Lecture Notes in Control and Information Sciences 246. Editor: M. Thoma

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1 Lecture Notes in Control and Information Sciences 246 Editor: M. Thoma

2 Springer London Berlin Heidelberg New York Barcelona Hong Kong Milan Paris Santa Clara Singapore Tokyo

3 Dirk Aeyels, Framboise Lamnabhi-Lagarrigue and Arjan van der Schaft (Eds) Stability and Stabilization of Nonlinear Systems Springer

4 Series Advisory Board A. Bensoussan M.J. Grimble P. Kokotovic A.B. Kurzhanski H. Kwakernaak J.L. Massey M.M. Morari Editors Dirk Aeyels, Professor Dept. of Systems Dynamics, Universiteit Gent, Technologiepark-Zwijnaarde 9, 9052 Gent, Belgium Francoise Lamnabhi-Lagarrigue, Docteur d'etat Laboratoire des Signaux et Systemes, CNRS, SUPELEC, GIF-SUR-YVETTE, France Arjan van der Schaft, Professor Dept. of Applied Mathematics, University of Twente, PO Box 217,7500 Enschede, The Netherlands ISBN Springer-Verlag London Berlin Heidelberg British Library Cataloguing in Publication Data Stability and stabilization of nonlinear systems: editors, D. Aeyels, F. Lamnabhi-Lagarrigue and A. van der Schaft. - (Lecture notes in control and information sciences) 1.Nonlinear systems 2.Nonlinear control theory I.Aeyels, D. II.Lamnabhi-Lagarrigue, F. (Francoise) IILSchaft, A.J. van der 629.8'36 ISBN Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress 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, mis 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 maybe made. Typesetting: Camera ready by contributors Printed and bound at the Athenaeum Press Ltd., Gateshead, Tyne & Wear 69/ Printed on acid-free paper SPIN

5 Preface The chapters of this book have been presented at the 1st Workshop of the Nonlinear Control Network*, which was held in Ghent, March 15,16, These contributions give an overview of some of the current and emerging trends in nonlinear systems and control theory. As editors of this book we would very much like to thank the speakers at this workshop for their stimulating presentations and for their efforts to bring this material to its current form, which we are sure will provide stimulating reading as well. Dirk Aeyels Franchise Lamnabhi-Lagarrigue Arjan J. van der Schaft 'The Nonlinear Control Network is a four year project within the framework of the European Commission's Training and Mobility of Researchers (TMR) Programme that started on December 1, There are nine partners involved: Dirk Aeyels Universiteit Gent Dirk.Asysls4rug.ac.be Belgium Alfonso Banos Universidad de Murcia abanostdif.um.es Spain Fritz Colonius Universitat Augsburg ColoniusCmath.uni-augsburg.de Germany Alberto Isidori Universita di Roma isidoricgiannutri.caspur.it Italy Francoise Lamnabhi-Lagarrigue Centre National de la Recherche Scientifique lamnabhiclss.supslsc.fr France (coordinator) David H. Owens University of Sheffield D.H.OnensCshsffield.ac.uk England Arjan J. van der Schaft Universiteit Twente a.j.vanderschaftcmath.ut8ents.nl The Netherlands Fatima Silva Leite Universidade de Coimbra fleitecmat.uc.pt Portugal John Tsinias National Technical University of Athens jtsincmath.ntua.gr Greece url: Nonlinear Control Network

6 Contents 1. Disturbance attenuation for discrete-time feedforward nonlinear systems 1 T. Ahmed-Ali, F. Mazenc and F. Lamnabhi-Lagamgue 1.1 Introduction Inverse optimal control Disturbance attenuation for discrete-time systems Feedforward discrete-time nonlinear systems 11 References Further results on decoupling with stability for Hamiltonian systems 19 A. Astolfi and L. Menini 2.1 Introduction and motivations Background on Hamiltonian systems Some geometric properties Decoupling with stability by dynamic state-feedback Proof of proposition Examples Conclusions 49 References Issues in modelling and control of mass balance systems.. 53 G. Bastin 3.1 Introduction Mass balance systems Positivity Conservation of mass Hamiltonian representation Compartmental representation Special case : inflow controlled systems Example 1 : a biochemical process Example 2 : a grinding process Bounded input - bounded state Systems without inflows 64

