nonlinear synthesis: proceedings of a iiasa workshop held in sopron, hungary june 1989

Progress 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 Editors

Alexander Kurzhansky

Nonlinear Synthesis

Proceedings of a nASA Workshop held in Sopron, Hungary June 1989

Springer Science+Business Media, LLC 1991

Christopher 1. Bymes Department of Systems Science

and Mathematics Washington University St. Louis, MO 63130, U.S.A.

Alexander Kurzhansky International Institute for

Applied Systems Analysis A-2361 Laxenburg 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

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

CIP

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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 ........ .

Invariant Manifolds, Zero Dynamics and Stability H. W. Knobloch and D. Flockerzi .....

Tracking Control for Robotic Manipulators by Local Linear Feedback H. P. Kobayashi . . . . . . . . . . . . .

Synthesis of Control Systems Under Uncertainty Conditions (Game Theory)

112

. 132

... 141

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

Factorization of Nonlinear Systems H. Nijmeijer ............. .

On the Approximation of Set-Valued Mappings in a Uniform (Chebyshev) Metric

.200

. 214

M. S. Nikolskii ............ . . . . . . . . . 224

Estimation of a Guaranteed Result in Nonlinear Differential Games of Encounter A. G. Pashkov ............ .

Limit Sets of Trajectories N. N. Petrov .....

Nonlinear Systems with Impulsive and Generalized Function Controls A. V. Sarychev ......... .

Extremal Trajectories, Small-time Reachable Sets and Local Feedback Synthesis: A Synopsis of the Three-dimensional Case H. Schlittler ............ .

Regularity Properties of the Minimum-time Map G. Stefani ............... .

Optimal Synthesis Containing Chattering Arcs and Singular Arcs of the Second Order M. I. Zelikin and V. F. Borosov

The Invariants of Optimal Synthesis L. F. Zelikina .

Author's Index

vi

.232

.242

..... 244

.258

.270

.283

.297

.305

PREFACE

In its broadest sense, nonlinear synthesis involves in fact the synthesis of sometimes sophisticated 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 invariance 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 regulation 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'ganization 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 Chairman Department of Systems Science and Mathematics Washington University St. Louis, MO USA

Alexander B. Kurzhanski Chairman Department of Systems and Decision Sciences International Institute for Systems Analysis Laxenburg, Austria

Agrachev, A.A., 1

Angelini, M., 68

Aubin, J.-P., 12

Bejczy, A.K., 99

Borosov, V.F., 283

Byrnes, C. I., 34

De Luca, A., 68

Dayawansa, W.P., 53

Di Masi, G.B., 88

Ding,X., 99

Fliess, M., 112

Flockerzi, D., 132

Frankowska, H., 12

Isidori, A., 34

Knobloch, H.W., 132

Kobayashi, H.P., 141

Kuntzevich, V.M., 156

Author's Index

Kurzhansky, A., 169

Martin, C.F., 53,185

Messager, F., 112

Monaco, S., 200

Nijrneijer, H., 214

Nikolskii, M.S., 224

Norrnand-Cyrot, D., 200

Pashkov, A.G., 232

Petrov, NN., 242

Sarychev, A.V., 244

Schattler, H., 258

Soernadi, A., 185

Stefani, G., 270

Tarn, T.J., 99

Valyi, I., 169

Vakhrarneev, S.A., 1

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:

(1.1)

m m

L.:( -1)m-i bi(M) ~ L.:( _1)m-i ci (J), ;=0 ;=0

b;(M) ~ Ci(J), i = 0,1, ... , m, dimM

X(M) = L.: (_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 Hessian 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]):

(C)

if infzEsll \/ l(x)lIz = 0,

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 set r = {x E M I I(x) ~ a} and bi(r) the i-th Betti number ofthe space r. 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

L(-1)m-1bi(r) :s 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

(1.2) Z = A(x) + L 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.

AGRACHEVANDVAKHRAMEEV 3

With the system (1.2) one can relate the input-output map F~o,T : L2 [0, T) ..... M assigning the right endpoint x(T) = x(T; xo , u(·» of the trajectory x(·) of system (1.2) to an arbitrary admissible control u(·).

If for every Xo E M, T > 0 the rank of this map is constant (i.e. the dimension of the image ImF~o,T( u) of the differential F~o,T( u) doesn't depend on u(·), then the system (1.2) is called a system of constant rank. This class of systems was introduced in [1] in the context of studying conditions for an extremal control to be bang-bang. The full theory on such systems can be found in [8].

These systems have a lot of remarkable properties, in particular the reachable set

is a smooth submanifold in M. There exists sufficient conditions characterizing this class of systems.

They can be described as follows. Let in some neighbourhood of each point Xo E M the condition of finite definitness hold, i.e. there exists an integer s ~ 0 such that

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

and the following bang-bang condition holds i, j = 1, ... , m, k = 0,1, ... ,

k m

(1.4) [Bi' adk ABj] = L L a~; ado ABj3 0=0(3=1

with the functions a::j3, a~; being smooth in that neighborhood. Then the system (1.2) is of constant rank.

The condition of finite definiteness one can replace by the real analycity condition. The general necessary and sufficient conditions are described in [1], [2] [8].

The following proposition is true.

Proposition 1.1 Let the system (J.2) be of constant rank and Xo E M, T > o be given. Then for an arbitrary smooth submanifold N C M that is transversal to the reachable set n~o (T) of this system, the set

is the Hilbert submanifold in L2[0, T]. If N is a closed submanifold in M then 1i can be equipped with the structure of a complete Riemannian manifold.

Let us now introduce the class of functionals

(1.5) f(u) = iT <p(x, u)dt

defined on the set of trajectories of the system (1.2). These are smooth functions on 1t for which the condition (C) is valid.

Proposition 1.2 Suppose that 0;

2. l<Px(x, u)-<Px(y,u)I+I-<Pu(x, u)-<Pu(Y, u)1 ::; Llx-YI,L = const > 0;

9. l<Px(x, u)1 + l<Pu(x, u)1 ::; b(x) + alu!, where a = const > 0 and b(x) ::; 0 is bounded on bounded subsets of Rd;

4. -(<Puu(x, u)(,() ;::: ll(x)I(1 2 , where 0 ::; Il(x) is bounded away from zero on compact subsets of Rd.

Then the functional (1.5) is a smooth function on 1t satisfying (C) condition by Palais and Smale. 1

We have no place to deal with the concrete examples of the theory. One can find these examples in [10].

It is necessary to note that for the infinite dimensional case the finite dimensional theory is also important. Namely, in the simplest case when the path space

PII = {u(.) E L~[O, T] I x(T; xQu(,)) = y}

is contract able for every yEN the homology groups of 1t and N n 'Rxo (T) conincide (and for its calculation the finite dimensional theory is used.) In the general case the homology groups of 1t can be calculated by means of spectral sequences if the homology groups of N n'Rxo(T) and PII are known.

Taking into account the previous remark, we first construct the Morse theory for the spaces with singularities in the finite dimensional case.

1 Note that the (0) condition by Palais and Smale is valid for the function J iff it is valid for the function - J. The sign "-" in the conditions 1), 4) of proposition 1.2 appears since we consider the "co-index" form of the Morse inequalities.


2 Morse theory for manifolds with corners (the finite dimensional case)

Manifolds with corners are objects which can be described locally in terms of a system of smooth nonlinear inequalities.

Let M be a compact finite dimensional smooth Riemannian manifold. The closed subset V C M is called a submanifold with corners iff for every Xo E V there exists a coordinate chart (o,ro' cp) of M and a convex polyhedral cone Kco C Rn , n = dim M with vertex at the origin such that cp(xo) = 0 and

We can define the notion of tangent cone T:r: V to V at the point x E V in the usual way. This cone is isomorphic to the cone ]{:r: and the set-valued map x 1-+ T:r: V: M -+ 2TM is lower semicontinuous. Moreover, there exists a complete integrable distribution TI:r:' x EM, of constant dimension such that

- def TI:r: = T:r:V = T:r:V - T:r: V \Ix E V.

Hence there exists a smooth manifold of minimal dimension that contains the submanifold V with corners. If T:r:M = fxV for all x E V then we say that sub manifold V with corners is solid. In the following we consider only solid sub manifolds with corners.

The maximal (with respect to inclusion) smooth submanifold reM is called the open face of submanifold V with corners iff reV. Every point of submanifold V with corners in a unique open face of maximal dimension. Moreover, the family {r} of all open faces defines on V the structure of stratified space (in the sense of Whitney).

Let f : V -+ R be the smooth function on submanifold V with corners (i.e. f is the restriction to V of some smooth function j : M -+ R.) The point Xo E V is called the critical point of this function if the differential, d:r:.J, of function f at point Xo belongs to the negative polar cone T;o V to the tangent cone T:r: o V at the same point, i.e. the following relation holds:

The critical point Xo is called a nondegenerate critical point iff (i) d:r:of E reI int T;o V; (ii) the point Xo is a nondegenerate critical point (in the usual sense) for

the restriction fir of function f to the open face r of V, having maximal dimension and containing this point in its relative interior.

As in classical theory the non degenerate critical points are isolated.


The real number c E R is called the critical value of the function I : V -+ R iff the set r 1 (C) = {Z E V I I( z) = C} contains the critical points. All other values are called regular.

In the following by Morse function we mean smooth functions which have only nondegerate critical points.

Now, our goal is to study the change of homotopic type of the level sets r = {z E V I I(z) ~ a} when the parameter a E R varies. For that purpose we need the important notion of £ransversality which generalizes the corresponding notion of smooth analysis to the case considered.

Let N be another smooth manifold, WeN be the submanifold with corners in N and the smooth map g : V -+ N is given. We call this map transversal to W iff for all Z E V, g( z) E W, the following condition holds:

-gu:T:c V + Tg(:c) W = Tg(:c)N.

Here g.:c is the differential (or tangent map) of g : V -+ N. The following transversality theorem is true:

Theorem 2.1 Let It, t E R, be the family of smooth maps from V to N smoothly depending on parameter! E R, and each of these maps transversal to submanilold WeN with corners. Then for arbitrary t', til E R the sets f t"7 1 (W) and f t"7,1 (W) are of the same homotopic type.

In fact, we can state the strong conclusion: there exist smooth maps

(not "onto" ) such that the maps FoG and Go F are smoothly isotopic to the identity. 2

Let us give a sketch of the proof of this theorem. We can assume that t' = 0, til = O. First we show that there exists a flow Pt , t E R, on manifold M such that Pt/;l(W) C ft- 1(W) for all t, 0 ~ t ~ 1 , i.e. the condition fo(z) E W implies the relation ft(Pt(z)) E W, 0 ~ t ~ 1. For an arbitrary flow Pt , t E R , on M we have the relation

where X t , t E R, is the (nonstationary) smooth vector field on M which generates this flow. Since :tft{Pt(z)) E Tj,(Pt(:c»N , we can use the transversality condition, and find nonstationary smooth vector field X t , t E R, on M such that the corresponding flow Pt , t E R, verifies the condition

2i.e. there exist the extensions P and G on M of the maps F and G such that FoG and Go F are smoothly isotopic to identity idM' Let us remember that the smooth map h : M _ M is smoothly isotopic to identity iff there exists the smooth map H : [0,1] X M _ M such that HI = H(t •. ) is a diffeomorfism Cor all t,O $ t $ 1, and Ho = idM,H1 = h.


d d/'(P,(x» E Tj,(Pt(zo»W Vt,O $ t $ 1.

Hence, using the condition fo(x) E W, we have f,(P,(x)) E W Vt,O $ t $ 1 . Replacing the family f" t E R, by the family It-" t E R, and repeating the previous reasoning, we find the flow Q" t E R, on M such that

Qt!ll(W) c f1':t(W), Vt,O $ t $ 1

Now, putting F = Pl and G = Ql we finished the proof. From Theorem 2.1 we immediately receive the following result: If on

the interval [a, b] there are no critical values of the function f, then the sets r and fb are of the same homotopic type.

For the proof let us consider the family ft = f - t, t E R, and the

submanifold with corners W = R, ~f {s I s ~ O} C R. Since in [a, b) there are no critical values of f, then this family is transversal to W. So, using the theorem 2.1, we have the sets f;;l(R+) = r .!b-l(R+) = fb are of the same homotopic type.

It is necessary to emphasize that Theorem 2.1 is true for such closed subsets V C M which have (convex) tangent cones Tzo V with only two properties: (1) the set-valued map x 1-+ Tzo V : M -+ 2TM is lower semi-

- def continuous; (2) the planes Tzo V = Tzo V - Tzo V, x E V, are of constant dimensions. This remark permits a more general definition of submanifold with corners to be given. Namely, call the closed subset V C M as submanifold with corners iff for every x E V there exists the coordinate chart (Ozo,1/J) of manifold M, the convex polyhedral cone Kzo C Rm with vertex at the origiri and the smooth map F : Rn -+ R m, n = dim M, that preserves the origin and is transversal to Kzo such that 1/J(x) = 0 and 1/J(Ozo n V) = F-l(Kzo)

With this definition the class of submanifolds with corners becomes closed under operation of transversal intersection. In the present paper we do not consider these objects. The theory of "general" submanifolds with corners, the smooth control systems on such submanifolds. Morse theory and other topics will appear in a forthcoming paper. Here we only note that a great deal of the results in geometrical control theory are also true for the smooth control systems on such manifolds with corners.

Now let us consider the case when interval [a, b) contains the critical values of the smooth function f : V -+ R. Changing the homotopic type of the level sets as in classical theory can be described in terms of co-indexes (positive indexes) of critical points which correspond to those critical values.

In the present context the co-index of critical point Xo E V is defined as co-index of Xo for the restriction fir of function f to the open face reV of maximal dimension which contains Xo in its relative interior.

The following theorem is true.

Theorem 2.2 Let f be the Morse function on submanifold V with corners and c, a < c < b, the unique critical value of this function on [a, b]; Xl. •.• , Xj:

the correspondent critical points and rt. ... , rj: the co-indexes of these points. Then the level set r have the homotopic type of the set fb with the cells Drl, ... , Dr., have been attached. In particular, V has a cell complex structure.

This theorem can be proved using the stratified Morse theory [4]. However the authors have proved this theorem without using this theory.

From Theorem 2.2 one can derive the Morse inequalities following Milnor approach [5]. However for deriving the Morse inequality it is sufficient to only have some variant of Morse lemma (see [10)). Let us formulate the "homotopic" variant of this lemma.

Let f be the smooth function on convex polyhedral cone KeRn with vertex at origin f(O) = ° and zero the non degenerate critical points of this function. Put P = K n (-K) and let Q be the orthogonal complement of P : P ED Q = Rn. Let us represent the point x E Rn in the form x = (p, q),p E P, q E Q, and define the function i: k -+ R by the formula

f"( ) of(O, 0) 02f(0, 0)( ) p, q = oq q + oqop p, p .

Theorem 2.3 There exists a neighbourhood U of origin in Rn such that the sets {(p, q) E K nUl i(p, q) ~ O} and {(p, q) E K nUl f(p, q) ~ O} are of the same homotopic type.

Theorem 2.3 can be proved by means of the transversality Theorem 2.1. From Theorem 2.3 (or Theorem 2.2) follows the main result of the present paper:

Theorem 2.4 Let f be the Morse function on submanifold V with corners. Then inequalities (1.1) hold with M replaced by V.

3 Palais-Smale theory for manifolds with corners

The following abstract theory can be applied in the study of optimal control problems with inequality constraints on the right end point of a trajectory.

Let the finite dimensional submanifold V C M with corners, Hilbert manifold 11. and a surjective submersion F : M -+ M be given. The inverse image V = p-l(V) is called a Hilbert submanifold with corners.


Since locally V is a convex polyhedral cone, the Hilbert submanifold V with corners can be represented as a direct product of convex polyhedral cone and a Hilbert space. Hence, the elementary geometry of Hilbert submanifold with corners is similar to those of finite dimensional submanifolds V with corners which modelled V. In particular, the tangent cone Tx V to Hilbert submanifold V with corners have the following properties: (1) the set-valued map x 1--+ Tx V : M - 2TM is lower semicontinuous; (2) the

space 'fxV d~f Tx V - Tx V, x E V , are of the constant co-dimension in TxM. Hilbert submanifolds with corners are also stratified spaces in the Whitney sense: this stratification is given by means of open faces. The open lace of Hilbert submanifold V with corners is the maximal (with respect to inclusion) submanifold .J C M such that .J C V. Of course, the open faces of V can be described in terms of open faces of the finite dimensional submanifold V with corners, which modelled V. Every point x E V is contained in a unique open face .J which has the minimal co-dimension in M.

In the following we assume that manifold M is complete Riemannian manilold. By (-, ·)x we denote the Riemannian structure on M and set IIplix = V{p,p)x,p E TxM.

The following analog of the (C) condition by Palais and Smale is an important consideration. To formulate this condition let us introduce the function m : V - R+ by the formula:

m(x) = sup {\lI(x),Xx)x. X"ET"V IIX,,11x:9

Generalized (Cj Condition: if S C V is the subset on which the function I : V - R is bounded and

inf m(x) = 0, xES

then there exists the critical point of the function I which belongs to ti, the closure in M the subset S. Note that if V = M we have the usual (C) condition by Palais and Smale.

For smooth function I : V - R on Hilbert submanifold V with corners as in finite dimensional case one can define the notions of critical points, nodegenerate critical points, its index and co-index, critical values, etc. For example, the point Xo E V is called the critical point of function I : V - R iff the differential dxJ of this function at point Xo satisfies the condition:


By Morse function we now mean the smooth function which has only non degenerate critical points and satisfies the generalized (C) condition.

The following theorem is true:

Theorem 3.1 Let V be the connected Hilbert submanifold with corners in complete Riemannian manifold M, and f : V -+ R be bounded from above Morse function on V. Then f achieves on V its maximal values.

Theorem 3.2 If f : V -+ R is bounded from above Morse function on Hilbert submanifold V with corners then the following Morse inequalities are valid: 'tim = 0,1,2, ... , a E R,

m m

I:(-I)m-ibi(r) ~ I:(_1)m-i c;(I) x(r) = I:(-I)ici (l). ;=0 ;=0 ;

In particular for all m = 0,1, .,.

Here cm (I) is the number of critical points of the function f lying on the level set r and having finite co-index i. The full proof of these statements can be found in [10].

Now let us consider the control system (1.2).

Proposition 3.1 Let (1.2) be the control system of constant rank and V C M is the submanifold with corners such that V C 'R-:co(T). Then the set V = F;"~T(V) is the Hilbert submanifold with corners in Lr[O, T].

Proposition 3.2 Let the conditions of Proposition 1.2 be valid. Then the functional (1.5) is a smooth function on V which satisfies the generalized (C) condition.

Here F:Co,T is the input-output map of system (1.2).

Bibliography

[1] Agrachev, A.A. and Vakhrameev, S.A. Nonlinear control systems of constant rank, and the bang-bang conditions for extremal controls. Soviet Math. Ookl. Vol. 30 (1984) No.3, pp. 620-624.

[2] Agrachev, A.A. and Vakhrameev, S.A. The linear on the control systems of constant rank, and the bang-bang conditions for extremal control. Uspeki mat. nauk. Vol. 41 (1986) No.6, pp. 163-164 (in Russian)


[3] Agrachev, A.A. and Vakhrameev, S.A. Morse Theory in Optimal Control and Mathematical Programming. Proc. Int. Soviet-Poland Workshop, Minsk, May 16-19, 1989. Minsk (1989) pp. 7-8.

[4] Goresky, M. and MacPherson, R. Stratified Morse Theory. Springer (1988), New York, etc.

[5] Milnor, J. Morse Theory. Princeton Univ. Press (1963),Princeton, New Jersey, USA.

[6] Palais, R. and Smale, S. A generalized Morse theory. Bull. Amer. Math. Soc. Vol. 79. (1964) pp. 165-171.

[7] Palais, R. Morse Theory on Hilbert manifold. Topology. Vol. 2 (1963) pp. 165-171.

[8] Vakhrameev, S.A. The smooth control systems of constant rank and the linearized systems. Itogi Nauki: Sov. Prohl. Mat. Nov. dost., Vol. 35 (1989) VINITI, Moscow, pp. 135-178. (Engl. transl. to appear in J. Soviet Math.)

[9] Vakhrameev, S.A. Palais-Smale theory for manifolds with corners. I. The case of finite co-dimension. Uspeki mat. nauk. to appear.

[10] Vakhrameev, S.A. Hilbert manifolds with corners of finite codimension and optimal control theory. (in preparation).

A.A. Agrachev, S.A. Vakhrameev All Union Institute of Scientific and Technical Information VINITI Baltijskaya ul. 14 Moscow 125219 USSR

Viability Kernel of Control Systems

J.-P. AUBIN & H. FRANKOWSKA

Abstract. Existence of viable (controlled invariant) solutions of a control system is investigated by using the concept of viability kernel of a set (largest closed controlled invariant subset.) Results are exploited to prove convergence of a modified version of the zero dynamics algorithm to a closed viability domain. This is needed to obtain dynamical feedbacks, i.e., differential equations governing the evolution of viable controls. Among such differential equations, the one leading to heavy solutions (in the sense of heavy trends), governing evolution of controls with minimal velocity is singled out.

Introd uction

Let us consider two finite dimensional vector-spaces: the state space X and the control space Z and let K be a subset of X. We define the control system (/, U) by a set-valued map U : K"-+ Z associating with each state x the set U( x) of feasible controls (in general state-dependent) and by a single-valued map f : Graph(U) 1--4 X describing dynamics of the system:

(i) for almost all t ~ 0, x'(t) = f(x(t),u(t)), u(t) E U(x(t))

Viable solutions in K of the above system are the ones which satisfy x(t) E K for all t ~ o. Recall that the contingent cone to K at x E K is the set TK(X) = {v E X lliminfh-+O+ dK(x + hv)/h = O}, where dK(·) denotes the distance function to K (see [4, Chapter 4].) The regulation map Ru associates with every state x E K the subset of controls u E U (x) such that the corresponding velocity is contingent to K at x:

v x E K, Ru(x) := {u E U(x) I f(x,u) E TK(X)}

The Viability Theorem states in essence that under adequate assumptions, for any Xo E K, there exists a viable solution to the

VIABILITY KERNEL 13

control problem (i) starting at Xo if and only if Ru( x) :f. 0 for any x E K. (This last property enjoyed by K is called controlled invariance and K is called a viability domain.) Furthermore, the viable solutions are regulated by controls satisfying the regulation law

for almost all t 2: 0, u(t) E Ru(x(t))

In this paper, we are looking for a system of differential equations or a differential inclusion governing the evolution of both viable states and controls. We are particularly interested by heavy solutions, which are evolutions where controls evolve with minimal velocity.

We set K = Dom(U). If a viable control u(.) is absolutely continuous, then "differentiating" the regulation law yields

(ii) for almost all t 2: 0, u'(t) E DRu(x(t),u(t))(J(x(t),u(t)))

where DRu(x, v) denotes the contingent derivative of the set-valued map Ru at (x, v) defined by Graph(DRu(x, v)) = TGraph(Ru) (x, v) (see [4, Chapter 5].) This is the second half of the system of differential inclusions we are looking for. Observe that this new differential inclusion has a meaning whenever the state-control pair (x(.), u(·)) remains in the graph of Ru. Fortunately, by the very definition of the contingent derivative, the graph of Ru is a viability domain of the new system (i), (ii). Unfortunately, as soon as viability constraints involve inequalities, there is no hope for the graph of the contingent cone, and thus, for the graph of the regulation map, to be closed, so that, the Viability Theorem cannot apply. However, if the contingent derivative of U obeys a growth condition:

(9) V (x, u) E Graph(U), inf II vII ~ c(lIull + Ilxll + 1) vEDU(x,u)(!(x,u))

then there exists an absolutely continuous solution (x(·), u(.)) of (i) verifying

(iii) for almost all t 2: 0, lIu'(t)11 ~ c(llu(t)1I + Ilx(t)1I + 1)

So, a strategy to overcome the above difficulty is to introduce the a priori growth condition (9) and to look for the viability kernel of (i.e., the largest closed viability domain contained in) Graph(U) of

14 AUBIN AND FRANKOWSKA

the system of differential inclusions (i), (iii). Such a viability kernel does exist (see Theorem 1.6 below.) If we regard this viability kernel as the closed graph of a (possibly empty) set-valued map denoted by R& : X "-+ Z, then we infer that whenever the initial state Xo is chosen in Dom(R&) and the initial control Uo in R&(xo), there exists a solution (x(·), u(·)) to the system (i) such that for almost all t ~ 0

u'(t) E Gc(x(t), u(t)) := DR&(x(t), u(t))(f(x(t), u(t)))

Instead of looking for closed loop control selections of the regulation map Ru, we shall look for selections g(.,.) of the set-valued map G c( ., .) defined above, called dynamical closed-loops.

Naturally, under adequate assumptions, Michael's Theorem implies the existence of a continuous dynamical closed loop. But under the same assumptions, we can take as dynamical closed-loop the minimal selection g0(-'.) defined by IIgO(x,u)1I = minveGc(x,u) IIvll, which, in general, is not continuous, but still yields smooth viable control-state solutions to the system of differential equations

x'(t) = f(x(t), u(t)) & u'(t) = gO(x(t), u(t))

called heavy viable solutions, (heavy in the sense of heavy trends) introduced in [2], [3]. They are the ones for which the control evolves with minimal velocity. In the case of the usual differential inclusion x' E F(x), where the controls are the velocities, they are the solutions wi th minimal acceleration (or maximal inertia.)

Heavy viable solutions obey the inertia principle: "keep the controls constant as long as they provide viable solutions".

Indeed, if zero belongs to Gc(X(tl), U(tl)), then the control will remain equal to u(tt) as long as for t ~ tt, a solution x(·) to the differential equation x'(t) = f(x(t), U(tl)) satisfies the condition 0 E Gc(x(t), U(tl)).

The outline of the paper is as follows. In Section 1 we recall few definitions and provide an upper estimate of the viability kernel. In Section 2 we compute the viability kernel of a descriptor system.

We already mentioned that in general, the graph of the contingent cone map TKC·) is not closed. To compensate the lack of this property, we suggest in Section 3 to replace the contingent cone TKCX) by the subset TJ« x) of directions v E TK( x) such that there exists a

VIABILITY KERNEL 15

measurable function x"(.) bounded by a constant c and satisfying for all t ~ 0, x(t):= x + tv + f~(t - T)X"(T)dT E K. The subset Tk(x), which can be interpreted as global contingent set, enjoys properties that the contingent cone may not have. In particular, the graph of the set-valued map TkO is closed. We next apply these properties to obtain the convergence of a modified version of the Bymes-Isidori zero dynamics algorithml to a closed viability domain (instead of the viability kernel) of an arbitrary closed set. Finally, heavy viable solutions are studied in Section 4.

1 Viability Kernel

Consider a finite dimensional vector space X and denote by B the closed unit ball in X. We say that a set-valued map F : X "-+- X is a Peano map if it is upper semi continuous with compact convex images and with linear growth on Dom(F) =I 0. We refer to [4] for definitions and calculus of set-valued maps.

A subset K C X is called a viability domain of F if for every x E K, F(x) n TK(x) =10.

Theorem 1.1 (Viability Theorem) Consider a Peano map F : X "-+- X and a closed subset K eX. If K is a viability domain, then for all initial state Xo E K, there exists a solution viable in K (on [0, oeD to the differential inclusion

(1) for almost all t ~ 0, x'(t) E F(x(t», x(o) = Xo

Corollary 1.2 Let F : X "-+- X be a closed set-valued map and <p : X X X 1-+ R+ be a continuous function having linear growth and

v x E Dom(F), V v E F(x), DF(x,v)(v) n r,o(x,v)B =I 0

Then, for any Vo E F(xo), there exists a solution x(·) to the inclusion (1) such that x'O is absolutely continuous and x'(O) = Vo.

The Viability Theorem can be applied to study existence of solutions to (1) in the presence of equality constraints:

lwhich is a generalization of the structure algorithm introduced by Silverman in [30] a.nd Ba.sile & Marro in [8] for linear control systems.


Corollary 1.3 Consider a Peano map F : X ~ X and a differentiable map h = (hI, ... , hm ) : X 1-+ R m such that for every x E ker( h) and 1 ~ k ~ m, h~(x) t O. If

'V x E kerCh), :1 y E F(x) verifying 'V 1 ~ k ~ m, h~(x)y = 0

then for all Xo E ker( h) there exists a solution x(·) to {1} defined on [O,oo[ and satisfying the equality constraint h( x(·)) == O.

The proof follows from the Viability Theorem, the closedness of the constraint set ]( = ker( h) and

Proposition 1.4 Consider a map h = (hI, ... , hm) : X 1-+ R m differentiable at x E ]( := kerCh). If for every 1 ~ k ~ m, h~(x) t 0, then TK(X) = ker(h'(x)).

Proof - We claim that for some v E X and for all 1 ~ k ~ m, h~(x)v t O. Indeed there exists VI satisfying h~ (X)Vl t o. Assume that we already know that for some Vs E X and all 1 ~ k ~ s ~ m - 1, h~(x)vs t O. If h~+l (x)vs t 0, then set v s+! = vs' Otherwise consider Vs+! E X such that h~+! (x)vs+! t 0 and let E > 0 be so small that for v s+! = Vs + EVs+! and all 1 ~ k ~ s, h~(x)vs+! t O. This and an induction argument prove our claim.

It is not restrictive to assume that for every 1 ~ k ~ m, h'(x)v> O. Indeed define g = (gl, ... ,gm) : X 1-+ Rm by

if h~(x)v > 0 otherwise

Then for every 1 ~ k ~ m, g~(x)v > O. On the other hand kerCh) = ker(g) and ker g' (x) = ker h' (x). Thus we may replace h by g.

Set IC = {x E XI hex) ~ O}, ](+ = {x E XI h(x) ;::: O}. Then ](_ U](+ = X and ]( = IC n ](+. By [29, Theorem 1] TK(X) = TK_ (x) n TK+ (x). On the other hand [4, Proposition 4.3.7] yields that TK+(X) = {v E XI 'V 1 ~ k ~ m, h~(x)v;::: O}. Replacing h by -h we also have TK_(X) = {v E XI 'V 1 ~ k ~ m, h~(x)v ~ O} and the result follows.

Q.E.D.

When the assumptions of the Viability Theorem are not satisfied we look for the largest closed viability domain contained in ](.

VIABILITY KERNEL 17

Definition 1.5 (Viability Kernel [5]) Let /( be a subset of X and F : X ~ X be a set-valued map. We call the largest closed viability domain of /( contained in /( (when it exists) the viability kernel of /( and denote it by Viabp (/().

Theorem 1.6 ([5]) Consider a Peano map F : X ~ X and a closed set /( eX. Then the viability kernel of /( exists (possibly empty) and is the subset of all initial states Xo such that there exists at least one solution to (1) viable in /( on [0,00[.

Remark - When /( := h-1(0) is defined by equality constraints (where h : X f-+ Y is an observation map), the viability kernel of h-1(0) is related to the global zero dynamics set. See the series of papers [23,10,11,12,14] and also [21, Chapter 6] dealing with the local zero dynamics submanifolds. Under some technical assumptions it can be obtained via the zero dynamics algorithm. 0

Proposition 1.7 Let F : X ~ X be a convex set-valued map and /( C X be a convex compact set. Then Viabp(/() is convex.

It results from CaratModory's Theorem and the following simple

Proposition 1.8 If/( C X is convex, then for all Vi E TK(Xi), Ai ~

0, 1 ~ i ~ k with 2:7=1 Ai = 1, we have 2:7=1 AiVi E TK(2:7=1 AiXi).

A natural question does arise how to obtain the viability kernel. We provide next an algorithm leading to its closed upper estimate.

Let F: X ~ X be a set-valued map and /( C X. Set /(0 = /(,

V n ~ 0, /(n+1 = closure {x E /(n I F(x) n TKn(x) =I- 0} c /(n

Define the closed set leX) = nn>O /(n.

Theorem 1.9 We always have Viabp (/() C /(00' Furthermore if x : [0, T] f-+ /( is a solution to (1) for some T > 0, Xo E /(, then for every t E [O,T], x(t) E /(00' Consequently all solutions to (1) viable in /( remain in the set /(00'

Proof - Consider Xo E Viabp(/() C /(0' Since TViabFK(xo) C TKo(xo), we deduce that Xo E /(1' Thus Viabp(/() C /(1. This and the induction argument end the proof of the first statement. To prove

the second one, consider a solution x(·) as above. Then for almost every t E [0, T], x'(t) E TK(X(t)) n F(x(t)). This and continuity of x(·) yield that for every t E [O,T], x(t) E K I . Applying again the induction argument we end the proof.

Q.E.D.

In general the viability kernel is strictly smaller than Koo. In the next section we show that for linear (singular or not) systems with K equal to a subspace, Koo is the viability kernel. Furthermore the above algorithm converge in a finite number of steps: Koo = Kn-I, where n denotes the dimension of X.

Proposition 1.8 yields

Corollary 1.10 Let F : X ~ X be a convex set-valued map and K C X be a convex set. Then lor every 1 S n S 00, Kn is convex.

The above results can be applied to obtain merely absolutely continuous viable solutions. We investigate next control systems for which there exist viable CI-solutions.

Let us consider a finite dimensional vector space Z and a control system (J, U) defined by a set-valued map U : X ~ Z and a singlevalued map I : Graph(U) ~ X, where X is regarded as the state space, Z the control space, I as a description of the dynamics and U as the a priori feedback. The evolution of a state-control solution (x(·), u(·)) viable in Graph( U) is governed by

(2) x'(t) = I(x(t), u(t)), u(t) E U(x(t))

We say that the pair (x(.), u( .)) is smooth if both x(·) and u(·) are absolutely continuous and that it is <p-smooth ifin addition for almost all t ~ 0, lIu'(t)1I S <p(x(t), u(t)), where <p : X X Z ~ R+ is a given function. In this case x(·) is continuously differentiable whenever the map of dynamics I is continuous.

We obtain smooth solutions to (2) by setting a bound to the growth of the evolution of controls. For that purpose, we associate with this control system and with any nonnegative continuous function Graph( U) 3 (x, u) ~ <p( x, u) with linear growth the system

(3) x'(t) = I(x(t), u(t)), u'(t) E <p(x(t), u(t))B

VIABILITY KERNEL 19

Observe that any solution (x(·),u(·» to (3) (viable in Graph(U» is a cp-smooth solution to the control system (2).

We thus deduce from the Viability Theorem applied to the system (3) on the graph of U the following Regularity Theorem:

Theorem 1.11 Assume that U is closed and f, cp are continuous with linear growth. Then the following two statements are equivalent:

i) For any initial state Xo E Dom(U) and control Uo E U(xo), there exists a cp-smooth state-control solution (x(·), u(·» to the control system (2) starting at (xo, uo).

ii) The set-valued map U satisfies

(4) V(x,u) E Graph(U), DU(x,u)(J(x,u»ncp(x,u)B ::I 0

Proof- Observe that the set-valued map (x,u) ~ {f(x,un x cp( x, u)B is upper semi continuous with compact convex values and linear growth. By the Viability Theorem i) holds true if and only if Graph(U) is a viability domain of this map, i.e., if and only if for all (x, u) E Graph(U), TGraph(u/x, u)n( {f(x, un xcp(x,u)B) ::10. By the very definition of the contmgent derivative of U, this is condition ii) of our theorem.

Q.E.D.

The assumption ii) of the above theorem is very strong, since it requires property (4) to be satisfied for all controls u·of U (x) (so that we have a solution for every initial control chosen in U(xo).) We may very well be content with the existence of a smooth solution for only some initial control in U(xo).

So, we can relax the problem by looking for the largest closed set-valued feedback map whose graph is the set of all initial statecontrols yielding smooth solutions to the control system (2). This amounts to studying the viability kernel of Graph( U) for the map (x, u) ~ {f( x, u)} X cp( x, u)B and leads us to introduce the following

Definition 1.12 (cp-growth regulation map) Denote by R't; the set-valued map whose graph is the viability kernel of Graph(U) for the map (x,u) ~ {f(x,un X cp(x,u)B. We call it the cp-growth regulation map of the control system (2).

We thus deduce from Theorem 1.6 the following result on the existence of smooth viable solutions.

Theorem 1.13 Assume that U is closed and f, <p are continuous with linear growth. Then for any initial state Xo E Dom(RU) and any initial control Uo E RU(xo), there exists a <p-smooth state-control solution (x(.),u(.)) to the control system (2) starting at (xo,uo). Furthermore, for all t 2: 0, u(t) E RtJ(x(t)).

Remark - Observe that the graph of RU is also the viability kernel of Graph( Ru) for the map {f (x, u)} X <pC x, u)B and that the regulation maps RU are increasing with <po 0

The case when <p = ° is particularly interesting, because it determines areas where the evolution of the control is constant:

Corollary 1.14 For every u E U, the subset (RY,)-I(U) is the viability kernel of U-l(u) for the map x 1-+ f(x,u).

Naturally, when (RY, t 1 (u) is reduced to a point, this point is an equilibrium of the system.

2 Viability Kernel of Descriptor systems

Let E, A E .C(Rn, Rm), U be a finite dimensional vector space and BE c(u,Rm). Consider the control system

(5) Ex' = Ax + Bu, u E U

An absolutely continuous map x : [0, T] 1-+ Rn is called a solution to (5) corresponding to a measurable control u : [0, T] 1-+ U if for almost all t E [0, T], Ex'(t) = Ax(t) + Bu(t).

Denote by E, B the images of E and B respectively and for every y E Rm set E-l(y) = {x E Rnl Ex = y}. We introduce the setvalued map F : Rn ~ Rn defined by F(x) = E-I(Ax + B). From [16] we know that solutions to (5) and to the differential inclusion

(6) x' E F(x)

do coincide. To obtain the viability kernel of the above system we introduce a decreasing family of subspaces:

K = Rn, KI = A-I(E + B), ... , Kk+1 = A-1(EKk + B)

VIABILITY KERNEL 21

Since they are subspaces of the n-dimensional space, Q := Kn - 1 = nk>l Kk. Therefore Q is equal to Koo introduced in the Section 1. By-Theorem 1.9, ViabFK C Q. Furthermore A-l(EQ + B) = Q.

Theorem 2.1 The space Q is the viability kernel ofRn for the map F, i.e. Q = ViabF(Rn ). Furthermore for every Xo E Q there exists a COO-solution to (5) defined on [O,oo[ and satisfying x(O) = Xo.

Proof - Indeed fix x E Q = A-1(EQ + B) and y E Q such that Ax E Ey + B. Hence y E E-1(Ax + B) and Q is a viability domain of F. The second statement follows from Theorem 2.3 given below.

Q.E.D.

Remark - This allows to construct the viability kernel of R n.

When K is a subspace of Rn, then it is enough to consider a linear operator C E £(R n, R n) such that K = ker C and the new system

Ex' = Ax + Bu, Cx = 0, u E U

The above result yields its viability kernel, which is equal to ViabF(K). The algorithm still converge in the number of steps::; n - 1. 0

We show next that the implicit system (5) may be reduced to an explicit one. Consider the set-valued map :F : Q ~ Q defined by :F(x) := E-1(Ax + B) n Q = F(x) n Q and let the map V : Q 1-+ Q and the subspace V C Q be defined by

v x E Q, Vx E :F(x), IIVxll = min lIyll, V = E-1(B)nQ = :F(O) 1IE.F(x)

Proposition 2.2 V is a linear operator from Q into itself. Furthermore for every x E Q, :F(x) = Vx + V.

Proof - By Theorem 2.1, for all x E Q, :F(x) "I 0. Since Graph(:F) is a subspace, Vx+ V c :F(x) + V = :F(x)+:F(O) C :F(x). To prove the equality, consider y E :F(x) C Q. Then Ey E Ax + B, EVx E Ax+B and therefore y-V(x) E Q and E(y-Vx) E B. This yields that y - Vx E V. It remains to show that V is linear. The element Vx being the orthogonal projection of zero onto the affine space Vx + V, V(Q) C V.i. Fix x, y E Q. Then

-EVx E -Ax + B, -EVy E -Ay + B, EV(x + y) E Ax + Ay + B


Adding these inclusions we get E(V(x + y) -Vx -Vy) E B. Thus V(x + y) -Vx -Vy E V n Vol and V(x + y) = Vx + Vy. Finally V is homogeneous, because .r(AX) = E-1(AAx + B) n Q = A.r(X).

Q.E.D.

Consider the control system

(7) x'(t) = Vx(t) + vet), vet) E V, x(t) E Q

Theorem 2.3 Solutions to (5) and (7) do coincide. Let B+ denote the orthogonal right inverse of B. Then the map

( ) _ { B+(EVx - Ax) u x - 0

if x E Q if not

is a regulation low of the system (5) such that for every Xo E Q there exists a Coo - solution to the singular system

(8) Ex' = Ax + Bu(x), x(O) = Xo

defined on [O,oo[ and evolving with minimal velocity, the so called slow solution (see [15]). It is unique if and only ifker En Q = {a}.

Proof - Theorems 1.9,2.1 and [16] yield that solutions to (5) and to the differential inclusion

(9) x' E .r(x)

coincide. Applying Proposition 2.2 we end the proof of the first statement. Vx being the element of the minimal norm in .r(x), the solution x(.) to the linear system x' = Vx starting at Xo E Q is a slow solution to (9). It is also a solution to the singular system (8). To prove the latter statement, observe that (8) may be written as:

(10) Ex' = EVx, x(O) = Xo

So the solution of (8) is unique if and only if so is the solution of (10). But this happens whenever zero is the only solution to x' E Vx + ker(E) n Q, starting at zero. Consequently ker(E)nQ = {a}.

Q.E.D.

VIABILITY KERNEL 23

Define recursively .1"1(0) = .1"(0), ... ,.1"k+1(0) = .1"(.1"k(O». The theory of nonsingular linear control systems applied to (7) yields

Corollary 2.4 For every T > 0 the reachable set of (5) at time T from the initial state zero is equal to .1"n(o). Furthermore reO) is the largest subspace such that for all x}, X2 E reO) and T > 0 there exists a COO-solution x(.) to (5) such that x(O) = x}, x(T) = X2.

Observe that results of this section allow to deal with the singular system (5) even in the case when solution corresponding to given control and initial condition is not unique. We investigate next necessary and sufficient conditions for uniqueness.

We say that (5) enjoys the uniqueness if to every measurable control u : [0, T] ~ U, T > 0 and Xo ERn corresponds at most one solution of the singular system (5) starting at Xo.

Theorem 2.5 Consider the subspace P = (A-l E)n(Rn). The following statements are equivalent :

i) The system (5) enjoys the uniqueness property ii) ker E n P = { 0 }

Proof - Observe that i) is equivalent to: x(·) == 0 is the only solution to the linear system Ex' = Ax starting at zero. By Corollary 2.4 applied with B = 0, the reachable set of this linear system from zero at time T > 0 is equal to reO), where .1"(x) = E-1(Ax) n P. Thus (5) enjoys the uniqueness property if and only if {O} = .1"n(o) :::> .1"(0) = ker En P. Hence i) implies ii). Conversely assume that ker En P = {O}. Then .1"(0) = {O} and therefore reO) = {O}.

Q.E.D

Remark - The above implies in particular the (well known) uniqueness of solution in the case when (E,A) is a regular pencil, i.e. when m = n and det(sE - A) is not identically zero. Indeed in this case we can change the basis in such way that (5) decomposes into two subsystems

u E U

where (x}, X2) E Rk X RS, k + s = n and the matrix N is nilpotent. So the uniqueness would follow if the control system Nx~ =


X2 + B2u, u E U enjoys the uniqueness. But in this case P (Id-1 N)8(R8) = N8(R8) = {O}.

3 A Viability Domain Algorithm

As we already mentioned in Section 1, in general the set Koo is larger than the viability kernel. There exist two reasons for this phenomenon. The first comes from the lack of regularity of the contingent cone: the map TK(') in general is not closed. The second one comes from the calculus of tangent cones: contingent cone to the intersection of a decreasing family of sets may be smaller than the intersection of contingent cones. This is why we study in this section some regular subsets of contingent cones.

Definition 3.1 Let K c X be closed and c > 0 be a positive constant. Denote by Tk( x) the subset of elements v E TK( x) such that there exists a measurable function x"(.) bounded by c and satisfying

V t ~ 0, x + tv + fo\t - r)x"(r)dr E K

We introduce the Peano map F from X x X to itself defined by F(x, v) := {v} x cB. The map t 1-+ x(t) := xo+tvo+ I~(t-r)x"(r)dr where IIx"(r)1I :5 c is a solution to the differential inclusion x" E cB and (x(·), x' (.)) is a solution to the differential inclusion

(x'(t),v'(t)) E F(x(t),v(t)), x(O) = Xo, v(O) = Vo

We remark at once that Graph(Tk(')) is the viability kernel of Graph(TK(')) for the set-valued map (x,v) ~ {v} x cB and observe that 0 E Tk(x) for all x E K.

Example It is not difficult to verify that for K := [0,1]

{R+ifX=O

TK(X) = R if x E ]0,1[ R_ if x = 1

and the global contingent set is equal to

(11) V x E [0,1], Tk(x) = [-v'2cx, v'2cx]

VIABILITY KERNEL 25

Proposition 3.2 The set-valued map x ~ Tk(x) is closed. Let KU := lim sUPn-+oo Kn denote the upper limit of a sequence of closed subsets Kn. Then the upper limit of Graph(TkJ·» is contained in Graph(TK.(·))·

Proof- Consider the set-valued map F(x,v) = {v} x cB and an element (x,v) of the upper limit of Graph(Tkn (·». Then (x,v) is the limit of a subsequence (xn", vn,,) E ViabF(Graph(TKn " (.))), so that there exist solutions xn" (.) to the differential inclusions

viable in Kn". Taking a subsequence and keeping the same notations we may assume, using [4, Theorem 7.2.2] that xn ,,(·) converge uniformly on compact sets to some continuously differentiable map x(·) satisfying x(O) = x, x'(O) = v and x" E cB. Since xn,,(t) E Kn" for all t ~ 0, x([O, ooD c KU. Therefore, for all t ~ 0, x'(t) E TKI(x(t» and the pair (x(·),x'(·)) is a solution which is viable in Graph (TKI (.)). Consequently, v E TKa(x).

Q.E.D.

Obviously, if CI ~ C2, then TIl c TIt. We also observe that

v x E K, V v E Tk(x), DTk(x, v)(v) n cB =I 0

Proposition 3.3 Let Y be a finite dimensional space, A E C(X, Y) be a linear operator and K eX, M c Y be closed subsets. Then

V x E K, A(Tk(x)) C T~~~~(Ax)

and thus for all x E A-I(M), TA-1(M)(x) c A-I (T~AII(Ax»). Furthermore, if A is surjective, then there exists p > 0 such that

Proof - Let v E Tk(x). Then there exists a solution x(·) to x" E cB viable in K and satisfying (x(O),x'(O)) = (x,v). Then yet) := A(x(t» is solution to the differential inclusion y" E cA(B) C cIlAIIB, viable in A(K), such that (y(O), y'(O)) = (Ax, Av). The


second statement follows by taking K := A-l(M). To prove the last one, consider w E TM(Ax) and a viable in M map

y(t) := y + tw + lot (t - r)y"(r)dr, \\ y"(r) \\ ~ c

Since A is surjective, there exist a constant p > 0 and solutions x and v to the equations Ax = y and Av = w satisfying inequalities IIxll ~ pllyll and IIvll ~ pllwll. Furthermore, by [4, Theorem 8.2.9] there exists a measurable solution z(.) to the equation Az( r) = y" (r) satisfying IIz(r)1I ~ plly"(r)1I ~ pc. Then x(t) := x + tv + fJ(t -r )z( r )dr is a solution to the differential inclusion IIx"lI ~ pc which is viable in A -l(M).

Q.E.D.

Consider a closed subset K C X and a set-valued map F: X ~ X. Define the set K 00 as in Section 1. We already know that ViabF(K) C Koo. In general the equality does not hold. For instance for K:= [0,1] X R and the set-valued map F(x,v):= {v} X cB we have Koo = K. On the other hand the viability kernel is the graph of TkO given by (11).

Thanks to Proposition 3.2, by replacing contingent cones TK( x) by the subsets Tk(x) in this algorithm, we can prove that such modified version converges to a closed viability domain.

Let us fix c > 0 and set K8 := K. We introduce the set-valued map Rg(x):= F(x)nTj(c(x) and set Kf:= Dom(Rg). If the subsets

o Kf have been defined up to n, we set

R~(x) := F(x) n TK:;(x), K~+1:= Dom(R~)

Proposition 3.4 Let K be closed and F : K ~ X be upper semicontinuous with compact values. Then K~ := ni>O Kf is a closed viability domain of F: -

Consequently if K is compact, then either Kf is empty for some i or K~ is a nonempty closed viability domain of F.

VIABILITY KERNEL 27

Proof - We claim that for every i ~ 1 the subset Kf is closed. Indeed if Xn E Kf converge to x, any sequence of elements Vn E F(xn ) n Tic? (xn ) lying in a compact set has a subsequence

.-1 converging to some v. Since the graphs of F and Tk!' (.) are closed,

.-1 we infer that v E F(x) n Tk!' (x), i.e., that x belongs to Kf. Thus

.-1 the sets Kf's form a decreasing sequence of closed subsets of K. Let x be chosen in K~ and Vn E F(x) n Tkc(x). Since the vn's remain .. in the compact subset F(x), a subsequence of Vn converges to some v. But (x, Vn) E Graph(Tkc(')) and from Proposition 3.2 we deduce .. that (x, v) belongs to the graph of Tkc •

00

4 Heavy Viable Solutions

Consider a control system (I, U) which has a nontrivial <p-growth regulation map RU for some continuous cp ~ 0 with linear growth.

Proposition 4.1 The cp-smooth state-control solutions (x(.),u(.)) to the control system (2) are also solutions to the system

(12) x'(t) = f(x(t), u(t)), u'(t) E DRU(x(t), u(t))(I(x(t), u(t)))

Conversely every solution (x(·),u(·)) to (12) is a smooth solution to (2).

Proof - Indeed, the absolutely continuous function (x(·), u(·)) takes its values into Graph(RU)' Thus for almost all t ~ 0

(x'(t), u'(t)) E TGraph(R&) (x(t), u(t)) = Graph(DRU(x(t), u(t)))

We then replace x'(t) by f(x(t), u(t)). The converse holds true because (12) makes sense only if (x(t), u(t)) E Graph(RU) c Graph(U).

Q.E.D.

The question arises whether we can construct selection procedures of the control component of the system (12). It is convenient for this purpose to introduce the following definition.

Definition 4.2 (Dynamical Closed Loop) A selection 9 from the contingent derivative of the <p-regulation map R'{, in the direction f defined by

(13) V (x, u) E Graph(R'f,), g(x, u) E DR'f,(x, u)(J(x, u»

is called a dynamical closed loop. The system of differential equations

(14) x'(t) = f(x(t),u(t)), u'(t) = g(x(t),u(t))

is called the associated closed loop differential system.

Clearly every solution to (14) is also a solution to (12). Therefore, a dynamical closed loop being given, solutions to system of ordinary differential equations (14) (if any) are smooth state-control solutions of the initial control problem (2). Such solutions do exist when 9 is continuous (and if such is the case, they will be continuously differentiable.) But they may also exist when 9 is no longer continuous (see [15,2,3,7]). This is the case for instance when g( x, u) is the element of minimal norm in DR'f,(x, u)(J(x, u».

Definition 4.3 Consider a continuous function 1.1' : Graph(U) 1-+

R+ with linear growth. We say that a control system (I, U) is 1.1'

dynamically regular if

( J the <p-regulation map R'{, is sleek and 3 , > 0 such that 15'l V v E X, sup(x,u)eGraph(R&) infweDR&(x,u)(v) IIwll :s; ,lIvll

Assumption (15) imply that the map (x,u,v) "'" DR'f,(x,u,v) is lower semi continuous on Graph(R'f,) X X (see [4, Proposition 5.1.6] for more details.) Thus the existence of continuously differentiable state-control solutions follows from Michael's Theorem (see [1].)

Theorem 4.4 Assume that U is closed and that f, 1.1' are continuous and have linear growth. If the control system (J, U) is <p-dynamically regular, then there exists a continuous dynamical closed loop g. The associated closed loop differential system (14) regulates continuously differentiable state-control solutions to (2) defined on [0,00[.

Since we do not know constructive ways to built continuous dynamical closed loops, we investigate some explicit dynamical closed loops such that the differential system (14) does possess solutions.

VIABILITY KERNEL 29

Definition 4.5 (Heavy Viable Solutions) Denote by g:( x, u) the element of minimal norm in DR'ij(x, u)(I(x, u)). Solutions to the associated closed loop differential system

x'(t) = f(x(t), u(t)), u'(t) = g:(x(t), u(t))

are called <p-heavy viable solutions to the control system (I, U).

Clearly for all (x,u), IIg:(x,u)II ~ <p(x,u).

Theorem 4.6 (Heavy Viable Solutions) Let us assume that U is closed and that f, <p are continuous and have linear growth. If the control system (I, U) is <p-dynamically regular, then for any initial state-control pair (xo, uo) in Graph(R,{;}, there exists a <p-heavy viable solution to the control system (2).

The reason why this theorem holds true is that the minimal selection is obtained through the selection procedure of a set-valued map we are about to describe.

Let F : X "-+- Y be a set-valued map with closed convex values and let m(F(x)) E F(x) denote the element of minimal norm in F(x). Consider the set-valued map SF: X"-+- Y defined by

(16) v x E X, SF(X) = IIm(F(x)) II B

where B denotes the closed unit ball in Y. Observe that the graph of the minimal selection is equal to Graph(F) n Graph(SF). Therefore, the minimal selection is obtained through a general selection procedure defined as follows (see [2,3]):

Definition 4.7 (Selection Procedure) A selection procedure of a set-valued map F : X"-+- Y is a closed set-valued map SF: X"-+- Y satisfying SF(X) n F(x) =J 0 for all x E Dom(F).

Another example of a selection procedure comes from optimization:

Proposition 4.8 Consider a lower semicontinuous set-valued map F : X "-+- Y with compact nonempty values. Let V : Graph( F) 1-+ R be continuous. Then the set-valued map SF defined by:

SF(X) := {Y E Y I V(x,y) ~ min V(X,y')} y/EF(:z:)

is a selection procedure of F.

To simplify notations, set

Gcp(x, u) := DR'{;(x, U)(f(X, u))

Theorem 4.9 We posit the assumptions of Theorem 1.13. Let SG<p : Graph( R'{;) "-+ X be a selection procedure with convex values and linear growth of the set-valued map Gcp. Then, for any initial state (xo, uo) E Graph( R'{; ), there exists a state-control solution to the associated closed loop system

(17) x' = f(x, u), u' E Gcp(x, u) n SG<p(x, u)

defined on [O,oo[ and starting at (xo,uo). In particular, if for any (x, u) E Graph( R'{;), the intersection

Gcp(x,u) n SG<p(x,u) = {s(DR'{;(x,u)(f(x,u)))}

is a singleton, then there exists a state-control solution defined on [O,oo[ and starting at (xo, uo) to the associated closed loop system

x'(t) = f(x(t), u(t)) , u'(t) = s (DR'{;(x(t), u( t))(f(x(t), u(t))))

Proof - Consider the system

(18) x' = f(x,u), u' E SG<p(x,u), (x,u) E Graph(R'{;)

Since the selection procedure SG<p has a closed graph and convex values, the right-hand side is an upper semicontinuous set-valued map with nonempty compact convex images and with linear growth. On the other hand Graph( R'{;) is a viability domain of the map {f( x, u)} X SG<p (x, u). Therefore, the Viability Theorem can be applied. For any initial state-control (xo, uo) E Graph(R,{;), there exists a solution (x(·),u(·)) to (18) which is viable in Graph(R'{;). Consequently, for almost all t ~ 0, the pair (x'(t), u'(t)) belongs to the contingent cone to the graph of R'{; at (x(t), u(t)), which is the graph of the contingent derivative DR'{;(x(t), u(t)). In other words, for almost all t ~ 0, u'(t) E Gcp(x(t),u(t)). Hence, the state-control pair (x(.),u(.)) is a solution to (17).

Q.E.D. Proof (of Theorem 4.6) - By [4, Lemma 9.3.1] the map

(x, u) 1-+ Ilg~(x, u)11 is upper semicontinuous. It has a linear growth

on Graph(R'{;). Thus the set-valued map (x,u) "-+ Ilg~(x,u)1I B is a selection procedure satisfying the assumptions of Theorem 4.9.

VIABILITY KERNEL 31

References

[1] AUBIN J.-P. & CELLINA A. (1984) DIFFER.ENTIAL INCL USIONS, Springer-Verlag

[2] AUBIN J.-P. & FRANKOWSKA H. (1984) Trajectoires lourdes de systemes contro/es, Comptes-Rendus de l'Academie des Sciences, PARIS, Serie 1, 298, 521-524

[3] AUBIN J.-P. & FRANKOWSKA H. (1985) Heavy viable trajectories of controlled systems, Annales de l'Institut Henri Poincare, Analyse Non Lineaire, 2, 371-395

[4] AUBIN J.-P. & FRANKOWSKA H. (1990) SET-VALUED ANALYSIS, Systems and Control: Foundations and Applications, Birkhauser, Boston, Basel

[5] AUBIN J.-P. (1989) Smallest Lyapunov functions of differential inclusions, J. Diff. & Integral Eqs., 2

[6] AUBIN J.-P. (1990) A survey of viability theory, SIAM J. Control & Optim.

