Table Of ContentAlgebraic and Geometric Methods in
Nonlinear Control Theory
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science. Amsterdam. The Netherlands
Editorial Board:
F. CALOGERO, Universita deg/i Studi di Roma. Italy
Yu. I. MANIN. St~klov Institute of Mathematics. Moscow. U.S.S.R.
A. H. G. RINNOOY KAN. Erasmus University. Rotterdam. The Netherlands
G.-c. ROTA. M.I. T .. Cambridge. Mass .• U.S.A.
Algebraic and Geotnetric
Methodsin
Nonlinear Control Theory
edited by
M. Fliess
Laboratoire des Signaux et Systemes,
CNRS - Ecole Sup~rieure d' Electricit~,
Gif-sur-Yvette, France
and
M. Hazewinkel
Centre for Mathematics and Computer Science,
Amsterdam, The Netherlands
...
D. Reidel Publishing Company
A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP "
Dordrecht / Boston / Lancaster / Tokyo
library of Congress Cataloging in Publication Data
Algebraic and geometric methods in nonlinear control theory.
(Mathematics and its applications)
Includes index.
1. Control theory. 2. Nonlino;:ar theories. I. Fliess, M. (Michel),
1945- . II. Hazewinkel, Michie!. III. Series: Mathematics and its
applications (D. Reidel Publishing Company)
QA402.3.A4534 1985 629.8'312 86-13073
ISBN-13: 978-94-010-8593-9 e-ISBN-13: 978-94-009-4706-1
DOl: 10.1 007/978-94-009-4706-1
Published by D. Reidel Publishing Company
P.O. Box 17, 3300 AA Dordrecht, Holland
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers,
101 Philip Drive, Assinippi Park, Norwell, MA 02061, U.S.A.
In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.O. Box 322, 3300 AH Dordrecht, Holland
All Rights Reserved
© 1986 by D. Reidel Publishing Company, Dordrecht, Holland
Softcover reprint of the hardcover 1s t edition 1986
No part of the material protected by this copyright notice may be reproduced or utilized
in any form or by any means, electronic or mechanical, including photocopying,
recording or by any information storage and retrieval system,
without written permission from the copyright owner
SERIES EDITOR'S PREFACE
Approach your problems from the right end It isn't that they can't see the solution. It is
and begin with the answers. Then one day, that they can't see the problem.
perhaps you will find the final question.
G.K. Chesterton. The Scandal of Father
'The Hermit Clad in Crane Feathers' in R. Brown 'The point"of a Pin'.
van GuIik's The Chinese Maze Murders.
Growing specialization and diversification have brought a host of monographs and
textbooks on increasingly specialized topics. However, the "tree" of knowledge of
mathematics and related fields does not grow only by putting forth new branches. It
also happens, quite often in fact, that branches which were thought to be completely
disparate are suddenly seen to be related.
Further, the kind and level of sophistication of mathematics applied in various
sciences has changed drastically in recent years: measure theory is used (non
trivially) in regional and theoretical economics; algebraic geometry interacts with
physics; ihe Minkowsky lemma, coding theory and the structure of water meet one
another in packing and covering theory; quantum fields, crystal defects and
mathematical programming profit from homotopy theory; Lie algebras ·are relevant
to filtering; and prediction and electrical engineering can use Stein spaces. And in
addition to this there are such new emerging subdisciplines as "experimental
mathematics", "CFD", "completely integrable systems", "chaos, synergetics and
large-scale order", which are almost impossible to fit into the existing classification
schemes. They draw upon widely different sections of mathematics. This pro
gramme, Mathematics and Its Applications, is devoted to new emerging
(sub)disciplines and to such (new) interrelations as exempla gratia:
- a central concept which plays an important role in several different mathematical
and/ or scientific specialized areas;
- new applications of the results and ideas from one area of scientific endeavour
into another;
- influences which the results, problems and concepts of one field of enquiry have
and have had on the development of another.
The Mathematics and Its Applications programme tries to make available a careful
selection of books which fit the philosophy outlined above. With such books, which
are stimulating rather than definitive, intriguing rather than encyclopaedic, we hope
to contribute something towards better communication among the practitioners in
diversified fields.
v
vi SERIES EDITOR'S PREFACE
System and control theory is a branch of applicable mathematics which studies
dynamic phenomena with both the ideas of external forces and partial
observations-only explicitly incorporated. That immediately totally changes many
of the questions one is naturally led to ask. It also generates new and different
mathematical problems to study and solve. This is already abundantly clear in the
deceptively simple looking case of linear systems x =Ax + Bu, y = ex, and becomes
even more evident for nonlinear ones x = f (x, u), y = h (x). Linear systems are com
fortable. A great deal is known and rewards/ outputs are proportional to
efforts/inputs. Eminently fair. The world, however, is nonlinear. This complicates
things and may seem daunting. The well-prepared engineer and mathematician,
however, rejoices in this fact and makes it work to his advantage. This book will
help.
The unreasonable effectiveness of mathemat As long as algebra and geometry proceeded
ics in science ... along separate paths, their advance was slow
and their applications limited.
Eugene Wigner But when these sciences joined company
they drew from each other fresh vitality and
Well, if you know of a better 'ole, go to it. thenceforward marched on at a rapid pace
towards perfection.
Bruce Bairnsfather
Joseph Louis Lagrange.
What is now proved was once only ima-
gined.
