fast for a given sampling time. Otherwise, t h e errors of the numerical c o m p u t a t i o n of the differential equations c o n s i d e r a b l y increase. T h e observer errors using observer 4 with d e a d - b e a t design are s h o w n in Figure 6. For t > 2, t h e y are nearly identical to the case w i t h slow d y n a m i c s whereas the continuous-time observer with Ai,c = - 1 2 0 leads to higher observer errors (see Figure 7). In addition, the c o r r e s p o n d i n g discrete-time eigenvalue is e -12~ = 0.391 a n d therefore the convergence rate is smaller c o m p a r e d to a d e a d - b e a t design.

508


0.8

1t

0.6 t-r 9

113

'

,~

O.4 0.2 t ~ 0 -O.2 -0.4

i

' ~t

i!

'

Ii

'

i'i i

' !

~

-0.6

'II~!

-0.8

V 0

el - e2 e3 ...... ....... '

5

10

',e

15

20

t

25

FIGURE 6. Observer error using discrete-time observer 4 with

30

Ai,d = 0

1 0 er

-1 -2 -3

i!

-4

ii

el - -

ii

ii

e2 ......

ii

-5 -6

0

I

I

I

I

I

5

10

15

20

25

t

30

FIGURE 7. Observer error using discretized continuous-time observer with Ai,c : -120

6.5

Concluding Remarks

T h e simulations have shown that the continuous-time observer leads to better results if the error dynamics are 'slow'. However, since the sampling time has to be chosen with respect to the observer dynamics, the eigenvalues of the observer dynamics must lie within a certain range or the sampling time has to be decreased. A discrete-time observer even allows a dead-beat design because the sampling time has to be chosen only with respect to the system dynamics. This point is very i m p o r t a n t if high convergence rates are desired. However, a crucial point is finding a discrete-time representation of the system with satisfying accuracy. In the considered case, only a coarse discretization was possible. ghrther simulations have shown t h a t the discrete-time observer for this


509

example is much more sensitive to noise but less sensitive to parameter uncertainties.

7

Discussion and Conclusions

In this chapter a discrete-time observer design via nonlinear observability form and extended nonlinear observer form using additional past output values is presented. Like the design procedure presented in [2] and [10], it does not require a diffeomorphic system function and the transformation can be calculated without solving a system of partial differential equations. In addition, the main problem of [2] and [10], the restrictive condition that the Hessian matrix of the nonlinear function appearing in observability form has to be diagonal, does not occur. Every strongly locally observable system is state equivalent to a system in EOF. The observer design was applied to a problem in the field of communication in discrete-time and to a synchronization problem in the continuoustime context. For the latter, the performance of the discrete-time observer depends on the accuracy of the system discretization. 8

REFERENCES [1] P. Badola, S. S. Tambe, and B. D. Kulkarni. Driving systems with chaotic signals. Physical Review A, 46(10):6735-6737, 1992. [2] M. Brodmann. Beobachterentwurf.fiir nichtlineare zeitdiskrete Systeme. Number 416 of line 8 in VDI-Fortschrittberichte. VDI-Verlag, Dt~sseldorf, 1994. Dissertation, Universit~it Hannover. [3] S.-T. Chung and J. W. Grizzle. Sampled-data observer error linearization. Automatica, 26(6):997-1007, 1990. [4] G. Ciccarella, M. Dalla Morn, and A. Germani. A robust observer for discrete time nonlinear systems. Systems ~ Control Letters, 24:291300, 1995. [5] H. Huijberts, T. Lilge, and H. Nijmeijer. A control perspective on synchronization and the takens-aeyels-sauer recontruction theorem. Accepted for publication in Physical Review E, 1998. [6] R. Ingenbleek. Beobachtbarkeit und Beobachterentwur.f fiir zeitdiskrete nichtlineare Systeme. Number 03/93 in Forschungsberichte. Universit,it Duisburg, Duisburg, 1993. [7] R. Ingenbleek. Zustandsbeobachter fiir zeitdiskrete nichtlineare Systeme - Geometrische Analyse und Synthese. Number 527 of line 8 in

510

2. Nonlinear Discrete-Time Observers for Synchronization Problems VDI-Fortschrittberichte. VDI-Verlag, Dttsseldorf, 1996. Dissertation, Universit~tt Duisburg.

