Nonlinear and Adaptive Control: NCN4 2001 (Lecture Notes in Control and Information Sciences)

Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari 281 Springer Berlin Heidelberg NewYo...

Author: Alan S.I. Zinober | David H. Owens

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Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari

281

Springer Berlin Heidelberg NewYork Hong Kong London Milan Paris Tokyo

Alan Zinober, David Owens (Eds.)

Nonlinear and Adaptive Control NCN4 2001 With 88 Figures

13

Series Advisory Board A. Bensoussan · P. Fleming · M.J. Grimble · P. Kokotovic · A.B. Kurzhanski · H. Kwakernaak · J.N. Tsitsiklis

Editors Professor Alan Zinober The University of Sheffield Departement of AppliedMathematics Western Bank Sheffield S10 2TN UK Professor David Owens The University of Sheffield Dept. of Automatic Control and Systems Engineering Mappin Street Sheffield S1 3JD UK Library of Congress Cataloging-in-Publication Data Nonlinear and Adadtive Control (4th : 2001 : University of Sheffield) NCN4 2001 / Alan Zinober, David Owens (eds.). p. cm - (Lecture notes in control and information sciences ; 281) Contains 31 papers based on talk delivered at the 4th Workshop of the Nonlinear Control Network, held at the University of Sheffield, 25–28 June, 2001. ISBN 3-540-43240-X 1. Nonlinear control theory--Congresses. I. Zinober, A.S.I. (Alan S.I.) II. Owens, D. H. (David H.) III. Title. IV. Series. QA402.35 .W67 2001 629.8’36--dc21

ISBN 3-540-43240-X

2002026832

Springer-Verlag Berlin Heidelberg New York

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Preface

This volume contain thirty-one papers based on talks delivered at the 4th Workshop of the Nonlinear Control Network (http://www.shef/∼NCN4), held at The University of Sheﬃeld, 25–28 June, 2001. We would like to emphasize one peculiarity of our ﬁeld: nonlinear control is an example of a theory situated at a crossroad between mathematics and engineering science. Due to this position, nonlinear control has its roots in both ﬁelds and, as we deeply believe, can bring new ideas and new results for both domains. The book reﬂects very well this “double character” of nonlinear control theory: the reader will ﬁnd in it results which cover a wide variety of problems: starting from pure mathematics, through its applications to nonlinear feedback design, and all the way to recent industrial advances. The following delivered invited talks at the Workshop : Frank Allgower, Alessandro Astolﬁ, Riccardo Marino Francoise Lamnabhi-Lagarrigue, Wei Lin and Malcolm Smith Altogether the book contains 31 papers and therefore it is impossible to mention here all the discussed topics. We wish to thank all invited speakers and contributors to the 4th Nonlinear Control Network Workshop for making this conference an outstanding intellectual celebration of the area of Nonlinear Control. We would like to thank the TMR Program for the ﬁnancial support which, in particular, helped numerous young researchers in attending the Workshop. Many thanks to our local administrative helpers; Nirvana Bloor, Marianna Keray, Lynda Harrison, Ali Jafari Koshkouei, Russ Mills and Alex Price. Also, greatly appreciated thanks are extended to Ali Jafari Koshkouei for his very patient and attentive ﬁnal checking of the complete manuscript.

Alan Zinober Sheﬃeld, UK November 2001

David Owens

Contents

Invariant manifolds, asymptotic immersion and the (adaptive) stabilization of nonlinear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Alessandro Astolﬁ, Romeo Ortega 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Asymptotic stabilization via immersion and invariance . . . . . . . . . . . . 4 3 Robustiﬁcation vis ` a vis high order dynamics . . . . . . . . . . . . . . . . . . . . 6 4 Adaptive control via immersion and invariance . . . . . . . . . . . . . . . . . . . 9 5 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Robust control of a nonlinear ﬂuidics problem . . . . . . . . . . . . . . . . Lubimor Baramov, Owen Tutty, Eric Rogers 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Mathematical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Control design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Results and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal control of the Schr¨ odinger equation with two or three levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ugo Boscain, Gr´egoire Charlot, Jean-Paul Gauthier 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The two levels case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The resonant problem with three levels . . . . . . . . . . . . . . . . . . . . . . . . . 4 The counter intuitive solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strong robustness in adaptive control . . . . . . . . . . . . . . . . . . . . . . . . . Maria Cadic, Jan Willem Polderman 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 General setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Strong robustness and strongly robust stability radius . . . . . . . . . . . . 4 Existence of strongly robust sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Strong robustness and open balls of systems . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 22 28 29 31

33 33 35 37 41 42 42 45 45 46 47 49 50 54 54

VIII

Contents

On composition of Dirac structures and its implications for control by interconnection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joaqu´ın Cervera, Arjan J. van der Schaft, Alfonso Ban ˜ os 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Parametrization of kernel/image representations . . . . . . . . . . . . . . . . . 3 Dirac structures composition expression . . . . . . . . . . . . . . . . . . . . . . . . . 4 Application example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adaptive nonlinear excitation control of synchronous generators . . Gilney Damm, Riccardo Marino, Fran¸coise Lamnabhi-Lagarrigue 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Dynamical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Adaptive controller and main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameter identiﬁcation for nonlinear pneumatic cylinder actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N.A. Daw, J. Wang, Q.H. Wu, J. Chen, and Y. Zhao 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Mathematical model of pneumatic actuators . . . . . . . . . . . . . . . . . . . . . 3 Static friction measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Description of Genetic Algorithms for parameter identiﬁcation . . . . . 5 Experiment and simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The quasi-inﬁnite horizon approach to nonlinear model predictive control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rolf Findeisen, Frank Allg¨ ower 1 Introduction to nonlinear model predictive control . . . . . . . . . . . . . . . 2 Quasi-inﬁnite horizon NMPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Some results on output feedback NMPC . . . . . . . . . . . . . . . . . . . . . . . . 4 Some remarks on the eﬃcient numerical solution of NMPC . . . . . . . . 5 Application example–real-time feasibility of NMPC . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 56 58 61 62 63 65 65 66 69 74 75 76 77 77 78 81 82 83 86 87 89 89 94 98 102 103 104 105

State and parameter identiﬁcation for nonlinear uncertain systems using variable structure theory . . . . . . . . . . . . . . . . . . . . . . . 109 Fabienne Floret-Pontet, Fran¸coise Lamnabhi-Lagarrigue 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 2 State identiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Contents

3 Parameter identiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Simultaneous state and parameter identiﬁcation . . . . . . . . . . . . . . . . . 5 Simulations results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stabilizing memoryless controllers and controls with memory for a class of uncertain, bilinear systems with discrete and distributed delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.P. Goodall 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries and nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Stability results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sliding mode control of a full-bridge unity power factor rectiﬁer . Robert Grin ˜´ o, Enric Fossas, Domingo Biel 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Steady-state and zero-dynamics analysis . . . . . . . . . . . . . . . . . . . . . . . . 4 Control design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Parameter analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flatness - based control of the induction drive minimising energy dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Veit Hagenmeyer, Andrea Ranftl, Emmanuel Delaleau 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Model of the motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Control of the reduced order model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Tracking control of the complete model . . . . . . . . . . . . . . . . . . . . . . . . . 5 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IX

115 118 119 127 128

129 129 130 130 131 133 136 137 138 139 139 140 141 143 144 146 148 148 149 149 150 152 155 157 159 159

Exact feedforward linearisation based on diﬀerential ﬂatness: The SISO case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Veit Hagenmeyer, Emmanuel Delaleau 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 2 Exact feedforward linearisation based on diﬀerential ﬂatness . . . . . . . 162

X

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3 Exact feedforward linearisation and control law design . . . . . . . . . . . . 4 Structure of the error equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Extended PID control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

164 165 166 166 170

Local controlled dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efthimios Kappos 1 Control system basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Accessible sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Control-transverse geometric objects . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Variations of transverse objects and the Conley index . . . . . . . . . . . . . 5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171

Adaptive feedback passivity of nonlinear systems with sliding mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ali J. Koshkouei, Alan S. I. Zinober 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Passivation of nonlinear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Sliding mode of passive systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Sliding mode of linear systems with passivation . . . . . . . . . . . . . . . . . . 5 Adaptive passivation of nonlinear systems . . . . . . . . . . . . . . . . . . . . . . . 6 Example: Gravity-ﬂow/pipeline system . . . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear algebraic tools for discrete-time nonlinear control systems with Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ¨ Kotta, M. T˜ U. onso 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Linear algebraic framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Solutions of the control problems in terms of the Hk subspaces . . . . 4 Symbolic implementation using Mathematica . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adaptive compensation of multiple sinusoidal disturbances with unknown frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Riccardo Marino, Giovanni L. Santosuosso, Patrizio Tomei 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Adaptive compensation design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Extension to m sinusoidal disturbances . . . . . . . . . . . . . . . . . . . . . . . . . 5 An illustrative example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171 173 176 179 180 180 181 181 182 184 186 190 191 193 194 195 195 196 197 202 204 207 207 209 213 217 220 224 224

Contents

Backstepping algorithms for a class of disturbed nonlinear nontriangular systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Russell E. Mills, Ali J. Koshkouei, Alan S. I. Zinober 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Applicable class of systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 DAB algorithm for disturbed systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Combined sliding backstepping control of disturbed systems . . . . . . . 5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonminimum phase output tracking using sliding modes . . . . . Govert Monsees, Jacquelien Scherpen 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Method of dichotomies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Bounded preview time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Bounded preview time with sliding mode control . . . . . . . . . . . . . . . . . 5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Goursat distributions not strongly nilpotent in dimensions not exceeding seven . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Piotr Mormul 1 Introduction: geometric classes (basic geometry) of Goursat germs . 2 Kumpera–Ruiz algebras. Theorems of the paper . . . . . . . . . . . . . . . . . 3 Nilpotency orders of KR algebras – proof of Thm. 1 . . . . . . . . . . . . . . 4 Nilpotent approximations of Goursat germs often are not Goursat (proof of Theorem 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stabilizability for boundary-value control systems using symbolic calculus of pseudo-diﬀerential operators . . . . . . . . . . . . . Markku T. Nihtil¨ a, Jouko Tervo 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 State space equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 On transfer function analysis of coercive problems . . . . . . . . . . . . . . . . 4 Transfer function manipulations by symbols . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XI

227 227 228 228 234 236 237 237 239 239 241 244 245 246 247 248 249 249 251 254 258 260 263 263 265 268 269 272

Multi-periodic nonlinear repetitive control: Feedback stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 David H. Owens, Liangmin Li, Steve P. Banks 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 2 Lyapunov stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

XII

Contents

3 Asymptotical stability of multi-periodic repetitive control for a class of nonlinear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Exponential stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perturbed hybrid systems, applications in control theory . . . . . Christophe Prieur 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Perturbed hybrid systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Applications in control theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Global stabilization of nonlinear systems: A continuous feedback framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chunjiang Qian, Wei Lin 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Global strong stability in a continuous framework . . . . . . . . . . . . . . . . 3 Construction of non-Lipschitz continuous stabilizers . . . . . . . . . . . . . . 4 Global strong stabilization of cascade systems . . . . . . . . . . . . . . . . . . . 5 Control of an underactuated mechanical system . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On computation of the logarithm of the Chen-Fliess series for nonlinear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eug´enio M. Rocha 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Terminology and basic facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Sketch of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

278 280 280 282 283 285 285 286 292 294 295 295 297 301 308 310 313 317 317 318 320 323 326

On the eﬃcient computation of higher order maps adkf g(x) using Taylor arithmetic and the Campbell-Baker-Hausdorﬀ formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Klaus R¨ obenack 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 2 Automatic diﬀerentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 3 Series expansion of state-space systems and Lie derivatives . . . . . . . . 330 4 Tangent system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 5 Lie Brackets and the Cambell-Baker-Hausdorﬀ formula . . . . . . . . . . . 332 6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

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Nonlinear M.I.S.O. direct-inverse compensation for robust performance speed control of an S.I. engine . . . . . . . . . . . . . . . . . . 337 A.Thomas Shenton, Anthemios P. Petridis 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 2 Nonlinear engine model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 3 Direct-inverse linearisation and quasi-linearisation . . . . . . . . . . . . . . . . 341 4 The inverse compensators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 5 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 Exact delayed reconstructors in nonlinear discrete-time systems control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hebertt Sira-Ram´ırez, Pierre Rouchon 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 State reconstruction in nonlinear observable systems . . . . . . . . . . . . . . 3 The non-holonomic car system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalized PID control of the average boost converter circuit model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hebertt Sira-Ram´ırez, Gerardo Silva-Navarro 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 An average model of a boost converter . . . . . . . . . . . . . . . . . . . . . . . . . . 3 GPID regulation around a nominal trajectory . . . . . . . . . . . . . . . . . . . 4 Conclusions and suggestions for further research . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

351 351 352 356 359 360 361 361 362 364 369 369

Sliding mode observers for robust fault reconstruction in nonlinear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 Chee Pin Tan, Christopher Edwards 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 3 Robust reconstruction of actuator faults . . . . . . . . . . . . . . . . . . . . . . . . 376 4 Designing the observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 5 Robust reconstruction of sensor faults . . . . . . . . . . . . . . . . . . . . . . . . . . 379 6 An example: A nonlinear crane system . . . . . . . . . . . . . . . . . . . . . . . . . . 380 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 Real-time trajectory generation for ﬂat systems with constraints . . Johannes von L¨ owis and Joachim Rudolph 1 Introduction and problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Trajectory generation algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

385 385 387 388 389 392 394

Invariant manifolds, asymptotic immersion and the (adaptive) stabilization of nonlinear systems Alessandro Astolﬁ1,2 and Romeo Ortega3 1

2

3

Dip. di Elettronica e Informazione, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy Dept. of Electrical and Electronic Engineering, Imperial College, Exhibition Road, London SW7 2BT, England Laboratoire des Signaux et Syst´emes, Supelec, Plateau de Moulon, 91192 Gif-sur-Yvette, France

Abstract. A new method to design asymptotically stabilizing and adaptive control laws for nonlinear systems is presented. The method relies upon the notions of system immersion and manifold invariance and does not require the knowledge of a (control) Lyapunov function. The construction of the stabilizing control laws resembles the construction used in nonlinear regulator theory to derive the (invariant) output zeroing manifold and its friend. The method is well suited in situations where we know a stabilizing controller of a nominal reduced order model, which we would like to robustify with respect to high order dynamics. This is achieved by designing a control law that immerses the full system dynamics into the reduced order one. We also show that in this new framework the adaptive control problem can be formulated from a new perspective that, under some suitable structural assumptions, allows to modify the classical certainty equivalent controller and derive parameter update laws such that stabilization is achieved. It is interesting to note that our construction does not require a linear parameterization, furthermore, viewed from a Lyapunov perspective, it provides a procedure to add cross terms between the parameter estimates and the plant states. The method is illustrated with several practical examples.

1

Introduction

The problems of stabilization and adaptive control of nonlinear systems have been widely studied in the last years and several constructive or conceptually constructive methodologies have been proposed, see e.g. the monographs [14,24,10,16] for a summary of the state of the art. Most of the nonlinear stabilization methods rely on the use of (control) Lyapunov functions either in the synthesis of the controller or in the analysis of the closed loop system. For systems with Lagrangian or Hamiltonian structures Lyapunov functions are replaced by storage functions with passivity being the sought–after property [18]. Alternatively, the input–to–state stability point of view [25], the concept of nonlinear gain functions and the nonlinear version of the small A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 1-20, 2003. Springer-Verlag Berlin Heidelberg 2003

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gain theorem [27] have been used in the study of cascaded or interconnected systems. At the same time the local/global theory of output regulation has been developed and systematized [5]. This relies upon the solution of the so-called Francis-Byrnes-Isidori (FBI) equations: a set of partial diﬀerential equations (PDEs) that must be solved to compute a solution to the regulator problem. It must be noted that the solution of the nonlinear regulator problem requires ultimately the solution of a (output feedback) stabilization problem, whereas, to the best of the authors’ knowledge, the tools and concepts exploited in the theory of output regulation have not yet been used in the solution of standard stabilization problems. In the present work we take a new look at the nonlinear stabilization and adaptive control problems. More precisely we make use of two classical tools of nonlinear regulator theory and of geometric nonlinear control to design stabilizing and adaptive control laws for general nonlinear systems: (system) Immersion and (manifold) Invariance. For this reason we call the new methods I&I stabilization and I&I adaptive control. The concept of invariance has been widely used in control theory. The development of linear and nonlinear geometric control theory (see [29,17,8] for a comprehensive introduction) has shown that invariant subspaces, and their nonlinear equivalent, invariant distributions, play a fundamental role in the solution of many design problems. The notions of slow and fast invariant manifolds, which naturally appear in singularly perturbed systems, were used for stabilization [13] and analysis of slow adaptation systems [22]. Relatively recently, it has also been discovered that the notion of invariant manifolds is crucial in the design of stabilizing control laws for classes of nonlinear systems. More precisely, the theory of the center manifold [6] has been instrumental in the design of stabilizing control laws for systems with non-controllable linear approximation, see e.g. [1], whereas the notion of zero dynamics and the strongly related notion of zeroing manifold have been exploited in several local and global stabilization methods, including passivity based control, backstepping and forwarding. The notion of immersion has also a longstanding tradition in control theory. Its basic idea is to immerse the system under consideration into a system with prespeciﬁed properties. For example, the classical problem of immersion of a generic nonlinear system into a linear and controllable system by means of static or dynamics state feedback has been extensively studied, see [17,8] for further detail. State observation has traditionally being formulated in terms of system immersion, see [11] for a recent application. More recently, the notion of immersion has been used in the nonlinear regulator theory to derive necessary and suﬃcient conditions for robust regulation. In [5,9] it is shown that robust regulation is achievable provided that the exosystem can be immersed into a linear and observable system.

Immersion and invariance

3

Instrumental for the developments of this paper is to recast stabilization in terms of system immersion. Towards this end, consider the system x˙ = f (x, u) and the basic stabilization problem of ﬁnding (whenever possible) a state feedback control law u = u(x) such that the closed loop system is locally (globally) asymptotically stable. This is analogous to seek a control law u = u(x), a target dynamical system ξ˙ = α(ξ) which is locally (globally) asymptotically stable, and a mapping x = π(ξ) such that f (π(ξ), u(π(ξ))) =

∂π (ξ)α(ξ), ∂ξ

i.e. any trajectory x(t) of the closed loop system x˙ = f (x, u(x)) is the image through the mapping π(·) of a trajectory of the target system. Note that the mapping π:ξ→x is in general an immersion, i.e. the rank of π is equal to the dimension of ξ. We stress that Lyapunov based design methods are somewhat dual to the approach (informally) described above. As a matter of fact, in Lyapunov design one seeks a function V (x), which is positive deﬁnite (and proper, if global stability is required), and which is such that the autonomous system V˙ = α(V ) is locally (globally) asymptotically stable. Note that the function V : x → I, where I is an interval of the real axis, is a submersion and the target dynamics, namely the dynamics of the Lyapunov function, are one dimensional. From the above discussion it is obvious that the concept of immersion requires the selection of a target dynamical system. This is in general a nontrivial task, as the solvability of the underlying control design problem depends upon such a selection. For general nonlinear systems the classical target dynamics are linear, and a complete theory in this direction has been developed both for continuous and discrete time systems [17,8]. For physical systems the choice of linear target dynamics is not necessarily the most suitable one because, on one hand, workable designs should respect the constraints imposed by the physical structure. On the other hand, it is well– known that most physical systems are not feedback linearizable. Fortunately, in many cases of practical interest it is possible to identify a natural (not necessarily linear) target dynamics. For instance, for systems admitting a slow/fast decomposition—which usually appears in applications where actuator dynamics or bending modes must be taken into account—a physically reasonable selection for the target dynamics is the slow (rigid) subsystem, for which we assume known a stabilizing controller. In all these examples the application of the I&I method may be interpreted as a procedure to robustify, with respect to some high order dynamics, a controller derived from a low

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A. Astolﬁ and R. Ortega

order model. Other physical situations include unbalanced electrical systems where their regulated balanced representation is an obvious choice, whereas for AC drives the so–called ﬁeld oriented behavior is a natural selection for the target dynamics. I&I is also applicable in adaptive control, where a sensible target dynamics candidate is the closed–loop system that would result if we applied the known parameters controller. Clearly, in this case the target dynamics is only partially known but, as we show in the paper, under some suitable structural assumptions it is still possible to modify the classical certainty equivalent controller and derive parameter update laws such that adaptive stabilization is achieved. It is interesting to note that—in principle—our procedure does not require a linear parameterization. Also, viewed from a Lyapunov function perspective, I&I provides a procedure to add cross terms between the parameter estimates and the plant states. (It is worth mentioning that the unnatural linear parameterization assumption and our inability to generate non–separable Lyapunov functions have been the major Gordian knots that have stymied the practical application of adaptive control.) In this work, the I&I method is employed to design stabilizing and adaptive control laws for academic and physical examples in some of the situations described above. In Section 2 the general theory is presented, namely a set of suﬃcient conditions for the construction of local (global) stabilizing control laws for general nonlinear aﬃne systems. These results are then used in Section 3 to treat examples of actuator dynamics and ﬂexible modes. Section 4 presents the formulation—within the framework of I&I—of the adaptive control problem. Finally, Section 5 gives some summarizing remarks and suggestions for future work.

2

Asymptotic stabilization via immersion and invariance

The present section contains the basic theoretical results of the paper, namely a set of suﬃcient conditions for the construction of globally asymptotically stabilizing static state feedback control laws for general, control aﬃne, nonlinear systems. Note however, that similar considerations can be done for dynamic output feedback and nonaﬃne systems, while local versions follow mutatis mutandis. Theorem 1. Consider the system1 x˙ = f (x) + g(x)u, 1

(1)

Throughout the paper, if not otherwise stated, it is assumed that all functions and mappings are C ∞ . Note however, that all results can be derived under much weaker regularity assumptions.

Immersion and invariance

5

with state x ∈ IRn and control u ∈ IRm , with an equilibrium point x∗ ∈ IRn to be stabilized. Let p < n and assume we can ﬁnd mappings α(·) : IRp → IRp

π(·) : IRp → IRn

φ(·) : IRn → IRn−p

c(·) : IRp → IRm

ψ(·, ·) : IRn×(n−p) → IRm

such that: (H1) (Target system) The system ξ˙ = α(ξ),

(2)

with state ξ ∈ IRp , has a globally asymptotically stable equilibrium at ξ∗ ∈ IRp and x∗ = π(ξ∗ ). (H2) (Immersion condition) For all ξ ∈ IRp f (π(ξ)) + g(π(ξ))c(ξ) =

∂π α(ξ). ∂ξ

(3)

(H3) (Implicit manifold) The following set identity holds {x ∈ IRn | φ(x) = 0} = {x ∈ IRn | x = π(ξ), ξ ∈ IRp }.

(4)

(H4) (Manifold attractivity and trajectory boundedness) The (parameterized) system z˙ =

∂φ (f (x) + g(x)ψ(x, z)) ∂x

(5)

with state z, has a globally asymptotically stable equilibrium at zero uniformly in x. Further, the trajectories of the system x˙ = f (x) + g(x)ψ(x, z)

(6)

are bounded for all t ∈ [0, ∞). Then, x∗ is a globally asymptotically stable equilibrium of the closed loop system x˙ = f (x) + g(x)ψ(x, φ(x)). Remark 1. The result summarized in Theorem 1 lends itself to the following interpretation. Given the system (1) and the target dynamical system (2) ﬁnd, if possible, a manifold M, described implicitly by {x ∈ IRn | φ(x) = 0}, and in parameterized form by {x ∈ IRn | x = π(ξ), ξ ∈ IRp }, which can be rendered invariant and asymptotically stable, and such that the (well deﬁned) restriction of the closed loop system to M is described by ξ˙ = α(ξ). Notice, however, that we do not propose to apply the control u = c(ξ) that renders the manifold invariant, instead we design a control law u = ψ(x, z) that drives to zero the oﬀ–the–manifold coordinate z and keeps the system trajectories bounded.

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A. Astolﬁ and R. Ortega

Remark 2. It is obvious that the existence, and the properties, of the various mappings depends upon the selection of the target dynamics. As already observed, this selection is crucial and has to be done following physical and system theoretic considerations, as shown, via examples in the following sections. Remark 3. In Theorem 1 a stabilizing control law is derived starting from the selection of a target (asymptotically stable) dynamical system. A diﬀerent perspective can be taken: given the mapping x = π(ξ), hence the mapping z = φ(x), ﬁnd (if possible) a control law which renders the manifold z = 0 invariant and asymptotically stable, and a globally asymptotically stable vector ﬁeld ξ˙ = α(ξ) such that the FBI equation (3) holds. If this goal is achieved the system (1) with output z = φ(x) is (globally) minimum phase. It is therefore apparent that the result in Theorem 1 can be regarded as the dual of the classical stabilization methods based on the construction of passive or minimum phase outputs, see [4,18] and the survey paper [3]. We conclude this section with a deﬁnition, which will be used in the rest of the paper to provide concise statements. Deﬁnition 1. A system described by equations of the form (1) is said to be I&I-stabilizable with target dynamics ξ˙ = α(ξ) if the hypotheses (H1) to (H4) of Theorem 1 are satisﬁed.

3

Robustiﬁcation vis ` a vis high order dynamics

In this section we present two simple examples of application of I&I stabilization where we know a stabilizing controller of a nominal reduced order model, and we would like to robustify it with respect to some high order dynamics. First, we consider a levitated system with actuator dynamics, then we treat a robot manipulator with joint ﬂexibilities. Our objective with these simple exercises—that can be solved with many other existing techniques—is to illustrate the procedure. It should be noted, however, that in Section 4 we use the I&I technique to solve the long standing open problem of adaptive visual servoing. 3.1

Magnetic levitation system

Consider a magnetic levitation system consisting of an iron ball in a vertical magnetic ﬁeld created by a single electromagnet. We adopt the standard assumption of unsaturated ﬂux; that is, λ = L(θ)i, where λ is the ﬂux, θ is the diﬀerence between the position of the center of the ball and its nominal position, with the θ–axis oriented downward, i is the current, and

Immersion and invariance

7

L(θ) denotes the value of the inductance. The dynamics of the system is obtained by invoking Kirchhoﬀ’s voltage law and Newton’s second law as λ˙ + R2 i = w mθ¨ = F − mg where w is the voltage applied to the electromagnet, m is the mass of the ball, R2 is the coil resistance, and F is the force created by the electromagnet, which is given by F =

1 ∂L(θ) 2 i . 2 ∂θ

A suitable approximation for the inductance (in the domain −∞ < θ < 1) is k , where k is some positive constant that depends on the number L(θ) = 1−θ of coil turns, and we have normalized the nominal gap to one. If we assume that the manipulated variable is w, then the (equilibrium point corresponding to θ = θ∗ of the) aforementioned model of the levitated ball system can be asymptotically stabilized with nonlinear static state feedback derived using, for instance, feedback linearization [8], backstepping techniques [14] or interconnection and damping assignment control [19], see [28] for a comparative study. For the sake of completeness we give below the controller obtained using the latter approach 1 ˙ w = R2 i − Kp [ (1 − θ)i − 2kmg + θ − θ∗ ] − Kv θ. k where Kp , Kv are positive gains. In medium–to–high power applications the voltage w is generated using a rectiﬁer that includes a capacitance. The dynamics of this actuator can be described by the RC circuit shown in Fig 1, where the actual control voltage is u, while R1 and C model the parasitic resistance and capacitance, respectively. The full order model of the levitated ball system, including the actuator dynamics, is then given by  R2   x˙ 1 = − (1 − x3 )x1 + x2   k     1 1 1   (1 − x3 )x1 − x2 + u  x˙ 2 = − Ck R C R 1 1C (7) Σ:  1   x˙ 3 = x4   m       x˙ 4 = 1 x21 − mg 2k ˙ with VC where we have deﬁned the state variables as x = col(λ, VC , θ, mθ), the voltage across the capacitor. Instead of re-deriving a new controller, which could be a time consuming task, we might want to simply modify the one we already have to make it

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A. Astolﬁ and R. Ortega

R2

R1 u

C

L

Fig. 1. Electrical circuit of the levitated ball and the actuator.

robust with respect to the actuator dynamics. This is a typical scenario where I&I stabilization can provide a workable design. In particular, we have the following proposition. Proposition 1. The full order model of the levitated ball system (7) is I&I stabilizable (at a constant equilibrium corresponding to x3 = x3∗ ) with target dynamics2  R2   ξ˙1 = − (1 − ξ2 )ξ1 + w(ξ)   k    1 ˙ ΣT : ξ2 = ξ3 (8)  m     1 2   ξ˙3 = ξ − mg 2k 1 where w is a stabilizing state feedback for ΣT . Remark 4. The full order system Σ is a port–controlled Hamiltonian system, that is the class for which the interconnection and damping assignment technique has been developed [19], however, the inclusion of the actuator dynamics stymies the application of the method. On the other hand, we have seen above that the I&I approach trivially yields a solution. 3.2

Flexible joints robot

As a second example of robustiﬁcation we consider the problem of global regulation of the n–degrees of freedom ﬂexible joint robot model q1 + C(q1 , q˙1 )q˙1 + g(q1 ) + K(q1 − q2 ) = 0 D(q1 )¨ (9) Σ: J q¨2 + K(q2 − q1 ) = u 2

The target dynamics ΣT is the slow model of Σ taking as small parameter the parasitic time constant R1 C.

Immersion and invariance

9

where q1 , q2 ∈ IRn are the link and motor shaft angles, respectively, D = DT > 0 is the inertia matrix of the rigid arm, J is the constant diagonal inertia matrix of actuators, C is the matrix related to the Coriolis and centrifugal terms, g is the gravity vector of the rigid arm, K = diag{ki } > 0 is the joint stiﬀness matrix, and u is the n dimensional vector of torques3 . Proposition 2. The ﬂexible joint robot model (9) is globally I&I stabilizable (at a constant equilibrium corresponding to q1 = q1∗ ) with target dynamics ξ˙ = ξ2 ΣT : ˙1 (10) ξ2 = −D(ξ1 )−1 [C(ξ1 , ξ2 )ξ2 + g(ξ1 ) − w(ξ)] where w is a stabilizing state feedback for ΣT . Remark 5. A similar result can be derived for the case of global tracking. The only modiﬁcation is that, in this case, w will also be a function of t via the reference trajectory. It is interesting to note that the control law can be computed knowing only the ﬁrst and second derivatives of the link reference trajectory. This should be contrasted with other existing methods where the third and fourth order derivative are also needed—see, e.g. Section 3 of [18]. Remark 6. In [26] the composite control approach is used to derive approximate feedback linearizing asymptotically stabilizing controllers for the full– inertia model. In the case of block–diagonal inertia considered here the slow manifold equations can be exactly solved and the stabilization is global. It is interesting to note that, as in the previous example, the resulting control law is diﬀerent from the I&I stabilizer proposed here. Remark 7. The target dynamics ΣT is not the rigid model obtained from a singular perturbation reduction of the full model Σ with small parameters 1 ki . In the latter model the inertia matrix of ΣT is D + J, and not simply D as here, see [26]. Our motivation to choose this target dynamics is clear noting that, if we take the rigid model resulting from a singular perturbation reduction the solution of the FBI equations leads to π3 = ξ1 + K −1 (w − ξ˙2 ), signiﬁcantly complicating the subsequent analysis.

4

Adaptive control via immersion and invariance

In this section we show how the general I&I theory of Section 2 can be used to develop a novel framework for adaptive stabilization of nonlinear systems. A key step for our developments is to add to the classical certainty–equivalent control a new term that, together with the parameter update law, will be designed to satisfy the conditions of I&I. In particular, this new term will shape the manifold into which the adaptive system will be immersed. For the sake 3

See, e.g., [18] for further details on the model and a review of the recent literature.

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of clarity we will consider in detail the case of linear parameterizations, for which a systematic and constructive procedure is available. The nonlinearly parameterized case will only be illustrated with a prototype robotic vision problem, which has attracted the attention of several researchers in the last years, and proved unsolvable with existing adaptive techniques. 4.1

Problem formulation

We consider the problem of stabilization of systems of the form (1) under the following assumption. (H5) (Adaptive stabilizability) There exists a parameterized function Ψ (x, θ), where θ ∈ IRq , such that for some unknown θ∗ ∈ IRq the system x˙ = f∗ (x) := f (x) + g(x)Ψ (x, θ∗ )

(11)

has a globally asymptotically stable equilibrium at x = x∗ . The I&I adaptive control problem is then formulated as follows. Deﬁnition 2. The system (1) with assumption (H5) is said to be adaptively I&I stabilizable if the system x˙ = f (x) + g(x)Ψ (x, θˆ + β1 (x)) Σ: (12) ˙ ˆ θˆ = β2 (x, θ) ˆ and “controls” β1 and β2 , is I&I stabilizable with with extended state x, θ, target dynamics ΣT : ξ˙ = f∗ (ξ) := f (ξ) + g(ξ)Ψ (ξ, θ∗ )

(13)

From the ﬁrst equation in (12) we see that in the I&I approach we depart from the certainty–equivalent philosophy and do not apply directly the estimate coming from the update law in the controller. It is important to recall that, in general, f (x), and possibly g(x), will depend on θ∗ and are therefore only partially known. 4.2

Classical approach revisited

To put in perspective the I&I approach—underscoring its new features— it is convenient to recall ﬁrst the “classical” procedure to address the adaptive control problem4 . First, a linear parameterization of the control is invariably adopted. That is, we assume: 4

We do not consider here the case of systems with special structures, in particular, triangular forms for which a highly specialized machinery has been developed, see e.g. [14], [16]. Also, we leave aside the question of the restrictiveness of all the assumptions involved.

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11

(H6) (Linearly parameterized control) The function Ψ (x, θ) may be written as Ψ (x, θ) = Ψ0 (x) + Ψ1 (x)θ

(14)

for some known functions Ψ0 (x) and Ψ1 (x). Second, to ensure that a suitable parameter estimator is realizable, we assume the existence of a Lyapunov function V (x) for the equilibrium x∗ of the ideal system x˙ = f∗ (x) such that ∂V ∂x g(x) is independent of the parameters, hence computable5 . Then, invoking the standard procedure, we propose a certainty– equivalent adaptive control of the form u = Ψ0 (x) + Ψ1 (x)θˆ T ˙ θˆ = −Γ Ψ1T (x) ∂V ∂x g(x) where Γ = Γ T > 0 is the adaptation gain. Denoting the parameter error as θ˜ := θˆ − θ∗ , we can deﬁne the error dynamics as x˙ = f∗ (x) + g(x)Ψ1 (x)θ˜ T

∂V ˙ θ˜ = −Γ Ψ1T (x) g(x) ∂x

(15)

We then postulate the classical separate Lyapunov function candidate ˜ ˜ = V (x) + 1 θ˜T Γ −1 θ, W (x, θ) 2 whose derivative is given by ˙ = ∂V f∗ (x) ≤ 0. W ∂x If the Lyapunov function V is strict, i.e. ∂V ∂x f∗ (x) is a negative deﬁnite function of x, then LaSalle’s invariance principle allows to conclude global regulation to x∗ of the systems state. If V (x) is not strict then it is necessary to add a detectability assumption, namely that, along the trajectories of the overall system (15), the following implication is true ∂V f∗ [x(t)] ≡ 0 ⇒ lim x(t) = x∗ t→∞ ∂x

(16)

See [21] for further details, and a procedure to overcome this obstacle in some cases. For the purposes of comparison with the I&I approach the following remarks are in order. 5

Notice that, in general, V (x) will depend on θ∗ .

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• The analysis relies on the construction of the error equations (15), a concept which is central to the whole adaptive control theory, and appears here at two diﬀerent levels. First, in the decomposition of the x–dynamics into a stable portion and a perturbation due to the parameter mismatch. Second, ˜˙ a construction that is possible in the deﬁnition of the estimator in terms of θ, because the true parameter is assumed constant. In the I&I formulation we will also write the x–dynamics in an error equation form but with the ideal system perturbed by the oﬀ–the–manifold coordinate z. Our second error equation will be precisely the dynamics of z that will be driven to zero by rendering the manifold attractive. • The procedure is based on the cancellation of the perturbation term in the Lyapunov function derivative. It is well–known that this cancellation is the source of many of the robustness problems of classical adaptive control (essentially due to the generation of equilibrium spaces). In the I&I formulation we will not try to cancel the perturbation term coming from z, but as we have seen above, only make it asymptotically vanishing. (Notice that, as pointed out in Section 2, in the I&I approach we do not even need to construct a Lyapunov function. Another advantage of the I&I method is that the detectability assumption required in the classical approach to handle non–strict Lyapunov functions is, obviously, not needed.) • The linear parameterization assumption is critical in the classical procedure. (In some cases it may be replaced by an integrability or a convexity condition, but this is very hard to verify for more than one uncertain parameter.) As we will see below, in adaptive I&I we do not require linearity, it will of course help to provide more constructive results. 4.3

Linearly parameterized control

We will consider ﬁrst the case of linearly parameterized control, that is, we assume the control has the form (14). We will present two propositions, the ﬁrst one is similar in spirit to the derivations presented in Subsection 4.2, where we do not make any explicit assumptions on the dependence of the system with respect to the unknown parameters, but assume instead a condition similar to the “realizability” of ∂V ∂x g(x). In the second result we explicitly assume that the systems drift vector ﬁeld is linearly dependent on θ∗ . Proposition 3. Assume we can ﬁnd a function β1 : IRn → IRm such that: 1 (H7) (Realizability) ∂β ∂x (x)f∗ (x), with f∗ (x) deﬁned in (11), is independent of the unknown parameters. (H8) (Stability) All trajectories of the error system

x˙ = f∗ (x) + g(x)Ψ1 (x)z ∂β1 z˙ = g(x)Ψ1 (x)z ∂x are bounded and satisfy limt→∞ x(t) = x∗ .

(17) (18)

Immersion and invariance

13

Under these conditions, the system (1) with assumptions (H5), (H6) is adaptively I&I stabilizable Proof. Similarly to the nonlinear stabilization problem we have to verify the conditions (H1)–(H3) of Theorem 1, however, instead of (H4) we will directly prove that—under the conditions of the theorem—we have global convergence of x to x∗ , with all signals bounded. First, (H1) is automatically satisﬁed from (H5). Second, for the immersion condition (3) we are looking for mappings π(ξ) and c(ξ), with

x c1 (ξ) πx (ξ) c(ξ) = , = π(ξ) = πθ (ξ) c2 (ξ) θˆ where we have introduced the partitions for notational convenience, that solve the FBI equations ∂πx f∗ (ξ) = f (πx (ξ))+g(πx (ξ)){Ψ0 (πx (ξ))+Ψ1 (πx (ξ))[πθ (ξ)+c1 (πx (ξ))]} ∂ξ ∂πθ f∗ (ξ) = c2 (πx (ξ), πθ (ξ)). ∂ξ For any function c1 (ξ) a solution for these equations is clearly given by πx (ξ) = ξ, πθ (ξ) = θ∗ − c1 (ξ) and c2 (ξ) deﬁned by the last identity. Setting β1 (ξ) = c1 (ξ) we get the implicit manifold condition (4) as ˆ := θˆ − θ∗ + β1 (x) = 0. φ(x, θ)

(19)

Now, the x–subsystem, obtained from (12), (14) as x˙ = f (x) + g(x)Ψ0 (x) + g(x)Ψ1 (x)(θˆ + β1 (x)), can be written in terms of the oﬀ–the–manifold coordinates z = φ, yielding the ﬁrst error equation (17). On the other hand, the oﬀ–the–manifold dynamics (5) is obtained diﬀerentiating (19), taking into account (17), (12), and the fact that θ∗ is constant, as z˙ = β2 (x) +

∂β1 (f∗ (x) + g(x)Ψ1 (x)z). ∂x

Now, using assumption (H7) we can select the parameter update law as β2 (x) = −

∂β1 f∗ (x) ∂x

(20)

This yields the second error equation (18). The proof is completed with assumption (H8). Remark 8. The realizability assumption (H7) is strictly weaker than the strict matching assumption [14], which requires that the uncertain parameters enter in the image of g(x). It is clear that in this case, we can always ﬁnd Ψ1 (x) such that f∗ (x) is independent of the parameters. It is also less restrictive

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than the assumption of realizability of ∂V ∂x g(x) (of the classical approach) since, in contrast to the I&I approach where β1 (x) is chosen by the designer, in the former case V (x) is essentially ﬁxed by the functions Ψ0 (x) and Ψ1 (x). It is true, however, that besides (H7) β1 (x) should also ensure stability of the error system. Remark 9. To study the stability of the error system (17), (18) we can replace (H8) by the following assumptions. (H8’) (Stability) For all x ∈ IRn , z ∈ IRq we have

T ∂β1 T ∂β1 g(x)Ψ1 (x) + g(x)Ψ1 (x) z+ Q(x, z) = z ∂x ∂x ∂V (f∗ (x) + g(x)Ψ1 (x)z) ≤ 0, ∂x where V (x) is a Lyapunov function for the target system x˙ = f∗ (x). (H8”) (Convergence) Q(x, z) = 0 ⇒ x = x∗ . The proof is completed considering the Lyapunov function V +|z|2 , and using LaSalle’s invariance principle. The key requirement here is the attractivity of the manifold, which is ensured with the (restrictive) condition that the matrix in square brackets in Q is negative semideﬁnite (uniformly in x). However, exploiting the particular structure of the system we can choose β1 to stabilize the error system under far weaker conditions. Remark 10. The inclusion of the term β1 in the control was instrumental to achieve the error equation (17). We reiterate the fact that the stabilization mechanism does not rely on cancellation of the perturbation term g(x)Ψ1 (x)z in this equation. The additional degree of freedom β1 (x) shapes the manifold (as seen in (19)) and renders it attractive (as it follows from stability of (18)). Remark 11. ¿From a Lyapunov analysis perspective, the I&I procedure automatically includes cross terms between plant states and estimated parameters, as done in Remark 9 with the function V (x) + 12 |θˆ − θ∗ + β1 (x)|2 . 4.4

Linearly parameterized control and system

As pointed out in Remark 8, we can always ﬁnd β1 (x) such that the realizability assumption (H7) is satisﬁed, however, β1 (x) should also verify the requirement of stabilization of the error equation, that may be diﬃcult in some practical cases. To overcome this obstacle we need to assume more structure on the uncertain system. In the next proposition we remove the realizability assumption by explicitly assuming a linear dependence on θ∗ of the vector ﬁeld f (x).

Proposition 4. Assume that

Immersion and invariance

15

(H7’) (Linearity) The vector ﬁeld f (x) can be written in the form f (x) = f0 (x) + f1 (x)θ∗ for some known functions f0 (x) and f1 (x). (H8’) (Stability) We can ﬁnd a function β1 : IRn → IRm such that all trajectories of the (new) error system x˙ = f∗ (x) + g(x)Ψ1 (x)z ∂β1 f1 (x)z z˙ = − ∂x

(21) (22)

are bounded and satisfy limt→∞ x(t) = x∗ . Then, the system (1) with assumptions (H5), (H6) is adaptively I&I stabilizable. Proof. In view of the previous derivations, we only need to prove that with assumption (H7’) we can generate the second error equation (22). This follows immediately from the calculations ∂β1 f (x) + g(x)Ψ0 (x) + g(x)Ψ1 (x)(θˆ + β1 (x)) z˙ = β2 (x) + ∂x ∂β1 f0 (x) + f1 (x)θ∗ + g(x)Ψ0 (x) + g(x)Ψ1 (x)(θˆ + β1 (x)) = β2 (x) + ∂x and the choice of the adaptation law ∂β1 f0 (x) + g(x)Ψ0 (x) + (f1 (x) + g(x)Ψ1 (x))(θˆ + β1 (x)) β2 (x) = − ∂x which, recalling that z = θˆ − θ∗ + β1 (x), clearly yields (22).6 4.5

Examples

To illustrate the (rather non–standard) I&I adaptive design procedure let us consider three simple examples. Example 1. Consider the stabilization to zero of an unstable ﬁrst order linear system x˙ = θ∗ x + u, with θ∗ > 0. In this (matched) case we can ﬁx a target dynamics independent of θ∗ , for instance as ξ˙ = −ξ. This choice yields the adaptive I&I control law u = −x − x(θˆ + β1 (x)). Selecting the update law 1 as (20), that is β2 (x) = ∂β ∂x x, yields the error equations (17), (18), which in this case become x˙ = −x − xz ∂β1 xz z˙ = − ∂x 6

We should note that, in contrast with (20), in this construction the update law ˆ β2 (x) is an explicit function of θ.

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The problem then boils down to ﬁnding a function β1 that stabilizes this system. The function β1 (x) = γ2 x2 , with γ > 0, clearly does the job, and ˙ yields the “classical” estimator θˆ = γx2 . Notice however that the I&I control u = −x − x(θˆ +

γ 2 x ), 2

automatically incorporates a cubic term in x that will speed–up the convergence. A Lyapunov function for f∗ (x) such that conditions (H8’) and (H8”) of Remark 9 hold is V (x) = 12 log(1 + x2 ), which yields Q(x, z) =

−x2 [(1 + x2 )z 2 + z + 1] ≤ 0. 1 + x2

Other, more practical choices, are immediately suggested, for instance β1 (x) =

γ log(1 + x2 ), 2

gives a normalized estimator ˙ θˆ = γ

x2 , 1 + x2

a feature that is desirable to regularize the signals. Example 2. Let us study the problem of adaptive I&I stabilization to a non– zero position (x1∗ , 0) of the basic pendulum x˙ 1 = x2 x˙ 2 = −θ∗ sin(x1 ) − (x1 − x1∗ ) − x2 + u where we assume that a PD controller has already been applied. The immersion dynamics can be chosen again independent of θ∗ , say as f∗ (x) = [x2 , −(x1 − x1∗ ) − x2 ]T , hence the adaptive I&I controller becomes u = sin(x1 )(θˆ + β1 (x)). The error equations take the form

0 z x˙ = f∗ (x) + sin(x1 ) ∂β1 sin(x1 )z z˙ = − ∂x2 which immediately suggests β1 (x) = −x2 sin(x1 ). The ﬁnal control and adaptation laws are u = θˆ sin(x1 ) − x2 sin2 (x1 ) ˙ θˆ = − ((x1 − x1∗ ) + x2 ) sin(x1 ) − x22 cos(x1 ). We should point out that, even in this simple (exhaustively studied) examples, adaptive I&I has generated new control and adaptation laws, and given place to new non–separate Lyapunov functions.

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17

Example 3. In the recent paper [23] a controller for the well–known boost converter system was proposed. Although robustness with respect to uncertain load was established, experimental evidence has shown that the scheme is sensitive to variations in the source voltage. We will show here that adaptive I&I stabilization, in the form presented in Proposition 4, allows to overcome this problem. The model of the boost converter is given by 1 x2 u + θ∗ C 1 1 x2 + x1 u x˙ 2 = − RC L where x1 is the inductance ﬂux, x2 is the charge in the capacitor voltage, and u is the continuous control signal, which represents the slew rate of a PWM circuit controlling the switch position in the converter. The positive constants C, L, R, θ∗ are the capacitance, inductance, load resistance, and voltage source, respectively. In [23] it is shown that the controller u = Ψ1 (x2 )θ∗ , with xα 2 Ψ1 (x2 ) = C α+1 and 0 < α < 1, asymptotically stabilizes the equilibrium x2∗ x∗ . If we try to apply the approach suggested in Proposition 3 we can easily see that there does not exist a function β1 (x) verifying, both, the realizability and the stability conditions. However, the systems drift vector ﬁeld is linearly parameterized, and some simple calculations prove that the condition (H8’) of Proposition 4 is satisﬁed with β1 (x) = x1 . This leads to the following adaptive I&I stabilizer x˙ 1 = −

xα 2 (θˆ + x1 ) xα+1 2∗ α+1

x2 ˙ˆ (θˆ + x1 ). θ = − 1− x2∗

u=C

4.6

Adaptive visual servoing: A nonlinearly parameterized problem

It this subsection we illustrate with a visual servoing problem how adaptive I&I stabilization can be applied in the nonlinearly parameterized case. We consider the visual servoing of planar robot manipulators under a ﬁxed– camera conﬁguration with unknown orientation7 . The control goal is to place the robot end-eﬀector over a desired static target by using a vision system equipped with a ﬁxed camera to ‘see’ the robot end-eﬀector and target. Invoking standard time–scale separation arguments we assume an inner fast loop for the robot velocity control, and concentrate on the kinematic problem where we must generate the references for the robot velocities. The robot is then described by a simple integrator q˙ = v, where q ∈ IR2 are the 7

We refer the interested reader to [7,12,15,2] for further detail on this problem.

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joint displacements and v the applied joint torques. We model the action of the camera as a static mapping from the joint positions q to the position (in pixels) of the robot tip in the image output denoted x ∈ IR2 . This mapping is described by [12] x = aeJθ∗ [k(q) − ϑ1 ] + ϑ2

(23)

where θ∗ is the orientation of the camera with respect to the robot frame, a > amin > 0 and ϑ1 , ϑ2 denote intrinsic camera parameters (scale factors, focal length and center oﬀset, respectively), k : IR2 → IR2 stands for the robot direct kinematics, and

0 −1 J= . 1 0 The direct kinematics yields k˙ = J (q)q, ˙ where J ∈ IR2×2 is the analytic robot Jacobian, which we assume nonsingular. Diﬀerentiating (23), and replacing the latter expression, we obtain the dynamic model of the overall system of interest x˙ = aeJθ∗ u,

(24)

where we have introduced the input change of coordinates u := J (q)v. The problem is then to ﬁnd u such that x → x∗ in spite of the lack of knowledge of a and θ∗ . The task is, of course, complicated by the highly nonlinear dependence on the unknown parameters, in particular θ∗ . To the best of our knowledge all existing results require some form of over–parameterization and/or persistence of excitation assumptions, which as is well–known, significantly degrades performance. Using adaptive I&I stabilization it is possible to prove the following result, see [2] for further detail. Proposition 5. Assume a strict lower bound amin on a is known. Then, the system (24) is adaptively I&I stabilizable with target dynamics ξ˙ = −aξ +ax∗ .

5

Conclusions and outlook

The stabilization problem for general nonlinear systems has been addressed from a new perspective. It has been shown that the classical notions of invariance and immersion, together with tools from the nonlinear regulator theory, can be used to design globally stabilizing control laws for general nonlinear systems. The proposed approach is well suitable in applications where we can deﬁne a—structurally compatible—“desired” reduced order dynamics. We have explored the applicability of the method to adaptive stabilization, even in the case of nonlinearly parameterized systems where classical adaptive control is severely limited. It is clear from the construction of the

Immersion and invariance

19

adaptive I&I control laws that, besides the classical “integral action” of the parameter estimator, through the action of β1 we have introduced in the control law a “proportional” term. In fact, in some cases, I&I adaptation reduces to classical certainty equivalent control with an estimation law consisting of integral and proportional terms, that in early references was called PI adaptation. Although it was widely recognized that PI update laws were superior to purely integral adaptation, except for the notable exception of output error identiﬁcation, their performance improvement was never clearly established, see e.g. [20] for a tutorial account of these developments. The I&I framework contributes then to put PI adaptation in its proper perspective.

References 1. D. Aeyels. Stabilization of a class of nonlinear systems by smooth feedback control. Systems and Control Letters, 5:289–294, 1985. 2. A. Astolﬁ, L. Hsu, M. Netto, and R. Ortega. A solution to the adaptive visual servoing problem. In Conference on Robotics and Automation, Seoul, Korea, pages 743–748, 2001. 3. A. Astolﬁ, R. Ortega, and R. Sepulchre. Stabilization and disturbance attenuation of nonlinear systems using dissipativity theory. In Control of Complex Systems, K. Astrom, P. Albertos, M. Blanke, A. Isidori, W. Schaufelberger, R. Sanz Eds., pages 39–75. Springer Verlag, 2000. 4. C. I. Byrnes, A. Isidori, and J. C. Willems. Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems. IEEE Trans. Autom. Control, 36:1228–1240, November 1991. 5. C.I. Byrnes, F. Delli Priscoli, and A. Isidori. Output regulation of uncertain nonlinear systems. Birkhauser, 1997. 6. J. Carr. Applications of center manifold theory. Springer Verlag, 1998. 7. S. Hutchinson, G.D. Hager, and P.I. Corke. A tutorial on visual servo control. IEEE Trans. on Robotics and Automation, 12:651–670, October 1996. 8. A. Isidori. Nonlinear Control Systems, Third Edition. Springer Verlag, 1995. 9. A. Isidori. Remark on the problem of semiglobal nonlinear output regulation. IEEE Trans. Autom. Control, 42:1734–1738, December 1997. 10. A. Isidori. Nonlinear Control Systems II. Springer Verlag, 1999. 11. N. Kazantzis and C. Kravaris. Nonlinear observer design using lyapunov’s auxiliary theorem. Systems and Control Letters, 34:241–247, 1998. 12. R. Kelly. Robust asymptotically stable visual servoing of planar robots. IEEE Trans. on Robotics and Automation, 12:759–766, October 1996. 13. P.V. Kokotovic. Recent trends in feedback design: an overview. Automatica, pages 225–236, 1985. 14. M. Krstic, I. Kanellakopoulos, and P.K. Kokotovic. Nonlinear and Adaptive Control Design. John Wiley and Sons, 1995. 15. E. Lefebre, R. Kelly, R. Ortega, and H. Nijmeijer. On adaptive calibration for visual servoing. In IFAC Symp Nonlinear Control Systems Design, Enschede, NL, pages 1–3, 1998. 16. R. Marino and P. Tomei. Nonlinear control design. Geometric, adaptive and robust. Prentice Hall, 1995.

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17. H. Nijmeijer and A. J. Van der Schaft. Nonlinear Dynamical Control Systems. Springer Verlag, 1989. 18. R. Ortega, A. Loria, P.J. Nicklasson, and H. Sira-Ramirez. Passivity based control of Euler-Lagrange systems. Springer Verlag, 1998. 19. R. Ortega, A. van der Schaft, I. Mareels, and B. Maschke. Putting energy back in control. IEEE Control Syst. Magazine, to appear. 20. R. Ortega and T. Yu. Robustness of adaptive controllers: A survey. Automatica (Survey paper), 25(5):651–677, Sept. 1989. 21. E. Panteley, R. Ortega, and P. Moya. Overcoming the obstacle of detectability in certainty equivalent adaptive control. Int. report, Laboratoire des Signaux et Syst´emes, 2000. 22. B. Riedle and P.V. Kokotovic. Integral manifolds of slow adaptation. IEEE Trans. Autom. Control, pages 316–324, 1986. 23. H. Rodriguez, R. Ortega, G. Escobar, and N. Barabanov. A robustly stable output feedback saturated controller for the boost DC-to-DC converter. Systems and Control Letters, 40:1–8, 2000. 24. R. Sepulchre, M. Jankovic, and P.K. Kokotovic. Constructive Nonlinear Control. Springer Verlag, 1996. 25. E.D. Sontag. On characterizations of the input-to-state stability property. Systems and Control Letters, 24:351–359, 1995. 26. M. Spong, K. Khorasani, and P.V. Kokotovic. An integral manifold approach to the feedback control of ﬂexible joint robots. IEEE J Robotics and Automation, pages 291–300, 1987. 27. A. R. Teel. A nonlinear small gain theorem for the analysis of control systems with saturation. IEEE Trans. Autom. Control, 41:1256–1270, September 1996. 28. M. Torres and R. Ortega. Feedback linearization, integrator backstepping and passivity–based controller design: A comparison example. In Perspective in Control: Theory and Application, D. Normand–Cyrot Editor. Springer Verlag, London, 1998. 29. W. M. Wonham. Linear Multivariable Control. A geometric approach. Springer Verlag, 1985.

Robust control of a nonlinear ﬂuidics problem Lubimor Baramov1 , Owen Tutty2 , and Eric Rogers3 1

2

3

Honeywell Prague Laboratory, Prague, Czech Republic, Email [email protected] School of Engineering Sciences, University of Southampton, Southampton, UK, Email: [email protected] Department of Electronics and Computer Science, University of Southampton, UK, Email: [email protected]

Abstract. This chapter deals with ﬁnite-dimensional boundary control of the linearized 2D ﬂow between two inﬁnite parallel planes. Surface transpiration along a few regularly spaced sections of the bottom wall is used to control the ﬂow. Measurements from several discrete, suitably placed shear-stress sensors provide the feedback. Unlike other studies in this area, the ﬂow is not assumed to be periodic, and spatially growing ﬂows are considered. An H∞ control scheme is designed to guarantee stability for the model set and to reduce the wall-shear stress at the channel wall. This design has been tested by simulations with a nonlinear Navier-Stokes PDE solver.

1

Introduction

Recently ﬂow control has attracted considerable attention in the ﬂuids research community. The systems dealt with in these problems are, in control terms, very complex, nonlinear and inﬁnite dimensional, even if the ﬂuid ﬂow is comparatively simple. Plane Poiseuille ﬂow, i.e. ﬂow between two inﬁnite parallel plates, is one of the simplest and best understood cases of ﬂuid dynamics. Controlling this ﬂow is, however, still a challenging problem, even if it is assumed that deviations from the steady-state are small enough for the governing equations to be linearized. It has become a benchmark problem for developing control algorithms for ﬂuid ﬂows and was considered in [1–3] among others. All these references make the assumption that the ﬂow is spatially periodic in the streamwise direction, and hence the Fourier-Galerkin decomposition can be used to obtain independent dynamics for each respective Fourier mode. In this chapter we shall not assume periodicity and we shall deal with control of a spatially growing ﬂow. This type of ﬂow is of considerable practical importance. For example, standard transition prediction methods are based on spatially growing disturbances. Control of such ﬂows has not been considered before, to the best of our knowledge. We consider again boundary control in the form of blowing/suction panels and discrete-points measurements. Speciﬁcally, we shall use four regularly spaced blowing/suction panels A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 21-31, 2003. Springer-Verlag Berlin Heidelberg 2003

22

L. Baramov, O. Tutty, and E. Rogers

and ﬁve sensors—which is suﬃcient to reduce the wall-shear stress signiﬁcantly from the actuation/sensing area downstream. In this setting we cannot apply streamwise Fourier decomposition and use ﬁnite diﬀerences instead. By a technique based on the Redheﬀer star product we obtain (pointwise) frequency-response data of the ﬂow. A low-order model is then ﬁtted on these data. A modeling uncertainty is estimated using the frequency-domain model validation ideas of [4]. Finally, a robustly stable H∞ controller was designed for this model set to reduce spatially growing components of the ﬂow ﬁeld and the control design tested in simulations with a nonlinear Navier-Stokes solver in the loop.

2

Mathematical models

We consider a planar ﬂow in an inﬁnite channel of ﬁxed height. The ﬂow is non-dimensionalised using the channel half-height and the center-line velocity. We consider a coordinate system as in Fig 1. Let p(x, y, t) be the pressure and, u(x, y, t) and v(x, y, t) be the velocities in the direction of x and y axes, respectively. The steady base ﬂow is given by p = −2x/Re, u = 1 − y 2 =: U (y) and v = 0 where Re is the Reynolds number. Assume that the quantities pˆ(x, y, t) ≡ p+2x/Re, u ˆ(x, y, t) ≡ u−U , and vˆ(x, y, t) ≡ v are small so ﬂow is governed by the linearized Navier-Stokes equations. The boundary conditions are u ˆ (x, 1, t) = vˆ(x, 1, t) = 0, u ˆ(x, −1, t) = 0 and vˆ(x, −1, t) = −q(t)(dl(x)/dx). The last condition describes the wall-normal blowing/suction. The function l(x) represents the geometric conﬁguration of the blowing and suction elements. The function q(t) modiﬁes the blowing/suction according to the control law–it is the normalized suction velocity through the wall. In the case of several independent actuators we may consider q to be an m-dimensional column vector and l an (1, m)-dimensional matrix.

y 1 -1 Fig. 1. Co-ordinate System

x

Robust control of a nonlinear ﬂuidics problem

23

To write the Navier-Stokes equations into the form used in this study, we use the so-called modiﬁed stream function Φ(x, y, t) as in [1] and write ∂Φ ∂Φ ∂l ∂f ,− (1) (ˆ u, vˆ) = + q l ,− f ∂y ∂x ∂y ∂x where f (y) is any smooth function which satisﬁes f (−1) = 1 and f (1) = ∂f (y) = 0. The linearized Navier-Stokes equations now become ∂y y=−1,1

1 4 ∂ 2 ∂ 2 d2 U ∂Φ ∇ Φ = −U ∇ Φ+ + ∇ Φ − q∆(f ˙ l) ∂t ∂x dy 2 ∂x Re ∂ 2 1 4 d2 U dl −q U ∇ (f l) − − ∇ (f l) f ∂x dy 2 dx Re

(2)

with boundary conditions Φ = ∂Φ/∂y = 0 for y = ±1. As an output we use the streamwise shear component at a point xn on the lower wall given by 2 ∂ Φ ∂ 2 f +q 2l (3) zn = ∂y 2 ∂y y=−1,x=xn To obtain a ﬁnite-dimensional approximation of the above description we use, as in [1,3], a Chebyshev expansion in y. As we do not impose periodicity as in [2,1,3], we use ﬁnite-diﬀerences in x. The x-coordinate is discretized by regularly spaced samples {xn , n = . . . , −1, 0, 1, . . .} and set δ = xn − xn−1 . The function Φ is represented, at these points, as Φ(xn , y, t) ≈

M +4

ξnm (t)Γm (y)

n = . . . , −1, 0, 1, . . .

(4)

m=0

where Γm (y) are Chebyshev polynomials. In order to enforce the boundary conditions, we must have that M +4

ξnm Γm ,

m=0

∂Γm ∂y

y=±1

= 0, n = . . . , −1, 0, 1, . . .

(5)

and the functions ξnM +1 , .., ξnM +4 can be expressed in terms of ξn0 , .., ξnM . The partial derivatives ∂ k /∂xk k = 1, . . . , 4 are approximated by standard second order central diﬀerence formulae. A Galerkin procedure is now used to produce for the n-th grid point a set of 2(M + 1) ordinary ﬁrst order equations which can be written as x˙ n = Axn + B1n q + B2n q˙ + B−2 xn−2 + B−1 xn−1 ˆ−1 x˙ n−1 + B+1 xn+1 + B ˆ+1 x˙ n+1 + B+2 xn+2 +B

where xn = ξn0 · · · ξnM

T

.

(6)

24

L. Baramov, O. Tutty, and E. Rogers

The output approximating shear at the n-th gridpoint becomes zn =

N

Cxn + Dn q

(7)

n=−N

To condense the notation, we combine the equations for a set of 5 grid points as n n x˙ sn = As xsn + B1n vib + B2 vif

where 

xsn





xn  ..  =  . , xn+4

(8)

n vib

 xn−2  xn−1     =  x˙ n−1  ,  q  q˙

 xn+6 =  xn+5  x˙ n+5 

n vif

(9)

Here the subscripts ib and if denote input-back and input-forward, respectively. The output feedback and feedforward vectors for this system are denoted by     xn+1 xn+3  xn   xn+4      n n    x ˙ vof =  (10) vob =  n  ,  x˙ n+4   q   q  q˙ q˙ These equations approximate the behavior of a channel segment over 5 grid points, i.e. of length 4δ. We can conveniently form the transfer function matrix n n Gn (s) mapping Laplace transform images of vib , vif to the Laplace images n n of vob , vof . To get a model of two adjacent segments of total length 9δ we use a feedback interconnection of Gn (s) and Gn+5 (s) through a Redheﬀer Star Product, (see e.g. [5]) and denoted by Gn ∗ Gn+5 . Then, a long stretch of the channel ﬂow is obtained as a chain of star products, Gn−5N1 (s) ∗ . . . ∗ Gn (s) ∗ Gn+5 (s) ∗ . . . ∗ Gn+5N2 (s)

(11)

Theoretically, a state-space representation of a channel could be constructed this way as well, but this not necessary here. Instead, we shall store only frequency domain data, so the above cascade of star products is computed pointwise, for a ﬁnite set of {jωi , i = 1, . . . K}, where the star-product over complex matrices is a simple operation. Low-order transfer functions from the boundary input to the shear output will be ﬁtted on this frequency-domain data. The boundary conditions at the up and down-stream ends of the chann−5N1 n−5N1 to vib nel can be described as transfer function matrices from vob

Robust control of a nonlinear ﬂuidics problem

25

n+5N2 n+5N1 and vof to vif , respectively. In our case, we shall leave them zero and consider a long enough channel so that the conditions at its ends have an negligible eﬀect on the relevant mid-section. Note here that if we conn−5N1 n+5N2 n+5N1 n−5N1 with vof and vif with vob we would obtain nected vib the periodic case, studied in [1–3]. We assume that the dynamics of the actuator is described by

x˙ p = Ap xp + Bp u, q = Cp xp

(12)

where u is the control input, and here we take Ap = −1, Bp = 100 and Cp = 1. We take Re = 104 . For the controlled boundary condition, we take dl(x)/dx as a rectangular pulse of width 0.5π and height 1. Note that we must avoid the ends of the panels when placing shear sensors because of the singularities in the solutions at these points. The grid size was chosen as δ = π/50 which produces frequency output data nearly identical to those of δ = π/100 and π/200, except for the segments containing singularities. The scaling involving the π factor is due to the fact that one of the fundamental wavenumbers has length 2π, and the Chebyshev order is taken as M = 40. An analysis in [3] (where the periodic case was treated) shows that this is suﬃciently accurate for frequency range up to ω ≈ 1. The control design developed below is within this range.

Fig. 2. Magnitude plot for shear along the channel ﬂow. Dashed line shows the position of the blowing/suction panel.

26

L. Baramov, O. Tutty, and E. Rogers

The transfer function matrix from the pump input u to the wall-shear stress is denoted by Guz (s, x) where s is the Laplace transform variable and x the coordinate. Its frequency domain representation Guz (jω, x) is obtained by a procedure based on a string of star products discussed above. The data available are for x ∈ [0, 100π], sampled at intervals ﬁve-times the grid length. To eliminate the eﬀect of up- and down-stream boundary conditions, we added another 100π of channel to each end. The blowing/suction takes place at x ∈ [5.5, 6]π (i.e. dl(x)/dx = 1 on this interval and zero elsewhere). A plot of shear magnitudes |Guz (jω, x)| is given in Fig 2. In accordance with intuition, the actuator has very little aﬀect on the upstream but that on down-stream shear is signiﬁcant. The control-to-shear gain grows in a narrow band ω ≈ [0.17, 0.27]. The maximum growth rate is at ω ≈ 0.22, which has growth of approximately 1.088 every 2π. These results are as expected from the underlying physics of the ﬂow. Elsewhere, the gain is (for large x) nearly constant (high frequencies) or slowly decreasing (low frequencies). For positions close to the panels, the shear output is very sensitive to x which means that it is very important to place the sensors accurately. Up to now we have been discussing the single-input-single-output system; however, with no loss of generality, we can obtain the multivariable system frequency-domain data directly from the scalar case. Consider a set of k sensors regularly spaced in intervals of δs and a set of m panels with a distance δp between their centers, we can get the transfer function matrix Guz (s) = [Guz (s, x + (i − 1)δs + (1 − j)δp )]i=1,...,k; j=1,...m where the scalar function Guz (s, x) is as above and x is now ﬁxed. In what follows we shall consider δs = δp = π, k = m = 4. This separation between panels/sensors is chosen so that the controller can most eﬃciently attack the disturbance of 2π wavelength. The constant x was chosen as x = 6.2π which means that sensors are placed at a distance of 0.2π downstream from the panels. Each of the above transfer function matrices is chosen as an Nij -order proper rational function. Generally, the further is the distance between the i-th sensor and j-th panel, the more complex this function is. To obtain an approximation of the i-th row we can use weighted least squares ﬁtting. The frequency-dependent weight was chosen as |(30jω + 1)/(2.5jω + 1)2 | which emphasizes the frequency range of [0.03, .4] which contains the range of spatial growth. The orders were chosen as 6,8,12,12 for the respective rows of Guz which ﬁts the data fairly accurately. The transfer functions were converted to state-space representations which were used later in the control design. During this procedure we have to ensure that the resulting models are stable, in accordance with the behaviour of the real system (in this setting the ﬂow trajectories grow spatially but not temporaly for ﬁxed x). If the approximation order is unnecessarily high, the procedure may result in models with nearly cancelled unstable zero-pole pairs. The uncertainty representation we use is the left-coprime factor uncertainty (see e.g. [5]). Let the low-order representation of the ﬂow obtained

Robust control of a nonlinear ﬂuidics problem

27

ˆ uz (s), and its left-coprime facfrom the ﬁtting procedure be denoted by G −1 ˆ torization by Guz (s) = M (s) N (s). One possible choice is the normalized left-coprime factorization and here we consider the model set {(M + W1 ∆1 W2 )−1 (N + W1 ∆2 W3 ) | ∆1 , ∆2 stable, [∆1 , ∆2 ]∞ < 1}

(13)

where W1 , W 2, W3 are stable and minimum phase weighting matrices. First, we shall consider the following problem: Given ﬁxed weights and set of frequency points {ωi }ki=1 , are the data Guz (jωi ), i = 1, . . . K consistent with the model (13)? As is well known, even if the model is consistent, i.e. there is a feasible uncertainty such that −1

Guz (jωi ) = (M (jωi ) + W1 ∆1 W2 (jωi )) (N (jωi ) + W1 ∆2 W3 (jωi )) , i = 1, . . . K

(14)

it does not imply that the model (13) is valid—no ﬁnite set of data can validate a model. We can only suppose that if the model is not invalidated for a suﬃciently large and carefully chosen data set, it is very likely valid, and this problem is (rather inaccurately) termed “model validation”. It is, in a more general setting, and under the presence of measurement noise, solved in [4]. The main result is that checking consistency of a frequency-domain noise corrupted data set with a general class of uncertain models requires the solution of a set of K independent linear matrix inequalities (LMI’s). In the following, for simplicity, we restrict attention to diagonal weights; moreover, W1 will be constant. For our speciﬁc problem we take W2 to be a band-pass ﬁlter whose frequency range will contain the interval of spatial growth, approx 0.17–0.27. Conversely, W3 is a band-stop ﬁlter over these frequencies. The order of the latter ﬁlter as well as its maximum gain/minimum gain ratio is higher than those for W2 . This is because we a priori assume that the low-order model is more accurate in the growth interval than elsewhere (which was enforced by the weighting in the least-square ﬁtting). In this setting, H∞ control syntheses will yield control restricted to this narrow frequency range. This is not a serious restriction—elsewhere the disturbances are not spatially growing and hence relatively harmless. The frequency set consists of 320 unevenly distributed points and 240 of them are concentrated between 0.1 and 1. To get tighter bounds, we can apply a minimization procedure based on W1 (s) = W1 (s)Q1 , W2 (s) = Q2 W2 (s) and W3 (s) = Q3 W3 (s), and invoke the results of [4]. Here, W1 , W2 , W3 are weights for which the model (13) is not invalidated and Q1 , Q2 and Q3 are constant diagonal matrices. For control system design we need a disturbance model. Although its accuracy does not aﬀect the closed-loop stability (and hence its error is not included to the overall modeling uncertainty) it is important for performance. First we consider this problem: If we measure the shear disturbance at a point x, can we estimate the value of shear at x+∆x at the same time? The answer

28

L. Baramov, O. Tutty, and E. Rogers

is yes, provided it is far downstream from its source and ∆x is close to the basic wavelengths for the fast growing components (i.e. approx 2π) (or their integer multiples). Let F2π (s, x) =

Guz (s, x + 2π) Guz (s, x)

(15)

Then from a Bode plot of (15) we ﬁt a stable 4th-order transfer function Fˆ2π (s) to frequency-domain data in this region. Assuming that zi is shear disturbance measured at x = xi , Fˆ2π (s)zi is our estimate for shear at xi + 2π. Time-domain simulations showed that this simple model estimates the advance of the waves remarkably well, regardless of the base value of x. It follows from the sensor conﬁguration proposed in the introduction here that we need to estimate the disturbance also at x + π. This is not possible with the same accuracy as in the previous case due to the fact that the stronger way than (15). We magnitude of Fπ (jω, x) depends on x in a much ﬁt a low-order rational function Fˆπ (s) on − Fˆ2π (s) to obtain at least a good estimate for the phase. For a more eﬃcient disturbance attenuation we put one additional sensor 4π upstream from the ﬁrst of the four sensors already used. This is far enough upstream from all panels so that the eﬀect of wall transpiration on this sensor’s measurements is negligible. It measures directly the oncoming ﬂow disturbance (assuming the measurement noise is negligible in the relevant frequency range). The model of Fˆ4π (s) is computed in the same way as that for Fˆ2π (s). The overall disturbance transfer matrix from the shear measured far upstream to disturbance components at the other sensors is then given by F (s) = Fˆ4π (s) 1 Fˆπ (s)

3

Fˆ2π (s)

Fˆ2π (s)Fˆπ (s)

T (16)

Control design

The control conﬁguration in the H∞ control setting is shown in Fig 3. There are two sets of external inputs—w1 disturbs the coprime-factor model, and w2 which generates the ﬂow disturbance measured at the upstream sensor. This signal is then ﬁltered through W4 (s), which passes only frequencies in the growth region, and then enters the disturbance model F (s) discussed in the previous section. After this, it is fed to the controller as measurement y1 . The measurements from the other sensors are in the other measurement y2 . The penalized outputs z1 , z2 are ﬁltered shear measurements and controls, respectively. The weights W1 , W2 , W3 were discussed in the previous section and we chose W1 (s) = Q1 , W2 (s) = Q2 w2 (s), W3 (s) = Q3 w3 (s), where Q1 , Q2 , Q3 are constant diagonal scaling matrices. Speciﬁcally, w2 (s) = {2.8 · 104 (s + 2.135)(s + 0.0248)(s2 + 2.136s+

Robust control of a nonlinear ﬂuidics problem

w2 W4 w1

29

F

W1

M -1

N

W2

z1 z2

W3

u

y1

y2

K Fig. 3. Control Conﬁguration.

4.7197)(s2 + 0.0239s + 6 · 10−4 )}/{(s2 + 0.2847s + 0.2) (s2 + 0.36s + 0.0529)(s2 + 0.0753s + 0.014)}, w3 (s) = 5.12 · 105 W4 (s)−3 , Q1 = 10−3 diag([2.2, 3.1, 3.2, 2.4]), Q2 = diag([2.4, 2, .4, 2, 3.27]), Q3 = diag([1.95, 2.5, 1.72, 1.44]) Finally, W4 (s) =

2(s2 + 1.349s + 1.007)(s2 + 0.0649s + 0.0023) . (s2 + 0.0382s + .0569)(s2 + 0.0325s + 0.0412)

For this weight combination, the model set (13) is consistent with our extensive data set. To guarantee robust stability for this set of plants it is suﬃcient for the H∞ performance index to be less than 1. For the above problem formulation we found a controller which guarantees the H∞ performance index γ = 0.96. It was computed by the hinfric routine in the MATLAB LMI toolbox. The plant order is 127, where 38 is the order of the model describing the relation between control and shear, and the order of the disturbance model is 13. The rest belong to the weights. This will result in a controller of the same order, which is very high for implementation. However, it can be readily reduced by the Hankel-optimal reduction procedure (see e.g. [5]) to 30 which is practically feasible, without performance degradation. Reduction down to 20 results in a slight, but acceptable, performance degradation.

4

Results and conclusions

In this section we consider the full-order controller where in the frequencydomain, the disturbance was simulated as the action of a blowing-suction

30

L. Baramov, O. Tutty, and E. Rogers

panel placed about 80π upstream from the panels. Figure 4 shows the magnitude of the shear disturbance with control near the actuation for the frequencies around the growth interval. The shear magnitude is signiﬁcantly reduced, with around an order of magnitude decrease in the size of the disturbance downstream of the actuation compared with the uncontrolled case. Note that the design directly penalizes shear only at the four points where the downstream sensors are placed.

Fig. 4. Controlled shear response.

The controller was tested by incorporating it in a closed-loop simulation with a 2D non-linear Navier-Stokes solver. This code is an implicit ﬁnite diﬀerence code which is fourth order in space and second order in time. Full details of the code can be found in [6]. The disturbance at the inlet (the upstream end of the channel) was generated from (a combination of) Orr-Sommerfeld modes. Here it is assumed that the disturbance to the ﬂow has the form ψ1 = Re [φ(y) exp(j(αx − ωt))]

(17)

where ψ1 is the perturbation streamfunction, φ is its complex amplitude, and α and ω are the wave number and frequency of the disturbance. φ is assumed to be suﬃciently small that the governing Navier-Stokes equations can be linearized. Hence φ satisﬁes a fourth order ODE (the Orr-Sommerfeld equation) with homogeneous boundary conditions φ = ∂φ/∂y = 0 at y =

Robust control of a nonlinear ﬂuidics problem

31

±1. This problem is a eigenvalue problem, where for our application the Reynolds number of the ﬂow and the frequency of the disturbance ω are chosen, and the complex wavenumber α = αr + jαj is the eigenvalue which is calculated as part of the solution. In fact, the value of ω is adjusted to give the desired real part of the wave number αr , which determines the wavelength of the disturbance. The complex part of α, αj , gives the growth rate of the disturbance. The disturbance was then generated at the inlet from (17) by setting x = 0 and adding the time varying perturbation to Poiseuille ﬂow. Two diﬀerent disturbances where considered. The ﬁrst has a single OrrSommerfeld mode with αr = 1, giving a disturbance which grows slowly in x. The second was a combination of ﬁve Orr-Sommerfeld modes with αr = 0.8, 0.9, 1.0, 1.1 and 1.2., which has two unstable modes, two stable, and one (close to) neutrally stable. Further, each of the modes was given the same magnitude so that the total disturbance had the form of a wave packet which is modulated as it propagates downstream. A detailed discussion of the results of these simulation studies can be found in [7]. Here it suﬃces to note that very good agreement with the theory has been achieved.

References 1. Joshi, S. S., Speyer, J. L. and Kim, J. (1999) Finite dimensional optimal control of Poiseuille ﬂow. Journal of Guidance, Control and Dynamics. 22, 340–348. 2. Bewley, T. R. and Liu, S. (1998) Optimal and robust control and estimation of linear paths to transition. Journal of Fluid Mechanics. 365, 305–349. 3. Baramov, L., Tutty, O.R., and Rogers, E. (2000) Robust control of plane Poiseuille ﬂow. AIAA Paper 2000-2684. 4. Chien, J. (1997) Frequency-domain tests for validation of linear fractional uncertain models. IEEE Transactions on Automatic Control, 42, 748–760. 5. Zhou, K. and Doyle, J. C. (1998), Essentials of Robust Control, Prentice-Hall. 6. Tutty, O. R. and Pedley, T. J. (1993) Oscillatory ﬂow in a stepped channel. Journal of Fluid Mechanics. 247, 179–204. 7. Baramov, L., Tutty, O.R., and Rogers, E. (2001) H∞ control for non-periodic planar channel ﬂows. IEEE International Conference on Decision and Control, to appear.

Optimal control of the Schr¨ odinger equation with two or three levels Ugo Boscain, Gr´egoire Charlot, and Jean-Paul Gauthier Universit´e de Bourgogne, D´epartement de Math´emathiques, 9, Avenue Alain Savary B.P. 47870-21078 DIJON, E-mail [email protected] Abstract. In this paper, we present how techniques of “control theory”, “subRiemannian geometry” and “singular Riemannian geometry” can be applied to some classical problems of quantum mechanics and yield improvements to some previous results.

1

Introduction

A quantum system with a ﬁnite number of (distinct) levels in interaction with an electromagnetic ﬁeld is described by the following Schr¨odinger equation (in a system of units such that = 1): idψ(t)/dt = Hψ where H = H0 +V (t), H0 = Diag(E1 , E2 , ..., En ), 0 < E1 < E2 < ... < En , ψ(.) : R → Cn and V (t) is a time dependent n × n Hermitian matrix that we assume, as usual, to be completely non–diagonal (i.e. (V (t))i,i = 0, i = 1, ..., n) and diﬀerent from zero only between times t0 and t1 . In the following we will assume t0 = 0. Writing ψ(t) = c1 (t)ϕ1 + c2 (t)ϕ2 + ... + cn (t)ϕn , where {ϕi }i=1,...,n , denote the elements of the canonical basis of Cn , we have, from the fact that H is Hermitian, that |c1 (t)|2 + |c2 (t)|2 + ... + |cn (t)|2 = const. This constant has to be taken equal to 1 in order to obtain the right probabilistic interpretation. For t < t0 and t > t1 , |ci (t)|2 is the probability of measuring energy Ei . d |ci (t)|2 = 0 for t < 0 and t > t1 . Notice the important fact that dt Usually in this kind of problems only “close levels” are coupled. This means that only the upper and lower diagonal elements of V (t) are diﬀerent from zero. Our problem can be stated in the following way. Assume |c1 (t)|2 = 1 for t < 0. We want to choose a suitable matrix V (t), such that we have |cj (t)|2 = 1 for time t > t1 , where j ∈ 2, ..., n is some ﬁxed label. V (t) describe n − 1 lasers in interaction with the system. We want to do this minimizing the amplitude of the laser pulses in a sense that will be made clear later on. This problem is clearly a control problem on the real 2n − 1 dimensional sphere in R2n (or on the complex sphere in Cn ). In the sequel, the following well known fact will be essential in eliminating the drift from the control system associated with the Schr¨odinger equation. = H(t)ψ(t), let Assume that ψ(t) satisfy the Schr¨odinger equation i dψ(t) dt U (t) be a unitary time dependent matrix and set ψ(t) = U (t)ψ (t). We have that ψ (t) satisfy the Schr¨odinger equation: i dψdt(t) = H (t)ψ (t) with the new A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 33-43, 2003. Springer-Verlag Berlin Heidelberg 2003

34

U. Boscain, G. Charlot, and J.-P. Gauthier

Hamiltonian H = U −1 HU − iU −1 dU dt . In the following, we will choose U (t) diagonal. As a consequence, if we write ψ(t) = c1 (t)ϕ1 +c2 (t)ϕ2 +...+cn (t)ϕn , and ψ (t) = c1 (t)ϕ1 + c2 (t)ϕ2 + ... + cn (t)ϕn , then, |ci (t)|2 = |ci (t)|2 , i = 1, 2, ..., n, that is H and H describe the same physical system. In Section 2 and 3, we treat the n = 2 and n = 3 cases. As we will describe in details both problems can be stated as control problems that are linear with respect to the control (or ”distributional control problems”): x˙ = u1 F1 + u2 F2 , where x ∈ S 3 for n = 2 and x ∈ S 5 for n = 3. Then it is very natural to treat this problem as a sub-Riemannian problem, that is t using the cost 0 1 u21 + u22 dt. The reason of choosing this cost is twofold. First this is “essentially” the ”area” of the laser pulses. Second this cost is time–reparameterization invariant. This turns out to be a crucial point since very smooth controls (i.e. with u˙ 1 and u˙ 2 small) are strongly preferred in practice. Here, with such a cost one can always reparameterize the time of the optimal solution in order to obtain the required smoothness. In Section 2 we treat the case n = 2 in the most general setting, in the sense that we control both the amplitude and the phase of the laser pulses. We get that controlling both the amplitude and the phase of the laser pulses guarantees the complete controllability of the system but does not add anything to the optimal strategy. The optimal strategy could be realized with a laser pulse in resonance (see below for details). These fact are well known in physics. Here, they are consequences of purely geometric considerations. In Section 3 we treat the n = 3 case with controls in resonance. We prove ﬁrst that the system is not at all controllable: on the contrary, the orbit under consideration is a two dimensional sphere, and the control problem reduces to a degenerate Riemannian problem on this sphere. We compute the geodesic joining the initial state the ﬁnal state, and we give an example of time reparameterized optimal controls. These results are new and are interesting both for practical applications and from a mathematical point of view. The results show control strategies that do not look like the classical strategy in this type of problems, but they are very ”natural”. In Section 4 we consider the same control system as in Section 3, but we are interested now with trajectories, going from state 1 to state 3 on the interval [0, t1 ], such that |c2 (t)| ≤ δ for each t ∈ [0, t1 ] and some ﬁxed δ > 0 small. This means that the probability of measure energy E2 during all the interval [0, t1 ] is always less or equal than δ 2 . Solution of this kind are generated by controls in the so called “counter intuitive order”. Roughly speaking this means that to go from state 1 to 3, one has to send ﬁrst photons of energy E3 − E2 and then photons of energy E2 − E1 . This problem is very interesting for applications and for its connection to the solution of the Schr¨odinger equation in the so called “adiabatic approximation”, see [5]. The solution we propose is based upon the idea (standard in control theory) of ”tracking a reference trajectory”. It looks interesting here, because it cares

Optimal control of the Schr¨odinger equation

35

explicitly about the ﬁnal state, while the classical methods just cares about it via the constraint |c2 (t)| ≤ δ.

2

The two levels case

In this Section, we study a quantum system with 2 levels, in interaction with a laser for which we control both the amplitude and the phase: dψ(t) E1 Ω(t) = H(t)ψ, H(t) = i , (1) Ω ∗ (t) E2 dt Here the control is Ω(t) : R → C (that describes the laser). Our aim is to transfer all the population from level E1 to level E2 minimizing the t laser amplitude 0 1 |Ω(t)| dt. Writing ψ(t) = c1 (t)ϕ1 + c2 (t)ϕ2 , where ϕ1 = (1, 0), ϕ2 = (0, 1), we start from any point satisfying |c1 (0)|2 = 1, and our target is deﬁned by |c2 (t1 )|2 = 1. In this case the drift is eliminated by the transformation ψ → U (t)ψ with U (t) = Diag(e−iE1 t , e−iE2 t ). The new Hamiltonian is (dropping the prime from now on): 0 u(t) 0 ei(E1 −E2 )t Ω(t) = , (2) H= u∗ (t) 0 0 e−i(E1 −E2 )t Ω ∗ (t) where we have deﬁned u(t) := ei(E1 −E2 )t Ω(t), to eliminate the explicit dependence on the time. This new Hamiltonian clearly gives rise to a driftless (or ”distributional”) control system, while the original Hamiltonian (1) had a drift term. Notice that since we have assumed Ω ∈ C, this simpliﬁcation work without any additional hypothesis on Ω. Surprisingly, the fact that the optimal strategy has the laser in resonance (i.e. Ω(t) = b(t)eiωt , ω = E2 −E1 ) will be obtained as a consequence. The Schr¨odinger equation corresponding to the Hamiltonian given by formula (2) is equivalent to the system of ODE for the ci ’s: c˙1 = −iu(t)c2 , c2 = −iu∗ (t)c1 . Setting c1 = x1 + ix2 , c2 = x3 + ix4 , u = u1 + iu2 these equations become x˙ = u1 F1 + u2 F2 where x = (x1 , x2 , x3 , x4 ), F1 = (x4 , −x3 , x2 , −x1 ),F2 = x3 , x4 , −x1 , −x2 ), and the functional to be minit mized is now 0 1 u21 + u22 dt. In these new variable the condition |c1 (t)|2 + 4 |c2 (t)|2 = 1 is i=1 x2i (t) = 1, so in fact x ∈ S 3 . The initial condition is now 1 := {x ∈ S 3 : x21 + x22 = 1} and the target is one point on the circle Sin t 1 3 2 2 Sf in := {x ∈ S : x3 + x4 = 1}. The problem of minimizing 0 1 u21 + u22 dt is a classical sub-Riemannian problem on S 3 , which is contact, as we shall see immediately. By setting F3 = [F1 , F2 ], one can check that [F2 , F3 ] = 4F1 , [F3 , F1 ] = 4F2 , hence Lie(F ) = su(2) ∼ so(3). Let F := {F1 , F2 }. Since F is an analytic family of vector ﬁelds on an analytic manifold, we can use the Hermann-Nagano Theorem (see for instance

36

U. Boscain, G. Charlot, and J.-P. Gauthier

[6]). In this case, it says that the orbit is an analytic submanifold of S 3 of dimension given by Liex (F ) where x is any point of the orbit. Let n(x) := rankx (F1 , F2 , F3 ), x ∈ S 3 be the rank of the distribution. We have n(x) = 3. It follows easily that the control system is completely controllable. Notice that, the control system is invariant under the transformation generated by the Lie bracket [F1 , F2 ] = F3 . This means that all the initial conditions x0 ∈ Sin are equivalent. Moreover this sub-Riemannian problem is not generic at all: ﬁrst it is isoperimetric (it has a symmetry), second, even among isoperimetric SR problems, it is non generic in the sense that the main basic sub-Riemannian invariant vanishes (see [1,2,4]). Now we will be able to ﬁnd optimal trajectories joining our boundary conditions without making any computation. In fact F1 andF2 are two orthog4 onal vectors for the standard metric of S 3 i.e. F1 · F2 = i=1 (F1 )i (F2 )i = 0. Hence, the length of an admissible curve is just its standard Riemannain length on S 3 . Therefore, if we ﬁnd an admissible trajectory going from the state 1 to the state 2, which is a minimizing geodesic for the Riemannian metric on S 3 , then it is also a minimizer for our sub-Riemannian problem. Now, each integral curve of the vector ﬁeld b(cos(α)F1 + sin(α)F2 (b ∈ R, α ∈ [0, 2π]) are such admissible trajectories. It follows that: Theorem 1 Every constant control of the form: u1 = b cos(α), u2 = b sin(α), b ∈ R α ∈ [0, 2π] is optimal and the target is reached at time t1 = π/(2b). t Now since the functional 0 1 u21 + u22 dt, is invariant under time reparameterizations, one can take b to be a function of the time so that every optimal control has the form given by Ω(t) = b(t) exp(i[(E2 − E1 )t + α]), where α is an arbitrary constant, b(.) : R → R+ is a real function with compact support t [0, t1 ] and satisfying 0 1 b(t) = π/2. 2.1

Another interpretation: the Hopf ﬁbration and the Isoarea Problem

The Hopf ﬁbration (see for instance [7]), π : S 3 → S 2 is deﬁned by2 the the variables (x , x , x , x ), xi = 1 following map. Let S 3 be described by 1 2 3 4 2 zi = 1/4. We have: and S 2 by the variables (z1 , z2 , z3 ), 1 (z1 , z2 , z3 ) = π(x1 , x2 , x3 , x4 ) = ( (x21 +x22 −x23 −x24 ), x1 x3 −x2 x4 , x2 x3 +x1 x4 ). 2 The Hopf ﬁbration gives to S 3 the structure of a principal bundle with base S 2 and ﬁber S 1 ∼ U (1). The following proposition (that one easily checks) shows why the Hopf ﬁbration is connected to our problem: Proposition 1 Let F3 := [F1 , F2 ], and π the Hopf ﬁbration deﬁned above. Then F3 ∈ Ker(dπ).

Optimal control of the Schr¨odinger equation

37

Notice that F3 is the generator of the symmetry that “transport” along Sin and Sf in that means the following. If x0 ∈ Sin (resp. x0 ∈ Sf in ) then the orbit O(x0 ) of F3 coincide with Sin (resp. Sf in ). From Proposition 1 it follows that Sin and Sf in shrink into two points through π. In particular these points are respectively the opposite points (z1 , z2 , z3 ) = (± 12 , 0, 0). Notice that we have a one parameter family of geodesics connecting Sin and Sf in , since their images under π are opposite points on S 2 . In fact, the sub-Riemannian problem we have, is as follows: The distribution (transversal to the ﬁbers of the Hopf ﬁbration) deﬁnes a connection over this (circle) principal bundle. It is easily seen that the curvature form of this connection is just the pull back (by the bundle projection) of the volume form of the euclidean metric on S 2 . As a consequence (see [2]), our sub-Riemannian problem correspond to the “isoarea problem” on the Riemannian sphere S 2 : given two points (antipodal on S 2 in our case), and any ﬁxed curve x(.) joining these two points, ﬁnd another curve y(.), joining also the two points, such that the length of y(.) is minimal, and the area encircled by the curves x(.), y(.) has a given value. Our (very special) solutions exhibited above, are in fact geodesics of the euclidean metric on S 2 : they correspond to the choice of geodesics for the curve x(.), and to the zero value of the area. A consequence of this isoperimetric situation is: Proposition 2 The only optimal trajectories from Sin to Sf in are the trajectories corresponding to the controls given by Theorem 1.

3

The resonant problem with three levels

In this section, we study a 3–levels quantum system, with only close levels coupled, controlled by two laser pulses in resonance i.e. with frequencies ω1 = E2 − E1 , ω2 = E3 − E2 , being E1 , E2 , E3 the three energy levels:   E1 u1 (t)eiω1 t 0 dψ(t) E2 u2 (t)eiω2 t  , = H(t)ψ, H(t) =  u1 (t)e−iω1 t (3) i dt −iω2 t 0 u2 (t)e E3 Here the controls are u1 (.), u2 (.) : R → R (the laser pulses amplitudes). Our aim is to transfer all the population from the state with energy E1 t to the state with energy E3 minimizing the length 0 1 u21 + u22 dt. Writing ψ(t) = c1 (t)ϕ1 + c2 (t)ϕ2 + c3 (t)ϕ3 , where ϕ1 = (1, 0, 0), ϕ2 = (0, 1, 0), ϕ3 = (0, 0, 1), we start from one point satisfying |c1 (0)|2 = 1, and our target is deﬁned by |c3 (t1 )|2 = 1. In this case to eliminate the drift we should use the matrix U (t) = Diag(e−iE1 t , e−iE2 t , e−iE3 t ). The new Hamiltonian is then (in the following

38

U. Boscain, G. Charlot, and J.-P. Gauthier

we drop the prime): H=

 0 u1 (t) 0  u1 (t) 0 u2 (t)  . 0 u2 (t) 0 

(4)

Notice that in this case (i.e. with real controls), to obtain this strong simpliﬁcation it is essential to use lasers in resonance. The Schr¨odinger equation corresponding to the Hamiltonian given by formula (4) is equivalent to the system of ODE for the ci : c˙1 = −iu1 (t)c2 , c˙2 = −i(u1 (t)c1 +u2 (t)c3 ), c˙3 = −iu2 (t)c2 . Setting c1 = x1 +ix2 , c2 = x4 −ix3 , c3 = x5 + ix6 , these equations become: x˙ = u1 F1 + u2 F2 , where x ∈ S 5 and F1 and F2 can be easily computed. The initial condition is now one 1 := {x ∈ S 5 : x21 + x22 = 1} and the target is point on the circle Sin t1 2 1 5 2 2 Sf in := {x ∈ S : x5 + x6 = 1}. With the choice 0 u1 + u22 dt, of the functional to be minimized, our problem looks like a classical sub-Riemannian problem on S 5 , but, as we shall see, it is very degenerate. Similarly to the two levels case, by setting F3 = [F1 , F2 ], one can check that [F2 , F3 ] = F1 , [F3 , F1 ] = F2 , hence Lie(F ) = su(2) ∼ so(3). Let n(x) := rankx (F1 , F2 , F3 ), x ∈ S 5 be the rank of the distribution. We have n(x) = 2 if x ∈ Q and n(x) = 3 if x ∈ S 5 \ Q, where Q is the subset of S 5 deﬁned by the following equations: x3 x6 − x4 x5 = 0,

x1 x6 − x2 x5 = 0,

x3 x2 − x4 x1 = 0.

(5)

Notice that if x1 , x3 , x5 are all diﬀerent from zero, equations (5), are equivalent to: x2 /x1 = x4 /x3 = x6 /x5 Now since every initial condition lies in Q 1 ∈ Q), from the Hermann-Nagano Theorem it follows that for each (i.e. Sin 1 , the orbit O(x0 ) is an analytic two dimensional submanifold of x0 ∈ Sin S 5 . More precisely deﬁning x0 (α) as the initial condition corresponding to x1 (0) = cos(α), x2 (0) = sin(α), where α ∈ [0, 2π[, we get the following: Theorem 2 The orbit O(x0 (α)), α ∈ [0, 2π[ is the two dimensional sphere. In particular O(x0 (0)) is deﬁned by the equation x21 + x23 + x25 = 1. Notice that ﬁxed x0 (α), one can reach only two points of the ﬁnal target. For instance if α = 0 i.e. x1 (0) = 1, we can reach Sf1in only in the two points x5 = ±1. Let us now study

the structure of Q. From the Hermann–Nagano Theorem we have Q ⊇ x0 ∈S 1 Ox0 , but one can easily check that in fact in

Q = x0 ∈S 1 Ox0 . Let σ be the antipodal involution of S 1 × S 2 , that is, in σ(α, p) = (α + π, −p). The involution σ has no ﬁxed point, and is orientation reversing. Also, clearly, by Theorem 2, our orbits are σ stable. Hence: Q = (S 1 × S 2 )/∼ where x ∼ x if σ(x) = σ(x ). Therefore, it is not hard to see that Q is (the only) non-orientable sphere-bundle over S 1 . Due to the invariance under the transformation generated by F3 , all the points of Sin can be considered equivalently. In the following we will study the

Optimal control of the Schr¨odinger equation

39

optimal control problem on the orbit O(x0 (0)), that is the sphere of equation x21 + x23 + x25 = 1. In the sequel, we will still call F1 and F2 the restrictions of F1 and F2 to this sphere. In order to get labels for coordinates corresponding to quantum states, and in order to have F2 pointing in the positive direction from the point x3 = 1, we deﬁne y1 = x1 , y2 = x3 , y3 = −x5 . The control system under consideration is then (y˙ 1 , y˙ 2 , y˙ 3 ) = u1 F1 + u2 F2 where F1 = (−y2 , y1 , 0), F2 = (0, −y3 , y2 ). The vector ﬁelds are plotted on the following picture:

y1 :=x1 y2 :=x 3 y :=-x 5

State 1

Integral Curves of F

y

1

1

F1 State 3

3

State 2

State 2

y F

2

F2

2

Integral Curves of F

2

F

1

State 3

y

3

State 1

Fig. 1 The initial condition is the point y1 = 1. The state number 1 (resp. 2, 3) correspond to the points y1 = ±1 (resp. y2 = ±1, y3 = ±1). The state two can be reached from the state one using only F1 and the state three can be reached from the state two using only F2 (dotted lines). However, the state three cannot be reached from state one using a trajectory contained in the circle of equation y2 = 0 (i.e. y12 + y32 = 1), since no piece of this circle is an admissible trajectory. This is due to the fact that F1 is collinear to F2 on this circle, and not tangent to it. Let us describe the orbit in spherical coordinates: y1 = cos(θ) cos(φ), y2 = sin(θ), y3 = cos(θ) sin(φ). We have: F1 G1 cos(φ) sin(φ) G1 = ∂ θ =R where R := , F2 G2 G2 = tan(θ)∂φ − sin(φ) cos(φ) Since R ∈ SO(2), the couple (G1 , G2 ) is a new orthonormal frame for the sub˙ φ) ˙ = Riemannian length. For this new frame the control systemis then: ( θ, t1 v12 + v22 dt. The v1 G1 + v2 G2 , and the functional to be minimized is 0 relation between u1 , u2 and v1 , v2 is obtained from u1 F1 +u2 F2 ≡ v1 G1 +v2 G2 .