7 VIII Contents 3.12 A fundamental control problem Robust output feedback control of BIBS minimum phase systems Robust state feedback stabilisation of the total mass Concluding remarks 70 References Control of dynamic bifurcations 75 N. Berglund and K.R. Schneider 4.1 Introduction Dynamic bifurcations One-dimensional center manifold Two-dimensional center manifold 83 References Extension of Popov criterion to time-varying nonlinearities: LMI, frequential and graphical conditions 95 P.-A. Bliman 5.1 Introduction A LMI criterion for rational systems A frequency domain criterion and its graphical interpretation A weaker frequency domain criterion and its interpretation in Popov plane Extensions and remarks Computation of stability margin for a 3rd order rational system Extended sketch of the proofs 108 References Uniqueness of control sets for perturbations of linear systems 115 F. Colonius and M. Spadini 6.1 Introduction Problem formulation and examples Periodic solutions of linear systems and control sets Nonlinear perturbations 127 References Design of control Lyapunov functions for "Jurdjevic-Quinn" systems 137 L. Faubourg and J.-B. Pomet 7.1 Introduction Jurdjevic-Quinn systems Reshaping Lyapunov functions Example: The TORA system Conclusion 148

8 Contents IX References Bifurcation analysis of a power factor precompensator F. Gordillo, G. Escobar and J. Aracil 8.1 Introduction Problem formulation Dynamical harmonic balance Analysis of the approximated model Conclusions 163 References Stabilization by sampled and discrete feedback with positive sampling rate 165 L. Griine 9.1 Introduction Setup and definitions Stability concepts for sampled systems A counterexample to stabilizability with positive sampling ratel Homogeneous systems Examples 178 References Linear controllers for tracking chained-form systems 183 E. Lefeber, A. Robertsson and H. Nijmeijer 10.1 Introduction Preliminaries and problem formulation The state feedback problem An observer The output feedback problem Conclusions 197 References Asymptotic methods in stability analysis and control 201 L. Moreau and D. Aeyels 11.1 Introduction Preliminaries Practical stability Convergence results on an infinite time scale Homogeneous systems 210 References Robust point-stabilization of nonlinear affine control systems 215 P. Morin and C. Samson 12.1 Introduction 215

9 X Contents 12.2 Sufficient conditions for exponential and robust stabilization Control design Control laws for a dynamic extension 233 References Stabilization of port-controlled Hamiltonian systems via energy balancing 239 R. Ortega, A.J. van der Schaft and B.M. Maschke 13.1 Introduction Port controlled Hamiltonian systems Controller design procedure Stabilization via energy-balancing Casimir functions method Concluding remarks and future research 256 References Invariant tracking and stabilization: problem formulation and examples 261 P. Rouchon and J. Rudolph 14.1 Introduction Invariant tracking and stabilization The nonholonomic car A chemical reactor Conclusion 272 References Control of mechanical structures by piezoelectric actuators and sensors 275 K. Schlacher and A. Kugi 15.1 Introduction Some remarks on Lagrangian systems Control of Lagrangian systems The mechanical model Control of beams Plates Conclusion 291 References A novel impedance grasping strategy as a generalized hamiltonian system 293 S. Stramigioli 16.1 Introduction Background Controllable springs Physical controller structure 296

10 Contents XI 16.5 The controlled hand Implementation of the control scheme Grasping strategies Conclusions 323 References A nonsmooth hybrid maximum principle 325 H. Sussmann 17.1 Introduction Hybrid optimal control problems The general form of the maximum principle A version involving classical differentials Outline of the proof The case of a nondifferentiable reference vector field and Lagrangian 353 References A converse Lyapunov theorem for robust exponential stochastic stability 355 J. Tsinias and J. Spiliotis 18.1 Introduction Notions of exponential robust stability Main results Lyapunov description of expiss Remarks for deterministic systems 373 References LMIs for robust stable neural model-based control 375 B. Warns, M. A. Botto, T. van den Boom and J. Sd da Costa 19.1 Introduction Control setting Implementation with a feed-forward neural network Concluding remarks 386 References 386