Table Of ContentProgress in Systems and Control Theory
Volume 9
Series Editor
Christopher I. Byrnes, Washington University
Associate Editors
S.-1. Amari, University of Tokyo
B.D.O. Anderson, Australian National University, Canberra
Karl Johan Astr6m, Lund Institute of Technology, Sweden
Jean-Pierre Aubin, CEREMADE, Paris
H.T. Banks, University of Southern California, Los Angeles
John S. Baras, University of Maryland, College Park
A. Bensoussan, INRIA, Paris
John Burns, Virginia Polytechnic Institute, Blacksburg
Han-Fu Chen, Beijing University
M.H.A. Davis, Imperial College of Science and Technology, London
Wendell Fleming, Brown University, Providence, Rhode Island
Michel Fliess, CNRS-ESE, Gif-sur-Yvette, France
Keith Glover, University of Cambridge, England
Diederich Hinrichsen, University of Bremen, Federal Republic of Germany
Alberto Isidori, University of Rome
B. Jakubczyk, Polish Academy of Sciences, Warsaw
Hidenori Kimura, University of Osaka, Japan
Arthur J. Krener, University of California, Davis
H. Kunita, Kyushu University, Japan
Alexandre Kurzhansky, IIASA, Laxenburg, Austria and Academy of Sciences,
U.S.S.R.
Harold J. Kushner, Brown University, Providence, Rhode Island
Anders Lindquist, Royal Institute of Technology, Stockholm
Andrzej Manitius, George Mason University, Fairfax, Virginia
Clyde F. Martin, Texas Tech University, Lubbock, Texas
Sanjoy Mitter, Massachusetts Institute of Technology, Cambridge
Giorgio Picci, University of Padova, Italy
Boris Pshenichnyj, Glushkov Institute of Cybernetics, Kiev
H.J. Sussmann, Rutgers University, New Brunswick, New Jersey
T.J. Tarn, Washington University, St. Louis, Missouri
V.M. Tikhomirov, Institute for Problems in Mechanics, Moscow
Pravin P. Varaiya, University of California, Berkeley
Jan C. Willems, University of Groningen, The Netherlands
W.M. Wonham, University of Toronto
Christopher 1. Byrnes Alexander Kurzhansky
Editors
Nonlinear Synthesis
Proceedings of a nASA Workshop
held in Sopron, Hungary
June 1989
Springer Science+Business Media, LLC 1991
Christopher 1. Bymes Alexander Kurzhansky
Department of Systems Science International Institute for
and Mathematics Applied Systems Analysis
Washington University A-2361 Laxenburg
St. Louis, MO 63130, U.S.A. Austria
Library of Congress Cataloging-in-Publication Data
Nonlinear synthesis I edited by Christopher 1. Bymes and Alexander
Kurzhansky.
p. cm. --(Progress in systems and control theory : v. 9)
Proceedings of the Sopron Conference on Nonlinear Synthesis, held
June 1989, in Sopron, Hungary and sponsored by SDS.
ISBN 978-0-8176-3484-1 ISBN 978-1-4757-2135-5 (eBook)
DOI 10.1007/978-1-4757-2135-5
1. Automatic control--Congresses. 2. Nonlinear systems--
Congresses. 1. Bymes, Christopher 1., 1949-
ll. Kurzhansky, A.B. ID. Sopron Conference on Nonlinear
Synthesis (1989) IV. International Institute for Applied Systems
Analysis. Dept. of Systems and Decision Sciences. V. Series.
TJ212.2.N56 1991
629.8--dc20 91-21274
CIP
Printed on acid-free paper.
e
Springer Science+Business Media New York 1991
Originally published by Birkhlluser Boston in 1991
Softcover reprint ofthe hardcover Ist edition 1991
All rights reserved. No part of this publication may be reproduced, stored in a retrieval
system,or transmilted.in any formorby any means,electronic, mechanical, photocopying,
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addressed directly to Springer Science+Business Media, LLC .
ISBN 978-0-8176-3484-1
Camera-ready text prepared by the authors.