40

U. Boscain, G. Charlot, and J.-P. Gauthier

The metric deﬁned by the frame (G1 , G2 ) is a singular metric. Indeed when θ = 0 we have G2 = 0, that is exactly on the circle y2 = 0. Notice that the singularity of the metric for θ = π/2 is only due to the choice of the coordinate system. Let us compute the geodesics using the Maximum Principle. Let P = ∗ (Pθ , Pφ ) ∈ Tθ,φ M . By deﬁnition the Hamiltonian is H(θ, φ, Pθ , Pφ , v1 , v2 ) =< P, v1 G1 + v2 G2 > +λ(v12 + v22 ). It is easily checked that, as for the Riemannian case, we can always assume λ = 0 (there are no abnormal extremals) and we can normalize λ = −1/2. Extremal controls are computed from the maximum condition and we get v1 = Pθ , v2 = Pφ tan(θ). Hence, we get that the extremals are projections on the (θ, φ) space, of integral curves of the Hamiltonian vector ﬁeld corresponding to the following Hamiltonian HM = 12 (Pθ2 + (tan(θ)Pφ )2 ). Setting a := Pφ , the Hamiltonian equations are: θ˙ = Pθ , φ˙ = a tan2 (θ), P˙ θ = −a2 tan(θ)(1 + tan2 (θ)). This Hamiltonian system is Liouville integrable since we have two independent and commuting constants of the motion HM and Pφ = a. Anyway we are interested to ﬁnd the geodesic without the parameterization, i.e. arelation between θ and φ. ˙ θ˙ = φ/P ˙ θ = ±a tan2 (θ)/ 1 − a2 tan2 (θ), where we We have dφ/dθ = φ/ have normalized HM = 1/2 (that corresponds to a time rescaling). So we have the two families of solution parameterized by the value of a: φ± a (θ) = ± arctan

√ a 1 + a2 a sin(θ) − √ sin(θ) (6) arctan Ξ(θ, a) Ξ(θ, a) 1 + a2

1 2 2 (1 − a + (1 + + family φa with a > 0

where Ξ(θ, a) :=

a2 ) cos(2θ)). To ﬁx the ideas, let us con-

sider only the and suppose θ ≥ 0, the other cases being symmetric. Let us call this family φa (θ). Expression (6) deﬁne φa (θ) in the interval [0, θ¯a [ where θ¯a := arctan(1/a). Notice that limθ→θ¯a Ξ[θ, a] = 0 so: √ limθ→θ¯a φa (θ) = π2 1 − a/ 1 + a2 . In the following we consider φa (θ) deﬁned in [0, θ¯a ] where by deﬁnition φa (θ¯a ) is the value given by the limit above. Indeed from the symmetries of the system, the whole relation between θ and φ is given by two branches: φ1a (θ) = φa (θ), φ2a (θ) = 2φa (θ¯a ) − φa (θ). The geodesic reaching √ the target is the one satisfying φa (θ¯a ) = π/4 and it corresponds to a := 1/ 3. The numerical value of the minimal cost is ∼ 2.7207. Notice that by considering both signs in formula (6) (or equivalently a positive and negative) and θ ∈ [−π, π] one gets four equivalent optimal trajectories reaching the state 3. Finally the set of geodesics parameterized by a allows easily to compute an optimal synthesis for the problem. One gets an explicit expression of the controls reaching the ﬁnal point θ = 0, φ = π/2, as function of the time, by choosing a parameterization for the π 2 curve θ(t). For example taking θ(t) = 3(−1+e 9 ) −1 + exp(−36t(t − t1 )/t1 , one gets the control showed in the following picture:

Optimal control of the Schr¨odinger equation

41

2 1.75 1.5

u1

1.25

u

2

1 0.75 0.5 0.25 1

4

2

3

4

The counter intuitive solutions

For practical applications, trajectories going from the state one to the state three, that are good approximations of the non admissible trajectory, are also interesting. We recall that the non admissible trajectory is contained in the circle y2 = 0. In this section we study a trajectory in which φ(t) is monotonously increasing between 0 and π/2, and θ(t) ≤ ε, for every t ∈ [0, t1 ], for some small ε > 0 ﬁxed. This means that the probability of measure energy E2 during all the interval [0, t1 ] is always less or equal than sin2 (ε) = ε2 + O(ε4 ). It is well know [3,5] that the non admissible trajectory can be approximated using controls u1 (t) and u2 (t) in the so called ”counterintuitive” order:

u2

u1 t

In the following we show how to build a trajectory satisfying condition θ(t) ≤ ε, and connecting exactly the points P1 deﬁned by (φ, θ) = (0, 0) and P2 deﬁned by (φ, θ) = (π/2, 0). The idea is to tracking a trajectory connecting these two points taking care of the constraints that we have on the derivatives at P1 and P2 . More precisely we should ﬁnd a function θ(φ) such that the following holds. Let φ1 (θ) (resp. φ2 (θ)) be the inverse function of θ(φ) in a neighborhood of P1 (resp. P2 ). Since F1 and F2 vanish respectively at points P1 and P2 , we must have dφ1 /dθ|θ=0 = 0, dφ2 /dθ|θ=0 = 0. A possible choice is the symmetric function θ(φ) = ε π4 φ( π2 − φ), that reaches the value ε only at the point φ = π4 . Now we have to choose a parameterization (θ(t), φ(t)). To have continuous controls satisfying u1 (0) = 0, u2 (0) = 0, u1 (t1 ) = 0, u2 (t1 ) = 0 we must have θ˙1 (t)|0 = 0, θ˙2 (t)|t1 = 0. From the equations above we get φ¨1 (t)|0 = 0, φ¨2 (t)|t1 = 0. A possible choice to get controls having zero derivatives at the initial and ﬁnal points is: φ(t) = 13 11 9 8 7 2 4 5 6 t12 t1 t 6006 π 13 − 2 + 15 t11 t1 − 2 t10 t1 3 + 5 t 3t1 − 3 t 4t1 + t 7t1 /t13 1 .

42

U. Boscain, G. Charlot, and J.-P. Gauthier

˙ Controls u1 (t) and u2 (t) are computed with relations v1 (t) = θ(t), v2 (t) = ˙ φ(t)/tan(θ(t)) and u1 F1 + u2 F2 ≡ v1 G1 + v2 G2 . Notice that for very small values of ε one get very big values of the controls. We would like to stress the fact that, the trajectory obtained with this tracking, reaches exactly the ﬁnal target for any ﬁxed value of ε. While in the ”counter intuitive” strategies used in literature, if εc is the maximum value reached by θ, then the ﬁnal target is reached with an error smaller than εc but diﬀerent from zero. In the last picture the tracking solution corresponding to the expression of φ(t) above for ε = 2/19, and t1 = 4 and a typical strategy used in litera2 2 ture: u1 (t) = −5/e16 + 13/e(−4+1.5 t) , u2 (t) = −5/e9 + 13/e(−2.7+1.5 t) are compared. Notice that the pulses have similar area. Moreover notice that the trajectory corresponding to the controls above (obtained integrating numerically the Schr¨odinger equation) reaches negative values of θ. Tracking solution

θ

14 12

u

10 8

0.19

1

u2

6

0.11

4

0.04

2 1

2

3

4

φ

5

Target

Fig. 2

5

Conclusion

In summary, we have shown how optimal controls for two and three level models can be constructed on the basis of geometric arguments. The optimal trajectories appear as geodesics (Riemannian or singular Riemannian) on two dimensional spheres. Furthermore, besides the optimal control strategies, the standard tracking technique allows us to analyse a method used in recent experiments. It leads to an improvement that allows to reach the target state precisely.

Optimal control of the Schr¨odinger equation

43

References 1. A. Agrachev, (1996) Exponential mappings for contact subriemannian structures , J. Dynam. Control Syst. 2, 321–358. 2. A. Agrachev, J.P. Gauthier, (1999) On the Dido Problem and Plane Isoperimetric Problems, Acta Applicandae Mathematicae 57, 287–338. 3. C.E. Carroll and F.T. Hioe, (1990) Analytic Solutions for three-state system with overlapping pulses, Phys. Rev. A, 42, 1522–1531. 4. H. Chakir, J.P. Gauthier, I. Kupka, (1996) Small subriemannian balls on R 3 , J. Dynam. Control Syst. 2, 359-421. 5. K. Bergmann, H. Theuer, and B. W. Shore, (1998) Coherent population transfer among quantum states of atoms and molecules, Rev. Mod. Phys. 70, 1003-1026. 6. V. Jurdjevic, (1997) Geometric Control Theory, Cambridge University Press. 7. R. Montgomery, A tour of subriemannian geometry, book in preparation. 8. L.S. Pontryagin, V. Boltianski, R. Gamkrelidze and E. Mitchtchenko, (1961) The Mathematical Theory of Optimal Processes, John Wiley and Sons, Inc.

Strong robustness in adaptive control Maria Cadic1 and Jan Willem Polderman2 1

2

University of Twente, P.O. box 217, 7500 AE Enschede, The Netherlands [email protected] University of Twente, P.O. box 217, 7500 AE Enschede, The Netherlands [email protected]

Abstract. This paper considers the issue of strongly robust stabilization of a set of linear, controllable, time-invariant and SISO systems of degree n ≥ 1. First of all the notion of Strong Robustness is introduced. Then we prove the existence of an open strongly robust spherical neighborhood around any system within the studied class. Next, balls of systems are considered and we give a suﬃcient and a necessary conditions on their radius to ensure their strong robustness. These results are illustrated by the example of ﬁrst order pole placement design.

1

Introduction

When dealing with adaptive control of an unknown system, certainty equivalence may be used if the uncertainty on the parameters describing the system is small enough to at least design a stabilizing controller based on the knowledge of this uncertainty. However, if we have no knowledge on the system, using a certainty equivalence type of strategy might result in a control system showing undesirable transients. To overcome this problem, we may use a multi-phase adaptive control based on at least two phases. In the ﬁrst phase, the emphasis is put on identiﬁcation. Once the parameter uncertainties have been decreased to a certain level, we can switch to the second phase putting most of the eﬀort on control. The problem is to distinguish these phases in a quantitative way. Our idea is to remain in the ﬁrst phase until we identify a region containing the true system such that wherever our estimated system is located in this region and wherever the true system actually is, we can stabilize this system on the basis of our estimate. Otherwise stated, we remain in this ﬁrst phase until we identify a set S containing the actual system such that the controller based on any system of S stabilizes any other system of S. Sets having this property are said to be Strongly Robust and this paper focuses on their characterization. This paper is organized as follows. Section 2 ﬁrst settles the general setup of the problem. In Sect.3, we deﬁne the concepts of Strongly Robust sets of systems and Strongly Robust Stability Radius; ﬁnally we derive a criterion for Strong Robustness. Then in Sect.4, we prove that around any given system, there exists an open strongly robust spherical neighborhood. This allows us in Sect.5 to work on non-trivial balls of systems about which we derive suﬃcient and necessary conditions ensuring their strong robustness. A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 45-54, 2003. Springer-Verlag Berlin Heidelberg 2003

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M. Cadic and J.W. Polderman

Next, our main interest lies in an algorithm testing the strong robustness of a region containing the system to be controlled. Indeed if it is strongly robust, we know that the controller designed on the basis of any other system of this region stabilizes the true system. The construction of such an algorithm is the subject of Sect.6.

2

General setup

In this part we deﬁne the class Pn of systems we are going to study, the associated class of controllers, and some additional notation. Class Pn of studied systems. Pn consists of SISO systems that are linear, time invariant with order n ≥ 1, described by: d d )y = B( )u (1) dt dt n−1 n−1 where A(ξ) = ξ n + i=0 ai ξ i and B(ξ) = i=0 bi ξ i are real polynomials. Moreover, we suppose these systems to be controllable, i.e., for each system described by (1), A(ξ) and B(ξ) are coprime. We identify each system in Pn with its parameters, i.e., P = (an−1 , · · · , a0 , bn−1 , · · · , b0 )T , so that Pn can be seen as a subset of R2n . A(

Controllers. The control objective itself is left unspeciﬁed. We assume that for each controllable system of the form (1), there exists a unique controller such that the control objective is satisﬁed. The controllers that we consider are of the form: d d (2) C( )u = −D( )y dt dt n−2 n−1 where C(ξ) = ξ n−1 + i=0 ci ξ i and D(ξ) = i=0 di ξ i are real polynomials. Formally, there exists a map φ : Pn → R2n−1 P → φ(P ) = (cn−2 , · · · , c0 , dn−1 , · · · , d0 )T

(3)

such that the closed-loop system (P, φ(P )) satisﬁes the control objective. The resulting closed-loop characteristic polynomial is of degree 2n − 1 and is assumed to be strictly Hurwitz. We denote it by: χP,φ(P ) (ξ) = A(ξ)C(ξ) + B(ξ)D(ξ) .

(4)

Finally, we assume that φ is continuous. Pole Placement and Linear Quadratic control fulﬁll these requirements.

Strong robustness in adaptive control

47

Notation. ∀P 0 ∈ Pn , we denote by: • B(P 0 , r) = {P ∈ Pn : d(P, P 0 ) < r} the open ball of systems in Pn , where d(., .) denotes the Euclidean distance; • ∂B(P 0 , r) = {P ∈ Pn : d(P, P 0 ) = r} the boundary of B(P 0 , r); • SP 0 = {P ∈ Pn |(P, φ(P 0 )) is asymptotically stable} the robust stability set around P 0 ; • rP 0 = max≥0 {|B(P 0 , ) ⊂ SP 0 } the robust stability radius around P 0 ; • For all R(ξ) = r0 + r1 ξ + r2 ξ 2 + · · · ∈ R[ξ], Re (ξ) = r0 − r2 ξ 2 + r4 ξ 4 − · · · and Ro (ξ) = r1 − r3 ξ 3 + r5 ξ 5 − · · · .

3

Strong robustness and strongly robust stability radius

We now deﬁne the notion of Strong Robustness. Definition 3.1 (Strongly Robust set) S ⊂ Pn is strongly robust if for any systems P1 , P2 in S, P1 ∈ SP2 . Equivalently, S is strongly robust if the controller based on any P1 ∈ S stabilizes any P2 ∈ S. Similarly to the notion of Robustness in the classical sense, we deﬁne the strongly robust stability radius around a given system P 0 ∈ Pn as the radius of the largest strongly robust ball centered in P 0 : Definition 3.2 (Strongly robust stability radius) The strongly robust stability radius around the system P 0 is deﬁned by ρP 0 = inf { : ∃P ∈ ∂B(P 0 , )|B(P 0 , ) ⊂ SP } . ≥0

(6)

A criterion for Strong Robustness follows directly from Deﬁnition 31: Theorem 3.3 A set S ⊂ Pn is strongly robust ⇔ S ⊂ P ∈S SP . Proof Denote by S a subset of Pn . From Deﬁnition 31, S is strongly robust if and only if ∀P, P ∈ S, then P ∈ SP . Hence S is strongly robust if and only if ∀P, P ∈ S, S ⊂ SP , therefore if S ⊂ P ∈S SP . Example 3.4 Let us now investigate Strong Robustness in the case of ﬁrst order pole placement design. Let α ∈ R+ ∗ denote the desired closed-loop pole. P1 consists of systems (a0 , b0 )T described by: d y(t) + a0 y(t) = b0 u(t) . dt

48

M. Cadic and J.W. Polderman

The associated feedback controllers (1, d0 )T lead to control laws of the form: u(t) = −d0 y(t). We obtain: SP 0 = {(a0 , b0 )T ∈ P1 |a0 − b0

α + a00 > 0} . b00

(8)

and S⊂

SP 0 ⇔ ∀(a0 , b0 )T ∈ S, ∀(a00 , b00 )T ∈ S, a0 > b0

P 0 ∈S

α + a00 . b00

(9)

From a geometrical point of view, (8) is equivalent to: S ∈ SP 0 ⇔ S is in the right half-plane with boundary L00 , where L0 denotes the line going trough (−α, 0)T and (a00 , b00 )T in the parameter space, and L00 denotes its parallel going trough (0, 0)T . And (9) can be formulated as follows: S is strongly robust ⇔ S is in the strip with boundaries LT0 and LT , where LT denotes the line tangent to S going through (−α, 0)T , and LT0 its parallel going through (0, 0)T . These results are depicted in Fig. 1 where S1 is a strongly robust set of system, whereas S2 is not. From this example,

Fig. 1. Strong robustness for the ﬁrst order case, pole located in α.

it appears also that a set can be strongly robust with respect to a certain control objective, while it is not with respect to another one. Suppose indeed that α becomes closer to the b axis, then there exists a value (denoted by αm in Fig. 1) below which the set S1 is not strongly robust anymore.

Strong robustness in adaptive control

4

49

Existence of strongly robust sets

We are now going to prove the existence of an open strongly robust spherical neighborhood around any system P 0 ∈ Pn . We ﬁrst prove that for a given P 0 ∈ Pn , any controller close enough to φ(P 0 ) also stabilizes any system in B(P 0 , rP 0 ). Lemma 4.1 ¯ is For a given system P 0 ∈ Pn , ∃P 0 > 0 : if |φ¯ − φ(P 0 )| < P 0 , then (P, φ) − 0 − asymptotically stable for any P ∈ B(P , rP 0 ) where rP 0 < rP 0 . Proof We ﬁrst recall that a polynomial p(ξ) = p0 +p1 ξ +· · ·+pn ξ n is strictly Hurwitz if and only if n functions {Fi (pn , · · · , p1 , p0 )}i=1...n are strictly positive (Routh-Hurwitz criterion) [1]. These functions are moreover continuous with respect to pn , · · · , p0 . Now, for any P, P ∈ Pn , to check if the system (P, φ(P )) is asymptotically stable is equivalent to check whether the polynomial χP,φ(P ) (ξ) is strictly Hurwitz; therefore, it is also equivalent to check whether a ﬁnite number ﬁnite number Nn > 0 of functions {Fi (X, Y )}i=1..n are strictly positive, where X and Y are the parameters vectors corresponding to P and φ(P ) respectively, and where the functions Fi are continuous with respect to X and Y . Let K denote the ball B(X 0 , rP−0 ), K is a compact set. Let the functions Fi take values on (K, φ(K). We know that Y 0 is such that ∀X ∈ K, Fi (X, Y 0 ) > 0. Let’s ﬁrst prove that the functions Fi (X, .) are non-negative for any X ∈ K in a complete open neighborhood of Y 0 . Otherwise stated, let us prove the result: (S1): for any i = 1..n, there exists i > 0 such that if |Y − Y 0 | ≤ i , then > 0 for all X such that |X − X 0 | ≤ rP−0 . Fi (X, Y ) Suppose that (S1) is not true. Then ∃i ≤ n such that ∀ > 0, ∃X ∈ K and ∃Y such that |Y − Y 0 | ≤ with Fi (X, Y ) ≤ 0. In particular, ∀k > 0, ∃(Xk , Yk ) with |Yn − Y 0 | ≤ k1 and Xk ∈ K such that Fi (Xk , Yk ) ≤ 0. Since the sequence of {(Xk , Yk )k∈N } is bounded, it has a converging subsequence, say: ¯ Y 0 ). lim (Xkp , Ykp ) = (X,

p→∞

(12)

¯ ∈ K. Because the sequence {Xk }k∈N is deﬁned on the compact set K, X Now, because of the continuity of the function Fi , we obtain: ¯ Y 0) , lim Fi (Xkp , Ykp ) = Fi (X,

p→∞

(13)

therefore ¯ Y 0) ≤ 0 . Fi (X,

(14)

50

M. Cadic and J.W. Polderman

But we know that Y 0 is such that ∀X ∈ K, Fi (X, Y0 ) > 0. Therefore ¯ Y0 ) > 0 . Fi (X,

(15)

Equations (14) and (15) yield a contradiction, therefore we proved the following result: (S2): ∀i = 1..n, ∃i > 0 such that if |Y − Y 0 | ≤ i , then Fi (X, Y ) > 0 ∀X such that |X − X 0 | ≤ rP−0 . We deﬁne P 0 = min {i | S2 holds} .

(16)

i=1..n

Hence P 0 is such that: if |Y − Y 0 | ≤ P 0 , then Fi (X, Y ) > 0 for all X ∈ K. Therefore, we proved that if |φ(P ) − φ(P 0 )| ≤ P 0 , then Fi (P, φ(P )) > 0 for all X ∈ B(X 0 , rP−0 . It means that P 0 is such that any controller in B(φ(P 0 ), P 0 ) stabilizes any system in a compact subset of B(P 0 , rP 0 ). This concludes the proof of Lemma 41. We can now prove the existence of Strongly Robust sets. Theorem 4.2 (Existence of Strongly robust set) For any system P 0 in Pn , there exists an open spherical neighborhood of P 0 contained in SP 0 which is strongly robust. Proof Fix P 0 ∈ Pn . φ(P 0 ) is such that the system (P 0 , φ(P 0 )) is asymptotically stable. φ is continuous, therefore there exists 0 < δ < rP 0 such that if |P − P 0 | ≤ δ, then |φ(P ) − φ(P 0 )| < P 0 , for P 0 deﬁned in (16). Consequently, for any P ∈ B(P 0 , δ), φ(P ) ∈ B(φ(P 0 ), P 0 ). Referring to Lemma 41, we conclude that for any P ∈ B(P 0 , δ), φ(P ) stabilizes the ball B(P 0 , rP 0 ). In particular, for any P ∈ B(P 0 , δ), φ(P ) stabilizes B(P 0 , δ). Hence, B(P 0 , δ) is strongly robust. This concludes the proof of Theorem 42.

5 5.1

Strong robustness and open balls of systems A necessary and a suﬃcient condition for strong robustness

We now focus on open balls of systems and give a suﬃcient and a necessary condition on their radius to ensure strong robustness. Theorem 5.1 1. For any open ball S ⊂ Pn , if its radius is such that r(S) ≤ then S is strongly robust.

1 2

minP ∈S rP ,

Strong robustness in adaptive control

51

2. Conversely, if an open ball S ⊂ Pn is strongly robust, then its radius must be such that r(S) ≤ 12 minP ∈∂(S) d¯p , where ∂(S) is the boundary of S, and d¯P = maxP ∈SP |P − P |, ∀P ∈ Pn . Proof 1. Suppose that r(S) ≤ 12 minP ∈S rP . ∀P ∈ S : ∀P ∈ S, |P − P | < 2r(S). So, ∀P ∈ S, |P − P | < minP ∈S rP ≤ rP . Hence, ∀P, P ∈ S, P ∈ SP . Consequently, S is strongly robust; this proves the ﬁrst point. 2. Let S ⊂ Pn be strongly robust. Then, ∀P ∈ S, ∀P ∈ S, P ∈ SP ⇒ ∀P ∈ S, |P − P | ≤ d¯P , ⇒ max , |P − P | ≤ d¯P , P ∈S

, |P − P | ≤ d¯P . ⇒ max P ∈∂(S)

Therefore, ∀P ∈ ∂(S) we have 2r(S) = maxP ∈∂(S) |P − P | ≤ d¯P , or: r(S) ≤ 12 minP ∈∂(S) d¯P . The second point is then veriﬁed. 5.2

Calculation of the strongly robust stability radius

Robust stability radius. We ﬁrst concentrate on the calculation of rP 0 , for a ﬁxed system P 0 ∈ Pn . These results follows the work of H. Chapellat and S. P. Bhattacharyya [2]. We ﬁx P 0 = (a0n−1 , · · · , a00 , b0n−1 , · · · , b00 )T ∈ Pn . The controller based on P 0 is then φ(P 0 ) = (c0n−1 , · · · , c00 , d0n−1 , · · · , d00 )T , leading to the closed-loop characteristic polynomial χP 0 ,φ(P 0 ) (ξ) = A0 (ξ)C 0 (ξ) + B 0 (ξ)D0 (ξ). Now, in order to ﬁnd the radius ρP 0 of the largest stability ball centered in P 0 , we assume that P 0 is subject to variations of its parameters. In other words, we look at the systems P such that |P − P 0 | < ρP 0 yielding an asymptotically stable closed-loop system (P, φ(P 0 )). Similarly as in [2] the two following sets Π0 (P 0 ) and Πω (P 0 ) for any ω > 0 are deﬁned as follows: Π0 (P 0 ) = {P ∈ Pn : χP,φ(P 0 ) has a root in 0} ; Πω (P 0 ) = {P ∈ Pn : χP,φ(P 0 ) (ξ) = (ξ 2 + ω 2 )δ(ξ), ∀δ(ξ) ∈ R[ξ]}; We can show that Π0 (P 0 ) = ∅, for any system P 0 ∈ Pn . We denote by l0 (P 0 ) 0 and lω (P ) the distances from P 0 to Π0 (P 0 ) and Πω (P 0 ) respectively: l0 (P 0 ) =

min

P ∈Π0 (P 0 )

|P − P 0 | ;

minP ∈Πω (P 0 ) |P − P 0 | lω (P 0 ) = ∞

if Πω (P 0 ) = ∅ if Πω (P 0 ) = ∅

52

M. Cadic and J.W. Polderman

Finally, we deﬁne l(P 0 ) = inf ω>0 lω (P 0 ). In [3] it is shown that the largest ball centered at P 0 such that for any P in this ball (P, φ(P 0 )) is asymptotically stable has a radius rP 0 equal to: rP 0 = min{l0 (P 0 ), l(P 0 )} .

(17)

Moreover in [3] some explicit algorithms computing l0 (P 0 ) and l(P 0 ) are given. They are summarized as follows: l02 (P 0 ) =

(b00 d00 + a00 c00 )2 ; (c00 )2 + (d00 )2

(18)

lω2 (P 0 ) =

λ21 [[Z 1 ]]2 + λ22 [[Z 2 ]]2 − 2λ1 λ2

Z 1 , Z 2 , [[Z 1 ]]2 [[Z 2 ]]2 −

Z 1 , Z 2 2

(19)

where, using the notation introduced in Sect.2: λ1 = (D0 )e (ω)(B 0 )e (ω) − ω 2 (D0 )o (ω)(B 0 )o (ω) + (C 0 )e (ω)(A0 )e (ω) − ω 2 (C 0 )o (ω)(A0 )o (ω) , λ2 = (D0 )e (ω)(B 0 )o (ω) + (D0 )o (ω)(B 0 )e (ω) + (C 0 )e (ω)(A0 )o (ω) + (C 0 )o (ω)(A0 )e (ω) , Z 1 = ((D0 )e (ω)P1 (ξ) + (D0 )o (ω)P2 (ξ), (C 0 )e (ω)P1 (ξ) + (C 0 )o (ω)P2 (ξ)) , Z 2 = ((D0 )e (ω)P2 (ξ) − ω 2 (D0 )o (ω)P1 (ξ), (C 0 )e (ω)P2 (ξ) − ω 2 (C 0 )o (ω)P1 (ξ)) , with • if n = 2l > 0: P1 (ξ) = ξ − ω 2 ξ 3 + · · · + (−1)l−1 ω 2l−2 ξ 2l−1 , P2 (ξ) = 1 − ω 2 ξ 2 + · · · + (−1)l ω 2l ξ 2l , • if n = 2l + 1: P1 (ξ) = ζ − ξ 2 ξ 3 + · · · + (−1)l ω 2l ξ 2l+1 , P2 (ξ) = 1 − ω 2 ξ 2 + · · · + (−1)l ω 2l ξ 2l . Strongly Robust Stability Radius: Algorithm. Referring to Theorem 51, if S denotes a ball of systems in Pn with a radius such that r(S) ≤

1 min{min(l0 (P ), l(P ))} 2 P ∈S

where l0 (P ) and l(P ) are computed in (18) and (19), then it is strongly robust. Let’s examine how it applies to the ﬁrst order pole placement design.

Strong robustness in adaptive control

5.3

53

First order pole placement design

Let’s apply these results to the case of ﬁrst order pole placement design (see Example 34). Fix S ⊂ P1 an open ball of systems. For any P ∈ P1 , we obtain that: lω (P ) = ∞, ∀ω > 0 and l0 (P ) =

−α|b0 | b20

+ (a0 + α)2

;

Therefore, if r(S) ≤

min

(a0 ,b0 )T ∈S

−α|b0 | 1 , 2 b20 + (a0 + α)2

(21)

then S is strongly robust. This can be reformulated as follows: if the diameter of S is less or equal than the Euclidean distance between the two lines L0T and LT previously deﬁned (see Example 34), then S is strongly robust. This result is consistent with our previous result: S is a strongly robust disc of systems if it lies in the region in between the two lines L0T and LT . In Fig. 2 we illustrate these results; S1 is a strongly robust disc of systems whereas S2 is not.

Fig. 2. Strongly robust spheres for the ﬁrst order case, pole located in α.

5.4

Computational aspect

The computation of the distance l(P 0 ) for a given P 0 ∈ Pn requires the minimization of lω (P 0 ) over all ω > 0. Unfortunately, the complexity of this expression of lω (P 0 ) increases dramatically with the order n of the system to be controlled. Indeed, already for n = 2, we should minimize a rational function in ω with a numerator of degree 12 in and with a denominator of degree 16.

54

M. Cadic and J.W. Polderman

In order to build an algorithm tractable even for higher orders, our idea is then to approximate the balls of systems by boxes of systems. In this way, the coeﬃcients parameterizing our system are now independent from each other, which simpliﬁes considerably the minimization problem. In this algorithm some Kharitonov-like results are used in order to check that the controller based on any system in the original box of systems leads to an asymptotically stable closed-loop system when applied to any other system in this box. Some partial results have been already obtained, but we did not expose them in this paper due to space limitation. They will appear in a short time delay in a Memorandum of the Faculty of Mathematical Sciences, University of Twente, The Netherlands.

6

Conclusion

In an Adaptive Control framework, the notion of Strong Robustness presents attractive features. Indeed, if in the ﬁrst phase, we can identify a strongly robust set of systems containing the system to be controlled, then we are sure that this sytem is stabilizable on the basis on any of its estimate lying in this strongly robust set, although some uncertainties on the parameters still remain. In this paper, we proposed an algorithm testing the Strong Robustness of a given ball of systems. However its computational complexity is not manageable when applied to systems of order n > 1. Another solution, approximating these balls by orthotopes of systems, should allow us to test Strong Robustness of a given box of system, even in the high order case. This alternative algorithm is still under investigation.

References 1. Polderman, J. W., Willems, J. C. (1998) Introduction to Mathematical Systems Theory. Springer-Verlag, New York, Inc. 2. Chapellat, H., Bhattacharyya, S. P. (1998) Robust Stability and Stabilization of Interval Plants. Robustness in Identiﬁcation and Control, International Workshop on Robustness in Identiﬁcation and Control edited by Milanese M., Tempo R., and Vicino A. 207–229. 3. Biernacki, R. M., Hwang H., Bhattacharyya, S. P. (1987) ”Robust Stability with Structured Real Parameter Perturbations”, IEEE Transactions Automatic Control, Vol. AC-32, No. 6, 495–506.

On composition of Dirac structures and its implications for control by interconnection Joaqu´ın Cervera1 , Arjan J. van der Schaft2 , and Alfonso Ba˜ nos1 1 2

Universidad de Murcia, Departamento de Inform´atica y Sistemas, Spain Fac. of Mathematical Sciences, University of Twente, Enschede, The Netherlands

Abstract. Network modeling of complex physical systems leads to a class of nonlinear systems, called Port-Controlled Hamiltonian Systems (PCH systems). These systems are geometrically deﬁned by a state space manifold of energy variables, a power-conserving interconnection formalized as a Dirac structure, together with the total stored energy and a resistive structure. Basic features of these systems include their compositionality properties (a power-conserving interconnection of PCH systems is again a PCH system), and their stability and stabilizability properties exploiting the energy function and the Casimir functions. In the present paper we further elaborate on the compositionality properties of Dirac structures by providing an explicit parametrization of all achievable closedloop Dirac structures in terms of their constituent parts. Amongst others this opens up the way to a complete characterization of the class of PCH systems which are stabilizable by interconnection with a PCH controller.

1

Introduction

Network modeling of complex physical systems (possibly containing components from diﬀerent physical domains) leads to a class of nonlinear systems, called port-controlled Hamiltonian systems, see e.g. [2,9,3,4,1,8]. Portcontrolled Hamiltonian systems are deﬁned by a Dirac structure (formalizing the power-conserving interconnection structure of the system), an energy function (the Hamiltonian), and a resistive structure. Key property of Dirac structures is that the power-conserving composition of Dirac structures again deﬁnes a Dirac structure, see [4,7]. This implies that any power-conserving interconnection of port-controlled Hamiltonian systems is also a port-controlled Hamiltonian system, with Dirac structure being the composition of the Dirac structures of its constituent parts and Hamiltonian the sum of the Hamiltonians. As a result power-conserving interconnections (in particular classical feedback interconnections) of port-controlled Hamiltonian systems can be studied to a large extent in terms of the composition of their Dirac structures. In particular the feedback interconnection of a given plant port-controlled Hamiltonian system with a yet to be speciﬁed port-controlled Hamiltonian controller system can be studied from the point of view of the composition of a given plant Dirac structure with a controller Dirac structure. Preliminary results concerning the achievable ”closed-loop” Dirac structures have been A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 55-63, 2003. Springer-Verlag Berlin Heidelberg 2003

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obtained in [7], but much more work is needed here since the parametrization of achievable Dirac structures obtained in [7] is still implicit. In particular, it does not yield an explicit parametrization of the obtainable Casimir functions of the closed-loop system, which is crucial for the passivity-based control of the port-controlled Hamiltonian system plant system, see e.g. [10,1,8,6]. Another point of view on Dirac structures is given by the fact that (constant) Dirac structures are in one-to-one correspondance with orthonormal maps, if one switches from power-variables (e.g., voltages and currents, or forces and velocities) to scattering variables, see [5,8]. This means that in principle all compositionality issues of Dirac structures can be also studied from this point of view. Since in general a Dirac structure is only given by an (implicit) relation (instead of a map) this may have important advantages. In the present paper we provide an explicit correspondance between kernel/image representations of Dirac structures and orthonormal maps. Furthermore, we show that the standard composition of two Dirac structures corresponds in scattering variables to the Redheﬀer star product of the corresponding orthonormal maps. Clearly, these results open up the way to give explicit formulae for composed Dirac structures and their Casimir functions, but on the other hand, they also provide a new angle to the classical study of feedback interconnections of systems in scattering variables. Indeed, using the classical relation between the small-gain property and passivity (see e.g. [8]) we may transfer properties of control interconnections in the small-gain/scattering framework to the passivity/Hamiltonian framework, and vice-versa. In particular, results concerning the parametrization of H∞ controllers are expected to be transferable to the control of port-controlled Hamiltonian systems.

2

Parametrization of kernel/image representations

Given a constant Dirac structure D there are inﬁnite (F, E) pairs representing it in kernel/image representation, but only one O matrix in scattering representation. The following theorem establishes the characterization of the set of (F, E) pairs representing D based on this unique scattering representation. Theorem 1 The set of (F, E) pairs representing a given Dirac structure D on F in kernel/image representation is given by (1) (F, E) : F = M (I + OT ), E = M (I − OT ) where O is the scattering representation for D and M is any l × l invertible matrix (l = dim F ). Proof. Any (F, E) pair corresponding to D on F can be expressed as F =A−B (2) E =A+B

On composition of Dirac structures

57

in terms of the pair of l × l invertible matrices (A, B) given by A = (E + F )/2 B = (E − F )/2

(3)

The proof for invertibility of A and B can be found in [8]. The orthonormal matrix O, scattering representation for D, can also be expressed in terms of A and B O = (F − E)−1 (F + E) = (−2B)−1 (2A) = −B −1 A .

(4)

From which it can be concluded A = −BO .

(5)

Using (5) in (2) F and E can be expressed as F = −B(I + O) E = B(I − O)

(6)

Given a kernel/image representation (F, E) for D any other (F , E ) is given by F = CF , E = CE, being C any l × l invertible matrix. This comes from the fact that, considering kernel representation, D = ker[E|F ] = ker C[E|F ] = ker[CE|CF ] = ker(E , F ) .

(7)

Using C = −OT B −1 in (6) this new representation for D is found F = I + OT E = I − OT

(8)

Taking (8) as base or canonical representation and using (7) the parametrization given in (1) is obtained.

Remark 1 Theorem 1, together with the result O = (F − E)−1 (F + E)

(9)

given in [8] and which deﬁnes the expression of the scattering representation O for a Dirac structure D given by a kernel/image representation (F, E), allows bidirectional conversion between scattering and kernel/image representations.

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J. Cervera, A.J. van der Schaft, and A. Ba˜ nos

Theorem 1 can be used to obtain the characterization of the Casimir functions of an implicit PCHS whose Dirac structure D (on F ) is given by its scattering representation O. Let OS be the submatrix of O composed by its ﬁrst s rows. Using (8),   I|S| I|S| − OST1 T   0 − OS = ES = . (10) −OST2 0 Proposition 1 Casimir functions C(x) of an implicit PCHS with s states whose Dirac structure is given by O are those whose gradients g(x) = ∂C ∂x (x) are such that g(x) = OST1 g(x) (11) 0 = OST2 g(x) or, in other words, such that g(x) is a ﬁxed point of OST1 and belongs to ker OST2 . Proof. It is inmediate from (10).

3

Dirac structures composition expression

In this section the expression for the scattering representation of the composition of two Dirac structures, in terms of the scattering representations of both individual Dirac structures, is given. Consider Dirac structures DA on F1 × F2 and DB on F2 × F3 (F2 is the space of shared ﬂow variables), with their connections splitted as B B ((f1 , e1 ), (f2A , eA 2 )) and ((f2 , e2 ), (f3 , e3 )) respectively (Fig. 1). It can be

f1

e1

DA

f 2 A f 2B e2A

e 2B

DB

f3

e3

Fig. 1. Dirac structures DA and DB and their connections split.

readily seen that the composition of two Dirac structures corresponds, in scattering representation (Fig. 2), to Redheﬀer star product. This fact implies v2A = z2B

(12)

z2A

(13)

=

v2B

On composition of Dirac structures

v1

z1

DA

z2A = v2B

v2A = z2B

DB

v3

59

z3

Fig. 2. Composition of DA and DB , DA ||DB , in scattering representation.

Theorem 2 The scattering representation for DAB = DA ||DB , in terms of OA and OB representing DA and DB respectively, is given by OAB = OAB11 + OAB12 OAB22 OAB21 where

(14)

1

OAB11 = OAB12 =

OA11 0 0 OB11 OA12 0 0 OB12

(15) (16)

(I − OB22 OA22 )−1 0 0 (I − OA22 OB22 )−1 OA21 0 = 0 OB21

OAB22 = OAB21

and this decomposition of OA and OB is used z1 v1 v1 OA11 OA12 = O = A OA21 OA22 z2A v2A v2A B B B OB11 OB12 v2 v2 z2 = OB = z3 v3 OB21 OB22 v3

OB22 I I OA22

(17) (18)

(19) (20)

Proof. Using (12), (13), (19) and (20) these expressions for z2A and z2B , depending only on external connections, can be obtained z2A = (I − OA22 OB22 )−1 (OA21 v1 + OA22 OB21 v3 ) z2B 1

= (I − OB22 OA22 )

−1

(OB21 v3 + OB22 OA21 v1 )

(21) (22)

In this version of the theorem invertibility of (I −OB22 OA22 ) and (I −OA22 OB22 ) is assumed. Some comments on this topic after the proof.

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J. Cervera, A.J. van der Schaft, and A. Ba˜ nos

Substituting (22) and (21) in (19) and (20) yields this expression for OAB = OA ||OB , which can be rewritten as (14)

OAB = −1 − OA22 )−1 OA21 OA12 (I − OB22 OA22 )−1 OB21 OA11 + OA12 (OB 22 −1 −1 OB12 (I − OA22 OB22 ) OA21 OB11 + OB12 (OA − OB22 )−1 OB21 22

(23)

In the particular case in which for one of the individual Dirac structures, say DB , all its ports are connected to the other one, i.e. dim F3 = 0, OB = OB11 , and (14) reduces to 2 −1 OAB = OA11 + OA12 (OB − OA22 )−1 OA21 .

(24)

Both in (14) and (24) invertibility of certain terms is being assumed. The authors have been working on the characterization of cases in which this assumption is not satisﬁed. It seems this situation has to do with the fact that these are not useful combinations, in the sense that in these conﬁgurations there are not used connections (connections not aﬀecting neither being aﬀected by the environment), being these conﬁgurations structures achievable with lower dimensional DA and DB . In Fig. 3 it is shown an example of such a situation (corresponding to the simpliﬁed case, (3)). Dirac structures

v1a

v1b

z1a

z1b

v2Aa

z2Ba

v2Ab

z2Bb

z2Aa

v2Ba

z2Ab

v 2Bb

Fig. 3. Example of composition of Dirac structures with not used connections.

to be composed are given by  1 0 OA11 OA12 OA = = 0 OA21 OA22 0

00 00 01 10

 0 1 ; 0 0

OB = OB11 =

10 01

Anyway, the authors are working on an alternative theorem and associated expression which do not need this assumption at all, and that they hope to publish soon. 2

−1 Invertibility of (OB − OA22 ) is assumed.

On composition of Dirac structures

4

61

Application example

One of the applications of the results presented in this work consists of a systematic method to characterize the Casimir functions of the implicit PCHS corresponding to the composition of a given plant PCHS, P , and a certain controller PCHS, C. Furthermore, this idea could be extended to the description of all the Casimir functions achievable by a proper choice of C. The method consists of these steps: • If DP or DC , Dirac structures corresponding to P and C, are expressed in kernel/image representation, obtain their scattering representation, OP and OC , using (9). • Obtain OP C from OP and OC using (14). • Compute Casimirs from OP C , using Prop. 1. This method will be illustrated with the plant-controller system represented in Fig. 4, where plant PCHS is just a mass m and controller PCHS is composed by another mass, mC , together with two springs, with constants k and kC , and a damper, with damping constant b. DP and DC are given by   1 2 2 OP11 OP12 OP = = (1/3)  −2 −1 2  OP21 OP22 −2 2 −1   3 4 2 4 2  −4 −3 2 4 2    OC11 OC12 2 −2 −1 −2 −6  = (1/7)  OC =   OC21 OC22  −4 4 2 −3 2  −2 2 −6 2 1 (ﬁrst step is not necessary in this case). There are ﬁve states in the resul-

kc

k mc

m

e

b CONTROLLER PCHS

PLANT PCHS

Fig. 4. Example of plant-controller PCHS.

ting system, (q, p, ∆qC , pC , ∆q), corresponding respectively to plant position, mass m momentum, relative displacement of spring with kC constant (respect

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J. Cervera, A.J. van der Schaft, and A. Ba˜ nos

to its equilibrium position), mass mC momentum and relative displacement of spring with k constant. e is the external force applied to the plant. Second step yields 

OP C

3 8  −8 −2   −2 2 = (1/11)   −2 2   −6 6 −2 2

 −2 2 −6 2 −2 2 −6 2   5 6 4 6  −6 −5 4 6   4 −4 1 −4  −6 6 4 −5

After third step, this is the space where Casimirs’ gradients should lie ker ES = span{(1, 0, −1, 0, −1)} An example of Casimir satisfying this constraint is C(x) = ∆q − (q − ∆qC ).

5

Conclusions

In the present work it has been shown how the composition of two Dirac structures is equivalent in scattering representation to Redheﬀer star product. Using this fact it has been obtained a ﬁrst expression for the scattering representation of the composition of two Dirac structures, assumming invertibility of certain terms, assumption which excludes some cases and which is aimed to be avoided in a future version of this expression. Because of the oneto-one correspondance between Dirac structures and orthonormal matrices, this expression also gives a parametrization of achievable closed loop Dirac structures for a given ﬁrst Dirac structure to be combined with a second one freely chosen. Finally, it has also been obtained the expression which connects scattering representation with kernel/image representation. As a next step a second expression for composition of Dirac structures, without invertibility assumption, is being looked for. Also the meaning of this noninvertibility is being studied. After this, parametrization of Casimir functions achievable by interconnection will be further explored. With this results, one of the ﬁrst objectives will be the characterization of the PCHS’s stabilizable by interconnection.

Acknowledgements This work has been partially supported by Fundaci´ on S´eneca, Centro de Coordinaci´ on de la Investigaci´ on, under program Becas de Formaci´ on del Personal Investigador.

On composition of Dirac structures

63

References 1. M. Dalsmo & A.J. van der Schaft, “On representations and integrability of mathematical structures in energy-conserving physical systems”, SIAM J. Control and Optimization, 37, pp. 54-91, 1999. 2. B.M. Maschke, A.J. van der Schaft & P.C. Breedveld, “An intrinsic Hamiltonian formulation of network dynamics: Non-standard Poisson structures and gyrators”, J. Franklin Inst., 329, pp. 923-966, 1992. 3. B.M. Maschke, A.J. van der Schaft, “Interconnection of systems: the network paradigm”, Proc. 35th IEEE CDC, Kobe, Japan, pp. 207-212, 1996. 4. B.M. Maschke, A.J. van der Schaft, “Interconnected mechanical systems, Parts I and II”, pp. 1–30 in Modelling and Control of Mechanical Systems, Editors A. Astolﬁ, D.J.N. Limebeer, C. Melchiorri, A. Tornamb`e, R.B. Vinter, Imperial College Press, London, 1997. 5. B. Maschke, A.J. van der Schaft, “Scattering representation of Dirac structures and interconnection in network models”, in Mathematical Theory of Networks and Systems, A. Beghi, L. Finesso, G. Picci (Eds.), Il Poligrafo, Padua, pp.305– 308, 1998. 6. R. Ortega, A.J. van der Schaft, I. Mareels, & B.M. Maschke, “Putting energy back in control”, Control Systems Magazine, 21, pp. 18–33, 2001. 7. A.J. van der Schaft, “Interconnection and geometry”, in The Mathematics of Systems and Control, From Intelligent Control to Behavioral Systems (eds. J.W. Polderman, H.L. Trentelman), Groningen, 1999. 8. A.J. van der Schaft, L2 -Gain and Passivity Techniques in Nonlinear Control, 2nd revised and enlarged edition, Springer-Verlag, Springer Communications and Control Engineering series, p. xvi+249, London, 2000 (ﬁrst edition Lect. Notes in Control and Inf. Sciences, vol. 218, Springer-Verlag, Berlin, 1996). 9. A.J. van der Schaft & B.M. Maschke, “The Hamiltonian formulation of energy conserving physical systems with external ports”, Archiv f¨ ur Elektronik ¨ und Ubertragungstechnik, 49, pp. 362-371, 1995. 10. S. Stramigioli, B.M. Maschke & A.J. van der Schaft, “Passive output feedback and port interconnection”, in Proc. 4th IFAC NOLCOS, Enschede, pp. 613-618, 1998.

Adaptive nonlinear excitation control of synchronous generators Gilney Damm1 , Riccardo Marino2 , and Fran¸coise Lamnabhi-Lagarrigue1 1

2

Laboratoire des Signaux et Syst`emes, CNRS Sup´elec, 3, rue Joliot-Curie 91192 Gif-sur-Yvette Cedex, France [email protected], [email protected] Dipartimento di Ingegneria Elettronica, Universit`a di Roma Tor Vergata, via di Tor Vergata 110 00133 Rome, Italy [email protected]

Abstract. In this paper, continuing the line of our previous works, a nonlinear adaptive excitation control is designed for a synchronous generator modeled by a standard third order model on the basis of the physically available measurements of relative angular speed, active electric power and terminal voltage. The power angle, which is a crucial variable for the excitation control, is not assumed to be available for feedback, as mechanical power is also considered as an unknown variable. The feedback control is supposed to achieve transient stabilization and voltage regulation when faults occur to the turbines so that the mechanical power may permanently take any (unknown) value within its physical bounds. Transient stabilization and voltage regulation are achieved by a nonlinear adaptive controller, which generates both on-line converging estimates of the mechanical power and a trajectory to be followed by the power angle that converges to the new equilibrium point compatible with the required terminal voltage. The main contributions here, compared with our previous works, is the use of on-line computation and tracking of equilibrium power angle, and the proof of exponential stability of the closed loop system for states and parameter estimates, instead of the previous asymptotical one.

1

Introduction

The problem of stabilization of power generators is a classical power systems and control systems problem. It has been approached for some time now in many works initially by classic control and linear modern control techniques with good results, but only locally valid. Recently this problem has been treated by nonlinear methods as Lyapunov techniques (see for instance [1], [2], [3]). Recently, feedback linearization techniques were proposed in [4], [5] and [6] to design stabilizing controls with the purpose of enlarging the stability region of the operating condition. Nonlinear adaptive controls are also proposed in [7] and [8]. The nonlinear feedback control algorithms so far proposed in the literature make use of power angle and mechanical power measurements which are A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 65-76, 2003. Springer-Verlag Berlin Heidelberg 2003

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G. Damm, R. Marino, and F. Lamnabhi-Lagarrigue

physically not available and have the diﬃculty of determining the faulted equilibrium value which is compatible with the required terminal voltage once the fault (mechanical or electrical failure) has occurred. Following the lines of our previous works [9] and [10] we make use of the standard third order model used in [8] (see [11] and [12]) to show that the terminal voltage, the relative angular speed and the active electric power (which are actually measurable and available for feedback) are state variables in the physical region of the state space In this paper, continuing our previous results, we study the zero dynamics of the system, with respect to the terminal voltage, for typical values to show the existence of two equilibrium points, one stable and one unstable. This is a motivation for the use of nonlinear control instead of the classical one computed using the approximate linearized model around the stable point. In our previous work [9], a nonlinear adaptive feedback control on the basis of physically available measurements (relative angular speed, active electric power and terminal voltage) was presented. There, when a perturbation occurred, the system was maintained in the old equilibrium point, no longer valid, causing wrong outputs, while the estimation of the new equilibrium point was made. Then, a trajectory to drive the system to the new equilibrium point was computed. In the present work, estimation of the new equilibrium point and computation of the trajectory that drives the system there are done on-line. Global exponential stability is guaranteed for the whole closed loop system to this new previously unknown equilibrium point. There is a considerable improvement with respect to the output errors with the on-line procedure, and robustness is guaranteed by the exponential stability.

2

Dynamical model

As in [9], we consider the simpliﬁed mechanical model expressed in per unit as δ˙ = ω ω˙ = −

ωs D ω+ (Pm − Pe ) H H

(1)

where: δ(rad) is the power angle of the generator relative to the angle of the inﬁnite bus rotating at synchronous speed ωs ; ω(rad/s) is the angular speed of the generator relative to the synchronous speed ωs i.e. ω = ωg − ωs with ωg being the generator angular speed; H(s) is the per unit inertia constant; D(p.u.) is the per unit damping constant; Pm (p.u.) is the per unit mechanical input power; Pe (p.u.) is the per unit active electric power delivered by the generator to the inﬁnite bus. Note that the expression ωs2 /ωg is simpliﬁed as

Adaptive nonlinear excitation control of synchronous generators

67

ωs2 /ωg ωs in the right-hand side of (1). The active and reactive powers are given by Vs Eq sin(δ) Xds Vs V2 Q= Eq cos(δ) − s Xds Xds

Pe =

(2) (3)

where: Eq (p.u.) is the quadrature’s EMF; Vs (p.u.) is the voltage at the inﬁnite bus; Xds = XT + 12 XL + Xd (p.u.) is the total reactance which takes into account Xd (p.u.), the generator direct axis reactance, XL (p.u.), the transmission line reactance, and XT (p.u.), the reactance of the transformer, and

the deﬁnition XS = XT + 12 XL . The quadrature EMF, Eq , and the transient quadrature EMF, Eq , are related by

Eq =

Xds Xd − Xd Vs cos(δ) Eq − Xds Xds

(4)

while the dynamics of Eq are given by dEq 1 = (Kc uf − Eq ) dt Td0

(5)

in which: Xds = XT + 12 XL + Xd (p.u.) with Xd denoting the generator direct axis transient reactance; uf (p.u.) is the input to the (SCR) ampliﬁer of the generator; Kc is the gain of the excitation ampliﬁer; Td0 (s) is the direct axis short circuit time constant. Especially because Pe is measurable while Eq is not, it is convenient to express the state space model using (δ, ω, Pe ) as states, which are equivalent states as long as the power angle δ remains in the open set 0 < δ < π, as follows.

δ˙ = ω ωs D ω− (Pe − Pm ) H H Vs 1 1 Vs P˙e = − Pe + sin(δ)[Kc uf + Td0 (Xd − Xd ) ω sin(δ)] Td0 Td0 Xds Xds ω˙ = −

Pe ω cot(δ)} + Td0

(6) in which (δ, ω, Pe ) is the state and uf is the control input.

68

G. Damm, R. Marino, and F. Lamnabhi-Lagarrigue

Note that when δ is near 0 or near π the eﬀect of the input uf on the overall dynamics is greatly reduced. Note also that here we have introduced the notation Td0 =

Xds Td0 Xds

The generator terminal voltage modulus is given by Vt =

Xs2 Pe2 2Xs Xd Xd2 Vs2 + Pe cot(δ) + 2 2 2 X Xds Vs sin (δ) ds

12

which is the output of the system to be regulated to its reference value Vtr = 1(p.u.) 2.1

Power Angle

The power angle is not measurable and is also not a physical variable to be regulated; the only physical variable to be regulated is the output Vt , while (Vt , ω, Pe ) are measured and are available for feedback action. As a matter of fact (Vt , ω, Pe ) is an equivalent state for the model (6) (as proved in [9]) δ = arccotg

Vs X s Pe

Xd Vs + − Xds

Vt2

X2 − s2 Pe2 Vs

(7)

If the parameters (Vs , Xs , Xd , Xds ) are known, state measurements are available. From (7) it follows that in order to regulate the terminal voltage Vt to its reference value (Vtr = 1(p.u.)) δ should be regulated to δs = arccotg

Vs X s Pm

Xd Vs + − Xds

Vtr2

X2 2 − s2 Pm Vs

(8)

From a physical viewpoint the natural choice of state variables is (Vt , ω, Pe ) which are measurable. The state feedback control task is to make the stability region of the stable equilibrium point (Vtr , 0, Pm ) as large as possible. In fact the parameter Pm may abruptly change to an unknown faulted value Pmf due to turbine failures so that (Vtr , 0, Pm ) may not belong to the region of attraction of the faulted equilibrium point (Vtr , 0, Pmf ). The state feedback control should be design so that typical turbine failures do not cause instabilities and consequently loss of synchronism and inability to achieve voltage regulation.

Adaptive nonlinear excitation control of synchronous generators

69

A reduction from Pm to (Pm )f of the mechanical power generated by the turbine, changes the operating condition: the new operating condition (δs )f is the solution of −

(Pm )f sin(δ)f =0 + Pm sin(δs )

and since (Pm )f is typically unknown, the corresponding new stable operating condition (δs )f is also unknown. 2.2

Zero Dynamics

If we regulate the voltage output (Vt ) to its reference value (Vtr ), the zero dynamics will be given by δ˙ = ω ωs Xd D ω+ (Pm + V 2 sin(δ) cos(δ) H H Xs Xds s X2 Vs sin(δ) Vtr2 − 2d Vs2 sin2 (δ) − Xs Xds

ω˙ = −

which are very complex, and for some initial conditions or parameters values may become unstable. If we use the values deﬁned in [9] we may plot ω˙ as a function of ω and δ, there will then be two points of δ that satisfy the equilibrium of the zero dynamics. These points (the two real ones) are δ = 1.26 and δ = 2.96. Linearizing the system around each one of these two points, we will have as eigenvalues the pairs [−0.31 − 7.58I, −0.31 + 7.58I] and [−13.86, +13.86] respectively. Thus we have shown the existence of two equilibrium points, one stable and one unstable. There will then be an attraction region for the stable one. If one is driven out of this region (by initial conditions or by a fault), the controller will not act regulating ω and δ and the system will become unstable. This shows that using the output error voltage as the only error signal may be dangerous as one may regulate this voltage and loose stability.

3

Adaptive controller and main result

In this section we present the calculation of the adaptive controller as in [9]. Our main result is then to prove the global exponential stability of the whole system, with parameter exponential convergence. The model (6) is rewritten substituting Pm by θ(t) which is a possibly time-varying disturbance: this parameter is assumed to be unknown and to

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G. Damm, R. Marino, and F. Lamnabhi-Lagarrigue

belong to the compact set [θm , θM ]: where the lower and upper bounds θm , θM are known. Let f (θ, x) be a C 3 reference signal to be tracked. Deﬁne (λ1 > 0)

˜ = δ(t) − f (θ, x) δ(t) ω ∗ = −λ1 δ˜ + f (θ,˙ x) ω ˜ = ω − ω ∗ = ω + λ1 δ˜ − f (θ,˙ x) so that the ﬁrst two equations in (6) are rewritten as

˙ δ˜ = −λ1 δ˜ + ω ˜ ωs D ¨ x) (θ(t) − Pe ) − λ21 δ˜ + λ1 ω ω ˜˙ = − ω + ˜ − f (θ, H H Deﬁne (λ2 > 0, k > 0) the reference signal for Pe as Pe∗ =

H ωs

1 ωs 2 D ¨ x) + λ2 ω ˜ − f (θ, ˜ + δ˜ + k ω ˜ + θˆ − ω − λ21 δ˜ + λ1 ω H 4 H

while θˆ is an estimate of θ = Pm and P˜e = Pe − Pe∗ so that (6) may be ˆ rewritten as (θ˜ = θ − θ) ˙ δ˜ = −λ1 δ˜ + ω ˜

ωs ˜ k ωs 2 ωs ˜ Pe − ˜− ω ˜+ ω ˜˙ = −δ˜ − λ2 ω θ H 4 H H 1 Vs (Xd − Xd )Vs2 sin(δ)K u + ω sin2 (δ) + Pe ω cot(δ) P˜˙e = − Pe + c f Td0 Xds Td0 Xds Xds D H 2 − ˜) −λ1 + 1 + λ1 (−λ1 δ˜ + ω ωs H k ωs 2 ωs D D 2˜ ¨ ˜ − Pe − f (θ, x) − ω − λ1 δ + λ1 ω + − + λ1 + λ2 + H 4 H H H D k ωs 2 ˆ ˆ˙ − − + λ1 + λ2 + θ−θ H 4 H k ωs 2 ˜ D ¨ H ¨˙ D f (θ, x) + f (θ, x) θ+ − − + λ1 + λ2 + H 4 H ωs ωs Deﬁning λ3 > 0, we then propose the control law

Adaptive nonlinear excitation control of synchronous generators

71

Xds Td0 φ0 Vs Kc sin(δ) 1 (Xd − Xd ) 2 φ0 = Pe − Vs ω sin2 (δ) − Pe ω cot(δ) Td0 Xds Xds D H 2 + ˜) −λ1 + 1 + λ1 (−λ1 δ˜ + ω ωs H k ωs 2 ωs D D 2˜ ¨ ˜ − Pe − f (θ, x) − ω − λ1 δ + λ1 ω + − + λ1 + λ2 + H 4 H H H 2 D k ωs ˙ + − + λ1 + λ2 + θˆ + θˆ H 4 H 2 k k ωs 2 D ¨ ωs H ¨˙ D P˜e − f (θ, − x) − f (θ, x) − λ3 P˜e + ˜ω − + λ1 + λ2 + 4 H 4 H ωs ωs H

uf =

then, the closed loop system becomes ˜˙δ = −λ1 δ˜ + ω ˜

ωs ˜ k ωs 2 ωs ˜ Pe − ˜− ω ˜+ ω ˜˙ = −δ˜ − λ2 ω θ H 4 H H ωs D k ωs 2 ˜ ω ˜ − λ3 P˜e − − + λ1 + λ2 + P˜˙e = θ H H 4 H 2 k k ωs 2 D P˜e − − + λ1 + λ2 + 4 H 4 H

(9)

The adaptation law is (γ is a positive adaptation gain) ˙ θˆ = γP roj

D k ωs 2 ωs P˜e − λ1 − λ2 − +ω ˜ , θˆ H 4 H H

(10)

ˆ is the smooth projection algorithm introduced in [13] where P roj(y, θ) ˆ = y, ˆ ≤0 P roj(y, θ) if p(θ) ˆ = y, ˆ ≥ 0 and gradp(θ), ˆ y ≤ 0 P roj(y, θ) if p(θ) ˆ = [1 − p(θ)|grad ˆ ˆ P roj(y, θ) p(θ)|], otherwise (11) with

p(θ) =

(θ −

θM +θm 2 m ) − ( θM −θ ) 2 2 θM −θm 2 + 2( 2 )

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for an arbitrary positive constant which guarantees in particular that: ˆ ≤ θM + i) θm − ≤ θ(t) ˆ ≤ |y| ii) |P roj(y, θ)| ˆ roj(y, θ) ˆ ≥ (θ − θ)y ˆ iii) (θ − θ)P Consider the function

W =

1 ˜2 2 (δ + ω ˜ 2 + P˜e ) 2

(12)

whose time derivative, according to (9), is

. k ωs 2 2 ωs 2 W = −λ1 δ˜2 − λ2 ω ˜ 2 − λ3 P˜e + ω ˜ θ˜ − ω ˜ H 4 H 2 D k ωs 2 ˜ ˜ k k ωs 2 D 2 P˜e − − + λ1 + λ2 + θ Pe − − + λ1 + λ2 + H 4 H 4 H 4 H Completing the squares, we obtain the inequality 2 2 ˙ ≤ −λ1 δ˜2 − λ2 ω W ˜ 2 − λ3 P˜e + θ˜2 k

(13)

which guarantees arbitrary L∞ robustness from the parameter error θ˜ to the ˜ω tracking errors δ, ˜ , P˜e . The projection algorithms (11) guarantee that θ˜ is bounded, and, by ˙ ˜ ω virtue of (12) and (13), that δ, ˜ and P˜e are bounded. Therefore, θˆ is bounded. Integrating (13), we have for every t ≥ t0 ≥ 0

t

2

(λ1 δ˜2 + λ2 ω ˜ 2 + λ3 P˜e )dτ +

− t0

2 k

t

θ˜2 dτ ≥ W (t) − W (t0 ) t0

Since W (t) ≥ 0 and, by virtue of the projection algorithm (11), ˜ ≤ θM − θm + θ(t) it follows that

t

(λ1 δ˜2 + λ2 ω ˜ 2 )dτ ≤ W (t0 ) + t0

2 (θM − θm + )2 (t − t0 ) k

Adaptive nonlinear excitation control of synchronous generators

73

which, if W (t0 ) = 0 (i.e. t0 is a time before the occurrence of the fault), ˜ caused implies arbitrary L2 attenuation (by a factor k) of the errors δ˜ and ω by the fault. To analyze the asymptotic behavior of the adaptive control, we consider the function

V =

1 ˜2 1 1 ˜2 2 (δ + ω ˜ 2 + P˜e ) + θ 2 2γ

The projection estimation algorithm (11) is designed so that the time derivative of V satisﬁes 2 V˙ ≤ −λ1 δ˜2 − λ2 ω ˜ 2 − λ3 P˜e

(14)

Integrating (14), we have

t

2

(λ1 δ˜2 + λ2 ω ˜ 2 + λ3 P˜e )dτ ≤ V (0) − V (∞) < ∞

limt→∞ t0

˜˙ ω From the boundedness of δ, ˜˙ and P˜˙e , and Barbalat’s Lemma (see [14], [15]) it follows that   ˜ δ(t)   ˜ (t) limt→∞ ω =0 P˜e (t) We may now rewrite the closed loop system following the normal form: x ˜˙ = A˜ x + Ω T θ˜ ˙ θ˜ = −ΛΩ x ˜ which leads to:    1 0 −λ1 0  ˜ +  ωs  θ˜ x ˜˙ =  −1 − (λ2 + c2 ) − ωHs H x ωs k 2 −c1 0 − λ3 + 4 c1 H ˙ ˜ θ˜ = −γ 0 ωHs −c1 x 

(15)

where c1 and c2 , as c3 on next equation, are constants. And then computing: ΩΩ T =

ωs2 + c21 = c3 > 0 H2

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We then may show by persistency of excitation that x ˜ and θ˜ will be globally (for the model validity region) exponentially stable, then all error signals go exponentially to zero, for all C 3 f (θ, x). This is also valid then for the particular case where f (θ, x) = δr where δr is given by equation (8). But, since δr is a one-to-one smooth function of θ, it will converge to the correct equilibrium value δs as θ˜ converges to 0, i.e. the reference trajectory will converge to the unknown equilibrium point and then lim t→∞ (δ −δs ) = 0.

4

Simulation results

In this section some simulation results are given with reference to the eightmachine power system network reported in [5] with the following data: ωs = 314.159 rad/s D = 5 p.u. H = 8s Kc = 1 Xd = 1.863 p.u. Td0 = 6.9s Xd = 0.257 p.u. XT = 0.127 p.u. XL = 0.4853 p.u. The operating point is δs = 72o , Pm = 0.9 p.u., ω0 = 0 to which corresponds Vt = 1 p.u., with Vs = 1 p.u.. It was considered a fast reduction of the mechanical input power, and simulated according to the following sequences 1. The system is in pre-faulted state. 2. At t = 0.5s the mechanical input power begins to decrease. 3. At t = 1.5s the mechanical input power is 50% of the initial value. The simulations were carried out using as control parameters λi = 20 1 ≤ i ≤ 3 γ=1 k = 0.1 Fig. 1.a1) shows that the calculated power angle matches perfectly the real one. It also shows that the trajectory for the power angle (δr ) goes smoothly to its ﬁnal value, and that δ follows it perfectly. In Fig. 1.a2) one may see that the estimation of the mechanical power is accurate, it may be very fast if we change the parameters, and specially if larger errors are accepted for the state and output variables. This may be understood by looking at equation (10). The electrical power is also correctly driven to the mechanical one as we see in Fig. 1.a3. The same may be observed in Fig. 1.b1) for the rotor velocity. Fig. 1.b2) shows how the output voltage drops during the fault, and goes to its correct value when the system is driven to the correct equilibrium point. If estimation was not correct, there would be a steady state error. Finally, one can see in Fig. 1.b3) that the control signal is very smooth and is kept inside the prescribed bounds. Note that during all time, errors are very small. They can be made even smaller by increasing the parameter k.

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(a) (b) Fig. 1. a1) Real δ (-), Calculated δ (-.), δr (- -) a2) Pm (-),θˆ (- -) a3) Pm (-), Pe (- -) b1) ω b2) Vt b3) Control signal

5

Conclusions

In a previous work we have computed the zero dynamics of the system with respect to the terminal voltage having then obtained a highly nonlinear second order dynamics. Based on typical values, we show here that there is one stable and one unstable points, and then, an attraction region for the stable one. This is a motivation to be concerned with all the state vector and not only with the output voltage since, even keeping it regulated to its reference value one may ﬁnd instability for the whole system. It is also a motivation for the nonlinear control as the system may always be driven to an unstable point where a linear control, specially one designed using the linearized system around the stable point, will not be able to stabilize it. Finally, using the same controller as in previous works, we prove the exponential stability of the closed loop system. We also prove that the estimate of the parameter converges exponentially to its true value. The system may be driven arbitrarily fast to the new equilibrium point. The only restriction will be the magnitude of the control signal and the accepted error signal. Our present research includes the problem of transmission line failure. We have also started the procedure to do practical implementations to verify ours simulations. The multi-machine problem will then be the next step.

Acknowledgments The ﬁrst author would like to acknowledge the ﬁnancial support of CAPES Foundation.

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References 1. Siddiquee, M., W. (1968) Transient stability of an a.c. generator by Lyapunov direct method. Int. J. of Control 8, 131-144 2. Pai, M., A. and Rai, V. (1974) Lyapunov-Popov stability analysis of a synchronous machine with ﬂux decay and voltage regulator. Int. J. of Control 19, 817-826 3. Marino, R. and Nicosia, S. (1974) Hamiltonian-type Lyapunov functions. Int. J. of Control 19, 817-826 4. Marino, R. (1984) An example of nonlinear regulator. IEEE Trans. Automatic Control 29, 276-279 5. Gao, L., Chen, L., Fan, Y. and Ma, H. (1992) A nonlinear control design for power systems. Automatica 28, 975-979 6. Wang, Y., Hill, D. J., Middleton, R. H. and Gao, L. (1993) Transient stability enhancement and voltage regulation of power systems. IEEE Trans. Power Systems 8, 620-627 7. Bazanella, A., Silva, A. S., Kokotovic, P. (1997) Lyapunov design of excitation control for synchronous machines. Proc. 36th IEEE - CDC, San Diego, CA 8. Wang, Y., Hill, D. J., Middleton, R. H. and Gao, L. (1994) Transient stabilization of power systems with an adaptive control law. Automatica 30, 1409-1413 9. Marino, R., Damm, G.R., Lamnabhi-Lagarrigue, F. (2000) Adaptive Nonlinear Excitation Control of Synchronous Generators with Unknown Mechanical Power. book - Nonlinear Control in the Year 2000 - Springer–Verlang 10. Damm, G.R., Lamnabhi-Lagarrigue, F., Marino, R. (2001) Adaptive Nonlinear Excitation Control of Synchronous Generators with Unknown Mechanical Power. 1st IFAC Symposium on System Structure and Control, Prague, Czech Republic 11. Bergen, A. R. (1989). Power Systems Analysis. Prentice Hall, Englewood Cliﬀs, NJ 12. Wang, Y. and Hill, D. J. (1996) Robust nonlinear coordinated control of power systems. Automatica 32, 611-618 13. Pomet, J. and Praly, L. (1992) Adaptive nonlinear regulation: estimation from the Lyapunov equation. IEEE Trans. Automatic Control 37, 729-740 14. Narendra, K. S. and Annaswamy, A. M. (1989). Adaptive Systems. Prentice Hall, Englewood Cliﬀs, NJ 15. Marino, R. and Tomei, P. (1995). Nonlinear Control Design - Geometric, Adaptive and Robust. Prentice Hall, Hemel Hempstead

Parameter identiﬁcation for nonlinear pneumatic cylinder actuators N.A. Daw1 , J. Wang1 , Q.H. Wu1 , J. Chen2 , and Y. Zhao2 1

2

Department of Electrical Engineering and Electronics, University of Liverpool, Brownlow Hill, Liverpool L69 3GJ, UK. E-mail: [email protected] Department of Mechanical Engineering, Shandong University of Science and Technology, P.R. China

Abstract. Pneumatic actuators exhibit highly nonlinear characteristics due to air compressibility, signiﬁcant friction presence and the nonlinearities of control valves. The unknown nonlinear parameters can not be directly measured once the actuators have been manufactured and assembled, which causes a great diﬃculty in pneumatic system modeling and control. A learning algorithm has been developed in this paper to identify the unknown pneumatic system parameters. The algorithm is initially developed and tested using the data generated by simulations. Then the algorithm has been extended onto the parameter identiﬁcation using the data obtained from the real system measurement. The results revealed the characteristics of uneven distribution of friction parameters which are position, velocity, moving direction dependent. The results obtained in the paper can provide the manufacturers with the observation to the characteristics inside pneumatic cylinders.

1

Introduction

Pneumatic actuators are widely employed for position and speed control applications when cheap, clean, simple and safe operating conditions are required. In recent years, low cost microprocessors and pneumatic components become available in the market, which made it possible to adopt more sophisticated control strategies in pneumatic system control. Hence, investigations have been initiated for employing pneumatic actuators to accomplish more sophisticated motion control tasks [3], [5], [7], [9], and [10]. However, pneumatic actuators exhibit highly nonlinear behaviours that are associated with compressibility of air and complex friction characteristics [8]. The friction forces are mainly caused by their sliding seals which are used to prevent air leakage and they are unevenly distributed along pneumatic cylinders. The complicated nonlinear frictions cause a great diﬃculty to drive the pneumatic actuators to perform accurate position control. To address the problems, better understanding of system nonlinear characteristics is required. Therefore, an identiﬁcation method is developed in this paper in order to estimate the unevenly distributed dynamic friction parameters along the pneumatic cylinders. The pneumatic actuator system model has been obtained and validated in the authors’ previous work [12]. Based on the model, a Genetic Algorithm A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 77-88, 2003. Springer-Verlag Berlin Heidelberg 2003

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has been developed to identify the unknown friction parameters. The algorithm is initially tested in a simulation environment using the data purely generated by simulations. Once the algorithm has been proved reliable and robust, the work is moved onto parameter identiﬁcation using the data obtained from the real system measurement. A test rig to host both rodded and rodless cylinders was set up in the research laboratory at the University of Liverpool. The test rig is a PC based control/data acquisition system, which consists of the supporting frame, a PC, a proportional ﬂow control valve, rodded and rodless cylinders, a position sensor, two pressure sensors, an air compressor, a stabiliser, interface circuits between the computer and the pneumatic components. The open-loop system responses under diﬀerent conditions have been obtained using this test rig. The results revealed the characteristics of uneven distribution of frictions parameters which are position, velocity, moving direction dependent. The results obtained in the paper can provide the manufacturers with observations on the characteristics inside cylinders.

2

Mathematical model of pneumatic actuators

The following nomenclature will be used in the paper: a, b - Subscripts for inlet and outlet chambers respectively; A - Ram area (m2 ); Cd - Discharge coeﬃcient; Ff - Viscous friction coeﬃcient; k - Speciﬁc heat coeﬃcient; l - Stroke length (m); m - Mass ﬂow rate (Kg/s); M - Payload (Kg); Pd - Down stream pressure (P si); Pe - Exhaust Pressure (P si); Ps - Supply pressure (P si); Pu Up stream pressure (P si); R - Universal gas constant; Ts - Supply temperature (K); V - Volume (m3 ); w - Port width (m); Xa,b - Spool displacement of Valves (m); x - Load position (m). The constants appeared in the system model are listed below: k = 1 1.4, Cd = 0.8, Ps = 6 × 105 N/m2 , Pe = 1 × 105 N/m2 , Ck =

2 k−1

k+1 k+1 k−1

2

= 3.864, Cr =

2 k+1

k k−1

= 0.528,

Ts = 293K, R = 287 J/Kg K . An analysis of dynamic behaviour of a pneumatic system usually requires individual mathematical descriptions of the dynamics of the three component parts of the system: (i) the valve, (ii) the actuator, and (iii) the load. Such an analysis is presented below with reference to the co-ordinate system illustrated in Figure 1.

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Fig. 1 Co-ordinate system of a pneumatic cylinder

2.1

Flow relationships for control valves

With the assumption of constant supply and exhaust pressures, the mass ﬂow rates across the two control ports of the control valves can be regarded as a function of the valve displacement and the chamber pressure for references, see [1] and [4]. According to the standard oriﬁce theory, the mass ﬂow rate through the valve oriﬁce takes the form 1

m = Cd C0 wXPu f˜(Pr )/Tu2

(1)

where Tu is the up-stream temperature, Pr = Pd /Pu is the ratio between the down-stream and up-stream pressures at the oriﬁce and f˜(Pr ) =

Patm /Pu < Pr ≤ Cr

1, 2/k Ck [Pr

−

(k+1)/k 1 Pr ]2 ,

(2)

C r < Pr < 1

It can be shown that the function f˜(.) and its derivative are continuous with respect to Pr . For the convenience of the analysis, the following functions are introduced √ Ps f˜(Pa /Ps )/ Ts , A is a drive chamber (3) f (Pa , Ps , Pe ) = √ Pa f˜(Pe /Pa )/ Ta , B is a drive chamber and f (Pb , Ps , Pe ) =

√ Pb f˜(Pe /Pb )/ Tb , A is a drive chamber √ Ps f˜(Pb /Ps )/ Ts , B is a drive chamber

(4)

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Dynamic relationship within the control chambers

From [1], [3], and [4], a model for the mass ﬂow into each of the cylinder chambers can be obtained from the energy conservation equation for the control volume bounded by the cylinder and piston. m=

1 ˙ 1 PV P V˙ + RTs kRTs

(5)

For the cylinder Chambers A and B, substituting (1) - (4) into (5), the following equations can be derived

2.3

k V˙a k P˙a = − Pa + RTs Cd C0 wa Xa f (Pa , Ps , Pe ) Va Va

(6)

k V˙b k P˙b = Pb + RTs Cd C0 wb Xb f (Pb , Ps , Pe ) Vb Vb

(7)

Load dynamics

From the Newton’s Law, load dynamics can be described by Aa Pa − Ab Pb − Kf x˙ − Ks−c (x)S(x, ˙ Pa , P b ) = M x ¨

(8)

where x˙ represents velocity, x ¨ acceleration, Kf viscous friction coeﬃcient, Ks−c (x) the combination of static and Coulomb friction forces which are position and velocity dependent. Function S(x, ˙ Pa , Pb ) is described below   (Ab Pb ) − (Aa Pa ), x˙ = 0, |Aa Pa − Ab Pb | ≤ Ks (x) S(x, ˙ Pa , Pb ) := (9)  ˙ x˙ = 0 Kc (x)sign(x), For the case Ks (x) = Ks (a constant) and Kc (x) = Kc (a constant, Kc is normally smaller than Ks ), Ks−c (x) ˙ is shown in Figure 2. The chamber volumes Va and Vb are deﬁned as Va = Aa (l/2 + x + ∆) and Vb = Ab (l/2 − x + ∆),

(10)

where ∆ can be considered as an equivalent extra length to the cylinder. Let x1 = x, x2 = x, ˙ x3 = Pa , x4 = Pb , u1 = Xa , and u2 = Xb then a state-space system model is obtained. x˙1 = x2

(11)

x˙2 = (−Kf x2 − Ks−c S(x2 , x3 , x4 ) + Aa x3 − Ab x4 )/M

(12)

x˙3 = −

kx3 kRTs x2 + Cd C0 wa f (x3 , Ps , Pe )u1 (13) (l/2 + x1 + ∆) Aa (l/2 + x1 + ∆)

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Fig. 2 Combination of frictions

x˙4 =

kx4 kRTs x2 + Cd C0 wb f (x4 , Ps , Pe )u2 (l/2 − x1 + ∆) Ab (l/2 − x1 + ∆)

(14)

where x1 ∈ [−l/2, l/2], x3 ∈ [Pe , Ps ], x4 ∈ [Pe , Ps ], and normally wa = wb = w. If a single ﬁve port valve is adopted in the system, u1 = −u2 . For a rodless cylinder, Aa = Ab = A.

3

Static friction measurement

From the system model described in Section 2, there exist mainly two types of frictions in pneumatic actuators, namely, static and dynamic frictions. Once the actuators are manufactured and assembled, it is impossible to obtain the frictions through direct measurement due to some technical diﬃculties. In practice, the cylinder chamber pressures and piston velocity can be measured using the low cost sensors. So it is possible to identify the friction characteristics through indirect measurement and estimation. The method used to measure the static friction along the pneumatic cylinder is based on the load dynamics (8). The static friction force can be obtained by calculating the driving forces at the critical time instant when the piston is at a standstill and just before the piston starts moving, i.e. x˙ = 0 by the following formula, Ks−c (x) = Aa Pa − Ab Pb ,

(15)

where Pa and Pb are the chamber pressures at the moment when and just before the piston starts moving. If the chamber pressures are assumed to have a uniform distribution inside the cylinder chambers, the chamber pressures can be obtained by the pressure sensors mounted at the locations of inlet/outlet ports. Experimental study of static friction forces has been contacted in the research laboratory. The experiment conditions for rodless cylinder test are: (1) Cylinder: 1(m), φ = 32mm, Festo cylinder; (2) Valve: 5 ports , 1/8 ﬂow

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capacity, Festo proportional valve; (3)Velocity sensor: BEC encoder; (4) Pressure sensor: Piezo, Festo. Using the above listed components, the friction distribution along a rodless cylinder has been found as shown in Figure 3. Figure 3 revealed that the static friction forces are piston position and moving direction dependent.

Fig. 3 Static friction force distribution along a rodless cylinder

4

Description of Genetic Algorithms for parameter identiﬁcation

Compared with the static friction measurement, it is more diﬃcult to obtain the dynamic friction forces. Genetic Algorithms (GA) [6] have been chosen as a tool for dynamic friction identiﬁcation. GAOT , a GA Matlab Toolbox, is chosen for implementation of the GA system parameter identiﬁcation algorithm. The main purpose of parameter identiﬁcation is to obtain the friction coeﬃcient Kf .The structure of the algorithm is as follows: (1) Start: input the piston velocity d(n), n = 1, 2, ..., N obtained from the experiment/simulation as the benchmark data; (2) Step1: set up the initial conditions of the pneumatic cylinders; (3) Step2: obtain the updated population value of Kf for GA; (4) Step3: conduct simulation using the newly estimated value Kf under the same conditions as the experiments/ simulation performed to obtain the benchmark data; (5) Step4: save the simulation results to S(n) (n = 1, 2, ..., N );

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(6) Step5: calculate the errors between d(n) and S(n), and save the diﬀerences to e(n); (7) Step6: calculate the Mean Square Errors, MSE and convert MSE to the ﬁtness; (8) Step7: if the ﬁtness does not converge to a stable value, go to Step 2; If the ﬁtness converges to a stable value, go to Step 8; (9) Step8: return the optimal value of Kf and stop the process. In order to evaluate how close the simulation output is to the output obtained from the benchmark data, the Mean Square Errors is used in the algorithm as a cost function J, which is deﬁned by M SE = E[e2i (n)], where E is the statistical expectation. The MATLAB functions provided by GAOT can only work with the cases to ﬁnd the maximum values of ﬁtness functions. Therefore, the cost function is then converted to the evaluation function as G/(1 + M SE) = G/(1 + E([d(n) − s(n)]2 )), where G represents the gain to enlarge the diﬀerence between the good and bad solutions and enhance the survival chance of good solutions.

5 5.1

Experiment and simulation results Algorithm test by simulation

Before the algorithm described in Section 4 can be used for system parameter identiﬁcation, it has been tested using the data generated from the simulation model. The process of testing is as follows: (1) Start: choose a value of Kf and substitute the value into the system model; (2) Step1: run simulation to obtain the open-loop system step input dynamic responses and the system outputs are stored into a data array d(n) as a benchmark for identiﬁcation; (3) Then: follow the step 2 - step 8 described in Section 4; A typical test simulation result for a rodless cylinder is shown in Figure 5. The initial conditions and system parameters for the simulation illustrated in Figure 4 are: cylinder bore size φ = 32mm, cylinder length l = 1000mm, supply compressed air pressure Ps = 4.5bar, valve opening |Xa | = |Xb | = 4mm , initial position x = 500mm, initial velocity x˙ = 0 , initial chamber pressures Pa = 2.8bar and Pb = 2.6bar. In the simulation results, the friction coeﬃcient has been set up as 15.4. 5.2

Identiﬁcation of friction coeﬃcient using GA

A test rig has been set up in the authors’ laboratory. The test rig can host rodded cylinder, rodless cylinder and air motors. A typical test data example

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Fig. 4 Velocity obtained from the simulation (Solid line: benchmark;Broken line: simulation results obtained from GA identiﬁcation)

Fig. 5 Typical dynamic response of rodless cylinder

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is shown in Figure 7. The algorithm described in Section 4 has been tested using a set of simulation data. The friction coeﬃcient Kf will be estimated or identiﬁed using the algorithm in that section. GAOT Toolbox has diﬀerent options for each process or function. Diﬀerent choices from the options can lead to diﬀerent performance of GA, for example, speed and accuracy. Table 1 summaries the options chosen in friction coeﬃcient identiﬁcation. These options may not be the best but are conventional. In order to choose a

Table 1. Options from GAOT Options Functions or values Cost function M SE = E[d(n) − d(n)]2 Evaluation function 1000/(1 + M SE) Range of search space Determined according to diﬀerent cases Encoding method Floating point encoding Population size 20 strings Initial population Generated by ”initializega.m” Reproduction strategy Normalized Geometric selection Selection of parameters 0.1,i.e.the probability to select the best one Crossover strategy Arithmetic crossover Mutation strategy Non-uniform mutation Terminate function ”maxGenTerm.m” Maximum generation 80 generation(stop if exceed)

suitable maximum generation, a series of tests with diﬀerent values of maximum generations have been carried out. The percentage of convergence to the optimum with respect to diﬀerent maximum generation values is shown in Figure 6. The results of parameter identiﬁcation are shown in Figures 7 and

Fig. 6 Percentage of convergence of the algorithm

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8 when the piston moves to diﬀerent directions. The static friction forces are replaced by the average friction. From Figures 7 - 10, it can be seen that the Table 2. Parameters

The piston moves to the The piston moves to the positive direction negative direction Supply pressure of air 4.3 bar 4.3 bar Static friction Ks 79.0 N 77.1 N Start position(m) -.037 0.41 Initial velocity (m/s) 0.0 0.0 Initial chamber pressure Pa 2.27 bar 2.61 bar Initial chamber pressure Pb 2.85 bar 1.85 bar Valve spool displacement (m) 0.0007 -0.0007 Search space for Kf [10 30] [20 60] Best ﬁt friction coeﬃcient Kf 15.4 43.3

simulation results agreed with the data measured from the experiments very well. And also, it can be seen that the dynamic friction forces are diﬀerent for diﬀerent directions of the piston movement.

Fig. 7 Open loop system step input response from the experiment

6

Conclusion

A parameter identiﬁcation method based on Genetic Algorithms was proposed in the paper. Both simulation and experimental results have demonstrated the eﬃciency of the method. The results revealed that the friction forces (static and dynamic)are piston position, and moving direction dependent. From the work reported in the paper, we believe that further work needs to be done on identiﬁcation of distributed dynamic friction forces.

Parameter identiﬁcation for nonlinear pneumatic cylinder actuators

Fig. 8 Open loop system step input response from the simulation

Fig. 9 Open loop system step input response from the experiment

Fig. 10 Open loop system step input response from the simulation

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References 1. Anderson, B.W., (1976) The Analysis and Design of Pneumatic Systems,Robert E. Krieger Puplishing Co., INC, New York, USA. 2. Armstrong-Helouvry, S., P. Dupont, and C. Canudas De Wit, (1994) A survey of model, analysis tool and compensation methodes for the control of machines with friction, Automatica, 30, 1083-1183. 3. Ben-Dov, D. and S.E. Salcudean, (1995) A force-controlled pneumatic actuator, IEEE Transactions on Robotics and Automation, 11, 906-911. 4. Blackburn, J.F., G. Reethof and J.L. Shearer, (1960) Fluid Power Control, The Technology Press and J. Wiley Inc. New York. 5. Drakunov, S., G.D. Hanchin, W.C. Su and U. Ozguner, (1997) Nonlinear control of a rodless pneumatic servoactuator, or sliding modes versus Coulomb friction, Actumatica, 33, 1401-1408. 6. Goldberg, D. E., (1989) , Genetic Algorithms in search, optimation, and machine learning, Addison-Wesley Publishing Company, Inc, USA. 7. McDonell, B.W. and J.E. Bobrow, (1993) Adaptive tracking control of an air powered robot actuator, ASME journal of Dynamic Systems, Measurement, and Control, 115, 427-433. 8. Moore, P.R., J. Pu and R. Harrison, (1993)Progression of servo pneumatics towards advanced applications, in Fluid Power Circuit, Component and System Design, Edited by K. Edge & C. Burrows, Published by Research Studies Press, ISBN 0 86380 139 0, 347-365. 9. Sesmat, S., S. Scavarda and X. Lin-shi, (1995) Veriﬁcation of electropneumatic servovalve size using non-linear control theory applied to cylinder position tracking, The Proceeding of the 4th Scandinavian International Conference on Fluid Power, Tampere, Finland, 1, 504-511. 10. Van Varseveld, R.B. and G.M. Bone, (1997) Accurate position control of a pneumatic actuator using on/oﬀ solenoid valves, IEEE/ASME Transactions on Mechatronics, 2, 195-204. 11. Wang, J., J. Pu, P.R. Moore and Z. Zhang, (1998) Modelling study and servocontrol of air motor systems, International Journal of Control,71, 459-476. 12. Wang, J., D.J.D. Wang, P.R. Moore, and J. Pu, (2001) Modelling study, analysis and robust servo control of pneumatic cylinder actuator systems, IEE Proceedings on Control Theory and Applications, 148, 35-42.

The quasi-inﬁnite horizon approach to nonlinear model predictive control Rolf Findeisen and Frank Allg¨ower Institute for Systems Theory in Engineering, University of Stuttgart, 70550 Stuttgart, Germany, {ﬁndeise,allgower}@ist.uni-stuttgart.de Abstract. In the past decade nonlinear model predictive control (NMPC) has witnessed steadily increasing attention from control theoretists as well as control practitioners. The practical interest is driven by the fact that today’s processes need to be operated under tighter performance speciﬁcations. At the same time more and more constraints, stemming for example from environmental and safety considerations, need to be satisﬁed. Often these demands can only be met when process nonlinearities and constraints are explicitly considered in the controller. This paper reviews one NMPC technique, often referred to as quasi-inﬁnite horizon NMPC (QIH-NMPC). An appealing feature of QIH-NMPC is the fact that a short control horizon can be realized, implying reduced computational load. At the same time the controller achieves favorable properties such as stability and performance. After introducing the basic concept of model predictive control, the key ideas behind QIH-NMPC are discussed and the resulting properties for the state feedback case are presented. Additionally some new results on output feedback NMPC using high-gain observers are given. With the use of a realistic process control example, it is demonstrated, that even fairly large problems can be considered using NMPC techniques if state-of-the-art optimization techniques are combined with eﬃcient NMPC schemes.

1

Introduction to nonlinear model predictive control

Model predictive control (MPC), also referred to as moving horizon control or receding horizon control, has become an attractive feedback strategy, especially for linear processes. Linear MPC refers to a family of MPC schemes in which linear models are used to predict the system dynamics, even though the dynamics of the closed-loop system is nonlinear due to the presence of constraints. Linear MPC approaches have found successful applications, especially in the process industries. A good overview on industrial linear MPC techniques can be found in [46], where more than 2200 applications in a very wide range from chemicals to aerospace industries are summarized. By now, linear MPC theory is quite mature. Important issues such as online computation, the interplay between modeling/identiﬁcation and control and system theoretic issues like stability are well addressed [32,42]. This paper focuses on the application of model predictive control techniques to nonlinear systems. It is not intended as a complete review of existing NMPC techniques. By now a couple of excellent reviews exist, see for A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 89-108, 2003. Springer-Verlag Berlin Heidelberg 2003

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example [38,1,15,9,42,47]. Instead we give a review for one speciﬁc NMPC technique, often referred to as quasi-inﬁnite horizon NMPC [10,8]. Quasi-inﬁnite horizon NMPC (QIH-NMPC) allows a (computationally) eﬃcient formulation of NMPC while guaranteeing stability and performance of the closed-loop. Many NMPC schemes based on similar ideas have been proposed [44,13,45,30]. A detailed discussion of the connections between these schemes can be found in [38,27]. These schemes allow for an eﬃcient formulation of NMPC. Thus they facilitate, together with well suited numerical solution techniques, the real-time application of NMPC [22,17]. Besides giving an overview of QIH-NMPC, some recent results with respect to the output-feedback problem for NMPC, as well as a numerically eﬃcient solution of the dynamic optimization problem resulting from QIHNMPC, are presented. The applicability of NMPC to real processes using these methods is shown by considering the control of a high purity distillation column. This paper is organized as follows. In Sect. 1.1 the basic principle of model predictive control is reviewed. The mathematical formulation of NMPC used in this paper is introduced in Sect. 1.2. Based on this formulation, Sect. 2 presents the quasi-inﬁnite horizon NMPC scheme. Furthermore, some notes on modiﬁcations and improvements regarding the robust case and special system classes, as well as connections to other NMPC schemes, are given. Section 3 contains some new results on the output-feedback NMPC problem. The results suggest that the performance of the state feedback scheme can be recovered in the output feedback case, if state observers with a suitable structure and a fast enough dynamics are used. Section 4 contains some remarks on the eﬃcient numerical solution of the open-loop optimal control problem that must be solved in NMPC. In Sect. 5 an example application of NMPC to a high purity distillation column is given. This shows, that using well suited optimization strategies together with the QIH-NMPC scheme allow real-time application of NMPC even with todays computing power. The ﬁnal conclusions and remarks on future research directions are given in Sect. 6. In the following, · denotes the Euclidean vector norm in Rn (where the dimension n follows from context) or the associated induced matrix norm. Vectors are denoted by boldface symbols. Whenever a semicolon “;” occurs in a function argument, the following symbols should be viewed as additional parameters, i.e. f (x; γ) means the value of the function f at x with the parameter γ. 1.1

The principle of nonlinear model predictive control

In general, the model predictive control problem is formulated as solving online a ﬁnite horizon open-loop optimal control problem subject to system dynamics and constraints involving states and controls. Figure 1 shows the basic principle of model predictive control. Based on measurements obtained

The quasi-inﬁnite horizon approach to NMPC Past

6- Future

91

Setpoint

¯ Predicted state x Closed-loop state x

¯ Open-loop input u

Closed-loop input u t t+δ

t + Tc

t + Tp

-

Control horizon Tc

-

Prediction horizon Tp

-

Fig. 1. Principle of model predictive control.

at time t, the controller predicts the future dynamic behavior of the system over a prediction horizon Tp and determines (over a control horizon Tc ≤ Tp ) the input such that a predetermined open-loop performance objective functional is optimized. If there were no disturbances and no model-plant mismatch, and if the optimization problem could be solved for inﬁnite horizons, then we could apply the input function found at time t = 0 to the system for all times t ≥ 0. However, this is not possible in general. Due to disturbances and model-plant mismatch, the true system behavior is diﬀerent from the predicted behavior. In order to incorporate some feedback mechanism, the open-loop manipulated input function obtained will be implemented only until the next measurement becomes available. We assume that this will be the case every δ sampling time-units. Using the new measurement, at time t + δ, the whole procedure – prediction and optimization – is repeated to ﬁnd a new input function, with the control and prediction horizons moving forward. Notice, that in Fig. 1 we assume the input parametrized as piecewise constant. For simplicity this is often done in practice, as discussed in Section 1.2. In general the open-loop input could be parametrized by a (ﬁnite) number of basis functions. 1.2

Mathematical formulation of NMPC

We consider the stabilization problem for a class of systems described by the following nonlinear set of diﬀerential equations ˙ x(t) = f (x(t), u(t)) ,

x(0) = x0

(1)

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subject to input constraints1 u(t) ∈ U , ∀t ≥ 0

(2)

where x(t) ∈ Rn and u(t) ∈ Rm denotes the vector of states and inputs, respectively. The set of feasible input values is denoted by U and we assume that U satisﬁes the following assumption: Assumption 1 U ⊂ Rm is compact and the origin is contained in the interior of U . In its simplest form, U is given as a box constraint: U := {u ∈ Rm |umin ≤ u ≤ umax }

(3)

where {umin }i ≤ 0 ≤ {umax }i . Here umin and umax are given constant vectors. With respect to the system we additionally assume, that: Assumption 2 The vector ﬁeld f : Rn × Rm → Rn is continuous and satisﬁes f (0, 0) = 0. In addition, it is locally Lipschitz continuous in x. Assumption 3 The system (1) has a unique continuous solution for any initial condition in the region of interest and any piecewise continuous and right continuous input function u(·) : [0, Tp ] → U . In order to distinguish clearly between the real system and the system model used to predict the future “within” the controller, we denote the internal vari¯, u ¯ ). Notice that the predicted ables in the controller by a bar (for example x values need not and generally will not be the same as the actual values. Usually, the ﬁnite horizon open-loop optimal control problem described above is mathematically formulated as follows 2 : Problem 1 Find ¯ (·); Tp ) min J(x(t), u

(4)

¯ (·) u

with:

t+Tp

¯ (τ ))dτ. F (¯ x(τ ), u

¯ (·); Tp ) := J(x(t), u

(5)

t

subject to: ¯˙ (τ ) = f (¯ ¯ (τ )) , x ¯ (t) = x(t) x x(τ ), u ¯ (τ ) ∈ U , ∀τ ∈ [t, t + Tp ] . u 1

2

(6a) (6b)

Only input constraints are considered in this paper. A generalization to include state constraints is easily possible. For simplicity we assume that the control and prediction horizon have the same length, Tc = Tp .

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¯ (·) is the solution of (6a) The bar denotes internal controller variables and x ¯ (·) : [0, Tp ] → U with initial condition x(t). driven by the input u The function F speciﬁes the desired control performance that can arise, for example, from economical and ecological considerations. The standard quadratic form is the simplest and most often used one: F (x, u) = (x − xs )T Q(x − xs ) + (u − us )T R(u − us ) ,

(7)

where xs and us denote given setpoints; Q and R denote positive deﬁnite, symmetric weighting matrices. In order for the desired reference (xs , us ) to be a feasible solution of Problem 1, us should be contained in the interior of U . As already stated in Assumption 2 we consider, without loss of generality that (xs , us ) = (0, 0) is the steady state that should be stabilized. Note the initial condition in (6a): The system model used to predict the future in the controller is initialized by the actual system states, i.e. the states must be either measured directly or estimated. For a numerical implementation, the input function is often parameterized ¯ (·) on [t, t + Tp ] satisﬁes in a step-shaped manner. As shown in Fig. 1, u ¯ (τ ) = const for τ ∈ [t + jδ, t + (j + 1)δ), j = 0, 1, · · · , Np , where Np = Tδp u and δ denotes the sampling period. In the following an optimal solution to the optimization problem (exis¯ ∗ (·; x(t), Tp ) : [t, t + Tp ] → U . The open-loop tence assumed) is denoted by u optimal control problem will be solved repeatedly at the sampling instances t = jδ, j = 0, 1, · · · , once new measurements are available . The closed-loop control is deﬁned by the optimal solution of Problem 1 at the sampling instants: ¯ ∗ (τ ; x(t), Tp ) , τ ∈ [t, δ] . u∗ (τ ) := u

(8)

Model-plant mismatch and disturbances are generally not considered in the optimization problem. At most, they are indirectly considered. Often their eﬀect is assumed to be constant over the prediction horizon. For example, ¯ an additional unknown but constant state vector d(t) ∈ Rq is introduced to cover the eﬀect of the disturbances and uncertainties: ¯ ¯˙ = f (¯ ¯ , d) x x, u ¯˙ = 0 . d

(9a) (9b)

¯ is estimated on-line. The value of d As mentioned, the open-loop optimal control Problem 1, that must be solved on-line, is often formulated in a ﬁnite horizon manner and the input function is parameterized ﬁnitely, in order to allow a (real-time) numerical solution of the nonlinear open-loop optimal control problem. It is clear, that the shorter the horizon is the less costly is also the solution of the on-line optimization problem. Thus it is desirable from a computational point of view to implement MPC schemes using short horizons. However, when a ﬁnite

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prediction horizon is used, the actual closed-loop input and state trajectories will diﬀer from the predicted open-loop trajectories, even if no model plant mismatch and no disturbances are present [1]. This obvious fact has two immediate consequences. Firstly, the actual goal to compute a feedback such that the performance objective over the inﬁnite horizon of the closed loop is minimized is not achieved. In general it is by no means true that a repeated minimization over a ﬁnite horizon objective in a receding horizon manner leads to an optimal solution for the inﬁnite horizon problem (with the same stage cost F ). In fact, the two solutions will in general diﬀer signiﬁcantly if a short horizon is chosen. Secondly, if the predicted and the actual trajectories diﬀer, there is no guarantee that the closed-loop system will be stable. It is indeed easy to construct examples for which the closed-loop becomes unstable if a (small) ﬁnite horizon is chosen. Hence, when using ﬁnite horizons in standard NMPC, the stage cost cannot be chosen simply based on the desired physical objectives. From this discussion it is clear that short horizons are desirable from a computational point of view, but long horizons are required for closed-loop stability and in order to achieve the desired performance in closed-loop. The concept of the quasi-inﬁnite horizon as discussed in the next section can be used to resolve this dilemma. It will be shown that by use of a suitable modiﬁcation of the optimal control problem to be solved, short horizons can be used without jeopardizing closed-loop stability and without sacriﬁcing closed-loop performance.

2

Quasi-inﬁnite horizon NMPC

Diﬀerent possibilities to achieve closed-loop stability for NMPC using ﬁnite horizon length have been proposed [31,39,37,45,30,10,14,27]. Most of these approaches modify the NMPC setup such that stability of the closed-loop can be guaranteed independent of the plant and performance speciﬁcations. This is usually achieved by adding suitable equality or inequality constraints and suitable additional terms to the cost functional to the standard setup. These additional constraints are usually not motivated by physical restrictions or desired performance requirements but have the sole purpose of enforcing stability of the closed-loop. Therefore, they are usually termed stability constraints [35,36]. NMPC formulations with a modiﬁed setup to achieve closed-loop stability independent of the choice of the performance parameters in the cost functional are usually referred to as NMPC approaches with guaranteed stability. We do not review NMPC schemes with guaranteed stability here (see for example [38,27] for a overview). Instead we present exemplary the so called quasi-inﬁnite horizon NMPC. In the quasi-inﬁnite horizon NMPC method [8,10] a terminal region constraint ¯ (t + Tp ) ∈ Ω x

(10)

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95

and a terminal penalty term E(¯ x(t + Tp )) are added to the standard setup. Note that the terminal penalty term is not a performance speciﬁcation that can be chosen freely. Rather E and the terminal region Ω in (10) are determined oﬀ-line such that the cost functional with terminal penalty term t+Tp ¯ (τ )) dτ + E(¯ ¯ (·), Tp ) = F (¯ x(τ ), u x(t + Tp )) (11) J (x(t), u t

gives an upper approximation of the inﬁnite horizon cost functional with stage cost F . Thus closed-loop performance over the inﬁnite horizon is addressed. Furthermore, as is shown later, stability is achieved, while only an optimization problem over a ﬁnite horizon must be solved. The resulting open-loop optimization problem is formulated as follows: Problem 2 [Quasi-inﬁnite Horizon NMPC]: Find ¯ (·); Tp ) min J(x(t), u

(12)

¯ (·) u

with:

t+Tp

¯ (·); Tp ) := J(x(t), u

¯ (τ ))dτ + E(¯ F (¯ x(τ ), u x(t + Tp )).

