Nonlinear Control Systems: An Algebraic Setting (Lecture Notes in Control and Information Sciences)

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

242

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

G. Conte, C.H. Moog and A.M. Perdon

Nonlinear Control Systems An Algebraic Setting

~ Springer

Series A d v i s o r y B o a r d A. B e n s o u s s a n • M.J. G r i m b l e • P. K o k o t o v i c • H. K w a k e r n a a k J.L. M a s s e y • Y.Z. T s y p k i n Authors Professor G. Conte, Universith of Ancona, Dipartimento di Elettronica ed Automatica, via Brecce Bianche, 60131 Ancona, Italy Dr C.H. Moog, IRCYN, Institut de Recherche en Cybern~tique de Nantes, 1 rue de la No6, BP 92101, 44321 Nantes Cedex 3, France Professor A.M. Perdon, Universith of Ancona, Dipartimento di Matematica "V. Volterra", via Brecce Bianche, 60131 Ancona, Italy ISBN 1-85233-151-8 Springer-Verlag London Berlin Heidelberg British Library Cataloguing in Publication Data Conte, G., 1951Nonlinear control systems : an algebraic setting. (Lecture notes in control and information sciences) 1.Nonlinear control theory 2.Nonlinear control theory - Mathematical models I.Title II.Moog, C. H. III.Perdon, A. M. 629.8'36 ISBN 1852331518 Library of Congress Cataloging-in-Publication Data Conte, G., 195 lNonlinear control systems : an algebraic setting / G. Conte, C.H. Moog and A.M. Perdon. p. cm. -- (Lecture notes in control and information science ; 242) Includes bibliographical references. ISBN 1-85233-151-8 (all<.paper) I. Nonlinear control theory. I. Moog, C. H., 1955- II. Perdon, A. M.III. Title. IV. Series. QA402.35.C65 1999 98-52362 629.8'312- -dc21 CIP Apart from an)' fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. © Springer-Verlag London Limited 1999 Printed in Great Britain 2nd printing 2001 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Typesetting: Camera ready by authors Printed and bound at the Athenaeum Press Ltd., Gateshead, Tyne & Wear 6913830-54321 Printed on acid-free paper SPIN 10790893

To F e d e r i c a P a o l a To R e n a n - A b h i n a v

Half of the story belongs to the one who tells the story, half belongs to the one who listens. (Montaigne) L'histoire appartient pour moiti6 k celui qui la raconte, et pour moiti6 ~ celui qui l'6coute. (Montaigne)

Preface The theory of nonlinear control systems owes large part of its modern development and success to the systematic use of differential geometric methods and tools. One of the first problems to be considered from the point of view of differential geometry was, at the beginning of the 70's, that of analyzing the controllability of a nonlinear systems. Early works on that topic ([102], [138], [139], [8]) highlighted the power and the potentiality of the differential geometric approach and motivated the interest of many researchers. During the 80's, the possibilities offered by the use of differential geometric techniques in the study of nonlinear control systems were largely exploited. One of the underlying leading ideas (see [68], [79]) was that of generalizing, to the greatest possible extent, the so-called geometric approach which had been first developed in the linear case (see[144], [5]). The research effort produced, in that period, many important results and it provided effective solutions to several control problems, like disturbance decoupling problems, noninteracting control problems and model matching problems. Excellent and comprehensive descriptions of the methodology and of the achieved results, together with meaningful examples of applications, can be found in [78] and in [112]. In the second half of the 80's the limits of the differential geometric approach started to be explored and to become known. In particular, it became clear that problems like system inversion or the synthesis of dynamic feedbacks could hardly be tackled with the - already - well established differential geometric methods. In the same period, the introduction of differential algebraic methods in the study of nonlinear control systems ([43]) offered a way to circumvent a

x

PREFACE

number of difficulties encountered up to that time. The use of differential algebraic concepts characterized, through the work of several authors, a novel approach, which has essentially an algebraic nature, and, at the same time, it provided additional tools for investigating old and new problems. Further results on problems pertaining to inversion, noninteracting control, realization and reduction to canonical forms were obtained in the following years and other were made achievable. Today, the use of an algebraic point of view in nonlinear control problems has gained popularity and diffusion. This motivates the present book, whose aim is to give an account of the algebraic approach to nonlinear system theory and of its development in the last years. Together with a number of results which are scattered in the literature, the reader will find in it a self contained, comprehensive description of techniques and tools which can enrich his equipment as a control theorist and can provide a solution to otherwise not easily tractable control problems. One of the distinctive characteristics which make the algebraic approach interesting and useful is its inherent simplicity. In comparison with the mathematical background needed for employing profitably differential geometric methods, the knowledge required for using the tools described in this book is, in facts, very limited. A significant example of this is offered by the way in which the notion of accessibility and the problem of linearization are dealt with. In both cases, a single tool, based on elementary differentiation of a function, namely the notion of relative degree, gives the key for carrying on a deep analysis and for characterizing relevant dynamical properties. From a didactic point of view, simplicity renders the algebraic approach a practicable and valid choice in teaching engineering courses on nonlinear control. The book emphasizes this aspect and is usable as a teaching aid. In addition, simplicity facilitates the development of efficient algorithmic procedures, which are relevant in solving concrete analysis and synthesis problems. Another positive quality of the algebraic approach is its wide applicability in the field of dynamical systems and control. Although only continuous-time systems are considered in the book, the tools and methods which are described apply successfully to a number of control problems concerning discrete-time nonlinear systems, as shown in [61], [4]. Applications to time-varying systems are also possible and, recent results have been obtained in dealing with time-

PREFACE

xi

delay systems (see [12, 107]). With respect to other general methodologies, then, the algebraic approach appears to be more versatile and capable of going to the heart of the problem. Only a basic knowledge of systems and control theory is required for reading the book, whose material is arranged in a self-explicatory way. The general setting and the fundamental notions are described and illustrated in the first part, entitled Methodology. Mathematical preliminaries are presented including notations from exterior differentiation. The system analysis completes this part and deals with fundamental properties as accessibility and observability. The structure algorithm and a canonical decomposition of the system are given as well. In the second part, entitled Applications to Control Problems, the tools and techniques of the algebraic approach are employed for solving a number of basic control problems, which are of practical interest in fields like robotics and control of general mechanical systems, as well as in process control. The solution to the Feedback Linearization Problem is given in terms of the accessibility filtration {7/k } introduced in Chapter 3. The Disturbance Decoupling Problem is solved using the subspace X fqy, namely the subspace which is observable independently from the input (Chapter 4). In the Noninteracting Control Problem and in the Model Matching Problem we use the output filtration {£k} and the structure algorithm (Chapter 5). Finally, in the third part, entitled Differential Algebra, differentially algebraic tools and concepts are introduced in an elementary, but comprehensive, way and the results of the first parts are revisited and analyzed from a differentially algebraic point of view. A conceptually powerful way of approaching the theory of nonlinear systems has been proposed at the end of the 80's in [43, 46, 48], introducing the use of differential algebra and differential-algebraic methods. In comparison with other approaches which employ differential geometric methods (see [78, 112] for a comprehensive description) or Volterra or generating series (see [42]), this one appears in particular capable of removing some drawbacks present in the notions of rank and it allows to characterize general invertible compensators. In addition, it provides useful insight and results in connection with various analysis and synthesis problems (see [39, 46, 48, 57, 120]). Although differential algebraic methods are better suited for dealing with systems described by means of polynomials or rational func-

xii

PREFACE

tions, extensions to more general cases (as suggested, for instance, in [44, 125]) are possible. Here, we give a brief introduction to the principal notions of differential algebra and we introduce a general notion of dynamical system in differential algebraic terms. Related system theoretic properties are described and the principal results obtained by the differential algebraic approach are mentioned without entering into the details, in order to help the reader in establishing a connection with the notions studied in the previous parts of this book. The authors acknowledge the NATO for its financial support by to a joint research project, under grant CRG 890101. Several results of this project are reported in this book. Finally the authors would like to thank J.W. Grizzle, M.D. di Benedetto for some joint research work which inspired this book. Valuable discussions with M. Fliess, A. Isidori, A. Glumineau, E. Aranda, J.B. Pomet, R. Andiarti, U. Kotta, Y.F. Zheng are acknowledged as well as the careful reading by L.A. M£rquez Martlnez and R. Pothin, and the help of E. Le Carpentier. Ancona, December 1998 Nantes, December 1998

Giuseppe Conte and Annamaria Perdon Claude Moog

Contents I

1

3

Methodology

1 3

Preliminaries 1.1 A n a l y t i c a n d M e r o m o r p h i c F u n c t i o n s . . . . . . . . . . . . . . . 1.2 C o n t r o l S y s t e m s . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Linear A l g e b r a i c S e t t i n g . . . . . . . . . . . . . . . . . . . . . . 1.3.1 O n e - f o r m s . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 T w o - f o r m s . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 s-forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 E x t e r i o r p r o d u c t . . . . . . . . . . . . . . . . . . . . . . 1.4 F r o b e n i u s T h e o r e m . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 E x a m p l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 7 8 9 10 12 12 13 15

Modelling 2.1 R e a l i z a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 S t a t e e l i m i n a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 E x a m p l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 20 24

Accessibility

27

3.1 3.2 3.3 3.4 3.5 3.6

27 28 28 30 32 33

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R e a c h a b i l i t y , c o n t r o l l a b i l i t y a n d accessibility . . . . . . . . . . . Autonomous Elements ....................... Accessible s y s t e m s . . . . . . . . . . . . . . . . . . . . . . . . . Controllability Canonical Form ..................

CONTENTS

xiv 3.7

C o n t r o l l a b i l i t y Indices

.......................