[8] W. Lee and K. Nam. Observer design for autonomous discrete-time nonlinear systems. Systems ~ Control Letters, 17:49-58, 1991. [9] T. Lilge. On observer design for nonlinear discrete-time systems. European Journal of Control, (4):306-319, 1998. [10] W. Lin and C. I. Byrnes. Remarks on linearization of discrete-time autonomous systems and nonlinear observer design. Systems ~ Control Letters, 25:31-40, 1995. [11] M. Loecher and E. R. Hunt. Control of high-dimensional chaos in systems with symmetry. Physical Review Letters, 79(1):63-66, 1997. [12] H. Nijmeijer. Observability of autonomous discrete time nonlinear systems: A geometric approach. Int. Journal of Control, 36(5):86774, Nov. 1982. [13] H. Nimeijer and I. Mareels. An observers look to synchronization. IEEE Transactions on Circuits and Systems, 44(10):882-90, 1997. [14] E. Ott, T. Sauer, and J. A. Yorke. Coping with Chaos. John Wiley & Sons, Inc., New York, 1994. [15] L. M. Pecora and T. L. Carroll. Synchronization in chaotic systems. Physical Review Letters, 64(8):821-824, 1990. [16] K. Pyragas. Generalized synchronization of chaos in directionally coupled chaotic systems. Physical Review E, 51(2):980 994, 1995. [17] K. Pyragas. Weak and strong synchronization of chaos. Physical Review E, 54(5):R4508-R4511, 1996. [18] T. Stojanovski, U. Parlitz, L. K., and R. Harris. Exploiting delay reconstruction for chaos synchronization. Physics Letters A, 233:355360, 1997. [19] Y. Zhang, M. Dai, W. Hua, Y. Ni, and G. Du. Digital communication by active-passive-decomposition synchronization in hyperchaotic systems. Physical Review E, 58(3):3022-3027, 1998.

Chaos Synchronization Ulrich Parlitz 1, Lutz Junge I and Ljupco Kocarev 2 1Drittes Physikalisches Institut, Universit~tt G6ttingen, Bttrgerstr. 42-44, D-37073 G6ttingen, Germany 2Department of Electrical Engineering, St Cyril and Methodius University, Skopje, P O Box 574, Macedonia

1

Introduction

Synchronization is a phenomenon of interest in many scientific areas ranging from celestial mechanics to laser physics, from electronics to communications, and from biophysics to neuroscience [1]. In particular, synchronization of chaotic dynamics [7, 4, 24] has attracted much attention during the last years because of its role in understanding the basic features of man-made and natural systems. Thus, for example, optical communication using chaotic waveforms demonstrated experimentally [8, 37, 38] and theoretically [3], is possible because of chaos synchronization between transmitter and receiver. On the other hand, the evidence of chaotic behavior in the brain [34] and the importance of synchronization in perceptive processes of mammals [33] indicate a possible role of chaos synchronization in neural ensembles [32] as well. The phenomenon of synchronization also occurs for uni-directionally coupled systems and in this case the driven system (or response system) may be viewed as a nonlinear observer of the driving system. Or, conversely, nonlinear observer theory may be used to construct pairs of uni-directionally coupled synchronizing systems. Such pairs may then be used for system and parameter estimation or for potential applications in communication systems, see [1] and [31]. In all these cases, the two coupled systems are (almost) identical and therefore identical synchronization occurs which means that the difference of drive and response state vectors converges to zero for t --~ oc. If two different systems are coupled, more sophisticated types of synchronization [5] may occur like generalized synchronization or phase synchronization that will be discussed in Sections 3 and 4. We shall begin with a presentation of synchronization phenomena of spatially extended systems that are given in terms of (chaotic) partial differential equations. For more information about chaos synchronization and related problems the reader is refered to the article collections in [31].

512

3. Chaos Synchronization

2 Synchronization of Spatially Extended Systems In the following we shall discuss identical synchronization of uni-directionally coupled spatially extended systems t h a t are described by a partial differential equation in the form

Ou F ( u , OU 02u O---t = cgx' Ox 2 ' ' ' " )'

x 9 [o,n]