[7] AUBIN J.-P. (to appear) VIABILITY THEOR.Y,

[8] BASILE G. & MARRO G. (1969) Controlled and conditional invariant subspaces in linear system theory, J. Optim. Theory Appl., 3, 296-315

[9] BYRNES C.1. & ANDERSON B.D.O. (1984) Output feedback and generic stabilizability , SIAM J. Control & Optim., 22, 362-379

[10] BYRNES C. & ISIDORI A. (to appear) The analysis and design of nonlinear feedback systems. I. Zero dynamics and global normal forms,

[11] BYRNES C. & ISIDORI A. (to appear) The analysis and design of nonlinear feedback systems. II. Global stabilization of minimum phase systems,

[12] BYRNES C. & ISIDORI A. (to appear) Feedback design from the zero dynamics point of view,


[13] BYRNES C. & ISIDORI A. (to appear) Output regulation of nonlinear systems,

[14] BYRNES C. & ISIDORI A. (this volume)

[15] FALCONE M. & SAINT-PIERRE P. (1987) Slow and quasi-slow solutions of differential inclusions, J. Nonlin. Anal., T.M.A., 3,367-377

[16] FRANKOWSKA H. (1990) On controllability and observability of implicit systems, Syst. & Control Letters

[17] FRANKOWSKA H. (1990) Some inverse mapping theorems, Annales de l'Institut Henri Poincare, Analyse Non Lineaire,3

[18] FRANKOWSKA H. (to appear) SET-VALUED ANALYSIS AND CONTROL THEORY

[19] HADDAD G. (1981) Monotone viable trajectories for functional differential inclusions, J. Diff. Eqs., 42, 1-24

[20] HADDAD G. (1981) Monotone trajectories of differential inclusions with memory, Israel J. Maths., 39, 38-100

[21] ISIDORI A. (1985) NONLINEAR CONTROL SYSTEMS: AN INTRODUCTION, Springer-Verlag Lecture Notes in Control and Information Sciences, Vol. 72

[22] KRENER A. & ISIDORI A. (1980) Nonlinear zero distributions, 19th IEEE Conf. Decision and Control

[23] KRENER A. J. & ISIDORI A. (1983) Linearization by output injection and nonlinear observers, Syst. & Control Letters, 3, 47-52

[24] KURZHANSKI A. B. & FILIPPOVA T. F. (1986) On viable solutions for uncertain systems, IIASA WP

[25] KURZHANSKI A. B. (1985) On the analytical description of the viable solutions of a controlled system, U spekhi Mat. Nauk, 4

VIABILITY KERNEL 33

[26] KURZHANSKI A. B. (1986) On the analytical properties of viability tubes of trajectories of differential systems, Doklady Acad. Nauk SSSR, 287, 1047-1050

[27] MARRO G. (1975) FONDAMENTI DI TEO RIA DEI SISTEMI, Patron Editore

[28] MONACO S. & NORMAND-CYROT D. (1988) Zero dynamics of sampled linear systems, Syst. & Control Letters

[29] QUINCAMPOIX M. (to appear) Frontier-es de domaines d'invariance et de viabiliti pour des inclusions differentielles avec contraintes, Comptes-Rendus de l'Academie des Sciences, PARIS, Serie 1,

[30] SILVERMAN L. M. (1969) Inversion of multivariable linear systems, IEEE Trans. Automatic Control, 14, 270-276

[31] WONHAM W.M. (1985) LINEAR MULTIVARIABLE CONTROL. A GEOMETRIC ApPROACH, Springer-Verlag

Jean-Pierre Aubin and Helene Frankowska CEREMADE, Universite Paris-Dauphine 75775 Paris, FRANCE

New Methods for Shaping the Response of a Nonlinear System

CHRlSTOPHER I. BYRNES, ALBERTO ISIDORI

1. Introduction

Shaping the response of a control system has long been a central problem in the analysis and design of feedback systems. The widespread use of both frequency domain techniques and state-space methods is at least in part due to the relative ease and intuitive content of these methods in addressing problems such as asymptotic tracking and disturbance attenuation for linear systems. Recently, a combination of methods drawn from geometric nonlinear control theory and from nonlinear dynamics was developed to give an admittedly unanticipated local solution to the nonlinear regulator problem, yielding necessary and sufficient conditions for nonlinear regulation for the class of detectable and stabilizable nonlinear systems ([1], [2]). In section 2, we state the basic nonlinear regulator problem and give conditions for solvability of the problem in terms of the solvability of a system of nonlinear partial differential equations. In the linear case, these "regulator equations" coincide with the linear equations derived by Francis [3] in his rather complete treatment of the linear multi variable regulator problem. The derivation of the nonlinear regulator equations and the consequent design of a, nonlinear controller repose on two essential problems: feedback stabilization for nonlinear systems, a research area currently enjoying intense activity and success, and an analysis of the "steady-state response" of a nonlinear system to a driving input. In section 3, we sketch our solution to the problem of existence of such a steady-state response using center manifold methods, from which the regulator equations can be derived mutatis mutandis. In section 4, we extend this analysis using the Hopf Bifurcation Theorem to problems involving a broader class of either driving signals or signals to be tracked, proving in particular that the nonlinear regulator scheme can also provide controllers which achieve asymptotic tracking, with internal stability, of stable limit cycles with sufficiently small amplitude. The existence of such a result was suggested to us by J. Grizzle, and also posed as a problem independently by H. J. Sussmann, both of whom we thank for their helpful remarks. In section 5, we survey the geometric existence theory for the regulator equations,

Research partially supported by grants from the AFOSR, the NSF and MURST

SHAPING THE RESPONSE OF A NONLINEAR SYSTEM 35

giving necessary and sufficient conditions for nonlinear regulation in terms of the zero dynamics of the system to be controlled and of the augmented system including the exogeneous system producing reference outputs and disturbances. This characterization can also be cast in terms of the solvability of a more intrinsic "Lyapunov POE", for which stable, unstable and center manifold theory provides a local existence theory. In the linear case, this Lyapunov equation was studied by Hautus who proved that well-posedness is equivalent to the assertion that no system transmission zero be a natural frequency of the exosystem, a rather intuitive frequency domain criterion. We also note that for an interesting class of problems, the controller designed by this method has the familiar form of a feedback and a feed forward term, both of which depend on the "off-line" solution of the Lyapunov POE. This structure is somewhat reminiscent of solutions obtained in the linear quadratic regulator problem, a fact which motivated the recent solution [4] of a class of nonlinear optimal control problems by feedback laws which involve the "off-line" solution of a "Riccati POE." This is illustrated in Section 6 for a simple quadratic performance measure for both the finite-time and semi-infinite horizon problems.

2. Output Regulation of Nonlinear Systems

An important problem in control theory is that of controlling a fixed plant in order to have its output tracking (or rejecting) reference (or disturbance) signals produced by some external generator (the exosystem). In a general muItivariable nonlinear setting, the problem in question can be formulated in the following terms. Consider a system modeled by equations of the form

(2.1)

:i:=!(:c,w,u)

tV = s(w) e = h(:c, w)

in which :c is the plant state vector, u the control input, w a vector of exogenous signals, e the output error, the difference between the actual plant output and its desired reference behavior, and !(.,., .), s(·), h(.,·) are smooth functions, defined in a neighborhood of a reference (equilibrium) point, namely (:c,w,u) = (0,0,0). The control action to (2.1) is to be provided by a dynamic compensator, which processes the output error e, generates the appropriate control input u, and is modeled by equations of the form

(2.2) i = 11(:, e) u = O(z)

36 BYRNES AND ISIDORI

in which '7{-, .), 0(-) are C le functions (for some integer k ~ 2) defined in a neighborhood of the equilibrium (z, e) = (0,0).

The purpose of the control is twofold: closed-loop stability and output regulation. More explicitly, we require

(S) Closed-loop stability: The interconnection of (2.1) and (2.2) with w = 0, i.e. the closed loop system

(2.3) x = f(x, O,O(z»

z = '7( z , h( x , 0» has a (locally) exponentially stable equilibrium at (x, z) = (0,0).

(R) Output regulation: For each initial condition (x(O), z(O), w(O» in a neighborhood of (0,0,0), the response x(t), z(t), w(t) of the closed loop system (2.1)-(2.2) satisfies

lim e(t) = 0, t-oo

Note that the composition (2.1)-(2.2) has the form

(2.4) x = f(x,w,O(z» z = '7(z,h(x,w»

tV = s(w)

i.e., that of a locally exponentially stable closed-loop system driven by the exosystem tV = s(w). Thus, if the latter is Lyapunov stable, then the equilibrium (x,z,w) = (0,0,0) of (2.4) is also Lyapunov stable and, for sufficiently small initial conditions, its trajectories remain in a neighborhood of (0, 0, 0) for all t ~ 0.

As in the case of linear systems, in order to identify appropriate necesary and sufficient conditions for the solvability of the output regulator problem, it is very reasonable to assume that the exosystem does not contain any asymptotically stable "subsystem". If this were the case, in fact, at least for some subset of initial states w(O), the output regulation property (R) would be a straightforward implication of the stability property (8). The property that tV = s(w) does not contain any asymptotically stable subsystem, together with the already assumed stability of its equilibrium w = 0, can be, for instance, given the form of the following hypothesis.

(H) Neutral Stability: The exosystem has a Lyapunov stable equilibrium at w = ° and, for some neighborhood W of w = 0, the set n of all


w-limit points of all trajectories which are initialized in W is such that nnw is dense in W.

Theorem 2.1 ([3}) Assume (H). Assume the plant is locally exponentially stabilizable and the composition plant - exosystem is exponentially detectable. Then, the output regulation problem can be solved if and only if the following pair of equations

(2.5) (hr ow s(w) = f(11'(w), W, c(w))

0=h(11'(w),w)

are solved by some C k mappings 11'(w),c(w) (satisfying 11'(0) = 0, c(O) = 0).

Remark 2.2 In the case of a linear system

(2.6) i: = Ax + Bu + Pw

w=Sw

e = Cx+Qw

this problem has been addressed and solved by several authors (among whom we mention - for instance - Davison [5], Francis and Wonham [6], Francis [5]) in the years 1972-77. In [5], Francis has shown that-if the plant is stabilizable and the composition plant-exosystem is detectable-a compensator of the form (2.2) yielding closed-loop stability and output regulation can be found if the pair of linear matrix equations

(2.5') lIS = All + Br + P

CII+Q = 0

is solvable, in the unknowns II and r. The solvability of this equation becomes also a necessary condition for the solvability of the linear regulator problem provided the exosystem is antistable, i.e. if the eigenvalues of the matrix S lie in the closed right-half complex plane. Of course, in the linear case (2.5) reduces to (2.5)' so that the partial differential equation in (2.5) is a nonlinear form of the Lyapunov equation of linear control. We will discuss the Lyapunov PDE further in sections 5-6.

Remark 2.3 Theorem 2.1 addresses the problem of output regulation using error measurement. In the case where the state of the system to be controlled and of the exosystem are available, there exists a similar result (see [3], [4]); viz., assuming (H) and the local exponential stabilizability of the plant, a necessary and sufficient condition for solvability of the output

regulation problem is the solvability of the regulator equations (2.5). The proofs of these two results are of course similar, reposing on the development of the notion of the "steady-state response" of a stable nonlinear system, to ("Lyapunov stable") driving signals. In section 3 we show how this notion may be derived naturally from the center manifold theorem. The case of error measurement also uses a version of the "internal model principle" ([6]), in an essential way.

Remark 2.4 The proof of the sufficiency of (3.1) is constructive. In fact, in [1] we have shown that a solution of the regulator problem can be obtained by choosing

(2.7)

where c(.) and 11'(-) are solutions of (2.5), k(·) is any feedback which exponentially stabilizes the equilibrium x = 0 of x = !( x, 0, k( x)), and Zl, Z2 are (exponential) asymptotic estimates of x, w, the state vector of the composition plant-exosystem, provided by an exponential detector

Zl = ryl(zl,z2,e) Z2 = ry2(zl,z2,e)

In the case of full state feedback we may take the estimates as identities Zl = x, Z2 = w. In particular, (2.7) has the form of a nonlinear feedback law involving a nonlinear gain, 11'( w), which may be determined "off-line" by solving a Lyapunov PDE-a situation quite reminiscent of linear quadratic regulator theory (see section 6).

3. Steady State Analysis for Nonlinear Systems

There are two essential ingredients in the analysis and design of the nonlinear regulators described in section 2: Feedback stabilization, which is comparatively well developed, and the analysis of the "steady-state response" of an asymptotically stable nonlinear system. We will first concentrate on the steady-state response to a sinusoid, 6sinwt. In this case, for 6 < < 00 the existence of a periodic trajectory has been examined classically by the method of harmonic balance, which can sometimes be enhanced with the use of describing function methods to determine a periodic orbit which is known to exist, and in some cases by averaging methods. In general, however, consider the controlled system

(3.1) x = !(x, u) u(t) = 6sinwt

where it is assumed that x = 0 is a (locally) exponentially stable equilibrium for the unforced system

(3.2) x=/(x,O)

There are two principal questions about (3.1) which can be very simply resolved using center manifold methods (see e.g. [7]):

(1) Does there exist Xo such that the traj~ctory of (3.1) initialized at x(O) = Xo is periodic, x(t) = x(t + T), of period T = 27r/w?

(2) Is {x(t)} an asymptotically stable periodic orbit?

We can rewrite the time-varying system as an augmented stationa:y system

x = /(x, r(6)) x E lR,n

(3.3) e = S6, 6 = [~~] E lR,2

whereS= [~ ~], 6 0 = [~] and r(6) =62 .

Thus, we have realized (3.1) as an exponentially stable system driven by

the output of a harmonic oscillator. We first note that ( ~~) = (~) is an

equilibrium for (3.3) and that 6 = 0 is an invariant sub manifold of (3.3) on which the restricted flow takes the form (3.2). Since the first approximation to (3.3) has n eigenvalues in C- and 2 eigenvalues on the imaginary axis, we see (locally) that

(1) 6 = 0 is the stable manifold for (3.3); (2) by the center manifold theorem there exists a complementary invari-

ant surface E defined by x = 7r( 6). Being the graph of a function of 6, E supports a totally periodic flow, i.e. for (x(O), 6(0)) E E

(x(t), 6(t)) = (7r(6(t)), 6(t)) = (7r(6(t + T)), 6(t + T))= (x(t + T), 6(t + T))

where T = 27r/w. Therefore, Xo = 7r ([~]) is an initial condition de

termining a periodic trajectory x(t), provided 6 « 00. Moreover, by the principle of asymptotic phase all trajectories x(t) initialized near 0 will tend to x(t) exponentially,

IIx(t) - x(t)1I < ce-6t , c,6 > O.

N.B. To say E=graph (7r) is invariant is to say that 7r satisfies the (Lyapunov) partial differential equation

(3.4) 07r 06 (S6) = /(7r(6), r(6))

7r(0) = 0


which should be compared to the regulator equations (2.5). Remark 3.1 A similar analysis has been used in [7] to define and analyze

the steady-state response of the nonlinear system (3.1), with (3.2) locally exponentially stable , to arbitrary signals produced by a Lyapunov stable exosystem for which the first approximation has spectrum lying entirely on the imaginary axis. We illustrate this in an example.

Example 3.2 Consider the periodically forced system

(3.5)

or equivalently

(3.6)

x = -x + lOcos t sin2t

x = -x + l(k"'IW~ WI = W2

W2 = -WI

A center manifold for (3.6) is the graph of the function

(3.7)

and is depicted in Figure 3.1, together with the corresponding periodic orbit on E . Figure 3.2 illustrates the principle of asymptotic phase. Figure 3.3 shows an asymptotically periodic trajectory.

Figure 3.1


Figure 3.2 Figure 3.3

4. Asymptotic Tracking of Stable Limit Cycles

The results discussed in Section 2 prove the existence of a local solution to the nonlinear output regulation problem, which is defined on a neighborhood of a given equilibrium point of the system (2.1). Indeed, a local version of center manifold theorem was used in the proof in order to show that the existence of a solution 7r(w),c(w) to the regulator equations (2.7) implies the existence of a solution of the output regulation problem. This sufficiency argument reposes on the fact that the density hypothesis (H) on the exosystem implies that the Jacobian ~Iw=o has its spectrum lying entirely on the imaginary axis, a fact which itself follows from the center manifold theorem. In fact, the full density hypothesis is only used to determine necessity of the solvability of the regulator equations.

This observation leads to the possibility of achieving tracking for signals generated by a more general class of exosystems, using the regulator theory developed in Section 2. In particular, one may wish to consider wider classes of exosystems which include, for instance, nonlinear oscillators with isolated, stable periodic trajectories. One typical example is a van der Pol


oscillator with a stable limit cycle, namely the system

Wl = W2

W2 = -Wl + c(W2 - w~)

with c > O. This system cannot be immediately treated by means of the methods

illustrated in Section 2, because its Jacobian matrix

has eigenvalues in the right-half plane. However, one can consider the parameter c as an additional state variable satisfying

i=O

thus obtaining, for the augmented exosystem, a Jacobian matrix with all eigenvalues on the imaginary axis at the augmented equilibrium (w, c) = (0,0).

Even though such an exosystem does not satisfy the density hypothesis (H), much of our previous analysis is still valid. If a compensator (2.2) satisfies the stability requirement (8), the closed loop system (2.4) has a center manifold M near (x, z, w) = (0,0,0), which is the graph of a C le

mapping W -+ (1r(w), u(w)). Since this manifold is locally attractive, if the second regulator equation is satisfied and if the amplitude of the stable limit cycle goes to zero with c, the output regulation requirement (R) is also achieved for some open set of initial data provided c is sufficiently small. In this sense, we can track sufficiently small amplitude, stable limit cycles.

Using a similar argument, in conjunction with the Hopf Bifurcation Theorem, we can solve output regulation problems for more general exogeneous systems having a possibly stable limit cycle with sufficiently small amplitude.

Theorem 4.1. Suppose the system (2.1) is exponentially stabilizable and consider the smooth, parameter-dependent i-dimensional exosystem

( 4.1) W = s(w,c), s(O,c) = O.

Suppose (4.1) satisfies the standard conditions (e.g. [8], p. 81) for a bifurcation to a limit cycle:

(1) :~ (0, 0) has i- 2 eigenvalues lying in C-;

(2) ;~(O,c) has 2 eigenvalues A(c), .\(c) for which ReA(c) ~ 0 for c ~ 0

and for which ReA(O) = OJ and

(3) oReA(c) I _ 0 oc e_O>

Then, a sufficient condition for the existence of an co > 0 and a solution for each c, 0 < c < co, of the output regulation problem, is the existence of function z = 1r(w), U = c(w) satisfying the regulator equations (2.5). Moreover, if the standard conditions for stability of the bifurcation limit cycle we(t) are satisfied then, for all Zo near 0, the state of the closed-loop system converges to the periodic reference signal 1r( We (t)).

5. A Geometric Existence Theory for the Regulator Equations

We now turn to the question of existence of a solution to the regulator equations (2.5). We also show, for the problem of asymptotic tracking with stability, that in a certain normal form the regulator equations take an especially appealing form, giving a nonlinear enhancement of both linear state-space and frequency domain feedback design.

For scalar input-output linear systems which are asymptotically stable, there is a classical design procedure for tracking a sinusoid, sin wt. Namely, one should drive the system by a sinusoid of the same frequency which will result in a steady state periodic response having the same frequency, but whose phase and amplitude must be adjusted to match sin wt. This procedure works, provided iw is not a zero of the system transfer function. This simple criteria also has a multivariable linear counterpart. Indeed, the equations (2.5)' lend themselves to the following immediate geometric interpretation which plays a central role in our nonlinear analysis and design: The graph of the linear mapping

z = IIw,

i.e. the vector space Z = {(z,w): z = IIw},

is a controlled invariant subspace of (2.6), and is contained in the kernel of the output map.

Merging (2.5a), and (2.5b), into a single equation yields a linear matrix equations of the form

(5.1) [~ ~] x- [~ ~] XS=- [~] where X = col(II, f). Equations of this type have been studied by Bautus in [9]. In the specific case of equations (5.1), he showed that a solution


exists for each value of (P, Q) if and only if the rank of the matrix

is equal to the number of its rows for each A which is an eigenvalue of S. The matrix R(A) is actually the well-known Rosenbrock's system matrix

of the plant. The property that this matrix has a rank equal to the number of its rows at some A E C corresponds exactly to the property that the plant is right-invertible, and the value of A at which its rank may drop are exactly the transmission zeros of the system. Thus, Hautus' results say that the regulator equations (2.5)' are solvable for each (P, Q) if and only if the plant is right-invertible and none of its transmission zeros coincide with an eigenvalue of the exosystem.

We now show how the results of Hautus extend to the nonlinear regulator problem, in a form involving a partial differential equation whose existence theory can be derived from the recent theory of nonlinear zeroes [10], [11]. The theory of nonlinear zeros has been developed in the last decade for systems which are affine in the control (see e.g. [10]-[16]) and for nonaffine systems, see [17]).

Intuitively, the nonlinear analogue of transmission zeros identifies the latter in terms of the internal dynamics of a system associated with the constraint that the output is identically zero for a certain interval of time. Quite generally, the zero dynamics is known to exist as a differential inclusion (see [15]-[16]). Under a mild regularity assumption, these dynamics can be expressed in the form of an autonomous dynamical system

x = /*(x)

defined on a smooth submanifold Z* of the plant state space. More precisely, Z* is a smooth connected manifold which contains the equilibrium z = 0, and satisfies

(1) h(x, 0) = ° for each x E Z* (2) for some smooth u*(x), the vector field /*(x) = f(x,O,u*(x)) is

tangent to Z* , (3) if Z' is any other connected submanifold which satisfies (1) and (2),

then Z' n u c Z* n U for some neighborhood U of x = 0.

The pair (Z*, 1*) is called the zero dynamics of the plant.

Example 5.1 (Asymptotic Tracking with Internal Stability) Suppose (2.1) is affine in the control u and, for simplicity of notation, suppose (2.1) is scalar input-scalar output. We shall consider the problem of asymptotic


tracking with internal stability; i.e. for (2.1) we consider the nonlinear regulator problem with zero disturbance:

(5.1)

x = I(x) + g(x)u

tV = sew) e = hex) - r(w)

Concerning (5.1), we maintain our assumptions that all vector fields and maps are Ck (k ~ 2), that I have an equilibrium at x = 0 in IRn, that s have an equilibrium at w = 0 in IRl and that the output h( x) and the reference output r( w) vanish at the respective equilibria.

In order to give a simple, but interesting, illustration of our existence criterion we further assume that the plant equations (5.1a) can be put in "normal form"

(5.2)

z = 10 ( z , Yl , . .. ,Yp)

Yl = Y2

where Yl = h(z, y;) is the system output. A sufficient condition for the local existence of this normal form in a neighborhood of x O = 0 is that the system have relative degree p, viz.

LgL~-lh(x) == 0, i = 1, ... ,p - 2

but

This condition is necessary if g(x) does not vanish in a neighborhood of 0 and if we ask only that LgLr1h(x) does not vanish identically. As in the linear case, for system with relative degree p, the output is related to the input through a chain of p integrators, as is explicitly demonstrated in the normal form (5.2). Using the normal form (5.2) one can also obtain coordinates in which the zero dynamics are explicitly represented. Namely, to impose the constraints Yl = Y == 0 is to impose the constraints Yi = y(i-l) = 0 on the state variables Yi, i = 1, ... ,p. This yields three objects:

(1) a constraint submanifold Z· defined via

(5.3) Yl = ... = Yp = OJ


(2) a control law rendering Z· invariant, defined in coordinate-free terms as

(5.4)

(3) maximality of Z· subject to controlled invariance and the constraint y= 0.

Moreover, the closed-loop system / + gu· restricted to Z· is just

(5.5) %=/o(z,O).

For a system in the normal form (5.2), the regulator equation must be solved for 1I'(w),c(w) where u = c(w) and

(5.6h (5.6h,i

First note that (5.6h,l and (2.5b) imply the identity

1I'2,l(W) = r(w)

so that 11'2,1 is known. More generally, we introduce the notation

(5.7) .( ) - (i-l)( ) - 8r i-l ( ) . - 1 1 r, w - r w - --a;;;-s w , ,- , ... , p +

We also note that the function ri(w) are known. In the light of (5.2), (5.6h,i yields, for i = 1, ... ,p - 1, the identities

1I'2,i = ri(w), i = 1, ... ,p

so that the functions 1I'2,i are uniquely determined by the regulator equations. Finally, (5.2) and (5.6h,p yield an explicit expression for c( w) as a ''feed-through" law

where 1I'1(W) is determined (via (5.6h) by solving the "off-line" Lyapunov partial differential equation

(5.9)


The Lyapunov partial differential equation (5.9) has, of course, a geometric interpretation. Indeed, treating the error as the output of the augmented system (5.1), to transform the plant equations to normal form is also to transform (5.1) to normal form in terms of the variables z and ei = Yi -ri, i = 1, ... ,po In this case, the zero dynamics of (5.1) may again be computed as evolving on the constraint submanifold el = ... = ep = 0 and is given in local coordinates as

(5.10) i = lo(z, ri)

tV = s(w)

We note that w = 0 is an invariant submanifold for (5.10) with restricted flow given by the plant zero dynamics (5.5) and the equation for an invariant complement is precisely (5.9). We note first that stable, unstable and center manifold theory can often be used to prove the existence of such a complement. Secondly, because for square, invertible linear systems the zero dynamics is a linear system with spectrum coinciding with the system transmission zeroes, in the linear case the standard criterion for well-posedness of the linear Lyapunov equation (5.9) is exactly the criterion of Hautus.

The existence criterion derived in Example 5.1 in terms of zero dynamics does not in fact require the existence of a relative degree and is not limited to asymptotic tracking. In fact, using the center manifold theorem, in [2] we proved a much more general result.

Theorem 5.2. Let (Z*, 1*) denote the zero dynamics 01 the plant and (Z;,1n those of the system (2.1). Then, the equations (2.5) can be solved by a pair of smooth mappings 11'( w) and c( w) if and only if the zero dynamics of (2.1) have the following properties:

(1) in a neighborhood of(x,w) = (0,0), the set M = Z; n(X X {O}) is a smooth submanifold

(2) there exists a smooth submanifold Z, of Z; which contains the point (x, w) = (0,0) and satisfies ToZ* = ToZ, EEl ToM

(3) Z, is locally invariant under I: and the restriction of I: to Z, is locally diffeomorphic to the vector field s( w) which characterizes the exosystem

(4) M is locally invariant under I: and the restriction of I: to M is locally diffeomophic to the vector field I*(z) which characterizes the zero dynamics of the plant.


6. Riccati Partial Differential Equations for Optimal Regulation

In Example 5.1, we derived an explicit form for a controller achieving asymptotic tracking with internal stability for a scalar input-scalar output system having relative degree p, viz.

(6.1) u(Z) = k(z -1I"1(W),Yi - ri(w» + c(w)

where k(z) is any exponentially stabilizing feedback law, where 1I"1(W) is determined by solving the "off-line" Lyapunov PDE (5.9) involving the system zero dynamics and the exosystem dynamics, and where c(w) is the "feedthrough" law (5.8). The form of (6.1), (5.8) and the dependence on the solution of an "off-line" differential equation is quite reminiscent of solutions to optimal stabilization and regulation problems via LQR methods involving a Riccati equation. Indeed in [4] it was observed that similar feedbck laws can in fact be derived for a general class of optimal control problems. Because of its classical relation to regulation and tracking problems for linear systems, we will present some of the local results in [4] which give closed-loop solutions to certain optimal control problems in Lagrange or Bolza form. These feedback laws involve the off-line solution of a class of nonlinear partial differential equations, for which there exists a corresponding geometric existence theory. We remark that a related partial differential equation has recently been discovered independently by Ben Artzi and Helton [18] in connection with the problem of factorizing nonlinear systems.

Following [4] we consider the time-varying Riccati PDE, where z E IRn, 1I"(z) E IRn and t E (O,T)

{)11" {)11" ()t = - ()z!t(z,1I"(z,t» + 12(z,1I"(z,t»

(6.2) 11"(0) = 0

We also consider the steady-state Riccati equation

(6.3) ()11" ()z!t(z,1I"(z» = 12(z, 1I"(z»

11"(0) = 0

A special but important case of (6.3), which was fundamental in our treatment of the nonlinear regulator problem is, of course, the Lyapunov PDE

()11" ()z!t(z) = 12(z, 1I"(z»

(6.4) 11"(0) = 0

As an illustration, consider the finite time horizon optimal control problem for the system

(6.5) x = f(x) + g(x)u, y = hex)

with performance measure

In [4] it is shown that if 71"( x, t) is a solution of the Riccati PDE

7I"(O,t) = 0, 7I"(x,T) = +Y',rQ(x)

then u.(x, t) = _g(x)T 7I"(x, t) is an optimal control for the unconstrained minimization problem, minJT (xo, u). Moreover, we have a basic local exis-

u tence result.

Theorem 6.1 [4] There exists an c « 00 so that for IIxll < c and t E [0, T] a solution 71" (x , t) of the Riccati PDE (6.9) exists.

Remark 6.2 The proof is related to the derivation sketched below for the infinite horizon problem. For more general performance indices and for global existence results we refer to [4]. The fact that an optimal closed-loop law exists locally for such problems was known, see e.g. [19]. In particular, Willemstein [19] extended previous work by Lukes [20] on the steady-state problem by developing a series expansion for u. (x, t). In this light, the sum of such a series satisfies the Riccati PDE.

There has also been an extensive literature on series expansions and smoothness properties of optimal control laws for infinite time problems such as

(6.8) minJ(xo, u), J(xo, u) = 100 lIy(t)1I2 + lIu(t)1I2dt u 0

(see e.g. Albrekht [21]-[22], Brunovsky [23], and Lukes [20], Zubov [21], Sain [22] and Yoshida and Loparo [23]). Again, the sum of such series can

be shown to be a e" function defining an optimal feedback control of the form

(6.9) u.(z) = _g(z)T 1I"(z)

using the stable manifold theorem ([23],[20]). Explicitly, if H(z,p, u) is the hamiltonian of this problem then by the Maximum Principle any optimal u must satisfy

for p some nontrivial solution of the adjoint equations. Defining H. via H.(z, p) = H(x, p, u.), following the Maximum Principle we consider the hamiltonian system

(6.10)

. oH. z = op . oH. p= - oz

Exponential detectability and stabilizability of (6.5) imply that (6.10) has an n-dimensional stable manifold W'(O) which is the graph of a function p = 1I"(z) , for which (6.9) is then a stabilizing optimal control law (see [23],[20]). We note that invariance of W'(O) under (6.10) implies that 1I"(x) is a solution to the steady-state Riccati PDE:

(6.11)

( Of)T 011" 011" T ohT OZ 1I"(z) - ozf(x) - ozg(z)g(z) 1I"(z) + OZ h(x)

_1I"(z)T ~!g(z)T 1I"(z) = 0, 11"(0) = 0

and stability of the flow on W'(O) implies that 11" satisfies the constraint

(6.12)

Theorem 6.3 [4] If (6.5) is exponentially stabilizable and detectable, there exists an € > 0 such that for all z, IIzll < €, then there exists a unique solution to (6.11)-(6.12) defining an optimal closed-loop feedback law (6.9).

Remark 6,4 Again we refer to [4] for results concerning more general performance measures and a more detailed proofs.


REFERENCES

1. C.I. Byrnes and A. Isidori, Reg'dation a.,mptotique de •• ,.teme. Aonlinelire6, C.R. Acad. Sci. Paris 309 (1989), 527-530.

2. A. lsidori and C.I. Byrnes, Output Regulation of Nonlinear S,.tem., IEEE Trans. Aut. Contr. AC-35 (1990), 131-140.

3. B.A. Francis, TAe Linear Multivaria6le Regulator Problem., SIAM J. Contr. Optimiz. 115 (1977),486-505.

4. C.I. Byrnes, "Some Partial Differential Equations Arising in Nonlinear Control, Computation and Control, II," (K. Bowers, J. Lund, eds.), Birkhaiiser-B08ton, to appear ..

5. E.J. Davison, The Output Control of Linear Time-Invariant Multi- Variable S,.tem witA Unmea.urable Ar6it7'llry Di"urbance., IEEE Trans. Aut Contr AC-l T (1912), 621-630.

6. B.A. Francis and W.M. Wonham, The Internal Model Principle for Linear Multivariable Regulator., J. Appl. Math. Optimiz. 2 (1975), 170-194.

7. C.I. Byrnes and A. !sidori, Stead, State Re.pon.e, Sepa7'lltion Principle and tAe Output Regulation of Nonlinear Sy.tem., Proceedings of the 28th IEEE Conference on Decision and Control, Tampa (1989),2247-2251.

8. J.E. Marsden and M. McCracken, ''The Hopf Bifurcation and Its Applications," Springer-Verlag, New York, Heidelberg, Berlin, 1976.

9. M. Hautus, Linear Matrix Equation. witA Application to tAe Regulator Problem, Outils and Modeles Mathematique pour l'Automatique ... (I.D.Landau ed.),C.N.R.S. (1983), 399-412.

10. C.I. Byrnes and A. Isidori, A Frequenc, Domain PAilo.opA, for Nonlinear S,.tem. witA Application. to Stabilization and Adaptive Control, Proc. of 23rd IEEE Conf. on Dec. and Control, Las Vegas, NV (1984).

11. A.J. Krener, A. Isidori, Nonlinear Zero Di.tribution., Proc. of the 19th IEEE cone. on Dec. and Control, Albuquerque (1980).

12. C.I. Byrnes and A. !sidori, "Heuristics for Nonlinear Control," in Modelling and Adaptive Control, (Proc. of the IIASA Conf. Sopron, July 1986, C.I. Byrnes, A. Kurzhansky, eds.), Springer-Verlag, Berlin, 1988, pp. 48-70.

13. A. Isidori and C. Moog, On tAe Nonlinear Equivalent of tAe Notion of Tran.mi",ion Zero., in Modelling and Adaptive Control, (Proc. of the IIASA Conf. Sopron, July 1986, C.I. Byrnes, A. Kurzhansky, eds.), Springer-Verlag, Berlin.

14. C.I. Byrnes and A. !sidori, Local Stabilization of Minimum-pAa.e Nonlinear Sy.tem., Systems and Control Letters 11 (1988), 9-17.

15. J.P. Aubin, C.1. Byrnes and A. lsidori, Viability Kernels, Controlled Invariance and Zero Dynamics for Nonlinear Systems in Analysis and Optimization of Systems, (Proc. of 9th Int'l Conf., Antibes, June 1990, A. Bensoussan and J.L. Lions, eds) (1990), 821-832. Springer-Verlag, Berlin.

16. J.P. Aubin and H. Frankowska, Viabilit, Kernel of Control Sylfem" Nonlinear Synthesis (C. I. Byrnes and A. Kurzhansky, eds). Birkhiuser-Boston, 1991.

17. X.M. Hu, Roh.t Stabilization of Nonlinear Control Sy.tem., Ph.D. dissertation (1989). Arizona State University.

18. A. Ben-Artzi and J.W. Helton, A Riccati Partial Differential Equation for Factoring Nonlinear S,.tem., preprint.

19. A.P. Willemstein, Optimal Regulation of Nonlinear S"tem, on a Finite Interval, SIAM J. Control Opt. 15 (1977), 1050-1069.

20. D.L. Lukes, Optimal Regulation of Non/ienar Dynamical Sylfem" SIAM J. Control and Opt. T (1969), 75-100.


21. E.G. Al'brekht, On the optimal,tabilization of nonlinear '!I,tem" J. Appl. Math. Mech. 25 (1962), 1254-1266.

22. E.G. Al'brekht, Optimal ,tabilization of nonlinear '!I,tem" Mathematical Notes, vol. 4, no. 2, The Ural Mathematical Society, The Ural State University of A. M. Gor'kil, Sverdiovsk (1963). In Russian.

23. P. Brunovsky, On optimal,tabilization of nonlinear '!I,temll, Mathematical Theory of Control, A. V. Balakrishnan and Lucien W. Neustadt, eds., Academic Press, New York and London (1967).

24. M.K. Sain (Ed), Applicationll of tenllor, to modelling and control, Control Systems Technical Report #38, Dept. of Elec. Eng., Notre Dame University (1985).

25. T. Yoshida and K.A. Loparo, Quadratic Regulator!l Theory for Anal!ltic Non-linear S!I,tem, with Additive Control, Automatica 25 (1989), 531-544.

Christopher I. Byrnes Alberto Isidori Department of Systems Science and Mathematics Washington University St. Louis, MO 63130 USA

Asymptotic Stabilization of Low Dimensional Systems

W. P. DAYAWANSA; C. F. MARTINt

Abstract

This paper studies the asymptotic stabilization of two and three dimensional nonlinear control systems. In the two dimensional case we review some of our recent work and in the three dimensional case we give some new sufficient conditions and necessary conditions.

1 Introduction

We consider the single input system,

x = I(x) + g(x)u (1.1)

where x E ~n , U is a scalar input, and I, 9 are C l vector fields. It is assumed that 1(0) = 0, g(O) f:. O. The system is said to be C" feedback stabilizable at the origin of ~n if there exists a real valued C" function a( x) defined on some small neighborhood ofthe origin in ~n such that x = I(x)+g(x)a(x) is locally asymptotically stable at O.

There has been much work done in the recent past on this problem. Prominent among them are the techniques based on center manifold theory, pioneered by Ayels [Ay1] and used effectively by Kokotovic and coauthors among others, the idea of zero dynamics introduced by Byrnes and Isidori [BIl,BI2] etc., and the topological obstructions derived by Brockett [Br1], Krosnosel'skii and Zabreiko [Krl], the work on continuous feedback stabilization by Sontag and Sussmann [SS1], Kawski [Ka1] etc.

An extremely important observation on asymptotic stabilization was made by R. Brockett [Brl]. For the moment let us consider (1.1) with arbitrary state space dimension n and arbitrary number of inputsm. Brockett proved that the following are necessary for stabilization of (1.1) with a Cl feedback function. (B1:) The uncontrollable eigenvalues of the linearized system should be in

the closed left half of the complex plane.

• Supported in part by NSF Grant #ECS-8802483. tSupported in part by NSA Grant #MDA904-85-HOO09.

54 DAYAW ANSA AND MARTIN

(B2:) (1.1) is locally asymptotically controllable to the origin i.e. For an arbitrary open neighborhood W of the origin there exist a neighborhood W of the origin and control u(·) such that for all XOfW the solution t t-+ x(t, xO, u(t» of (1.1) stays in U for all t > 0 and converges to the origin as t t-+ 00,

(B3:) The function (x, u) t-+ i(x) + g(x)u : lRn X lRm --+ lRn is locally onto at (0,0).

The key condition here is (B3), which shows that very interesting pathologies are possible. This condition follows from a theorem due to M. A. Krosnosel'skii and P. P. Zabreiko [Krl], which states that the index of a continous vector field in lRn at a locally asymptotically stable equilibrium point is equal to (_1)n. The focus of much of the research work on low dimensional cases has been on finding further necessary conditions and on finding rather strong sufficient conditions.

In section 2. of this paper we wil review our recent work on the two dimensional stabilization problem for real anlytic systems. In particular it will follow that (B3) is necessary and sufficient for CO stabilization. We will give some sufficient conditions for Cl stabilizability and Coo stabilizability. In section three we will derive some necessary conditions and some sufficient conditions for the asymptotic stabilizability of homogeneous polynomial systems i.e. f(x) is a homogeneous polynomial vector field and g(x) is a constant vector.

2 Stabilization of two dimensional systems

In this section we will review some of our recent work on the stabilization problem for two dimensional systems. Throughout we will assume that the system is real analytic.

Since g(O) # 0 in (1.1) we may assume without any loss of generality that the system has the form,

x = f(Xl,Xa)

za = u,

where f(O) = OJ Xl, xa E lR, u E lR and f is real analytic. The following theorem was proved in [DMK].

(2.2)

(2.3)

Theorem 2.1 Consider the system (2.1). The following conditions are equivalent.

(i) The system (hence (1.1)) is locally asymptotically stabilizable by CO feedback.

(ii) The Brockett condition (B9) is satisfied. (iii) For all f > 0 there exist pfB(O) n lR~ and qfB(O) n ~~ such

that f(p) < 0 and f(q) > O. (Here lR~ = {(Xl, Xa)IXl > O} and lR~ =

ASYMPTOTIC STABILIZATION OF LOW DIMENSIONAL SYSTEMS 55

{(Xl, x2)lxl < O} and Bf(O) denotes the Euclidean ball of radius E around the origin.

Remark 2.1 The stabilizing feedback can be found to be Holder continous.

Remark 2.2 : Prior to our work M. Kawski has shown that (see {Ka1} ) that small time local controllability is a sufficient condition for CO stabilization. Theorem 2.1 strengthens this result.

The Cl and Coo feedback stabilizability are much more subtle even in the two dimensional case. We derived some sufficient conditions in [DMK]. We first define two indices.

Since multiplication of f by a strictly positive function and coordinate changes do not affect stabilizability of (2.1), we may assume without any loss of generality that f is a Weierstrass polynomial, xr + a1(x2)x~-1 + ... + am (x2) and ai(O) = 0 , 1 $ i $ m. It is well known that the zero set of a Weierstrass polynomial can be written locally as the finite union of graphs of convergent rational power series X2 = 4>(X1) where Xl E [0, E) or Xl E (-E, 0] . Let us denote the positive rationals by Q+ and define,

A+ = hE Q+ I f(Xl, 4>(Xl» < 0 for all Xl E (0, E), for some e > O.

and for some convergent rational power series 4>(Xl) with leading 1

exponent equal to -} 'Y

A- = hE Q+ I f( -Xl, 4>(Xl) > 0 for all Xl E (0, E), for somee > 0

and for some convergent rational power series 4>(X1) with leading 1

exponent equal to -}. 'Y

Definition 2.1 The index of stabilizability of f is max{ inf {'Y}, inf {'Y}}' -yEA+ -yEA-

Definition 2.2 The fundamental stabilizability degree of f is the order of the zero of am (z2) at Z2 ::: O. The secondary stabilizability degree of f is the order 0/ the zero 0/ am-l(x2) at X2 ::: O.

Notation:

I Index of stabilizability of /

81 .- Fundamental stabilizability degree of /

82 Secondary stabilizability degree of f.

56 DAYAWANSAAND MARTIN

Theorem 2.2 The system (2.2) and hence (1.1)) is C 1-stabilizable if Sl > 21 -1

If Sl ~ 1 + 2S2 and Sl is odd, then (2.1) is CW stabilizable. If S1 < 1 + 2s2, then (2.1) is not Coo stabilizable.

3 Stabilization of homogeneous systems

In this section we consider a single input homogeneous system,

x=f(x)+bu (3.4)

where x E ~n, U E~, b is a real vector and f is a homogeneous polynomial vector field of some degree p i.e. f(>.x) = >.r f(x) for all x~n and>' > 0 .. For the most part we will be seeking to find a feedback function u = a(x) which is homogeneous of degree p along rays from the origin i.e. a(>.x) = >.Pa(x). For the sake of clarity henceforth we will use the term, positively homogeneous, to describe such functions. We remark that for this class of feedback the local and global stabilization are equivalent. Unless specified otherwise we will assume that f is C1.

The following theortem is due to Andreini, Bacciotti and Stefani [ABS]. Theorem 3.1 Consider the system,

i1 F(X1, X2)

i2 = U (3.5)

where (Xl, X2) E ~P x ~m, U E ~m, F is homogeneous of some odd degree p. The system is asymptotically stabilizable by homogeneous feedback of degree p if i1 = F(xl. 0) is asymptotically stable.

The following example captures the spirit of this theorem.

Example 3.1 Consider the system,

(3.6)

where p is an odd integer. We show that this system is asymptotically stabilizable. This is done by using an induction argument.

When n = 1, u = -x~ is a stabilizing feedback law and V(x) = !x~ is a Lyapounov function.

Suppose that for some n 2: 1 (3.6) admits a stabilizing feedback function u(x) = -(l(X1, ... ,Xn)P, where I is a linear function, and admits a quadratic Lyapounov function V(x) = ~xTQx. Let us consider the n + 1


dimensional case. First let us change coordinates as, Yi = :ei j i = I, ... , n and Yn+1 = :en+1 + 1(:e1, ... , :en). By applying the Holder's inequality and by using the Lyapounov function V(Y1, ... , Yn)+ ~Y~+1 it is easily seen that for large enough K, u = -K(l!n+1) is a stabilizing feedback function. This concludes the asymptotic stabilizability of {3.6}.

For the rest of the section we will focus on the stabilization problem for three dimensional homogeneous systems. Necessary and sufficient conditions for the asymptotic stability of three dimensional homogeneous systems were derived by Coleman in [Co] (see [Hal] also). Let us consider the system

z = F(:e) (3.7)

where :e E ~n and F is a positively homogeneous vector field (not necessarily polynomial) of degree p. One can derive an associated system on the

n - 1 dimensional sphere sn-1 by first writing an equation for ~ (11:11) as,

(3.8)

and then changing the time scale, in an 1I:e1l dependent way so that the

equation depends only on 11:11. Thus we obtain,

(3.9)

Coleman's theorem states the following.

Theorem 3.2 ([Co]): Let A denote the union of all equilibrium points and periodic orbits of {3.8} on sn-1. Let C denote the cone generated by C. Then the system {3.7} is asymptotically stable if and only if it is asymptotically stable when restricted to C.

This can be used to generalize the theorem of Andreini, Bacciotti and Stefani [ABS] as follows in the three dimensional case. This theorem was proven independently by M. Kawski (see [Ka2] ) also.

Theorem 3.3 Consider the positively homogeneous control system

iJ h(y,z)

z = u (3.10)

where y E ~2, Z E ~, u E ~ and h is positively homogeneous of degree p i.e. h(ay, az) = aPh(y, z) for all a E ~.

58 DAYAWANSA AND MARTIN

Suppose that there exist a Lipschitz continuous function z = ¢(y) : lR2 --4

lR which is a positively homogeneous of degree 1 such that the system

iJ = h(y, ¢(y)),

is asymptotically stable. Then there exists a Lipschitz continuous feedback function, u = a(y, z), which is homogeneous of degree p, such that the system,

y h(y, z)

a(y, z) (3.11)

is asymptotically stable.

Proof: After a small perturbation of ¢, we may assume that the function 'I/J = ¢Isl : S1 --4 lR is Coo. (Here 8 1 - denotes the standard unit circle in

lR2). Now let M denote the intersection of the positive cone (; ~f {(y, z) I z = ¢(y), Y E lR2 } and 8 2 • Let u : 8 2 --4 8 2 be a smooth diffeomorphism which preserves poles and moves points longitudinally such that uo'I/J(81 )

is the equator of 8 2 •

Now let, iJ = a(O) + b(O)u (3.12)

be the associated system on 8 2 , obtained by (3.10), as described in the introduction. Let qn and q8 denote the north and the south poles of S2 and let D be a band around the equator bounded by two latitudes and such that the inverse image of D under u contains the equator. Now first transform (3.11) by u to obtain,

~ = (u.au- 1) (f3) + (u.bu- 1) (f3)u

c(f3) + d(f3)u. (3.13)

Now find a smooth function 'Y : 8 2 --4 lR such that it has the following properties.

(P1) 'Y < 0 above D and 'Y > 0 below D

(P2) For all f3 E D, the positive limit set w(f3) of the solution of

/3 = c(f3) + d(fJh(fJ)

is contained in the equator. (In particular the equator is positively invariant) .

Now consider the feedback function,

Then it follows at once that C is an invariant cone of

iJ = h(y, z)

z = a(y,z) (3.14)

and that the system is asymptotically stable on C. Moreover all other invariant one or two dimensional cones meet 8 2 outside of 0'-1 0 D. Since za(y, z) < 0 outside of the cone generated by 0'-1 0 D it follows that the system is asymptotically stable on all such invariant cones. Hence by Coleman's theorem the asymptotic stability of (3.14) follows.

Q.E.D.

In view of this lemma, one can use known results on the stability of two dimensional homogeneous systems in order to derive sufficient conditions for asymptotic stabilization of three dimensional systems. The following theorem is of interest to us.

Theorem 3.4 ({Hal]): Consider the two dimensional system,

(3.15)

where f = [It, h]T is Lipschitz continuous and is positively homogeneous of degree p. The system is asymptotically stable if and only if one of the following is satisfied:

(i) The system does not have anyone dimensional invariant subspaces and

or

1211' cos e 11 (cos e, sin e) + sin e h (cos e, sin e) de 0 o cos e h( cos e, sin e) - sin e It (cos e, sin e) <

(ii) The restriction of the system to each of its one dimensional invariant subspaces is asymptotically stable.

As an application of theorems 3.2 and 3.3, let us consider the problem of stabilization of the angular velocity of a rigid body when only one of the control torques is available. This system has the structure,

i1 = a1 z 1z 2+ b1U

i2 = a2z1z3 + b2u

i3 = a3Z1Z2 + b3 u.

D. Ayels and M. Szafranski have shown in [AS] that this system is locally asymptotically stabilizable when no two principal moment of inertia


are equal. The case when two of the principal moment of inertia are equal (equivalently a1 = -a2) was the topic of study of the recent paper [SS2] by E. Sontag and H. J. Sussmann . They have shown that if none of the bi's are equal to zero, then indeed the system is globally stabilizable by smooth feedback. Below we show that the system is globally stabilizable by Lipschitz continous, positively homogeneous feedback.

It is easily seen that (see [SS2]) the problem can de reduced to the stabilization of,

:1:1 = :l:2:1:S

i2 = -:l:S:l:1 - b:l:~

is = u

where b is a nonzero constant. By theorem 3.2, if we can show that there is a Lipschitz continous function :l:s = ¢(:l:1' :l:s) which is positively homogeneous of degree 1, which stabilizes,

i1 = :l:2:1:S

i2 = -:1:1:1:S - b:l:~ (3.16)

then the desired conclusion follows. Without any loss of generality we assume that b > O. Since the stability

is preserved under multiplication of the vector field by strictly positive functions we will first consider,

i1 = :1:2

i2 = -:1:1- b:l:s (3.17)

and seek to find a strictly positive stabilizing Lipschitz continous feedback function :l:s = :l:S(:l:1, :1:2) which is positively homogeneous of degree one . Since asymptotic stability of a positively homogeneous system is robust under small purtubations by functions of the same degree of homogeneity, we can relax the requirement of strictly positiveness to positiveness. It is seen at once by using the Lyapounov function :I:~ + :I:~ that,

satisfies the requirements. This concludes the proof that the system is asymptotically stabilizable by globally Lipschitz continous feedback which is positively homogeneous of degree one.

Theorems 3.2 and 3.3 can be used to generate further sufficient conditions for the asymptotic stabilizability of positively homogeneous systems.


Let us consider the system

u (3.18)

where (Xl,X2,X3) ...... (l1,h)(x1,x2,x3) : ~3 ---. ~2 is a positively homogeneous function of some degree p.

Theorem 3.5 :Suppose that there exists a smooth function cp : 8 1 ---. ~ such that at least at one 00 E 8 1 , the vector (11,/2)T(cosOo,sinOo,cp(Oo» points radially inwards and at no points 00 E 8 1, the vector field (11, hf (cos 0, sin 0, cp( 0» points radially outwards. Then the system is asymptotically stabilizable.

Proof: By (ii) of theorem (3.3) the system

is asymptotically stable. Now the theorem follows from theorems 3.2 and 3.3. Q.E.D. The sufficient condition given in theorem 3.4 can be tested

quite easily by using the locus of zeros of a certain function. Note that the crucial properties in the theorem are satisfied by the roots of the equation

l1(x1,x2,X3) - x3h(x1,x2,x3) = o. (3.19)

Using homogeneity we rewrite (3.19) as

since l1(cosO,sinO,x3) - x3h(cosO,sinO,x3) = O. (3.20)

One can now draw the locus of the zeros of (3.20) against () E [0,211"] in a graph and decide at once the existence or nonexistence of a function 'P as desired.

Our next sufficient condition is applicable to to homogeneous polynomial systems of odd degree and relates to (i) of theorem (3.3).

Now we consider the generic case and rewrite (ii) in the form,

Xl x~ + gl(Xl, X2, X3)

u (3.21 )

where gl and g2 are homogeneous polynomials of odd degree p; gl does not contain z~ terms and gl and g2 do not contain z~ terms. A generic system can be written in this form after a suitable linear change of coordinates.

Theorem 3.6 Suppose that the function

takes either strictly positive values or strictly negative values. Then (9.21) is asymptotically stabilizable.

Proof: Let

and h(Zl, Z2, Z3) = -z~ + g2(Zl, Z2, Z3).

The objective here is to construct a "base" which is positively invariant and use it to establish the asymptotic stability. We will first consider the case when Rng(1J) C (0,00). Then the leading term of the polynomial 11(0,1, Z3) is of even power. Now it follows at once that there exists a neighborhood U = [11'/2 - e, 11'/2 + e] of 11'/2 such that,

11 (cos 0, sinO, Z3) > 0

for all Z3 E~, and all 0 E U. Similarly,

l1(cosO,sinO,z3) < 0 for all 0 E U + {11'} and all Z3 E~.

Let

{ h( cos 0, sin 0, Z3) I } ,,\ = max f ( O' 0 ) 0 E [11'/2 - e,1I'/2], Z3 E [0,00)

1 cos ,sm , X3

and Jl=max{-h(CoSO,~inO'O)IOE [311', 311' -e]}.

11 ( cos 0, sm 0, 0) 2 2

Existence of Jl is clear. Existence of ,,\ follows since

h(cosO,sinO,x3) [11' 11'] ~'---:-'--:.~"---''7 goes to - 00 uniformly in 0 E -2 - e, -2 11 (cos 0, sm 0, X3)

as Z3 goes to infinity. N ow define the angle 00 E (11'/2 - e, 11'/2] via the following construction.


Let us define 80 by,

80 = max { i - l, tan -1 (2 cos l + ~: : 2-\) sin l) } •

This choice of 80 can be explained via figure 1 rather easily. Let us start with an arbitrary 6 > 0 and draw a line of slope -\ through

(0, -6) until it meets the line with polar coordinate equal to 31r/2 + l at A. Now draw a line vertically upwards until it meets the line of slope m through (0,26) at B. The polar coordinate of this point of intersection

IS equal to tan -1 . • Of course one may need to . (2 COSl + (m + 2-\)sinl) SlDl

decrease l if necessary in order that the required intersection occur.

%. r slop" '" Tn B (0.26)

E'

6: \ \

\~ , . , \

I I

\ • A -lope == A

!0.-(3~/2l\ ---J c F slop" .. A D

Figu're 1

64 DAYAW ANSA AND MARTIN

Now lets define a line segment £1 and an angle 1 E (0, f) in the following way. Start from B and draw £ to be of very large negative slope until it hits the line (J = 37r/2 + fat C. Now draw a vertical line downwards until it hits the line (J = 37r /2 + f -1 at D. The choice of the slope of £ and 1 is made such that the line of slope m through D meets the negative z2-axis

( -36) at 0'-2- .

Let E = (0,26) and F = (0, -36/2). We will now define a Lipschitz continuous function Z3 = ¢(ZI, Z2) which

is homogeneous of degree 1 such that the system

(3.22)

is asymptotically stable. Let us first consider the line i. We fix cp to be

.. h h Ih(ZI, Z2,CP(ZI,Z2))1· I a large pOSItive constant L on l suc t at f ( ( )) IS a ways 1 XI, X2, cp Xl, X2

greater than the magnitude of the slope of i. This is obviously possible from the hypothesis on gl and g2. Vary cp smoothly from L at C to zero at D along CD. Set cp == ° on F D. Increase cp from ° at E to L smoothly along EB. Now use homogeneity to define cp on ~2. It is clear that one can construct a Lipschitz continuous function cp this way.

Now let us consider (3.22). It is clear that there aren't anyone dimensional invariant unstable subspaces, for by our construction the vector field [11, h]T points into the region EBCDF along the portion of the boundary which does not lie on the x2-axis. Suppose that there aren't anyone dimensional invariant stable subspaces either. Then the solution with initial condition (0,26) enters into EBCDF and cannot leave it on EBU BCDU DF and hence has to cross OF. But by homogeneity this now implies asymptotic stability. Now by theorem 3.2 the stabilizabilityof (3.21) follows.

In the case when Rng(T]) C (-00,0), one can do essentially the same construction in the left half plane instead of the right half plane as above. Q.E.D.

Now we discuss some topological aspects of the stabilization problem for the homogeneous three dimensional systems(3.1O). We focus on finding some stronger requirement of the Krosnosel'skii - Zabreiko theorem which cannot be captured by (B3).

For the sake of simplicity we will assume that h(x) only has isolated zeroes on the unit sphere S2. Let u = a(z) be a (not necessarily homogeneous) continous feedback function. Let ¢( x) = [( h( x)T , a( x )]T. Let S~ denotes a small enough ball in ~3 such that the origin is the only zero of ¢ on and inside S~. Let Z = {pfS~lh(p) = O}. Let deg(h,p,w) denotes the Brower degree of h with respect to p E S~ and W E ~2.


Theorem 3.7 : A necessary condition for the asymptotic stabilizability of 3.10 is that there exist W C Z such that upEwdeg(h,p,O) = -l.

Proof: Let 1/J = ,pI II ,p II: S: -+ sl and let deg(1/J,p, q) denotes the Brower degree of 1/J with respect to PES: and q E Sl. Then it is easily seen that deg(1/J,p, 1/J(p)) = sgn a(p)deg(h,p, 0) for all p E Z. Since a necessary condition for stabilizability is that index,p = EPES2 deg (1/J,p, [0,0, IV) = -1, the conclusion follows. • Q.E.D.

Some other necessary conditions which are similar in spirit appear in [Ka2] and [Cor].

4 Acknowledgements

The authors wish to thank Professor Chistopher Byrnes and Professor H. J. Sussmann for many helpful discussions related to the work reported here.

References

[BIl] C. 1. Byrnes and A. Isidori, "A Frequency Domain Philosophy for Nonlinear Systems," Proc. of 23rd IEEE Can/. on Decision and Control, Las Vegas, 1984, 1569-1573.

[BI2] C. 1. Byrnes and A. Isidori, "The Analysis and Design of Nonlinear Feedback Systems I, II: Zero Dynamics and Global Normal Forms", preprint.

[Br1] R. Brockett, "Asymptotic Stability and Feedback Stabilization" in Differential Geometric Control Theory, Birkhauser, Boston, 1983.

[Hal] W. Hahn, Stability of motion, Springer Verlag, NY, 1967.

[GH1] P. Griffiths, J. Harris, Principles of Algebraic Geometry, John Wiley and Sons, NY, 1978.

[Ka1] M. Kawski, "Stabilization of Nonlinear Systems in the Plane," Systems and Control Letters 12 (1989) 169 -175.

[Ka2] M. Kawski, "Homogeneous Feedback Laws in Dimension Three," Proceedings of the IEEE conference on CDC, Dec 1989 1370 -1376.

[Cor] J. M. Coron, " Necessary conditions for feedback stabilization" (preprint). bibitem[Ca1]Cal J. Carr, Applications of Center Manifold Theory, Springer Verlag, NY, 1981.

[LeI] S. Lefschetz, Algebraic Geometry, Princeton University Press, New Jersey, 1953.

[Hail] V. T.' Haimo, "An Algebraic Approach to Nonlinear Stabilization," Nonlinear Theory Methods and Applications, Vol. 10, No.7, 1986.

66 DAYAWANSAANDMARTIN

[AnI] A. Andreini, A. Bacciotti, G. Stefani, "Global Stabilizability of Homogeneous Vector Fields of Odd Degree," Systems and Control Letters, Vol. 10, 1988, 251-256. Systems," Systems and Control Letters, Vol. 2, 1982, 48-52.

[Ah1] L. V. Ahlfors, Complex Analysis, 2nd ed., McGraw Hill, NY, 1966.

[Krl] M. A. Krosnosel'skii and P. P. Zabreiko, Geometric Methods of Nonlinear Analysis, Springer Verlag, NY, 1984.

[Arl] Z. Artstein, "Stabilization with relaxed controls," Nonl. Anal, TMA 7 (1983): 1163-1173.

[BM1] W.M. Boothby, and R. Marino, , "Feedback stabilization of planar nonlinear systems," Systems and Control Letters 12(1989): 87-92.

[Ko1] D.E. Koditschek, "Adaptive techniques for mechanical systems," Proc. 5th. Yake Workshop on Adaptive Systems, pp. 259-265, Yale University, New Haven, 1987.

[Mal] R. Marino, "Feedback stabilization of single-input nonlinear systems," Systems and Control Letters 10(1988): 201-206.

[TS1] J. Tsinias, "Sufficient Lyapounov like conditions for stabilization," to appear in Mathematics of Control, Signals, and Systems.

[Sol] E.D. Sontag, "Further facts about input to state stabilization," Report 88-15, SYCON - Rutgers Center for Systems and Control, Dec 88.

[SSl] E.D. Sontag and H.J. Sussmann, "Remarks on continuous feedback," Proc. IEEE Conf. Decision and Control, Albuquerque, Dec. 1980, pp. 916-921.

[SS2] E. D. Sontag and H. J. Sussmann, "Further Comments on the Sta.bilizability of the angUlar velocity of a rigid body," Systems and Control Letters 12 (1989), 213-217.

[DMK] W. P. Dayawansa, C. Martin and G. Knowles, "Asymptotic stabilization of a class of smooth two dimensional systems," submitted to the SIAM J. on Control and Optimization.

[DM1] W.P. Dayawansa and C.F. Martin, "Two Examples of Stabilizable Second order Systems," Proceedings of the Montana Conference on Computation and Control, Montana State University, June 1988.

[DM2] W.P. Dayawansa and C.F. Martin, "Asymptotic Stabilization of Two Dimensional Real Analytic Systems," Systems and Control Letters, 12(1989 205-211.


[Ayl] D. Ayels, "Stabilization of a class of nonlinear systems by a smooth feedback," Systems and Control Letters 5 (1985), 181-191.

[Ay2] D. Ayels and M. Szafranski, "Comments on the Stabilizability of the angular velocity of a rigid body," Systems and Control Letters 10 (1988), 35-39.

[Sa] N. Samardzija, "Stability properties of autonomous homogeneous polynomial differential systems," J. Differential Eq., Vol. 48, 60-70, 1983.

[Co] C. Coleman, "Asymptotic stability in 3-space," Contributions to the Theory of Nonlinear Oscillations, Vol. V, Annals of Mathematics Studies, Vol. 45, eds. L. Cesari, J. P. LaSalle and S. Lefschetz, Princeton Univ. Press, 1960.

[AFl] E. B. Abed and J.B. Fu, "Lacal feedback stabilization and bifurcation control ,I, Bopf bifurcation" Systems and Control Letters 7, (1986) 11-17.

[AF2] E. B. Abed and J .B. Fu, "Lacal feedback stabilization and bifurcation control,II. Stationary bifurcation," Systems and Control Letters 8, (1987) 467-473.

W. P. Dayawansa Departmrnt of Electrical Engineering University of Maryland College park, Md 20742 and C. F. Martin Department of Mathematics Texas Tech University Lubbock, Tx 79409

Zero Dynamics in Robotic Systems

ALESSANDRO DE LUCA

Abstract. The notion of zero dynamics of a nonlinear system is used in the investigation of three classes of problems that arise in advanced robotics: control of robots in rigid contact with the environment, free motion control of manipulators with redundant degrees of freedom, and trajectory control of robot arms with flexible links. In each case, the internal dynamics present in the system when a proper output is constrained to be zero is characterized, and a physical interpretation of such dynamics is provided. Simple examples are worked out to show how this analysis supports the design of stabilizing controllers, and that existing results can be reviewed in the spirit of zero dynamics.