34

Observability 4.1 4.2 4.3 4.4 4.5 4.6 4.7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . T h e O b s e r v a b l e Space . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 A n o b s e r v a b i l i t y criterion . . . . . . . . . . . . . . . . . O b s e r v a b i l i t y C a n o n i c a l Form . . . . . . . . . . . . . . . . . . . O b s e r v a b i l i t y Indices . . . . . . . . . . . . . . . . . . . . . . . . Synthesis of observers . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 I n p u t - o u t p u t Injection and Linearization Problem . . . 4.7.2 Observer Design 1 . . . . . . . . . . . . . . . . . . . . . 4.7.3 S o l u t i o n to the L i n e a r i z a t i o n P r o b l e m by I n p u t - o u t p u t Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.4 G e n e r a l i z e d P r o b l e m . . . . . . . . . . . . . . . . . . . . 4.7.5 Observer Design 2 . . . . . . . . . . . . . . . . . . . . . 4.7.6 S o l u t i o n to Generalized P r o b l e m . . . . . . . . . . . . .

37 37 38 38 40 41 41 42 42 43 43 44 45 45 46

Systems Structure

51

5.1

52 52 53 56 58

5.2 5.3 5.4

S t r u c t u r a l Indices . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 S t r u c t u r e at infinity . . . . . . . . . . . . . . . . . . . . Structure Algorithm ........................ Invertibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zero d y n a m i c s . . . . . . . . . . . . . . . . . . . . . . . . . . .

System Transformations

59

6.1

Generalized State . . . . . . . . . . . . . . . . . . . . . . . . . .

59

6.2 6.3 6.4 6.5

Q u a s i - S t a t i c S t a t e Feedback . . . . . . . . . . . . . . . . . . . . Output Injection . . . . . . . . . . . . . . . . . . . . . . . . . . Canonical Form . . . . . . . . . . . . . . . . . Some e x a m p l e s . . . . . . . . . . . . . . . . . 6.5.1 Example . . . . . . . . . . . . . . . . . 6.5.2 E x a m p l e . . . . . . . . . . . . . . . . . . . . . . . . . . . G e n e r a l i z i n g the n o t i o n of o u t p u t injection Other Examples . . . . . . . . . . . . . . . . . . . . . . . . . .

61 62 63 68 68 69 70 74

6.6 6.7

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...........

CONTENTS

xv

II

77

7

Applications to Control Problems I n p u t / O u t p u t Linearization

79

7.1 7.2 7.3

79 80 81

Problem Statement ......................... Single o u t p u t case . . . . . . . . . . . . . . . . . . . . . . . . . M u l t i o u t p u t case . . . . . . . . . . . . . . . . . . . . . . . . . .

$ Noninteracting Control 8.1 8.2 8.3

N o n i n t e r a c t i n g control via regular static state feedback . . . . . N o n i n t e r a c t i n g control via q u a s i - s t a t i c s t a t e feedback . . . . . . N o n i n t e r a c t i n g control via regular d y n a m i c s t a t e feedback . . .

9 I n p u t / S t a t e Linearization 9.1

9.2

89 90 90

9.1.2 S o l u t i o n to the i n p u t / s t a t e l i n e a r i z a t i o n p r o b l e m . . . . D y n a m i c state feedback l i n e a r i z a t i o n . . . . . . . . . . . . . . . 9.2.1 Problem statement ..................... 9.2.2 S o l u t i o n to the d y n a m i c state feedback l i n e a r i z a t i o n . .

90 95 95 96

10.1 P r o b l e m S t a t e m e n t . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 S o l u t i o n to d i s t u r b a n c e decoupling . . . . . . . . . . . . . . . .

11 Model Matching

Differential

99 100 101

105

11.1 A special form of the inversion a l g o r i t h m . . . . . . . . . . . . 11.2 Model M a t c h i n g P r o b l e m . . . . . . . . . . . . . . . . . . . . . 11.3 Left F a c t o r i z a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Left factorization p r o b l e m (L.F.P.) . . . . . . . . . . . .

12

83 84 86

Static s t a t e feedback l i n e a r i z a t i o n . . . . . . . . . . . . . . . . . 9.1.1 Problem Statement .....................

10 Disturbance Decoupllng

III

83

Algebra

105 109 119 119

127

Differential Algebraic Methods in Nonlinear Systems T h e o r y 1 2 9 12.1 Basic n o t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129

CONTENTS

xvi 13 S t a t e s p a c e s y s t e m s 13.1 A differential algebraic point of view . . . . . . . . . . . . . . . 13.1.1 The basic differential ring . . . . . . . . . . . . . . . . . 13.1.2 Differential Algebraic Analysis . . . . . . . . . . . . . . 13.2 Explicit systems . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Further remarks . . . . . . . . . . . . . . . . . . . . . . . . . .

133

14 I n p u t - o u t p u t s y s t e m s 14.1 Inputs, states and outputs . . . . . . . . . . . . . . . . . . . . . 14.2 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143

133 134 135 136 140

143 146

15 A c c e s s i b i l i t y a n d o b s e r v a b i l i t y f r o m a d i f f e r e n t i a l l y a l g e b r a i c point o f v i e w 147

16 Control P r o b l e m s 16.1 Dynamic disturbance decoupling . . . . . . . . . . . . . . . . . 16.2 Left factorization problem . . . . . . . . . . . . . . . . . . . . . 16.3 Noninteracting control . . . . . . . . . . . . . . . . . . . . . . .

149 149 149 150

Bibliography

151

Index

165

Chapter 1

Preliminaries In dealing with nonlinear control systems, we are interested in considering situations in which the system theoretic properties at issue are generic, that is they hold on open and dense subsets of suitable domains of definition. From a practical point of view, system theoretic properties are very often characterized by conditions which require that specific functions, obtained from the system's coefficients, do not vanish. Since there are C ~ functions, like e.g. < 0 { e-1/~' ifx that are neither generically zero nor generically f(x) = O, i f z >_ 0 ' different from zero, the notion of generic property does not make sense, in general, for systems defined by means of C ~ functions. The situation is different if we restrict our attention to systems defined by means of analytic, or also meromorphic, functions and this motivates such a choice throughout the book.

1.1

Analytic and Meromorphic Functions

In order to specify the class of functions we will deal with, let us give the following Definition.

Definition 1.1.1 Let I C ~:~ be an open interval, a function f : I --~ ~:~ is said analytic at the point zo of I if it admits a Taylor series expansion in a neighborhood of zo.

4

C H A P T E R 1. P R E L I M I N A R I E S

We call analytic a function f : ~ --+ ~ which is analytic at any point of ~ . E x a m p l e s of analytic functions are given by p o l y n o m i a l functions, sin z and c o s z . R a t i o n a l functions are analytic at any point of their d o m a i n of definition. T h e function e x p ( - l / 1 0 x 2) : /R -+ /R is C ~ but not analytic at x = 0 - see Figure 1.1.

-! 1

10

8 6 4 2 0 -5

, -4

1

2

3

4

Figure 1.1: G r a p h of e x p ( - 1 / 1 0 z 2) T h e basic property of analytic functions we are interested in is stated in the following Proposition. Proposition

1 . 1 . 2 Let f : IR ~ IR be an analytic function, then either

(i) / is zero, or (ii) the zeros of f are isolated. A p o l y n o m i a l function has a finite n u m b e r of zeros which are isolated. The function f ( x ) = sin x has an infinite n u m b e r of isolated zeros located at z =

1.1. A N A L Y T I C

AND MEROMORPHIC

FUNCTIONS

5

krr for any positive or negative integer k. A typical example of non ~nalytic continuous function whose zeros are not isolated is the following. E x a m p l e 1.1.3 The function f ( x ) , defined by f(x)

=

sin(1/x) i f x • 0

f(0)

=

0

is not analytic since z = 0 is a point of accumulation for the zeros of f - see Figure 1.2.

-1 10, 10

51 ol

-2

-l

11// tlV 0

1

x

Figure 1.2: Graph of s i n ( l / z )

In the multivariable case, one can analogously consider an open domain D C /R n, n E j~V and the following Definition.

Definition 1.1.4 A function f : D ~ ~ is said analytic at zo E D if it admits a Taylor series expansion in a neighborhood of xo. f : H:l'~ --+ ~ is said analytic if it is analytic at any x E ~ n .

The generalization of Proposition 1.1.2 becomes

6

C H A P T E R 1. P R E L I M I N A R I E S

P r o p o s i t i o n 1.1.5 Let f : IRn --+ IR be an analytic function, then either

(i) f is zero, or (ii) the set of zeros of f has empty interior. E x a m p l e 1.1.6 Let f : /R 2 -4 ~ , be defined by f(Xl,X2) = X l - 1. It is easily seen that f is analytic. For any point P E ~ 2 which belongs to the set Z of zeros of f , there does not exist any/R2-neighborhood included in Z - see Figure 1.3. X2

P

}

set Z of zeros of f

Xl

Figure 1.3: Zeros of f ( z l , x2)

=

Zl

-

1

By Proposition 1.1.5, nonzero analytic functions are different from zero at the points of an open dense subset of j~n. Then it makes sense to define the generic rank of a matrix whose entries are analytic functions as the dimension of the maximum square submatrix having nonzero determinant. As the determinant is an analytic function, the generic rank coincides with the rank of the matrix at the points of an open dense subset o f / R '~. Moreover, the generic rank is greater than or equal to the rank at any point o f / R n.

1.2. CONTROL SYSTEMS

7

Analytic functions do not have, in general, analytic inverses. With the usual notions of sum, denoted by +, and of product, denoted by., the set of analytic functions from/R r to IR forms a ring, denoted by .At. The ring .A~ does not contain zero divisors, hence it is an integral domain. Note that C ~ functions also form a ring, but not an integral domain since the product of two nonzero { e-1/~, i f x < 0 a n d f 2 ( x ) _ _ { 0, ifx_<0 elements like fl(x) = 0, ifx > 0 e -1/*~, ifx > 0 is equal to zero. One can associate to the integral domain A~ the quotient field ]Or whose elements are pairs (f, g) of elements of Ar such that g r 0, modulo the equivalence relation ~R defined by (f, g) ~R (f', g') if and only if fg' = gf' Choosing a representative in the equivalence class, an element of K~ will be written as

I/g.

The sum + and product 9 of two such pairs are formally defined as

(s

+ (f~,g2) :--- (s

+ f2gl,glg2)

(fl,gl)" (f~,g2) :: (flf2,glg~) The elements of the quotient field/Or of the ring of analytic functions are called meromorphic functions. Any rational function is a meromorphic function, an other typical example is tan x = sin x~ cos x.