William Blake
Bussum, May 1986 Michiel Hazewinkel
TABLE OF CONTENTS
Series Editor's Preface v
Preface xi
CONTROLLABILITY, OBSERVABILITY, REALIZATION
AND OTHER STRUCTURAL PROPERTIES
B. JAKUBCZYK / Realization Theory for Nonlinear Systems;
Three Approaches 3
C. REUTENAUER / The Local Realization of Generating
Series of Finite Lie Rank 33
Z. BARTOSIEWICZ / Realizations of polynomial Systems 45
P. E. CROUCH and C. I. BYRNES / Symmetries and Local
Controllability 55
A. J. KRENER / The Intrinsic Geometry of Dynamic
Observations 77
H. KELLER and H. FRITZ / Design of Nonlinear
Observers by a Two-Step-Transformation 89
FEEDBACK SYNTHESIS AND LINEARIZATION
TECHNIQUES
H. NIJMEIJER / On the Input-Output Decoupling of
Nonlinear Systems 101
A. ISIDORI / Control of Nonlinear Systems Via
Dynamic State-Feedback 121
M. D. DI BENEDETTO / A Classification of Nonlinear
Systems Based on the Invariant Subdistribution
Algorithm 147
C. I. BYRNES and A. ISIDORI / Asymptotic Expansions,
Root-Loci and the Global Stability of
Nonlinear Feedback Systems 159
D. CLAUDE / Everything You Always Wanted to Know
About Linearization 181
viii TABLE OF CONTENTS
T. J. TARN, D. CHENG, and A. ISIDORI / Feedback
Linearization and Simultaneous Output Block
Decoupling of Nonlinear Systems 227
w.
M. BOOTHBY / Global Feedback Linearizability of
Locally Linearizable Systems 243
w.
RESPONDEK / Global Aspects of Linearization,
Equivalence to Polynomial Forms and
Decomposition of Nonlinear Control Systems 257
W. J. RUGH / The Extended-Linearization Approach for
Nonlinear Systems Problems 285
C. REBOULET, P. MOUYON, and C. CHAMPETIER / About the
Local Linearization of Nonlinear Systems 311
OPTIMAL CONTROL
H. J. SUSSMANN / Envelopes, Conjugate Points, and
Optimal Bang-Bang Extremals 325
I. A. K. KUPKA / Geometry of the Optimal Control 347
M. FLIESS and F. LAMNABHI-LAGARRIGUE / Volterra Series
and Optimal Control 371
A. J. VAN DER SCHAFT / Optimal Control and
Hamiltonian Input-Output Systems 389
DISCRETE-TIME SYSTEMS
S. MONACO and D. NORMAND-CYROT / Nonlinear Systems in
Discrete Time 411
J. W. GRIZZLE / Local Input-Output Decoupling of
Discrete Time Nonlinear Systems 431
E. D. SONTAG / Orbit Theorems and Sampling 441
VARIOUS OTHER THEORETICAL ASPECTS
A. M. BLOCH and C. I. BYRNES / An Infinite Dimensional
Variational Problem Arising in Estimation
Theory 487
R. SCHOTT / Iterated Stochastic Integrals in
Nonlinear Control Theory 499
C. HESPEL and G. JACOB / Approximation of Nonlinear
Systems by Bilinear Ones 511
TABLE OF CONTENTS ix
APPLICATIONS
R. MARINO / Feedback Linearization Techniques in
Robotics and Power Systems 523
B. D'ANDREA and J. LEVINE / C.A.D. for Nonlinear
Systems Decoupling, Perturbations Rejection
and Feedback Linearization with Applications
to the Dynamic Control of a Robot Arm 545
S. MONACO and S. STORNELLI / A Nonlinear Feedback
Control Law for Attitude Control 573
DADUGINETO / Identification of Different Discrete
Models of Continuous Non-linear Systems.
Application to Two Industrial Pilot Plants 597
S. BELGHITH, F. LAMNABHI-LAGARRIGUE, and M.-M. ROSSET /
Bang-Bang Solutions for a Class of Problems
Arising in Thermal Control 623
INDEX 633
A purely linear world would be a sad place to live in.
PREFACE
A dynamical system, in the sense that the word is used in mathematical system and
control theory, is something like
xf (x,u), y = h(x), (1)
i. e. we have a time evolution recipe for a quant~ty x which depends on additional
parameters u which can be used as inputs or controls and, in addition, the basic
quantity x is not directly observable: the only directly accessible quantities for
observations or measurements are certain derived numbers y=h(x). Intuitively x is
high dimensional and u and y live in fairly low dimensional spaces. Thus the concept
incorporates both the ideas of a possible presence of external forces and the idea of
observers.
Of course, there are a million variations on the evolution recipe encoded by (1):
equations in discrete time instead of continuous time, there can be delays, there can
be additional stochastic aspects such as noise and disturbances, etc.
The workhorse of applied system theory is no doubt the linear system
x = Ax + Bu. y = Cx. (2)
A lot is known about systems like (2) and they are immensely useful in many
applications, i.e. modeling situations. Morever, to a first approximation around a
reference trajectory, every system is linearizable if explicit time dependence of the
matrices A, Band C is allowed.
It is however, also totally clear that the study of linear systems alone is by no
means enough. First there is, of course, the fact that in many situations there is not
(and cannot be) a well-determined reference trajectory. More important in the long
run, even from the applications standpoint alone, is the observation that the purely
linear world is a sad one and an expensive one. The (feedback) controls needed to do
all kinds of things tend to be massive; there is a depressing, yes, linearity about it
which precludes using something like pressure-point control, where use is made of,
say, the bifurcation points of the parametrized family x = f(x, u), to steer things with
very small (changes in the) u. (Not that this is, at the moment, a workable,
implemented technique.) To appreciate what we are hinting at it might be good to
reflect that if electronics was purely linear, i.e. if it consisted basically of Ohm's law,
we would have no fun with transistors, television, memory cells, etc.; in fact you
would not be reading these lines.
xi