(13)

t

subject to: ¯˙ (τ ) = f (¯ ¯ (τ )) , x ¯ (t) = x(t) x x(τ ), u ¯ (τ ) ∈ U , τ ∈ [t, t + Tp ] u ¯ (t + Tp ) ∈ Ω x

(14a) (14b) (14c)

If the terminal penalty term E and the terminal region Ω are chosen suitably, stability of the closed-loop can be guaranteed. To derive the stability results we need that the following holds for the stage cost-function. Assumption 4 F : Rn×U → R is continuous in all arguments with F (0, 0) = 0 and F (x, u) > 0 ∀(x, u) ∈ Rn × U \{0, 0}. Given this assumption, the following result, which is a slight modiﬁcation of Theorem 4.1 in [7], can be established: Theorem 1. Suppose (a) that Assumptions 1-4 are satisﬁed, (b) E is C 1 with E(0, 0) = 0, Ω is closed and connected with the origin contained in Ω and there exists a continuous local control law k : Rn → Rm with k(0) = 0, such that: ∂E f (x, k(x)) + F (x, k(x)) ≤ 0, ∂x with k(x) ∈ U ∀x ∈ Ω

∀x ∈ Ω

(15)

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(c) the NMPC open-loop optimal control problem has a feasible solution for t = 0. Then for any sampling time 0 < δ < Tp the nominal closed-loop system is asymptotically stable with the region of attraction R being the set of states for which the open-loop optimal control problem has a feasible solution. A formal proof of Theorem 1 can be found in [7,9] and for a linear local controller as described below in [10]. Loosely speaking E is a local Lyapunov function of the system under the local control k(x) in Ω. As will be shown, equation (15) allows to upper bound the optimal inﬁnite horizon cost inside Ω by the cost resulting from a local feedback k(x), namely by E. Notice, that this result is nonlocal in nature, i.e. their exists a region of attraction R of at least the size of Ω. The region of attraction is given by all states for which the open-loop optimal control problem has a feasible solution. Obtaining a terminal penalty term E and a terminal region Ω that satisfy the conditions of Theorem 1 is not easy. In the case of a locally linear feedback law u = Kx and a quadratic objective functional with weighting matrices Q and R, the terminal penalty term can be chosen to be quadratic of the form E(x) = xT P x. For this case, a procedure to systematically compute the terminal region and a terminal penalty matrix oﬀ-line is available [10], assuming that the Jacobian linearization (A, B) of (1) is stabilizable, where ∂f ∂f (0, 0) and B := ∂u (0, 0). This procedure can be summarized as A := ∂x follows: Step 1 : Solve the linear control problem based on the Jacobian linearization (A, B) of (1) to obtain a locally stabilizing linear state feedback u = Kx. Step 2 : Choose a constant κ ∈ [0, ∞) satisfying κ < −λmax (AK ) and solve the Lyapunov equation T (AK + κI) P + P (AK + κI) = − Q + K T RK (16) to get a positive deﬁnite and symmetric P , where AK := A + BK. Step 3 : Find the largest possible α1 deﬁning a region Ω1 := {x ∈ Rn | xT P x ≤ α1 }

(17)

such that Kx ∈ U , for all x ∈ Ω1 . Step 4 : Find the largest possible α ∈ (0, α1 ] specifying a terminal region Ω := {x ∈ Rn | xT P x ≤ α}

(18)

such that the optimal value of the following optimization problem is non-positive: max{xT P ϕ(x)−κ · xT P x | xT P x ≤ α} x

(19)

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where ϕ(x) := f (x, Kx) − AK x. This procedure allows to calculate E and Ω if the linearization of the system at the origin is stabilizable. If the terminal penalty term and the terminal region are determined according to Theorem 1, the open-loop optimal trajectories found at each time instant approximate the optimal solution for the inﬁnite horizon problem. The following reasoning make this plausible: Consider an inﬁnite horizon cost functional deﬁned by ∞ ¯ (·)) := ¯ (τ )) dτ F (¯ x(τ ), u (20) J ∞ (x(t), u t

¯ (·) on [t, ∞). This cost functional can be split up into two parts with u ∞ t+Tp ¯ (·)) = min ¯ (τ )) dτ + F (¯ ¯ (τ )) dτ . (21) min J ∞(x(t), u F (¯ x(τ ), u x(τ ), u ¯ (·) u

¯ (·) u

t+Tp

t

The goal is to upper approximate the second term by a terminal penalty term E(¯ x(t+Tp ). Without further restrictions, this is not possible for general nonlinear systems. However, if we ensure that the trajectories of the closedloop system remain within some neighborhood of the origin (terminal region) for the time interval [t + Tp , ∞), then an upper bound on the second term can be found. One possibility is to determine the terminal region Ω such that a local state feedback law u = k(x) asymptotically stabilizes the nonlinear system and renders Ω positively invariant for the closed-loop. If an additional terminal inequality constraint x(t + Tp ) ∈ Ω (see (14c)) is added to Problem 1, then the second term of equation (21) can be upper bounded by the cost resulting from the application of this local controller u = k(x). Note that the predicted state will not leave Ω after t + Tp since u = k(x) renders Ω positively invariant. Furthermore the feasibility at the next sampling instance ¯ and replacing it by the nominal is guaranteed dismissing the ﬁrst part of u open-loop input resulting from the local controller. Requiring that x(t+Tp ) ∈ Ω and using the local controller for τ ∈ [t + Tp , ∞) we obtain: t+Tp

¯ (·)) ≤ min minJ ∞(x(t), u ¯ (·) u

¯ (·) u

∞

¯ (τ )) dτ + F (¯ F (¯ x(τ ), u x(τ ), k (¯ x(τ ))) dτ t

. (22)

t+Tp

If, furthermore, the terminal region Ω and the terminal penalty term are chosen according to condition b), integrating (15) leads to ∞ F (¯ x(τ ), k (¯ x(τ ))) dτ ≤ E (¯ x(t + Tp )) . (23) t+Tp

Substituting (23) into (22) we obtain ¯ (·)) ≤ min J(x(t), u ¯ (·); t + Tp ) . min J ∞ (x(t), u ¯ (·) u

¯ (·) u

(24)

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This implies that the optimal value of the ﬁnite horizon problem bounds that of the corresponding inﬁnite horizon problem. Thus, the prediction horizon can be thought of as extending quasi to inﬁnity. Equation (24) can be exploited to prove Theorem 1. Like in the dual-mode approach [40], the use of the terminal inequality constraint gives the quasi-inﬁnite horizon nonlinear MPC scheme computational advantages . Note also, that as for dual-mode NMPC, it is not necessary to ﬁnd optimal solutions of Problem 1 in order to guarantee stability. Feasibility also implies stability here [10,48]. Many generalizations and expansions of QIH-NMPC exist. For example discrete time variants can be found in [26,14]. If the nonlinear system is aﬃne in u and feedback linearizable, then a terminal penalty term can be determined such that (23) is exactly satisﬁed with equality [7], i.e. the inﬁnite horizon is recovered exactly. In [12,11,34] robust versions using a min-max formulation are proposed, while in [20] an expansion to index one DAE systems is considered. A variation of QIH-NMPC for the control of varying setpoints is given in [21,23]. So far we assumed that the full state is available for feedback. In general not all states are measured directly. A common approach is to still employ a state feedback NMPC and to use a state observer to obtain an estimate of the system states used in the NMPC. However, then in general little can be said about the stability of the closed-loop, since there exists no universal separation principle for nonlinear systems as it does for linear systems. In the following we brieﬂy review some results on combining NMPC schemes with high-gain observers to obtain semi-regional stability results for the closedloop system. A detailed description can be found in [28,29].

3

Some results on output feedback NMPC

For output feedback NMPC diﬀerent approaches have been proposed. However they either only guarantee local stability or fail in the case of disturbances and noise. In [41] a combination of a moving-horizon observer and a dual-mode NMPC [40] is proposed. Assuming certain observability assumptions hold and adding a contraction constraint to the moving horizon observer, it is shown that the closed-loop is (semi-regional) stable if no model/plant mismatch or disturbances are present. In [33,49] asymptotic stability for observer based nonlinear MPC for “weakly detectable” systems is derived. These results are, however, only of local nature, i.e. stability is only guaranteed for a suﬃciently small observer error. Here we propose to combine high-gain observers with NMPC to achieve semi-regional stability. If the observer gain is increased suﬃciently, the stability region and performance of the state feedback is recovered. The closed loop system is semi-regionally stable in the sense, that for any subset S of the region of attraction R of the state-feedback controller (compare Theorem 1)

The quasi-inﬁnite horizon approach to NMPC

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there exists an observer parameter (gain), such that S is contained in the region of attraction of the output-feedback controller. The results are based on “nonlinear separation principles [3,50]” and it is assumed, that the NMPC feedback is instantaneous (see below). We will limit the presentation to a special SISO systems class and only give the main result. The more general MIMO case and proofs can be found in [29,25]. We consider the stabilization of SISO systems of the following form: x˙ =Ax + bφ(x, u) y =x1 .

(25a) (25b)

with u(t) ∈ U ⊂ R and y(t) ∈ R. The output y is given by the ﬁrst state x1 . The n × n matrix A and the n × 1 vector b have the following form:   0 1 0 ··· 0 0 0 1 · · · 0  

T  ..  , b = 0 · · · 0 1 n×1 , (26a) A =  ...  .   0 · · · · · · 0 1 0 · · · · · · · · · 0 n×n Additional to the Assumptions 1-4 we assume, that: Assumption 5 The function φ : Rn × U → R is locally Lipschitz in its arguments over the domain of interest. Furthermore φ(0, 0) = 0 and φ is bounded in x everywhere. Note that global boundedness can in most cases be achieved by saturating φ outside a compact region of Rn of interest. The proposed output feedback controller consists of a high-gain observer to estimate the states and an instantaneous variant of the full state feedback QIH-NMPC controller as outlined in Sect. 2. By instantaneous we mean that the system input at all times (i.e. not only at the sampling instances) is given by the instantaneous solution of the open-loop optimal control problem: u(x(t)) := u (τ = 0 ; x(t), Tp ).

(27)

This feedback law diﬀers from the standard NMPC formulation in the sense that no open-loop input is implemented over a sampling time δ. Instead u(t) is considered as a “function” of x(t). To allow the subsequent result to hold, we have to require that the feedback resulting from the QIH-NMPC is locally Lipschitz. Assumption 6 The instantaneous state feedback (27) is locally Lipschitz. The observer used for state recovery is a high-gain observer [3,50,51] of the following form: ˆ˙ = Aˆ x x + bφ(ˆ x, u) + H(x1 − x ˆ1 )

(28)

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where H = α1 /, α2 /2 , . . . , αn /n . The αi ’s are chosen such that the polynomial sn +α1 sn−1 +· · ·+αn−1 s+αn = 0, is Hurwitz. Here 1 is the high-gain parameter and can be seen as a time scaling for the observer dynamics (28). A, b and φ are the same as in (25). Notice that the use of an observer makes it necessary that the input also be deﬁned (and bounded) for (estimated) states that are outside the feasible region of the state feedback controller. We simply deﬁne the open-loop input for x ∈ R as ﬁxed to an arbitrary value uf ∈ U : u(x) = uf ,

∀x ∈ R.

(29)

This together with the assumption that U is bounded separates the peaking of the observer from the controller/system [19]. Using the high-gain observer for state recovery, the following result, which establishes semi-regional stability of the closed-loop can be obtained [28,29]: Theorem 2. Assume that the conditions a)-c) of Theorem 1 and Assumption 5-6 hold. Let S be any compact set contained in the interior of R (region of attraction of the state feedback). Then there exists a (small enough) ∗ > 0 such that for all 0 < ≤ ∗ , the closed-loop system is asymptotically stable with a region of attraction of at least S. Further, the performance of the state feedback NMPC controller is recovered as decreases. By performance recovery it is meant that the diﬀerence between the trajectories of the state feedback and the output feedback can be made arbitrarily small by decreasing . The results show that the performance of the state feedback scheme can be recovered in the output feedback case, if a state observer with a suitable structure and a fast enough dynamics is used. Figure 3 shows the simulation result for an illustrative application of the proposed output feedback scheme to a two dimensional pendulum car system as depicted in Fig. 2 and presented in [28]. The angle between the pendulum and the vertical is denoted by z1 , while the angular velocity of the pendulum

z1 (t)

u(t)

Fig. 2. Sketch of the inverted pendulum/car system

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is given by z2 . The input u is the force applied to the car. The control objective is to stabilize the upright position of the pendulum. To achieve this objective, a QIH-NMPC scheme with and without (state-feedback case) a high-gain observer is used. For the results shown in Fig. 3 the pendulum 1 terminal region 0.5 border region of attraction state feedback NMPC

0 −0.5 −1

z2

−1.5 −2 −2.5 −3 −3.5 −4 −4.5 −1

ε=0.01 ε=0.05 ε=0.09 ε=0.1 state feedback −0.5

0

0.5

1

1.5

z1

Fig. 3. Phase plot of the pendulum angle (z1 ) and the angular velocity (z2 )

is initialized with an oﬀset from the upright position, while the high-gain observer is started with zero initial conditions. The ﬁgure shows the closed loop trajectories for state feedback QIH-NMPC controller and the outputfeedback controller with diﬀerent observer gains. The gray ellipsoid around the origin is the terminal region of the QIH-NMPC controller. The outer “ellipsoid” is an estimate of the region of attraction of the state-feedback controller. As can be seen for small enough values of the observer parameter the closed loop is stable. Furthermore the performance of the state feedback is recovered as tends to zero. More details can be found in [28]. The achieves that the recovery of the region of attraction and the performance of the state-feedback is possible up to any degree of exactness. In comparison to other existing output-feedback NMPC schemes [33,49] the proposed scheme is thus of non-local nature. However, the results are based on the assumption that the NMPC controller is time continuous/instantaneous. In practice, it is of course not possible to solve the nonlinear optimization problem instantaneously. Instead, it will be solved only at some sampling instants. A sampled version of the given result, in agreement with the “usual” sampled NMPC setup can be found in [25]. Notice also, that the use of a high

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gain observer is critical, if the output measurements are very noise, since the noise will be ampliﬁed due to the high gain nature of the observer.

4

Some remarks on the eﬃcient numerical solution of NMPC

Besides the problem of stability of the closed-loop and the output-feedback problem, the eﬃcient solution of the resulting open-loop optimal control problem is important for any application of NMPC to real processes. A real-time application of NMPC is possible [22,43] if: a) NMPC schemes that do not require a high computational load and do not sacriﬁce stability and performance, like QIH-NMPC, are used and b) the resulting structure of the open-loop optimization problem is taken into account during the numerical solution. For the numerical solution of Problem 2 diﬀerent possibilities exist, see for example [4,5,17,1]. An eﬃcient solution should: 1) take advantage of the special problem structure of the open-loop optimization problem, 2) reuse as much information as possible from the previous sampling interval in the next sampling interval. One possible dynamic optimization scheme that can be adapted to provide all these properties is the so-called direct multiple shooting approach [6]. We do not go into details here. For a complete description of a specially tailored version of multiple shooting to NMPC see [17,16]. Besides taking advantage of the problem structure, we only mention two key components in the optimization, both of which lead to a signiﬁcant decrease in solution time and refer to [17] for more details: • Initial Value Embedding Strategy: Optimization problems at subsequent sampling instants diﬀer only by diﬀerent initial values x(t). If one accepts an initial violation, the complete solution trajectory of the previous optimization problem can be used as an initial guess for the current problem. Furthermore, all problem functions, derivatives as well as an approximation of the Hessian matrix can be reused from the previous problem, so that the ﬁrst QP solution can be performed without any solution of the diﬀerential equation of the system. • Gauss-Newton Hessian Approximation: In case of a quadratic stage cost F and a quadratic ﬁnal state penalty E, it is possible to use a GaussNewton approach to calculate an excellent Hessian approximation. Such an approximation only needs ﬁrst order derivatives that are available from the gradient computations without any additional computational eﬀort.

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5

103

Application example–real-time feasibility of NMPC

To show that nonlinear predictive control can be applied to even rather large systems if eﬃcient NMPC schemes and special tailored numerical solution methods are used, we give some results from a real-time feasibility study of NMPC for a high-purity distillation column as presented in [17,18,43,2]. Figure 4 shows the in this study considered 40 tray high-purity distillation column for the separation of Methanol and n-Propanol. The binary mixture is

40

L

xD

28

F , xF 21 14

V

1

xB

Fig. 4. Scheme of the distillation column

fed in the column with ﬂow rate F and molar feed composition xF . Products are removed at the top and bottom of the column with concentrations xB and xD respectively. The column is considered in L/V conﬁguration, i.e. the liquid ﬂow rate L and the vapor ﬂow rate V are the control inputs. The control problem is to maintain the speciﬁcations on the product concentrations xB and xD . For control purposes, models of the system of diﬀerent complexity are available. As usual in distillation control, xB and xD are not controlled directly. Instead an inferential control scheme which controls the deviation of the concentrations on tray 14 and 28 from the setpoints is used, i.e. only the concentration deviations from the setpoint on trays 14 and 28 plus the inputs are penalized in the cost-function. The QIH-NMPC control scheme is used for control. The terminal region and terminal penalty term have been calculated as suggested in Sect. 2. In Table 1 the maximum and average CPU times necessary to solve one open-loop optimization problem for the QIH-NMPC scheme in case of a disturbance in xF with respect to diﬀerent model sizes are shown. Considering that the sampling time of the process control system connected to the distillation column is 30sec, the QIH-NMPC using the appropriate tool for op-

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Table 1. Comparison of the average and maximum CPU time in seconds necessary for the solution of one open-loop optimal control problem. The results are obtained using MUSCOD-II [5] and QIH-NMPC for models of diﬀerent size. The prediction horizon of is 10 minutes and a controller sampling time δ = 30sec is used model size 42 164

max 1.86s 6.21s

avrg 0.89s 2.48s

timization is even real-time feasible for the 164 th order model. Notice, that a straightforward solution of the optimal control problem for the 42 nd order model using the optimization-toolbox in Matlab needs in average 620sec to ﬁnd the solution and is hence not real-time implementable. Also a numerical approximation of the inﬁnite horizon problem by increasing the prediction horizon suﬃciently enough is not real-time feasible as shown in [24]. More details and simulation results for the distillation column example can be found in [2,17,43]. First experimental results on a pilot scale distillation column are given in [18]. The presented case study underpins, that NMPC can be applied in practice already nowadays, if eﬃcient numerical solution methods and eﬃcient NMPC formulations (like QIH-NMPC) are used.

6

Conclusions

The focus of this paper is a review of the quasi-inﬁnite horizon approach to NMPC. After introducing the basic principle of MPC we gave the mathematical formulation of QIH-NMPC and outlined some of the key properties. The main feature of QIH-NMPC is that it allows a (computationally) eﬃcient formulation while guaranteeing stability and good performance in comparison to other NMPC formulations. In practice not all states are available for measurement and so state observers must be used. Since no general separation principle exists for nonlinear systems, output feedback controllers using the observer information in general only guarantee local stability. To overcome this problem we reviewed some recent results on output feedback NMPC. As shown, the use of a high-gain observer allows recovery of performance and the region of attraction of the state feedback controller. Future work will consider generalizations of this result to the sampled case and a wider system class. Besides the state estimation problem, numerically eﬃcient solution algorithms must be used to allow for a real-time application of NMPC. As outlined in Sect. 4 and Sect. 5 eﬃcient solution methods for the resulting dynamic optimization problem in combination with eﬃcient NMPC formulations such as QIH-NMPC allow even nowadays for a computationally feasible application of NMPC to realistically sized problems.

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Acknowledgements The authors gratefully acknowledge Lars Imsland and Bjarne Foss from the Department of Engineering Cybernetics, NTNU Trondheim, Norway for the ongoing cooperation in the area of output-feedback NMPC. Major parts of Sect. 3 are based on this cooperation. Furthermore the authors would like to acknowledge the fruitful cooperation with Moritz Diehl, Hans Georg Bock and Johannes Schl¨oder from the Center for Scientiﬁc Computing (IWR) at the University of Heidelberg, Germany in the area of eﬃcient solution of NMPC as outlined in Sect. 4 and Sect. 5.

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State and parameter identiﬁcation for nonlinear uncertain systems using variable structure theory Fabienne Floret-Pontet and Fran¸coise Lamnabhi-Lagarrigue Laboratoire des Signaux et des Syst`emes, C.N.R.S.–SUPELEC–Universit´e de Paris-Sud, Plateau de Moulon, 3 Rue Joliot-Curie, 91 192 Gif-sur-Yvette C´edex, FRANCE (fabienne.floret,francoise.lamnabhi)@lss.supelec.fr Abstract. In this paper, we investigate a new identiﬁcation algorithm grounded on the Variable Structure theory in order to study parameter and state identiﬁcation for uncertain nonlinear systems in continuous time. On the one hand, we use a speciﬁc observer to estimate unknown states of the studied nonlinear system, based on the Variable Structure theory. On the other hand, the well-known chattering property and the invariance properties, characteristic of the Variable Structure theory, make possible the design of the parameter sliding identiﬁer without the use of the additional persisting excitation condition usually required for the input in order to guarantee the parameter convergence. Stability of the state observer, parameter identiﬁability of the plant and asymptotic convergence of estimated parameters to their nominal values are studied. Finally, we propose some simulations on a pendulum in order to point out the validity of the methodology even if the input u is not persistently exciting. These numerical simulations highlight the robustness of the state and parameter identiﬁcation with respect to signiﬁcant parameter uncertainties thanks to the use of the Variable Structure theory.

1

Introduction

This article is dedicated to the issue of simultaneous identiﬁcation of the vector state and parameter values of nonlinear systems. Concerning the issue of parameter estimation in linear context, is widely applied, especially regarding industrial plants. Indeed, the most famous method for linear identiﬁcation is based on the least squares algorithm [10]. Unfortunately, most of plants encountered in practice belong to nonlinear systems, which can only be adequately described by nonlinear models. As a consequence, we focus on the identiﬁcation of nonlinear plants, in other words, systems where the outputs could not be expressed linearly in terms of the unknown parameters. Identiﬁcation of nonlinear plants has not yet received a huge exposure. This is probably due to the diﬃculty of designing identiﬁcation algorithms that could be applied to a reasonably large class of nonlinear systems. One possible solution to nonlinear identiﬁcation consists to use a method grounded on the least squares algorithm [11]. This implies a linearization around some equilibrium point and automatically a restricted range of the study. Moreover, A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 109-128, 2003. Springer-Verlag Berlin Heidelberg 2003

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other techniques for nonlinear identiﬁcation have already been investigated such as Volterra-series approach [3] or the method of maximum-likelihood studied by [2]. Nevertheless, both of the previous methods are designed for speciﬁc nonlinear systems and assume that the whole vector state is measurable, which sometimes could be impossible to realize in practice because of the use of expensive sensors or the generation of unreliable noisy derivatives. As a consequence, our main goal is to introduce a new approach which would combine parameter identiﬁcation with estimated non-measurable states restored through an observer. Therefore, we would want to generalize to parameter identiﬁcation in nonlinear context the work by Utkin [17], [18] on parameter identiﬁcation for linear systems. Utkin [17], [18] proposes to design at the same time identiﬁcation of parameter values and state vector using sliding regimes for linear systems. In this speciﬁc context of Variable Structure theory, some techniques concerning parameter identiﬁcation of nonlinear systems have already been investigated [7], [1] or [21]. Nevertheless, the two latter methods assume that the whole vector state is measurable, which is often narrowly linked to an expensive cost in practice. Moreover, the references [1] and [21] deal with parameter identiﬁcation as a part of a controller. In this paper, we propose a new parameter identiﬁcation, as a part of a Variable Structure state observer. On the one hand, the Variable Structure theory is used in our algorithm to design the observer restoring the unknown states. On the other hand, the use of the well-known chattering property, an intrinsic phenomenon of the Variable Structure observer, makes possible the design of the parameter estimation law. For instance, the classical adaptive control requires an additional property (the persistent excitation) in order to ensure the parameter convergence. In a linear context, it is quite easy to verify the persistent excitation condition. Generally speaking, m parameters of a linear system are identiﬁed if there are at least m 2 frequencies applied to the input. However, in the nonlinear case, such simple relation may not be valid. Indeed, it is not clear how many frequencies are required to guarantee the persistent excitation of the input because it depends strongly on the studied nonlinear function. In this paper, the persistent excitation condition is advantageously replaced by the use of the chattering signal. Of course, we do not use directly the chattering signal. Indeed, this signal is generated by the fast switching of the discontinuous function around the sliding manifold. The high-frequency components of the chattering are most of the time undesirable because they may excite non-modeled high-frequency plant dynamics. To solve this problem, a ﬁrstorder ﬁlter is required in order to cut the undesirable high-frequency components. Finally, the chattering signal, ﬁrstly ﬁltered, in combination with the knowledge of all the vector state restored through the observer, provides the asymptotic convergence of the parameter estimation law. In section 2, a speciﬁc sliding observer for nonlinear systems where only the ﬁrst state is measurable is proposed. Thanks to the use of some Lyapunov

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candidate functions, one can prove the convergence in ﬁnite time to a neighborhood of zero for all state estimation errors. In section 3, the main contribution of our paper consists in the study of a new and simple parameter estimation law designed thanks to some characteristic and well-known properties of the sliding regimes. Firstly, we recall some notions on the global structural identiﬁability of nonlinear systems in order to guarantee the validity of the parameter identiﬁcation algorithm. Then, this new parameter estimation law is restricted for the moment to nonlinear systems linearly parameterized. Thanks to a Lyapunov function and to some well-known properties of the Variable Structure Theory, one can prove the asymptotic convergence to zero of parameter estimation errors. In section 4, we propose to compute at the same time the values of the parameter vector and the estimates of the state vector without the additional use of an extrinsic persistent signal. Simulations results in section 5 illustrate the use of the algorithm and highlight that the method is robust with respect to parameter uncertainties (constant and slowly-varying) and additive measurement noise thanks to the use of the Variable Structure theory.

2 2.1

State identiﬁcation Preliminaries: Nonlinear triangular systems

Let us assume the single output analytic system of the general form x˙ = φ (x, u, t) y = h(x)

(1)

with x ∈ Rn . Suppose that Ω is a open connected relatively compact subset of Rn . We assume that (1) is observable on Ω, it is equivalent to the requirement that the set of functions, the observation space of (1), deﬁned by {h, Lf h, · · · , Lif h, · · · } separates the points on Ω. (L denotes the Lie derivative operator). When the system described by (1) is observable, the map Ψ : Rn → Rn   h(x)  Lf h(x)   x→ ···  n−1 Lf h(x)

(2)

is almost everywhere regular (see for instance : [8]). Moreover, if we assume that Ψ deﬁned in (2) is a diﬀeomorphism from Ω onto Ψ (Ω), the system (1) can be written as  x˙ 1 = x2      x˙ 2 = x3 ... (3)   = f (x) + g(x)u x ˙  n   y = x1

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Let us suppose that the triangular system (3) is linear with respect to the input u. Moreover, if we assume that f (x) et g(x) are globally Lipschitz on Rn , the observability condition is still guaranteed. Finally, we suppose that the vector state x and the input u are bounded. Remark Regarding the design of the triangular system (3) obtained by simple transformations, x1 is the single measurable state. 2.2

Design of the state observer

Let us denote x ˆ the estimates of the vector state (3). The dynamics of the estimates, in other words the Variable Structure observer, could be written (see [5] and [16]) ˆ 2 + v1 x ˆ˙ 1 = x ... x ˆ˙ i = x ˆi+1 + vi ... ˙x ˆn = f (ˆ x) + g(ˆ x)u + vn with

(4)

v1 = k1 sign(x1 − x ˆ1 ) vi = ki sign(zi ) τi z˙i + zi = vi−1 for i = 2, 3, ..., n Considering equations (3) and (4), the following system concerning the state estimation errors is obtained ˆ1 e1 = x1 − x (5) e˙ 1 = e2 − k1 sign(e1 )  ˆi  ei = xi − x e˙ i = ei+1 − vi (6)  for i = 2, 3, ..., n − 1 ˆn en = xn − x (7) e˙ n = f (x) − f (ˆ x) + g(x)u − g(ˆ x)u − vn Proposition: For any initial condition x0 ∈ Rn , we can ﬁnd positive constants ki , i = 1, ..., n suﬃciently large and positive constants τi , i = 2, .., n suﬃciently small such that the state estimation errors ei i = 1, ..., n (5),(6),(7) converge to a neighborhood of zero in a ﬁnite time.

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113

Proof First Step Let us ﬁrst consider the candidate Lyapunov function V1 =

1 2 e 2 1

Taking into account (5), its derivative with respect to time is V˙ 1 = e1 e˙ 1 = e1 (e2 − k1 sign(e1 )) → V˙ 1 = e1 e2 − k1 |e1 | Therefore, V˙ 1 ≤ |e1 | |e2 |max − k1 |e1 | Finally, if the positive constant k1 is chosen such that k1 > |e2 |max V˙ 1 will be negative deﬁnite. A sliding mode is reached after a ﬁnite time, t1 [15]. The sliding manifold is deﬁned by e1 = 0; k1 > |e2 |max Moreover, thanks to the invariance property [4], the state estimation error e1 and its time derivative are such that: e1 (t) = 0 and e˙ 1 (t) converges to a neighborhood of zero

∀t ≥ t1

Finally, there exist a ﬁnite time t1 and a constant k1 satisfying x ˆ1 = x1 ∀t ≥ t1 |e˙ 1 (t)| ≤ ε1 where ε1 is small. Step i, i = 2, 3, ..., n − 1 Let us study now the following candidate Lyapunov function Vi =

1 2 e + Vi−1 2 i

By using the previous steps and especially (8), its time derivative is V˙ i ≤ ei e˙ i Consider (6). We have V˙ i ≤ ei (ei+1 − vi )

(8)

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Choosing τi suﬃciently small and using the invariance properties of the previous step, one can write V˙ i ≤ ei (ei+1 − ki sign(ei ))

(9)

Let us now re-write the equation (9) V˙ i ≤ |ei | |ei+1 |max − ki |ei | Therefore, if the positive constant ki is chosen such that ki > |ei+1 |max V˙ i will be negative deﬁnite. A sliding mode is reached after a ﬁnite time, such that ti > ti−1 . The sliding manifold is deﬁned by |ei (t)| ≤ κi ; ki > |ei+1 |max Moreover, taking into account the invariance conditions, the state estimation error ei and its time derivative are such that |ei (t)| ≤ κi |e˙ i (t)| ≤ εi ,

∀t ≥ ti

Finally, there exist a ﬁnite time ti and two constants ki and τi such |ˆ xi − xi | ≤ κi ∀t ≥ ti |e˙ i (t)| ≤ εi

(10)

where κi and εi are supposed to be small. Step n Suppose the complete Lyapunov function Vn =

1 2 e + Vn−1 2 n

By considering the previous steps and (10), its time derivative is V˙ n ≤ en e˙ n Taking into account (7), we obtain V˙ n ≤ en (f (x) − f (ˆ x) + g(x)u − g(ˆ x)u − vn ) Choosing τn suﬃciently small and using the invariance properties of the previous step, one can write V˙ n ≤ en (f (x) − f (ˆ x) + g(x)u − g(ˆ x)u − kn sign(en ))

(11)

State and parameter identiﬁcation using VS theory

115

Let us now suppose that the Lipschitz condition is veriﬁed regarding f and g and u supposed to be a bounded input. Therefore, the equation (11) is re-written in the following V˙ n ≤ |en | (α |en | − kn ) with α as a Lipschitz constant. Therefore, if the constant kn is chosen such that kn > α |en | V˙ n will be negative deﬁnite. A sliding mode is reached after a ﬁnite time, denoted tn > tn−1 . The sliding manifold is deﬁned by |en (t)| ≤ κn ; kn > α |en | Finally, there exist a ﬁnite time tn and two positive constants kn and τn such ∀t ≥ tn

|ˆ xn − xn | ≤ κn

where κn is assumed to be small.

3

Parameter identiﬁcation

In order to determine the elements of the functions f (x) and g(x) from the observed state vector x, we have to assume that the system is globally identiﬁable. Let us recall some basic notions on the global identiﬁability. In linear context, the study of the identiﬁability is grounded on the notion of the transfer function, see [19] and [20]. Let us assume that M (s, θ) and M (s, θ ) are two transfer functions associated with the linear model for θ and θ , values of the parameter vector. The linear model is said to be structurally globally identiﬁable if and only if M (s, θ) = M (s, θ ) ⇒ θ = θ Generally speaking, it is more complex to verify the global identiﬁability in nonlinear context. In this latter case, the notion of “generating power series“ is required. 3.1

Preliminaries: Global identiﬁability in nonlinear context

Let us consider the following class of nonlinear systems x˙ = f (x, u, θ) = f0 (x, θ) +

i=m i=1

fi (x, θ) ui

(12)

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where x ∈ Rn represents the vector of states, θ ∈ Rp is the vector of unknown parameters and f0 , fi , i = 1, . . . , m ∈ C ∞ . The ﬁrst step in parameter identiﬁcation issues is to guarantee the global structural identiﬁability property of the system (12), as it is deﬁned in [20]. This step is of algebraic nature only and formal computing algorithms for this class of systems could be found in [12] and [13]. Let us recall very brieﬂy the main result. Consider the generating power series < g > (see [6]) associated with (12). This is a non-commutative power series deﬁned by < g > = x(0) +

m

Fj(0) ...Fj(ν) x |x(0) zj(ν) ...zj(0)

ν≥0 j(0),...,j(ν)=0

where F0 and Fi , i = 0, ..., m are Lie-derivatives given by F0 =

j=n

f0 (x, θ)

∂ ∂xj

fi (x, θ)

∂ ∂xj

j=1

Fi =

j=n j=1

(13)

Because of the analytical property of the generating power series, see [6], two models described by (12) have exactly the same input-output behaviour for all times and inputs if and only if they are associated with the same generating power series. Denote by < g > (θ) and < g > (θ ) the generating power series associated with the model output for the values θ and θ of the parameter vector. The model described by (12) will be structurally globally identiﬁable if and only if, for almost any admissible value of θ, < g > (θ) =< g > (θ ) admits θ = θ as its unique admissible solution. Now, the identity of < g > (θ) and < g > (θ ) is equivalent to the identity of the coeﬃcients Fj(0) ...Fj(ν) x |x(0) (θ) and Fj(0) ...Fj(ν) x |x(0) (θ ). The formal computation of these coeﬃcients can be found in [12] and also for instance in the paper [14]. For our knowledge, there is no general result giving how many of these coeﬃcients we need to obtain exactly p non trivial and algebraic independent coeﬃcients. We have at our disposal only formal computing algorithms. We then consider the algebraic nonlinear system of p equations Fj(0) ...Fj(ν) x |x(0) (θ) = Fj(0) ...Fj(ν) x |x(0) (θ )

(14)

The system (12) is identiﬁable if and only if the system (14) has a unique solution θ = θ .

State and parameter identiﬁcation using VS theory

3.2

117

Parameter Identiﬁcation algorithm

Let us assume that we could re-write the equations (3) with the following form x˙ 1 = x2 ... x˙ n = θa a(x) + θb u

(15)

where θa and θb are constant parameter vectors with adequate dimensions, u is the input vector and the system (15) is supposed to be globally identiﬁable. By using the system (15), the following identiﬁcation algorithm is proposed ξ˙n = θˆa a(x) + θˆb u + v with the estimated parameter values ˙ θˆa = λvaT (x) ˙ θˆb = λvuT (t)

(16a) (16b)

where v is the signal in relation with sliding modes. v contains some chattering property and makes possible parameter identiﬁcation without the use of an additional condition such as persistent excitation. Remark : λ is a positive constant which determines the speed convergence of the parameter estimation law. Preliminaries Suppose that a sliding mode appears on s = xn − ξn = 0. Its mathematical description requires the knowledge of the equivalent vector veq = ∆θa a(x) + ∆θb u

(17)

which is the unique solution of s˙ = 0. Beforehand, we have deﬁned ∆θa = θa − θˆa ∆θb = θb − θˆb Moreover, let us deﬁne now ∆Θ = (∆θa , ∆θb ) and z=

a(x) u(t)

Therefore, the equation (17) is re-written in the following veq = ∆Θz

(18)

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Proof of the asymptotic parameter convergence Let us now consider the following Lyapunov function 1 ∆Θ∆ΘT 2

V =

˙ Suppose that parameters are constant. Compute ﬁrst ∆Θ. ˙ = −θˆ˙a , −θˆ˙b ∆Θ Taking into account equations (16a) and (16b), we have ˙ = λv −aT (x), −uT ∆Θ = −λvz T

(19)

Therefore by using (19), the derivative of the Lyapunov function V could be re-written with the following form V˙ = −λvz T ∆ΘT Thanks to the invariance properties of the sliding mode, the equality v = veq = ∆Θz (equation (18)) is satisﬁed on the sliding manifold s = 0. It implies V˙ = −λ||veq ||2 Finally, we conclude limt→∞ V = 0. This implies that limt→∞ ∆Θ = 0, in other words lim ∆θa = 0

t→∞

lim ∆θb = 0

t→∞

As a consequence, the estimated parameter vectors θˆa and θˆb converge asymptotically to the unknown values of θa and θb .

4

Simultaneous state and parameter identiﬁcation

The state estimation described in section 2 is now combined with the parameter identiﬁcation developed in section 3. Let us assume that we study the canonical system of the following form x˙ i = xi+1 for i = 1, ..., n − 1 ... x˙ n = θa a(x) + θb u y = x1

(20)

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119

wherexT = (x1 , ..., xn−1, xn ), aT = (a1 , ..., ap−1 , ap ), θa = θa1 , ..., θap−1 , θap , θb = θb1 , ..., θbm−1 , θbm , u is the input vector (u ∈ Rm ), θaj and θbk are scalar constant parameters which have to be determined while only the ﬁrst state x1 is measured. Moreover, we suppose that the system (20) is observable (a(x) is globally Lipschitz) and globally identiﬁable. The ﬁnal algorithm has the following form ˆi+1 + vi for i = 1, 2, ..., n − 1 x ˆ˙ i = x ... ˙x ˆn = θˆa a(ˆ x) + θˆb u + vn with v1 = k1 sign(x1 − x ˆ1 ) vi = ki sign(zi ) for i = 2, 3, ..., n τi z˙i + zi = vi−1 for i = 2, 3, ..., n and ˙ ˙ ˆ θa = λvn aT (ˆ x), i.e. θˆaj = λvn aj (ˆ x) ˙ ˙ˆ θb = λvn uT , i.e. θˆbk = λvn uk where x ˆT = (ˆ x1 , x ˆ2 , ..., x ˆn ), θˆa = θˆa1 , θˆa2 , ..., θˆap , θˆb = θˆb1 , θˆb2 , ..., θˆbm are respectively estimates of the vector state and of the parameter values. ki and τi are some positive constants designed in section 2 in order to make all the state estimation errors ei converge to a neighborhood of zero in a ﬁnite time.

5

Simulations results

5.1

Preliminaries

Model Consider the following nonlinear triangular system describing the behaviour of a pendulum [9] represented in Figure 1 x˙ 1 = x2 1 g k x˙ 2 = u − sin x1 − x2 2 ml l m = θ1 u + θ2 sin x1 + θ3 x2 y = x1

(21)

k where θ1 = ml1 2 , θ2 = − gl and θ3 = − m have to be determined and x1 is supposed to be the only available state. To summarize, two simultaneous goals have to be achieved

• restoring x2 through the measure of the ﬁrst state x1 • estimating the three estimated values of the unknown parameter vector.

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l

α

g

k : friction parameter

m

Fig. 1. Pendulum

Global identiﬁability In order to verify that the system is globally identiﬁable, in other words the system is described by an unique parameter vector, we apply the work of [12] on structurally globally identiﬁability for nonlinear systems. Let us consider the vector ﬁelds associated with the pendulum’s model regarding equations (13) ∂[ ] ∂[ ] + (θ2 sin x1 + θ3 x2 ) ∂x1 ∂x2 ∂[ ] F1 [ ] = θ1 ∂x2

F0 [ ] = x2

Let us compute the following Lie derivatives F1 (F0 (0, θ)[y]) F1 (F0 (F0 (0, θ)[y])) F1 (F0 (F0 (F0 (0, θ)[y]))) applied to the natural output y = x1 of the system. This leads to F1 (F0 (0, θ)[y]) = θ1 F1 (F0 (F0 (0, θ)[y])) = θ1 θ3 F1 (F0 (F0 (F0 (0, θ)[y]))) = θ1 θ32 + θ2 Applying the equality g < θ >= g < θ > leads to θ1 = θ1

θ1 θ3 = θ1 θ3 2 2 θ1 θ3 + θ2 = θ1 θ3 + θ2

(22)

State and parameter identiﬁcation using VS theory

121

where θ = (θ1 , θ2 , θ3 ) is an another triplet of parameters. We obtain a nonlinear algebraic system (22) with three independent equations (“as many equations as parameters “). By solving the algebraic system (22), we obtain θ1 = θ1

θ3 = θ3 θ2 = θ2 In other words, we have proven that the equality g < θ >= g < θ > implies the equality θ = θ where g represents the generating power series introduced in section 3. Finally, we can conclude that the pendulum is an identiﬁable system. 5.2

State and parameter identiﬁcation algorithm

Regarding the equations (20), the following equalities for the system (21) are veriﬁed θa = (θ2 , θ3 ) θb = θ1 aT = (sin(x1 ), x2 ) The state observer has the following structure x ˆ˙ 1 = x ˆ 2 + v1 ˙x ˆ1 + θˆ3 x ˆ 2 + v2 ˆ2 = θˆ1 u + θˆ2 sin x v1 = k1 sign (x1 − x ˆ1 ) v2 = k2 sign (z2 ) τ2 z˙2 + z2 = v1 The parameter identiﬁcation algorithm has the following form ˙ θˆ1 = λv2 u ˙ θˆ2 = λv2 sin x ˆ1 ˙ˆ ˆ2 θ3 = λv2 x Noiseless context We suppose that the reference values are such θ1 = 1000, θ2 = −0.5 and θ3 = −3. Concerning the state estimation errors, the results are shown in Figure 2. Observation errors for both states converge to zero in less than 0.5s. Moreover, the shape of the observation errors, switching around the horizontal axis, is characteristic of the discontinuous control used in the Variable Structure observer. In Figure 3, one can see the evolution of the three estimated parameters versus time. It is worth noticing that es-

122

F. Floret-Pontet and F. Lamnabhi-Lagarrigue −7

1.5

Observation error − First state

x 10

1 0.5 0 −0.5 −1 −1.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.7

0.8

0.9

1

Observation error − Second state 0.01

0.005

0

−0.005

−0.01

0

0.1

0.2

0.3

0.4

0.5 0.6 time (seconds)

Fig. 2. Observation errors for both states in noiseless context

Estimated θ1 1000 Nominal value : 1000 500

0

0

0.1

0.2

0.3

0.4

0.5 Estimated θ

0.6

0.7

0.8

0.9

1

2

10 Nominal value : −0.5 0 −10 −20

0

0.1

0.2

0.3

0.4

0.5 Estimated θ

0.6

0.7

0.8

0.9

1

3

40 Nominal value : −3 20 0 −20

0

0.1

0.2

0.3

0.4

0.5 0.6 time (seconds)

0.7

0.8

0.9

1

Fig. 3. Parameter identiﬁcation with the Variable Structure Identiﬁer (noiseless context)

timated parameter are close to their reference values in approximately 0.5s with no large overshooting. Moreover, the parameter convergence is achieved without the use of an additional condition (the persistent excitation) usually required regarding the input of the system. Here, the only condition required for the input u is its boundary. As a consequence, u in the Variable Structure identiﬁer is a single frequency sinusoidal signal. To point out this interesting

State and parameter identiﬁcation using VS theory

123

lack of extra persisting input, we deal now with the classical Least Squares algorithm. Comparative study To achieve this goal (the Least Squares algorithm), we assume that x1 is still the single available state. As a consequence, we have to take the derivative of x1 to obtain x2 . Moreover, in order to obtain the parameter convergence, a persistent excitation condition is required for the input u in the case of the Least Squares algorithm. We obtain the results shown in Figure 4. These results are obtained with the Least Squares algoEstimated θ1 1000

500

0

0

0.1

0.2

0.3

0.4

0.5 0.6 Estimated θ2

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5 Estimated θ

0.6

0.7

0.8

0.9

1

0.5 0.6 time (seconds)

0.7

0.8

0.9

1

−50

−150

−250

3

140 90 40 −10

0

0.1

0.2

0.3

0.4

Fig. 4. Parameter identiﬁcation with the Least Squares algorithm with exponential forgetting (noiseless context)

rithm with exponential forgetting which is more robust with respect to perturbations in comparison to the classic Least Squares algorithm. The speed of parameter convergence obtained in the VS identiﬁer is comparable to the one obtained with the Least Squares algorithm with exponential forgetting and almost better. Moreover, one can notice that the Least Squares algorithm generates some large overshoots compared to the ones obtained in the case of the VS identiﬁer. Finally, we could point out the high-frequency components obtained in Figure 3 devoted to the Variable Structure identiﬁer. They represent a direct consequence of the ﬁltered chattering signal. It is possible to decrease these high-frequency components by re-scheduling the ﬁrst-order ﬁlter. In that case, the speed of the parameter convergence would be larger. Generally speaking, we have to deal with a trade-oﬀ between the speed of the convergence and the high-frequency components in the estimated parameters. Concerning the Least Squares algorithm, the trade-oﬀ concerns the

124

F. Floret-Pontet and F. Lamnabhi-Lagarrigue

choice of the exponential forgetting factor modifying the rate of parameter convergence and the large overshoots. 5.3

Robustness study

Constant parameter perturbation Let us suppose that each parameter θ1 , θ2 and θ3 undergo at t1 = 0.8s, t2 = 1.6s and t3 = 2.4s (respectively) some bounded parameter uncertainties such θi = θiref for t ≤ ti for i = 1, 2, 3 θi = θiref + ∆θi for t > ti where ∆θ1 =

50 100 θ1ref ,

∆θ2 =

500 100 θ2ref

and ∆θ3 =

200 100 θ3ref .

In Figure 5,

Estimated θ

1

2000 1500 1000 500 0

0

0.5

1

1.5 Estimated θ

2

2.5

3

1.5 Estimated θ

2

2.5

3

1.5 time (seconds)

2

2.5

3

2

60 40 20 0 −20

0

0.5

1

3

40 20 0 −20

0

0.5

1

Fig. 5. Estimated parameters with successive constant parameter uncertainties VS Identiﬁer

one can see that estimated parameters θˆ1 , θˆ2 and θˆ3 are very close to their reference values θ1ref , θ2ref and θ3ref before and after the eﬀect of the perturbations. After the eﬀect of the perturbations, the algorithm takes approximately less than 0.5s to reach the new reference value. Therefore, simulation results have shown that the method is robust with respect to large and constant parameter uncertainties. In order to highlight the eﬀectiveness of the method proposed here, we would want to compare the Variable Structure identiﬁer with a classical identiﬁcation algorithm, the Least Squares algorithm in the same simulations conditions. A persisting input is now applied to the system and we obtain the results shown in Figure 6. In order to have a fair comparative study, the re-

State and parameter identiﬁcation using VS theory

125

Estimated θ1 2000 1500 1000 500 0

0

0.5

1

1.5 Estimated θ2

2

2.5

3

0

0.5

1

1.5 Estimated θ3

2

2.5

3

0

0.5

1

1.5 time (seconds)

2

2.5

3

60 40 20 0 −20 40 20 0 −20

Fig. 6. Estimated parameters with successive constant parameter uncertainties Least Squares algorithm with exponential forgetting

sults represented in Figure 6 are shown with the same scale as the one used in Figure 5 devoted to the Variable Structure identiﬁer. These results point out the better results obtained with the Variable identiﬁer compared to the Least Squares algorithm in terms of robustness and performances. Regarding the robustness, one can notice that the Variable Structure identiﬁer provides some smaller overshoots when there are some constant perturbations. Moreover, the rate of parameter convergence is smaller in the case of the Variable Structure identiﬁer in comparison to the Least Squares algorithm. Slowly-varying parameter perturbation Let us suppose that θ1 undergoes a sinusoidal parameter uncertainty such θ1 = θ1ref = θ1ref + ∆θ1 where ∆θ1 = θ51 sin(0.33t). In Figure 7, one can see that the estimated parameter θˆ1 is really close to its reference value θ1ref even if this perturbation is a slowly-varying one. Therefore, simulation results have shown that the method is more robust with respect to a signiﬁcant sinusoidal parameter uncertainty, in comparison to the result obtained in Figure 8 with the Least Squares algorithm. Noisy context In order to highlight the robustness of the methodology presented in this paper with respect to measurement noise, we add a white Gaussian noise with a magnitude of 5% of the original measured signal x1 .

126

F. Floret-Pontet and F. Lamnabhi-Lagarrigue Estimated θ1 1200

1000

800

600

400

200

0

0

5

10

15

20

25

time (seconds)

Fig. 7. Estimated parameter θˆ1 with a sinusoidal parameter uncertainty - VS Identiﬁer . . . : estimated parameter, - : reference parameter Estimated θ

1

1400

1200

1000

800

600

400

200

0

0

5

10

15

20

25

time (seconds)

Fig. 8. Estimated parameter θˆ1 with a sinusoidal parameter uncertainty - Least Squares algorithm with exponential forgetting

Therefore, the behaviour of the ﬁrst parameter is shown in Figure 9. The estimated parameter θˆ1 is really close to its reference value θ1ref even if the measurement undergoes a bounded perturbation (ﬁgure 9). It is worth noticing that the method rejects perturbations such as additive measurement noise.

State and parameter identiﬁcation using VS theory

127

Measured State with additive noise 0.05 0.04 0.03 0.02 0.01 0

0

0.5

1

1.5

2

2.5

3

2

2.5

3

Estimated θ

1

1200 1000 800 600 400 200 0

0

0.5

1

1.5 time (seconds)

Fig. 9. The measured state x1 with noise and estimated parameter θˆ1 in noisy context

Remark We do not present in this paper the results obtained with the Least Squares algorithm. Indeed, with this latter method, we have to take the derivative of x1 to obtain the value of x2 . It implies the generation of dirty derivatives (derivation of noise) and provides unreliable estimated parameters.

6

Conclusion

In this paper, a new approach for state and parameter identiﬁcation of a reasonably large class of nonlinear uncertain systems is investigated. Specifically, it could be applied to nonlinear systems linearly parameterized where the vector state is partially measurable. The unknown states are restored through a Variable Structure observer in order to avoid some dirty derivatives obtained in the case of the Least Squares algorithm for instance. This state observer is combined with a parameter identiﬁer in order to make estimated parameters converge as close as possible to their nominal values. The parameter estimation algorithm takes into account some intrinsic properties of the Variable Structure observer and simpliﬁes advantageously the design of this parameter law compared to more classical methodologies. Indeed, the method proposed in the paper does not require some extra persistent excitation regarding the input of the system in comparison to the classical Least Squares algorithm. Moreover, this Variable Structure identiﬁer rejects perturbations such as additive measurement noise and it is robust with respect to signiﬁcant parameter uncertainties, constant and slowly-varying parameter perturbations thanks to the use of the Variable Structure Theory.

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F. Floret-Pontet and F. Lamnabhi-Lagarrigue

References 1. T. Basar, G. Didinsky, Z.Pan (1996) A new class of identiﬁers for robust parameter identiﬁcation and control in uncertain systems. F. Garafalo and L. Glielmo (eds), Robust Control via structure and Lyapunov techniques, Springer-Verlag, pp 149–173. 2. G. Bastin, O. Bernard, L.B. Chen, V. Chotteau (1997) Identiﬁcation of mathematical models for bioprocess control. Proc. of the 11th Forum for Applied Biotechnology, pp. 1559–1572. 3. S.A. Billings (1980) Identiﬁcation of nonlinear systems - a survey, IEE PROC (127), pp. 272–285. 4. C.Canudas de Wit, J.J.E. Slotine (1998) Sliding observers for robot manipulators, Automatica 27, pp. 859–864. 5. S. Drakunov, V. Utkin (1995) Sliding modes observers. Tutorial, Proc. 34th IEEE Conference on Decision and Control, pp. 3376–3378. 6. M. Fliess, M. Hazewinkel (1985) Algebraic and Geometric Methods in Nonlinear Control Theory, Proc. Conf. Paris, Eds M. Hazewinkel. 7. F. Floret-Pontet, F. Lamnabhi-Lagarrigue (2001) Parameter identiﬁcation methodology using sliding regimes, International Journal of Control (to appear). 8. J.P. Gauthier, H. Hammouri, S. Othman (1992) A simple observer for Nonlinear systems, Applications to Bioreactors, IEEE Transactions on Automatic Control, vol 37, 6, pp 875–880. 9. K. Khalil (1992) Nonlinear systems, Macmillan Publishing Company, NewYork. 10. I.D. Landau (1998) Identiﬁcation et commande des syst`emes, Collection p´edagogique d’automatique, Herm`es, Paris. 11. I.D. Landau, B.D.O. Anderson, F. De Bruyne (2000) Algorithms for identiﬁcation of continuous time nonlinear systems: a passivity approach, in Nonlinear Control in the Year 2000, Eds. A. Isidori, F. Lamnabhi-Lagarrigue and W. Respondek, Springer Verlag, vol. 2, pp. 13–44. 12. Y. Lecourtier, F. Lamnabhi-Lagarrigue, E. Walter (1987) Volterra and generating power series approaches to identiﬁability testing, in Identiﬁability of parametric models, edited by E. Walter, Pergamon Press, pp. 50–66. 13. F. Ollivier (1990) Le probl`eme de l’identiﬁabilit´e structurelle globale: approche th´eorique, m´ethodes eﬀectives et bornes de complexit´e, Ecole Polytechnique. 14. F. Rotella, I. Zambettakis (1991) Generating power series for analytic systems, Algebraic Computing in Control, Edited by G. Jacob and F. LamnabhiLagarrigue, LNCIS 165, Springer-Verlag, pp. 279–287. 15. J.J. Slotine, Weiping Li (1991) Applied nonlinear control, Prentice Hall. 16. J.J. Slotine, J.K. Hedrick, E.A. Misawa (1987) On sliding observers for nonlinear systems, Trans. ASME, Journal of Dynamic Systems, Measurement and Control, 109, pp. 245–252. 17. V.I. Utkin (1992) Sliding modes in control optimization, Springer-Verlag. 18. V.I. Utkin (1981) Principles of identiﬁcation using sliding regimes, Sov. Phys. Dokl., 26(3), pp. 271–272. 19. E. Walter (1987) Identiﬁability of parametric models, Pergamon Press. 20. E. Walter, L. Pronzato (1994) Identiﬁcation de mod`eles param´etriques, Masson. 21. J. Xu, H. Hashimoto (1993) Parameter identiﬁcation methodologies based on variable structure control, Int. J. Control, vol. 57, no. 5, pp. 1207–1220.

Stabilizing memoryless controllers and controls with memory for a class of uncertain, bilinear systems with discrete and distributed delays D.P. Goodall Control Theory & Applications Centre, Coventry University, Priory Street, Coventry CV1 5FB, U.K.

Abstract. A class of robust feedback controls are designed to stabilize a class of uncertain, bilinear, delay systems containing multiple time-delays. Each delay system, in the class considered, is assumed to be of the retarded-type, and contains both discrete and distributed delays. A deterministic methodology based on Lyapunov theory and Lyapunov-Krasovski˘i functionals is utilized and feedback controllers are synthesized that will ensure, under appropriate hypotheses and satisfaction of appropriate stability criteria, a global uniform asymptotic stability property for the prescribed class of delay systems of the retarded-type.

1

Introduction

The principal objective in this investigation is to design robust feedback controls which will stabilize a class of uncertain, bilinear, time-delay systems containing multiple delays. The delay system is assumed to be of the retardedtype, and contains both discrete and distributed delays. The deterministic approach, used here, is to represent the uncertainty as a class of nonlinear perturbations inﬂuencing a known idealized (nominal) model. An assumption on the nominal model is speciﬁed in terms of an algebraic Riccati-type matrix equation relating to the model parameters. Moreover, the nonlinear perturbations are assumed to be state, delayed-state and input-dependent and/or time varying. There have been numerous studies on uncertain time-delay systems, for example, in [1], [6], [7], the nominal model is linear, whereas in [2], [3] and [8] the nominal model is assumed nonlinear. This investigation falls into the latter category. Note that [2], [3] and [8] do not consider distributed delays. Parametric uncertainty is not considered in this paper; instead, a priori bounding knowledge of the system uncertainty, in terms of growth conditions with respect to its arguments, is assumed. One feature of the controllers, employed to stabilize the class of uncertain systems, is that the gains depend explicity on the upper bounds of the uncertainty and, thus, robustness of the feedback controls is a consequence. Another feature is that the controllers may depend on delayed states, i.e. controls with memory. For such controls, practical implementation may not be straightforward and it may also be computationally expensive in the sense that the past history of A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 129-138, 2003. Springer-Verlag Berlin Heidelberg 2003

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the state needs to be stored. Therefore, this investigation will also consider the feasibility of using memoryless controllers (see [1] and [8]).

2

Preliminaries and nomenclature

First some mathematical notation is introduced. Let R := (−∞, ∞), R+ 0 := [0, ∞) and deﬁne the Euclidean inner product on the vector space Rn and the induced norm by · , · and ·, respectively. If X ⊂ Rp and Y ⊂ Rq , C(X ; Y) denotes the space of all continuous functions mapping Rp → Rq , whereas C 1 (X ; Y) ⊂ C(X ; Y) denotes the space of continuous functions which have continuous ﬁrst order partial derivatives. Introduce the notation xt = xt (r) := x(t + r) (r ∈ [−τ, 0], τ > 0) to denote the restriction of x(·) to the interval [t − τ, t] and note that if xt ∈ C([−τ, 0]; Rn ), then an appropriate norm is xt τ := supr∈[−τ,0] x(t + r). Additionally, deﬁne a set of bounded functions 0]; Rn ) to be QτA := δ ∈ C([−τ, 0]; Rn ) : δτ < A, 0 < in C([−τ, A < ∞ . Let Rp×p represent the set of real square matrices of order p×p and, if P ∈ Rp×p , P > 0 denotes that P is positive deﬁnite. Finally, let σmin (·) denote the minimum eigenvalue of a symmetric positive deﬁnite matrix.

3

Problem statement

The class of uncertain time-delay systems to be investigated consists of a nominal (known) bilinear time-delay system, of the retarded type, with discrete and distributed delays, augmented by unknown functions, having the following form: t uj (t) x(t) ˙ = Ax(t) + Bi x(t − ρi ) + Dk x(z) dz + i∈R

k∈S

t−τk

j∈M

+ pj (t, x(t), x(t − ρ1 ), . . . , x(t − ρr ), I(x, τ1 ), . . . , I(x, τs ), u(t)) Cj x(t) + q(t, x(t), x(t − ρ1 ), . . . , x(t − ρr ), I(x, τ1 ), . . . , I(x, τs ), u(t)) , (1) where, for m, n ∈ N, x(t) ∈ Rn and u(t) ∈ Rm (1 ≤ m ≤ n) is the control (or input) vector, ρi are r bounded discrete delays and τk are s bounded distributed delays which aﬀect the system, M := {1, 2, . . . , m}, t R := {1, 2, . . . , r}, S := {1, 2, . . . , s}, and I(x, τ ) := t−τ x(z) dz. The matrices A, Bi , Cj and Dk are assumed known and the functions pi and q, representing uncertainty, are unknown. With x0 ∈ Rn speciﬁed, φ ∈ C([−τ, 0]; Rn ) and φ(0) = x0 , the initial condition xt0 (θ) = φ(θ) ,

θ ∈ [−τ, 0] ,

where τ = max[ρi , τk ] , i,k

(2)

is assumed to hold. The functions pi : R ×Rn ×Rn ×. . .×Rn ×Rm → Rm and q : R × Rn × Rn × . . . × Rn × Rm → Rn , which are unknown and are assumed

Stabilizing controllers for uncertain bilinear time-delay systems

131

to be continuous, belong to a known non-empty class which comprise all possible system uncertainties, including any known time-dependent and/or nonlinear elements. Observe that pj represents the matched uncertainty and q the residual (or mismatched ) uncertainty. Subject to the initial condition (2), the main objective is to design a class of robust continuous state-feedbacks, F , which will globally stabilize system (1). Consider a general retarded functional diﬀerential equation of the form x(t) ˙ = h(t, xt ) , xt0 (θ) = φ(θ) ,

(t, xt ) ∈ R × QτA ,

t > t0 , τ > 0 ,

(3)

θ ∈ [−τ 0] with φ(0) = x . 0

(4)

where x(t) ∈ Rn , h has the properties (a) h(t, 0) = 0 for all t; (b) for any bounded set B ⊂ R × QτA , the closure of h(B) := {h(b) : b ∈ B} is compact; and the initial condition function satisﬁes φ ∈ C([−τ, 0]; Rn ). It is assumed that there exists at least one local solution to (3)–(4). Denote such a solution by x(·, t0 , φ). In the ensuing stability analysis the following lemmas, relating to system (3)–(4) and deduced from the appropriate references, will be invoked. Let S ⊂ Rn be a compact non-empty set, containing {0} and, moreover, let ds (a, S) := inf s∈S d(a, s), where d denotes some metric on Rn , denote the distance between a point a ∈ / S and the set S. Lemma 1 (Deduced from [5], Chapter 2, Theorem 2.7.15) Suppose that every solution x(·, t0 , φ) of (3)–(4) is bounded, then every solution can be extended to the interval [t0 − τ, ∞). Lemma 2 (Deduced from [5], Chapter 6, Theorem 6.2.22) Let Ω denote the set of scalar nondecreasing functions ω ¯ ∈ C(R; R) such that ω ¯ (ζ) > 0 for ¯ 1 (ζ) → ∞ ζ > 0 and ω ¯ (0) = 0. Suppose there exists ω ¯ i ∈ Ω, i = 1, 2, 3, with ω as ζ → ∞, and a functional W ∈ C 1 (R × QτA ; R+ 0 ) such that (a) for all (t, ψ) ∈ R × QτA ,

¯2 ω ¯ 1 (ds (ψ(0), S)) ≤ W (t, ψ) ≤ ω

sup ds (ψ(θ), S)

;

θ∈[−τ,0]

(b) along solutions to (3)–(4), ˙ (t, xt ) + ω W ¯ 3 (ds (x(t), S)) ≤ 0 , then S is a globally uniformly asymptotically stable invariant set.

4

Hypotheses

The nominal (known) bilinear system, with multiple time-delays, is described by t Bi x(t − ρi ) + Dk x(α) dα + uj (t)Cj x(t) , x(t) ˙ = Ax(t) + i∈R

k∈S

t−τk

j∈M

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and, for this system, the assumptions stated in Hypothesis 1 are assumed to hold. Hypothesis 1 For given symmetric Q > 0, Ri > 0 for i = 1, . . . , r, and Sk > 0 for k = 1, . . . , s, suppose there exist w1 , w2 ∈ {0, 1} and a unique symmetric P > 0 which is a solution to the algebraic Riccati-type matrix equation P A + AT P + w1

(Ri + P Bi Ri−1 BiT P )

i∈R

+ w2

τk (Sk + P Dk Sk−1 DkT P ) = −Q .

(5)

k∈S

Remarks. (i) It is well known (see [4]) that the existence of a symmetric P > 0, satisfying (5) with w1 = w2 = 1, guarantees that x = 0 is asymptotically stable under the dynamics of the associated drift system: x(t) ˙ = Ax(t) +

i∈R

Bi x(t − ρi ) +

k∈S

t

Dk

x(α) dα . t−τk

Other forms of (5) may be obtained using diﬀerent transformations, see [4] for more details. (ii) The main purpose of the parameters w1 and w2 is to investigate stability of an uncertain system when w1 = w2 = 0, as well as w1 = w2 = 1. With respect to the unknown uncertainty functions pi and q, two sets of bounding assumptions are considered. In the ﬁrst set, fairly general bounding assumptions are speciﬁed for the matched uncertainty in terms of the functions δj , dependent upon time, state, delayed states and an additive term involving the control. Hypothesis 2 For all (t, x, y1 , . . . , yr , z1 , . . . , zs , u) ∈ R ×Rn ×. . .×Rn ×Rm , (a) there exist known continuous functions δj : R×Rn ×. . .×Rn → R+ 0 , which are uniformly bounded with respect to t and locally uniformly bounded with respect to the arguments x, yi , zk and known continuous functions ξj : R × Rn × . . . × Rn → [0, ξ¯j ], with 0 ≤ ξ¯j < 1 for all j, such that |pj (t, x, y1 , . . . , yr , z1 , . . . , zs , u)| ≤ δj (t, x, y1 , . . . , yr , z1 , . . . , zs ) + ξj (t, x, y1 , . . . , yr , z1 , . . . , zs )|uj | ; (b) there exist known real constants α ˆ , βˆi , γˆk , a ˆ ≥ 0 and continuous func, uniformly bounded with respect to t tions δˆj : R × Rn × . . . × Rn → R+ 0 and locally uniformly bounded with respect to the arguments x, yi and

Stabilizing controllers for uncertain bilinear time-delay systems

133

zk , such that q(t, x, y1 , . . . , yr , z1 , . . . , zs , u) ≤ a ˆ+α ˆ x + +

γˆk zk +

k∈S

βˆi yi

i∈R

δˆj (t, x, y1 , . . . , yr , z1 , . . . , zs )| x, P Cj x | .

j∈M

A stronger set of assumptions is presented in Hypothesis 3. Hypothesis 3 For all (t, x, y1 , . . . , yr , z1 , . . . , zs , u) ∈ R ×Rn ×. . .×Rn ×Rm , (a) for all i ∈ R, j ∈ M and k ∈ S, there exists known real constants ˜ j ≥ 0, known continuous functions ξj : R × Rn → [0, ξ¯j ] with i , ˜j , κk , κ ¯ 0 ≤ ξj < 1 for all j, and known continuous functions αj : R × Rn → R+ 0, which are uniformly bounded with respect to t and locally uniformly bounded with respect to x, such that i yi |pj (t, x, y1 , . . . , yr , z1 , . . . , zs , u)| ≤ αj (t, x) + ˜j +κ ˜j

i∈R

κk zk + ξj (t, x)|uj | ;

k∈S

(b) there exists known real constants α ˆ , βˆi , γˆk , a ˆ ≥ 0 and continuous func, uniformly bounded with respect to t and locally tions α ˆ j : R × Rn → R+ 0 uniformly bounded with respect to x, such that q(t, x, y1 , . . . , yr , z1 , . . . , zs , u) ≤ a ˆ+α ˆ x + βˆi yi +

k∈S

γˆk zk +

i∈R

α ˆ j (t, x)| x, P Cj x | .

j∈M

Note that, in Hypothesis 3, the bounding functions αj in the matched uncertainty and the functions α ˆ j in the residual uncertainty only depend on t and x and not the delayed-states.

5

Stability results

In this section a suﬃcient condition for the stabilization of system (1)–(2) is presented, under appropriate hypotheses. Initially, the synthesis of robust controls with memory is considered . However, for such controls, practical implementation may not be straightforward and it may also be computationally expensive in the sense that past history of the states needs to be stored. Therefore, in addition, robust memoryless controllers are designed, and stability properties for the system are investigated.

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D.P. Goodall

Stabilization via feedbacks with memory

In this subsection a suﬃcient condition for the stabilization of system (1)– (2) is presented, assuming Hypotheses 1 and 2 hold. The class of feedback controls, with memory, which stabilize this system consist of the nonlinear functions u(t) = f (t, xt ) = [f1 (t, xt ) f2 (t, xt ) . . . fm (t, xt )]T , where (t, xt ) → fj (t, xt ) :=

−(1 − ξj (t, xt ))−1 πj2 (t, xt ) x(t), P Cj x(t) πj (t, xt )| x(t), P Cj x(t) | + χj x(t)

2

,

(6)

χj > 0 are design parameters, πj (t, xt ) = δj (t, xt ) + P x δˆj (t, xt ), and with abuse of notation, δj (t, xt ) := δj (t, x(t), x(t − ρ1 ), . . . , x(t − ρr ), I(x, τ1 ), . . . , I(x, τs )), δˆj (t, xt ) := δˆj (t, x(t), x(t − ρ1 ), . . . , x(t − ρr ), I(x, τ1 ), . . . , I(x, τs )), ξj (t, xt ) := ξj (t, x(t), x(t − ρ1 ), . . . , x(t − ρr ), I(x, τ1 ), . . . , I(x, τs )). Loosely speaking, the term involving ξj is designed to counteract input uncertainty and πj is a multiplicative gain component which is designed to counteract other uncertainty in the system. The structure of the controller, in this paper,

is of a similar form to those introduced in [9]. Let χ ˆ = j∈M χj and suppose 1 BiT P k˜ := σmin (Q) − α ˆ+ γˆk τk P − (1 − w1 ) βˆi + 2

− (1 − w2 )

i∈R

i∈R

k∈S

τk DkT P > 0 .

(7)

k∈S

Then χj can be selected so that ˆ>0. k1 := k˜ − χ

(8)

Thus, with design parameters χj satisfying (8), the following result can be deduced by invoking Lemmas 1 and 2 and utilising the Lyapunov-Krasovski˘i functional candidate 0 ψ(θ), (w1 Ri + λi In )ψ(θ) dθ θ→

v(ψ(θ)) := ψ(0), P ψ(0) + +

k∈S

0

τk

i∈R 0

−φ

−ρi

ψ(θ), (w2 Sk + νk In )ψ(θ) dθ dφ ,

where In is the n × n identity matrix and λi , νk ≥ 0 for all i ∈ R and k ∈ S. ˜ deﬁned in (7), satisﬁes Theorem 1 Suppose Hypotheses 1 - 2 hold. If k, k˜ > 0, then, with feedback control u(t) = f (t, xt ), deﬁned by (6) and with χj satisfying (8), the compact set A A1 ⊇ {0}, where A1 := x ∈ Rn : x ≤ k1−1 a ˆ P ,

Stabilizing controllers for uncertain bilinear time-delay systems

135

is a globally uniformly asymptotically stable invariant set for the class of systems (1) subject to the initial condition (2). In the next subsection, it is shown that a class of memoryless controllers can be used to stabilize the uncertain system (1)–(2) provided stronger hypotheses hold. 5.2

Stabilization via memoryless feedbacks

In this subsection, a class of memoryless controls is presented which stabilizes system (1)–(2) subject to stronger hypotheses; namely Hypotheses 1 and 3. The class of feedback controls, which are continuous for (t, x) ∈ R × Rn \ {0}, consists of the nonlinear memoryless functions u(t) = f˜(t, x(t)) = T f˜1 (t, x(t)) . . . f˜m (t, x(t)) , with (t, x) → f˜j (t, x) := − µj +

˜j2 (t, x) (1 − ξj (t, x))−1 π 2

π ˜j (t, x)| x, P Cj x | + χ ˜j x

x, P Cj x , (9)

where µj := µ(˜ j + κ ˜ j ), π ˜j (t, x) := αj (t, x) + P x α ˆ j (t, x), and the design parameters µ, χ ˜j > 0 are real.

Let χ ¯ = j∈M χ ˜j , ¯ = i∈R 2i , κ ¯ = k∈S κ2k , and deﬁne Γ := r¯

˜j + κ ¯

j∈M

k∈S

τk2

κ ˜j .

(10)

j∈M

Suppose k˜ > 0,then µ and χ ˜j are chosen to satisfy Γ >0. k˜ − χ ¯− 4µ

(11)

For example, suppose ˜j + κ ˜ j = 0 for some j ∈ M, then Γ > 0. Hence, µ can ˜ where 0 < η < 1 is arbitrary, so that be selected to satisfy µ > Γ/(4η k), ˜ ˜ k − Γ/(4µ) > (1 − η)k > 0. Thus, if Γ > 0, χ ˜j are then chosen to ensure (1 − η)k˜ − χ ¯ > 0. In the case when ˜j + κ ˜ j = 0 for all j ∈ M, µ > 0 is selected ˜ arbitrarily and χ ˜j are chosen to ensure χ ¯ < k. Introducing a real positive constant k2 , deﬁned to be k˜ − χ ¯, Γ =0, k2 := (1 − η)k˜ − χ ¯ , otherwise , and utilizing Hypotheses 1 and 3, together with (9), the following result can be deduced.

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Theorem 2 Suppose Hypotheses 1 and 3 hold. If k˜ > 0, then, with memo˜j ryless feedback u(t) = f˜(t, x(t)), where f˜j are deﬁned by (9) and µ and χ satisfy (10)–(11), the compact set A A2 ⊇ {0}, where A2 := x ∈ Rn : x ≤ k2−1 a ˆ P , is a globally uniformly asymptotically stable invariant set under the dynamics of system (1) subject to the initial condition (2). Remarks (i) The gains of the memoryless controllers may possibly be large. See the example in Sect. When w

1 = w 2 = 0 in the Riccati-type matrix

6. (ii) T B T P and τ appear in the stability D P equation, then terms k i k criterion. (iii) If ˆa = 0 then {0} is a globally uniformly asymptotically stable invariant set under the dynamics of (1)–(2). (iv) The methodologies used in subsections 5.1 and 5.2 can easily be adapted to account for time-varying delays (see, for example, [3]).

6

Example

The following example is adopted to illustrate the above theory. Suppose a system is subject to uncertainty and the uncertain dynamics are modelled as x(t) ˙ = −2x(t) + x(t − ρ1 ) + 14 x(t − ρ2 ) + u(t)x(t) +

+ x(t)p t, x(t), x(t − ρ1 ), x(t − ρ2 ),

+ q t, x(t − ρ1 ),

x(z) dz, u(t)

t

x(z) dz t−τ

t

x(z) dz, u(t) t−τ

t

1 2

,

t−τ

subject to the initial condition xt0 (θ) = 5, θ ∈ [−4, 0], where x(t), u(t) ∈ R, p, q are unknown functions, and ρ1 = 12 , ρ2 = 4, τ = 1 are delays. It is easy to verify that the conditions of Hypothesis 1 are satisﬁed with P = [2], Q = [1], R1 = [2], R2 = 12 , S1 = [1], and w1 = w2 = 1. For simulation purposes, suppose the matched and residual uncertainty are given by p(t, x, y1 , y2 , z, u) = 10 + 2x2 + y1 cos(t) − sin(y1 )y2 + sin(2t)z + 12 u sin(u)

q(t, y, z, u) = −0.05y cos(u) + 0.125 cos(y)z . Two cases will be considered. 1. Controls with memory The conditions of Hypothesis 2 hold with δ1 (t, x, y1 , y2 , z1 ) = 10 + 0.5x2 + ˆ = a, α ˆ = 0, βˆ1 = 0.05, βˆ2 = 0, γˆ1 = 0.125 and |y1 | + |y2 | + |z1 |, ξ1 = 0.5, a

Stabilizing controllers for uncertain bilinear time-delay systems

137

δˆ1 ≡ 0. The stability criterion (7) holds and, since k˜ = 0.15, χ1 is chosen to have the value 0.01. The controls, with memory, are designed to be f1 (t, xt ) =

−4π12 (t, xt ) , 2π1 (t, xt ) + 0.01

where π1 (t, xt ) = 10 + 0.5x2 + |xt (−ρ1 )| + |xt (−ρ2 )| + |I(x, τ )|. The matched uncertainty has been speciﬁcally chosen so that the open-loop system is unstable. The closed-loop response and control history are shown in Fig. 1. (a)

(b) 5

−10

x(t)

x(t) −20

4

−30 3 −40 2 −50 1 −60 0

−1 0

−70

4

8

12

16

20

−80 0

4

t

8

12

16

20

t

Fig. 1. Controls with memory: (a) Closed-loop response (b) Control history

2. Memoryless controls In this case, Hypothesis 3 is satisﬁed with α1 (t, x) = 10 + 0.5x2 , ˜1 = 1 = 1, ˆ = a, α ˆ = 0, βˆ1 = 0.05, βˆ2 = 0, γˆ1 = 0.125 and κ ˜ 1 = κ1 = 1, ξ1 = 0.5, a ˜1 are chosen to have the α ˆ 1 ≡ 0. Since Γ = 5, the design parameters µ and χ values 34 and 0.01, respectively. The controller is designed to be f1 (t, x) = −48x2 −

4˜ π12 (t, x) , 2˜ π1 (t, x) + 0.01

where π ˜1 (t, x) = 10 + 0.5x2 . The closed-loop response and control history are shown in Figs. 2. It is clear from Fig. 2 (b) that the control gains have increased drammatically, as compared with those for controls with memory (see Fig. 1 (b)). Also, the speed of response is slower using a memoryless controller, for this particular simulation (compare Fig. 1 (a) with Fig. 2 (a)).

7

Concluding remarks

In this paper, it has been shown that, under certain speciﬁed conditions, global uniform asymptotic stability of appropriate invariant compact sets, containing the state origin, for a class of imperfectly known bilinear timedelay systems of the retarded type, with both multiple discrete and multiple

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D.P. Goodall

(a)

(b) 5

0

x(t)

x(t) −200 4 −400 3

−600

−800

2

−1000 1 −1200

0 0

4

8

12

16

20

−1400 0

4

8

12

16

t

20

t

Fig. 2. Memoryless controls: (a) Closed-loop response (b) Control history

distributed time-delays, can be achieved through continuous feedbacks. It was ﬁrst of all demonstrated that the desired stability result could be attained using feedbacks with memory and, then, under a slightly more stringent uncertainty hypothesis, a class of memoryless feedbacks was synthesized that guaranteed global stability results for the prescribed class of systems.

References 1. Choi, H.H., Chung, M.J. (1995) Memoryless stabilization of uncertain dynamic systems with time-varying delayed states and controls. Automatica 31, 1349– 1351 2. Clarkson, I.D., Goodall, D.P. (2000) On the stabilizability of imperfectly known nonlinear delay systems of the neutral type. IEEE Trans. Automatic Control 45, 2326–2331 3. Goodall, D.P. (1998) Stability criteria for feedback-controlled, imperfectly known, bilinear systems with time-varying delay. Mathematics & Computers in Simulation 45, 279–289 4. Kolmanovskii, V.B., Richard, J.-P. (1999) Stability of some linear systems with delays. IEEE Trans. Automatic Control 44, 984–989 5. Michel, A.N., Wang, K. (1995) Qualitative Theory Of Dynamical Systems, Marcel Dekker, New York, U.S.A. 6. Moon, Y.S., Park, P., Kwon, W.H. (2001) Robust stabilization of uncertain input-delayed systems using reduction method. Automatica 37, 307–312 7. Sun, Y.-J., Hsieh, J.-G., Yang, H.-C. (1997) On the stability of uncertain systems with multiple time-varying delays. IEEE Trans. Automatic Control 42, 101–105 8. Thowsen, A. (1983) Uniform ultimate boundedness of the solutions of uncertain dynamic delay systems with state-dependent and memoryless feedback control. Int. J. Control 37, 1135–1143 9. Wu, H., Mizukami, K. (1993) Exponential stability of a class of nonlinear dynamical systems with uncertainties. Systems & Control Letters 21, 307–313

Sliding mode control of a full-bridge unity power factor rectiﬁer Robert Gri˜ n´o1 , Enric Fossas1 , and Domingo Biel2 1

2

Institut d’Organizaci´o i Control de Sistemes Industrials, Universitat Polit`ecnica de Catalunya, Barcelona, Spain. E-mail: {grino, fossas}@ioc.upc.es Dept. Enginyeria Electr`onica (EUPVG), Universitat Polit`ecnica de Catalunya, Vilanova i la Geltr´ u, Spain. E-mail: [email protected]

Abstract. This work studies the dynamics of a single-phase unity power factor fullbridge boost converter circuit and develops a nonlinear controller for the regulation of its output DC voltage, which keeps the input power factor close to unity. The controller has a two loop structure: the inner is a fast dynamic response loop with a sliding controller shaping the inductor input current of the converter, and the outer is a linear controlled slow dynamic response loop that regulates the output DC voltage. The squared value of the DC voltage is passed through a LTI notch ﬁlter to eliminate its ripple before using it in the outer control loop. This ﬁlter, consequently, allows one to expand the bandwidth of the loop and improves its dynamic response.

1

Introduction

In order to meet the requirements of the electrical quality standards (for example IEC 1000-3-2) for the input current of low-power equipment, it is necessary to perform the AC-DC conversion of the electrical power using switch-mode power converters [3]. Among these circuits, the most popular choice for medium and high power applications is the boost power converter operating in continuous conduction mode [1,6]. However, as it is known, it is diﬃcult to control these converters because of their non-minimum phase behaviour with respect to the output voltage. This fact is worsened by the basic control objective of the sinusoidal shape in the converter input current. This speciﬁcation has a non-standard form because it only imposes the shape of a signal and not its value as a function of time, which would ﬁt to a tracking problem. Also, there is another control objective: the mean value of the DC bus capacitor voltage must be regulated to a speciﬁed value. But, as it will be seen in the next section, the two control objectives must accomplished with only one control variable, and this fact implies a complex controller structure, in general, a two control loop topology. The paper is organized as follows: Section II presents the model of the bidirectional boost rectiﬁer and the control objectives. Section III discusses the steady-state and the zero-dynamics behaviour of the system, showing the converter input-output active power balance. Section IV develops a set of A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 139-148, 2003. Springer-Verlag Berlin Heidelberg 2003

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bounds for the system’s response based on the physical parameters of the system. Section V shows the design of the controller detailing each one of the two loops (current and voltage). In Section VI several simulations of the proposed control scheme are showed. Finally, Section VII summarizes the conclusions of the present work.

2

Problem formulation

2.1

Physical model of the boost converter

The averaged model of the boost converter (at the switching frequency) [2] is given by Lx˙ 1 = −ux2 − rx1 + vs 1 C x˙ 2 = ux1 − x2 R

(1) (2)

where x1 and x2 are the input inductor current and the output capacitor voltage variables, respectively; vs = E sin(ωr t) is the ideal sinusoidal source that represents the AC-line source; R is the DC-side resistive load; r is the parasitic resistance of the inductor; and L and C are the inductance and the capacitor of the converter. The control variable u takes its value in the closed real interval [−1, 1] and represents the averaged value of the PWM (pulse-width-modulated) control signal injected into the real system.

Fig. 1. Bidirectional boost active rectiﬁer converter.

In the actual implementation of the system it is assumed that the output voltage x2 , the input current x1 and the source voltage vs are available for measurement. In the following analysis, it will be interesting to deal with the DC component 1 of some variables that will be noted as ·0 . It is important 1

The DC component, or averaged function, of a periodic signal f (t) of period T t is calculated as f (t)0 T1 t−T f (τ )dτ .

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to remark that the system described by equations (1) and (2) can be seen as the interconnection of two subsystems with diﬀerent time constants. In particular, the dynamics of equation (2) is much slower than the dynamics of equation (1). This fact has led to the development of the classical control schemes for these systems consisting of two concentric control loops: the inner (fast) for shaping the inductor current, and the outer (slow) for regulating the output capacitor voltage. In this control architecture, the output of the outer loop controller acts as the modulating signal in an AM modulator, with carrier vs , whose output is the reference for the inner loop. The handicap of this control topology, caused by the slow outer voltage loop, is the need for big capacitors in the DC bus to prevent large overvoltages in case of great load perturbations. 2.2

Control objectives

The control objectives are: 1. The AC-DC converter must operate with a power factor close to one. This is achieved by ensuring that, in the steady-state, the inductor current x1 follows a sinusoidal signal with the same frequency and phase as the ACline voltage source vs , i.e. x1d = Id sin(ωr t). The value for Id should be calculated by the controller in order to accomplish the following objective, 2. The DC component of the output capacitor voltage x2 0 should be driven to the constant reference value x2 0d , where x2 0d > E in order to have boost behaviour. 3. The value of the DC bus capacitor must be as low as possible for cost reasons. This requirement implies that the controller should be able to reject large perturbations in the load with short transients to prevent overvoltages in the bus.

3

Steady-state and zero-dynamics analysis

If the state vector of the system (1)-(2) is ﬁxed assuming perfect control action, at the desired values (x1d = Id sin(ωr t), x2d = Vd = x2 0d ) and neglecting the higher order harmonics, an input-output active power balance [2] is performed resulting in: Pi = x1d vs − rx21d 0 = Po =

V2 x22d = d R R

1 (EId − rId2 ) 2

(3) (4)

Since the input active power must be equal to the output active power (Pi = Po ), then 1 V2 (EId − rId2 ) = d 2 R

(5)

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2Vd2 E E2 should hold. This equation has two solutions Id = 2r ± 4r 2 − rR which R are real if and only if VEd < 8r [2]. This condition is known as the boost E condition of the power converter. The smaller solution of (5), Id = 2r − 2 2 2V E d 4r 2 − rR , corresponds to a stable equilibrium and is the selected relation between the desired mean value of the DC capacitor (Vd ) and the amplitude of the desired inductor current (x1d = Id sin(ωr t)). As it is known, the bidirectional boost rectiﬁer has relative degree 1 regardless of the output, x1 or x2 . Besides this, it is also known that if the output is x2 , the system has a non-minimum phase behaviour. For this reason, this system is usually controlled through the current x1 . In this case, the system has a minimum phase behaviour, i.e. its zero-dynamics is stable. In order to verify this assertion, x1 is taken as the output of the system by ﬁxing its value to x1d = Id sin(ωr t) in equations (1)-(2) resulting in

u=

(E − rId ) sin(ωr t) − ωr LId cos(ωr t) x2

(6) 2

Id 2 sin(ωr t) cos(ωr t)ωr L Id 2 (sin(ωr t)) r dx2 =− − dt Cx2 Cx2 2

x2 Id (sin(ωr t)) E (7) − Cx2 CR where u and x2 are the control variable and the capacitor voltage, respectively, in the zero-dynamics. Then, equation (7) describes the behaviour of the zero-dynamics of the system. This equation is a Bernoulli ODE, but multiplying each side of the equation (7) by x2 and taking z = 12 x22 , we get the following linear ODE: +

2

Id 2 sin(ωr t) cos(ωr t)ωr L Id 2 (sin(ωr t)) r dz =− − dt C C 2 2z Id (sin(ωr t)) E − , (8) + C CR 2t whose solution is z(t) = f (t) + p(t) + K, where f (t) = 12 C1 exp(− RC ) is the vanishing (limt→∞ f (t) = 0) term corresponding to the ﬁrst order linear dynamics, p(t) = A sin(2ωr t) + B cos(2ωr t) is the oscillating term (at freV2 quency 2ωr ), and K = 2d is the constant term. It is worth noting that the V2

DC value of z(t) in steady-state is z0 = K = 2d , i.e. averaging z(t) with period T = ωπr in steady-state results in the mean value of the DC capacitor bus squared and divided by 2. The same result can be obtained averaging equation (8), 2z0 1 V2 2z0 dz0 = EId − rId2 − = d − dt 2C RC RC RC whose solution is z0 =

Vd2 2

2t + C1 exp(− RC ).

(9)

Sliding mode control of a full-bridge unity power factor rectiﬁer

4

143

Control design

This section is devoted to the design of both the control u and Id , since the latter operates as a control in a linear equation describing the dynamics of x22 /20 .

Î

Ü½

Î Ù

Î Á Ü½

Ù

Ü¾

Á

Ü¾

Fig. 2. Control scheme blocks diagram.

The control objectives can be written as follows: • x1 (t) = Id sin(ωr t) • z0 = 0.5Vg2 both requirements being demanded in steady-state, where z = 0.5x22 . As far as the ﬁrst objective is concerned, sliding control is proposed since it is appropriate due to its very nature for switching converters, and it will provide a controlled system robust with respect to load variations [4]. Thus, σ(x, t) = x1 − Id sin(ωr t) = 0 is considered as the switching surface. Note that its relative degree is one. Following the standard procedure [5], one has 1 [(E − rId ) sin(ωr t) − ωr LId cos(ωr t)] x2 −1 if σ(x, t) < 0 u= +1 if σ(x, t) > 0

ueq =

A necessary condition for sliding motion is x2 = 0; note that the dot product of the x-gradient of σ(x, t) and the control vector is −x2 /L which, in turn, will be assumed negative. Furthermore, −1 ≤ ueq (x, t) ≤ +1 deﬁnes the subset of σ(x, t) = 0 where sliding motion occurs. The substitution of the zero-dynamics in these inequalities results in the necessary conditions to be held by the plant parameters and will be considered in the next section. With regards to the second objective, the variable z0 is regulated to Vd2 /2 applying classical linear control design to equation (9) with r = 0, where

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Fig. 3. min2 {d(t)} − max2 {n(t)} as a function of the variables (L, C) in the range L ∈ [0.0005, 0.005], C ∈ [0.0005, 0.01].

Id acts as the control variable. This ordinary diﬀerential equation describes the zero-dynamics, i.e. the Ideal Sliding Dynamics. Taking the zero-dynamics as the dynamics of z = 0.5x22 makes sense because the current loop is much faster than the voltage one, as has already been pointed out. In addition, z(t) has a DC component and a fundamental harmonic at 2ωr which is removed through the linear notch ﬁlter H(s) =

s2

s2 + 4ωr2 + 4ξωr s + 4ωr2

(10)

A block diagram depicting this control scheme is shown in Fig. 2.

5

Parameter analysis

The plant parameters are important in the performance of the controlled converter. Some of them, such as the load R and the input voltage, can vary with time, or can be aﬀected by perturbations. Others, such as the inductance L and the capacitance C, are design parameters; that is to say, their values can be speciﬁed by the converter designer and can be assumed constant as long as the process takes. In this section, the inﬂuence of parameters L, C and R on the fulﬁlment of the control objectives is considered. Furthermore, a new speciﬁcation is taken into account; namely, an output voltage ripple lower or equal to 0.05 p.u. For the sake of simplicity, r = 0 has been chosen in this section. First, let it be assumed the steady-state for the input current, x1 = x1d = Id sin(ωr t). From equation (1), the steady-state for the product ux2 is given

Sliding mode control of a full-bridge unity power factor rectiﬁer

145

Fig. 4. lpu as a function of the variables (L, C) in the range L ∈ [0.0005, 0.005], C ∈ [0.0005, 0.01].

by uss x2ss = [E sin(ωr t) − ωr LId cos(ωr t)]

(11)

Second, as in the previous sections, the change of variable z = 0.5x22 in equation (2) results in dz 2z = ux2 x1 − (12) dt R Thus, from equations (11) and (12), the steady-state for the new variable z is V2 2LCVd2 ωr2 − E 2 zss (t) = d 1 + 2 cos(2ωr t) 2 E (1 + ωr2 R2 C 2 ) ωr (E 2 R2 C + 2LVd2 ) sin(2ω − t) (13) r E 2 R(1 + ωr2 R2 C 2 ) √ By substitution of x2ss = 2zss in equation (11), the steady-state for the input variable is obtained, which is a quotient of periodic signals. The numerator, n(t), is a pure sine of amplitude V 4 ω 2 L2 E 1 + 4 d 4r 2 E R C

and the denominator, d(t), is a periodic signal that oscillates between

4

1 + 4 ωr2 V4d L2 2

E R Vd 1 ± 1 + ωr2 R2 C 2

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A necessary condition to avoid saturation of the input variable, i.e. 0 < u < 1, is max{n(t)} < min{d(t)} Since both terms of the inequality are positive, it is equivalent to ωr2 Vd 4 L2 2 2 2 2 min {d(t)}−max {n(t)} = Vd − E 1 + 4 E 4 R2 ωr 2 Vd 4 L2 −1 − Vd4 1 + 4 (1 + ωr 2 R2 C 2 ) > 0 E 4 R2

(14)

Note that inequality (14) implies Vd > E, or equivalently, x2 d > E, recovering the boost character of this converter. The graph of min2 {d(t)}−max2 {n(t)} as a function of the variables (L, C) in the range L ∈ [0.0005, 0.005], C ∈ [0.0005, 0.01] is depicted in Fig. 3. As for the second speciﬁcation, from equation (13), the amplitude of the voltage ripple is given in p.u. by 



E 4 R2 + 4 ωr 2 Vd 4 L2

 −1 lpu = 1 + (15) E 4 R2 (1 + ωr 2 R2 C 2 ) The graph of lpu as a function of the variables (L, C) in the range L ∈ [0.0005, 0.005], C ∈ [0.0005, 0.01] is depicted in Fig. 4. It is worth noting that if (14) and lpu < 0.05 holds for L, C, E and R0 , it also holds for L, C, E and R ≥ R0 . Both inequalities provide us with restrictions to be held by the parameters E,L, C, R, ωr and Vd . Comments • From Figs. 3 and 4, L ≥ 0.001 and C ≥ 0.003√appear to be conditions for the speciﬁcations to be held, presuming E = 220 2V, R0 = 10 Ω, Vd = 400 V and ωr = 100π rad/s. • Note from the pictures the dominance of the value of the capacitor in the fulﬁlment of the plant parameter requirements.

6

Simulation results

The previous control design has been simulated in a single-phase√active ﬁlter with the following parameters: L = 1 mH, C = 4.7 mF, vs (t) = 220 2 sin(ωr t)V, ωr = 100π rad/s, Vd = 400 V. A pulsating function taking values R = 100 Ω and R = 10 Ω, respectively, has been considered as the resistive load. Fig. 5 shows the evolution of the input voltage vs , the input current x1 and the bus voltage x2 . A detail of the input current x1 in phase with the output voltage vs is shown in Fig. 6. Finally, in Fig. 7, the auxiliary variable z = 0.5 x22 is depicted together with the notch ﬁlter output, z being the input.

Sliding mode control of a full-bridge unity power factor rectiﬁer

147

Fig. 5. The input voltage vs (t) together with the input current x1 (t) and the output voltage x2 (t).

Fig. 6. Input current x1 (t) in phase with the input voltage vs (t).

Fig. 7. z(t) together with z0 .

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Conclusions

In this paper, a dynamic sliding-mode control scheme for a single phase AC/DC regulator system with unity power factor has been proposed. The design procedure presented in this work suggests a sliding surface dynamically deﬁned in order to fulﬁl two speciﬁcations in a single input system, namely: unity power factor and output voltage regulation. Fundamental to the regulation is a linear notch ﬁlter, eliminating the harmonics of the squared output voltage. A converter parameter design procedure that can be used to minimize the capacitor has also been proposed. The theoretical predictions have been validated by means of simulation results.

Acknowledgments This work has been partially supported by the Comisi´on Interministerial de Ciencia y Tecnologia (CICYT) under project DPI2000-1509-C03-02,03.

References 1. R. Boys and A. W. Green. Current-forced single-phase reversible rectiﬁer. IEE Proc., pt. B, 136, September 1989. 2. G. Escobar, D. Chevreau, R. Ortega, and E. Mendes. An adaptive passivitybased controller for a unity power factor rectiﬁer. IEEE Trans. on Control Systems Technology, 9(4):637–644, July 2001. 3. Ned Mohan, Tore M. Undeland, and William P. Robbins. Power Electronics. Converters, Applications, and Design. John Wiley & Sons, Inc., New York, second edition, 1995. 4. H. Sira-Ramirez. Sliding motions in bilinear switched networks. IEEE Trans. on Circuit and Systems, 34(8):919–933, August 1987. 5. V. I. Utkin. Sliding Modes and their Applications in Variable Structure Systems. Mir, Moscow, 1978. 6. E. Wernekinck, A. Kawamura, and R. Hoft. A high frequency ac/dc converter with unity power factor and minumum harmonic distortion. IEEE Trans. on Power Electronics, 6:364–370, July 1991.

Flatness-based control of the induction drive minimising energy dissipation Veit Hagenmeyer1 , Andrea Ranftl2 , and Emmanuel Delaleau3 1

2 3

Laboratoire des signaux et syst`emes, CNRS–Sup´elec–Universit´e Paris-Sud, 3 rue Joliot-Curie, 91 192 Gif-sur-Yvette, France [email protected] Student at Erlangen-N¨ urnberg University, Germany ´ Laboratoire d’´electricit´e signaux et robotique, Ecole normale sup´erieure de Cachan, 61 Avenue du Pr´esident Wilson, 94 235 Cachan, France [email protected]

Abstract. Due to the ﬂatness of the induction drive, a control scheme that achieves energy dissipation minimisation can be designed. It makes use of an on-line replaniﬁcation of desired trajectories by using the information of a fast converging load torque observer. Simulation results show the dynamical performance of the proposed control scheme.

1

Introduction

This paper shows that it is possible to recast the results of [1] on energy dissipation minimisation for the induction drive, originally obtained by optimal control, in the context of diﬀerential ﬂatness. This work can also be seen as a generalisation to the induction motor of the results attained for the DC motor, which were presented in a former NCN Workshop [7]. Energy loss minimisation in AC drives has been intensively studied for constant motor torques in a founding article by Kusko and Galler [9]. They gave a closed form expression for the optimal slip frequency minimising copper losses, if iron losses are neglected. Kirschen et al. [8] stated that, when motor torques are smaller than the rated value, motor eﬃciency can be greatly improved by adjusting the motor ﬂuxes. This is done by using a heuristic method to ﬁnd the optimal rotor ﬂux. Motivated by these results, Famouri and Cathay [4] minimised the copper losses by variation of the rotor ﬂux on the basis of a constant slip independent of the mechanical speed and the desired torque. In [2], Canudas de Wit and Seleme included iron losses in the calculations by respecting the magnetic energy stored in the motor whilst operation. This leads to a minimum-energy control for torque regulation which results also in varying rotor ﬂuxes. Thereby the steady-state operation points are functions of the desired torque. More recently, Canudas

Work partially supported by the TMR “Nonlinear Control Network” (#ERB FMRXCT-970137). One author (V.H.) was ﬁnancially supported by the German Academic Exchange Service (DAAD).

A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 149-160, 2003. Springer-Verlag Berlin Heidelberg 2003

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de Wit and Ramirez presented an optimal control design for current-fed induction motors in [1]. It treats for the ﬁrst time the transitions of desired motor torques and is optimal with respect to the stored magnetic energy and the copper losses while satisfying torque tracking control objectives. The diﬀerential ﬂatness property of the model of induction drive shows inherently that keeping the rotor ﬂux magnitude at a constant value, as done in the normal operation of the well known Field Oriented Control (FOC), is equivalent to neglecting the use of one degree of freedom for the control of the motor. Thus, the results of [1] can be recasted in the context of diﬀerential ﬂatness, which allows to design energy loss optimal reference trajectories for the ﬂux magnitude. The presented control scheme is valid for the motions of acceleration, of constant angular speed and of braking, which are all treated by the same method. The drive can be started from rest without any discontinuous jerk and without premagnetising the machine. An important point to notice is the on-line replaniﬁcation of desired trajectories by using a fast converging load torque observer. The stabilisation of the induction motor around the desired trajectories is achieved by a cascaded control structure. This structure avoids diﬃculties arising from dead times, which result from the architecture of power converters, and is nevertheless able to use the feed-forward information based on diﬀerential ﬂatness. The paper is organised as follows: After having brieﬂy exposed the mathematical model of the motor and its ﬂatness property in Sec. 2, we present the ﬂatness based control which achieves the minimisation of energy dissipation in Sec. 3 on the basis of the reduced model and in Sec. 4 for the complete model. The article is concluded in Sec. 5 by simulation results, which show the dynamical performance of the proposed control scheme.