1.2

Control Systems

In recent years, the problem of defining what a dynamical system is has been approached and discussed by many authors. This has lead to the development of the so called behavioral approach of [143] as well as to the generalized state space representation of [48]. Here, we do not enter into this debate, but we take the classical state space representation for granted. The dynamical systems which are mainly considered in this book are objects described by a system of first order differential equations of the form

E= { ~y = h(x)f(x)Wg(x)u

(1.1)

8

C H A P T E R 1. P R E L I M I N A R I E S

where the state x E ~ n , the input u E/R m, the output y E/R p and the entries of f, g, h are meromorphic functions. Systems of the above kind are useful in modelling, at least locally, many physical systems like robots, satellites, vehicles, industrial processes and so on [92, 131]. In the following, we will always assume that the following Assumption is satisfied. A s s u m p t i o n 1.2.1 Given a system E of the form (1.1), the matrix g(x) is such that rank 9 = m. The above assumption is not restrictive, since it only means that the independent inputs have independent actions on E. We will refer to the state space representation (1.1) as to the internal representation. External representations, that is without state variable, will be discussed in Chapter 2, together with their relation with internal representations.

1.3

Linear Algebraic Setting

Assuming that a nonlinear control system ~ of the form (1.1) is given, the objective of this Section is to construct an algebraic setting for defining and studying system theoretic properties of E. Our approach is built up by introducing the notion of differential form in an abstract and formal way. This choice is motivated by simplicity and by the fact that in the rest of the book we will be interested only in the abstract algebraic and formal properties of differential forms. Other less abstract treatments of the same topics, like those developed in [92], [75], [112], [t41] agree at a formal level with the one described here. Our approach has contact points with that of [17] and, in order to avoid technicalities, the reader is referred to [17] for proofs and technical constructions which are not found here. To begin with, denoting by n and m respectively the dimension of the state space and of the input space of the system E, let us consider the infinite set of real indeterminates C = { x i , i = l , . . . , n ; u ~ k ) , j = 1 . . . . , m , k > O}.

1.3. L I N E A R A L G E B R A I C S E T T I N G

9

For any positive integer r, we use the first r elements of C for denoting the coordinates of a point in ~ r . Hence, a function f r o m / R r to ~ , in particular an element of/Or, will be written as a function in the first r indeterminates of C. The usual partial derivative operators O/Oxi and O/Ou~ k) act naturally on the field 1Or of all meromorphic functions from ~ to N, which, for that reason, is said to be endowed with a differential field structure. Differential fields and their properties will be formally introduced and studied in Chapter 12. Here, it is sufficient for our aims to remark that, letting ~ denote the set theoretic union Ur g r , / c has an obvious field structure and, moreover, it can be endowed with a differential structure determined by the system E. In facts, since any element of/C is a meromorphic function depending on a finite subset of indeterminates of C and, consequently, can be in general denoted by F({xi, uJ.k)}), we can define a derivative operator 5, acting on K:, as follows:

Jxi - fi(x) + gi(x)u(~

all/= 1,..., n

junk) = "J"(k+l) fork >_ 0 and for allj = 1 , . . . , m n

i=1

j=l,...,m;k>O

The resulting differential field is the starting point for a number of constructions which will be used in charactering the system theoretic properties of E. 1.3.1

One-forms

We consider now the infinite set of symbols

dC = {dxi,i = 1 , . . . , n ; d u ~ k ) , j = 1 , . . . , m , k

>_ 0}

(1.2)

and we denote by s the vector space spanned over/C by the elements of de, namely g = spanpcdC. (1.3)

CHAPTER 1. PRELIMINARIES

10 Any element in g is a vector of the form

v = E Fidxi + i=l

E

Fjkdu5k)

j=l,...,rn;k~_O

where only a finite number of coefficients F are nonzero elements of/C.We can define now an operator from 1~ to g, which by abuse of notation will be denoted by d, in the following way: 71

dF({x,, u~k)})= E(cgF/Oxi)dxi + i=1

E

(cOF/OuJa))du~k)"

j=l,...,rn;k_>0

The elements of s will be called one-forms and we will say that v E g is an exact one-form if v = d F for some F E K. We will usually refer to d F as to the differential of F. E x a m p l e 1.3.1 Let F = sin(xlx2) E K:. Then d F = cos(xlx2) [x2dxl + xldx2] e g. The vector space of one-forms g can be endowed with a differential structure by defining a derivative operator A, in terms of the derivative operator 6 and of the differential operator d acting on K, as follows: Av

=

A ( E i ~ 1 Fidxi + E j = I .....,n;a>oFjkduJ a))

, ~rjku.~uj(k) q- Fjkd(~uJk))). =~-'~;=i(~Fidxi -t- Fid(~xi)) + Zv-,- ~ j = I ..... rn;k>_OI, 1.3.2

Two-forms

We consider now the infinite set of symbols AdO = {dxi A dxi,; duJ k) A duJ,k'); dxi A dusk); duJ k) A dxi; for i = 1 , . . . , n; i'=l,...,n;j=l,...,m;j'=l,...,m;k>_0;k'>0} and the vector space, that we denote by Ag, spanned over ]C by the elements of AdO. In Ag we consider the equivalence relation R spanned by the equalities

1.3. LINEAR ALGEBRAIC SETTING

11

da A dfl = -dfl A da.

(1.4)

The vector space Ag rood R will be denoted in the following by g(2). The elements of g(~) are called two-forms. We can define now an operator from g to g(2), that by abuse of notation will again be denoted by d, in the following way : n

dv

d ( E Fidxi + y~ i=1 j = 1,...,m k>O

=

FjkduJ.k))

' . - OFi ,. (k) = ,(" = 1E.... , (~Fx,v )dxi, Adzi +i = 1~ ... (ncgttjk))ClU: Adx/ I

iI = 1 , . . . , n

j= 1,...,m k>0

y ~ (-~x/)dxi A duJk)+ 1' OFjk ~ ,t ~cgFjk ) a u j , , (/r A i=l,...,n j= ...,~r~'

q-

j = l,...,m k>O

duJk))modR.

jl = l , . . . , m k>O,k'>O

A canonical representative of dv is given by

dv

=

v,(OF~ ~ ~

+

OFv. Ox-----i)dxi, A dxi

ON

i=

Z (Ou 1,...,n

OFjk)duJ k) A dzi

j = 1,...,m k>0

k>O kl>O

opj~

OFt,k,,. -)auj,(k'l A du}k)

CHAPTER 1. PRELIMINARIES

12

E x a m p l e 1.3.2 Let v = dxt - (xt/x2)dx2, then

dv = 0 - (1/x2)dXl A dx2. 1.3.3

s-forms

We need now to consider a more general construction of the same kind of that giving rise to g (2). To this aim, let us consider, for any integer s, the infinite set of symbols A"dC -" {d(0 Ad~l A . . . A d ~ , ; ( i E C;i = 0 , . . . , s } and the vector space, that we denote by A~s spanned over ]C by the elements of ASdC. In ASg we consider the equivalence relation R spanned by the equalities d(io Ad(i, A . . . A d(i, = (-1) ad(jo Ad(jl A . . . A d(j, where ~r is the signature of the permutation

j0

...

js

. The vector space

ASs rood R will be denoted in the following by g(s+l) its elements are called

(s + 1)-forms. By the constructions described above we obtain a set of vector spaces s g(s) which are related to the system N. These algebraic objects will be the basic tools for analyzing the system properties of ~ in Chapters 3 to 6 and for solving several design problems in the second Part of the book.

s

1.3.4

Exterior

product

The exterior product or wedge product of a p-form wl and a q-form w2, denoted by wl A w~, can now be defined as the (p + q)-form whose representative, if wl is represented as

wl=

Z

F'~}P>

i=l,...,k

with ~}P) E AP-ldC and a;2 is represented as

j=I,...,h

13

1.4. F R O B E N I U S T H E O R E M

with {Jq) E Aq-ldC, is given by

F;Cj{} ') A

t.O1 A {.02 =

i=l,...,k;j=l,...,h

It can easily be verified that the exterior product is associative, moreover it induces a map A : g(P) x E (q) --+ s given, quite obviously, by A(Wl, w2) = t.d 1 A ~d2.

1.4

Frobenius Theorem

In this Section we investigate the problem of checking the exactness of a given one-form. To begin with, let us first state some elementary facts. D e f i n i t i o n 1.4.1 A one.form v E s is closed if dr = O. P r o p o s i t i o n 1.4.2 Any exact one-form is closed. Proof. Consider a function ~o E /C of n variables, say & , . . - , &

Then, d~ = ~ i = 1 d(d~)

with {1 E C.

. d~i and

=

i,j

02

d{5 Ad{i d{i A d~j

=

0 |

Poincar~'s Lemma (see [17] for a proof) establishes that the converse of Proposition 1.4.2 is true only locally. L e m m a 1.4.3 P o i n c a r $ L e m m a Let v be a closed one-form in E. there exists locally ~o E IC such that v = d~v.

Then,

E x a m p l e 1.4.4 A typical example of closed form that is not exact is the following. In/~2, consider the closed one-form w = x 1 - ~ 2 dxl - x a - ~ d z 2 .

C H A P T E R 1. P R E L I M I N A R I E S

14

Locally, around any point (xl, x2) such that x2 r 0, w --- d [arctan(xl/x2)], and around any point (xl, x2) such t h a t x2 = 0 and xt r 0, w = d [arctan(-x2/xl)]. But there is no function ~0 such that ~ = d~o globally. A requirement weaker than exactness for a one-form v is t h a t of being colinear to an exact form, i.e. that there exists )~ and ~ in/C such that )~v = d p or, equivalently, such t h a t spanpc{v } = spanx:{d~}. The characterization of this property is a special case of the Frobenius Theorem that will be stated later. T h e o r e m 1.4.5 Given v E s spanx:{d~} if and only if

there exists a function ~, such that span~c{v} = dvAv=0.