987654321
CONTENTS
Preface vii
Author's Index viii
Morse Theory and Optimal Control Problems
A. A. Agrachev and S. A. Vakhrameev . . . . . . . . . . 1
Viability Kernel of Control Systems
f.-P. Aubin and H. Frankowska 12
New Methods for Shaping the Response of a Nonlinear System
C. I. Byrnes and A. Isidori ............... 34
Asymptotic Stabilization of Low Dimensional Systems
W. P. Dayawansa and C. F. Martin . . . . . . . . . . . . . 53
Zero Dynamics in Robotic Systems
A. De Luca ........ . . ...... 68
Adaptive Methods for Piecewise Linear Filtering
G. B. Di Masi and M. Angelini ....... ..... 88
Nonlinear Feedback Control for Flexible Robot Anns
x. Ding, T. f. Tarn, and A. K. Bejczy . . . . . ..... 99
Methods of Nonlinear Discontinuous Stabilization
M. Fliess and F. Messager ........ . 112
Invariant Manifolds, Zero Dynamics and Stability
H. W. Knobloch and D. Flockerzi ..... . 132
Tracking Control for Robotic Manipulators by
Local Linear Feedback
H. P. Kobayashi . . . . . . . . . . . . . ... 141
Synthesis of Control Systems Under Uncertainty Conditions
(Game Theory)
V. M. Kuntzevich .................... 156
v
Ellipsoidal Techniques for the Problem of Control Synthesis
A. B. Kurzhansky and I. Vdlyi ............... 169
Extened Gaussian Quadrature and the
Identification of Linear Systems
C. F. Martin and A. Soemadi ........... 185
Multirate Sampling and Zero Dynamics:
from Linear to Nonlinear
S. Monaco and D. Normand-Cyrot .200
Factorization of Nonlinear Systems
H. Nijmeijer ............. . . 214
On the Approximation of Set-Valued Mappings
in a Uniform (Chebyshev) Metric
M. S. Nikolskii ............ . . . . . . . . . 224
Estimation of a Guaranteed Result in Nonlinear
Differential Games of Encounter
A. G. Pashkov ............ . .232
Limit Sets of Trajectories
N. N. Petrov ..... .242
Nonlinear Systems with Impulsive and
Generalized Function Controls
A. V. Sarychev ......... . ..... 244
Extremal Trajectories, Small-time Reachable Sets and
Local Feedback Synthesis: A Synopsis of the
Three-dimensional Case
H. Schlittler ............ . .258
Regularity Properties of the Minimum-time Map
G. Stefani ............... . .270
Optimal Synthesis Containing Chattering Arcs and
Singular Arcs of the Second Order
M. I. Zelikin and V. F. Borosov .283
The Invariants of Optimal Synthesis
L. F. Zelikina . .297
Author's Index .305
vi
PREFACE
In its broadest sense, nonlinear synthesis involves in fact the synthesis of sometimes so
phisticated or complex control strategies with the aim of prescribing, or at least influencing,
the evolution of complex nonlinear systems. Nonlinear synthesis requires the development
of methodologies for modeling complex systems, for the analysis of nonlinear models, and
for the systematic design of control schemes or feedback laws which can achieve a wide
variety of prescribed objectives. The modeling, analysis and control of complex systems in
the face of uncertainty form on of the major components of the current research program
in the Department of Systems and Decision Sciences (SDS) at the International Institute
for Applied Systems Analysis (IIASA).
In June 1989, a IIASA workshop on Nonlinear Synthesis, sponsored by SDS, was held
in Sopron, Hungary. We are proud to present this volume as the proceedings of this
workshop, a workshop attened by prominent researchers in nonlinear systems from both
the East and the West. Since the promotion and encouragement of scientific cooperation
between researchers in the East and in the West is one of the goals at IIASA, we feel
the Sopron Conference on Nonlinear Synthesis was very successful. Moreover, we were
especially pleased by the impressive new advances presented at the workshop which, in
this volume, are now part of the conference record.
In particular, this volume contains some very original contributions to controlled invari
ance using viablity theory, to the control of nonlinear sampled-data systems, to control
synthesis for uncertain systems, to differential games, to feedback stabilization of nonlinear
systems, to identification and filtering, as well as some very recent advances in the regu
lation and optimal control of nonlinear systems. In addition, three papers on motion and
trajectory control for rigid and for flexible robots illustrate the application of new control
techniques and point in the direction of one of the most exciting new research challenges
in nonlinear synthesis, the control of nonlinear, distributed pammeter systems.
We would like to thank the participants and the Hungarian National Member Ol'gani
zation to IIASA for their time and effort in making the Sopron Conference on Nonlinear
Synthesis such a successful exchange of new ideas and techniques between EllSt and West.