2

Model of the motor

The physical properties which underly the dynamic behaviour of the induction motor have been studied thoroughly and can be found, for instance, in [10]. A state space model of this motor can be written as follows: dθ dt dω J dt dis Ls σ dt ˙ Tr (ρ˙ + αρ)

=ω np 2 ρ α˙ − f ω − τl Rr M ˙ eδ = us − Rs is − ρ˙ + δρ Lr −δ = −ρ + M e is =

(1a) (1b) (1c) (1d)

where θ and ω respectively denote rotor position and velocity, τl stands for the load torque (the disturbance input); is = isa + isb is the stator current1 , 1

After having performed the well known transformation of a three phases machine to a two phases equivalent one [10], it is possible to build complex variables asso-

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151

us = usa +usb is the stator voltage (the control input), and ψ s = ψra +ψrb = ρeδ is the rotor ﬂux. The rotor, stator and mutual inductances are expressed by Lr , Ls and M respectively, while the stator and rotor resistances are 2 Lr expressed by Rs and Rr ; Tr = R is the rotor time constant and σ = 1− LM r r Ls is the Blondel factor. The parameter J represents the moment of inertia, f stands for the viscous friction coeﬃcient, np is the number of pole pairs. The system of equations of the induction drive (1) is ﬂat 2 [13] (see also [3] for another presentation) with ﬂat output (α, θ) where α = δ − np θ. To simplify the development of the control law, we use, as usually done, a reduced order model for the induction drive in dq-coordinates3 with the two stator currents isd and isq as inputs (as it was ﬁrst done in the so-called Field Oriented Control (FOC), see [10]). Deﬁning ξ = TMr isq , the reduced state equations of (1) read then dθ =ω dt dω np τl f = ρξ − ω − dt JRr J J dρ 1 M = − ρ+ isd dt Tr Tr

(2a) (2b) (2c)

Notice that the reduced model (2) does not contain the second element α of the ﬂat output of the full model as an explicit variable. It can be calculated using α˙ = ξ/ρ

(3)

via simple integration under the knowledge of the trajectories t → ξ(t) and t → ρ(t). This fact will become important when working with the full model after the control of the reduced model (see Sec. 4). The reduced model (2) is structurally similar to the model of the separately excited DC drive [10], which has been proven to be ﬂat (see [7] for the control of the DC drive based on diﬀerential ﬂatness aiming energy losses minimisation). The ﬂat output of the reduced model is z = (ρ, θ), which can be checked by deriving the expressions of ω, isd and isq in terms of ρ, θ and their derivatives. Recall that the FOC method controls the reduced model of the drive (2) via the quadrature current as input ξ = TMr isq , since the latter directly

2 3

ciated to each electrical quantity by putting in the real part what concerns one phase (subscript a) and by putting what concerns the second phase (subscript b) in the imaginary part. Every complex variable is underlined; x and [x] represent respectively the conjugate and the imaginary part of the complex variable x. See [11] for an introduction to (diﬀerentially) ﬂat systems. The dq-coordinates are introduced by setting xdq = xd + xq = e−δ (xa + xb ). Notice that the dq-coordinates are those corresponding to the orientation of the rotor ﬂux since ψrdq = ρ.

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operates on the electromagnetic torque τm = Rpr ρξ. Thereby the absolute value of the rotor ﬂux ρ is kept constant (with the exception of ﬁeld weakening for higher velocities) by setting isd to a constant.

3

Control of the reduced order model

3.1

Nominal control

From the ﬂatness property, one obtains directly the nominal (open-loop) control of the reduced model with the stator currents i∗sd and i∗sq as inputs i∗sd = (Tr ρ˙ ∗ + ρ∗ )/M

(4a)

i∗sq

(4b)

∗

= Tr /M · ξ = Lr /(M np ) · (J θ¨∗ + f θ˙∗ + τˆl )/ρ∗

where the reference trajectories of the components of the ﬂat output t → z ∗ = (ρ∗ , θ∗ ) will be designed below not only in order to avoid singularities, i.e. that i∗sd and i∗sq remain everywhere deﬁned and bounded, but moreover to minimise energy dissipation (§ 3.3). The variable τˆl denotes an estimation (see below) of the value of the load torque. Notice that the injection of the estimation of the disturbance τˆl in (4b) facilitates the work of the closed loop controller. With this choice, the feedback has only to cope with the unknown transients of the disturbance as its instantaneous mean value τˆl is estimated on-line by an observer. 3.2

Observer for the load torque and the rotor ﬂux

Hence we go on in constructing a rapidly converging observer for the load torque τl using the prediction of ρ∗ and the measurement of isq and ω: dˆ τl = −l1 (ˆ ω − ω) dt np M ∗ τˆl f dˆ ω = ˆ − + l2 (ˆ ρ isq − ω ω − ω) dt JLr J J

(5a) (5b)

Subtracting (2b) from (5b), supposing the unknown behaviour of τl to be modeled by τ˙l = 0 and assuming, in view of control, that ρ ≈ ρ∗ results in the linear error dynamics d dt

0 1 = 2 −1/J

−l1 −f /J + l2

1 ˆ − ω) , (with 1 = τˆl − τl , 2 = ω 2 (6)

The observer gains l1 and l2 are chosen such that the error dynamics converge faster to zero than the dynamics of the observed drive. Using the knowledge of τˆl , we feedback on-line this information to the desired trajectory generator, and therefore to the nominal open loop control (see (4b)).

Flatness-based control of the induction drive

153

Since ρ will be used in the controller in § 3.4, we estimate ρˆ taking the measurement of the stator current isd into account. Using (2c) we ﬁnd 1 dˆ ρ M = − ρˆ + isd dt Tr Tr

(7)

1 3 Subtracting (2c) from (7) yields the linear error equation d dt = − Tr 3 where 3 = ρˆ−ρ. This error equation is exponentially stable with small time constant Tr . Considering furthermore, that the ﬂux is known at rest (at the initial point of time), we have ρ(0) = 0 which leads to 3 (0) = 0. Thus the estimator (7) is quite suﬃcient for the ongoing study (as for a faster converging observer of the rotor ﬂux see, for instance, [13]).

3.3

Reference trajectory generation

In view of the nominal control using the ﬂatness approach (§ 3.1), we have to design the reference trajectories t → z ∗ = (ρ∗ , θ∗ ) for both components of the ﬂat output. For the sake of conciseness, we will focus to the design of angular velocity control t → ω ∗ , the desired reference for θ∗ is then won by simple integration. The trajectory ω ∗ is chosen to be a spline which ensures a continuous jerk and passes through given speeds at given instants of time (see [7] for details). We will show, that it is possible to choose the reference trajectory of the rotor ﬂux ρ∗ as a function of the reference speed ω ∗ and the estimated load torque τˆl in order to achieve energy dissipation minimisation. We consider 4 PJ = Rr ir ir + Rs is is the total instantaneous power dissipated by the Joule’s eﬀect in the rotor and stator circuits of the motor. The stored magnetic energy is represented by W = 12 (Ls σis is + L1r ρ2 ) and reﬂects to some extend a measure of both the iron losses and the power factor (see [2]). Since in view of energy dissipation the sum H = PJ + W is a function of the variables of the full model (1), which is a ﬂat system, it is possible to express H in terms of the ﬂat output (α, θ) of the full model. Using the expressions of the currents ir and is leads to (see [3]) 1 Lr L2r 2 L2r 2 2 ˙ + 2 ρ˙ is is = 1 + 2 α˙ ρ + 2 ρρ (8a) M2 Rr Rr Rr 1 (8b) ir ir = 2 α˙ 2 ρ2 + ρ˙ 2 Rr From the fact, that the FOC method keeps ρ constant (in view of its fast converging dynamics in (2c)) to operate directly on the electromagnetic torque n τm = JRpr ρξ in (2b) (remember ξ = TMr isq ), and from ξ = ρα˙ deduced of (3) we get the following assumptions ρ ρ˙ ρα, ˙ ρ˙ (9) Tr 4

We denote by ir the rotor current.

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r Therefore we are able to neglect ρ˙ in (8a) and (8b). Moreover, using α˙ = τnmpR ρ2 (obtained from the ﬂatness of the complete model) we get an expression of the form

H = A ρ2 + B (τm /ρ)2

(10)

where A and B are constant depending of the parameters of the motor. ∗ = J ω˙ ∗ + f ω ∗ + τˆl The mechanical torque produced by the motor t → τm ∗ is ﬁxed for a given desired speed trajectory t → ω under a given assumed load torque behavior t → τˆl . Consequently, H in (10) can be thought as a function of ρ only. As H is the sum of two positive terms, one of them being an increasing function of ρ and the second one being a decreasing function of ρ, it admits a minimum w.r.t. ρ. Remembering that the torque produced n ˙ one sees that a given torque can be produced by the motor is τm = JRpr ρ2 α, by various choices of ρ and α. ˙ All these diﬀerent values of ρ and α˙ lead to diﬀerent values of H. By diﬀerentiation of H one obtains5 the following ∗ optimal ρ∗ and ρ˙ ∗ for a given τm τ˙ ∗ 1 ∗ | and ρ˙ ∗ = Γ m (11) ρ∗ = Γ |τm ∗| 2 |τm which leads to a constant slip speed α˙ ∗ =

Rr np Γ 2

(12)

We ﬁnd again the result of [1] in the context of diﬀerential ﬂatness. There it was shown, that the approximation via the assumptions (9) is valid in the ∗ opt ∗ ,τ˙ ∗ ||→0 |ρ −ρ sense: lim||¨τm | = 0 (where ρopt is the optimal rotor ﬂux under m no assumption). ∗ = 0 can be avoided by an appropriate The singularity of ρ˙ ∗ in (11) for τm choice of the degree of the polynomials deﬁning ω ∗ . This is the same study as in [7] and it is not repeated here. 3.4

Tracking control of the reduced model

The tracking control for the reduced model is simple with respect to its ∗∗ structure: the applied stator currents i∗∗ sd and isq are a sum of their nominal ∗ ∗ values isd and isq (4) and the output of a PI-controller respectively (see [12]). Considering (7) this leads to t ∗ ∗ = i + λ (ˆ ρ − ρ ) + λ (ˆ ρ − ρ∗ )dτ (13a) i∗∗ Pρ Iρ sd sd 0 t ∗ ∗ = i + λ (ω − ω ) + λ (ω − ω ∗ )dτ (13b) i∗∗ Pω Iω sq sq 0

Thereby the controller coeﬃcients have to be chosen appropriately. 5

Set Γ =

4

Lr Rr 2Lr Rs +Lr Ls −M 2 +2M 2 Rr n2 2Lr Rs +Lr Ls −M 2 +M pM

.

Flatness-based control of the induction drive

4

155

Tracking control of the complete model

For the tracking control of the complete model, we use a cascaded control structure. The angular velocity and the rotor ﬂux are controlled by an outer control loop using the two components of the stator current as inputs (§ 3.4); the latter are regulated by a faster inner control loop using the stator voltages as inputs. This structure avoids diﬃculties arising from dead times, which result from the architecture of power converters, and is nevertheless able to use the feed-forward information based on diﬀerential ﬂatness. From diﬀerential ﬂatness, the nominal control for the the stator voltages in dq-coordinates u∗sd and u∗sq can be deﬁned6 as d ∗∗ K ∗ ∗ ∗∗ ∗∗ d ∗ usd = σLs γisd − isq δ + isd − ρ (14a) dt dt Tr d ∗∗ d ∗ δ + i∗∗ + Knp ω ∗ ρ∗ (14b) u∗sq = σLs γi∗∗ sq + isd dt dt sq where δ ∗ = np θ∗ +

t

α˙ ∗ dτ

δ˙ ∗ = np ω ∗ + α˙ ∗

(15)

0

and α˙ ∗ is given by (12). From (13) we ﬁnd using (7) and (2b) di∗∗ di∗sd Tr M ρˆ ∗ ∗ sd = + ρ − ρ ) + λP ρ − + isd − ρ˙ (16a) λIρ (ˆ dt dt M Tr Tr di∗sq di∗sq np M ∗∗ f Tr τˆl ∗ ∗ = + ρˆi − ω − − ω˙ λIω (ω − ω ) + λP ω dt dt M JLr sq J J (16b) di∗

di∗

where dtsd and dtsq are calculated from (4). The tracking control for the inner control loop of the full model is again simple with respect to its structure: the stator voltages in dq-coordinates usd and usq are a sum of their nominal values u∗sd and u∗sq (4) and the output of a PI-controller respectively t ∗∗ ) + λ (i − i )dτ usd = u∗sd + σLs λP isd (isd − i∗∗ Iisd sd sd sd 0 t ∗∗ usq = u∗sq + σLs λP isq (isq − i∗∗ ) + λ (i − i )dτ Iisq sq sq sq

(17a) (17b)

0

To obtain the stator voltages in stator coordinates to be applied to the in−δ ∗ using (15). Thereby the controller duction drive, we calculate us = udq s e coeﬃcients have to be chosen appropriately. The complete control scheme is depicted in Fig. 1. 6

Set γ =

Rs σLs

+

Rr M 2 σLs L2 r

and K =

M σLs Lr

.

Trajectory generator

... ω ∗ , ω˙ ∗ , ω ¨ ∗, ω ∗

Fig. 1. Complete control scheme i∗sq , ρ∗

τˆl

Current generator

i∗sd , i∗sq , ω˙ ∗ , ρ∗

and flux observer

Load torque

ρˆ

Flatness based controller

ω

ω

∗∗ i∗∗ sd , isq

∗∗ i∗∗ sd , isd

ρˆ

usa , usb

isd , isq , ω

Voltage controller

AC Motor

τl

156 V. Hagenmeyer, A. Ranftl, and E. Delaleau

Flatness-based control of the induction drive

5

157

Simulation results

The parameters used for the simulations correspond to those identiﬁed on a real induction drive7 .

100 α,α (−−)

ω,ω* (−−)

100

50

*

50 0 −50

0

5

0 −50

10

0

5

10

0

5

10

0

5

10

0

5

10

0

5

10

1.5 hat

*

l

−2

ρ,ρ (−−),ρ

(:)

0

lhat

2

τ ,τ

(:)

4

0

5

1

0.5 0

10

4 **

isd,isd(−−),isd(:)

5 0

−5

*

sq sq

sq

i ,i* (−−),i** (:)

10

0

5

100 usd,usd(−−)

200

0

*

usq,u*sq(−−)

0

−2

10

400

0

−200

−100

0

5

10

−200 200

usa

200

usb

2

0

−200

0

5

10

0

−200

Fig. 2. Acceleration from rest, constant speed, and breaking under load torque 7

The considered motor is located in “Laboratoire de G´enie Electrique de Paris”, CNRS–Sup´elec, Gif-sur-Yvette, France and is devoted to a national benchmark leaded by the “Groupement de Recherche en Automatique”. The parameters are: M = 0.44 H, Lr = 0.47 H, Ls = 0.47 H, Rr = 3.6 Ω, Rs = 8.0 Ω, J = 0.06 kg m2 , f = 0.01 Nm/rad, np = 2. The ﬂatness based controller was established

158

V. Hagenmeyer, A. Ranftl, and E. Delaleau 50 α,α (−−)

50 0 −50

0

5

ρ,ρ*(−−),ρhat(:)

τl ,τlhat(:)

0 0

5

0 isd,i*sd(−−),i** (:) sd

isq,i*sq(−−),i** (:) sq

0 0

5

5

10

0

5

10

0

5

10

0

5

10

2 0

100 usd,u*sd(−−)

usq,u*sq(−−)

0

4

−2

10

400 200 0

0

−100

0

5

10

−200 200 usa

200

0

−200

10

1

10

5

−200

5

0.5

10

−5

0

1.5

2

−2

−50

10

4

usb

0

*

ω,ω* (−−)

100

0

5

10

0

−200

Fig. 3. Load torque perturbation and braking electrically

In Fig. 2, the drive is accelerated without load torque (τl = 0) from rest to a velocity of 70 rad/s in 2.5 s; at t = 5 s a constant load torque of half the nominal torque of the motor is applied; ﬁnally, the drive is braked back to rest in 3 s under the condition of τl > 0. One notes a good trajectory tracking and remarks the eﬀect of the on-line replaniﬁcation of the trajectory when the load torque is applied. The ﬂux is estimated by the observer (5a) established in § 3.4 and § 4, its coeﬃcients are λP ρ = −100 A/Wb , λIρ = −2500 A/(Wb s), λP ω = −125 (A s)/rad, λIω = −3906 A/rad, λP isd = −1910 V/(H A),λIisd = −1000000 V/(H A s), λP isq = −1910 V/(H A),λIisq = −1000000 V/(H A s). The observer gains are: l1 = −15000 Nm/rad and l2 = −6000 s−1 .

Flatness-based control of the induction drive

159

in § 3.2, which shows an excellent convergence behavior. The estimated load torque τˆl is used both in the desired trajectory generator and in the ﬂatness based controller (see Fig. 1). We remark that the controller corrects quickly the errors stemming from the unmodelled dynamics of τl . To show the performance of the developed ﬂatness based control technique minimising energy dissipation, we depict in Fig. 3 the response of the closed loop system to several load torque perturbations. We remark, that the control scheme ﬁnds an optimal balance between the rotor ﬂux and and its orientation for every point of operation consisting of the pair of desired velocity and given load torque. All states and the inputs stay within their given physical bounds. We see, that a kind of ﬁeld weakening is automatically imposed on the drive following our optimal desired trajectory design using the diﬀerential ﬂatness property: when accelerating the drive the ﬂux is led to its maximum (see (1b)) where the desired motor torque τm ∗ is also maximal. For maintaining a certain angular velocity the ﬂux is thereafter reduced to the necessary level.

6

Conclusion

The simulations depicted in Fig. 2 and Fig. 3 show the dynamical performance of the proposed control scheme. The stability of the closed loop can be established using a MIMO extension of [5,6]. Applications of the work reported here concern, for instance, traction drives like in high speed trains for which the induction machine is widely used.

References 1. C. Canudas de Wit and J. Ramirez. (1999) Optimal torque control for currentfed induction motors. IEEE Trans. Automatic Control 44, 1084–1089 2. C. Canudas de Wit and S.I. Seleme. (1997) Robust torque control design for induction motors: the minimum energy approach. Automatica 33, 63–79 3. E. Delaleau, J.-P. Louis, and R. Ortega. 2001 Modeling and control of induction motors. Int. J. Appl. Math. Comput. Sci., 11, 105–129 4. P. Famouri and J.J. Cathey. (1991) Loss minimization control of an induction motor drive. IEEE Trans. Industry Applications, 27, 32–37 5. V. Hagenmeyer. (2001) Nonlinear stability and robustness of an extended PIDcontrol based on diﬀerential ﬂatness. In: CD-Rom Prepr. of the 5th IFAC Nonlinear Control Systems Design Symposium, Saint-Petersburg (Russia) 6. V. Hagenmeyer and E. Delaleau. (2001) Exact feedforward linearisation based on diﬀerential ﬂatness: the SISO case. In this book 7. V. Hagenmeyer, P. Kohlrausch, and E. Delaleau. (2000) Flatness based control of the separately excited DC drive. In Lamnabhi-Lagarrigue F. Isidori A. and W. Respondek, editors, Nonlinear Control in the Year 2000 (volume 1), pages 439–451, Springer-Verlag, London 8. D.S. Kirschen, D.W. Novotny, and T.A. Lipo. (1987) Optimal eﬃciency control of an induction motor drive. IEEE Trans. Energy Conversion, 2, 70–76

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9. A. Kusko and D. Galler. (1983) Control means for minimization of losses in DC and DC motor drives. IEEE Trans. Industry Applications, 19, 561–570 10. W. Leonhard. (1997) Control of Electrical Drives. Springer-Verlag, Braunschweig, 2nd edition 11. M. Fliess, J. L´evine, Ph. Martin, and P. Rouchon. (1999) A Lie-B¨acklund approach to equivalence and ﬂatness of nonlinear systems. IEEE Trans. Automatic Control, 44, 922–937 12. M. K. Maaziz, M. Mendes, and P. Boucher. (2000) Nonlinear multivariable real-time control strategy of induction machines based on reference control and PI controllers. Proc. 13th Internat. Conference Electrical Machines (ICEM’00), Espoo (Finland) 13. Ph. Martin and P. Rouchon. (1996) Two remarks on induction motors. Differential ﬂatness and control of induction motors. In: Proc. Symposium on Control, Optimization and Supervision; Computational Engineering in Systems Applications IMACS Multiconference, Lille (France), 76–79

Exact feedforward linearisation based on diﬀerential ﬂatness: The SISO case Veit Hagenmeyer1 and Emmanuel Delaleau2 1

2

Laboratoire des signaux et syst`emes, CNRS–Sup´elec–Universit´e Paris-Sud, 3 rue Joliot-Curie, 91 192 Gif-sur-Yvette, France [email protected] ´ Laboratoire d’´electricit´e signaux et robotique, Ecole normale sup´erieure de Cachan, 61 Avenue du Pr´esident Wilson, 94 235 Cachan, France [email protected]

Abstract. The trajectory aspect of ﬂatness is considered in this article from a feedforward point of view. This leads to the introduction of the notion of exact feedforward linearisation based on diﬀerential ﬂatness.

1

Introduction

Diﬀerential ﬂatness has been presented in both the diﬀerential algebraic setting [3] and the diﬀerential geometric setting of inﬁnite jets and prolongations [4]. In both cases the motion planning has been addressed and ﬂatness has been shown to be a property which is related to the trajectories of a system. Moreover, it has been established that every ﬂat system is linearisable by endogenous feedback (a special class of dynamic feedback). Furthermore, ﬂatness has allowed to develop the control of many practical examples and led, in several cases, to industrial applications (see e.g. [14,8]). Even though exact feedback linearisation is an important and well-known subject, considering ﬂatness only in this context seems to be too restrictive. Hence, taking this argument as a motivation, the trajectory aspect of ﬂatness is presented in this article from a feedforward point of view. The notion of exact feedforward linearisation based on diﬀerential ﬂatness is introduced in order to emphasise in a new way, that the property of diﬀerential ﬂatness can also be considered to design control laws which do not exactly feedback linearise the nonlinear system. In its ﬁrst part it is shown, that applying a nominal feedforward input deduced from diﬀerential ﬂatness, when starting from the right initial condition, yields a linear system in Brunovsk´y form without closing the loop. Furthermore it is demonstrated, that if the initial condition, which is taken into consideration for the design of the nominal feedforward, is not equal but close to the right initial condition, then a unique solution in the vicinity of the solution of the aforementioned Brunovsk´y form exists for the nonlinear ﬂat system.

One author (V.H.) was ﬁnancially supported by the German Academic Exchange Service (DAAD).

A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 161-170, 2003. Springer-Verlag Berlin Heidelberg 2003

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Via exact feedforward linearisation based on diﬀerential ﬂatness, it is possible to indicate the point in the formula of the nominal feedforward signal at which the output of a feedback control should be added in view of stabilising deviations from the desired trajectory. Stability of this control scheme can be demonstrated when using a simple extended PID control for the feedback part. By considering Kelemen’s stability result, it can be shown that the absolute values of the control coeﬃcients have to be balanced with the velocity of the desired trajectory. Thus, given reasonable bounded initial errors, nonlinear ﬂat systems can be stabilised around given desired trajectories by applying exact feedforward linearisation and extended PID control. The paper is organised in the following manner: Sec. 2 presents the notion of exact feedforward linearisation based on diﬀerential ﬂatness. Using this result, Sec. 3 establishes a speciﬁc control law design methodology which leads to a speciﬁc error equation discussed in Sec. 4. Thereafter the control law design is illustrated by extended PID control which is presented in Sec. 5. In Sec. 6 Kelemen’s result is recalled as elaborated in [13] before stability for the proposed control strategy is shown.

2

Exact feedforward linearisation based on diﬀerential ﬂatness

Diﬀerential ﬂatness is a structural property of a class of multivariable nonlinear systems, for which, roughly speaking, all system variables can be written in terms of a set of speciﬁc variables (the so-called ﬂat outputs) and their derivatives. In this article only SISO ﬂat systems are studied for the sake of simplicity. Given the SISO nonlinear system ˙ x(t) = f (x(t), u(t)), x(0) = x0

(1)

where the time t ∈ IR, the state x(t) ∈ IRn , the input u(t) ∈ IR and the vector ﬁeld f : IRn × IR → T IRn is smooth. The system (1) is said to be (diﬀerentially) ﬂat [3,4] iﬀ there exists a ﬂat output z(t) ∈ IR, such that z = F (x)

(2) (n−1)

x = φ(z, z, ˙ ...,z ) (n) u = ψ(z, z, ˙ ...,z )

(3) (4)

These equations1 yield, that for every given trajectory of the ﬂat output t→ z(t), the evolution of all other variables of the system t → x(t) and t → u(t) are also given without integration of any system of diﬀerential equations. Moreover, given a suﬃciently smooth desired trajectory —called the nominal trajectory— for the ﬂat output t → z ∗ (t), (4) can be used to design the 1

The maximal number of derivatives of z in (3) and (4) respectively are due to the results of [9,1] (see more details below in the proof of Proposition 1).

Exact feedforward linearisation based on diﬀerential ﬂatness

163

corresponding feedforward u∗ (t) directly —called the nominal control. One gets u∗ (t) = ψ(z ∗ , z˙∗ , . . . , z ∗ (n) (t))

(5)

Exact feedforward linearisation based on diﬀerential ﬂatness is established by the following proposition. Proposition 1 If the desired trajectory of the ﬂat z ∗ (t) is consis output t → tent with the initial condition x0 , that is x0 = φ z ∗ (0), z˙∗ (0), . . . , z ∗ (n−1) (0) from (3), then, when applying the nominal feedforward (5) to the diﬀerentially ﬂat system given by (1), the latter is equivalent by change of coordinates to a linear system in Brunovsk´y form for all times. Moreover, if the desired trajectory of the ﬂat output z ∗ is not consistent close to the initial condition with the initial condition x0 , but x0 is suﬃciently (n−1) ∗ ∗ ∗ deﬁned by φ (z (0), z˙ (0), . . . , z (0) , then, when applying (5) to (1), there exists a unique solution of (1) at least for a ﬁnite time interval in the vicinity of the desired trajectory, which represents the solution of the aforementioned Brunovsk´y form. Proof. Considering the results of [9,1], it is easy to show, that every SISO ﬂat system can be represented as follows. Setting z = [z, z, ˙ . . . , z (n−1) ]T = [z1 , z2 , . . . , zn ]T the system (1) can be transformed via the well deﬁned diﬀeomorphism z = F (x)

(6)

where F = φ−1 (cf. (3)), into the control normal form z˙i (t) = zi+1 (t), i ∈ {1, . . . , n − 1} z˙n (t) = α(z(t), u(t))

(7)

where α(·, ·) is also smooth with respect to its arguments and z0 = z(0) = F (x0 ) from (6). In view of (7) it is evident, that (4) is the solution for u of 0 = α(z, u) − z˙n

(8)

with z˙n = z (n) . Hence one gets α(z, ψ(z, z˙n )) = z˙n

(9)

Applying the feedforward (5) to the diﬀerentially ﬂat system given by (1) is equivalent to the application of u∗ (t) of (5) to (7), which results in ζ˙i (t) = ζi+1 (t), i ∈ {1, . . . , n − 1} ζ˙n (t) = α ζ(t), ψ(z∗ (t), z˙n∗ (t))

(10)

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V. Hagenmeyer and E. Delaleau

and ζ 0 = ζ(0) = F (x0 ). The proof proceeds by studying the solution t → ϕ(t, ζ 0 ) of the non-autonomous system of diﬀerential equations (10). Since α(·, ·) is smooth, at least local existence and uniqueness of the solution ϕ(t, ζ 0 ) of (10) is given for t ∈ [0, τ1 ], τ1 > 0. To distinguish the two cases of consistent and non consistent initial conditions, the corresponding solutions ˜ ζ 0 ) respectively: are denoted by t → ϕ(t, ¯ z∗0 ) and t → ϕ(t, 1. The case of a consistent initial condition is deﬁned by ζ 0 = z∗0 = F (x0 ) (cf. (6)). It is evident that ϕ(t, ¯ z∗0 ) = z∗ (t), which can be veriﬁed using (10) and considering (9). But ϕ(t, ¯ z∗0 ) = z∗ (t) is also the solution of the following Brunovsk´ y form ζ¯˙i (t) = ζ¯i+1 (t), i ∈ {1, . . . , n − 1} ζ¯˙n (t) = z˙n∗ (t)

(11)

with ζ¯0 = z∗0 . Moreover, it is clear that the subset S of [0, ∞) such that (10) is equal to (11) is closed in [0, ∞). Reasoning as above, one shows that S is also open in [0, ∞). Since [0, ∞) is connected, it follows that S = [0, ∞). 2. The case of non consistent initial conditions is deﬁned by ζ 0 = z∗0 . Since ∗ the consistent case solution ϕ(t, ¯ z0 ) of (10) is equal to the designed desired ¯ z∗0 ) belongs to a certain D for all t ∈ [0, t1 ], z∗ (t), it is assured, that ϕ(t, where D ⊂ IRn is an open connected set and 0 < t1 < ∞. Then it can be shown by direct application of Theorem 2.6 of [11], that given > 0, there is δ > 0, such that if ||ζ 0 − z∗0 || < δ

(12)

then there is a unique solution ϕ(t, ˜ ζ 0 ) of (10) deﬁned on [0, t1 ], with the ˜ ζ 0 ) satisﬁes initial condition ϕ(0, ˜ ζ 0 ) = ζ˜0 = F (x0 ), and ϕ(t, ||ϕ(t, ˜ ζ 0 ) − z∗ (t)|| < , ∀t ∈ [0, t1 ]

(13)

The restriction to the time interval t ∈ [0, t1 ] is valid for structurally unstable systems. If the system is structurally stable, then the result holds for all times.

3

Exact feedforward linearisation and control law design

Taking into consideration the result of Proposition 1, the following control law design methodology does not exactly feedback linearise the system (as for feedback linearisation see for instance [7]) and avoids therefore the cancelling of “well-behaving” terms. It linearises the system exactly by feedforward when being on the desired trajectory and stabilises it around them. Thus, the control law to be designed consists in two parts 2 , a feedforward part (5) and 2

In the linear case, refer to the control structure advocated in [5].

Exact feedforward linearisation based on diﬀerential ﬂatness

165

a feedback part taking the tracking error into account. The structure of the combination of both parts is established in the following. Since in (11) the term z˙ n∗ (t) plays the role of the input to the Brunovsk´y form, the new input v is designed as v = z˙n∗ + Λ(e)

(14)

where the tracking error e = [e1 , e2 , . . . , en ]T is generally deﬁned as ei = zi − zi∗ , i = 1, . . . , n

(15)

and for the feedback part generally holds Λ(0) = 0

(16)

Thereby the control Λ(e) can be any type of control, be it of sliding mode type (see e.g. [15]), based on Lyapunov stability theory (see e.g. [2]), or classical PID (see e.g. [14]). The latter is studied in Sec. 5. The combination of (5) and (14) results in the following control structure 3 u = ψ(z∗ , v) = ψ(z∗ , z˙n∗ + Λ(e))

(17)

The advantage of this structure becomes evident in view of (9). One gets when being on the desired trajectory α(z∗ , ψ(z∗ , v)) = v

(18)

and therefore ∂α(z∗ , ψ(z∗ , v)) =1 ∂v

4

(19)

Structure of the error equation

To study the behaviour of the system (1) under the control law (17) in the vicinity of the desired trajectory, the control law (17) is applied to (7), which yields z˙i = zi+1 , i ∈ {1, . . . , n − 1} z˙n = α(z, ψ(z∗ , v)) = α z, ψ(z∗ , z˙n∗ + Λ(e))

(20)

Using (20) and (15), the corresponding tracking error system can be denoted as e˙ i = ei+1 , i ∈ {1, . . . , n − 1} 



e˙ n = α e + z∗ , ψ(z∗ , z˙n∗ + Λ(e)) − z˙n∗

(21)

v 3

˜ ∗ ) + v = ψ(z ˜ ∗ ) + z˙n∗ + Λ(e). Refer to [6] for the cases in which one gets u = ψ(z

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where e = [e1 , e2 , . . . , en ]T . The linearised system around the desired trajectory (e = 0) is then given by      0 0 1 0 ··· 0   0 0 1 · · · 0  0          .. . . .. .  + . ν ν · · · ν  e˙ δ =  . (22) 1 2 n  eδ . . .        0 0 0 · · · 1  0 1 γ1 γ2 γ3 · · · γn where (remember the arguments of α(z, u) in (7)) ∂α ∂zi ∂α γi = = ∂zi ∂ei e=0 ∂zi z=z∗ ∂zi ∂ei

= 1 in view of (15), and furthermore ∂α ∂u ∂v ∂v ∂Λ(e) = = νi = ∂u ∂v ∂ei e=0 ∂ei e=0 ∂ei e=0

since

since

5

(23)

∂α ∂u ∂u ∂v |e=0

=

∂α ∂v |e=0

(24)

= 1 in view of (19) and (21).

Extended PID control

When using a PID-like stabilisation around the desired trajectory, then Λ(e) in (14) can be written as

t

e1 (τ )dτ +

Λ(e) = λ0 0

k+1

λi ei (t)

(25)

i=1

where k is a ﬁxed integer in {0, . . . , n − 1}. Thus, the control structure given in (17) can be denoted by t k+1 ∗ ∗ ∗ e1 (τ )dτ + λi ei (t) (26) u = ψ(z , v) = ψ z , z˙n + λ0 0

i=1

This structure consists of a nonlinear combination of a nonlinear feedforward part based on diﬀerential ﬂatness and a simple linear feedback part of extended PIDk type.

6

Stability

In this section it is shown, that the proposed control strategy (26) is able to stabilise the system (1) around given desired trajectories z ∗ . For the sake of generality, full state information is applied to the PID k -part of the control, that is k = n − 1 in (25) for the sequel. Using this setting, comparability with the classic feedback linearisation approach is given.

Exact feedforward linearisation based on diﬀerential ﬂatness

6.1

167

A stability result by Kelemen

To study the stability of the proposed methodology, one makes use of a result which was primarily introduced by M. Kelemen in [10] and reinterpreted by [13] and [12]. In the following, the presentation of [13] is adopted. Given the system ˙ η(t) = g(η(t), ν(t)), η(0) = η 0 , t ≥ t0

(27)

where η(t) is the n × 1 state vector and ν(t) is the m × 1 input vector. We assume that • (H1): g : IRn × IRm → IRn is of class C 2 with respect to its arguments, • (H2): there is a bounded, open set Γ ⊂ IRm and a continuously diﬀerentiable function ξ : Γ¯ → IRn such that for each constant input value υ ∈ Γ , g(ξ(υ), υ) = 0, • (H3): there is a λ > 0 such that for each υ ∈ Γ , the eigenvalues of ∂g ∂η (ξ(υ), υ) have real parts no greater than −λ. These hypotheses guarantee that the system (27) has a manifold of exponentially stable constant equilibria, which we have chosen to parameterise by constant values of the input. In the sequel, we let · denote the (point-wise in time) Euclidean norm of a (time-varying) vector. Theorem 1 ([10], [13]) Suppose the system (27) satisﬁes (H1), (H2) and (H3). Then there is a ρ∗ > 0 such that given any ρ ∈ (0, ρ∗ ] and T > 0, there exists δ1 (ρ), δ2 (ρ, T ) > 0 for which the following property holds. If a continuously diﬀerentiable input t → ν(t) satisﬁes ν(t) ∈ Γ , T ≥ t0 , η 0 − ξ(ν(t0 )) < δ1

(28)

and 1 T

t+T

˙ )dτ < δ2 , t ≥ t0 ν(τ

(29)

t

then the corresponding solution of (27) satisﬁes η(t) − ξ(ν(t)) < ρ, t ≥ t0

(30)

In an important corollary of this theorem, the following properties are established. If, in addition to the conditions of the theorem, the input signal ˙ = 0, limt→∞ ν(t) = ν ∞ ∈ Γ , then the corresponsatisﬁes limt→∞ ν(t) ding solution of (27) satisﬁes limt→∞ η(t) = ξ(ν∞ ). Furthermore, if for some T1 > t0 , ν(t) = ν ∞ ∈ Γ for all t ≥ T1 , then η(T1 ) is in the domain of attraction of the exponentially stable equilibrium (ξ(ν ∞ ), ν ∞ ).

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6.2

Stability for the proposed control strategy

The t augmented tracking error Tsystem can be found after the deﬁnition e0 = e dτ and e = [e0 , e1 , . . . , en ] . Then one gets for (21) in view of (25) 0 1 e˙ i = ei+1 , i ∈ {0, . . . , n − 1} n ∗ ∗ ∗ λi ei ) − z˙n∗ = β (e, z∗ ) e˙ n = α e + z , ψ(z , z˙n +

(31)

i=0

where z∗ = [z ∗ , z˙∗ , . . . , z ∗ (n) ]T . This system can be written as e˙ = Υ (e, z∗ )

(32)

where the desired ﬂat output and its n derivatives z∗ play the role of an input to the tracking error system in e. Now, in comparing (32) with (27) (that is . . η = e and ν = z∗ ), the stability result of the proposed control strategy can be stated as follows. Proposition 2 There is a ρ∗ > 0 such that for all ρ ∈ (0, ρ∗ ] and T > 0 there exists δ1 (ρ), δ2 (ρ, T ) > 0 for which the following property holds. If a suﬃciently continuously diﬀerentiable desired trajectory z ∗ (t) satisﬁes z∗ (t) ∈ Γ ⊂ IRn+1 (Γ as deﬁned in (H2) of Theorem 1), T ≥ t0 , e(t0 ) < δ1

(33)

and 1 T

t+T

z˙ ∗ (τ )dτ < δ2 , t ≥ t0

(34)

t

then the corresponding solution e of (31) satisﬁes e(t) < ρ, t ≥ t0

(35)

that is the system (1) is stable under the tracking control law (26). Moreover, if in addition the desired trajectory satisﬁes lim z˙ ∗ (t) = 0,

t→∞

lim z∗ (t) = z∗∞ ∈ Γ

t→∞

(36)

then the corresponding solution of (31) satisﬁes lim e(t) = 0

t→∞

(37)

Furthermore, if for some T1 > t0 , z∗ (t) = z∗∞ ∈ Γ for all t ≥ T1 , then e(T1 ) ∗ ), is in the domain of attraction of the exponentially stable equilibrium (0, z∞ that is the system (1) is asymptotically stable under the tracking control law (26).

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169

Proof. Since Proposition 2 is based on Kelemen’s result as elaborated in [13], one proceeds by showing, that the hypotheses (H1)-(H3) of Theorem 1 given in Sec. 6.1 can always be fulﬁlled: • (H1): Straightforward. • (H2): The boundedness of the set Γ ⊂ IRn+1 can be assured by the fact, that only suﬃciently smooth bounded desired trajectories z ∗ (t) with bounded derivatives are studied. Therefore all point-wise in time inputs z∗ ∈ Γ = Γ¯ ⊂ IRn+1 . That Γ is closed in the present case and not open as in Kelemen’s theorem does not interfere with its applicability: the hypothesis of openness of Γ in [10,13] is due to the hypothesis of existence of a manifold composed of stable equilibria when applying constant inputs from Γ (see [10,13] for details). In the present case, the manifold of stable equilibria contracts to a single stable equilibrium (for its stability see (H3)): setting the input z∗ (t) ∈ Γ ⊂ IRn+1 of (31) point-wise in time as constant z∗ ∈ Γ ⊂ IRn+1 and studying the corresponding equilibria leads to the continuously diﬀerentiable function ξ : Γ¯ → IRn+1 , which is deﬁned by 0 = ξi+1 , i ∈ {0, . . . , n − 1} 0 = β (ξ(z∗ ), z∗ )

(38)

from (31). Coupling the relation between (4), (8) and (9) with (31) and (38) leads to the origin ξ(z∗ ) = 0 as the equilibrium. (0, z∗ ) by choosing • (H3): One is always able to act on the eigenvalues of ∂Υ ∂e the respective design parameters of the PID n−1 -part of the control law appro(0, z∗ ) being of the structure priately: comparing (32) with (21) leads to ∂Υ ∂e of the matrices given in (22). Thereby νi = λi , i ∈ {0, . . . , n} in view of (24). Thus it can always be assured, that there is a λ > 0 such that the eigenvalues (0, z∗ ) have real parts no greater than −λ. of ∂Υ ∂e In interpreting z∗ as the “slowly-varying” input of the tracking error system (31), one has to assure for its stability both desired trajectories which (0, z∗ ). are designed “not too fast” and the negativity of the eigenvalues of ∂Υ ∂e ∗ Since z is composed of the desired ﬂat output and its derivatives, it is ev(0, z∗ ) can be modiﬁed by ident, that the farther left the eigenvalues of ∂Υ ∂e the respective λi , the faster the desired trajectories can be designed. Therefore one remarks that the two diﬀerent tasks of choosing the velocity of the desired trajectory and of modifying the point-wise in time “poles” of the closed loop system have to be balanced carefully. Moreover, when consider(0, z∗ ) can be placed for a ing λi = λi (z∗ (t)), it is clear that the poles of ∂Υ ∂e desired characteristic polynomial. Furthermore, in the context of Proposition 2, stability can also be analysed for partial state feedback by setting the respective control law coeﬃcients λi = 0, i ∈ {k + 2, . . . , n} (as in (25)). The minimal number of derivative actions k to be necessary (but not implicitly suﬃcient) for stability can be determined from (22), (23) and (24) using the necessary condition for negativity of the eigenvalues, that is γi + νi < 0, i ∈ {0, . . . , n}.

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Acknowledgements The authors are very grateful to Richard Marquez for many important discussions concerning the philosophy of this article. They are moreover thankful to Thomas Meurer, Michael Zeitz and Paulo S´ergio Pereira da Silva for their respective advice regarding the proof of Proposition 1.

References 1. B. Charlet, J. L´evine, and R. Marino. (1989) On dynamic feedback linearization. Syst. Contr. Lett., 13, 143–151 2. A. Chelouah, E. Delaleau, Ph. Martin, and P. Rouchon. (1996) Diﬀerential ﬂatness and control of induction motors. In: Proc. Symposium on Control, Optimization and Supervision; Computational Engineering in Systems Applications IMACS Multiconference, Lille (France), 80–85 3. M. Fliess, J. L´evine, Ph. Martin, and P. Rouchon. (1995) Flatness and defect of non-linear systems: introductory theory and examples. Internat. J. Control, 61, 1327–1361 4. M. Fliess, J. L´evine, Ph. Martin, and P. Rouchon. (1999) A Lie-B¨acklund approach to equivalence and ﬂatness of nonlinear systems. IEEE Trans. Automatic Control, 44, 922–937 5. M. Fliess and R. Marquez. (2000) Continuous-time linear predictive control and ﬂatness: a module-theoretic setting with examples. Internat. J. Control, 73, 606–623 6. V. Hagenmeyer. (2001) Nonlinear stability and robustness of an extended PIDcontrol based on diﬀerential ﬂatness. In: CD-Rom Prepr. of the 5th IFAC Nonlinear Control Systems Design Symposium, Saint-Petersburg (Russia) 7. A. Isidori. (1995) Nonlinear Control Systems. 3rd edn. Springer-Verlag, Berlin 8. F. Jadot, Ph. Martin, and P. Rouchon. (2000) Industrial sensorless control of induction motors. In: A. Isidori, F. Lamnabhi-Lagarrigue, and W. Respondek, (Ed.), Nonlinear Control in the Year 2000 (volume 1), Springer, London, 535– 544 9. B. Jakubczyk and W. Respondek. (1980) On linearization of control systems. Bull. Acad. Pol. Sci., S´er. Sci. Math., 28, 517–522 10. M. Kelemen. (1986) A stability property. IEEE Trans. Automatic Control, 31, 766–768 11. H. K. Khalil. (1996) Nonlinear systems. 2nd Edn. Prentice-Hall, Upper Saddle River 12. H. K. Khalil and P. V. Kokotovic. (1991) On stability properties of nonlinear systems with slowly varying inputs. IEEE Trans. Automatic Control, 36, 229 13. A. L. Lawrence and W. J. Rugh. (1990) On a stability theorem for nonlinear systems with slowly varying inputs. IEEE Trans. Automatic Control, 35, 860– 864 14. J. L´evine. (1999) Are there new industrial perpectives in the control of mechanical systems? In: P.M. Frank (Ed.), Advances in Control (Highlights of ECC’99), Springer, London, 197–226 15. H. Sira-Ram´ırez. (2000) Sliding mode control of the PPR mobile robot with a ﬂexible joint. In: A. Isidori, F. Lamnabhi-Lagarrigue, and W. Respondek (Eds.) Nonlinear Control in the Year 2000 (volume 2), Springer, London, 421–442

Local controlled dynamics Efthimios Kappos University of Sheﬃeld, Sheﬃeld, U.K. Abstract. In this work, we study the local dynamics of control-aﬃne systems that are ‘typical’ in that in an open and dense set of the state space the state vector ﬁeld does not vanish. After establishing that the ‘local accessible sets’ of points in this set cannot contain a neighbourhood of the initial point, we introduce a number of new geometrical concepts useful in the analysis of local dynamics. The key notion is that of control-transverse objects: submanifolds or foliations transverse to the control distribution. These are shown to locally contain both the case of openloop and closed-loop controls. We use the Conley index to study variations of the invariant dynamics as the control-transverse object moves.

1

Control system basics

The aim of this paper is to study the controlled trajectories of a control system using a geometric approach. This approach relies on the separation of state space into a generically thin singular set and an open set of points where we cannot have equilibrium points (the regular points). A central role is played by foliations and submanifolds transverse to the control distribution. In common with previous work in geometric control, we take the view that a control system deﬁned on the state space manifold M n (M paracompact and smooth) is either a nonempty ‘collection of vector ﬁelds’, D ⊂ X (M ), or a point-wise nonempty closed subset of the tangent bundle, C ⊂ T M (with some suitable continuity or smoothness assumptions.) We shall mostly use the latter viewpoint. In this case, if π is the natural projection π : T M → M , ∅ we have required that π(C) = M or that the control set (or indicatrix) Cp = for all p ∈ M . In view of the various ‘bang-bang’ theorems, we shall assume that the control sets Cp are convex. ˜ Deﬁnition 1. An aﬃne distribution on the manifold M n is a pair (D, X), ˜ where D ⊂ T M is a proper sub-bundle of the tangent bundle T M and X is a section of the quotient bundle T M/D → M . (The rank of D is not assumed to be constant.) This way of deﬁning an aﬃne distribution is advantageous from the point of view of feedback equivalence in that it does not distinguish any speciﬁc ‘state dynamics’ (or drift vector ﬁeld) f , as the traditional coordinate form x˙ = f (x) + i ui gi (x) does. Deﬁnition 2. A control-aﬃne system is a control system C that is a ˜ and is such that, for all p ∈ M , the subset of some aﬃne distribution (D, X) control set Cp has nonempty interior in Dp . A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 171-180, 2003. Springer-Verlag Berlin Heidelberg 2003

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(Thus, a drop in the dimension of the control set is reﬂected in the dimension ˜ p .) of the subspace Dp and not in Cp as a subset of the aﬃne subspace X Note that, if we let m be the maximal dimension of the distribution D and use a local basis {g1 , . . . , gm } of vector ﬁelds for D, we arrive at the classical description of control-aﬃne systems. A control-aﬃne system is in some sense a ﬁrst approximation to a general control system around a ‘nominal’ section of the control system C (see [6].) Controllability questions lie at the heart of the mathematical theory of control systems. In this work, we are interested in studying the local accessible sets of control systems. We shall denote by Γ (C) the set of locally deﬁned, piecewise continuous sections of the ﬁbration C → M . These sections correspond to feedback or closed-loop controls. The deﬁnition of open-loop controls is a little more awkward, since the control sets depend on the state, but can be obtained by considering the set of pairs of maps deﬁned on a connected interval I ⊂ R: a piecewise continuous map U to the control ﬁbration C, together with a piecewise C 1 map xU : I → M such that the tangent map of xU satisﬁes dxU (t) = U (t) ∈ CU (t) , dt

∀t ∈ I.

We shall denote the set of all open-loop controls by ΓOL (C). Integral curves of local vector ﬁelds obtained from Γ (C) (or Γ (C)) will be called controlled trajectories. Deﬁnition 3. For an open set V ⊂ M , the local controlled dynamics of the control system C is the set of all control sections Γ |V (C). The local accessible set from the point p ∈ M , Ap (V ), is the set of all connected controlled trajectories through p and contained in V . Deﬁnition 4. The singular set Σ of the control system C is the closed set π(zM ∩C) of all points of p ∈ M such that 0 ∈ Cp (here, zM is the zero section of T M .) ˜ is the closed set The singular set Σ of the control-aﬃne system (D, X) of all points of M where the distribution D gives a subspace of T M , in other ˜ −1 (0). words Σ = X Generically, the singular set of a constant-rank control-aﬃne system is a smooth sub-manifold of M of dimension m and is hence a ‘thin set.’ Thus, most points in M are regular, in the sense that they are not in the singular set Σ. We shall see in this paper that the local accessibility problem is simpler near regular points and that the distinction between open-loop and closedloop controls is unnecessary there.

Local controlled dynamics

2 2.1

173

Accessible sets Accessibility and orbits of families of vector ﬁelds

We begin by looking at a familiar example that illustrates the diﬀerences between control systems with generic singular sets and systems where the whole state space is singular. Example: (Sussmann, [11], p.177) In R2 , deﬁne the singular control distri∂ in the closed half-space H − = {x ≤ 0} and by bution D by the span of ∂x ∂ ∂ the span of ∂x and φ ∂y , with φ = φ(x) > 0 in H + = {x > 0}; thus the distribution D is the whole of the tangent space in H + = {x > 0}. It is clear that if we start anywhere in H − , we can reach any other point in the plane by ﬁrst entering H + , achieve the correct vertical coordinate y and then drive to the desired point along a horizontal path. It follows that the whole of the plane R2 consists of a single orbit of D. This example is used by Sussmann to illustrate his Theorem 4.1, about orbits of a family of vector ﬁelds, suitably deﬁned, always being submanifolds of the state space manifold. Speciﬁcally, it shows that the ‘right’ distribution to use is not simply the controllability Lie algebra (the Lie algebra generated by all control sections), but a larger distribution, called PD , which is invariant under the ‘group’ of all possible local control diﬀeomorphisms. Thus, it highlights the important distinction between local and global controlled trajectories; the former involves the Lie-algebraic conditions, but the latter involves something more topological about the control system. This distinction will be a useful one to keep in mind in the sequel. Note that, if we are only allowed to move forward along the given vector ﬁelds, then the ‘neatness’ of the Sussmann-Stefan theory is completely lost: the orbits are not manifolds and, moreover, they overlap. The orbit starting from (−1, 0), for example, is the union of the line segment [−1, 0] on the x-axis, together with the quadrant {(x, y); x > 0, y ≥ 0}. Unlike in the Sussmann theory, we shall be interested in control systems with a generic singular set, so that most points of M are regular in the sense all possible control directions are non-zero. 2.2

Local accessible sets near regular points

Our ﬁrst result points out the crucial diﬀerence between systems where the state dynamics is zero and those with nontrivial state dynamics. It says that a very large control action is necessary if we are to have a chance to go backwards along the state vector ﬁeld locally (assuming this is possible, of course.) In other words, and this is the form the theorem takes, if the control action is bounded (as is the case in practice), then we cannot locally reach an open subset of some neighborhood of our starting point. In the case of an integrable control distribution this subset can be given explicitly: its boundary is a union of leaves of the foliation of D. This is

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our second result. If D is of codimension one, then the boundary is exactly the leaf of D through the starting point. If the codimension is greater than one, then there is more freedom in choosing the boundary. We assume that both M n and D have been provided with some riemannian metric (thus it makes sense to talk about norms and orthogonality.) Control-aﬃne systems will also be denoted in the more traditional form X + D, with D the control distribution and X the state dynamics. ˜ be a control-aﬃne system and let p ∈ M n \ Σ be Theorem 1. Let (D, X) ˜ a regular point, so that X(p) = 0. For K > 0 and V an open set containing p, deﬁne ΓK,V (D) to be the set of sections U ∈ Γ |V (D) such that supq∈V |U (q)| < K. Then, there is a neighborhood N of p and an open subset N ⊂ N such that a) p ∈ N and b) any trajectory of controlled dynamics X + U with u(p) ∈ U that is contained in the neighborhood N misses the open subset N . The second theorem does not impose a contraint on the control, but requires D to be integrable. Theorem 2. Suppose that p ∈ M n \ Σ is regular. Also suppose D is locally integrable near p. Then there is some neighborhood N of p and a local hypersurface Σ ⊂ N through p that is invariant under D and divides N into two open sets N+ and N− with common boundary Σ (so that N = N+ ∪ N− ) and are such that N+ is positively invariant for all controlled trajectories contained in N . Remark: The conclusions of these theorems hold irrespective of whether any Lie algebraic rank condition holds, such as the condition that the span of the {adkX D}k≥0 is of rank n at p. Proof of Theorem 1: The aim is to construct a ‘local Lyapunov function’ for all possible local control trajectories. The ﬁrst step is to set up a nonzero vector ﬁeld and obtain a ﬂow box for it. It is always possible to locally decompose T M n = D ⊕ D⊥ ; hence write / Dp , so X = X1 + X2 , with X1 ∈ D and X2 ∈ D⊥ . By assumption, X(p) ∈ X2 (p) = 0 and hence is nonzero locally in some neighborhood N0 of p. Pick a ﬂow box ψ : N1 → W ⊂ Rn (N1 ⊂ N0 ) so that ψ(p) = 0 and T Y = ψ∗ (X2 ) = (0, . . . , 0, 1) (see Arnol’d [1].) Choose an orthogonal basis for n R so that bn = Y (0) and ψ∗ (Dp ) = span {b1 , . . . , bm } (remember X2 ∈ D⊥ .) Write z1 , . . . , zn for the coordinate functions in this basis. Thus dzn (Y )(0) = 1. Then the one-form dzn pulls back to an exact form α = dβ in N1 and α(X2 )(p) = 1 (β = ψ ∗ (zn ).) The level sets {zn = constant} pull back to leaves of the local foliation of α. We compute LX+U β: LX+U β = α(X + U ) = α(X) + α(U ).

(1)

Now α(X)(p) = α(X2 )(p) = 1 and α(U )(p) = 0 since Dp ⊂ ker dβ by construction. Thus LX+U β(p) = 1 > 0.

Local controlled dynamics

175

Claim. There is some neighborhood N of p such that α(U )(p) < 1 for all p ∈ N. This is because α(Dp ) = 0 and the compact K–ball in the ﬁbres of D gives a continuous non-negative function γ : N1 → R: y → sup{|α(v)| ; v ∈ BK (0) ⊂ Dy }. One then takes N = γ −1 [0, 1) ∩ N1 . For y ∈ N we have α(U )(y) < 1 and so LX+U β > 0. This completes the proof of Theorem 1, since one easily checks that the set N in the statement of the theorem is the set where β < 0. Remark: As we take larger and larger controls, the neighborhood N will shrink and it is possible that controls will exist driving us downwards with respect to the local Lyapunov function β. Example: (Brockett, [3], p.182) Consider the control system   x˙ 1 = u1 x˙ 2 = u2 (2)  x˙ 3 = x2 u1 − x1 u2 Claim. There is no continuous stabilizing feedback law near 0. However, the local controllability condition is satisﬁed and every state can be driven to the origin using a smooth control function. Now suppose we have the system         x˙ 1 0 1 0  x˙ 2  =  0  +  0  u1 +  1  u2 = f + g1 u1 + g2 u2 x˙ 3 1 x2 −x1

(3)

so we have constant state dynamics pointing ‘upwards.’ Following the procedure

given in the above proof, we take the function β = x3 . Then α = 0 0 1 and we compute LX+U β = x˙ 3 = 1 + x2 u1 − x1 u2 .

T ˙ Taking our starting point to be x = 0 0 1 , we have β(x) = 1 > 0, which stays positive for nearby points, provided the control is bounded. Proof of Theorem 2: The basic idea is the same. We decompose X = X1 + X2 , with X1 ∈ D and X2 ∈ D⊥ and use a local ﬂow box ψ : N0 → W ⊂ R for X2 near p. The m-dimensional foliation of the integrable distribution D pushes forward to a foliation of W . Now the orthogonal complement D⊥ is a distribution —not integrable in general, but which can give us local transverse sections for the foliation of D. Thus, consider Σ0 to be a local (n − m)-dimensional manifold transverse to the distribution D and orthogonal to it at p, so that X2 (p) ∈ Σ0 . It intersects the zn = 0 plane along an (n − m − 1)-dimensional manifold Σ1 and the leaves through Σ1 give the desired D–invariant Σ. Theorem 2 follows, since dzn (Y ) > 0 in W . Remarks: Theorems 1 and 2 say that, locally near a point where X = 0, we cannot use the extra directions arising from the Lie brackets of X and

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the basis vector ﬁelds of D. The question is then: under what conditions are these extra directions available? One instance where this is possible is in the vicinity of an equlibrium point (again we emphasize that then Theorems 1 and 2 do not apply.) The reader should perhaps go back at this point to the instances where Lie bracket conditions lead to local accessibility and notice that they either apply in a neighborhood of an equilibrium point or to the case when X ∈ D (e.g Assumption 3.1(a) in [10], p.72.) The geometric viewpoint gives an appreciation of the global problem we shall be concerned with: even though locally we are constrained by the geometry of the control pair, globally these objects have nontrivial ‘twisting’. In the case of an equilibrium, for example, the state dynamics will be used to control in the complement of the control distribution. This can be best seen in the following linear example. Example: Consider the two-dimensional linear system x˙ 1 = −x1 + u x˙ 2 = x1 + x2 It is controllable, since −1 Ab = . 1 T

However, let us pick a point, such as x = (0, 1) , where Ax = 0 and Ax ∈ / D = span{b}. It should be clear that the ‘extra direction’ given by the Lie bracket is not available at x. In fact, as we expected from theorem 2, in a neighborhood of x, it is only possible to move ‘upwards’ –i.e. no trajectory can enter the lower half of a local plane. We know it should be possible to steer x to zero, but the question is how? In this simple two-dimensional case, it is easy to see that what we should do is steer to the left until the state dynamics starts pointing downwards and then drive towards the origin. Thus any stabilized trajectory through our chosen x must take a large ‘detour’ before heading towards zero.

3

Control-transverse geometric objects

Deﬁnition 5. A control-transverse manifold for the control-aﬃne sys˜ is a submanifold N ⊂ M that is transverse to D at all of its tem (D, X) points. Deﬁnition 6. A control-transverse foliation F is a foliation of M deﬁned in an open subset V ⊂ M that is everywhere transverse to D.

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These deﬁnitions can be generalized to the control system C, assuming a local dimension can be deﬁned. The importance of control-transverse geometric objects (submanifolds and foliations) lies in the following key facts: the control system C associates to each control-transverse object a unique dynamical system. The converse is also true, locally: all possible local dynamics of the control system are associated with a control-transverse object. The dynamics on the transverse object will be called control-transverse or feedback invariant dynamics, to accord with the use of this notion in previous control research. The geometry for the ﬁrst assertion is simple: the control-transverse object gives a splitting of the exact sequence 0 → D → T M → T M/D → 0 which means that any vector ﬁeld X (restricted to the relevant geometric object) is decomposed uniquely into components X1 + X2 , with X1 ∈ D and X2 ∈ T N , for the case of transverse manifolds (foliations are similar.) The feedback-invariant dynamics on N is then the vector ﬁeld X2 . 3.1

Transverse objects and control

Theorem 3. 1. Let the feedback control U : V → C be deﬁned on an open ˜ = 0 in V . set V ⊂ M , with X Then, there exists a foliation F in V , transverse to D and invariant under the local ﬂow of U and, furthermore, such that the feedback-invariant dynamics on the leaves of F coincide with the restriction of the control dynamics U to the leaves. 2. Let the open-loop control U : (− , ) → C and the initial condition / Σ. xU (0) = p ∈ M be given and suppose that p ∈ Then, there exists a local transverse foliation F containing the trajectory xU (−δ, δ)) for some δ ≤ . Proof: In both cases note that, by assumption, there are no equilibrium points of the controlled dynamics in V , resp. near p. For the ﬁrst conclusion of the Theorem, we have an autonomous ﬂow and the ﬂow box theorem applies, locally. Working in the parallelized ﬂow, it is easy to ﬁnd a foliation of codimension m transverse to the image of D and invariant under the controlled ﬂow. This is then mapped back to a foliation in some open set in V . ˜ For the second part, observe that the open condition X(p) = 0 holds in some neighbourhood of p and thus, locally, the trajectory is not contained in and has no tangency with D. We need to extend the given orbit to a manifold of dimension (n − m) transverse to D locally. Note that we cannot use continuous dependence on initial conditions directly, because we cannot guarantee that the resulting time-varying system will yield a ﬂow (a foliation by trajectories.)

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Instead, we use a local chart that trivializes the distribution D, so that it is mapped to the product Rm × W , with W ⊂ Rn−m . Fixing a basis {e1 , . . . , em } in Rm , we can use the velocity vectors of the trajectory, x˙ U (t), as the (m + 1)-th basis vector, along the trajectory; thus we can consider the trajectory in the trivializing patch as the (m+1)-th coordinate axis. After this straightening, it is a simple matter to foliate the trivializing neighbourhood by the levels of D; pulled back, we get the desired control-transverse foliation containing the control trajectory. 3.2

The geometric setting for control

Having established the local equivalence of controls (closed or open loop) and control-transverse objects, we are now ready to give a geometric formulation of the controlled dynamics problem for control-aﬃne systems. Given a manifold M n and a distribution Dm (0 < m < n), consider, for every open set V ⊂ M the set of all D-transverse foliations FD (V ) in V . We clearly have, for W ⊂ V , the restriction ρV,W : FD (V ) → FD (W ) and we can check the axioms for a pre-sheaf of sets. We can deﬁne germs of transverse manifolds by using these restrictions in the usual manner. At the level of ‘stalks’, the group of n × n matrices of the form AB (4) , A ∈ Gl(m, R), C ∈ Gl(n − m, R), B ∈ M m×(n−m) 0 C acts transitively on the set of all transverse subspaces to a given m-space. In light of Theorem 3, we are interested in the set of all possible controltransverse objects in M \ Σ. Let us remark that if the state space M is simply Rn , then any distribution of T M is globally trivializable. This is generally not the case in the open manifold M \ Σ, though. We feel strongly that this is a proﬁtable direction for future research. 3.3

Smooth feedback stabilization

Clearly, the singular set contains all possible equilibria of controlled dynamics. So far, we have only considered regular points and shown the adequacy of considering only sections of the control bundle, which correspond to controltransverse geometrical objects by the results of the previous section. Near a singular point, however, the transverse objects, while yielding a rich class of feedback dynamics, are not enough. It can be shown, for example, that the non-smooth controller used to stabilize the simple control example given by Kawski corresponds to a manifold that fails to be transverse on the singular set. If we restrict ourselves to smooth feedback controls, however, then we can prove an important relation between stabilization (see [2]) and existence of transverse submanifolds.

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Theorem 4. Let p ∈ Σ be a singular point. Then there is a smooth locally stabilizing feedback section U ∈ Γ (D) if and only if there exists a local control-transverse submanifold N with locally asymptotically stable dynamics. The proof can be found in [8]. As an example, the system

x˙ = −x + u y˙ = −4x3 + 2x + y

(5)

is checked to be locally smoothly stabilizable, but not globally so, since we cannot ﬁnd a global control-transverse manifold with asymptotically stable dynamics.

4

Variations of transverse objects and the Conley index

Here, we outline an approach to controlled dynamics that uses the topological theory of the Conley index. Generally speaking, if we move a control-transverse manifold, while it remains transverse, then we expect that the feedback invariant dynamics may change. It would be desirable to have a systematic way of examining these variations in these controlled dynamics. The use of the Conley index has the advantage that it gives a rather ‘coarse classiﬁcation’ of dynamics, making it easier to classify the control-transverse dynamics (coarsely!) Deﬁnition 7. A variation of a control-transverse object is a continuous family of transverse objects. Thus, for transverse submanifolds, for example, we assume that we are given a map ι : [−1, 1] × N → M such that ιt : N → M is an embedding transverse to D for all t ∈ [−1, 1] and ι0 gives the original embedding map for the transverse submanifold N . A compact variation of a control object is a variation for which there exists a compact subset K ⊂ N such that the variation does not move the set N \ K (resp. K ⊂ M and M \ K for foliations.) The precise deﬁnitions of isolated invariant sets S (maximal invariant sets in some open set containing them), isolating neighbourhoods N (closed sets with S in its interior and with no internal tangencies) and index pairs (N, N+ ) (isolating neighbourhood and its exit set) can be found in, for example, [4] or [6]. The Conley index of the isolated invariant set S is the topological type of the pointed space (N/N+ , [N+ ]). The study of the feedback-invariant dynamics obtained from variations of control objects is performed using the Conley index and transversality. Compact variations are easy to understand:

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Proposition 1. If N is a D-transverse manifold that contains the close, compact set K that is an isolating neighbourhood for the set SN , then any compact variation of N with respect to K yields isolated invariant sets St with the same Conley index. The proof consists of a continuation argument and the fact that the compact variation does not destroy the isolating property of K.

5

Concluding remarks

We have outlined an alternative approach to controlled dynamics, one that uses geometric objects transverse to the control set instead of the analytical and Lie-algebraic approaches that are dominant in the literature. The notion of zero dynamics is familiar enough in control theory, but it is less well known that it can be generalized to arbitrary transverse manifolds and that, furthermore, all local controlled dynamics are obtained in this way. We believe that this constitutes a proﬁtable avenue of research and one that can lead to non-trivial results about both local and global controlled dynamics.

References 1. Arnol’d, V.A. (1983) Ordinary Diﬀerential Equations, MIT Press. 2. Bacciotti, A. (1992) Local Stabilizability of Nonlinear Control Systems, World Scientiﬁc. 3. Brockett, R.W. (1983) Asymptotic Stability and Feedback Stabilization, 181– 191, in: Brockett et. al. eds: Diﬀerential Geometric Control Theory, Birkh¨ auser, Boston. 4. Conley, C. (1978) Isolated Invariant Sets and the Morse Index, C.B.M.S. Reg. Conf. Board in Math., No.38. 5. Hermann, R. (1963) On the Accessibility Problem in Control Theory, 325–332, in: LaSalle J. et. al. eds. Intern. Symp. Nonlinear Diﬀ. Equations, Academic Press. 6. Kappos, E. (2000) Global Controlled Dynamis, Book Draft. 7. Kappos, E. (1996) New Geometric and Topological Methods of Analysis of the Global Nonlinear Control Problem, Electronics Research Memorandum UCB/ERL M96/42, University of California, Berkeley. 8. Kappos, E. (2000) A Condition for Smooth Stabilization, Proceedings of the Maths of Signals and Systems Symposium, P´erpignan. 9. Kappos, E. (1995) The Role of Morse-Lyapunov Functions in the Design of Nonlinear Global Feedback Dynamics, Chapter 12, 249–267, in: A. Zinober ed. Variable Structure and Lyapunov Control, Springer Verlag. 10. Nijmeijer, H., van der Schaft, A. (1990) Nonlinear Dynamical Control Systems, Springer Verlag. 11. Sussmann, H.J. (1973) Orbits of Families of Vector Fields and Integrability of Distributions, Trans. AMS, 180, 171-188.

Adaptive feedback passivity of nonlinear systems with sliding mode Ali J. Koshkouei and Alan S.I. Zinober Department of Applied Mathematics, The University of Sheﬃeld, Sheﬃeld S10 2TN, UK Email: {a.koshkouei, a.zinober}@shef.ac.uk Abstract. Passivity of a class of nonlinear systems with unknown parameters is studied in this paper. There is a close connection between passivity and Lyapunov stability. This relationship can be shown by employing a storage function as a Lyapunov function. Passivity is the property stating that any storage energy in a system is not larger than the energy supplied to it from external sources. An appropriate update law is designed so that the new transformed system is passive. Sliding mode control is designed to maintain trajectories of a passive system on the sliding hyperplane and eventually to an equilibrium point on this surface.

1

Introduction

The link between Lyapunov stability and passivity increases the importance of passivity in the control area. In fact passivity is not only important because of this link, but also there is a relation between passivity and optimality [9]. Passivity has wide applications including electrical, mechanical and chemical process systems [8,9]. The passivity concept is a particular case of dissipativity, which has been addressed by Willems [13]. Passivity has been considered in recent years in many diﬀerent areas; the stability of feedback interconnected systems [2,3], applications to robotics and electro-mechanical systems [6,7], and the geometric approach to feedback equivalence [1]. Full details of passivity of nonlinear systems including concepts, stability and applications can be found in [9], and for Euler-Lagrange systems in [8]. The passivity of a general canonical form of nonlinear systems has been considered in [10], based upon the properties of projection operators. The system is converted into a generalized Hamiltonian system, and is passive whenever the appropriate symmetric matrix is negative deﬁnite. This work has been generalized to multivariable nonlinear systems [11] in which the transformed output is no longer the same as the original system. The new output is the product of a positive deﬁnite matrix and the output of the original system. An adaptive passivation procedure of SISO nonlinear systems based control has been studied in [4]. In this paper a nonlinear aﬃne system is transformed into a passive system via a transformed control. The resulting system is (strictly) passive regarding the diﬀerent choices of the rate function. The only assumption is a transversality condition, i.e. the directional derivative of the storage function in the A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 181-194, 2003. Springer-Verlag Berlin Heidelberg 2003

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input matrix direction is not zero [10]. This assumption is not restrictive because we can select a storage function suitably. Feedback passivation of controllable linear systems is considered as a special illustrative case of the results. Sliding mode control of passive systems is also studied. In Section 2 feedback passivation of nonlinear systems is considered. The sliding mode control of nonlinear passive systems is studied in Section 3. In this method the system is ﬁrst converted into a passive system with a new control via feedback transformation, and then sliding mode control is designed for the passive system. So the method beneﬁts from the properties of both approaches for designing the controller. In this paper we consider the output sliding mode but it can be easily extended to the general case by considering the function s instead of the system output, where s = 0 is an arbitrary sliding surface. Then, s acts like an output in the classical passivity concept. The sliding mode control of linear systems with passivation is discussed in Section 4. Feedback passivation is designed in Section 5 for parameterized nonlinear systems. Suitable estimates for unknown parameters are obtained so that the system is passive with respect to these estimates. In Section 6 a gravity-ﬂow/pipeline model illustrates the results. Conclusions are presented in Section 7.

2

Passivation of nonlinear systems

Consider the nonlinear system x˙ = f (x) + g(x)u y = h(x)

(1)

where x ∈ X ⊆ Rn is the state; u ∈ U ⊆ Rm the control input and y ∈ Y ⊆ Rm the output. Functions f and g are smooth on X . Assume that X is a pathwise connected open subset of Rn . The equilibrium point xe ∈ X satisﬁes f (xe ) + g(xe )ue = 0 with ue ∈ U constant. h is a smooth function deﬁned on X . Let us consider the well-known deﬁnitions [8,9]: Deﬁnition 1. (i) The system (1) is dissipative with respect to the supply rate w(u, y) : U × Y → R if there exists a storage function V : X → R+ such that T V (x(T )) ≤ V (x(t0 )) + w(u(t), y(t))dt (2) t0

for all u ∈ U , all T and x(t0 ) such that x(t) ∈ X for all t0 ≤ t ≤ T . (ii) The system (1) is said to be passive if it is dissipative with supply rate w(u, y) = uT y. (iii) The system (1) is said to be strictly passive if it is passive with supply rate w(u, y) = uT y − Ψ (t, x, u, y) with Ψ (t, x, u, y) a positive function.

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(iv) If there exists a positive function γ such that Ψ (t, x, u, y) = γy2 , the system is strictly output passive. Deﬁnition 2. Let V : X → R+ with V (0) = 0, be a smooth positive deﬁnite storage function. Deﬁne Lg V (x) =

∂V g(x) ∂xT

(3)

The following assumption is needed to design feedback passivation using this method. Assumption: It is assumed that Lg V (x) = 0 for all x ∈ X , and this condition is known as the transversality condition. Let Ω = {x ∈ X : g(x) = 0}. Assume that the control will be “turned oﬀ” near all x ∈ Ω where Lg V (x) = 0. We now follow a new approach. The following theorem yields a feedback transformation which transforms the system into a passive system with new control. The new control v can be designed in many ways including the backstepping approach or sliding control techniques, or a combination of both methods. Theorem 1. Consider system (1). Let Ψ (t, x, u, y) : R+ × X × U × Y → R+ be a positive real function. The following feedback transformation with a new external independent control input v LTg V (x) ∂V T (4) − T f (x) + h (x)v − Ψ (t, x, u, y) u= Lg V (x)2 ∂x transforms system (1) into LTg V (x) ∂V LTg V (x) T x˙ = I − g(x) h (x)v − Ψ (t, x, u, y) f (x)+g(x) Lg V (x)2 ∂xT Lg V (x)2 (5) which is strictly passive with dissipation rate Ψ . If Ψ = 0 the system is passive. With Ψ = γy2 , γ > 0, the system is strictly output passive. Proof. Consider the time derivative of V and substitute (4) to yield ∂V ∂V V˙ = T f (x) + g(x)u ∂x ∂xT LTg V (x) ∂V ∂V ∂V T = T f (x) + g(x) − T f (x) + h (x)v − Ψ (t, x, u, y) ∂x ∂xT Lg V (x)2 ∂x = hT (x)v − Ψ (t, x, u, y) Transformed system (5) is obtained by substituting (4) in (1).

(6)

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The system could be strictly passive with Ψ = 0. So the existence of the function Ψ is not a necessary condition for the strictly passivity of the system. From (6), the control (4) can compensate all nonlinearities in the systems. In many practical problems, compensation of only part of the nonlinearities is needed. Therefore the above theorem can be modiﬁed as follow. Theorem 2. Consider the system (1). Assume that f (x) = f1 (x) + f2 (x) ∂V where ∂x T f1 ≤ 0. Then the original control u with a new external independent control v LTg V (x) ∂V T f (x) + h (x)v − Ψ (t, x, u, y) (7) − u= 2 Lg V (x)2 ∂xT transforms system (1) into LTg V (x) T LTg V (x) ∂V h (x)v f (x)+g(x) x˙ = I −g(x) Lg V (x)2 ∂xT Lg V (x)2 LTg V (x) ∂V −g(x) f (x) Ψ − 1 Lg V (x)2 ∂xT which is passive with dissipation rate Ψ .

(8)

For simplicity and without loss of generality, for the rest of the paper, we assume that f1 (x) = 0, i.e. f (x) = f2 (x). When V = 12 x2 or V = 12 xT P x with a positive deﬁnite matrix P , then the system is (asymptotically) stable if (V˙ < 0) V˙ ≤ 0. From Theorem 1, it is clear that (V˙ ≤ 0) V˙ < 0 if and only if (hT (x)v ≤ Ψ (t, x, u, y)) hT (x)v < Ψ (t, x, u, y). This condition is satisﬁed even if Ψ = 0. When h(x) = 0, the system is LTg V (x) LTg V (x) ∂V Ψ (9) f (x) − g(x) x˙ = I − g(x) 2 T Lg V (x) ∂x Lg V (x)2 with the control v absent. So when h(x) = 0 after a certain time, the control v has no eﬀect on the system and the stability or asymptotic stability of the system depends only upon the control (4). Note that a system may be passive but have unstable zero dynamics [9]. This case may happen whenever the storage function V is not positive deﬁnite. In general, the storage function in the deﬁnitions of dissipativity and passivity is not necessarily a positive deﬁnite function. However, in this paper we focus only on storage functions with positive deﬁnite property as deﬁned in Deﬁnition 1. Therefore Theorem 1 with Deﬁnition 1 assures that a passive system has a stable zero dynamics.

3

Sliding mode of passive systems

Now a control is designed so that the system (5) is converted to a reduced order system. This system could be the zero dynamics of the nonlinear system. Here we consider the output sliding mode control but the approach can

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185

be easily extended to the general case by replacing s(x) with h(x), where s(x) = 0 is an arbitrary sliding surface. The rest of the theory remains intact. Therefore, in this method the system (1) is ﬁrst converted into a new system (5) via the feedback transformation (4). Then a sliding mode control is designed for system (5). Consider h(x) = 0 as a sliding hyperplane. Ideal sliding motion occurs when h(x) = 0. The system equation of the ideal sliding mode is LTg V (x) LTg V (x) ∂V Ψ (10) f (x) − g(x) x˙ = I − g(x) Lg V (x)2 ∂xT Lg V (x)2 and the control (4) is now us =

LTg V (x) Lg V (x)2

∂V − T f (x) − Ψ (t, x, u, y) ∂x

(11)

When ideal sliding mode occurs, it is required that on the manifold h(x) = 0, ∂h ∂h = feq = 0 with ∂t ∂x LTg V (x) ∂V LTg V (x) Ψ (t, x, u, y) feq = I − g(x) f (x) − g(x) Lg V (x)2 ∂xT Lg V (x)2 ∂h may not be zero outside ∂t the boundary layer. Assume that h(x) = 0 and hT (x)h(x) is invertible. It can be shown from (5) that the control

In a system with a boundary layer, h(x) and/or

vs = −h(x)(h(x)T h(x))−1 Lg V (x)

∂h g(x) ∂x

−1

∂h feq ∂x

∂h = 0. vs is not the equivalent control, i.e. the control when h(x) = 0. yields ∂t The equivalent control is veq = 0. However, vs ensures that the trajectories remain inside a neighbourhood of the sliding surface for future time. For the occurrence of a proper sliding mode, the trajectories may cross the sliding manifold repeatedly and remain in the boundary of the sliding manifold, and along this manifold they tend to an equilibrium point whenever the sliding zero dynamics of the system is stable. A control is designed next so that the trajectories tend to a neighbourhood of h(x) = 0, an attractive sliding region, and remain inside for future time. Consider the following control for the passive system (5) v = vs −Kh(x)(h(x)T h(x))−1 Lg V (x)

∂h g(x) ∂x

−1

∂h ∂h sgn ( )T h(x) (12) ∂x ∂x

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where K > 0 is a constant sliding mode gain. The control (12) enforces the system trajectories to the sliding manifold s = h(x) = 0, since ∂h x˙ sT s˙ = hT ∂x −1 ∂h ∂h ∂h T T ∂h =h g feq − g feq + Ksgn ( ) h ∂x ∂x ∂x ∂x ∂h ∂h sgn ( )T h ≤ 0 (13) = −KhT ∂x ∂x Substituting the control (12) into (5) yields −1 −1 ∂h ∂h ∂h ∂h ∂h x˙ = I − g g g sgn ( )T h feq − g ∂x ∂x ∂x ∂x ∂x

4

(14)

Sliding mode of linear systems with passivation

We consider a linear system for illustrative purposes in this section. It is wellknown that a strictly stable SISO controllable linear system is passive if and only if it is strictly positive real and these conditions are equivalent to the result of the Kalman-Yakubovich Lemma [12]. We do not intend to present an equivalent proposition to the above statement in this section. A new idea related to passivity and sliding mode is presented here. There are numerous methods for designing sliding mode control for linear systems. We design a sliding control using the feedback transformation (4) with a particular storage energy function. This method yields a control which diﬀers from traditional approaches. Consider the linear controllable canonical form     0 0 1 0 ... 0 0   0 0 1 . . . 0     x˙ =  . .. .. . . ..  x +  ..  u .  .. . . . .  −a0 −a1 −a2 . . . −an−1 y = c1 c2 . . . cn x

1

(15)

∂V = 0. We exclude the ∂xn choice V = 12 x2 because the equilibrium point of the system is 0 and the ∂V = 0. We will consider choice V = 12 x2 does not fulﬁl the condition ∂xn V = 12 s2 to assure the satisfaction of the transversality condition. Note that there are many choices to select the storage function V for linear system (15) so that V = 12 x2 . The control (4) with Ψ = 0 is now Let V (x) be a storage function, so that Lg V (x) =

u = a0 x1 +

n−1 i=1

(ai − ζi ) xi+1 +

y ∂V ∂xn

v

(16)

Adaptive feedback passivity of nonlinear systems with sliding mode

where ζi = The control  0 0  x˙ =  .  ..

∂V ∂xi

∂V / , ∂xn

187

i = 1, . . . , n − 1

(16) transforms the linear system (15) into    1 0 ... 0 0 0 0 1 ... 0  cx     ..  v .. .. . . ..  x + ∂V . . . . . 

(17)

∂xn

0 −ζ1 −ζ2 . . . −ζn−1

1 ∂V Note that if one selects the control w = cxv/ ∂x , then the system (17) n with the new control w, is in a Brunovsky controller form. So the feedback transformation (16) is able to transform the system (15) into a Brunovsky controller form without changing the state coordinates. Consider V (x) = 12 y 2 with ci > 0, i = 1, . . . , n. The control (16) is u = a0 x1 +

n−1 i=1

Then the system  0   0 1  x˙ =  . cn  ..

ci ai − cn

xi+1 +

1 v cn

(18)

is converted into

    0 0 ... 0 0  cn . . . 0      .. . . ..  x +  ..  v  .   . . . 1 0 −c1 −c2 . . . −cn−1 cn 0 .. .

(19)

Now a sliding control v is designed. Consider the sliding surface s = cx = 0. During the ideal sliding mode, s˙ = 0 which yields v = veq = 0. The new control v should be selected so that the system reaches a sliding mode in a ﬁnite time. Therefore v does not aﬀect the system during the sliding mode and inﬂuences only the reaching time of a sliding mode. In fact, the system (17) with v = 0, is the reduced order system. The reaching condition is cn sv < 0. Since cn > 0, one can select v = −(W s + Ksgn(s)) with W ≥ 0, K ≥ 0 and H + K = 0. This condition assures that the reaching condition is satisﬁed. The system in the sliding mode is   0 1 0 ... 0 0 0 1 ... 0    x .. .. .. . . .. (20) x˙ =  . .  . . .   c1 c2 cn−1 0− − ... − cn cn cn One eigenvalue of the reduced order system is zero and the n − 1 remaining eigenvalues are the roots of the equation cn z n−1 +cn−1 z n−2 +. . .+c2 z+c1 = 0. So for stability of the system it is required to select ci so that the auxiliary

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equation is Hurwitz. Since ci > 0, i = 1, . . . , n, the auxiliary equation is Hurwitz. If any of ci is nonpositive, the system (15) with control u is unstable despite the occurrence of a sliding mode. Then the zero dynamics with output y = cx is an unstable subsystem and therefore the linear system is a non-minimum phase system. A similar analysis can be undertaken for multivariable systems. 4.1

Example

Consider the simple pendulum ml2 θ¨ + kl2 θ˙ + mgl sin(θ) = T

(21)

where m, l, g, θ and k denote the mass of the bob, the length of the rod, the acceleration due to gravity, the angle subtended by the rod and the vertical axis, and the coeﬃcient of friction, respectively. T is a torque applied to the pendulum. It is desired to stabilize the pendulum at an angle α. Let x1 = θ − α,

x2 = θ˙

Then the system (21) can be written as x˙ 1 = x2 x˙ 2 = −a sin(x1 + α) − bx2 + hT

(22)

where a = g/l, b = k/m and h = 1/(ml2 ). The equilibrium point x ¯ 1 = 0, x ¯2 = 0 with Tf = a sin(α)/h is a steady value of T . Linearization of the system at the origin yields x˙ 1 = x2 x˙ 2 = −a sin(α)x1 − bx2 + hu

(23)

where u = T − Tf . Consider the sliding surface cx1 + x2 = 0 with c > 0. Now the control (16) is u = (a sin(α)x1 + (b − c)x2 + v) /h

(24)

The sliding control v for linear system (23) is v = −W (cx1 + x2 ) − Ksgn(cx1 + x2 )

(25)

where K ≥ 0, W ≥ 0 and K + W = 0. Substituting u (24) into the original nonlinear system (22) yields x˙ 1 = x2 x˙ 2 = −a sin(x1 + α) + a sin(α)x1 − cx2 + a sin(α) + v A sliding control v for the nonlinear system (26) is

(26)

Adaptive feedback passivity of nonlinear systems with sliding mode

189

. Angular velocity, θ

Angle, θ 0.7

0.8

0.6 0.6

0.5 0.4

0.4

0.3 0.2

0.2

0.1 0

0 0

1

2

3

0

1

Time

2

3

Time

Sliding function 0.5 0 −0.5 −1 −1.5 0

1

2

3

Time

Fig. 1. The responses of the system (22) with control (24)-(25)

. Angular velocity, θ

Angle, θ 0.7 0.8

0.6 0.5

0.6

0.4 0.4

0.3 0.2

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0 0

1

2

3

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0

1

2

3

Time

Sliding function 0.5 0 −0.5 −1 −1.5 0

1

2

3

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Fig. 2. The responses of the system (22) with control (24)-(27)

vn = a sin(x1 + α) − a sin(α)x1 − a sin(α) − W (cx1 + x2 ) − Ksgn(cx1 + x2 ) (27) The control (27) enforces the system (26) to the sliding mode. Since the system in the sliding mode, x˙ 1 + cx1 = 0, is stable, the control (25) with (24) stabilizes the system (23). Simulation results are shown in Figs. 1 and 2 for K = 2, W = 0, a = 2, b = 0.1, h = 1, c = 3, α = π/4. Note that the sliding control (25) was designed for the linear system (23) in which the

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control u is given by (24), while the sliding control vn (27) was designed for the nonlinear system (22). However, the behaviour of the nonlinear system (22) with controls (25) and (27), is quite similar as can be seen in Figs. 1 and 2.

5

Adaptive passivation of nonlinear systems

Now consider the parameterized SISO nonlinear system x˙ = f (x) + φ(x)θ + (g(x) + ψ(x)θ)u y = h(x)

(28)

where u ∈ U ⊆ R the control input and y ∈ Y ⊆ R the output. φ, ψ ∈ Rn×p are smooth functions and θ = [θ1 θ2 . . . θp ]T is a vector of unknown constant parameters. We obtain ˆ + (φ(x) + ψ(x)u)θ˜ x˙ = f (x) + φ(x)θˆ + (g(x) + ψ(x)θ)u

(29)

y = h(x) ˆ The nominal system of (29) is where θˆ is an estimate of θ and θ˜ = θ − θ. ˆ + gˆ(x, θ)u ˆ x˙ = fˆ(x, θ)

(30)

y = h(x) ˆ = f (x)+φ(x)θˆ and gˆ(x, θ) ˆ = g(x)+ψ(x)θ. ˆ According to Theorem with fˆ(x, θ) 1 it is a passive system via feedback transformation (4) by replacing f and g with fˆ and gˆ, respectively. Consider the Lyapunov function V and assume that the transversality condition Lgˆ V = 0 holds. The nominal passive system with new control input v is given by (5) by replacing fˆ and gˆ with f and g. Now consider the extended Lyapunov function 1 W = V + θ˜T Γ −1 θ˜ 2

(31)

where Γ is a positive deﬁnite matrix. Then ˆ˙ ˙ = ∂V x˙ + 1 θ˜T Γ −1 (−θ) W ∂x 2 1 ∂V ˆ˙ (φ + ψu) θ˜ + θ˜T Γ −1 (−θ) = hT v − Ψ + ∂x 2

T ∂V 1 ˜T −1 ˙ˆ T (φ + ψu) − θ Γ = h v−Ψ + θ Γ 2 ∂x The update estimate function is selected to be T ∂V ˙ T θˆ = Γ (φ + ψu) ∂x

(32)

(33)

Adaptive feedback passivity of nonlinear systems with sliding mode

191

and the last term of equation (32) is eliminated to yield ˙ = hT v − Ψ W Therefore the feedback transformation LTgˆ V (x) ∂V ˆ ˆ T u= − T f (x, θ) + h (x)v − Ψ (t, x, u, y) Lgˆ V (x)2 ∂x

(34)

(35)

transforms the system into LTgˆ V (x) ∂V LTgˆ V (x) T ˆ ˆ(x, θ) ˆ + gˆ(x, θ) ˆ x˙ = I − gˆ(x, θ) h (x)v f Lgˆ V (x)2 ∂xT Lgˆ V (x)2 LTgˆ V (x) ∂V ˆ ˆ T + φ(x) + ψ(x) − T f (x, θ) + h (x)v − Ψ (t, x, u, y) θ˜ Lgˆ V (x)2 ∂x ˆ −ˆ g (x, θ)

LTgˆ V (x) Ψ (t, x, u, y) Lgˆ V (x)2

(36)

This system is a passive system and the control v can be designed in many diﬀerent ways.

6

Example: Gravity-ﬂow/pipeline system

Consider the following gravity-ﬂow/pipeline system including an elementary static model for an “equal percentage value” [5] Ap g Kf 2 x2 − x L ρA2p 1 1 FCmax α−(1−u) − x1 x˙ 2 = At y = x2 x˙ 1 = −

(37)

with x1 x2 FCmax g L Kf ρ Ap At α u

: : : : : : : : : : :

volumetric ﬂow rate of liquid leaving the tank height of the liquid in the tank maximum value of the volumetric rate of ﬂuid entering the tank gravitational acceleration constant the pipe length friction of the liquid density of the liquid cross sectional area of the pipe cross sectional area of the tank rangeability parameter of the value control input, taking values in the closed interval [0, 1].

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The equilibrium point of the system (37) is X1 = FCmax α−(1−U ) ;

X2 =

LKf 2 X gρA3p 1

corresponding to a constant value U ∈ [0, 1]. The operating region of the system is R2+ . Using the auxiliary control w = FCmax α−(1−u) and assuming that the friction factor θ = Kf is unknown, the system (37) becomes Ap g 1 2 x2 − x θ L ρA2p 1 1 x˙ 2 = (w − x1 ) At y = x2 x˙ 1 = −

(38)

Assume θˆ is an estimate of θ. The system (38) can be expressed as follows Ap g 1 2ˆ 1 2 ˆ x2 − x θ− x (θ − θ) L ρA2p 1 ρA2p 1 1 x˙ 2 = (w − x1 ) At y = x2

x˙ 1 = −

and therefore A g − Lp x2 − ρA1 2 x21 θˆ ˆ ˆ p , f (x, θ) = − A1t x1

(39)

ˆ = gˆ(x, θ)

0 1 At

,

Φ=−

1 2 x ρA2p 1

It is desired that the states track the constant X = [X1, X2 ]T . The associated Lyapunov function is 1 ˆ2 Γ (40) (x1 − X1 )2 + x22 + (θ − θ) V = 2 x2 Assume that Lg V (x) = = 0. The update function is At 1 2 ˙ θˆ = −Γ x (x1 − X1 ) ρA2p 1 and the passive control is At (x1 − X1 ) Ap g 1 2ˆ At Ψ x2 − w=− x θ + x1 + At v − x2 L ρA2p 1 x2

(41)

(42)

The control v can be designed in various ways. Since the zero dynamics of the system is stable, the passive system is also stable regardless of selection

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of the control v. The control v can be designed by the sliding mode technique which has been discussed in Section 3. Using (12), the control v is given by 1 2ˆ x1 − X1 Ap g Ψ x2 − 2 x1 θ + − Ksgn (x2 − X2 ) (43) v= x2 l Ap ρ x2 Favourable simulation results are shown in Fig. 3 for g = 9.81, l = 914, ρ = 998, At = 10.5, Ap = 0.653, α = 9.3, FCmax = 2.5, K = 0.1, Γ = 10.2 and θ = 4.4739. The desired equilibrium for U = 0.9 is X1 = 2, X2 = 6. Since the unknown parameter θ is positive, the value of θ does not aﬀect the stability of the zero dynamics of the system. Therefore, the control w = x1 − β(x2 − X2 ) − Ksgn (x2 − X2 )

(44)

with β > 0 and K ≥ 0, stabilizes the system even if K = 0. Note that the control (42) with (43) yields the same structure as (44). However, the control v in (42) can be designed in many diﬀerent ways. This example is a particular case because the control u controls only the state x2 . With output y = x1 , the volumetric ﬂow rate of the liquid leaving the tank, the system does not have zero dynamics and this method yields the appropriate control with a suitable estimate of the unknown parameter. So this method is applicable for the systems with or without zero dynamics with a given relative degree. Volumetric flow rate of liquid leaving the tank 2.2

Height of liquid 7 6 5

1.6

4

x1

x2

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Estimate of friction factor

200 Time

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u

^θ

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0.7

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300

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Fig. 3. A gravity-ﬂow tank/pipeline

7

Conclusions

Passivity based control has been studied for aﬃne nonlinear systems. In this method, the system is transformed into a new system, the so-called passive

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system via control feedback. Sliding mode control has been designed for the passive system. The system is ﬁrst converted into a passive system with a new auxiliary control, and then sliding mode control (the auxiliary control) is designed. The method has been studied to design a sliding mode control for a linear system as a special case. Feedback passivation and suitable parameter estimates have been described for a class of parameterized nonlinear systems. A gravity-ﬂow/pipeline system has been presented to demonstrate the results.

References 1. Byrnes, C. I., Isidori, A., Willems, J. C. (1991) Passivity, feedback equivalence and global stabilization of minimum phase nonlinear systems. IEEE Trans. Automat. Control, 36, 1228–1240 2. Hill, D., Moylan, P. (1976) The stability of nonlinear dissipative systems. IEEE Trans. Automat. Control, 21, 708–711 3. Hill, D. and Moylan, P. (1997) Stability results for nonlinear feedback systems. Automatica, 13, 377–382 4. R´ıos-Bol´ıvar, M., Acosta-Contreras, V., Sira-Ram´irez, H. (2000) Adaptive passivation of a class of uncertain nonlinear system. Proc. 39th Conference on Decision and Control, Sydney 5. R´ıos-Bol´ıvar, M., Zinober, A. S. I. (1999) Dynamical adaptive sliding mode control of observable minimum phase uncertain nonlinear systems. In: “Variable Structure Systems: Variable structure systems, sliding mode and nonlinear ¨ uner), Springer-Verlag, London, 211-236 control, ( Editors: Young and Ozg¨ 6. Ortega, R. (1991) Passivity properties for stabilizing of cascaded nonlinear systems. Automatica, 27, 423–424 7. Ortega, R., Spong, M. (1989) Adaptive motion control of rigid robots: a tutorial. Automatica, 25, 877–888 8. Ortega, R., Nicklasson, P., Sira-Ram´ırez, H. (1998) The Passivity Based Control of Euler-Lagrange Systems. Springer-Verlag, London 9. Sepulchre, R., Jankovi´c, M., Kokotovi´c, P. (1997) Constructive Nonlinear Control. Springer-Verlag, London 10. Sira-Ram´ırez, H. (1998) A general canonical form for feedback passivity of nonlinear systems. Int. J. Control, 71, 891–905 11. Sira-Ram´ırez, H., R´ıos-Bol´ıvar, M. (1999) Feedback passivity of nonlinear multivariable systems. Proc. 14th World Congress of IFAC, Beijing, 73–78 12. Slotine, J.-J. E., Li, W. (1991) Applied nonlinear control. Prentice Hall, London 13. Willems, J. C.(1971) The analysis of feedback systems. MIT Press, Cambridge, MA

Linear algebraic tools for discrete-time nonlinear control systems with Mathematica ¨ Kotta and M. T˜onso U. Institute of Cybernetics, Tallinn Technical University, Akadeemia tee 21, 12618 Tallinn, Estonia tel: +372 620 4153, fax: +372 620 4151, email: [email protected], maris@staﬀ.ttu.ee

Abstract. This paper presents a contribution to the development of symbolic computation tools for discrete-time nonlinear control systems. A set of functions is developed in Mathematica 4.0 on the basis of linear algebraic approach that allows the solution of several modelling, analysis and synthesis problems. In all these problems, a certain sequence of subspaces, associated to a control system and based on the classiﬁcation of one-forms according to their relative degree, provides the solution.

1

Introduction

The paper describes an attempt to building a software package using the (linear) algebraic approach [1] to provide the basic tools for modelling, analysis and synthesis of nonlinear discrete-time systems. The characteristics which make the linear algebraic approach useful, are its wide applicability (with proper modiﬁcations it applies both in discrete-time and continuous-time cases), inherent simplicity (the procedures are transparent and most of them can be straightforwardly implemented in Mathematica) and perhaps, most importantly, its universality. The latter is demonstrated in the paper by a certain sequence of the subspaces associated to the control system, and based on the classiﬁcation of one-forms according to their relative degree. This single tool provides the key for solving seven diﬀerent control problems and a software package in Mathematica is developed around this tool. It includes procedures for • checking (forward) accessibility and decomposing the state equations into the accessible and the non-accessible subsystems • transforming the system equations into the normal form which is a good starting point for applying the inversion-based control algorithms and for computing zero dynamics • transforming the state equations into the controller canonical form and static state feedback linearization (full and partial) • transforming the state equations into the prime form, using the state feedback, state and output diﬀeomorphisms A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 195-205, 2003. Springer-Verlag Berlin Heidelberg 2003

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• checking irreducibility of the higher order input-output (i/o) diﬀerence equations and reducing them into the equivalent minimal form, checking the (transfer) equivalence of two or more systems • ﬁnding the minimal realization of the i/o equation in the classical state space form • transforming the generalized state equations into the classical state space form or lowering the order of the input shifts in the generalized equations The implementation of the last three problems has been described elsewhere [2,3] and will be not repeated here. In [4] the Maple 5 procedures have been described for checking accessibility and static state feedback linearizability conditions. However, the procedures, described in [4] do not allow to ﬁnd the state transformation as well the feedback which makes the closed-loop system linear. The basic ingredients of our procedures are (i) ﬁnding the backward shift operator, (ii) checking integrability and (iii) integration the set of integrable one-forms. Though it is easy to check the integrability property with the help of the Frobenius theorem, it can be extremely diﬃcult to ﬁnd the integrable basis and to integrate the required one-forms to ﬁnd the state transformation even for medium-size, medium-complexity examples. The reason is that Mathematica has almost no facilities for those tasks. To improve the capabilities of Mathematica in this respect, we have implemented an additional function IntegrateOneForms. This function replaces integration of the set of one-forms by solution of the sequence of linear homogeneous PDEs.

2

Linear algebraic framework

Consider a discrete-time single-input single-output nonlinear system Σ described by the equation x(t + 1) = f (x(t), u(t)),

y(t) = h(x(t))

(1)

where u ∈ U ⊂ IRm is the input variable, y ∈ Y ⊂ IRp is the output variable, x ∈ X, an open subset of IRn , is the state variable, f : X × U → X and h : X → Y are the real analytic functions. In order to be able to use mathematical tools from the linear algebraic framework, we assume that the following assumptions hold generically for system (1) throughout the paper: A1 f (x, u) is a submersion, i. e. rank[ ∂f (x, u)/∂(x, u)] = n, A2 rank[∂f /∂u] = m A3 rank dh = p. We follow the notation of [5]. Let K denote the ﬁeld of meromorphic functions in a ﬁnite number of variables {x(0), u(t), t ≥ 0}. The forward-shift operator δ : K → K is deﬁned by δζ(x(t), u(t)) = ζ(f (x(t), u(t)), u(t + 1)). Under A1 the pair (K, δ) is a diﬀerence ﬁeld [5], and up to an isomorphism, there exists a unique diﬀerence ﬁeld (K∗ , δ ∗ ), called the inversive closure of (K, δ), such that K ⊂ K∗ , δ ∗ : K∗ → K∗ is an automorphism and the

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restriction of δ ∗ to K equals δ. In [5] an explicit construction of (K∗ , δ ∗ ) is given, and in particular, an algorithm to compute the backward shift operator δ −1 : K → K. By abuse of notation, hereinafter we use the same symbol to denote (K, δ) and its inversive closure. Over the ﬁeld K one can deﬁne a diﬀerence vector space E := spanK {dϕ | ϕ ∈ K}. The operators δ and δ −1 induce the forward-shift operator ∆ : E → E and the backward-shift operator ∆−1 : E → E by ai dϕi → δai d(δϕi ), ai , ϕi ∈ K i

i

i

ai dϕi →

(δ −1 ai )dδ −1 ϕi ), ai , ϕi ∈ K.

i

The relative degree r of a one-form ω ∈ X = spanK { dx(0)} is deﬁned to spanK {dx(0)}. If such an integer does be the least integer such that ∆r ω ∈ not exist, we set r = ∞. A sequence of subspaces {Hk } of E is deﬁned for k ≥ 1 by [5]: H1 = spanK {dx(0)}, Hk+1 = spanK {ω ∈ Hk | ∆ω ∈ Hk }.

(2)

The Hk ’s can alternatively be deﬁned by Hk+1 = ∆−1 (Hk ∩ ∆Hk ), k ≥ 1.It is clear that sequence (2) is decreasing. Denote by k ∗ the least integer such that H1 ⊃ H2 ⊃ . . . Hk∗ ⊃ Hk∗ +1 = Hk∗ +2 = . . . = H∞ The subspaces Hk contain the one-forms whose relative degree is equal to k or greater than k, and are proved to be invariant under the state space diﬀeomorphism, and the regular static state feedback. The sequence of subspaces {Hk } plays a key role in the solution of the seven control problems, listed in the introduction.