(2.1)

with spatial length L. In this chapter we consider only one dimensional P D E ' s . Generalizations for higher dimensional systems are straightforward. For this class of systems we define: Two spatially extended systems are called synchronized, if their states u(x, t) and v ( x , t ) converge to each other in the whole spatial domain, i.e. if Vx E [0, L] : l i m t ~ I[u(x,t) - v(x,t)[[ = 0 As in the case of low dimensional systems there exists an invariant m a n ifold u = v (also called synchronization manifold), whose stability properties determine the occurence of stable synchronization. If the transverse system w • = ~1 ( u - v) has an asymptotically stable fixed point at the origin, then this manifold is asymptotically stable and synchronization occurs. Indeed, all known techniques (see [1], [31]) for verifying synchronization such as necessary criteria like negative conditional Lyapunov exponents or sufficient criteria like Lyapunov functions and stability of unstable periodic orbits, can be generalized and can in principle be applied to spatially extended systems, too. On the other hand, the generalization of the coupling techniques used for low dimensional systems is not so straightforward. A coupling along the whole spatial axis is possible for numerical simulations but m a y turn out to be impractical or even impossible for experiments. A similar argument holds for local pinning coupling schemes that are used for synchronizing coupled m a p lattices (CMLs)[19, 15, 16, 11, 9]. These schemes use coupling in points which is not only practically impossible but also in some sense useless for PDE's. An alternative is the sensor coupling scheme, introduced in [20], which generalizes the pinning schemes to systems with continuous space variables. The idea is, t h a t typical experimental measurement devices have a finite resolution l and measure local spatial averages of the desired quantity. T h e left plot of Figure 1 shows the concept of the sensor coupling scheme and the right plot illustrates the notion of a measured sensor signal. According to [20] we want to call these elements sensors. Each sensor measures a scalar time-series of the form

3. Chaos Synchronization --1.-

,

d

513 ,

L

111

I12

V

U3

space

FIGURE 1. Principle of the sensor coupling scheme. Left: sketch, right: visualization of three sensor time series measured from spatio-temporal chaos.

~.(t)

=

1 fnd+l/2 u ( x , t ) d x ,

7 Jn~-u2

n=l,...,N

(2.2)

which is averaged over a width 1. Because of the exponential decrease of spatial correlations in spatially extended chaotic systems, we need several but a finite number N of coupling signals that contain all the necessary information to reconstruct the whole state in the synchronization process. Therefore the N sensors are distributed with equal distance d = L (for periodic b o u n d a r y conditions) along the spatial axis. Numerical investigations have shown t h a t the equidistant arrangement is nearly optimal for systems with extensive chaos [13]. Now we have to choose a coupling scheme t h a t will be applied locally using the sensor signals as driving forces. To do this we measure in the driven syst e m N sensor signals at the same positions and apply a diffusive coupling t e r m with coupling strength c f ( u , v) = { e(~'~ 0- V~)

:: elsend - I/2 < x < nd + I/2

(2.3)

at each sensor position. As an example we shall examine now the one dimensional complex Ginzburg-Landau model 0U

0--/= #u - (1 - ia)lul2u + (1 + i/3)Au,

u 9 [0, L]

(2.4)

with periodic boundary conditions. This equation possesses uniform travelling wave solutions. For 1 - a/3 < 0 they become unstable and different

514


I!

1.04

1.03

400

400

1.04

1.03

1.02

1.02

300

300 1.01

200

1.01

2OO 1.00

100 0

0.99

~il

1.00

:iii

100

0.99

I

o

20 4 0 6 0 8 0 1 0 0

0

20

40

60

80

100

v

F I G U R E 2. Synchronization of two Ginzburg-Landau equations in the phase turbulent regime (Left: drive, right: response) using N = 15 sensors with width I = 3 and coupling strength e = 0.2. The amplitudes of drive and response P D E are grey scaled. t y p e s of t u r b u l e n c e occur. I n t h e following we will c o n s i d e r two p a r a m e ter sets, # -- 1.0, a -- 2.0,/3 = 0.7 c o r r e s p o n d i n g to p h a s e t u r b u l e n c e a n d tt = 1.0, a = 2.0,/3 = 1.2 which yields defect t u r b u l e n c e . In b o t h cases extensire chaos is o b s e r v e d a n d tile L y a p u n o v d i m e n s i o n DL of t h e u n d e r l y i n g a t t r a c t o r increases w i t h t h e s y s t e m size L like DL ~ 0.102L for p h a s e t u r b u l e n c e a n d w i t h DL ~ 0.332L for defect t u r b u l e n c e . In o r d e r to achieve s y n c h r o n i z a t i o n we d r i v e an i d e n t i c a l c o p y of (2.4) using N sensor signals (2.3) t h a t are a p p l i e d in i n t e r v a l s of w i d t h 1.