Proof of the necessity. Since d~, E span~c{dv}, there exists a nonzero function (~ such that c~v = d~. Hence a v is exact, that is d ( a v ) = 0. Since 0 = d(c~v) = d o A v + a d v one has 0 = d((~v) A v = 0 + c ~ d v A v which yields the desired result. The proof of sufficiency is not elementary and for that the reader is referred

to [17].

I

A function o~ as in Theorem 1.4.5 is called an integrating ]actor. To generalize the result of T h e o r e m 1.4.5, let us introduce the following Definition. D e f i n i t i o n 1.4.6 A subspace V C g is said exact, or closed, or integrable if

V has a basis which consists of closed forms only. Theorem

1.4.7 F r o b e n l u s T h e o r e m

Let V = s p a n ~ c { ~ l , . . . , w r } be a sub-

space of g. V is closed if and only if d~i A a ; 1 A . . . A ~ E x a m p l e 1.4.8

= 0, for a n y i = 1 , . . . , r .

1.5.

15

EXAMPLES

9 The one-form ca = xldXl + x2dx2 is closed according to Definition 1.4.1. In fact, ca = 89 + x~). 9 The one-form ca = dxl + xldz~ is not closed since dw = dxl A dx~ ~s 0. However, the vector space span;c{ca } is integrable since dw A ca = 0 and one may choose the integrating factor a = l/x1. Frobenius Theorem is often stated, in the literature about nonlinear systems, in terms of vector fields. The version given here is a dual version which fits more naturally our formalism.

1.5

Examples

The algebraic formalism of one-forms is mainly devised for facilitating computations involving gradients and Jacobian matrices. Note, in particular, that the differential dy of the output of the control system (1.1) is a vector in C: dy = ~xldXl + . . . +

dx,~

The differential of the k - t h time derivative of y is in s as well. E x a m p l e 1.5.1 The vector Wl -= dxl E C may be thought, with respect to the basis dC, as the row vector [ 1 0 0 ... ]. This row vector is the Jacobian P

/~

~

" " ~ of the function [ ( x ) =

L

xl, which belongs to/C, and it is

.t

obviously exact. E x a m p l e 1.5.2 Let us consider the one form ~2 = z 3 ~ I = x 3 d x l E g, which may be identified with the row vector w2 = [ x3 0 0 ... ]. The question of the exactness of w~ yields the following integrability problem: does there exist a f u n c t i o n e E ~ s u c h t h a t w 2 =

k

~

1

r exists, then necessarily it solves the system ~

~

X3

=

0

~

2

~

3

...

J

~ If s u c h a

CHAPTER 1. PRELIMINARIES

16

Using second order derivatives, one concludes that there is no solution since

0 0r 0 0r Oza Ox~ - 1 # Oxl Oxa

o

Examples of this kind motivate a formal study of second order derivatives and give rise to the notion of two-form, that generalize somehow Hessian matrices made of second order partial derivatives. E x a m p l e 1.5.3 Let v = (1/x2)dzl - (xl/x22)dx2. To check the closure (or

0 - _ ~ . The two-form dv embodies these computations: x2

dv -

Oz20 z21 dx2 A dxl + __._~_2 dzl A dx

(1.5)

-- -(1/x~)dx2 A dxl - (1/x~)dx, A dx2.

(1.6)

Now, the closure of v results from the fact that 0z-'~'~~ =

0 ~X] 1 0Xl

or, since in

g(2) one has dxl Adx2 = - d x 2 Adz1, from being dv = 0. In fact, v = d(xl/x2). In the case of Example 1.5.1 and of Example 1.5.2, one has respectively, dwl = d(dxl) = 0

dw2 = d(x3dzl) = dx3 A dxl + x3d(dxl) = dx3 A dxl where dw2 # 0 displays the fact that w2 is not exact. In fact, as a linear combination of the symbols dxl A dx3 and dxa A dxl, dw2 reads as: dw2 =

0

1

$

candidate for

candidate for 0

0 0

Since ~

dxl A dx3+

$

dx3 A dxl

= ~za~zl f o r a n y ~ a n d d x a A d x 3 = - d x 3 A d x l ,

dw2r

formalizes the fact that there is no function r E ~ such that w2 = de.

Chapter 2

Modelling Control systems may be described in several ways. Physics often yields inputoutput descriptions in terms of higher order differential equations. Motivated by the analysis of such situations, a typical control theoretic problem is to restate such descriptions in terms of coupled first order differential equations, so embodying the so-called state space representation. For linear systems, it is possible to switch from one representation to the other. In this Chapter, we argue the existence of state equations for input-output descriptions (realization problem) and conversely, that of input-output descriptions for state space representations (state elimination problem). In that respect, the Laplace transform plays a key role for linear systems. Although such a symbolic computation is not available for nonlinear systems, it is possible to treat directly the two above mentioned problems.

2.1

Realization

In this Section we are interested in the transformation from the input-output, or external, representation of a system into a state-space, or internal, representation. Namely, in what is called the realization problem. Here, we only recall some results from [48] obtained quite naturally by working on the input-output equations, which yield a so-called generalized realization. At the end of this

18

CHAPTER

2.

MODELLING

section, we refer to the literature for results on classical (instead of generalized) realizations. Consider a system of p input-output differential equations Fi(y,...,y(k),u,...,u

0))=0

(2.1)

i = l,...,p

where the Fi's are meromorphic functions. A standing assumption will be that the F ' s are independent. More precisely, letting

O(F,) r 0},

k~ -- max{j such t h a t ~

(2.2)

0(y~)

we assume that 0 _< k i Assumption

< oo.

In addition, we also make the following

2.1.1 O(FI,...,

Fp)

rank0(y~kl),..., yp(kp)) -- p"

(2.3)

An easy (generalized) state space representation can be constructed by introducing the state vector ( x l , . . . , x k , + + k ~ ) defined by

(xl,

9 '

.,xkl)

(yl, ' - " , Y l k,-1, )

-

-

-

9

(xk~+k~+...+k,_l+l,...,xk,+

+k~)

=

(2.4)

(yp,...,yp

kp

-1),

which yields the following implicit representation Xl

ik,-1 ~'kl'i-'"-l-kp-1 F(xl,

. . . , xkl,

--

x2

=

xk,

~

Xki+...+kp

. . . , xk,+...+ap,

(2.5)

xal, . . . , xa,+...+kp,

u , . . ., u (s)) = 0

2.1. R E A L I Z A T I O N

19

If some ki = 0, then some line is empty in (2.4). Assumption (2.1.1) allows us to write, at least locally, 2:1

~

~2

xk,-1

=

xk,

Xkxq'...+kp

=

~,p(z, u,/~,..,u(~))

Xk,

---- ~I(X,U,/t,.'',~/(s))

(2.6)

Equations (2.6) form a generalized state space realization of the system described by (2.1), where the term generalized accounts for the presence of the derivatives of u. In general, the application of the Implicit Function Theorem does not guarantee that ~ 1 , . . . , Up are meromorphic functions. E x a m p l e 2.1.2 Consider the input output equation y2 = y + u. The above procedure yields the implicit representation

{ j:2 y

= .:

x+u x

or, locally, one of the following explicit realizations, depending whether y > 0 ory
(x

{x y

=

=

x

(2.8)

Remark that the right hand side is not meromorphic at the origin. In general, the above procedure does not yield a realization as (1.1). In particular, systems with non empty zero dynamics [80] (with transmission zeroes, if the system is linear) will always be realized in a generalized sense by this elementary procedure. In other words, the presence of derivatives of u in the right hand side of (2.5) attests somehow the presence of a non trivial finite zero dynamics. However, the realization (2.5) may be transformed under suitable

CHAPTER 2. MODELLING

20

hypotheses to a realization containing no time-derivatives of u. Conditions for ensuring the existence of such a realization are fully characterized in [28] and are not reported here (see [24, 41, 50, 83, 84, 134, 135, 142]). E x a m p l e 2.1.3 Consider the following linear example, which has a transmission zero:

~=u+i~ The generalized realization which is derived reads Xl y

:

x2

----

X1

Notions of controllability/accessibility and observability are reported in Chapters 3 and 4. With respect to these notions the above realization (2.6) is in general only observable but not necessarily accessible. In this sense it is not minimal.

2.2

State

elimination

Let us consider now the problem of going from an internal state space representation to an input-output external representation. It will be shown in the following that for an affine state space system ~ of the form (1.1) there always exists, at least locally, a set of input-output differential equations. There are no additional difficulties in considering a generalized system, i.e. a system of the form

{

x

y

= =

y(x,u,...,u(,))

h(x,u,...,uO))

(2.9)

whose state equations depend on a finite number of time derivatives of the input, where f and h are assumed to be analytic, x E ~ n and u E /~rn. Given system Z of the form (2.9) the problem is to find, if possible, an inputoutput differential equation as

F(y, y , . . . , y(k), u,/~,..., u (~)) =- 0 such that any pair (y(t), u(t)) solution of (2.9), satisfies (2.10).

(2.10)

2.2, STATE ELIMINATION

21

T h e o r e m 2.2.1 Given system ~ of the form (2.9), where f and h are analytic functions, then there exist an integer V and an open dense subset V of j~+rn~ such that in the neighborhood of any point of V, there exists an input-output representation of the system of the form (2.9).

Proof The first step in constructing an input-output representation consists in applying a suitable change of coordinates. To this aim let us denote by sl the minimum non negative integer such that rank0(hl""

h~sz-1)) 0(hi,. h~sx)) 9' = rank " "' Ox Ox

If Ohl/Oz - 0 we define sl - O. Analogously for 1 < j _< p, lets us denote by sj the minimum integer such that rank0(hl,...,h[S~-l);...;hj,...,hJ Oz

"~- t ))

= rankO(hl,...,h~"-l);...;hj,...,hJ Ox

si))

If hj_l(~J-,-U) = r a n k O ' h( a ' .. .,'~j-1 h(~j_,-a), hi) Ox Ox we define sj = 0. Write K -- sl + ... + sp. The vector rankO.hl, -..,(

S = (hl,...,h]l-1,...,hp,...,hv"-1), where hj does not appear if sj = 0, satisfies the following relation rank [0~-zS] = K ,

for almost every x.