3 3.1

Solutions of the control problems in terms of the Hk subspaces Accessibility and the accessible subspace

Theorem 1. [5] The following statements are equivalent: (i) The nonlinear system is strongly accessible (ii) H∞ = {0} H∞ contains the one-forms with inﬁnite relative degree so that these oneforms will never be inﬂuenced by the input of the system. For system (1), the subspace H∞ is a nonaccessible subspace, and the factor space Xa := X /H∞ such that Xa ⊕ H∞ = X precisely describes the accessible part of the system. Although Hk are, in general, not completely integrable, i. e. they do not admit the basis which consists only of closed forms, the limit H∞ turns out to be completely integrable [5]. There exist locally r functions, say ζ1 , . . . , ζr with inﬁnite relative degree so that H∞ = spanK {dζ1 , . . . , dζr }. Since H∞ is invariant under applying forward shift operator, one has in particular

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ζ 1 (t + 1) = f 1 (ζ 1 (t)) for some function f 1 and ζ 1 = (ζ1 , . . . , ζr ). Now, since Xa is also completely integrable, choosing ζ 2 = (ζr+1 , . . . , ζn ) from Xa = spanK {dζr+1 , . . . , dζn }, we get the accessible subsystem ζ 2 (t + 1) = f 2 (ζ(t), u(t)).

Example 1 f = {x1 [t](x3 [t]2 + 1)2 , x2 [t](x3 [t]2 + 1)3 , x3 [t] + u[t]}; Xt = {x1 [t], x2 [t], x3 [t]}; Ut = {u[t]}; stEq = StateSpace[f , Xt, Ut, t]; Accessibility[stEq] False newEq = AccessibilityDecomposition[stEq, ζ# [t]&]; EquationForm[PowerExpand[newEq]] ζ1 [1 + t] == −(−1)1/3 ζ1 [t] ζ2 [1 + t] == ζ2 [t](1 + ζ3 [t]2 )2 ζ3 [1 + t] == u[t] + ζ3 [t] 3.2

Normal form

The system description in the so-called normal form ζi1 (k + 1) = ζi2 (k) .. . ζi,ri −1 (k + 1) = ζi,ri (k + 1) = η(k + 1) = yi (k) =

ζi,ri (k) Ψi (ζ(k), η(k), u(k)), γ(ζ(k), η(k), u(k)) ζi1 (k)

(3) i = 1, . . . , p

is a convenient starting point for applying several control algorithms like output tracking, stabilization etc. Unlike the continuous-time case, the condition ∂γ(ζ(k), η(k), u(k))/∂u(k) = 0.

(4)

is, in general, not satisﬁed in the discrete-time case which will imply that the zero dynamics cannot be deﬁned independently of the applied control. Under the condition that H2 is completely integrable [6], new state coordinates can be found such that (4) is satisﬁed. Note that H2 is the subspace of one-forms whose relative degrees are greater than or equal to two, which implies that their time-shift does not depend on control. To ﬁnd the required state coordinates, note that the basis of H2 can be chosen as (recall that dim H2 = n − p) H2 = spanK {dζi1 (0), . . . , dζi,ri −1 (0), i = 1, . . . , p, ω1 (0), . . . , ωn−r1 −...−rp (0)} and if H2 is integrable we can take dηi (t) = ωi (t), i = 1, . . . , n − r1 − . . . − rp .

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Example 2 f = {x3 [t], u1 [t]u2 [t] + x2 [t], x1 [t]x4 [t] + u2 [t], x5 [t] + u1 [t], x3 [t]x5 [t]}; Xt = {x1 [t], x2 [t], x3 [t], x4 [t], x5 [t]}; Ut = {u1 [t], u2 [t]}; h = {x1 [t], x2 [t]}; Yt = {y1 [t], y2 [t]}; stEq = StateSpace[f , Xt, Ut, h, Yt]; TransFormabilityToNormalForm[stEq] False 3.3

Controller canonical form and static state feedback linearization

The nonlinear controller canonical form zi1 (t + 1) = zi2 (t) .. . ziki −1 (t + 1) = ziki (t) ziki (t + 1) = f˜i (z(t), u(t)),

(5) i = 1, . . . , m

can be deﬁned in analogy to the corresponding linear form. Equations (5) can be linearized by a regular static state feedback u(t) = ϕ(z(t), v(t)),

rankK [∂ϕ(·)/∂v] = m

(6)

into the linear controllable system. The applicability of this design method is the characteristic property of the nonlinear controller canonical form. System (1) is said to be linerizable by static state feedback (6) if there exist a state diﬀeomorphism z(t) = Φ(x(t))

(7)

and a regular static state feedback, such that in new coordinates, the compensated system reads (5) with f˜i (·) = vi (t). Theorem 2. [5] System (1) is transformable via state transformation into the controller canonical form (linearizable by regular static state feedback) iﬀ (i) H∞ = {0}, (ii) For 1 ≤ k ≤ k ∗ , Hk is completely integrable So, in the nonlinear case, not every accessible system can be transformed into controller canonical form. Calculation of the Hk subspaces yields a set of one-forms which deﬁne the inﬁnitesimal Brunovsky canonical form. In order to transform the system into the controller canonical form, the one-forms must be integrated. The latter is possible only in case when all Hk ’s are completely integrable.

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Suppose H∞ = {0}. Let Wk∗ be a basis for Hk∗ . By deﬁnition, Wk∗ and ∆Wk∗ are in Hk∗ −1 . Since Wk∗ and ∆Wk∗ are linearly independent, it is always possible to choose a set (possibly empty) Wk∗ −1 such that Wk∗ ∪ ∆Wk∗ ∪ Wk−1 is a basis for Hk∗ −1 . Repeating this procedure k ∗ − 1 times we obtain {ω1 (0), . . . , ωm (0)} = Wk∗ ∪ . . . ∪ W1 . Let ωi = dϕi (recall that by complete integrability of Hk ’s, ωi can be chosen to be exact) and deﬁne zij = δ j−1 ϕi (x), 1 ≤ i ≤ m, 1 ≤ j ≤ ki = {k | ωi ∈ Wk } In coordinates zij system (1) becomes as (5). The fact that the forms ∆j wi are independent implies that the matrix [∂ f˜i /∂uj ]ij has full rank. Therefore, the static state feedback (6) where f˜i (z(t), ϕ(z(t), v(t))) = vi (t), i = 1, . . . , p, brings the system into the linear form. When system (1) cannot be fully linearized using static state feedback, it is possible to characterize the largest feedback linearizable subsystem in terms of the sequence of subspaces Hk and in terms of its bottom derived system [5]. The function PartialLinearization has been implemented but will be not described neither demonstrated here because the lack of space.

Example 3 f = {u1 [t], x3 [t] + u1 [t], x4 [t], x7 [t]u2 [t], x6 [t]u1 [t], u2 [t], x5 [t]u3 [t], x3 [t] + x8 [t]}; Xt = #[t]&/@{x1 , x2 , x3 , x3 , x5 , x6 , x7 , x8 }; Ut = #[t]&/@{u1 , u2 , u3 }; stEq = StateSpace[f , Xt, Ut, t]; Linearizability[stEq] True lin = Linearization[stEq, z# [t]&]; EquationForm[lin] z1 [1 + t] == z2 [t] z2 [1 + t] == z3 [t] z3 [1 + t] == z4 [t] z4 [1 + t] == v1 [t] z5 [1 + t] == z6 [t] z6 [1 + t] == z7 [t] z7 [1 + t] == v2 [t] z8 [1 + t] == v3 [t] v1 [t] → z4 [t] + u2 [t]z7 [t] u3 [t](−z2 [t] + z3 [t])z8 [t] v2 [t] → z5 [t] v3 [t] → u1 [t] 3.4

Equivalence to prime systems

The system (1) is said to be equivalent to the prime system if there exist a state space diﬀeomorphism (7), a regular static state feedback (6) and an

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output space diﬀeomorphism y˜ = Ψ (y) such that in the new coordinates the compensated system reads (5) with fi (·) = vi (t) and with the output equation yi (t) = zi1 (t), 1 ≤ i ≤ m. Deﬁne F0 = spanK {dh(x(0))}. Theorem 3. [7] System (1) is equivalent to prime system iﬀ the following conditions are satisﬁed: 1. 2. 3. 4.

H∞ = {0}; For k = 1, . . . , k ∗ , Hk is completely integrable; For k = 1, . . . , k ∗ , Hk = Hk ∩ F0 + ∆Hk+1 ; For k = 1, . . . , k ∗ , Hk ∩ F0 is completely integrable.

If compared to the feedback linearization problem, the crucial step in the equivalence to prime system is the requirement that Hk ’s can be (and should be) constructed using only one-forms deﬁned by the output functions and their forward time-shifts. In particular, the forms ω1 , . . . , ωm in Wk∗ ∪. . .∪W1 can be written as a linear combination of dhj (x)   m m ωi = aij (h(x))dhj (x) = h∗  aij (y)dyj  = h∗ (ηi ). j=1

j=1

The output diﬀeomorphism y˜ = Φ(y) is deﬁned by dΨi (y) = ηi (note that by complete integrability ηi can be chosen to be exact). Example 4 (state equations are same as in Example 3) h = {x1 [t] + x8 [t], x2 [t], x1 [t]x3 [t]/x5 [t]}; Yt = #[t]&/@{y1 , y2 , y3 }; stEq = StateSpace[f , Xt, Ut, t, h, Yt]; EquivalenceToPrimeSystem[stEq, PrintInfo → True] True ˜# [t]&]; {pf , rules} = PrimeForm[stEq, ξ# [t]&, v# [t]&, y EquationForm[pf ]; TableForm[rules] ξ1 [1 + t] == ξ2 [t] ξ2 [1 + t] == ξ3 [t] ξ3 [1 + t] == ξ4 [t] ξ4 [1 + t] == v1 [t] ξ5 [1 + t] == ξ6 [t] ξ6 [1 + t] == ξ7 [t] ξ7 [1 + t] == v2 [t] ξ8 [1 + t] == v3 [t] y2 [t] == ξ5 [t] ~ y3 [t] == ξ8 [t] y1 [t] == ξ1 [t] ~ ~ y1 [t] → y1 [t] − y2 [t] ~ ~2 [t] → y3 [t] y y3 [t] → y1 [t] ~ v1 [t] → ξ4 [t] + u2 [t]ξ7 [t] u3 [t](ξ2 [t] + ξ3 [t])(ξ2 [t] − ξ8 [t]) v2 [t] → ξ5 [t] v3 [t] → u1 [t] + ξ3 [t]

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Example 5 (same as example 2) EquivalenceToPrimeSystem[stEq,PrintInfo → True] The subspace H2 is not completely integrable. Holds inequality H2 = H2 ∩ F0 + ∆H3 Holds inequality H3 = H3 ∩ F0 + ∆H4 False

4 4.1

Symbolic implementation using mathematica Problems related to the computation of the backward shift

The main ingredient of all our procedures is the calculation of the sequence of the subspaces Hk , associated with the control system. For explicit construction of the subspaces Hk we need to calculate the backward shift. For that purpose we have developed Mathematica function BackwardShift which relies on the use of Mathematica function Solve. Note that although the choice of a function deﬁning δ −1 is not unique, each possible choice brings up a ﬁeld extension that is isomorphic to K∗ /K. Of course, diﬀerent choices may aﬀect the complexity of computations, and the proper choice is not straightforward. In order to avoid the calculation of the backward shift one can use in some problems instead of the sequence of subspaces {Hk } a related sequence of decreasing subspaces {Ik } introduced in [4] and deﬁned by I1 = spanK {dx(0), Ik+1 = Ik ∩ ∆Ik , k ≥ 1. Lemma 1. Ik = ∆k−1 Hk and I∞ = H∞ . Ik is completely integrable iﬀ Hk is completely integrable. The drawback of the sequence of new subspaces Ik if compared to Hk ’s is that the sequence {Ik } can be used only to check the necessary and suﬃcient conditions about the existence of state and output transformation and feedback, but not for their construction. For that purpose, one needs the sequence {Hk }. 4.2

Integrability conditions

We will say that ω ∈ E is an exact one-form if ω = dF for some F ∈ K. A one-form ν for which dν = 0 is said to be closed. It is well-known that exact forms are closed, while closed forms are only locally exact. It is easy to check the integrability property with the help of the Frobenius theorem and the latter can be straightforwardly implemented in Mathematica. Theorem 4. (Frobenius). Let V = spanK { ω1 , . . . , ωs } be a subspace of E. V is closed if and only if dωi ∧ ω1 ∧ . . . ∧ ωs = 0, for any i = 1, . . . , s.

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203

Problems related to ﬁnding the integrating factor and integrating the one-forms

The third important ingredient of all the procedures is to ﬁnd the integrable basis of certain subspace (which exists if this subspace is completely integrable), and to integrate the integrable one-forms. Assume that we have computed the subspace Hs+2 = spanK {ω1 , . . . , ωn } and Frobenius condition is satisﬁed for one-forms ω1 , . . . , ωn . As a next step we try to transform the basis vectors into the set of exact one-forms. There is no hope to ﬁnd a general solution by solving a system of PDE’s. Note also that Hs+2 is a vector space, so we can change the basis by linear transformations: add one element to the other and multiply an element with an arbitrary expression diﬀerent from zero. Integrating the set of one-forms is a very diﬃcult problem and in Mathematica almost no ready to use functions are available for the above problem. To improve the capabilities of Mathematica in this respect, we have implemented an additional function IntegrateOneForms which bases on the algorithm described below [8]. Despite this extension, the solution of the set of the partial diﬀerential equations is still often unsuccessful. Consider a set of linearly independent one-forms ω1 , . . . , ωn−s (1 ≤ s ≤ n − 1) in n-dimensional diﬀerence vector space ω1 ≡

a11 dx1 + a12 dx2 + . . . + a1n dxn .. .

=0

dx1 + an−s dx2 + . . . + an−s dxn = 0 ωn−s ≡ an−s n 1 2 Assume that the variables x1 , . . . , xs are independent and xs+1 , . . . , xn are the unknown functions of x1 , . . . , xs , i. e. s+1 s+1 dxs+1 = bs+1 dxs 1 dx1 + b2 dx2 + . . . + bs .. .

(8)

dxn = bn1 dx1 + bn2 dx2 + . . . + bns dxs , where the coeﬃcients bji are the functions of x1 , . . . , xn . In case the set of one-forms ω1 , . . . , ωn−s satisﬁes the Frobenius condition, one can solve (8) for xs+1 , . . . , xn . System (8) is equivalent to a following system of PDE-s: X1 (f ) ≡

∂f ∂x1

∂f n ∂f + bs+1 1 ∂xs+1 + . . . + b1 ∂xn = 0

.. . Xs (f ) ≡

∂f ∂xs

(9)

∂f n ∂f + bs+1 s ∂xs+1 + . . . + bs ∂xn = 0

where f is an unknown function. Let ϕs+1 , . . . , ϕn be any n − s algebraically independent solutions of the ﬁrst PDE. Deﬁne new variables yi as follows:

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ϕs+1 (x1 , . . . , xn ) = ys+1 , . . . , ϕn (x1 , . . . , xn ) = yn . It is possible to show that the new unknowns ys+1 , . . . , yn satisfy the following system of equations: dys+1 = X1 (ϕs+1 )dx1 + X2 (ϕs+1 )dx2 + . . . + Xs (ϕs+1 )dxs .. .

(10)

dyn = X1 (ϕn )dx1 + X2 (ϕn )dx2 + . . . + Xs (ϕn )dxs Since X1 (ϕs+i ) = 0, i = 1, . . . , n − s, the coeﬃcients of dx1 are zeros in the above system. The other coeﬃcients do not depend on x1 . If the variable x1 does not vanish, the one-forms ω1 , . . . , ωs do not satisfy the Frobenius condition. Thus, we have got a new system s+1 dxs dys+1 = cs+1 1 dx2 + . . . + cs .. .

(11)

dyn = cn2 dx2 + . . . + cns dxs , where cji = cji (x2 , . . . , xs , ys+1 , . . . , yn ). The last system has the same form as (8), but the number of variables has decreased. We can repeat the process of reducing the variables until we reach the system dxs , . . . , dun = hns dxs dus+1 = hs+1 s

(12)

where hji = hji (xs , us+1 , . . . , un ). The last system has n − s solutions, depending on xs and n − s variables, deﬁned on the previous step. Finally, we have to substitute variables step-by-step backward: u → . . . → y → x. The implementation of the above algorithm requires solving a single PDE which is the most sensitive step and may cause the failure of the program.

References 1. Conte G., Moog C. H., Perdon A. M. (1999) Nonlinear Control Systems. Lecture Notes in Control and Inf. Sci. Springer, London ¨ T˜ 2. Kotta U., onso M. (1999) Transfer Equivalence and Realization of Nonlinear Higher Order Input/Output Diﬀerence Equations Using Mathematica. J of Circuits, Systems and Computers, 9:23–35 ¨ T˜ 3. Kotta U., onso M. (2000) On the Implementation of a Nonlinear Realization Algorithm Using Mathematica. In: First Conference on Control and Selforganization in Nonlinear Systems at Bialystok, 21–22 4. Aranda-Bricaire E. (1996) Computer Algebra Analysis of Discrete-Time Nonlinear Systems. In: the 13th IFAC World Congress at San Fransisco, USA, 306–309 ¨ Moog C. (1996) Linearization of Discrete-Time 5. Aranda-Bricaire E., Kotta U., Systems. SIAM J Control and Optimization 34:1999–2023

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¨ (2000) Comments on “On the Discrete-Time Normal Form”. IEEE 6. Kotta U. Transactions on Automatic Control 45:2197 ¨ (1998) Equivalence of Discrete-Time Nonlinear 7. Aranda-Bricaire E., Kotta U. Systems to Prime Systems. J of Mathematical Systems, Estimation, and Control, 8:471–474 (Summary), Full manuscript published electronically, 12pp, 387 054 bytes. Retrieval code: 53142 8. G¨ unter N. M. (1934) Integration of the First Order Partial Diﬀerential Equations. Gosudarstvennoe Tehniko-Teoreticheskoe Izdatelstvo, Leningrad, Moskva

Adaptive compensation of multiple sinusoidal disturbances with unknown frequencies Riccardo Marino1 , Giovanni L. Santosuosso2 and Patrizio Tomei3 1

2

3

Dipartimento Ingegneria Elettronica, Universit`a di Roma ‘Tor Vergata’, Via di Tor Vergata 110, 00133 Roma, Italy; [email protected] Dipartimento Ingegneria Elettronica, Universit`a di Roma ‘Tor Vergata’, Via di Tor Vergata 110, 00133 Roma, Italy; [email protected] Dipartimento Ingegneria Elettronica, Universit`a di Roma ‘Tor Vergata’, Via di Tor Vergata 110, 00133 Roma, Italy; [email protected]

Abstract. It is addressed the problem of designing an output feedback compensator which drives to zero the state of a system aﬀected by two additive noisy biased sinusoidal disturbances with unknown bias, magnitudes, phases and frequencies. The problem is solved for a linear, asymptotically stable, observable system of order n with known parameters by a [3n + 15] -order compensator. The regulating scheme contains exponentially convergent estimates of the biased sinusoidal disturbances and of its parameters, including frequencies. The algorithm is generalized to the case of an arbitrary number m of sinusoidal disturbances, with unknown parameters, yielding a [n(m + 1) + 2m2 + 3m + 1]-order compensator.

1

Introduction

In the last decades the problem of the adaptive compensation of sinusoidal disturbances has attracted a considerable attention in the control community. This problem is of interest in active noise and vibration control ([5], [14], [26]), torque ripple cancellation in hybrid step motors (see [6]), eccentricity compensation ([23], [9], [3]) and in adaptive regulation when the signal to be tracked yr is a biased sinusoid with unknown frequency. An early approach to the problem, if the frequencies of the disturbances are known, is based on the internal model principle (see [11], [8], [7]) according to which the deterministic exogenous signals are treated as outputs of homogeneous dynamic exosystems excited by nonzero initial conditions. In this framework, the disturbance eﬀect on the plant response is completely eliminated if the exosystem model is suitably reduplicated in the feedback path of the closed loop system. In recent years the regulation of linear systems in the case of an unbiased sinusoidal disturbance with unknown frequency, no additive noise has been studied in a series of papers (see [4], [23], [2], [5], and [10]). In particular in [2] two schemes (a direct one and an indirect one) are presented and analyzed: while the direct scheme is local in the initial condition of the frequency estimate, the indirect one (see also [10] for a recent improvement) allows for larger initial conditions; on the other hand only the direct scheme guarantees exact disturbance compensation. Related work can be found in A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 207-226, 2003. Springer-Verlag Berlin Heidelberg 2003

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[18], [19], [20], in which the system transfer function is supposed to be minimum phase with known coeﬃcients in [18], and unknown coeﬃcients in [19] (see [20] for extensions to nonlinear systems), and in [24] where the regulation problem is addressed for minimum phase nonlinear systems with unknown parameters in the exosystem. In this paper it is considered the linear single-input, single-output, observable system     −an−1 1 · · · 0 bn−1  .. .. . . ..   .   . . . x˙ =  .  x +  ..  (u + δ)  −a1 0 · · · 1  (1) b0 −a0 0 · · · 0 = F x + g(u + δ) y = 1 0 · · · 0 x = hx in which x ∈ n is the state, u ∈ 1 is the control input, δ ∈ 1 is a modelled disturbance; the output y ∈ 1 , the only measured variable, is to be regulated to zero. The disturbance input δ(t) is modelled as two biased sinusoids δ(t) = A0 + A1 sin(ω1 t + φ1 ) + A2 sin(ω2 t + φ2 )

(2)

with unknown constant magnitude Ai > 0, i = 1, 2, frequency ωi > 0, phase φi , and bias A0 . We assume that: (H1) all coeﬃcients ai , bi , 0 ≤ i ≤ n − 1 are known; (H2) the system is asymptotically stable, that is the polynomial a(s) = sn + an−1 sn−1 + ... +a1 s + a0 has all its roots with negative real part; (H3) b(s) = bn−1 sn−1 + ... +b1 s + b0 has no roots on the imaginary axis, so that the system is allowed to be non-minimum phase. We pose the global adaptive disturbance compensation problem, that is the design of a dynamic output feedback control which, for any initial condition of the system and of the compensator and for any unknown constant values A0 , Ai , ωi , φi , i = 1, 2 guarantees that the output y(t) along with the state x(t) are driven exponentially to zero. In this paper a compensator of order (3n + 15) is proposed which solves the posed problem by using the adaptive observers developed in [15], [17]. In particular, the unknown biased sinusoidal disturbance (2) and its constant parameters are exponentially estimated. The design procedure is generalized to the case of m additive sinusoids with unknown frequencies yielding a [n(m + 1) + 2m2 + 3m + 1]-order compensator. The paper is organized as follows; in Section 2 the adaptive compensation algorithm for two biased sinusoids it is described; in Section 3 is shown the global asymptotic stability of the overall closed loop error system and in Section 4 the dynamic regulation scheme is generalized to an arbitrary number of sinusoidal disturbances.

Adaptive compensation of sinusoidal disturbances

2

209

Adaptive compensation design

The biased sinusoidal disturbance δ(t) = A0 + A1 sin(ω1 t + φ1 ) + A2 sin(ω2 t + φ2 ) is modeled by the ﬁfth order linear exosystem w˙ 1 w˙ 2 w˙ 3 w˙ 4 A˙ 0 δ

= = = = = =

w2

− ω12 + ω22 w1 + w3 −θ1 w1 + w3 w4

− ω12 ω22 w1 −θ2 w1 0 A0 + w1

(3)

with θ1 = ω12 + ω22 > 0, θ2 = ω12 ω22 > 0 unknown positive parameters and A0 (0) = A0 , w1 (0) = A1 sin φ1 +A2 sin φ2 , w2 (0) = A1 ω1 cos φ1 +A2 ω2 cos φ2 , w3 (0) = A1 ω22 sin φ1 + A2 ω12 sin φ2 , w4 (0) = A1 ω22 ω1 cos φ1 + A2 ω12 ω2 cos φ2 , unknown initial conditions. The overall system (1), (3) is rewritten as 

   bn−1 0 0 0 bn−1 bn−1    .. .. .. .. ..   x   ..  x˙ F   . . . . .     .   w˙ 1   w    1 b0 0 0 0 b0     b0       w2    w˙ 2   0 100 0   + 0   =0  u  w3    w˙ 3      0     0 −θ1 0 1 0 0     w4    w˙ 4   0  0 001 0  0     A˙ 0  0 −θ2 0 0 0 0  A0  0  0 0 000 0 0 T y = 1 0 · · · 0 xT , w1 , w2 , w3 , w4 , A0

(4)

which is observable for every θ1 > 0, θ2 > 0, since by assumption (H3) b(s) has no roots on the imaginary axis. The observable system (4), is transformed into the observer canonical form

2 ai y + ¯bi u ¯0 y + ¯b0 u + i=1 θi −¯ ζ˙ = Ac ζ − a y = Cc ζ with 

0  ..  Ac =  . 0 0

1 ··· .. . . . . 0 ··· 0 ···

 0 ..  . , 1 0

Cc = 1 0 · · · 0 ,

(5)

210

R. Marino, G.L. Santosuosso, and P. Tomei T

a¯0 = [an−1 , . . . , a0 , 0, 0, 0, 0, 0] , T

a ¯1 = [0, 1, an−1 , . . . , a0 , 0, 0, 0] , T

a ¯2 = [0, 0, 0, 1, an−1 , . . . , a0 , 0] , ¯b0 = [bn−1 , . . . , b0 , 0, 0, 0, 0, 0]T , ¯b1 = [0, 0, bn−1 , . . . , b0 , 0, 0, 0]T , ¯b2 = [0, 0, 0, 0, bn−1 , . . . , b0 , 0]T ,

by the linear transformation (which is nonsingular for all θ1 > 0, θ2 > 0)

ζ1 ζ2 ζ3 ζ4

= = = =

ζ5 = ζi =

ζn+1 = ζn+2 = ζn+3 =

x1 , x2 + bn−1 w1 + bn−1 A0 , θ1 x1 + x3 + bn−2 w1 + bn−1 w2 + bn−2 A0 , θ1 x2 + x4 + bn−3 w1 + bn−2 w2 + bn−1 w3 +bn−3 A0 + θ1 bn−1 A0 , θ1 x3 + θ2 x1 + x5 + bn−4 w1 + bn−3 w2 +bn−2 w3 + bn−1 w4 + (bn−4 + θ1 bn−2 ) A0 , 4 θ1 xi−2 + θ2 xi−4 + xi + j=1 bn+j−i wj + (bn+1−i + θ1 bn+3−i + θ2 bn+5−i ) A0 , 6 ≤ i ≤ n, 4 θ1 xn−1 + θ2 xn−3 + j=1 bj−1 wj + (b0 + θ1 b2 + θ2 b4 ) A0 , 4 θ1 xn + θ2 xn−2 + j=2 bj−2 wj + (θ1 b1 + θ2 b3 ) A0 , 4 θ2 xn−1 + j=3 bj−3 wj + (θ1 b0 + θ2 b2 ) A0 ,

(6)

ζn+4 = θ2 xn + b0 w4 + θ2 b1 A0 , ζn+5 = θ2 b0 A0 ,

T

which is expressed, by setting ζ = [ζ1 , ζ2 , . . . , ζn+5 ]T , w = [w1 , w2 , w3 , w4 ] in matrix form as

T ζ = T (θ1 , θ2 ) xT wT A0

(7)

Adaptive compensation of sinusoidal disturbances

211

where T =  1 0 0 1   θ1 0   0 θ1   θ2 0    0 θ2    .. ..  . .   0 ···    0 ···   0 ···   0 ···   0 ···

0 0 1 0 θ1

0 0 0 1 0

··· ··· ··· ··· ···

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0

0 0

0 0 0

0 0 0 0

bn−1 bn−2 bn−1 bn−3 bn−2 bn−1 bn−4 bn−3 bn−2 bn−1

0 θ1 · · · 0 0 0 bn−5 bn−4 bn−3 bn−2 .. . ···

.. . . .. . . . · · · · · · θ1 . · · · · · · .. 0 · · · · · · · · · θ2 ··· ··· ··· 0 ··· ··· ··· 0 0 ··· ··· ··· ··· 0

.. . 0

.. . 1

.. . b1

.. . b2

.. . b3

.. . b4

θ1 0 θ2 0 0

0 b0 θ1 0 0 0 θ2 0 0 0

b1 b0 0 0 0

b2 b1 b0 0 0

b3 b2 b1 b0 0

0



 bn−1   bn−2   bn−3 + θ1 bn−1   bn−4 + θ1 bn−2  2  bn−5 + θi bn−5+2i   i=1  . ..  .  b1 + θ1 b3 + θ2 b5    b0 + θ1 b2 + θ2 b4    θ1 b1 + θ2 b3   θ1 b0 + θ2 b2   θ2 b1 θ2 b0

Chosen any vector d = [1, dn+3 , . . . , d0 ]T ∈ n+5 such that all the roots of d(s) = sn+4 + dn+3 sn+3 + · · · + d0 have negative real part, and deﬁned   −dn+3 1 0 · · · 0 0  −dn+2 0 1 · · · 0 0     .. .. .. . . .. ..   . . . . . . D= ,  −d2 0 0 · · · 1 0     −d1 0 0 · · · 0 1  −d0 0 0 · · · 0 0 consider the ﬁltered transformation (ξi ∈ n+4 , µi ∈ , i = 1, 2) (see [16])

ai y + ¯bi u ξ˙i = Dξ + [0, In+4 ] −¯ (8) µi = 1 0 · · · 0 ξi i = 1, 2,

0 (9) z=ζ− ξ1 θ1 + ξ2 θ2 which transforms system (5) into an “adaptive observer” form (z ∈ n+5 ) ¯0 y + ¯b0 u + d (µ1 θ1 + µ2 θ2 ) z˙ = Ac z − a y = Cc z. The inverse of the mapping (7), (9) can be expressed as  

x 0  w  = T −1 (θ1 , θ2 ) z + ξ1 θ1 + ξ2 θ2 A0

(10)

(11)

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for all θ1 > 0, θ2 > 0. Following [17], [15] and [1] (see also [16]), an adaptive observer for (10) is given by ¯0 y + ¯b0 u + d µ1 θˆ1 + µ2 θˆ2 + K(y − Cc zˆ) zˆ˙ = Ac zˆ − a ˙ (12) θˆ1 = γ1 q1 (t)(y − Cc zˆ) ˙ θˆ2 = γ2 q2 (t)(y − Cc zˆ) in which K = (Ac + λI)d

(13)

with λ > 0 a design parameter and γi > 0, i = 1, 2 the adaptation gains. The projection operators qi (t) i = 1, 2, are introduced to prevent the estimate θˆi (t) from being equal to zero since θi > 0, and are deﬁned as  µi (t) if θmi <θˆi     µ (t) if θ ≥θˆ and (y − C zˆ) µ (t)θˆ ≥ 0 i mi i c i i (14) qi (t) = ˆ  1 − p ( θ) µ (t) otherwise i i    for i = 1, 2, ˆ = where θmi are lower bounds for θi , pi (θ)

2 − θˆi2 θmi

2 , and εi with 2 − (θ θmi mi − εi ) i = 1, 2 are design parameters smaller than θmi . The operators qi (t) are continuous; setting θ˜i = θi − θˆi , we have (see [22]) that if θmi ≤θˆi (t0 ), then for all t ≥ t0 ,

θmi − εi ≤ θˆi , |qi (t)| ≤ |µi (t)| . Notice that if θmi ≤θi , and also θmi ≤θˆi (t0 ), then for all t ≥ t0 , θi − θˆi qi (t) (y − Cc zˆ) ≥ θi − θˆi µi (t) (y − Cc zˆ) ; i = 1, 2. State estimates for system (4) are deﬁned from (11) as  

x ˆ 0 w ˆ  = T −1 (θˆ1 , θˆ2 ) zˆ + , ξ1 θˆ1 + ξ2 θˆ2 Aˆ0

(15) (16)

(17)

(18)

on the basis the estimates zˆ, θˆ1 , θˆ2 provided by the observer (12). The inverse map (18) is well deﬁned since θˆi (t) > θmi − εi , i = 1, 2, for all t ≥ t0 , and the matrix T (θ1 , θ2 ) in (7) is invertible for all θ1 > 0, θ2 > 0. Finally, on the basis of the estimate w1 of the two sinusoids and of the estimate of the bias A0 given in (18), the compensating control is deﬁned as u = −w ˆ1 zˆ, θˆ1 , θˆ2 , ξ1 , ξ2 − Aˆ0 zˆ, θˆ1 , θˆ2 , ξ1 , ξ2 . (19)

Adaptive compensation of sinusoidal disturbances

213

In conclusion the overall adaptive compensation algorithm is given by (8), (12) (14), (18), (19): its order is 3n + 15 and its design parameters are the n + 4 coeﬃcients dn+3 , .... d0 , in the ﬁlters (8), the scalar parameters λ in (13), γi , i = 1, 2 in (12) and θmi , εi , i = 1, 2 in (14).

3

Stability analysis

The ﬁrst step in the stability analysis is to show that the signals µi (t), i = 1, 2 deﬁned in (8) can be also expressed as the output of asymptotically stable exosystems which are independent of u(t) and are driven by the disturbance δ(t). To show this property, deﬁne ηi ∈ n+4 , i = 1, 2 as     0 0  In  0         (20) M1 =  ηi = ξi − Mi x; i = 1, 2;  0  ; M2 =  0  , 0   In  0 0 so that by virtue of (1) the ﬁlter (8) yielding the regressor terms µi (t), i = 1, 2 in (10) are equivalently generated by ¯ η˙ i = Dη i − [0, In+4 ] bi δ(t) µi = 1 · · · 0 ηi , i = 1, 2,

(21) T

with initial condition ηi (t0 ) = ξi (t0 ) − Mi x(t0 ); i = 1, 2. Set µ = [µ1 , µ2 ] . T By the arguments in [25], Theorem 2.7.2, the time varying vector µ = [µ1 , µ2 ] + is persistently exciting, i.e. there exist two positive reals T, k ∈ , such that t+T µ(τ )µT (τ )dτ ≥ kI2 for all t ≥ t0 , (22) t

since the signal δ(t) is “suﬃciently rich of order 2”, the transfer vector function

1 0 · · · 0 (D − sIn+4 )−1 [0, In+4 ] ¯b1

Hµ (s) = 1 0 · · · 0 (D − sIn+4 )−1 [0, In+4 ] ¯b2 is stable and Hµ (jω1 ), Hµ (jω2 ) are linearly independent. Deﬁne now ηr,i ∈ n+4 , i = 1, 2 as η˙ r,i = Dηr,i − [0, In+4 ] ¯bi δ(t)

(23)

with ηr,i (t0 ) = 0, which are reference signals for the ﬁlters (21). Set η˜i = T T ˆ ηi − ηr,i , i = 1, 2, z˜ = z − zˆ, θ = [θ1 , θ2 ] , θˆ = θˆ1 , θˆ2 , θ˜ = θ − θ, η˜ =

η˜1 γ 0 q (t) ¯ = D 0 . ; Γ = 1 ; q(t) = 1 ; D η˜2 q2 (t) 0 γ2 0 D

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We now show that the overall closed loop error dynamics can be expressed as ¯ η˜ η˜˙ = D (24) z˜˙ = A˜ z + dµT (t)θ˜ (25) ˙˜ θ = −Γ q(t)Cc z˜  ˆ x  x˙ = F + g θ˜T Π1 (θ, θ) (26) ˆ η + Π3 (θ, θ)˜ ˆ z + Π4 (θ, θ, ˆ t)θ˜  +g θ˜T Π2 (θ, θ)˜ ˆ j = 1, 2, 3 are continuous matrices of their arguments for where Πj (θ, θ), ˆ ˆ t) is a matrix with entries that are all θi > 0, θi > 0, i = 1, 2 and Π4 (θ, θ, ˆ continuous functions of θ, θ for all θi > 0, θˆi > 0, i = 1, 2 which are bounded for any t ≥ t0 , provided that θ and θˆ are bounded. The dynamics of the ﬁlter error trajectories (24) can be derived from (21), (23), while (10), (12) yield the observer error trajectories (25). In order to derive (26 ), recall from (11) and (20) that the dynamic compensator variables ξ and zˆ can be expressed as ξ1 = η1 + M1 x = η˜1 + ηr,1 + M1 x ηr,2 + ξ2 = η2 + M2 x = η˜2 +   M2 x x 2 0   − i=1 zˆ = z − z˜ = T (θ1 , θ2 ) w θi − z˜. η˜i + ηr,i + Mi x A0

(27)

The input error δ(t) + u(t) = w1 (t) − w ˆ1 (t) + A0 −Aˆ0 (t) by virtue of (11) and (18) together with ( 27), becomes (see [21]) ˆ + θ˜T Π2 (θ, θ)˜ ˆ η + Π3 (θ, θ)˜ ˆ z + Π4 (θ, θ, ˆ t)θ˜ δ(t) + u(t) = θ˜T Π1 (θ, θ)x

(28)

ˆ j = 1, 2, 3, along with Π4 (θ, θ, ˆ t) are the matrices in (26). where Πj (θ, θ), In order to analyze the convergence property of the overall closed loop system, note that system (24) is clearly globally asymptotically stable. To show that (25) is globally asymptotically stable, notice that the triple (Cc , A, d) is strictly positive real, so that by Meyer-Kalman-Yakubovic Lemma (see for instance [16]), there exists a symmetric positive deﬁnite matrix P satisfying AT P + P A = −lT l − Q Pd =

(29)

CcT ,

for a positive real , a vector l, and a symmetric positive deﬁnite matrix Q. Consider the parameter dependent candidate Lyapunov function 2 ˜ t) = 1 z˜T P z˜ + 1 θ˜T Γ −1 θ˜ + p¯ ¯ θ˜ − µ(t)dT z˜(t) W (˜ z , θ, , Q(t) 2 2

(30)

Adaptive compensation of sinusoidal disturbances

215

proposed in [17], where p¯ is a positive scalar parameter to be deﬁned later ¯ and Q(t) is generated following [13] by the ﬁlter ¯ dQ(t) ¯ + d2 µ(t)µT (t), = −Q(t) dt

(31)

¯ 0 ) = d e−T kI, where k is the integer deﬁned in (22). If cm ≤ with Q(t d µ(t) ≤ cM for all t ≥ t0 , where cm , cM are suitable positive reals then it is straightforward to deduce that 2

¯ > d2 ke−2T I, for all t ≥ t0 , c2M I ≥ Q(t) 2 2 2 2 ˜ t) ≤α2 ˜ z + θ˜ ≤ W (˜ z , θ, z + θ˜ , α1 ˜

(32) (33)

where 1 max λ1 , . . . , λn+5 , γ1−1 , γ2−1 + 2¯ pc2M max c2M , 1 2 1 α1 = min λ1 , . . . , λn+5 , γ1−1 , γ2−1 , 2 and λ1 , . . . , λn+5 are the eigenvalues of P. By computing the time derivative ˜ t) we have of W (˜ z (t),θ(t), α2 =

˙ (t) = 1 z˜T P A + AT P z˜ + θ˜T (µ(t) − q(t)) Cc z˜ W 2 T T ˙ T ˜ Q ˜˙ ¯˙ θ+ ¯ θ−µ(t)d ¯ θ˜ − µ(t)dT z˜ Q +2¯ p Q z˜ . z˜ − µ(t)d ˙ By (17) θ˜T (µ(t) − q(t)) Cc z˜ ≤ 0 and by recalling (31) we have dW 2 ≤ −α3 ˜ z dt T T ¯ θ˜ − QΓ ¯ q(t)Cc z˜ + µ(t)dT A˜ ¯ θ˜ − µ(t)dT z˜ −Q +2¯ p Q z − µ(t)d ˙ z˜ where α3 is a positive real such that lT l + Q ≥ α3 I2 . By adding and subT ¯ θ˜ − µ(t)dT Cc µ(t)dT Cc z˜ in the right hand side of previous tracting 2¯ p Q inequality, we obtain 2 dW ¯˜ 2 θ − µ(t)dT z˜ ≤ −α3 ˜ z − 2¯ p Q dt T

T ¯ θ˜ − µ(t)dT z˜ ¯ q(t)Cc − µ(t)dT A − µ(t)d −µ(t)dT − QΓ z˜. +2¯ p Q ˙ (34) Since by virtue of (21) µ(t), µ(t) ˙ are bounded, then there exist a positive real α4 ∈ + , such that T 2 ¯ q(t)Cc − µ(t)dT A − µ(t)d α4 ≥ −µ(t)dT − QΓ ˙ ,

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for all t ≥ t0 . By “completing the squares” we have T ¯ q(t)Cc ¯ θ˜ − µ(t)dT z˜ −µ(t)dT − QΓ 2¯ p Q 2

p¯ ¯ ˜ 2 T θ − µ(t)dT z˜ +2¯ z˜ ≤ Q −µ(t)dT A − µ(t)d ˙ pα4 ˜ z , 2 2 2 2 ¯ ˜ ¯˜ θ . θ − µ(t)dT z˜ ≤ p¯ µ(t)dT z˜ − 12 Q −¯ p Q

(35)

(36)

Substituting (35), (36) in (34) and rearranging terms we obtain 2 2 ¯˜ 2 ˙ ≤ α2 b1 ˜ W θ − µ(t)dT z˜ , z +α2 b2 θ˜ +¯ pb3 Q where

b1 = α12 −α3 + p¯ 2α4 + c2M 4 b2 = − 2αp¯2 k 2 e−4T d b3 = − 12 .

3 Note that for any p¯ < 2α4α+c 2 , we have bi < 0, for i = 1, 2, 3. By setting κ1 = M mini=1,2,3 {|bi |}, we conclude that

˙ ≤ −κ1 W. W Inequality (37) along with (33) yield 2 α2 ˜ 2 ˜ 2 2 ˜ z (t) + θ(t) ≤ ˜ z (t0 ) + θ(t0 ) e−κ1 (t−t0 ) α1

(37)

(38)

which implies that system (25) is globally exponentially stable. By the arguˆ ments above it follows that the entries of the matrices Πi (θ, θ(t)), i = 1, ..., 3, ˆ t) are bounded for all t ≥ 0. By recalling that F is Hurwitz and Π4 (θ, θ(t), by (38) the time varying system ˆ x˙ v = F + g θ˜T (t)Π1 (θ, θ(t)) xv ˜ 0 ) ∈ 2 . Therefore the system (26) is globally exponentially stable for any θ(t is a globally exponentially stable system driven by the input ˆ η + Π3 (θ, θ)˜ ˆ z + Π4 (θ, θ, ˆ t)θ. ˜ u∗ (t) = θ˜T Π2 (θ, θ)˜ ˜ Since Π2 (·), Π3 (·), Π4 (·), are bounded functions and z˜(t), θ(t), η˜(t) converge ∗ exponentially to zero for any initial condition, then u (t) converges exponentially to zero and, in turn, (26) is also globally exponentially stable for any initial condition. To summarize the following statement has been proved. Proposition 1. Consider system (1), together with the dynamic output-feedback compensator (8), (12), (19). Assume that hypotheses (H1), (H2), (H3) hold. Then the overall closed loop error system (24), (25), (26) is globally exponentially stable and in particular y(t) and x(t) tend exponentially to zero.

Adaptive compensation of sinusoidal disturbances

4

217

Extension to m sinusoidal disturbances

The arguments above can be generalized when the disturbance is a biased sum of m sinusoids, with m arbitrary positive real. In this case the disturbance m Ai sin(ωi t + φi ) can be modelled by the (2m+1)−order linear δ(t) = A0 + exosystem

w˙ A˙ 0

i=1

=

S0 0 0

w A0

(39)

δ = A0 + 1 0 · · · 0 w with 

···  ···   · ··   · ·· S=  .  · · · · · · · · · · · · · · · ..   0 0 0 0 0 ··· −θm 0 0 0 0 · · · 0 −θ1 0 −θ2

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

 0 0  0  0  ..  .  1 0

and suitable initial conditions. The overall system (1), (39) is rewritten as 

bn−1 ..   .   F  x˙ b 0   w˙  =  0   . A˙ 0  ..  0 0

0

   0 bn−1 bn−1 .. ..   .  . .      ..     0 · · · 0 b0  x  b0    u 0  w  +   0    ..  A0   . S .  .  .   0 0 0 ··· 0 0 0 .. .

··· .. .

(40)

T y = 1 0 · · · 0 xT wT A0 which is observable for every θi > 0, i = 1, . . . m, since by assumption (H3) T b(s) has no roots on the imaginary axis. Set θ = [θ1 , . . . θm ] . The observable system ( 40) is transformed into the observer canonical form

m ai y + ¯bi u ¯0 y + ¯b0 u + i=1 θi −¯ ζ˙ = Ac ζ − a y = Cc ζ

(41)

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where ζ ∈ 2m+n+1 , and T

a ¯0 = [an−1 , . . . , a0 , 0, 0, 0, 0, . . . , 0, 0] , T a ¯1 = [0, 1, an−1 , . . . , a0 , 0, 0, . . . , 0, 0] , T a ¯2 = [0, 0, 0, 1, an−1 , . . . , a0 , 0, . . . , 0] , ... T a ¯m = [0, 0, 0, . . . , 0, 1, an−1 , . . . , a0 , 0] , ¯b0 = [bn−1 , . . . , b0 , 0, 0, 0, 0, . . . , 0, 0]T , ¯b1 = [0, 0, bn−1 , . . . , b0 , 0, 0, . . . , 0, 0]T , ¯b2 = [0, 0, 0, 0, bn−1 , . . . , b0 , 0, . . . , 0]T , ... ¯bm = [0, 0, 0, . . . , 0, 1, bn−1 , . . . , b0 , 0]T , 

0  ..  Ac =  . 0 0

1 ··· .. . . . . 0 ··· 0 ···

 0 ..  . , 1 0

Cc = 1 0 · · · 0 ,

by a suitable linear transformation (which is nonsingular for all θ1 > 0, . . . θm > 0) expressed in matrix form as T ζ = T (θ) xT wT A0

(42)

where the columns of the matrix T (θ) = [t1 (θ), t2 (θ), . . . , tn+2m+1 (θ)] are t1 (θ) = t2 (θ) = ... ... tn (θ) = tn+1 = tn+2 = ... ... tn+2m = tn+2m+1 (θ) =

T

[1, 0, θ1 , 0, θ2 , . . . , 0, θm , 0, 0, . . . , 0, 0] , T [0, 1, 0, θ1 , 0, θ2 , . . . , 0, θm , 0, . . . , 0, 0] , ... T [0, 0, . . . , 0, 1, 0, θ1 , 0, θ2 , . . . , 0, θm , 0] , T [0, bn−1 , bn−2 , . . . , b0 , 0, 0, . . . , 0, 0, 0, 0] , T [0, 0, bn−1 , bn−2 , . . . , b0 , 0, 0, . . . , 0, 0, 0] , ... T [0, 0, 0, . . . , 0, 0, 0, bn−1 , bn−2 , . . . , b0 , 0] , m−1 tn+1 + i=1 θi tn+2i+1 + θm t¯,

with T t¯ = [0, 0, 0, . . . , 0, 0, 0, bn−1 , bn−2 , . . . , b0 ] .

Chosen any vector d = [1, dn+2m−1 , . . . , d0 ]T ∈ n+2m+1 such that all the roots of d(s) = sn+2m + dn+2m−1 sn+2m−1 + · · · + d0 have negative real part,

Adaptive compensation of sinusoidal disturbances

and deﬁned  −dn+2m−1  −dn+2m−2   ..  . D=  −d 2   −d1 −d0

219

 00 0 0  .. ..  . . ; 0 0 ··· 1 0  0 0 ··· 0 1 0 0 ··· 0 0 1 0 ··· 0 1 ··· .. .. . . . . .

consider the ﬁltered coordinate change as in [16] (ξi ∈ n+2m , µi ∈ , 1 ≤ i ≤ m)

ai y + ¯bi u ξ˙i = Dξ + [0, In+2m ] −¯ (43) µi = 1 0 · · · 0 ξi i = 1, 2, . . . m

0 (44) z=ζ− m i=1 ξi θi which transforms system (41) into an “adaptive observer” form (z ∈ n+2m+1 ) m ¯0 y + ¯b0 u + d ( i=1 µi θi ) z˙ = Ac z − a (45) y = Cc z. The inverse of the mapping (42), can be expressed as  

x  w  = T −1 (θ) z + m 0 i=1 ξi θi A0

(46)

for all θi > 0, 1 ≤ i ≤ m. Following [17], [15] and [1] (see also [16]), an adaptive observer for (45) is given by m ˆi + K(y − Cc zˆ) zˆ˙ = Ac zˆ − a ¯0 y + ¯b0 u + d µ θ i i=1 ˙ (47) ˆ θi = γi qi (t)(y − Cc zˆ) i = 1, 2, . . . , m (with zˆ ∈ n+2m+1 ) in which K = (Ac + λI)d, where λ > 0 is a design parameter and γi > 0, with 1 ≤ i ≤ m, are the adaptation gains. The projection operators qi (t), 1 ≤ i ≤ m are introduced to prevent the estimate θˆi (t) from being equal to zero since θi > 0 and are deﬁned as  µi (t) if θmi <θˆi     ˆ ˆ µ ≥θi and (y − Cc zˆ) µi (t)θi ≥ 0 i (t) if θmi qi (t) = (48) ˆ  1 − pi (θ) µi (t) otherwise,    i = 1, 2, . . . , m

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ˆ = where θmi , 1 ≤ i ≤ m are lower bounds for θi , pi (θ)

2 − θˆi2 θmi

2, 2 − (θ θmi mi − εi ) and εi are design parameters smaller than θmi . State estimates for system (40) are deﬁned on the basis of (47), (11) as  

x ˆ 0 −1 ˆ  w ˆ = T (θ) zˆ + m , (49) ˆ i=1 ξi θi Aˆ0

T where θˆ = θˆ1 , . . . θˆm . The inverse map (49) is well deﬁned since θˆi (t) > θmi − εi i = 1, 2, . . . , m, for all t ≥ t0 , and the matrix T (θ) in (42) is invertible for all θi > 0, i = 1, 2, . . . , m. Finally, on the basis of the estimates of the sum of sinusoids w1 and of the bias A0 given in (49), the compensating control law is deﬁned as ˆ ξ1 , . . . , ξm − Aˆ0 zˆ, θ, ˆ ξ1 , . . . , ξm . (50) u = −w ˆ1 zˆ, θ, In conclusion the overall adaptive compensation algorithm is given by (43), (47) (48), (49), (50): its order is n(m + 1) + 2m2 + 3m + 1 and its design parameters are the n + 2m coeﬃcients dn+2m−1 , .... d0 , in the ﬁlter (43), the scalar parameters λ, γi , with i = 1, 2, . . . , m in (47), θmi and εi in (48). By using the same arguments in Section 3 it can be shown that the overall closed loop error system is globally asymptotically stable.

5

An illustrative example

The proposed disturbance compensation method is illustrated in this section for a speciﬁc second order system with a disturbance input δ(t) modelled as in (2). In this example we set x ∈ 2 and (1)-(3) becomes       −a1 1 b1 0 0 0 b1   b1 x˙ x      w˙ 1   −a0 0 b0 0 0 0 b0   w1   b0     0 0 0 1 0 0 0    0   w˙ 2         =  0 0 −θ1 0 1 0 0   w2  +  0  u  w˙ 3       (51)    0 0 0 0 0 1 0   w3   0    w4     w˙ 4    0 0 −θ2 0 0 0 0  0 A0 A˙ 0 0 0 0 000 0 0 T T y = 1 0 · · · 0 x , w 1 , w 2 , w 3 , w 4 , A0 . System (51) (which is observable for every θ1 > 0, θ2 > 0), is transformed into the observer canonical form

2 ai y + ¯bi u ¯0 y + ¯b0 u + i=1 θi −¯ ζ˙ = Ac ζ − a (52) y = Cc ζ

Adaptive compensation of sinusoidal disturbances

221

with T a ¯0 = [a1 , a0 , 0, 0, 0, 0, 0] , T a ¯1 = [0, 1, a1 , a0 , 0, 0, 0] , T a ¯2 = [0, 0, 0, 1, a1 , a0 , 0] , ¯b0 = [b1 , b0 , 0, 0, 0, 0, 0]T , ¯b1 = [0, 0, b1 , b0 , 0, 0, 0]T , ¯b2 = [0, 0, 0, 0, b1 , b0 , 0]T , by the linear transformation ζ = T (θ) where   1 0 0 0 0 0 0  0 1 b1 0 0 0 b1     θ1 0 b0 b1 0 0 b0     T (θ) =   0 θ1 0 b0 b1 0 θ1 b1  .  θ2 0 0 0 b0 b1 θ1 b0     0 θ2 0 0 0 b0 θ2 b1  0 0 0 0 0 0 θ2 b0 The ﬁlters (8) become    ξ˙11 −d5 1 0  ξ˙    21   −d4 0 1 ˙    ξ31   −d3 0 0 ˙ =  ξ41   −d2 0 0 ˙   −d1 0 0  ξ51  −d0 0 0 ˙ξ61 µ1 = ξ11   ξ˙12 −d5 1  ξ˙    22   −d4 0 ˙    ξ32   −d3 0 ˙ =  ξ42   −d2 0 ˙   −d1 0  ξ52  −d0 0 ˙ξ62 µ2 = ξ12 .

0 0 1 0 0 0

0 0 0 1 0 0

        1 ξ11 0 0  ξ21    a1   b1   0                 0   ξ31  + −  a0  y +  b0  u  ξ41    0  0  0         0  1   ξ51    0  0 ξ61 0 0

(53)

0 0 1 0 0 0

0 0 0 1 0 0

        0 ξ12 0 0  ξ22    0  0  0                 0   ξ32  + −  1  y +  0  u  ξ42    a1   b1   0          b0   1   ξ52    a0  0 ξ62 0 0

(54)



0 1 0 0 0 0

The adaptive observer (12) is 2 ¯0 y + ¯b0 u + d µ θ zˆ˙ = Ac zˆ − a i i i=1 +K(y − Cc zˆ) ˙ ˆ = θ0 θˆ = γq(t)(y − Cc zˆ), θ(0)

(55)

with K = [k6 , . . . , k0 ]T deﬁned as in (13) and q(t) deﬁned as in (14). The inverse of T (θ) is given by T −1 (θ) = −1 (θ) [t1 , t2 , . . . , t7 ]

(56)

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where (θ) = b41 θ2 + b40 + b20 θ1 b21 θ2 b0 , and     θ2 b0 b41 θ2 + b40 + b20 θ1 b21 0 5     θ2 b1 b20 b20 θ1 + θ2 b21

   θ22 b0

 2  2 2 2    b θ b b θ + θ b b θ θ + θ b −b 2 1    2 21 0 20 1 0 22 2 0 1 2 2 1 2  2    t1 =   −θ2 b1 b0 θ1 b0 +2 θ42 b1 θ1 − b0 θ2  , t2 =  −b0 θ2 b20 θ31 + θ2 b1  ,     θ2 b0 b1 −θ     2 b0

    −θ22 b40 −θ22 b0 b1 b20 θ1 + θ2 b21 0 0       0 0 0   θ2 b21 b30     −θ2 b40 b1 −b31 θ2 b20       4 2 2 3      θ b θ b b −θ b b 2 1 0 2 1 0      22 0

3 4  2     −θ b b θ b b θ b b θ + θ b , t , t t3 =  = =  2 1 0 02 12 2 2 1  4  3 2 20  5  2 22 1 0 2  ,       b θ b b −θ b b b θ + θ b 2 1 0   1 2 22 02   0 2 30 2 1 1     −θ2 b1 b0     b1 θ2 b0 θ22 b30 b1 0 0 0     0 0 5     b41 θ2 b0 1 θ2     −b

3 2 2 2     −b1 θ2 b0 −b0 b0 + θ1 b1     2 2 3 .    θ2 b1 b0 −b1 θ2 b0 t6 =  , t7 =   

 −θ2 b1 b0 b20 + θ1 b21   −b20 θ12 b21 + b20 θ1 − θ2 b21    2 2  

  b0 θ2 b0 + θ1 b21   −θ2 b1 b0 b20 + θ1 b21 4 4 2 2 0 b1 θ2 + b0 + b0 θ1 b1 According to (19), the compensator is u=− w ˆ1 (t) + Aˆ0 (t) (57) where by virtue of (18) and (56) 1 −b20 θˆ2 b20 θˆ1 + θˆ2 b21 zˆ1 w ˆ1 = b41 θˆ2 + b40 + b20 θˆ1 b21 +θˆ2 b1 b0 b20 θˆ1 + θˆ2 b21 zˆ2 + θˆ1 ξ11 + θˆ2 ξ12 +θˆ2 b40 zˆ3 + θˆ1 ξ21 + θˆ2 ξ22 − θˆ2 b1 b30 zˆ4 + θˆ1 ξ31 + θˆ2 ξ32 +θˆ2 b21 b20 zˆ5 + θˆ1 ξ41 + θˆ2 ξ42 − b31 θˆ2 b0 zˆ6 + θˆ1 ξ51 + θˆ2 ξ52 −b20 b20 + θˆ1 b21 zˆ7 + θˆ1 ξ61 + θˆ2 ξ62 1 zˆ7 + θˆ1 ξ61 + θˆ2 ξ62 . Aˆ0 = θˆ2 b0 We assume the disturbance (2) to be δ(t) = sin(3t) + sin(23t) + 0.75 (so that θ1 = 538 and θ2 = 4761) and the parameters in system (1) to be a0 = 150, a1 = 25, b0 = 10, b1 = −100. In the dynamic control law (53), (54), (55) we set d = [60, 1500, 20000, 150000, 600000, 1000000]T

Adaptive compensation of sinusoidal disturbances

223

λ = 400, γ1 = 109 , γ2 = 1011 , θm1 = 2, θm2 = 2, ε1 = 1. ε2 = 1. The system (51) has been simulated starting from zero initial conditions; the compensator (53)-(54)-(57) is activated at time t = 0, with zero initial conditions except for θˆ1 (0) = 10, and θˆ2 (0) = 103 . The simulation results are reported in Figures 1, 2, 3. Figure 1 shows the plot of the system output y(t) for 0 ≤ t ≤ 5. Figure 2 shows the time history of the functions $ ' % %ˆ & θ1 (t) − θˆ12 (t) − 4θˆ2 (t) ω ˆ 1 (t) = 2 $ ' % %ˆ & θ1 (t) + θˆ12 (t) − 4θˆ2 (t) ω ˆ 2 (t) = 2 estimates respectively of the frequencies ω1 and ω2 , with ω1 < ω2 , along with Aˆ0 (t), estimate of A0 = 0.75. Figure 3 displays the time history of the disturbance δ(t) together with −u(t), the opposite of the control input (57), for 0 ≤ t ≤ 5. Notice that −u(t) quickly reproduces the unknown disturbance δ(t) within 1 second of simulation, thus allowing the system output y(t) to converge to zero. 1.5

1

system output y(t)

0.5

0

−0.5

−1

−1.5

−2

0

0.5

1

1.5

2

2.5 time t

3

3.5

Fig. 1. System output y(t), for 0 ≤ t ≤ 5.

4

4.5

5

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R. Marino, G.L. Santosuosso, and P. Tomei

est. of ω1

6 4 2 0

0

0.5

1

1.5

2

2.5 time t

3

3.5

4

4.5

5

0

0.5

1

1.5

2

2.5 time t

3

3.5

4

4.5

5

0

0.5

1

1.5

2

2.5 time t

3

3.5

4

4.5

5

40

est. of ω

2

30 20 10 0

est. of A0

4 2 0

−2 −4

Fig. 2. Above ω ˆ 1 (t), estimate of ω1 = 3; in the center ω ˆ 2 (t), estimate of ω2 = 23; below Aˆ0 (t), estimate of A0 = 0.75, for 0 ≤ t ≤ 5.

δ (t) (dashed line) and −u(t) (solid line)

4

2

0

−2

−4

−6

0

0.5

1

1.5

2

2.5 time t

3

3.5

4

4.5

5

Fig. 3. Disturbance δ(t) (dashed line) together with −u(t) (solid line), for 0 ≤ t ≤ 5.

Adaptive compensation of sinusoidal disturbances

6

225

Conclusions

A [3n + 15]-order compensator has been designed to reject two biased sinusoidal disturbances with unknown bias, phases, amplitudes and frequencies for a linear, asymptotically stable system, which is allowed to be nonminimum phase. The closed loop error system is shown to be globally exponentially stable and in particular exponentially convergent estimates of the disturbances and of their frequencies are provided. The algorithm is generalized to the case of an arbitrary number m of sinusoidal disturbances, with unknown parameters, yielding a [n(m+1)+2m 2 +3m+1] -order compensator. The regulation strategy presented in this note improves previous results on this subject since it is a globally convergent algorithm for systems allowed to be non-minimum phase.

References 1. Bastin, G. and M. Gevers (1988) Stable adaptive observers for nonlinear time varying systems. IEEE Trans. Automat. Contr. 33, 1054–1058. 2. Bodson, M and S. C. Douglas (1997) Adaptive algorithms for the rejection of periodic disturbances with unknown frequencies. Automatica, 33, 2213–2221. 3. Canudas De Wit, C. and L. Praly (2000) Adaptive eccentricity compensation. IEEE Trans. Contr. Syst. Technology, 8, 757–766. 4. Bodson, M A. Sacks and P. Khosla (1994) Harmonic generation in adaptive feedforward cancellation schemes. IEEE Trans. Automatic Contr., 39, pp. 1939–1944. 5. Bodson, M. (1999) A discussion of Chaplin and Smith’s patent for the cancellation of repetitive vibration. IEEE Trans. Automat. Contr., 44, 2221–2225. 6. Chen, D. and B. Paden (1993) Adaptive linearization of hybrid step motors: stability analysis”. IEEE Trans. Automat. Contr., 38, 874–887. 7. Davidson, E. J. (1976) The robust control of a servomechanism problem for linear time invariant multivariable systems. IEEE Trans. Automatic Control, 21, 25–34. 8. Francis, B. A. and W. M. Wonham (1975) The internal model principle for linear multivariable regulators. Appl. Math. Optim., 2 , 170–194. 9. Garimella S. S. and K. Srinivasan (1996) Application of repetitive control to eccentricity compensation in rolling”. J. Dyn. Syst., Meas., Contr., 118, 657– 664. 10. Hsu, L. R. Ortega and G. Damm (1999) A globally convergent frequency estimator. IEEE Trans. Automat. Contr., 44, 698–713. 11. Johnson, C. D. (1971) Accommodation of external disturbances in linear regulator and servomechanism problems. IEEE Trans. Automatic Control, 16, 635–644.

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12. Khalil, H.K. (1996) Nonlinear Systems. Prentice Hall, Upper Saddle River, (NJ), 2-nd edition. 13. Kreisselmeier, (1977) Adaptive observers with exponential rate of convergence. IEEE Trans. Automat. Contr., 22, 2–8. 14. Kuo, S. M. and D. R. Morgan (1996) Active noise control systems. New York, Wiley. 15. Marino R. and P. Tomei (1992) Global adaptive observers for nonlinear systems via ﬁltered transformations. IEEE Trans. Automatic Control, 37, 1239–1245. 16. Marino, R., and P. Tomei (1995) Nonlinear Control Design - Geometric, Adaptive and Robust. Prentice Hall, Hemel Hempstead. 17. Marino, R., G. Santosuosso and P. Tomei (2001) Robust adaptive observers for nonlinear systems with bounded disturbances. IEEE Trans. Automat. Contr., 46, 967–972. 18. Nikiforov, V.O. (1996) Adaptive servocompensation of input disturbances. Proc. 13th World IFAC Congr. 175–180, San Francisco, USA. 19. Nikiforov, V.O. (1997) Adaptive controller rejecting uncertain deterministic disturbances in SISO Systems. Proc. European Control Conf. Bruxelles. 20. Nikiforov, V.O. (1998) Adaptive nonlinear tracking with complete compensation of unknown disturbances. European Journal of Control, 4, 132–139. 21. Nijmeijer, H. and A. J. Van Der Shaft (1990) Nonlinear dynamical control systems. Springer Verlag, New York. 22. Pomet, J.B. and L. Praly (1992) Adaptive nonlinear regulation: estimation from the Lyapunov equation. IEEE Trans. Automat. Contr., 37, 729–740. 23. Sacks, A. M. Bodson and P. Khosla (1996) Experimental results of adaptive periodic disturbance cancellation in a high performance magnetic disk drive. ASME J. Dynamic Systems Measurements and Control, 118, 416–424. 24. Serrani, A, Isidori, A, and L. Marconi (2000) Semiglobal nonlinear output regulation with adaptive internal model. IEEE 39th Conf. on Decision and Control, Sydney, 2, 1649–1654. 25. Sastry, S. and M. Bodson (1989) Adaptive control- stability, convergence and robustness. Prentice Hall, Englewood Cliﬀs, NJ. 26. Shoureshi R. and P. Knurek, (1996) Automotive application of a hybrid active noise and vibration control. IEEE Contr. Syst. Mag., 16, 72–78.

Backstepping algorithms for a class of disturbed nonlinear nontriangular systems Russell E. Mills, Ali J. Koshkouei and Alan S. I. Zinober Department of Applied Mathematics, The University of Sheﬃeld, Sheﬃeld S10 2TN, UK, Email: {r.e.mills, a.koshkouei, a.zinober}@shef.ac.uk

Abstract. This paper presents a method for the design of dynamical adaptive nonlinear controllers for regulation and tracking of a class of observable minimum phase uncertain nonlinear systems. This method also allows one to design non-adaptive controllers for systems without uncertainty, and dynamical adaptive sliding mode controllers (ASMC). The design procedure employs the basic ideas of the adaptive backstepping algorithm with tuning functions via input-output linearization with or without sliding mode control, and is applicable to both triangular and nontriangular systems with bounded disturbances and unmodelled dynamics. The control law produced uses the bounds of the disturbances but requires no other knowledge of these uncertainties.

1

Introduction

The backstepping algorithm [4] was an important advance in the ﬁeld of adaptive control. Previous Lyapunov approaches required systems to be satisfy the matching, or extended matching [1], conditions. Backstepping was applicable to nonlinear systems, aﬃne in control, for which the output is a linearising function. This allows the system to be transformed into a triangular form, either parametric pure feedback (PPF) or parametric strict feedback (PSF). It used an error variable system in which the ﬁrst variable was the error between the output y = x1 and the desired output yr . It also used tuning functions to design an update law for the estimates of the unknown parameters. The algorithm has been improved upon in two separate ways. Firstly, dynamical backstepping enlarged the class of applicable systems from triangular to nontriangular forms [7]. This was done by using a dynamic input-output linearisation technique, and considering derivatives of the control to be new state variables. The original algorithm was also extended to systems in the parametric semi-strict feedback (PSSF) form [3]. This allowed measurement noise and unmodelled dynamics to be considered as external disturbances. In this paper we use both of these approaches to design a backstepping controller for aﬃne nonlinear systems with unmodelled or external disturbances [2]. We use a dynamical input-output linearisation and assume that there is a well-deﬁned relative degree ρ. A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 227-238, 2003. Springer-Verlag Berlin Heidelberg 2003

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Applicable class of systems

Consider the nonlinear nontriangular disturbed system x(t) ˙ = f (x) + φ(x)θ + (g(x) + ψ(x)θ) u + η(x, w, t) y(t) = h(x)

(1)

where x ∈ Rn is the state; u, y ∈ R the input and output respectively; and θ = [θ1 , . . . , θp ]T is a vector of unknown parameters. f, g and the columns of the matrices φ, ψ ∈ Rn×p are smooth vector ﬁelds on Rn ; and h is a smooth scalar function also deﬁned Rn . η(x, w, t) ∈ Rn are unknown nonlinear scalar functions including all the disturbances and unmodelled dynamics. w is an uncertain time-varying parameter. It is assumed that the relative degree of (1) with respect to u is 1 ≤ ρ ≤ n. Assumptions 1. The functions η(x, w, t) are bounded by known positive functions q(x) |ηi (x, w, t)| ≤ qi (x),

1≤i≤n

(2)

2. The unperturbed system is observable and minimum-phase.

3

DAB algorithm for disturbed systems

For systems of the form (1), the general problem is to track adaptively a bounded reference signal yr (t), which is smooth and has bounded derivatives up to n-th order, in the presence of unknown constant parameters θ and disturbances η(x, w, t). A truncated description is presented below. Full details can be accessed in [2] and [5]. Step 1 Deﬁne the ﬁrst error variable as the output tracking error ˆ (0) − yr z1 = y − yr = h

(3)

ˆ (0) = h. Then with h ˆ (0) ∂h [f + φθ + (g + ψθ) v1 + η] − y˙ r ∂x ∂h ∂h ˆ (0) ˆ (0) ˆ (0) ∂h f +φθˆ + g+ψ θˆ v1 + (φ+ψv1 ) θ− θˆ + η− y˙ r = ∂x ∂x ∂x ˆ (1) − y˙ r + ω1 θ˜ + ξ1 =h (4)

z˙1 =

Backstepping for disturbed systems

229

ˆ are estimates of the unknown parameters θ, θ˜ = θ − θˆ the paramwhere θ(t) eter estimate error, v1 = u, and ˆ (0) ˆ v1 ) = ∂ h ˆ (1) (x, θ, h f + φθˆ + g + ψ θˆ v1 ∂x ˆ (0) ∂h (φ + ψv1 ) ω1 = ∂x n ˆ (0) ˆ (0) ∂h ∂h ξ1 = η= ηj ∂x ∂xj j=1 Consider the Lyapunov function V1 =

1 2 1 ˜T −1 ˜ z + θ Γ θ 2 1 2

where Γ ∈ Rp×p is a symmetric positive deﬁnite matrix. The derivative of V1 is ˙ V˙ 1 = z1 z˙1 + θ˜T Γ −1 −θˆ ˆ (1) − y˙ r + ξ1 + θ˜T Γ −1 Γ ω T z1 − θˆ˙ = z1 h 1 ˆ (1) as a virtual Set the ﬁrst tuning function τ1 = Γ ω1T z1 . If we consider h ˆ (1) = −α1 + y˙ r , where control in (4) and set h α1 (x, t) = (c1 + ζ1 ) z1 , c1 > 0 2 n ˆ (0) n at ∂ h ζ1 = e qj2 4 ∂x j j=1 then the system is stabilised. However, as this result is not valid for all time, we deﬁne the second error variable as the diﬀerence between the two expressions ˆ (1) + α1 − y˙ r z2 = h giving the closed loop derivatives z˙1 = −c1 z1 + z2 + ω1 θ˜ + ξ1 − ζ1 z1

˙ V˙ 1 = −c1 z12 + z1 z2 + ξ1 z1 − ζ1 z12 + θ˜T Γ −1 τ1 − θˆ Proceeding iteratively, we obtain the k-th step. Step k, 1 ≤ k ≤ n − 1 We have ˆ (k−1) + αk−1 − y (k−1) zk = h r

(5)

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R.E. Mills, A.J. Koshkouei, and A.S.I. Zinober

where ωk = τk = Γ

ˆ (k−1) ∂αk−1 ∂h + ∂x ∂x k

(φ + ψ v1 )

ωjT zj

(6)

(7)

j=1

∂h ˆ (k−1) ˆ (k−1) ˆ (k−1) ∂h ˆ (k) = ∂ h τk + f + φθˆ + g + ψ θˆ v1 + h ∂x ∂t ∂ θˆ k−ρ k−1 (k−1) (j−1) ∂h ∂h ˆ ˆ + Γ ωkT vj+1 + zj ˆ ∂v j ∂ θ j=1 j=2 ∂α ∂αk−1 k−1 f + φθˆ + g + ψ θˆ v1 + τk αk = zk−1 + ck zk + ∂x ∂θ k−ρ k−1 ∂αj−1 ∂αk−1 ∂αk−1 + Γ ωkT + ζk zk + vj+1 + zj ˆ ∂t ∂v j ∂θ i=1 j=3 2 k ˆ (k−1) n ∂αk−1 ∂h + qj2 ζk = eat 4 ∂x ∂x j j j=1 with ck > 0. The time derivative of zk is ˆ (k−1) ˆ (k−1) ˙ ∂ h ˆ (k−1) ∂h ∂h [f + φθ + (g + ψθ) v1 + η] + θˆ + ∂x ∂t ∂ θˆ ∂αk−1 ∂αk−1 ˆ˙ ∂αk−1 [f + φθ + (g + ψθ) v1 + η] + + θ+ ∂x ∂t ∂ θˆ k−ρ k−ρ ∂h ∂αk−1 ˆ (k−1) + vj+1 + vj+1 − yr(k) ∂v ∂v j j j=1 j=1 ∂α ∂α ˙ ∂αk−1 k−1 k−1 ˆ ˆ (k) + f + φθˆ + g + ψ θˆ v1 + =h θ+ ∂x ∂t ∂ θˆ k−ρ (k−1) ∂αk−1 ˆ ∂h ˙ + vj+1 − yr(k) + θˆ − τk + ωk θ˜ ˆ ∂v j ∂ θ j=1

z˙k =

+ξk −

k−1 j=2

zj

ˆ (j−1) ∂h Γ ωk ∂ θˆ

where v1 = u, v2 = u, ˙ . . . , vj = u(j−1) .

(8)

(9)

(10)

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231

Augment the Lyapunov function with a term quadratic in the new error variable 1 Vk = Vk−1 + zk2 2 k 1 2 1 ˜T −1 ˜ = z + θ Γ θ 2 j=1 j 2 This has derivative V˙ k = V˙ k−1 + zk z˙k =−

k−1



k−1

cj zj2 + 

j=1

j=2

 ∂αj−1 ˙ ˆ (j−1) k−1 ∂h  θˆ − τk−1 + zj zj ∂ θˆ ∂ θˆ j=3

ˆ g + ψ θˆ v1 ˆ (k) + ∂αk−1 f + φθ+ ξj zj − ζj zj2 +zk zk−1 + h + ∂x j=1 k−1

+

k−ρ j=1

+

ˆ (k−1) ˙ ∂αk−1 ∂h vj+1 − yr(k) + θˆ − τk + ωk θ˜ ∂vj ∂ θˆ

k−1 ˆ (j−1) ∂αk−1 ˆ˙ ∂αk−1 ∂ h − Γ ωkT zj θ+ ˆ ∂t ∂θ ∂θ j=2 ˆ (k−1) ∂αk−1 ∂h ˙ + η + θ˜T Γ −1 τk−1 − θˆ + ∂x ∂x

We deﬁne the new error variable as the diﬀerence ˆ (k) + αk − y (k) zk+1 = h r

(11)

This gives the closed loop derivative of the error variable as ˆ (k−1) ∂αk−1 ˆ˙ ∂h + θ − τk z˙k = −zk−1 − ck zk + zk+1 + ∂ θˆ ∂ θˆ   k−1 ∂h ∂αj−1 ˆ (j−1) k−1  Γ ωkT + ξk − ζk zk + zj zj +ωk θ˜ −  ∂ θˆ ∂ θˆ

j=2

j=3

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and with this, the derivative of the Lyapunov function is   k−1 k−1 ∂h ∂αj−1 ˙ ˆ (j−1) k−1  θˆ − τk−1 + V˙ k = − cj zj2 +  zj zj ∂ θˆ ∂ θˆ j=1

+

j=2

k−1

j=3

˙ ξj zj − ζj zj2 + θ˜T Γ −1 τk−1 − θˆ + zk

j=1



+ωk θ˜ + ξk − ζk zk − 

k−1 j=2

[ −c z +z k k

k+1

 k−1 (j−1) ˆ ∂h ∂αj−1  + Γ ωkT zj zj ˆ ˆ ∂θ ∂θ j=3

]

ˆ (k) ∂αk ˆ˙ ∂h + θ − τk ∂ θˆ ∂ θˆ   k k k ˆ (j−1) ∂ h ∂α ˙ j−1  ˆ + θ − τk =− cj zj2 + zk zk+1 +  zj zj ∂ θˆ ∂ θˆ +

j=1

+

k

j=2

j=3

˙ ξj zj − ζj zj2 + θ˜T Γ −1 τk − θˆ

j=1

Step n ˙ At this step, the actual update law θˆ = τn and the dynamical controller are obtained. ˆ (n−1) + αn−1 − yr(n−1) , so We have zn = h ˆ (n−1) ˆ (n−1) ˙ ∂ h ˆ (n−1) ∂h ∂h [f + φθ + (g + ψθ) v1 + η] + θˆ + ∂x ∂t ∂ θˆ ∂αn−1 ∂αn−1 ˆ˙ ∂αn−1 [f + φθ + (g + ψθ) v1 + η] + + θ+ ∂x ∂t ∂ θˆ n−ρ n−ρ ∂h ∂αn−1 ˆ (n−1) + vj+1 + vj+1 − yr(n) ∂v ∂v j j j=1 j=1 ˆ (n−1) ∂αn−1 ∂h f + φθˆ + g + ψ θˆ v1 − yr(n) + ωn θ˜ + = ∂x ∂x n−ρ ∂h ˆ (n−1) ˆ (n−1) ∂αn−1 ∂αn−1 ∂h + vj+1 + + + ∂vj ∂vj ∂t ∂t j=1 ˆ (n−1) ∂αn−1 ˆ˙ ∂h + θ + ξn + ˆ ∂θ ∂ θˆ

z˙n =

with ωn , ξn found from substituting k = n in (6) and (10), respectively.

Backstepping for disturbed systems

233

Augmenting the Lyapunov function with a term quadratic in zn 1 (n + 1) −at e Vn = Vn−1 + zn2 + 2 2a n 1 2 1 ˜T −1 ˜ (n + 1) −at e = z + θ Γ θ+ 2 j=1 j 2 2a the derivative of the ﬁnal Lyapunov function is (n + 1) −at e V˙ n = V˙ n−1 + zn z˙n − 2   n−1 n−1 ∂h ∂αj−1 ˙ ˆ (j−1) n−1  θˆ − τn−1 + =− cj zj2 +  zj zj ∂ θˆ ∂ θˆ j=1

+

j=2

n−1

j=3

˙ ξj zj − ζj zj2 + θ˜T Γ −1 τn−1 − θˆ

j=1

ˆ (n−1) ∂αn−1 ∂h f + φθˆ + g + ψ θˆ v1 + +zn zn−1 + ∂x ∂x n−ρ ∂h ˆ (n−1) ˆ (n−1) ∂αn−1 ∂αn−1 ∂h + + − yr(n) vj+1 + + ∂v ∂v ∂t ∂t j j j=1 ˆ (n−1) ∂αn−1 ˆ˙ ∂h (n + 1) −at ˜ e + θ + ωn θ + ξn − + ˆ ˆ 2 ∂θ ∂θ

We ﬁnd the control law u = v1 from the diﬀerential system v˙ 1 = v2 v˙ 2 = v3 .. . v˙ n−ρ =

−1 ˆ (n−1)

∂h ∂αn−1 + ∂vn−ρ ∂vn−ρ

ˆ (n−1) ∂αn−1 ∂h ˆ g+ψ θˆ v1 f +φθ+ + ∂x ∂x

(12)

  n−1 n−1 (n−1) (j−1) ˆ ˆ ∂h ∂αn−1  ∂h ∂αj−1  + + + + Γ ωnT zj zj ˆ ˆ ∂t ∂t ∂ θ ∂ θ j=2 j=3 ˆ (n−1) ∂αn−1 ∂h + τn + zn−1 + cn zn − yr(n) + ˆ ∂θ ∂ θˆ  n−ρ−2 ˆ (n−1) ∂αn−1 ∂h + + vj+1 + ξn zn  ∂v ∂v j j j=1

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with cn > 0. Choosing the parameter estimate update law to be n ˙ ωj zj θˆ = τn = Γ j=1

we get V˙ n = −

n

n

˙ ξj zj − ζj zj2 ci zi2 + θ˜T Γ −1 τn − θˆ +

i=1



+

j=1 n j=2

≤−

n

ˆ (j−1) ∂h + zj ∂ θˆ

n j=3

zj



(n + 1) ∂αj−1  ˆ˙ e−at θ − τn − 2 ∂ θˆ

ci zi2

i=1

Taking Wn =

n

2 i=1 ci zi

and integrating

t

Wn (z(s))ds ≤ Vn (0, z(0)) − Vn (t, z(t)) 0

t Therefore the integral 0 Wn (z(s))ds exists and is ﬁnite. Hence, by Barbalat’s Lemma, limt→∞ Wn = 0. So zi → 0 as t → ∞ and the z system is stable. In particular, y − yr → 0 i.e. the output tracks the reference signal.

4

Combined sliding backstepping control of disturbed systems

Robustness can be added to the disturbed DAB algorithm by having sliding mode control at the ﬁnal stage. This disturbed DAB-SMC algorithm generates the error variables in the same way as the disturbed DAB algorithm. At the n-th step, design the sliding surface σ = k1 z1 + k2 z2 + . . . + kn−1 zn−1 + zn = 0 where ki > 0 are designed such that the polynomial in the complex variable s p(s) = k1 + k2 s + . . . + kn−1 sn−2 + sn−1 = 0 is Hurwitz. The ﬁnal Lyapunov function is selected to be 1 (n − 1) −at Vn = Vn−1 + σ 2 + e 2 2a n−1 1 2 1 2 1 ˜T −1 ˜ (n − 1) −at e = z + σ + θ Γ θ+ 2 i=1 i 2 2 2a

Backstepping for disturbed systems

235

with derivative (n − 1) −at e V˙ n = V˙ n−1 + σ σ˙ − 2   n−1 n−1 ∂h ∂αj−1 ˙ ˆ (j−1) n−1  θˆ − τn−1 + =− ci zi2 + zn−1 zn +  zj zj ∂ θˆ ∂ θˆ i=1 j=2 j=3 +

n−1

˙ ξj zj − ζj zj2 + θ˜T Γ −1 τn−1 − θˆ

j=1

+σ

ˆ (n−1) ∂h ∂αn−1 + ∂x ∂x

f + φθˆ + g + ψ θˆ v1

n−ρ−1 ˆ (n−1) ˆ (n−1) ∂αn−1 ∂h ∂αn−1 ∂h + + + + vj+1 ∂t ∂t ∂vj ∂vj j=1 ˆ (n−1) ∂αn−1 ˆ˙ ∂h + θ − yr(n) + ωn θ˜ + ξn + ∂ θˆ ∂ θˆ n−1 ˆ (i−1) ∂αi−1 ˙ ∂h θˆ − τi +ωi θ˜ + + ki −zi−1 − ci zi + zi+1 + ∂θ ∂ θˆ i=1    i−1 i−1 ˆ (j−1) ∂ h ∂α j−1  Γ ωiT  − (n − 1) e−at +ξi −ζi zi −  + zj zj ˆ ∂θ 2 ∂θ j=2 j=3 The dynamic controller can be found by solving ˆ (n−1) ∂αn−1 ∂h f + φθˆ + g + ψ θˆ v1 − yr(n) + zn−1 + ∂x ∂x   n−1 n−1 ∂h ∂αj−1 ˆ (j−1) n−1 T T  Γ ωn + + + zj zj ki ωi ∂ θˆ ∂ θˆ j=2

j=3

n−ρ−1 ˆ (n−1) ∂h

i=1

ˆ (n−1) ∂αn−1 ∂h ∂αn−1 + τn + vj+1 + + ∂vj ∂vj ∂ θˆ ∂ θˆ j=1 n−1 ˆ (i−1) ∂αi−1 ∂h + + ki −zi−1 −(ci + ζi ) zi + zi+1 + (τn −τi ) ∂θ ∂ θˆ i=1    i−1 i−1 ˆ (j−1) ˆ (n−1) ∂αn−1 ∂ h ∂α ∂h j−1  −  + + Γ ωiT  + zj zj ∂θ ∂t ∂t ∂ θˆ j=2 j=3

= −λσ − β sign σ − ζn σ − sign(σ)

n−1 i=1

ki νi

(13)

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where i ˆ (i−1) ∂αi−1 ∂h νi = + qj ∂x ∂x j=1 λ ≥ 0 and β > 0. This gives V˙ n = −

n−1

ci zi2 − zn−1

i=1

n−1

ki zi +

i=1

+σ ξn − ζn σ+

n−1

n−1

ξj zj − ζj zj2

j=1

ki (ξi −νi sign(σ))−λσ+β sign σ −

i=1

(n − 1) −at e 2

T

≤ − [z1 . . . zn−1 ] Q [z1 . . . zn−1 ] − λ σ 2 − β |σ| ≤0 where Q ∈ R(n−1)×(n−1) 

c1 0  Q= .  ..

0 c2 .. .

... ... .. .

0 0 .. .

   >0 

(14)

k1 k2 . . . kn−1 + cn−1 This guarantees the existence of the sliding mode and asymptotic stability of the system.

5

Example

Consider the model of a dynamically ﬁeld-controlled DC motor [6] x˙ 1 = −θ1 x1 + θ2 x2 u + θ3 + η x˙ 2 = −θ4 x2 + θ5 x1 u y = x2 where x1 is the armature current and x2 is the angular velocity of the rotating axis. In [6], the undisturbed system was controlled. Here, a noise term η is added, where |η| ≤ x21 . Notice that this system is not in the parametric semi-strict feedback form.

Backstepping for disturbed systems

237

The output y can be regulated to zero by applying the DAB-EXT algorithm. z1 = x2 ω1 = [0, 0, 0, − x2 , x1 u] τ1 = Γ ω1T z1 ˆ (1) = −θˆ4 x2 + θˆ5 x1 u h α1 = c1 z1 z2 = −θˆ4 x2 + θˆ5 x1 u + c1 z1 ω2 = −ux1 θˆ5 , − u2 x2 θˆ5 , uθˆ5 , − (c1 − θˆ4 )x2 , (c1 − θˆ4 )x1 u

τ2 = Γ ω1T z1 + ω2T z2 2 1 ζ2 = eat uθˆ5 x21 2 −1 ˆ ˆ u˙ = θ5 u −θ1 x1 − θˆ2 x2 u + θˆ3 + c1 − θˆ4 −θˆ4 x2 + θˆ5 x1 u x1 θˆ5

[

+ζ2 z2 + ω1T τ2 + z1 + c2 z2

]

With this choice of dynamic controller, the system is made stable. This can be seen by plotting the system responses against time. Taking ˆ = [1, 0, 1, 2, 3]T , u(0) = 1; • initial conditions x(0) = [3, 2]T , θ(0) • design parameters c1 = c2 = 2, Γ = 0.1I; and • simulation unknowns θ = [2, −3, 0.25, 1.6, 2]T and η = x21 cos(16πt) the results are plotted in Figure 1.

6

Conclusions

Sliding mode control is a robust control method design and adaptive backstepping is an adaptive control design method. In this paper, the control design has beneﬁted from both design approaches. Backstepping control and sliding backstepping control were developed for a class of nonlinear systems with unmodelled or external disturbances. An illustrative example shows the eﬀectiveness of the design.

R.E. Mills, A.J. Koshkouei, and A.S.I. Zinober State variable x1

3 2 1 1

2

3

1 0.8 0.6 0.4

0

1

2

1.2

1

0

1

2

2

0

1

2

3

0

1

2

3

0

1

2

3

0

1

2

3

0.2

0

1.9 1.8

3

3.1

1

3.05

3

0

0.4

3

1.4

1

Estimate of theta2

0

2

Estimate of theta4

0

Control law

Estimate of theta5

Estimate of theta3

Estimate of theta1

State variable x1

238

0

1

2

3

0.5 0 −0.5

Fig. 1. Responses for disturbed nontriangular system

References 1. Kanellakopoulos, I., Kokotovi´c, P. V. , Marino, R. (1991) An extended direct scheme for robust adaptive nonlinear control. Automatica, 27, 247–255 2. Koshkouei, A. J., Mills, R. E., Zinober, A. S. I. (2001) Adaptive backstepping control. In: Variable Structure Systems: Towards the 21st Century, Eds: X. Yu and J.-X. Xu, Springer-Verlag, London. 3. Koshkouei, A. J., Zinober, A. S. I. (1999) Adaptive sliding backstepping control of nonlinear semi-strict feedback form systems. In: Proceedings of the 7th IEEE Mediterranean Control Conference, Haifa, Israel. 4. Krsti´c, M., Kanellakopoulos, I., Kokotovi´c, P. V. (1992) Adaptive nonlinear control without overparametrization. System and Control Letters, 19, 177–185. 5. Mills, R. E. (2001) Robust backstepping control of nonlinear uncertain systems, PhD Dissertation, The University of Sheﬃeld, UK. 6. Rios-Bol´ıvar, M. (1997) Adaptive backstepping and sliding mode control of uncertain nonlinear systems, PhD Dissertation, The University of Sheﬃeld, UK. 7. Rios-Bol´ıvar, M., Sira-Ram´ırez, H., Zinober, A. S. I. (1994) Output tracking control via adaptive input-output linearization: A backstepping approach. In: Proc. 34th CDC, New Orleans, USA. 2, 1579–1584

Nonminimum phase output tracking using sliding modes Govert Monsees and Jacquelien Scherpen Delft University of Technology, Faculty of Information Technology and Systems, Control Systems Engineering, PO Box 5031, 2600 GA Delft, The Netherlands {G.Monsees, J.M.A.Scherpen}@ITS.TUDelft.nl Abstract. This paper introduces a stable feedforward controller which gives approximate tracking for discrete time nonminimum phase linear systems by only using a bounded preview of the desired output signal. The methods presented in this paper are based on the method of Dichotomies, which is adapted to the bounded preview time. Special attention is given to marginally stable zeros, which were not permitted in previous publications. It is shown that the error due to the bounded preview time is bounded, and decreasing while increasing the preview time. The performance of the feedforward controller is then further increased by adding a sliding mode controller within the feedforward controller. A simulation example demonstrates the applicability of the proposed controllers.

1

Introduction

Precision output tracking of sampled data systems still forms an important control problem. To obtain both good tracking and robustness, the control problem is typically split into two parts. The ﬁrst part is the computation of the feedforward controller which ideally results in perfect tracking for the nominal system. The second part of the control problem is the design of a feed-back controller which is added to the system to increase robustness against parameter variations and disturbances. Whereas output tracking for minimum phase systems is relatively easy to achieve, nonminimum phase systems still lead to severe problems. The main problem encountered for nonminimum phase systems is the fact that the inverse of the system is unstable, leading to an unstable feed-forward controller. It is well known that a stable feedforward controller, resulting in perfect tracking of a nonminimum phase system, can be constructed in the case where the desired output signal is fully known in advance. The topic of this paper is the design of a stable feedforward controller which only uses a bounded preview of the desired output signal. We consider the target tracking problem for the system: x[k + 1] = Ax[k] + Bu[k]

(1a)

y[k] = Cx[k] + Du[k]

(1b)

with x ∈ R , y ∈ R , u ∈ R , where p ≥ m. The system is assumed to be controllable, observable, and stable. The latter condition can always be n

p

m

A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 239-248, 2003. Springer-Verlag Berlin Heidelberg 2003

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met by adding a feedback loop which stabilizes the system. Now the problem of target tracking can be deﬁned as generating a bounded input u[·] which results in an output y[·] which is equal to the desired output yd [·]. We will call this generated input the desired input ud [·]. In this paper, we will generate the input signal in a feedforward controller and hence we are aiming at generating an input signal which gives perfect tracking for the model. To ensure that the actual system will track the desired output as well a feedback loop could be added to reduce the sensitivity to disturbances and model errors. This is, however, outside the scope of this paper. We assume that, either by identiﬁcation or by white box modeling, the matrices (A, B, C, D) are available. Before we go into the details of the feedforward controller design, we mention that for nonsquare systems (i.e. p > m) perfect tracking is not possible for an arbitrary desired output. Therefore, we deﬁne the square system: x[k + 1] = Ax[k] + Bu[k] z[k] = C x[k] + D u[k]

(2a) (2b)

where the output z[·] is deﬁned by z[·] = P y[·], P ∈ Rm×p and Rank {P } = m. Similarly, the desired output for the square system can be found from zd [·] = P yd [·]. The projection matrix P can be used as a trade-oﬀ of importance of each entry of the output vector. In the special case that yd [·] is tractable (i.e. there exists some input ud [·] resulting in yd [·]), tracking of zd [·] automatically leads to the tracking of yd [·]. In Section 2 a description is given of the feedforward controller design technique called the Method of Dichotomies (MD) introduced in [1] for nonlinear continuous time systems. The method was translated to the discrete time linear case in for example [2]. As can be seen in Section 2, the method relies on a preview of the desired output. For minimum phase systems the preview time is given by the relative degree of the system. However, for nonminimum phase systems the preview time is inﬁnite, i.e. the target signal should be known entirely in advance. In Section 3 we consider the case where the preview time is bounded while still larger then the relative degree of the nonminimum phase system. The presented approach is inspired by the method given in [4] for the Steering Along the Zeros Control (SAZC) method, which was introduced in [5]. Section 3 shows that the tracking error due to the ﬁnite preview time is bounded and decreasing while increasing the preview time. Then Section 4 introduces a feedforward controller structure which increases the tracking performance considerably. This is accomplished by including a sliding mode controller within the feedforward controller. The simulation example in Section 5 demonstrates the applicability of the derived ﬁnite preview time controllers for nonminimum phase systems. Finally Section 6 presents the conclusions.

Nonminimum phase output tracking using sliding modes

2

241

Method of dichotomies

The Method of Dichotomies (MD) was introduced in [1] for nonlinear continuous time systems, and extended to the class of linear discrete time systems in for example [2]. In both cases, marginally stable zero dynamics (i.e. zeros on the unit circle for linear discrete time systems) were not permitted. In this section the MD is presented, where the case of marginally stable zero dynamics are considered as well. By shifting the output z[·] deﬁned in equation (2b) forward in time by T with each ri being an integer the relative degree r (r = r1 r2 . . . rm number) we obtain the equation: ˜ ˜ z[k + r] = Cx[k] + Du[k]

(3)

By z[k + r] it is meant that the ith entry of z[k] is shifted by ri steps forward ˜ are deﬁned by: in time. The matrices C˜ and D     c1 Ar1 c1 Ar1 −1 B  c2 Ar2   c2 Ar2 −1 B              ˜ = C˜ =  .  D  .. .     .  .        rm −1 rm B cp A cp A T By deﬁnition, the relative degree r is such that for any r = [r1 . . . rm ] ˜ with at least one component ri < ri , the matrix D computed from r is not ˜ computed from r is invertible. invertible, while D Obviously the following input signal ud leads to perfect tracking of the desired output zd with a delay r:

˜ −1 zd [k + r] − Cx[k] ˜ ud [k] = D (4)

With the above control law, we can write for the closed-loop dynamics:

˜ −1 zd [k + r] ˜ −1 C˜ x[k] + B D x[k + 1] = A − B D (5) The poles of the closed-loop system correspond to the eigenvalues of the ˜ which also contains the zeros of the system. Hence, ˜ −1 C, matrix A˜ = A − B D for non-minimum phase systems, the above direct inversion technique does not lead to a stable closed-loop system. By the nonsingular transformation Tb : ζ[k] = Tb x[k] η[k]

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we can bring the ζ[k + 1] = η[k + 1]

system (5) into the Brunovsky canonical form [3]: ˜1 A˜11 0 B ζ[k] + ˜ zd [k + r] η[k] A˜21 A˜22 B2 ζ[k] z[k] = C˜1 0 η[k]

(6a) (6b)

with: T ζ[k] = z1 [k] z1 [k + 1] . . . z1 [k + r1 − 1] . . . zm [k] . . . zm [k + rm − 1] (7) The poles of the closed-loop system (6a) are given the eigenvalues of the

by m matrices A˜11 and A˜22 . The matrix A˜11 has rΣ = i=1 ri eigenvalues located at the origin. The remaining n − rΣ eigenvalues, corresponding to those of A˜22 , are located at the zeros of the output z[·]. The variable zd [·] in equation (6a) is completely speciﬁed and also ζ[·] can be substituted by ζd [·] (ζd [·] can be found by replacing all entries zi [·] in equation (7) by zi,d [·]) and hence is completely speciﬁed. However, η[·], representing the zero dynamics, still has to be found. The description for the zero dynamics can be extracted from equation (6a): ˜2 zd [k + r] + A˜21 ζd [k] η[k + 1] = A˜22 η[k] + B

(8)

We deﬁne the transformation matrix Tη such that:     ηs [·] As 0 0  ηm [·]  = Tη η[·]  0 Am 0  = Tη A˜22 Tη−1 , ηu [·] 0 0 Au where As contains all strictly stable eigenvalues of A22 , Am contains all marginally stable eigenvalues of A22 , and Au contains all strictly unstable eigenvalues of A22 . Applying the transformation Tη to the zero dynamics given in equation (6a) results in: ˜2,s zd [k + r] + A˜21,s ζd [k] ηs [k + 1] = As ηs [k] + B ˜2,m zd [k + r] + A˜21,m ζd [k] ηm [k + 1] = Am ηm [k] + B

(9b)

˜2,u zd [k + r] + A˜21,u ζd [k] ηu [k + 1] = Au ηu [k] + B

(9c)

(9a)

The solutions for equations (9a) and (9b) can be found by simple simulation starting from some known initial ηs [ki ] and ηm [ki ]. Since the eigenvalues of the matrix Au are unstable this method cannot be employed to obtain the solution for equation (9c). However, if the ﬁnal value ηu [kf ] is known, the solution for equation (9c) can be found by solving it backward in time. Summarizing, we can solve equations (9a), (9b) and (9c) under the following conditions:

Nonminimum phase output tracking using sliding modes

243

I The initial and ﬁnal states x[−∞] and x[∞] (and consequently the zero dynamics) are known to be zero. II The desired output trajectory zd [·] is bounded and has ﬁnite support. III The desired output trajectory zd [·] is such that it stabilizes the marginally stable zero dynamics. Conditions II and III are additional conditions which guarantee that the boundary conditions given in condition I exist for the given desired output trajectory. Keeping these conditions in mind, the solutions for ηs [k], ηm [k], and ηu [k] can then be found to be: k−1

ηs [k] =

˜2,s zd [i + r] + A˜21,s ζd [i] B A(k−i) s

(10a)

˜2,m zd [i + r] + A˜21,m ζd [i] B A(k−i) m

(10b)

˜2,u zd [i + r] + A˜21,u ζd [i] B A−(i+1−k) u

(10c)

i=−∞ k−1

ηm [k] =

i=−∞ ∞

ηu [k] = −

i=k

Condition III is illustrated for the use of one zero at z = 1 in the following example. Example 1. Suppose that the system under consideration has one marginally stable zero at z = 1. Furthermore let us consider that the system started at the initial state x[−∞] = 0, therefore ηm [−∞] = 0. From equation (10b) the ﬁnal value ηm [∞] can be found to be: ηm [∞] =

∞

˜2,m zd [i + r] + A˜21,m ζd [i] B

i=−∞

By deﬁnition (see equation (7)), ζd [·] is constructed by the delayed components of zd [·]. Therefore it can easily be seen that ηm [∞] is a linear combina∞ tion of

∞ i=−∞ zd [i]. To guarantee that ηm [∞] returns to zero it is suﬃcient that i=−∞ zd [i] = 0. With the generated zero dynamics (equations (10a), (10b) and (10c)), the desired state can be constructed as:    ζd [k]   ηs [k]   (11) xd [k] = Tb−1   Tη−1  ηm [k]   ηu [k] Substituting the above desired state into equation (4) results in the desired input signal. For the model, the desired input signal leads to perfect tracking. From the equations (10a), (10b), and (10c), it can be seen that the developed feedforward controller is acausal. Computing the stable and marginally

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stable part requires the desired output max{ri }i=1...m time steps in advance. However, the computation of the unstable part depends on the total future of the desired output trajectory. The next section considers the case where only a bounded preview time of the desired output trajectory is available.