Ov

0--t = # v - (1 - ia)lvl2v + (1 + il3)Av + f ( u , v)

(2.5)

N o t e t h a t this is a local c o n t r o l t e c h n i q u e a n d t h e d r i v e n s y s t e m (2.5) evolves freely b e t w e e n t h e sensor locations. F i g u r e 2 shows t h e s y n c h r o n i z a t i o n of d r i v e (left) a n d r e s p o n s e (right) in t h e p h a s e t u r b u l e n t r e g i m e . F o r this e x a m p l e we u s e d N -- 15 e q u a l l y s p a c e d sensors w i t h w i d t h l = 3 to s y n c h r o n i z e two G i n z b u r g - L a n d a u e q u a t i o n s w i t h l e n g t h L = 100. A t t = 170 t h e c o u p l i n g is s w i t c h e d on a n d t h e r e s p o n s e s y s t e m q u i c k l y converges to t h e s y n c h r o n i z e d s t a t e . I n t h e b e g i n n i n g of t h e c o u p l i n g t h e p e r t u r b a t i o n i n t r o d u c e d t h r o u g h t h e sensors signals i n d u c e s a p e r i o d i c p a t t e r n , which d e c a y s very fast d u e t o t h e s y n c h r o n i z a t i o n . S i m i l a r results have b e e n o b t a i n e d for defect t u r b u l e n c e . If t h e s e n s o r c o u p l i n g is a p p l i e d o n l y l o c a l l y one m a y o b s e r v e local synchronization as it is shown in Fig. 3. R e p l a c i n g t h e d r i v i n g sensor signals b y v a n i s h i n g sig-


0

20

40

60

80

100

x

0

20

40

60

80

515

100

x

FIGURE 3. Local synchronization of defect turbulence of the Ginzburg-Landau equations. The left figure shows the dynamics of the response system and the right figure the synchronization error between drive and response that vanishes (dark areas) in those intervals where sensors are placed (N = 2*6, l = 2, e = 2.0). rials one m a y also suppress (locally) the chaotic oscillations of the response system and stabilize the homogeneous state. Furthermore, the sensor coupling has also been succesfully applied to a pair of Kuramoto-Sivashinski equations and provides nonlinear observers that can be used for estimating p a r a m e t e r s of P D E s from time series [13].

3

Generalized Synchronization

If a pair of very similar or even identical systems is coupled one m a y observe identical synchronization (IS) where the difference of the state vectors of b o t h systems converges to zero, even in the case of chaotic dynamics. This kind of synchronization, however, cannot be expected for coupled systems that are of completely different origin (e.g., an electrical circuit coupled to a mechanical system). W h a t does "synchronization" mean in such a more general case? Periodic systems are usually called synchronized if either their phases or frequencies are locked. For chaotic systems, however, the notions of "frequency" or "phase" are in general not well defined and can thus not be used for characterizing synchronization (except for some class of chaotic systems where a phase variable can be introduced to quantify chaotic phase synchronization t h a t will be discussed in the next section). In this Section we present different notions of generalized synchronization (GS) t h a t have been proposed during the last years [4, 29, 2, 17, 26, 6, 12, 35, 21]. Basically two types of generalized synchronization of uni-directionally coupled systems have been investigated in the literature so far. In its strongest form GS leads to the existence of a function t h a t m a p s (asymptotically for t --* oc) states of the drive system to states of the response

516


system. In this case the chaotic dynamics of the response system can be predicted from the drive system. W h e t h e r such a function exists and whether it is continuous or even smooth depends on the features of the drive and response system [6, 12, 35]. As an example consider a m-dimensional (chaotic) dynamical system x n+l = f ( x n)

(3.6)

t h a t drives the following one-dimensional system

yn+l = byn + cos(27rx~)

(3.7)

with b < 1. Is is easy to see that 0 1 > #2 the coefficients ci can for large i by approximated by ci ~ c o n s t . . (b/#2) ~. Therefore, the first derivative of g diverges for b > #2 (or in t e r m s of

3. Chaos Synchronization (a)

517

i O.B

0.8

0.6

0.6 89 0.4-

x2

0.4

0.2 0

0 0

0,2

0.4

0.6 x 1

0.8

1

0

0.2

0.4

0.g

0_8

1

x I

FIGURE 4. Grey scaled plot of the response state y vs. the (xl, x2) coordinates of the driving cat map. (a) b = 0.01 (b) b = 0.4. Both figures have been computed by iterating the dynamical systems (3.7), (3.11) and (3.12) and transients have been discarded.