1. Introduction

In recent years robotics has served as an exciting field of application for advanced findings in nonlinear control theory. Several interesting control problems have been posed and solved for nonlinear systems which are linear in the control input u and in the disturbance input z,

(1) x = f(x) + g(x)u + r(x)z, y = h(x),

where x E IRn , u E IRm , z E JR!, y E JRt, f and the columns gi and ri of matrices g and r are smooth vector fields, and h is a smooth output vector function. In particular, necessary and sufficient conditions have been found for the problems of feedback linearization, input-output noninteraction, disturbance decoupling, and full (i.e. state plus output) linearization, using a static state-feedback law of the form

(2) u = O'(x) + (3(x)v, with (3(x) nonsingular,

as control input and, when needed, a transformation in the state-space i = w(x) (see [1], and the references therein).

This general framework is well suited to robotics, as to many other mechatronics systems, for two main reasons. First, dynamic models of

ZERO DYNAMICS IN ROBOTIC SYSTEMS 69

articulated manipulators present strong, although smooth, nonlinearities in the state, but are always linear in the inputs. Mechanical sources of nonlinear dynamics are the large changes of apparent inertia in different arm postures and the dependency of gravity forces on the configuration. Second, in standard robot control problems, the relation between applied input forces and controlled outputs - the joint variables or the end-effector pose - fully displays the typical interactions of multi variable systems.

To set up a common background, let us recast some standard results for conventional robots, namely those manipulators constituted by open kinematic chains of rigid bodies, connected by N rotational or prismatic joints, and with an independent actuator driving each degree of freedom. Also, N is less or equal to 6, the maximum number of parameters needed for specifying an arbitrary position and orientation of the end-effector in free space. Defining a vector q E JRN of generalized coordinates (e.g. the joint variables), the Lagrangian of the system L = T - U + m T q, is formed from the kinetic energy T( q, q) = iq TB( q)q, the potential energy U (q), and the nonconservative forces m performing work on q. Applying the principle of least action [2], the equations of motion for conventional robots are obtained as

(3) B( q)q + c( q, q) + e( q) = m,

where B is the positive definite symmetric inertia matrix, c is the Coriolis and centrifugal vector, and e is the gravitational vector. The components of the last two terms have the explicit expressions

(4) ( .) _ 1 . T (8hi (8hi ) T 8B) . Ci q, q - 2q 8q + oq - Oqi q,

for i = 1, ... , N, being hi the ith column of B. State equations in the form (1) (with z = 0, viz. l = 0) are readily obtained from (3), setting x = (q,q) = (xp,xv ) E JR2N and u = m E JRN:

(5)

with n( q, q) c( q, q) + e( q) for compactness. Thus, n = 2N, m = N. As in (1), an output equation can be associated to this dynamic system, typically in the form

(6) { q, Y = k(q),

joint space output cartesian space output

where p = k( 1) is the so-called direct kinematics of the arm, a nonlinear mapping k:JR _JRN in the conventional square case (p=m=N). When

70 ALESSANDRO DE LUCA

the full end-effector pose is considered in vector p, a minimal representation should be used for orientation, like Euler or roll-pitch-yaw angles.

In current industrial robots, the nonlinear effects present in the state equations (5) are masked by the adoption of torque transmission elements with a large reduction ratio [3]. However, the demand for accurate endeffector force control, extreme precision in the whole workspace, and high speed in performing automatic tasks, has led to the introduction of a new generation of direct-drive arms, i.e. with unitary transmission ratios. For these robots, full consideration of the complex nonlinear and interacting dynamics is unavoidable. Thus, use of standard (approximate) linearization procedures is not adequate, and the motion control problem really asks for nonlinear feedback laws in the form (2). Nonetheless, researchers have soon discovered that, even for direct-drive robots, the general picture is not as bad as it may seem. In fact, in their conventional setting, robot arms satisfy all the conditions needed for solving the aforementioned control problems, in particular full linearization and input-output noninteraction. It is easy to see that the suitable feedback and state transformation for the joint output case is

(7) u = n(x) + B(xp)v, 'li(x) = x,

while for the cartesian output is

(8) u = n(x) + B(xp)J-l(xp) [-j(x)xv + v], 'li(x) = [J~!:)~v] , where J(q) = 8k/8q is the (square) Jacobian of the direct kinematics. These control laws achieve state and output linearization, in the proper coordinates, and input-output decoupling at the same time, as can be checked by direct inspection. Such a striking result has been labeled in many different ways in robotics, namely as computed torque, inverse dynamics approach, resolved-acceleration method, or operational space control. In the cartesian space, slight variations may be encountered depending on the chosen representation of end-effector orientation, but - apart from different algorithmic singularities in the induced Jacobian J - a basically unique approach can be resumed [4,5].

In this paper, our purpose is to show that there is more than just 'computed torque' in advanced robotic control problems. There are still plenty of situations needing for an useful transfer of new results from nonlinear control theory. Recently, the notion of zero dynamics of a system, i.e. the internal dynamics consistent with the constraint that the system output is zero for all times, has been stated in a precise way also in the nonlinear case [6], providing a convenient tool for the analysis of several control problems and for the generation of powerful results. To mention a few, it has been shown in [7] that invertible systems with no zero dynamics can always be fully linearized by means of dynamic state-feedback.

Exponentially stable zero dynamics allows exact or, at least, asymptotic reproduction of output trajectories [8]. In this case, the system is usually referred to as being minimum phase. Finally, in the regulation problem for nonlinear systems, solvability conditions can be restated in terms of zero dynamics of the system [9].

In the following, the role of zero dynamics will be investigated with reference to three relevant problems in robotics: control of robots in rigid contact with the environment, free motion control of manipulators with redundant degrees of freedom, and trajectory control of robot arms with flexible links. Increasing interest is being devoted to these robotic applications, which are indeed more complex than conventional ones, but much work has still to be done for obtaining control schemes which perform in a very satisfactory way. Our contribution here is to indicate how the notion of zero dynamics can be used for reinterpreting known results, for providing alternate solutions to challenging problems, or in the definition of new control strategies. Although most of the arguments are presented through examples, we will try to put each case study in the wider perspective of the pertinent robotic field. For completeness, general concepts related to zero dynamics and a computational algorithm are recalled first.

2. Zero dynamics of nonlinear systems

Consider the class of nonlinear systems that are not fully linearizable by feedback. For these systems, the feasibility of a number of control laws is strictly related to the stability properties of a particular dynamics which depends upon the specific control problem faced. When seeking for exact reproduction of output trajectories by means of system inversion, the minimal inverse dynamics is of concern. For input-output noninteracting problems (as well as for disturbance decoupling), the critical issue stands in the dynamics of maximal dimension which can be rendered unobservable via feedback. On the other hand, for solving local stabilization problems using smooth static state-feedback, one should investigate the properties of the internal dynamics when the output is forced to zero. For linear and for nonlinear single-input single-output invertible systems, it is known that the dynamics of the minimal inverse, the dynamics associated with maximal loss of observability under feedback, and the closed-loop dynamics obtained when zeroing the output, are in fact coincident. This equivalence is no longer true for general nonlinear systems [10]. However, these three notions collapse into the same one when the decoupling matrix of the system is nonsingular (or is full row rank, when p < m). We shall keep this assumption from now on.

In any case, the stability requirement for the closed-loop system plays the major role in validating any of the previous control designs. Taking advantage of the assumed equivalence of the above three notions, we will focus only on the derivation of the zero dynamics of a given system. Besides,

this is also simpler from a computational point of view, as will be evident in the considered robotic examples.

A coordinate-free algorithm has been given in [8] for computing both the zero dynamics manifold M*, i.e. the set of states that may be assumed by the nonlinear system when the output is constrained to zero, and the zero dynamics vector field f* (x) 1M", i.e. (the restriction of) a vector field which is always tangent to the manifold M*. Let Xe be a regular point in the sense of [8], and assume f(xe) = 0 and h(xe) = O. Denoting by Tx(M) the tangent space at x to M, a sequence {Md of manifolds can be computed as:

step 0: Mo = h-l(O);

step k: in a neighborhood Uk-l(Xe) such that Uk-l n Mk-l is smooth,

At a step k* < n, this algorithm converges to M* := M k ". Moreover, there exists a smooth state-feedback u = u*(x) such that f*(x) := f(x) + g(x)u*(x) is tangent to M*, and

(9) x* = f*(x*), with x* E M* => y = h(x*(t)) == O.

Since the vector field f*(x) is tangent to M*, the restriction f*(x)IM" is a well-defined vector field on M*. Note that, in local coordinates, the zero dynamics algorithm is similar to the so-called structure algorithm [11]. Moreover, it can be generalized to time-varying constrained outputs and to non-square (p < m) systems, in which case u* is not unique.

3. Robots in constrained motion

In many industrial tasks, the robot end-effector is required to move in contact with an environmental surface. In such situations, one is interested in controlling motion along selected directions and exchanged forces in some orthogonal ones, thus leading to an hybrid control scheme. The task is usually denoted as compliant, when some elasticity is assumed at the contact point. This compliant dynamics is due to the non-ideal rigidity of environment and robot, and to the natural deformation of the force/torque sensing device, when present. Experimental evidence shows that the most critical cases arise when the contact stiffness is very high, with the occurrence of an unstable chattering behavior. This motivates the investigation of the limit case, when the contact is perfectly rigid and the end-effector is actually constrained to a given hypersurface.

A general i-dimensional holonomic constraint [2] will be denoted as (p) = 0, being p the end-effector generalized coordinates. Using the arm direct kinematics, this constraint can be rewritten as

(10) (p) = (k( q)) = 0,


For the constrained robotic system, the right hand side of the overall dynamics (3) becomes, by the principle of virtual work,

(11) T a~ T

m = u + J (q)( ap) F,

In (11), u is the control input at the joints (m = N), while F is the generalized constraint force arising at the end-effector. Only compatible forces are generated within this formulation, and these can be interpreted as Lagrange multipliers associated to the given constraint [12]. If simple contact is desired, i.e. with no reaction forces arising at the contact point, vector F plays the role of a disturbance for the system. Accordingly, setting again x = (q, ci), n = 2N, and with the i-dimensional vector z = F, the state equations assume exactly the form (1). In a similar way, when a contact force F d =F 0 is desired, the disturbance becomes z = F - F d.

Starting from equation (3) with (11), the following questions will be addressed:

• how do we compute input torques u for staying on the surface ~ = O?

• how do we express the dynamic behavior on ~ = O?

A direct answer is provided through the derivation of the zero dynamics of this robotic system, by taking as output to be constrained to zero

(12) y = b( q) = ~(k( q»,

thus having p = i in (1). From

(13) . ab. T()' y = aq q = q q, y = T( q)q + T( q, ci)ci,

using the robot dynamic model (3), it is found that y explicitly depends on the input u in a nonsingular way, provided that the i x N constrained Jacobian T = (a~/ap)J is of full rank l. Note that matrix T in (13) contains information about the constraint curvature. The zero dynamics manifold M* is obtained by setting y = y = 0:

(14)

From y = 0, the state-feedback law which keeps the dynamic flow of the closed-loop system tangent to M* is (15) u* = n(q, ci) - B(q)Tt (q)T( q, ci)ci + B( q) [I - Tt( q)T(q)]v - TT(q)F,

= a(x) + (J(x)v + ,(x)z,

where Tt = TT(TTT)-l is the pseudoinverse of T. This static statefeedback law contains also a disturbance measurement term, i.e. a feedback


from the force sensing device. Also, the new control input v is premultiplied by the projection matrix into the null space of T. Applying (15), the closedloop dynamics becomes

(16) q = -Tt(q)T(q, q)q + [I - Tt(q)T(q)]v,

which is a set of N second-order differential equations. Since the projection mat~ix has rank N -i, this will also be the actual number of independent control inputs, out of the N components of v. When the initial conditions are specified on Moo, all solutions of (16) will 'live' in the lower dimensional manifold Moo onto which the dynamical system is projected.

To display correctly the zero dynamics, the vector field roo, implicitly defined within (16), has to be restricted to Moo. Thus, instead of projecting, we should reduce equations. In order to do so, a proper change of coordinates is needed:

(17)

where qr E JRN -l (a subpart of q) is used as a local parametrization of the constraint surface. Then, the reduced set of 2(N - i) first order equations expressing the zero dynamics of the robot system in constrained motion is

(18)

being a the reduction to Moo of the forcing term v in (16). This dynamics describes in terms of joint coordinates the motion of the robot end-effector on the surface 4> = O. Input a can be used to stabilize this motion, independently of the constraint forces acting in the span of the rows of (04)/ op ) T.

This interpretation is in the same spirit of hybrid control schemes [13], in which dynamic decoupling of force and velocity control loops can be obtained via a nonlinear feedback similar to (15). Also, the proposed approach is consistent with the results derived in [12]. An example will further illustrate the above concepts.

Example. Consider a planar RP (cylindric) robot arm, moving the end-effector in rigid contact with a circular surface of radius r, as in Fig. 1. In this case, N = 2, i = 1, and

showing that the end-effector constraint maps into a linear one in the joint space. The dynamic model of this robot arm is (20)

[ bu(q2) 0] [~l] + [n1(q2,41:th)] = [Ul] + [-Q2S1 o b22 Q2 n2(Q2, Qt} U2 Cl


with the usual notation Si = sin qi, Ci = cos qi . The scalar value F, the force acting in the direction normal to the constraint surface, is directly available from a sensor measurement.

Fig. 1 - A planar RP robot in constrained motion

Simplifications are introduced when the constraint is expressed in joint coordinates, since

Then, the nonlinear feedback control (15) takes the form

(22)

assuming that the desired contact force is zero. This control law becomes a linear one when evaluated on M* = {(q,q) E m4: q2 - r = O,(h = O}, with ui = bU(r)vl, u; = -F. The closed-loop dynamics (16) collapses into q = [vlO]T so that, selecting qr = ql and a = VI, the zero dynamics reduces simply to iit = Vl . Note that in this case ql parametrizes globally the constraint surface. _

4. Redundant robot arms

Kinematic redundancy in robot arms is a relative concept . A robot is said to be redundant when the number N of degrees of freedom (viz. of joints) is larger than p, the number of coordinates strictly needed for describing a


given compatible task. For instance, a planar 3-dof arm is redundant for the end-effector positioning task, but it is not if also the orientation around an axis normal to the plane is of concern. The 'primary' task is often specified in terms of the robot end-effector variables, as in this example. Therefore, it is convenient to define these task variables as characterizing outputs for the redundant system.

Introduction of redundancy increases arm dexterity, allowing collision avoidance with workspace obstacles or enabling to comply with joint range limits. Most important, it provides the manipulator with the capability of avoiding singular configurations, where the kinematics Jacobian looses full rank. A nice analytic feature is that one can associate significant performance indices to all these problems, e.g. the minimum distance of the arm from an obstacle. The robot additional degrees of freedom can then be used for optimizing these performance criteria during local or global motion. Similarly, one can exploit redundancy for the satisfaction of secondary or 'augmented' tasks. A review of the most common resolution schemes can be found in [14]. On the other hand, the kinematic transformation of end-effector paths into joint-space paths for redundant arms is not straightforward, due to the non-existence of a closed-form solution or, equivalently, to the presence of an infinite number of admissible inverse solutions. This complexity is inherited also at the dynamic and control levels. The fact that different arm postures may correspond to the same end-effector location induces some specific undesirable issues that can be classified as follows:

• non-repeatability: a cyclic behavior of the task variables may not correspond to a cyclic behavior of the joint variables;

• self-motions: a non-zero joint velocity may still be present though the end-effector is fixed at a given point, even for nonsingular arm configurations.

These two problems depend on the particular resolution strategy that is being used. If redundancy is controlled at a kinematic level using a homogeneous law of the form q = H( q)p, necessary and sufficient conditions for obtaining a repeatable motion have been stated in [15], requiring the involutivity of the columns of H. Note that a repeatable behavior can be considered as a stability property in the large.

Here, we will be mainly interested in the control of self-motions. In particular, we would like to design a control law which stabilizes the internal arm configuration around a desired equilibrium, while keeping the endeffector at the fixed location p. The notion of zero dynamics is again helpful in finding a solution to this problem. For, define as output

(23) y = h( q) = k( q) - p, with p fixed,

and since

(24) y=J(q)q, y=J(q)q+j(q,q)q,


an explicit dependence of y on u is found through the dynamic model B( q)q + n( q, q) = u. Setting y = y = 0, the zero dynamics manifold follows as

(25) M * {( .) JR2N [k(q) - p] } = q, q E : J ( q)q = 0 .

From y = 0, all control inputs making M* invariant have the form

(26) u* = n( q, q) - B(q)Jt (q)j(q, q)q + B(q) [I - Jt(q)J(q)]v,

which yields in the closed loop

(27) q = -Jt(q)j(q, q)q + [I - Jt(q)J(q)]v,

where the second term on the right hand side is a joint acceleration vector lying in the null space of the Jacobian J. Note that the closed-loop system is described by purely kinematic equations, because dynamic terms have already been cancelled via feedback. Although equation (27) - just as (16) - does not describe correctly the zero dynamics of the redundant system, yet it is a suitable basis for the design of an external input v which stabilizes the arm around the desired equilibrium configuration (q, q) = (qd, 0). Since qd may not be consistent with the fixed p (i.e. k( qd) :I p), it is reasonable to define a projected state error

(28) Kp >0,

in which any suitable gain scaling matrix may be chosen for Kp, e.g. the identity. This error term will be zeroed using the following result, which provides thus a stabilizing control law for self-motions of redundant arms.

Theorem. For the dynamics (27), the choice

(29) v = -Kpq + KEE - jtJ[Kp(qd - q) - q],

with KE > 0, is such that the projected error E in (28) asymptotically tends to zero.

Proof. Define a Lyapunov candidate as V = ~ETE, and note that

and

(31)


Since the projection matrix is idempotent, using ET[I - JtJ] = ET, and applying (29) gives

(32) v = ETt = -ET(v + Kpq) - ET[jtJ + Jtj] [Kp(qd - q) - q]

= -ETKEE ~ O.

Note that the assumption of full row rank for J is never needed, contrary to the arguments used in [16]. Q.E.D.

The obtained stabilization is performed according to projection rules. However, if the zero dynamics concept is exploited in full, one can show that stabilization may be achieved working only within the reduced zero dynamics submanifold. This will be illustrated in the following example.

Example. Consider a planar PPR-robot arm (N = 3) with the third link of length £ (see Fig. 2). For the task of positioning the end-effector

(33)

this robot is redundant.

y

1

Fig. 2 - A planar PPR robot arm

The Jacobian matrix

(34) J = [1 0 -£S3] o 1 le3

x

is always of full rank, and so its pseudoinverse Jt and the projection matrix


We first show that, taking v = 0 instead of (29) in (27), an undesirable limit cycle may be induced as a self-motion of the arm. In fact, the closed-loop dynamics becomes in this case

(36)

so that, entering at t = to a fixed location p = p = (PZ,PII)' with p = 0, and having, say, q3(tO) = 0 but q3(tO) = -y i= 0, would give

q1 (t) = pz -l cos -y(t - to),

(37) q2(t) = PII -l sin -y(t - to),

q3(t) = -y(t - to),

i.e. an endless harmonic motion of the two prismatic joints. On the other hand, there is a stabilizing strategy to be pursued which is also more direct than (29). The idea is to reparametrize the robot joint coordinates in terms of q3 and q3 only

(38) q1 = pz -lc3 = gl(q3), q2 = PII -l83 = g2(q3), q1 = lq383 = g3(q3, q3), q2 = -lq3c3 = g4(q3, q3),

so that the zero dynamics manifold can be characterized as (39) M'" = {(q, ci) E1R6: q1 =gl (q3), q2 =g2(Q3), q1 =g3(Q3, q3), q2 =g4(Q3, q3)}

while the zero dynamics is just q3 = V3. This will be globally stabilized at the value Qd3 by choosing V3 = Kp(Qd3 - Q3) - Ktl Q3, with Kp, Kti both positive. Although kinematically simple, this robot is not a trivial case. For instance, it is interesting to remark that solving redundancy by pseudoinversion, i.e. choosing ci = Jt (q)p, does not yield a repeatable solution. The necessary and sufficient condition is that the columns jl of Jt, seen as vector fields, are involutive. However, this is not the case since

(40)


where [jI ,j~ ] = (oj~/ oq)jI - (ojI / oq)j~ is the Lie bracket of the two vector fields [1] .•

The proposed stabilizing strategy for self-motions could be generalized to the case i> :# 0, and used for keeping under control the robot joint velocity. We conclude this section by noting that the idea of reparametrization of redundant robots and of reduction to a lower-dimensional manifold has been explored also for optimization purposes in [17,18].

5. Robots with flexible elements

There are two types of possible deformation in robots, namely joint elasticity and link flexibility. Joint elasticity is introduced by transmission elements like harmonic drives, belts, or long shafts, and is the basic source of vibration in industrial arms with massive links [19]. The dynamic model of robots with elastic joints is obtained by doubling the number of generalized coordinates, using one variable for the actuator position and a different one for the link position. The resulting model is described by 2N second order differential equations of the form (3), but with only half of the components of m available for control (i.e. m = [INxN ONxNFu, up to a row permutation). Link flexibility, instead, is non negligible for long and/or lightweight arms, like the Space Shuttle Remote Manipulator. Although deflection is distributed in nature, finite-dimensional dynamic models are usually derived, by representing each link as an Euler beam with proper boundary conditions and limiting to Ne the number of modal functions in the associated small deformation eigenvalue problem. Alternatively, the methods of finite elements or of assumed modes can be used for directly approximating the link deflection [20]. Using the Lagrangian approach, the coupling of N rigid motion equations with Ne flexible ones will result in a nonlinear dynamic model that is still in the form (3), but now with m = [INxN ONxNeFu.

Various motion control objectives can be pursued in the presence of flexibility, ranging from point-to-point control with vibrational damping to accurate trajectory tracking. Indeed, trajectories specified at the actuator level produce a different and oscillatory behavior at the end-effector level, both for the elastic joint and for the flexible link case. In spite of this analogy, the trajectory tracking control problem is completely different in the two cases. The basic issue is summarized in the following question:

• is it possible for robots with flexible elements to find an input torque so to exactly reproduce a desired (smooth) end-effector trajectory?

This can be restated as finding whether the computed torque method can be extended also to flexible robot arms. The answer is always positive for robots with joint elasticity. When controlling the end-effector or the link position - both outputs being beyond the elasticity - of these robots, one is dealing with an invertible nonlinear system having no zero dynamics.


Therefore, by the use of static [21] or, when needed, dynamic [22] inversionbased state-feedback, a fully linear closed-loop system can be obtained, equivalent to a set of independent strings of input-output integrators of length greater than or equal to four. These results closely mimic the ones obtained for conventional rigid robots.

In the case of link flexibility, the trajectory tracking problem is much more involved. Instead of following a general but cumbersome formalism, we will focus on a one-link planar flexible arm, modeling just one deflection mode (Ne = 1). This finite-dimensional model, although of reduced-order, is still a representative one in the sense that it displays the same basic control properties of more accurate and/or distributed models. It will be shown that the zero dynamics is useful in the study of the tracking problem, and it is crucial in the design of outputs for which exact trajectory reproduction can be achieved in a stable fashion via inversion control. In particular, the zero dynamics analysis will provide a constructive answer to the question of finding the input that produces a desired end-effector trajectory.

Example. Consider the one-link flexible arm of Fig. 3, moving on the horizontal plane. Following the general modeling technique proposed in [23], based on the Ritz-Kantorovitch expansion for approximating link deformation, we assume here second-degree polynomials as basis functions. Imposing geometric boundary conditions of the 'clamped' type at the link base, it turns out that a parabolic shape is sufficient to describe the pure bending deformation of the link.

Fig. 3 - A one-link flexible robot arm

Let f be the length of the uniform link and m its mass, 10 the inertia of the hub, mp and 1p the mass and inertia of a payload located at the tip, and u the input torque. The angular position ql of the link base and the deflection q2 at the tip point are chosen as generalized coordinates. If the analysis is not limited to small deflections, a nonlinear dynamic model is


obtained in the standard form

( 41) [b1~~:2) :~~] [~~] + [C2?q~~2J1~1 +q~q2] = [~] U,

with elements of the inertia matrix B( q2) and Coriolis and centrifugal terms given by

(42) bll (q2) = a + bqi, b12 = C, b22 = d,

C1(q2,q1,q2) = 2bq2Q1Q2, C2(q2,Qd = -bq2Q~,

where

1 2 2 1 a = 10 + ami + Ip + mpi, b = 5'm + mp,

1 2 1 4 C = 4"mi + IIp + mpi, d = 5'm + i2Ip + mp.

(43)

In (41), k = 4EI/i3 is the link elasticity coefficient, with Young modulus E and link cross sectional inertia I. State equations are derived by setting x = (q1, q2, Q1, Q2) E JR4. For trajectory control, a scalar output can be conveniently defined as the angular position of a generic point along the link, as seen from the base. For simplicity, a linearized version of this output will be used. Its parametric expression is

(44) A E [0,1].

For A = 0, the output is the joint angle, while for A = 1, the output is the angular position of the tip. Being interested in tracking end-effector trajectories, one should consider mainly y(I). Unfortunately, with this choice inversion control, which is the strategy used to guarantee exact tracking, leads in general to an unstable closed-loop behavior. This is because of the presence of an unstable zero dynamics that limits the application of pure inversion control, and is consistent with the usual non-minimum phase characteristics of the transfer function from joint torque to tip position in linear dynamic models of one-link flexible arms [24]. However, the actual situation is more tricky and it is interesting to see for what values of A, i.e. for which output, it is still possible to reproduce exactly a trajectory. For this purpose, an explicit expression should be derived for the system zero dynamics. The relative degree of output (44) is two, except for a particular parameter value Ao which will be characterized later. Then, the synthesis of an inversion-based control is accomplished deriving twice the output, setting jj = v, and solving for u:

.. b12 - ~bll(q2) . U=C1(Q2,ql,q2)- b Ab (C2(Q2,Qd+ kQ2)

22 - I 12

det B(Q2) + A V = a(x) + f3(x)v. b22 - Ib12

(45)

Using the linear change of coordinates i = w(x) = (y, ii, q2, Q2), the closedloop equations become

(46)

It is easy to see that the zero dynamics, restricted to the manifold M* = {( q, q) e R': q1 = - iQ2, Q1 = - H2}, represents in this case the dynamics of the flexible variable Q2, evaluated for y(t) == 0:

(47)

The stability properties of this two-dimensional dynamics can be studied for different values of '\. Whenever the zero dynamics will be found to be asymptotically stable, the following choice for v

(48)

will guarantee asymptotic tracking of Yd(t), or even exact tracking for matched initial conditions. In particular, for ,\ = 0, the linear zero dynamics 9.2 = -(k/d)Q2 is found, having two complex poles on the imaginary axis. The presence of some structural damping in the model would force exponential stability. It follows immediately that joint trajectories can be always tracked in a stable fashion. This is a general conclusion for robots with flexible links, holding even in the multi-link, multi-modal case. For ,\ > 0, it can be assumed that the (state-dependent) coefficient of q2 in the numerator of (47) remains positive, as it is when testing stability in the first approximation. Therefore, the properties of the zero dynamics will depend only on the sign of the denominator A, a function of the parameter '\. The stability condition can be rewritten more explicitly as

) ( ,\ 1) Ip (49 A('\) = 4' - '5 m + (,\ - l)mp + (2,\ - 4) (2 < O.

In the absence of a payload (mp = Ip = 0), asymptotic stability is obtained for all ,\ e [0,4/5). Then, the inversion controller (45), with (48), will 'stiffen' the behavior of any output point which is up to four fifth of the link length (, letting it trace the desired trajectory while keeping the arm deformation bounded. On the other hand, choosing an output associated to ,\ > 4/5 will lead to an unbounded state evolution, once inversion control is applied. The transition from stable to unstable behavior occurs at that link point corresponding to '\0 = 4/5, where the relative degree is larger than two. This particular point can be physically visualized, in terms of in phase or out of phase motion. In fact, with the undeformed arm initially

at rest, this point will have zero acceleration at time t = 0+ in response to a step input torque applied at time t = 0- at the joint, and will separate positive from negative acceleration points. However, note that the location of such a point along the link depends on the mechanical characteristics of the arm. If the robot arm is loaded with a concentrated tip mass mp = kl m (Ip = 0), then

(50) 20k1 + 4

~(~) < 0 {::::::> 0:5 ~ < 20k1 + 5 < 1,

and by increasing the payload mass via kl' the transition point will move closer to the tip, although never reaching it. On the other hand, assuming the payload mass negligible (mp ~ 0) with respect to its inertia Ip = k2m£2, one has

(51) 80k2 +4

~ < 0 {::::::> 0:5 ~ < 40k2 + 5 '

so that the same positive benefit is obtained for a sufficiently large k2 •

Moreover, in this case the critical point ~o becomes greater than unity for k2 > 0.025. As a consequence, by mechanically increasing tip inertia, also an end-effector trajectory can be exactly reproduced in a stable fashion using inversion control. Stated differently, the tip point is not anymore a non-minimum phase output, according to the stability achieved for the associated zero dynamics. _

Combining the above two ideal analyses leads to similar results for a real payload, with both non-zero mass and inertia. This formal result is also confirmed by common experience: in fact, it is much easier to control the end-point motion of a flexible arm when this is subject to heavy (relative to the link mass) loading at the tip. The numerical simulations reported in [25] for a slightly different model of the flexible arm, display the effects of different feasible choices for~. Performance of the resulting controllers are evaluated in terms of tip motion accuracy and control effort. When ~o < 1, increasing ~ in the feasible range [0, ~o) produces remarkable improvements in the end-effector trajectory tracking.

A final comment is in order about the occurrence of instability in case of non-minimum phase output. When the zero-dynamics is unstable, inverse control substantially leads to an unbounded state evolution in the closed loop. However, it is possible to show that there exists a particular initial condition for the arm, depending on the desired trajectory, which still guarantees an overall bounded evolution under pure inversion control. Computation of this initial condition is a by-product of the output regulation theory for nonlinear systems [9]. Further analysis and numerical results for a flexible robot can be found in [26]. We just note here that, if the arm is in a different initial state (e.g. typically undeformed), only the

regulator approach is capable of achieving asymptotic output tracking with bounded internal state. However, if a non-causal solution is admitted [27], one may also find an input torque to be applied for t < 0, i.e. before the start of the actual trajectory, so to lead the system in the required initial condition at time t = o.

6. Conclusions

Robotics proposes several interesting problems where advanced nonlinear control techniques find a natural application. Beside feedback linearization for conventional rigid manipUlators or exact linearization via dynamic feedback for robots with joint elasticity, we believe that use of more recently developed nonlinear tools could also lead to similar relevant results. We have shown here that an important role can be recognized for the notion of zero dynamics of a system.

Zero dynamics was investigated here in relation to some special robotic control problems. Robots in constrained maneuvers were revisited in this key, interpreting the zero dynamics as the description of the end-effector motion on the constraining surface. This provides also a basis for understanding the intrinsic decoupling achievable in the hybrid control of normal force and of tangential velocity. The problem of controlling self-motions in redundant arms, treated from the point of view of zero dynamics, led to the statement of a new stabilization result. The given analysis supports the conclusion that, for stabilizing purposes, it is sufficient to work in the reduced space of the extra degree of freedoms. In the trajectory control problem for robot arms with flexible links, the stability condition for the zero dynamics was found to be a clean guide in selecting system outputs to be used for inversion. In particular, a set of alternate control strategies can be generated by re-Iocating the output point within a feasible range along the link. The chosen approach proved helpful also for showing that the non-minimum phase property of the end-effector control problem strongly depends on the mechanical characteristics of the flexible arm - an aspect which is often overlooked.

We finally remark that the scope of the obtained results is not limited to the relatively simple case studies presented. Ultimately, the unifying perspective offered by the notion of zero dynamics has provided a deeper understanding of the considered robotic control problems.

Acknowledgements

This paper is based on work supported by the Ministero dell'Universita e della Ricerca Scientifica e TecnoJogica under 40% funds and by the Consiglio NazionaJe deJJe Ricerche, grant no. 89.00521.67 (Progetto FinaJizzato Robotica).


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[10] A. Isidori and C.H. Moog, "On the nonlinear equivalent of the notion of transmission zeros," in Modelling and Adaptive Control, C.I. Byrnes and A. Kurzhanski Eds., Lecture Notes in Control and Information Sciences, vol. 105, pp. 146-158, Springer Verlag, 1988.

[11] S.N. Singh, "A modified algorithm for invertibility in nonlinear systems," IEEE Trans. on Automatic Control, vol. AC-26, no. 2, pp. 595-598, 1981.

[12] N.H. McClamroch and D. Wang, "Feedback stabilization and tracking in constrained robots," IEEE 1Tans. on Automatic Control, vol. AC-33, no. 5, pp. 419-426, 1988.

[13] A. De Luca, C. Manes, and F. Nicolo, "A task space decoupling approach to hybrid control of manipulators," Proc. 2nd IFAC Symp. on Robot Control (SYROCO'88) (Karlsruhe, FRG, Oct. 5-7, 1988), pp. 157-162.

[14] D.N. Nenchev, "Redundancy resolution through local optimization: a review," J. of Robotic Systems, vol. 6, no. 6, pp. 769-798, 1989.

[15] T. Shamir and Y. Yomdin, "Repeatability of redundant manipulators: mathematical solution of the problem," IEEE Trans. on Automatic Control, vol. AC-33, no. 11, pp. 1004-1009, 1988.

[16] P. Hsu, J. Hauser, and S. Sastry, "Dynamic control of redundant ma-


nipulators," J. of Robotic Systems, vol. 6, no. 2, pp. 133-148, 1989. [17] A. De Luca and G. Oriolo, "The reduced gradient method for solv

ing redundancy in robot arms," Prepr. 11th IFAC World Congress (Tallinn, Estonia, Aug. 13-17, 1990), vol. 9, pp. 143-148.

[18] A. De Luca and G. Oriolo, "Efficient dynamic resolution of robot redundancy," Proc. 1990 American Control Con/. (San Diego, CA, May 23-25, 1990), pp. 221-227.

[19] M.C. Good, L.M. Sweet, and K.L. Strobel, "Dynamic models for control system design of integrated robot and drive systems," ASME J. of Dynamic Systems, Measurement, and Control, vol. 107, no. 3, pp. 53-59,1985.

[20] L. Meirovitch, Analytical Methods in Vibrations, Macmillan, New York, 1967.

[21] M.W. Spong, "Modeling and control of elastic joint robots," ASME J. of Dynamic Systems, Measurement, and Control, vol. 109, no. 3, pp. 310-319, 1987.

[22] A. De Luca, "Dynamic control of robots with joint elasticity," Proc. 1988 IEEE Int. Con/. on Robotics and Automation (Philadelphia, PA, Apr. 24-29, 1988), pp. 152-158.

[23] S. Nicosia, P. Tomei, and A. Tornambe, "Dynamic modelling of flexible robot manipulators," Proc. 1986 IEEE Int. Con/. on Robotics and Automation, (San Francisco, CA, Apr. 7-10, 1986), pp. 365-372.

[24] R.B. Cannon, Jr. and E. Schmitz, "Initial experiments on the endpoint control of a flexible one-link robot," Int. J. of Robotics Research, vol. 3, no. 3, pp. 62-75, 1984.

[25] A. De Luca, P. Lucibello, and G. Ulivi, "Inversion techniques for trajectory control of flexible robot arms," J. of Robotic Systems, vol. 6, no. 4, pp. 325-344, 1989.

[26] A. De Luca, L. Lanari, and G. Ulivi, "Nonlinear regulation of endeffector motion for a flexible robot arm," in New Trends in Systems Theory, G. Conte, A.M. Perdon, B. Wyman Eds., Birkhauser, Boston, to appear.

[27] E. Bayo, "A finite-element approach to control the end-point motion of a single-link flexible robot," J. of Robotic Systems, vol. 4, no. 1, pp. 63-75, 1985.

Alessandro De Luca Dipartimento di Informatica e Sistemistica Universita degli Studi di Roma "La Sapienza" Via Eudossiana 18, 00184 Roma, Italy

Adaptive methods for

piecewise linear filtering

GIOV ANN! B. 01 MASI and MARINA ANGELINI

Abstract. A nonlinear discrete-time stochastic dynamical system is considered with piecewise linear drift coefficients and whose initial condition and disturbances are distributed according to fmite mixtures of normal distributions. In particular the normal components of the mixtures relative to the state process have variances which vanish with a parameter E.

For such system the nonlinear filtering problem is studied. It is shown that a suitable linear adaptive filtering problem can be constructed whose solution coincide, for vanishing E, with that of the original nonlinear problem. The use of measure transformation techniques allows the derivation of the results under milder condition than those assumed so far in a similar context.

1. Introduction

On a probability space {C,F,P} we consider a discrete-time nonlinear partially observable process described by the state and observation equations

(1.a)

(1.b)

PIECEWISE LINEAR FILTERING 89

where t = O ..... T. XtE Rn. YtE Rm. nX and nY denote noise processes. The filtering problem consists in the evaluation for a suitable function f of the conditional expectation E{f(x01f"Ytl. where :FYt:= a{yo •... ytl. In model (1) the functions at and Ct are piecewise affme. namely

N l\ (x) = L [At (i)x + Bt (i)] I1ti(x) (2.a)

i=I

N ct (x) = L [Ct (i)x + Dt (i)] I1ti(x) (2.b)

i=I

with {1ti:i = I •... ,N} a finite partition of Rn into polyhedra. It is also assumed that the noises are of the form

(3.a)

(3.b)

where Vt and Wt are independent standard white Gaussian noises. ext and eYt are unobserved finite-valued random processes and e is a small parameter that clearly influences only the state dynamics and initial condition. Notice that the noise processes and the initial condition xo are distributed according to finite mixtures of normal densities. In fact. denoting by e xi (i=I •...• NX) and eyi (i=I •... NY) the possible values of eXt and 9Yt respectively. assumed with probabilities

we have

W ntX - L(XxiN (.;jlxt(exi).e2l:xt(exi»

i=I (4.a)

90 DI MASI AND ANGELINI

NY ntY - LayiN(';IlYt(ayi),:D't(ayi»

i=l (4.b)

where the symbol - stands for "is distributed according to" , N(';Il,I:) denotes the normal density with mean vector Il and covariance matrix I: and I:xt=crXt(crx0', I:Yt=crYt(crY0'. Notice that only the variances relative to the state process, more precisely those of the initial condition XO=nxO and of the state noise nXt, depend on the small parameter t, while those relative to the observation noise do not. Furthermore the mutual dependence of the random variables xo, nXt and nY t is related to the mutual dependence of a:= (aXo, ... ,a"T,aYo, ... ,aYT) so that various dependence structures are possible according to the various choices of the joint distribution

In particular, it is possible to deal with the case when the noises nXt and nYt are sequences of independent, non-gaussian random variables. It is also worth remarking that the model considered here is, conditionally on a, the usual model for a discrete-time linear system corrupted by white noise. The introduction of the random vector a allows noise distributions given in terms of mixtures of normal distributions.This, together with the piecewise linear character of the mappings at and Ct allow the interpretation of the problem considered here as an approximation to rather general nonlinear problems. Piecewise linear systems have recently received increasing attention in the context of nonlinear filtering problems [1,3,5,6,10,11] and of stochastic control problems [4] The present paper can be considered as a generalization to [3] with crucial assumptions on the regularity of the coefficients at and Ct dropped. Our aim here will be to study the connections between the solution to problem (1) and the solution to a suitably constructed (bayesian-)adaptive linear filtering problem, namely a problem relative to a linear model with random coefficients (see (13) below). In particular it will be shown that for vanishing e, the optimal solutions to the piecewise linear and to the adaptive linear problems tend to coincide with the optimal solution to a suitable finite-state limit problem. In what follows the notation will indicate a generic dependence on a irrespective of the variables actually involved.


2. Adaptive linear and limit problems

In this section we shall fIrst be concerned with the derivation of the adaptive linear model associated to the original model (1). Second, we shall construct a limit filtering problem whose optimal solution will be approached (for vanishing e) by the solutions of both the original problem and the adaptive problem previously constructed. The technique used here to construct the models we shall be concerned with, is based on measure transformations in discrete time. More precisely, on the probability space {Q,1''po},we start with the reference problem defmed by the family of models parametrized by e

which corresponds to model (1) with Yt =nYt. We shall now associate to the state process X£t a limit process ~t and an adaptive process Xli£t obtained by suitably approximating X£t.We shall for the moment assume that the mappings at and Ct are continuous; in Remark 2 we shall discuss how to extend the results to the discontinuous case. Under this assumption the a.s. limit of the process X£t is given by the process ~t defIned by the equation

We consider now the process

N

11t = L i I1ti (~t) i=l

(6)

(1)

which at each time t provides the element of the partition {1ti: i=I, ... ,N} where ~t lies, so that ~t admits also the representation

(6.a)

where At and Bt are as in (2.a). Notice that the process ~t and consequently 11t depend only on 9 so that we can defme A at(9) and Bat(9) by


(8.a)

and analogously we define

(8.b)

where Ct and Ot are given in (2.b). We now consider the family of processes Xil£t defmed by the adaptive linear model

We shall now derive all the relevant observation models by a suitable absolutely continuous change of probability measures. Oefine the a-algebra Ft:=a{9,vt,wt}, and consider the probability measures P£, pil£ and pI whose Radon-Nikodym derivative with respect to PO have restrictions on F t given respectively by

dJl€\ t A£t= dPo :Tt = !1 exp {cs'(x£s)(D's(9»-1 [Ys -~s(9)] -

-1/2 cs'(x£s)(~Y(9»-lcs(x£s)) (1O.a)

dPli£1 t Ail£t= dPO Ft = !1 exp{(Cas(9)xil£s +oas(9»'(IYs(9»-1[ys -J.LYs(9)]

-1/2 (cas(9)xil£s +oas(9»'(l'.sY(9»-1(cas(9)xil£s+oas(9))} (10.b)

dPII t Alt= dPO 1't = !1 exp {cs'(;s)(IYs(9»-1 [Ys -J.l.Ys(9)] -

-1/2 Cs'(;S>(~Y(9»-lCs(;s)} (10.c)

The following proposition, which is the discrete-time analog of Girsanov theorem, shows that under the three measures defined by (lO.a,b,c) the observation process Yt admits different representations.The proof can be obtained by direct calculation of the suitable probability densities; a proof providing more


insight into the measure transformation approach can be found for example in [4].

Proposition 1. Under the measures pE, paE and pi, the random vector Band the process Vt have the same distribution as under PO. The observation process Yt has under pE the representation

(11.0)

where wet is (PE,Fi)-standard gaussian white noise independent of {vJ and B. Under paE it has the representation

(l1.b)

where wOEt is (paE ft)-standard gaussian white noise independent of {vJ and B. Finally, under pi, Yt has the representation

(l1.c)

where wit is (pl,l't)-standard gaussian white noise independent of {vJ and B.

Remark 1. Notice that on the spaces characterized by the measures p£,P8£ and pI we can now consider three different filtering problems having the same observation process. More precisely, under P£ we have the problem with state X£t and observation Yt as given by (11.a), namely

This can be considered as the original problem (I) and its solution can be represented using the formula for conditional expectation under a change of measure [9], which is here more conveniently written as

(12.c)


where

t U\= II exp -1/2 ([Ys -~s(9) -cs(x£s)]'(~Y(9»-1 [Ys -~s(9) -cs(x£s)]} (12.d)

s=O

Under pa£ we can consider the state Xli£t with observation Yt as given by (l1.b). This is the adaptive problem

Analogously to (12.c), we have for its solution

t LII£t= II exp -1/2{[Ys-~s(9)-cas(9)xll£s-Das(9)],(I.sY(9»-1

s=O

(13.c)

Finally, under the measure pI we can consider the limit problem defined by the state ~t and the observation as given in (11.c), namely

~t+l = ~(~J + J.1\+l(9)

Yt = Ct(~U +J.1Yt(9) + crYt(9)wIt

and we have

t Lit = II exp -1/2{[ys -J.LYs(9)-cs(~s)]'(l:sy(e»-1[ys -J.1Ys(9)-cs(~s)]}

S=O

(14.a)

(14.b)

(14.c)

(14.d)


3. Asymptotic approximation

In this section we show that for e-/'O the solutions to the adaptive problem (13) and to the original problem (12) converge to the solution to the limit problem (14) . Therefore, for small e, any of the above problems can be interpreted as an approximation to the others. Also notice that both the adaptive linear problem and the limit problem admit an explicitly computable, finitedimensional solution. In fact the limit problem is finite-state and its solution can be obtained as in [7,13]. On the other hand the adaptive problem is a conditional (on 9) linear problem, parametrized by the fmite-valued parameter 9; an algorithm for its solution can be found in [3] (see also [2,12]. The results of this section then show that the solution (12.c) to the original nonlinear flltering problem admits two explicitly computable, finite-dimensional, asymptotic

f

approximations, namely the adaptive linear approximation (13.c) and the limit approximation (14.c).

Theorem 1. Let f be piecewise continuous and with polynomial growth. then for eJo

E£{f(x£t) / FltJ ~ El{f(~t) / FltJ

Ea£{f(xfJ£t) / FltJ ~ El{f(~t) / FltJ

and (15.a)

(15.b)

Proof. We shall first prove (15.a). Using the representation (12.c), it is enough to show that as e-/'O

(16)

Notice first that for e-/'O and for each ro EO f(xEt(ro» converges to f(~t(ro». Analogously c(xEt(ro» converges to c(~t(ro». Therefore. using (12.d) and (14.d) it is clear that as £-/,O we have for all ro E 0

We now show that I Utf(xEt> I is majorized by a Po-integrable function. In fact taking into account that by (12.d) I U t lSI and using the polynomial growth of f. we have for suitable constants Fl.F2 and positive integer k


On the other hand. by the piecewise linear nature of atO and the finiteness of e we have from (12.a)

for £:s; I and suitable Hl.H2.H3. It is then easy to show by induction that for £ :s; I and each t = O.I ..... T we have

t

ILEtf(xE01:s;at+L~ts I Vs Ik S=O

with at and ~ts suitable constants and where the right hand side is a Pointegrable random variable. Using the conditional form of Lebesgue dominated convergence theorem. (16) follows immediately. Convergence (IS.b) is proved in a perfectly analogous way.

Q.E.D.

Remark 2. Theorem I can be extended to a discontinuous situation. namely when at(') and CtO in (12.a,b) are piecewise linear but not necessarily continuous and fO in (12.c) is of polynomial growth but only piecewise continuous. This situation is apparently important also in connection with the study of estimation properties of chaotic systems; in fact some classes of such systems of particular relevance in applications are characterized by a piecewise linear dynamics (see e.g. [8]). The crucial point· for the extension to the discontinuous situation is the construction of the adaptive linear model. Denoting as before by ~t the a.s.limit of the process XEt and letting gO be a generic piecewise continuous function, the limiting operations on g(xE0 lead to the quantities g(~t+) and g(~d. It is then convenient, instead of referring to ~t to resort directly to a limiting state process Vt assuming the formal values ~t- and ~t+ with suitable probabilities. Notice however that because of the discontinuity of atO in (12.a), the evolution ~t and also that ofVt do not depend only on the random variable e but also on the entire history of the noise process vt = {VO, ... ,vtl. The adaptive linear model can then be constructed by a procedure similar to the one followed in the continuous situation, but it is characterized by a parameter depending on e and vt. The adaptive model is then, conditionally on this parameter, a linear


model with nongaussian initial condition and disturbances, and in general it does not admit a rmite-dimensional solution. On the contrary, the limit model can be constructed in the usual way and because of its finite state space it can be solved by a finite-dimensional procedure. Consequently, in the discontinuous situation the solution to the limit problem is the only explicitly computable, finite-dimensional, asymptotic approximation to the given problem (12).

REFERENCES

[1] V.Benes and 1. Karatzas, Filtering for Piecewise Linear Drift and Observation, Proc. 20th Conf. on Dec. and Control (1981), 583 - 589.

[2] G.B. Di Masi and W.I. Runggaldier, On measure transformations for combined filtering and parameter estimation in discrete time, Sys.&Control Letters 2 (1982), 57 - 62.

[3] G.B. Di Masi and W.J.Runggaldier, Asymptotic Analysisfor Piecewise Linear Filtering, in Analysis and Optimization of Systems (A. Bensoussan and J.L. Lions eds.), Springer Verlag, L.N.in Control and Info Sci. 111, 1988, 753-759.

[4] G.B. Di Masi, W.I. Runggaldier, Piecewise linear stochastic control with partial observations, Proc. Imperial College Workshop on Applied Stochastic Analysis (M.H.A. Davis and R.I. Elliott eds.), Gordon & Breach, Stochastic Monograph Series, New York (to appear).

[5] W.H.Fleming, D. Ji and E. Pardoux,(1988) Piecewise Linear Filtering with Small Observation Noise., in Analysis and Optimization of Systems (A. Bensoussan and J.L. Lions eds.), Springer Verlag, L.N.in Control and Info Sci. 111, 1988, 752-759.

[6] A.E.Kolessa, Recursive Filtering Algorithms for Systems with Piecewise Linear Nonlinearities, Avtom. Telemekh .. 5 (1986),48-55 (English Translation: 480-486).

[7] R.S. Liptser and A.N. Shiryayev, Statistics of random processes, Springer-Verlag, New York, 1978.

[8] A.Lasota and M.C. Mackey, Probabilistic properties of deterministic systems, Cambridge University Press, Cambridge, 1985.

[9] M. Loeve, Probability theory, Van Nostrand Reihold Company, New York,1963.


[10] E. Pardoux and C. Savona, Piecewise Linear Filtering, in Stochastic Differential Systems, Stochastic Control Theory and Applications (W.H. Fleming and P.L. Lions eds.), Springer Verlag, IMA Volume in Mathematics and Applications 10, 1987.

[11] C. Savona, C. Approximate Nonlinear Filtering for Piecewise Linear Systems. Systems and Control Letters 11 (1988), 327-332.

[12] F.L. Sims, D.G. Lainiotis and D.T. Magill, Recursive algorithm for the calculation of the adaptive Kalman filter weighting coefficients (Correspondence), IEEE Trans.Autom.Control AC·14 (1969), 215-218.

[13] W.M. Wonham, Some applications of stochastic differential equations to optimal nonlinear filtering, SIAM J. on Control 2 (1965),347 - 369.

Giovanni B Di Masi UniversitA di Padova Dipartimento di Matematica Pura ed Applicata Via Belzoni, 7 35131 Padova, Italy

also with CNR - LADSEB

Marina Angelini Via A. Aleardi, 49 30172 Venezia - Mestre, Italy

Nonlinear Feedback Control for Flexible Robot Arms

Xuru Ding T. J. Tarn A. K. Bejczy

Modeling and control of robot arms with nonnegligible elastical deformations have invited researches in the recent years, due to the demands on robot arms which are of lighter weight, move faster and consume less energy. In this paper, the authors present a nonlinear, distributed-parameter dynamic model for a two-link robot arm, derived using Hamilton's principle. The dynamic model is then transformed into state-space expression, which gives an infinite-dimensional dynamic system. Position of the tip of the robot arm is chosen as the output of the dynamic system. A nonlinear feedback law is proposed to achieve input-output linearization and decoupling, which is an extension of input-output linearization and decoupling of finite-dimensional dynamic systems. Stabilizability of the system under the feedback is then studied in light of the concept of zero dynamics and perturbation theory of infinitedimensional dynamic system.

Dynamic Model

In previous publications, the authors derived the dynamic model for a large class of flexible robot arms under very general assumptions [1]. In this paper, however, we shall limit our discussions mainly to a typical two-link flexible robot arm. The robot arm consists of two thin beams connected in series (Figure 1 and Figure 2). Rotation of each beam is controlled by an actuator. The motion of the robot arm is confined to a plane, and so is the deflection of each link. The deflection at a point on the neutral axis of a link, denoted by wi'

is a function of the rigid position of that point x. and time t. Suppose that the shear strain and

1

the longitudinal strain are nE!gligible and the deflection is small compared to the length of the

lOO DING, TARNANDBEJCZY

link. Hook's law is assumed for linear elasticity. The strain energy of the robot arm is given by

1 2 L

2 i-l

2 2

f [a wi (xi' t)] Ei I. 2 dx.

Link i 1 ax. 1 1

where E. is the Young's modulus of link i. I. is 1 1

the inertia of the cross section of link i. Notice that the deflection of link i, wi' is expressed

with respect to the local coordinate system of link i, Xi -Yi ·

In order to obtain both the kinetic energy and the potential energy of the whole system, the kinematics of the robot arm has to be considered. Adopting the homogeneous coordinates provided by the Denavit-Hartenberg four-parameter representation and the frame structure by Richard Paul [2] for the undeformed robot arm, incorporating the effects of elastical deformation of each link, we obtained the homogeneous transformation matrices from X.-Y. (i=l,2) to Xo-

1 1

YO' denoted by T .. The entries of T. are functions 1 1

of both the joint angles 0i (i=l,2) and deflections

at the end of each link. With the help of these homogeneous transformation matrices, position and velocity of an infinitesimal mass element of a link can be expressed with respect to the world coordinate system XO-YO attached to the base of the

robot arm, and hence the kinetic energy and potential (including gravitational and elastic) energy can be calculated. Hamilton's principle reads

~t

f (oL + oW) dt = 0 o

where "0" denotes variation, L denotes the Lagrangian of the system which is the difference between the kinetic energy and the potential energy, and W is the work done by the actuators. Variations of the Lagrangian and the work done were calculated and integration by parts with respect to time and the spacial variable xi was carried out to

FEEDBACK CONTROL FOR FLEXIBLE ROBOT ARMS 101

obtain a set of integro-partial-differential equations and the boundary conditions associated with them. A condensed form of the dynamic model is the following:

[0 1 0]

2 d r l T'--=

1 dt2

(la)

[0 1 0]

(lb)

(2a)

(2b)

(2c)

(2d)

and

102

A.= ~

[°501 sinO.

~

0

[C~S£16 s~n£16

0

DING, TARNANDBEJCZY

-sinO. Li cos01J ~

cosO. L.sinO. ~ ~ ~

0 1

i=1,2

-sin£16

£~2J cos£16 0

In the above, £12(t)=w1 (x1 , t)lx -0 is the 1-

* * translational displacement of the origin of X1 -Yl to the origin of X1 -Y1 and £16(t) is the rotational

* * angle from Xl - Yl to X1 -Y1 and we have tan £16=

aw1 (x1 ,t) aX1 Ix1=0 (Figure 2). Notice that the

integro-partia1-differentia1 equations are in hybrid variables, i.e., joint angles and deflections. The boundary conditions associated with the point where the two links are connected, Eqn (2c) and (2d), are in the form of dynamic equations, which can degradate to simpler constraints under certain arm configurations.

This dynamic model includes the inertia forces, centrifugal and Coriolis forces due to the joint angle motion as well as the deflection along the links. It describes the vibration of the beams


with moving boundary points whose motions are actuated by external forces applied at the joints. This alone determines the strong coupling among the variables and the nonlinear feature of the dynamic model. Let ql(xl,t)-wl(xl,t), qz(xz,t)=wz(xz,t),

q3(t)=Ol(t), q4(t)=OZ(t)+£16(t), qS(t)=£lZ(t), q6=

aWl awz at ' q7-at ' qS-Ol' qg=oZ(t)+£16' qlO=£lZ' Then the dynamic model is transformed into its statespace expression,

Z q - f(q) + L gi(q) u i

i=l (4)

with boundary conditions, where q is the tendimensional vector of state variable, including joint angle, deflection and their time derivatives; f and g are (nonlinear) operators acting on q, u.

~

(i=l,Z) is the torque applied by the actuator to

the ith joint. Let the output of the system be the tip position (end-effector position if there is one) of the robot arm with respect to the world coordinate system, denoted by y, and

Yj = hj(q) , i=l, Z (S)

where h. is a nonlinear operator. Note that since J

y is a point measurement, h. includes evaluating J

components of q at certain points. This introduces difficulty in insuring the continuity of the operator hj' j=l, Z.

To circumvent this problem, Sobolev spaces are utilized as the state space [4]. This basically introduces a new norm to the state space which then guarantees the smoothness of the operators.

The state-space expression of the flexible robot arm gives a nonlinear infinite-dimensional system in hybrid variables and with two point inputs and two point outputs.

Nonlinear Control

Consider the input-output relation of the flexible robot arm represented by Eqns. (4) and

104 DING, TARN AND BEJCZY

(5). Analogous to the case of finite-dimensional nonlinear dynamic system, we define the problem of input-output linearization (by static statefeedback) to be that of finding a state

transformation ~: q ~ [~] and a feedback

u = a(q) + ~(q)v

such that the closed-loop system in the new variable becomes

" "

y == Cr " " "

(6)

where A, B, C are matrices, v is a new reference -1 input vector, ~ and ~ are both smooth mappings up

to some order >1. Likewise, define the problem of input-output decoupling (via static state-feedback) to be that of finding ~ and u in the above forms such that the new input v and the output are noninteracting. The words "static state-feedback" emphasize the fact that the values of the input u at time t depends on the values at t of the state q and of v. On the other hand, a dynamic feedback means that the value of u at time t depends on the values at t of the state q, and of v, and on the value of an additional auxiliary state vector z -(zl' ... ,zk)· In this case,

u a(q,z) + ~(q,z)v • z 1(q,Z) + S,q,z)v. (7)

In this paper, it is the dynamic state-feedback that we shall employ to achieve input-output linearization and decoupling.

Using the notation of Lie derivatives of smooth operators, "relative degree" rand "decoupling matrix" A(q) can be defined and calculated for the infinite-dimensional system (4) and (5) as follows:

rl-l rl-l rl-l L Lf hl(q) L Lf h2(q) L Lf hl(q) gl g2 gm

A(q)= r 2-l r 2-l r 2-l

L Lf hl(q) L Lf h2(q) L Lf hl(q) gl g2 gm

r -1 r -1 r -1 L L m h2 (q) L L m h (q) ... L L m h (q gl f g2 f m g f m m

where m = number of inputs = number of outputs, r = [r l , r 2 ,···,rm], r.

J is the integer such that

r. -1 L L~.=O for all k<r.-l and l~i~m but L LfJ h.~O, gi J J gi J

for some i. If A(q) is nonsingular, r is called the relative degree of the system. It is then proven that the input-output decoupling problem for system

(4) and (5) can be solved locally around q by static o

state-feedback (7) with nonsingular ~(q) if and only

if the decoupling matrix A(q) is nonsingular at q . o

In addition, if this is the case, the input-output linearization problem will be solved simultaneously.

The scenario of this kind of input-output linearization and decoupling is to differentiate each output variable for necessary times until at least one input variable appears explicitly, and all the input variables can be solved by inverting those explicit expressions. This technique was applied to rigid-body robot arms [3]. The difference here is that the system is an infinite-dimensional one.

For the two-link flexible robot arm (4), calculation shows that the decoupling matrix is always singular, due to the fact that u2 appears

explicitly in the second-order time derivatives of both components of y, while ul does not appear in

either of them. Therefore, static state-feedback cannot achieve input-output linearization and decoupling for the chosen output variables. To solve this problem, an integrator is added to the second


input channel so that u2 is integrated before it

enters the system (Figure 3). The extended system with the integrator is described by

[ ~ ] == [f(q) +og2(q)Z] + [g~ (q) ~] [~~]

- f(q,z) + g(q)u (8)

and Yj = hj(q), where z is the output of the added

integrator. The decoupling matrix of the extended system (8) is

rl-l _ rl-l_

[Lgl Lf hI (q) Lg2 Lf h2 ( q) J

A(q)= r 2-l _ r 2-l_ L- L- h (q) L- L- h (q) gl f 2 g2 f 2

where q- [i] is the state vector of the extended

system, and r l -3, r 2==3. The decoupling matrix A(q)

of the extended system (8) is nonsingular generically. This enables the input-output linearization and decoupling via the feedback of the extended state (q, z)':

u - a(q, z) + ~(q, z)v (9)

where a(q, z) __ (A)-I

~(q, z) == (A)-I.