3

Bounded preview time

In this section we consider the case where the preview time, i.e. the time that the desired output signal is known in advance, is limited. Therefore we will adapt the method of Dichotomies which results in the Method of Dichotomies with Bounded Preview time (MD+BP). We will assume that the number of time steps that the desired output signal is known in advance is larger then the largest entry of the relative degree vector. Then it can be seen that equations (10a) and (10b) are not aﬀected by this in the sense that all required information to compute ηs [k] and ηm [k] are available at time k. This is not the case for the unstable zero dynamics represented by equation (10c). Setting all unknown terms of zd [k + i] ∀ i > Tpre to zero we come to the following estimate of ηu [k]:

Tpre

ηˆu [k] = −

˜2,u zd [i + r] + A˜21,u ζd [i] Au−(i+1−k) B

(12)

i=k

The total preview time is given by Ttot = Tpre + max{ri }i=1...m . The error of the above estimation can be found to be: epre [k] = −

∞

˜2,u zd [i + r] + A˜21,u ζd [i] Au−(i+1−k) B

(13)

i=Tpre +1

By boundedness of the desired output signal zd [·] and the stable eigenvalues of the matrix A−1 u , the above error signal is bounded as well. Furthermore, it is easy to see that the estimation error decreases while increasing the preview time Tpre . In the limit case Tpre → ∞, the error epre becomes zero. ˆd [·] and the tracking error We can ﬁnd for the error state ex [·] = xd [·] − x ey [·] = yd [·] − yˆd [·]:    0    0 ˜ −1  −1 ˜ −1 CT  (14a) ex [k + 1] = Aex [k] + B D b T    0 η epre [k]    0    0 ˜ −1  −1 ˜ −1 CT  ey [k] = Cex [k] + DD (14b) b T    0 η epre [k]

Nonminimum phase output tracking using sliding modes

245

The above error system is driven by the error signal epre [·]. It was already concluded that epre [·] is bounded and by assumption the system (A, B, C, D) is stable. Therefore, the tracking error ey [·] is bounded as well. As can be seen in the simulation example of Section 5, the presented method for a bounded preview time works very well for a small preview time. For even shorter preview times the performance decreases considerably. Therefore, the next section presents the combination of a sliding mode controller with the above procedure. Remark 1. In this section the desired output trajectory was estimated to be zero after the preview horizon represented by Ttot . Of course, depending on the application, more sophisticated approximation schemes could be employed such as polynomial extrapolation.

4

Bounded preview time with sliding mode control

MD+BP, introduced in the previous section, results in an estimate of the desired state xd [·]. Since the estimation error is made in the zero dynamics, the ˆd [·]. Unfortunately, the generated state exhibits the property that yd [·] = C x computed input signal by MD+BP does not result in the estimated desired state signal. Therefore, we simulate the model within the feedforward controller to update the achieved state. The estimated desired state is used as the target signal for a state based sliding mode controller which controls the simulated model. The resulting controller is called the Method of Dichotomies with Bounded Preview time and Sliding Mode Control (MD+BP+SMC). The scheme is presented in Figure 1, where the shaded box represents the total feedforward controller. For an introduction to discrete-time sliding mode

Fig. 1. MD+BP+SMC schematic. The shaded box represents the boundaries of the feedforward controller.

controller we refer to for example [6]. We deﬁne the switching function as: σ[k] = S(˜ xd [k] − x ˆd [k])

(15)

with σ ∈ Rm and S ∈ Rm×n . It is assumed that the design parameter S is chosen such that in sliding mode (i.e. σ[·] = 0) the simulated model is stable.

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For the design procedure of S we refer to for example [6]. The variable x ˜ d [·] represents the estimated desired state obtained by MD+BP. The variable ˆxd [·] is the state of the simulated model. Deﬁning the reaching law σ[k+1] = Φσ[k] (where the matrix Φ ∈ Rm×m has all its eigenvalues within the unit circle) we can determine the control law from equations (15) and (2a) to be: ˜d [k + 1] − SAˆ xd [k] − Φσ[k]) uf f [k] = (SB)−1 (S x

(16)

The above equation relies on the estimated desired state at time k + 1, therefore the total preview time is increased by one time step (i.e. Ttot = Tpre + max{r} + 1). In the next section the applicability of the above proposed structure will be demonstrated with a simple simulation example.

5

Simulation

In this section we compare the tracking performance of the MD+BP (Section 3) and MD+BP+SMC (Section 4). We consider the system:     0.9997 −0.0003 0.0050 0 0.0005  0.0027 0.9123 0.0004 0.0044   0.0850     x[k + 1] =   −0.1057 −0.2074 0.9831 0.0163  x[k] +  0.3131  u[k] 1.0233 −33.498 0.1635 0.7515 32.4748 y[k] = 1 0 0 0 x[k] The above system has relative degree r = 1, zeros {−3.54, 0.969, −0.260} and poles {0.8323 ± 0.373i, 0.991 ± 0.022i}. The system is stable, controllable, observable, nonminimum phase, and square. The sampling time is given by Ts = 5ms. The parameters for the sliding mode controller are given by Φ = 0 and S = [0.93 − 2.33 − 0.48 − 0.99]. Figure 2 presents the target signal. In

yd[k]

0.5

0

−0.5 −1

−0.5

0

0.5

1

1.5

2

2.5

3

time [s]

Fig. 2. Desired . output trajectory yd

Figure 3 the xy-plots are presented for MD+BP and MD+BP+SMC, for the same total preview time Ttot = 4Ts . The ﬁgures show that the performances

Nonminimum phase output tracking using sliding modes

MD+BP+SMC

1

1

0.5

0.5

yMD+BP+SMC

yMD+BP

MD+BP

0

−0.5

−1 −1

247

0

−0.5

−0.5

0

0.5

1

−1 −1

y

−0.5

0

0.5

1

y

d

d

Fig. 3. xy-plot of achieved tracking. The diagonal dashed represents perfect tracking.

of MD+BP+SMC is better the MD+BP. Another tracking criterion is the Variance Accounted For (VAF), deﬁned by: V ar(r1 − r2 ) V AF (r1 , r2 ) = 1 − 100% V ar(r1 ) Computing the VAF for the three cases results in: V AF (yd , yM D+BP ) = 75.1%

V AF (yd , yM D+BP +SM C ) = 98.6%

In Figure 4 the VAF is displayed for a total preview time varying from Ttot = 2Ts to Ttot = 7Ts . There it can be seen that MD+BP+SMC already has a VAF of 99% for a total preview time of Ttot = 4Ts , where the MD+BP requires a minimum total preview time of Ttot = 6Ts to obtain a VAF which is better then 99%.

6

Conclusions

In this paper the problem of nonminimum phase output tracking was considered for linear discrete time systems. Furthermore, it was assumed that there is only a limited amount of information available in advance of the desired output signal. Therefore, the method of Dichotomies was extended to the bounded preview time problem. It was shown that the tracking error due to the bounded preview time is bounded and that it is decreasing while increasing the preview time. The performance of the feedforward controller was considerably increased by the inclusion of a sliding mode controller in the feedforward path. Even in the case of extremely short preview times, the performance of the proposed feedforward controller setup is still acceptable. A simulation example shows the applicability of the proposed controller setup. Despite a relatively short preview time, the proposed feedforward controllers

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Fig. 4. Variance Accounted For (VAF) versus total preview time (Ttot ) for the MD+BP and the MD+BP+SMC.

still lead to approximate tracking. The inclusion of a sliding mode controller improves the tracking performance even further.

References 1. Devasia, S., Chen, D., and Paden, B. (1996) Nonlinear inversion-based output tracking. IEEE Transactions on Automatic Control, 41(7):930–942, July. 2. George, K., Verhaegen, M., and Scherpen, J. M. A. (1999) A Systematic and Numerically Eﬃcient Procedure for Stable Dynamic Model Inversion of LTI Systems. Proceedings of the 38th IEEE Conference on Decision and Control, 1881–1886. 3. Isidori, A. (1995) Nonlinear Control Systems. Springer-Verlag, London, U.K., 3 edition. 4. Marconi, L., Marro, G., and Melchiorri, C. (2001) A solution for almost perfect tracking of nonminimum-phase, discrete-time linear systems. International Journal of Control, 74(5):496–506. 5. Marro, G., and Hamano, F. (1996) Using preactuation to eliminate tracking error in feedback control of nonminimum-phase systems. Proceedings of the IEEE Conference on Decision and Control, Kobe, Japan, 4:4549–4551. 6. Monsees, G., and Scherpen, J. M. A. (2001) Discrete-Time Sliding Mode Control with a Disturbance Estimator. Proceedings of the European Control Conference 2001, 3270–3275. ¨ uner, U. ¨ (1999) A control engineer’s guide to 7. Young, K. D., Utkin, V., and Ozg¨ sliding mode control. IEEE Transactions on Control Systems Technology, 7(3).

Goursat distributions not strongly nilpotent in dimensions not exceeding seven Piotr Mormul Institute of Mathematics, Warsaw University, Banacha 2, 02–097 Warsaw, Poland Email: mormul@ mimuw.edu.pl Abstract. We ﬁrst give a proof of a result announced in [16] that Goursat distributions of arbitrary corank (= length of the associated ﬂag of consecutive Lie squares of a G. distribution) locally possess nilpotent bases (i. e., bases generating over R nilpotent Lie algebras) of explicitly computable orders of nilpotency of the induced Lie algebras (KR algebras). We say that G. distributions are locally weakly nilpotent in the sense of [11]. Recalling that the germs of such distributions are stratiﬁed into geometric classes of Jean, Montgomery and Zhitomirskii, in certain geometric classes termed tangential, the computed nilpotency orders of KR algebras turn out to coincide with the nonholonomy degrees, computed by Jean, at the reference points for germs. In the tangential classes, then, the nilpotency orders of KR algebras are minimal among all possible nilpotent bases. Secondly, in dimension 6 and 7, two smallest dimensions in which not all geometric classes are tangential, we show that all G. germs in the non-tangential classes are not strongly nilpotent in the sense of [2].

1

Introduction: Geometric classes (basic geometry) of Goursat germs

Goursat distributions are special rank–2 subbundles D in the tangent bundle T M to an (r + 2)–dimensional manifold M s. t. the Lie square [D, D] of D is rank–3 everywhere, the Lie square of [D, D] is everywhere rank–4, and so on, the ranks of consecutive Lie squares growing very slowly – always by 1 – until reaching r + 2 (the entire T M ) in altogether r steps. A Goursat ﬂag of length r, F , is any such D together with the nested sequence of its consecutive Lie squares through T M inclusively. The members of F are indexed – this is important – by their coranks; ‘a next’ means ‘a Lie square root of’. (If D is a general distribution, then via consecutive Lie squares produced is the big ﬂag of D.) The work by Kumpera and Ruiz [9], [13] revealed singularities (singularities of the 1st order, to be precise) hidden in the seemingly regular world of Goursat objects. First version of a geometric characterization of those singularities can be found in [5], p. 455 (cf. also [8], p. 506). After that the paper [12] brought in plenty of new information on Goursat, although encoded in the language of a kinematic model of cars pulling strings of passive trailers, trailers indexed backwards starting from the last (r − 1 trailers giving rise to A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 249-261, 2003. Springer-Verlag Berlin Heidelberg 2003

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corank–r Goursat distributions, which was clear from the recurrence formulas describing the kinematics of the model). Jean proposed a neat partition of the conﬁguration space into ‘regions’, or else: a stratiﬁcation in terms of √ certain critical values a1 = π2 , a2 = π4 , a3 = arctan 22 , a4 , . . . of the angles in that model, with strata of codimensions ranging from 0 to the number of trailers less one. As a matter of fact, the singularities of 1st and also higher orders were thus implicitly deﬁned! The accent in Jean’s work was put on the small growth vector,1 very diﬃcult to compute and the same within a given stratum. That partition is even sometimes called the stratiﬁcation by [small] growth vectors. But this is misleading, and more important than the vectors are singularities of Goursat objects of all orders implicitly present in [12]. Jean’s partition could now be termed – cf. two paragraphs down – the stratiﬁcation by diﬀerent basic geometries. In a canonical geometric way, singularities were deﬁned and used only in [15], including those of higher orders (see in [15]: Ex. 2, the list of singularities for length 4 on p. 473, Ex. 3, and Ex. ‘s + 1’ on p. 475), although the letter coding recalled below was not included in [15]. (In between [12] and [15], in [7], Chap. 6, Jean strata were given r-cipher codes using three ciphers 1, 2, 3 – prototypes of the present codes.) This notwithstanding, by the middle of 1999 Montgomery and Zhitomirskii already brought to surface the true geometrical meaning of Jean’s strata. They called it the basic geometry of a Goursat germ, and the strata – geometric classes of Goursat germs, and encoded them by words consisting of letters instead of ciphers: G (instead of 2), S (instead of 3), and T (instead of 1). That contribution, explained to us by Zhitomirskii in the middle of 1999, has been recalled in Sect. 1.3 of [18]. Since then the codes for geometric classes of corank–r Goursat germs are r–letter words such that two ﬁrst letters are always G and never a T goes directly after a G. In the briefest r´esum´e, letters S in a stratum word encode the positions of the right angles ±a1 in an instantaneous conﬁguration of trailers. In the equivalent geometric language, they indicate all ﬂag’s members being, at a point, in the 1st order singular positions. That is, coalescing with the Cauchy– characteristic directions of the square of the square of such member. Letter(s) T going after an S encode tangent positions of consecutive ﬂag’s members. In the trailers’ model, again equivalently, such letters T encode the appearances of consecutive critical angles ±a2 , ±a3 , . . . in an instantaneous conﬁguration. The absence of further tangent positions at the reference point means a code ﬁlled with G’s until the next S in it (or until the end; cf. Sect. 1.3 of [18] for all details, including the realization problem for codes). 1

The small growth vector of a distribution D at a point p is the sequence of linear dimensions at p of modules of vector ﬁelds: V1 = D, Vi+1 = Vi + [D, Vi ], i = 1, 2, . . . For D completely nonholonomic this depending on p vector always ends with the value dim M ; its length is then called the degree of nonholonomy of D at p, dNH , while D = V1 ⊂ V2 ⊂ · · · is called the small ﬂag of D.

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By elementary combinatorics, there are F2r−3 (Fibonacci number) codes of geometric classes for corank r. Thus, for length 2 there is but one class GG, for length 3 – only GGG and GGS, for length 4 – GGGG, GGSG, GGST, GGSS, GGGS. The classes are, obviously, pairwise disjoint and invariant under the action of the local diﬀeomorphisms between manifolds. Deﬁnition 1. For a ﬁxed corank, the union of all geometric classes with letters S in ﬁxed positions in the codes is called, after [15], a Kumpera-Ruiz class of Goursat germs of that corank. Consequently, there are ‘only’ 2r−2 Kumpera-Ruiz classes of germs of corank r. The simplest (and the fattest) among them is GG. . . G with r letters G, featuring no singularities. The classical works of von Weber [22] and E. Cartan [6] dealt locally with Goursat ﬂags precisely in these classes, and gave an (indexed by r) series of local models for them. These are so-called chained systems – the germs at 0 ∈ Rr+2 (x1 , x2 , . . . , xr+2 ) of ∂r+2 , ∂1 + x3 ∂2 + x4 ∂3 + · · · + xr+2 ∂r+1 (C) (here and in the sequel we skip writing ‘span’ before a set of vector ﬁeld generators).

2

Kumpera–Ruiz algebras. Theorems of the paper

Since the 80’s – [1], [10], [11], [14], [19], [20] – there is interest in control theory, for inst. in the motion planning problem (and recently also in subRiemannian geometry, cf. [2], [3], [21]) in nilpotent bases for distributions associated to control systems linear in controls. That is, in bases generating over R nilpotent Lie algebras. Deﬁnition 2. In a nilpotent Lie algebra, the minimal number of Lie multiplications yielding always zero is called its order of nilpotency (or, shorter, ‘nilpotency order’), see [14], p. 239. In 2000 we proved (cf. Rem. 1 in [16]) that Goursat distributions locally belong to this distinguished class – by giving explicit formulas2 for the order of nilpotency of the real algebras induced by produced bases. In fact Theorem 1. Goursat distributions of corank r, locally around a point belonging to a Jean stratum C, possess nilpotent bases (X, Y) of nilpotency order dr , where dr is the last term in the sequence d1 , d2 , . . . , dr deﬁned only in function of the Kumpera-Ruiz class subsuming C: d1 = 2, d2 = 3, dj+2 = dj +dj+1 when the (j +2)-th letter in C is S, and otherwise dj+2 = 2dj+1 −dj . Note that, in [20], a result stating just the nilpotency of the same algebras – the existence of nilpotency orders – is given. Thm. 1 is proved in Sect. 3. 2

presented during the workshop Math. Control Th. and Robotics, Trieste 2000

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In the proof, speciﬁcally for Goursat germs (when written in particularly suited coordinates prompting a basis) constructed are certain key integers that lead, in two steps, to the answer dr . Upon inspection, they turn out to be the weights that can be naturally associated, as for inst. in [1], to the small ﬂag of any completely non-holonomic distribution germ. How to ﬁnd such bases? It appears that a basis naturally prompted by any Kumpera-Ruiz coordinates (or, better, glasses) x1 , . . . , xr+2 for a Goursat germ does. The essence of the result in [13], given in the notation of vector 1

2

ﬁelds, is as follows: when C starts with s letters G, one puts Y = ∂1 , Y = 1

s+1

s

Y + x3 ∂2 , . . . , Y = Y + xs+2 ∂s+1 . When s < r, then the (s + 1)th letter in m

C is S. More generally, if the mth letter in C is S, and Y is already deﬁned, then m+1

Y

m

= xm+2 Y + ∂m+1 .

But there can also be T’s or G’s after an S. If the mth letter in C is not S, m

and Y is already deﬁned, then m+1

Y

m = Y + cm+2 + xm+2 ∂m+1 ,

where a real constant cm+2 is not absolutely free but • equal to 0 when the mth letter in C is T, • not equal to 0 when the mth letter is G going directly after a string ST. . . T (or after a short string S). In the sequel we will write shortly X m+2 = cm+2 + xm+2 . Now, on putting r+1

X = ∂r+2 and Y = Y , and understanding (X, Y) as the germ at 0 ∈ Rr+2 , Theorem 2 ([13]). Any Goursat germ D of corank r on a manifold of dimension r + 2, sitting in a geometric class C, can be put (in certain local coordinates) in a form D = (X, Y), with certain constants in the ﬁeld Y corresponding to G’s past the ﬁrst S in C. Deﬁnition 3. The real Lie algebra generated by X and Y we call shortly the KR algebra of D. This algebra does not depend on the choice of coordinates in Thm. 2, although so do the constants in Y. Its nilpotency order we denote henceforth by OKR . (A short analysis shows that this algebra depends solely on the KumperaRuiz class of D.) Deﬁnition 4. We will call tangential the geometric classes whose codes possess letters G only in the beginning, before the ﬁrst S (if any) in the code. Tangential (non-tangential) germs are those sitting in the tangential (resp., non-tangential) classes.

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Example 1. Up to dimension 5 all geometric classes are tangential. The ﬁrst, and unique non-tangential class in dimension 6 is GGSG. In dimension 7 there are eight tangential classes and ﬁve non-tangential: GGSGG, GGSTG, GGGSG, GGSSG, GGSGS. All these non-tangential classes are addressed in Thm. 3 below. Observation 1. In every KR presentation (X, Y) of a Goursat germ D there is no non-zero constant ⇐⇒ the geometric class of D is tangential. So the KR presentations for tangential germs are unique – such germs are easily given local models. How to compute the degree of nonholonomy of Goursat germs? The answer is already given in [12], because the car systems are universal models – [15], Appendix D, and [20] – for germs of Goursat distributions (i. e., for the Goursat objects understood locally). Namely, as is observed in [15], the car model with r − 1 trailers, being r times Cartan prolongation of the tangent bundle to a plane, is automatically modelling all corank–r Goursat germs. We note also that Thm. 4.1 in [5], attributed by Bryant to Cartan, basically covers all these observations. Proposition 1 ([12]). The nonholonomy degree dNH of any germ in the geometric class C equals the last term br in the sequence b1 , b2 , . . . , br deﬁned only in terms of C: b1 = 2, b2 = 3, bj+2 = bj + bj+1 when the (j + 2)-th letter in C is S, bj+2 = 2bj+1 − bj when the (j + 2)-th letter in C is T, bj+2 = 1 + bj+1 when the (j + 2)-th letter in C is G. Comparison of these formulas for the nonholonomy degree dNH with the ones for nilpotency order OKR given in Thm. 1 yields quickly that in each ﬁxed geometric class C • dNH ≤ OKR , • dNH = OKR only when C is tangential. In the communication [17] a complex question has been put forward if the KR algebras is the last word for the Goursat distributions: whether there exist better nilpotent bases with lower nilpotency orders. For the germs in tangential classes – of course not because OKR − 1 Lie multiplications (i. e., OKR = dNH Lie factors – vector ﬁelds) is still necessary to achieve the construction of the small ﬂag at the reference point. For the tangential germs KR bases are then optimal. Up to dimension r + 2 = 5 all geometric classes are tangential. Therefore, this question (bridging algebra and geometry) is not void from dimension 6 onwards. Cf. also Conjecture at the paper end. Deﬁnition 5. A distribution germ D is strongly nilpotent when it is equiv alent to its own nilpotent approximation D

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(cf. [2], p. 23; the hat notation for nilpotent approximations is also used in [3], where that object is shown to be well deﬁned – to be independent of the chosen coordinates). Example 2. It follows from the proof of Thm. 1 in Sect. 3 below that the Goursat germs in the tangential classes coincide with their nilpotent approximations, hence are strongly nilpotent. Such are, for inst., generic Goursat germs in the von Weber–Cartan classes GGG. . . G. We have found, in dimensions not exceeding seven, all not strongly nilpotent G. germs: it turns out that such are, simply, all non-tangential germs. In the dimension 6, we deal with the unique class GGSG that is non-tangential; in dimension 7 – with ﬁve existing such classes (cf. Ex. 1). In the wording of the result we also bring out the weights wi that have been invoked and used in the proof of Thm. 1. They are determined by the small growth vector, hence are ﬁxed within each geometric class. These weights are the departure point for the computation of nilpotent approximations. Theorem 3. A. In dimension 6, the Goursat germs in the geometric class geometric class w1 w2 w3 w4 w5 w6 = dNH OKR GGSG 1 1 2 3 4 6 7 are not strongly nilpotent. B. In dimension 7, the Goursat germs in the geometric classes listed in the table geometric class GGSGG GGSTG GGGSG GGSSG GGSGS

w1 1 1 1 1 1

w2 1 1 1 1 1

w3 2 2 2 2 2

w4 3 3 3 3 3

w5 4 4 4 4 5

w6 w7 = dNH OKR 5 7 9 5 8 9 6 8 10 6 9 11 7 11 12

are not strongly nilpotent. This theorem is proved in Sect. 4. The nilpotent approximations of germs in all listed classes simply turn out not to be Goursat objects. The weak form of nilpotency – possession of a nilpotent basis, discussed in general for the ﬁrst time in [11] and pertinent to all Goursat germs (Thm. 1 in the present text, and [20]) – appears thus much less restrictive than the strong form of nilpotency of a distribution germ.

3

Nilpotency orders of KR algebras – Proof of Thm. 1

The statement of Thm. 1 will ﬁrstly be justiﬁed for tangential germs D (see Def. 4). Recall that – Obs. 1 – for such D all possible constants in any its KR presentation (X, Y) are zero, and the value dr equals – by Prop. 1 – the

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nonholonomy degree dNH of D at the reference point. In fact, the (chosen) KR coordinates x1 , x2 , . . . , xr+2 will turn out to be certain privileged or: adapted coordinates for D in the sense of Def. 4.13 in [3], and (X, Y), when watched in these coordinates, – to coincide with its nilpotent approximation, yielding automatically OKR = dNH = dr . Secondly, and quickly, the same formula for OKR will be extended to the KR algebras of non-tangential germs in the Kumpera-Ruiz class (Def. 1) determined by the tangential germ from the ﬁrst step. First step. We are going to propose certain – very natural for KR presentations of Goursat – weights w(xl ) and w(∂l ) = − w(xl ), l = 1, 2, . . . , r + 2 that will eventually turn out to be the weights deﬁned for general completely non-holonomic distribution germs in [1], [3], [4], [10]. When any weights are attached to both variables and versors, diﬀerent terms in expansions of arbitrary vector ﬁelds are also given their weights (like in [1], p. 215). Our proposal is such that X and all terms in Y are of constant weight −1, which we write down as w(X) = w(Y) = −1 .

(1)

This claimed homogeneity of the polynomial (and involved) ﬁeld Y is the key ingredient in the proof. The deﬁnition of weights is recursive from r + 2 backwards to 1; we remember that in this step there is no constants. In the beginning we declare r+1 w Y = w(∂r+2 ) = −1, hence declare also w(xr+2 ) = 1. Assume now, for m+1 1 ≤ m ≤ r, that w Y < 0 and w(xm+2 ) > 0 are already deﬁned. The recursive deﬁnition depends on the positions of letters S in C, determining the way of prolonging the sequence of vector ﬁelds in question. m+1

(•)

If Y

m

= Y + xm+2 ∂m+1 then we put

m m+1 w Y =w Y , m+1

(••) If Y

w(∂m+1 ) = w

m+1 Y − w(xm+2 ) .

m

= xm+2 Y + ∂m+1 then we put

m m+1 w Y = w Y − w(xm+2 ) ,

w(∂m+1 ) = w

m+1 Y .

2 At the last step (m = 1), w Y = w(∂1 + x3 ∂2 ) < 0 and w(x3 ) > 0 are 1 2 2 assumed known, making w(∂1 ) = w Y = w Y and w(∂2 ) = w Y − w(x3 ) known. In the outcome, w(xr+2 ), w(xr+1 ), . . . , w(x3 ), and w(x2 ) = − w(∂2 ), w(x1 ) = − w(∂1 ) are all deﬁned in such a way that (1) is guaranteed. Observe that two variables, xr+2 and xj s. t. Y(0) = ∂j (when C = GG . . . G – model (C)

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– it is just x1 ), have weight 1, and all the remaining variables have weights bigger than 1. Fact. x1 , x2 , . . . , xr+2 are linearly adapted at 0 and their weights in the sense of Def. 4.10 in [3] are just the weights deﬁned above in the present proof. Pf. Consider the small ﬂag {Vi } of D, V1 = D (cf. footnote 1). It is established in [12] that for distributions modelling the car systems (and hence for all Goursat distributions – see the paragraph after Obs. 1) the small growth vectors have only jumps ≤ 1: Jean’s functions β n , n = the number of trailers, are strictly increasing – cf. [12], Thm. 3.1. Let us focus on the small ﬂag of (X, Y) at 0 and follow the notation of [3]. For certain positive integers wi (‘Bella¨ıche’s weights’) we have here Vwi−1 (0) = Vwi −1 (0) Vwi (0) , with dim Vwi (0) = i for i = 3, 4, . . . , r + 2. The weights w1 and w2 are just 1 because V1 (0) is two-dimensional.3 Observe that w3 = 2 and w4 = 3 independently of the germ D under consideration, and that wr+2 is but the nonholonomy degree of D. Clearly, (2) V1 (0) = ∂r+2 , ∂j . We intend to show by induction on i that, for certain permutation k3 , . . . , kr+2 of indices from 1 through r + 1 save j, (3) Vwi (0) = ∂r+2 , ∂j , ∂k3 , . . . , ∂ki for i = 3, 4, . . . , r + 2. This will precisely mean that x1 , x2 , . . . , xr+2 are linearly adapted at 0. The statement about the weights will follow automatically from the argument going to be used. So assume that (4) Vwi−1 (0) = ∂r+2 , ∂j , ∂k3 , . . . , ∂ki−1 for certain 3 ≤ i ≤ r + 2. Note that for i = 3 this assumption is just (2), and the smallest weights w1 = w2 = 1 of [3] are indeed our lowest weights w(xr+2 ) and w(xj ). From (1), any Lie product over X and Y using wi factors is homogeneous of our weight −wi . Because of this and of the absence of constants in the KR 3

Some explanations are in order. These integers wi are ﬁrst associated in [3] to the small ﬂag at a point, and only later attached, as weights, to any linearly adapted coordinates. When the small growth vector has jumps bigger than 1, those weights wi appear in groups of equal values. For Goursat, however, these are, except for r−1 = dNH . the ﬁrst group of w1 and w2 , the singletons w3 = β3r−1 , . . . , wr+2 = βr+2

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presentation, the products of wi factors from among X’s and Y’s can have at 0 non-zero ∂m – components only for m s. t. w(xm ) = wi . By the same reason, ∂r+2 , ∂j , ∂k3 , . . . , ∂ki−1 – produced earlier in the small ﬂag at 0 – have weights bigger than −wi . Hence all such potential indices m are outside the set {r + 2, j, k3 , . . . , ki−1 }. These products are solely responsible for dim Vwi (0) > dim Vwi−1 (0); hence there are such coordinates xm . On the other hand, if there were more than one such coordinates then the image of the projection Vwi (0) −→ Vwi (0)/Vwi−1 (0) ⊂ ∂m , m : w(xm ) = wi (cf. (4) ) would be a proper subspace of ∂m , m : w(xm ) = wi . The longer Lie products of X and Y, due to their higher negative weight, would never ﬁll in this gap contradicting the complete nonholonomy of D. So there is just one variable xm s. t. w(xm ) = wi ; moreover, this variable has the property that Vwi (0)/Vwi−1 (0) = (∂m ). This coupled with (4), upon putting ki = m, gives (3) for this i. Fact is now proved by induction.4 In view of Fact, the nonholonomic orders of the variables xl do not exceed their weights, cf. [3], p. 35. But any nonholonomic derivative of xl , over X and Y, of order s < w(xl ) is, again by (1), a homogeneous function of weight w(xl ) − s > 0, hence vanishes at 0 (the absence of constants is crucial also at this point). Thus the nonholonomic orders are equal to weights and the KR coordinates are adapted for D around 0. In these coordinates, always by (1), the ﬁelds X and Y coincide with their nilpotent components. The ﬁrst step is done. Second step. By Obs. 1, an arbitrarily ﬁxed KR presentation of a nontangential germ features certain constants. Whenever the case (•) applies after at least one case (••), instead of certain coordinates xm+2 there appear now shifted variables X m+2 = xm+2 + cm+2 . Lie brackets over X and Y are invariably polynomial vector ﬁelds, only now in certain variables x and certain X. What has been the conclusion in the ﬁrst step? That certain Lie product(s) over X and Y of OKR factors still yields(yield) the naked versor (numerical coef.)∂N having the highest negative weight (it is always ∂2 , which does not matter), and that any product of OKR + 1 factors yields identically zero. All these polynomial computations and results look identically in the now mixted variables x and X, leading to the same value OKR for the nilpotency order of a KR algebra of any non-tangential germ within the same Kumpera-Ruiz class. Theorem 1 is proved. 4

Note that the absence of constants is cardinal in Fact. KR coordinates for nontangential germs are not linearly adapted, let alone adapted.

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Nilpotent approximations of Goursat germs often are not Goursat (proof of Theorem 3)

Part A. By Thm. 2, the KR presentations for germs D in GGSG are (X, Y) = ∂6 , x5 ∂1 + x3 x5 ∂2 + x4 x5 ∂3 + ∂4 + X 6 ∂5 ,

(5)

0. After a rescaling we can assume for simplicity where X 6 = c6 +x6 and c6 = c6 = 1 (but beware: only few of all KR constants can be normalized, cf. Rem. 1 below). The weights wi for the germ D are given in the wording of theorem. Knowing them, the eﬀective procedure in [3] improves the KR coordinates x1 , . . . , x6 (not even being linearly adapted at 0, cf. footnote 5) to certain adapted coordinates z 1 = x6 , z2 = x4 , z3 = x5 − x4 , z4 = x1 − 12 (x4 )2 , 1 (x4 )5 z5 = x3 − 13 (x4 )3 , z6 = x2 − 15 (i. e., linearly adapted and having at 0 the non-holonomic orders wi ). In these coordinates the vector ﬁelds in (5) assume the form (X, Y) =

1 ∂1 , ∂2 + z1 ∂3 + z3 ∂4 + z2 z3 ∂5 + z2 z5 + z23 z3 + z3 z5 ∂6 . 3

We stick to the notation of [3] and repeat that a hat means the nilpotent component – all terms of weight −1 in the expansion, whenever adapted coordinates are in use. The generator X has only the nilpotent component, + Y(0) , where Y(0) = z3 z5 ∂6 is an important addition of weight while Y = Y alone is defective. To see it 2 + 4 − 6 = 0. Without it, the nilpotent part D clearly, we pass to better adapted coordinates z1 , z2 , z3 , z4 −z2 z3 , z5 − 12 z22 z3 , 1 4 takes the form (we use the same z6 − 12 z22 z5 + 24 z2 z3 . In these coordinates D letters for the new variables) Y) = ∂1 , ∂2 + z1 ∂3 − z1 z2 ∂4 − 1 z1 z 2 ∂5 + 1 z1 z 4 ∂6 , (X, 2 2 2 24 Y’s have and one sees that all products of at least two factors from among X’s, no ∂1 – and ∂2 – components and depend only on z1 , z2 . Hence the products of the big ﬂag coincides with such products vanish and, for the distribution D, at 0 is the same as that of the small one! But the small growth vector of D D (the basic property of nilpotent approximations, see [1]), hence it is [2–6] and D have diﬀerent big (an easy computation using (5) ). Consequently, D growth vectors at 0 (the latter, being Goursat, has [2–6] ) and as such are not equivalent. Part A is proved. Part B. For germs in the classes GGSGG, GGSTG, GGGSG, GGSSG there do work the arguments used in dimension 6 (naturally, some shortcuts can be introduced to the general Bella¨ıche procedure of arriving at variables adapted for a given distribution germ). To give more details, we choose the

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last class from this subgroup. Then we treat separately the last non-tangential class GGSGS in dimension 7, being resistant to these arguments. For a germ D sitting in GGSSG, here is a KR presentation (Thm. 2) (X, Y) = ∂7 , x5 x6 ∂1 + x3 x5 x6 ∂2 + x4 x5 x6 ∂3 + x6 ∂4 + ∂5 + X 7 ∂6 , with X 7 = c7 +x7 and c7 = 0. After a rescaling, also here we can assume c7 = 1. Using the wi ’s that show up in this case and shortcutting the polynomial procedure [3] of transforming x1 , . . . , x7 into certain adapted z1 , . . . , z7 , in these latter we obtain (X, Y) = ∂1 , ∂2 + z1 ∂3 + z3 ∂4 + z2 z3 ∂5 + z22 z4 + 12 z23 z3 + z2 z3 z4 ∂6 1 6 + z22 z6 + 10 z2 z3 + z2 z3 z6 ∂7 . Clearly visible is the addition part Y(0) = z2 z3 z4 ∂6 + z2 z3 z6 ∂7 without which the nilpotent remainder is geometrically deﬁcient. Indeed, working only with the latter and escaping to better, also adapted, z1 , z2 , z3 , z4 −z2 z3 , z5 − 12 z22 z3 , 1 4 1 6 1 z2 z3 , z7 − 13 z23 z6 + 18 z2 z4 + 630 z27 z3 , we have it in the form z6 − 13 z23 z4 − 24 (same letters for simplicity) Y) = ∂1 , ∂2 + z1 ∂3 − z1 z2 ∂4 − 1 z1 z 2 ∂5 − 1 z1 z 4 ∂6 + 1 z1 z 7 ∂7 . (X, 2 2 2 24 2 630 (6) The dependence only on z1 , z2 is decisive, as previously (and as in other classes in dimension 7 save GGSGS). It implies, again, that the big growth vector of (6) at 0 is equal to the small growth vector of D at 0, i. e., to [2–7] has at 0 ∈ R6 a much slower big growth vector than D. (cf. [12] ). Thus D As regards the last in the table class GGSGS, one can invariably pass from KR presentations (X, Y) = ∂7 , x5 x7 ∂1 + x3 x5 x7 ∂2 + x4 x5 x7 ∂3 + x7 ∂4 + X 6 x7 ∂5 + ∂6 , X 6 = c6 + x6 , c6 = 0 (normalizable to 1), to the presentation `a la Bella¨ıche ∂2 + z1 ∂3 + z1 z2 ∂4 + z1 z4 ∂5 + z1 z3 z4 ∂6 (X, Y) = ∂1 , (7) + z1 z3 z6 + 13 z1 z33 z4 + z1 z4 z6 ∂7 in certain adapted z1 , . . . , z7 . But the analogy ends here. After removing in (7) the term z1 z4 z6 ∂7 of weight 1 + 3 + 7 − 11 = 0, the remaining nilpotent Y) cannot be simpliﬁed to two variables because its big approximation (X, vector at 0 is not the small one of (X, Y). Yet it is not the Goursat vector, Y) is neither. A direct computation of it gives the answer [2–7]. Hence ( X, not Goursat. Theorem 3 is proved. Remark 1. By now we have shown that Goursat germs in many other nontangential geometric classes are not strongly nilpotent. Namely, in

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• all Kumpera-Ruiz classes (Def. 1) having only one letter S in the codes (in all dimensions – this subsumes Part A and 3/5 of Part B in Thm. 3), • GGSGSG in dimension 8, • in the Kumpera-Ruiz class ∗∗S∗SS∗ in dimension 9, • GGSGSGSG in dimension 10. The long computations leading to this last point show under way that an invariant parameter (module) of the local classiﬁcation of Goursat residing in GGSGSGSG disappears on the level of nilpotent approximations. The same can be said about several other, known to this day, examples of moduli of Goursat objects. Theorem 3 together with the experiments summarized above give grounds to put forward Conjecture. The Goursat germs in all non-tangential geometric classes are not strongly nilpotent: their nilpotent approximations are not equivalent to the germs themselves. Or, equivalently (cf. Ex. 2), for Goursat germs of any corank ‘not strongly nilpotent’ is a synonym of ‘non-tangential’. Note ﬁnally that stronger than this conjecture is a (probably diﬃcult, if easier than the question mentioned in Sect. 2) hypothesis that all nontangential germs do not admit local bases generating nilpotent Lie algebras of relevant nilpotency orders dNH . This statement is stronger because every nilpotent approximation admits by its construction a nilpotent basis of relevant order dNH .

References 1. Agrachev, A. A., Gamkrelidze, R. V., Sarychev, A. V. (1989) Local invariants of smooth control systems. Acta Appl. Math. 14, 191–237. 2. Agrachev, A. A., Gauthier, J–P. On subanalyticity of Carnot–Caratheodory distances. Preprint 25/2000/M, SISSA, Trieste. 3. Bella¨ıche, A. (1996) The tangent space in sub-Riemannian geometry. In: Bella¨ıche, A., Risler, J–J. (Eds.) Sub-Riemannian Geometry. Birkh¨auser, Basel Berlin, 1–78. 4. Bianchini, R. M., Stefani, G. (1990) Graded approximations and controllability along a trajectory. SIAM J. Control Optim. 28, 903–924. 5. Bryant, R., Hsu, L. (1993) Rigidity of integral curves of rank 2 distributions. Invent. math. 114, 435–461. 6. Cartan, E. (1914) Sur l’´equivalence absolue de certains syst`emes d’´equations diﬀ´erentielles et sur certaines familles de courbes. Bull. Soc. math. France XLII, 12–48. 7. Cheaito, M., Mormul, P. (1999) Rank–2 distributions satisfying the Goursat condition: all their local models in dimension 7 and 8. ESAIM: Control, Optimisation and Calculus of Variations ( URL: http://www.emath.fr/cocv/ ) 4, 137–158. 8. Cheaito, M., Mormul, P., Pasillas–L´epine, W., Respondek, W. (1998) On local classiﬁcation of Goursat structures. C. R. Acad. Sci. Paris 327, s´er. I, 503–508.

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9. Giaro, A., Kumpera, A., Ruiz, C. (1978) Sur la lecture correcte d’un r´esultat d’Elie Cartan. C. R. Acad. Sci. Paris 287, s´er. I, 241–244. 10. Hermes, H. (1986) Nilpotent approximation of control systems and distributions. SIAM J. Control Optimiz. 24, 731–736. 11. Hermes, H., Lundell, A., Sullivan, D. (1984) Nilpotent bases for distributions and control systems. J. Diﬀ. Eqns 55, 385–400. 12. Jean, F. (1996) The car with n trailers: characterisation of the singular conﬁgurations. ESAIM: Control, Optimisation and Calculus of Variations ( URL: http://www.emath.fr/cocv/ ) 1, 241–266. 13. Kumpera, A., Ruiz, C. (1982) Sur l’´equivalence locale des syst`emes de Pfaﬀ en drapeau. In: Gherardelli, F. (Ed.) Monge –Amp`ere Equations and Related Topics. Inst. Alta Math. F. Severi, Rome, 201–248. 14. Laﬀerriere, G., Sussmann, H. J. (1993) A diﬀerential geometric approach to motion planning. In: Canny, F. J., Li, Z. (Eds.) Nonholonomic Motion Planning, Kluwer, Dordrecht London, 235–270. 15. Montgomery, R., Zhitomirskii, M. (2001) Geometric approach to Goursat ﬂags. Ann. Inst. H. Poincar´e – AN 18, 459–493. 16. Mormul, P. (2000) Goursat ﬂags: classiﬁcation of codimension–one singularities. J. Dyn. Control Syst. 6, 311–330. 17. Mormul, P. (2000) The car with n trailers locally possesses a nilpotent basis. Workshop Math. Control Theory and Robotics, Trieste. 18. Mormul, P. (2001) Simple codimension–two singularities of Goursat ﬂags I: one ﬂag’s member in singular position. Preprint 01-01 (39) Inst. of Math., Warsaw University, available at http://www.mimuw.edu.pl/wydzial/raport mat.html 19. Murray, R. M. (1994) Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems. Math. Control Signals Syst. 7, 58–75. 20. Pasillas–L´epine, W., Respondek, W. (2001) Nilpotentization of the kinematics of the n-trailer system at singular points and motion planning through the singular locus. Int. J. Control 74, 628–637. 21. Sachkov, Y. (2000) On nilpotent sub-Riemannian (2, 3, 5) problem. Workshop Math. Control Theory and Robotics, Trieste. 22. Von Weber, E. (1898) Zur Invariantentheorie der Systeme Pfaﬀ’scher Gleichungen. Berichte Ges. Leipzig, Math–Phys. Classe L, 207–229.

Stabilizability for boundary-value control systems using symbolic calculus of pseudo-diﬀerential operators Markku T. Nihtil¨a and Jouko Tervo University of Kuopio, Department of Computer Science and Applied Mathematics, P.O.Box 1627, FIN-70211 Kuopio, Finland {Markku.Nihtila, Jouko.Tervo}@uku.fi Abstract. The paper considers control systems which are constructed from certain pseudo-diﬀerential and boundary-value operators. The system contains the derivative control which is after a change of variables included in many such situations where the control variables (originally) appear in the boundary condition. Suﬃcient analytical conditions for some stability properties of the transfer function are shown. In addition, we describe some methods to compute the transfer function by means of symbolic calculus. An illustrative example is considered as well.

1

Introduction

We consider some basic ideas to analyze the transfer function of linear control systems where the operators are appropriate pseudo-diﬀerential and boundary-value operators. We express our formalism tailored for the boundary-value problems related to parabolic partial diﬀerential operators (PDOs). Pseudo-diﬀerential operator calculus is usually taken to provide a framework for partial diﬀerential operator problems with a larger freedom of manipulation. One is able to form operator algebras which give eﬀective tools to study a large variety of models in physics, engineering and applied sciences. Recently one has contributed fairly complete algebras also for problems of parabolic type [7–9,13,14]. The calculus with the help of symbols, so-called (parameter-dependent) symbolic calculus, gives rules to form compositions, adjoints, generalized (right or left) inverses and compatibility conditions. One is able to study (microlocal) properties of resolvents and e.g. fractional powers as well as semigroups generated by these operators can be more completely covered. The resolvent is (under suitable assumptions) in the same algebra of operators as the system operator itself. This gives better possibilities than the conventional PDO algebras to treat controllability and stability problems by analytical and algebraic (such as described in [11]) methods. Furthermore, the trace operator type inputs/outputs can be treated within these algebras. Finally, one can manipulate the given systems to get more convenient analytical or numerical properties; for example, by eliminating one or more unknows. A well known and typical example is the linearized Navier-Stokes equation, see [6], [8]. A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 263-273, 2003. Springer-Verlag Berlin Heidelberg 2003

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In the stability problems one may use among others algebraic analysis in the desirable ring of transfer matrices. The basic idea is to study the module theoretic properties (such as torsioness, ﬂatness, coherence, projectivity, freeness) of the modules corresponding the transfer matrix. One has obtained a large variety of necessary and suﬃcient algebraic properties whether the system is stabilizable or not [4,12,15,18]. Using these methods one is also able to study the existence of (doubly) coprime factorizations and Youla parametrizations which are useful in practical controller design. The properties of a transfer function such as its inclusion to the Hardy or Callier-Desoer algebras are important in the stabilization problems, e.g. in the construction of a plant compensator of the given system. The stable plant compensator enables the design of various kind of stable feedback controllers e.g. for several tracking and disturbance rejection problems [18]. In applications the the main task is to classify the transfer function that is, one must make clear whether it is in the desired ring (or algebra) of transfer functions. The transfer function analysis can be applied only for linear problems. However, the stability or stabilization problems for nonlinear cases can be often reduced to the corresponding problems for the linearized systems. For example, for certain semilinear parabolic problems one has obtained even characterization: The problem is (stable) stabilizable if and only if the underlying linear problem is stabilizable [1]. In this paper we formulate stability properties of the linear control systems of the form ∂u ∂v = Av + B1 u + B2 ; y = D1 v + D2 u ∂t ∂t where the operators are constructed employing the above-mentioned pseudodiﬀerential and boundary-value operators. We give some preliminary ideas to analyze the transfer function G(λ) = D2 + D1 (λI − A)−1 (B1 + λB2 ) in the framework of symbolic calculus. We present some analytical results based on coercivity (which is closely related to dissipativity) of the state operator. A well-known potential tool to show the coercivity is the (symbolic) calculus of pseudo-diﬀerential operators [7]. In addition, we present an idea to handle the transfer function more explicitly with the help of symbols. 1.1

Basic notations

Let G be an open bounded set in Rn and let ∆ be an interval in R. We assume that the closure G is a diﬀerentiable manifold with boundary. The spaces C ∞ (G), C ∞ (∂G) and C ∞ (G × ∆) are, correspondingly, the collections of smooth functions G → R, ∂G → R and G × ∆ → R. C0∞ (G) is the space of test functions. Furthermore, we deﬁne C0∞ (G × R+ ) = φ ∈ C ∞ (Rn+ ) | ∃ ψ ∈ C0∞ (Rn ) s.t. φ = ψ|G ×R+ (1)

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where G is an open subset in Rn−1 . Let X be a normed space. The space C l (∆, X) denotes the l times diﬀerentiable functions ∆ → X. In C ∞ (G)N N we deﬁne an inner product v1 , v2 L2 (G)N = j=1 G v1j (x)v 2j (x)dx. The N Lebesque spaces Lp (G) are standardly deﬁned. The Sobolev spaces H k (G) = W 2,k (G), k ∈ N0 , and more generally W p,k (G), k ∈ N0 , 1 ≤ p < ∞, their inner products and norms are also standardly deﬁned. Let u ˆ (or F u) denote the Fourier transform u ˆ (ξ) = Rn u(x)e−ix,ξ dξ for n a function u in the Lebesque space L1 (R ). We denote by ρ(A) the resolvent set and by σ(A) the spectrum of an operator A. Let C = {λ ∈ C| λ > } and similarly let C = {λ ∈ C| λ ≥ } . The ˆ and for the Hardy algebra deﬁnitions for Callier-Desoer classes Aˆ− (β), B(β) H∞ (β) of bounded analytic functions on Cβ are found e.g. in [5]. The space of m × n-matrices is M (m × n) equipped with the usual Frobenius norm.

2 2.1

State space equation System operators

We formulate the (generalized) boundary control systems with the help of pseudo-diﬀerential and boundary-value operators [7,9,13]. Let S m (G × Rn ) be a space of functions a ∈ C ∞ (G×Rn ) such that for any compact set K ⊂ G and α, β ∈ Nn0 there exists a constant Cα,β,K > 0 such that |∂xβ ∂ξα a(x, ξ)| ≤ Cα,β,K (1 + |ξ|)m−|α| for all ξ ∈ Rn , x ∈ K.

(2)

In the case where we have no boundary considerations, the (classical) pseudodiﬀerential operator A : C0∞ (G) → C ∞ (G) is deﬁned by −n ix,ξ ˆ Aφ(x) = (2π) a(x, ξ)φ(ξ)e dξ. (3) Rn

When the boundary is included in the considerations the deﬁnition of pseudodiﬀerential operator contains some modiﬁcations and we need several other classes of operators. It is suﬃcient to explain the classes only in the local case that is, in the case when the closure G is replaced with a set G × R+ where G is an open interval of Rn−1 and where the boundary ∂G is replaced with a set G . The reduction to that case is standard but technically tedious and we omit it. The operator L is a pseudo-diﬀerential operator C ∞ (G) → C ∞ (G). Locally, this means that for φ ∈ C0∞ (G × R+ ) + φ)(ξ)eix,ξ dξ Lφ(x) = r+ (2π)−n

(x, ξ)(e (4) Rn

with a symbol in the class Um (m is the order of the pseudo-diﬀerential operator). The symbol class Um and the symbol classes below have been explained

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e.g. in [7,9,13]. Above r+ refers to the restriction operator r+ f = f|G ×R+ and e+ refers to the extension by zero from G × R+ on G × R . The boundary condition is formulated with the help of a so-called trace operator T . The operator T is a trace operator C ∞ (G) → C ∞ (∂G) with a symbol in the class Tr,d (d is the order and r is the class of the trace operator). Locally this means that T is decomposed into two operators T φ(x ) = T1 φ(x ) + T2 φ(x ). The ﬁrst operator T1 is of the form j ∂ T1 φ(x ) = Sj φ(·, 0) (x ) (5) j ∂x n 0≤j≤r−1 where Sj are pseudo-diﬀerential operators in Rn−1 of order d − j. The second operator T2 is of the form ∞ T2 φ(x ) = (2π)−(n+1) eix ,ξ t(x , xn , ξ )F˜ φ(ξ , xn )dxn dξ (6) Rn−1

0

for φ ∈ C0∞ (G × R+ ) where F˜ is the partial Fourier transform F˜ φ(ξ , xn ) = Rn−1 φ(x , xn )e−ix ,ξ dx . The operator K is a potential (or Poisson) operator C ∞ (∂G) → C ∞ (G) with symbol in the class Km (m is the order of the potential operator). Locally this means that for ϕ ∈ C0∞ (G ) K is of the form Kϕ(x) = (2π)−n+1 k(x , xn , ξ )ϕ(ξ ˆ )eix ,ξ dξ . (7) Rn−1

The operator Q is a so-called (singular) Green operator C ∞ (G) → C ∞ (G) with symbol in the class Br,d (d is the order and r is the class of the singular Green operator). Locally this means that Q is of the form Qφ(x) = Q1 φ(x) + Q2 φ(x), φ ∈ C0 (G × R+ ). Here the ﬁrst operator is of the form j ∂ Q1 φ(x) = Kj φ(·, 0) (x) (8) ∂xjn 0≤j≤r−1 where Kj are potential operators of order d − j. The second operator Q2 , for φ ∈ C0∞ (G × R+ ), is of the form ∞ Q2 φ(x) = (2π)1−n eix ,ξ q(x , xn , yn , ξ )F˜ φ(ξ , yn )dyn dξ (9) Rn−1

2.2

0

State space operator

Let v1 , ..., vN be indeterminates in C 1 (G × ∆). Denote v = (v1 , . . . , vN )T . Furthermore, let Lij : C ∞ (G) → C ∞ (G), i, j = 1, ..., N

(10)

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be pseudo-diﬀerential operators and let Tlj : C ∞ (G) → C ∞ (∂G), j = 1, ..., N, l = 1, ..., m

(11)

be trace operators. Finally, suppose that Qij : C ∞ (G) → C ∞ (G), i, j = 1, ..., N are (singular) Green operators. Deﬁne a N × N -type matrix A by    L11 · · · L1N  .. ..  +  A = (Aij ) =  ... . .   LN 1 · · · LN N

 Q11 · · · Q1N .. .. ..  . . .  QN 1 · · · QN N

Similarily, deﬁne a m × N -type matrix T = (Tlj ) by   T11 · · · T1N  .. ..  . T = (Tlj ) =  ... . . 

(12)

(13)

(14)

Tm1 · · · TmN We deﬁne a linear operator A˜ : C ∞ (G)N → C ∞ (G)N by the requirements ˜ = {v ∈ C ∞ (G)N | (Tlj )v = 0} D(A) ˜ = (Aij )v. Av

(15a) (15b)

˜ Suppose that A : L2 (G)N → L2 (G)N is a closed extension of A. 2.3

The control system

Let X1 and X2 be Hilbert spaces and let B1 : X1 → L2 (G)N , B2 : X1 → L2 (G)N , D2 : X1 → X2 .

(16)

be (bounded) linear operators. In addition, we assume that D1 : L2 (G)N → X2 is a A-bounded linear operator that is, D(A) ⊂ D(D1 ) and for all v ∈ D(A) ||D1 v||X2 ≤ C ||Av||L2 (G)N + ||v||L2 (G)N . (17) In the following we consider controls u ∈ C 1 (∆, X1 ). Using the operators above we can formulate the control system ∂v ∂u = Av + B1 u + B2 ; y = D1 v + D2 u. ∂t ∂t

(18)

In system (18) v ∈ C 1 (∆, L2 (G)N ) is the state variable, u is the control variable and y ∈ C(∆, X2 ) is the output variable.

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On transfer function analysis of coercive problems

At ﬁrst we consider some analytical results for the transfer function. We assume that v(0) = u(0) = 0. The transfer function G = G(λ) with respect to input u and output y is given by (see e.g. [3,5,6]) G(λ) = D2 + D1 (λI − A)−1 (B1 + λB2 )

(19)

where we need the assumption that D1 is A-bounded. We assume that X1 and X2 are ﬁnite-dimensional inner product spaces. Then G(λ) ∈ M (q × p) when dim X1 = p, dim X2 = q. We use the Frobenius norm in M (q × p) which is equivalent to the operator norm deﬁned in the space of linear operators X1 → X2 . Suppose that H is a Hilbert space such that H ⊂ L2 (G)N and that ||v||L2 (G)N ≤ C1 ||v||H .

(20)

We make the following assumptions for all v, u ∈ H |−Av, uL2 (G)N | ≤ C||v||H ||u||H

(21)

−Av, vL2 (G)N ≥ c||v||2H .

(22)

The boundedness assumption (21) is usually easy to verify. The condition (22) is called coercivity of the operator −A and it is much more diﬃcult to treat. The vastly known assumption is the strong ellipticity of the underlying boundary-value system which implies coercivity estimates with H = H 1 (G)N after tedious (symbolic) computations [2,7,16]. However in many cases the coercivity can be proved also for non-elliptic problems. The application of symbolic calculus of pseudo-diﬀerential operators is a very eﬀective modern method in the veriﬁcation of estimates (21) and (22), see [7]. The operator A is called dissipative if Av, vL2 (G)N ≤ 0, v ∈ D(A).

(23)

Thus the concept of coercivity is closely related to the dissipativity: When c = 0 the operator −A is coercive if and only if the operator A is dissipative. In the following we formulate a theorem without proof. The proof will be given elsewhere. Theorem 1. Suppose that the operator A satisﬁes the estimates (21) and (22). Furthermore, suppose alternatively that 1.

||D1 v||X2 ≤ C||v||H , v ∈ D(A), or

(24)

2.

||D1 v||X2 ≤ C||v||L2 (G)N , v ∈ D(A).

(25)

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Let := 2Cc 1 . Then the transfer function G(λ) is analytic in C− and corresponding to the alternatives above it has the following properties 1.

G(λ) = D2 + G1 (λ) + λG2 (λ), ||Gj (λ)|| ≤

2.

G(λ) = D2 + G1 (λ) + λG2 (λ), ||Gj (λ)|| ≤

M 1

(1 + |λ|) 2

, λ ∈ C− (26)

M , λ ∈ C− . 1 + |λ|

(27)

In particular, in the case of Assumption 2, G(λ) ∈ H∞ (− ).

(28)

Combining Corollary 2.2.3 and Theorem 7.3.2 of [5] we see that when G(λ) is analytic in C− and the condition ||λG(λ)|| ≤ C, λ ∈ C−

(29)

is valid, then G(λ) ∈ Aˆ− (0). Especially, in the case where D1 satisﬁes the assumption 2, Theorem 1 implies that G1 (λ), G2 (λ) ∈ Aˆ− (0). In [10] one has shown that instead of (29) actually the weaker condition ||λ1/2+d G(λ)|| ≤ C, λ ∈ C−

(30)

where d > 0 is suﬃcient to quarantee that G(λ) ∈ Aˆ− (0). Applying Theorem1 one gets various suﬃcient conditions for the property G(λ) ∈ Aˆ− (0) but we leave them here.

4

Transfer function manipulations by symbols

In this section we outline some methods how it is possible more explicitly to analyze transfer functions using symbols of operators. One is able to get very reﬁned qualitative information by applying the so-called parameter-dependent symbolic calculation. We use the terminology of [7] which we shall not recall here. We assume that the pseudo-diﬀerential operator (Lij ) is of order m and the singular Green operator (Qij ) is of order m and of class r ≤ m. In addition, we with the assume that the boundary-value system (Tlj ) isnormal associated λI − (Lij ) is parameterorder m. Finally, we suppose that the system (Tlj ) elliptic on the ray λ = reiθ0 , r ≥ 0. Then there exists r0 ≥ 0, > 0 such that the resolvent (λI − A)−1 is analytic with respect to λ in a truncated sector Wr0 , = {λ ∈ C| |λ| ≥ r0 and |Arg λ − θ0 | ≤ }.

(31)

In addition, the resolvent (λI − A)−1 is of the form (λI − A)−1 f = Pλ f + Rλ f.

(32)

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In (32) the operator Pλ is a parameter-dependent pseudo-diﬀerential operator of order −m that is, locally the (matrix) elements of Pλ have the form + φ)(ξ)eiξ,x dξ Pij,λ φ(x) = r+ (2π)−n pij (x, ξ, λ)(e (33) Rn

where pij (·, ·, λ) ∈ U−m . The second operator Rλ in (32) is a parameterdependent singular Green operator of order −m and of class 0 that is, locally the (matrix) elements of Rλ have the form ∞ eix ,ξ rij (x , xn , yn , ξ , λ)F˜ φ(ξ , yn )dyn dξ Rij,λ φ(x) = (2π)1−n Rn−1

0

as outlined in section 2. The transfer function G(λ) can be calculated from G(λ) = D2 + D1 (Pλ + Rλ )(B1 + λB2 ).

(34)

In the calculation of (34) one is often able to use the symbolic (composition) calculus since frequently the operators Dj , Bj are in the algebra described here. The symbols pij (x, ξ, λ) and rij (x , xn , yn , ξ , λ) have more reﬁned properties of parameter-dependent symbols. For example, they have power series estimates where the terms satisfy certain (quasi)homogeneity conditions and they satisfy useful estimates. The following example illustrates the abstract formulations developed above and gives practical understanding to the operators under consideration. Example. Let L(D) = a∂x2 +b∂x +c be a PDO with constant coeﬃcients and let d1 , d2 be constants. Furthermore, let G =]0, 1[ ⊂ R, ∆ = ]0, ∞[. Consider the system ∂v = L(D)v + q(x)u ∂t ∂x v(0, t) + d1 v(0, t) = 0, ∂x v(1, t) + d2 v(1, t) = 0 y = D1 v = v(1, t)

(35a) (35b)

where q ∈ C 2 (G) and u ∈ C 1 (∆). v is the state variable and u is the control variable. The transfer function is G(λ) = D1 (λI − A)−1 q = ((λI − A)−1 q)(1).

(36)

We proceed formally in calculating G(λ). A. At ﬁrst we compute the resolvent (λI − A)−1 and compare the result to the decomposition (32). We must calculate the expression v = vλ := (λI − A)−1 f . We ﬁnd that (λI − A)v = f and then λv−(a

∂v ∂v ∂v ∂2v (0)+d1 v(0) = 0, (1)+d2 v(1) = 0. (37) +b +cv) = f, 2 ∂x ∂x ∂x ∂x

Stabilizability using pseudo-diﬀerential operators

From the ﬁrst of equations (37) we get 1 1 ∂v ∂2v −iξx + f (ξ). + cv) e dx = f e−iξx dx = e λv − (a 2 + b ∂x ∂x 0 0

271

(38)

Applying partial integration and by taking into account the boundary conditions we obtain + f (ξ) g1 (ξ) g2 (ξ) e + v(ξ) = e − v(1) − v(0). λ − (ξ) λ − (ξ) λ − (ξ)

(39)

where (ξ) = −aξ 2 + bi ξ + c is the symbol of L(D) and where g1 (ξ) = (ad2 − b − ai ξ)e−iξ , g2 (ξ) = b − ad1 + ai ξ. Hence by the Fourier inverse formula e+ f (ξ) iξx + −1 e dξ − g1 (λ, x)v(1) − g2 (λ, x)v(0) e v(x) = (2π) R λ − (ξ) where

(41)

gk (ξ) iξx e dξ, k = 1, 2 R λ − (ξ) e + f (ξ) Denote (Pλ f )(x) = (2π)−1 R λ−(ξ) eiξx dξ. From (41) we obtain gk (λ, x) = (2π)−1

(40)

v(1) + g1 (λ, 1)v(1) + g2 (λ, 1)v(0) = (Pλ f )(1) v(0) + g1 (λ, 0)v(1) + g2 (λ, 0)v(0) = (Pλ f )(0).

(42)

(43)

From the equations (43) we can solve v(0) =

g1 (λ, 0)(Pλ f )(1) − [1 + g1 (λ, 1)](Pλ f )(0) , W (λ)

(44)

v(1) =

g2 (λ, 1)(Pλ f )(0) − [1 + g2 (λ, 0)](Pλ f )(1) W (λ)

(45)

where W (λ) = g1 (λ, 0)g2 (λ, 1) − [1 + g1 (λ, 1)][1 + g2 (λ, 0)]. By substitution of (44,45) to (41) we ﬁnd that (λI − A)−1 f = v = Pλ f + Rλ f where (as above) Pλ is a pseudo-diﬀerential operator 1 + −1 iξx + (46) (2π) (Pλ φ)(x) = r e φ(ξ)e dξ R λ − (ξ) and Rλ is (can be shown to be) a singular Green operator g1 (x, λ) {[1 + g2 (λ, 0)](Pλ φ)(1) − g2 (λ, 1)(Pλ φ)(0)} W (λ) g2 (x, λ) {g1 (λ, 0)(Pλ φ)(1) − [1 + g1 (λ, 1)](Pλ φ)(0)}. − W (λ)

(Rλ φ)(x) =

(47)

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B. The transfer function can be immediately computed by G(λ) = v(1) =

g2 (λ, 1)(Pλ q)(0) − [1 + g2 (λ, 0)](Pλ q)(1) . W (λ)

(48)

The previous example can be applied to certain linearized bioreactor models [17], and we will give some related simulations for the controller design elsewhere. Acknowledgement. This work is supported by the Academy of Finland, Reseach Council for Natural Sciences and Engineering via the framework programme Mathematical Methods and Modelling in the Sciences (MaDaMe) in the project Computational Bioprocess Engineering (No: 49938).

References 1. Amann, H. (1989) Feedback stabilization of linear and semilinear parabolic systems. In: Cl´ement et al. (Eds.) Semigroup Theory and Applications, Lecture Notes Pure Appl. Math. 116. M. Dekker, New York, 21-57 2. Amann, H. (1995) Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory. Birkh¨auser, Basel 3. Banks, S.P. (1983) State-space and Frequency-domain Methods in the Control of Distributed Parameter Systems. P. Peregrinus, London 4. Callier, F.M., Desoer, C.A. (1982) Multivariable Feedback Systems. Springer, Berlin 5. Curtain, R.F., Zwart, H.J. (1995) An Introduction to Inﬁnite-Dimensional Linear Systems Theory. Springer, New York 6. Fattorini, H.O. (1999) Inﬁnite Dimensional Optimization and Control Theory. Cambridge University Press, Cambridge 7. Grubb, G. (1986) Functional Calculus of Pseudo-Diﬀerential Boundary Problems. Birkh¨ auser, Boston 8. Grubb, G. (1991) Parabolic Pseudo-Diﬀerential Boundary Problems and Applications. In: Cattabriga L., Rodino, L. (Eds.) Microlocal Analysis and Applications, Lecture Notes in Math., Vol. 1495, Springer, Berlin, 46-117 9. Krainer, T. (2000) Parabolic Pseudodiﬀerential Operators and Long-Time Asymptotics of Solutions. PhD Thesis, University of Potsdam, Potsdam 10. Mossaheb, S. (1980) On the existence of right-coprime factorization for functions meromorphic in a half-plane. IEEE Trans. Aut. Control AC-25, 550–551 11. Pommaret, J.F. (2001) Controllability of nonlinear multidimensional control systems. In: Respondek, W. et al. (Eds.) Nonlinear Control of the Year 2000, Proc. NCN Conference, Vol. 2., Springer, Berlin, 245-255 12. Quadrat, A. (2000) Internal stabilization of coherent control systems. Preprint, 1-16 13. Rempel, S., Schulze, B.-W. (1982) Index Theory of Elliptic Boundary Problems. Akademie-Verlag, Berlin 14. Shulze, B.-W. (1998) Boundary Value Problems and Singular PseudoDiﬀerential Operators. Wiley

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15. Sule, V.R. (1998) Feedback stabilization over commutative rings: The matrix case. SIAM J. Control and Optimiz. 32, 1675–1695 16. Tanabe, H. (1979) Equations of Evolution. Pitman, London 17. Tervo, J., Nihtil¨a, M. (2000) Exponential stability of a nonlinear distributed parameter system. Zeitschrift f¨ ur Analysis und ihre Anwendungen 19, 77-93 18. Vidyasagar, M. (1985) Control System Synthesis. A Factorization Approach. MIT Press, Cambridge

Multi-periodic nonlinear repetitive control: Feedback stability analysis David H. Owens, Liangmin Li, and Steve P. Banks Department of Automatic Control and Systems Engineering, The University of Sheﬃeld, Mappin Street, Sheﬃeld S1 3JD, UK Email: [email protected], [email protected], [email protected] Abstract. In this paper the stability of multi-periodic repetitive control problem,where two or more periods exist in the reference and disturbance signals, is m studied. A Lyapunov analysis is used to prove Lm 2 (0, ∞) ∩L∞ (0, ∞) stability for a class of passive nonlinear systems subject to a class of nonlinear perturbations. A proof of exponential stability under a strictly positive real condition is provided.

1

Introduction

Inoue et al. [4,5] pioneered repetitive control problem based on internal model principle in 1980’s [2]. Repetitive control was used to track/reject periodic reference/disturbance signals to a high accuracy, which was demanded in a number of situations, such as vibration control, marine systems, and rotating machinery, etc. [3,6,7]. If the reference/disturbance contains only one fundamental frequency, single periodic repetitive control, which has been studied extensively in the past two decades, can be very eﬀective. However, in many cases, the reference/disturbance may contain diﬀerent fundamental frequencies, whose ratio can be irrational. Since single periodic repetitive control will fail in these situations, a so-called multi-periodic repetitive control scheme will be needed. In the previous work by the authors [8], MIMO multi-periodic linear repetitive systems were studied using Lyapunov techniques. It was proved there that the stability of the multi-periodic repetitive control system is guaranteed if the plant satisﬁes a positive real condition. Robustness was introduced in a simple manner by using a constant weighting ﬁlter. In this paper the previous results are ﬁrst brieﬂy reviewed, then extensions to a class of nonlinear system are given. Also a proof of exponential stability is given under a strictly positive real condition. A simulation example is presented to illustrate the procedure.

2

Lyapunov stability analysis

The Lyapunov stability analysis is given based on the main result of [8]. The multi-periodic repetitive control system is shown in Figure (1), where the A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 275-283, 2003. Springer-Verlag Berlin Heidelberg 2003

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D.H. Owens, L.M. Li and S.P. Banks

Fig. 1. Multi-periodic repetitive control system

multi-periodic repetitive controller M (s) =

m i=1

αi I 1−Wi e−sτi

is a convex linear

combination of single-periodic repetitive control elements. That is,

m

αi = 1,

i=1

αi > 0 and τi , i = 1, ..., m (the periods of the components of the external signals reference r and disturbance d ) are assumed known. The plant G is ﬁnite-dimensional, linear time-invariant and its transfer function G(s) is a p × p matrix. Plant structure is an important component in ensuring closed-loop stability. For completeness, the deﬁnition of a positive real linear system is given below: Deﬁnition 1. [1]: The system G is said to be positive real(PR) if 1). All elements of its transfer function G(s) are analytic in Re[s] > 0 , 2). G(s) is real for real positive s , and 3). G(s) +G∗ (s) ≥ 0 for Re[s] > 0 . where the superscript ∗ denotes complex conjugate transposition. Deﬁnition 2. [1] The system G is said to be strictly positive real(SPR)if 1). All elements of its transfer function G(s) are analytic in Re[s] > 0 , 2). G(s) is real for real positive s , and 3). ∃ > 0 such that G(s − ) +G∗ (s − ) ≥ 0 for any > 0 . The following is the well-known positive real lemma. Lemma 1 (Positive Real Lemma). [1]: Assume

Repetitive and nonlinear

x˙ = Ax + Bu y = Cx, x(0) = x0

277

(1) (2)

is a minimum realization of G . Then G(s) is positive real if and only if there exist matrices 0 < P = P T ∈ Rn×n and 0 ≤ LLT = Q = QT ∈ Rn×n such that

P A + AT P = −Q

(3)

P B = CT

(4)

Here the superscript

T

denotes transposition and L has full row rank.

The main result of stability analysis can be stated below. Theorem 1. Suppose that both the reference r and the disturbance d are zero and that |Wi | < 1, 1 ≤ i ≤ m. Suppose that G is positive real and strictly proper. Then the multi-periodic repetitive system in Figure (1) is globally asymptotically stable in the sense that the output signal y ∈ Lm 2 (0, ∞), the state x(·) ∈ Ln∞ (0, ∞) and Lx(·)∈ Lk2 (0, ∞) . Proof: The system

G

has the form

x˙ = Ax + B(v + d) y = Cx,

(5)

x(0) = x0

From Figure (1) it follows that

v(t) =

m

m

αi zi ,

i=1

αi = 1

i=1

zi (t) = Wi zi (t − τi ) + e(t) = Wi zi (t − τi ) − y(t)

(6)

Now introducing a positive deﬁnite Lyapunov function V of the form V = xT P x +

m i=1

t 2

zi (θ) dθ

αi t−τi

(7)

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D.H. Owens, L.M. Li and S.P. Banks

By diﬀerentiating V along solutions and using the positive real lemma V˙ = x˙ T P x + xT P x˙ +

m

2

2

αi [zi (t) − zi (t − τi ) ]

i=1

= −xT LLT x + 2y T v(t) m 2 2 αi [zi (t) − zi (t − τi ) ] + i=1 T

2

= −x LLT x − y m 2 2 2 + αi [y + 2y T zi (t) + zi (t) − zi (t − τi ) ]

(8)

i=1 m

Notice that from (6) the last term in (8) is −

αi (1 − Wi 2 ) zi (t − τi )

2

i=1

and notice that Wi ≤ 1 , thus (8) becomes 2 V˙ = −xT LLT x − y m 2 − αi (1 − Wi 2 ) zi (t − τi ) < 0

(9)

i=1

Integrating (9) and using (7) and the positivity of V yield t 0 ≤ V (0) = V +

t T

x LL xdt + 0

+

t m 0

2

y dt

T

0 2

αi (1 − Wi 2 ) zi (t − τi ) dt < ∞

i=1

from which Lx(·) ∈ Lk2 (0, ∞) , x(·) ∈ Ln∞ (0, ∞) and y(·) ∈ Lm 2 (0, ∞) which proves the result.

3

Asymptotical stability of multi-periodic repetitive control for a class of nonlinear systems

The Lyapunov stability analysis conducted in section 2 can be easily extended to a class of nonlinear systems S described by

x˙ = Ax + Bv + f (x) y = Cx

(10)

Repetitive and nonlinear

279

which satisﬁes P A + AT P = −Q P B = CT T x P f (x) ≤ 0

(11)

where 0 < P = P T ∈ Rn×n and 0 ≤ LLT = Q = QT ∈ Rn×n Remark: the nonlinear system (10) under condition (11) is a passive system, see deﬁnition of passivity and proof of passivity of (10)-(11)in Appendix. The result of stability analysis for nonlinear system (10) can be stated below. Theorem 2. Suppose that both reference r and disturbance d are identically zero and that |Wi | < 1, 1 ≤ i ≤ m. Then the multi-periodic nonlinear repetitive system is globally asymptotically stable in the sense that the output n k signal y ∈ Lm 2 (0, ∞), the state x(·) ∈ L∞ ( 0, ∞), Lx(·) ∈ L2 (0, ∞) and T x P f (x) ∈ L1 (0, ∞) . Proof: The proof is similar to that in section 2 and is outlined below. Now introducing a positive deﬁnite Lyapunov function V of the form of (8) By diﬀerentiating V along solutions and using the condition (11) V˙ = x˙ T P x + xT P x˙ +

m

2

2

αi [zi (t) − zi (t − τi ) ]

i=1

= −x LL x + 2y v(t) m 2 2 + αi [zi (t) − zi (t − τi ) ] + 2xT P f (x) T

T

T

(12)

i=1

From which it is seen that the expression for V˙ diﬀers from that obtained in the linear case by the addition of the term 2xT P f (x). As a consequence, the same procedure yields t 0 ≤ V (0) = V +

t T

0

+

0

y dt −

x LL xdt +

t m

t 2

T

0

2xT P f (x)dt 0

2

αi (1 − Wi 2 ) zi (t − τi ) dt < ∞

i=1

As all terms are positive, it follows that Lx(·) ∈ Lk2 (0, ∞), x(·) ∈ Ln∞ (0, ∞), T y(·) ∈ Lm 2 (0, ∞) and x P f (x) ∈ L1 (0, ∞), which proves the result.

280

4

D.H. Owens, L.M. Li and S.P. Banks

Exponential stability

In the earlier linear multi-periodic study [8] it has been proved that when the transfer function G(s) is strictly positive real, the tracking error decays in an exponential manner. Those results can also be extended to the class of nonlinear system whose plant is passive, described by (10), as illustrated below, subject to the condition that the linear part of this plant G(s) ∼ (A, B, C) is strictly positive real. Suppose Re(G(s))≥ 0 where Re(s) ≥ −, > 0, which is just Re(G(s − ))≥0 with Re(s ) ≥ 0, if we denote s = s + . By denoting f˜(t) = et f (t), equation (10)has the form x ˜˙ = (A + εI)˜ x + Bu ˜ + et f (x) y˜ = C x ˜

(13)

As (A+I, B, C) is PR, P = P T > 0 and Q = QT ≥ 0 can be constructed. It follows that the same is true for (A, B, C) with Q replaced by Q + 2P > 0. Note also that m m αi v˜i , αi = 1 u ˜(t) = i=1

i=1

zi (t − τi ) + e˜(t) v˜i (t) = (Wi eτi )˜

(14)

so that introducing the positive deﬁnite Lyapunov function V used previously and and diﬀerentiating V along solutions yield 2 x − ˜ y + 2˜ xT P et f (x) V˙ = −x˜T Q˜ m 2 − αi (1 − Wi 2 e2t ) z˜i (t − τi )

(15)

i=1

Notice that 2˜ xT P et f (x) = 2e2t xT P f (x) ≤ 0. Integrating and re-arranging, with the choice (W.L.O.G) of > 0 such that maxi |Wi eτi | < 1, yield the observation that x ˜ ( t ) is uniformly bounded. As a consequence, ∃ M > 0 such that ˜ x(t) ≤ M → x(t) ≤ M e−t ∀ t > 0 which indicates exponential stability of the state.

5

Simulation

The nonlinear system S under control is described by x˙ = Ax + Bv + f (x) y = Cx −1 0 where A = 0 −3

1 B= 1

(16)

Repetitive and nonlinear

281

C = 1 1 and f (x) = −x1 3 0 It can be easily veriﬁed that (16) satisfying (11), so that the system (16) is passive and the repetitive control system with the controller shown in Figure (1) should be asymptotically stable, which is veriﬁed by the simulation result shown in Figure (2), where the reference r = 0, with the initial condition x1 (0) = 0.3 and x2 (0) = 0.5. Next the reference r is set as a mix of harmonics of two fundamental frequencies 0.27Hz and 1.1Hz, that is, r = r1 + r2 where r1 = sin ω1 t + sin 3ω1 t + 0.7 sin 5ω1 t r2 = 2 sin ω2 t + sin 3ω2 t with ω1 = 2π ∗ 0.27,

ω2 = 2π ∗ 1.1

The initial conditions are zero. The constant ﬁlterings are chosen as W1 = W2 = 0.98. A pure gain K = 100 is included in the forward path. The tracking error is shown in Figure (3).

0.4

0.3

e(t)

0.2

0.1

0

−0.1

−0.2 0

5

10

15

20 t(second)

25

30

35

40

Fig. 2. Tracking error e(t) when r = 0 and initial conditions are non-zero

282

D.H. Owens, L.M. Li and S.P. Banks 0.25 0.2 0.15 0.1

e(t)

0.05 0 −0.05 −0.1 −0.15 −0.2 0

5

10

15

20 t(second)

25

30

35

40

Fig. 3. Tracking error e(t) when r = 0 and zero initial conditions

6

Conclusions

Multi-periodic repetitive control system is studied for a class of nonlinear systems. These results are extensions of single and linear periodic repetitive control problems. The stability is analysed in the sense of Lyapunov stability and it has been shown that asymptotic stability and exponential stability is guaranteed if the plant is passive. The simulation example shows that the new control system can be very eﬀective.

Acknowledgements The authors would like to acknowledge the funding support from the EPSRC grant No. GR/M94106 and the EU TMR Nonlinear Control Network project FRMX-CT97-0137(DG-12-BDCN).

Repetitive and nonlinear

283

Appendix Deﬁnition: A input-output dynamic system is called passive if and only if t T u ydt > 0, for all input-output pair (u, y) for all t ≥ 0, where the initial 0

condition is zero. Then we have the following result: The nonlinear system S described by (10) is passive if it satisﬁes assumption (11) . Proof: Introduce a positive deﬁnite function V1 = xT P x Diﬀerentiating V1 along the solution of x yields V˙1 = −xT Qx + 2xT P f (x) + 2uT y which means 2uT y ≥ V˙ 1

(17)

because −xT Qx + 2xT P f (x) ≤ 0 Integrating both side of (17) and noticing the positivity of V1 and V1 (x) = t 0 for x = 0, we have uT ydt ≥ 1/2[V1 − V1 (0)] > 0 which means that S is passive.

0

References 1. B. D. O. Anderson and S. Vongpanitherd:Network analysis and synthesis: A modern system theory approach, Prentice Hall, Englewood Cliﬀs, NJ, (1973) 2. B. A. Francis and W. M. Wonham:The internal model principal for linear multivariable regulators, Appl. Math. Opt., Vol. 2, pp107-194, (1975) 3. S. S. Garimella and K. Srinivasan: Application of repetitive control to eccentricity compensation in rolling Journal of dynamic systems measurement and control-Transactions of the ASME, Vol. 118, No. 4, pp657-664,(1996) 4. T. Inoue, M. Nakano and S. Iwai:High accuracy control of servomechanism for repeated contouring , Proc. 10th Annual Symp. Incremental Motion Control Systems and Devices, pp258-292, (1981) 5. T. Inoue, M. Nakano, T. Kubo, S. Matsumoto and H. Baba:High accuracy control of a proton synchrotron magnet power supply, Proc. IFAC 8th World Congress, pp216-221, (1981) 6. K. Kaneko and R. Horowitz:Repetitive and adaptive control of robot manipulators with velocity estimation, IEEE Trans. on robotics and automation, Vol. 13, No. 2, pp204-217, (1997) 7. Y. Kobayashi, T. Kimura and S. Yanabe:Robust speed control of ultrasonic motor based on Hinf control with repetitive compensator, JSME international journal Series C., Vol. 42, No. 4, pp884-890, (1999) 8. D. H. Owens, L. M. Li and S. P. Banks: MIMO Multi-periodic Repetitive Control System:stability analysis, Proc. ECC 2001,pp3393-3397, Sept., Porto, Portugal, (2001)

Perturbed hybrid systems, applications in control theory Christophe Prieur Laboratoire d’analyse num´erique et EDP, Universit´e Paris-Sud, bˆ atiment 425, 91405 Orsay, France, [email protected] Abstract. We study a class of perturbed hybrid systems, i.e. dynamical systems with a mixed continuous/discrete state in presence of disturbances. We introduce a natural notion of trajectories, but it is very sensitive to noise. Therefore we deﬁne a new notion of trajectories and, to investigate the sensitivity, we enlarge this class of trajectories. Finally we consider two problems in control theory (the uniting problem and the problem of the robust stabilization of asymptotically controllable systems) which have no solution in terms of (dis)continuous controller in presence of disturbances and we give a solution with a robust hybrid controller.

1

Introduction

Let us consider the nonlinear control system of the form: x˙ = F (x, u) .

(1)

We want to ﬁnd a feedback law such that the closed-loop system x˙ = F (x, u(x)) ,

(2)

has some desirable properties (e.g. a minimum-time property, an asymptotic stable equilibrium). But we are often unable to ﬁnd continuous feedback law ensuring good properties (e.g. a minimum-time property, see [4, Example 1], or a stabilization, see [8]). Therefore, even if the control system (1) is continuous with respect to the variable (x, u), the diﬀerential equation (2) has a discontinuous right-hand side. Thus we may have a strong sensitivity with respect to small noise and therefore very surprising behaviour of trajectories (e.g. non convergence of some trajectories, existence of new equilibrium, ...) if we try to compute numerically or to implement this feedback in a physical system. To avoid this sensitivity, we have to consider a more general class of trajectories than the class of Carath´eodory trajectories and to solve the problem of the construction of a feedback law satisfying the desirable properties for (1) for this largest class of trajectories. This new problem can be very diﬃcult or impossible to solve. See [4, Example 2] where the same optimization problem (or [9, Artstein’s circles] for a stabilization problem) is studied for two diﬀerent classes of trajectories and yields two solutions which are completely diﬀerent. A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 285-294, 2003. Springer-Verlag Berlin Heidelberg 2003

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C. Prieur

In this case, it is natural to consider a new class of controllers which allows more easily an insensitivity to noises. More precisely, we introduce the class of dynamic feedback laws with a mixed continuous/discrete state as follows (see e.g. [2]) u = k(x, sd )

,

sd = kd (x, s− d) ,

(3)

where sd evolves in a discrete set D, k (resp. kd ): IRn × D → IRm (resp. IRn × D → D), and s− d is deﬁned, at this stage only formally as, s− d (t) = lim sd (s) .

(4)

s
This motivates us to study the control system (1) in closed loop with (3) which has the following form: x˙ = f (x, sd )

,

sd = kd (x, s− d) .

(5)

We say that this system is hybrid because we have a dynamic equation for a mixed continuous/discrete variable. We deﬁne the most natural notion of perturbed trajectories of a hybrid system and study their properties (see Section 2). In Section 2.1 we study an example of a perturbed hybrid system and we prove that its trajectories are very sensitive to perturbations, and yields unpleasant properties of the trajectories. Therefore we introduce a new notion of solutions in Section 2.2 satisfying good properties. We enlarge this class of trajectories to investigate the sensitivity to noise. Finally we consider some problems in control theory which have no solution in terms of continuous and discontinuous feedback laws with a class of trajectories which is suﬃciently large to take into account the small disturbances (see Section 3) and we give a solution of these problems in terms of hybrid controller.

2

Perturbed hybrid systems

In this section we are concerned with a diﬀerential system of the form (5). Assume that sd is in a discrete set D. We equip D with the discrete topology. Assume that f : IRn × D → IRn is locall Lipschitz and that kd : IRn × D → D is a function. The operator s− d is deﬁned, at this stage only formally, as (4). The most natural notion of a trajectory of (5) is the following (see [2] and references therein): Deﬁnition 1 Let T > 0 and (x0 , s0 ) be in IRn × D. We say that (X, Sd ) : [0, T ) → IRn × D is a trajectory of (5) if we have the following properties: 1. The function X is absolutely continuous on [0, T ) and satisﬁes, for almost ˙ every t in [0, T ), X(t) = f (X(t), Sd (t)). 2. The function Sd is right-continuous on [0, T ). 3. For all t in (0, T ), where Sd− (t) exists, we have Sd (t) = kd (X(t), Sd− (t)).

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Now we discuss the meaning of the notion of the initial condition of a trajectory. To understand this, note that a necessary and suﬃcient condition to make the concatenation of two trajectories X− and X+ of an ordinary diﬀerential equation without hybrid term (i.e. (5) without sd ) deﬁned respectively on (−T, 0) and [0, T ), where T > 0, is X− (0) = X+ (0), i.e. the equality of the initial conditions. In the context of (5), we note that we can concatenate (X− , Sd,− ) and (X+ , Sd,+ ) if we have − kd (X− (0), Sd,− (0)) = Sd,+ (0)

,

X− (0) = X+ (0) .

(6)

− (0), which is a priori not deﬁned, be equal to It follows that if we let Sd,+ − Sd,− (0), then (6) yields a constraint on Sd,+ (0). But initial conditions are usually free of constraints. Hence this motivates us to call initial condition of a trajectory the couple (X(0), Sd− (0)). Therefore we introduce the following

Deﬁnition 2 Let T > 0 and (x0 , s0 ) in IRn × D. We say that (X, Sd ) : [0, T ) → IRn ×D is a trajectory of (5) with initial condition (x0 , s0 ) if (X, Sd ) is a trajectory of (5) in the sense of Deﬁnition 1 and if the following equalities hold X(0) = x0

,

Sd (0) = kd (X(0), Sd− (0)) .

(7)

In this paper we are interested in the study of the sensitivity of the trajectories to the noise. Therefore we introduce a positive real number, ρ, and two unknown functions, e and d, such that: supt≥0 |e(t)| ≤ ρ

,

esssupt≥0 |d(t)| ≤ ρ ,

(8)

where supJ (g) and esssupJ (g) denote respectively the bound and the essential bound on J of a function g. In the following we study the perturbed hybrid system: x˙ = f (x + e, sd ) + d ,

sd = kd (x + e, s− d) .

(9)

Let us introduce the following deﬁnition: Deﬁnition 3 A function (X, Sd ): [0, T ) → IRn × D is said to have a switch at time t ∈ [0, T ) if Sdm (t) := {s, ∃tn →≤ t, Sd (tn ) → s} = {s, ∃tn →≥ t, Sd (tn ) → s} =: Sdp (t) . To locate the points where a trajectory has a switch we introduce, for all (i, j) in D, Ci→j = {x, kd (x, i) = j} , Σi→j = (Ci→j + B(0, ρ)) ∩ (Ci→i + B(0, ρ)) ∩ (Cj→j + B(0, ρ)) ,

(10) (11)

where B(0, ρ) denotes the ball of IRn whose center is 0 and whose radius is ρ.

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Proposition 1 Let t be a time such that a trajectory (X, Sd ) of (9) has a switch and i = j in D be such that i ∈ Sdm (t) and j ∈ Sdp (t). Then we have X(t) ∈ Σi→j . Due to the space limitation, we do not give here the proof of this proposition, but we can ﬁnd an analogous result in [7]. Proposition 1 has many applications. For example assume that there exist i, j and k in D such that d(Σi→j , Σj→k ) > 0, where d denotes the usual metric in IRn . Then we can prove very easily that for all trajectories (X, Sd ), such that we have three consecutive switches and such that the value of Sd is successively i, j and then k, we have a strictly positive mimimum time between the switches which is independent of the trajectory. We use this mimimum time to prove some properties as in [6] and [7]. But we note in the next section that the properties of the trajectories can be very sensitive to small perturbations and that this notion of trajectory is not suitable. 2.1

The Artstein’s circles

In this section we study the Artstein’s circles with a hysteresis. We prove that, in presence of small perturbations, we do not have pleasant properties of the trajectories. It motivates us to introduce an other notion of trajectories in the next section. The Artstein’s circles have been ﬁrst introduced in [1]. It yields a good example to study the sensitivity to noises (see e.g [9,5]). It is deﬁned by: x˙ 1 = u(−x21 + x22 )

,

x˙ 2 = −2ux1 x2 ,

(12)

where the parameter u is in IR. The integral curves are (see Figure 1) the origin, all circles centered on the x2 -axis and tangent to the x1 -axis and the x1 -axis. With u > 0 the circle is followed clockwise if x2 > 0 and anticlockwise if x2 < 0. Let us introduce the following sets (see Figure 1) π π Γ1 = {x ∈ IR2 \ {(0, 0)} : − < θ < } , 4 4 3π 3π Γ−1 = {x ∈ IR2 \ {(0, 0)} : < θ ≤ π or − π < θ < − } , 4 4 where θ denotes the polar angle of x = 0 in (−π, π]. Let us study the hysteresis of ((12) with u = 1) and of ((12) with u = −1) where u : IR2 × {−1, 1} → {−1, 1} is deﬁned by: u(x, sd ) = sd , ∀(x, sd ) ∈ IR2 × {−1, 1} , and the function kd : IR2 × {−1, 1} → {−1, 1} by   1 if x ∈ Γ1 , ∀ sd ∈ {−1, 1}, kd (x, sd ) = −1 if x ∈ Γ−1 ,  sd else .

(13)

(14)

Perturbed hybrid systems, applications in control theory

u = −1

x2

289

u=1

Γ−1

Γ1 u = −1

u=1 x1

u = −1

u=1

Fig. 1. The integral curves of the Artstein’s circles and the sets deﬁning the hysteresis.

Let us consider a trajectory (X, Sd ) of the system (12)-(14) with initial conditions (x0 , s0 ). It is easy to prove that only three cases may occur: 1. If x0 is in Γ1 , then, for all t in [0, +∞), Sd (t) = 1 and X is a Carath´eodory trajectory of x˙ 1 = −x21 + x22

,

x˙ 2 = −2x1 x2 .

(15)

2. If x0 is in Γ−1 , then, for all t in [0, +∞), Sd (t) = −1 and X is a Carath´eodory trajectory of x˙ 1 = x21 − x22

,

x˙ 2 = 2x1 x2 .

(16)

3. If x is in IR2 \ (Γ1 ∪ Γ−1 ), then, for all t in [0, +∞), Sd (t) = s0 , and X is a Carath´eodory trajectory of (15) if s0 = 1 or of (16) if s0 = −1. Therefore we do not have any switch of the trajectories and the controller (13)-(14) has overlapped the systems (15) and (16) on IR2 \ (Γ1 ∪ Γ−1 ). Let us choose now a positive real number ρ > 0, and let us consider e = (e1 , e2 ): IR≥0 → IR2 satisfying (8). The system (12)-(14) perturbed with these measurement noise is x˙ 1 = sd (−(x1 + e1 )2 + (x2 + e2 )2 ) , x˙ 2 = −2sd (x1 + e1 )(x2 + e2 ) , sd = kd (x + e, s− d) .

(17)

Let Σ1→−1 and Σ−1→1 be the sets associated to the real number ρ and deﬁned by (11). Let x0 be diﬀerent from (0, 0) and with a polar angle 3π 4 . Let Ξ be a Carath´eodory trajectory of (15) deﬁned on [0, T1 ) with T1 > 0.

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Let 0 < T2 < T1 such that, for all t in [0, T2 ], Ξ(t) is in B(0, ρ2 ). Let E ∈ IR2 , |E| < ρ be such that, ρ (18) x + E ∈ Γ−1 , ∀x ∈ B(Ξ(T2 ) ) . 4 Proposition 2 There exists an initial condition in IR2 × D, a noise e: IR≥0 → IR2 and a trajectory of (17) maximally deﬁned on [0, T ), with T < +∞ and limt→T |X(t)| < +∞. Proof of Proposition 2 Let us consider the initial condition (x0 , 1). Let IR2 be such that e: IR≥0 → e(t) = 0, ∀t ∈ [0, T2 ]

,

e(t) = E, ∀t > T2 .

(19)

On [0, T2 ), the system (17) is not perturbed, therefore we are in the case 3. Thus there exists a trajectory (X, Sd ) of (17) with initial condition (x0 , s0 ) and deﬁned on [0, T2 ), and, for all t in [0, T2 ), we have Sd (t) = 1 and X(t) = Ξ(t). Let us prove by exhibiting a contradiction that this trajectory is not deﬁned for time larger than T2 . Assume the contrary and that Sd is deﬁned on [T2 , T3 ) with T2 < T3 < T1 . Then, at time T2 , due to (19) the trajectory is unperturbed and Sd (T2 ) = 1. Due to Deﬁnition 2, Sd is right-continuous and there exists T2 < T4 < T3 , such that, ∀t ∈ [T2 , T4 ), Sd (t) = 1 .

(20)

But due to (18) and (19), there exists T2 < T5 < T4 , such that, for t ∈ [T2 , T5 ], we have X(t)+e(t) is in Γ−1 and thus, with (9) and (14) we have, Sd (t) = −1 for t in (T2 , T5 ]. This contradicts (20). This ends the proof. Proposition 2 is very surprising because, for an ordinary diﬀerential equation, a trajectory, which is maximally deﬁned on a ﬁnite domain, blows up, i.e. tends to inﬁnity at the boundary of the domain of deﬁnition. Therefore our notion of trajectories is not suitable anymore for the perturbed hybrid systems and we need to enlarge the class of considered trajectories. Thus we introduce a subset RC strictly include in IR2 × {−1, 1}, and we ask for the trajectories to be right-continuous only on RC. Now we are able to give a new notion of trajectories of any perturbed hybrid systems. 2.2

A new notion of trajectories

In this section we consider the system (9). We deﬁne a new notion of trajectories with suitable properties. This study yields applications in control theory of Section 3. Let ρ > 0 and (e, d) be two unknown functions satisfying (8), and, for i = j in D, Σi→j be the sets deﬁned by (11). We are now in position to introduce a new notion of trajectories of (9). For all i in D, we introduce Σi→j × {i}) . RC i = IRn × D \ ( j∈D,j=i

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291

Deﬁnition 4 Let T > 0 and (x0 , s0 ) be in IRn × D. We say that (X, Sd ) : [0, T ) → IRn × D is a RC-trajectory of (9) with initial condition (x0 , s0 ) if: 1. The function X is absolutely continuous on [0, T ) and satisﬁes, for almost ˙ every t in [0, T ), X(t) = f (X(t) + e(t), Sd (t)) + d(t). 2. Let i in D and t in [0, T ) such that Sd (t) = i. If (X(t), Sd (t)) is in RC i then the function Sd is right-continuous. 3. For all t in (0, T ) such that Sd− (t) exists, we have Sd (t) = kd (X(t) + e(t), Sd− (t)). 4. We have (7). We can prove that we have the following properties: Proposition 3 Let t be such that we have a switch of (X, Sd ) at time t, a RC-trajectory of (9), and i = j be such that i ∈ Sdm (t) and j ∈ Sdp (t). Then we have X(t) ∈ Σi→j . This proposition is the analogous of Proposition 1 for the trajectories given by Deﬁnition 2. We have also a result of existence: Proposition 4 For all initial conditions (x0 , s0 ) in IRn × D, there exists a RC-trajectory of (9) with this initial condition deﬁned on [0, T ) with T > 0. Finally, let us claim a result of blow up for RC-trajectories whose domain of deﬁnition is ﬁnite. This result was false for the trajectories given by Deﬁnition 2 (see Proposition 2). Proposition 5 Let (X, Sd ) be a RC-trajectory of (9) maximally deﬁned on [0, T ) with T < +∞. We have lim |X(t)| = +∞. t→T

This properties are well-known in the context of ordinary diﬀerential equations. In this paper we are interested in a notion of trajectory which is robust with respect to disturbances. For this reason we introduce the notion of generalized trajectory (see [4]) for perturbed hybrid systems. Deﬁnition 5 Let T > 0 and x0 be in IRn . We say that X : [0, T ) → IRn is a generalized trajectory of (9) with initial condition x0 if X(0) = x0 and, for all compact subset J of [0, T ), there exists a sequence (X n , Sdn ) of RCtrajectories of x˙ = f (x + e + en , sd ) + d + dn

,

sd = kd (x + e + en , s− d) ,

(21)

deﬁned on J and satisfying lim (esssupJ |dn | = 0 + supJ |en | = 0 + esssupJ |X n − X|) = 0 .

n→+∞

Let us introduce now the π-trajectories of (9) which are the polygonal approximations of Carath´eodory trajectories. In the context of control systems (as in Section 3) this notion is an accurate model of the process used in computer control. See [7,8] and references therein. Let us ﬁrst introduce a sampling schedule π of [0, +∞), π = {0 = t0 < t1 < · · · < t∞ = +∞}, whose diameter is deﬁned by d(π) = supi∈IN (ti+1 −ti ).

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Deﬁnition 6 Let T > 0 and (x0 , s0 ) be in IRn × D, we say that (X, Sd ): [0, T ) → IRn × D is a π-trajectory of (9) on [0, T ) with initial condition (x0 , s0 ) if 1. The function X is absolutely continuous on [0, T ) and satisﬁes, for almost every t in [0, T ) and for all i inN , ˙ X(t) = f (X(ti ) + e(ti ), Sd (ti )) + d(ti ) . 2. We have, for all t in [t0 , min(t1 , T )), Sd (t) = Sd (t0 ), and, for all i inN>0 and for all t in [min(ti , T ), min(ti+1 , T )), Sd (t) = kd (X(ti ) + e(ti ), Sd (ti−1 )) . 3. We have (7). We deﬁne the notion of Euler trajectories (see [8]) in our context: Deﬁnition 7 Let T > 0 and x0 be in IRn . We say that X : [0, T ) → IRn is a Euler trajectory of (9) on [0, T ) with initial condition x0 , if, for all compact subset J of [0, T ), there exists a sequence of sampling schedule π n of IR and a sequence (X n , Sdn ) of π n -trajectories of (9) deﬁned on J such that lim esssupJ |X n − X| + d(π n ) = 0 ,

n→∞

and such that we have X(t0 ) = x0 . We can prove that, for this three notions of trajectories, we have the analogous of Propositions 3, 4 and 5.

3

Applications in control theory

In this section we apply the general ideas contained in Section 2. We study two diﬀerent problems in control theory which have no solution in terms of (dis)continuous controllers, and we prove that we can ﬁnd a hybrid controller of the form (3) which solves these problems. For complete proofs see [6] and [7] respectively. 3.1

The uniting problem

We consider control systems for which we know two stabilizing controllers. One is globally asymptotically stabilizing, the other one is only locally asymptotically stabilizing but for some reasons (e.g. optimality, exponential convergence), we insist on using it in a neighbourhood of the origin. See [6] for more details. We deﬁne the uniting problem as the problem of the existence of a controller: 1) being equal to the local controller in a ﬁxed neighbourhood of the origin, 2) being equal to the global one outside of a compact set, 3) such

Perturbed hybrid systems, applications in control theory

293

that the origin of the corresponding closed-loop system is a globally asymptotically stable equilibrium i.e. such that we have the following two properties: 1) stability of the origin of the closed-loop system and 2) convergence to this point of all the solutions. We prove the existence of an obstruction (see [6]) to have a continuous solution of this uniting problem. Therefore we need to consider the class of discontinuous controllers but we exhibit a system for which there exists no (dis)continuous static controller solution of this uniting problem for all generalized trajectories (see Deﬁnition 5). This negative result leads us to reformulate our problem using hybrid controllers as (3). We prove a positive result in [6]: Theorem 1 We can exhibit a hybrid controller of the form (3) which is solution of the uniting problem for the RC-trajectories and the generalized trajectories. 3.2

Robust stabilization of asymptotic controllable systems

Here we study asymptotic controllable systems, i.e. control systems such that, for all initial conditions, there exists a time-varying controller such that we have a stability property and convergence to the origin of the trajectory with this initial condition (see [7] for more details). We are looking for a state feedback such that the origin of the closed-loop system is an asymptotic controllable system. Let us note that the origin of the system (12) is an asymptotic controllable equilibrium but there does not exists a continuous state feedback u(x) such that the origin of the closed-loop system is a globally asymptotic controllable equilibrium (see [8]). Therefore we must consider discontinuous controllers to stabilize all asymptotically controllable systems. The following property is proved in [3]: Any asymptotically controllable systems can be asymptotically stabilized by a discontinuous controller. In that paper, by trajectories of (2) in closed-loop with a discontinuous controller, the authors mean the π-trajectories. The controller in [3] is robust to actuator and external disturbances (i.e. all systems perturbed by small actuator and external disturbances are asymptotically stable) but are not robust to arbitrary small measurement noises. In [9] and [8, sec. 5.4], E.D. Sontag proves the existence, for all asymptotic controllable systems, of a sampling feedback which allows the origin to be a robust global asymptotic equilibrium for all π-trajectories with a sampling rate suﬃciently slow. The author proves also that the origin of the system (12) in closed-loop with a static discontinuous feedback is not an asymptotically stable equilibrium for the π-trajectories with any fast sampling rate. Therefore the problem of the stabilization of asymptotically controllable systems for the π-trajectories and the Euler trajectories perturbed by a suﬃciently small noise has no solution in terms of (dis)continuous controller.