Lyapunov exponents In(b) > ln(/~2) = A2). In this case the function g is essentially a Weierstrass function. Figure 4 shows how the function g looses its smoothness when b is increased from 0.01 = b #2. Typically a function exists if the response system is asymptotically stable when driven by the coupling signal and no subharmonie entrainment occurs [21]. If, for example, a periodic orbit of the drive entrains a stable periodic orbit of the response with twice the period (i.e. T o : TR = 1 : 2) then any point on the attractor of the drive is m a p p e d to two points on the response orbit and in this case there exists a relation but not a function. This multivaluedness always occurs for subharTnonic periodic entrainment with TD < TR. Note t h a t identical synchronization implies GS in any diffeomorphic equivalent coordinate system. On the other hand, if GS is observed between two dynamical systems with a diffeomorphic function this diffeomorphism can be used to perform a change of the response coordinate system such t h a t in the new coordinate system the response system synchronizes identically with the drive system. To find evidence for the existence of a (continuous) function relating states of the drive to states of the response one m a y apply nearest neighbors statistics [29]. This approach for identifying generalized synchronization can be applied to uni- and bi-directionally coupled systems if the original (physical) state spaces of drive and response are accessible. If only (scalar) time series from the drive and the response system can be sampled, t h e n delay embedding [30] m a y be used to investigate neighbourhood relations in the corresponding reconstructed state spaces. In this case, however, only generalized synchronization of un/-directionally coupled systems can be detected by predicting the (reconstructed) state of the response system using

518


a time series from the drive system. A prediction of the evolution of the drive system based on data from the response system is always possible (i.e. with and without generalized synchronization), because (almost) any time series measured at the response system may also be viewed as a time series from the combined systems drive ~z response and may thus be used to reconstruct and predict the dynamics of drive and response. In this sense a time series based test provides no information about GS in the case of bi-directionally coupled systems. In such a case where drive and response are not related by a function, a second, weaker notion of GS may apply that assumes only asymptotic stability of the response system but not the existence of a function mapping states of the drive to states of the response system [2, 21]. This type of GS can be verified using the so-called auxiliary system m e t h o d where two identical copies of the response system are driven by the same driving signal. Identical synchronization of both response systems indicates GS in the weaker sense. Note that using the auxiliary systems approach one may also observe nonidentical (i.e. generalized) synchronization of identical systems that fail to synchronize identically. Current research in the field of generalized synchronization focuses on the question whether the different phenomena and approaches for characterizing (generalized) synchronization can be unified in a mathematically rigorous sense using the notion of non'really hyperbolic invariant manifolds that are smooth and persistent under perturbations of the system(s) [39, 10].

4

Phase Synchronization

Another generalization of the notion of identical synchronization is the phenomenon of phase synchronization (PS) [36, 27, 22, 25, 18]. It can easily be observed when a well defined phase variable can be identiffed in both coupled systems. This can be done heuristically for strange attractors that spiral around some particular point (or "hole") in a twodimensional projection of the attractor, like the R~ssler attractor shown in Fig. 5. In such a case, a phase angle r can be defined that de- or increases monotonically. Phase synchronization of two coupled systems occurs if the difference [r (t) --r between the corresponding phases is bounded by some constant. A more general definition includes rational relations Inr - rnr ] ( const for arbitrary integers n and m. Using the phase angle r one may define a mean rotation frequency = limt_~ r and in the case of PS, this mean rotation frequencies of the drive and the response system coincide, i.e., also for chaotic systems PS leads to the frequency entrainment known from coupled periodic oscillations. The amplitudes of both systems remain in this case completely uncorrelated [27].


519

This phenomenon may be used in technical or experimental applications where a coherent superposition of several o u t p u t channels is desired. In more abstract terms PS occurs when a zero Lyapunov exponent of the response system becomes negative. This leads to a reduction of the degree of freedom of the response system in the direction of the flow. For systems where a phase variable can be defined the direction of the flow coincides in general with the coordinate t h a t is described by the phase variable. A zero LE t h a t becomes negative reflects in this sense a restriction t h a t is imposed on the motion of the phase variable. If the zero LE t h a t decreases is the largest LE of the response system then phase synchronization occurs together with GS [22]. If there exist, however, in addition to the formerly zero LE, other LEs which are and remain positive, PS occurs but no GS. This scenario for the onset of PS m a y be observed for a sinusoidally driven RSssler system [36]:

21

=

0 . 4 -~ X l ( X 2 - 8 . 5 )

22

=

--Xl -- X3 + a cos(t)

X3

"~-

X2

(4.13)

+ 0.15x3.