It will be established in Chapter 4 that the case K < n corresponds to non observable systems. In this ease, there exist analytic functions g l ( x ) , . . . , gn-g(x) such that the matrix

j = O(S, g l , . . . , g n - K ) Oz

CHAPTER

22

2. M O D E L L I N G

has full rank n. Then the system of equations

h i ( x , u , . . . , u (c~)) h~'a-1)(x, u , . . . , u ( " + s ' - 1)) XsI+I

:

h2(x, u , . . . ,

u (a))

(2.11) xs1+s2

:

h~-l)(~,u,...,u(~+s~-~))

xs1+s2+...+~p

:

h~SP-1)(x, u, .., u("+~p-1))

Xsl+s2+...+sp+i

:

gi(x,u,...,u

(~))

i = 1,...,n - K

is of the form F i ( x , ~ c , u , . . . , u (w)) = 0, i = 1 . . . , n with

O(F~,..., F,)/O(~,.

., ~n) = J.

To avoid the introduction of new notations, it is not restrictive to assume 3' _> max{~ + si - 1, i = 1 , . . . , p}. The determinant of J is an analytic function whose set of zeros has empty interior, so there exists an open dense subset V of IR n+m'~ such that det J is different from zero at every point of V and the Implicit Function Theorem applies. Therefore there exist n functions

xi=r

(7))

for

1
which define a local diffeomorphism r parametrized by u , . . . , u('Y):

= r

(2.12)

23

2.2. S T A T E E L I M I N A T I O N

By applying the change of coordinates induced by (2.12), the system (2.9) becomes: L

Xl x2

=

x,2 {c4

"-~

~ Xsl+2

=

Xs 1 2

Xsx+l

L

Xsl+s~

=

Xsl+...+sp

=

(2.13)

2,

L

Xsl+...+sp+i

.

=

gi(~),u,...,u(~))

Y2

=

Xsl+l

yp

=

Xst+...+Sv_l+ 1

Yl

i= 1,...,n-

K

~

In the neighborhood of any point where det J ~ 0 we have also

Oh~s~) Ox

{Oh10h~ s~-l) Oh~ E spanjc

oX'''"

Ox

' Ox"'"

Ohls~-l) } 0------~

so that Oh} ~)

=

[cl ... c~,+...+~, 0 ... O]ej = 0

j > sl+'"+si

where ej is the j - th column of the identity matrix. Therefore the functions hJSJ)(r u , . . . , u (~)) depend only on ;~1,..., ;~s,+...Ts/'

24

CHAPTER

2.

MODELLING

Since the following identities hold Yl :

:~1,

~)1 = ~:2, 9 9 y~r) = ~ l + r

forr=0,...,sl-

1

~Sl.{.....~t.~j_l.~.l Yj : ~:~1+...+sj-1+2,.--, y~r) = s fort = 0,...,sj yj

=

--

1, j = 2 , . . . , p .

From (2.13) we get the input/output relations we were looking for. In fact we have:

yl ~,) :

h~')(~(yl,yl,...,~',-')),,,...,~(~))

~('p) =

h(/~)(~(yl,...,yl s,-')

,..

.,yp,...,,p9 ( ' P - 1)),~,...

, u(~)) |

The input-output equations (2.t4) are not uniquely defined since, for instance, if K is less than n, different choices of the functions g ~ ( x , u , . . . , u ('~)) produce a different system (2.11). Instead of { s l , . . . , sp}, it is possible to use the observability indices as defined in Chapter 4 to derive an analogous input-output equation.

2.3

Examples

E x a m p l e 2.3.1 For the system x2

=

ul

9v3 Yl

: =

u2 Xl

Y2 =

x2,

2.3. E X A M P L E S

25

we h a v e Yl "- X3Ul, Yl : U2Ul + X3Ul a n d finally /)1 = u 2 u l + ( ~ ) l / u l ) u l .

The last equation holds at every point in which ul r 0. For the second output we have immediately ~)2 = ul. The following example shows that for more general non linear system, where does not appear explicitly such as

F(x, x, u , . . . , u (~))

(2.15)

the method described above cannot be applied. E x a m p l e 2.3.2 Consider the system (x - u) 2 = 0 y = ~

(2.16) (2.17)

The implicit function theorem does not apply to equation (2.16) to obtain x, since for every x and every u we have ~(& - u)2/Ox ~ O. By the way, for this example an input/output relation is given by ( 3 - u) 2 = 0 or b y

(~) -

u) = 0

Results similar to the ones described above may be found in [140]. A state elimination method which yields global results is studied in [39].

Chapter 3

Accessibility 3.1

Introduction

A basic notion in control systems theory is the notion of reachable state and of controllability. Controllability concerns the possibility of steering the system from a state xl to another state x2. For linear systems controllability is a structural property, in the sense that any linear system can be split into a controllable subsystem and an autonomous one. This is not the case for nonlinear systems as the examples in Section 3.2 show. The structural property that in the nonlinear case play a role similar to that of controllability in the linear case and can be given a similar characterization is the accessibility property. For illustrating the notions of reachability, controllability and accessibility we will consider, in this Chapter, a system without outputs of the form

= f(x) +g(x)u where x E ~ n , u E / R m, the entries of f(x) and g(x) are elements of K.

(3.1)

28

3.2

C H A P T E R 3. A C C E S S I B I L I T Y

Examples

The following examples illustrate some classical pathologies steaming from the fact that, in general, one can not expect a nonlinear system to be controllable from any initial state to any final state. E x a m p l e 3.2.1 Consider the following one dimensional system (3.2)

~: = xu

This system is clearly not controllable from zero: if the initial state is located at the origin, then the trajectory x(0, u(t)) of (3.2) remains at the origin for any input function u(t). From an initial state x0 different from the origin, it is not possible to reach a state xl such that XlXo < 0 since, for any continuous function u(t), the trajectory should pass through the origin and remain there. However, any point xl such that XlX0 > 0 is reachable from x0 # 0. E x a m p l e 3.2.2

xl = &2 "-

u

(3.3)

This system is not controllable from any initial point x0, in the sense that any neighborhood of x0 contains a point xl which can not be reached from x0. Figure 3.1 below shows the set of points which are reachable from x0.

3.3

Reachability, controllability and accessibility

The following definitions formalize the phenomena displayed by the introductory examples. D e f i n i t i o n 3.3.1 For a nonlinear system ~ of the form (3.1), the state xl is said to be reachable from the state xo if there exists a finite time T and a Lebesgue measurable function u(t) : [0, T] --+ ~ m , such that x(xo, u, T) = xl.

3.3. REACHABILITY, CONTROLLABILITY AND ACCESSIBILITY

29

X2

x0

ipoints

Figure 3.1: Reachable set of system (3.3) For the system (3.2), the state xl = 2 is reachable from x0 = 1, whereas x2 = - 1 is not. D e f i n i t i o n 3.3.2 A system ~ of the form (3. l) is said to be controllable at xo if there exists a neighborhood V of xo, such that any state xl in V is reachable from xo. The system (3.2) is controllable at any state different from the origin, whereas there does not exist any state x0 in ~ 2 at which the system (3.3) is controllable. D e f i n i t i o n 3.3.3 A system ~ of the form (3.1) is said to be accessible at xo if the set of reachable points from xo contains an open subset of ~ n As illustrated in Figure 3.1, the system (3.3) is accessible in this sense at any point of/~2. From [138], the accessibility of a nonlinear analytic system at a given point x0 of the state space ~ is characterized by the fact that the so-called strong accessibility distribution s has dimension n at that point. The existence of an open and dense submanifold of the state space ~'~ at whose points the system is accessible is then characterized by the fact that the strong accessibility distribution s has generically dimension n . From an algebraic point of view, accessibility will be characterized in the next Section using the concepts of autonomous element introduced in [120].

30

C H A P T E R 3. A C C E S S I B I L I T Y

3.4

Autonomous Elements

Let X denote the subspace of g given by X = span~:{dxi, i = 1 , . . . , n} .

Definition 3.4.1 A function ~ in IC~, is said to be an autonomous element for a system E of the form (3. I) if there exists an integer v and a non zero meromorphic function F so that F(ta, ~b, ..., ~p(")) = 0

(3.4)

where ~ = ~ip.

Definition 3.4.2 The relative degree r of a function ~ in ]C~ such that d~ E X is given by r = m i n { k E ~ , ] span~:{d~, ...,d~ (k)} • X} (3.5) Proposition 3.4.3 If a function ~ in IC~ is an autonomous element for a system E of the form (3. I) then

(i)

e X

(ii) ~ has infinite relative degree. Proof: Any vector w E C and which does not belong to X satisfies dim span~: {w, ...,r (k-U} = k

(3.6)

for any k > 1. From Definition 3.4.1, this is not. true for w = dip and k = t, + 1. This ends the proof of statement (i). If ip has finite relative degree, then dimspan~:{d~, ..., d~ (k-l)} = k

(3.7)

for any k >_ 1. This contradicts Definition 3.4.1. I The notion of autonomous element can be defined also in the context of nonexact forms.

3.4. A U T O N O M O U S E L E M E N T S

31

D e f i n i t i o n 3.4.4 A one form w in ~' is said to be an autonomous element for a system E of the form (3.1) if there exists an integer u and meromorphic function coefficients ai in I~, for i = 1 , . . . , u, so that mow + . . . + a , w (~) = 0

(3.8)

D e f i n i t i o n 3.4.5 The relative degree v of a one form w in X is given by r = min{k E IW, [ spanlc{w,

...,w (k)} r X}

(3.9)

P r o p o s i t i o n 3.4.6 A one form w in X is an autonomous element if and only if it has infinite relative de9ree. Proof: only if part: Assume that w in 2" has infinite relative degree. Since d i m X = n, there exists k, 0 < k < n such that dim spantc{w, ..., w(k)} = k

(3.10)

This yields that w is autonomous. if part: By contradiction, show that if w has finite relative degree, then it is not autonomous. As a matter of fact, if w has finite relative degree, then dimspantc{w , ...,w (k-l)} = k for any k > 1. This completes the proof. It is now straightforward to prove

(3.11) II

P r o p o s i t i o n 3.4.7 The function ~ E IC and the one form d~ have the same relative degree. As a consequence, ~ is autonomous if and only if d~ is autonomous. P r o p o s i t i o n 3.4.8 The set at of autonomous elements of g is a subspace of s Proof." Using Proposition 3.4.6, the proof becomes straightforward. Consider two vectors in A, their sum still has infinite relative degree. The same holds for the product of an element in at by a scalar function in/C~.