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But we prove the following theorem in [7] (see [7] for a precise statement) Theorem 2 If (1) is asymptotically controllable then there exists a hybrid controller of the form (3) which makes the origin a globally asymptotically stable equilibrium and which is robust to measurement noises, actuator errors and external disturbances, i.e. for all π-trajectories and all Euler trajectories perturbed by suﬃciently small perturbations.

References 1. Artstein Z. (1983) Stabilization with relaxed controls. Nonlin. Th., Meth., App. 7, 1163–1173 2. Branicky M.S. (1998) Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Autom. Control 43 (4), 475–482 3. Clarke F.H., Ledyaev Yu.S., Sontag E.D., Subbotin A.I. (1997) Asymptotic controllability implies feedback stabilization. IEEE Trans. Autom. Control 42, 1394–1407 4. Hermes H. (1967) Discontinuous vector ﬁelds and feedback control. In Diﬀerential Equations and Dynamic Systems, (Hale J.K. and La Salle J.P., eds.), Academic Press, New York London, 155–165 5. Prieur C. (2000) A Robust globally asymptotically stabilizing feedback: the example of the Artstein’s circles. In Nonlinear Control in the Year 2000 (Isidori A. et all, eds.), Springer Verlag, London, 279–3000 6. Prieur C. (2001) Uniting local and global controllers with robustness to vanishing noise. Math. Control Signals Systems 14, 143–172 7. Prieur C. (2001) Asymptotic controllability and robust asymptotic stabilizability. Preprint 2001-03 Universit´e Paris-Sud, France 8. Sontag E.D. (1999) Stability and stabilization: discontinuities and the eﬀect of disturbances. In Nonlinear Analysis, Diﬀerential Equations, and Control, (Proc. NATO Advanced Study Institute, Montreal, Jul/Aug 1998; Clarke F.H., Stern R.J., eds.), Kluwer, 551–598 9. Sontag E.D. (1999) Clocks and insensitivity to small measurement errors. ESAIM: COCV, www.emath.fr/cocv/ 4, 537–557

Global stabilization of nonlinear systems: A continuous feedback framework Chunjiang Qian1 and Wei Lin2 1 2

The University of Texas at San Antonio, San Antonio, TX 78249, USA Case Western Reserve University, Cleveland, OH 44106, USA

Abstract. This article develops a continuous feedback framework for global stabilization of inherently nonlinear systems that may not be stabilized by any smooth feedback, even locally. Suﬃcient conditions are given for the existence of continuous but non-smooth state feedback control laws that achieve global strong stability. A systematic design method which combines homogeneous systems theory with the idea of adding a power integrator is presented for the explicit construction of C 0 globally stabilizing controllers. The signiﬁcance of this new framework is illustrated by solving a variety of open nonlinear control problems that cannot be dealt with by existing methods.

1

Introduction

Without doubt global asymptotic stabilization of nonlinear systems is one of the most important topics in nonlinear control theory, which has received and is increasingly receiving a great deal of attention. Over the past decade, with the aid of the diﬀerential geometric approach [4,12,13,23], tremendous progress has been made towards the development of systematic design methodologies for the control of nonlinear systems via smooth state feedback. By comparison, less progress has been achieved in the design of continuous controllers for nonlinear systems. The major reason for studying the continuous feedback stabilization is that many physical systems of practical importance such as a class of underactuated mechanical systems [28] are inherently nonlinear and cannot be stabilized, even locally, by any smooth feedback. The simplest example may be the scalar system—x˙ = u3 + x—which is unstabilizable by any smooth 1 feedback but continuously stabilizable [29], e.g. by u = −(2x) 3 . A more subtle example, which has been studied in the literature [15,16], is the planar system x˙ 1 = x32 + x1 , x˙ 2 = u. Since the linearized system has an uncontrollable mode whose eigenvalue is positive, by the necessary condition (i) in [3] no C r (r ≥ 1) controllers stabilize the planar system. To overcome such topological obstruction which may occur in smooth feedback stabilization, several elegant continuous feedback control strategies

This work was supported in part by the U.S. NSF under Grants ECS-9875273, ECS-9906218, and DMS-9972045. Corresponding author: Prof. Wei Lin (e-mail: [email protected])

A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 295-315, 2003. Springer-Verlag Berlin Heidelberg 2003

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have been proposed for lower-dimensional systems. For example, using the notions of homogeneous approximation and homogeneity with respect to a family of dilations [11], Kawski [15,16] proved that every STLC (small-time locally controllable [31,32]) aﬃne system in the plane is locally asymptotically stabilizable by continuous state feedback. An algorithm for the construction of H¨older continuous stabilizing controllers was also given, resulting in the ﬁrst solution to the stabilization problem for x˙ 1 = x32 + x1 , x˙ 2 = u. The idea of using continuous feedback to achieve local asymptotic stabilization, together with the powerful concepts such as homogeneous approximation [11] and homogeneity with respect to a family of dilations [10,11,15,16], has completely opened the door to a much deeper understanding of feedback stabilization of inherently nonlinear systems, leading to various exciting developments in the area of asymptotic stabilization of nonlinear systems with uncontrollable unstable linearization (i.e. Jacobian linearization has uncontrollable modes associated with eigenvalues on the right-half plane) [2,5– 8,10,11,27]. For analytic aﬃne systems in the plane, an important result was obtained by Dayawansa, Martin and Knowles [8], which provides necessary and suﬃcient conditions for local asymptotic stabilization by continuous feedback. For details about the other developments in the area of continuous feedback stabilization of lower dimensional systems, we refer the reader to Dayawansa’s survey paper [7] and Bacciotti’s book [2] as well as references therein. Most of the results reviewed so far for two or three dimensional aﬃne systems, e.g. [2,7–9,15,16], are basically local in nature, due to the use of nilpotent or homogeneous approximation. For higher-dimensional systems, fewer results are available. One of them is the work [6], where the existence of local C 0 stabilizers was proved for a class of triangular systems, by using a homogeneous approximation and Hermes’ robust stability theorem for homogeneous systems [10]. An important issue on how to design a local continuous stabilizer remained open but was studied later in [5]. In the work [33], Tsinias generalized the results of [6] to a class of nontriangular systems and showed that global stabilization can be achieved via C 0 dynamic state compensators. However, the result in [33] is also an existence theorem because its proof relies on Artstein’s theorem [1] and on partition unity arguments [33]. Recently, the paper [34] showed that a chain of power integrators perturbed by a linear vector-ﬁeld in a suitable form is homogeneous with respect to certain dilation, and hence is globally stabilizable by C 0 homogeneous state feedback. In a diﬀerent direction, we have investigated the problem of smooth feedback stabilization for a class of high-order systems with null linearization [21,22]. Although the systems under consideration are neither feedback linearizable nor aﬃne in the control variable, it was proved that they are globally stabilizable by smooth state feedback [21,22], under a set of assumptions which can be viewed as a “high-order” version of the feedback linearizable condition. A globally stabilizing C ∞ controller was explicitly constructed for

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high-order nonlinear systems, by a novel design method called adding a power integrator [21,22]. Realizing that continuous feedback design might be a natural strategy to overcome the topological obstruction (i.e. the necessary condition (i) collected in [3,16]), we shall concentrate in this work on the problem of achieving global strong stabilization (in the sense of Kurzweil [18]) via continuous state feedback, for a class of highly nonlinear systems that cannot be dealt with by existing methods. The main purpose of the paper is to address two important issues: (i) when does there exist a continuous controller u = u(x) that renders the trivial solution x = 0 of the nonlinear system globally strongly stable (GSS)? (ii) is it possible to develop a machinery for the explicit construction of a C 0 GSS controller? Throughout this paper, we shall provide answers to these questions, by developing a continuous feedback design approach to global strong stabilization of nonlinear systems. Speciﬁcally, using a homogeneous-like Lyapunov function inspired by [34] combined with the idea of adding a power integrator [21,22], we present a new feedback design tool that enables us to solve the problem of global strong stabilization, for a large class of nonlinear systems with uncontrollable unstable linearization. Suﬃcient conditions are given under which it is possible to prove, while no smooth controllers exist, the existence of continuous GSS state feedback control laws. The proof is constructive and accomplished by repeatedly using an adding a power integrator technique, which simultaneously generates a C 0 global stabilizer as well as a C 1 control Lyapunov function.

2

Global strong stability in a continuous framework

It is well-known that the classical Lyapunov stability theory (see, for instance, [9,17]) and the concepts such as stability and asymptotic stability in the sense of Lyapunov can only be applied to a nonlinear diﬀerential equation having a unique solution. In the case when the system x˙ = f (t, x) is only continuous and does not satisfy the Lipschitz condition, the conventional Lyapunov stability theory is not applicable because x˙ = f (t, x) may have inﬁnite many solutions. Therefore, new concepts on asymptotic stability and Lyapunov stability theory must be introduced in the continuous framework. In this section, we review a concept related to the notion of global strong stability (GSS) introduced by Kurzweil [18]. We also introduce Lyapunov’s second theorem and converse Lyapunov theorem on global strong stability, which were proved by Kurzweil [18] in the quite general framework, for a nonlinear system x˙ = f (t, x),

t ∈ IR,

x ∈ IRn ,

(1)

with f : IR × IRn → IRn being a continuous function of (t, x) and f (t, 0) = 0 ∀t ∈ IR, without requiring uniqueness of the trajectories of system (1) from any initial condition.

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Deﬁnition 2.1 (pp. 69 in [18]) The trivial solution x = 0 of (1) is said to be globally strongly stable (GSS) if there are two functions B : (0, +∞) → (0, +∞) and T : (0, +∞) × (0, +∞) → (0, +∞) with B being increasing and lims→0 B(s) = 0, such that ∀α > 0 and ∀ε > 0, for every solution x(t) of (1) deﬁned on [0, t1 ), 0 < t1 ≤ +∞ with x(0) ≤ α, there is a solution z(t) of (1) deﬁned on [0, +∞) satisfying (i) (ii) (iii)

z(t) = x(t), t ∈ [0, t1 ); z(t) ≤ B(α), ∀t ≥ 0; z(t) < ε, ∀t ≥ T (α, ε).

As we shall see in the sequel, the notion of global strong stability will serve as a starting point for the systematic development of a continuous feedback design approach. Using this notion, Kurzweil proved [18] that Lyapunov’s second theorem remains true under the condition that the function f (t, x) is continuous. Indeed, he presented in [18] the following modiﬁcation of Lyapunov’s second theorem from which one can conclude that under certain conditions all solutions of the continuous system (1) tend to zero. Theorem 2.2 (Kurzweil, pp.23 in [18]) Suppose there exist a C 1 function V : IR × IRn → IR, a C 0 function U1 : IRn → IR, which is positive deﬁnite and proper, and C 0 functions Ui : IRn → IR, i = 2, 3, which are positive deﬁnite, such that U1 (x) ≤ V (t, x) ≤ U2 (x) ∂V ∂V + f (t, x) ≤ −U3 (x). ∂t ∂x Then, the trivial solution x = 0 of system (1) is globally strongly stable.

(2)

This theorem will prove to be instrumental, throughout the remainder of this paper, in establishing various global strong stabilization results by continuous state feedback, for highly nonlinear control systems that are not smoothly stabilizable. Notably, Theorem 2.2 is analogous to Lyapunov’s second theorem and contains the case of the so-called global asymptotic stability when the solution of system (1) is unique. Intuitively speaking, Theorem 2.2 implies that every solution x(t) of the continuous system (1) from any initial condition x(0) is globally stable and eventually converges to zero, no matter whether system (1) has a unique solution or not. This intuition can be better understood from the following example, which illustrates that a non-Lipschitz continuous system can have inﬁnite many solutions even if the trivial solution x = 0 is globally strongly stable. Example 2.3 Consider a C 0 planar system 1

x˙ 1 = −x13 2

1

x˙ 2 = x13 x23 − x2 whose vector ﬁeld is not Lipschitz in any neighborhood of the origin.

(3)

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First of all, note that system (3) is GSS at x = 0. The conclusion follows from solving the diﬀerential equation (3). In fact, a direct calculation shows that starting from any initial condition, the solution trajectory x1 (t) is unique and described by  2 3/2 2  0 ≤ t < 32 x13 (0) sgn(x1 (0)) x13 (0) − 23 t x1 (t) = (4) 2  0 t ≥ 32 x13 (0).

0, the corresponding solution is When x2 (0) =  2 2 3/2 2 2  2 −3t 3 3 3  sgn(x x (0)) e (0) − x (0) − 1 + x (0) + 1 − t  2 2 1 1 3   2   3 3  0 ≤ t < 2 x1 (0)

3/2 2 x2 (t) = 2 2 3  −t x1 (0) 3 3  sgn(x2 (0))e + x2 (0) − x1 (0) − 1 e      2  t ≥ 32 x13 (0) (5) which is unique. However, when x2 (0) = 0, x2 (t) has inﬁnite many solutions (e.g. x1 (0) = 1, x2 (0) = 0) and can be expressed as x2 (t) = ±φ(t),  0 0≤t
3/2 2   2 2  2   e−t ex13 (0) − e 3 c (x13 (0) + 1 − 23 c) t ≥ 32 x13 (0)

(6)

2

where c is an arbitrary real number in [0, 32 x13 (0)]. From (4), it is clear that x1 (t) converges to zero in a ﬁnite time. Conse2

quently, all the solutions x2 (t) tend to zero exponentially after t ≥ 32 x13 (0), although the solution of (3) is not unique. On the other hand, using the following C 1 Lyapunov function V (x1 , x2 ) =

1 2 3 4/3 x + x , 2 1 4 2

it is easy to prove that 1 4/3 4/3 V˙ ≤ − (x1 + x2 ). 2 By Theorem 2.2, system (3) is GSS at (x1 , x2 ) = (0, 0), which conﬁrms the previous conclusion.

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A simulation result shown in Fig. 1 further veriﬁes our theoretic analysis. In fact, all solutions of the continuous system (3) tend to zero although (3) has inﬁnite many solutions.

0.8

Trajectories

x1 0.6

0.4

0.2

x

2

0

−0.2

0

1

2

3 Time

4 Time

5

6

7

Fig. 1. State trajectories of system (3) with x1 (0) = 1 and x2 (0) = 0

Theorem 2.2 provides suﬃcient conditions for the continuous system (1) to be GSS. In [18], Kurzweil also proved that the converse of Theorem 2.2 is also true. Speciﬁcally, he showed that global strong stability, together with the continuity of f (t, x), implies the existence of a smooth converse Lyapunov function. Theorem 2.4 (Kurzweil, pp. 58 in [18]) If the trivial solution x = 0 of system (1) is globally strongly stable, then there exist a C ∞ function V : IR × IRn → IR, a C 0 function U1 : IRn → IR, which is positive deﬁnite and proper, and C 0 functions Ui : IRn → IR, i = 2, 3, which are positive deﬁnite, such that the inequality (2) is satisﬁed. In the case of autonomous systems, a combination of Theorems 2.2 and 2.4 leads to the following result. Theorem 2.5 Consider a diﬀerential equation x˙ = f (x),

x ∈ IRn ,

(7)

with f : IRn → IRn being a continuous function and f (0) = 0. The trivial solution x = 0 of system (7) is globally strongly stable if and only if there exists a C ∞ (IRn , IR) Lyapunov function V (x), which is positive deﬁnite and proper, such that ∂V f (x) < 0, ∂x

∀x ∈ IRn − {0}.

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According to Theorem 2.5, system (3) must have a C ∞ Lyapunov function whose derivative along the trajectories of (3) is negative deﬁnite. In fact, this is the case and a smooth Lyapunov function is given, for instance, by V (x1 , x2 ) = 12 (x21 + x22 ) + 14 x41 , whose derivative is such that 1 4/3 10/3 V˙ (3) ≤ − (2x1 + 2x1 + x22 ) < 0 3

3

∀(x1 , x2 ) = 0.

Construction of non-Lipschitz continuous stabilizers

We present in this section an iterative design approach that combines the theory of homogeneous systems [10,11,15,16,34] with the idea of adding a power integrator [21,24], for the explicit construction of C 0 controllers which globally stabilize a signiﬁcant class of nonlinear systems that are usually impossible to stabilize by any smooth feedback. For the sake of simplicity, throughout this section we focus our attention on the nonlinear system x˙ 1 = d1 (t)xp21 + f1 (t, x1 , x2 ) x˙ 2 = d2 (t)xp32 + f2 (t, x1 , x2 , x3 ) .. . x˙ n = dn (t)upn + fn (t, x1 , · · · , xn , u),

(8)

where x = (x1 , · · · , xn )T ∈ IRn and u ∈ IR are the system state and the control input, respectively. For i = 1, · · · , n, pi is an odd positive integer, fi : IR × IRi+1 → IR, is a C 0 function with fi (t, 0, · · · , 0) = 0 ∀t ∈ IR, and di (t) is a C 0 function of time t, which represents an unknown time varying parameter. To investigate the global stabilization problem, we make the following assumptions. A3.1 that

For i = 1, · · · , n, there are positive real numbers λi and µi such 0 < λi ≤ di (t) ≤ µi .

A3.2

For i = 1, · · · , n,

fi (t, x1 , · · · , xi , xi+1 ) =

p i −1

xji+1 ai,j (t, x1 , · · · , xi )

(9)

j=0

where xn+1 = u. Moreover, there is a smooth function γi,j (x1 , · · · , xi ) ≥ 0 such that for j = 0, · · · , pi − 1 |ai,j (t, x1 , · · · , xi )| ≤ (|x1 | + · · · + |xi |)γi,j (x1 , · · · , xi ).

(10)

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Remark 3.3 In the case when ai,j (t, x1 , · · · , xi ) is independent of t (i.e. ai,j (t, x1 , · · · , xi ) ≡ ai,j (x1 , · · · , xi )), if the function ai,j (x1 , · · · , xi ) is C 1 and ai,j (0, · · · , 0) = 0, by the Taylor expansion theorem there always exists a smooth function γi,j (x1 , · · · , xi ) ≥ 0 satisfying the property (10). Under A3.1-A3.2, the existence of continuous GSS controllers for the nonlinear system (8) can be guaranteed, as shown in the following theorem which is one of the main results of this paper. Moreover, a systematic design algorithm for the explicit construction of C 0 controllers can also be developed. Theorem 3.4 For a nonlinear system (8) satisfying A3.1-A3.2, there is a continuous controller u = u(x) with u(0) = 0, which renders the trivial solution x = 0 of (8) globally strongly stable. Sketched Proof. The proof is carried out by using an adding a power integrator method—a machinery which enables one to simultaneously constructs a C 1 Lyapunov function which is positive deﬁnite and proper, as well as a µ := continuous stabilizer. To begin with, deﬁne λ := mini=1,··· ,n {λi }, maxi=1,··· ,n {µi }. x21 2 .

Initial Step. Choose V1 (x1 ) = the trajectories of (8) is

Then, the time derivative of V1 along

V˙ 1 (x1 ) = x1 [d1 (t)xp21 + f1 (t, x1 , x2 )] .

(11)

Using A3.2 and Young’s inequality, it is not diﬃcult to show that |f1 (t, x1 , x2 )| ≤

λ|x2 |p1 + |x1 |γ1 (x1 ), 2

(12)

where γ1 (x1 ) > 0 is an appropriate smooth function. Substituting (12) into (11) yields λ|x2 | V˙ 1 (x1 ) ≤ x1 d1 (t)xp21 + |x1 | 2

p1

+ x21 γ1 (x1 ).

Then, the continuous virtual controller x∗2 deﬁned by

1/p1 2(n + γ1 (x1 )) 1/p x∗2 = − x1 := − (x1 β1 (x1 )) 1 , λ where β1 (x1 ) > 0 is smooth, is such that λ|x2 | V˙ 1 (x1 ) ≤ −nx21 + x1 d1 (t)xp21 + |x1 | 2

p1

− x1

1 λx∗p 2 2

1 Since −x1 x∗p ≥ 0 and λ ≤ d1 (t) ≤ µ, we have 2

λ p1 1 1 V˙ 1 (x1 ) ≤ −nx21 + d1 (t)x1 [xp21 − x∗p − |x∗p 2 ] + |x1 | [|x2 | 2 |] 2 λ 1 with c := µ + . ≤ −nx21 + c|x1 (xp21 − x∗p 2 )|, 2

(13)

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303

Inductive Step. Suppose at step k−1, there are a C 1 Lyapunov function Vk−1 : IRk−1 → IR, which is positive deﬁnite and proper, and a set of C 0 virtual controllers x∗1 , · · · , x∗k , deﬁned by x∗1 x∗2 p1

=0 = −ξ1 β1 (x1 ) .. .

x∗k p1 ···pk−1 = −ξk−1 βk−1 (x1 , · · · , xk−1 )

ξ1 = x1 − x∗1 , ξ2 = xp21 − x∗2 p1 .. .

ξk = xk p1 ···pk−1 − x∗k p1 ···pk−1 , (14)

with β1 (x1 ) > 0, · · · , βk−1 (x1 , · · · , xk−1 ) > 0, being smooth, such that V˙ k−1 ≤ −[n − k + 2]

k−1

ξl2

1 2− p0 p1 ···p

pk−1 k−2 ∗ pk−1 xk + c ξk−1 − xk , p0 = 1 (15)

l=1

Obviously, (15) reduces to the inequality (13) when k = 2. Since p0 is identical to one, in what follows we simply omit p0 in (15). We claim that (15) also holds at step k. For, consider the Lyapunov function Vk : IRk → IR, deﬁned by

Vk (x1 , · · · , xk ) = Vk−1 (x1 , · · · , xk−1 ) + Wk (x1 , · · · , xk ) xk p1 ···p 2− p ···p1 k−1 1 k−1 ds. s Wk (x1 , · · · , xk ) = − x∗k p1 ···pk−1

(16)

x∗ k

It can be proved [25] that Vk (x1 , · · · , xk ) thus deﬁned is C 1 , positive deﬁnite and proper. Moreover, for l = 1, · · · , k − 1,  1  ∂Wk = ξ 2− p1 ···pk−1 k ∂xk (17) 1 p ···p  ∂Wk = −bk ∂x∗k 1 k−1 x∗k (sp1 ···pk−1 − x∗ p1 ···pk−1 )1− p1 ···pk−1 ds. ∂xl

∂xl

xk

k

1 . where bk = 2 − p1 ···p k−1 With the help of (17), we deduce from (15) that 2− p1 ···p1 k−2 pk−1 2 2 ∗ pk−1 ˙ Vk ≤ −(n − k + 2)(ξ1 + · · · + ξk−1 ) + c ξk−1 xk − xk 1 1 ···pk−1

2− p

+ξk

∂Wk k−1 k dk (t)xpk+1 + fk (t, x1 , · · · , xk+1 ) + x˙ l . (18) ∂xl l=1

Next we estimate each term on the right hand side of (18). First, it is not diﬃcult to deduce that (e.g. see [25]) 2− p ···p1

p ξ 2 c ξk−1 1 k−2 xkk−1 − x∗k pk−1 ≤ k−1 + ξk2 ck , ck > 0. (19) 3

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C. Qian and W. Lin

Similar to (12), it can be proved that k−1 2 1 2− p1 ···p1 k−1 λ 2− p ···p pk l=1 ξl ξ 1 k−1 |x ≤ |ξ + ξk2 ρ˜k (·), (20) f (·) | | + k k k+1 k 2 3 where ρ˜k (x1 , · · · , xk ) is a non-negative smooth function. To estimate the last term in (18), we observe from (17) and (14) that ∗ p1 ···pk−1 ∂Wk ≤ ak |ξk | ∂xk , ak > 0, f or l = 1, · · · , k − 1. ∂xl ∂xl With this in mind, one has k−1 2 k−1 k−1 p ···p ξ ∂Wk ∂x∗k 1 k−1 x˙ l ≤ ak |ξk | x˙ l ≤ l=1 l + ξk2 ρ¯k (·), ∂xl ∂xl 3 l=1

(21)

l=1

where ρ¯k (x1 , · · · , xk ) ≥ 0 is a smooth function. Substituting the estimates (19), (20) and (21) into (18), we arrive at V˙ k ≤ −(n − k + 1)

k−1

1 1 ···pk−1

2− p

ξl2 + ξk

k dk (t)xpk+1

l=1 1 λ 2− + |ξk | p1 ···pk−1 |xk+1 |pk + ξk2 [ck + ρ˜k (·) + ρ¯k (·)] 2

Hence, there exists a C 0 virtual controller of the form 1

x∗k+1 = − (ξk βk (x1 , · · · , xk )) p1 ···pk

1 = − (xk p1 ···pk−1 − x∗k p1 ···pk−1 )βk (x1 , · · · , xk ) p1 ···pk , (22) p1 ···pk−1 with βk (·) := λ2 [n − k + 1 + ck + ρ˜k (·) + ρ¯k (·)] > 0 being smooth, such that V˙ k ≤ −(n − k + 1)

k

1 1 ···pk−1

2− p

ξl2 + ξk

k dk (t)xpk+1

l=1 1 λ λ 2− p ···p1 2− k + |ξk | p1 ···pk−1 |xk+1 |pk − ξk 1 k−1 x∗p k+1 2 2 2− p1 ···p1 k−1 pk ∗pk 2 2 ≤ −(n − k + 1)(ξ1 + · · · + ξk ) + c ξk (xk+1 − xk+1 )

2− p

1 ···p

k The last inequality follows from A3.1 and the fact that −ξk 1 k−1 x∗p k+1 ≥ 0. The induction argument shows that (15) holds for k = n + 1. Hence, at the last step choosing 1

u = xn+1 = x∗n+1 = − (ξn βn (x1 , · · · , xn )) p1 ···pn

(23)

Global stabilization of nonlinear systems

305

yields

V˙ n ≤ − x21 + (x2 p1 − x∗2 p1 )2 + · · · + (xn p1 ···pn−1 − x∗n p1 ···pn−1 )2 .

(24)

where Vn (x1 , · · · , xn ) is a C 1 positive deﬁnite and proper Lyapunov function of the form (16). By Theorem 2.2, the trivial solution x = 0 of system (8)–(23) is globally strongly stable. Using Theorem 3.4, it is possible to obtain a number of important global strong stabilization results for a class of highly nonlinear systems whose global stabilization problem has been remained open and unsolved so far. The ﬁrst result is a direct consequence of Theorem 3.4. Corollary 3.5 For the time-varying nonlinear system x˙ i =

i di (t, x, u)xpi+1

+

p i −1

xji+1 ai,j (t, x, u), i = 1, · · · , n, xn+1 := u, (25)

j=0

suppose there is a smooth function γi,j (x1 , · · · , xi ) ≥ 0, such that |ai,j (t, x, u)| ≤ (|x1 |+· · ·+|xi |)γi,j (x1 , · · · , xi ),

0 ≤ j ≤ pi −1,

1 ≤ i ≤ n.

Moreover, there exist smooth functions λi (x1 , · · · , xi ) and µi (x1 , · · · , xi+1 ) satisfying 0 < λi (x1 , · · · , xi ) ≤ di (t, x, u) ≤ µi (x1 , · · · , xi , xi+1 ), i = 1, · · · , n. Then, system (25) is globally strongly stabilizable by continuous state feedback. The second result is a straightforward combination of Theorem 3.4 and Remark 3.3. Corollary 3.6 Consider a time-invariant nonlinear system of the form x˙ 1 = xp21 +

p 1 −1

xj2 a1,j (x1 )

j=0

x˙ 2 = xp32 +

p 2 −1

xj3 a2,j (x1 , x2 )

j=0

.. . x˙ n = upn +

p n −1

uj an,j (x1 , · · · , xn ),

(26)

j=0

where the functions ai,j (x1 , · · · , xi ), i = 1, · · · , n, j = 0, · · · , pi − 1, with ai,j (0, · · · , 0) = 0, are continuously diﬀerentiable (i.e. C 1 ). Then, there is a continuous static state feedback control law that globally stabilizes the trivial solution x = 0 of (26).

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Observe that the nonlinear system (26) contains a chain of power integrators by a lower-triangular vector ﬁeld as its special case, i.e. pi −1 perturbed j x a (x , · · · , xi ) ≡ ai,0 (x1 , · · · , xi ), for i = 1, · · · , n. By Coroli,j 1 i+1 j=0 lary 4.9, the only condition for achieving global strong stabilization is that ai,0 (x1 , · · · , xi ) is C 1 and ai,0 (0, · · · , 0) = 0. Therefore, Corollary 3.6 has the following important consequence that was proved in [24], without imposing any growth condition. Corollary 3.7 [24] Every chain of odd power integrators perturbed by a C 1 lower-triangular vector ﬁeld is globally strongly stabilizable by continuous state feedback. We now discuss, via two examples, how the continuous feedback stabilization theory developed so far can be used to design C 0 GSS controllers, for a family of planar systems that are not smoothly stabilizable. Example 3.8 Consider an aﬃne system of the form x˙ 1 = xp2 + xp−2 ap−2 (x1 ) + · · · + x2 a1 (x1 ) + a0 (x1 ) 2 x˙ 2 = v

(27)

where a0 (x1 ), a1 (x1 ), · · · , ap−2 (x1 ) are smooth functions, with a0 (0) = a1 (0) = · · · = ap−2 (0) = 0, and p ≥ 1 is an integer. This planar system is representative of a class of two-dimensional aﬃne systems. In fact, Jakubczyk and Respondek [14] proved that every smooth aﬃne system in the plane, i.e. ξ˙ = f (ξ) + g(ξ)u, is feedback equivalent (via a change of coordinates and smooth state feedback) to (27) if g(0) and adpf g(0) are linearly independent. In other words, (27) is a normal form of two-dimensional aﬃne systems when rank [g(0), adpf g(0)] = 2. In the case of p = 3, the local feedback stabilization of (27) was discussed in [2,14]. It has been known that in general there are no smooth state feedback laws that stabilize system (27) because its Jacobian linearization may contain an uncontrollable and unstable mode. Here we are interested in the global stabilization problem of (27) with p ≥ 1 being an odd integer. In this case, global stabilization of (27) is an open problem that remains unsolved (even in the simple case where p = 3). We remind the reader that in addition to the non-existence of a smooth controller, system (27) is not in a triangular form which makes it exceptionally diﬃcult to stabilize. On the other hand, observe that the planar system is a special case of (26). Using Corollary 3.6, it is immediate to conclude that global stabilization of (27) is always achievable by continuous state feedback, without imposing any growth condition on (27). Moreover, a continuous controller can be explicitly designed via an adding a power integrator approach (i.e. the constructive algorithm in the proof of Theorem 3.4). For illustration, let us examine the planar system (27) with p = 3 and a1 (x1 ) = a0 (x1 ) = x1 , i.e. x˙ 1 = x32 + x1 x2 + x1 x˙ 2 = u

(28)

Global stabilization of nonlinear systems

307

which is not smoothly stabilizable. Moreover, system (28) is neither in a triangular form nor homogeneous. By the homogeneous approximation [10,15,16,27], only local stabilization can be achieved by homogeneous state feedback. However, the problem of global stabilization does not seem to be solvable by existing methods. Following the constructive proof of Theorem 3.4, one can design the continuous controller

3(7 + x21 )2 (17 + x21 − 4x2 )2 14x21 + u = − 18 + 3 4

1 21 + x21 3 3 x2 + x1 4 (29)

which renders the trivial solution (x1 , x2 ) = (0, 0) of system (28) globally strongly stable. The conclusion can be veriﬁed using the C 1 Lyapunov function x2 3 53 1 2 s − x∗3 ds V2 (x1 , x2 ) = x1 + 2 ∗ 2 x2 x1 2 with x∗3 2 = − 4 (x1 + 21). In fact, a direct calculation gives

2 x1 V˙ 2 |(28)−(29) ≤ −x21 − x32 + (x21 + 21) < 0 4

∀(x1 , x2 ) = 0.

(30)

The next example illustrates an interesting application of Corollary 3.5. It also highlights an important role played by the unknown time-varying parameters di (t, x, u) in system (25). Example 3.9 Consider the time-invariant planar system x˙ 1 = x32 (1 + sin2 u) + x1 x˙ 2 = u,

(31)

which is obviously not stabilizable by any smooth state feedback. For this system, d1 (t, x, u) = 1 + sin2 u. Thus, A3.1 is satisﬁed as 1 ≤ d1 (t, x, u) ≤ 2. By Corollary 3.5, (31) is globally stabilizable by continuous state feedback. In fact, it is straightforward to verify that using the Lyapunov function x2 5 1 x21 3 + V2 = (s3 − x∗3 2 ) ds 2 25 x∗2 with x∗2 = −(3x1 ) 3 , one can explicitly construct a C 0 stabilizing controller 1

1

u = −100(x32 + 3x1 ) 3 ,

(32)

which renders 2 3 3 V˙ 2 ≤ −x21 − x2 + 3x1 < 0 25

∀(x1 , x2 ) = 0.

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This shows that system (31) is globally stabilizable at the origin by (32).

Another nice application of Corollary 3.5 is that the planar system x˙ 1 = x32 (

1 + sin2 u) + x1 + x52 , 1 + x21

x˙ 2 = u

is also globally stabilizable by continuous state feedback, simply because 2 1 1 1 2 2 d1 (t, x, u) = 1+x 2 + sin u + x2 and 1+x2 ≤ d1 (·) ≤ 1 + 1+x2 + x2 . 1

4

1

1

Global strong stabilization of cascade systems

We now discusses how the feedback stabilization results developed so far can be extended to a larger class of cascade nonlinear systems of the form z˙ = f0 (z, x1 ) x˙ 1 = d1 (t)xp21 + f1 (t, z, x1 , x2 ) x˙ 2 = d2 (t)xp32 + f2 (t, z, x1 , x2 , x3 ) .. . x˙ r = dr (t)upr + fr (t, z, x1 , · · · , xr , u),

(33)

where u ∈ IR is the control input, z ∈ IRn−r and pi , i = 1, · · · , r, are arbitrary odd positive integers. The functions fi : IR × IRn−r+i+1 → IR, i = 1, · · · , r, are C 0 with fi (t, 0, 0, · · · , 0) = 0 and f0 : IRn−r+1 → IRn−r is a continuously diﬀerentiable function with f0 (0, 0) = 0. A4.1 There are a C 1 function x1 = x∗1 (z) with x∗1 (0) = 0, and a C 2 Lyapunov function V0 (z) which is positive deﬁnite and proper, such that z˙ = f0 (z, x∗1 (z)) satisﬁes ∂V0 f0 (z, x∗1 (z)) ≤ −z2 α(z), ∂z

α(z) > 0.

(34)

The next two conditions are a slight modiﬁcation of A3.1-A3.2. A4.2 For i = 1, · · · , r, there are real numbers λi and µi such that 0 < λi ≤ di (t) ≤ µi . A4.3

For i = 1, · · · , r,

fi (t, z, x1 , · · · , xi , xi+1 ) =

p i −1

xji+1 ai,j (t, z, x1 , · · · , xi )

(35)

j=0

where xr+1 = u. Moreover, there are smooth functions γi,j (z, x1 , · · · , xi ) ≥ 0, such that for 0 ≤ j ≤ pi − 1, 1 ≤ i ≤ r the following holds |ai,j (t, z, x1 , · · · , xi )| ≤ (||z|| + |x1 | + · · · + |xi |)γi,j (z, x1 , · · · , xi ).

Global stabilization of nonlinear systems

309

Under the above-mentioned conditions, we are able to prove the following global stabilization theorem for the cascade system (33). Theorem 4.4 Suppose a cascade system (33) satisﬁes A4.1-A4.3. Then, there exists a continuous controller u = u(z, x1 , · · · , xr ) with u(0, 0, · · · , 0) = 0, which globally stabilizes system (33). The proof of Theorem 4.4 is similar to that of Theorem 3.4, and hence is omitted for the reason of space. The reader is referred to [25] for details. As a consequence of Theorem 4.4, we arrive at the following important conclusion which can be viewed as an extension of Corollary 3.6. Corollary 4.5 [25] Consider a time-invariant cascade system described by equations of the form z˙ = f0 (z, x1 ) x˙ 1 = xp21 +

p 1 −1

xj2 a1,j (z, x1 )

j=0

x˙ 2 = xp32 +

p 2 −1

xj3 a2,j (z, x1 , x2 )

j=0

.. . x˙ r = upr +

p r −1

uj ar,j (z, x1 , · · · , xr ),

(36)

j=0

where the functions ai,j (z, x1 , · · · , xi ) with ai,j(0, · · · , 0) = 0, i = 1, · · · , r, j = 0, · · · , pi −1, are C 1 . Under A4.1, there exists a C 0 controller u(z, x1 , · · · , xr ) that globally stabilizes the trivial solution (z, x1 , · · · , xr ) = 0 of (36). Using Corollary 4.5, it is easy to see, for example, that the cascade system z˙ = x1 + x21 z x˙ 1 = x32 + x1 (ezx1 + x2 ) x˙ 2 = u

(37)

is globally stabilizable by C 0 state feedback, although it is even not locally −z smoothly stabilizable. In fact, choose x∗1 = 2(1+z 2 ) and a positive deﬁnite and 4 2 proper Lyapunov function V0 (z) = z + 2z . Then, a direct calculation shows that A4.1 is fulﬁlled. Since system (37) is in the form of (36), the conclusion follows from Corollary 4.5. So far A4.1 has been employed for the technical convenience. However, it should be pointed out that A4.1 can actually be relaxed, as illustrated by the following example.

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Example 4.6 Consider the cascade system z˙ = z 3 − 2z 2 y 3 y˙ = u3 + y.

(38)

for which A4.1 is apparently not fulﬁlled, simply because there does not exist any C 1 controller y = y ∗ (z) rendering the z-subsystem z˙ = z 3 − 2z 2 y 3 LES. However, following the design procedure of Theorem 4.4, it is still possible to design a C 0 global stabilizer for (38). First, choose a C 0 virtual controller y ∗ = −z 1/3 and y 3 5/3 z2 + W (z, y), W (z, y) = s − y ∗3 ds. V (z, y) = 2 y∗ Let ξ = y 3 − y ∗3 = y 3 + z. Then, the C 0 non-Lipschitz feedback controller 1/3 1/3 1 1 u = −y − ξ 7/3 ( + c) = −y − (y 3 + z)7/3 ( + c) 2 2 is such that 1 V˙ ≤ − (z 4 + ξ 4 ) 2

5

for some

c > 0.

Control of an underactuated mechanical system

The proposed continuous feedback design method is now applied to a benchmark nonlinear system [28], which represents a class of underactuated mechanical systems that are exceptionally diﬃcult to control. The mechanical system consists of a mass m1 on a horizontal smooth surface and an inverted pendulum m2 supported by a massless rod as shown in Figure 2. The mass is interconnected to the wall by a linear spring and to the inverted pendulum by a nonlinear spring which has cubic force-deformation relation. Let x be the displacement of mass m1 and let θ be the angle of the pendulum from the vertical such that at x = 0 and θ = 0, the springs are unstretched. A control force acts on m1 . The system has two degrees of freedom and is underactuated. The equations of motion for the system are ks g (x − l sin θ)3 cos θ θ¨ = sin θ + l m2 l k ks u x ¨=− x− (x − l sin θ)3 + m1 m1 m1

(39) (40)

Deﬁne the smooth change of coordinates x1 = θ,

x2 = x˙ 1 ,

x3 = (x − l sin θ) 3

ks cos θ, m2 l

x4 = x˙ 3 ,

(41)

Global stabilization of nonlinear systems

x u

F=-kx

311

F=-ksy3 m2

m1

g

l

6

F

Fig. 2. An Underactuated System with Weak Coupling

˙ x, x) on (θ, θ, ˙ ∈ (− π2 , π2 ) × IR3 . Then, the inverse mapping is given by x3 + l sin x1 , θ = x1 , θ˙ = x2 , x = ks 3 cos x 1 m2 l x4 x3 x2 sin x1 + + lx2 cos x1 , x˙ = ks 3 3 3 mks2 l cos4 x1 cos x 1 m2 l which is well-deﬁned and smooth on (x1 , x2 , x3 , x4 ) ∈ (− π2 , π2 ) × IR3 . The coordinate transformation (41), together with the smooth state feedback v + 1 x2 x4 tan x1 x3 (x33 + gl sin x1 ) + x4 x2 u = m1 3 + m1 tan x1 ks 3 3 3 mks2 l cos x1 m2 l cos x1   2 1 1 + sin x x3 1 3 +m1 x3 x22 +k + l sin x1  ks ks 3 3 2 3 cos x1 m2 l cos x1 cos x 1 m2 l m l g 2 +x33 + m1 l (x33 + sin x1 ) cos x1 − x22 sin x1 cos x1 l deﬁned on (x1 , x2 , x3 , x4 ) ∈ (− π2 , π2 ) × IR3 , transforms system (39)-(40) into the state space model x˙ 1 = x2 x˙ 2 = x33 + x˙ 3 = x4 x˙ 4 = v.

g sin x1 l (42)

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Since the Jacobian linearization of (42) has an uncontrollable mode associated with a positive eigenvalue, there is no C 1 static state feedback law which stabilizes the system at the origin [3]. In the recent paper [28], the global stabilization problem was solved by continuous state feedback for the homogeneous system g (43) x˙ 1 = x2 , x˙ 2 = x33 + x1 , x˙ 3 = x4 , x˙ 4 = v l which is a homogeneous approximation of system (42). Therefore, by the Hermes’ theorem on robust stability of homogeneous systems [10,27], there is a C 0 state feedback control law which makes the origin of (42) locally asymptotically stable. In what follows, we brieﬂy illustrate how a C 0 global stabilizer can be explicitly constructed for system (42), using the continuous feedback design method presented in the previous sections. For the computational convenience, we ﬁrst normalize the parameter gl in (42) by assuming gl = 14 . Then, choose V1 (x1 ) = 18 x21 . Obviously, the virtual controller x∗2 = − 12 x1 renders V˙ 1 (x1 ) = − 18 x21 + x41 ξ2 with ξ2 = x2 − x∗2 = x2 + 12 x1 . Consider the Lyapunov function V2 (x1 , x2 ) = V1 (x1 ) + 12 ξ22 . Then, it is easy to verify that the C 0 virtual controller 13 3 1 x∗3 = − (x2 + x1 ) (44) 2 2 is such that 1 3 V˙ 2 ≤ − x21 − ξ22 + ξ2 [x33 − x∗3 3 ]. 16 4

(45)

3 3 Next we deﬁne ξ3 = x33 − x∗3 3 = x3 + 2 ξ2 and consider the Lyapunov function x3 53 2 s3 − x∗3 3 ds. V3 (x1 , x2 , x3 ) = V2 (x1 , x2 ) + 15 x∗3

A direct calculation shows that V3 (·) is C 1 , positive deﬁnite and proper. Moreover, it follows from (45) that 3 x2 2 5 V˙ 3 ≤ − 1 − ξ22 + ξ33 x4 + 18ξ32 . 32 16 15 1

Thus, the virtual controller x∗4 = −18ξ33 leads to 3 x2 1 2 5 V˙ 3 ≤ − 1 − ξ22 − ξ32 + ξ33 [x4 − x∗4 ]. 32 16 4 15 In the last step, we add one more integrator and construct x4 53 s3 − x∗4 3 ds. V4 (x1 , · · · , x4 ) = V3 (x1 , x2 , x3 ) + x∗ 4

(46)

Global stabilization of nonlinear systems

313

Denote ξ4 = x34 − x∗3 4 . Similar to Step 3, it can be shown that 5 1 x2 1 V˙ 4 ≤ − 1 − ξ22 − ξ32 + ξ43 v + aξ42 + bξ42 64 16 16

where a and b are two positive real numbers. Now it is not diﬃcult to conclude that a C 0 state feedback law of the form

1 3 1 3 3 3 3 v = −ξ4 (1 + a + b), ξ4 = x4 + 18 x3 + (x2 + x1 ) , (47) 2 2 makes the origin of system (42) globally strongly stable. Indeed, 2 2 1 x2 1 1 3 1 V˙ 4 ≤ − 1 − x2 + x1 − x33 + (x2 + x1 ) 64 16 2 16 2 2

2 3 1 − x34 + 183 x33 + (x2 + x1 ) . 2 2 In addition, the Lyapunov function V4 is C 1 and satisﬁes V4 ≥

(x2 + x1 /2)2 (x3 − x∗3 (x1 , x2 ))6 x21 (x4 − x∗4 (x1 , x2 , x3 ))6 + + + . 5 5 8 2 45 · 4 3 6 43 6

That is, V4 is positive deﬁnite and proper.

References 1. Z. Artstein, Stabilization with relaxed controls, Nonlinear Analysis, TMA-7, (1983), 1163-1173. 2. A. Bacciotti, Local stabilizability of nonlinear control systems, World Scientiﬁc, 1992. 3. R.W. Brockett, Asymptotic stability and feedback stabilization, in Diﬀerential Geometric Control Theory, R.W. Brockett, R.S. Millman and H.J. Sussmann, Eds., Birk¨ auser, Basel-Boston (1983), 181–191. 4. C.I. Byrnes and A. Isidori, Asymptotic stabilization of minimum phase nonlinear systems, IEEE Trans. Automat. Contr., vol. 36 (1991), 1122–1137. 5. S. Celikovsky and E. Aranda-Bricaire, Constructive non-smooth stabilization of triangular systems, Syst. Contr. Lett., Vol. 36 (1999), 21-37. 6. J. M. Coron and L. Praly, Adding an integrator for the stabilization problem, Syst. Contr. Lett., Vol. 17 (1991), 89-104. 7. W. P. Dayawansa, Recent advances in the stabilization problem for low dimensional systems, Proc. of 2nd IFAC NOLCOS, Bordeaux (1992), 1-8. 8. W. P. Dayawansa, C. F. Martin and G. Knowles, Asymptotic stabilization of a class of smooth two dimensional systems, SIAM. J. Contr. and Optim., Vol. 28 (1990), 1321-1349. 9. W. Hahn, Stability of Motion, Springer-Verlag (1967). 10. H. Hermes, Homogeneous coordinates and continuous asymptotically stabilizing feedback controls, in Diﬀerential Equations: Stability and Control, S. Elaydi Ed., Marcel Dekker, New York, (1991), 249-260.

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11. H. Hermes, Nilpotent and high-order approximations of vector ﬁeld systems, SIAM. Review, Vol. 33 (1991), 238-264. 12. A. Isidori, Nonlinear Control Systems, 3rd eds, New York: Springer-Verlag, 1995. 13. A. Isidori, Nonlinear Control Systems II, New York: Springer-Verlag, 1999. 14. B. Jakubczyk and W. Respondek, Feedback equivalence of planar systems and stability, Robust Control of Linear Systems and Nonlinear Control, Kaashoek M.A. et al. eds., (Birkh¨auser, 1990), 447-456. 15. M. Kawski, Homogeneous stabilizing feedback laws, Control Theory and Advanced Technology, Vol. 6 (1990), 497-516. 16. M. Kawski, Stabilization of nonlinear systems in the plane, Syst. Contr. Lett., Vol. 12 (1989), 169-175. 17. H. Khalil, Nonlinear systems, Macmillan Publishing Company, New York, 1992. 18. J. Kurzweil, On the inversion of Lyapunov’s second theorem on the stability of motion, American Mathematical Society Translations, Series 2, 24 (1956), 19-77. 19. W. Lin, Global robust stabilization of minimum-phase nonlinear systems with uncertainty, Automatica, Vol. 33 (1997), 453-462. 20. W. Lin and C. Qian, Semi-global robust stabilization of nonlinear systems by partial state and output feedback, Proc. of the 37th IEEE CDC, Tampa, pp. 3105-3110 (1998). 21. W. Lin and C. Qian, Adding one power integrator: a tool for global stabilization of high order lower-triangular systems, Syst. Contr. Lett., Vol. 39, 339-351 (2000). 22. W. Lin and C. Qian, Robust regulation of a chain of power integrators perturbed by a lower-triangular vector ﬁeld, Int. J. of Robust and Nonlinear Control Vol. 10, pp. 397-421 (2000). 23. H. Nijmeijer and A. J. van der Schaft, Nonlinear dynamical control systems, Springer-Verlag, 1990. 24. C. Qian and W. Lin, Non-Lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization, Proc. of the 39th IEEE CDC, Sydney, pp. 1655-1660 (2000). Also, Syst. Contr. Lett. Vol. 42, No. 3, pp. 185-200 (2001). 25. C. Qian and W. Lin, A continuous feedback approach to global strong stabilization of nonlinear systems, IEEE Trans. Automa. Contr. Vol. 46, No. 7 (2001), 1061-1079. 26. C. Qian, W. Lin, and W. P. Dayawansa, Smooth feedback, global stabilization and disturbance attenuation of nonlinear systems with uncontrollable linearization, SIAM J. Contr. Optimiz. Vol. 40, No.1 (2002) 191-210, electronically published on May 31, 2001. 27. L. Rosier, Homogeneous Lyapunov function for homogeneous continuous vector ﬁeld, Syst. Contr. Lett., Vol. 19 (1992), 467-473. 28. C. Rui, M. Reyhanoglu, I. Kolmanovsky, S. Cho and N. H. McClamroch, Nonsmooth stabilization of an underactuated unstable two degrees of freedom mechanical system, Proc. of the 36th IEEE CDC, San Diego, pp. 3998-4003 (1997). 29. E.D. Sontag, Feedback stabilization of nonlinear systems, in: M.K. Kaashoek et al eds., Robust Control of Linear Systems and Nonlinear control (Birkh¨auser, 1990), 61–81. 30. E.D. Sontag, A “universal” construction of Artstein’s theorem on nonlinear stabilization, Syst. Contr. Lett., Vol. 13 (1989), 117-123.

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31. G. Stefani, Polynomial approximations to control systems and local controllability, Proc. of the 24th IEEE CDC, Florida, (1985), 33-38. 32. H. Sussmann, A general theorem on local controllability, SIAM. J. Contr. and Optim., Vol. 25 (1987), 158-194. 33. J. Tsinias, Global extension of coron-praly theorem on stabilization for triangular systems, European Control Conference, (1997), 1834-1839. 34. M. Tzamtzi and J. Tsinias, Explicit formulas of feedback stabilizers for a class of triangular systems with uncontrollable linearization, Syst. Contr. Lett., Vol. 38 (1999), 115-126.

On computation of the logarithm of the Chen-Fliess series for nonlinear systems Eug´enio M. Rocha Departamento de Matem´atica, University of Aveiro. 3810–193 Aveiro, Portugal. [email protected] Abstract. We obtain expressions for the logarithm of the Chen-Fliess series for a nonlinear control system. This logarithm provides an alternative to Sussmann’s exponential product expansion of the Chen-Fliess series. We also formulate a rule for generating the coeﬃcient of any Lie brackets in the logarithm expansion.

1

Introduction

Consider the nonlinear (analytic) control system x(t) ˙ =

m

uν (t)X ν (x(t))

(1)

ν=1

deﬁned in a real (analytic) manifold M. The Chen-Fliess series represents trajectories of the above control system and has been discussed in [4,5,7]. It is a kind of noncommutative analogue of the Taylor series, with iterated primitives of controls as coeﬃcients. By virtue of the Ree’s theorem, the ChenFliess series can be represented as the exponential of a Lie series (coordinates of ﬁrst kind ) – so we can talk about the logarithm of the Chen-Fliess series – or as a product of exponentials of Lie elements (coordinates of second kind ), also called Sussmann’s exponential product expansion. H.J. Sussmann, ﬁrst in [12], and later M. Kawski and H.J. Sussmann, in [7], provided explicit formulas for coordinates of second kind. This tool turned out to be very usefull for analysis and design of control systems, e.g. for reachability, controllability and motion planning algorithms. In the case of coordinates of ﬁrst kind, M. Kawski in [6] provided an algorithm for its computation with respect to a Hall set. This algorithm provides rather complex expressions for the coeﬃcients. In fact, one of the reasons for such complexity was the use of Hall sets, whereas these sets are well ﬁtted for coordinates of second kind. Using computer algebra tools, M. Kawski also presented formulas for the iterated primitives associated with the Lie brackets of order ≤ 5 for two-input systems. In our work we provide explicit formulas for the coeﬃcients of the logarithm with respect to a larger set than a Hall set. Hence linearly dependent Lie brackets will appear in the expansion. We also present a rather simple rule A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 317-326, 2003. Springer-Verlag Berlin Heidelberg 2003

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E.M. Rocha

for computing the coeﬃcients, without any use of previous coeﬃcients only the information of the respective formal Lie bracket. The method employed in this work relies essentialy in three piers: the paper [1] of A.A. Agrachev and R.V. Gamkrelidze, where they present the foundations of chronological calculus, computations of the variation of a ﬂow provided in [3], and the knowledge of the algebras involved – namelly, symmetrization of chronological algebras (in the sense of Agrachev et al., [2]) and anti-symetrization of Zinbiel algebras (in the sense of J.-M. Lemaire, [8]). We notice that, although the formal existence has already been ensured, we don’t study the convergence of the series. At this moment we can only speculate about the usefulness of this formula, although the notion of the logarithm has already appeared in some works, e.g. the study of stability properties of ODE and stabilizability of control systems [11], and recent works of M. Kawski. This will be a subject of further study. The next section establishes the basic terminology and algebraic relations. In this publication we only give a brief description of the relations and properties of the algebras used. The following section presents the main result and a short example. We ﬁnish with some remarks and a sketch of the proof.

2

Terminology and basic facts

Let G be a real Lie group with Lie algebra G on M. For P ∈ G and X, Y ∈ G, we use Ad(P ) to denote the inner automorphism of G : Ad(P )X = P ◦X◦P −1 , and ad Y the inner derivation of G : (ad Y ) X = [X, Y ]. By a nonstationary vector ﬁeld we mean a family Xt ∈ C ∞ (IR, G) and a nonstationary ﬂow a family Pt ∈ C ∞ (IR, G) – viewed with the compact-open C ∞ topology. Let ℵ be a set, which we call an alphabet, whose elements are letters – usually ℵ = {1, . . . , m}. A tree on the alphabet ℵ is a bracketed ﬁnite sequence of elements of ℵ. Let M (ℵ) be the free magma constructed on ℵ, i.e. the free non-associative structure over the alphabet ℵ. Elements of M (ℵ) are all binary trees over ℵ. This structure has a natural graduation and any element w of length (weight) |w| ≥ 2, can be decomposed as w = l(w)r(w), where l(w) and r(w) denote the elements of M (ℵ) that verify |l(w)| < |w|, |r(w)| < |w| and are called the left and the right factors of w. Let Lib(ℵ) denote the free non-associative IR-algebra of the set ℵ. This module admits M (ℵ) as basis. If A is an algebra, every mapping of ℵ into A can be extended uniquely to a homomorphism of Lib(ℵ) into A. Let k denote the bigger integer smaller than k. Construct the set of trees H ⊂ M (ℵ), in the classic way, by induction on the integer n sets Hn ⊂ M n (ℵ) and a total ordering on these sets: 1. We write H1 = ℵ and give it a total ordering. 2. The set H2 consists of the products xy with x, y ∈ ℵ and x < y. We give it a total ordering.

Logarithm of Chen-Fliess series

319

3. Let n ≥ 3 be such that the totally ordered sets H1 , . . . , Hn−1 are already n−1 deﬁned. In the case, n is odd, deﬁne Hn = i=12 Hi × Hn−i , otherwise, n−1 n is even, deﬁne Hn = i=12 Hi × Hn−i ∪ Yn with Yn = {w1 w2 ∈ Hn/2 × Hn/2 : w1 < w2 }. In either cases, give it a total ordering induced by the order of all Hj , in the following way: let w, w ∈ Hn , w < w if and only if |l(w)| < |l(w )| or |l(w)| = |l(w )| and l(w) < l(w ) or l(w) = l(w ) and r(w) < r(w ). We write H = ∪n≥1 Hn ; we give H the total ordering deﬁned thus: w < w if and only if |w| < |w | or |w| = |w | = n and w < w in the set Hn . Clearly, this set is much larger than a P. Hall set over ℵ. If an algebra admits a derivation D and a linear mapping I, such that D(I(xy)) = I(D(xy)) = xy, we denote it a (D, I)-algebra. Let AsR , LieR , CommR denote categories of (D, I)-algebras, respectively: associative, Lie, and commutative and associative. As examples consider respectively the vector space of nonstationary vector ﬁelds that vanish at zero with the composition product, the same vector space with the Lie product, and the ring of locally absolutely continuous functions that vanish at zero with the usual pointwise product of functions. For any (D, I)-algebra R deﬁne a new (right) algebra structure by the product x y = I(x · D(y)), where · is the R algebra product, we call it a I-D algebra over R. The integration by parts formula relates right I-D algebras with left I-D algebras, deﬁned as expected. An I-D algebra over an algebra of LieR is a (right) chronological algebra with product ≡ , in the sense of Agrachev et al. [2], that satisﬁes the identity X (Y Z) = [X, Y ] Z + Y (X Z).

(2)

This is nothing else than the result of applying the Jacobi identity to X (Y Z). If the right chronological algebra product is R and the left product is L then X R Y = −Y L X. On the other hand, an I-D algebra over an algebra of CommR is a (right) Zinbiel algebra with product ∗ ≡ , that satisﬁes the relation f ∗ (g ∗ h) = (f · g) ∗ h,

(3)

where · is the product algebra of CommR . The relation is simply the application of the integration by parts formula, and implies f ∗ (g ∗ h) = g ∗ (f ∗ h). These algebras satisfy the identity f ∗R g = g ∗L f . The denomination of Zinbiel algebra is proposed by J.-M. Lemaire (see [8]), although in some works these algebras are designated as chronological algebras, see [7,6]. From any algebra we can construct its symmetrization algebra by introducing into the algebra the symmetric product x y = x · y + y · x, and its anti-symmetrization algebra by the skew-symmetric product xy = x·y−y·x. To obtain formulas for the logarithm we are interested in the algebra A that

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comes from a chronological algebra by applying the symmetric product, obtaining the product x

y = −I(D− (x y)), and the algebra B that arises from a Zinbiel algebra by applying the skew-symmetric product, obtaining the product x y = −I(D− (x y)), where D− (xy) = D(x)y − xD(y). All this relations between categories of algebras can be assembled into the following diagram - Comm R inc - ASR

- LieR ` e

` e

..

?

Zinbiel

?

Chrono ..

?

B

?

A

One of the main tools in this work will be the use of the operator notations of chronological calculus, which can be found in [1,3].

3

Main result

To deﬁne the logarithm of a nonstationary ﬂow, let us consider the nonlinear system x˙ = F (t, x),

(4) −→

where Ft ≡ F (t, ·) is a complete nonstationary vector ﬁeld. Let Pt = exp t F dτ be the nonstationary ﬂow generated by it – the family of diﬀeomor0 τ phisms satisfying the operator diﬀerential equation dPt = Pt ◦ Ft , P0 = Id. dt

(5)

The logarithm Λ(t) of Pt is a nonstationary vector ﬁeld that veriﬁes the equality eΛ(t) = Pt . The Chen-Fliess series represents Pt in the particular control-linear case: Ft = ν∈ℵ uν X ν . In the view of system (1), we denote by X ℵ the set of vector ﬁelds {X i }i∈ℵ t and by αℵ the set primitives of controls {αi }i∈ℵ (i.e. αi (t) = 0 ui (τ ) dτ ). Let AC(αℵ ) denote the commutative and associative algebra of locally absolutely continuous functions generated by the set of functions αℵ (deﬁned on [0, +∞[ and vanishing at zero) endowed with the usual product of functions. Introduce in AC(αℵ ) the product ∗, resulting a (right) Zinbiel algebra with product t ∂ βτ dτ, ατ (6) (α· ∗ β· )t = ∂τ 0

Logarithm of Chen-Fliess series

321

αt , βt ∈ AC(αℵ ) – denoted by ZA. Let B be the anti-symmetrization of ZA with product t d d (α· β· )t = ατ βτ − βτ ατ dτ. (7) dτ dτ 0 The constant function 1 is a left anti-unit, that is (1 α· )t = −(α· 1)t = αt . Although this product is anti-commutative it does not satisfy any independent identity involving products of three variables (such as associativity, Jacobi, Zinbiel,...) – the type of the algebra is not a quadratic operad, and we don’t know any non-trivial identity involving higher number of variables. Observe that the symmetrization of the ZA product is a shuﬄe product which, by previous reasoning, is nothing else than the usual product of functions. From B we deﬁne AB as the IR-algebra whose generators are all elements of B with the usual pointwise product of functions. Let E1 be the canonical homomorphism from H to B, that is E1 : H → B is deﬁned as E1 (w) = αw , for w ∈ H1 , and E1 (w) = E1 (l(w)) E1 (r(w)), otherwise. In the same way, let E2 be the canonical homomorphism from H to G(X ℵ ) deﬁned as expected: E2 (w) = X w , for w ∈ H1 , and E2 (w) = [E2 (l(w)), E2 (r(w))], otherwise. Also, deﬁne the multi-function sub : H → 2AB recursively by sub[w] = {E1 (w)}, for |w| ≤ 2, and sub[w] = {E1 (w)} ∪ {x · y : x ∈ sub[l(w)], y ∈ sub[r(w)]}, otherwise – where · denote the usual product of functions. Theorem 1. For the perturbed nonlinear control system uν X ν , x˙ =

(8)

ν∈ℵ

where ℵ is a ﬁnite alphabet, the perturbed Chen-Fliess logarithm is of the form Λ (t) =

∞

ξ

ξ=1

where

¯ ζ(w) =

¯ ζ(w) ⊗ E2 (w),

(9)

w∈Hξ

cw∗ w∗ ,

(10)

w∗ ∈sub[w]

for cw∗ ∈ IR. Corollary 1. Suppose existence of the perturbed Chen-Fliess logarithm. Then the Chen-Fliess logarithm for the system (1) is of the form ¯ (11) ζ(w) ⊗ E2 (w), Λ(t) = w∈H

¯ where ζ(w) is given in the theorem.

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One can derive a simple rule for determining the logarithm coeﬃcient of a chosen Lie bracket in H: in the expression of the given Lie bracket t replace each vector ﬁeld X v by the respective primitive of control αv (t) = 0 uv (τ )dτ and each Lie product by the product, then the logarithm coeﬃcient is the weighted sum of all possible substitutions of products by pointwise products, including none – where the bracket structure is preserved and no substitution is made if less then two products exist. Consider the case ℵ = {a, b}. For example to obtain the coeﬃcient of the vector ﬁeld [X b , [X a , [X a , X b ]]], consider (αb (αa (αa αb ))) and compute all allowed decompositions – by substitution of the product by the pointwise product – {αb (αa (αa αb )), αb (αa (αa αb )), αb αa (αa αb )}; the logarithm coeﬃcient is the weighted sum of these factors. Therefore, the ﬁrst seven terms of the logarithm would be

+ + + + + + +

1 Λ(t) = αa X a + αb X b + (αa αb ) [X a , X b ] 2 1 a 1 (α (αa αb )) − αa (αa αb ) [X a , [X a , X b ]] 3 12 1 b 1 b a (α (αa αb )) − α (α αb ) [X b , [X a , X b ]] 3 12 1 a 1 a a a a (α (α (α αb ))) − α (α (αa αb )) [X a , [X a , [X a , X b ]]] 4 12 1 a 1 a b (α (αb (αa αb ))) − α (α (αa αb )) [X a , [X b , [X a , X b ]]] 4 12 1 b 1 b a (α (αa (αa αb ))) − α (α (αa αb )) [X b , [X a , [X a , X b ]]] 4 12 1 b 1 b b b a b a b (α (α (α α ))) − α (α (α α )) [X b , [X b , [X a , X b ]]] 4 12 ...

where we omitted 0 (αa )2 (αa αb ), 0 (αb )2 (αa αb ) and 0 αa αb (αa αb ) – formulas for scalars cw∗ can be found in the proof. Observe that, using an algorithm presented in [9], we can write elements of H as a linear combination of elements of a Hall set. Therefore, we can ﬁnd expressions for logarithm coeﬃcients with respect to a Hall set, although such expressions will be of very high complexity and will lose the structure presented in this work. We can derive formulas involving controls in a more explicit way. Let C be the anti-symmetrization of the Zinbiel algebra of controls (measurable t functions) with product (ua · ub )t = ubt 0 uaτ dτ . Denote the product of C by

. If g : B → C is the expected homomorphism between B and C, then any

Logarithm of Chen-Fliess series

x ∈ B is equal to

323

g(x). So, for example,

τ τ 1 t ub (θ)dθ − ub (τ ) ua (θ)dθ dτ ; ζ¯ ab = ua (τ ) 2 0 0 0 t t 1 1 a a b a (u (u u ))τ dτ − (ua ub )τ dτ = ζ¯ a(ab) = u 3 0 12 0 τ τ τ 1 t ub (τ ) ua (θ)dθ ua (θ)dθ − ua (τ ) ub (θ)dθ = 3 0 0 0 0 τ τ θ ua (θ)dθ + ua (τ ) ua (θ) ub (ν)dν − ub (θ) 0

0

0

t τ 1 a ua (ν)dνdθ dτ − ub (θ)dθ u (t) ua (τ ) 12 0 0 0 τ −ub (τ ) ua (θ)dθ dτ = 0 t 1 1 b 5 2 a = u ζ(a) − u ζ(a)ζ(b) + ζ(ab) dτ. 4 6 0 4

θ

As pointed out by M. Kawski, the representation of coordinates of ﬁrst kind by a recursive formula is not apropriate. Although there is the simple recursive formula xw w = c xw ∗ xw , c ∈ IR, for coordinates of second kind.

4

Sketch of the proof

Since the proof is relatively technical ([10]), we here only sketch the main ideas. Introduce a small parameter in (4) coming to x˙ = Ft (x). We will ﬁnd −→ t relations for the logarithm Λ (t) of exp 0 Fτ dτ that imply Λ (t)

e

−→

= exp

t

Fτ dτ.

(12)

0 −→ t The map Φt () = exp 0 Fτ dτ denotes the family of nonstationary ﬂows on M which depends on ∈ IR. The logarithm of the Chen-Fliess series is the particular case when Ft = ν∈ℵ uν X ν and = 1. The idea is simple, deﬁning Λ0 = 0 we ensure that (12) is valid for = 0, and determine Λ from the diﬀerentiation of (12) with respect to . Let us start with the diﬀerentiation of the right hand side of equation (12). A.A. Agrachev et al. [3] obtained invariant “bracket” formula for the tangent vector to the curve → q0 ◦ Φt () in M at point q0 ◦ Φt (0), in the case where arbitrary singularities take place. By the variation of constants formula and diﬀerentiation of Φt () with respect to we have

∂ Φt () = Zt () ◦ Φt (), ∂

(13)

324

E.M. Rocha

where Zt () is a time-varying vector ﬁeld depending absolutely continuously t on time t; Zt (0) = 0 Fτ dτ . The vector ﬁeld Zt () has been called ’angular velocity’ of the ﬂow (see [3]), and satisﬁes the equation ∂ Zt () = (Z· () Z· ())t , ∂

t where is the chronological product (X· Y· )t = 0 [Xτ , ∂/∂τ Yτ ]dτ , whose algebra is the algebra of locally absolutely continuous on t nonstationary vector ﬁelds vanishing at zero. Representing homogeneous elements of this chronological algebra with only one generator Zt (0) in terms of the symmetrized product of the algebra product and using properties of their subalgebras we arrive to the following lemma. +∞ (k) Lemma 1. Let Zt () = k=0 k Zt . Then, for k ≥ 0, (k) ρ(w) (w ⊗ w) ∈ B ⊗ G(X ℵ ), Zt =

(14)

w∈Hk+1

where the scalar ρ(w) ∈ IR is given by the map ρ : H → IR deﬁned recursively by ρ(w) = 1, w ∈ H1 ∪ H2 , and |w| − 2 ρ(w) = ρ(l(w))ρ(r(w)). |l(w)| − 1 Now, in order to diﬀerentiate eΛ , we must use the formal variant of the variation formula for quasistationary vector ﬁelds (see [1]), obtaining ∂ Λ e = ∂

1

eτ ad Λ dτ

0

Therefore, using (13), 0

1

eτ ad Λ dτ

∂ Λ ◦ eΛ . ∂

∂ Λ ∂ e

=

∂ ∂

−→

exp

(15) t 0

Fτ dτ implies

∂ Λ = Zt (). ∂

(16)

From the application of the inverse function Γ (ζ) = we obtain the formal diﬀerential equation ∂ Λ = Γ (adΛ )Zt (). ∂

ζ eζ −1

of

1 0

eτ ζ dτ =

eζ −1 ζ ,

(17)

σ as S σ = {σ} ∪ subsets Consider S of M (IN), for σ ∈ IN, deﬁned σ−1 σ−1 2 2 S i × S σ−i for σ odd, and, S σ = {σ} ∪ S i × S σ−i ∪ Qσ i=1 i=1 σ

σ

for σeven – where Qσ = {v1 v2 ∈ S 2 × S 2 : v1 < v2 }. As before, let ∞ S = σ=1 S σ . For any tree v ∈ S, let the image of the map |·|R : M (IN) → IN be the right-deep length of the tree v recursively deﬁned by |v|R = 1 if v ∈ S

Logarithm of Chen-Fliess series

325

and |v|R = 1 + |r(v)|R . Deﬁne the map s[·] : M (IN) → IN as the sum of all elements of the foliage of the given tree. Also, any tree v ∈ S has a unique right factorization in trees of S j , for j < σ, v = v1 (v2 (· · · vp ))) where vp ∈ IN and p = |v|R . We deﬁne for a tree v ∈ S the polynomial Pv ∈ Lib(IN) as Pv = v if v ∈ IN and Pv = Pl(v) Pr(v) − Pr(v) Pl(v) . For any tree v ∈ S σ the scalar coeﬃcient of v is the image of the map : M (IN) → IR : v → ¯(Pv ), where 1 if |v|R = 1, the map ¯ : Lib(IN) → IR is deﬁned recursively as ¯(v) = s[v] and, for v = v1 (v2 (· · · vp ))),

¯(v) =

1 Bp−1

¯(v1 ) · · · ¯(vp−1 ), s[v] (p − 1)!

(18)

extend it linearly to Lib(IN), where Bk are Bernoulli numbers. Let G(Z) be the Lie algebra generated by {Z (0) , Z (1) , . . . } and EV : Lib(IN) → G(Z) be the unique homomorphism extension of the map IN → G(Z) : n → Z (n−1) . Let M (M (ℵ)) be the free magma over the set M (ℵ), with formal product, denoted by ’·’. Hence, M (H) ⊂ M (M (ℵ)). Since all Z (k) ∈ B ⊗ G(X ℵ ), by the properties of the Lie product, G(Z) ⊆ AB ⊗ G(X ℵ ), and [E2 (Hi ), E2 (Hj )] = E2 (Hi+j ), i ≤ j, imply EV (v) ∈ AB ⊗ E2 (Hn ), v ∈ S n . Let Lib(H) denote IR M (H) and Eb : Lib(H) → AB the extension of E1 : H → B. Hence, for w1 , w2 ∈ H, [Eb(w1 ) ⊗ E2 (w1 ), Eb(w2 ) ⊗ E2 (w2 )] = Eb(w1 · w2 ) ⊗ E2 (w1 w2 ) = Eb(l(w) · r(w)) ⊗ E2 (w), with w = w1 w2 ∈ H. Deﬁne the injective multifunction q : S → 2M (H) as q[σ] = Hσ , σ ∈ IN, and q[v1 v2 ] = q[v1 ] · q[v2 ]. Let sub−1 : M (H) → H transform any w∗ ∈ M (H) into a w ∈ H by substitution of all products of M (M (ℵ)) by products of M (ℵ). Clearly, sub−1 is a non injective homomorphism. Also, deﬁne ρ¯(·) : M (M (ℵ)) → IR ρ(r(w∗ )). as ρ¯(w∗ ) = ρ(w∗ ) if w∗ ∈ M (ℵ), and ρ¯(w∗ ) = ρ¯(l(w∗ ))¯ Lemma 2. With the above deﬁnitions in mind, we have

(v)EV (v) ∈ G(Z) Λ(σ) =

(19)

v∈S σ

and EV (v) =

ρ¯(w∗ )Eb(w∗ ) ⊗ E2 (sub−1 (w∗ )).

(20)

w∗ ∈q[v]

For w ∈ Hσ , consider the set Vw = {(v, w∗ ) ∈ S |w| × M (H) : sub−1 (w∗ ) = w, w∗ ∈ q[v]}. There exists only one v ∈ S for every w∗ , such that w∗ ∈ q[v]. So, q has the inverse map q −1 : M (M (ℵ)) → M (IN) deﬁned as q −1 [w∗ ] = |w∗ |M (ℵ) if w∗ ∈ M (ℵ), and q −1 [w∗ ] = q −1 [l(w∗ )] × q −1 [r(w∗ )]. Observe the existence of w∗ ∈ M (M (ℵ)) for which q −1 [w∗ ] ∈ S. Recall that we want to write Vw explicitly in order to w. Observing that, for i = 1, 2, sub−1 (w∗ ) ∈ Hi ⇒ q −1 [w∗ ] = i, we can write the “generalized” inverse multi-function of sub−1 as the multi-function sub : M (ℵ) → 2M (M (ℵ)) deﬁned by sub[w] = {w}

326

E.M. Rocha

if |w| ≤ 2, and sub[w] = {w} ∪ sub[l(w)] · sub[r(w)], otherwise. By induction on n,

q −1 [w∗ ] = S n , (21) w∗ ∈ w∈Hn sub[w] shows that q −1 and sub are compatible with the graduations of S and H. Acknowledgement. This work is part of the author’s PhD project which is carried out at the University of Aveiro – Portugal, and has been partially supported by PRODEP and by a program of pluriannual ﬁnancing of FCT, Portugal – R&D Unit “Mathematics and Applications” of the University of Aveiro. The author is grateful to his PhD supervisor, A.V. Sarychev, for stimulating discussions regarding the subject.

References 1. Agrachev A.A., Gamkrelidze R.V. (1978) Exponential representation of ﬂows and chronological calculus. Matem. Sbornik 107, 467-532. English transl. in: Math. USSR Sbornik 35 (1979), 727-785. 2. Agrachev A.A., Gamkrelidze R.V. (1979) Chronological algebras and nonstationary vector ﬁelds. Journal Soviet Mathematics 17, 1650-1675 (in Russian). 3. Agrachev A.A., Gamkrelidze R.V., Sarychev A.V. (1989) Local invariants of smooth control systems. Acta Applicandae Mathematicae 14, 191-237. 4. Chen K.T. (1957) Integration of paths, geometric invariants and a generalized Baker-Hausdorﬀ formula. Annals of Mathematics 65, 163-178. 5. Fliess M. (1989) Fonctionelles causales nonlin´eaires et indetermin´ees noncommutatives. Acta Appl. Math. 15, 3-40. 6. Kawski M. (2000) Calculating the logarithm of the Chen Fliess series. International Journal Control (submitted) (also: elec. proc. MTNS 2000). 7. Kawski M., Sussmann H.J. (1997) Noncommutative power series and formal Lie-algebraic techniques in nonlinear control theory. in: Helmke U., Pr¨atzelWolters D., Zerz E. Eds. Systems and Linear Algebra. Teubner, Stuttgart, 111-128. 8. Loday J-L. (2001) Dialgebras. in: Dialgebras and related operads. Springer Lecture Notes in Math. 1763, 7-66. 9. Reutenauer C. (1993) Free Lie algebras. London Mathematical Society Monographs - new series 7, Oxford Science Publications. 10. Rocha E.M. (2001) On computation of the logarithm of the Chen-Fliess series for nonlinear systems. CM01/I-08, Departamento de matem´atica, Universidade de Aveiro. 11. Sarychev A.V. (2000) Stability criteria for time-periodic systems via high-order averaging techniques. in: Isidori A., Lamnabhi-Lagarrique F., Respondek W. Eds., Nonlinear Control in the Year 2000, 2, Springer-Verlag, Berlin et al, 365377. 12. Sussmann H.J. (1986) A product expansion of the Chen series. In Theory and Applications of Nonlinear Control Systems, Byrnes C.I., Lindquist A. eds., Elsevier, North-Holland.

On the eﬃcient computation of higher order maps adkf g(x) using Taylor arithmetic and the Campbell-Baker-Hausdorﬀ formula Klaus R¨obenack Institut f¨ ur Regelungs- und Steuerungstheorie, TU Dresden, Mommsenstr. 13, D-01062 Dresden, Germany Abstract. The computation of adkf g requires derivatives of f and g up to order k. For small dimensions, the Lie brackets can be computed with computer algebra packages. The application to non-trivial systems is limited due to a burden of symbolic computations involved. The author proposes a method to compute function values of adkf g using automatic diﬀerentiation.

1

Introduction

During the last two decades, many methods for controller design for systems with smooth nonlinearities have been developed [14,22,13]. Quite a few of these methods are based on diﬀerential-geometric concepts using Lie derivatives and Lie brackets. For smooth functions f , g : Rn → Rn and h : Rn → Rp , Lie derivatives Lkf h(x) are deﬁned by the recursion Lkf h(x) =

∂Lk−1 h(x) f ∂x

f (x) with

L0f h(x) = h(x) .

The Lie bracket [f , g] is deﬁned by [f , g](x) =

∂g ∂f f (x) − g(x) . ∂x ∂x

The vector ﬁelds adkf g(x) are deﬁned recursively by adkf g(x) = [f , adk−1 g](x) with f

ad0f g(x) = g(x) .

Among other thinks, these maps adkf g are used to check the controllablitiy rank condition [12] or in observer design [17] for state-space systems x˙ = f (x) + g(x)u,

y = h(x) .

Up to now, these maps adkf g required by various algorithms for controller and observer design were computed symbolically using computer algebra packages (see [19,20,26,21,18]). Even though these methods have been applied successfully to diﬀerent real-world systems, their use is limited due A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 327-336, 2003. Springer-Verlag Berlin Heidelberg 2003

328

K. R¨ obenack

to the amount of symbolical computations involved [8]. The authors suggest a new approach which is based on automatic diﬀerentiation. In Sect. 2, we introduce the reader to the concept of automatic diﬀerentiation and Taylor arithmetic. The use of Taylor arithmetic for the series expansion of autonomous ODEs and the computation of Lie derivatives is described in Sect. 3. The series expansion of the associated tangent systems in discussed in Sect. 4. The author presents a new method to calculate values of the maps adkf g in Sect. 5. In Sect. 6, this method is applied to an example.

2

Automatic diﬀerentiation

Derivatives of a function F are needed in many diﬀerent areas of control and automation, e.g. for sensitivity analysis. Conventionally, these derivatives are computed symbolically with computed algebra packages such as Mathematica [28], Maple [7] or MuPAD [23]. Unfortunately, symbolic diﬀerentiation usually has the problem of expression growth due to product, quotient and chain rule. Furthermore, symbolic diﬀerentiation often entails a repeated evaluation of common subexpressions. Moreover, the function under consideration may not be described by an explizity given mathematical expression but by an algorithm containing loops or conditional statements. An other way to obtain derivative values of a function F is the divided diﬀerence approach. In this case, the value F (x) is approximated by diﬀerence quotients such as (F (x + h) − F (x))/h or (F (x + h) − F (x − h))/(2h). Diﬀerence quotients do not provide accurate values due to cancellation and truncation errors. For higher order derivatives, these accuracy problems become acute. These drawbacks of symbolic or numeric diﬀerentiation can be avoided with automatic or algorithmic diﬀerentiation [10]. Similar to the symbolic diﬀerentiation, elementary diﬀerentiation rules like sum, product, quotient and chain rule will be applied systematically. The main diﬀerence is, that software packages for automatic diﬀerentiation use ﬂoating point numbers instead of symbolic expressions. In contrast to divided diﬀerences approach, automatic diﬀerentiation incurs no truncation errors at all. Usually, automatic diﬀerentiation is only used to compute Gradients, Jacobians, or Hessians. In this paper, we will use univariate Taylor series to compute higher order derivatives. Consider a d times continuously diﬀerentiable function F : Rn → Rm . Let x be a curve given by x (t) = x0 + x1 t + x2 t2 + · · · + xd td + O(td+1 )

(1)

with vector-valued coeﬃcients x0 , . . . , xd ∈ Rn . The function F maps the curve x into a curve z. We can express z by the truncated Taylor series z (t) = z 0 + z 1 t + z 2 t2 + · · · + z d td + O(td+1 ) k 1 ∂ z(t) with z k = k! . k ∂t t=0

(2)

On the eﬃcient computation of higher order maps adkf g(x)

329

Each Taylor coeﬃcient z k ∈ Rm is determined by the coeﬃcients x0 , . . . , xk . In fact, we have the following relations [11]: z0 z1 z2 z3

= = = =

F (x0 ), F (x0 )x1 , F (x0 )x2 + 12 F (x0 )x1 x1 , F (x0 )x3 +F (x0 )x1 x2 + 16 F (x0 )x1 x1 x1 ,

(3)

We have already used tensor notation to describe the coeﬃcients z 2 and z 3 . Note that we do not have to compute the high dimensional derivative tensors 2 3 d F (x0 ) ∈ Rm×n , F (x0 ) ∈ Rm×n , . . . , F (d) (x0 ) ∈ Rm×n . Using automatic diﬀerentiation, each Taylor coeﬃcients z k can be obtained directly from the Taylor coeﬃcients x0 , . . . , xk . In modern programming computer languages such as C++, one may simply replace the ﬂoating point type (e.g. double) by a new class (say Tdouble) which contains not only one value by all (d + 1) Taylor coeﬃcients: const int d=...; class Tdouble { public: double coeff[d+1]; } Now, one has only to overload elementary functions (e.g. sin, cos, exp) and operations (e.g. +, −, ∗, /). Some examples of the associated calculations are listed in Table 1 (see [1] and [10, Chap. 10.2]). Using operator overloading, the Taylor coeﬃcients of composite functions will be computed with these modiﬁed rules. These rules are already implemented in Automatic Diﬀerentiation packages such as ADOL-C [11] and TADIFF [1], where both tools use C++. In contrast to symbolic diﬀerentiation, the computational eﬀort (time, memory requirement) to evaluate the Taylor coeﬃcients z 0 , . . . , z d by automatic diﬀerentiation increases only quadratically (not exponentially) with respect to d (see [10]). Up to this point, the derivative values (i.e., the Taylopr coeﬃcients) are computed almost simultaneously with the function values itself. More precisely, the calculation of derivative values takes place in the same order as the calculation of function values. For this reason, the diﬀerentiation method described up to now is called forward mode of automatic diﬀerentiation. In the so-called reverse mode, the derivative values are computed in reverse order compared to the original program ﬂow. In fact, the reverse mode can be seen as a generalization of the backprobagation algorithm used in neuronal networks [15]. Details of the reverse mode are presented in [10]. The reverse mode of automatic diﬀerentiation can be used for an eﬃcient computation of the matrices A0 , . . . , Ad ∈ Rm×n of the Jacobian path F (x (t)) = A0 + A1 t + A2 t2 + · · · + Ad td + O(td+1 ) .

330

K. R¨ obenack Table 1. Computation of Taylor coeﬃcients

Operations z = x±y z = x·y z = x/y

Taylor Coeﬃcients zk = xk ± yk k zk = xi yk−i i=0 k zk = y10 xk − yi zk−i k−1

z = exp(x)

zk =

1 k

z = ln(x)

zk =

1 x0

zk =

1 2z0

z=

√

z

Conditions

i=0

y0 = 0

for k ≥ 1,

i=1

(k − i) zi xk−i

k−1 i zi xk−i xk − k1 i=1 k−1 zi zk−i xk −

for k ≥ 1, z0 = exp(x0 ) for k ≥ 1, z0 = ln(x0 ), x0 > 0 for k ≥ 1, z0 =

√

x0 ,

x0 > 0

i=1

The matrices A0 , . . . , Ad are partial derivatives of the Taylor coeﬃcients of the curves x and z. There holds the following identity [5]: ∂z j ∂z j−i Aj−i if j ≥ i , = = (4) 0 otherwise . ∂xi ∂x0

3

Series expansion of state-space systems and Lie derivatives

In this section we will employ automatic diﬀerentiation to compute the function values of Lie derivatives Lkf h(x0 ). Consider the autonomous ODE x˙ = f (x),

x(0) = x0 .

(5)

Simlutaneously, we will look at f as a map from the curve x into a curve z, i.e., z(t) ≡ f (x(t)), where the curves are given by truncated Taylor series (1) and (2), respectively. For known Taylor coeﬃcients x0 , x1 , . . . , xi of x with 0 ≤ i ≤ d, the Taylor coeﬃcients z 0 , z 1 , . . . , z i of z can be computed with automatic diﬀerentiation as described in Sect. 2. If the curve x is assumed ˙ to be the solution of (5), the curve z has to fulﬁl the identity z(t) ≡ x(t). This identity can be used to get to next Taylor coeﬃcient (i.e., xi+1 ) of the solution curve x: xi+1 =

zi . i+1

(6)

Let us start with an initial value x0 ∈ Rn . To get z 0 , we simply have to evaluate f , i.e., z 0 = f (x0 ). Using (6), we obtain x1 . Now, the Taylor arithmetic of automatic diﬀerentiation can be used to calculate z 1 . Again, we apply (6) to compute x2 . Finally, we obtain the Taylor coeﬃcients x0 . . . . , xd

On the eﬃcient computation of higher order maps adkf g(x)

331

of the solution of (5). In fact, this expansion method has been used to develop a whole class of new integration methods for ODEs, see [6,3,4,9]. Next, we will take the output map y(t) = h (x(t))

(7)

into consideration. The Taylor coeﬃcients y 0 , . . . , y d ∈ Rp of the output curve y(t) = y 0 + y 1 t + y 2 t2 + · · · + y d td + O(td+1 )

(8)

can be computed using the Taylor arithmetic. On the other hand, the output curve y can be expressed in terms of Lie derivatives Lkf h(x0 ) (see [13]): y(t) =

∞

Lkf h(x0 )

k=0

tk . k!

(9)

Comparing (8) and (9), we get Lkf h(x0 ) = k ! y k . With this equation we are able to compute the values of Lie derivatives Lkf h(x0 ) using automatic diﬀerentiation (see [25,24]).

4

Tangent system

˙ Let us consider the soluation curve x of (5) together with the curve z = x. The Jacobians Aj−i = ∂z j /∂xi ∈ Rn×n between two Taylor coeﬃcients z j and xi (see (4)) can be obtained from automatic diﬀerentiation tools such as ADOL-C [11]. Using (6), we obtain the partial derivate between a Taylor coeﬃcient xj and a preceding Taylor coeﬃcient xi (0 ≤ i < j) of the solution curve x: ∂xj 1 ∂z j−1 1 = = Aj−i−1 . ∂xi j ∂xi j Taking the dependencies between Taylor coeﬃcients into account, the total derivatives Xk = dxk /dx0 ∈ Rn×n can be accumulated as follows (see [9]) 1 dz k 1 ∂z k dxj 1 = = Ak−j Xj , k + 1 dx0 k + 1 j=0 ∂xj dx0 k + 1 j=0 k

Xk+1 =

k

(10)

where X0 = dx0 /dx0 = I is the identity matrix. The matrices Xk are the coeﬃcient matrices of the series expansion X(t) = X0 + X1 t + X2 t2 + · · ·

332

K. R¨ obenack

of the solution of the tangent system ˙ X(t) = f (x(t)) · X(t),

X(0) = I ,

(11)

where x is the solution of the inital value problem (5), see [10]. Now, let us take the dual tangent system ˙ Z(t) = −Z(t) · f (x(t)),

Z(0) = I

(12)

into consideration. Similar as above (see (10)), one can obtain an accumulation rule for the coeﬃcients Zk ∈ Rn×n of the solution Z(t) = Z0 + Z1 t + Z2 t2 + · · · of (12): 1 Zi Ak−i , k + 1 i=0 k

Zk+1 = −

5

Z0 = I .

(13)

Lie Brackets and the Cambell-Baker-Hausdorﬀ formula

Let Φft (·), and Φgt (·) denote the ﬂows generated by f and g, respectively. The map Adt,f g(x) is deﬁned by ∂Φf f −t (z) Adt,f g(x) = · g Φ (x) t ∂z z=Φf (14) t (x) ∂ Φf−t ◦ Φgs ◦ Φft (x) = ∂s s=0

There is the following relation between adkf g(·) and Adt,f g(·) (see [16]): Adt,f g(x0 ) =

∞

adkf g(x0 )

k=0

tk . k!

If one rewrites (14) in terms of this series expansion, one gets the CampbellBaker-Hausdorﬀ formula (e.g. see [13]): ∞ ∂Φf−t (z) tk k · g Φft (x0 ) . adf g(x0 ) = (15) k! ∂z k=0 z=Φf (x ) 0

t (∗) The term (∗) is the solution of the initial value problem (12) in connection with (5), i.e., we have ∂Φf−t (z) Z(t) = (16) ∂z f z=Φt (x0 )

On the eﬃcient computation of higher order maps adkf g(x)

333

(e.g. see [27, p. 168]). Next, we will turn our attention to the remaining part of (15). For this reason we take the initial value problem x˙ = f (x),

˜ = g(x), y

x(0) = x0

(17)

˜ d ∈ Rn of ˜0, . . . , y into account. The Taylor coeﬃcients y ˜ (t) = g Φft (x0 ) = y ˜0 + y ˜1t + · · · + y ˜ d td + O(td+1 ) y can be calculated using the method explained in Sect. 3. If we write the Cambell-Baker-Hausdorﬀ formula (15) in terms of the dth order truncated ˜ , we obtain Taylor series Z and y ∞

adf g(x0 )

k=0

tk ˜ (t) + O(td+1 ). = Z(t) · y k!

˜ (t) can be obtained by a genThe Taylor coeﬃcients of the product Z(t) · y eralization of the product rule (see Tab. 1): ˜ 0 + (Z0 y ˜ 1 + Z1 y ˜0) t + · · · + ˜ (t) = Z0 y Z(t) · y

k

˜ k−i Zi y

tk + · · ·

i=0

After all we get an expression of the values adkf g(x0 ) based on Taylorcoeﬃcients computed by automatic diﬀerentiation: adkf g(x0 ) = k!

k

˜ k−i Zi · y

for

0 ≤ k ≤ d.

(18)

i=0

Figure 1 shows the whole computation scheme to calculate function values of adkf g(x0 ).

6

Example

Consider the simple example system     exp(x2 ) 0 x˙ =  krx1 + x22  +  exp(x2 )  u krx1 − x2 0

f (x) g(x)

(19)

with the 3-dimensional state-space vector x = (x1 , x2 , x3 )T (see [13, p. 151]). We want to calculate the values of the maps adkf g at x0 = (1, 2, 3)T employing automatic diﬀerentiation. For the computation we used ADOL-C 1.8 in

334

K. R¨ obenack

Fig. 1. Computation scheme for ad kf g(x0)

connection with the GNU C++ compiler gcc 2.95.3. With the initial value x0 we obtained the following expansion of the ﬂow x(t) = Φft (x0 ) of (17):           0 0 1 0 0 x(t) ≈  2  +  5  t +  10  t2 +  21.667  t3 +  46.667  t4 . −2.5 3 −1 −3.333 −5.417 ˜ , we are only interested in the expression As for y exp(x2 (t)) ≈ 7.389 + 36.945 t + 166.254 t2 + 683.488 t3 + 2630.81 t4 . The matrices Ak and Zk computed in the reverse mode and by the accumulation rule (13), respectively, are shown in Tab. 2. Moreover, Tab. 2 shows the values of adkf g(x0 ) for k = 0, . . . , 4 computed with automatic diﬀerentiation as described in Sect. 5. To verify our results, we computed the maps adkf g for k = 0, . . . , 3 symbolically:   exp(x2 ) ad0f g(x) =  exp(x2 )  0   (x1 + x22 ) exp(x2 ) ad1f g(x) =  (x1 − 2x2 + x22 − 1) exp(x2 )  0   (x1 + x22 )(x1 + 2x2 + x22 ) exp(x2 ) ad2f g(x) =  (−4x1 + 2x2 − 2x1 x2 + x21 − 2x32 + x42 + 2x1 x22 ) exp(x2 )  (2x2 + 1) exp(x2 )

On the eﬃcient computation of higher order maps adkf g(x)

335

Table 2. Values of Ak , Zk and adkf g(x0 ) k 0

1

2

3

4

Ak  0 0 0 1 4 0 1 −1 0   0 0 0  0 10 0  0 0 0   0 0 0  0 20 0  0 0 0   0 0 0  0 43.333 0  0 0 0   0 0 0  0 93.333 0  0 0 0 



 Zk  100 0 1 0 001   0 0 0  −1 −4 0  −1 1 0   0 0 0  2 3 0 −0.5 −2 0

0 0  −1 2.667 0.667 −0.667  0 0  −0.667 −1 0.167 0.667

 0 0 0  0 0 0

adkf g(x 0)  7.389  7.389  0   36.945  0  0   332.508  36.945  −36.945   4100.93  295.562  −554.179   63139.5  2253.66  −9827.44

The expression of ad3f g(x) was already to large to ﬁt in the column. A numerical evaluation of these symbolical expressions for x0 = (1, 2, 3)T yields the result shown in Tab. 2 obtained with automatic diﬀerentiation.

References 1. C. Bendtsen and O. Stauning. TADIFF, a ﬂexible C++ package for automatic diﬀerentiation. Technical Report IMM-REP-1997-07, TU of Denmark, Dept. of Mathematical Modelling, Lungby, 1997. 2. A. Bensoussan and J. L. Lions, editors. Analysis and Optimization of Systems, Part 2, volume 63 of Lecture Notes in Control and Information Science. Springer, 1984. 3. Y. F. Chang. Automatic solution of diﬀerential equations. In D. L. Colton and R. P. Gilbert, editors, Constructive and Computational Methods for Diﬀerential and Integral Equations, volume 430 of Lecture Notes in Mathematics, pp. 61–94. Springer Verlag, New York, 1974. 4. Y. F. Chang. The ATOMCC toolbox. BYTE, 11(4):215–224, 1986. 5. B. Christianson. Reverse accumulation and accurate rounding error estimates for Taylor series. Optimization Methods and Software, 1:81–94, 1992. 6. G. F. Corliss and Y. F. Chang. Solving ordinary diﬀerential equations using Taylor series. ACM Trans. Math. Software, 8:114–144, 1982. 7. J.-M. Cornil and P. Testud. An Introduction to Maple V. Springer, 2001. 8. B. de Jager. The use of symbolic computation in nonlinear control: is it viable? IEEE Trans. on Automatic Control, AC-40(1):84–89, 1995. 9. A. Griewank. ODE solving via automatic diﬀerentiation and rational prediction. In D. F. Griﬃths and G. A. Watson, editors, Numerical Analysis 1995, volume 344 of Pitman Research Notes in Mathematics Series. Addison-Wesley, 1995.

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10. A. Griewank. Evaluating Derivatives — Principles and Techniques of Algorithmic Diﬀerentiation, volume 19 of Frontiers in Applied Mathematics. SIAM, 2000. 11. A. Griewank, D. Juedes, and J. Utke. A package for automatic diﬀerentiation of algorithms written in C/C++. ACM Trans. Math. Software, 22:131–167, 1996. 12. R. Hermann and A. J. Krener. Nonlinear controllability and observability. IEEE Trans. on Automatic Control, AC-22(5):728–740, 1977. 13. A. Isidori. Nonlinear Control Systems: An Introduction. Springer, 3rd edition, 1995. 14. B. Jakubczyk, W. Respondek, and K. Tchon, editors. Geomatric Theory of Nonlinear Control Systems. Technical University of Wroclaw, 1985. 15. D. Juedes and K. Balakrishnan. Generalized neuronal networks, computational diﬀerentiation, and evolution. In M. Berz, editor, Proc. of the Second International Workshop, pp. 273–285. SIAM, 1996. 16. A. J. Krener. (adf,g ), (adf,g ) and locally (adf,g ) invariant and controllability distributions. SIAM J. Control and Optimization, 23(4):523–524, 1985. 17. A. J. Krener and A. Isidori. Linearization by output injection and nonlinear observers. Systems & Control Letters, 3:47–52, 1983. 18. A. Kugi, K. Schlacher, and R. Novaki. Symbolic Computation for the Analysis and Sythesis of Nonlinear Control Systems, volume IV of Software for Electrical Engineering, Analysis and Design, pp. 255–264. WIT-Press, 1999. 19. R. Marino and G. Cesareo. Nonlinear control theory and symbolic algebraic manipulation. In Mathematical Theory of Networks and Systems, Proc. of MTNS’83, Beer Sheva, Israel, June 20-24, 1983, volume 58 of Lecture Notes in Control and Information Science, pp. 725–740. Springer, 1984. 20. R. Marino and G. Cesareo. The use of symbolic computation for power system stabilization: An example of computer aided design. In Bensoussan and Lions [2], pp. 598–611. 21. Neil Munro, editor. Symbolic methods in control system analysis and design. IEE, 1999. 22. H. Nijmeijer and A. J. van der Schaft. Nonlinear Dynamical Control systems. Springer, 1990. 23. W. Oevel, F. Postel, G. R¨ uscher, and St. Wehrmeier. Das MuPAD Tutorium. Springer, 1999. Deutsche Ausgabe. 24. K. R¨ obenack and K. J. Reinschke. A eﬃcient method to compute Lie derivatives and the observability matrix for nonlinear systems. In Proc. 2000 International Symposium on Nonlinear Theory and its Applications (NOLTA’2000), Dresden, Sept. 17-21, volume 2, pp. 625–628, 2000. 25. K. R¨ obenack and K. J. Reinschke. Reglerentwurf mit Hilfe des Automatischen Diﬀerenzierens. Automatisierungstechnik, 48(2):60–66, 2000. 26. R. Rothfuß, J. Schaﬀner, and M. Zeitz. Rechnergest¨ utzte Analyse und Synthese nichtlinearer Systeme. In S. Engell, editor, Nichtlineare Regelungen: Methoden, Werkzeuge, Anwendungen, volume 1026 of VDI-Berichte, p. 267—291. VDIVerlag, D¨ usseldorf, 1993. 27. E. D. Sontag. Mathematical Control Theory, volume 6 of Texts in Applied Mathematics. Springer-Verlag, 2nd edition, 1998. 28. S. Wolfram. The MATHEMATICA Book. Cambridge University Press, 1999.

Nonlinear M.I.S.O. direct-inverse compensation for robust performance speed control of an S.I. engine A. Thomas Shenton1 and Anthemios P. Petridis 2 1

2

University of Liverpool, Department of Engineering, Liverpool, L69 3GH, UK, E-mail [email protected] Ford Motor Company, Dunton Research and Engineering Centre, Basildon, Essex, SS15 6EE, UK

Abstract. Nonlinear inverse compensation is developed for an SI engine MISO system from an identiﬁed plant model. The proposed application of the inverse is similar to the use of inverse SISO systems in one of Horowitz’s nonlinear QFT methods. For robust control the nonlinearity is characterised as an unstructured uncertainty. The objective of the inverse system is linearisation to reduce the size of the uncertainty necessary to account for the system nonlinearity. An increase in achievable robust performance with any linear robust performance control design technique may then be obtained. The novel feature of the MISO strategies proposed is that they provide additional design control over the control eﬀorts. The method is experimentally validated by application to the automotive SI engine idle speed control problem using an electric dynamometer.