Figure 5 shows the onset of PS when the driving amplitude a exceeds some critical value of ac ~ 0.4. The solid gray lines belong to the chaotic attractor of the driven system (4.13) and the black dots are plotted at times tn = n27~ yielding a stroboscopic phase portrait. As can be seen in Fig. 5a these dots are scattered on the chaotic a t t r a c t o r if the driving foice is too weak, indicating no fixed phase relation of the chaotic oscillation with respect to the driving signal. Figure 5b shows the distribution near the onset of PS where the dots already s t a r t to form a cluster. If the amplitude a is sufficiently high, phase sychronization occurs as can be seen for an amplitude of a = 0.7 in Fig. 5c. This transition can also be studied in terms of the Lyapunov exponents of the response system. Figure 6a shows the two largest exponents A1 and A2 of the RSssler system (4.13) plotted in dependence on the coupling p a r a m e t e r a. For a > 0.4 the zero exponent starts decreasing while A1 remains positive. The driven system thus looses a degree of freedom although it stays chaotic. This degree of freedom is associated with the zero Lyapunov exponent A2, i.e., with the (tangential) direction of the trajectories. This direction, however, is exactly the direction of the spiraling motion around the "hole" in the attractor t h a t was used for introducing a phase variable.

520

3. C h a o s S y n c h r o n i z a t i o n

(a) I0

0 x3

-10

'-lOl ....

Oi . . . . x2

irO '

,

(b) I0

0 x 3

-10

'

t

.

.

.

-I0

.

i

.

.

0

.

.

!

,

,

1o

x 2

(o) 10

0 x 3

-10

'-10' . . . .

O~ . . . .

1'0 '

'

'

x 2

F I G U R E 5. P h a s e s y n c h r o n i z a t i o n of a p e r i o d i c a l l y d r i v e n R0ssler s y s t e m (4.13) (a) a = 0.1, no PS; (b) a = 0.5, o n s e t of PS; (c) a = 0.7, full PS.


521

(a) 0.1 0.05 Ak 0 --0.05 --0.1 &

(b)

o:6

o:8

150 e

100

50

0

0:2

0:4

a

F I G U R E 6. (a) The two largest Lyapunov exponents A1 and A2, and (b) the mean synchronization error e (4.14) of a pair of identical R~ssler response systems (4.13) vs. the driving amplitude a.

Figure 6b shows the m u t u a l averaged s y n c h r o n i z a t i o n error e

e=

1 Y ~ ~=111x(tn) - ~(tn)ll2

(4.14)

of two identical R6ssler systems t h a t are driven by the same sinusoidal signal and sampled with t n : n " 27r/25. For GS such a c o m p a r i s o n with an auxiliary s y s t e m would result in an a s y m p t o t i c a l l y vanishing error, b u t here b o t h response systems are chaotic a n d t h e PS leads only to a decrease of e by a factor of a b o u t two. In this sense PS leads to a constructive interference of chaotic response signals t h a t has also been observed in m e a n field variables of arrays of slightly different response systems which were driven by a c o m m o n signal [25, 18]. A n o t h e r p h e n o m e n o n t h a t is closely related to PS is lag synchronization t h a t was observed recently by R o s e n b l u m et al. [28] a n d leads to synchronization with some time delay between drive a n d response.

522

5


Conclusions

In this chapter we have addressed specific topics and examples of "chaos synchronization": synchronization of spatially extended systems (PDEs), generalized synchronization and phase synchronization. In particular synchronization phenomena of uni-directionally coupled identical systems (here: pairs of PDEs) are very closely related to questions of observability and observer design. If the underlying dynamics is chaotic, intermittent breakdown of synchronization may occur if not not all of the unstable orbits which form the skeleton of the chaotic attractor are synchronized (for details see [1] and references cited therein). This phenomenon can be excluded if proper (global) stability conditions can be established (for example, in terms of Lyapunov functions). Another topic that is worth mentioning are potentim applications of (chaos) synchronization like system identification and data encryption. For system identification or parameter estimation a model equation is varied until it synchronizes with a given time series. This approach has been successfully applied to maps, coupled ODE's and coupled PDEs in order to estimate some free parameters [23, 14]. The advantage of chaotic dynamics for this task is the fact that a larger part of the state space is explored compared to periodic solutions. The second potential application mentioned above, "chaos communication", was actually for m a n y researchers the main motivation to study synchronization mechanisms of uni-directionally coupled systems. The basic idea is to transmit a modulated chaotic signal and to use synchronization for recovering the message at the receiver (see [37, 38] for a recent fast optical implementation and [1] for other examples). Whether this approach can really compete with standard cryptographie is still an open question but some special applications seem possible.