Let ¢: [i] -+ [~] be given by


r 1 "" hI' r 2 "" L_hl(-hl ) , 2 ..

r3 = L_h1(-hl ) f f

r 4 - h2' - L_h2(-h2)' 2

rS r6 - L_h2(-h2) f f

'71 - ql' '72 "" q2' '73 - q6

'74 - q7' '7S - qlO· (10)

Both ~ and its inverse ~-l are smooth for generic arm configurations and hence ~ qualifies as a state transformation.

Under the state transformation (10) and the feedback (9), the extended system is decomposed into two subsystems

" "

The first subsystem of (11) is linear, controllable, of dimension six and is input-output linearized and decoupled, while the second subsystem is nonlinear,


describes the vibration along the links and will be influenced by the first subsystem.

A feedback of the form ,.. ,.. v ,. Fr + v

can be used to stabilize the linear subsystem, ,.. ,..,.. provided that the eigenvalues of (A + BF) are all in the left-half complex plane and the deflection along ,.. the links are small. v can be designed to set the tip of the robot arm at desired positions.

In order to study the stability of the full system under the feedback control, consider the second subsystem in (11) which represents the vibratory motion along the links when the position, velocity and acceleration of the tip point are regulated in certain fashion by v through the first subsystem.

If v is such that it keeps the tip at the same position for all t~O, the corresponding subsystem (ii) is called zero dynamics of the system which are governing dynamic equations of the beam vibrations under the constraint that the tip point is kept fixed by external forces. By studying the stability of this zero dynamics and applying perturbation theory of infinite-dimensional systems, conclusion of local stability of the full system is reached in the presence of viscous damping and structural damping.

To summarize, a nonlinear feedback is used to control the position of the tip of a flexible robot arm, while the vibration along the links are stabilized. The feedback depends on joint angles, deformation, their velocities and accelerations.

References:

[1] X. Ding, T. J. Tarn and A. K. Bejczy, "On the Modeling of Flexible Robot Arms," Progress in Robotics and Intelligent Systems, C. Y. Ho and G. Zobrist, Editors, to be published.

[2] R. P. Paul, "Robot Manipulators: Mathematics, Programming and Control," MIT Press 1981.


[3] T. J. Tarn, A. K. Bejczy, A. Isidori and Y. Chen, "Nonlinear Feedback in Robot Arm Control," Proceedings, 23rd IEEE Conference on Decision and Control, Las Vegas, Nevada, Dec. 1984.

[4] X. Ding, T. J. Tarn and A. K. Bejczy, "Nonlinear Feedback Control of Flexible Robot Arms," Progress in Robotics and Intelligent Systems, C. Y. Ho and G. Zobrist, Editors, to be published.

Acknowledgement. This research was supported in part by the National Science Foundation under grants ECS-8515899, DMC-8615963.

Xuru Ding General Motors Corporation 30001 Van Dyke Avenue Warren, MI 48090

T.J. Tarn Washington University Campus Box 1040 St. Louis, MO 63130

A.K. Bejczy Jet Propulsion Laboratory California Institute of Technology Pasadena, CA 91109

110 DING, TARNANDBEJCZY

(JoInt. are clamped)

Figure 1. Coordinate Systems of a Two-Joint Flexible Robot Arm

Lz -----~

Figure 2. Variables in the Dynamic Model


--------~.~I~_s ____ ~-------.-- z

Figure 3. Adding Integrator to Input Channel

METHODS OF NONLINEAR DISCONTINUOUS

STABILIZATION

Michel FLIESS, Fran90is MESSAGER1

ABSTRACT : We review, with two examples, some recent advances in

nonlinear discontinuous control.

1. INTRODUCTION

Since Sussmann [17], it has been well-known that as opposed to the

time-invariant linear case, controllability of a nonlinear system

does not imply the possibility of stabilization by a smooth feedback.

Although it is still a very natural and challenging problem to

stabilize a nonlinear system with smooth control as done by Byrnes and

Isidori [4,5], via zero dynamics, here we will try to achieve this aim

by introducing discontinuties on the control variables. This kind of

control has existed for a long time with relay (Tsypkin [18]), sliding

modes (Utkin [19]) and, of course, optimal bang-bang control. Through

numbers of papers, from Arstein [2], Sontag [16], Hermes [11], Fliess,

Chantre, Abu e1 Ata and Coic [10], it is interesting to see that many

authors have tried to rehabilitate this kind of control.

We present here two examples of systems which were shown not to be

stabilizable by smooth feedback :

1 Thesis work supported by a CIFRE convention between the CNRS end AML.

NONLINEAR DISCONTINUOUS STABILIZATION 113

The first one, taken from Aeyels [1], is treated by introducing a

generalized controller canonical form [8] to define a kind of

linearizing feedback. By choosing the right parameters we are able to

express a discontinuous stabilizing state feedback. The introduction

of switching curves permits to take into account the structure of the

system.

The second one, taken from Brockett [3], is solved by purely geometric

means, i.e. by partitioning the state space.

2. FIRST EXAMPLE

The generalized controller canonical form is obtained by methods

resulting from differential algebra. (See [7,8] to have some

explanations of differential algebra applied to nonlinear systems).

2.1. THE GENERALIZED CONTROLLER CANONICAL FORM (GCCF)

2.1.1. Introduction of Dynamics (cf. [8])

Let k be a given differential ground field of characteristic zero. Let

k be the differential field generated by k and the elements of a

finite set u = (~1' ... , urn) of differential quantities. A dynamics is

a finitely generated differentially algebraic extension K/k.

The (non-differential) transcendence degree of K/k is finite, say

n. Take a finite set ~ = (~1' ... , ~v), v ~ n, of elements in K, which

contains a transcendence basis of K/k. Each of the deri vati ves

~l' ... '~V are k-algebraically dependent over ~:

Ay ( ~v' ~, u, ':', ... , u (Q) = 0 ,

114 FLIESS AND MESSAGER

where AI' ... ' Ay are polynomials over k. We should stress that the

above differential equations are implicit. From an implicit function

theorem type argument, we get explicit differential equations

~l = a l (~, u, U, ... , u (U))

~v = ~ (~, u, U, ... , u (U) ) •

The explicit form is only locally valid, i.e., in domains where the

Jacobian matrix has full rank.

We like to think of ~ as a generalized state or a state for short. The

integer v is called its dimension. A minimal (generalized) state,

i. e., a state of minimal dimension, is a transcendence basis of

K/k; its dimension is n. Such a state is characterized by the

k-algebraic independence of its components.

Take two minimal states x = (xl' ... ' x n ), x component of x is k-algebraically dependent over the components of

x. There exist thus polynomials PI' ... ' Pn over k so that:

• (U)_ u, ... , u ) - 0

- • (U)_ Pn (xn' x, u, u f ••• , U ) - 0,

because (by minimality) also the components of x are k

algebraically dependent over those of x.

2.1.2. Differential primitive element

The theorem of the primitive element can be extended to a

differential setting (cf. [13]). The theorem of the differential

primitive element states that there exists a single element ~ e K,

which is a differential primitive element, such that K = k<u,~>

(i.e. K is differentially generated by k and ~) .

The transcendence degree n of K/k is the smallest integer such as

~(n) is k-algebraically dependent of ~,~, ... ,~(n-l). Set (n-l)

ql - ~, ... , qn - ~ . So, q = (ql'···' qn) is a transcendence basis

of K/k, which yields the equations :

. . ( ) C(qn,q,u,u, ... ,u n ) o

This is what we call a generalized global controller canonical form

(cf. [8]) (where C is a polynomial over k). We may locally obtain a

generalized local controller canonical form :

. . ( ) qn - c(q,u,u, ... ,u n )

This GCCF is also used by Sira-Ramirez in [15] to stabilize the

descent of a thrusted vehicle on the surface of a planet with non

negligible atmospheric resistance.


2.2. THE DISCONTINUOUS STABILIZATION METHOD (see also [9])

Let us consider the n-dimensional dynamic

{c fl (~,u)

(E)

fn(~'u)

where ~= (~l'''''~n) and u = (ul'".,um). We suppose that fl'".,f n are

polynomial functions with real coefficients.

The study of the last paragraph shows the existence of a differential

primitive element Xl leading to the GCCF :

where x

. . ( ) G(xn,x,u,u, ... ,u m )

(Xl'" .,xn), and therefore to

= X n

Xn = g(x,u,~, ... ,u(m»

o

which is, in general, just locally valid. The equality

(E) . ( ) g(x,u,u,,, .,u m )

defines a linearizing feeback, where the ai'S and the bi's are real

constants and v = (vl' ... ,vp) is the new input. In order to stabilize


the system we solve (E) with the input u. The feedback is obtained by

putting u in the system (~g') during a short sampling period ~t.

2.3. EXAMPLE

The following system :

x + y3

f(x,y) + v

with a single input v and f(O,O)

feedback (Aeyels (1).

0, is not stabilizable by smooth

2.3.1. Fixst Study

By a first smooth feedback as u = f(x,y) + v, (~) becomes

(~') x + y3

= u

Here, we have to reveal 4 zones depending on the sign of x and x, thus

depending on the state position in regard of the curve x + y3 = 0.

y

B' -- ~ A

A' x

.> A A' - +

3 0 - x y =

B

Remark : The arrows, in the figure, represent the way of the free

state evolution in the plane.

The plane is divided into

A { (x,y) " ]R2 / x > 0, x + y3 > OJ

B { (x,y) " ]R2 / x ;::: 0, x + y3 ~ OJ

A' { (x,y) E ]R2 / x < 0, x + y3 < OJ

B' = {(x,y) E ]R2 / x ~ 0, x + y3 ;::: OJ

If (x,y) " BUB', x and x are opposed, Aeyels [1] showed that there

exists an input u driving the state (x,y) from zone BUB' to the

origin.

If (x,y) e A U A', x and x have the same sign, the system could not be

stabilized directly. With the same argument as above, there exists an

input driving the state (x,y) from zone A (resp. A') to

(resp. B').

2.3.2. Choice and computation of the input

2 .3.2 . 1. Zones A and B

Ix.:.") e A

zone B

We easily check that (x - u) is a differential primitive element.

Setting

- u

we obtain the following GCCF

(l:gl) {~, -x2

x2 = -u + (x2+u ) + 3u (X2+~ - {X1+U ))2/3

The equality

defines a linearizing dynamic state feedback, where a 1, a 2 and bare

real constants and w the new input. For obvious reasons of

stabilization, a l and a 2 will be taken negative and w the new input as

zero.

We consider (E l ) as an equation in u, u et u which is solved at the

sampled period t - n~t with

We can compute the input as a Taylor expansion (truncated at 3rd

order) u (t) - At2 + Bt + C and latch it in the system for the next

period n~t < t < (n+l)~t. So the state leaves the A zone and x

still has the same sign.

(x.yl • B

In ordre to stabilize (I') we will impose to the x component a

linear evolution, as x x. So we impose the state following the

curve 2x + y3=O since :

x - - x <=> x + y3 = - x <=> 2x + y3 0

This curve appears as a switching curve. If the state is above this

curve we keep the last control which will reach the state to the other

side of the switching curve.

B' Y -_....;. __ + __ :--___ x

-----x + y3= 0

3

B 2x + y = 0

When (x,y) crosses this curve we change the kind of control to find

one which can reach the state in the neighbourhood of this curve and

favorize the stabilization of y. Here we choose the primitive element

(x + u). Setting :

{Yl x + u

(F2) + y3 Y2 Yl = x + u

we obtain the following GCCF

(Ig2 ) {y, - Y2

Y2 = u + (Y2-u ) + 3u (Y2-~ - (Y1-u) ) 2/3

The equality

defines a linearizing dynamic state feeback, where a 1, a 2 and bare

real constants and w the new input. For obvious reasons of

stabilization, a 1 and a 2 will be taken as negative reals and w the new

input as zero.

We consider (E 2) as an equation in u, u et u which is solved at the

sampled period t = ndt with

u = (Y1 - Y2)1/3

u = Y2 - Y1 + (Y1 - Y2)1/3

u = alYl + a2Y2 - Y1 + (Y1 - Y2)1/3

We can compute the input as a Taylor development (truncated at 3rd

order) u (t) = At2 + Bt + C and put it in the system for the next

period ndt < t < (n+1)dt. The state goes across the switching curve by

stabilization of y.

Remark : Kawski [12] used a similar control, with a fractional power

of the state and, therefore, with an infinite derivative at the

origin.


The concatenation of those two controls imposes the state to reach a

neighbourhood of the origin. In fact while in the B zone, the state

arrives in a neighbourhood of the origin, the control used can, if we

take a sampled period which is too long, makes it leave this area as

shown in the following figure

v A

o

That is why the state is not asymptotically stable at the origin but

just in a neighbourhood of it. It is clear that, from a practical

point of view, this is not an important restriction.

2.3.2.2. zones A I and B'

The inputs computed in section a are symmetrical with respect to the

origin. Thus, we conserve in zone A' (resp. B ') the controls of zone

A (resp. B).

2.3.3. Algorithmic

The logic of computation does not depend on the region where the state

takes place. Only the equations change to allow us to compute the

control. The computing algorithm is defined as follows

1°_ Sampling of (1:') at the t = n~t instant

state (x,y).

acquisition of the

2°_ Computation of the virtual state (1:g1 ) by using the (F 1)

formulas after a new initialization of the control.

3°- Identification of A, Band C parameters of the control u (t) by

solving (E i ).


4°_ Application of this control in (k') during a short sampled period

~t. The processing is started again from the first step, one ~t after,

at the t = (n+1)~t instant.

2.3.4. Results of simulation

The following curves illustrate two simulations defined by the

parameters

~t = 0.02 s.

-10 and the sampled period is

The first simulation (see figures 1 and 2) starts at (x,y) = (5,10).

The second simulation (see figures 3 and 4) starts at (x,y) = (0,-

10) .

y

100 10

80

60

40

0t-------~--------r_----t__r~ x 20 90

-2

-4 Y

-6 -20

Figure 1 Figure 2

y


3. SECOND EXAMPLE (see also [14])

We propose a switching control criterion in our stabilization approach

through discontinuous feedbacks. The introduction of this criterion

allows us to adapt the inputs according to the kind of evolution

adopted by the system. rts determination comes from the idea that we

want to highlight zones where the system has different evolutions,

which is usual in nonlinear situations.

We illustrate this technique through an application taken from

Brockett [3], which was shown not to be stabilizable through smooth

feedbacks.

3.1. 'l'HE BROCltE'l''l' EXAMPLE

Brockett [3] introduced two theorems which, based on smooth feedbacks,

allowed him to determine the stability of a general system defined as:

:it f(x,u)

where xeJR n , ueJRm and f(xo'O) = 0 with Xo as an equilibrium point.

A first consequence can be summarized as follows there exists a

bi-dimensional smooth control which makes the rigid body stable (see

also Aeye1s [1] and Byrnes and Isidori [5]).

He also defined, using these two theorems, "a counter example to the

oft repeated conjecture asserting that a reasonable form of local

controllability implies the existence of a stabilizing control law".

This example is presented in the following manner :

There exists no smooth control law: (u,v)

makes the origin asymptotically stable for

(:E) {; ~: z = xv - yu

(u(x,y,z),v(x,y,z» which


Giving up the requirements of a smooth control, we can determine, as

Sussmann foresaw ten years ago [17], a stabilizing control for a

system which could not be stabilized by smooth feedback.

Remarks : This system has a very particular nonlinear structure, as is

shown in the following properties :

- every triplet (x,y,z) e 2 3 is an equilibrium point of the system

(~). Indeed, zero entries u and v suffices to stop the evolution of

the system.

as it is shown below, the speed, the acceleration and all

derivatives of this system have the same structure. A derivation of

the system requires only the substitution of inputs with their

derivatives

f" = u {"" (2)

u (2) (3) (2)

etc ... y v y v (3) (2) (2)

(2) z xv - yu z xv - yu

Except at the origin, we could also control the state and all of its

derivatives using a command such as the one developed in [9]. As it is

shown below, this has no utility because a piecewise constant control

allows us to stabilize the system.

3.2. PRJ:LIMINARY STUDY

System (1:) presents two independent inputs u and v, each acting

linearly on a different couple of the state :

u-+(x,z)

v -+ (y,z)

The kind of control to be calculated is chosen according to a

criterion which is a function of the state. Such a criterion is

defined in order to partition the state space between evolution zones

of the system. It can be formulated as fOllow :


One of the two inputs will be consecrated to the stabilization of the

state component, x or y, whichever's absolute value is the higher,

whereas the other will be consecrated to stabilize the last

component z.

We have revealed two switching planes, whose equations are

Ixl-Iyl

or o o

Here is the projection on the (x,y) plane.

x

3.3. CHOICE AND COMPUTATION OF THE CONTROLS

we shall distinguish the cases according to the Zl or Z2 zones. As has

been specified in the criterion statement in the preceding paragraph,

the controls will be computed in the Zl zone (resp. Z2) to favour the

stabilization at the origin of the state couple (x,z) (resp. (y,z».

Zl zone

A simple way of stabilizing, or rather of heading for a stabilization

at the origin, in the Zl zone, is to choose the inputs so as to obtain

x - -x and z =-z

126

which yield

FLIESS AND MESSAGER

U !;II: -x and v -~ x '" -y

z X

It is clear that numerical problems will occur when the state z is

considerably greater than x. Indeed, contrary to the z component, when

the first component is close to the origin the input v will diverge.

These problems will be investigated in the chapter on singular cases.

Z2 zQne

Like in the Zl zone, we choose the inputs so as to obtain

y - -y and z =-z

which yield

v = -y and u '" ~ = -x +!. y y

As above, if the component y is considerably greater than z, the input

u will diverge. These problems will also be investigated in the

chapter on singular cases.

3.4. COMPUTATION METHOD

3.4.1. Singular cases

AS seen before, the control risks of divergence occurs when, in Zl

(resp. Z2), z becomes considerably greater than y (resp. x). These

"danger zones" may be defined as:

Z3 Zl n { (x,y,z) e JR3/

Z4 - Z2 n { (x,y,z) e JR3/ I .=. y

If we consider the controls computed for the triplets of Z3 and Z4,

the problem is that u and v have been chosen in order to stabilize the

last component z. Now, according to the hypothesis about the situation

of the state, this component is not on the same order of magnitude as


the other two. We also seem to control the system with inputs close to

Dirac's impulse, which implies that the system evolves in bounds (cf.

[6]) •

We are confronted with a pure numerical analysis problem which is hard

to solve on a digital computer. Indeed, the integration of

differential equations on constant time scales, with different orders

of magnitude variables was, and still is, critical. We will avoid this

problem by taking an upper-bound of the control in these two zones.

On the other hand, every triplet (x,y, z) included in the two

half-lines defined as

{X y z ~ 0

o

implies

{~ = u

v

o

Because the last component is no longer accessible, (L) is therefore

uncontrollable. In order to find again a full-accessibility situation,

it is sufficient to force the control to u ~ 0 and v ~ 0 for a finite

time.

3.4.2. Algorithmic

The logical sequence of computation does not depend on the region

where the state (x,y,z) takes place, except in the particular case of

the line x = y = 0, which will be considered as a little cylinder of

small radius r, and of infinite length. Only the equations change to

allow us the control computation. The computing algorithm is defined

as follows:

1°_ Sampling of (~) at the t = nat instant acquisition of the state

(x,y,z) .


2°_ Indentification of the zone where the state belongs and

computation of the corresponding controls u and v.

3°_ Application of those constant and continuous controls in (E)

during a sampled period At. The processing is started again

from the first step, one At after, at the t = (n+1)At instant.

3.5. Rll:St1L'l' 01' SIMULA'l'ION

The following curves illustrate two simulations defined by the

parameters :

The first simulation (see figures 5 and 6) starts at

z - 1 and the sampled period is : At - 0.05 s.

x = 1, Y = 1,

The second simulation (see figures 7 and 8) starts at x 0.1,

y ~ -0.1, z = -10 and the sampled period is : At = 0.05 s.

Z 1.2 Y

1.0

1.0 0.8

0.8 0.6

0.4 0.6

0.2 0.4

\ 0.0 0.2 -0.2

0.0 -0.4

0 10 20 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

t x

Figure 5 Figure 6

0 z y 1.2

-2

-4 0.6

-6 0.0

-8

-10 -0.6 0 2 6 8 10 -1 0 2 3 5 6

t x

Figure 7 Figure 8


4. CONCLOSION

This work still is under progress. Several other examples are being

calculated, such as the rigid body steering, using a single gas jet

along two principal axes [14].

5. BIBLIOGRAPHY

[1] D. AEYELS : Stabilization of a class of nonlinear systems by a

smooth feedback control. Systems & Control Letters, 5 (1985)

289-294.

[2] Z. ARSTEIN : Stabilization with relaxed controls. Nonlinear

AnalysiS, Theory, Method & Application, 7 (1983) 1163-1173.

[3] R.W. BROCKETT Asymptotic stability and feedback

stabilization. "Differential Geometric Control Theory", R. W.

Brockett, R.S. Millman and H.J. Sussmann eds, Birkhauser,

Boston, 1983, pp. 181-191.

[4] C.I. BYRNES et A. ISIDORI Regulation asymptotique des

systemes non lineaires. C. R. Acad. Sc. Paris, 1-309 (1989)

527-530.

[5] C.I. BYRNES at A. ISIDORI : New results and examples in

nonlinear feedback stabilization. Systems & Control Letters, 12

(1989) 437-442.

[6] M. FLIESS Fonctionnelles causales non lineaires et

indeterminees non commutatives. Bull. Soc. math. France, 109

(1981) 3-40.


[7] M. I'LIESS : Automatique et corps differentiels. Forum Math.,

1 (1989) 227-238.

[8] M. I'LIBSS : Generalized controller canonical forms for linear

and nonlinear dynamics. IEEE Trans. Automat. Control, 35 (1990)

994-1001.

[9] M. I'LIESS and 1'. MESSAGER Vers une stabilisation non

lineaire discontinue. "Analysis and Optimization of Systems", A.

Bensoussan and J .L. Lions eds. Lect. Notes Control Inform.,

144, pp. 778-787, Springer, Berlin, 1990.

[10] M. I'LIESS, P. CHAN'l'RE, S. ABU EL A'l'A and A. COlC

Discontinuous predictive control, inversion and singularities.

Application to a heat exchanger. "Analysis and Optimization of

Systems", A. Bensoussan and J.L. Lions eds. Lect. Notes Control

Inform., 144, pp. 851-860, Springer, Berlin, 1990.

[11] H. HERMES : On the synthesis of a stabilizing feedback control

via algebraic methods. SIAM J. Control and Optimization, 18

(1980) 352-361.

[12] M. ltAWSItI : Stabilization of nonlinear systems in the plane.

Systems & Control Letters, 12 (1989) 169-175.

[13] E.R. ItOLCHIR : Differential Algebra and Algebraic Groups.

Academic Press, New York, 1973.

[14] 1'. MESSAGER : Two example of discontinuous stabilization.

Proc. Conf. "Analysis of Controlled Dynamical Systems", Lyon,

1990.

[15] H. SIRA-RAMIREZ : Nonlinear dynamically feedback controlled

descent on a non atmosphere-free planet

algebraic approach. Submitted for publication.

A differential


[16] B.O. SONTAG : Nonlinear regulation The piecewise linear

approach, IEEE Transactions on Automatic Control, 26 (1981)

346-357.

[17] H.J. SQSSMANN Subanalytic sets and feedback control.

Journal of Differential Equations, 31 (1979) 31-52.

[18] Ya. Z. TSYPlCIN : Relay Control Systems (translated from the

Russian), Cambridge University Press, Cambridge, 1984.

[19] V.I. UTlCIN : Sliding Modes and their Applications in Variable

Structure Systems (translated from the Russian). Mir, Moscow,

1978.

MI!', FM: Laboratoire des Signaux & Systemes. CNRS-ESE, Plateau de

Moulon 91192 Gif-sur-Yvette Cedex (France).

Tel : (33-1) 69.41. 80.40 - Fax: (33-1) 69.41. 30.60

I!'M: Applications Mathematiques et Logiciel. 6, rue Amedee Bollee

92500 Rueil-Malmaison (France).

Tel: (33-1) 47.49.14.00 - Fax: (33-1) 47.51.10.89

Invariant Manifolds, Zero Dynamics and Stability

Hans W. Knobloch and Dietrich Flockerzi*

1. Introduction

We consider ordinary differential equations which can be written as coupled pairs

x = g{t,x,y), iJ = h{t,x,y) (1.1)

with x E 1Rn and y E 1Rm. It will be tacitly assumed throughout this paper that g, h and all mappings - as sand w - which appear in the sequel are everywhere defined smooth eN -functions of their respective variables for some appropriate integer N ~ 1. That solutions of an ordinary differential equation exist on a given (finite) time interval will also be taken for granted. Concerning (1.1) our basic assumption is

h(t,x,O)=O (1.2)

so that y = ° represents a global invariant manifold for the system (1.1). The differential equation

x=g(t,x,O) (1.3)

then describes the dynamics which prevail within this basic invariant manifold. For shortness we refer to (1.3) as to the differential equation of the "zero dynamics" for (1.1) with (1.2).

The question we are interested in concerns the possible relations between the zero dynamics (1.3) and the dynamics of the full system (1.1). To be more specific, let us consider a set M in the (t, x)-space which is positive invariant with respect to (1.3). We introduce its cross sections

Wt = {x E 1Rn : (t, x) E M}

and assume that these are bounded sets in the x-space. So the diameters

diam Wt := sup{llx - YII : X,y E Wtl

·This work has been supported by Deutsche Forschungsgemeinschaft Kn 164/3-3.

INVARIANT MANIFOLDS 133

are well-defined. A problem which has been raised (and answered in principle) in [3] can be phrased in this way:

Given some positive time T, specify a set Eo

of initial conditions in the (t, x) - space

and a constant C = C(T) such that (1.4)

(x(O), y(O)) E Eo implies for t E [0, T] :

x(t) E Wt , Ily(t)1I $ C diamWt .

In presenting the background material in Section 2 we restate the solution of problem (1.4). Its implications to questions of stability and of estimating the region of attraction for the equilibrium of differential equations of the form

if = f(t, y) (1.5)

with f(t, 0) = 0 will be considered in Section 3. We close this introduction with a notational remark. For an integrable

function p : [0, T] -+ R we define

p(t):= exp(-lt p(s)ds), p(t,r):= p(t)p-l(r) (1.6)

for 0 $ r, t $ T.

2. Review of Background Material

The basis of our approach is the notion of bifurcating invariant manifolds as introduced in [3, Section 4]. Note that in contrast to [3] we work here in forward time direction on the interval I = [0, T], T > O. Given two smooth maps

(2.1)

we put Wo := {x ERn: w(x) $ O}. (2.2a)

Since M is supposed to be positive invariant it is clear that the cross sections Wt for tEl can be characterized as follows:

Wt is the set of points in x - space which

can be reached at time t along trajectories

of (1.3) starting at time 0 in Woo

(2.2b)

The explicit formula of the constant C(T) in (1.4) which will be given in this paper involves positive numbers il:T(X) and rT(x) depending on the

134 KNOBLOCH AND FLOCKERZI

terminal time T and on the initial points x E Wo. An explanation of these quantities has been given in [3]. We wish to restate it in our present setting where the bifurcating invariant manifold evolves forward in time from the given initial manifold

Eo := {(x, y) : y = s(x)} (2.3)

at time t = O. To this purpose let us consider the solution (x(.), y(.)) of (1.1) on I which

satisfies the initial conditions

x(O) = x E Wo, y(O) = s(x). (2.4)

The variational equation of (1.1) along this solution can be written in the form

with

tV = A(t)w + A1(t)Z, i = B1(t)W + B(t)z (2.5a)

A(t) = D.:g(.), A1(t) = Dyg(.), B1(t) = D.:h(.), B(t) = Dyh(.)

(2.5b)

where the right-hand sides are to be evaluated along the solution(t,x(t)Y(t)). Let clh(t, r) be the principal matrix solution of tV = A(t)w and let <1>2(t, r) be the one of z = B(t)z. We assume that the following inequalities hold true for all t, rEI with integrable functions 0:, p, (3 on I and positive numbers 1'1,1'2, d1 and d2 :

lI<1>l(t, r)1I ~ 1'la(t, r) for 0 ~ t ~ r ~ T, (2.6a)

II <1>2(t , r)1I ~ 1'2~(t, r) for 0 ~ r ~ t ~ T, (2.6b)

o:(t) < p(t) < (3(t) (2.6c)

d1 ~ sup{(p(t) - 0:(t))-1I1A1(t)1I : t E I}, (2.6d)

d2 ~ sup{(,8(t) - p(t))-1I1B1(t)1I : t E I}. (2.6e)

For the definition of a,~ we refer to (1.6). Recalling that all the above mentioned quantities depend on the intiial data (2.4) and on the underlying time interval I = [0, T] we define

Theorem 2.1 Under the hypotheses

(1) s(x) = 0 ifw(x) ~ 0, (2) Wo is compact with nonempty interior, (3) ICT(X) < 1 and IID.:s(x)lIrT(X) < 1 for all x E Wo

INVARIANT MANIFOLDS

there exists a eN -mapping S : [ x IRn -+ IRm with these properties:

(a) S(O, x) = sex) for all x E WO,

Set, x) = 0 for all x ¢ Wt,

135

(b) The set E = ((t,x,y) : t E [,x E Wt,y = S(t,x)} is a locally invariant "bifurcating manifold". With increasing time tmjectories of (1. I} can leave E only through the hyperplane t = T,

(c) IID~S(t,x)1I ~ (TT with

(1T:= sup 'Yl'Y2{(1- 1I:)-ld2 + (1- 1I:)-2I1D~s(x)II[1- rIlD~s(x)II1-1} ~ewo

where the quantities on the right-hand side depend on x E Wo and the terminal time T (cf (2.6), (2.7)).

Except for a time scaling the statements (a) and (b) are identical to the ones of Theorem 5.1 in [3]. Part (c) follows from Corollary 3.2 of [3] if one takes (2.6c) into account.

Corollary 2.2 Under the hypotheses of Theorem 2.1 every solution x(.),y(.) of (1. I} on [ with initial conditions (2.4) satisfies

(2.8)

3. Stability with Peaking

Given an ordinary differential equation

v=f(t,y),tE[=[O,T], YEIRm , (3.1)

with the trivial solution y = 0 we assume that the principal matrix solution ~o(t, T) of the variational equation

z = Bo(t)z, Bo(t):= Dyf(t, 0), (3.2)

along y = 0 to satisfies

lI~o(t, T)II ~ 'YPo(t, T) for 0 ~ T ~ t ~ T (3.3)

with some 'Y ~ 1 and some continuous function Po on [ (see (1.6». Asymptotic stability of the equilibrium can be inferred from asymptotic properties of Po in case [ = [0,00), e.g. from the statement

lim ! r Po(s)ds exists and is positive. t-oo t io (3.4)

As a mere stability criterion (3.3) and (3.4) do not render a better service as the standard criterion for asymptotic stability in the first approximation


(when /30 is taken to be a positive constant). The situation is different when it comes to the problem of estimating the region of attraction of the equilibrium. By using a time-variant /30 one may be able to account for peaking, i.e. for the tendency of a trajectory starting at time 0 close to the equilibrium to deviate during an initial period of time before approaching it. again as t tends to infinity.

In this section we will elaborate stability criteria for time-variant /30 based on the results on "zero dynamics", in particular on Corollary 2.2. The technical details will be presented at a later occasion. To this end we embed (3.1) into a system of the form (1.1-2) that is adapted to the particular solution y(.) of (3.1) with initial value yeO) =~. We define

h(t, y) := f(t, y) - Bo(t)y

and determine a solution k(t, y) = Ky + k2(t, y) of

k(t, y) - (Dyk(t, y))y = h(t, y),

k2(t,0) = 0 and Dyk2(t, 0) = O.

By the method of characteristics k2 is found to be given by

The constant (m, m)-matrix J{ will be adjusted to y(.) by the choice

(3.5a)

(3.5b)

(3.5c)

(3.5d)

The initial value problem (3.1) with yeO) = Yo can then be written as

iJ = f(t, y) = k(t, y) + pet, y)y, yeO) = Yo, (3.6a)

with pet, y) := Bo(t) - Dyk(t, y) (3.6b)

satisfying P(O,~) = 0 and pet, 0) = Bo(t) - K. We denote the derivative of pet, y) along the differential equation (3.1) by P'(t, y). Introducing some continuous function no : I --+ lR with

/3o(t) - no(t) 2: () > 0, t E I, (3.7)

for some constant () and taking the (m, m)-matrix X as a new variable we study

x = -no(t)X + ao(t)P*(t,x,y), X(O) = 0,

iJ = k(t, y) + ao1(t)Xy, yeO) = Yo, (3.8)


where P"(t, X, y) denotes the time derivative of pet, y) along (3.8). We note that the solution of (3.8) with initial value (Xo, YO) = (0, e) is given by (iio(t)P(t,y(t)),y(t)) =: (X(t),y(t)) since P*(t,X(t),y(t)) is equal to PI(t, Yet)). System (3.8) is of the form (1.1) with the basic invariant manifold y = 0 as required by (1.2). The "zero dynamics" are determined by system (1.3) which now reads

x = -ao(t)X + iio(t)Bo(t).

In order to stay strictly within the notation of Section 1 we associate to the (m,m)-matrix X = (Xij) and to the (m,m)-matrix valued (Pij)(t,X,y) the m2-vector valued

z=(Xu , ... ,Xml' ... ,Xlm, ... ,Xmm)T

and

Thus, taking in (2.2a)

Wo(o) = {z : IIzll =:; o/2}

the diameters of the cross sections Wt in (2.2b) satisfy

diam Wt = iio(t)o.

For an application of Corollary (2.2) to the solution (i(t), yet)) of

z = -ao(t)z + iio(t)p"(t, z, y), z(O) = Zo,

iJ = k(t, y) + iiOl(t)Y z, y(O)yO,

with (zo, Yo) = (O,e) and

( ) mxm2 Y = ImYl, ... , ImYm E 1R

(3.9)

we need to establish the set-up of Section 2. In particular we need to define an initial manifold I;o passing through (zo,Yo) = (O,e) (cf. (2.3)). To this end we fix a positive number;; and choose in (2.1) a smooth s(.) from the z-space into the y-space with

s(z) = 0 for z ¢ Wo(o),

IIDxs(x)11 =:; ;; for all z,

s(O) = e.

(3.l0a)

(3.l0b)

(a.l0c)


This can be done by an appropriate choice of 6 = 6(e) in such a way that

(3.10d)

is fulfilled with some X,X from the interval (0,1/2). Next we need to establish the relations (2.6-7) and to compute an upper bound for (1T in the estimate (2.8). A rather involved argumentation leads to the following result.

Lemma 3.1 Let YO be the solution of (3.1) on [ with initial value y(O) = e and let the function f3o(.) of (3. 2-3) satisfy (3. 7) for some continuous ao (.). Moreover assume lIell to be sufficiently small in dependence of'Y and y and o and

bounds for II DyJII , IID;iyJII, IID~ityJII on Vt := {y : Iiyli ::; 'Yiio(t)}, t E [(i,j = 1, ... ,m).

(3.11)

Then there exists a 6 > 0 such that on [ the solution y(.) of (3.1) satisfies the following estimates:

The proof of Lemma 3.1 is based on Corollary 2.2 and hence on the relations (2.5-7) for system (3.9). Here we confine ourselves to point out that the matrices in the corresponding variational equation (2.5) have the following properties:

(1) A(t) = -ao(t)[ + K(2)(t,y(t»Y(t)y where the entries of the (mm,m)matrix K(2) are second order partial derivatives of the components of k.

(2) B(t) = Dyk(t, y(t» + iiOl X(t) so that B(t) reduces to Bo(t) along the solution (x(t), y(t» of (3.9).

(3) Bl(t) = iiol(t)Y(t). (4) IIAl(t)1I ::; iio(t).\(t) along the solution of (3.9) with initial value

(0,0) for some function .\(t) depending on (3.11).

It is clear that bounds for the partial derivatives of k can be expressed in terms of bounds for the partial derivatives of f via the relations (3.5). The above relations for A, B, Al and Bl allow us to choose u( u ::; 1/12) and 6 sufficiently small (independently from each other) so that the quantities KT(X) and rT(X)i1 are both not greater than 1/2 for all x E Wo(6). This fact-together with (3.10) and Corollary 2.2-leads us to the estimates in (3.12).

We want to point out that Lemma 3.1 can also be understood to hold on the infinite time interval [0,00) whenever its hypotheses are valid on this interval.


Qualitatively the above lemma is similar to standard results which can be found e.g. in [1,§6] or [2, Chap. III]. Quantitatively however it goes beyond what concerns an explicit description of the dependence on the data, in particular on (3.11). Thereby it offers itself as a tool for an iteration procedure. As an example of what can be gained we mention the following theorem which we formulate for third order polynomial systems.

We consider the special case of system (3.1)

iJ = I(t, y) = Bo(t)y + F2(t, y) + Fa(t, V), yeO) = ey. with 0:::; e :::; e., lIy.1I = 1,

(3.13)

for tEl = [0,11 where F2 and Fa are second and third order polynomials in y respectively. By ~(tj e) with ~(Oj e) = I we denote the principal matrix solution of the variational equation

i = B(tje)z, B(tje) := D"/(t, y(tje» (3.14)

along the solution y(tje) of the initial value problem (3.13). We assume that there exist positive constants ; ~ 1 and () and continuous functions a(.jO),,8(.jO): I - R with

1I~(tj 0)11 :::; ;P(tj 0) for 0:::; T :::; t :::; T, ,8(tj 0) - a(tj 0) ~ 2() > 0, tEl.

With a constant ;' > ; we assume the existence of continuous functions a(tje),,8(tje) with

,8(tje) - a(tje) ~ (J > 0

,8(tje) = ,8(tj 0) - ;';a2(t) foe P(tj J.')dJ.'

- 12;;';'x-1aa(t) foe P(tjJ.') foJJ ii(tjp)dpdJ.'

(3.15a)

(3.15b)

for (t,e) E I x [O,e.]. Hereby the functions a2(.), aa(.) stand for continuous upper bounds for IID:"F2(t, 0)11 and liD:"" Fa(t, 0)11 respectively. By Lemma 3.1 one arrives at the following result.

Theorem 3.2 From the above set-up one has the following estimates for (9.19) and (9.14):

lIy(tje)1I :::; 12;'x-1 foe ii(tjJ.')dJ.',

1I~(tj e)1I :::; ;P(tj e)

for tEl and e E [0, e.].


In a forthcoming paper (of the second author) the solvability of the integral equation (3.15) is investigated. There it is shown that for quadratic systems (3.13) the corresponding analog of Theorem 3.2 allows one to conclude by an iterative process that the region of attraction of the equilibrium contains a ball which is about twice as big as the one which can be obtained by standard methods.

REFERENCES

1. L. Cesari, "Asymptotic Behavior and Stability Problems in Ordinary Differential Equations," 2nd edition, Springer Verlag, Berlin, 1963.

2. W.A. Coppel, 'Stability and Asymptotic Behavior of Differential Equations,"Heath Math. Monographs, D.C. Heath and Company, Boston, 1965.

3. H.W. Knobloch, Invariant ManiJoldllJor Ordinary Differential Equations, to appear in: Proceedings of the UAB International Conference on Differential Equations and Mathematical Physics, March 15-21 (1990).

H.W. Knobloch and Dietrich Flockeni Mathematisches Institut Universitat Wiirzburg Am Hubland 0-8700 Wiirzburg Germany

Tracking control for Robotic Manipulators

by Local Linear Feedback

HISATO P. KOBAYASHI!

1. Introduction

Robotic manipulators are widely used in industries. Recently as actuators for theses manipulators, DD motors (direct drive motors) are gradually taking places of conventional geared DC motors, because of maintenance free, high-speed response, high power, high precision, clearness and quietness.

On the other hand, We can not neglect the reciprocal torque caused by the movements of the another links, while geared DC motors decrease mutual effect caused by mechanical connection and let each link be able to be operated independently. we have to treat the whole system as multiinput multi-output nonlinear system. There are excellent studies, (e.g. [1]-[5]), for controlling nonlinear systems, but their control strategies are not so simple.

The normal controllers for conventional robotic manipulator controllers only consist of the motor drivers which include local linear feedback loop and supervising computers which generate sequences of target positions.

Therefore it is a still open problem whether such a poor architecture can accomplish to drive the complicated nonlinear system. The prospect is not gloomy, because robotic manipulators have some profitable properties: for example the inertia matrix is positive definite [7] and the decoupled drive systems can be designed comparatively easily [8][9].

Kawamura, Miyazaki and Arimoto[lO] showed that local PD feedbacks with enough large gains can make trackings with sufficient accuracy. These results are very useful in practical sense, but they do not treat exact tracking problems.

In this paper, we consider the exact tracking problem and derive a result similar as linear systems, that is 'a local linear feedback control with integrators can accomplish trackings, and the number of integrators is equal

1 Supported in part by Alexander von Humboldt Foundation

142 HISATO KOBAYASHI

to the order of the reference trajectory.' We also show experimentally that a conventional simple controller can accomplish the tracking control of a robotic manipulator.

2. Formulation and Problem Description

Let us consider the following system; this dynamical system represents almost every robotic manipulator systems.

(1) d2

F(-itz,z) + G(-itz,z)u -z = d2t

z(O) = Zo

1,z(O) = Zoo

where z means an n-dimensional position vector and -itz is its velocity vector. u is an n-dimensional control torque vector. These values are restricted in the following regions.

( d) E XC Rnx2 iHz,z

u E OC~

Definition 1 (Admissible Control) The control function u' (t), t E [0,1'] is called as admissible control in class el:, when it satisfies the following conditions during [O,T].

Moreover ul(t), t E [0,1'] satisfies,

dl: -ul(t) = 0 dl:t

then we call it as admissible control in class 01: .

It is impossible for robotic systems to track to any arbitrarily given trajectory. Thus we restrict the class of reference trajectories, instead of considering the invertibility condition.[2]

Definition 2 (Possible Trajectory) A trajectory zl(t), t E [0,1'], which is started from the initial point zl (0) = zoo, t. zl (0) = Zo and driven by an admissible control in class 01:, is called as possible trajectory in class TOI: .

TRACKING CONTROL FOR ROBOTIC MANIPULATORS 143

A trajectory with the same condition but driven by an admissible control in class Cle is called as possible trajectory in class TC Ie •

Let sand i be differential operator tt and integral operator I~ ·dt. Let us define the difference between the reference trajectory and the real state.

se ft(:c - :cl )

e :c - :cl

i e = 1t (:c - :cl)dt

(-k )2e l' 1t (:c - :cl)dt dt

The robotic manipulator system (1) is naturally feedback linearizable system. ( e.g. [1]) Thus there exist a nonlinear feedback u = o:(:c, ft:c) +f3(:c, tt:c)v and there exists a diffeomorphism z = 4>(:c) which transforms

the closed loop system -Gt:c = (F( tt:C':C )+G( *:c, :c) .o:( tt:c,:c »+G( tt :c, :c). f3(tt:c, :c)v into a linear system -Gtz =A1z + A2 t, z + Bv. Moreover, in this case G( tt:c, :c) is always invertible, thus we can find a direct nonlinear feedback u = G( tt:c, :c )-1 (v - F( *:c, :c» which derives simple closed system d2 ifItz = v.

If we use this kind of nonlinear feedback control, we can easily solve the robot tracking problem as a linear tracking problem. However in this paper, we try to make a tracking only by using linear feedback control. Our problem is described as follows.

PROBLEM: For any given possible trajectory :cl (t), t E [0, T], does there exist linear feedback control u = pose + P1e+P2je+" '+Pq( j )q-1e which makes :c(t) = :cl (t) and it :c(t) = it:cl (t) on time period [O,T] ?

This problem includes some derived sub-problems: Are the feedback gains scalar or matrix ?, (if they are scalar or diagonal matrix, then each input can be made only by local state values); What is the relation between the class of trajectory and the order of feedback ?; Are there any relieved control which makes 1 :cl(t) - :c(t) 1< €, 1 it (:cl(t) - :c(t» 1< €? and so on.

3. Theoretical Results

The answers of the problems are given by the following three theorems.

Theorem 1 If F(f,z,z) is bounded in X, G(f,z,z) is always positive definite in X and G( f,z, z) > G. > 0, then there exists a feedback control which accomplishes the given tracking. More precisely, when the given possible trajectory is in class TO Ic then a linear local feedback control represented as

1 1 Ic 11. = pose + Ple + P2Se + ... + Plc+l( S) e

can accomplish the given tracking, where Po, Pl, ... ,Pic+! are scalar constants.

In this theorem we must notice the following three matters.

• the positiveness of G( f,z, z) always holds in the case of robotic rnanipulators.[7]

• If the class of the trajectory is TO Ic then the degree of integral feedback is k.

• As Pi, i = 0, ... , k + 1 is scalar, the feedback is local, namely the i-th input Ui consists of only the i-th variables Xi and its derivative and integrals. Ui = PO(ftXi - ftz!) + Pl(Zi - z~) + P2(f Zi - J z~) + ...

Theorem 2 If F(f,z,z) is bounded in X, G(f,z,z) is always positive

definite in X, G(f,z,z) > G. > 0, and the possible trajectory is in class TC Ic then there exists a local linear feedback control

u=pose+Ple+P2ie+···+P'+l(i)'e, for any I, O$I<k

which makes for arbitrarily small e,

I zl(t) - z(t) I < e,

d I dt (zl(t) - z(t)) I < e,

The degree of feedback control is arbitrarily during (0, k). Increase of feedback gain implies decrease of the value e. 1= ° implies the conventional

PD control, and this case agrees with the result in [10], that PD controllers with enough large gains can accomplish the tracking practically.

Theorem 3 If the feedback control,

u = pose + Pie + P2te + ... + P'+i(t)'e,

accomplices the tracking then the scalar multiplied control,

0" ~ 1,

can also accomplish the tracking. Moreover if 0" is enough large, The feedback control is robust against the fluctuation of F(ftx,x) and G(ftX,x).

Indeed, the feedback control does not depend on the values of F( tt x, x) and

G(ftx,x). It only depends on the positiveness ofG(ttx,x).

4. Proofs for Theorems

Before starting the proofs, we prepare an important lemma which is based on linear regulator theory. Let matrices A( T) and B«) be as follows:

A(T) [ Omxn Imxm ] A .. (T)

B«) = [ Omxn ] B .. «)

where, m = k . n, A(T) E R(m+n)x(m+n), A .. (T) E Rnx(m+n), B«) E R(m+n)xn and B .. «) E Rnxn.

Let P be a «k + 1) . n) x «k + 1). n) -dimensional symmetric constant matrix whose elements are n-dimensional identity matrices with scalar multipliers, and which satisfies the conditions (2) and (3).

(2)

(3)

P=

Pk,kInxn Pk-i,kInxn

P > 0

Pk,k-iInxn Pk-i,k-iInxn

Pi,k-iInxn Pk-iInxn

Pk,oInxn Pk-i,oInxn

Pi,oInxn PoInxn

Pd def [Omxn Imxm] p- i [ ~::: ] < 0

146 HISATOKOBAYASHI

Lemma If A.(T) is uniformly bounded for all T and B.(e) is always positive definite for all e, B.(e) > Bmin > 0, then there exist a constant (To and a constant negative definite matrix -Q such that,

V(T,(T > (To,

(4) (A(T) - (TB(e) [p"Inxn p,,-lInxn ... polnxn] f P

+P(A(T) - (TB(e) [p"Inxn p,,-lInxn .,. poInxn])

< -Q < 0

Proof Since B.(e) is invertible, the following identity holds.

(5) (TB«) [p"Inxn p,,-lInxn ... Polnxn]

= (TB«)B.«)-l B«? P

Thus the left hand side of the inequality in the lemma is calculated as follows;

(6) (A(T) - (TB«) [p"Inxn plr:- 1lnxn ... polnxn])T P

+P(A(T) - (TB«) [p"Inxn p,,-lInxn ... Polnxn])

= P { P-l(A(T) _ (TB«)B.«)-l B«)T) +(A(T) - (TB«)B.«)-l B«)T)p-l}p

= p [ Pd +* pI * ] p -2(TB.«)

> P [ Pd +* pI * ] P -2(TBmin

Since the leading dominant part is negative definite, we can know the above matrix is negative definite for enough large (T by using the lemma in [11].

Q.E.D.

Next we define an n(q + 1) dimensional expanded state vector and an expanded system by (q - 1) times integrating state vector.

(7) x=


where each Xi, i = 0, ... , q is n dimensional vector and f,Xi = Xi-l, i = 1, ... ,q,Xl =z.

(8) d -X= dt

X g_ l

X g_ 2

Xo F(Xo, Xt) + G(Xo, Xt)u

where, Xo(O) = Zo

Xl(O) = Zoo

Xi(O) = 0, i = 2, ... , q =

Corresponding to the expanded system, we define an expanded reference system which is made by the possible trajectory.

(9)

xf xA

where each X!, i = 0, ... , q is n dimensional vector and f,X! = xLl' i = 1, ... ,q,xf =zl.

(10) ::"XI = dt

where, XUO) = Xf(O) =

xA F(xA, xl) + G(xg, xDu'

Zo xl (0) = 0 i = 2, ... ,q

Zoo = We define a vector E as the difference between the expanded system

and the expanded reference system.

Eq Xg _Xl 9 I

Eq- l X g_l - X q_l

(11) E= = El xl-xl Eo Xo-xA


where E is equal to e as defined before.

(12) .!!..E= dt

Eq_ 1

Eq_ 2

Eo F(Xo, Xl) - F(Xt,Xn + G(Xo,Xt}u - G(X~,xDu'

where E;(O) = 0, i = 0, ... ,q.

We consider the stability problem of the error system (12). We make clear whether it is possible to let the error system be asymptotically stable by linear feedback.

Proof We expand the function F and G around the reference trajectory (xt, X~).

(13) F(Xo,Xt}

(14)

F(x~,xf) + Fl(X~, xf; Eo, El )

+F2(X~, xf; Eo, El ) + ... F(X~, Xf) + Ao(Xt, Xf)Eo

+Al(X~, Xf)El + FH(X~, xf; Eo, Et} F(31) + Ao(31)Eo +Al(31)El + FH(31; Eo, Et}

G(31)ul + A~(31, ul)Eo

+A~(31,ul)El +GH(31,ul ;Eo,El)

where Fj and Gj are j-th order functions of (Eo, El ) and FH and GH are combinations of higher order functions. Ao, At, A~ and Ai are Jacobian matrices. 3 1 and 3 are introduced for simple description.

(15)

(16)

:;:1 d.!,f (Xl Xl) - 0' 1

,..., def (Xo, Xt)


By using (13), the difference system (12) can be rewritten as follows.

Let P be a « k + 1) . n) x « k + 1) . n )-dimensional symmetric constant matrix which satisfies the lemma conditions (2) and (3), and whose last n rows are as follows:

[Plc 1nxn PIc-l1nxn . .. p1lnxn Polnxn

By using this matrix P, we make a linear feedback law;

q

(18) u = -u 2)Pi . Ei) i=O

Then the closed loop system becomes;

(19) :!..E dt

where,

(20)

+ Onxn

FH(SI; Eo, Ed + G(SI)ul

Inxn No

Ni = { A;(SI) -:.. uPiG(S) .... i = 0,1 -upiG(,::.I) ... t = 2, ... , q


We subtract the bias from E and introduce a new variable E* .

Eq ul Eq_1 ul(l)

(21) E· = 1 (fpq ul(q-2)

El On

Eo On

The modified error system is represented as an equation of E· .

Onxn

(22) !!..E* = dt Onxn Nq

1

Inxn Onxn

Nq- 1

On

On + ul(q-l)

E* 1

Eo

On On Nqu' FH(SI; Eo, E1) + G(SI)ul

From the expansion (14), the following equation holds.

(23)

= G(SI)ul + A~(SI, ul)Eo

+A~(SI, ul)El + GH(S', uR; Eo, E1)

Thus the error system (22) can be reduced to;

(24) !!..E" = dt

Onxn Inxn Nl + A~(SI, ul) No + A~(SI, ul)


x E; E" 1

Eo

+

q

= A(SI, ul)E" - B(S){O' L(Pi . Et)}

+

i=O

On __ l_ul(q-l)

qp,

On FH(S'i E~, E{) + GH(SI, U1i Eo, En

Where A(S', ul ) and B(S) be as follows.

(26) B(S) =

where

Onxn Onxn Inxn Onxn Onxn A"l(SI,ul ) A"o(SI,ul )

Onxn

Onxn G(S)

A"l(SI,ul ) = Al(SI)+A~(SI,u')

A"o(S', ul ) = Ao(SI) + A~(SI, ul )

Let us consider stability of this closed system. We introduce the following Liapunov function, where P is the positive definite matrix considered before. (27) L(E") = E"T PE"


The time derivative of (27) along the trajectory of (24) is described as follows.

(28) ! L(E*) = E*T { A(SI, ull p + P A(SI, ul )

[ I I I ]T B(::::)Tp - q Pk nxn Pk-l nxn . .. Po nxn -

- qPB(S) [Pk1nxn Pk-l1nxn ... polnxn] } E*

On

On __ l_ul(q-l)

"Pq On

F (\:II'EI EI)+G (\:I' ul'E* E*) H ~, 0' 1 H ~, , 0' 1

If we choose q as an enough large value, from the Lemma, there exists negative definite matrix -Q, and we can get the following inequality.

(29) .!!..L(E*) = _E*TQE* dt

On __ l_uU(q-l)

"Pq On

F (\:II. E' EI) + G (\:I' u,· E* E*) H ~, 0, 1 H -, , 0' 1

The fact that the reference trajectory is in class TCq- 1 means ul(q-l) = O. Since FH and GH are higher order term of Eo and Ei, the system is asymptotically stable in the neighborhood of the origin. The initial state of the modified error system is at the origin, then E* always remains at the origin. As Eo = Eo and Ei = El , the error (Eo, Ed = 0, and it means the linear feedback control can completely accomplish the tracking of the original system. Even if u,(q-l) f- 0, we can hold the state of the error system by using enough large q.

Q.E.D.

5. Experimental Verification

Let us verify the above theoretical result experimentally. The robotic manipulator used here is a 2-link scalar type manipulator with direct drive


motors. The state equation is presented by (30).

(30) iF d2tZ = F(z,z) + G(z)u

z = [:~] = [:~] u = [:~ ] F(z,z) [ !t(z,z) ]

h(z,z)

where,

!t(z,z) = {(m2L~ + J2)(2z1 + Z2)m2L1L2Z2 + m2L1L2z~ .(m2L~ + J2 + m2L1L2 sin Z2)} sinz2/d(z2)

h(z,z) = {- (L2 + L1 cosz2)(2z1 + z2)m~L1L~z2 - (m1L~ + J1 + m2L~ + 2m2L1L2 cos Z2 + J2 +m2L~). m2L1L2(Z1)2 } sinz2/d(z2)

gl(Z) = (m2L~ + J2)/d(Z2)

g2(Z) = -m2L~ - J2 - m2L1L2 cos Z2/d(Z2)

g3(Z) = (m1L~ + J1 + m2L~ + 2m2L1L2 cos Z2

+J2 + m2L~)/d(Z2) d(Z2) = (mlL~ + Jl + m2Ln(m2L~ + J2) - m~L~L~ cos Z2

The constants are as follows:

Lo = O.05[m] L1 = 0.3[m] L2 = O.l[m] m1 = 217.1[kg] m2 = 10.5[kg] J1 = 8.15[kg/m2] h = O.32[kg/m2]

The admissible control is the closed region described as:

-U1 ::; Ul ::; Ul -U2 ::; U2 ::; U2

UI = 160[Nm] U2 = 30[Nm]

We calculate a possible trajectory by a using bang-bang type control previously. Since this type of control belongs to class 0 1 piecewisely, the possible trajectory is a set of possible trajectories of the class TOI. Thus, from Theorem 1, the PID control

1 u = Pose + PIe + P2-e

s


can accomplish the tracking practically. The experiment was done by using a conventional architecture. We

fed the calculated possible trajectory to the drivers as a pulse sequence command through the supervisory computer. The result is shown in Figure 1. We know the conventional PID controller can play an enough role in this type of nonlinear tracking problem.

Figure 1. Experimental Result

REFERENCES

[1] T .J .Tarn, Zuofeng Li, Nonlinear robot arm control through third order motor model, Preprints of IFAC Symposium on Robot Control, 1988, 6.1-6.6


[2] R.M.Hirschorn, Invertibility of nonlinear control systems, SIAM J. Control and Optimization, vo1.17, no.2, 1979, 289-297

[3] R.M.Hirschorn, J .H.Davis, Global output tracking for nonlinear systems, SIAM J. Control and Optimization, vo1.26, no.6, 1988, 1321-1330

[4] D.G.Taylor, P.V.Kokotovic, R.Marino, I.Kanellakopoulos, Adaptive Regulation of nonlinear systems with unmodeled dynamics, IEEE trans. Automatic Control, vo1.34, no.4, 1989, 405-412

[5] C.I.Byrnes, A.Isidori, Global feedback stabilization of nonlinear systems, Proc. 24th Conference on Decision and Control, 1985, 1031-1037

[6] C.I.Byrnes, A.Isidori, Local stabilization of minimum-phase nonlinear systems, Systems & Control Letters, 11, 1988, 9-17

[7] S.Arimoto, F.Miyazaki, Stability and robustness of PID feedback control for robot manipulators of sensory capability, Robotics Research, MIT Press, 1984,783-799

[8] P.Kiriazov, P.Marinov, On the decoupled drive systems design of industrial robots, Bulgarian Academy of Science Theoretical and Applied Mechanics, 18, NO.4, 1987, 25-29

[9] P.Kiriazov, P.Marinov, On the independent dynamics controllability of manipulator systems, Bulgarian Academy of Science Theoretical and Applied Mechanics, 20, NO.1, 1989, 19-23

[10] S.Kawamura, F.Miyazaki, S.Arimoto, Is a local PD feedback control law effective for trajectory tracking of robot motion?, IEEE Conference on Robotics and Automation, 1988, 1335-1340

[11] Hisato Kobayashi, E.Shimemura, Some properties of optimal regulators and their applications, Int.J .Control, vo1.33, no.4, 1981, 587-599

Hisato P. Kobayashi Department of Electrical Engineering Hosei University Koganei, 184 Tokyo, JAPAN

Synthesis of Control Systems Under Uncertainty Conditions (Game Theory)

V.M. Kuntzevich

Abstract

For controlled plants, subjected to uncontrolled constrained (by specified a priori estimates) disturbances, whose precise parameter values are unknown and for which only their a priori estimates are specified in the form of their belonging to convex sets the minimax problem of cotrol synthesis is formulated. To improve the quality of control the adaptive approach to the solution of a control problem is used. By virtue of the adopted assumptions about the character of uncontrolled disturbances the identifcation procedure used for constructing the adaptive control system generates a sequence of guaranteed estimates of the vector of parameters in the form of its belonging to convex sets. The efficiency of the obtained control system is illustrated by the results of digital simulation.

1 Introduction

Let a class of controlled plants be specified by a motion equation of the form

(1) x = i(X, u, L, F), t ~ 0, o

X(O) = X,

where X, U and F are vectors of state, control and uncontrolled disturbance, respectively, iO is a specified vector function.

Since hereafter it is assumed that a sequence of controls Un = U(nT), where T is a. period of quantification, n is a number of steps, is generated by a computer it is more convenient to consider the motion of control system (1) only at discrete time moments and to handle the difference equation of dynamics

(2) o

n = O,I, ... ,Xo = X,

where CJ(·) is a specified m-dimensional vector function, and the rest of notation is the same as earlier.

GAME THEORY 157

Let then for Fn only a priori estimate of the form

(3) 'v' ~ 0,

be specified where :F is a convex set. A similar a priori estimate is specified also for the vector of parameters

(4) L eCo,

where C is a given convex set. It is obvious that under such constrained a priori estimates of vectors Fn and L any problems of the analysis and synthesis for system (2) are incorrect and therefore they require some additional definition.

Below we shall proceed from the fact that a designer of the control system knows only a priori estimates (3), (4) about the undefined values. Under these conditions there are no other alternatives except for the use of the game approach to formulation of the control problem (see, e.g., [1]-[4]).

2 Statement of synthesis problem

Let us consider the simplest problem of control synthesis when a function of specific losses

(5)

is specified and it is necessary to minimize its value at each step by the choice of control

(6) Un eU 'tin ~ 0,

where U is a given convex set. According to the abovesaid about the necessity of the additional def

inition of synthesis problems for system (2) under conditions (3), (4) we finally formulate the problem of optimal control synthesis in the form

(7) n = 0,1, ....