32

CHAPTER 3. ACCESSIBILITY

3.5

Accessible s y s t e m s

Let us state now formally the following Definition.

Definition 3.5.1 The system (3.1) is said to satisfy the strong accessibility condition if ,4 = 0

(3.12)

or, equivalently, there does not exist any nonzero autonomous element in/C. A practical Criterion for evaluating accessibility is given as follows.

Accessibility Criterion: Computation of ,4 Let us define a filtration of g, i.e. a sequence of subspaces {7/h} of g such that each 7/k, for k > 0, is the set of all one-forms with relative degree at least k. The sequence is defined by induction as follows: Ho

=

span~c{dx,du},

~t'~j

:

{~ E qF/j_l [ ~ E ~'~j--1}.

It is clear that this sequence is a decreasing one, i.e. g D 7/0 D 7tl D 7/2 D - 9 9 and that, at the first step, we have 7/1

=

spanx:{dx}.

An easy consequence of the construction is the following.

Proposition 3.5.2 7/k is the space of one-forms which have relative degree greater than or equal to k. Furthermore, there exists an integer k* > 0 such that: 7/k DT/k+1, f o r k < k * , 7/k'q-1 : 7 / k ' + 2 -" ' ' " -- 7/co

By definition, it follows that A = 7/oo. The existence of the integer k* comes from the fact that each 7/k is a finite dimensional/(:-vector space so that, at each step either the dimension decreases by at least one or 7/k+1 = 7/k. Systems which satisfy the strong accessibility condition get now an easy algebraic characterization [3].

3.6. CONTROLLABILITY CANONICAL FORM

33

T h e o r e m 3.5.3 The system (3. i) satisfies the strong accessibility condition if and only if ?'/o~ = 0 (3.13) The condition (3.13) is locally equivalent to the fact that the strong accessibility distribution s spans the whole tangent bundle T M to the state manifold M, where the strong accessibility distribution s is defined as the limit of a filtration 0CA1C...CA~C...CTM. of involutive distributions Ak given by

Ak = g + adlg + ... + ad)g The remarkable require to work easier than that required and no

3.6

fact is that the condition given by Theorem 3.5.3 does not with exact forms only. The practical construction of ~ k is of Ak, since a low number of purely algebraic computations is involutivity condition need to be considered.

Controllability Canonical Form

Although 7{k is in general not closed, i.e. it does not admit a basis which consists only of closed forms, the limit A = H ~ turns out to be closed. This follows from the fact that locally A•163

and it is stated in the following Proposition ([78]).

Proposition 3.6.1 Let {a;1, ...,wr} be a basis for A, then dwi AWl A...AtOm = 0, 1 < i < r

(3.14)

From Frobenius Theorem, there exist locally r functions, say (1, ..., (~, with infinite relative degree so that A -- spanjc{d~l, ..., d(r}

CHAPTER 3. ACCESSIBILITY

34

Since A is invariant under time differentiation, one has in particular

~

=

1~(~,...,~)

: ~r

=

(3.15) h(~l,'",~r)

Now, choosing n - r arbitrary functions, say ~ r + l , ' ' ', &~, so that X = spanK:{d~l, ... , d~,~} where X denotes span~: {dx}, one derives a representation, called controllability canonical form, of system (3.1) of the form

~1 = ]1(~1,'",~) 9 ~r ~r+l

= :

~,, =

fr((1,'",(~) /r+l(~l,''',~n)-t-gr+i(~l,''',~n)

:,(~,...,~,)

(3.16) u

+a,(~,...,~,,),,

Remark that, as a consequence of the state elimination results in section 2.2, the 41, 9 9 (~ are autonomous elements which satisfy a differential equation of order less than or equal to r.

3.'/"

Controllability

Indices

Define, now, hi = d i m ' / / / - dim~di+l, for i >_ 1 Moreover, hk. is non zero and hk = 0, for any k > k*. The set of controllability indices { k ~ , . . . , k~} of system (3.1) are defined as the dual set of { h 0 , . . . , hk.} by means of the relations hi = card{k;lk; _> i}

k; = card{hdhi >_j} for j = 1 , . . . , m . In particular, k* = m a x { k ~ , . . . , k~n}.

35

3.7. CONTROLLABILITY INDICES

P r o p o s i t i o n 3.7.1 For system (3.1) one has k~ + . . . + k,n = n - dimA

(3.17)

Note that in the nonlinear setting the controllability indices describe only the structure of the accessible subsystem. It is not possible in general to display them as in the linear case by means of a Brunovsky canonical form. E x a m p l e 3.7.2 Consider the unicycle described in Figure 3.2 below whose

X2

Xl

Figure 3.,2: The unicycle

36

C H A P T E R 3. A C C E S S I B I L I T Y

state representation is

COSX3 U1 ] &=

sin x3 ul

9

tt2 Compute 7/1 = 7/2 = 7/3=0.

span~c{dx} s p a n j c { ( s i n x 3 ) d x l - (cos xa)dx2}

T h e controllability indices are c o m p u t e d as follows, hi = 2, h~ = 1, h3 -- 0 , . . . and k~ = 2, k~ = 1. However there does not exist any change of coordinates t h a t gives rise to a representation containing a Brunovsky block of dimension 2. T h e s y s t e m is accessible, there does not exist any a u t o n o m o u s element. Example

3 . 7 . 3 Consider

X4

I 1 00 x3 X4

0

ul)

U2 Xn--1

xn 0

(3.18)

0 1

then, c o m p u t e 7/2 --- spanpc{x3dxl - d x 2 , . . . , xndXl - d x n - 1 } and m o r e generally, for 2 < k < n - 1, 7/k = spanpc{x3dxl - d x 2 , . . . , x , ~ - k d x l - d x n - k + l } , 7/~ = 7ioo = 0 Thus, hi -- 2, h2 = 1, h3 = 1 , . . . , h,~, = 1, hn = 0 and k~ = n - 1, k~ = 1.

Chapter 4

Observability 4.1

Introduction

The notion of observability of a system, linear or nonlinear, concerns the possibility of recovering the state x(t) from the knowledge of the measured output y(t), the input u(t) and, possibly, a finite number of their time derivatives y(k)(t), k > 0 and uq)(t), l >_ 0. The structural property which can be easily characterized in a nonlinear framework concerns the existence of an open and dense submanifold of the state space ~ n around whose points the system is locally observable. Thus, the situation is quite similar to the one pertaining to controllability. For illustrating the notions of observability we will consider, in this Chapter, a system of the form

{x

= f(~)+g(~)~ y

=

h(x)

(4.1)

where z E ~ n , u E / R m, y E ~P and the entries of f, g and h are meromorphic functions of x. We will sometimes ignore the presence of inputs in (4.1).

38

4.2

C H A P T E R 4.

OBSERVABILITY

Examples

E x a m p l e 4.2.1 = Y =

0 x~

(4.2)

Clearly, this system is not observable in a neighborhood of the origin. It is not possible, in fact, to distinguish the positive or the negative value of the state just from the knowledge of the output. If one has the additional information that the state belongs to an open neighborhood of some point x0 which does not contain the origin, then the value x of the state may be deduced from the value of the output y: x=sgn(xo)x/~ where sgn(.) denotes the sign function. Then, the system (4.2) is locally observable around any point different from the origin. E x a m p l e 4.2.2 Let q be an integer, q _> 1. --

x2q

Y =

x2

(4.3)

As in Example (4.2), the knowledge of y at time t is not sufficient to deduce the value of x at time t. However, y = 2x 2q+1 and so, 0,+ x=

4.3

if y = 0 , ify~0

Observability

Complementary studies of observability of nonlinear systems may be found in [40, 66, 116, 146, 147]. The notion of observability which gets a nice characterization involving only the structure of the system is the so-called local weak observability. It is defined using the notion of (in-)distinguishable states [66].

4.3. O B S E R V A B I L I T Y

39

D e f i n i t i o n 4.3.1 Two states xl(to) and z2(to) are said to be indistinguishable if, the corresponding outputs yl (to, x 1(to), u(t) ) and y2 (to, x2(to), u(t)) are equal for any t > to and any admissible input function u(t). E x a m p l e 4.3.2 Consider the unicycle in Example 3.7.2: -

[ cosx3 ul ] sin x3 Ul U2

with output y = zl. Let x 1 = (0, 0, 0) and z 2 = (0, z, 0). The states z 1 and z 2 are indistinguishable for any z E / R since

y l ( t o , z l , u ( t ) ) = y2(to, z2, u(t)) =

f; (J;) cos

u2(a)dtr

ul(r)dr.

This definition of indistinguishable states induces an equivalence relation on /R n. Denote by I ( z ~ the equivalence class, or coset, of x ~ 6 / R n. Two states may however be distinguished after a long (even infinite) time. Thus, the definition of observability has to be strengthened for practical purposes. It is achieved constraining the state trajectory to remain in a given neighborhood and yields a notion of local indistinguishability, or//-indistinguishability. D e f i n i t i o n 4.3.3 Let U be a subset of l~ n. Two states x 1 E / / and z 2 E / / are said to be U-indistinguishable

9 if the corresponding outputs yl and y~ are equal for any t >_ to and any admissible input function u(t) 9 and if the trajectories za(t) and z2(t) remain i n / / f o r any t > to. This definition of//-indistinguishability does no more induce any equivalence relation on /Rn since it is no more transitive [66]. Denote however by I u ( z ~ the set of states which are indistinguishable from x0. It yields the following definition for local weak observability. D e f i n i t i o n 4.3.4 The state z ~ is l~ally weakly observable if there exists an open neighborhood )2 of x ~ such that for any open neighborhood bl of :co contained in );, I u ( x ~ = {x~

40

C H A P T E R 4. O B S E R V A B I L I T Y

D e f i n i t i o n 4.3.5 The system (4.1) is said to be locally weakly observable if there exists an open dense subset M of IFIn such that any x ~ ~ M is locally weakly observable. In the rest of this book, local weak observability will simply be termed observability, In Chapter 15, an alternative definition of observability will be given, Both definitions express the fact that the state x can be recovered as a function of the output y, the input u and a finite number of their time derivatives.