Acknowledgments This work was supported by the German Science Foundation (DFG grant Pa 643/1-1) and a binational German-Macedonian grant (MAK-004-96). 6

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Vol. 199: Cohen, G.; Quadrat, J.-P. (Eds) 1lth Intemational Conference on Analysis and Optimization of Systems Discrete Event Systems: Sophia-Antipolis, June 15-16-17, 1994 648 pp. 1994 [3-540-19896-2] Vol. 200: Yoshikawa, T.; Miyazaki, F. (Eds)

Experimental Robotics II1:The 3rd Intemational Symposium, Kyoto, Japan, October 28-30, 1993 624 pp. 1994 [3-540-19905-5] Vol. 201: Kogan, J. Robust Stability and Convexity 192 pp. 1994 [3-540-19919-5] Vol. 202: Francis, B.A.; Tannenbaum, A.R.

(Eds) Feedback Control, Nonlinear Systems, and Complexity 288 pp. 1995 [3-540-19943-8] Vol. 203: Popkov, Y.S.

Vol. 194: Cao, Xi-Ren Realization Probabilities: The Dynamics of Queuing Systems 336 pp. 1993 [3-540-19872-5]

Macrosystems Theory and its Applications: Equilibrium Models 344 pp. 1995 [3-540-19955-1]

Vol. 204: Takahashi, S.; Takahara, Y. Logical Approach to Systems Theory 192 pp. 1995 [3-540-19956-X] Vol. 205: Kotta, U. Inversion Method in the Discrete-time Nonlinear Control Systems Synthesis Problems 168 pp. 1995 [3-540-19966-7] Vol. 206: Aganovic, Z.; Gajic, Z. Linear Optimal Control of Bilinear Systems with Applications to Singular Perturbations and Weak Coupling 133 pp. 1995 [3-540-19976-4] Vol. 207: Gabasov, R.; Kirillova, F.M.; Pdschepova, S.V. Optimal Feedback Control 224 pp. 1995 [3-540-19991-8] Vol. 208: Khalil, H.K.; Chow, J.H.; Ioannou, P.A. (Eds) Proceedings of Workshop on Advances inControl and its Applications 300 pp. 1995 [3-540-19993-4] Vol. 209: Foias, C.; Ozbay, H.; Tannenbaum, A. Robust Control of Infinite Dimensional Systems: Frequency Domain Methods 230 pp. 1995 [3-540-19994-2] Vol. 210: De Wilde, P. Neural Network Models: An Analysis 154 pp. 1996 [3-540-19995-0]

Vol. 211: Gawronski, W. Balanced Control of Flexible Structures 280 pp. 1996 [3-540-76017-2] Vol. 212: Sanchez, A. Formal Specification and Synthesis of Procedural Controllers for Process Systems 248 pp. 1996 [3-540-76021-0] VoI. 213: Patra, A.; Rao, G.P. General Hybrid Orthogonal Functions and their Applications in Systems and Control 144 pp. 1996 [3-540-76039-3]

Vol. 214: Yin, G.; Zhang, Q. (Eds) Recent Advances in Control and Optimization of Manufacturing Systems 240 pp. 1996 [3-540-76055-5] Vol. 215: Bonivento, C.; Marro, G.; Zanasi, R. (Eds) Colloquium on Automatic Control 240 pp. 1996 [3-540-76060-1]

Vol. 216: Kulhav~, R. Recursive Nonlinear Estimation: A Geometric Approach 244 pp. 1996 [3-540-76063-6] Vol. 217: Garofalo, F.; Glielmo, L. (Eds) Robust Control via Variable Structure and Lyapunov Techniques 336 pp. 1996 [3-540-76067-9] Vol. 218: van der Schaft, A. L2 Gain and Passivity Techniques in Nonlinear Control 176 pp. 1996 [3-540-76074-1] Vol. 219: Berger, M.-O.; Deriche, R.; Herlin, I.; Jaffr6, J.; Morel, J.-M (Eds) ICAOS '96: 12th International Conference on Analysis and Optimization of Systems Images, Wavelets and PDEs: Pads, June 26-28 1996 378 pp. 1996 [3-540-76076-8] Vol. 220: Brogliato, B. Nonsmooth Impact Mechanics: Models, Dynamics and Control 420 pp. 1996 [3-540-76079-2]