Note that if the a priori estimate Co of parameters of (2) is rather rough, then, despite the mathematical correctness of the solution of the synthesis problem, the final results of control will be rather bad. This situation can be radically improved only with the help of the adaptive control system which at each step will solve not only the control problem properly but also the problem of identification of the vector L, i.e., of construction of a sequence Cn of estimates of vector L such that

(8)

158 V.M. KUNTZEVICH

In this case the control Un will be determined not by the solution of the problem (7) but by the solution of a sequence of problems

(9) n = 0,1,2, ... ,

where

3 Calculation of guaranteed estimates in parametric identification problems

Let at the n-th step there be an estimate

(10)

For simplicity we assume that the state vector Xn is available for direct measurement and is measured without any noise. Then at the (n + 1)-th step after the measurement of vector Xn from (2) we obtain the estimate

(11) L E .en+1 = {L : Xn+1 - cJ(XnUn, L, Fn) = O}.

The estimate (2) is also used when constructing the set .en+! in the space of parameters {L} by relation (11). Then from (10) and (11) we finally obtain the a posteriori estimate in the form

(12)

For a more detailed analysis of the general procedure of set identification (12) we narrow down the class of systems (2) and below we shall consider only a class of linear stationary systems, i.e., we assume that (2) has a form

(13) Xn+1 = A(L)Xn + B(L)un + Gin.

Here we assume that Un and In are, respectively, scalar control and disturbance, A(.) is an (m x m) matrix, B(·) and G are m-dimensional vectors.

For system (13) it is convenient to consider the matrix A and vectors B and G reduced to the canonical form, i.e.

(14) A(.) = II Ol~:-lll' B(.) = II b~ II ' c=II~II· where I m - 1 is an (m - 1) x (m - 1) unitary matrix.

In this case the parameter vector L of system (13) has a maximal dimensionality and equals

GAME THEORY 159

(15)

Then from (13) and (14) we obtain that the mth equation of (13) has the form

(16)

where Zm,n+l is the mth element of vector X, and, therefore, for the set In+! we obtain

(17) L E In+l = {L : A~Xn + bmun + In - zm,n+! = OJ.

For scalar disturbance In the estimate (3) is rewritten in the form

(18) In E:F 'In ~ 0,

where :F is a given set (interval). For the relation (17) the recurrent procedure of parametric identification

(12) admits the obvious geometrical interpretation (see Fig. 1). If the initial set Co is a convex polytope then by virtue of the linearity of

equation (12) (see also (17» the whole sequence of estimates Cn generated by procedure (12) will be also a sequence of convex polytopes since this class of sets is closed with respect to the operation of intersection.

It is appropriate to mention here that in the last years the method for constructing guaranteed estimates has received wide application (see, e.g., [5]-[10]) when solving problems of determination of the state vector and the parameter vector or when solving these problems simultaneously (see, e.g., [11]). However A.B. Kurzhanski and L.F. Chernousko as well as G.M. Bakan and his colleagues (see, e.g., [12]-[14]) in their works use roughened upper bounds obtained when approximating the polytopes by ellipsoids instead of exact estimates obtained for linear systems from (12) in the form of convex polytopes.

For executing the operation of intersection of the polytope Cn with two hyperplanes defined by equation (16) and interval (whose solution algorithms are described in [4]) the information about the polytope Cn must be stored in a computer memory, e.g., in the form of the matrix of its vertices

where Li,n is a vector of coordinates of the i-th vertex of the polytope, Nn

is the number of its vertices. Storage of matrix Gn in a computer memory is connected with two

inconveniences: 1) the dimensionali ty of the (N n x m) matrix is variable and this number is unknown beforehand; 2) at sufficiently large dimensionality

160 V.M. KUNTZEVICH

~~----~~------------------~t1

(m ~ 0) there arise certain difficulties due to the limited capacity of the main memory.

The method of ellipsoids is free from these drawbacks but it should be remembered that there the computational procedure is simplified at the expense of the roughening of the obtained estimates. From the above it follows that apparently both methods have their most preferable fields of application.

The solution of the considered problem of parametric identification of the linear (by parameters) system is a sequential solution of the system of linear algebraic equations with uncertainty in its right-hand side, i.e., in "classic" notation, the system of the form

(19) AX=B,

where

(20) BeB

GAME THEORY 161

B is a specified convex set (polytope), and (19) a priori estimate of solution is specified in the form

X EXo.

For more detail see [9], [10]. From the analysis of general properties of the system (19), (20) it is easy

to obtain the necessary and sufficient conditions under which, despite the uncertainty in the right-hand side of the system, its solution is a one-point set.

The given general scheme of solving the problem of parametric identification can be generalized and extended to the class of systems nonlinear by parameters which in this case is "plunged" into the problem of searching for a set of solutions of the system of nonlinear equations with undefined parameters.

So, let a sequence of equations

(21) Cf'n(X, G) = 0, n = 1,2, . .. ,N,

be specified where X C Rm , m ~ N is an unknown vector to be defined for which its a priori estimate

(22) X E Xo

is given, Xo is a specified convex set, G C Rk is a vector of undefined parameters for which its a priori estimate

(23) GEe

is known, Cf'n(-) is a nonlinear (at least with respect to X) scalar function whose properties will be specified below. (Everywhere below, the subscript n in (21) will be omitted for simplicity).

By a set X C Rm of solutions of equation (21), as above, will be meant a set of such vectors X for each of which there will be a vector G satisfying (23) such that for the pair (X,G), (21) is satisfied. '

The refined set Xl of solutions of equation (21) under conditions (22), (23) obviously will assume the form

(24)

As applied to system (21) when considering it as a sequence of equations, the refined set of solutions takes the form similar to (12), i.e.

(25)

The set of solutions (21) under condition (23) has the form of a "curvilinear band" in the space of solutions and formally can be represented in the form

162 V.M. KUNTZEVICH

(26)

Large dimensionality of m, complexity of configuration of the set :1:0

and complexity of the function (21) makes it expedient to perform the decomposition of the problem with respect to vector X. For this we select from the vector X a set of its components Y of dimensionality P. The remaining T = m - p components of vector we denote by Z, i.e. introduce the notation XT = (yT, ZT).

Let a priori data X about solution be represented in the form Xo = Yo x Zo where Y E Yo c RP, Z E Zo C RT.

Denote by

(27) Yl = JIy(Xd

(respectively: Zl = JIz(Xd) the projection of the set Xl C Rm onto the subspace RP, i.e., the "y" subspace (respectively: RT, the "z" subspace).

The upper bound approximating the set Xl will be searched in the form of Cartesian product of sets YI x Zl' The search for YI by calculating (26) with subsequent projection onto the subspace RP is a rather complex problem in many cases. So we use the method of projections suggested below which performs the decomposition of the initial problem into a set of simpler problems.

We define the set

(28) y = {y C RPI3Z E Zo,C E C: ¢(.) = OJ.

Then the following theorem defines the required estimate Y:

Theorem 1 If a priori estimate of solution Xo of equation (21) under condition (23) is representable in the form Yo x Zo then the projection of solution Xl onto the subspace RP is defined by the expression

(29) Yl = Y nyo

where the set Y is specified by formula (28).

The proof of the theorem is not given because of limited space. Thus, the method of projections makes it possible to obtain exact esti

mates of solution projections when operating in a smaller subspace. For a two-dimensional system when Y and Z are scalars the geometric interpretation of the method of projections is given in Fig. 2.

An evident description of the set YI according to (29) presents certain difficulties since the curvilinear ban (26) is bounded by nonlinear surfaces. So we consider the estimates of the set Xl described by more simple surfaces - planes. Equation (21) can be represented in the form

(30) aleX, C)YI + a2(X, C)Y2 + ... + apeX, C)yp - b(X, C) = 0,

GAME THEORY 163

z

~o~--~-+---------.--~~--~y

Fig. 2. Illustration of the method of sequential projections.

where Yi are components of vector Y, i E l,p, aiO and b(.) and b(·) over the whole domain of their definition (Xo x C). After this the problem of searching for solution of equation (30) is reduced to the problem of the form

(31)

(32)

ATy +b= O.

A E A,b E B,

where interval sets .A and B are specified. Here aT = (AT, b). The definition of solution of the equations of the form (31) under condi

tions (32) has been already described in [9], [10] and it is, generally speaking, a non-convex set Y in the space RP represented in the form of unification of the finite number of polytopes. Each of these polytopes is described in the explicit form by the system of p + 2 linear inequalities and is located in its orthant, and here only two inequalities are characteristic of the

164 V.M. KUNTZEVICH

polytope, and p inequalities define the corresponding orthant. Taking into account the a priori estimate of the required solution, i.e., the set Xo, the upper bound Xl for the set of solutions X of equation (3) under condition (32) is finally obtained in the form

(33)

Note here that since Y belongs to the subspace RP of the space R m

then the intersection (33) is understood in the sense of intersection in this subspace since without this stipulation we must write

From the practical point of view it is convenient to perform intersections (33) in each orthant separately. In this form the intersection procedure itself is an integration, into one system, of conditions describing Xo with two inequalities specifying the polyhedron and p inequalities defining the considered orthant.

It has been noted above that there is an arbitrariness in the method of quasilinearizing projections when selecting from the vector X the components which form the vector Y. Here there is the refinement of only y components of vector X with respect to a posteriori information, the connection equation (21). It is natural to use this equation for refining the remaining components by the method of linearizing projections. This is the essence of the method of sequential linearizing projections.

We divide vector X into q sets of components yl, ... , yq of Pl, ... ,Pq dimensionalities, respectively, where L:ip; = m. We sequentially apply q

times the method of linearizing projections to equation (21) with selection of components yl, ... , yq respectively. The obtained estimates yi, i E 1, qPi of projections of a set of solutions Xl onto the subspace RP; when intersecting with a priori information Xo about solution give the upper bound

(34)

of the set of solutions Xl itself in subspace Rm.

A principal importance of the described method of sequential linearizing projections consists, primarily, in its linearizing possibilities. Its "decomposing" property which makes it possible to successfully overcome "the damnation of dimensionality" is no less important.

GAME THEORY

4 Solution of minimax problem and analysis of properties of synthesized control system

165

Now after description of the procedure of parametric identification of the considered linear dynamic system (13) as well as its extension to the case of systems nonlinear by parameters, we continue to consider the principal purpose of synthesis: calculation of optimal control from the solution of problem (9). Here it is necessary to define concretely a form of the function of specific losses w(.). The form of this function is defined by the value of the synthesis problem. As the function w(.), we can use, in particular, the first difference 6.vn of the system (13), the square (module) of the difference between the controlled "output" of the system, i.e., coordinate Xm,n+l,

and its "standard" value in systems with a standard model, etc. Consider here the problem of optimal stabilization of the system (13). Assume the function w(.) in the form

(35)

Solution of minimax problems of the form (9) in the general case is a very complex problem. In the given case its solution may be substantially simplified taking into account a concrete form of the function w(·) as well as of a character of restrictions imposed upon vector L.

At first consider the simplest case when f = <p and when there are no restrictions upon the scalar control, i.e., when U = Rl. Then Theorem 2 proved in [4] is valid.

Theorem 2 The optimal control in the problem

(36)

where en is a convex polytope, and LT = (A;', bm ), is a unique root of the equation

(37)

Note that here the problem min{·}(max{-}) is nonlinear programming problem solved by standard methods.

The root of equation (37) is determined by standard iterative procedures.

The presence ofrestrictions upon the control, e.g., in the form Iunl :$ 6. does not change the general scheme of solution of the synthesis problem. Indeed, it is easy to show that in this case the optimal control ~n has the form ~n = 6.sign ~n where ~n, as before, is the root of equation (37).

Theorem 3 is proved in [4].

166 V.M. KUNTZEVICH

Theorem 3 In the linear system (13), (14) with optimal contro/~n defined from solution of the problem (37) the procedure of parametric identification

. . (12), (17) terminates in (m+ 1) steps in obtaining one-point set Cm+l = L

• where L is a true value of the parameter vector L.

We cannot present the proof of Theorem 3 because of the shortage of space, so we give only the principal idea of this proof. For this we introduce the following notations

Then it is easy to show that the sequential application of the procedure (12), (17) is equivalent to solution of a system of equations

ZNL = XN.

But if det ZN :f:. 0, we obtain

It may be shown that the solution of equation (37) is the nonlinear function Xn , i.e.,

~n = ¢(Xn ) :f. CT X n .

and this provides the fulfillment of condition det ZN :f:. o. Theorem 3 establishes an important property of control systems con

structed on the basis of the recurrent procedure of identification (12) and control defined from solution of problem (37): a sequence of controls obtained from solution only of the problem of minimization of the function of specific losses (35) permits at the same time of obtaining solution of the parametric identification problem.

The obvious consequence of definition of the precise value of the parameter vector in the result of identification consists of obtaining from the solution of the problem (36) for the optimal control

which provides the asymptotic stability of the system (13), (14). It is natural that in the case when :F :f:. ¢ the identification procedure

in the general case does not terminate in obtaining the one-point set and

GAME THEORY 167

its final result will be to obtain an unimprovable estimate of parameters. However, as it has been shown in [4] the optimal control here defined from solution of the problem (9), (34) in combination with the procedure of recurrent identification (12), (17) provides dissipativity of the synthesized control system under all initial conditions Xo and a priori estimate .co.

5 Conclusion

The presented method of obtaining non-roughened estimates of the parameter vector with small modifications is extended to the class of linear but non-stationary systems for which the parameter vector Ln = var with one stipulation that the rate of their change, i.e., ALn = Ln+1 - Ln is limited and this limitation is a priori specified for a system designer.

The other direction of generalization of the obtained results is related to consideration of the most widely encountered case when the state vector Xn is not directly measured but its mixture with a limited noise, i.e.

where 'v'n;::: 0

is measured. Here Z is a specified constrained convex set. In this case, obviously, there arises the necessity for solving the problem

of simultaneous construction of the guaranteed estimates of both the vector of state and the vector of system parameters if only its a priori estimate (4) is specified. A modification of the described recurrent procedure of polytope intersection makes it possible to obtain the solution of this problem.

A change-over to the class of linear systems with vector control does not introduce any principal changes into the procedure of set recurrent identification but substantially complicates the solution of the minimax problem in searching for the vector of optimal control.

The problems of optimal control synthesis for nonlinear (by the state vector and by the control vector) systems are completely unstudied. However the general scheme of the procedure of set identification for nonlinear (by parameters) systems has been considered above in outline.

References

[1] Krasovski N.N. Game problems about rendezvous of motions. N auka, Moscow, 1970.

[2] Kurzhanski A.B. Control and observation under conditions of uncertainty. Nauka, Moscow, 1977.

168 V.M. KUNTZEVICH

[3] Chernous'ko F.L., Melikjan A.A. Game problems of control and search. Nauka, Moscow, 1973.

[4] Kuntzevich V.M., Lychak M.M. Synthesis of optimal and adaptive control systems. Game approach. Naukova dumka, Kiev, 1985.

[5] Fomin, V.N., Fradkov A.L., Yakubovich V.A. Adaptive dynamic object control. Nauka, Moscow, 1981.

[6] Fomin V.N. Methods of control of linear discrete objects. Leningrad University Publishing House, 1985.

[7] Chernous'ko F .L. Estimation of phase state of dynamic systems. Method of ellipsoids. Nauka, Moscow, 1988.

[8] Kurzhanski A.B. Identification theory of guaranteed estimates. Working Paper IIASA, Austria, 1989.

[9] Kuntzevich V.M., Lychak M.M., Nikitenko A.S. (1988a) Obtaining the estimate in the form of sets in parametric identification problem. 8th IFAC/IFORS Symposium Beijing, China, August 27-31, 1988, pp. 1237-1241.

V.M. Kuntzevich V.M. Glushkov Institute of Cybernetics Academy of Sciences of the Ukrainian SSR 252207 Kiev 207 USSR

Ellipsoidal Techniques for the Problem of Control Synthesis

A.B. KURZHANSKI and I. V ALYI

Introduction

This paper introduces a technique for solving the problem of control synthesis for linear systems with constraints on the controls. Taking a scheme based on the notion of extremal aiming strategies of N. N. Krasovski, the present paper concentrates on constructive solutions generated through ellipsoidal-valued calculus and related approximation techniques for setvalued maps. Namely, the primary problem which originally requires an application of set-valued analysis is substituted by one which is based on ellipsoidal-valued functions. This yields constructive schemes applicable to algorithmic procedures and simulation with computer graphics.

1 The Problem of Control Synthesis

Consider a control system

(1.1) i: = f(t, z, u), z E nn, u E n r , to $ t $ tl,

with controls u being subjected to a constraint

u E P(t),

where P(t) is a continuous set-valued function with values P(t) E convnn

(the set of all convex compact subsets of nn). The function f(t, z, u) is such that the respective set-valued map

F(t, z) = {Uf(t, z, u) : u E P(t)}

is continuous in t and upper-semicontinuous in z. Let M E convnn be a given set. The problem of control synthesis will consist in specifying a setvalued function U = U(t, z), (U(t, z) ~ P(t)) - "the synthesizing control strategy" - which would ensure that all the solutions z(t, T, ZT) = z[t] to the equation (1.2) i: E f(t, z, U(t, z»,

170 KURZHANSKY AND V A.LYI

that start at some given position {T,ZT}, (T E [to,t1], ZT = Z(T», would reach the terminal set M at the given instant of time t = t1 - provided ZT E W( T, M), where the solvability set W( T, M) is the set of states from which the solution to the problem does exist at all. Here we kept the notation f for the set-valued function defined as f(t, z,U) = {Uf(t, z, '11.) : 'II. E U}.

We presume W(T, M) 1= ¢, 'Vi E [to, t1)'

The strategy U(t, z) must belong to a class U of feasible feedback strategies, which would ensure that the synthesized system (a differential inclusion) does have a solution defined throughout the interval [to, t1]'

We now recall a technique that allows to determine U(t, z) once the problem satisfies some preassigned conditions that will be listed below.

For a given instant T E [to, t1) consider the "largest" set W( T, M) of states Z(T) = ZT from which the problem of control synthesis is resolvable in a given class U. Having defined W( T, M) for any instant T, we come to a set-valued function

W[T] = W(T,M), to $ T $ t1; W[t1] = M.

The following simplest conditions, [2], ensure that the function W[T] is convex compact valued and continuous in t.

Lemma 1.1 Assume that the set-valued mapping :F(t, z) is upper semicontinuous in Z for all t, continuous in t, with .1'(t, z) E convn." and

1I:F(t, z)1I $ k· h(t)

for some k > 0 and h(t) integrable on [to, t1]' Also assume that the graph

gr.1' = Ht, z) : t E [to, t1], Z E .1'(t, z)}

of the mapping :F(t, z) is convex. Then the set W[t] E convnn for t E [to, t1] and the function W[t] is

continuous in t.

We further assume that W[T] E convnn. The Synthesizing Strategy is defined then as the following set-valued

map

(1 3) U(t ) - { P(t) if Z E W[t] . ,Z - {'II.: f(t,z,u) = 8tp(-f!l1 :F(t, x»} if Z ~ W[t].

Here f!J = f!J(t,z) is a unit vector that resolves the problem

(tJ, z) - p(tJ I W[t]) = max { (i, z) - p(i I W[t]) : lIill $ 1 },

ELLIPSOIDAL TECHNIQUES 171

where symbol p(t I W) = max{(t, x) : x E W} stands for the support function of set Wand 8/.g(t, t) denotes the subdifferential of g(t, t) in the variable t.

Strategy U(t,x) reflects the rule of "extremal aiming" introduced by N.N. Krasovski [1]. Particularly, it indicates that with x FI. W[t] one has to choose the unit vector _to that is directed from x to so, namely _to = (SO - x)lIsO - xll- 1 , where SO is the metric projection of x onto W[t]. After that, U(t, x) is defined as the set of points uO E P(t) each of which satisfies the "maximum" condition:

(1.4) (-tJ, f(t, x, uO)) = max{( -tJ, f(t, x, u)) : u E P(t)},

so that U(t,x) = {uO}. The latter procedures are summarized in (1.3).

Lemma 1.2 Once the conditions of Lemma 1.1 are satisfied and the system (1.1) is linear in u, the following assertions are true:

(i) The set-valued map U(t, x) is convex compact-valued, continuous in t and upper semicontinuous in x. This secures the existence of solutions to the differential inclusion

X E f(t, x,U(t, x)).

(ii) If XT E W[T], T E [to, td, then any solution x[t] to the system

xEf(t,x,U(t,x», X(T)=XT' T~t~tl'

satisfies the inclusion x[t] E W[t], (W[t l ] = M).

It is obvious that the crucial element for constructing the synthesized control strategy U(t, x) is the set-valued function WIt]. It is therefore important to define an evolution equation for W[tJ, [2].

Lemma 1.3 Under the conditions of Lemma 1.1 the set-valued function W[t] satisfies the evolution equation

(1.5) lim h(W[t - 00], U{(x - oo:F(t, x)) : x E W[t]}) = 0, to ~ t ~ tl <7_+0

with boundary condition W[t1] = M.

Here h(W', W") is the Hausdorff distance between W', W". (Namely, h(W', W") = max{h+(W', W"), h_(W' , W")} where h+(W' , W") = min{r > o : W' ~ w" + rS}, h_ (W', W") = h+ (W", W') are the Hausdorff semi distances and S is the unit ball in nn.)

172 KURZHANSKY AND VALYI

The conditions of Lemmas 1.1 and 1.2 are clearly satisfied for a linear system (1.6) x = A(t)x + u, U E pet).

The evolution equation (1.5) for determining W[t] then turns to be as follows (1.7) lim O'-lh(W[t - 0'], (I - A(t)O')W[t] - O'P(t» = 0

<7-+0

(here I is the unit matrix), and

(1.8)

The aim of this paper is to demonstrate that this theory could be converted into constructive relations that allow algorithmization and online computer simulation. This could be achieved by introducing a calculus for ellipsoidal-valued functions that would serve to approximate the set-valued functions ofthe theory of the above, (also see [3], §§ 10-12).

It is important to observe that the relations given in the sequel do allow an exact approximation of the solution to the primary problem through ellipsoidal approximations.

We will further concentrate on the linear system (1.6). By substituting z = Set, tt}x and returning to the old notation, without any loss of generality it could be transformed into

(1.9) x = U, U E pet), x(tt} EM,

where x E 'Rn , pet), M E conv'Rn, the function pet) is continuous in t and the matrix valued function S(t,tl) E 'Rnxn is the solution to the equation

2 The Ellipsoidal Techniques

In this paper we do not elaborate on the ellipsoidal calculus in whole but do indicate the necessary amount of techniques for the specific problem of control synthesis.

We will start with the assumption that pet) is an ellipsoidal-valued function and that set M is an ellipsoid. Namely

pet) = &(p(t), P(t»,

M = &(m,M).

where the notations are such that the support function is

pet I tea, Q» = (t, a) + (t, Qt)l/2.


With det Q =F 0 this is equivalent to the inequality

£(a,Q) = {x En: (x - a)'Q-1(x - a) $ 1}.

Therefore a stands for the center of the ellipsoid and Q ~ 0 for the symmetric matrix that determines its configuration.

With sets £(p(t), P(t», £(m, M) being given we are to determine the tube W[t] for t $ t1 under the boundary condition W[t1] = M = £(m, M). According to the above, the set-valued function W[t] satisfies the evolution equation (2.1) lim O'-1h(W[t - 0'], W[t] - O'£(p(t), P(t))) = 0,

0"-++0

(2.2) W[td = £(m, M).

Obviously

(2.3) i tl

W[t] = £(m, M) - t £(p(r), P(r»dr,

so that W[t] is similar to the attainability domain for system (1.6) but here it is taken in backward time; W[t] is the set of all states Xt from which it is possible to steer system (1.6) to the set £(m, M) in time t1 - t with open loop control

u(r) E P(r), r E [t,t1]'

It is clear that although £(m, M), £(p(t), P(t» are ellipsoids, the set W[t], in general, is not an ellipsoid.

Therefore the first problem that does arise here is as follows: is it possible to approximate W[t], both externally and internally, with ellipsoidalvalued functions?

The answer to the question is affirmative as will be shown in the sequel. We will first state the results for A(t) 1= 0 in (1.6).

Consider the inclusion

(2.4) Z E A(t)x + £(p(t), P(t»,

x(tI) E £(m, M)

with W[r] = W(r, M) being the set of all states X T from which there exists an open-loop control u(t) E £(p(t), P(t» that steers the solution from X T

into &(m, M). Denote z(t) E nn to be the solution to the equation

(2.5) i(t) = A(t)z(t) + p(t),

z(tI) = m,

174 KURZHANSKY AND V ALYI

and QH(t) E 'R-nxn to be the solution to the matrix equation

= A(t)QH(t) + QH(t)A'(t) -- H- 1(t)[H(t)QH(t)H'(t)p/2[H(t)P(t)H'(t)p/2 H-l(t) -

- H- 1(t)[H(t)P(t)H'(t)p/2[H(t)QH(t)H'(t)p/2H- 1(t),

QH(tt} = M,

where H(t) is a continuous matrix valued function

HO : [T, tl] -+ 'R-nxn

with invertible values (the set of all such functions will be denoted as H).

Theorem 2.1 (Internal Approximation)

(i) The following inclusion is true

(2.7)

whatever is the function HO E H.

(ii) The following equality is true

(2.8) U t(z(r), QH(r» = W[rJ, H(-)eH

where the symbolIC stands for the closure of set IC.

Further on, denote Qq(t) to be the solution to the equation

(2.9) Qq(t) = A(t)Qq(t) + Qq(t)A'(t) - q-l(t)Qq(t) - q(t)P(t),

Qq(tt} = M,

where q(t) > 0 is a continuous scalar function:

q(.) : [T, td -+ (0,00)

(the class of such functions will be denoted as Q).

Theorem 2.2 (External Approximation)

(i) The following inclusion is true

(2.10) W[T] ~ t(Z(T),Qq(T»

whatever is the function q(.) E Q.

ELLIPSOIDAL TECHNIQUES

(ii) The following equality is true

(2.11) W[T] = n £(Z(T), Qq(T)). qUeQ

175

Equations (2.6) (2.9) are obviously simplified under the condition A(t) == o (we further presume that it holds). It is therefore clear that the set-valued function W[t] satisfies the inclusions

(2.12) £- [t] = £(z(t), QH(t)) ~ W[t] ~ £(z(t), Qq(t)) = £+[t]

to ~ t ~ tl

whatever are the functions H(.) E H, q(.) E Q. Since W[t] is the solution to the evolution equation (2.1) the next ques

tion arises: do there exist any two types of evolution equations whose solutions would be £- [t] and £+ [t] respectively?

The answer to this question is given in the following assertion: Consider the evolution equation

(2.13) lim u-1h+(£[t - u], Crt] - u£(p(t), P(t))) = 0 11-++0

t[td = £(m, M).

We will say that function £+[t] is a solution to equation (2.13) if it satisfies (2.13) almost everywhere and if it is ellipsoidal-valued (!).

Also consider the evolution equation

(2.14) lim u-1h_(£[t - u], £[t] - u£(p(t) , P(t))) = 0 11-++0

£[td = £(m, M).

We will define E- [t] to be a solution to equation (2.14) if it

• satisfies (2.14) almost everywhere,

• is ellipsoidal-valued and

• is also a maximal solution to (2.14).

The latter means that there exists no other ellipsoidal-valued solution t'[t] to (2.14) such that £/[t];2 E-[t] and £/[t] 't E-[t].

Each ofthe equations (2.13), (2.14) has a nonunique solution.

Lemma 2.1 Whatever are the solutions £+ [t], E- [t] to the evolution equations (2.13), (2.14), the following inclusions are true

£_ [t] ~ W[t] ~ £+ [t].


Lemma 2.2 Each o/the ellipsoidal-valued/unctions£-[t] = £(Z(t),QH(t», (H(·) E H) is a solution £_[t] to equation (2.14).

Lemma 2.3 Each o/the ellipsoidal-valued/unctions £+[t] = £(z(t), Qq(t)), (q(.) E Q) is a solution £+[t] to equation (2.13).

To conclude this section we underline that the tube W[t] can be exactly approximated by ellipsoids - both internally and externally - according to relations (2.8), (2.11). To achieve the exact approximation it is necessary in general to use an infinite variety of ellipsoids (actually, a countable set). The given approach, (see also [4]), therefore goes beyond the suggestions of [5] and [6], where the sums of two or more convex sets were approximated by one ellipsoid.

The ellipsoidal approximations will now be used to devise a synthesized control strategy for solving the problem of the above. This strategy will guarantee the attainability of the terminal set M in prescribed time.

3 Synthesized Strategies for Guaranteed Control

The idea of constructing the synthesizing strategy U(t, x) for the problem of the above was that U(t, x) should ensure that all the solutions x[t] = z(t, r, ZT) to the equation

:i: E U(t, z),

with initial state z[r] = ZT E W[r], would satisfy the inclusion

z[t] E W[t], r ~ t ~ t1

and would therefore ensure z[td E M. We will now substitute W[t] by one of its internal approximations £_[t].

The conjecture is that once W[t] is substituted by £_ [t], we should just copy the scheme of Section 1, constructing a strategy U_(t, x) such that for every solution z_ [t] = z_ (t, r, ZT) that satisfies equation

(3.1)

the following inclusion would be true

(3.2)

and therefore


It will be proven that once the approximation C_ [t] is selected "appropriately" , the desired strategy U_ (t, z) may be constructed again according to the scheme of(1.3), except that W[t] will now be substituted by C_ [t], namely

_ { C(p(t), P(t» if z e C_ [t] (3.3) U(t, z) - p(t) _ P(t)£O(lO, P(t)lO)-1/2 if z ¢ C_ [t],

where £0 = 8I1:d(z,C_[t]) at point z = z(t), that is the unit vector that solves the problem

(3.4) (£0, z) - p(£O I C_ [t]) = max{(l, z) - p(ll C_[t]) : IIlil $ I}.

The latter problem may be solved with more detail (since C_[t] is an ellipsoid). Indeed, if sO is the solution to the minimization problem

(3.5) sO = argmin{II(z - s)II : s e C_[t], z = z(t)}

then we can take

in (3.3).

Lemma 3.1 Consider a nondegenerate ellipsoid C = C(a, Q) and a vector z ¢ C(a, Q), then the subgradient £0 = 8I1:d(z, C(a, Q» can be expressed through £0 = x - SO Iliz - sOli,

sO = (I + AQ-1)-1(x - a) + a,

where A > 0 is the unique root of the equation h(>..) = 0, with

Assume a = O. Then the necessary conditions of optimality for the minization problem

liz - sll = min, (s, Q- 1s) $ 1

are reduced to the equation

-x + s + >"Q-1 s = 0

where>.. is to be calculated as the root of the equation h(>") = 0, (a = 0). Since it is assumed that x ¢ C(O, Q), we have h(O) > O. With>" - 00

we also have

This yields h('\) < 0, ,\ ~ ,\. for some ,\. > o. The equation h('\) = 0 therefore has a root ,\0 > O. The root ,\0 is

unique ~ince direct calculation gives h'(,\) < 0 with ,\ > O. The case a # 0 can now be given through a direct shift x -+ x-a. We will now prove that the ellipsoidal valued strategy U_(t, x) of (3.3)

does solve the problem of control synthesis, provided we start from a point X T = x(r) E £[r], r $ t $ tl.

Indeed, assume x E £_ [r] and x[t] = x(t, r, xT ) to be the respective trajectory. We will demonstrate that once x[t] is a solution to equation

(3.6)

we will have x[t] E £_[t], r $ t $ tl,

(With isolated trajectory x[t] given, it is clearly driven by a unique control u[t] = x(t) a.e. such that u[t] E P(t)).

Suppose, on the contrary, that the distance d(x[t.],£_[t.]) > 0 for some value t. > r.

Since x[r] E £_ [r] and since d[t] = d(x[t], L[t]) is differentiable, there exists a point t •• E (T, t.] such that

(3.7) d dt d[tll t=t •• > 0, d[t •• ] > O.

Calculating

d[t] = max{(f,x(t)) - p(f I £_[t]): IIfll $ I}

we observe d d dtd[t] = dt [(~,x(t))-p(fOI£_[t])]

and since fO is a unique maximiser,

!d[t] = (~,x(t)) - :tp(fOI£-[t]) =

= (to, u[t]) - ! [(to, z(t)) + (to, Q(t)tO)1/2]

where £_[t] = £(z(t), Q(t)). For a fixed function H(.) we have L[t] = £(Z(t),QH(t)) ,where z(t),

QH(t) satisfy the system (2.5), (2.6), (A(t) == 0). Substituting this into the relation for the derivative of d[t] and remembering the rule for differentiating a maximum of a variety of functions

! d[t] = (~, u[t]) - (fO, p(t)) - ~(~, QH(t)~)-1/2.

ELLIPSOIDAL TECHNIQUES

. (tJ, A-l(t)([A(t)QH(t)A(t)Jl/2[(A(t)P(t)A'(t)p/2+

+[A(t)P(t)A' (t)F/2[A(t)Q H(t)A' (t)F/2)A'-1 (t)tJ)

or due to the Bunyakovsky-Schwartz inequality

~ d[t] ~ (tJ, u[t]) - (fO, pet)) - (fO, QH(t)tJ)1/2,

where u[t] E £(p(t), pet))

and u[t] E U_(t, x).

For the case x f/. &-(z(t), QH(t)) the latter inequality gives us

~d[t]1 = 0 dt t=t ••

which contradicts with (3.7). What follows is the assertion

179

Theorem 3.1 Define an internal approximation &- [t] = &-(z(t), QH(t)) with given parametrization H(t) of {2.6}. Once X(T) E &-[T] and the synthesizing strategy is U_(t, x) of {3.3}, the following inclusion is true:

x[t] E &- [t], T ~ t ~ tl,

and therefore x[ttJ E £(m, M).

The ellipsoidal synthesis thus gives a solution strategy U_(t,x) for any internal approximation £_ [t] = &- (z(t), QH(t)).

With x f/. £_[t], the function U_(t,x) is single-valued, whilst with x E &- [t] it is multivalued (U_ (t, x) = &- [t]) being therefore upper-semi continuous in x, measureable in t and ensuring the existence of a solution to the differential inclusion (3.6).

We will now proceed with numerical examples that demonstrate the constructive nature of the solutions obtained above.

4 A Numerical Example

We take system (2.4) to be 4 dimensional. Let the initial position {to, Xta}

be given by

180 KURZHANSKY AND VALYI

at the initial moment to = 0, the target set M = C(m, M) by

m=(D and

Coo D

010 M==. 0 0 1

000

at the final moment t1 = 5. We consider a case when the right hand side is constant:

( 0 1

-1 0 A(t) ==. 0 0

o 0

o 0) o 0 o 1 '

-4 0

describing the position and speed of two independent oscillators. The restriction u(t) E C(p(t), P(t» on the control u, is also defined by time independent constraints:

P(t) ==. (~ ~ H), o 0 0 1

so that the controls couple the system. Therefore the class of feasible strategies is such that

U = {U(t, x) : U(t, x) ~ C(p(t), P(t»)}.

The results to be presented here we obtain by way of discretization. We divide the interval [0,5] into 100 subintervals of equal lengths, and use the discretized version of (2.6). Instead of the set valued control strategy (3.3) we apply a single valued selection:

{ p(t) ih E C_ [t] (4.1) u(t, x) = p(t) _ P(t)fO(£O, P(t)fO)-1/2 ih ¢ C_ [t].

again in its discrete version. We calculate the parameters of the ellipsoid C_ [t] = C_(z(t), QH(t» by

chosing H(t) = p1/2(t), t E [0,5]


in (2.6). The calculations give the following internal ellipsoidal estimate L [0] =

£(z(O), QH(O)) of the solvability set W(O, M):

( 4.2371 ) 1.2342

z(O) = -2.6043 ' -3.1370

and

(31.1385 0 0

Q (0) = 0 31.1385 0 H 0 0 12.1845

o 0 2.3611

o ) o 2.3611 .

44.1236

Now, as is easy to check, Xo E £_ [0] and therefore Theorem 3.1 is applicable, implying that the control strategy of (3.3) steers the solution of (3.6) into M.

As the ellipsoids appearing in this problem are four dimensional, we present their two dimensional projections. The figures are divided into four windows, and each shows projections of the original ellipsoids onto the planes spanned by the first and second, third and fourth, first and third, and second and fourth coordinate axes, in a clockwise order starting from bottom left. The drawn segments of coordinate axes corresponding to the state variables range from -10.0642 to 10.0642. The skew axis in Figure 1 is time, ranging from 0 to 5.

Figure 1 shows the graph of the ellipsoidal valued map L [t], t E [0,5] and of the solution of

(4.2) x(t) = A(t)x(t) + u(t, x(t)),

x(O) = Xo

where we use u(t, x) of (4.1). Figure 2 shows the target set M = £(m, M), (projections appearing as

circles), the solvability set L [0] = £(z(O), QH(O)) at the initial moment t = 0, and the trajectory of the solution of (4.2). .


Figure 1: 'lUbe of ellipsoidal solvability sets and graph of solution


Targat Problem

Figure 2: Target set, initial ellipsoidal solvablity set and trajectory in phase space


References

[1] Krasovski, N.N. The Control of a Dynamic System, Nauka, Moscow, 1986.

[2] Kurzhanski, A.B., Nikonov, O. I. Funnel Equations and Multivalued Integration Problems for Control Synthesis, in: B. Jakubczyk, K. Malanowski, W. Respondek Eds. Perspectives in Control Theory, Progress in Systems and Control Theory, Vol. 2, Birkhauser, Boston, 1990. pp. 143-153.

[3] Kurzhanski, A.B. Control and Observation under Conditions of Uncertainty, Nauka, Moscow, 1977.

[4] Kurzhanski, A.B., Valyi, I. Set Valued Solutions to Control Problems and Their Approximations, in: A. Bensoussan, J. L. Lions Eds. Analysis and Optimization of Systems, Lecture Notes in Control and Information Systems, Vol 111, Springer Verlag, 1988. pp. 775-785.

[5] Schweppe, F.C. Uncertain Dynamic Systems, Prentice Hall Inc., Englewood Cliffs, N.J., 1973.

[6] Chernousko, F. L. Estimation of the Phase State of Dynamical Systems, Nauka, Moscow, 1988.

Extended Gaussian Quadrature and the

Identification of Linear Systems

CLYDE F. MARTINI and ANDREAS SOEMADI

1. Introduction

The problems of parameter identification for linear systems and the problem of Gaussian quadrature would seem at first thought t,o be quite distinct. However for single input single output Iineltr syst.ems the problem of identification is readily seen to be equivalent to a moment problem and likewise it can be seen that the problem of Gaussian qlHldrltt.ure is a standard moment problem. The method of Prony was used by Ammar, Dayawansa and Martin in [2] to construct numerically satisfact.ory methods for exponential interpolation which is seen to be a system ident.ification problem.

In this paper we show that an extended version of Gaussian quadrature in which the values of various derivatives are used is similar to the problem of the identification of linear systems in which the output function is of the form of the sum of products of polynomials and exponentials. The solution of the system identification problem is found in [2] and for the purposes of space will not be included here. In this pltper we will conr.cntrat.e on the solution of the Gaussian quadrature problem as an extended moment problem using a generalization of Prony's method.

The parallels between linear systems theory and problems of interpolation and quadrature are striking. An important quest,ion which remltins unanswered is whether or not these parallels extend to nonlinear problems in system theory. The answer will probably lie in whet.hf'r or not there are nonlinear versions of such things as the moment problem.

2. Generalized Gaussian Quadrature

Consider the following quadrature problem. Assume that w(.r) > 0 and

1 Supported in NASA Grant ,NAG2-89, NSF Grant ,DMS 8005334 and NSA Grant ,MDA904-90-H-4009

186 MARTIN AND SOEMADI

is integrable on an interval [a, b]. A set of weights, Wij, and nodes, Xj, is to be found such that

16 r nj ti<i) J(x.) w(x)J(x)dx = L: L: Wij d (i/

a ;=li=O X

is exact for polynomial of maximal degree. It is now possible to do the following calculations

(1)

Consider the following notations:

y, = 16 w(x)x'dx,

ii. = (y" Y(.+1)'···, Y(,+N_l»)T,

where r

N = ~)ni + 1), i=1

and finally,

QUADRATURE

with WO;

WNjj

Now define the matrix: A, = (r'lIr,,"",r'r)'

where

X~ J

82:(,-1) j

x~+l J (8 + l)xj

r,; = x,+2 J (8 + 2)xj+1

, withj= 1,2,"',r,

n";-I( ') '-"I 1=0 8 - I Xj

'n";-I( + 1 ') ,-";+1 1=0 8 - I Xi

n"i- I ( + 2 ') ,-";+2 1=0 8 - I xJ

xj+N-I (8 + N - l)xj+N-2 '" a

187

and a = n~~;1(8+(N -1) - i)x~-";+(N-I), Thus, the system of equations (1) can be replaced by the N X Iv system of equations, U, = A, tV, Furthermore, define Il,/ to be diagonal matrix with entries (xj,xrl, .. "xt";) and let

1 8

xJ (8 + l)xJ

I',; = x~ J (8 + 2)x1

Observe that r,; = I',;Il,i'

fi"/-1( ') 1=0 8 - I

fi"/-l( 1 ') 1=0 8 + - t Xi

fi";-l( + 2 ') 2 1=0 8 - t Xi

n"/-I( N 1 ') N-I 1=0 8 + - - I Xj

Consider the coefficients of the entries of the m-th row of the matrices r,; and roj , where m = 1,2, .. " N, The coefficients of the m-th row of rO j is

(1, (m -1), (m - 1)(m - 2)"", [em -1)(m - 2)", (m - "i)]),


while the m-th row of f"j is

(I, (s+m-I), (s+m-I)(s+m-2), ... , [(s+m-I)(s+m-2) ... (s+m-nj)]).

Polynomials are to be determined to "reduce" the second quantity to the first. That is, for example, we are seeking polynomials Pu(s), PI3(S), P23(S) to reduce the second and third entries, i.e., determine those polynomials such that

(s + m - 1) + Pu( s) . 1 = m - 1

(s + m - 1)(s + m - 2) + Pt3(S) ·1 + P23(s)(m - 1) = (m - 1)(m - 2).

It can be shown that in general

where

Pu{s) = -s{s - 1)(s - 2) .. . (s - Ie + 2)

n-t

Pnl:{s) = L{*){-I)i Pu(s + (n -1- i», 1+0

( n-l ) (*) = i '

Ie> nand n = l,2,3, .. ·,nj. Now construct nj many square matrices of size (nj + 1), each of which

is of the following (orm

MI(s) =

100

010

o 0

o 0

o 0

o 0

o o

1 P(/_l)'(S) 0

010

o o 1

o o o

o o

o o o

1

where

QUADRATURE

PlI(S)

P(I-I)I

1

o

is on the It" column, and I = 2,3, .. " (nj + 1). Then let

nj+1

Mj(s) = II M,(S), 1=1

189

with MI(s) being the identity matrix. Also, let Vj be a diagonal matrix 'th t' (1 -I -2 -n;) WI en nes ,Xj' Xj ,"', Xj .

The following are some examples of Mj(s): for nj = 0, A{j(s) = (1), for nj = 1,

MU(s) = ( : ~s ),

for nj = 2,

Mi(.) = ( : -s s(s+ I) )

1 -2s ,

0 1

and for nj = 3,

1 -s s(s + 1) -s(s + 1)(s + 2)

0 1 -2s 3s(s + 1) Mj(s) =

0 0 1 -3s

0 0 0 1

Likewise for ni = 4,

1 -a a(a + 1) -a(a + 1)(a + 2) a(a + l)(a + 2)(a + 3)

0 1 -2a 3a(a + 1) -4s(a + 1)(a + 2)

Mi(a) = 0 0 1 -3a 68(8 + 1)

0 0 0 1 -48

0 0 0 0 1

and for ni = 5,

1 -a S(8 + 1) -2(8 + 1)(8 + 2) 0' a

0 1 -2s 3S(8 + 1) (3 b

0 0 1 -38 r c Mi(8) =

0 0 0 1 6 d

0 0 0 0 1 f

0 0 0 0 0 1

where 0' = a(8 + 1)(8 + 2)(s + 3), (3 = -4S(8 + 1)(8 + 2), r = 6S(8 + 1), 6 =

-4s, a = -sea + 1)(8 + 2)(a + 3)(8 + 4), b = 5S(8 + l)(s + 2)(8 + 3), c = -10s(8 + 1)(8 + 2), d = 108(s + I). f = -5s.

Using induction on ni' one can show for a square matrix Mi(8), the entries are

o 1

if Ie > 1

if Ie = 1

(-1)'-1: n~:I:-l(8+ i) if Ie < 1 ( 1-1 ) Ie -1 1_0

Observe that r'jMi(s)Di = fOr Thus,

f,; = foAMi(s)Di]-lA,;

and

QUADRATURE 191

At this stage, we wish to show that [MJ(3 + 1)]-1 Ali (3) is independent of 3. The following lemmas will be used.

Lemma 1 Let j and k be p03itive integer3. Let n be an, fired but arbitrary positive integer. If (j - k) > 0, then

j _ k _ 1 = (n + 1) E (n + 1)(n +?) .. ',(n + i - k) i=Hl (. - k).

( . _ k) ~ n(n + 1) ... (n + i - (k + 1» n + J L..J (. _ k)' .

i=i+1 •.

Lemma 2 Let j and k be positive integers such that (j - k) e Z+, then for all non-negative integers 3,

1 ~ 3(3 + 1) ... (3 + i - (k + 1» _ + L..J (. L), -

i=i+1 • - /I; •

(2) (3 + 1)(, + 2)· .. (3 + j - (k + 1»(. _ k) (j-k)! J.

The proofs of the two lemmas are inductive and will be ommittcd.

Theorem 1 Let MJ(3) be a square matriz of size (nJ + 1) by (nJ + 1) with entne3 defined b,

o 1

if k > j

if k = j

(-I)i-i( j-l )8(8+1)"'(8+ i -(k+l» ifk j

aiJ = 1 if k = j

(-I)i-ia=Hi if k < j

Then Mi(8)A = Mi(8+ I).

Proof: Consider Mi(s)A. The kj entries of Mi(s)A, with k = 1,2,.··, ";+ 1 and j = 1,2, ... ,"; + 1, are

if k > j

if k = j

if k < j

Thus, in order to prove the theorem, it suffices to show that

(3) ; E ml:i(8)oi; = ml:;(s + 1) i=1:

holds for all non-negative integer s whenever k < j. Consider the left-hand side (LIIS) of equation (3);

;-1 LHSo/(3) = ol:i + E mH(s)oi; + ml:;(s)

1=1:+1

= (-1);-1:0 - k)! (k - I)!

;-1 (-ly-I:(; _ I)! i-(Hl) (j -I)! + L (I: _ 1)'(" _ 1:)' II (s + I) C - I)'

i=l:+l . • . 1=0 •.

Thus for all s and whenever k < j, we are to show that

(-ly-1: (j -I)! (1 + ~ s(s + 1) .. . (s + i - (I: + 1») _ (I: - I)! L- (i - 1:)' -

1=1:+1 .

QUADRATURE

But the right-hand side of the previous equality is equal to

'-i (i-I)! . (-1)' (k -1!(j _ k)!«8 + 1)(8 + 2)·· '(8 + J - k»-

'-i (j-l)! . (-I)' (k _ 1)!(j _ k)!(S(8 + 1) .. '(8 + J - (k + 1))).

Hence it is to be shown that

or

1 + I: 8(8 + 1) ... (8 + i - (k + 1» = i=l:+1 (i - k)!

(S + 1)(8 + 2) .. '(8 + j - k) - s(s + 1)·· .(s + j - (k + 1» (j-k)!

j-1 1 + L S( 8 + 1) ... (8 + i - (k + 1» =

i=1:+1 (i - k)!

(s + 1)( s + 2) ... (s + j - (k + 1» ( . _ k) (j-k)! J,

for all non-negative integers s whenever j > k.

193

The previous lemma can now be used to finish the proof of the theorem.

Corollary 1 (Mj(s + 1»-1 Mj(s) is independent of s.

Lemma 3 Let I be the identity matrix of size nj + 1 bY"j + 1. lf A is the square matrix given in Theorem 1, then the index of nilpotency of A - I is nj + 1.

Proof: The lemma will be proven by using induction on the size of the matrix A.

For nj = 1, (A is a 2 x 2 square matrix) the lemma is obviously true. For nj = k, (A is a k + 1 x k + 1 square matrix), suppose that

(A - I)1:+1 = 0 and (A - 1)1: :f O.

Then it is to be shown that for nj = k + 1, (A is a k + 2 x k + 2 square matrix,

(A - 1)1:+2 = 0 and (A - I)Hl :f O.


Denote the matrix of size Ie + 2 x Ie + 2, by A·. Then

where

Notice that

= (~ I (A -01)· B )

#; 0

and

= O.

Now let us int.roduce some more notation. Let M9s) be the block diagonal matrix

DIAG(M1(s), ... , Mr(s»,

D the block diagonal matrix

DIAG(D1,···, Dr),

and o6(s) the block diagonal matrix

DI AG(06," .. " ~'r)'

Theorem 2 The matrix A,+! A; 1 is independent of s and it is also similar

QUADRATURE

to the matrir J I of the form

ZI 1 0

o ZI 1 0

o

o

o 0 Z2 1 0

o 0 0 Zr 1

o 0

Proof: Since A5 = Ao(M(s)D)-16(s), then

A.+1A;1 = Ao(M(s + I)D)-16(s + 1)6 -t(s)M(s)DAol

= AoD-l M-l(S + I)DM(s)(AoD-l)-t.

195

Thus the matrix A.+1A;1 is similar to the matrix M-l(s + l)DM(s). Furthermore, such a matrix M-l(s + l)DM(s) can be partitioned into a block diagonal matrix. The j-th block of the above diagonal block matrix is

(Mi(s + 1»-1 DiMi(s),

which is equal to zj(Mi(s+ 1»-1 Mi(s), and by Corollary 1 is independent of s. lienee, the matrix A.+tA;1 is independent of s. Furthermore the elements of the matrix (Mi(s + 1»-1 DjMi(s) are

if k > I

if Ie = I, ,

if k < I

where Clel are some real constants. Thus the characteristic polynomial, P(z), of the matrix

(M(s + 1»-1 DM(s)

is (z - ZI)"I+t(Z - Z2)"2+1 ... (z - zr)"r+1, Now use Lemma 3 to conclude that (M(s + 1»-1 DM(s) has a block diagonal matrix Jordan representa..


tion, the diRgonRI entries of which rue of the form

%j 1 0 0 0 0

o %j 1 0 0 0

000 ~ 1 0

000

000

%j 1

o %;

Therefore, the matrix A.+!A;l is similar to the mRtrix J. Now that A,+!Ai is independent of s, one CRn obtain a recursive rela

tion between fl.+! and fl. from which a method f computing the nodes %j

can be ge~erated.

Theorem 3 It is stlfficient that the nodes %j be the roots of a polynomial that annihilates %1 for maximal k.

Proof: Since fl. = A.tV and y.+1 = A.+! w then

fl.+l = A.+!A;lfl •.

Notice that the shifted nature of A.+!A;l implies that A,+lA;l is of companion form. Suppose such a form is represented by C, where

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

c= 0 0 0 1 0

0 0 0 0 1

Qo Qt Q2 QN-t

and the Q~S are real numbers. consider the last row of fl.+!. Then it follows that

N-l E QiY.+i = y.+N· i=O

QUADRATURE 197

Referring to the definition of YI:, the following equality,

holds. The equality implies the sufficiency of Xj to be the roots of the polynomial E~=-ol O'l:xl: - x N , which would annihilate x' for maximal s.

Recall that E;:OI O'iY,+i = 11a+N. where 8 is non-negative integer. Thus

N-l

L O'iYi = YN i=O

N-l

L O'iYl+i = Yt+N i=O

N-l

L O'iY2+i = Y2+N i=O

N-l

L O'iYN-l+i = Y2N-l· i=O

Therefore, it must be the case that;; = (0'0,0'1, ... , 0' N _.)T is the solution to

Yo Yt YN-l YN

Yl Y2 Y3 YN YN+l

Y2 Y3 YN+l YN+2

YN-l YN YN+1 Y2N-2 Y2N-l

The weights of the quadra.ture rule can be obtained hy solving

Yo = Aow,

for W. A numerically stable method concerning this procedure was developed by Ammar, Dayawansa, and Martin [2].

Finally, to conclude this chapter we are to obtain the degree of precision of such a Gaussian quadrature, described by the beginning of this chapt,er. To do so, we refer the reader to the theorems of Popoviciu [7] and of Karlin [6].

Theorem.. For any prescribed positive odd integers, < J'i >~=1' there , t d' t t· k l' (b) d lb· i=l,j=Pi- 1 eXls IS ance i = In a, an rea num ers < aij >i=l,j=O

such that • i Pi-1

RU) = 1 /(z)w(x)dz - L L a;;/;(t;) a i=1 ;=0

vanishes for f E 1'n-1o (set 0/ polyn.omials 0/ degree ~ n - 1), where n = E~==1 J'i + k,

The uniqueness of such nodes, it, and weights, at;, W8.'! proven by KItrJin. Moreover, he also showed that for any prescribed even integers Iti, provided that J'i = I for all i = 1,2, ... , k, then quadrature rule is exact for all f EP,i-l.

Corollary 2 The Gaussian quadrature problem described at the beginning of this chapter has a degree of precision, if n; is even, of N + r-l. Jl owever, if n; is odd, then the degree 0/ precision is (I + l)r - 1, provided that n; = I Jorallj= 1,2, ... ,r.

3. Conclusion

To solve the generalized Gaussian quadrature prohlem described a.t the beginning of this chapter, the following procedure can be used.

1. Find Ii = (ao, at.'··, aN-I), from

Vo Yl V2 YN-l YN

VI YN VNtt

Y2 Va

YN-l YN YN+1 Y2N-2 V2N-l

2. Find the roots of the polynomial E~;Ol O'iZl - zN. There are r distinct roots which will be the nodes of the generalized Gaussian quadrature.

QUADRATURE 199

3. To obtain the weights of the generalized Gaussian Quadrature, solve fio = Aow, for w.

4. The degree of exactness of the generalized Gaussian quadrature can be determined by using Corollary 2.

A numerically stable method concerning step 1 and step 3 can be found in the paper of Ammar, Dayawansa, and Martin [2].

This procedure is another alternative in solving the generalized Gaus.'!ian quadrature. Different procedures can be found in the papers of Thran [3], Popoviciu [7], Karlin and Pinkus [6].

References

[1] IIildebrand, F.B., Introduction to Numerical Analysis, New York: McGraw-Hili Book Co., 1956.

[2] Ammar, G., Dayawansa, W.P., Martin, C., Exponential Interpolation: Theory and Numerical Algorithms, preprint.

[3] Thran, P., On the Theory 0/ the Mechanical Quadrature, Acta Sci. Math., Szeged. 12, Par. A, 1950, pp. 30-37.

[4] de Prony, R., Essai experimental et analytiqtte, J. Ecole Poly tech., Vol. 1, 1785, pp. 24-76.

[5] Gantmacher, F.R., Matrix Theory, Vol. 1, New York: ChelRea Publishing Company, 1977.

[6] Karlin, S., Pinkus, A., Studies in Spline Functions and Approximation Theory, New York: Academic Press, Inc., 1976.

[7] Popoviciu, T., Asurpa unei generalizari a/ormtllei de intrgrnre ntlmer;ca a lui gauss, Acad. R.P. Romine Fil. Iasi Studii Cere. Sti. 6, 1955, pp. 29-57.

Clyde F. Martin Texas Tech University Lubbock, Texas, USA

Andreas Soemadi Iowa State University Ames, Iowa, USA

Multirate Sampling and Zero Dynamics: from linear to nonlinear

S. Monaco and D. Nonnand-Cyrot

Abstract

It has recently been shown that the concept of zero dynamics plays a central role in the design of some nonlinear control systems. As in the linear context, where the property of stable zeros is necessary in the use of direct design techniques, in the solution of nonlinear control problems such as input-output linearization, tracking or input-output decoupling, the stability of part or of the whole zero dynamics constitutes a basic requirement

When solving the above-mentioned nonlinear control problems by means of a digital scheme, where the design of the control law is based on the sampled model of the plant, some extra problems appear since the zero dynamics stability is not preserved under sampling. In fact, for small sampling intervals, the zero dynamics of the sampled model is always unstable if the relative degree of the plant is greater than one.

The purpose of this paper is to show how this drawback can be avoided by using a discretization technique on a time scale on the output which is a multiple of the time scale on the control (multirate sampling), the order being equal to the relative degree of the continuous given single input-single output model.

Multirate discretization techniques are known in the literature; in the linear case this technique at order n allows the arbitrary positioning of zeros of the sampled transfer function. A different point of view is taken here where the proposed multirate sampled model results in a square system of dimension equal to the multirate order. The paper studies the properties of the zero dynamics of the multirate sampled model of a given nonlinear plant in the SISO and MIMO cases. It is shown that the multirate control strategies based on such a sampling technique allows us to obtain results which maintain and also improve the performances of the continuous control scheme.

1. Introduction

The role played by the zeros or transmission zeros of a scalar or multivariable linear

process in several design techniques is well known. The relevance of the notion of zero

dynamics in nonlinear control theory was immediately clear from its introduction in the

literature [3-7]. In fact, the main part of synthesis methods available for linear analytic

control systems involves total or partial cancellation of the zero dynamics (e.g., input

output linearization, stabilization, tracking, disturbance and input-output decoupling). The

stability of the zero dynamics minimum phase property is a major issue in the solution of

these problems.

The loss of stability of the zero dynamics under sampling is one of the main problems when dealing with digital control [1, 10].

MUL TIRATE SAMPLING AND ZERO DYNAMICS 201

To overcome that problem, suitable multirate digital control strategies were

introduced in [9, II] to maintain the minimum phase property of the continuous plant,

thus allowing for a corresponding digital solution to any existing continuous one, at least

for the class of single input-single output nonlinear systems. The idea of multirate

sampling at an order equal to the dimension of the system, proposed in [2] as a method of

discretization which makes it possible to assign the zero of the pulsed transfer function,

was also used recently to solve the digital linearization of a nonlinear continuous plant [5].

The purpose of this paper is twofold: to show how multirate digital control can be

fruitfully used to solve control problems for non minimum phase systems which do not

admit a simple continuous solution, and to introduce multirate digital control for the class

of multivariable nonlinear systems.

To understand the idea which is at the basis of our approach to multirate digital

feedback, let us consider the following simple control problem. Given a scalar continuous

linear plant represented by matrices (A,B,C) and a reference output Yr(t), design a digital

feedback on the sampled values of the state such that the output of the control system

coincides with Yr at the sampling instants t=kB. Denoting by (AD,BD,C) the representation

of the sampled plant over time intervals of amplitude B, it is easily verified that the zero

order holding of the discrete control

uD(k) = - (CBD)-l (CAD x(kB) + yrC(k+ I)B»

is well defined for almost all small B and solves the posed problem. It is well known

that under such a control law the dynamics of the whole control system (which is the right

inverse of the given one) is characterized by the zeroes of the sampled plant; so the

stability of the zeroes must hold.

This fact represents an obstruction in the design of digital controllers since unstable

zeroes appear frequently under discretization. Taking into account that the sampled

transfer function has (n-l) zeroes, and limiting our analysis to the more usual cases in

which B is small enough, the sampled plant has m zeroes which maintain the properties of

their corresponding in the continuous plant while the others (n-m-I) are unstables if n

m>2 ([I]).

To overcame these problems we propose a digital control law which is maintained

constant over time subintervals of the sampling (and observation) period; in this way one

obtains a sampled model with more independent inputs which can be settled to guarantee

the stability of the control system.

Let us return to our tracking problem and assume that the continuous plant be

minimum phase. We admit our control to change (n-m) times, between two sampling

instants, in order to impose that the sampled values of the output and its first (n-m-l)

derivatives coincide with Yr and its first (n-m-I) derivatives at time t=(k+ I )B. Such a

control law gives a solution to the digital tracking problem with stability, since the

202 MONACO AND NORMAND-CYROT

minimum phase property is maintained ([9), moreover we get a "smooth" tracking (since

we imposed the coincidence of the first (n-m-l) time derivatives).