4.4

The

Observable

Space

Let us denote by X , / / a n d y the spaces defined, respectively, by X = spanK:{dx}, U = span]c{du(J),j > 0} and y = Ul>0y~, where y i = spanx:{dy(J) ' 0 < j < i}. The chain of subspaces

OCOoC01C02C...COk

C...

(4.4)

where Ok := A' N (3;k + / / ) is called the observability filtration In the special case of linear systems, O~ reads as Ok = spanx:{Cdx, C A d x . . . . , CA~-~dx}. D e f i n i t i o n 4.4.1 The subspaee X f) (J2 + U) is called the observable space of the system (.~.1), The observable space can be computed as the limit of the observability filtration (4.4). This limit will be denoted by O~ and obviously we have

o ~ = x n (y + u). E x a m p l e 4.4.2 ~1

-~-

X2 "Jr X3tt

[c2

-=-

Xl

~3

=-

x2

y

=

Xl

4.5. OBSERVABILITY CANONICAL FORM

41

Compute ~1= x2 + Xau and y(2) = xl + xai* + x2u. So, O0 = XN (y0 + U )

=

spanlc{dxl}

O 1 - - X ("1 ( Y 1 "lt-/4r

--"

span~z{dxl,dX2+udx3}

O~=xn(y~+u)

=

eV

Thus, 0oo = X. 4.4.1

An observability

criterion

The general result characterizing the observability of system (4.1) follows from [66]. T h e o r e m 4.4.3 System (4-i) is observable if and only if O~ = X

(4.5)

Condition (4.5) is equivalent to the observability rank condition:

ranklc [O(Y, ~J, . " .,Y(n-1)) ] b~

----- n .

Thus, it reduces to the standard Kalman's criterion for observability in the special case of linear systems.

4.5

Observability

Canonical

Form

One shows that Or is closed and then, it has a basis {dxl, ..., dx~}. Complete the set { x l , . . . , x r } to a basis { x l , . . . , x r , x r + l , . . . , X n } of X. Then, in these coordinates, the system reads as Xl

~--- f l ( X l ,

x~

= f~(~x,...,x~) +9~(x~,...,~,.)u

Xr+l

y

' 9 9 , X r ) -}" ffl ( X l , 9 9 9 , X r ) U

---- f r + l (X) + f f r + l ( X ) U

=

h(xl,...,*r)

(4.6)

42

CHAPTER

4.6

4.

OBSERVABILITY

Observability Indices

Define 0.i = dim O i - i Oi- 2

for i > 1. The dual list { s l , . . . , sp} of the list {o'1, 0"~,...} is defined as the list of the observability indices of system (4.1) by si = card{0"j > i} Conversely, Cry = card{si > j}. The integer 0"j represents the number of observability indices si which are greater than or equal to j. P r o p o s i t i o n 4.6.1 F o r s y s t e m (4.1), one has sl + . . . + s v = dimOoo

(4.7)

E x a m p l e 4.6.2 Consider the unicycle in Example 3.7.2 whose outputs are the coordinates (xl, z2): Yl " - X l Y2 -" X2

One computes (9o = span~:{dzl,dz2} and O1 = X. Thus, 0"1 = 2, cr2 = 1 and 81 :

2, 8 2 :

1.

A canonical form is given in chapter 6 which displays a splitting decomposition of system (4.1) into observable blocks whose dimensions equal the observability indices. This canonical form is obtained thanks to so-called generalized transformations.

4.7

Synthesis of observers

In this section, we give some results on the synthesis of a nonlinear observer for system (4.1) which is based on one hand on its linearization via output injection and a state space coordinates transformation and on the other hand on standard Luenberger observer design performed on the linearized system.

4.7. SYNTHESIS OF OBSERVERS

43

The use of an observer is necessary whenever the state employed in a feedback loop is not directly measurable. In contrast to the linear situation, observability of the system is necessary but not sufficient to assure the possibility of finding a suitable observer. This is due to the necessity of an inherent linear structure, which is characterized in the rest of this section. If the additional conditions which allow to design an observer with such structure are not fulfilled, it is possible to investigate alternative design techniques, giving rise to high gain observers [56] or to sliding mode observers [130] e.g., which are not considered here. Consider the system (4.1) and assume that it is single output (p - 1) and that it is observable. As a consequence, it has a single observability index which equals n. 4.7.1

Input-output

Injection

and

Linearization

Problem

Given the system (4.1), we want to find, if possible, a state space coordinate transformation (~l,...,~n) = r such that rank~--~Cx= n

(4.8)

and

(4.9) Y = ~1 The terms ~'i in (4.9) are a special form of input-output injection. A more general definition will be given in Chapter 6. 4.7.2

Observer

Design

1

System (4.9) has the form y

=

Cr

44

CHAPTER 4. OBSERVABILITY

where (C,A) is a pair of constant matrices in canonical observer form. An estimate ~ of the state ( is obtained from the following system: =

A~

+

T(Y, u)

+

K(~I - y)

(4.10)

where K is chosen so that the eigenvalues of matrix A+ K C are in the open left half complex plane. Thus, the estimation error d - ~ is asymptotically stable. From assumption (4.8), locally x = r , ~ ) and an estimate & for the original state x of system (4.1) is given by

~--- (~-l(~l,-..,~n).

4.7.3

Solution to the Linearization Problem by Inputoutput

Injection

The provided solution goes through the state elimination of system (4.1) as it has been presented in section 2.2. Let

y(n) = F(y, ~I, . . ., y ( n - 1 ) U , / l , . . . , U(~)) be the input-output equation of system (4.1). Define the following sequence of differential one-forms. ~0 := 0. Define the following induction for k : 1 , . . . , n: Fk : : Fk_ -

(4.11) Set F0 : :

F,

(4.12)

OFk ~ OF~ ~k = OY('~-k) dY + j=tZ-~OuJn--k) duj

(4.13)

If d~0k ~ 0, stop! If dwk = 0, then let ~k (Y, u) be a solution of: '~ 0~k d u ., ,~ dy+~--~-z--j=l

=

~ok

=

F.

for l < k < n - 1 ,

oy

Notation: Adu stands for Adul A du2 A 9- 9A dum. A necessary and sufficient condition,of the existence of a state coordinates transformation r is given in the next theorem:

4.7. S Y N T H E S I S OF OBSERVERS

45

T h e o r e m 4.7.1 The nonlinear system (4.1) is locally equivalent to the system (4.9) under a state coordinates transformation ~ = r /f and only if: d~k = 0

(4.14)

.for 1 < k < n, where the wk's are defined by (4.13). Theorem 4.7.1 is a special case of Theorem 4.7.2 below which will be proved in the following section.

4.7.4

Generalized Problem

Given system (4.1), the problem considered in section 4.7.1 can be generalized by looking for a generalized state space coordinate transformation (~1,..., ~n) = r u,/~,..., u(')) such that rank-~- = n and

41 =

(4.15)

~2+~z(y,u,/L,...,uO))

o-1 : 4o =

,u(,))

(4.16)

~n(y,u,u,...,u(')) y = (1 The terms ~i in (4.16) are a special form of input-output injection. An observer can be computed from the form (4.16) as well. 4.7.5

Observer

Design

2

System (4.16) has the form

4 = y=

A~+~(y,u,i~,...,u(')) C~

where (C,A) is a pair of constant matrices in canonical observer form9 An estimate ~ of the state ~ is obtained from the following system: = a ~ + ~(y, u, u , . . . , u (,)) + I((~l - y)

(4.17)

CHAPTER 4. OBSERVABILITY

46

where K is chosen so that the eigenvalues of matrix A + KC are in the open left half complex plane. Thus, the estimation error ~ - ( is asymptotically stable. From assumption (4.15), locally x = r ,~n, u, u , . . . , u0)) and an estimate & for the original state x of system (4.1) is given by

4.7.6

S o l u t i o n to G e n e r a l i z e d P r o b l e m

The solution is stated in terms of the input-output equation (4.11). Define the following sequence of differential one-forms. Set F0 := F, ~0 := 0. Define the following induction for k = 1 , . . . , n: := F

wa = ~

OFk dy

_I - G : [ m

+1

(4.18)

OFk

+ j=IZ OuCh-k+,)duJ .)

(4.19)

If dwk A du A d u . . . A d u ('-1) r 0, stop! If dwa A du A d u . . 9A du0-1) = 0, then let ~k(Y, u, ~ , - . . , u (~)) be a solution of:

O:;.dy+ ~

j=l

O~____2__du(,) k

Oju(S)

3

= Wk

for l < k < n - 1 ,

~n(y,u,i~,'",u (~)) = Fn N o t a t i o n : Adu (l) stands for Adu~0 A du~t) A . . . A du~ ). A necessary and sufficient condition of the existence of a generalized state coordinates transformation r is given in the next theorem:

The nonlinear system (4.1) is locally equivalent to the system (4.16) under a generalized state coordinates transformation "- r if and only if:

Theorem4.7.2

dwk A d u A d u . . . A d u

0-1) = 0

for 1 < k < n, where the wk's are defined by (4.19).

(4.20)

4.7.

SYNTHESIS

47

OF OBSERVERS

Before proving Theorem 4.7.2, let us consider an illustrating example. E x a m p l e 4.7.3 [C 1

--

X2

&2 -

0

y

=

9U 2

(4.21)

Xl

Compute the input-output differential equation: F

:

y(2)

2.0-/~

(4.22)

it

For s = 0 or s = 1, Conditions (4.20) are not fulfilled. Set s = 2, and apply the above procedure. One computes Fz := 2ili~/u. The differential form wl is derived from (4.19): wl

OF1

=

OFz

. dy+ O~) ~du -u- d y .