Vol. 221: Kelkar, A.; Joshi, S. Control of Nonlinear Multibody Flexible Space Structures 160 pp. 1996 [3-540-76093-8] Vol. 222: Morse, A.S. Control Using Logic-Based Switching 288 pp. 1997 [3-540-76097-0] Vol. 223: Khatib, O.; Salisbury, J.K. Experimental Robotics IV: The 4th International Symposium, Stanford, Califomia, June 30 - July 2, 1995 596 pp. 1997 [3-540-76133-0]

Vol. 224: Magni, J.-F.; Bennani, S.; Terlouw, J. (Eds) Robust Flight Control: A Design Challenge 664 pp. 1997 [3-540-76151-9] Vol. 225: Poznyak, A.S.; Najim, K. Learning Automata and Stochastic Optimization 219 pp. 1997 [3-540-76154-3] Vol. 226: Cooperman, G.; Michler, G.; Vinck, H. (Eds) Workshop on High Performance Computing and Gigabit Local Area Networks 248 pp. 1997 [3-540-76169-1]

Vol. 227: Tarbouriech, S.; Garcia, G. (Eds) Control of Uncertain Systems with Bounded Inputs 203 pp. 1997 [3-540-76183-7] Vol. 228: Dugard, L.; Verriest, E.l. (Eds) Stability and Control of Time-delay Systems 344 pp. 1998 [3-540-76193-4]

Vol. 234: Arena, P.; Fortuna, L.; Muscato, G.; Xibilia, M.G. Neural Networks in Multidimensional Domains: Fundamentals and New Trends in Modelling and Control 179 pp. 1998 [1-85233-006-6] Vol. 235: Chen, B.M. Hoo Control and Its Applications 361 pp. 1998 [1-85233-026-0] Vol. 236: de Almeida, A.T.; KhatJb, O. (Eds) Autonomous Robotic Systems 283 pp. 1998 [1-85233-036-8] Vol. 237: Kreigman, D.J.; Hagar, G.D.; Morse, A.S. (Eds) The Confluence of Vision and Control 304 pp. 1998 [1-85233-025-2] Vol. 238: Ella, N. ; Dahleh, M.A. Computational Methods for Controller Design 200 pp. 1998 [1-85233-075-9]

Vol. 229: Laumond, J.-P. (Ed.) Robot Motion Planning and Control 360 pp. 1998 [3-540-76219-1]

Vol. 239: Wang, Q.G.; Lee, T.H.; Tan, K.K. Finite Spectrum Assignment for Time-Delay Systems 200 pp. 1998 [1-85233-065-1]

Vol. 230: Siciliano, B.; Valavanis, K.P. (Eds) Control Problems in Robotics and Automation 328 pp. 1998 [3-540-76220-5]

Vol. 240: Lin, Z. Low Gain Feedback 376 pp. 1999 [1-85233-081-3]

Vol. 231: Emel'yanov, S.V.; Burovoi, I.A.; Levada, F.Yu. Control of Indefinite Nonlinear Dynamic Systems 196 pp. 1998 [3-540-76245-0]

VoL 241: Yamamoto, Y.; Hara S. Learning, Control and Hybrid Systems 472 pp. 1999 [1-85233-076-7]

Vol. 232: Casals, A.; de Almeida, A.T. (Eds) Experimental Robotics V: The Fifth International Symposium Barcelona, Catalonia, June 15-18, 1997 190 pp. 1998 [3-540-76218-3] Vol. 233: Chiacchio, P.; Chiavedni, S. (Eds) Complex Robotic Systems 189 pp. 1998 [3-540-76265-5]

Vol. 242: Conte, G.; Moog, C.H.; Perdon A.M. Nonlinear Control Systems 192 pp. 1999 [1-85233-151-8] Vol. 243: Tzafestas, S.G.; Schmidt, G. (Eds) Progress in Systems and Robot Analysis and Control Design 624 pp. 1999 [1-85233-123-2]

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