If the plant is not minimUnl phase, a multirate control for solving the digital tracking

problem with stability can be designed arguing as in the follows: let W(s) =N+(s)N-(s) I D(s), where N+ denotes the factor of N whose zeroes have positive real part; moreover let

y'(t) = C' x(t) a "dummy" output function such that W'(s) = N-(s) ID(s). Use a multirate

of order (n-m-) > (n-m) such that y' and its (n-m--l) derivatives are fixed to impose: (i)

the sampled values of the output and its first (n-m-l) derivatives coincide with Yr and its

first (n-m-l) derivatives; (ii) the last (m-m-) derivatives of y' be zero. In the particular

case of yr(t) = const and y'(t) such that W(s) = K I D(s) we obtain dead beat output

regulation with flat response.

This simple idea, based on the fact that the problem can be solved by a suitable

multirate control law which tracks the reference output without cancelling the unstable

factor of the zero linear dynamics, is applied, in section 4., to the stabilization of a

nonlinear nonminimum phase plant.

2. Preliminaries on multirate sampling and zero dynamics

Let us consider a single input - single output continuous system of the form

it(t) = f(x(t» + g(x(t» u(t)

yet) = h(x(t»

(2.1)

(2.2)

where the state x belongs to an open subset M of An, U e R, f and g are analytic vector fields defined on M and the output function h is analytic on M. Denoting by Ll- the Lie

derivative of the function A along the vector field 't, we recall the following definition.

D(!initjQn 21, The continuous system (2.1, 2.2) has relative degree r < n if:

x e U an open dense subset of M, Ix indicates the evaluation of the function at x.

Let us assume the control u(t) constant over intervals of amplitude % and denote by

ui(k) the corresponding constant value over the interval [k8 + (i-~)8 , k8 + i~ [

for i=l, ... ,r, and t = k8.

MULTIRATE SAMPLING AND ZERO DYNAMICS 203

It is now possible to defme the multirate sampled system of order r associated to the

continuous system (2.1), (2.2). As introduced in [9, 11], it is composed of the sampled

state dynamics of (2.1) over r intervals of amplitude ~ and the r-l ftrSt derivatives of the

output function (2.2).

Definition 2.2. The multirate sampled system of order r associated to the continuous

model (2.1), (2.2) is described by the following equations

x(k+l) = F&(x(k), ul(k), ... , uf(k»

Yi(k) = ~(x(k» = Lri-1h/X(k) for i=I, ... ,r •

with

(2.3) (2.4)

(2.5)

~UILa) In equation (2.5), ef denotes the flow associated to the solution of the equation (2.1) and Id the identity function.

After performing expansions in terms of the successive powers of ui in the

exponential terms appearing in (2.5) and adequately regrouping homogeneous terms, we

can rewrite (2.5) as follows: f

F&(., ul, ... , ul) = F&O(.) + L uf F&i(.) + O(y2) (2.6) i=l

where O(u2) represents the nonlinear action of the control on the state dynamics. We

easily compute the expressions:

F&O(x) = e&l.f (ld)/x (2.7) (i-l).hf &! (f-i)&4

Fi(x) = e f (L;p; Lfi W La) e f (Id)/x pSI p.

(2.8)

with the shuffle product "w" recursively defined as follows:

Compact expressions for the terms appearing in O(1l2) can be obtained by using

204 MONACOANDNORMAND~OT

combinatory formulae related to the exponential expansion. A more detailed presentation

of this can be found in (11). In the present context it is important to emphasize the

following facts.

Remarks

(i) In the square multi-input. multi-output sampled system (2.3). (2.4). the relative degree ri associated to each output Yi is equal to 1 for i = 1 •...• r. This implies that

\l\su(i=l;r; )dj = r and enables us to maintain the dimension of the zero dynamics as

developed in [11).

(ii) After some lengthy but easy calculation. it can be shown. as noted in [11). that

the input decoupling square matrix A(x). of order r. associated to the sampled system

(2.3). (2.4) is defined by:

(2.9)

Considering the matrix coefficients of lower power in ~. we recover the Vandermonde

matrix. which is non singular. multiplied by LgLtlh/x. which differs from zero. This

proves the non singularity of the sampled decoupling matrix. O(&-i+2) contains terms of

order superior or equal to r-i+2 in ~.

(iii) The zero dynamics associated to the sampled system (2.3). (2.4) maintains. for

small ~. the properties of the zero dynamics associated to the original continuous model

(2.1). (2.2). This fact discussed in [9) is easily understood when considering these two

zero dynamics according to their respective definitions.

The continuous model Given a SISO continuous system of relative degree r. an equilibrium point xe

satisfying h(xe) = O. we denote by L(xe) the n - r dimensional surface defined by

L(xe) = [x E M I Lihlx = o. i = O ..... r-l)

We briefly recall that the defmition of zero dynamics proposed in [3-7) corresponds to the (n-r) dimensional state dynamics restricted to L(xel when a feedback control law assigning

the output and its first r-l derivatives to zero is applied.

MULTIRAlE SAMPLING AND ZERO DYNAMICS 205

The sampled system

By analogy to the continuous situation. the zero dynamics can be defined [10] as the

restriction to L(xe) of the (n-r) dimensional state dynamics when a feedback control

strategy which forces the r outputs Yi to zero is applied.

The use of a multirate control strategy is fundamental to force the vector output to

zero. and to maintain this value. With a single rate. only one derivative of the output can

be annulled.

3. Digital stabilization of non minimum phase plants

In this section we will show the possibility of solving stabilization or output

tracking problems by means of digital control when the continuous plant is not minimum

phase and state feedback linearizable. The idea of using multirate digital control strategies

is based on the fact that. due to the nonsingularity of the decoupling matrix A of the

multirate sampled model (2.3). (2.4). it is possible to impose desired values to the output

and its first (r-l) derivatives by state feedback. We easily show that the expressions for

the successive output functions Yi = Li-1h for i = I •...• r can be written as follows:

(3.1)

The non singularity of the decoupling matrix A(x) in (2.9) which can be rewritten as

[ d SlIi(x. u1 •.•.• ur) / ] A..(x) = .

IJ dol u=o !=1...r J=I ... r

is a necessary and sufficient condition for the existence of a digital feedback strategy

which brings the outputs to fixed values. The condition 5 sufficiently small ensures the

convergence of the series (3.1).

Lernma3-1.

Assuming the system (2.1). (2.2) state feedback linearizable. then there exist

analytic functions cp : M ~ Rand H : Rn-r+i ~ R such that

(3.2)

206

Proof From the hypothesis of state feedback linearizability, the existence of a function cp :

M ~ R which has relative degree n follows. By performing the well-known coordinates transformation: ,= (cp, LfP, ... , Ltlcp)T, [6], we obtain:

~ = ~+1 ~ = a(z) + b(z) u y:::; Ii(z). (3.3)

Since the relative indice of the system is r, simple computations show that Ii depends on the first (n-r+ 1) new coordinates only. •

Remqrks (i) Noting in the previous lemma that the function H depends on cp and its first (n

r-l) derivatives, we can rewrite this function as: H(D, cp) where D denotes the operator of

derivation with respect to time.

We recall that, xe being an equilibrium point such that h(xe> :::; 0, the zero dynamics

is characterized by the property y(t) = 0, t ~ 0; we immediately conclude that the differential equation

initialized at ~ = (cp(xo)' ... , Lrn-rcp(xo»T (xo e zero dynamics surface) gives an implicit

description of the zero dynamics of the system. (ii) The result stated in Lemma 3.1 is a nonlinear counterpart of the well-known fact

that for a linear controllable SISO system the numerator of the transfer function can be expressed as the application of a polynomial of degree m in the operator D to a linear function of the state which has relative degree equal to n:

y = ex = p(D) cpx.

Proposition 3.1. If the function H which defines the zero dynamics of the feedback

linearizable continuous plant admits a factorization of the form:

such that the autonomous differential equation

MULTIRATE SAMPLING AND ZERO DYNAMICS

~(D. ~)=O

is asymptotically stable at ~ = O. then there e;Kists a multirate control law which stabUizes

the system.

Sketch o/proof. The proof is based on the fact that, assuming as dummy output y'(t) = h'(x) = ~(D. cp), the new system has relative degree r' > r and has an asymptotically

stable zero dynamics. By applying a multirate digital dead beat control strategy which

fixes

y'(x+l) = O. yo(k)(k+l) = O. k = 1 •..•• r' - 1.

the state evolutions of the sampled model are maintained on the stable zerQ surface

dynamics.

Remarks (i) When dealing with linear plants the factorization in Proposition 3.1 always edsts

and corresponds to the decomposition of the numerator of the transfer function with

respect to its stable and unstable factors. In that case a multirate of order r' = n • m

(m- = degree of the stable factor) can be used to obtain asymptotic stabilization. (ii) The factorization of the zero dynamics in proposition 3.1 corresponds to the

edstence of a function, namely y' = ~(D. ~), which has relative degree r' > r, its "zero

dynamics" is stable and its "zero dynamics surface" is contained in the real one, i.e .• that

one which is associated to the real output. The previous remark (i) gives the idea for computing such a function in the linear case; we must now consider the system described by the transfer function where the unstable zeros are cancelled.

Similar arguments can be used to obtain "digital tracking" when the continuous plant

is not minimum phase.

4. Multirate samplin, for square multi-input - multi-output systems

Let us consider the MIMO continuous system

m x;'(t) = f(x(t» + L gj(x(t» uj(t)

j~l

Yj(t) = hj(x(t», i = 1 •... , m; Xo e Rn

(4.1)

(4.2)

where f, gi : M ~ Rn are analytic vector fields and the output functions hi are analytic on

M. Let us assume that the relative indices associated to each output Yi are defined (Le.,

ri < 00, i = 1, ... , m) and that the continuous zero dynamics is built by applying the zero

dynamics algorithm proposed in [3-7].

The purpose of this section is to compute a multirate sampled model the zero

dynamics of which preserves the continuous one. For this we extend the method

proposed in section 2 by assuming that each control ui is constant over time intervals of

amplitude Si = !. for i = 1, ... , m and denoting uij for j = 1, ... , mi its value.We then 1

consider the corresponding multirate sampled state dynamics of order ill, with ill the

minimum common multiple of the mi, i = I, ... , m. We obtain the state equation

x(k+ 1) = F~(x(k), ui(k), ... , u~j(k), ... , u~ (k), ... , u'::tm)

m

The first questions which arise are how to choose M = L mi ~ n dummy outputs i=1

and the multirate order mj on each input in such a way that the resulting square MlMO

sampled system will be characterized by a zero dynamics which reproduces the properties

of the continuous one. Following the notion of relative indices associated to the outputs,

we define the set of "relative indices associated to the inputs" of the continuous model

(4.1, 4.2).

Drjinition 4.1. Pi is the relative indice associated to ui for i = 1, ... , m if

'ltj = 1, ... , m,

(Pi represents the minimum order of derivative of the output vector affected by the

ith control u).

Remarks (i) The equality r = P obviously holds in the SISO case.

(ii) It is clear that Pi ~ n and that Pi = 00 means that the ith input will never affect

any output.

The "input set of relative indices" is said to be well defined if Pi < 00 for

i = 1, ... , m. After an eventual reordering of the outputs, we can assume that PI :l!: ... :l!: Pm and r1 :l!: ••• :l!: rm with Pm = rm' The following proposition can be proven.

Praposition 4.1. Assuming that the input set and output set of relative indices are defined and that the continuous zero dynamics is uniquely defined, then there exists a unique choice for the multirate orders on the inputs which enables us to preserve, for small a, the

properties of the continuous zero dynamics.

Remarks

(i) We say that the zero dynamics is uniquely defined if the algorithm for computing this zero dynamics stops and if the continuous feedback law u = ')'(x) which defines this

zero dynamics is unique.

(ii) When the zero dynamics is not uniquely defined, the multirate strategy is also

not unique.

Sketch of proof When the continuous system has an invertible decoupling matrix, the

situation is a straightforward extension of the SISO case and the choices for the outputs and the multirate order ~ are very intuitive. In fact, after a possible reordering of the

m

inputs and outputs, we fix the L rj output functions as ~'+1 = L~hj for j = 1, ...• m and j=1 J

ij = 0, ... , rj - 1 and set mj = rj. Generalizing the techniques of expansion recalled in section 2, it is easy to verify

that the decoupling matrix corresponding to this square sampled system is non singular m

and that the (n - L ri) dimensional zero dynamics preserves the properties of the i=1

continuous one.

In the more general situation which occurs when the continuous decoupling matrix

is not invertible but has a "uniquely defined" zero dynamics. we must take into account

the successive steps of the algorithm. In fact, in such a case. the zero dynamics algorithm

allows the construction of dummy output functions with respect to which the decoupling

matrix is invertible. We thus revert to a known situation. Denoting by }.;1' •••• A.m these output functions and assuming the output and input

sets of relative indices are ordered as r't :l!: ... :l!: r'm and P't :l!: ••• :l!: P 1m with

m m L r'o > L r., we easily choose the multirate order on the inputs uj as r'j = m'j for j=l J j=l J


m

j = 1 •...• m and the I, r'j output functions as ~'+1 = ~A.j for j = 1 •...• m and ij = O. j=l 1

.• r'j-l.

It is a matter of calculation to verify that the resulting multirate square sampled m

system admits a non singular decoupling matrix and a (m - I, r'j) dimensional zero j=l

dynamics which preserves the properties of the continuous one.

To conclude this section we show how multirate control strategies can be designed

to solve. in terms of static state feedbacks. control problems which admit either dynamic

state feedback solutions or are not solvable in the continuous time case. This fact is

illustrated below by two simple linear examples for which the decoupling problem is

unsolved.

EXlllI/llie 1.

Let us consider the linear system

it = Ax + By. y = Cx

(0 1 0] with A = 0 -10 •

101 (00] 0 1 1

B= ~~ . C=(111)

It can easily be verified that r 1 = r2 = 1 and that the decoupling matrix U :;: [ ~ ~ ] is

singular; the decoupling problem thus does not admit a continuous static state feedback

solution. Performing the preliminary invertible feedback law

and setting v2 :;: z and z;' :;: wI' we obtain the extended system SE

y=Cx

We note that. with the new control vector (wI' v2). the corresponding output

relative indices are r~ = ~ :;: 2. The decoupling matrix UE = [ ! ~~ ] is non singular and

the decoupling problem is thus solvable.


A static state digital solution can easily be computed. Let us note that S is fully

linearizable with respect to the two dummy outputs

We obtain r\ = 2 and r'2 = 1 and the invertible decoupling matrix U' = [~ ~ ] Moreover, in this case we easily compute p\ = 2 and P'2 = 1. This enables us to fix a

double rate on ul and a single rate on u2. We have:

x(k+ 1) = A6x(k) + A6 b~u~(k) + b~~(k) + b~ u2(k); 2" 2" 2"

YI(k) = xl(k); Y2(k) = x2(k); Y3(k) = x3(k»

6

with A6 = e6A and b~ = J e'tAbi d't, i = 1,2. o

The existence of a static digital control law which brings, in one sampluing time cS, the three dummy outputs YI' Y2 and Y3 respectively to v2 - v!' ° and VI and therefore the

two real outputs YI and Y2 respectively to VI and v2 follows from the non singularity of

the sampled decoupling matrix

Example 2,

Let us consider the linear system

it = A'x + B'lL; y = Cx

with ( 010] (00] A' = ° -10 , B' = 10 , 1 ° 1 01

( 120) C'= 100 .

The corresponding matrix of transfer functions is given by:

[ 1+2s 0] S'(s+l)

S(S~l) ° .


We note that the conttol u2 never affects the output vector of S' and thus the decoupling

problem cannot be solved with respect to external conttols.

We show below that a digital solution exists. In fact, if we consider the two dummy outputs Y 1 = Xl and y 2 = x2' a double rate on uI, the decoupling problem is solvable by

means of a static digital conttollaw designed on the following sampled system

Y2(k) =~(k)

It is clearly sufficient to compute a feedback which brings in one sampling time the

two dummy outputs YI and Y2 to v2 and ~ (VI - v2) respectively, which means the two

real outputs YI and Y2 to VI and v2 respectively. The existence of such a feedback is

linked to the rank equal to two in the following matrix

(A'II b~, b~). 2" 2" 2"

Concluding remarks

In this paper we show how multirate sampling can be used to preserve the

properties of the zero dynamics of a MIMO continuous plant. The multirate conttol

strategies proposed correspond to the digital version of continuous time solutions.

It is also shown how multirate conttol strategies can be more powerful than

continuous time strategies when usual continuous solutions do not exist. This possibility

is illustrated in the stabilization problems of non minimum phase SISO plants and in the

decoupling of MIMO plants.

These ideas, which are not yet fully understood in the linear context either, will be

the object of further studies.

References

[1] K.T. AstrOm, P. Hagander and J. Sternby, Zeros of sampled systems, Automatica, 20, pp.21-30, 1984.

[2] K.T. AstrOm and B. Wittenmark, Computer Controlled Systems, Theory and Design, Prentice-Hall, 1984.


[3] C. Byrnes and A. Isidori, Local stabilization of critically minimum phase nonlinear systems, Syst and Contr. Letters, 11, pp.9-17, 1988.

[4] S.T. Glad, Output dead-beat controlfor nonlinear systems with one zero at infinity, Syst. and Contr. Letters, 9, pp.249-255, 1987.

[5] J .W. Grizzle and P. V. Kokotovic, Feedback linearization of sampled data systems, IEEE Trans. Aut. Contr., 33, pp.857-859, 1988.

[6] A. Isidori, Nonlinear Control Systems, an Introduction, Lect. Notes in Cont. and Info. ScL, 72, Springer Verlag, 1985.

[7] A. Isidori and C. Moog, On the nonlinear equivalent of the notion of transmission zeros, in: C.I. Byrnes and A.H. Kurszanski, Eds., Modeling and Adaptive Control, Springer, Berlin, New York, 1988.

[8] S. Monaco and D. Normand-Cyrot, Minimum phase nonlinear discrete time systems and feedback stabilization, Proc. 26th IEEE CDC, Los Angeles, 1987.

[9] S. Monaco and D. Normand-Cyrot, Sur la commande digitale d'un systeme non liniaire d dephasage minimal, Lect. Notes in Cont. and Info. Sci. (A. Bensoussan and J.L. Lions, cds.), III, Springer-Verlag, Berlin, pp.193-205, 1988.

[10] S. Monaco and D. Normand-Cyrot, Zero dynamics of sampled nonlinear systems, Syst. and Contr. Letters, 11, pp.229-234, 1988.

[11] S. Monaco and D. Normand-Cyrot, Multirate sampling and nonlinear digital control, Rap. D.I.S. University of Rome "La Sapienza", submitted for pub., 1988.

Salvatore MONACO Dipartimento di Ingegneria Elettrica,

UniversitA di L'Aquila, 67040 L'Aquila, ITALY.

Dorothee NORMAND-CYROT Laboratoire des Signaux et Systemes, CNRS-ESE, 91192 Gif-sur-Yvette, FRANCE.

(1)

FACTORIZATION OF NONLINEAR SYSTEMS

HENK NIJHEIJER

Abstract

We introduce a new concept of factor distribution for a nonlineat system as a new tool for studying the decomposition of such a system. The idea of factorizing generalizes a similar idea from linear system theory as well as the notion of controlled invariance for nonlinear systems.

1. Introduction

Consider a smooth nonlinear control system L locally described as

{ Yx - f(x.u)

- h(x.u)

where x - (xl •...• Xn ) are local coordinates of a smooth manifold 11. u ~ (u1 •...• um) are the controls belonging to some neighborhood U in Rm. f is a smooth mapping in both its arguments and the outputs Y - (Yl' ·.·.Yp ) are defined via the output map h which is also smooth in x and u. The purpose of this paper is to discuss some preliminary results on a general theory of decomposing the system (1) into two (or more) lower dimensional systems of a similar form. Of course. when studying this problem we need to specifl what we mean by saying that L is the composition of two systems. say Land P. The basic idea we will use throughout is that we want to obtain the system L as a lower dimensional system ~ together with a dynamic compensator P. That is. we search for a system ~ locally given as

{: - [(x.v)

(2) - h(x.v)

and a "x-parametrized" system P

{ ~ -rp(z.x.u)

(3) a(z.x.u)

so that the precompensated system. to be denoted as Po~. equals L. and thus (2) and (3) provide in local coordinates (x.z) for M another description of the system (1). In case we have L - po~ we will call f a factor system of L.

The study of the above decomposition or factorization problem for a

FACI'ORIZATION OF NONLINEAR SYSTEMS 215

general nonlinear system is motivated for at least three different reasons. Firstly, our decomposition problem extends the decomposition theory for nonlinear systems as described in [Kr, Re, Fll, Fl2, NvdS) where parallel, series and cascade decompositions are characterized. Though not very explicit in these references, one important reason to discuss such decompositions is, undoubtly, that control problems for decomposed systems are in some sense simpler than for the original system. Of course, this holds true as long as we believe in an implicit assumption that it becomes an easier task to solve control problems for lower dimensional (factor) systems. A simple example illustrates this point of view. Various control. problems for simple robot manipulators are based on the computed torque method, thereby neglecting that in a first step the torques applied at each joint are not controls but outputs generated by, for instance, electric motors.

A second motivation for studying factorizations comes from linear system theory. A complete characterization of a factor system for a linear finite-dimensional system x - Ax + Bu, y = ex + Du is given in [Sc), thereby extending earlier results from [Wi, Tr). An interesting complement to finite dimensional factorization forms the idea of "aggregation" of an infinite dimensional linear system as was studied in [Pi). One of the first approaches in discussing some kind of factorization of nonlinear systems was given [AS) where the idea of "reduction" was used in the study of local controllability and optimal control problems.

A third line of inspiration for our factorizations forms the (differential) geometric approach in nonlinear control theory. In the last decade a lot of the linear geometric theory, cf. [Wo), has been generalized to a nonlinear context, see e.g. [Is, NvdS). Essential tools in this frame are the so-called (linear) controlled invariant subspaces or (nonlinear) controlled invariant distributions. More recently, in the linear theory the notion of almost controlled invariance has been introduced for studying approximate or "almost" solutions of specific control problems, cf. [Wi, Tr). Our idea of factorization generalizes the concept of almost controlled invariance to nonlinear systems and therefore can perhaps be useful in solving almost synthesis problems for nonlinear systems. In this aspect it is an interesting question to see to what extent the nice results on nonlinear almost disturbance decoupling of [MRvdS) fit into our idea of factorization. In the present paper we mainly concentrate on the second aspect of factorizing a nonlinear system and leave further implications related to the other issues for future research, see also [Nij).

The organization of the paper is as follows. In section 2 we introduce the notion of factor distribution and we discuss several aspects about factorization. In section 3 we describe linear factor subspaces for a linear system and in section 4 we present partial results on factorization of a single input nonlinear system. Section 5 contains the conclusions.

2. Factor distributions

Consider as in section 1 the smooth system on H

f(x,u) (1)

h(x,u)

We define a class of admissible controls ~ as follows. Let UO, ... ,~ be a family of open neighborhoods in IRm. We will call a smooth input

216 HENK NIJMEIJER

u(·) = (u1(·), ... ,um (·»: IR ... lR m admissible if u(i)(t) E Ui , i - O, ... ,n t E IR m, provided that the time evolution of x(·) is defined for all positive and negative times (completeness). In this way the system (1), or better the dynamical equation

(4) x - f(x,u)

induces a set of smooth trajectories on M defined as

(5) BM - {x(·): IR ... MI 3u(·) E'II s.t. x(t) - f(x(t),u(t»lft E IR}

In the sequel we will introduce the factorization of (1) as a factorization of the trajectories (5) of (4). Suppose that D is an involutive constant dimensional distribution on M for which M - M (mod D) is again a smooth (Hausdorff) manifold. Note that this implies the existence of a smooth projection ~: M ... M. The distribution D is called a global factor distribution if there exists a smooth system, to be called the factor system

(6) x - [(x,w)

on M with input space W so that

(7)

where BM denotes the set of smooth trajectories on M of the system (6).

Essentially (7) says that the projection of each trajectory in (5), i.e. ~x(·), is a trajectory of a system living on M. SO, given x(·) E BM ,

there exists u(·) E'II and Xo EM such that x(·) is precisely the solution of (4), initialized at Xo and there should exist an input w(·) E W so that the solution x(t) of (6) initialized at ~(xo) coincides with ~x(t), t E IR. The above definition has a global nature as M - M (mod D) is required to be a smooth manifold. Without this assumption we say that an involutive constant dimensional distribution is a factor distribution if locally about each point Xo in M the equality (7) holds true on a neighborhood of Xo respectively ~(xo) and where the system (6) is defined on this neighborhood of ~(xo). We remark that our definition of a factor distribution is completely independent from the outputs of the system. Of course in a further decomposition theory for the system one would possibly require that the distribution D is contained in ker dxh(·, .). Given the definition of a factor distribution, we need to study the following questions. (i) Do there exist factor distributions? (ii) What are factor distributions for a linear system x - Ax + Bu ? (iii) Given a factor distribution D, does there exist a natural relation

between the inputs w of the factor system (6) and the original inputs u in the dynamics (4)?

(iv) Characterize - if possible - all factor distributions for (4). Question (ii) will be completely solved in the next section. Some partial result regarding the problems (iii) and (iv) will be given in section 4. Before that we first address the existence issue raised in problem (i).

Examp~e 1 Consider on 1R3 the linear controllable system xl - xz ' with smooth IR-valued input functions u and let i = 1,2,3. Note that 1R3 (mod Di ) = IR z , i - 1,2,3. Dl distributions and Dz is not. The factor system for Dl

x2 x3 ' X3 = u Di - span{ajaxil, and D3 are factor

FACTORIZATION OF NONLINEAR SYSTEMS 217

is given as Xl - Xz • Xz - wand wand u are related via the trivial law

w - u. The factor system for D3 is given as xl - xZ '. Xz - wand the

connection between the inputs u and w is given via w - u. or better z - u. W - z. Observe that the smoothness assumption on the inputs u and w is very transparent here; any input u is in a one-to-one correspondence with an input wand an initial state zoo Finally. to see that Dz is not a factor. we note that the trajectories modulo Dz are given as traj ectories of xl - vI' X3 - v 2 with the input constraint VI - vz • which is obviously not a class of admissible controls of the type we allow for. 0

Example 1 shows that. at least for linear systems. there do exist factor distributions. In fact. without any further investigations we can show that any controlled invariant distribution is a factor distribution. Recall. see [Is. NvdS]. that an involutive constant dimensional distribution D is controlled invariant if there exists a regular static state feedback u - a(x.v). i.e. a(x •. ): U ~ U is a diffeomorphism for each X E 11. so that [f(·. v) .D] c D for all v E U. where f(x.v):- f(x.a(x.v».

Proposition 2 Any controlled invariant distribution D for the system (4) is a factor distribution for (4).

Proof First observe that the property of being a factor distribution for (4) is invariant under regular static state feedback (cf. the definition of factor distribution). Thus we may equally well assume that the distribution D is invariant under (4). Using Frobenius' local coordinates for D. i.e. coordinates for which D - span{8j8xl •...• 8j8xk l. we obtain from the invariance of D that

(8) f(x.u) - [ fl (Xl' ..•• Xn • Ul ••.•• Urn) ]

£(Xk + 1 •...• ~ .U1 •...• Um)

where f1 and fZ are respectively k- and (n-k)-dimensional vectors. Writing x - (Xk + 1 ••••• xn ) the factor system is then given as

(9)

which indeed shows that D is a factor distribution. o

With proposition 2 we have identified a large class of factor distributions. However. as we will see in the next sections. there are also factor distributions which are not controlled invariant.

3. Linear factorization

In this section we discuss the factorization of a linear system ~

(10) x - Ax + Bu

where x E ~n and u E ~m (note that as in the previous section we ignore an output equation y - ex + Du). Throughout we assume that the class ~ of admissible controls for (10) consists of all smooth functions u: ~ ~ ~m and we let B~ denote all the corresponding smooth trajectories

218 HENK NIJMEIJER

of (10). Factorization of the linear system (10) was first introduced in [Wi) and later on in [Tr]. The complete characterization of linear factors as will be given here is due to [Sc). A linear subspace V c ~n is called a factor subspace for (10) if the set

(11) L(mod V) - (x(·)(mod V)I x(·) E BLI

forms the set of trajectories of a smooth system

(12)

on ~n (mod V). If this is the case L(mod V) is called the factor system determined by Land V. Letting IT: ~n ~ ~n(mod V) be the canonical proj ection along V, the factor system L(mod V) contains precisely all output trajectories of the system

(13) {ic-

y - rrx

Ax + Bu

and so the set of smooth outputs (y(·)l of (13) should coincide with the state trajectories of (12). Before characterizing all possible linear factors we need one further definition. A subspace V c ~n is called an almost controlled invariant subspace for (10) if

(14)

where Vo is a controlled invariant subspace for the system (10) and the subspaces Bi form a chain, i.e. 1m B ~ Bo ~ B1 ~ ... ~ Bn- 1 , and AF - A + BF. We then have the following result, cf. [Sc].

Theorem 3 Consider the linear system (10). A subspace V is a factor subspace if and only if the subspace V is an almost controlled invariant subspace.

Remark Although it is quite natural to discuss linear factors for the linear system (10), it should be noted that the above theorem does not say anything about factor distributions for (10). In fact there exist (local) factor distributions for (10) resulting in a nonlinear factor system. The main reason to discuss linear factors for (10) is that the corresponding factor system is again linear.

The above theorem fully characterizes all possible linear factors and thus (partially) answers question (ii) of section 2. Of course, even in this linear context question (iii) is relevant, i. e. what is the relation between the controls u of (10) and the controls w of the factor system (12). Using Theorem 3 and the explicit characterization (14) one may show that for a single input controllable system (10) this connection is given as a dynamic precompensator for the system (12), i.e.

(15) { z = Fz +

u - Kz +

Gx + Hw

LX + Hw

with dim(z) - dim V. The general case will be treated in a future publication.

FAcroRIZATION OF NONLINEAR SYSTEMS 219

4. Factor distributions for single input nonlinear systems

In this section we discuss factor distributions for single input nonlinear systems. Specifically we will see if a result as given in Theorem 3 extends in some way to a nonlinear context. In Proposition 2 we have seen that controlled invariant distributions are indeed factors, so, motivated by Theorem 3 and (14) our next interest is in the analogue of a chain Bo + AFBl + ... + ArIBn _ 1 . Consider the affine nonlinear system (with no constraints on the inputs)

m (16) X - f(x) + E gi(X)Ui

1-1

and let G - span{gl'" .,gm}' Then we have

Proposition 4 Consider the system (16) and assume the distribution G is invo1utive and constant dimensional. Then G is a factor distribution for (16).

Proof Take around an arbitrary point Xo Frobenius' coordinates so that G - span{alax1 ••••• alaxm }. With a suitably defined feedback U - a(x) + P(x)v the system (16) takes the form

i 1 .... ,m (17)

i - m+l .... ,n

Then, modulo G the system (17) takes the form

(18)

with inputs w1 •... ,wm taking arbitrary values in a cubic neighborhood of (x10, ... ,Xmo), and so indeed G is a (local) factor for (16). (Note that we have used as in Proposition 2 that factorizing is a feedback invariant.) 0

Remarks (i) An analogous result has been proven in [AS]. (ii) The procedure sketched in the above proof is the converse of making an arbitrary nonlinear system x - f(x,u) affine in the control by adding integrators ui - Wi' i - l, ...• m. The linking between the systems (16) and (18), or more precisely between the inputs U and W is given by the compensator U - a(x) + P(x)v, with %i - Vi and wi - Z1' i - l, ... ,m.

Like in the linear case (taking the chain Bo - 1m B. B1 - O. i-l.2 •...• n-l) , we may conclude that the "input distribution" G 1s a factor distribution provided it is involutive and constant dimensional. Clearly, except for single input affine systems, this forms a severe restriction to the nonlinear generalization of the result of Theorem 3. As indeed for a single input nonlinear system the distribution G is always a factor (as long as it is constant dimensional), the next question is what about the nonlinear version of B + AFB + ... ? As a nonlinear counterpart for such a linear subspace we propose for a single input system a distribution defined as follows. Let k E ~ and define

(19) Dk - inv.clos.span{ad~g, j - 0,1, ... ,k-l}

where inv.clos. stands for taking the involutive closure and where the single input vectorfield in (16) is written as g. Note that for a single input linear system Dk corresponds to the linear subspace

220 HENK NIJMEIJER

span(b + Ab + ... + Ak-1b) which in the notations used in (14) corresponds to the chain Bo - Bl - Bk- 1 - span b. Bk - Bk+1 ... - 0 and an arbitrary feedback matrix F. (This implies that in (14). disregarding controlled invariant subspaces Vo. there are only finitely many (- n) factors.) The next theorem shows when a distribution Dk is a factor.

Theorem 5 Consider a single output affine nonlinear $ystem (16). Let k E ~ and suppose that the distribution Dk given by (19) is constant dimensional. Then Dk is a factor distribution for (16) if and only if

(20)

Remark The condition (20) means that the distribution generated by all the vectorfields [f.X) with X E Dk is either equal to Dk or has dimension equal to Dk plus one.

Proof Suppose the distribution Dk has dimension p. Take around an arbitrary point Xo Frobenius' coordinates so that Dk - span! 8/8x1 •... ... • 8/8xp }. Now the condition (20) implies that the (n-p)xp matrix

~ ~ 8x1 8xp

(21) (x) - F(x)

has rank 0 or 1. Clearly. rank F(x) - 0 corresponds with the situation that the distribution Dk is invariant under the vectorfield f. As by definition g E Dk it follows that Dk is an invariant distribution and it is therefore according to Propos i tion 2 a fac tor. On the other hand. when rank F(x) - 1. there exists a function

(22)

with (8h/8x1 , ... ,8h/8xp) ~ (0 •... ,0), so that

(23) [ ~P+l(h(Xl •...• xn),xp+l, ...• xn)l

En (h(x1.···,xn).xp+1.···.Xn)

where the right-hand side of (23) explicitly depends upon y. Interpreting (22) as an output we note that system (16.22) has a finite relative degree p, with p s k. That is the p-th time-derivative of the output y depends explicitly upon u. But that implies that the system (16.22) is right-invertible. cf. [RN) and so by allowing arbitrary smooth inputs u in (16) and varying the initial state we obtain arbitrary smooth functions y. taking values in some neighborhood about h(x10 •... ,Xno). That is. we may interprete these outputs y as inputs and we conclude from (23) that this defines a factor system on Rn(mod D). 0

(24)

Example 6 Consider on R5 the system

x4 - Q4(XZ 'X3 ,x.,xS )

Xs - Q,(XZ 'x3 ,x.,xS )

FACTORIZATION OF NONLINEAR SYSTEMS 221

Let k = 2, then D2 - span(8/8xl , ... ,8/8x3 ) and the factorization condition (20) means that the Jacobian of the function o«x) = (0<4 (X) ,0<5 (X») with respect to Xl' X2 and x3 has rank ° or l. However, for generic functions 0<4 and 0<5 this rank equals 2 and so generically Dl is not a factor distribution. For instance when 0<4 (X) - x2 and 0<5 (X) - x3 , this Jacobian has rank 2 and the factor system would be described by the ~quations x4 - wi and X5 - w2 with the further requirement that (wl )2 - w2 ' which is clearly not a class of admissible inputs in our context. 0

Let us note that for a single input linear system the condition (20) is equivalent to the fact that Im(b;Ab;··· iAkb)mod(Im b; ... ;Ak-1b) has dimension ° or 1. We conclude that, in contrast to the linear situation, the distribution Dk is a factor distribution only in very exceptional cases. On the other hand, as in Proposition 4, the relation between the input of the factor system on H(mod Dk ) and the original system is given via a simple dynamic precompensator. Namely, as we have seen in the proof of Theorem 5, the factor system on H(mod Dk ) is

locally given as x - f(y,x), where x = (xp + l "" ,xn ), and f and y follow <P) P p-l from (22,23). Moreover, we have that y = Lfh(x) + LgL f h(x)u where

P-l LgL, hex) ,. 0, and so, defining

u - -(LgLrlh(x) rlL~h(x) + LgL~-lh(x)v, we

the state feedback

obtain y<P) = v, which

precisely describes the relation between the input of the factor system (= y) and the feedback modified input v.

Summarizing so far we have obtained two kinds of factor distributions, namely the controlled invariant distributions and those identified in Theorem 5. An easy calculation shows that the sum of a controlled invariant distribution D and a distribution Dk as defined in (19) is again involutive and it will satisfy the conditions of Theorem 5, provided the distribution Dk does. So distributions D with D - D + Dk where D is controlled invariant and Dk satisfies the condition (20) are also factor distributions. It is our conjecture that this exactly identifies all possible factors for a single input nonlinear system. So far we have provided partial answers to the problems mentioned in section 2. We conclude with another specific question. Suppose we are dealing again with an affine single input nonlinear system and assume for some k E IN the distribution Dk is a factor distribution. The question is then under which conditions is the resulting factor system feedback equivalent to a controllable linear system? Or, slightly more general in the terminology of section 1, when can the system (16) be thought of as a controllable linear system together with a nonlinear dynamic precompensator? In terms of the distributions Di , the answer is rather immediate. Recall, see [JR, vdS] that a single input system (16) is feedback equivalent to a controllable linear system if and only if the distributions Dj satisfy dim Dj - j, j = 1, ... ,no Using a similar argument we obtain the following result.

Theorem 7 Consider a single input affine nonlinear system (16) and assume that for some k the distribution Dk is a factor distribution. The factor system on H(mod Dk ) is feedback equivalent to a controllable linear system if and only if

(25) for i - 0, ... ,dim H(mod Dk )

222 HENK NIJMElJER

5, Conclusions

We have introduced the new concept of factor distributions for nonlinear systems as a new tool for studying the decomposition of a nonlinear system. This idea of factorizing, is very much inspired by a similar idea in linear system theory, originated in [Wi) and further developed in [ScI, and generalizes earlier results on the decomposition of a nonlinear system. The here presented analysis is by no means complete and further research is needed for a full understanding of nonlinear factorization.

REFERENCES

[AS) A.A. Agrachev 6. A.V. Sarychev, "On reduction of a smooth system linear in the control", Math. USSR Sbornik 58, pp. 15-30, (1987) .

[Fll) M. Fliess, "Decompositions et cascades des systemes automatiques et feuilletages invariants", Bull. Soc. Math. France 113, pp. 285-293, (1985).

[F12) M. Fliess, "Cascade decompositions of nonlinear systems, foliations and ideals of transitive Lie algebras", Systems 6. Control Lett. 5, pp. 263-265, (1985).

[Is) A. Isidori, Nonlinear control systems, 2nd edition, Springer Verlag, New York, (1989).

[JR) B. Jakubczyk 6. W. Respondek, "On linearization of control systems", Bull. Acad. Polon. Sci. (Math.), 28, pp. 517-522, (1980).

[Kr) A.J. Krener, "A decomposition theory for differentiable systems", SIAM Jnl. Control 6. Optimiz. 15, pp. 813-829, (1977).

[MRvdSj R. Marino, W. Respondek 6. A,J. van der Schaft, "Almost disturbance decoupling for single-input single-output nonlinear systems", IEEE Trans. Aut. Contr. 34, pp. 1013-1017, (1989).

[Nij] H. Nijmeijer, "On the theory of nonlinear control systems", in Three Decades of Mathematical System Theory (eds. H. Nijmeijer 6. J.M. Schumacher) LNCIS 135, Springer Verlag, Berlin, pp. 339-357, (1989).

[NvdS] H. Nijmeijer 6. A.J. van der Schaft, Nonlinear dynamical control systems, Springer Verlag, New York, (1990).

[Pi] G. Picci, "Aggregation of linear systems in a completely deterministic framework", in Three Decades of Mathematical System Theory (eds. H. Nijmeijer 6. J.M. Schumacher) LNCIS 135, Springer Verlag, Berlin, pp. 358-381, (1989).

[Re) W. Respondek, "On decomposition of nonlinear control systems", Syst. Contr. Lett. 1, pp. 301-308, (1982).

[RN) W. Respondek 6. H. Nijmeijer, "On local right-invertibility of nonlinear control systems", C-TAT 4, pp. 325-348, (1988).

[Sc) J.M. Schumacher, "Linear system representations", in Three Decades of Mathematical System Theory (eds. H. Nijmeijer 6. J .M. Schumacher) LNCIS 135, Springer Verlag, Berlin, pp. 382-409, (1989).

[vdS) A.J. van der Schaft, "Linearization and input-output decoupling for general nonlinear systems", Syst. Contr. Lett. 5, pp. 27-33, (1984).

[Tr] H.L. Trentelman, Almost invariant subspaces and high gain feedback, CWI Tract 29, Amsterdam (1986).

[Wi) J .C. Willems, "Almost invariant subspaces: an approach to high gain feedback design - part I, almost controlled invariant subspaces", IEEE Trans. Automat. Contr. 26, pp. 235-252, (1981),

FACfORIZATION OF NONLINEAR SYSTEMS 223

[Wo} W.M. Wonham, Linear multivariable control: a geometric approach, Springer Verlag, Berlin, (1979).

H. Nijmeijer Department of Applied Mathematics University of Twente P.o. Box 217 7500 AE Enschede The Netherlands

On the Approximation of Set-Valued Mappings in a Uniform ( Chebyshev) Metric

M.S. Nikolskii

Abstt·act

The theory of set-valued mapping (SVM) has essential applications and stimuluses in different domains of mathematics (see for example [1], [2]). Convex-valued continuous SVM form an important class. It is natural, by analogy with classical analysis, to consider the problem of approximating such mappings in uniform metric by SVM which have a simple structure.

1 Notation and definitions

Let Ric be real euclidean arithmetical space of dimension J(. The elements of Rk are ordered set of J( numbers which we shall write in column.

The inner product in Ric is defined by the formula:

I.: (z, y) = 2: ZiYi

i=l

Let f(v(RIc) denote the complete metric space of non-void convex compacta from Ric with Hausdorff metric h(., -) (see [1], [2]). Let A C Ric be a non-empty set. The support function W(A, w) is defined for W E Ric by the next formula:

W(A, w) = sup(a, w}. aEA

The modulus of non-empty set A C RI.: is defined as:

IAI = sup lal, aEA

where lal is modulus (length) of vector a E RI.:. The algebraic sum A + B of sets A, B from Ric is defined by the formula:

A + B = {c = a + b : a E A, bE B}.

SET-VALUED MAPPINGS IN A CHEBYSHEV METRIC 225

We shall denote by 2:;:'1 Ai the algebraic sums of sets Ai C Rk. The set

AA = {c = Aa : a E A}

is called the product of the number A and the set A C Rk. The set

LA = {c = La : a E A}

is called the product of the pxk-matrix L and the set A C Rk. About properties of these operations see for example [1], [2]. We mention

some of it:

1. A + B = UaEA(a + B) = UbEB(A + b);

2. A+B=B+A;

3. A + (B + C) = (A + B) + C;

4. A+ {OJ = A;

5. a(,BA) = (a,8)A;

6. l·A = A;

7. O· A = {OJ for A =f ¢;

8. a(A + B) = aA + aB;

9. (a + ,B)A C aA + ,BA;

10. (a + ,B)A = aA +,BA for a ~ O,,B ~ 0, A E Kv(Rk);

11. h(2:;:'l AiAi, 2:;:'1 AiBi) ~ 2:;:'1 IAilh(Ai, Bd,

where Ai E R1 ,Ai,Bi E Kv(Rk);

12. h(aA,,BA) ~ la - ,BI·IAI, where A E Kv(Rk).

The metric space Kv(Rk) is invariant relatively of all three operations (see above), therefore convex analysis, in particular the apparatus of support functions (see for example [2]), is useful for utilization of these operations in Kv(Rk ).

2 The statement of the approximation problem

Let us consider SVM

226 M.S. NIKOLSKII

where V is the non-empty compactum from RI. The space C(U, Kv(R1c» of such SVM is the complete metric space with metric

(1)

We shall consider C(U, Kv(R 1c » as the metric space with uniform (Chebyshev) metric (1). In studying of the space C(U, Kv(R 1c» there is an interesting question: what of SVM is it naturally consider as simply made? In [3] the author considered simple SVMs called finite-generated.

Definition 1. The SVM PN : U -+ K v( Rm) of the type

N

(2) PN(X) = L: Ji(x)Ai ;=1

is called a finitely-generated SVM of dimension N if in (2) N 2: 1, Ji(X) is continuous scalar function on U and Ai E K v( Rm)

For applications it is interesting the class of Q-polynomials which are defined in such a way. Let p is nonnegative integer. Let us consider the elementary scalar polynomials

T(i1, ... ,il,X) = (Xdl(X2)i~ ... (XI)il

of every kind where ij 2: 0 and L~=1 ij $ P. Denote the number of such polynomials for a given p by N(p). It is obvious that these polynomials can be numbered: Tl(X), ... , TN(P)(X).

Let us consider the matrix function

(3)

where E is unit matrix of order nl.

Definition 2. The SVM Qp : U -+ /{ v( Rm) of the type

(4)

is called by Q-polynomial of dimension p if in (4) the matrix function Lp (x) is defined by the formula (3), p 2: 0 and B is non-void convex compactum from RN(p)m

For applications it is interesting also the subclass of S-polynomials from the class of finte-generated SVM.

Definition 3. The SVM SPIJ : U --+ Kv(Rm) of the type

q

(5) Spq(X) = ~ fpi(x )Ai i=l

is called by S-polynomial of dimension p x q if in (5) p ~ 0, q ~ 1, fpi(x) is a scalar polynomial of Xl, ... ,XI, of degree ~ p,Ai E Kv(Rm).

Lemma 1 The S-polynomial of dimension p x q is the Q-polynomial of dimension p.

From lemma 1 follows:

Lemma 2 The S-polynomials belongs to the set of Q-polynomials.

Remark 1. The Qp-polynomial is not always Spq-polynomial for sufficient by q. For example the Ql polynomial (see(4)) for m = 1, U = [-1,1], B = {b E R2 : Ibl ~ I} is not Slq-polynomial for every q ~ 1.

3 The theorems of Weierstrass type

The theorems of Weierstrass type take place for classes of finite-generated SVM, Q-polynomials and S-polynomials. This fact is consequence of the next theorem (see the lemma 2 in previous point).

Theorem 1 Let 0 E C(U, Kv(Rm)). Then for every c > 0 exist such p ~ 0, q ~ 1 and Spq-polynomiai (5) that

p(O, Spq) ~ c.

4 The existence of the best approximation

The question about existence of the best approximation in uniform metric in class Qp-polynomials for fixed p ~ 0 is very interesting.

Theorem 2 Let the interior of the compactum V be nonvoid in RI. Then for every SVM 0 E C(U,I<v(Rm) and fixed p ~ 0 the Qp-poiynomiai of the best approximation exists.

Remark 2. It is interesting to consider the question about the existence of best approximation in uniform metric for 0 E C(U, Kv(Rm)) in the class of finite-generated SVM for fixed N ~ 1 and in the class Spq-polynomials of fixed dimension p x q.

228 M.S. NIKOLSKII

5 The necessary and sufficient conditions for optimality in the class of Qp-polynomials

In connection with Theorem 2 consider the question about obtaining necessary conditions of optimality for Qp-polynomials which gives the best approximation in uniform metric among of all Qp-polynomials (p is fixed) for given SVM

Note that for A, B, E Kv(n:n), WE Rm

h(A,B) = max IW(A, W - W(B, w)l. 1"t1=1

Lemma 3 The Qp-polynomial of the best approximation is the minimal element among all Qp-polynomials for functional

(6) f(Qp) = max (W(11(x), w) - W(Qp(x), W»2 zEu,I"tI=l

For variation Qp we consider Qp-polynomial Qp)..:

'VxEU,

where A E [O,l],E C RN(p)m and generates Qp(x) (see (4», Bo is an arbitrary element from Kv(RN(p)m). The scalar function

(7)

corresponds to Qp).. for A E [0,1] , where

Qp(x) = Lp(x)iJ, Q;(x) = Lp(x)Bo.

It is not difficult to see that the function g(A) reaches the minimal value for A = 0, in the point A = 0 g(A) has right derivative g~(O) ~ 0 (see (6), (7) and Theorems 3.2 from [4]). By means of the theorem 3.2 from [4] it is possible to write some expression for g~(O). In this way we get the next theorem.

Theorem 3 Assume that Qp-polynomial of best approximation Qp satisfy the following necessary conditions.

(8)


where C is the set of pairs (x, W), x E R', Iwl = 1, for a maximum in (6). Then

Remark 9. It is natural to call these necessary conditions of optimality by conditions of Kolmogorov type (see [5]).

By the contrary method, using the convexity offunction g(..\), the next theorem is proved.

Theorem 4 Let the conditiosn (8) be fulfilled for given 0 E C(U, K v( Rm» and given Qp-polynomial Qp. Then Qp is the Qp-polynomial of the best approximation.

6 The case U = [0,1] In this case it can be used for effective approximation the generalized polynomials of Bernstein (see [3]):

Bn(t) = t c!tk(1- tt-kO(~), k=O

where t E [0, 1],C! is the binomial coefficient, n = 1,2, ..... By analogy with the proof of Popovichik's theorem (see [6]), the next

theorem can be proved.

Theorem 5 The next inequalities are fulfilled for SVM

o E C(U, Kv(Rm» for n = 1,2, ... :

3 1 p(O, Bn) ~ 2w(y'n)'

where wet) is modulus of continuity of 0, that is "It E [0,1]

wet) = sup h(O(td,0(t2». Itl-t~l$t tlhE[O,lj

7 Approximation by constant set-valued mappings in uniform (Chebyshev) metric

Let us consider the question about approximation of the elements 0 E C(U, Kv(Rm» by constant SVM, that is SVM A: U -+ Kv(Rm) for which it is fulfilled the equality:

230 M.S. NIKOLSKII

'VzEU,

In this case the question about the best approximation reduces to the minimization of the function

f(P) = maxh(P, 0(%» xEU

on the metric space K v( Rm ).

Theorem 6 The function f(P) is continuous and reaches its minimal value on Kv(Rm).

Theorem 7 The nen conditions are fulfilled for minimizing compactum

(9)

po E /{v(Rm) :

max [(W(O(z), \Ii) - W(PO, \Ii». (W(PO, \Ii) - W(P, \Ii»] ~ 0 (x.9)ED

where D is the set of pairs (%, \Ii), x E U, l\lil = 1, maximizing the function

(W(O(z), \Ii) - W(PO, \Ii»2.

Theorem 8 Let the conditions (9) be fulfilled for given 0 E C(U, /{v(Rm» and po E Kv(Rm). Then po is the minimizing element of the function f(P) on Kv(Rm).

Example 1. Let O(z) = (l-z)A+xB for z E [0,1], where A, BE Kv(Rm). It can be proved that the convex compactum P l = ~(A + B) is the minimizing element for f(P) on /{v(Rm).

Example t. Let us consider the previous example for

m = 2,A = {y E R2: Yl = 0, IY21 $ l},B = {y E R2: Iyd $ I,Y2 = O}.

In this example Pl = {y E R2 : IYll $ ~,IY21 $ Hand f(PI) = ~. Let

where co denotes the operation of convexification. It can be proved that every compactu~ P E Kv(R2) is the minimizing element for f(P) on Kv(R2) if P2 C P C P l .

Consequently, in general the case the best approximation is not unique.


Bibliography

[1] Borisovich, Ju. G.; Gelman, B. D.; Myshkis A. D.; Obuhovski, V.V. Set-valued mappings. In: Mathematical Analysis, VINITI, Moscow, 1982, series: Results of Science and Engineering, Vol. 19 (in Russian).

[2] Blagodatskih, V. I. and Filipov A. V. Differential Inclusions and Optimal Control. Proceedings of Mathematical Institute of the Academy of Science. 1985. Vol. 169 pp. 194-252 (in Russian).

[3] Vitale R. A. Approximation of Convex Set-Valued Functions. J. Approx. Theory, 1979, Vol. 26, No.4. pp. 301-316.

[4] Pschenichniy, B. N. Necessary Conditions of Extremum. Nauka, Moscow, 1982 (in Russian).

[5] Collatz L. and Krabs W. Approximationstheorie. Tschebyscbeff'sche Approximation mit Anwendungen. Stuttgart: Teubner, 1973.

[6] Natanson, I. P. Constructive Theory of Functions. State Publ. House TTL, Moscow and Leningrad 1949 (in Russian).

M.S. Nikolskii Steklov Institute of Mathematics Academy of Sciences of the USSR Moscow USSR

Estimation of a Guaranteed Result in Nonlinear Differential Games of Encounter

A.G. Pashkov

Abstract

A nonlinear differential game of encounter is considered. Estimation of the function of a guaranteed result (u - stable function) is proposed. The problem is considered in the framework of formalization [1, 2].

The paper is related to the works done in [3-16]. It is assumed that Hamiltonians of initial (nonlinear) and auxiliary systems are connected by special relationships. Unlike [11,12], in this paper the u-stable function for the auxiliary system is nondifferentiable. Conditions imposed on the connection between Hamiltonians of initial and auxiliary systems are not so severe as they were in [13].

An example, where the function of the guaranteed result is constructed by means of proposed approach is given.

1 The basic system

We consider the conflict-controlled system

(1) Z = I(t,z,u,v), v E P E RP, v E Q E RII,

where t E [to, 0], z E R!', P and Q are compact sets. The function I [to, 0] x Rn x P x Q is assumed to be continuous and satisfying the Lipschitz condition

sup II/(t, z(I), u, v) - I(t, z(2), u, v)1I x IIz(l) - z(2)11- 1 < 00 (t,~(;),U,tI)

Here z(i) E G(Z(I) ::f; z(2»), where G is an arbitrary bounded region in Rn. The motion of the system (1) is constructed on the time interval [to,O].

The payoff functional has the form

(2) '}'(z(·)) = O'(z(O))

where function 0' : Rn --+ Rl satisfies the Lipschitz condition. Formalization of the game (1), (2) is given in [1,2].

NONLINEAR DIFFERENTIAL GAMES OF ENCOUNTER 233

Let wo : [to, 0] x Rn -+ R1 be a value function in the game (1), (2). We know [2] that function wo(t, x) satisfies the Lipschitz condition

and boundary condition

(3) wo(O, x) = oo(x).

2 The first auxiliary system

Let us introduce the auxiliary system

Here t E [to, 0], x E Rn, P1 and Q1 are compact sets, matrix functions A(t), B(t) and C(t) are assumed to be continuous.

Let an u-stable function wO(t, x) : [to, 0] x Rn -+ R1 be given for the auxiliary system (2), satisfying the condition

3 The second auxiliary system

Let us consider another auxiliary game. Let us write

(6) ¢(t, x) = max IH(t, x, s) - H1(t, x, s)l. ,es·

Here S· = {S ERn: 11511 $ I}; H(t, x, s) and H1(t, x, s) are Hamiltonians of initial (1) and auxiliary systems respectively.

Let us consider the second auxiliary differential game

(7) :i: = A(t)x + B(t)U1 + C(t)Vl + ¢(t, x)v, U1 E P1, VI E Ql

where A(t). B(t) and C(t) are given by the same relationships as those used in the system (4), function ¢(t,x) satisfies the relationship (6), v is n-dimensional vector, whilst IIvll $ 1. We will assume that the constraints imposed on the controls U1, V1 and the payoff functional in the game (7) are all the same as in the game (4), (2).

Remark 1. The Hamiltonian of the system (7)

(8) H1(t, x, s) + ¢(t, x)1I811

234 A.G. PASHKOV

and the Hamiltonian of system (1) according to (6) are connected by the inequality

(9) H1(t,:I:, s) + ¢(t, x)IISIl ~ H(t, x, s)

where (t, x, s) E [to, 8] x G x Rn.

4 Statement of the problem

It is required to construct an u-stable function wHt, x) : [to,8] x Rn ::}

R1 for initial nonlinear differential game (1), (2), satisfying the Lipschitz condition and boundary condition

w~(B,x) = u(x).

We will search the function wHt,x) in the form

(10)

Here wO(t,x) is an u-stable function for the auxiliary game (4), (5) (value function in particular), Awl(t, x) and Aw(t) are some differentiable functions. Below we will give the relationships for these functions.

We will assume that

(11)

where wP(t, x) (1 = 1,2) are given smooth functions. Let us introduce the notations

We assume that the function Awl (t, x) is differentiable and satisfies the relationships


The differentiable function Ll\ll(t) satisfies the inequality

oLl\ll(t) at ~

~ sup {minmax(.Ab1 + (1 - A)b2+ ('>',z)e[o,ljxG Ul til

(13) + gradzLlw1(t, x), A(t)x + B(t)Ul + C(t)Vl)-

- minmax(Ab1 + (1 - A)b2, A(t)x + B(t)Ul + C(t)Vl)}, u, til

~\lI(O) = 0.

5 Solution of the problem

We know [3] that an upper conjugate derivative of the function wHt, x) is expressed as

(14) D·w~(t,x)l(s) = sup ((s,h) - o_w~(t,x)l(h)) heRn

where o_wHt, x)l(h) is a lower derivative of the function wHt, x) by the direction (1, h) at the point (t, x)

o_w~(t, x)(h) = liminf {w~(t + 6, x + 6h) - w~(t, x)6- 1 } . 610

For any wHt, x) E Lip, (t, x) E [to, B] x G thefunction h -+ 0_ wHt, x)(h) assumes finite values and satisfies the Lipschitz condition in the whole space Rn. From differentiability of the functions wHt, x) and ~ \lI(t) it follows that the upper conjugate derivative D*w~(t, x)(s) has the form

D·w~(t, x)(s) = (15) O~wl(t x) d~\lI(t)

= D·wO(t, x)(s - gradz~wl(t, x)) - ot' - dt .

Let us obtain an expression for the upper derivative of the function wO(t, x). If in the position (t,x) E [to, 0] x G of the game (4) the function wO(t,x) is differentiable, i.e., maximum in (11) is achieved for i = 1 or i = 2, then

(16) D*wO(t,x)(s)= - at Ifs-gra zWi t,x { awp(t,x). _ d O( )

00 if S:f gradzwp(t,x).

If the position {t., x.} is such that the function wO(t, x) is nondifferentiable in this position then the following equalities hold

(17)

236 A.G. PASHKOV

For these positions the conjugate derivative has the form

(18) D* o(t )( ) _ { -[Aal + (1 - A)a2] if s = Abl + (1 - A)b2 w *,z* S - 00 if s=/=Abl +(I-A)b2.

where 0 ~ A ~ 1. Substituting s-gradx~wl(t, z) into (5.5) instead of s, we have

(19) D*wO(t, z)(s - gradx~wl(t, x» =

{ -[Aal + (1- A)a2], if s - gradx~wl(t, x) = Abl + (1- A)b2 +00, if s - gradx~wl(t, x) =/= Abl + (1- A)b2.

where 0 ~ A ~ 1.

6 Outline of proof

For the solution of the initial problem we have to prove, that the function wHt, z) defined by expressions (10)-(13) is a u-stable function for initial nonlinear game (1), (2). To do this, according to Theorem 1 from [5] we have to prove the inequality

(20) D*w~(t, x)(s) 2: H(t, x, s)

for all (t,z,s) E [to x 8] x G x Rn. It follows from Remark 1 that it is sufficient to prove the inequality

(21) D*w~(t, z)(s) 2: Hl(t, x, s) + ¢(t, z)lIsll. It follows from (15) that to establish the last inequality it is sufficient

to verify that

D*wO(t, z)(s - gradx~wl(t, x»-

a~wl(t, x) d~ \lI(t) - at - dt 2: Hl(t, x, s) + ¢(t, z)lIsll·

(22)

Using u-stability of the function wO(t, x) and inequalities (12), (13) we can prove inequality (22) and hence the u-stability of the function wHt, x) for initial nonlinear differential game (1), (2).

7 Consequences

Remark 2. Let us denote by w*(t, x) the function of the guaranteed result in the problem (1), (2) defined according to the theorem in [13]

(23) w*(t, z) = wO(t, x) + c* x (8 - t) x m


where WO(t, x) is the value function for auxiliary game (4), (5), m is the Lipschitz constant on the x of the function wO(t, x). Taking (11), (18) from this paper and (1.7), (1.9) from paper [13] into account we find the constant e* in the following form

(24) e* max {IH(t,x,s)-H1(t,x,s)1 sup lI-Xb1 +(I--X)b211} (t,x,,)E AE[O,l]

[to,9]xGxS·

(e* max {¢(t,x) sup lI-Xb 1 + (1- -X)b2 11}) (t,x)E AE[O,l]

[t,9]xG

Thus for all (t, x) E [to, 8] x G for which the following inequality holds

d~\lI(t) (25) me· 2: sup lI-Xb1 + (1 - -X)b2 + gradx~wl(t, x)II¢(t, x) - d

AE[O,l] t

the relation

(26) w~(t, x) ::; w*(t, x)

is fulfilled at all (t, x) E [to, 8] x G.