Christopher I. Byrnes Alexander B. Kurzhanski
Chairman Chairman
Department of Systems Science Department of Systems and
and Mathematics Decision Sciences
Washington University International Institute for
St. Louis, MO USA Systems Analysis
Laxenburg, Austria
Author's Index
Agrachev, A.A., 1 Kurzhansky, A., 169
Angelini, M., 68 Martin, C.F., 53,185
Aubin, J.-P., 12 Messager, F., 112
Bejczy, A.K., 99 Monaco, S., 200
Borosov, V.F., 283 Nijrneijer, H., 214
Byrnes, C. I., 34 Nikolskii, M.S., 224
De Luca, A., 68 Norrnand-Cyrot, D., 200
Dayawansa, W.P., 53 Pashkov, A.G., 232
Di Masi, G.B., 88 Petrov, NN., 242
Ding,X., 99 Sarychev, A.V., 244
Fliess, M., 112 Schattler, H., 258
Flockerzi, D., 132 Soernadi, A., 185
Frankowska, H., 12 Stefani, G., 270
Isidori, A., 34 Tarn, T.J., 99
Knobloch, H.W., 132 Valyi, I., 169
Kobayashi, H.P., 141 Vakhrarneev, S.A., 1
Kuntzevich, V.M., 156 Zelikin, M.I., 283
Zelikina, L.F., 305
Morse Theory and Optimal Control Problems
A.A. Agrachev and S.A. Vakhrameev
1 Introduction
It is well known that Morse Theory is a very flexible tool for dealing with
nonlinear problems of analysis and topological problems. The main purpose
of the present paper is to describe a modification of this theory which can
be used for the study of optimal control problems. The necessity of such
a modification is related to the fact that for these problems the inequality
constraints are typical (for example, control constraints, phase constraints,
etc.) The inequalities destroy the smooth structure and hence the necessity
to construct the theory for spaces with singularities. We encounter this
situation in the case of optimal control problems.
However, note that the classical theory also can be used for the study
of some optimal control problems. So we begin by pointing out this class
of problems.
Let us remember that the basic facts of Morse theory are so called Morse
inequalities which relate the topological characteristic of manifold M with
the number of critical points of a smooth function f : M ~ R with definite
index (or co-index).
Let b;(M) be the i-th Betti number of the Riemannian manifold M,
X(M) be the (homological) Euler characteristic of M, f : M ~ R be a
Morse function on M ( i.e. the smooth function with compact level sets
r = {x E M I f(x) ~ a} which has only non degenerate critical points),
Ci(J) be the number of critical points of this function of co-index i. Then
for arbitrary m = 0,1, ... ,dimM, the following inequalities hold:
Lm. :( Lm. :(
-1)m-ibi(M) ~ _1)m-ici(J),
;=0 ;=0
(1.1) b;(M) ~ Ci(J), i = 0,1, ... , m,
dimM
L.:
X(M) = (_1)ic;(J).
;=0
Let us remember that the index (co-index) of the critical point Xo of the
function f is the maximal dimension of subspaces in TXoM where the Hes
sian Hxo(J) is negatively (positively) defined. Of course, inequalities (1.1)
2 MORSE THEORY AND OPTIMAL CONTROL PROBLEMS
can be written in the usual "index" form if we replace the function 1 by
-I·
Similar inequalities are valid if M is an infinite dimensional manifold
and 1 a smooth function with only non degenerate critical points satisfying
condition (C) (by Palais and Smale see [6], [7]):
if infzEsll \/ l(x)lIz = 0,
(C)
then there exists the critical point p of the function
1 which belongs to the closure S of the set S .
Here IIpliz = V(p,p}z and (., ·}z is the Riemann structure on M. More
precisely, let Ci(f) be the number of critical points of 1 lying on the level
sre.t r = {x E M I I(x) ~ a} and bi(r) the i-th Betti number ofthe space
Then, if 1 is bounded from above, satisfying the Palais-Smale condition
(C) and having only nondegenerate critical points, then the following Morse
inequalities holds:
m m
:s
L(-1)m-1bi(r) L(-1)m-ici(f), X(r) - L(-1)ici(f)
i=O i=O i
(where X(r) is the homological Euler characteristics of r ). In these
inequalities the number ci(f) denotes the number of critical points of 1 in
r
with finite co-index. In particular, for all m = 0,1,2, ....
i = 0,1,2, ... , m
The last relations are valid even in the case when 1 is not bounded from
above. In this case Ci(f) and bi(r) may be infinite.
Now let us consider the smooth control system
m
L
(1.2) Z = A(x) + UiBi(X), x E M,u E Rm,
i=l
on the smooth manifold M isometrically embedded into the Euclidean space
Rd. The smooth vector fields A, Bi, i = 1, ... , m, can be identified with 0'
dimensional vector functions. We propose that these functions satisfy the
following growth conditions:
i = 1, ... ,m,
where kl' k2 = const ~ 0; (., .) is the inner product in Rd and I . I is the
corresponding Euclidean norm in Rd.