1

Introduction

This paper presents the theory and experimental implementation of a novel multiple-input single-output (MISO) nonlinear direct-inverse robust performance technique. The method is applied to the speed control of 1.6L portinjected automotive spark-ignition (SI) engine using the Liverpool University low-inertia electric dynamometer facility [1]. The multiple control inputs are the spark advance/retard angle and the air-bleed valve duty-cycle. The controlled single output is the engine crankshaft speed. The dynamics of this type of engine system includes pure time delays as well as nonlinear uncertain dynamics [2]. This engine control application is primarily a tracking or regulation problem. In addition to the signiﬁcant uncertainty associated with disturbances and plant characteristics and to the demanding performance targets, severe constraints exist on control-eﬀort due to possible saturation. The paper addresses how the additional degrees of freedom in the MISO structure may be exploited to design robustly performant controllers for the nonlinear uncertain plant subject to these control-eﬀort constraints.

A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 337-349, 2003. Springer-Verlag Berlin Heidelberg 2003

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Although linear robust performance techniques can often be applied directly to nonlinear systems by treating nonlinearity as uncertainty, this approach can be excessively conservative. In robust performance control the ability to handle uncertainty must always be traded oﬀ with achievable performance. Direct inverse compensation [3] [4], signiﬁcantly reduces the cost of this trade-oﬀ by reducing the component of uncertainty necessary to account for system nonlinearity. Accordingly in direct nonlinear control, the input-output transmission is linearised (or at least made more linear) by using the inverse of the minimum phase factor of the system (without time delay). This is either as a pre (right-inverse) nonlinear ﬁlter or as a post (left-inverse) nonlinear ﬁlter. Linear techniques are then used to obtain robust performance on the resulting system. The possibly non-causal nonlinear ﬁlter combines with a causal linear ﬁlter to be a causal nonlinear control compensator. The key advantage of such direct-inverse control over other comparable nonlinear techniques such as linearising feedback and nonlinear observer systems is that no inner-loop control is required. This means that robust performance of the complete system is addressed directly in the linear ﬁlter design. However with this approach a choice must be made to use either a preﬁlter or a post-ﬁlter. Using a pre-ﬁlter the demand (input to the pre-ﬁlter) to plant output transmission is linearised. In the feedback conﬁguration this gives a linear path between both the demand and error signal and the controlled plant output. The demand and error to control-eﬀort path is then nonlinear. Alternatively a post ﬁlter structure gives a linear transmission between demand and error to control-eﬀort. However, the demand and error to controlled output is then nonlinear. These nonlinear paths do not allow the linear robust performance techniques to directly design the linear compensator for both the output tracking or regulation and the control eﬀort constraint. As with many robust performance and inverse techniques the direct inverse method is currently most fully developed for SISO or square MIMO systems. The key idea in this paper is to exploit the additional freedom of the MISO structure, which is characteristic of the automotive engine control application, to obtain better design control over the nonlinear paths. Using a pre-ﬁlter the direct design for the linearised demand and error to output transmission is complemented by improved design control over the nonlinear control-eﬀort paths. In this paper a nonlinear state-space model of the engine dynamics is identiﬁed and validated from experimental dynamometer tests. A MISO inverse compensator system is constructed and this itself is validated by simulation.

M.I.S.O. direct-inverse compensation of an S.I. engine

339

The compensator is implemented on the engine to establish the reduction in uncertainty required to account for the system nonlinearity by means of linear identiﬁcation. A nonlinear inversion technique similar to Horowitz’s nonlinear plant inversion QFT technique is presented in this paper. In Horowitz’s technique the QFT templates are obtained based on parametric uncertainty of the nonlinear plant model, however, in the present work, linear spot-point models are obtained at diﬀerent operating points on the open loop system resulting in templates which arise as circles in the Nyquist chart. The spread of these models is represented as uncertainty about the resulting nominal linear model. An inversion is carried out on the minimum phase part of a nonlinear identiﬁed model of the engine. The resulting inverted engine model is applied as a pre-ﬁlter to the engine system which serves to cancel out the nonlinear engine dynamics thus linearising the combined system. The degree to which the linearisation is successful can be determined by the closeness of the linear spot-point models obtained at diﬀerent operating points on the open loop system. Open loop testing of the engine in combination with the inverse pre-ﬁlter enables linear identiﬁcation of the linearised plant dynamics. The uncertainty associated with the identiﬁed linear models arising from using linear models to represent a nonlinear system, is thereby reduced. Subsequent linear robust controller design is possible using robust linear methods such as [5], [6], [7], [8] A phenomenologically structured multiple-input single-output (MISO) nonlinear state-space engine model for a Ford 1.6 litre SI engine identiﬁed on an electric dynamometer is used as a basis for the inversion. Previous experimental testing based on a SISO system was carried out by the authors [4] [9] in which the possible reductions in LTIE uncertainty was experimentally quantiﬁed. In this paper an alternative inverse system approach is presented. The potential for additional robustness and performance with nonlinear inverse compensation is established through the experimental determination of the vector margins using an electric engine dynamometer.

2

Nonlinear engine model

The model structure selected to represent the Ford Zetec 1.6L engine ABV to speed response is based on the nonlinear phenomenological engine model described in [10] but simpliﬁed to contain the most prominent engine dynamics for this work. The model form is a set of continuous nonlinear state-space diﬀerential and output equations with unknown parameter coeﬃcients. The ﬁnal implementation of the controller is as a discrete ﬁxed sample time system using the Matlab Simulink Real-Time-Workshop numerical discretisation process. A SISO model was previously used in [4], however the MISO model structure identiﬁed in this paper includes a pure time delay to account for the intake-to-combustion stroke. The resulting engine model equations are

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A.T. Shenton and A.P. Petridis

in the form of two nonlinear state-space equations and an output equation inclusive of the delay, with identiﬁed parameter coeﬃcients as follows: x˙ 1 = p1 + p2 x2 + p3 x2 x1 + p4 x2 x21 + p5 u1 x˙ 2 = p6 + p7 x2 + p8 x2 x1 + p9 x2 x21 + p10 x22 + p11 u2 + p12 u22 + p13 u2 x2 y(t) = x2 (t − Td ).

(1) (2) (3)

The two state variables x1 and x2 represent the intake manifold pressure and the engine speed respectively. u1 is the duty cycle input to the air-bleed valve (ABV), u2 is the spark-advance angle (SPK), p1,2,...,13 are the model parameters, and Td is the time delay between the inputs and output which acts on the engine output speed y. Disturbance torque load is known to act as an additive (lagged step) disturbance on the second state equation. For identiﬁcation of the appropriate inverse, the input-output data record was shifted to account for the time delay by 5 sample times which was the delay identiﬁed by carrying out step response tests. For a nominal sample time of Ts = 33.2 ms, the delay term was found to be Td = 5Ts . A ﬂoating point genetic algorithm [11] minimising least squares errors was used to identify the remaining parameters pi . The resulting parameters are given in Table 1. Table 1. Identiﬁed parameter coeﬃcients Parameter

Value

p1

56.995×103

p2

709.7754

p3

−12.1×10−3

p4

50.151×10−9

p5

−553.840×103

p6

438.1427

p7

3.072

p8

−72.466×10−6

p9

61.531×10−12

p10

747.160×10−6

p11

3.4441

p12

−900.2×10−3

p13

22.7×10−3

Td

5×33.2−3

M.I.S.O. direct-inverse compensation of an S.I. engine

341

4

x 10

Measured P Simulated P

7

Pa

6.5

6

5.5 0

10

20

30 Time (s)

40

50

60

Measured N Simulated N

1600

RPM

1400 1200 1000 800 600 400 0

10

20

30 Time (s)

40

50

60

Fig. 1. Validation of identiﬁed model with unseen data. (a) Inlet manifold pressure. (b) Engine speed

To validate the parameters, the model was simulated and compared to engine data (diﬀerent to that used in the identiﬁcation) as shown in Fig. 1 for the two engine states, intake manifold pressure and engine speed.

3

Direct-inverse linearisation and quasi-linearisation

Horowitz’s nonlinear QFT inverse (direct-inverse) method [3] was chosen as a basis for the inverse compensation rather than other commonly used linearisation methods such as feedback linearisation. Reasons for this are that techniques such as feedback linearisation surrender complete control over the control eﬀort and make direct design for minimum control eﬀort diﬃcult. Another reason is the issue of the ‘robust inner’ and ‘robust outer’ feedback loops associated with a linearising feedback system. That is, while it may be possible to guarantee robust stability for each loop individually, when the ﬁnal system is attached to the plant robust stability for the complete system is not automatically guaranteed. This is not an issue with single loop feedback controllers using a direct inverse system. Furthermore, it is natural to include feedforward control in the direct inverse system since the input is

342

A.T. Shenton and A.P. Petridis

the desired output trajectory. Direct inversion also allows open loop testing enabling linear identiﬁcation of the compensated system to verify the amount of linearisation actually achieved. Also, when using a direct inverse approach such as the nonlinear QFT technique, explicit design for robust stability and robust performance is allowed. In the nonlinear QFT method, a nonlinear compensator and a linear controller are implemented in series preceding the plant. Linearisation is achieved since the approximate inverse of the mean or nominal plant dynamics cancels out the nonlinear plant dynamics. The LTIE plant uncertainty due to nonlinearity which limits the controller design process, is thereby reduced. Here the inverse system for control is required to take x2 as the demand signal (set-point) and determine u1 and u2 to achieve this. In previous work it was established that a full inverse system could be achieved by u1 (the ABV duty-cycle) alone so it is clear that there are an inﬁnite number ways to achieve MISO control. Evidently we should exploit the extra degree of freedom introduced by use of u2 as a control signal. Linear Quasi−Linear

∆ yd

e

−

K1 K2

y1 y2

y0

u1

ΛG

u2

G

z −n

∆y

Fig. 2. MISO Direct inverse control scheme

The central idea of this paper is to add a further constraint on the inputs u1 and u2 so that some additional linearity in the system can be obtained. The motivation for this is to allow better control over the magnitude of the control signals |u1 | or |u2 | which have tight upper and lower allowed limits. With any direct inverse system compensation it is of course straightforward to apply one’s favourite robust linear controller design technique around the inverse compensated system. The input to output paths are linearised. However the complete input to control eﬀort transmission is in general nonlinear and then linear techniques optimising weighting functions on the control effort sensitivity function (closed loop transfer from demand to plant input control eﬀort) cannot be used. In the inverse compensators considered here, it is required to obtain improved design control over the control signals applied directly to the plant and the plant output itself by the combination of the control inputs on the compensator. More exactly in practice on the engine for economy and emissions reasons it is required to work with desired control signal set-points of

M.I.S.O. direct-inverse compensation of an S.I. engine

343

u1 and u2 , for a regulated output set-point y 0 = x02 , we will require to regulate about optimal control eﬀorts u01 and u02 . In our scheme this is done by constraining the deviations from these set points, ∆u1 and ∆u2 by specifying y 0 as two inputs to the inverse system y1 and y2 so that u1 = u01 + ∆u1 u2 = u02 + ∆u2 y = y 0 + ∆y.

(4) (5) (6)

We set up the inverse system with the two inputs y1 and y2 so that ∆y = y1 + y2 .

(7)

Two cases of this quasi linearisation scheme (Fig 2) can be considered: • coordinating the control signals to obtain linearity in the ratio of the control eﬀorts pre-ﬁlter and post-ﬁlter (coordinated control); • pre-ﬁlter input-output linearisation in one selected channel by additional nonlinear action in the other channel (selective control). Coordinated control In this case the idea is to make the ratio of the control eﬀorts track the ratio of the pre-ﬁlter inputs. We require y1 ∆u2 = y2 ∆u1 .

(8)

The idea here is that to maximally exploit say u1 , we generally require as high a gain as is feasible in this channel. Design control over the ratio ∆u1 : ∆u2 allows the designer to balance the amount of use of these control eﬀorts. If, for example, |u2 | is found to be too large then the gain in the linear controller feeding the system via y2 is reduced and the gain in the linear controller operating though y1 is increased to maintain the desired y. Selective control In this alternative scheme we exactly linearise one of the channels, that is either the y1 to ∆u1 or y2 to ∆u2 transmission. We then simply require ∆u1 = αy1

(9)

or ∆u2 = αy2 as selected, where α is any chosen linear operator. The selected channel is then linearised by the action of the other. This approach is useful where the selected channel is signiﬁcantly more prone to saturation than the linearising channel (actually the case with the spark-advance angle channel with respect to the air-bleed valve channel). Clearly the range over which linearisation is achievable in the selected channel is constrained by the bandwidth of the linearising channel.

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10 . x1

x1 1 2

. x2

x2

3

y

11 u2

5

∆y ∆u2

y2 8

u1

4

6

∆u1

7 y1

Fig. 3. Coordinated control directed constraint diagram

4

The inverse compensators

This inverse compensator is conveniently determined by the solution of a constraint graph. This scheme may be thought of as a two-vertex typed graph where the vertices labelled by circles represent the dynamic system variables and the vertices labelled by rectangular boxes represent the constraint relations (in this case equation nos.). To determine this the system equations are conveniently ﬁrst expressed as an undirected constraint graph. The directed version is then established by causality assignment in the manner of bond graph methods [12]. With the additional deﬁnitional equations t x˙ 1 dt = x1 (10) 0 t x˙ 2 dt = x2 (11) 0

we have all the necessary system variables and constraints. The solution to the directed version of the constraint graph for the coordinated control inverse ﬁlter with y1 and y2 as input and u1 and u2 as

M.I.S.O. direct-inverse compensation of an S.I. engine

345

10 . x1

x1 1 2

. x2

x2

3

y

11

6 ∆y

y2

u2

7 u1

4

∆u 1

9

y1

Fig. 4. Selective control directed constraint diagram

output is shown in Fig. 3. It can be seen that it is necessary to establish u1 by feedback within the inverse ﬁlter. To avoid algebraic loops in the real-time implementation it was found possible to introduce a small time lag in this loop without signiﬁcant degradation in simulated ﬁlter performance. The solution to the directed version of the constraint graph for the selective control inverse ﬁlter, again with y1 and y2 as input and u1 and u2 as output is shown in Fig. 4. The switching conditions between the appropriate solutions are established for x1 by the physical necessity for x1 > 0 and for u1 , by simulation. An open loop simulation was used to validate the recovery performance of the coordinated control inverse compensator. Test signals passing through a forward model G drive the inverse compensator Λ which itself pre-compensates another model of G. The test signals and those recovered by the inverse compensation are shown, together with the ratios of the driving and those of the recovered signals in Fig. 5. Slight deviations due to initial conditions can be seen in the initial few seconds and during high frequency perturbations.

A.T. Shenton and A.P. Petridis 0.55

1000

0.5

800

0.45

u (DTY)

1200

u Actual 1 u1 Recovered

0.4

1

600

2

x (RPM)

346

400

0.35 x Actual (Lagged) 2 x2 Recovered

200 0

0

5

10 Time (s)

15

0.3 0.25

20

25

0

5

10 Time (s)

15

20

1.5 u2 Actual u2 Recovered

20

y1 / y2 Actual u1 / u2 Recovered

1 Ratio

u2 (deg)

15 10

0.5

5 0 0 −5

0

5

10 Time (s)

15

20

−0.5

0

5

10 Time (s)

15

20

Fig. 5. Signals recovery of coordinated control inverse compensator

4.1

Identiﬁcation of linear models

This work was conducted with the low inertia electric engine dynamometer at the University of Liverpool [1] on a Ford Zetec 1.6L with 4 valves per cylinder and sequential port fuel injection. The dynamometer is interfaced using dSPACE-AUTOBOX and MATLAB Real Time Workshop (RTW). The nonlinear compensator network was implemented in the RTW Simulink environment by in eﬀect replacing the engine plant model P with the actual engine. To excite a wide range of frequencies, pseudo random binary sequence (PRBS) test signals were used as the input for identiﬁcation between the ABV and the engine speed as output. Resulting linear models were characterised by coeﬃcients of rational transfer functions with time delay. These were identiﬁed using the ARMAX algorithm in the System Identiﬁcation toolbox in MATLAB [13]. A collection of models were identiﬁed at diﬀerent spot point operating conditions representing the idle speed envelope.

M.I.S.O. direct-inverse compensation of an S.I. engine

5

347

Experimental results

Finally we assess the eﬀectiveness of the coordinated inverse control compensator in reducing the gain and phase uncertainty of the system. Since it is possible to reduce uncertainty at the expense of reducing the dynamic performance of the system, for an engineering evaluation it is also necessary to determine the eﬀect of compensation on the general system dynamics around the important crossover frequencies. y1 −> y without compensation

y2 −> y without compensation

VM=0.275

0

VM

−0.2 Imaginary axis

Imaginary axis

−0.2 −0.4 −0.6 −0.8 −1 −1.2

−0.6 −0.8 −1

−1.4 −1.5

−1

−0.5 Real axis

0

y1 −> y with compensation VM=0.750

−1.5

VM 0

−1

−0.5 Real axis

0

y2 −> y with compensation VM=0.654

−0.2 Imaginary axis

−0.2 Imaginary axis

−0.4

−1.2

−1.4

0

VM=0.268

0

−0.4 −0.6 −0.8 −1 −1.2

−0.4 −0.6 −0.8 −1 −1.2

−1.4

−1.4 −1.5

−1

−0.5 Real axis

0

−1.5

−1

−0.5 Real axis

0

Fig. 6. Experimental vector margins for loops y1 → y and y2 → y, with and without coordinated direct inverse compensation

Around the known crossover frequencies of the engine (between 1 and 4 rad/s) the plant uncertainty is highly unstructured. The uncertain frequency response in the Nyquist diagram is therefore well represented by uncertainty disks. These are ﬁtted at discrete frequencies on the transfer function loci. In determination of the beneﬁcial or detrimental eﬀects of any compensation on the uncertain dynamics it is necessary to compare the original and modiﬁed uncertainty in relative terms at each frequency. To achieve this, any certain (i.e. non-uncertain) non-minimum phase compensator may, for comparison purposes, be applied to the open loop (uncertain) system. This is accept-

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able since the controller can always later remove the exact dynamics of the compensator. Accordingly, to compare the vector margin (VM) at the same frequency a lag compensator was applied to the uncompensated uncertain Nyquist loci [9]. Disks spanning the frequency range 1.3 to 3.9 rad/s are shown in Fig. 6 indicating the relative degree of robust stability of the open loop system by means of the respective VMs. The VM for the y1 → y and y2 → y channel is compared at the nominal locus frequency of 2.13 rad/s where for the uncompensated system, VM=0.275 and VM=0.268 compared to VM=0.750 and VM=0.654 respectively for the inverse compensated disks.

6

Conclusions

The conclusions are: • A nonlinear MISO inverse compensation methodology has been presented. This was applied to the dynamics of an SI engine for the linear robust control problem of idle speed regulation. • In order to develop the nonlinear inverse of the engine system, a nonlinear phenomenologically based state-space model of the engine was used. The model represents the dominant engine dynamics between input via the air-bleed valve (ABV) and spark-advance angle (SPK) and the outputs intake manifold pressure and engine speed. • The inverse system is relaxed from being a true inverse by inverting only the minimum-phase factor of the nonlinear model. The linear controller design stage is required to ensure that the controller has the required excess of poles over zeros. • The inverse system is developed by means of transforming the statespace nonlinear plant equations into a constraint graph and reassigning the system causality. • Collections of LTI models were identiﬁed using an ARMAX approach, each model identiﬁed at a diﬀerent spot point operating condition. These models collectively represent the nonlinearity inherent in the engine system and are suitable for robust linear control system design. • The ﬁnal inverse compensated system was compared to the uncompensated system by experimental tests carried out on an engine dynamometer. Improved linearity was shown by reduced uncertainty regions in the linear models especially near the likely system crossover frequencies. The capability for superior robust performance was thereby experimentally demonstrated. • It was demonstrated that real-time control using the input-output inverse is in fact possible using modern computing digital signal processing hardware and software.

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References 1. R.E. Dorey, D. Maclay et al. (1995), Advanced powertrain control strategies. IFAC Workshop on Automotive Control, Ascona, Switzerland 144–149 2. B. Hariri, A.T. Shenton et al. (1998) Parameter identiﬁcation, estimation and dynamometer validation of the nonlinear dynamics of an automotive spark-ignition engine, Journal of Vibration and Control 4(1) 47–59 3. I. Horowitz (1981) Improvement in quantitative non-linear feedback design by cancellation, Int. J. Control 34 547–560 4. A.P. Petridis, A.T. Shenton (2000) Non-linear inverse compensation of an SI engine by system identiﬁcation for robust performance control. Inverse Problems in Engineering 8, 163–176 5. G.J. Balas, J.C. Doyle et al. (1991) µ-Analysis and synthesis toolbox, The MathWorks Inc. 6. V. Besson, A.T. Shenton (2000) Interactive parameter-space design for robust performance of MISO control systems, IEEE Trans. on Automatic Control 45(10), 1917–1924 7. C. Borghesani, Y. Chait et al. (1994) Quantitative feedback theory toolbox, The MathWorks Inc. 8. R.Y. Chiang, M.G. Safonov (1992) Robust control toolbox, The MathWorks Inc. 9. A.P. Petridis (2000) Non-linear robust control of S.I. engines, PhD thesis, Liverpool University 10. B. Hariri (1996) Modelling and identiﬁcation of SI engines for control system design, PhD thesis, Liverpool University 11. C.R. Houck, J.A. Joines et al. (1995) A genetic algorithm for function optimization: a Matlab implementation, North Carolina State University 12. D.C. Karnopp, D.L. Margolis et al. (1990) Systems dynamics: A uniﬁed approach, Wiley Interscience 13. L. Ljung (1991) System identiﬁcation toolbox, The MathWorks Inc.

Exact delayed reconstructors in nonlinear discrete-time systems control Hebertt Sira-Ram´ırez1 and Pierre Rouchon2 Cinvestav-I.P.N1 Dept. Ingenier´ıa El´ectrica, Secci´ on Mecatr´ onica Avenida IPN, # 2508, Col. San Pedro Zacatenco, A.P. 14740 07300 M´exico, D.F., M´exico. Centre Automatique et Syst`emes2 ` Ecole des Mines de Paris 35 Rue Saint-Honor´e 77305 Fontainebleau, Cedex, France

Abstract. In this article we show that the n-dimensional state of an observable discrete-time nonlinear SISO system can always be exactly synthesized by means of a “structural reconstructor” which only requires knowledge of a ﬁnite number of delayed inputs and delayed outputs. This fact, when combined with the diﬀerence ﬂatness of the system, results in an eﬀective systematic feedback control scheme which avoids the need for traditional asymptotic state observers.

1

Introduction

Availability of the state vector in the synthesis of model-based designed feedback control laws is a crucial assumption, or requirement, needed in achieving the desired closed loop behavior of a given dynamic system. The lack of knowledge of the state vector, due to the necessarily limited character of measurements on the system variables, may sometimes be replaced by the use of a complementary dynamic system, called an asymptotic state observer, whose state trajectories are guaranteed to converge towards those of the original plant, irrespectively of its arbitrary initial state values. The literature on asymptotic state observers for linear and nonlinear systems, of continuous or discrete-time nature, is vast and certainly out of the scope of this article for a fair review. Nevertheless, for the interested reader, we brieﬂy mention some important contributions made in the past, all in the realm of discrete-time nonlinear systems (DTNLS), which are relevant to the problem of observer design, observability and feedback control of this important class of systems. The work of Grizzle [3], Jakubczyk and Sontag [4] and Monaco and Normad-Cyrot [5] and Fliess [1], all deal with fundamental aspects of the description of DTNLS and the relevance of particular analysis tools. The work of Aranda-Bricaire et al [6] devotes special attention to the problem of feedback linearizability of DTNLS, which we also use in the control aspects related to this work. The reader is also referred to the book recently edited by A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 351-360, 2003. Springer-Verlag Berlin Heidelberg 2003

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Nijmeijer and Fossen for a glimpse of the current state of the art in nonlinear systems observer design (see [7]). In this article, we present an approach, based on exact state reconstructors, to the problem of controlling constructible DTNLS. State reconstructors are based on accurate knowledge of only past inputs and outputs. This fact, which is the outcome of the diﬀerence algebra approach to the study of observability in nonlinear discrete time systems, was advocated in the work of M. Fliess (see Fliess [2]) thirteen years ago. In that work, exact state reconstruction is recognized as an algebraic elimination problem. Somehow, this idea never led, to our knowledge, to the development of particular application examples dealing with control systems design. We emphasize that the exact state reconstruction approach is fundamentally diﬀerent from the traditional asymptotic observer approach in the sense that an exact, either immediate or “dead-beat”, recovery of the true state trajectories is always guaranteed under the assumption of constructibility of the system, a weaker condition than that of observability. The state reconstruction, in an n dimensional discrete-time nonlinear system, is exact from the initial time on provided a ﬁnite string, of length n−1, of past values of the applied inputs and the corresponding outputs are remembered, or stored. If such a delayed input and output information is not yet available at the initial time, the exact state reconstruction still takes place at the end of the next n−1 steps, provided the applied inputs and corresponding outputs are stored from this initial time onwards. The exact reconstruction is then independent of the initialization values arbitrarily assigned to the unavailable past inputs and past outputs in the reconstructor expression. In the latter case, the convergence of the reconstructor is, of course, also independent of any design gains or of a particularly desired asymptotic estimation error dynamics. Section 2 establishes the main result of the article, which basically proves, through elementary considerations, that an observable nonlinear discretetime system is also constructible. Section 3 is devoted to present a controller design example for the non-holonomic car model. The system presented is exactly discretized, from the original continuous-time version, and an approximate feedback linearizable version of the system model is adopted for dynamic feedback controller design purposes. The feedback performance results, tested on the full model, are illustrated by means of digital computer simulations. Section 4 presents the conclusions and suggests some topics for further research.

2

State reconstruction in nonlinear observable systems

Consider the following n-dimensional MIMO nonlinear system with k ∈ {0, 1, 2, ...}, xk+1 = f (xk , uk ), xk ∈ Rn , uk ∈ Rm yk = h(xk ), yk ∈ Rp

(1)

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We concentrate, for simplicity sake, on the local analysis. A global analysis can be achieved with additional technical assumptions. Note that the computations based on this analysis provide, for the nonlinear example in next section, almost global reconstruction formulae. 2.1

Basic assumptions

1. We assume that the strings of applied inputs and obtained outputs, prior to k = 0, are known from the time 1 − n on. In other words, yk and uk , for 1 − n < k < 0 are known. 2. The system is assumed to lie around an equilibrium point (xe , ue , ye ), i.e., xe = f (xe , ue ), ye = h(xe ). 3. The system (1) is assumed to be locally observable around the constant operating point (xe , ue , ye ). This means that the Jacobian matrix ∂{yk , yk+1 , ..., yk+(n−1) } ∂xk

(2)

evaluated at the constant operating point (xe , ue ) is full column rank n. 2.2

Notation

We use the delay operator δ to express the fact that δφk = φk−1 , and, correspondingly, the advance operator is denoted by δ −1 . The expression, δ µ φk , for any positive µ, stands for the identity δ µ φk = φk−µ and, similarly, δ −µ φk = φk+µ . The underlined symbol δ, as in, δ µ φk , stands for the collection: {φk−1 , φk−2 , ..., φk−µ }, i.e. δ µ φk = {δφk , ..., δ µ φk }. Evidently, δ 0 = δ 0 = Id and δ 1 = δ. On the other hand, δ −µ φk stands for the collection, {φk , φk+1 , ..., φk+µ } = {φk , δ −1 φk , ..., δ −µ φk }. Note that the system equation (1) is equivalent to: xk = δf (xk , uk ) = f (δxk , δuk ) = f (xk−1 , uk−1 ) Since, in turn, one may write: xk−1 = f (xk−2 , uk−2 ) = f (δxk−1 , δuk−1 ) = f (δ 2 xk , δ 2 uk ) it is clear that xk = f (f (δ 2 xk , δ 2 uk ), δuk ). We denote this last quantity by f (2) (δ 2 xk , δ 2 uk ). The expression f (µ) (δ µ xk , δ µ uk ), for µ > 0, should be clear from the recursion: f (i) (δ i xk , δ i uk ) = f (f (i−1) (δ i−1 xk , δ i−1 uk ), δuk ) f (1) (δxk , δuk ) = f (δxk , δuk )

(3)

The operators δ and δ satisfy the following relation δ i δ −i φk = {φk , δ i φk }

(4)

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Similar expressions may be deﬁned for the advances of states. xk+1 = δ −1 xk = f (xk , uk ) = f [1] (xk , δ 0 uk ) xk+2 = δ −2 xk = f (f (xk , uk ), uk+1 ) = f [2] (xk δ −1 uk ) xk+3 = f (f [2] (xk δ −1 uk ), uk+2 ) = f [3] (xk , δ −2 uk ) .. . xk+i = f [i] (xk , δ −(i−1) uk )

(5)

We set f [0] (xk δ −1 uk ) = xk 2.3

An exact delayed input output state reconstructor

Using the system state equation in (1) in an iterative fashion, one ﬁnds: xk = δf (xk , uk ) = f (δxk , δuk ) xk = f (δ(f (δxk , δuk )), δuk ) = f (f (δ 2 xk , δ 2 uk ), δuk ) = f (2) (δ 2 xk , δ 2 uk ) .. . xk = f (n−1) (δ n−1 xk , δ n−1 uk )

(6)

The elements in a ﬁnite sequence of advances of the output signal, yk , are found to be given by yk = h(xk ) = h(f [0] (xk δ −1 uk )) yk+1 = δ −1 h(xk ) = h(δ −1 xk ) = h(f (xk , uk )) = (h ◦ f [1] )(xk , δ 0 uk ) yk+2 = δ −1 (h ◦ f (xk , uk )) = h(δ −1 f (xk , uk )) = h(f (f (xk , uk ), δ −1 uk )) = (h ◦ f [2] )(xk , δ −1 uk ) .. . yk+(n−1) = (h ◦ f [n−1] )(xk , δ −(n−2) uk ) In the following proposition we show that a locally observable system is always locally constructible. The converse is not necessarily true. Proposition 1. Under assumptions 1, 2 and 3, the system is locally constructible, i.e. there exists, locally around (xe , ue , ye ), a map ϕ (not necessarily unique) such that the state xk of the system can be exactly expressed

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in terms of the output and a ﬁnite string of previously applied inputs and obtained outputs, in the form: xk = ϕ(yk , yk−1 , ..., yk−(n−1) , uk−1 , ..., uk−(n−1) ), k > 0

(7)

provided the string of inputs and outputs {yk , uk } for −n + 1 < k < 0 is completely known.

Proof According to the constant rank theorem and the stated hypothesis, it follows that there exists a mapping Φ such that the solution xk of the np equations     h(xk ) yk  yk+1   (h ◦ f [1] )(xk , δ 0 uk )      (8)  =  .. ..     . . yk+(n−1)

(h ◦ f [n−1] )(xk , δ n−2 uk )

can be expressed via a function Φ (not necessarily unique) as follows: xk = Φ(δ −(n−1) yk , δ −(n−2) uk ).

(9)

This just consists of an extraction of n equations having a full rank Jacobian versus xk . Such extraction is not unique and thus Φ is also non unique. If we take n − 1 delays in this expression we clearly obtain: δ n−1 xk = Φ(δ n−1 δ 1−n yk , δ n−1 δ 2−n uk ) = Φ(yk , δ n−1 yk , δ n−1 uk )

(10)

Using (10) in the last expression of equation (6) we have: xk = f (n−1) (δ n−1 xk , δ n−1 uk ) = f (n−1) (Φ(δ n−1 yk , δ n−1 uk ), δ n−1 uk ) = ϕ(yk , δ n−1 yk , δ n−1 uk )

(11)

The result follows. The previous proposition allows for an exact local “delayed input-output parameterization” of the state vector at time k for locally observable systems. Examples below show that, following the same line of development, such “delayed input-output parameterization” can be almost global in practice.

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The non-holonomic car system

Consider the following (kinematic) model of the non-holonomic car system x˙ = v cos θ y˙ = v sin θ v (12) θ˙ = tan ϕ L where v is the forward velocity, acting as a control input and ϕ is the control input representing the angular direction of the front wheels with respect to the main axis of the car. The angle, θ, is the orientation angle with respect to the x-axis. The quantities x and y are the position coordinates of the rear axis of the car, which are the only measurable outputs. The parameter L is the length between the front and rear axes of the car. We deﬁne the auxiliary input ω as ω = (v/L) tan ϕ. Deﬁning the complex variable, z = x + jy, we obtain z˙ = v exp(jθ) θ˙ = ω η=z

(13)

where η denotes the measurable position outputs. An exact discretization of the complex system (13) follows by considering constant control inputs v and ω in an arbitrary time interval [t0 , t], and then proceeding to integrate the resulting diﬀerential equations. We obtain: exp jwk T − 1 zk+1 = zk + vk T exp jθk jwk T θk+1 = θk + ωk T ηk = zk (14) where zk = z(tk ), θk = θ(tk ), and vk = v(tk ) = v, ω(tk ) = ωk = ω and T = tk+1 − tk . 0. Following the procedure outlined in System (14) is observable for vk = the proof of Proposition 1, we ﬁrst rewrite the system dynamics as exp jωk−1 T − 1

exp jθk−1 jωk−1 T θk = θk−1 + ωk−1 T zk = zk−1 + vk−1

(15)

The state of the system in terms of advances of the inputs and the outputs is given by zk = ηk

π θk = arg(ηk+1 − ηk ) − arg exp(jωk T ) − 1 + 2 (16)

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Combining equations (15) and (16), we obtain an exact delayed inputoutput parameterization of the state of the discretized system (14) in the following terms zk = ηk θk = arg(ηk − ηk−1 ) − arg(exp(jωk−1 T ) − 1) +

π + ωk−1 T 2

(17)

The exact delayed reconstructor (17) will be used for feedback purposes. 3.1

Feedback controller design based on approximate ﬂatness

As the continuous time system (13) is diﬀerentially ﬂat, its exact discretization (14) is diﬀerence ﬂat, but with a very diﬀerent and not easy to use ﬂat output. In order to obtain a suitable, and simpler, controller for the system, we proceed to approximate the exactly discretized system (14) by a more useful diﬀerence ﬂat system. This is achieved by assuming ωk T to be suﬃciently small, thus yielding the approximation, exp jωk T ≈ 1 + jωk T . We obtain the following system, which entirely coincides with the Euler discretization of system (13), zk+1 = zk + T vk exp jθk θk+1 = θk + ωk T ηk = zk

(18)

The system (18) is also observable for vk = 0 and evidently diﬀerence ﬂat, with ﬂat outputs given by by the measurable outputs zk . We can thus express all system variables in terms of the complex output z and some of its advances. 1 | zk+1 − zk | vk = T θk = arg(zk+1 − zk ) 1 ωk = [arg(zk+2 − zk+1 ) − arg(zk+1 − zk )] T (19) The expressions in (19) are useful in obtaining a feedback controller. Note that in order to have an invertible relation between the largest advance of the complex ﬂat output z and the control inputs, we must introduce an extension to the system input vk , by deﬁning it as an auxiliary state, ξk , and proceed to consider the following (complex) dynamic input coordinate transformation: vk = ξk 1 | uk − zk+1 | ξk+1 = T 1 ωk = [arg(uk − zk+1 ) − arg(zk+1 − zk )] T

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with uk being a new complex control input. The transformed system is seen to be equivalent to the following linear system zk+2 = uk

(20)

The speciﬁcation of a prescribed trajectory for the position variables as zk∗ yields the following auxiliary controller: ∗ ∗ − k1 (zk+1 − zk+1 ) − k2 (zk − zk∗ ) uk = zk+2 ∗ ∗ = zk+2 − k1 (zk + T ξk exp jθk − zk+1 ) − k2 (zk − zk∗ )

(21) The dynamic feedback controller is then obtained as vk = ξk 1 ∗ ∗ | zk+2 ξk+1 = − k2 (zk − zk∗ ) − zk+1 T ∗ −(1 + k1 )(zk + T ξk exp jθk − zk+1 )| ∗ 1 ∗ arg zk+2 − k2 (zk − zk∗ ) − zk+1 ωk = T

∗ −(1 + k1 )(zk + T ξk exp jθk − zk+1 ) − θk (22) For the implementation of the designed dynamic feedback controller (22) on the exactly discretized system (14), we use the previously obtained state reconstructor (17). Knowledge of the car position at time k = 0, and the applied control inputs at time k = −1 results in a dynamic feedback controller capable of satisfactorily tracking the prescribed trajectory. 3.2

Simulation results

We prescribe a 3-leaved rose as a desired trajectory in the (x, y) plane. This function is described in polar coordinates as: ρ = a cos(mϑ)

(23)

where a is the radius of the circle in which the rose is inscribed and ϑ, the angle of a representative point of the rose in the plane (x, y). The integer m represents the number of “leaves” of the rose. We set the time parameterization of the angle ϑ as a linear growing function of time of the form: ϑ(t) = p + q(t − t0 ), with p and q being suitable constant parameters. Figures 1 and 2 show the performance of the approximate ﬂatness-based dynamic feedback controller implemented on the exactly discretized system (14). The exact state reconstructor (17) was used, in the controller implementation, providing it with precise knowledge of the prior values of the

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inputs and the outputs. The controller gains were set to be k1 = −1.3 and k2 = 0.5825. This choice of gains placed the roots of the closed loop characteristic polynomial of the linearized tracking error system at the values z = 0.65 ± 0.4j. The sampling time was set to T = 0.4. For generating the 3-leaved rose ﬁgure, the values a = 10, m = 3, p = π/2, q = 0.05 were used. In order to test the controller performance, we set the initial position values x0 , y0 far away from the origin, at the values x0 = 14, y0 = 3, with an initial orientation of the car given by the angle 2π [rad].

Fig. 1. Trajectories of non-holonomic car controlled with a state reconstructor having perfect knowledge of inputs and outputs prior to k = 0.

4

Conclusions

In this article we have presented an approach to the problem of controlling a nonlinear discrete time system without measurements of all the components of the state vector. The approach is based on using an exact state reconstructor which requires only knowledge of inputs, outputs and a ﬁnite string of delayed applied inputs and obtained outputs. We have tested the “observer-less” control scheme in connection with ﬂatness based controllers for a typical nonlinear system: a discretized non-holonomic multivariable car. The performance of the proposed feedback controller scheme, based on the exact delayed resconstructor, was shown to be good and with quite natural recovery features, specially in those cases where knowledge of applied inputs and corresponding obtained outputs, prior to the initial instant of time, was not allowed and the reconstructor had to be arbitrarily initialized. An important aspect is that of providing robustness to the proposed exact reconstructor-based feedback, for the case of external perturbation inputs and

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Fig. 2. Performance of controlled car following a “3-leaved rose” trajectory

other classes of uncertain inﬂuences on the given plant. This will be the topic of future work.

References 1. M. Fliess, “Automatique en Temps Discrete et Alg´ebre aux Diﬀer´erences, ”Forum Mathematicum, Vol. 2, pp. 213-232, 1990. 2. M. Fliess, “Quelques remarkes sur les observateurs non-lineaires,” 11eme Colloque GRETSI sur le traitement du Signal et des Images, Nice, June 1-5, 1987. 3. J. W. Grizzle, “A linear algebraic framework for the analysis of discrete time nonlinear systems,” SIAM J. on Control and Optimization, Vol. 31, pp. 10261044, 1993. 4. B. Jakubczyk and E. D. Sontag, “Controllablitiy of nonlinear discrete time systems; a Lie algebraic approach” SIAM J. Control and Optimization, Vol. 28, 1990. 5. S. Monaco and D. Normand-Cyrot, “Invariant distributions for discrete time nonlinear systems” Systems and Control Letters, Vol. 5, pp. 191-196, 1985. ¨ Kotta and C.H. Moog, “Linearization of Discrete-Time 6. E. Aranda-Bricaire, U. Systems” SIAM J. of Control and Optimization, Vol.34, No. 6, pp. 1999-2923, 1996. 7. H. Nijmeijer and T.I. Fossen, New Directions in Nonlinear Observer Design, Lecture Notes in Control and Information Sciences, Vol. 244, Springer-Verglag, London, 1999.

Generalized PID control of the average boost converter circuit model Hebertt Sira-Ram´ırez and Gerardo Silva-Navarro1 CINVESTAV-IPN Dept. Ingenier´ıa El´ectrica, Secci´ on Mecatr´ onica Avenida IPN, # 2508, Col. San Pedro Zacatenco, A.P. 14740 07300 M´exico, D.F., M´exico. Abstract. In this article we examine, in the context of equilibrium-to-equilibrium reference trajectory tracking, the Generalized Proportional Integral Derivative (GPID) control of a nonlinear average model of a DC-to-DC power converter of the “Boost” type. The design approach relies on the converter’s tangent linearization model and Lyapunov stability theory. The performance of the feedback controlled nonlinear system is evaluated by means of digital computer simulations including large, unmodelled, load parameter variations.

1

Introduction

Generalized Proportional-Integral-Derivative (GPID) control was introduced by Prof. M. Fliess and his coworkers (See [2] and [3]) in the context of linear time-invariant controllable systems. GPID control sidesteps the need for the traditional asymptotic state observers and directly proceeds to use, in a previously designed state feedback control law, structural state estimates in place of the actual state variables. These structural estimates are based on integral reconstructors requiring only inputs, outputs, and iterated integrals of such available signals. The eﬀect of the neglected initial states is suitably compensated by means of a suﬃciently large number of additional iterated integral output error, or integral input error, control actions. The method has enormous interest from a theoretical viewpoint and ties in with the algebraic module theoretic approach to linear systems (see Fliess [1]) and the theory of algebraic localizations (see also Fliess et al ([4]) ). GPID control can be extended to linear delay systems and it has been extended to particular instances of nonlinear systems in Sira-Ram´ırez et al ([11]). In this article, we present a study of the relevance of the GPID control for the trajectory tracking in an average nonlinear model of a DC-to-DC power converter of the “Boost” type. Our developments are cast in the context of linearized average models around nominal state reference trajectories accomplishing a desired equilibrium-to-equilibrium transfer. In order to demonstrate the ﬂexibility of the GPID controller implementation approach in accomodating for several state feedback controller design techniques, we use a Lyapunov-based controllers in the state feedback controller design. The A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 361-371, 2003. Springer-Verlag Berlin Heidelberg 2003

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performance of the GPID feedback controlled system is evaluated by means of digital computer simulations. Section 2 deals with the generalities of the average nonlinear model of a Boost converter. We particularly emphasize the ﬂatness of the system and the minimum and non-minimum phase character of its state variables. Section 3 uses a time-varying linearized model of the boost converter, valid around a nominal state and input trajectory, oﬀ-line speciﬁed on the basis of the nonlinear system ﬂatness. The nominal state trajectory is speciﬁed to achieve a typical equilibrium-to-equilibrium transfer taking place in ﬁnite time. We implement, via the GPID approach a Lyapunov-based state feedback controller achieving asymptotic stability to zero of the state tracking error. We illustrate the robust performance of the designed GPID controllers by means of computer simulations including large unmodelled load parameter variations.

2

An average model of a boost converter

Consider the boost converter circuit, shown in Figure 1. The system is described by the set of equations LI˙ = −uv + E v C v˙ = uI − R

(1)

where I represents the inductor current and v is the output capacitor voltage. The control input u, representing the switch position function, is a discretevalued signal taking values in the set {0, 1}. The system parameters are constituted by: L, which is the inductance of the input circuit; C the capacitance of the output ﬁlter and R, the output load resistance. The external voltage source has the constant value E.

Fig. 1. The boost converter circuit

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We introduce the following state normalization and time scale transformation: L t I v (2) x1 = , x2 = , τ = √ E C E LC The normalized model is thus given by: x˙ 1 = −ux2 + 1 x2 x˙ 2 = ux1 − Q y = x2

(3)

where now, with an abuse of notation, the “ ˙ ” represents derivation with respect to the normalized time, τ . The variable x1 is the normalized inductor current, x2 is the normalized output voltage and u, still represents the switch position function. The constant system parameters are all comprised now in the circuit “quality” parameter, denoted by Q and given by the strictly positive quantity, R C/L. It is assumed that the only system variable available for measurement is the output capacitor voltage x2 . The average state model of the boost converter circuit, extensively used in the literature, may be directly obtained from the model (3) by simply identifying the switch position function u with the duty ratio function, denoted by µ, which is now a function restricted to take values in the closed interval [0, 1]. The average normalized inductor current and capacitor voltage are denoted, respectively by z1 and z2 . We thus deal, from now on, with the following average normalized system equations which admit a physical interpretation, in terms of controlled voltage and current sources:

z˙1 = −µz2 + 1 z2 z˙2 = µz1 − Q y = z2 2.1

(4)

Properties of the average normalized model

The average system (4) is diﬀerentially ﬂat, with ﬂat output given by the total normalized stored energy F =

1 2 z1 + z22 2

(5)

Indeed, all system variables can be parameterized, modulo physical considerations, in terms of the ﬂat output F and its ﬁrst order time derivative

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F˙ . Q 1 z1 = − + 2 2

Q2 + 4(QF˙ + 2F ) Q Q2 ˙ + z2 = −QF − Q2 + 4(QF˙ + 2F ) 2 2 1 2 1 − F˙ + Q2 + 4(QF˙ + 2F ) − F¨ µ= 2 Q Q z2 1 + Q z1

(6)

An important property of the average model concerns the nature of the zero dynamics associated with the individual normalized average state variables. The variable z2 is a non-minimum phase output while the variable z1 is a minimum phase output (See Sira-Ram´ırez and Lischinsky-Arenas [9]). For this reason, in order to avoid internal instability problems, the feedback regulation of the average voltage, z2 , is usually carried out in an indirect fashion in terms of a corresponding regulation of z1 . It is also clear, from the strictly positive character of the ﬂat output, F , and the assumption that, pointwise, F˙ (t) > −(2/Q)F (t), the average state and the average input variables are all strictly positive signals. Henceforth, we concentrate in solving the stabilization and trajectory tracking problems for the described average normalized model of the Boost converter circuit. It is implied that a feedback solution of the average problem can be readily implemented, modulo some well-known approximation errors, on the actual switched system (1) by means of a suitable high frequency Pulse-Width-Modulated (PWM) feedback control scheme (see [8] and [10]).

3

GPID regulation around a nominal trajectory

Suppose it is desired to achieve an equilibrium to equilibrium transfer for the non-minimum phase variable z2 , within a given ﬁnite interval of time [t0 , t1 ]. This problem is suitably transformed into a problem of adequately controlling the total stored energy F between the two corresponding equilibrium values. For this, note that if it is required to transfer the normalized capacitor voltage between the constant equilibrium values, z 2 (t0 ) and z 2 (t1 ), the corresponding equilibrium values for the ﬂat output, F (t0 ), F (t1 ), are given, according to (6), by

2 1 z 22 (t0 ) 1 F (t0 ) = + z 22 (t0 ) 2 Q 2

2 1 z 22 (t1 ) 1 F (t1 ) = + z 22 (t1 ) (7) 2 Q 2 Thus, a nominal ﬂat output trajectory F ∗ (t) can be prescribed which smoothly and monotonically interpolates between the equilibrium values,

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F (t0 ), and F (t1 ). Note that this oﬀ-line planned prescription of the ﬂat output, immediately renders, via use of (6) the nominal (open loop) normalized state and control input trajectories, z1∗ (t), z2∗ (t), µ∗(t), without integrating any diﬀerential equations. 3.1

A time-varying linearized model around the nominal stabilizing trajectory

The jacobian linearization around the prescribed nominal stabilizing trajectory, characterized by the functions z1∗ (t), z2∗ (t) and µ∗ (t), is readily obtained to be z˙1δ = −µ∗ (t)z2δ − z2∗ (t)µδ z˙2δ = µ∗ (t)z1δ + z1∗ (t)µδ −

1 z2δ Q

yδ = z2δ

(8)

This linear time-varying system is written in matrix form: z˙ δ = A(t)zδ + b(t)µδ , yδ = c(t)zδ , as ∗ d z1δ 0 −µ∗ (t) z1δ −z2 (t) µδ = + 1 µ∗ (t) − Q z2δ z1∗ (t) dt z2δ

(9)

The linearized system is uniformly controllable for all physically feasible trajectories ( z1∗ (t) > 0, z2∗ (t) > 0 ). The controllability matrix is computed, according to the formula developed by Silverman and Meadows [7], C(t) = [ b(t), (A(t) −

∗ 1 ∗ −z2 (t) − Q z2 (t) d )b(t) ] = 1 ∗ z1 (t) z1∗ (t) −1 − Q dt

(10)

The system loses controllability around the conditions: z2∗ (t) = 0 and = −Q 2 , which are not physically signiﬁcant. The system is thus uniformly controllable in the region of the state space of interest. Hence, the system is diﬀerentially ﬂat, with ﬂat output given by a time-varying linear combination of the state variables. In this case, such a ﬂat output is given by: z1∗ (t)

Fδ = z1∗ (t)z1δ + z2∗ (t)z2δ

(11)

which represents the incremental normalized energy around its nominal value: F ∗ (t) = (1/2)([ z1∗ (t) ]2 + [ z2∗ (t) ]2 ). The ﬂat output time derivative represents the incremental consumed power and it is given by 2z ∗ (t) F˙ δ = z1δ − 2 z2δ Q

(12)

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Indeed, all the incremental system variables are diﬀerentially parameterizable in terms of Fδ and its time derivatives, z1δ = z2δ =

1 1+

2 ∗ z (t) Q 1

2 Fδ + F˙δ Q

1

z2∗ (t) 1 +

µδ = −

z2∗ (t)

2 ∗ z (t) Q 1

Fδ − z1∗ (t)F˙δ

1 2 ∗ z (t) Q 1

2 ∗ z2 (t)µ∗ (t)z1δ Q

1+

2 4 + µ∗ (t) 1 + z1∗ (t) − 2 z2∗ (t) z2δ + F¨δ Q Q

4 ∗ ∗ 1 2 ∗ 4 ∗ 1 ∗ z µ + (1 + ) − z = z µ Fδ 2 2 1 2 Q2 z2∗ Q Q2 2 ∗ z2∗ (t) 1 + Q z1 (t)

2 ∗ ∗ 2 ∗ 4 ∗ z1∗ ˙ ¨ Fδ + Fδ + z2 µ − µ∗ 1 + z1 − 2 z2 (13) Q Q Q z2∗

The linearized system, (9), is also uniformly observable from the output yδ since, according to the Silverman-Meadows test [7], we obtain the following observability matrix: d 0 µ∗ (t) O(t) = cT ; (A − )T cT = (14) 1 1 −Q dt and the system is seen to loose observability whenever the nominal duty ratio function satisﬁes µ∗ (t) = 0. It is clear, then, that the linearized system (9) is constructible and the unmeasured state can be expressed in terms of integrals of linear time-varying combinations of the incremental input µδ and the incremental output yδ . Indeed, from the linearized system equations (8) we have, t [ µ∗ (σ)yδ (σ) + z2∗ (σ)µδ (σ)]dσ z1δ (t) = − 0

z2δ (t) = yδ (t)

(15)

The relation linking the structural estimate of the incremental normalized inductor current, z1δ , with its actual value, z1δ , is given by z1δ = z1δ + z1δ (0)

(16)

Similarly, the relations between the structural estimates of the incremental ﬂat output and of its ﬁrst order time derivative, and their actual values are given, according to (11), (12), by, Fδ = Fδ + z1∗ (t)z1δ (0) ˙ + z (0) F˙ = F δ

δ

1δ

(17)

GPID control of boost converter

3.2

367

A Lyapunov based controller

Consider the time-invariant Lyapunov function candidate, V (z1δ , z2δ ) given by. V (z1δ , z2δ ) =

1 2 2 z1δ + z2δ 2

The time derivative of the Lyapunov function, along the trajectories of the linearized system yield 2

z V˙ (z1δ , z2δ , µ, t) = − 2δ + (z1∗ (t)z2δ − z2∗ (t)z1δ ) µδ Q

(18)

The time-varying controller µδ = −γ (z1∗ (t)z2δ − z2∗ (t)z1δ )

(19)

with γ > 0, results in a negative deﬁnite time derivative of V (z1δ , z2δ ) along the closed-loop controlled trajectories of the system, 2

z 2 V˙ (z1δ , z2δ , t) = − 2δ − γ (z1∗ (t)z2δ − z2∗ (t)z1δ ) Q

(20)

The closed loop system (8), (19) is then an exponentially asymptotically stable linear time-varying system for any strictly positive design parameter γ. Use of the structural estimate for z1δ , given by (15), in the Lyapunov based controller, (19) leads, after appropriate integral output error compensation, to the equivalent closed-loop incremental system: z˙1δ = −γ[z2∗ (t)]2 z1δ − [µ∗ (t) − γz1∗ (t)z2∗ (t)]z2δ + γ[z2∗ (t)]2 (z10δ − kζδ ) 1 z˙2δ = [µ∗ (t) + γz1∗ (t)z2∗ (t)]z1δ − ( + γ[z1∗ (t)]2 )z2δ Q +γz1∗ (t)z2∗ (t)(z10δ − kζδ ) ζ˙δ = z2δ (21) Letting ρδ = (z10δ − kζδ ), we obtain the following matrix representation of the closed loop system:      −γ[z2∗ (t)]2 −[µ∗ (t) − γz1∗ (t)z2∗ (t)] γ[z2∗ (t)]2 z z1δ d  1δ   ∗ 1 + γ[z1∗ (t)]2 ) γz1∗ (t)z2∗ (t)   z2δ  z2δ = [µ (t) + γz1∗ (t)z2∗ (t)] −( Q dt ρδ ρδ 0 −k 0 (22)

The point-wise eigenvalues of the closed loop system, (22), are guaranteed to be bounded away from the imaginary axis, in the open left portion of the

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complex plane, provided the constant gain k is chosen to satisfy the rather conservative, but feasible, condition: 0
1 Qγz2∗ max (t)

It is relatively straightforward then to show that the closed loop system, (22), satisﬁes the hypothesis in the following Theorem due to Rugh [6]. This, as usual, entitles a suﬃciently slow and suﬃciently diﬀerentiable nominal equilibrium to equilibrium transfer trajectory for the ﬂat output. The closed loop system, (22), is then seen to be exponentially asymptotically stable. Theorem 1. (Rugh, [6], [pp. 135-138]). Suppose that for the linear time-varying system x˙ = A(t)x, the matrix A(t) is continuously diﬀerentiable and there exist ﬁnite positive constants α and δ, such that, for all t, A(t) ≤ α, and every pointwise eigenvalue of A(t) satisﬁes Re[λ(t)] ≤ −δ. Then, there exist a positive constant β such that ˙ if the time derivative of A(t) satisﬁes A(t) ≤ β, for all t, the state equation is uniformly exponentially stable. 3.3

Simulation results

It was desired to take the actual output capacitor voltage from an initial value of x∗2 (t0 ) = 30 [Volts] to a ﬁnal value x∗2 (t1 ) = 60 [Volts]. Corresponding to this desired output voltage transfer, we speciﬁed a nominal trajectory for the ﬂat output, F ∗ (t), smoothly joining its initial value of, F ∗ (t0 ) = Finitial = 0.049 [Watts] with the ﬁnal value of F ∗ (t1 ) = Ff inal = 0.676 Watts where t0 = 0.07 [s] and t1 = 0.10 [s]. We used the following B´ezier polynomial for the nominal, desired, ﬂat output trajectory, (23) F ∗ (t) = Finitial + (Ff inal − Finitial ) 21 − 35∆ + 15∆2 ∆5 with ∆ = (t − t0 )/(t1 − t0 ). Corresponding to the speciﬁed ﬂat output nominal trajectory, F ∗ (t), we obtained the (non-normalized) average nominal values of the inductor current, x∗1 (t), the capacitor voltage x∗2 (t), and the nominal duty ratio control input function µ∗ (t). The nominal state and input trajectories are shown, along with the corresponding actual state and input GPI feedback controlled responses, in Figure 2. The controller parameters were set to be γ = 0.9 and the parameters of the desired closed loop linear system for the ﬂat output, in normalized coordinates, were set to be: ξ = 0.9 and ωn = 2. In order to test the robustness of the time-varying GPI control scheme, we hypothesized an unmodelled sudden, and permanent, variation of the load resistance value R. The variation was allowed to be of 300% above its nominal value of R = 30Ω. We allowed this unmodelled variation to occur at time t = 0.16 [s]. As in the stabilization case, the inductor current is severely

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Fig. 2. Generalized PID controlled responses around an equilibrium to equilibrium transfer trajectory

Fig. 3. Generalized time-varying PID controlled responses to an unmodelled, and permanent, load change

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aﬀected by this variation, but the normalized output voltage automatically recovers the desired constant reference value. The simulation results in Figure 3 clearly demonstrates the robustness of the proposed controller.

4

Conclusions and suggestions for further research

In this article, we have presented a GPID control scheme for stabilizing trajectory tracking tasks in an average nonlinear model of a DC-to-DC power converter of the “Boost” type. Eventhough the presented developments are cast in the context of linearized time-varying average models, the ideas can be extended to the full nonlinear case. The ﬂexibility of the GPID controller approach to accommodate to any linear state feedback controller design technique, has been illustrated by using Lyapunov-based controllers. The performance of the GPID feedback controlled systems was evaluated by means of digital computer simulations with highly satisfactory results.

References 1. Fliess M., (1990) Some Basic Structural Properties of Generalized Linear Systems. Syst. and Contr. Let. 15, 391–396. 2. Fliess, M. (2000) “Sur des Pensers Nouveaux Faisons des Vers Anciens”. In Actes Conf´erence Internationale Francophone d’Automatique (CIFA-2000), Lille. France, July 2000. 3. Fliess M., Marquez R., and Delaleau E., (2000) State Feedbacks without Asymptotic Observers and Generalized PID regulators. Nonlinear Control in the Year 2000, A. Isidori, F. Lamnabhi-Lagarrigue, W. Respondek, (Eds), Lecture Notes in Control and Information Sciences 258. 367–384, Springer, London. 4. Fliess M., Marquez R., Delaleau E., Sira-Ramirez H., (2001) Correcteurs Proportionnels-Int´egraux G´en´eralis´es. ESAIM: Control, Optimization and Calculus of Variations (to appear). 5. Marquez R., Delaleau E., and Fliess M., (2000) Commande par PID G´en´eralis´e ´ ` d’un Moteur Electrique sans Capteur M´ecanique. In Actes ConfErence Internationale Francophone d’Automatique (CIFA-2000). Lille, France. 6. Rugh W., (1996). Linear System Theory. (2nd Edition), Prentice Hall, Upper Saddle River, N.J. 7. Silverman L. M., (1966) Transformation of Time-Variable Systems to Canonical (Phase-Variable) Form. IEEE Transactions on Automatic Control, 11, 300–303. 8. Sira-Ram´ırez H., (1989) A Geometric Approach to Pulse-Width-Modulated Control in Nonlinear Dynamical Systems. IEEE Transactions on Automatic Control. 34, 184–187. 9. Sira-Ram´ırez H., and Lischinsky-Arenas P., (1991) The Diﬀerential Algebraic Approach in Nonlinear Dynamical Compensator Design for Dc-to-Dc Power Converters. International Journal of Control, 54 111–134.

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10. Sira-Ram´ırez H., P´erez–Moreno R., Ortega R., and Garc´ıa–Esteban M., (1997) Passivity–Based Controllers for the Stabilization of DC–to–DC Power Converters. Automatica 33 – . 11. Sira-Ram´ırez H., M´ arquez R., and Fliess M., (2001) Generalized PID Sliding Mode Control of DC-to-DC Power Converters. IFAC Symposium on System Structure, Prague, Czek Republic.

Sliding mode observers for robust fault reconstruction in nonlinear systems Chee Pin Tan and Christopher Edwards Engineering Department, Leicester University, University Road, LE1 7RH, U.K. Email - [email protected], [email protected] Abstract. This paper describes a method for designing sliding mode observers for detection and reconstruction of faults, that is robust against system uncertainty. The method seeks to design an observer which minimises the eﬀect of the uncertainty on the reconstruction of the faults, in an L2 sense. A nonlinear model of a gantry crane will be used to demonstrate the method and its eﬀectiveness.

1

Introduction

The fundamental purpose of a Fault Detection & Isolation (FDI) scheme is to generate an alarm when a fault occurs and also to identify the nature and location of the fault. Many FDI methods are observer based: the plant output is compared with the output of an observer designed from a model of the system, and the discrepancy is used to form a residual. Using this residual, a decision is made as to whether a fault is present. However, the model of the system about which the observer is designed will possess uncertainties. These uncertainties could cause the FDI scheme to trigger a false alarm when there are no faults, or even worse, mask the eﬀect of a fault, which may go undetected. Hence, there is a need for robust FDI schemes which are robust to model uncertainties. Much has been done in the area of robust FDI. Examples of schemes using linear observers appear in [1,8,10]. FDI schemes have also been developed using nonlinear approaches, in particular, sliding mode observers. In [7,14], sliding mode observers were used to generate residuals. In this paper, rather than generate residuals, a nonlinear sliding mode observer based strategy will be explored which seeks to reconstruct the fault signals [4]. The sliding mode observer under consideration feeds back the output error through a discontinuous switched term which is intended to induce a sliding motion in the state estimation error space. It was argued in [4] that by appropriate processing of the so-called equivalent output error injection signal required to maintain sliding, information about the faults could be obtained. This paper builds on the work of [4] by introducing explicitly an uncertainty representation into the design framework. A new design method is presented for the observer gains which seeks to minimise the L2 gain between the uncertainty and the fault reconstruction signal. The eﬃcacy of this method will be demonstrated with a nonlinear model of a crane. A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 373-383, 2003. Springer-Verlag Berlin Heidelberg 2003

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Preliminaries

This section introduces the preliminaries and background ideas necessary for the work presented in this paper. Consider the uncertain dynamical system x(t) ˙ = Ax(t) + Bu(t) + F fi (t, u) + M ξ(t, y, u) y(t) = Cx(t)

(1a) (1b)

where A ∈ Rn×n , B ∈ Rn×m , C ∈ Rp×n , F ∈ Rn×q and M ∈ Rn×k where n > p ≥ q. Assume that the matrices C and F are full rank and the function fi : R+ × Rm → Rq is unknown but bounded so that fi (t, u) ≤ α(t, u) where α : R+ × Rm → R+ is a known function. The signal fi (t, u) represents an actuator fault. The map ξ : R+ × Rp × Rm → Rk encapsulates any uncertainty or nonlinearities present. It is assumed to be unknown but bounded subject to ξ(t, y, u) < β where the positive scalar β is known. Edwards & Spurgeon [3] have proven that if p > q, rank(CF ) = q and the invariant zeros of (A, F, C) lie in the left half plane, then there exists a change of co-ordinates in which the system triple (A, F, C) and M have the following structure : 0 M1 A11 A12 F = C= 0 T M= (2) A= A21 A22 M2 Fo where A11 ∈ R(n−p)×(n−p) , Fo ∈ Rq×q is non-singular, T ∈ Rp×p is orthogonal and M1 ∈ R(n−p)×k . Deﬁne A211 as the top p − q rows of A21 . By construction, the pair (A11 , A211 ) is detectable and the unobservable modes of (A11 , A211 ) are the invariant zeros of (A, F, C) [3]. Also for convenience, deﬁne F2 ∈ Rp×q as the bottom p rows of F (which therefore includes Fo ). Edwards & Spurgeon [3] propose a state observer of the form z(t) ˙ = Az(t) + Bu(t) − Gl ey (t) + Gn ν

(3)

The matrix Gn in the co-ordinate system in (2), and the discontinuous output error injection vector ν, are respectively deﬁned by P e −ρ(t, y, u) Poo eyy if ey = 0 −LT T Gn = and ν = (4) T T 0 otherwise where L = Lo 0 with Lo ∈ R(n−p)×(p−q) , ey := Cz − y and Po ∈ Rp×p is a symmetric positive deﬁnite (s.p.d.) matrix which will be formally deﬁned later. The scalar function ρ : R+ × Rp × Rm → R+ will also be described formally later but it must represent an upper bound on the magnitude of the fault signal plus the uncertainty. The matrices Lo , Po and Gl are to be determined. In [11], the system associated with the state estimation error e := z − x was analysed in the case when ξ = 0 and ρ = CF α(t, u) + ηo where ηo is a positive scalar. The following result was proved:

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Proposition 1. Suppose there exists a s.p.d. matrix P , with the structure P1 P1 L P = (5) > 0, P1 ∈ R(n−p)×(n−p) , P2 ∈ Rp×p LT P 1 P 2 + L T P 1 L that satisﬁes P (A − Gl C) + (A − Gl C)T P < 0, then if Po := T P2 T T , the error e(t) is quadratically stable. Furthermore, sliding occurs in ﬁnite time on S = {e : Ce = 0} governed by the system matrix A11 + Lo A211 . This result will now be generalised to the case of uncertain systems as given in (1a), using ideas similar to those in [9]. From (1a) and (3), and deﬁning Ao = A − Gl C, the state estimation error satisﬁes e(t) ˙ = Ao e(t) + Gn ν − F fi (t, u) − M ξ(t, y, u)

(6)

Suppose there exists a s.p.d. matrix P which satisﬁes the requirements of Proposition 1. Deﬁne scalars µ0 = −λmax (P Ao + ATo P ) and µ1 = P M . Suppose the scalar gain function in (4) satisﬁes ρ ≥ CF α(t, u) + ηo , where ηo > 0 then the following holds: Lemma 1. The state estimation error e(t) in (6) is ultimately bounded with respect to the set Ω = {e : e < (2µ1 β/µ0 ) + } where > 0 is an arbitrarily small positive scalar. Proof. Ultimate boundedness with respect to Ω can be shown using the functional V (e) = eT P e and employing arguments similar to [9]. Lemma 1 will now be used to prove the main result of this section; that for an appropriate choice of ρ a sliding motion can be induced on S = {e : Ce = 0}. First introduce a co-ordinate change as in [3]; let TL : e → eL where In−p L TL := (7) 0 T In this new co-ordinate system, the triple (A, F, C) and M will be in the form A11 A12 M1 0 A= M= F= C = 0 Ip (8) A21 A22 M2 F2 where A11 = A11 + Lo A211 , F2 = T F2 and M2 = T M2 . The nonlinear gain matrix and the Lyapunov matrix will respectively be T and P = (TL −1 )T P (TL −1 ) = diag{P1 , Po } (9) Gn = 0 Ip The fact that P is block diagonal Lyapunov matrix for A − Gl C implies that A11 is stable and hence the sliding motion is stable [3]. In the new co-ordinates e˙ L (t) = Ao eL (t) + Gn ν − F fi (t, u) − Mξ(t, y, u)

(10)

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where Ao = A − Gl C. Partitioning eL conformably with (8) e˙ 1 (t) = A11 e1 (t) + (A12 − Gl,1 )ey (t) − M1 ξ(t, y, u) e˙ y (t) = A21 e1 (t) + (A22 − Gl,2 )ey (t) − F2 fi (t, u) − M2 ξ(t, y, u) + ν

(11a) (11b)

where Gl,1 and Gl,2 represent appropriate partitions of Gl . Proposition 2. An ideal sliding motion takes place on S = {e : ey = 0} in ﬁnite time if the function ρ from (4) satisﬁes (for a positive scalar ηo ) ρ ≥ 2A21 µ1 β/µo + M2 β + F2 α + ηo

(12)

Proof. Consider a Lyapunov function Vs = eTy Po ey . Because P from (9) is a block diagonal Lyapunov matrix for (A − Gl C), it follows that Po (A22 − Gl,2 ) + (A22 − Gl,2 )T Po < 0 and hence V˙ s ≤ 2eTy Po (A21 e1 − F2 fi − M2 ξ) − 2ρPo ey ≤ −2Po ey (ρ − A21 e1 − F2 fi − M2 ξ) + . From Lemma 1, in ﬁnite time e(t) ∈ Ω which implies e1 < 2µ1 β/µo √ ˙ s ≤ −2ηo Po ey ≤ −2ηo η Vs Therefore from the deﬁnition of ρ in (12), V where η := λmin (Po ). This proves that the output estimation error ey will reach zero in ﬁnite time, and a sliding motion takes place.

3

Robust reconstruction of actuator faults

In this section the sliding mode observer will be analysed with regard to its ability to robustly reconstruct the fault fi (t, u) despite the presence of the uncertainty ξ. The analysis will be performed under the condition p > q. Assuming a sliding mode observer has been designed, and that a sliding motion has been achieved, then ey = e˙ y = 0 and (11a) - (11b) become e˙ 1 (t) = A11 e1 (t) − M1 ξ(t, y, u) νeq = −A21 e1 (t) + F2 fi (t, u) + M2 ξ(t, y, u)

(14a) (14b)

where νeq is the equivalent output error injection term (the natural analogue of the concept of the equivalent control [13]) required to maintain a sliding motion. The signal νeq is computable online and can be approximated by νδ = −ρ(t, y, u)

Po ey Po ey + δ

(15)

where δ is a small positive scalar that governs the degree of accuracy; for details see [4]. From (14b), νeq depends on, or equivalently gives information

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about the fault fi . The case where there is no uncertainty was investigated in [4]. In the case when ξ = 0 the attempted reconstruction of fi will be corrupted by the exogenous signal ξ. Ideally the objective here is to choose L and a scaling of νeq to minimise the eﬀect of ξ on the fault reconstruction. Deﬁne a would-be reconstruction signal fˆi = W T T νeq where W := W1 Fo−1 and W1 ∈ Rq×(p−q) represents design freedom. It follows from (14a) - (14b) that ˆ fˆi (t) = fi (t, u) + G(s)ξ where the transfer function ˆ G(s) := W T T A21 (sI − A11 )−1 M1 + W T T M2 The objective now is to reconstruct the fault fi (t, u) whilst minimising the eﬀect of ξ(t, y, u). Using the Bounded Real lemma [2], the L2 gain from the exogenous signal ξ to fˆi will not exceed γ if the following holds   Pˆ A11 + AT11 Pˆ −Pˆ M1 −(W T T A21 )T  −MT Pˆ (16) −γI (W T T M2 )T  < 0 1 T T −W T A21 W T M2 −γI where the scalar γ > 0 and Pˆ ∈ R(n−p)×(n−p) is s.p.d. The objective is to ﬁnd Pˆ , L and W to minimise γ subject to (16) and Pˆ > 0. However this must be done in conjunction with satisfying the requirements of obtaining a suitable sliding mode observer as expressed in Proposition 2. Writing P from equation (5) as P11 P12 >0 (17) P = T P12 P22 where P11 ∈ R(n−p)×(n−p) and P12 := P121 0 with P121 ∈ R(n−p)×(p−q) , it follows there is a one-to-one correspondence between the variables (P11 , P12 , P22 ) and (P1 , L, P2 ) since −1 T −1 P1 = P11 , L = P11 P12 , P2 = P22 − P12 P11 P12

(18)

Choosing Pˆ = P11 , it follows Pˆ A11 = P11 A11 + P12 A21 , Pˆ M1 = P11 M1 + P12 M2

(19)

From (19), it follows that (16) is aﬃne with respect to the variables P11 , P12 , W1 and γ and suggests that convex optimisation techniques are appropriate.

4

Designing the observer

This section will present a method to design the sliding mode observer gains Lo , Gl and Po to induce the inequality (16). Speciﬁcally in this section it is proposed that the linear gain Gl from (3) be chosen to satisfy   P Ao + ATo P P (Gl D − Bd ) E T  (Gl D − Bd )T P −γo I HT  < 0 (20) E H −γo I

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where P is given in (17) and the four matrices Bd ∈ Rn×(p+k) , D ∈ Rp×(p+k) , H ∈ Rq×(p+k) and E ∈ Rq×n . The scalar γo > 0. Further assume : B d = 0 M , D = D 1 0 , H = 0 H2

(21)

where D1 ∈ Rp×p is a non-singular user deﬁned parameter and H2 ∈ Rq×k depends on W1 . If a feasible solution to (17) and (20) exists then the requirements of Proposition 2 will be fulﬁlled (since (20) implies P Ao + ATo P < 0). Hence the choice of Gl , the gain matrix L from (17) which follows once P is speciﬁed, Gn from (4) and Po constitute a sliding mode observer design. Proposition 3. Inequality (20) is feasible if and only if   P A + AT P − γo C T (DDT )−1 C −P Bd E T  −BdT P −γo I H T  < 0 E H −γo I

(22)

where an appropriate choice of Gl is given by Gl = γo P −1 C T (DD T )−1

(23)

Proof. This follows from algebraic manipulation, for details see [12].

The idea is now to relate (22) to (16). By using the partitions for Bd , D and H from (21), it is straightforward to show that a necessary condition for (22) to hold is that   T − (P11 M1 + P12 M2 ) E1T P11 A11 +AT11 P11 +P12 A21 +AT21 P12  −(P11 M1 + P12 M2 )T −γo I H2T  < 0 (24) E1 H2 −γo I since (24) is ‘embedded’ in inequality (22). Choosing E1 = −W A21 and H2 = W M2 will yield the same inequality as (16). The design method can now be summarised to be Minimise γ with respect to the variables P and W1 subject to (22), (16) and (17), where γo > 0 is an a-priori user-deﬁned scalar. Remarks - Let γmin be the minimum value of γ that satisﬁes (16), then, since (16) is a ‘sub-block’ of (22), γmin ≤ γo always holds. The optimisation problem above is convex in its variables and standard Linear Matrix Inequality software, such as [6], can be used to synthesise numerically γ, P and W1 . Once P has been determined, L can be determined from (18), Gl from (23), Gn from (4), and Po from Proposition 1.

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For a given Bd , D, E and H, inequality (20) can be viewed as resulting from an H∞ ﬁltering problem (page 462 of [15]), the idea being to minimise the eﬀect of ξ on z (see Figure 1: notation taken from [15]). However, here, E and H are regarded as design variables, which in particular depend onW . Once sliding is established, the choice of the linear gainGl is technically not relevant since the linear output error injection termGl ey ≡ 0 because ey ≡ 0.   exogenous - A Bd signal  E −H  +measured f - F∞ (s) - ? C D z − output Fig. 1. The H∞ ﬁltering problem.

5

Robust reconstruction of sensor faults

In this section, the actuator fault reconstruction method in §3 will be modiﬁed to enable robust sensor fault reconstruction in the presence of uncertainty. In this case, the system under consideration is the following: x(t) ˙ = Ax(t) + Bu(t) + M ξ(t, y, u) y(t) = Cx(t) + N fo (t)

(25a) (25b)

where fo ∈ Rr is the vector of sensor faults, N ∈ Rp×r where rank(N ) = r and r ≤ p. A physical interpretation of this is that some of the sensors are assumed to be perfect. The objective is to transform this problem so that the method described in §3 and §4 can be employed to robustly reconstruct fo (t). Consider a new state zf ∈ Rp that is a ﬁltered version of y, satisfying z˙f (t) = −Af zf (t) + Af Cx(t) + Af N fo (t)

(26)

where −Af ∈ Rp×p is a stable (ﬁlter) matrix. Equations (25a) - (26) can be combined to form an augmented state-space system of order n + p 0 M x(t) ˙ A 0 B x(t) u(t)+ fo (t)+ ξ(t, y, u) (27a) = + Af N 0 z˙f (t) Af C −Af zf (t) 0 Aa

x(t) zf (t)= 0 Ip zf (t)

Ba

Fa

Ma

(27b)

Ca

Equations (27a) - (27b) are in the form of (1a) - (1b), and treat the sensor fault as an actuator fault. As described in §3, an observer driven by the signal zf can be designed, replacing (A, F, C, M ) with (Aa , Fa , Ca , Ma ) respectively. From the general sliding mode observer theory in [3], an appropriate sliding mode observer exists if

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• rank(Ca Fa ) = r • any invariant zeros of (Aa , Fa , Ca ) lie in the left half plane Since rank(N ) = r, the ﬁrst condition is satisﬁed since Ca Fa = Af N and Af is invertible. An expression pertaining to the invariant zeros of (Aa , Fa , Ca ) will now be derived in terms of the system block matrices. Proposition 4. The invariant zeros of (Aa , Fa , Ca ) ⊆ λ(A). If λi ∈ λ(A), then λi is an invariant zero if and only if λi I − A11 −A12 N rank = n−p+r (28) −A12 λi N − A22 N

Proof. See [12]

6

An example: A nonlinear crane system

The robust fault reconstruction scheme presented in this paper will now be demonstrated with an example, which is a nonlinear model of a crane, taken from [5]. The equations of motion that govern the crane are: xcosθ = 0 (I + ml2 )θ¨ + cθ˙ + mglsinθ + ml¨ ¨ ˙ (mt + m)¨ x + bx˙ + mlθcosθ − mlθ2 sinθ = u

(29a) (29b)

The variable x represents the displacement of the truck and θ represents the angular displacement of the load from the (downward) vertical. The model parameters used in the following simulations are given by mt = 3.2kg, m = 0.535kg, b = 6.2kg/s, c = 0.009kgm2 , I = 0.062kgm2 , g = 9.81m/s2 and l = 0.35m. ˙ x, x) Choosing (θ, θ, ˙ as the state vector and using standard small angle approximations, the nonlinear equations (29a) - (29b) can be written in the form of (1a) - (1b) and (25a) - (25b), where ξ represents the lumped residual nonlinearities resulting from the linearisation. The state-space matrices are 

   0 0 0  −0.4248  0 2.6340    F =B=  0 1.0000  0 0 −1.7977 0.2899     0 0 10  7.5033  0  M = N = 0 0  0 0 01 0 0.2677

0 1.0000  −15.5660 −0.0731 A=  0 0 0.8138 0.0038 

 1000 C = 0 0 1 0 0001

The choice of the matrix N implies the second sensor (representing the truck position) is not faulty. For the simulations which follow a simple phase advance controller was designed based on the measurements of θ and x.

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Robust reconstruction of actuator faults

Specifying D1 = I3 and γo = 1, the synthesis procedure yields γ = 0.9239. Due to space constraints, the Gl , Gn , Po and W1 obtained will not be shown. For this simulation, the parameters of νδ in (15) are ρ = 50 and δ = 10−4 . Figure 2 shows how the observer robustly reconstructs the actuator fault.

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Fig. 2. The left subﬁgure shows a fault at the actuator, and the right subﬁgure shows the reconstruction via the observer

6.2

Robust reconstruction of sensor faults

Specifying Af = 20I3 , D1,a = 10I3 and γo,a = 10 and constructing the augmented state space matrices from (27a) - (27b), the synthesis procedure yields γa = 9.2269. In this example, it can be veriﬁed that (Aa , Fa , Ca ) does not possess invariant zeros. Hence the fact that A has an eigenvalue at the origin does not present any diﬃculties. In the simulation that follows, ρa = 50 and δa = 5×10−5 . Figures 3 and 4 show the observer reconstructing the sensor faults. 0.12

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Fig. 3. The left subﬁgure shows a fault at the ﬁrst sensor, and the right subﬁgure shows the reconstruction via the augmented observer

Figure 3 shows some slight imperfections in the reconstruction. This is to be expected since the value of γa is signiﬁcantly larger than the L2 gain for the actuator fault reconstruction.

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Fig. 4. The left subﬁgure shows a fault at the third sensor, and the right subﬁgure shows the reconstruction via the augmented observer

7

Conclusion

This paper has proposed a method for robust reconstruction of faults using sliding mode observers that minimise the eﬀect of system uncertainty on the reconstruction. The method was applied directly to the reconstruction of actuator faults, and subsequently extended to the case of sensor faults, by the use of suitable ﬁlters which enable the sensor faults to be treated as actuator faults. Both methods were demonstrated with a nonlinear crane example.

References 1. Chen, J., Zhang, H. (1991) Robust detection of faulty actuators via unknown input observers. Int. J. Systems Science, 22, 1829–1839 2. Chilali, M., Gahinet, P. (1996) H∞ design with pole placement constraints: an LMI approach. IEEE Trans. Aut. Control, 41, 358–367 3. Edwards, C., Spurgeon, S. K. (1994) On the development of discontinuous observers. Int. J. Control, 59, 1211–1229 4. Edwards, C., Spurgeon, S. K., Patton, R. J. (2000) Sliding mode observers for fault detection and isolation. Automatica, 36, 541–553 5. Franklin, G. F., Powell, J. D., Enami-Naeni, A. (1994) Feedback control of dynamic systems. Addison-Wesley 6. Gahinet, P., Nemirovski, A., Laub, A. J., Chilali, M. (1995) LMI Control Toolbox, Users Guide. The MathWorks, Inc. 7. Hermans, F. J. J., Zarrop, M. B. (1996) Sliding mode observers for robust sensor monitoring. Proc. of 13th IFAC World Congress, 211–216 8. Hou, M., Patton, R. J. (1996) An LMI approach to H− /H∞ fault detection observers. Proc. of the UKACC Int. Conf. on Control, 305–310 9. Koshkouei, A. J., Zinober, A. S. I. (2000) Sliding mode controller-observer design for multivariable linear systems with unmatched uncertainty. Kybernetika, 36, 95–115 10. Patton, R. J., Chen, J. (1992) Robust fault detection of jet engine sensor systems using eigenstructure assignment. J. Guidance, Control and Dynamics, 15, 1491–1497 11. Tan, C. P., Edwards, C. (2000) An LMI approach for designing sliding mode observers. Proc. of IEEE Conf. on Decision and Control, 2587–2592

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12. Tan, C. P., Edwards, C. (2001) Sliding mode observers for robust detection and reconstruction of actuator and sensor faults. Technical Report, Department of Engineering 00-8, Leicester University 13. Utkin, V. I. (1992) Sliding Modes in Control Optimization. Springer-Verlag 14. Yang, H., Saif, M. (1995) Fault detection in a class of nonlinear systems via adaptive sliding observer. Proc. of IEEE Int. Conf. on Systems, Man and Cybernetics, 2199–2204 15. Zhou, K., Doyle, J., Glover, K. (1995) Robust and Optimal Control. PrenticeHall.

Real-time trajectory generation for ﬂat systems with constraints Johannes von L¨owis and Joachim Rudolph Institut f¨ ur Regelungs- und Steuerungstheorie, Technische Universit¨at Dresden, D–01062 Dresden, Germany Abstract. A new real-time trajectory generation scheme for diﬀerentially ﬂat systems with constraints is proposed. Diﬀerential ﬂatness is exploited for both the construction of trajectories as well as for making sure the constructed trajectories satisfy the constraints of the system. The main feature of the approach is that rather good tracking for varying desired behaviour is possible. The reference trajectory is constructed by concatenating trajectory pieces the endpoints of which not necessarily correspond to equilibria. Special care has to be taken to make sure that such trajectory pieces can be continued without violating constraints.

1

Introduction and problem formulation

A typical situation which requires real-time trajectory generation is depicted in Fig. 1: The system is controlled by a trajectory tracking controller stabilizing its motion around a given reference trajectory yref . The system output y should follow the desired behaviour yd which is speciﬁed on-line by a human operator for instance. This means the desired behaviour is not known in advance. Since the desired behaviour can be almost arbitrary it cannot be used as a reference trajectory immediately. For, this could lead to actuator saturation or to violation of other constraints. Therefore, a device is useful that from the desired behaviour generates a reference trajectory which, in a sense, follows the desired behaviour in such a way that the constraints are satisﬁed. The tracking control problem is largely simpliﬁed if the system to be controlled is diﬀerentially ﬂat and its ﬂat output is the quantity for which the desired behaviour is speciﬁed. Nevertheless, due to constraints, the trajectory generation is still a diﬃcult task. In order to formulate the trajectory generation problem we use the notion of eventually constant signals, introduced by van Nieuwstadt and Murray [5], [6] for the same purpose: A signal s : IR → IR is called eventually constant if there is a t1 such that s(t) = sc = const. for all t > t1 . For any1 eventually constant yd the trajectory generator is supposed to generate a reference trajectory yref that converges to yd , i.e., limt→∞ (yref (t) − yd (t)) = 0. 1

In order to be able to track a constant reference it must be assumed that this is possible without violating the constraints.

A. Zinober and D. Owens (Eds.): Nonlinear and Adaptive Control, LNCIS 281, pp. 385-394, 2003. Springer-Verlag Berlin Heidelberg 2003

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yd

yref t

t

tracking trajectory generator yref , y˙ ref , . . . controller generates admissible desired measurement input behaviour reference trajectory yref controlled system joystick-like device output y yd

Fig. 1. Typical situation which requires real-time trajectory generation

Another desirable property of a trajectory generator is that it achieves ‘good’ tracking for (already) admissible desired behaviour yd , i.e., if the desired behaviour can be tracked without violating the constraints, we wish to generate yref following yd closely. The trajectory is constructed such that good tracking of already admissible desired behaviour will be achieved in many situations. However, since there is no formal deﬁnition of ‘good tracking for already admissible desired behaviour’, we cannot guarantee such a property for the proposed scheme. Flatness-based approaches split the tracking control problem into trajectory generation and stabilization. A work that deals especially with online trajectory generation is [6]. The trajectory generator proposed there is able to generate trajectories converging to eventually constant signals. The trajectories are found as parameterizations of a suitable set of basis functions and constraints are not taken into account. Trajectory generation for systems with linear inequality constraints is investigated in [1]. For this class of systems, in order to parameterize the set of basis functions one has to solve a set of linear inequalities (for this problem there exist standard algorithms). Another way to tackle the tracking control problem for constrained systems is by use of model-predictive schemes. The so-called reference governor, presented in [2], can be used for tracking of piecewise constant desired behaviour (setpoints). It bears some similarity to our approach, in the sense that a reference governor ﬁlters the desired behaviour before feeding it as a reference to a pre-stabilized system. The ﬁltering involves repeatedly simulating the pre-stabilized system in order to check for which reference the constraints are satisﬁed. The proposed new scheme allows us to achieve good tracking for nonconstant desired behaviour while respecting constraints and avoiding integration of diﬀerential equations.

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We shortly recall the notion of diﬀerential ﬂatness [3]. A system given as a set of implicit diﬀerential equations in the system variables z = (z1 , . . . , zn ) ˙ . . . , z (µ) ) = 0 with i = 1, . . . , q is said to be diﬀerentially of the form fi (z, z, ˙ . . . , z (ν) ) of the ﬂat if there exists a function (y1 , . . . , ym ) = y = φ(z, z, system variables and ﬁnitely many of their time derivatives, the so-called ﬂat output, for which the following holds: ˙ . . . , y (µ) ) = 0 relating the com• There is no diﬀerential equation h(y, y, ponents of y, i.e., trajectories t → y(t) for the ﬂat output can be freely assigned; • Such trajectories completely determine the motion of the system, i.e., all system variables z can be expressed in terms of y and ﬁnitely many of its time derivatives: ˙ . . . , y (γ−1) ). z = ψ(y, y,

2

(1)

Deﬁnitions

In this section we give deﬁnitions of the notions needed to describe the trajectory generator. We restrict our discussion to the case m = 1, i.e., y is a scalar function. The desired behaviour yd is a function of time. It is speciﬁed, e.g., by a human operator using a joystick-like device as depicted in Fig. 1. At time instant t we are given the desired value yd (t). The trajectory generator is supposed to generate a reference trajectory yref which, in some sense, follows the desired behaviour. In addition, the reference trajectory yref can be tracked by the system without violating the constraints which are assumed to be of the form ˙ ), . . . , y (γ−1) (τ )) ≤ ψi,max ψi,min ≤ zi = ψi (y(τ ), y(τ for all τ ∈ IR. Since the system under consideration is assumed to be diﬀerentially ﬂat, the trajectories of all system variables are determined by a trajectory yref for the ﬂat output. Their computation involves derivatives of yref up to order γ − 1, cf. (1). It is clear that for continuous zi the trajectory for y must be at least (γ − 1) times diﬀerentiable. To t → y(t) we associate the ﬂag y¯ := (y, y, ˙ . . . , y (γ−1) ). The desired behaviour yd is not known in advance. Therefore, the reference trajectory is constructed by successively concatenating trajectory pieces yref,k: [tk , tk + Tf,k ] → IR with deﬁnition intervals [tk , tk + Tf,k ]. The trajectory pieces for the ﬂat output are chosen such that the corresponding trajectories for the system variables zi satisfy the (inequality) constraints (γ−1)

yref,k (τ )) ≤ ψi,max ψi,min ≤ zi = ψi (yref,k (τ ), . . . , y˙ ref,k (τ )) = ψi (¯

(2)

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for i = 1, . . . , α and for all τ ∈ [tk , tk + Tf,k ]. Such a trajectory piece is said to be admissible. A trajectory piece y : [t0 , t1 ] → IR with y(τ ) = y0 = const. (for all τ ∈ [t0 , t1 ]) is an equilibrium. It corresponds to a ﬂag (y0 , 0, . . . , 0). The set y0 = (y0 , 0, . . . , 0) ∈ IRγ : ψi,min ≤ ψi (¯ y0 ) ≤ ψi,max , i = 1, . . . , α} Y0 = {¯ of all equilibria satisfying the constraints is called the set of admissible equilibria. For a trajectory piece y: [t0 , t1 ] → IR we deﬁne y¯(t0 ) := lim t→t0 y¯(t) and t>t0

y¯(t1 ) := lim t→t1 y¯(t). Furthermore, we say: y begins in the ﬂag y¯(t0 ), and t
ends in the ﬂag y¯(t1 ) whereas y¯(t0 ) is the starting point and y¯(t1 ) is the endpoint of y. A trajectory piece y1 reaches the trajectory piece y2 at time t if y¯1 (t) = y¯2 (t). In the trajectory generation algorithm (see Sect. 4.2) all trajectory pieces are parameterized with a ﬁnite set of basis functions. A trajectory piece y: [t0 , t1 ] → IR is said to be constructible, if we can compute it using the basis functions in the algorithm of our choice. We say y: [t0 , t1 ] → IR can be continued, if there is a constructible and admissible ycont: [t1 , t2 ] → IR which begins in y¯(t1 ) and ends in the equilibrium 2 (y(t1 ), 0, . . . , 0). Continuability of y is a property depending only on the endpoint y¯(t1 ). The continuability issue does not arise for trajectory generation schemes where every trajectory piece ends in an equilibrium.

3

Assumptions

In order to be able to ﬁnd a reference trajectory yref converging to eventually constant yd it is necessary to restrict the range of desired values yd (t) such that (yd (t), 0, . . . , 0) ∈ Y0 . Furthermore, it is assumed that, for all y¯1 , y¯2 ∈ Y0 , we can construct a trajectory (piece) y: [0, Tf ] → IR connecting y¯1 , y¯2 in time Tf ≤ Tf,max , i.e., y¯(0) = y¯1 and y¯(Tf ) = y¯2 . By suitable preprocessing we obtain a certain number nd of time derivatives of yd at each tk : (nd )

y¯d (tk ) = (yd (tk ), y˙ d (tk ), . . . , yd

(tk )).

This can be achieved by ﬁltering the (original) desired behaviour with a ‘fast’ ﬁlter (compared to the desired system dynamics). Based on y¯d (tk ) the desired 2

This deﬁnition of continuability is tailored for use with the algorithms of Sect. 4. In principle, it would be suﬃcient to request continuability in the sense that ‘there exists an admissible trajectory piece ending in some admissible equilibrium’. Since the trajectory generator must be able to construct such a continuing trajectory piece ycont , we deﬁne continuability as given above.

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behaviour of the (near) future can be predicted. Depending on the number ik of time derivatives taken into account, extrapolations χiy¯kd (tk ) with (i )

t → χiy¯kd (tk ) (t) := yd (tk ) + y˙ d (tk ) (t − tk ) + · · · +

yd k (tk ) (t − tk )ik ik !

(3)

and degree ik ≤ nd are obtained.

4 4.1

Trajectory generation algorithm Basic idea

The trajectory generator is a discrete-time device constructing a continuoustime trajectory yref for the ﬂat output (cf. Fig. 2). The trajectory yref is constructed by successively concatenating trajectory pieces of the form yref,k: [tk , tk + Tf,k ] → IR. The piecewise concatenation is necessary because the desired behaviour yd is not known in advance. Therefore the trajectory must be updated frequently in order to account for changes in the desired behaviour. The trajectory updates are performed at discrete sampling instances tk with a sampling period Ts .

i

extrapolation χy¯kd (tk )

yd

yref,k: [tk , tk + Tf,k ] → IR connects y¯ref,k−1 (tk ) and χ ¯yi¯kd (tk ) (tk + Tf,k )

yref,k−1 tk

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Fig. 2. The reference trajectory yref for the ﬂat output is constructed by successively concatenating trajectory pieces yref,k

4.2

Trajectory generator

Alg. 1 (given below) solves the real-time trajectory generation problem. The term ‘real-time’ refers to the fact that not the complete knowledge of the desired behaviour is required for computing the reference trajectory but only information known up to time tk . Whether the necessary computations can be performed fast enough depends on the system dynamics, the performance to be achieved, and the hard- and software used.

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The algorithm is executed at each tk . The current trajectory piece yref,k−1 is replaced by yref,k taking into account the current desired behaviour (represented by y¯d (tk )). For ‘reasonable’ desired behaviour the attempt to ﬁnd an updated trajectory yref,k reaching a current extrapolation of the desired behaviour χiy¯kd (tk ) will succeed (cf. line 1 of Alg. 1). In case it fails we ﬁrst try to continue using the remainder of yref,k−1 , and if this is not possible because the endpoint of yref,k−1 is reached, yref,k is chosen as the continuation of yref,k−1 to the equilibrium (yref,k−1 (tk−1 + Tf,k−1 ), 0, . . . , 0).

ÁÊ

ÁÊ

ÁÊ ! " ! # $ % & % ' ! ( ) % & * + + , & ( - .

4.3

Trajectory update

An important step in Alg. 1 is the attempt to ﬁnd an updated trajectory reaching an extrapolation of the current desired behaviour. It is detailed in Alg. 2. Alg. 2 aims at ﬁnding a trajectory piece reaching a current extrapolation χiy¯d (tk ) of the desired behaviour of highest possible degree i in shortest possible time Tf,k . This is motivated as follows: If the desired behaviour is admissible it can be expected that its extrapolation is the better the more derivatives are taken into account. It is also expected, that the extrapolation becomes the less meaningful the farther in the future tk + Tf,k lies. Before a trajectory piece y connecting the ﬂags y¯(tk ) = y¯ref,k−1 (tk ) and ¯iy¯d (tk ) (tk + Tf ) is constructed3 it is checked (in line 5 of Alg. 2) y¯(tk + Tf ) = χ whether the restriction of χiy¯d (tk ) to [tk , tk +Tf ] can be continued. (If this is not 3

N For trajectory pieces y = i=1 φi ai written as a linear combination of basis functions φi : [tk , tk + Tf ] → IR : t → φi (t) with i = 1, . . . , N the construction can be done by solving a system of linear equations in the coeﬃcients ai . For more sophisticated trajectory constructions see, e.g., [5].

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the case the constructed trajectory piece cannot be continued either.) Fig. 3 shows some trajectory pieces (dotted lines) connecting the current trajectory with extrapolations of diﬀerent degrees in diﬀerent times.

ÁÊ

ÁÊ

! ! " ! # " " $ ! # % & " ' # $ ! ( % ) ! ' ! ' * +

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, - . "/ "" "% "&

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The transition time Tf is increased up to Tf,max (cf. line 4 of Alg. 2). By this it is guaranteed that for eventually constant desired behaviour yd the generated reference trajectory yref has the property that there is a t∗ such that ∀t > t∗ : yd (t) = yref (t). Only integer multiples of the sampling period Ts are taken as endpoints tk +Tf,k of deﬁnition intervals for trajectory pieces. This guarantees that each deﬁnition interval ends at a time instance at which a trajectory update will be performed. Because all system variables can be expressed in terms of the ﬂat output and its time derivatives, a simple way to check whether a trajectory piece yref,k : [tk , tk + Tf,k ] → IR is admissible (cf. line 7 of Alg. 2) is to insert it at suﬃciently many τi ∈ [tk , tk + Tf,k ] into (2). Sometimes the computational burden of this procedure can be reduced by exploiting the special structure of the system equations, e.g., if the trajectory pieces are polynomial in t and the ψi are polynomial in y and its time derivatives.

392

J. von L¨owis and J. Rudolph reaching extrapolation χy2¯d (tk )

reaching extrapolation χy1¯d (tk ) reaching extrapolation χy0¯d (tk )

yd yref tk

tk

t

tk

t

t

Fig. 3. In the process of updating yref,k it is attempted to reach diﬀerent extrapolations of the desired behaviour yd

5

Example

In this section we show that the proposed scheme performs well. We use the example of a magnetically levitated ball in order to compare our approach to the reference governor [2]. U-magnet coil i

s0 y

ferromagnetic ball with mass m

g

Fig. 4. The model of the magnetically levitated ball is diﬀerentially ﬂat with the ball postion y as a ﬂat output 2

Consider the model m y¨ = −λ (s0 i+y)2 + m g of the magnetically levitated ball sketched in Fig. 4. The input current i can be expressed in terms of y, y, ˙ 2 . Some of the system variables, namely the and y¨ as i = m (g − y ¨ )(s + y) 0 λ ball position y, the input current i, and the acceleration y¨ are constrained as follows: 0A ≤

y ) ≤ 15.0 mm −0.3 mm ≤ y =: ψ1 (¯ m (g − y¨)(s0 + y)2 =: ψ2 (¯ y ) ≤ 10 A λ −∞ ≤ y¨ =: ψ3 (¯ y ) < g.

Real-time trajectory generation for ﬂat systems

393

The parameters λ = 5·10−6 N m2 /A2 , m = 0.2 kg, and s0 = 0.5 mm are taken from [4]. By an input transformation introducing a new input v, the system can be brought to the form y¨ = v. The ﬂatness-based tracking controller then reads v = vFB = y¨ref − k1 (y˙ − y˙ ref ) − k0 (y − yref ). The stabilizing controller for use with the reference governor reads v = vRG = −k1 y˙ − k0 (y − w) with w the desired behaviour ‘ﬁltered’ by the reference governor. The gains are chosen as k1 = −λ1 − λ2 and k0 = λ1 λ2 with λ1 = −110 s−1 and λ2 = −120 s−1 .

15

y in mm

10

5

0

0

0.5

1

1.5

1

1.5

1

1.5

t in s 12 10

i in A

8 6 4 2 0

0

0.5

t in s 12 10

i in A

8 6 4 2 0

0

0.5

t in s

Fig. 5. Tracking of mostly admissible desired behaviour (solid line ——) with the proposed trajectory generator (dash-dotted line − · −) and the reference governor (dashed line − − −): position (top), input current with proposed trajectory generator (middle) and with reference governor (bottom)

In Fig. 5 simulation results for tracking a mostly admissible desired behaviour are shown. The output of the controlled system is depicted in the top plot for both the proposed ﬂatness-based trajectory generator and for the reference governor. The input currents required to achieve the motion are displayed in the middle and the bottom plot (see also Fig. 6).

394

J. von L¨owis and J. Rudolph

From Fig. 5 it can be seen that with the proposed scheme the system output y follows the desired behaviour better than with the reference governor. For both schemes the input current respects the limits. The same holds for the acceleration y¨ (not shown in the ﬁgure). The ‘chattering’ that can be seen in the current plots is related to the discrete-time nature of the reference update. For the simulation the sampling period was chosen as Ts = 8 ms, i.e., every 8 ms the reference trajectory yref or the reference w, respectively, are updated. This explains the ‘step-like’ changes in the current signals (cf. Fig. 6). 10

i in A

8

6

4

2

0 0.3

0.31

0.32

0.33

0.34

0.35

0.36

0.37

0.38

0.39

0.4

t in s

Fig. 6. Magniﬁcation of current plots of Fig. 5: proposed trajectory generator (dashdotted line − · −) and reference governor (dashed line − − −)

References 1. S. K. Agrawal, N. Faiz, and R. M. Murray. Feasible trajectories of linear dynamic systems with inequality constraints using higher-order representations. In Proceedings of the 14th triennal world congress, Beijing, China, 1999. IFAC. 2. A. Bemporad. Reference governor for constrained nonlinear systems. IEEE Trans. on Automatic Control, AC-43(3):415–419, 1998. 3. M. Fliess, J. L´evine, P. Martin, and P. Rouchon. Flatness and defect of non-linear systems: introductory theory and examples. Int. J. Control, 61(6):1327–1361, 1995. 4. J. L´evine, J. Lottin, and J.-Ch. Ponsart. A nonlinear approach to the control of magnetic bearings. IEEE Trans. on Control Systems Technology, 4(5):524–544, 1996. 5. M. J. van Nieuwstadt. Trajectory Generation for Nonlinear Control Systems. Dissertation, California Institute of Technology, Pasadena, California, 1997. 6. M. J. van Nieuwstadt and R. M. Murray. Real time trajectory generation for differentially ﬂat systems. International Journal of Robust and Nonlinear Control, 8(11):995–1020, 1998.

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