Remark 3. If ¢(t, x) == ¢(t) then ~\lI(t) == O.

8 Example

We will consider a problem of encounter of two objects (pursuers) with the third (evader). The motion of the pursuers Pi(y~i), y~i) is described by the equations

(27)

.(i) (i) Yl = Y3 ,

(i) Y4 ,

iJ~i) = -(0: + Cl«y~i)2 + «y~i)2)1/2y~i) + Uli )

iJ~i) = -(0: + cl «y~i)2 + «y~i)2)1/2y~i) + u~i)

(i) (i) u(i) {O, 0, U 1 ,u2 }, (i = 1,2).

Here u(i) is a control vector of the pursuer Pi satisfying the constraint

(28)

0: > 0 is a coefficient of linear friction, Cl > 0 is a coefficient of quadratic friction. We will assume that Cl is a small quantity.

238 A.G. PASHKOV

The evader E( %1, %2) moves in accordance with the equation

%1 = %3, %3 = -{3Z3 + VI,

(29) %2 = %4, Z4 = -{3Z4 + v2·

Here v is a control vector of the evader E satisfying the constraint

(30)

We suppose that

(31)

The time instant t = () at which the game terminates is fixed. The payoff functional is a continuous function

(32)

where

From (32) it follows that the payoff functional is nonconvex.

Statement of the problem. It is required for any initial position of the game (27)-(32) to construct an u-stable function of the guaranteed result. To solve this problem we introduce an auxiliary linear differential game for objects governed by equations from the well-known "testing example" of Pontriagin [9].

The motion of the pursuers Pi(y~i), y~i» (i = 1,2) in the auxiliary game is described by the equations

.(i) _ (i) .(i) (i) + l(i) Y1 - Y3' Y3 = -aY3 u1 '

(33) Y·(i) _ y(i) y.(i) _ _ rvy(i) + u1(i) 2-4' 4- .... 42'

(u~(i)f + (u~(i)f :$ Jl~ > 0 (i = 1,2).

The evader in the auxiliary game moves in accordance with the equation

(34) %1 = Z3, Z3 = -{3z3 + v~ %2 = Z4, Z4 = -.{3Z4 + v~ (vD2 + (vD2 :$ IIf > O.

The payoff of the auxiliary game is specified by the continuous function (32). We suppose that "l ~ Jll>lIt/{3 ~ Jlt/a. The value function e:o(t,z)


of the game (33), (34) was obtained in [15]. Making the change of variables according to the formulae

Z1 = 1 Y1,

_ 1 Z2 - Y2, X3 = Z1. Z4 = Z2,

(35) z5 = 1 Ya, X6 = Y~, X7 = Z3, Xs = Z4,

Z9 = 2 Y1, ZlO=Y5, Xll = Y~, X12 = Y~,

we can write the functions ~o(t, x) and u(x(O)) in new variables. Taking (6) and (35) into account we have

(36) ¢(t, z) = max IH(t, x, s) - H1(t, z, s)1 = ~1¢·(t, z) .es·

where

(37)

Taking (12), (36) and (37) into account it can be shown that the function ~w1(t, z) satisfies the equality

(38) ~w1(t, z) = ~1(2 + a2(t) + b2(t)/12¢*(t, x)(O - t)

where a(t) = (1- exp(-O'(O - t)))0'-1,b(t) = (1- exp(-,B(O - t))),B-1. According to [15] the function co(t, x) has the form

(39)

where, taking (35) into account we have

~~(t, z) = «X2 - qt)2 + (xt}2)1/2 - r(t)

(40) qi = X4 ± «R(t))2 - (X3)2)1/2 (i = 1,2).

In the last expression i = 1 corresponds to the sign "+" and i = 2 corresponds to the sign "-". Besides

R(t) = v1,B-1(0 - t - a(t)), l'(t):;:: 11-10'-1(0 - t - b(t)).

The function ~\}I(t) (with accuracy O(ct}) satisfies the equality

(41) ~\}I(t) = O.

Thus, the function w~(t, x) has the form (with accuracy O(~d)

(42) w~(t, z) = ~o(t, x) + ~wl(t, z).

Here the functions co(t, z) and ~w1(t, x) are defined by formulae (39), (38) respectively.

240 A.G. PASHKOV

Remark 4. From Remark 2 it follows that the guaranteed result (42) will be less than the result of the solution of this example according to [13] for all positions where the inequality (25) is satisfied.

References

[1] Krasovskii, N.N., Subbotin, A.I. Game-Theoretical Control Problems. New York, Springer-Verlag, 1988,518 pp.

[2] Subbotin, A.I., Chentzov, A.G. Optimization of A Guaranteed Result in Control Problems. Moscow, Nauka, 1981, 287 pp. (in Russian)

[3] Subbotin, A.I., Taras'yev, A.M. Conjugate Derivatives of the Payoff Function of a Differential Game. Dokl. Akad. Nauk SSSR, 283, no. 3, pp. 559-564, 1985 (in Russian).

[4] Subbotin, A.I., Taras'yev, A.M. Stability properties of the value function of a differential game and viscosity solution of Hamilton-Jacobi equations. Probl. Control and Inform. Theory, vol. 15, no 6, pp. 451-463, 1986.

[5] Barron, E.N., Evans, L.C., Yentsen, R. Viscosity solution of Isaacs equations and differential games with Lipschitz controls. J. Different. Equat., vol. 53, no. 2, pp. 213-233, 1984.

[6] Crandall, M.G., Lions, P.L. Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc., vol. 277, no. 1, pp. 1-42, 1983.

[7] Lions, P.1., Souganidis, P.E. Differential games, optimal control and differential derivatives of viscosity solutions of Bellman's and Isaacs equations. SIAM J. Control and Optimiz., vol. 23, no. 4, pp. 566-583, 1985.

[8] Krasovskii, N.N. Differential games. Approximation and formal models. Mathematicheskii Sbornik, vol. 107, no. 4, pp. 541-571, 1978 (in Russian).

[9] Pontryagin, 1.S. Linear differential pursuit game. Math.sb., vol. 112, no. 3, pp. 307-330, 1980 (in Russian).

[10] Isaacs, R. Differential Games. New York, John Wiley and Sons, 479 pp., 1965.

[11] Pashkov, A.G. On the sufficient condition for nonlinear positional games of encounter. PMM, 40,1, pp.168-171, 1976.


[12] Pashkov, A.G. On an approach to solving certain nonlinear positional differential games. Izv. Akad. Nauk SSSR, Tekhn.Kibernetika, 1, pp. 17-22, 1979.

[13] Pashkov, A.G. Comparison of the solutions of linear and nonlinear positional differential games of encounter. PMM, 50, 4, pp. 551-560, 1986.

[14] Levchenkov, A.Yu., Pashkov, A.G., A Game of optimal approach with two inertial pursuers and a noninertial evader. PMM, 49, 4, pp. 536-547, 1985.

[15] Pashkov, A.G., Terekhov, S.D. Differential approach game involving two dynamic objects and a third. Izv. AN SSSR, MTT, vol. 21, no. 3, pp. 66-71, 1986.

[16] Pashkov, A.G., Terekhov, S.D. A differential game of approach with two pursuers and one evader. J. Optimiz. Theory and Applic., vol. 55,no. 2,pp. 303-311,1987.

A.G. Pashkov Institute for Problems of Mechanics USSR Academy of Sciences Moscow USSR

Limit Sets of Trajectories

N.N. Petrov

Dynamical polysystems defined on closed manifolds and consisting of finite number of smooth vector fields are considered. w-limit sets of their trajectories are studied. If a polysystem consists of one vector field we have an old and difficult problem in the Qualitative Theory of differential equations. In the general case it is known that for any trajectory the w-limit set exists, is compact and connected. In the general topology such sets are called continuums. There are no other restrictions on the structure of the wlimit set because it may be proved that any continuum on a manifold is the w-limit of a polysystem, that is, the w-limit set for one of its trajectories. A polysystem is said to be universal if any continuum on the manifold is its w-limit set. It is proved that if a polysystem V is universal then the polysystem 1) is also universal. Thus, an universal polysystem has a trajectory defined on the real line with prescribed behavior at +00 and -00. Now consider the following problem: what is the minimal number of vector fields which form an universal polysystem?

The answer is given by the following assertion: this number is equal to 2 if the manifold is Sl and equal to 3 in the other cases ([1]). The triplet of vector fields is as follows. It is necessary to take a pair of analytic vector fields in a general position constructed by Lobry and their sum with the sign minus. The proof is based on the concept of N -local controllability which is called now small time local controllability. This concept plays an important role in problems of synthesis. The observation mentioned above stimulated the investigation of the following problem: what is the possible structure of w-limit sets of polysystems consisting of two vector fields? This problem was studied by 1.8. Han. Our paper on this problem will appear in the journal "Differential equations". The main result may be stated as follows. Any continuum on the manifold is the w-limit set of a polysystem consisting of two COO-smooth vector fields, a pair of analytic vector fields that may not exist. In the analytic case the following theorem holds.

The w-limit set of a polysystem defined on a two-dimensional manifold and consisting of two analytic vector fields either is a point or contains an arc. This theorem gives a topological restriction on the w-limit set because it is known that there are non-trivial continuums (that is continuums having more than one point) which contain no arcs. In high dimensions the

LIMIT SETS OF TRAJECTORIES 243

assertion stated above is not true. There exists a polysystem defined on the three-dimensional manifold and consisting of two analytic vector fields such that one of its w-limit sets contains more than one point but contains no arcs ([2]).

References

[1] N.S. Rettiev. On limit sets of trajectories of control systems. VINITI, N1085-79 Den.

[2] N.N. Petrov, LS. Han. On limit sets oftrajectories of control systems. VINITI, N2358-B87.

N.N. Petrov University of Leningrad USSR

Nonlinear Systems with Impulsive and Generalized Function Controls

Andrej v. Sarychev

1 Introd uction

A smooth affine control system

r

X=IT+Lg~(X)Ui' xERn, uiER, i=l

or

(1.1)

is considered and the question we are interested in is: how the generalized functions can be employed as controls in this system?

It is well known that for 1.1 being a generalized function, the trajectory of system (1.1) may not exist in the traditional sense. Actually, substituting for example, the delta-function 8( T - TO) in place of 1.11 we shall evidently obtain trajectory xO jumping at TO. Transforming the differential equation (1.1) into the integral one we will have to integrate in the right-hand side the function IT (x( T)) being discontinuous at TO with respect to the atomic measure 8( T - TO) located just at the point of discontinuity. As it is well known this operation is not correctly defined. Some approaches based on the different definitions of such an integration are surveyed in [1, ch. 1.].

The present paper contains another approach to the construction of generalized trajectories of the system (1.1). Our idea (close to the one developed in [2-5]) is to furnish the space U of "ordinary" controls (for example functions u( T) from Ll[O, Tj)) and the space of trajectories with some weak (the weaker the better) topologies, for which the input-output map

is still uniformly continuous. Then may extend this map continuously onto the completion of U, containing generalized functions along with the ordinary ones.

NONLINEAR SYSTEMS 245

Our approach is based on a formula which for a given "ordinary" control uO expresses trajectories of the system (1.1) in terms of vO, being the primitive (or indefinite integral) of uO. This formula allows us to define generalized trajectories ofsystem (1.1) for the controls being the generalized derivatives of the bounded measurable functions and also to express the trajectory explicitly in terms of the generalized primitive of the control. We also consider in Section 6 the generalized time-optimal problem for the system (1.1) the generalized controls. Reducing this problem to an "ordinary" one we obtain the necessary conditions of optimality.

This paper is much related to the paper [6]. One will find there some details which are omitted here.

2 Basic definitions and terminology

Here we shall list some terms and definitions (see [7,8] for the details). Let Coo (Rn) be algebra of infinitely differentiable functions on Rn. We identify a point z E Rn with the multiplicative functional X : ¢ -+ X 0 ¢ = ¢(X) on this algebra. The diffeomorphism P : Rn -+ Rn is identified with the automorphism P : r/JO -+ P 0 ¢ = ¢(P()) of coo(Rn). The space of diffeomorphisms of Rn is denoted Diff( Rn). For any z E Rn and P E Diff Rn the composition z 0 P is again the multiplicative functional on Coo(Rn), i.e. the point P(z) E Rn. Finally vector fields are derivations of coo(Rn), or R-linear mappings of COO (Rn) into itself, meeting the Leibnitz rule:

X(¢¢) = (X¢)¢ + ¢(X¢).

The Lie bracket [X, Y] introduces the Lie algebra structure in the space Vect (Rn) of vector fields. For P E Diff( Rn) Ad P denotes inner automorphism of the Lie algebra Vect (Rn):

VX E VectRn AdPX = PoX Op-1 j

for any Y E VectRn, ad Y denotes inner derivation of VectRn:

adYX == [Y,X].

We consider in coo(Rn) the Whitney topology of convergence of all derivatives on compact subsets of Rn. If AT is a one-parameter family of operators on coo(Rn) , then the properties of absolute continuity, measurability, local integratibility, etc., of this family w.r.t. the parameter T is interpreted in weak sense:

AT possesses property (*) w.r.t. Tiff Vr/J E Coo(Rn)AT ¢ posseses this property w.r.t. T.

246 ANDREJ V. SARYCHEV

If Pr is an absolutely continuous family of diffeomorphisus then pr- 1 0

dPddr is locally integrable family of derivations of coo(Rn) which is called a time-dependent vector field Xr on Rn. The equality Pr-1dPr/dr = Xr implies the differential equation

(2.1)

The solution Pr of this equation is denoted (see [7]) as exp J; Xrdr and is called a right chronological exponential of X r . If it exists for all t E R, then it is called a flow on Rn. If Xr is time-independent (Xr = X), then Pr is denoted as erX .

Let us adduce the variation of constants formula for the right chronological exponential (see [7,8]):

exp 1t (Xr + Yr) dr = exp 1t exp for adXedeYrdr 0 exp fot Xrdr. (2.2)

The first term in the right-hand side is called a perturbation flow. The operator exponential exp J; ad Xe de is (parameterized by r) family of inner

automorphisms Ad exp J; Xede meeting the operator differential equation

ddr (exp 1t adXede) = (exp for adXede) adXr.

Further we shall need the formula of differentiating the exponential w.r.t. a parameter. According to [7,8].

_eX(~) = eX(~) e'1adX(~)_dTJ. o 11 oX op 0 op

3 Integration by parts in chronological exponential

Let us consider at first a linear in the control system

z = Xr(z)u(r), rE[O,T],xERn, u E R,

(2.3)

(3.1)

where u is a scalar control, uO E L1 [0, T], and Xr is an nonstationary vector field on Rn, being COO-smooth w.r.t. x and locally integrable w.r.t. r. Iffor given uO the solution of (3.1) exists for t E [0, t] we denote it (see Section 2)

(3.2)

In the following proposition we obtain for Pt an expression containing vO = J~ u(e)de instead of uO·


Proposition 3.1 (Integration by parts in chronological exponential: noncommutative case). Let the vector field X., be the absolutely continuous w.r.t. r, control uO belongs to L1 [0,11, function vO be a primitive of uO (d/drv(r) = u(r) a.e.}. Then if the solution (9.£) of (9.1) exists for t E [0,11 the following equality holds

Pt = exp 1t X.,u(r)dr

= exp 1t (_11 e(adX~II(")X.,v(r)de) droeXtll(t)

(3.3)

Remark. Diffeomorphism eXtll(t) = exp J; Xtv(t)de appearing in (3.3) is produced by time-independent vector field Xtv(t) with t fixed.

The proof goes by direct calculation. Let us consider the following composition of two diffeomorphisms:

Qt = exp 1t X.,u( r)dr 0 e-Xtll(t).

Differentiating Q w.r.t. t, we obtain

and

~ Qt = ~ (exp l' X.,u( r)dr) 0 e-Xtll(t) +

+exp 1t X.,u(r)dr o ~ (e-Xtll(t»)

= exp l' X.,u( r)dr 0 Xtu(t) 0 e-Xtll(t) +

+ exp 1t X.,u(r)dr 0 e-Xtll(t)

[1 e(ad Xtll(t)de ~ (-X,v(t» = .10 dt

= exp l' X.,u( r)dr 0 e-Xtll (') 0

(XtU(t) -11

e(adXtll(t) (Xtv(t») de -11

e(adXtll(t) (Xtu(t)) de)

248 ANDREJV. SARYCHEV

Since e{adXtv(t)Xt = Xt ,

~ Qt = Qt 0 ( _11 e{ad Xtv(t) Xtv(t)de)

Hence

exp 1t Xru(r)dr 0 e-Xtv(t) =

= Qt = eXp 1t (_11 e{adXTv(t)c:!r Xrv(t)de) dr

and acting on both sides of this equality by the exponential eXtv(t) we obtain (3.2).

The reasoning for calling (3.3) the integration by parts formula is the following one. If for every r E [0, t] the vector fields Xr and Xr commute, i.e. [Xr, X r] = 0, then for every r E (0, t] e{adXTv(t)Xr = Xr and the previous proposition implies the following:

Proposition 3.2 If in addition to the assumptions of Proposition 3.1 the Lie bracket [Xr,Xr] vanishes for almost allTE [O,t], then

exp 1t Xru( r)dr = exp 1t (-XrV( r») dr 0 eX,v(t) (3.4)

where vO is a primitive of uO (for example vO = J oU(7J)d7J ).

Comparing (3.4) with classical integration by parts formula

1t Xru(r)dr = 1t Xrdv(r) = -1t Xrv(r)dr + Xtv(t),

we see, that (3.4) is its multiplicative analogon. The previous results may be restated for the multi-input systems. Let

us consider a control system r

x = Xr(x)u(r) = L:X~(x)ui(r), (3.5) i=1

where Xr = IIX; .. . X;II, u(r) = (ul(r), ... , ur(r». Let for a given uO E L;;"[O, T]

(3.6)

be the solution of (3.5). To make integration by parts in (3.6) we need one additional and very important assumption: for almost every r E [0, t) and every V1, V2 ERr the vector fields XrV1 and XrV2 commute. Note that this assumption is equivalent to the Frobenius condition of integrability for the vector distribution span {Xrvlv ERr} with r fixed.


Proposition 3.3 If the assumptions of Proposition 9.1 are fulfilled for the vector fields X; and the controls ui(r)(i = 1, ... , r) and for every V1, V2 e RT' and almost every r e [0, t] the Lie bracket [XrV1' X1'V2] vanishes, then

exp lot X1'u(r)dr = exp lot (-101 e(adXrtl(1') X1'v(r))de) dr 0 eXttl(t)

(3.7) where vO is a primitive of uO. If in addition for every veRT' and almost every r e [0, t] the Lie bracket [X1'V, X1'v] vanishes, then

exp lot X1'u(r)dr=exp Iof(-X1'V(r))dr 0 eXttl(f). (3.8)

The proof is similar to the one of Proposition 3.1. Using the variation of constants formula (2.2) one can obtain other

versions of multiplicative "integration by parts formula" , for example one for the affine control system (1.1 ').

Proposition 3.4 Let the nonstationary vector fields f1',g~(i = 1, .. . r) be COO-smooth w.r.t. x, while f1' be locally intergable and G1' be absolutely continuous w.r.t. r. Let the vector field G1'V1 and G1'V2 be commuting for every V1, V2 e RT' and almost every r e [0, t] (Frobenius condition). If for a given control uO e Ll[O, t] the solution of equation (1.1 ')

i: = fT(x) +GT(x)u, x e Rn,u e RT',G1' = IIg~ .. ·g~1I

being denoted as Pf = exp J~(J1' + GTu(r))dr exists on [0,1'], then

exp 1t (IT +GTu(r))dr=

:exp Iof (eadGrtl(T)f1' -101 e(adGrtl(1')G1'V(r)de) droeGttl(f)

(3.9) where vO is a primative of uO.

4 Generalized inputs, generalized outputs and generalized input-output map

Let us consider an "ordinary" control or input uO e L![O, 1'] of the system (1.1). Substituting this input into the right-hand side of (1.1) one obtains the differential equation. If the solution of this equation exists on [0,1'], then denoted as Pt = exp J~(lT + G1'u(r))dr it is called an output of the system (1.1) on [0,11. So u ~ P is an input-output map (I/O map) for


the system (1.1') not the individual trajectory but the family of diffeomorphisms Pt being the totality of all trajectories.

Let us consider a control uO E L1[O, T]. From now the term "primitive of uO " denotes always vO = J~ u(TJ)dTJ· Let Qt(v) be the right-hand side of (3.9). According to (3.9) an output Pt(u()) coincides with Qt(v()) provided that vO is the primitive of uO. Let us consider a mapping

uO -+ vO -+ Qt( v())

If uO is from L1[O, T], then vO belongs to the space ACr[O, T] of absolutely continuous vector functions. Certainly any topology T introduced in ACr [0, T] induces by virtue of J some topology in the space L1 [0, T] : U is open in L1[O, T] iff J(U ~ ACr[O, T] is open in topology T. We want to set some (the weaker the better) topology in ACr[O, T] and also some topology in the space of outputs P providing uniform continuity of the mapping vO -+ Qt(v()). As far as ACr[O,T] is included in Ll[O,T] we may consider in ACr[O, T] the topology produced by Li:-norm. Obviously in this case ACr[O, T] is incomplete and dense in L1[O, T]. This topology of ACr[O, T] induces (by virtue of J) in the space Li: [0, T] the topology which is defined by a following norm

As far as ACr[O, T] is dense in L1[O, T], the completion of the space L1[O, T] of "ordinary" controls in DLi:-norm coincides with the space of generalized functions being the generalized derivatives offunctions from LHO, T]. This space will be denoted W!:l.1[O, T].

As to the space of outputs P, we define its topology by the family of seminorms lIP-II! K' where ,

IIP-II!,K = lT IlPrll.,KdT, ( 4.1)

and for given sand K and P E Diff Rn the seminorm IIPII.,K defines in the space of diffeomorphisms the topology of uniform convegence of all the derivatives up to the s-th order on the compactum K. The family of semi norms II . II.,K defines in the space of diffeomorphisms the Whitney topology of convergence of all derivatives on the compact sets. The space of outputs P. furnishes with the family of seminorms (4.1) becomes a Frechet space: the metrics in this space will be denoted LlCx'

Let us consider a set Ua of inputs uO such that the primitives of uO E Ua

are jointly uniformly bounded by constant a on [0, T]:

'Vt E [0, T]'Vu E Ua Iv(t)1 = 11t u(T)dTI :5 a.


Returning to the mapping uO -+ vO -+ Q( v()) we may say at first that from the results of [7] and [6] it follows that the mapping vO -+ Q( v()) being restricted on JUO/ is uniformly continuous in Li-metrics of vO's and L}C~-metrics of P. 'so This implies (see[6]) the following:

Proposition 4.1 The I/O map uO -+ P., when restricted on UO/ is uniformly continuous in DL'i -metrics of uO's and described above L}C~metrics of P. 's.

Standard reasoning allows to extend I/O map continuously on the completion of U" in Li-metrics. To describe this completion let us note firstly that J(U,,) is a set of absolutely continuous functions being uniformly bounded by a. Its completion in L'i-metrics coincides with the ball II"

in L~[O, T] : II" = {vOl vrai sup Iv(r)1 ::::; a}. Hence the completion of U" in DL'i-metrics consists of generalized derivatives offunctions from II. Evidently formula (3.9) defines the extended I/O map for generalized controls u E U" in terms of generalized primitives vO E L~ [0, T] of these controls. Since a is arbitrary then formula (3.9) defines I/O map for every u being the generalized derivative of an essentially bounded measurable function vO, i.e. for every u from W~1 00' So we may summarize the aforesaid in the following theorem. .

Theorem 4.1 Let us consider affine control system (1.1):

If for every VI, V2 E R r and almost every t E [0, T] the vector fields GrVI

and G r V2 commute, then for every u = (Ul,"" U,.) E W~ 1 00 being the generalized derivative of vO E L~[O, T] formula (3.9) .

exp itUr + Gru(r»dr) =

= exp it (ead GrtJ(r) fr -i1 e{adGrtJ(r)Crv(r)de)dr 0 eG,tJ(t)

defines the generalized output P. of system {i. 1). The generalized input/ output map of this system is continuous in DL'i -metrics of the generalized input space W~I.oo and L}C~-metrics of generalized output space.

5 Examples: impulsive controls

Let us consider again an affine control system (1.1')

and discuss the use of previous results for studying impulsive controls in the system (1.1). The term "impulsive control" means a linear combination ~f delta-functions located at different points of time axis.

Example 5.1. (Finite number of impulses). Let us consider an impulsive control of a kind: U = IT:l uic5(r - ri), where c5(r - ri) is the deltafunction located at the point ri, Ui E Rr (i = 1, ... ,JI) Let 0 = ro :$ rl :$ ... :$ rJl :$ T. Evidently U is the generalized derivative of the piecewise constant function v( r) = E~1 uih( r - ri) (here hO is the right-continuous Heavyside function). On the interval rm :$ r < rm+l function vO is equal to Vm = E~1 Ui, while v(O) = o. As long as vO is piecewise constant then expression (3.9) may be transformed to the form

Pt = fi exp 1.r~1 (eadGrll.-l fr -11 eead Grll'_1 GrVi-1 de) dro

exp 1~ (eadGrllmfr_11 eeadGrllmGTvmde)droeGtllm,

for rm:$ t < rm+1

(5.1)

Using the variation of constants formula we may obtain for v = const the equality

exp jf'/ (-11 eeadGrllGTVd~) dr = e-G(IIoeG~II

and then transform (see [6]) (5.1) to the form

As we see the output of the system (1.1) in this case is piecewise absolutely continuous w.r.t. r; it consists of trajectories of the field fr and jumps at points of impulses ri(i = 1, ... ,JI). The i-th jump is done along the trajectory of the field GT • Ui during the unit time interval.

Example 5.2. (Countable number of impulses.) Let us consider the generalized control of the kind U = E:l Uic5( r - ri), ro = 0, ri E [0, T), ri < Ti+l (i = 0,1, ... ), liIDi-+oo T; = f :$ T. We suppose that the se-. ,,00 ... I "JI d nes L....i=1 Ui converges, I.e. Its partla sums L....i=1 Ui ten to some vector V E Rr , when N -+ +00. Evidently if we want to calculate Pt for t < r, then we are in the scope of the previous example, because in this case only finite number of impulses precede t. If t ~ r, then it is quite natural to


suppose, that by analogy with (5.2):

In order to give sense to the infinite noncommutative composition in the right-hand side of (5.3) let us consider the sequence of diffeomorphisms

In [6] it is shown that the sequence p.ltf tends in Whitney topology (or C~-metrics) to the diffeomorphism

P = PT = (exp 1o"(eadGrV(T)fT - 101 eeadGrV(T)GTV(T)de) dToeGpV(p),

where VO = 2::1 Uih( T - Ti) is a piecewise constant function (the convergence of the functional series 2::1 Uih( T - Ti) ensue from the convergence of 2::1 Ui).

Thus we gave sense to the infinite composition in (5.3). The trajectory of (1.1) driven by the impulsive control 2::1 UiC(T-Ti) is piecewise absolutely continuous with countable number of jumps.

The last thing we want to note here is that in the scope of our approach we may consider the generalized derivatives of functions having uncountable set of discontinuities as control for system (1.1). In particular it means, that we may use impulsive controls with uncountable number of impulses. If the generalized primitive of such an "impulsive" control is known, then Theorem 4.1 gives a possibility to calculate corresponding output of system (1.1).

6 Generalized time-optimal problem for {1.1}: its reduction to the classical one. First-order optimality conditions

Let us consider again the control system (1.1) with the initial point of trajectories being fixed

z(O) = Zo, (6.1)

Everywhere below we suppose the conditions of Theorem 4.1 being satisfied. According to this theorem for a generalized control U E W: 100 (u =

I

254 ANDREJV. SARYCHEV

DvO, vO E L~ [0, T]) the generalized trajectory zO of (6.1) is defined as z(t) = Pt(zo), where Pt is set in (3.9). Since Pt is only locally integrable w.r.t. t, then t ..... z(t) is also only locally integrable and doesn't have any definite value at a point to. Nevertheless we will define the set of points being attainable by system (6.1) during the time interval [0, t] by means of given generalized control u E w~ 1 ,00'

Definition 6.1. Point x is attainable by system (6.1) during the timeinterval [0, t] by means of a given generalized control U E W~1,00' if there exists a sequence of "ordinary" controls umO E Li[O, T]) tending to U in DL1-metrics such that if zmO are the trajectories of (6.1) produced by controls umO, then limzm(t) = x.

The set of such points is denoted Axo (u : [0, t]). Theorem 4.1 implies that any point from Axo(u : [0, t]) pertains to

the integral manifold Cx,(Ot) of the integrable vector distribution at = span {GTvlv ERr} which passes through the point

Xt(v()) = Zo 0 exp if (eadGrV(T) fT -11 e€adGrV(T)GTv(r)de)dr,

where vO E L~ [0, T]) is generalized primitive of U. So Axo (Uj [0, t] ~ ex. (ad. Actually the results of [9] imply the stronger fact.

Proposition 6.1 Attainable set Axo ( Uj [0, t] of system (6.1) coincides with ex,(Gt ).

Omitting the proof of the proposition we shall only outline its idea. Turning to the Theorem 4.1 or formula (3.9) we see, that moving from Zo

to Zt is defined by the integral behaviour on [0, t] of function vO being the generalized primitive of control u, while the moving along ex. (at) is defined by the value vO at the endpoint. One may easily construct the sequence of absolutely continuous functions vmO E Acr [0, T]), such that:

(1) vm(O) = OJ (2) vmO ..... vO in metrics of L1[O, t])j (3) all the vmO's have any preassigned value V at the endpoint t.

In this case the controls umO = dvmjdr produce trajectories zmO ending at the points Xt(vm())eG•v tending to the point Xt(v()) oeG,v, when m ..... 00.

Compelling V to run the whole Rr we shall prove that any point of ex, (at) is obtainable. Remark that during the given time interval [0, t], starting from a given point Zo and using given generalized control u E W~1 00 we may attain not a single point as usual, but any point of r-dimen'sional manifold ex, (Gt ).

Let U ~ W~1 00 be a set of generalized controls. The set AXo(Uj [0, t]) = UUEuAxo ( Uj [0, tj) will be called the U-attainable set of system (6.1) during the time-interval [0, t].

Let us set generalized time-optimal problem for the system (6.1) imposing the endpoint condition

(6.2)

and demanding minimality of t

t -+ min. (6.3)

The condition (6.2) corresponds to the fixed endpoint condition x(t) = Xi

in classical setting.

Definition 6.2. The generalized control u E W': loo is locally optimal in generalized time-optimality problem (6.1) - (6.3)' iff Axo(u; [0, m :) Xi , and there exists a IS-neighbourhood U6 of u in DL'i -metrics such that 'Vt < i Xl ',l, AXo(U6; [0, tD (Xi is not attainable in time t less than i by means of any control u from U6.

Let us return to theorem 4.1 and formula (3.9) and consider the classical setting of time-optimality problem for the control system

:i: = eadGrv fT _11 e{adGrVGTvde, x(O) = Xo (6.4)

with endpoint condition of a kind

(6.5)

Theorem 4.1 and Proposition 6.1 joined with the study we performed above implies the following result.

Theorem 6.1 A generalized control u E W': l 00 is locally optimal in generalized time-optimality problem {6.1}-{6.3} iff its generalized primitive tiO E L~[O,tl is L'i-locaIlY optimal in the time-optimality problem {6.3}-{6.5}.

Sketch of the proof. If u is not locally time-optimal on [0, t], then in any DLi-neighborhood of u there exists U E W[,oo and i < t such that Xl E Axo Cit; [0, ~). Let tlO E L~ [0, tl be the generalized primitive of U. Then according to the Theorem 4.1 control tlO steers system (6.4) in time i to the integral manifold C x; (G i) of the distribution G i and hence meets the condition (6.5) for i < i. Thus in any L'i-neighborhood of vO there exists control tlO steering system (6.4) to the target manifold CX1(Gi ) in time i < t. We have presented the if part of the proof, the only if part is similar.

Thus the generalized time-optimality problem can be reduced to the traditional setting of this problem for the nonlinear in the control system


(6.4) with "sliding endpoint condition" (6.5). The generalized primitives of the generalized controls being optimal for (6.1)-(6.3) are in turn optimal for (6.3)-(6.5).

Since the first-order optimality conditions for the problem (6.3)-(6.5) are known ([10]), then using the previous reduction we may state the first-order optimality conditions for the generalized time-optimality problem (6.1)-(6.3). In order to formulate these conditions we shall define the Hamiltonian 1i as

Then theorem 6.2 and classical results of [10] imply

Theorem 6.2 If a generalized control U E W~l 00 is DLi -locally optimal in the generalized time optimality problem {6.1}-{6.3} and tiO E L~[O,t] is a generalized primitive of il, then there exists the pair of absolutely continuous functions (iO,?,b()) such that the triple (iO, ?,b0, ti())

{1} satisfies the canonical equations

. o1i(x,t/J,v(r),r) x = ot/J '

~ = o1i(x,t/J,v(r),r) ox

with the Hamiltonian {6.6},

{2} meets the stationary condition

81i _ =0 ov(i(r),t/J(r),ti(r),r)

{3} meets the transversality conditions at the endpoint (x(t), t/J(t), t)

"Iv E Rr (?,b(i), G~x)(i)v} = O~ "Ii = 1, ... , r(?,b(i), gf (x(t))} = O.

Bibliography

[1] Filippov, A.F. Differential equations with discontinuous right-hand side. Moscow, Nauka, 1985 (in Russian).

[2] Krasnoselski, M.A., Pokrovskij, A.V. Vibrostable differential equations with continuous right-hand side. Annals of Moscow Mathematical Society, 1972, Vol. 27, pp. 93-112 (in Russian)


[3] Sussmann, B.J. On the gap between deterministic and stochastic ordinary differential equations. Annals of Probability, 1978, Vol. 6, pp. 19-41.

[4] Orlov, Yu. V. Variational analysis of optimal systems with measurelike controls. In. Avtomatika i Telemekhanika, 1987, No.2, pp. 26-32 (in Russian). English transl. in Automation and Remote Control, 1987, No.2.

[5] Bressan, A. On differential systems with impulsive controls. Rend. Sem. Mat. Univ. Padova, 1987, Vol. 78, pp. 227-235.

[6] Sarychev, A.V. The integral representation of trajectories for the control system with generalized right-hand side. In Differentsialnije Uravnenija, 1988, Vol. 24, pp. 1551-1564 (in Russian). English translation in Differential Equations, 1988, Vol. 24, 1021-1031.

[7] Agrachev, A.A., Gamkrelidze, R.V. The exponential representation of flows and chronological calculus. Matem. Sbornik, 1978, Vol. 107, pp. 467-532 (in Russian). English translation in Math. USSR Sbornik, 1979, Vol. 35, pp. 727-785.

[8] Agrachev, A.A., Gamkrelidze, R.V. Sarychev, A.V. Local Invariants of Smooth Control Systems. Acta Applicandae Mathematicae, 1989, Vol. 14, pp. 191-237.

[9] Argrachev, A.A., Sarychev, A.V. On reduction of smooth affine control system. Matern. Sbornik, 1986, Vol. 130, pp. 18-34 (in Russian). English translation in Math. USSR Sbornik, 1987, Vol. 35, pp. 15-30.

[10] Pontryagin, L.S. et a1. Mathematical Theory of Optimal Processes. Interscience Publishers, 1962.

Andrej V. Sarychev Institute of Control Problems Profsojuznaja ul. 65 117342 Moscow USSR

Extremal Trajectories, Small-time Reachable Sets and Local Feedback

Synthesis: a Synopsis of the Three-dimensional Case

HEINZ SCHATTLER

1. Introduction

In any optimization problem three basic questions have to be answered. Does an optimal solution exist? How can one restrict the candidates for optimality by way of necessary conditions? Is a candidate found in this way indeed optimal (in a local and/or global sense)? For problems on function spaces by now fairly general existence results are known (see, for instance, Cesari [6]) which cover a wide range of realistic problem situations. The theories of necessary and sufficient conditions for optimality, on the other hand, lack a similar completeness of results. For optimal control problems, the Pontryagin Maximum Principle [11] gives first order necessary conditions for optimality. Several higher order conditions for optimality are known as well (cf. Krener [8], Knobloch [7] and the many references therein), but they mainly deal with special situations, like the generalized Legendre-Clebsch condition for singular arcs. Typically the necessary conditions will not suffice to single out the optimal control. In fact, in many cases there exists a significant gap between the structure of extremals (i.e. trajectories which satisfy the necessary conditions for optimality) and the structure of optimal trajectories in a regular synthesis. Roughly speaking, a regular synthesis consists of a family of extremals with the property that a unique extremal trajectory starts from every point of the state-space and which satisfies certain technical conditions which allow to prove that the corresponding feedback control is indeed optimal.

In the eighties significant progress has been made in closing this gap between necessary and sufficient conditions. Indeed, through a combined effort of several researchers this gap has been completely eliminated for the problem of locally stabilizing an equilibrium point of a nonlinear 3-dimensional system under generic conditions. The techniques developed for tackling this problem have a much wider range of applicability. The 3-dimensional problem only takes the role of a very important test case. However, its significance goes well beyond that of a test-case, but this example offers for the first time a possible explanation for the reason why there is such a

Supported in part by the National Science Foundation under grant No. DMS-8820413

A SYNOPSIS OF THE THREE-DIMENSIONAL CASE 259

gap between necessary and sufficient conditions. For this problem we now know precisely which trajectories are optimal in the local (variational) sense of calculus of variations, and which ones are optimal near the equilibrium point. And these two classes of trajectories do not coincide. But only geometric arguments which fall outside the realm of necessary conditions for local optimality allow to distinguish these classes. So apart from being a test situation for newly developed necessary conditions for optimality, also the structure of the synthesis itself has great significance for optimal control problems in general.

In this brief note we attempt to give a synopsis of some of the new techniques developed and how they complement each other to solve the 3-dimensional problem. These results have already been reported in the literature and we refer to these papers for proofs and mathematical details. Here we only try to illustrate the main ideas and the main implications of the results.

2. The Structure of Extremals: necessary conditions for optimality

We consider systems of the form

(1) L : x = f(x) + g(x)u, lui ~ 1,

where f and g are smooth or analytic vector fields. We denote the vector fields f - g and f + g by X and Y respectively. Admissible controls are Lebesgue-measurable functions which take values in the interval-l ~ u ~ 1 almost everywhere. The topic we study is time-optimal control between points.

Brunovsky's results on regular synthesis in the late seventies [4], [5] made it clear how large the gap in optimal control between necessary and sufficient conditions for optimality really was. In these papers Brunovsky significantly weakened the strong technical conditions of the original definition of a regular synthesis given by Boltyansky [2], but at the same time the main intuition that optimal controls should be at least 'piece-wise'regular (as it is experienced in all the standard textbook examples) was maintained. This could be done by using the powerful theory of subanalytic sets to deal with some of the technical problems due to discontinuities in the control (respectively value-function). However, this theory is only applicable if the extremals under consideration satisfy a local regularity condition of the following type: (BNS) for every point p in the state space there exists a neighborhood U of p, a time T> 0 and an integer N = N(U, T) with the property that any extremal trajectory which lies in U and is of duration ~ T, is a concatenation of at most N integral curves of analytic vector fields. For linear systems with a polyhedral control set this is the famous

260 HEINZ SCHATTLER

bang-bang theorem [4], but for nonlinear systems very few results of this type are known[14].

It therefore seemed rational to start by investigating low-dimensional problems. For systems in the plane this problem was completely solved by Sussmann in a series of papers [16]-[18] which deal both with general analytic vector fields and with smooth vector fields in the nonsingular case. However, the reasoning which proved the property (BNS) heavily relied on Stokes theorem in the plane and had no straight-foward extension to higher dimensions. Since then several constructions have been carried out which all obtain higher-order necessary conditions for optimality which specifically aim to prove the local regularity property (BNS). Typically calculations have been restricted to 3-dimensional problems, but some of these approaches are valid in general. We briefly discuss the most important ones.

The first significant result in dimension 3 is due to Bressan [3] for the problem to steer points time-optimally to an equilibrium p of f under the following assumption (A): the triplets (g, [f,g], [f + g, [f, g]]) and (g, [f, g], [fg, [f, g]]) consist of linearly independent vectors at p. ([.,.] denotes the Liebracket of vector fields). If the vectors [f + g, [f, g]] (p) and [I - g, [I, g]] (p) point to opposite sides of the plane spanned by g(p) and [I, g](p), then locally optimal trajectories behave as for a 3-dimensionallinear system, i.e. are bang-bang with 2 switchings. This can be seen by a simple application of the Maximum Principle. But if [f + g, [f, g]](p) and [f - g, [I, g]](p) point to the same side, the structure of the system is truly nonlinear. Now the first order necessary conditions allow for bang-bang trajectories with an arbitrary number of switchings on time intervals of arbitrarily small lengths. Using a nilpotent approximation near the equilibrium point, Bressan could eliminate bang-bang trajectories with 3 or more switchings by comparing trajectories of the form petlYet2Xet3Y with trajectories of the form pe$lX e$2Y e$3 X and then using a directional convexity property of the small-time reachable set. (We use exponential notation for the flow of vector fields and we let the diffeomorphisms act on the right, i.e. petX denotes the point obtained by following the integral curve of X through p for t units of time.)

We used similar ideas in [12] to construct bang-bang trajectories of different switching orders which steer a point p to the same point q by solving the equation

(2)

for (tt, t2, t3) as smooth functions of (81, 82, T). This frees the construction of the equilibrium point assumption on f, but other conditions have to be imposed to guarantee the local solvability of equation (2). The independence of the vectors f, 9 and [f, g] at a reference point p is a sufficient condition (and p is assumed to lie in a sufficiently small neighborhood of p). By


computing an asymptotic expansion for the difference in time along the traj ectories,

we obtained second order necessary conditions which excluded the optimality of bang-bang trajectories with more than 2 switchings under assumption (A). This expansion also allows to analyze cases when (A) is violated, but where conditions on higher-order brackets are imposed [12]. Assuming that I, g and [I, g] are independent, we showed that property (BNS) holds generically (within this class) for optimal bang-bang trajectories.

A very important tool in this construction was a generalization of the classical concept of conjugate points to optimal control problems given by Suss mann [15]. For our considerations here, it suffices to say that the points corresponding to 3 consecutive switchings of an extremal bang-bang trajectory are conjugate (in this sense) under assumption (A). In [15] Sussmann constructs a surface of conjugate points (i.e. for instance third switching points of bang-bang extremals in our case) which under certain technical assumptions can be used to exclude the optimality of trajectories beyond the conjugate point. This theory is applicable to the general case of a control system x = I(x, u), uM, Uf.U under rather weak, but technical regularity conditions. If one wants to exclude the optimality of a reference trajectory, then this construction has to be validated for the particular situation. For a 3-dimensional system of the form (1) Sussmann recently has done this under assumption (A) [19], thus showing that local optimality ceases for bang-bang trajectories at the third switching.

The same result had been obtained earlier also by Agrachev and Gamkrelidze [1] as a particular application of a theory which generalizes the classical concept of the Morse-index to extremals of optimal control problems. Like Sussmann's construction, this theory is applicable for general systems. Results about particular situations are obtained by explicit computations of the index. In dimension three this has been done assuming condition (A). But it is a highly nontrivial problem to perform these calculations for a general situation.

These results have led to a complete understanding of extremal trajectories for a three-dimensional sytem of the form (1) near a reference point p where assumption (A) is valid. The behavior of extremal trajectories which contain singular arcs (if such exist) is rather simple and only concatenations of the form BSB (an integral curve of the vector field X or Y followed by a singular arc and another integral curve of X or Y) can be optimal. This easily follows from the Maximum principle. The structure of extremal bangbang trajectories is much more complicated in that the Maximum principle allows for an arbitrary number of switchings in time intervals of arbitrarily small lengths. Sophisticated methods had to be developed to eliminate this possibility. Several new higher order necessary conditions for optimality

262 HEINZ SCHATTLER

have been developed which all exclude the optimality of extremal bangbang trajectories after the third switching point (some requiring additional hypothesis over (A)). So the only possible candidates for optimal controls are of the structure BSB or BBB. Indeed, as will also become clear in the next section, each of these trajectories is a local minimum in the sense of Calculus of Variations, i.e. when compared only with trajectories which have the property that they stay within a sufficiently small neighborhood of the reference trajectory in state space.

The reader might find it an interesting, though possibly frustrating learning experience to try and piece together a synthesis of optimal trajectories for the problem of stabilizing an equilibrium point based alone on this in fact complete information about the extremal trajectories. The difficulty lies in the fact that even the precise local structure of extremal trajectories is generally not sufficient to give a local synthesis of optimal trajectories near a reference point, but global properties of the extremal trajectories matter. These can be taken into account by explicitly considering smalltime reachable sets. The feedback synthesis of optimal controls for the three-dimensional system beautifully illustrates this fact.

3. From the structure of the small-time reachable set to a local time-optimal feedback synthesis

If r is a trajectory which steers a point p to a point q time optimally in time T, and if we add time to the system as extra coordinate, then the trajectory r lies in the boundary of the small-time reachable set from p for all times t < T. Hence knowledge about the small-time reachable set may be used to restrict the structure of time-optimal controls.

As a simple case, consider the standard textbook example of steering points time-optimally to zero for the double-integrator. Or, what amounts to be the same, consider the local structure of time-optimal controls to an equilibrium point p of 1 in the plane assuming g(p) and [I, 9 ](p) are independent. It is well-known (and follows from an easy application of the maximum principle) that optimal controls are bang-bang with at most one switching and that a local synthesis can be obtained by integrating these controls backward from p. This result can also be derived from Lobry's results [10] on the structure of the small-time reachable set from a point p for a 3-dimensional system where it is assumed that I, 9 and [I, g] are independent at p. For, if we add time as extra coordinate to the two-dimensional system, these assumptions are valid. The local synthesis of the two-dimensional system can be obtained by projecting the boundaries of the time-t-reachable sets in dimension three into the original state space. Of course, since this problem can be easily solved by more standard arguments, this is not the approach typically taken. However, already in dimension three, the problem of finding a local time-optimal synthesis

becomes highly nontrivial and it was this point of view which led to a solution of the problem. For a three-dimensional system which satisfies condition (A) at p, in [9] the precise structure of the small-time reachable set from p for the exte'lded four-dimensional system was given and in [13] a local time-optimal syntl • .;si:. to p was constructed by projecting the boundaries of the time-t-reachable sets into the original three-dimensional state space. Contrary to the planar case where this method turns out to be equivalent to standard variational techniques, the three-dimensional case shows that the geometric approach differs significantly from local variational methods in that it takes into account the global behavior (relative to a neighborhood of a reference point) of all trajectories. As the three-dimensional synthesis will show, the structure of extremal trajectories alone fails to lead to a regular synthesis if some of these trajectories, which are optimal relative to a neighborhood of the trajectory, are no longer optimal relative to a neighborhood of the reference point. Variational techniques cannot detect this, but phenomena of this type do show up in the small-time reachable set.

In this section we briefly want to highlight some of these results. Complete proofs can be found in [9] and [13]. We assume condition (A) holds at p and we also restrict attention to the case when optimal singular arcs exist. This case is characterized by d(p) < 0, where d is defined near p by the representation

(3) [X, [X, Y]] = aX + bY + c[X, Y] + drY, [X, Y]].

Extremal trajectories which contain a singular arc have at most the structure BSB. It is shown in [9] that the collection of all trajectories of this type forms a stratified three-dimensional set r * which lies in the boundary of the small-time reachable set. Topologically, the set r * is a disc with boundary consisting of all the points reached by bang-bang trajectories with at most one switching. It is also shown in [9] that r * can be "closed" by another stratified three-dimensional set r* which entirely consists of bang-bang trajectories and also has all the points reached by bang-bang trajectories with at most one switching as its relative boundary. And this is the only intersection of r* with r *. The small-time reachable set is the set enclosed between these stratified hypersurfaces.

The structure of bang-bang trajectories in r* constitutes the nontrivial aspect of this construction. We know that extremal bang-bang trajectories have at most two switchings. However, if r is such a trajectory, then the times along the individual arcs are not arbitrary, but the length of the middle arc determines the time until the next junction. It therefore restricts the lengths of the first and third leg by what we call conjugate point relations. As it is shown in [9], the extremal BBB-trajectories are given by

r- = {pe 31X e32Y el3X : Si small, nonnegative;

264

and

HEINZ SCHA TILER

S2 is unrestricted; 0'(Sl,S2) ~ 0, U(S2,S3) ~ O} N

f+ ::::: {pet1 Y et2X et3Y : ti small, nonnegative;

t2 is unrestricted; r(t1, t2) ~ 0, f(t2' t3) ~ O} N

where the nro sets of the functions ~, u, :!: and f define the conjugate

points. The optimality of trajectories whose times violate one of these inequalities can be excluded by the maximum principle. First order approximations for these functions can easily be computed as done in [9] and we have, for instance,

U (S2, S3) ::::: - S2 - S3 d(q2) + 0(S2), S::::: Sl + S2 + S3 q2 ::::: pe~1X e~lY

Proposition [9]: The hypersurfaces r- and r+ intersect transversally in a two-dimensional surface t. There exists a smooth function tt defined on the set D ::::: {(t1, t2): ti positive, small and such that :!:(t1,t2) > O} with the property that

t ::::: {pettYetlXet;Y: (t1,t2)fD}

Modulo higher order terms, t[ is given by

We call the surface t the cutlocus of bang-bang trajectories with two switchings. It is computed as the set of non-trivial solutions to the equation

(3)

Let us briefly indicate how this it done. Owing to the independence condition (A) we can rewrite both sides of equation (3) near p in canonical coordinates (of the second kind) as

Comparing coordinates we obtain the following four equations:

(i) Sl + S3 + 0(S3) ::::: t2 + 0(T3)

(ii) S2 + 0(S3) ::::: t1 + t2 + 0(T3)

(iii) s182(1 + O(S))) ::::: t2t3(1 + O(T))

(iv) 81S2(81d + 82 + O(S)) ::::: t2h(2t1 + t2d + t3 + O(T))


where S = 81 + 82 + 83 and T = t1 + t2 + t3. Dividing equation (iv) by equation (iii) we get

By the implicit function theorem, equations (i), (ii) and (iv') can be solved uniquely in terms of 8 or t. We have, for instance

~t1 + t2 + O(T2)

t1 + t3 + O(T3)

-~t1 + O(T2) d

Now substitute these functions for 8 into equation (iii) to obtain

In general, the quadratic terms need not dominate the cubic remainders. If however,the times ti satisfy a relation of the type

then this equation can be solved for t3 as

It easily follows therefore that the intersection of f- and f+ extends beyond the surface of Y X-trajectories, which is characterized by t3 = 0, if we allow negative times. By using the solutions of (i), (ii) and (iv) for t as a function of 8, it can be shown that r similarly extends beyond the XYboundary stratum as well. The curves t and 'Y of intersection of r with the

,.. surfaces of XY- and Y X -trajectories are precisely the curves of conjugate points characterized by the conditions r(t2, t3) = 0 (respectively ~(81' 82) = 0) and 1:(t1,t2) = 0 (respectively U(81,82) = 0). A qualitative sketch of

t-, t+ and of the cutlocus restricted to a time-slice is given in Fig. 1.

266 HEINZ SCHA. TILER

Fig. 1:

XY XY

x Y x Y + + -

YX YX

XY

r" x r Y +

YX ./

The cut-locus r is the most important structure of the small-time reachable set. Bang-bang trajectories with two switchings, but terminated at the cut-locus (which occurs prior to the third switching) together form the stratified hypersurface r* which 'closes'the stratified hypersurface r *. Recall that r. contains the trajectories which have singular subarcs. As is shown in [9], the small-time reachable set is the set of points enclosed between these two stratified hypersurfaces and the trajectories in r * and rare precisely the trajectories in the boundary ofthe small-time reachable set. Since only these trajectories can be time-optimal near the reference point p, it follows that optimality of bang-bang trajectories with two switchings near p ceases already at the cut-locus, not only at the third junction. At the same time, this construction also proves that bang-bang trajectories with two switchings are in fact strong local minima up to the third junction. Trajectories of the type XY X lose optimality at the cutlocus because of faster trajectories of the type Y XY. These trajectories, however, will not lie in a sufficiently small neighborhood ofaXY X reference trajectory. (The construction ofthe small-time reachable set takes into account the structure of all extremals and the statement above is an immediate consequence of the geometric properties of the small-time reachable set as constructed in [9].) This shows that local optimality of a trajectory in the sense of calculus of variations does not imply that the trajectory is also optimal in a neighborhood of the initial point. The latter is relevant for a regular synthesis. Based on the precise knowledge of the small-time reachable set, for


the problem of steering points time-optimally to an equilibrium point f of p in dimension three, in [13] a local regular synthesis of an optimal feedback control is constructed near p. (In addition to condition (A) we also assume Id(p) I ::ft 1, which implies that the system is small-time locally controllable at p.) The synthesis is obtained by integrating backward from p trajectories of the type BSB for arbitrary (but small) times along the individual arcs and by integrating backward bang-bang trajectories with two switchings until they reach the cutlocus. Equivalently, the boundaries of the time-treachable sets of the extended four-dimensional system are projected into the original three-dimensional state-space. It is proven in [13] that these sets 'foliate' a neighborhood of p and that the result is a regular synthesis in the sense of Boltyansky [2]. As a corollary, it follows that the trajectories in the boundary of the small-time reachable set are precisely the trajectories which are optimal near p.

4. Conclusion

For the problem of time-optimal control in dimension three a precise understanding of the structure of both extremal and optimal trajectories near a reference point p where condition (A) holds has emerged during the eighties. Extremal trajectories are concatenations of a bang arc, followed by a singular arc and then by another bang arc or they are bang-bang trajectories with at most two switchings. Whereas the trajectories which have a singular arc are optimal in a neighborhood of p, this is not the case for all bang-bang trajectories with two switchings. These trajectories are a strong relative minimum in the sense of calculus of variations up to the third junction, but optimality in a neighborhood of p ceases already at the cut-locus. This phenomenon is of a global nature and cannot be detected by infinitesimal variational techniques. It is an inherent drawback of variational techniques that the reference trajectory can only be compared to other trajectories close by. An explicit construction of the small-time reachable set, on the other hand, takes into account all possible trajectories. It is therefore fundamentally different in nature from variational methods. Clearly, variational techniques are of the utmost importance to reduce the class of extremal trajectories via necessary conditions for optimality. At the same time, this three-dimensional case shows that variational techniques alone may fail to reveal the correct structure of locally optimal trajectories. Using varitional techniques only it is impossible to reduce the class of trajectories any further than to those trajectories which are strong local extrema. As the example shows, this may not be good enough. Even in small neighborhoods of reference points, the global structure of extremals (relative to this neighborhood) has to be taken into account. An explicit construction of the small-time reachable set (if possible) does this.

These considerations also offer a plausible explanation for the gap between

268 HEINZ SCHATTLER

necessary and sufficient conditions in optimal control. All the known necessary conditions for optimality (like the Maximum-principle, the LegendreClebsch condition etc.) are variational in nature, whereas the sufficient conditions require to patch together locally optimal trajectories. Variational methods in general do not suffice to find locally optimal trajectories, but the structures of extremals away from the reference trajectory must be taken into account as well. The importance of doing this should not be underestimated. Intersections of surfaces of trajectories of different structures (such as the cut-locus of bang-bang trajectories with two switchings for the generic three-dimensional problem) are the norm in dimensions ~ 3, not the exception. For instance, the cut-locus of bang-bang trajectories plays an equally prominent role in the time-optimal control for the generic fourdimensional problem. More complicated cut-loci (such as between surfaces of bang-bang trajectories with surfaces of trajectories which contain singular arcs) appear in higher dimensions. Only if these loci can be analyzed, is it possible to construct a regular synthesis and thus solve the control problem. For this the global structure of extremals must be taken into account, as it was done for the three-dimensional system by construction of the small-time reachable set.

REFERENCES

[1). A. A. Agrachev, R. V. Gamkrelidze, Symplectic Geometry for Optimal Control, in: Nonlinear Controllability and Optimal Control, H. Sussmann (Ed.), Marcel Dekker.

[2). V. G. Boltyansky, Sufficient conditions for optimality and the justification of the dynamic programming method, SIAM J. Control 4 (1966), 326-361.

[3). A. Bressan, The generic local time-optimal stabilizing controls in dimension 3, SIAM J. Control and Optimization 24 (1986), 177-190.

[4). P. Brunovsky, Every normal linear system has a regular time-optimal synthesis, Math. Slovaca 28 (1978),81-100.

[5). P. Brunovsky, Existence of regular synthesis for general control problems, J. Differential Equations 38 (1980), 317-343.

[6). Cesari, "Optimization: Theory and Applications," Springer Verlag, New York, 1982.

[7). H. W. Knobloch, Higher Order necessary conditions in optimal control theory, Lecture Notes in Control and Information Sciences 34 (1981), Springer Verlag, Berlin.

[8). A. Krener, The higher order maximum principle and its application to singular extremals, SIAM J. Control and Optimization 15 (1977), 256-293.

[9). A. Krener, H. Schiittler, The structure of small-time reachable sets in low dimensions, SIAM J. Control and Optimization 27 (1989), 120-147.

(10). C. Lobry, Contrtilabilite des Systemes nonlineaires, SIAM J. Control 8 (1970), 573-605.

[11). L. S. Pontryagin, V. G. Boltyansky, R. V. Gamkrelidze, and E. F. Mishchenko, "Mathematical Theory of Optimal Processes," Wiley Interscience, New York, 1962.

(12). H. Schiittler, On the local Structure of time-optimal bang-bang trajectories in R3, SIAM J. Control and Optimization 26 (1988), 186-204.

[13). H. Schiittler, A local feedback synthesis of time-optimal stabilizing controls in dimension three, Mathematics of Control, Signals and Systems (MCSS) (to appear).


[14]. H. Sussmann, A bang-bang theorem with bounds on the number of switchings, SIAM J. Control and Optimization 17 (1979),629-651.

[15]. H. Sussmann, Envelopes, conjugate points and optimal bang-bang extremals, in: Proceedings of the 1985 Paris Conference on Nonlinear Systems (1986), M. Fliess, M. Hazewinkel (Eds.) Reidel Publishing Co., The Netherlands.

[16]. H. Sussmann, The structure of time-optimal trajectories for single-input systems in the plane: the Coo nonsingular case, SIAM J. Control and Optimization 25 (1987), 433-465.

[17]. H. Sussmann, The structure oj time-optimal trajectories Jor single-input systems in the plane: the general real analytic case, SIAM J. Control and Optimization 25 (1987), 869-904.

[18]. H. Sussmann, Regular synthesis Jor time-optimal control oj single-input real analytic systemll in the plane, SIAM J. Control and Optimization 25 (1987), 1145-1162.

[19]. H. Sussmann, Envelopes, high-order optimality conditions and Lie brackets, Proceedings ofthe 28th Conference on Decision and Control (December 198~, 1107-1112, Tampa, Florida.

Department of Systems Science and Mathematics, Washington University, St. Louis, Mo, USA

Regularity properties

of the minimum-time map

GIANNA STEFANI

1. Introduction

The aim of this paper is to give a survey on some known results concerning the regularity properties of the minimum-time map around an equilibrium point of a control system and to discuss the links of these properties with the viscosity solutions of the Hamilton Jacobi Bellman equation. For sake of simplicity let us consider a control system on R" defined by:

m (I;) x = f(x,u) :: fo(x) + Eujfj(x) , x(O)=xo

j=l where the fj's are Coo vector fields and the control map u=(ull ... ,um) belongs to the class CU of the integrable maps with values in the set

{l = { (Wt,·",wm) e Rm : Iwjl::::;1 , i=I,···,m }.

Moreover we assume that the initial point Xo is an equilibrium point of the system, i.e.

(HO)

and that the dimension at Xo of the Lie Algebra L::Lie{fo,·· .,fm} associated to the system is maximum, i.e.

(HI)

Let t -+ x(t,xo,u) denote the solution of (I;) relative to the control u and define the reachable sets by

R.v(xo,t)={x(t,xo,u) : u eCU} , R.v(xo)=U R.v(xo,t). t>o

Classically the minimum-time problem is the problem of minimizing the time that is necessary to reach the target point Xo from a given point x. Since we deal with time-independent systems,

REGULARITY PROPERTIES 271

changing fj in -fj, i=O,···,m, the problem is changed in the problem of reaching a given point x from the fixed initial point xo' For this reason we define the minimum-time function II: R" - R by:

( ) II(X) = {. +00 if X is not reached . 1.1 Inf { t : x E RJ;(xo,t)} , otherwIse.

By definition it is easy to see that the continuity of II at Xo is equivalent to the following property called autoaccessibility [1] , normal local controllability [19], [7], small time local controllability [28], and that we shall call symply local controllability [8]

(LC) Xo E intRJ;(xo,t) , V t > O.

Set

To = sup {t : RJ;(xo,t) is bounded} and Ro = U RJ;(xo,t). t<TO

It is not difficult to see that II is continuous on Ro iff (LC) holds true, [20] . Moreover estimates on the "size" of the reachable set at time t yield Holder continuity properties for the minimum-time map. Namely the following result due essentially to N.N. Petrov [21] holds true.

Lemma 1.1. The minimum-time map II is locally Holder continuous with exponent 1/ r on Ro if and only if there are positive real numbers 1 and p such that

B(xo,ptr ) ~ RJ;(xo,t) , V t<1

where B(xo,ptr ) is the ball centered at Xo whose radius is ptr. If this is the case we write /

-D,l r II E .l11oC •

Remark 1.2. Notice that To may be strictly less than +00 and Ro may be not the whole reachable set. For example consider the system on R given by

X=X2+U , x(O)=O.

For such a system we obtain RJ;(xo) =(-1,+00) Ro=(-1+2/(I+ell'), +00).

, To=7r/2 and

It is known [10] that if II is a C1 map, then it is a solution of the Hamilton Jacobi Bellman (HJB) equation

(HJB) H(DII(x) ,x) = 0

272 GIANNA STEFANI

where the Hamiltonian is given by

H(p ,x)=max(p ,fo(x) + E Wi fi(X»)-I= WEn i=l

=(p,fo(X) )+i~ll(p,fi(X) )1-1. (1.2)

Notice that v(x) is the minimum-time map for reaching x from Xo so that the HJB equation is the equation stated above and it is not the usual one H( -Dv(x),x)=O.

Normally v is not C1, in fact only local Holder continuity can be expected. For example v is locally Lipschitz continuous if and only if the following strong condition is satisfied [27], [19], [20]

span {f1(xo), ... ,fm(xo)} = R".

Recently some Authors, [11], [14], [2], [12] have proved that if v is continuous, then it is a viscosity solution of (HJB). The interest to the viscosity solutions is essentially due to the fact that the theory provides numerical methods and strong approximation results both for discrete and continuous approximating systems. Namely, under suitable hypotheses, if a sequence HE of Hamiltonians converges to H, then the corresponding sequence of solutions VE of HE(Dvdx),x)=O converges to v [11], [3], [5]. Moreover the Holder continuity of v yields estimates for the rate of convergence of VE [5], [4]. Some unicity results have been obtained in [2], [12], [6] under the assumption that f is globally Lipschitz continuous on R" uniformely with respect to wE O. Such a condition seems a little too strong to study for example polynomial systems. However in [12] it is proved the following local result for which only the uniform Lipschitz continuity in a neighbourhood of Xo is used.

Theorem 1.3. Let V be a neighbourhood of Xo. If v is a continuous solution (in the viscosity sense) of

H(Dv(x) ,x) = 0 v(xo)=O v(x»O

V xE V\{xo}

V xE V\{xo}

then there is a possibly smaller neighbourhood of Xo where v coincides with the minimum-time map. We refer to such a v as a positive viscosity solution of (HIB).

By the above result each sufficient (necessary) condition for local controllability becomes a sufficient (necessary) condition for the


existence of positive viscosity solutions of the HJB equation.

2. Continuity

The geometric theory provides many results based on the Lie Algebra associated to the system. It would be interesting to read these results as sufficient (necessary) conditions for the existence of positive viscosity solutions of (HJB) directly on the equation.

In [28] a subset of L is pointed out as a set of possible obstructions to local controllability. Most of controllability results say that if the "obstructions are neutralized" and (H1) holds true, then 1/ is continuous, see [28], [9] and the references therein. To state some examples of these conditions let us introduce some notations. For sake of symplicity let us choose as set of obstructions the set ~ of those brackets which contain fo an odd number of times and each fl ,' .. ,fm an even number of times. To be more precise let X be a bracket, let lxii, i=O,···,m, denote its "length with respect to i", i.e. the number of times that fj appears in X. Denote also "the total length" of X by m

IXI=E IXIi . j=O

For example l[fo,fl]lo=l and l[fo,fl ]I=2. We set

~ = {X : IXlo is even Ixii is odd i=l, .. ·,m}

Remark 2.1. The above definition is not precise. Namely some vector field can be obtained by different brackets. For the exact definition see [28] and [9].

Remark 2.2 - The set of obstructions could be restricted, see [28], [9]. In any case it is an open question which are the real obstructions to small time local controllability.

Let :r j and Lj be recursively defined by

:rl=span{ad} fk : j~O , k=l,···,m} , LI =span{fo," .,fm} • o

:rj=:rj-I+[:rj-I,11] , Lj=Lj_1 +[Lj_1,LI] , and set

e=11+span{actif [fhA] : j~O , h,k=l,···,m}. o

Consider the following assumptions

(H2)

274 GIANNA STEFANI

(H3)

(H4)

V h even (:rhnc:8)(xo)~:rh-l(xo)

V h (Lhnc:8)(Xo)~Lh_l(xO).

The results in [28] and [8] imply that each of the previous assumptions (together with (111» implies that v is continuous. In view of Theorem 1.3 we can state the following

Theorem 2.3. Let (H1) hold true. Each of the assumptions (H2) , (H3) , (H4) is a sufficient condition for the existence of a positive viscosity solution of (HJB).

Remark 2.4. Notice that the condition (H2) implies (H1).

Let us quote as an example a result of necessary condition of local controllability which gives also a necessary condition for the existence of positive viscosity solution of (HJB) in a neighbourhood of xo. Other conditions which are more technical can be found in [7], [26] and [15].

Theorem 2.5. ([27], [24]) Let m=1. A necessary condition for v to be continuous is that

V h even aJjl fo(xo) e :rh-1(xO}·

3. Holder continuity

For every condition of local controllability there is a technique which gives also the Holder continuity of v. For example condition (H4) implies that v is Holder continuous with exponent 1fr, where r=min{i : Lj(xo)=Rn}. (H2) and (H3) imply also local Holder continuity, but some other information is needed to determine r. Namely the length of the brackets involved in a given condition is a crucial information, see the subsequent Sec.4. In particular the length of the brackets spanning Rn gives a "maximum" for the regularity of v. To be more precise the following necessary condition holds true.

Theorem 3.1. [23] If v e .Hf~~/r then Lr(zo)=Rn•

The above condition is not sufficient as the following example shows.

Ezample 3.2. Consider the following system on RS. xl=u x2=xl x3=x2

X4=x3+fX~ xS=x4

Xl(O)=··· =XS(O)=O

If £=0 the system is a linear one so that vE H~:/S [17]. If £;f 0 the system is still locally controllable since it is its linearization, Ls(O)=R", but

In fact

d HO•1/(6-6) r •• 1: V It= loc lor any positive (J.

t tOT

X4(t) = £ J x~(s)ds + J J J xl(s)dsdTdO ~ 000 0

t t ~ £ J x~(s)ds - tS/ 2 (J x~(s)ds y/2.

o 0

Therefore if x4(t)<0, then

t (J x~(s)ds )1/2 < (l/£)tS/2.

o As a consequence

xs(t) ~ £((X~(S)dSdO - (r(T1/2(JTX~(S)dsl/2dTdOdS > 00 000 0

> _(1/£)tS/ 2 t7 / 2 dO = -(1/£)t6 •

The last i~equality implies that RL'(O,t) cannot contain a ball of radius pt6- for any positive 6 and the statement is proved taking into account Lemma 1.1. Using the techniques in [24] and [9] it is possible to prove that

HO•1/(6+6) r •• 1: V E loc , lor any positive !J.

Remark 3.3. Looking at the above example we can affirm that the minimum-time map of a system may be less regular than the one of its linearization. On the other hand v can obviously be more regular than the minimum-time map of the linearized system, in fact a system may be locally controllable and its linearization may be not.

The assumption

(H5)

276 GIANNA STEFANI

is a necessary and sufficient condition for liE H~~/r in particular cases. Namely the following results hold true.

Theorem 3.4.

a) II is locally Lipschitz iff L1(xo)=R" [27], [19],[21].

b) II e H~~/2 iff L2(xo)=R" [18].

Theorem 3.5. [17], [22] If either the system is symmetrc , i.e. fo=O, or the system is odd, i.e. fo is odd and /t,· .. ,fm are even, then (.85) is a necessary and sufficient condition for

JJ,l/r II e liloc •

Notice that the linear systems are odd systems.

In [12] a symmetric system such that L2(xo)=R" i~ c7f3idered and it is proved that any solution of (HJB) belongs to HI~~ . Using the geometric theory we can prove the same result for any affine system (see Theorem 3.4) and we can generalize the result for symmetric systems (see Theorem 3.5). As a matter of fact we can prove more : a solution does exist.

If r~3 , the condition (HS) is no longer sufficient to get Holder continuity with exponent 1/r. The reason is that between the brackets of length greather than 2 there are some obstructions to local controllability which can possibly reduce the size of the reachable set. In Example 3.2 the obstruction is the nonvanishing bracket

adl fo eL3 1

which gives rise t~ the term x2 on the 4th component and prevents II

to belong to H~~ 5 •

4. An useful technique

A crucial technique in studing local controllability is the theory of graded vector spaces and the possibility of choosing a graded structure in connection with the properties of the Lie Algebra L (see [25], [9] and the references therein). In this section we want to illustrate these techniques by two examples in order to suggest which kind of results one can expect. As examples we shall prove Theorem 3.1 and the following result:

Theorem 4.1. If the conditions (H4) and (HS) hold true, then

REGULARITY PROPERTIES

....o,l/r II E llioc •

277

Let us recall the basic results on graded approximations applied to the spaces L1's defined in Sec.2. For a more extensive exposition see [9]. Let

r=min{i: Lj(Xo)=R"} and mj=dimLj(xo), j=1,.··,r.

The first crucial result is the existence of a chart adapted to the "filtration" L1~···~LI~··· . Namely a chart at Xo with the properties a} and b} stated in the following Lemma 4.2.

Lemma 4.2. [25] [9] There exists a chart '11= ('11\ •.• ,'11") at Xo such that for each i = O, ••• ,r

a) Lj(xo) = span { ll(xO) , .•. , ~(xo) } k '11 8'11 J

b) ~l"'~S''1I (xo) = 0 , Vs'5.i , Vi1,···,isE{O,···,m}, V k > mj.

The adapted chart is not unique . Applying the algorithm described in [24] we can get an adapted chart by means of a polynomial change of coordinates whose inverse is also polynomial.

Let y~(t,xo,u) denote the ith component of the solution of (1::) in the adapted chart. It is known [13], [27] that

1 r m 1 r yx(t,xo,u)=xo+ E E fk ···fk.·y (xo)p(t,u,k1,·· .,kj)+ oCt }.

j=l k1 .. kj =O 1 :J

t tj t2

p(t,u,k1,·· .,kj}= J uk.(tj} J .. -J Uk/t1} dt1·· .dtj , O:J 0 0

where

and we have defined uo(t)=1. By the properties of the adapted chart stated in Lemma 4.2 we obtain for each iE{mj-1+1, ... ,mj}

1 r m 1 r yx(t,xo,u}=xo+ E. E fk ... fks·y (xo}p(t,u,k1, .. ·,ks)+ oCt }.

S=J k1.:k;=O 1

Therefore, since lu(t}I'5.1 for all t, then we h~ve p(t,u,k1,·· .,kj} =O(t'}

and hence

(3.1) y~(t,xo,u)-xo=O(~) ViE{mj-1+1, ... ,mj}'

By means of the adapted chart we give "a graded st~ucture" ~o R" (see [9]). Roughly speacking we giye the "weight" wl=j to yl if i belongs to {nj-1+1, ... ,nj} (i.e. if yl is a coordinate relative to the

278 GIANNA STEFANI

brackets of length j). As a consequence, for each multiindex a = (all ... ,an), the weight of the monomial ya == (y1)a1 ••• (yn)an is defined by : n .

CW'(ya) = E wla .• . I

Let tk denote the ith compo~en~ of the vector field fk in the adapted chart y. The minimum .weight of the monomials which appear in the asymptotic expansion of tk is wr1, [9]. Define

IPk =sum of the m<?nomials of weight equal to wr1 in the asymptotic expansion of tk

IPk = vector fi«:ld whose components in the adapted chart y are given by the IP~ 'so

The subsequent system (A) defined by the IPk's is called graded approximating system.

m (A) x = lPo(x) + E UjlPi(x) , x(O) =xo'

i=1 The approximating system has several properties. For example

(A) is locally controllable iff Xo is in~erior to the reachable set at time t=1. Moreover the ith component y~(t,xo,u) of the solution of (A) gives the principal part of the ith component of the solution of the original system with respect to time. Namely

y~(t,xo,u) =y~ (t,xo,u) +o(t Wi), see [25].

Other properties can be better pointed out through the dilations both on state and controls. The dilations on Rn are defined through the adapted chart by

yi 06£ = £ Wi yl.

Moreover given a control u defined on some interval [O,T] , we define its dilations 6£u on [O,£T] by

£ (t)= (t)={u(t/£) if £,c0 u£u _u£ - 0 if £=0 •

The solutions (in the adapted chart) y E and y A of the system and its approximation are related by

(3.2) 6iYE(ft,Xo,ud=YA(t,xO'u)+O(£),

see [9].

Proof of Theorem 3.1. Suppose that (H5) does not hold. If L(xo),cRn, then II is not even continuous. Otherwise the equality (3.1)


implies that if i~mr+l, then

ly~(t'Xo,u)-XoI~Ktr+l for some constant K, for all uE'U and for all t sufficiently small. Therefore Rl1(xo,t) cannot contain a ball of radius ptr. Lemma 1.1 completes the proof. 0

Proof of Theorem 4.1. By using the degree theory it is not difficult to show that from (3.2) the following result follows. IfxoEintRA(xo,l) ,then there is p such that

B(XO,pfr) ~R l1(XO,f) for sufficiently small f. On the other hand condition (H4) implies that the obstructions of system (A) vanish at Xo [9], therefore Xo is interior to each reachable set of the approximating system (A). Lemma 1.1 completes the proof. 0

Remark 4.2. In [16] an example is given in which an obstruction can be neutralized only with controls having an increasing number of switchings. For this example the graded space theory cannot help. However also in this case the minimum-time map is Holder continuous.

5. Final remarks and open questions

To my knowledge for each condition giving II continuous there is a technique which gives also II Holder continuous with some exponent. But is it true the following statement?

II is continuous iff it is Holder continuous .

The geometric theory provides many conditions for II to be Holder continuous. Which kind of meaning they have in terms of solutions of HJB equation? With respect to that, is the theory of graded vector spaces a crucial tool for studing the existence of Holder continuous solutions of HJB equations as it is in studing local controllability?

Which kind of role play the obstructions with respect to the HJB equation?

The sufficient (necessary) conditions provided by the geometric theory are stable under suitable perturbations. For example conditions (H4) and (HS) are stable if we perturb the vector fields with terms of order r+l, in fact (H4) and (HS) depend only on the values at Xo of the fj's and their derivatives up to order r. May we have similar results

280 GIANNA STEFANI

for the viscosity solutions of (HJB)? For example does the "Linearization Principle" hold for HJB equations? To be more precise set

afo A= ax (xo) and bj=fj(xo)·

It is well known that if m

i=Ax+Eujbj j=l

is locally controllable, then also the original system is. In view of Theorem 1.3, the above result can be stated as follows. If

(Dv(x) ,Ax)+j~ll(DV(X)' bj)I-1=0

has a continuous positive viscosity solution in a neighbourhood of Xo , then also

(Dv(x) ,fo(x) )+j~ll(DV(X) ,fj(x) )1-1=0

has a continuous positive viscosity solution in a neighbourhood of xo'

In some cases (for example linear systems) the minimum-time map v is analytic but in a "thin" set . Can the viscosity solutions theory give results in this direction?

To conclude it seems that at the present state of the knowledge control theory can give tools for better understanding the possibility of finding viscosity solutions, rather than viceversa.

5. References

[1] Bacciotti A, Aspetti topologici del problema del tempo minimo in Convegno internazionale su equazioni differenziali ordinarie ed equazioni funzionali" R. Conti, O. Sestini, O. Villari ed.s (1978), 423-432.

(2} Bardi M., A boundary value problem for the minimum-time function, SIAM J. Control and Optimization 27 (1989), 776-785.

[3] Bardi M. Falcone M., An approximation scheme for the minimumtime function, to appear in SIAM J. Control and Optimization.

[4] Bardi M. Falcone M., Discrete approximation of the minimum-time function for systems with regular optimal trajectories, to appear in Proceedings of 9th Int. Conference in Analysis and optimization of systems, Antibes 1990.

[5] Bardi M. Sartori C., Approximation and regular perturbations of


optimal control problems via Hamilton-Jacobi theory, preprint. [6] Bardi M. Soravia P., Hamilton-Jacobi equations with singular

boundary conditions on a free boundary and applications to differential games to appear in Trans. of AMS.

[7] Bianchini R.M. & Stefani G. - "Normal local controllability of order one" Int.J. Control, 39 (1984), 701-714 .

[8] Bianchini R.M., Stefani G., Sufficient conditions of local controllability in Proceedings of the 25th IEEE Conference on Decision and Control, Athens (1985), 967 -970.

[9] Bianchini R.M., Stefani G., Graded structures and local controllability along a reference trajectory, to appear in SIAM J. Control and Optimization 28 (1990).

[10] Boltyanskii V.G., Mathematical methods of optimal control, Balskrishnan-Neustadt Series, Holt Rinehart and Winston, New York 1971.

[11] Crandall M., Lions P.L. Viscosity solutions of the HamiltonJacobi equation, Trans. AMS, 227 (1983), 1-42.

[12] Evans L.C., James M.R., The Hamilton-Jacobi-Bellman equation for time-optimal control, SIAM J. Control and Optimization 27 (1989), 1477 -1479.

[13] Fliess M., Functonelles causales non lineairs et indeterminees non commutatives, Bull. Soc. Math. France, 109 (1981), 3-40.

[14] Hermes H., Feedback synthesis and positive local solutions to the Hamilton-Jacobi-Bellman equations, in Analysis and optimization of controls of nonlinear systems, North Holland 1988, 155-164.

[15] Kawski M., A new necessary condition for local controllability in Differential Geometry: The interface between pure and applied mathematics, M. Luksic, C.F. Martin, W. Shadwick eds., AMS Contem. Math. series 68 (1987), 143-156.

[16] Kawski M., Control variations with an increasing number of switchings, Bull. AMS 18 (1988), 149-152.

[17] Liverovskii A.A., Some properties of Bellman's function for linear and symmetric polysistems, (in russian) Differential'nye Uravnenija, 16 (1980), 413-423.

[18] Liverovskii A.A., Holder conditons for Bellman's function, (in russian) Differential'nye Uravnenija, 13 (1977), 413-423.

[19] Petrov N.N., Local controllability of autonomous systems (in russian), Differential'nye Uravnenija 4 (1968), 1218-1232.

[20] Petrov N.N., The continuity of Bellman's generalized function (in russian) Differencial'nye Uravnenija 6 (1970), 373-374.

[21] Petrov N.N., On the Bellman's function for the time-optimal

282 GIANNA STEFANI

process problem, (in russian) PPM 34 (1970), 820-826. [22] Stefani G., Local controllability of order p, in Workshop on

differential equations and control theory, Bucarest 1983, 171-182.

[23] Stefani G., Local properties of nonlinear control systems, in Proceedings of International school on applications of geometric methods to nonlinear systems, Bierotuvize 1984, 219-226.

[24] Stefani G., On the local controllability of a scalar input control system in Theory and applications of nonlinear control systems, Byrnes and Lindquist eds, North Holland 1986, 167-179.

[25] Stefani G., Polynomial approximations to control systems and local controllability, in Proc. 25th IEEE Conference on Decision and Control, Ft. Lauderdale (1985), 33-38.

[26] Stefani G., A sufficient condition for extremality, in Analysis and optimization of systems Lect. Notes in Control and Informations Sciences 111, Spriger Verlag New York 1988, 270-281.

[27] Sussmann H.J., A sufficient condition for local controllability, SIAM J. Control and Optimization 16 (1978), 790- 802.

[28] Sussmann H.J., A general theorem on local controllability, SIAM J. Control and Optimization 25 (1987), 158-194.

Gianna Stefani Dipartimento di Matematica. e Applicazioni Via Mezzocannone, 8 80134 Napoli - Italia [email protected]

Optimal Synthesis Containing Chattering Arcs and Singular Arcs of the Second Order

M.1. Zelikin and V.F. Borosov

In most optimal control problems, in which the optimal controls can be explicitly constructed, they are piecewise analytic with a finite number of discontinuity points (called switching points). Meanwhile there is an old example, proposed by A.T. Fuller, in which the optimal controls have an in~ finite number of switchings on the finite~time interval. Such a phenomenon has been called "chattering". Being published in 1961 [1], Fuller's example aroused some interest but soon was forgotten. About twelve years later there was risen the second wave of interest in this phenomenon. Several examples with the optimal chattering trajectories have been found [2-7]. In the recent years we observe the third wave of interest due to the intensive attempts to understand a real content of the notion of regular synthesis, and especially due to the remarkable work of J. Cupka [8], who proved that in the case of general position for some class of discontinuous Hamiltonian systems in the dimensions more or equal than 12 there exists at least one chattering trajectory.

The chattering is closely related with the generalized Legendre-KlebschKelley necessary condition of optimality of the singular arcs and with the notion of their order. Namely, the optimal nonsingular arc can conjugate the singular one of even order only with infinitely many switchings [9].

The following analysis shows the behavior of all the chattering arcs and the structure of the Lagrangian manifolds, which contain these arcs for the n-dimensional discontinuous Hamiltonian systems in the neighborhood of the manifold of the singular trajectories of second order. We denote this singular manifold by S. In general, the codimension of S equals 4.

It is proved that for each point w E S there exist two mutually tangent 2-dimensional manifolds M! and M;;; which consist of the chattering arcs. The trajectories of M! tend to w in the finite time with the infinite number of switchings; likewise but in the retrogressive time the trajectories of M;;; tend to w. Union of these manifolds M+ ::: UwMt (M;;; ::: UwM;;;) constitute the stable (unstable) manifold for S. It is worth noting that the set of all switching points of the stable (or unstable) manifold appears to be a smooth manifold except for the points of S itself.

Thus M+ and M- are two bundles with base Sand 2-dimensional

284 ZELIKIN AND BOROSOV

fibers. The asymptotical behavior of the trajectories in each fiber is the same as in Fuller's problem. These bundles define the structure of all the trajectories of the Hamiltonian system in question in the neighborhood of the singular manifold S. The Lagrangian manifolds containing the chattering arcs can be described as the inverse image of the projections (defined by these bundles) of a Lagrangian submanifold of S. As soon as Hamiltonian systems under consideration correspond to the affine in the control optimal problems we may construct the chattering-synthesis for a wide class of optimal problems. The switching points of such a synthesis form the piecewise smooth manifold and the chattering is conditioned by the fact that each concrete optimal trajectory intersects this manifold infinitely many times.

One has to remark that the usual procedure of the constructing of the optimal synthesis for these problems by the backward integrating of the Hamiltonian system of Pontriagins's maximum principle is unapplicable due to the lack of a nonzero-length interval of continuity of the control on a chattering arc at the point of the joining with the singular manifold S.

Using the developed method, the optimal synthesis is constructed for a number of examples applied significance. In particular, the optimal chattering-synthesis for the time-optimal control problem of the nonlionear two-links manipulator (robot), for Lawden's problem of the control of the rocket's moving in the Newtonian field of gravity, for the problem of the optimal resource allocation in mathematical economics and for some other problems is found.

I. We consider:

Problem 1.

(1) mInImIZe F = iT [/o(x) + u/I(x)] dt

on the solutions of the system of equations

(2) x = soo(x) + US01(X).

This problem is affine in the scalar control u E [-1,1]. Here x E nnj the functions Ii : nn -+ n, SOi : nn -+ n (i = 0,1) are assumed to be sufficiently smooth. The problem is defined by the choice of the initial conditions x(o) = Xo and the smooth terminal manifold x(T) E M, M C nn. The admissible controls are measurable functions, the admissible trajectories are absolutely continuous functions.

We will say that x(t) is the chattering are, if the corresponding control u(t) has infinitely many switchings on the finite time interval of this arc.

The Pontriagin maximum principle [10] for Problem 1 gives

(3) iJ = I· grad (Ho(y) + UH1(Y)) j u = sgn H1(y).

OPTIMAL SYNTHESIS CONTAINING CHA TIERING ARCS 285

Here y = (1/;, x) E n2n ; 1= (~ -OE), where E is the unit (n x n)-matrix;

Hi(Y) = tP'Pi(X) - >'Ofi(X) (i = 0,1). By the solution of the discontinuous systems (3) will be meant the solutions in the sense of A.F. Filippov [11]. We call the trajectory y(t) of the system (3) the singular one on the interval (to,tt), if Hl(y(t» = 0 for t E (to,tI). The number q will be called the intrinsic order of the singular trajectory y(t) if

(4) k = 0,1, ... ,2q = 1,

for all y from some open neighborhood of the trajectory y(t). If the relations (4) are valid only at points of the trajectory y(t) itself, then q is called the local order of the singular trajectory. It could be shown that q is integer [9]. Kelley's necessary condition for y(t) being optimal is

(5) o d2q I (-1)q ---2 Hl(y) :S O. ou dt q (3)

If the singular optimal trajectory has the second (or even any) intrinsic order and the strengthened Kelley condition is fulfilled, then it cannot join the nonsingular piecewise smooth trajectory with a jump of the control at the point of junction [9]. Consequently, if the junction occurs, the nonsingular arc must contain infinitely many points of switching.

Let us denote by

the Poisson-bracket of the functions Ho and H l . Suppose that the system (3) has the singular arc y(t) of second intrinsic order, on which the functions Zi = ad~ol Hl (i = 1,2,3,4) are functionally independent. It means that

the Jacobi matrix II%;" has a full rank. So we can find the functions

w = (Wl' ... , W2n-4) such that det IID(z, w)j D(y)lIly(t) =1= O. Changing the variables y --+ (z, w), the system (3) takes the form:

(6) Zl = Z2, Z2 = Z3, Z3 = Z4, Z4 = a(z, w) + u(3(z, w), tV = i(Z, w, u), u = sgnzi.

From the definition and the assumption that y( t) is of the second order, we have (3ly(t) =1= O. Suppose that the strengthened Kelley condition on y(t)

is satisfied, i.e. f3(0, w) < O. To allow the movement along the singular trajectories with lui ~ 1 let us suppose that la(O, w)1 < -f3(0, w).

Actually we deal with much more general systems than that reducing to (6). We want to include in consideration Hamiltonian systems with the singular trajectories of the local second order. With that end in view, we disturb the right-hand sides of the equations (6).

II. Consider the system of differential equations with the discontinuous right-hand sides

Zl = Z2 + h(z,w,u), Z2 = Z3 + f2(z,w,u), (7) Z3 = Z4 + 13(z, w, u), Z4 = a(w) + uf3(w) + 14(z, w, u),

w=g(z,w,u), U=sgnz1.

Here, z = (Zl,Z2,Z3,Z4) E n4, w = (W1, ... ,Wm) E nm, and li(z,w,u), (i = 1,2,3,4) have "the higher order" than the previous terms of the righthand sides of (7). For the exact definition of the order, see assumption 1 below. The dependence of the functions a and f3 on z in the system (6) is included here in ther term 14(z, w, u).

Regarding the principle parts of the first four equations of the system (7), we come to the consideration of the following nonsymmetrical Fuller problem.

ProbleIll 2.

subject to the restrictions

(8) :i; = y, y = u E [b, 1] (b < 0), x(O) = Xo, y(O) = Yo, x(T) = Xl, y(T) = Yi.

This problem was investigated by C. Marshal [2]. We remind his results in our terms and make some additions.

Problem 2 admits the group of symmetry, which is homeomorphic to the multiplicative group of the positive numbers:

(9) u(t) -+ u(t/K.), y(t) -+ K.y(t/K.), x(t) -+ K. 2 x(t/K.), 1/J~(t) -+ K.31/J~(t/K.), 1/Jy(t) -+ K.4 1/Jy(t/K.).

Hence, 1/J~ and 1/Jy are the conjugate variables to X and y. Problem 2 has the singular solution X = 0, y = 0, u = 0 of second order. 1fT is sufficiently large, the optimal trajectory goes to the origin then does stay at that point and finally goes to the terminal point. The first and the last stages of this process, i.e. junction with the singular are, can be described as the

synthesis on the (:z:, y)-plane. The first synthesis is defined by the switching curve

(10) { :z: = Ao(b)y2 for y < 0 (AO E (0, ~)), :Z:=Al(b)y2 fory>O (AIE(ib'O)).

'a-

v.= ,

Fig. 1

The second synthesis is defined by the switching curve

(11) { :z: = -Al(b)y2 for y < 0,

:z: = -Ao(b)y2 for y> O.

'd

~~------t-----.-------H-+--4-_

Fig. 2

It can be shown that the functions ~i(b) are smooth. Now we return to the system (7). In accordance with (9) we define the

action of the group g" on'R4

g,,(z) = (1I:4Z1' 1I:3Z2, 1I:2Z3, II:Z4) •

Assumption 1. The functions fi(z,w,u) (i = 1,2,3,4) have the higher order than the corresponding right-hand sides of (6) in the sense of this generalized homogeneity:

(12) -I' fi(g"z, w, u) 1m . < 00 ,,_+0 11:5- 1

It follows that fiCO, w, u) = 0, (i = 1,2,3,4) and thus there is the family of the singular trajectories of the system (7), which fill the plane S = {(z, w)lz = O}. The controls on these trajectories are u = -o:(w)/{J(w). Each trajectory of this family has the second local order. It will be considered the €-neighborhood: Izl < € of S, where € > ° is sufficiently small.

Theorem 1 Consider the region 0" on S, where the inequalities (J(w) < 0, 10:( w) I < (J( w), and Assumption 1 are valid. Then for each point w E 0"

there exist two 2-dimensional piecewise smooth manifolds M;t and M;;; consisting of the chattering solutions of the system (7) such that each solution of M;t in the finite time tends to w making a countable switching and each solution of M;;; likewise tends to w, only the time gets reversed. The switching curve /';t on Mj; (and /';;; on M;;;) is smooth except for the point w. The (m + I)-dimensional switching submanifold r+ = Uweu/';t (and r- = Uweu/';;;) is smooth except for the points of S. The (m + 2)dimensional stable manifold M+ = UweuMj; (and the unstable manifold M- = UweuM;;;) is smooth except for the points of rand S.

The full proof of this theorem is somewhat lengthy due to the rather cumbersome but necessary computations. It will be published in "Proceedings of Steklov Mathematical Institute". Here we try to explain the most essential items of this proof.

We consider the Poincare mapping CJ.) : So -+ So of the surface So = {(z, w)lzl = O} on itself. By definition, this mapping transfers the point (Zo, Wo) E So into the point (ZI' WI) E So of the first intersection of the trajectory, starting at (zo, wo) with So. To find the chattering arcs which join S, we seek for the invariant sets relative to CJ.).

Let us consider the point (z = 0, w = 0"). Taking into account the group g" of symmetry of the principle part of (7) we make a generalize blowing-up procedure to resolve the singularity of CJ.) at this point.

(13) Z4 = 11:, Z3 = ~1I:2, Z2 = /111:3 , W = 0" + VII:.

Here instead of the point (z = 0, w = 0') it is attached (m + 2)-dimensional plane I), = 0; in the region Z4 =F 0, the mapping ~ is one to one; and the points {Z4 = 0, Zj =F ° for some j =F 4} do not have the inverse images.

Let (zo,wo) E So; (Zl,Wt) = ~(zo,wo); T(zo,wo) be the moment of the first intersection of the trajectory, starting at (zo, wo) with So. To rewrite the mapping ~ in the coordinate {1)"A,I',v} we denote (1),1, A1,1'1,Vt) = ~(I),O, Ao, J.lo, va) and To = T(zo, WO)I),OTO. We have

1),1 1),0 (1 + (0'(0') + u~(O')) TO) + 1C00(1C0),

A1l),f = 1C5 (AO + TO + ~T6 (0'(0') + u~(O'))) + 1C50(1C0) ,

J.lll),~ K~ (1'0 + AoTo + ~T6 + ~Tg (0'(0') + u~(O'))) + K~O(ICO),

Vll),l KO (va + Tg(O, 0', u)) + 1),00(1),0),

where the function TO = TO(ICO, Ao, J.lo, va) has to be found from the equation

Let (K2,A2, J.l2, V2) = ~(1),1,A1,J.ll,Vt); T1 = T(Zl,W1) = K1Tl. We take the initial values (KO, Ao, J.lo, vo) such that u = -1 for t E (0, To) and u = +1 for t E (To, To + T1).Then

(14)

I),i+l = l),i(1 + AiTi) + KiO(lCi),

Ai+l = (Ai + Ti + ~AiTn(1 + Am)-2 + 0(1),,),

J.li+l = (J.l + Ai T, + ~Tl + ~AiT.s)(1 + A;Ti)-3 + O(Ki),

Vi+l = (v + Tig(O, 0', ui))(1 + A;Ti)-1 + 0(1),;).

The value of Ti is found from the equation

(15)

Here i = 0,1; Uo = -1, U1 = 1; Ao = 0'(0') - ~(O') > 0, Al = 0'(0') + ~(O') < 0. Without loss of generality we can put Ao = 1, A1 = b < 0, because (14), (15) can be reduced to this case by the change

I A-2 J.l = J.l a ,

It is convenient to consider the mapping ~2, which is defined by iteration of the formulae (14), (15) for all the initial values of (1C0, AO, 1'0, vo). ~2 coincides with ~2 in the region

(16) KOTO > 0, K1 Tl > 0, J.loK~ < 0, 1'11C~ > 0.

The fixed points of ~2 are found from the system

/C = 0,

A = (A + 1"0 + !1"~)(1 + 1"0)-2 + 1"1 + !b1"l) (1 + b1"1)-2,

(17) 1'::; (I' + A1"o + i1"~ + 11"3)(1 + 1"0)-3+ + 1"1 (A + 1"0 + i1"~)(1 + 1"0)-2 + i1"l + 1b1"l) . (1 + b1"d-3 ,

v ::; (v + 1"Og(O, 0", -1»(1 + 1"0)-1 + 1"lg(O, 0",1» (1 + b1"l)-l.

The functions 1"0 ::; 1"O(A, 1'), 1"1 ::; 1"1 (A, 1') are found from the equations

I' + !A01"O + 11"~ + i41"3 ::; 0,

(18) (I' + A1"O + i1"~ + 11"3)(1 + 1"0)-3+ + i1"l(AO + 1"0 + 11"~)(1 + 1"0)-2 + 11"1 + i4b1"f::; O.

We show that the mapping ~2 have in the plane /C ::; 0 exactly two fixed points which coincide with the fixed points of the analogous mapping for the nonsymmetric Fuller problem (8). Since we know these points we are able to calculate the linear part of ~2 •

For the first point M+::; (/C::; O,A+,J.I.+,V+) this linear part is equal to

go 0 0 0 8>'2 8>'2 0 * {)AO {)J.I.0

-2 {)J.I.2 {)J.I.2 D4iM+ ::;

* 0 {)AO {)J.I.0

1 * 0 0 -Em

go

Here go = (1 + 1"0)(1 + b1"l) E (0,1); Em is the unit (m x m)-matrix and stars stand for functions of no importance. It is shown that the eigenvalues of the matrix

( :~: :~:) {)J.I.2 {)J.I.2

{)AO {)J.I.0 (.\+ ,p+)

are more than 1 in the absolute values. By the theorem about the invariant manifold of the diffeomorphism [12]

there exists C1-curve, which is invariant relative to the mapping ~2 and tangents to the eigenvector of D with the eigenvalue go. Since at the point M+ we have 1"+ < 0, 1'+ > 0 the branch of the invariant curve, which lies in the halfspace /C < 0, satisfy the conditions (16). We let 11 ::; 4110,

r = ro u r1. The mapping ~2 smoothly depends on (1 and hence the curve r has the form

(19)

where i = 0 for I\: < 0 and i = 1 for I\: > O. It is shown that the functions Ai, J'i, Vi are smooth in (1. Let us denote by r; the inverse image of r( (1) by the mapping (13), and r+ = Ut7ESr;.

Taking into account that ~ 1,,=0 = Em we can describe the manifold r by the formulae Z2 = M(z4' w)z:, Z3 = A(z4' w)z~, where the functions M,A are smooth for Z4 > 0 and Z4 < 0 and they could be smoothly continuated to Z4 = O.

Let m; be the family of the trajectories with the initial data belonging to r;.

Likewise for the second fix point M- = (I\: = 0,A-1,J'-1) we find the unstable invariant curve r- «(1) the switching surface r- and the family m; of the trajectories which tend to the point (0, (1) in the retrogressive time.

III. Let us have the system (3) which can be reduced by the change (1/J,:r:) -+ (z, w) to the system (7), m = 2n - 4. By Theorem 1 the stable manifold E+ (and the unstable one E-) is a bundle with the projection p+ : E+ -+ S (p- : E- -+ S) and 2-dimensional fibers m~ (m;;;).

The manifold M C 'R,2n n {H = O} is called Lagrangian (in a broad sense) iffor any piecewise-smooth closed curve reM we have f.., 1/Jd:r: = O. (In this definition we omit the demand on the dimension of M.)

We will say that Problem 1 is regular if Pontriagin's maximum principle is valid with Ao = 1.

Theorem 2 Suppose that Problem 1 is regular. Let M C S be Lagrangian manifold. Then (p+)-l M is Lagrangian manifold.

Proof. Given an arbitrary piecewise smooth closed curve r c (p+)-l(M) n r) \ S we define on r the coordinate (I\:, (1) by substitution of (19) in (13). Then if ~2(1\:, (1) = (1\:1, (11) then

(20)

It is easy to see that (p+)-l M is piecewise-smooth. We have J.., 1/Jd:r: = 1. _/)~ 1/Jd:r: by the theorem about the integral invariant of PoincareC~an'" and the condition A = 1.

Let rn = ~rn-1' n = 2,3, .... In view of (20) rn -+ p+r for n -+ 00 in the C1-norm.

It follows that J.., 1/Jd:r: = J.., .. 1/Jd:r: = 1;,.., 1/Jd:r: = O. Let C be the shift of r in (p+)-l M along the trajectories of (7).


Once again using the Poincare-Cart an invariant integral we became that Ie 'l/Jdx = o. As soon as any C C (p+)-l M may be constructed in such a way, the theorem is proved. Q.E.D.

With the help of the preceeding construction the wide class of the locally optimal chattering syntheses for the various boundary conditions in Problem 1 could be built. The trajectory x(.) is called locally optimal in Problem 1 if there exists a neighborhood U of x(.) such that J(x(.» ~ J(x(·)) for any admissible trajectory x(·) which belongs to U. For instance, one can build the locally optimal chattering synthesis in the following manner.

Let Nl C S be a manifold such that

1) dimNl =n-3.

2) The vector of the singular velocity on S is not tangential to Nl : (0, «(3 - a)g(O, u, 1) + «(3 + a)g(O, u, -1)) /2(3) rJ.1(o,q)Nl .

3) The restriction of the differential form L~=l 'l/Jidxi to Nl is equal to zero: L?=l'I/JidxiIN1 = O.

Let us consider all the singular trajectories with the initial values at Nl on the sufficiently small interval of time (-6,6). The manifold which consist from these trajectories will be denoted by N 2 • Let 7r be the projection: 7r( 'I/J, x) = x. By m* we denote the set of trajectories from m+ which lie on E* = (p+)-lm2, r* = E* n r.

Theorem 3 Suppose that Problem 1 is regular. Let the corresponding Hamiltonian system (2) be reduced to (5)-(6). Let the target in Problem 1 M = 7r Nl be the smooth, connected manifold.

Suppose that the restriction 7rlno\p is regular mapping, which could

be regularily continued on r* \ N2 and 7rlr*\N2 is regular, which could be regularily continuated on N 2 •

Then in the small neighborhood of M the projections of trajectories m* on the space (x) form the locally optimal synthesis in Problem 1.

Proof. It follows from the definition of E* that (p+)-l (N2) is the piecewise smooth manifold. Under the conditions of the theorem the projections of the trajectories of the family m* do not intersect each other and cover an open neighborhood U of the manifold M. It follow's from Theorem 2 that (p+)-l N2 is the Lagrangian manifold.

Let x(·) be any admissible trajectory in Problem 1 with the initial conditions Xo E U and x* (-) be the trajectory of the family m* with the same initial conditions. Let T* be the time of hitching Nl along x* (-), T be such


a time along x(-). Using Pontriagin's maximum principle with Ao = 1, we have

T" T

[ f(x*(t»dt = 1 t/Jdx = 1 t/Jdx:$ [ f(x(t»dt, 10 zoo z(·) 10 where f(x) = fo(x) + u/t(x). Thus x*(·) is locally optimal. Q.E.D.

Remark. Theorems 2 and 3 allow us to construct the optimal chattering syntheses for the more general boundary conditions, if we use the trajectories of the family m- also. It is easy to find synthesis in which point move at first along the arc of the family m+, then along the singular arc and finally along the arc of the family m - .

IV. As an example of the use of the above-stated theory, we give the solution of the famous Lawden problem of the control the rocket in the central Newtonian field of gravity.

Problem 3. To minimize the time r - inf of the hitching the manifold M in the phase space of the system

{X = U, if = ~ cosO - J'X(X2 + y2)-3/2,

(21) Y = V, V = ~sinO - J'X(X2 + y2)-3/2,

m=-~,

X(O) = X o, U(O) = Uo, Y(O) = Yo, V(O) = Vo, m(O) = mo. Here the controls are the rocket thrust U E [0,1] and its angle 8; J.t and

c are some positive constants (see, for instance, [3]). We put

r (X2 + y2)1/2,

X (XU + YV)(X2 + y2)-1/2,

Y (XV + YU)(X2 + y2)-1/2,

tp 0 - arccos [X(X2 + y 2)-1/2] .

Then Problem 3 is reduced to the time optimal problem r - inf

{ r=x :i:=..!!..COSlll-..I!..+r.: , m T r3 r '

Y· = ..!!.. sin III - !1l. m = -!! m T r' c

(22)

u E [0,1]' tp E [0,211"], r(O) = ro, x(O) = xo, y(O) = Yo, m(O) = mo, (r(r), x(r), y(r), m(r» E M. Pontriagin's maximum principle for this prob-

lem has the form

(23)

· (2/J y2) zy tPr = - r3 + r2 tPz - r2 tP'I/ I · y tPz = -tPr + -tP'I/ I r · 2y z

tP'I/ = --tPz + -tP'l/I r r

tbm = - U (cos <P • tPz + sin <p • tP'I/) . m

The controls U, <p are to be found from the condition of maximum

(24) max u [tPz cos <p + tP'I/ sin <p - tPm] = U [tPz cos <p + tP'I/ sin <p _ tPm] . U,rp C C

We shall consider only nonsingular relative to <p solutions of (22)-(24): tP~ + tP~ =F O. One has

cos <p = tPz (tP~ + tP:) - i I sin <p = tP'I/ (tP~ + tP:) -1/2 .

First of all we find the singular (relative to u) solutions of (22)-(24):

H = Ho+uH1

where

Ho = ztPr + (_l!... + y2) tPz _ zy tP'l/I r2 r r

HI = ...!:.. (tP2 + tP2)1/2 - !tPm. m z '1/ C

Let us denote ZI = HI. By differentiating ZI one finds successively

il = Z2 / m 2 (ZI + ~tPm) I Z2 = -tPrtPz - ;tPztP'I/ + ;tP:;

_ .1.2 .1.2 (2/J y2) zy (z2 /J) 2. Z2=Z3, Z3-o/r+o/z 3'+2' -22tPZtP'I/+ 2'-3' tP'l/I r r r r r

i3 = Z4, Z4 = /Jr-4 (-8rtPrtPz - 6ztP! + lOytPztP'I/ - ztP;) ;

i4 = -4zr-1z4 + /Jr-4(4ztPrtPz -18ytPztP'I/ + 8rtP~)+ + (22/Jr- 2 - 34y2r-l )tP! + (/Jr- 2 + 9y2r-l - 2z2r-l )tP;+

3/JtPz(3tP~ - 2tP~) + u mr4( tP~ + tP:)l/2

We see that the singular trajectories have the second local order. The manifold S of the singular trajectories is defined by the equations ZI =

Z2 = Z3 = Z4 = 0 in 8-dimensional space (1/;r, 1/;:c, 1/;y , 1/;m, r, x, y, m). At points of S we find after some work

Y _- ±(21/;; + 31/;~h/1/;; - 21/;; . ~P 1/;YV1/;; - 21/;; {fa 1/;r = ± . -,

(31/;; - 21/;;)V1/;; + 1/;; r' V1/;; + 1/;; r3

u = _ pm1/;:c (1201/;~ + 1081/;:1/;~ + 661/;;1/;; - 271/;:)

r2 V1/;; + 1/;; (31/;; - 21/;;)3

The necessary Kelley condition isolate the region 1/;:c < 0, 1/;~ - 21/;; > O. The Jacobian of the change (1/;m'1/;r'x'y) .... (Zl,Z2,Z3,Z4) is equal to

Thus the manifold S of the singular arcs is four-dimensional and has a regular projection on the three-dimensional sub-manifold of the space (r, x, y, m). The choice of any two-dimensional Lagrangian sub-manifold Q C S define local optimal chattering synthesis of Lawden's problem.

References

[1] Fuller, A.T., Relay Control Systems Optimized for Various Performance Criteria, Proceedings of the IFAC Congress, Moscow, Vol. 1, Butterworth Publishers, London, 1961.

[2] Marchal, C., Chattering Arcs and Chattering Controls, Journal of Optimization Theory and Applications, Vol. 11, No.5, 1973.

[3] Marchal, C., Second-Order Test in Optimization Theories, Journal of Optimization Theory and Applications, Vol. 15, No.6, 1975.

[4] Dorling, C.M. and E.P. Ryan, Minimization of Non-quadratic Cost Functional for Third Order Saturating Systems, International Journal of Control, Vol. 34, No.2, 1981.

[5] Brunovsky, P. and J. Mallet-Paret, Switchings of Optimal Controls and the Equation y(4) + lylOSigny = 0, 0 < ex < 1, Casopis pro Pestovani Matematiky, roc. 110 (1985), Praha.


[6] Telesnin, V.R. Ob Optimizatii Perechodnych iscagenij, Trudy MlAN, Vol. 166 (1) (in Russian).

[7] Borschchevski, M.Z., and LV. Ioslovitch, K Zadache Optimalnogo po Bystrodejstviju Tormogenija Vrashchenija Osesymmetrichnogo Tverdogo Tela, Prikladnaja Mathematica i Mechanica, Vol. 49 (1), 1975 (in Russian).

[8] Kupka, I., Geometric Theory of Extremals. Fuller's Phenomenon. Proceedings of the 24-th Conference on Decision and Control, Ft. Lauderdale, Fl., v.2, 1985.

[9] Kelley, H.J., R.E. Kopp, and M.G. Moyer, Singular extremals, in Topics in Optimization (ed. Leitmann), New York, 1967.

[10] Pontriagin, L.S., V.G. Boltjanskii, R.V. Gamkrelidze, and E.F. Mischenko, The Mathematical Theory of Optimal Processes, John Wiley and Sons (Interscience Publishers), New York, 1962.

[11] Filippov, A.F., Differentialnye Uravnenija s Razryvnoj Pravoj Chastiju, Nauka, Moscow, 1985 (in Russian).

[12] Hartman, P., Ordinary Differential Equations, John Wiley and Sons, New York-London-Sidney, 1964.

M.1. Zelikin and V.F. Borosov Moscow University USSR

The Invariants of Optimal Synthesis

L.F. Zelikina

In this article we deal with the term "invariant of optimal synthesis." By this term we mean geometrical rather than algebraic invariant. To explain, let us consider the simplest case

(1) { % = ulF(x, y),

iJ = u2F(x, y),

Ul + U2 = 1, Uj ~ 0

T-inf,

x(O) = Xo, yeO = Yo,

(x(T), y(T» E M,

(i = 1,2); x> 0, y> O.

Here, F(x,y) > 0, ~;(x,y) > 0, ~~(x,y) > 0, M is the smooth manifold,

M E 'R~. For various types of targets M we have the following pictures of optimal

synthesis.

1. M= {(x,Y)E'R~ Ix=X>O}

7 -... ... ~

b1' fr. .. d ..

Fig. 1

298 L.F. ZELIKINA

2. M= {(z,v)e'R.~ Iv=Y>O}

, ,.v ~"'-. ....... .. .. ... ,

\

" r: a

Fig. 2

3. M = {(z,y) e 'R.~ I F(z,y) = F> O}

a

Fig. 3

One can see that the syntheses into the marked rectangles coincide. Is this an accidental coincidence or a display oC some general law? We assert that the latter is true. Namely, when building the synthesis in the retrogressive time from the target and using the bang-bang trajectories, vacuous places remain. We show that they have to be filled with a synthesis, which is universal Cor all the targets and depend only on the controlled system and Cunctional. This synthesis we call the universal structure. It is an invariant.

(2)

THE INVARIANTS OF OPTIMAL SYNTHESIS

I. Given the system

z = f(t,%,u) %E'R.nj uEU,

let us fix some synthesis u = u(%).

299

Definition 1. The set WE 'R.n+1 is said to be universal if, for any (to, %0) E W, thereexists To > to such that for any T E (to, To) there exists a neighborhood UtO ,II0 of the point (to, %0) such that for any (t, %) E UtO ,II0 we have 1to,1I0(T) E W. (See [1].)

Here, 1to,1I0(t) is the trajectory of the synthesis u(%) with the initial condition (to, %0).

This definition formalizes the following properties of the universal set

a) The universal set is the attractive setj

b) The universal set consists of the trajectories;

c) For any (tm,%m) -+ (to, %0) E W the time of hitching W along the 1t m ,II m O tends to zero.

Given the universal set W for the synthesis u = u(%) we denote by \tv the domain of attraction for W.

Definition 2. The triple { w, IV, h} } is called the universal structure for

the synthesis u(%) if for any (to, %0) E IV the corresponding trajectory 1to,~oO hits the universal set W.

II. Consider the ooptimal control problem for the system

(3) Zi = uiF(%), (i = 1, ... ,n),

where % E 'R.i., F : 'R.i. -+ 'R.~, F E C3('R.i.), :~ > 0, u E U {u I L:?=i Ui = l,ui ~ OJ. (4) R(T, %(T» -+ inf,

where T is the time at which the phase point % reaches M for the first time. Here R : 'R.++i -+ 'R.i, R E C i('R.++1), M is a smooth manifold (M is a target).

Let us denote

300 L.F. ZELIKINA

{ n 18F 8F } Di = Z E'R.+ -8 > m~ -8 ' , Zi l;ta ZI

8:it (::1 -::J 8:i. (::1 -::J A· .-'I"",'. - 8:i. (8!~_1 -::.)

1

and let 6.t ... ,i. be the algebraic adjunct to the s-th element of the last row of the determinant.

Theorem 1 Description of the universal structure for Pontrjagin's maximum principle of the system (3)-(4). (See [2]).

Let the function F(z) be such that

i) the set Vt, ... ,n = { Z E 'R.~ I ::. = ... = :::.} is not empty in n~;

ii) for any {ii, ... ,ik} E {1,n},(k = 2, ... ,n) and s E {i1, ... ,id the condition sign6.it, ... ,ilo = (_1)I:-t holds at the points of the set Vi., ... ,ik'

Then the sets Vi., ... ,i. are (n - k + I)-dimensional universal manifolds. The set W in Definition 2 has the following form: the closure of (n - J)dimensional manifolds Vi 1 i 2 - there are c~ of them - intersect on the onedimensional manifold Vt, ... ,n, subdividing the space 'R.~ into domains Di

(there are n of them); W = 'R.~; the universal structure is defined by the synthesis

"j(O) = { 0, j ::f: i,

if z E Di 1, j = i,

"j(o) = { 0 jE{il, ... ,i£:}

A~ . if z E Vi . '11"0'"" j E {i1, ... ,i£:},

'1, ···,'Ir A . ,

'1,···t·'

Definition 3. The trajectory .y(t) is said to be regular if for the plane [( passing through the end point P of .y(t), P E .y(t) n M which is parallel to any coordinate plane if the point P is a nonsingular point of M n J( and grad(R(t, Z)IMnK)(p) ::f: O.

The universal structure of optimal synthesis of the optimal control problem (3) is invariant to functionals from the class (4), i.e. the following result holds.

THE INVARIANTS OF OPTIMAL SYNTHESIS 301

Theorem 2 For any functional of the form (4) if some arc of the optimal regular trajectory .y(t) moves on the universal manifold Vl ..... " then the initial part of the optimal trajectory .y(t) is the path of the universal structure and it attains the universal manifold Vl ..... " in the shortest time.

Theorem 2 gives the method for constructing the synthesis of optimal trajectories for systems of high dimension. In order to do this for any functional ofform (4) at the beginning the universal structure will be found. Then one has to construct switching surfaces by integration of the equativl1s of Pontrjagin's maximum principle of the problem (3)-(4) from the target M. Such trajectories are the bang-bang ones. There is the region in 'R.+ which does not contain bang-bang trajectories. This region will be filled by the universal structure.

To give a visual demonstration we consider the problem (3)-(4) for n = 3 (in 'R.+') and F(X) = XIX2X3. The universal structure in this case has the following form:

Fig. 4

We have three half-planes

which intersect at the line V123 = {x I Xl = X2 = X3}' If the initial point lies below V23UVl 3, i.e. in the domain D3 , we use U3 = 1 and move upward until the hitching V23UVl3' At the surface V23 we use {U2 = U3 = ~, UI = O} and move in the plane Xl = const until the hitching V123. Then~ we move along V123 with Ul = U2 = U3 = !. By the exchange of the indices we became the analogous types of movements.

302 L.F. ZELIKINA

Now we show how to use this universal structure to build the optimal synthesis.

Consider the time-optimal problem for the system %i = Uj:Z:l:Z:2:Z:S, (i = 1,2,3), U E U = {u I E~=1 Ui = I, Ui ~ o} with the target :Z:s(T) = c.

If we integrate the Hamiltonian system of Pontrjagin's maximum principle retrogressively, starting from the target and using the bang-bang trajectories we became the region which does not contain the bang-bang trajectories. This emply domain is shown in Fig. 5.

Fig. Ii

It is easy to see that the equations of switching surfaces are :Z:1 = :Z:sln(c/z3)j Z2 = z31n(c/z3)' Each surface (for instance ZI = zsln(c/:z:s» is a cylinder, which is defined only for :Z:1 < cleo lIence the switching is impossible, if

Now let us understand what part oCthe switching surface :Z:1 = :z:sln(c/:z:s) is really taking part in the synthesis. On the target we have 1/11 = 1/12 = 0, 1/1s = 1. Consequently, U3 = I, and 1; (1/11 - 1/12) = 1/13:Z:3(:Z:2 - :z:t}. Hence if ZI < Z2 then 1/11 > 1/12 and we take only the part of the switching surface :z: 1 = :z:sln(c/:z:s) on which:Z:1 < Z2. (Analogously on the surface:Z:2 = :z:sln(c/:z:s)

THE INVARIANTS OF OPTIMAL SYNTHESIS 303

we take only the part Xl > X2.) After the switching we have

(.,p1 - .,p2) = .,p1 X3(X2 - Xl) and (.,p1 - .,p3) = .,p1X2(X3 - xt}.

As soon as at the switching point we have .,p1 = .,p3,.,p1 > .,p2 the inequality .,p1 > max(.,p2, .,p3) is true always after switching and no other switchings occur.

As soon as the surface Xl = x3In(c/x3) has two branches, we'take only the upper one. Thus we find all the bang-bang trajectories, and the empty domain has to be filled with the universal structure.

Another example ([3)) of the univers.al structure is that one which does not satisfy the conditions of the regular synthesis

T- inf

Zi = UiF(X), (5) U e U = {u IE?=l Ui = 1, E?=1(ui)2 ~ p2},

xi(T) = a (i = 1, ... ,n).

Here, X E n~, I/Vii ~ p ~ 1/.;n::::T, F(x) is invariant under the rotation about the straight line V = {x I Xl = X2 = ... = Xn} and

F(x) I"~ ~i=con.t is a concave function that reaches its maximum at L..".=l

points of the ray V.

(6)

Description of optimal synthesis:

if Xo e V then 'Y~o (-) E V

if Xo rt. V then 'Y~o(-) belongs to the straight line going from Xo to the ray V and intersecting V at the same angle Q = arctan vp2n-l

Fig. 6

304 L.F. ZELIKINA

This synthesis is not a regular one because the following two closely related conditions of regular synthesis [4] are not satisfied:

1. There exists a cellular partition of phase space such that the function u( x) is continuous and continuously differentiable on each cell and it can be continued to a continuously differentiable function on a neighborhood of the cell closure.

2. When an extremal path goes from cell to cell, the dimensions of two "neighboring" cells differ from each other by not more than unity.

In this example, optimal paths from an n-dimensional cell "descending" to a one-dimensional cell.

In this case we also have invariant of optimal synthesis, namely, for the optimal control problem (5) the universal structure (6) is invariant to functionals from the class (4).

References

[1] Zelikina, L.F., Universal manifold and turnpike theorems for a class of optimal control problems, Do};;l. Akad. Nauk SSSR, Vol. 224, No. 1, 1975. (English translation in Soviet Math. Dokl. 16 1975.)

[2] Zelikina, L.F., High dimensional synthesis and turnpike theorems for optimal control problems, in V.I. Arkin, editor, Probasbilistic Control Problems in Economics, Nauka, Moscow, 1977, pp. 33-114 (in Russian).

[3] Zelikina, L.F., On optimal control problems with nonregular synthesis, All- Union Con/. Dynamical Control, Abstracts of Reports, Sverdlovsk, 1979, p. 114 (in Russian).

[4] Boltjanskii, V.G., Mathematical methods of optimal control, 2nd rev., augm. ed., Nauka, Moscow, 1969. (English translation of 1st ed., Holt, Reinhart and Winston, 1971.)

L.F. Zelikina CEMI USSR

Progress in Systems and Control Theory

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PSCT2 Perspectives in Control Theory: Proceedings of the Sielpia Conference, Sielpia, Poland, 1988 B. Ja1cubczyk. K. Malanowski. and W. Respondek

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PSCT4 Robust Control of Linear Systems and Nonlinear Control: Proceedings of the International Symposium MTNS, Volume II M. A. Kaashoek. J. H. van Schuppen. and A. C. M. Ran

PSCT5 Signal Processing, Scattering and Operator Theory, and Numerical Methods: Proceedings of the International Symposium MTNS-89, Volume m M. A. Kaashoek. J. H. van Schuppen. and A. C. M. Ran

PSCT6 Control of Uncertain Systems: Proceedings of an International Workshop, Bremen, West Germany, June 1989 D. Hinrichsen and B. Mdrtensson

PSCf7 New Trends in Systems Theory: Proceedings of the Universita di Genova - The Ohio State University Joint Conference, July 9-11,1990 G. Conte. A. M. Perdon. and B. Wyman

PSCf8 Analysis of Controlled Dynamical Systems: Proceedings of a Conference in Lyon, France, July 1990 B. Bonnard. B. Bride. J. P. Gauthier and I. Kupka

PSCf9 Nonlinear Synthesis: Proceedings of a I1ASA Workshop held in Sopron, Hungary, June 1989 Christopher I. Byrnes and Alexander Kurzhansky

PSCflO Modeling, Estimation and Control of Systems with Uncertainty: Proceedings of a Conference held in Sopron, Hungary, September 1990 Giovanni B. Di Masi. Andrea Gombani and Alexander B. Kurzhansky

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