=

..

(4.23)

Condition (4.20) of Theorem 4.7.2 is fulfilled for k = 1. dwl A du Adit = 0. Function ~t (Y, u,/L, fi) is derived from (4.20): ~1 (Y, u, ~/, ii) -

2y.

(4.24)

For k = 2, one gets P2 = 2(it2/u 2 - i i / u ) , y. The differential form w2 is: OP2 dy

=

OP2 ,..

2(u~ _ ~ ) d y _ ? a f t

(4.25)

and verifies the condition of Theorem 4.7.2. Function ~2(Y, u,/t, ii) is derived from (4.20): ~2

=

2 ( ~/t2 -

/~) u "y

(4.26)

C H A P T E R 4. O B S E R V A B I L I T Y

48

System (4.2I) is then locally equivalent to the system:

~1 =

(2+ 2y-___~ U ii

/~2 Y :

(4.27)

{1

The generalized state coordinates transformation is ~ = (xl, x2u ~ - 2xl 9i*/u).

Proof of Theorem 4.7.2. Necessity Suppose that there exists a generalized state coordinates transformation = {(x, u,/~,..-, u ('-1)) which transforms system (4.1) into (4.16). Thus, F, which is equal to y(n) can be written as F = ~O1n - l ) Or- ~ n - 2 ) nt_... _1_~On = Fo. One gets FI:

FI

= F0 0u~,----7 ~

+ o1(y,..., y(~-

, u,.

9 -, u ( ' ~ - ~ + ~ ) )

The differential form w~ is given by (4.19): 501

-_

O&

Oy(n_l) dY"k ~dy+~'~

k

0&

(~)

~_ (n-l+w) dUj j=l ouj 0~~ du (s)

j----1 0U} ~)

3

T h e condition dwl A du A d~) A 99- A du ('-1) = 0 is then satisfied. The proof for steps 2 < k < n follows the same lines than for k = 1.

Sufficiency If the condition of Theorem (4.7.2) is satisfied, then let ~o(y, u,/L,-.., u(')) be given by (4.20). From the definition of ~01,..., ~',-1, one has

~i = ~i+1 + ~oi, for i = 1 , . . . , n -

1

4.7. SYNTHESIS OF OBSERVERS

49

Compute ~n:

L = y(") . .~(.1_).1 . . ~(L)2 . .

~l-(n-~)

(4.28)

From (4.20) and from the definition (4.18) and (4.20) of the functions F~ and ~ (1 < i < n), y('~) reads as: y(n) = ~ § ~a(1)1 + ... + ~'1 ~('~-~) and the result follows:

(4.29)

Chapter 5

Systems Structure Our study of accessibility was based on the notion of relative degree of a single function which may be viewed as an output. In this interpretation on the relative degree, it represents the delay existing between the control input and the output function: more precisely, it is the order of differentiation which has to be applied to the output to have explicit dependence on the input. This corresponds to the structure at infinity in the single output case. More generally, the structure at infinity of a nonlinear system describes, roughly speaking, the delay structure existing between the input and the output in the multivariable case. The structure at infinity of a nonlinear system will be formally defined in this Chapter and studied together with an algorithm for its computation. The algorithm which gives the structure at infinity is known also as inversion algorithm, since, for invertible systems, it expresses the input as a function of the output, its time derivatives and possibly some states. Thus it may be viewed as a control algorithm which directly computes the input necessary for generating a desired output function. In practice, control methods like trajectory tracking or computed torque control in robotics are special cases of application of the inversion algorithm.

CHAPTER 5. SYSTEMS STRUCTURE

52

5.1 5.1.1

Structural Structure

Indices at infinity

Given a system

E = { ky :: f(x)h(z)+g(z)u

(5.1)

where z E / / T ~, u E JT~m , y E / R p and the entries of f, g and h are meromorphic functions of z, one can naturally associate to ~ the chain of subspaces G0 C G1 C ... C Gn of G defined by G0 =

spanlc{dz}

(5.2) Gn = D e f i n i t i o n 5.1.1 {~k,k = 1 , . . . , n }

spantc {dz, d~t,..., dy(n) }.

Given the chain of vector spaces (5.2) the list of integers defined by crk = dimtc

& Gk-1

(5.3)

is called the s t r u c t u r e at i n f i n i t y of The list given by (5.3) contains structural information on the system that plays a crucial role in the solution of many control problems ([20], [37] and [117]). The list {sk,k = 1 , . . . , n } defined by 8 1 - - 0"1,

8k = O'k--O'k_l, k - ~ 2 , . . . , n

(5.4)

describes the zero at infinity of ~ as follows. sl is the number of zeros at infinity whose order equals 1; si is the number of zeros at infinity whose order is less than or equal to i. The list of orders of the zeros at infinity {n~} is derived by duality from the list {si}: s, = card{n+~ I n~ _< i}

5.2. S T R U C T U R E A L G O R I T H M

53

Note that the relative degree of a single-output Yi equals the order of the zero at infinity of the single-output system { x Yi

= =

f(x)+g(xlu hi(x)

An algorithm for computing the structure at infinity is discussed in Section 5.2. D e f i n i t i o n 5.1.2 The essential order hie of the scalar output component Yi is defined by

ni~ = min{k E iS/ I ) ,.. .,dy(n)} } dy}k) ~ spanx:{dx,d~),... ,ay, - ~ k - 1 ) . ,uyj#i,dy(k+l ..(k)

5.2

Structure

(5.5)

Algorithm

The structure algorithm, or Singh's inversion algorithm, is a fundamental tool in the study of nonlinear dynamical systems. It results from a sequence of papers initiated by Silverman's inversion algorithm for linear time-invariant systems [128], which received a first generalization to nonlinear systems in [67]. The extension in [129] is presented under the following form in [37]. Given the system (5.1), Step 1 Compute

Oh(x)

-

=

0x (I(x)+g(x)u)

al(x)+bl(x)u

Define Pl = rank bl(x). Permute, if necessary, the components of the output so that the Pl first rows of bl(x) are linearly independent over )U. Denote ~1 the vector consisting of the first Pl rows of y and denote by Yl the other p - pl rows, so that

Since the last rows of bl(x) are linearly dependent upon the first Pl rows, we can write L Yl = al(X) -~- bl(x)u

54

C H A P T E R 5. S Y S T E M S S T R U C T U R E

where the last equation is affine in Yl. Finally, define/)l(z) := bl(x). Step k + l

- ... , ~i ~) , yl ~) have been defined so that Suppose that in Steps i through k, Yl, L Yl

=

(~I(X) + b l ( X ) u

. . . . . .

+~(~,

yl ~)

=

~,

,Yk-1

. . . , ~,?-~),

. . . ,

~i~)(~,~l,...,~i"),.

~Yk-1]

~i~))~,

,~i~))

Namely, t)}j) and 9(k) are meromorphic functions in K;. Suppose also that the matrix/~k := [~T, 9 " ", ~T bk] T has full rank equal to Pk. Then compute k

~(k+l) = 0~)(k) Ox

k O:(k)

i = l j = i IOY}j)

'

and write it as

y~k+l)

---- ak+l(X,~]}j) ' 1 < i < k , i < j <_ k + 1) +bk+l(x, ~)}J), 1 < i < k, i <_ j <_ k)u.

T T , and Define Bk+l := [B~Tk , bk+l]

pk+i := rankBk+l ^(k+l)

Permute, if necessary, the components oi Yk so that the first flk+l rows of Bk+l are linearly independent. Decompose ~(k+l) as ,3(k+l) =-

:(k+l) / Yk+l Yk+l

55

5.2. S T R U C T U R E A L G O R I T H M

where ~(k+l) k+l consists of the first (Pk+l - Pk) rows. Since the last rows of Bk+l are linearly dependent on the first Pk+l rows, one can write L

=

•)(k+•) k+l •(k+l) k+l

=

--

5k+t(x,~} j ) , l < i < k , i < j < k + l ) __ -~-bk-bl(~, ~ffj), 1 < i < k , i < j < k)u fl(k+l)(x, f l } i ) , l < i < k + l , i < j < k + l ) k + l

--

--

- T , bk+l] ~T T 9 Finally, set Bk+l :~--"[Bk The algorithm stops when

rank[0(y, Yl,-.., ~)(~_+1))]/Ox = rank[0(y, b l , . . . , ~)(k))]/0x. The orders of the zeros at infinity, as well as the essential orders, may be computed from the following inversion equations for k > 1: "",

+bk(x, f h , . . . , ~ 1

"'',Yk-1

'Yk-1)

,'',Yk-1

(5.6)

~u

The list of the orders of the zeros at infinity is given by the list of the smallest orders of differentiation of yi, i = 1, . . . , p that occur in (5.6). The essential order ni~ equals the largest order of differentiation of Yi in (59149 Actually, the structure algorithm computes a basis for the chain of subspaces (5.2). T h e o r e m 5.2.1 [37] A basis for gk is given by {dx,dy 1, ' " , d~k), " ' ' , ~~l~(k-1) .-(1r .-(k)~ k - 1 ,oY~-I,0Yk Proof. Note that for k = 1, {dx,dy 1} is a set of generators of s since d~l E span~: {dx, d~l }. Since bl (x) is full row rank, the set also consists of independent vectors and it is thus a basis for s The rest of the proof may be done by induction and is reported in [37]. | It follows from Theorem 59 that the lists {sk,k = 1 , . . . , n } and {Pk, k = 1 , . . . , n} coincide. In particular Pn is the rank of the system E.

CHAPTER 5. SYSTEMS STRUCTURE

56

E x a m p l e 5.2.2 Consider the unicycle in Example 4.6.2. The inversion algorithm yields

31

=

Y2 =

(cos x3)ul

(5.7)

(sin x3)ul = (tan x3)yl.

(5.8)

The second step of the algorithm gives

i~ = (tan ~3)ih + (1/cos 2 ~3)ylu2 and the algorithm terminates. The input is then obtained as

ul

=

91/cosx3

us

--

[~-

(tan x3)~)l]cos 2 x3

(5.9)

yl The space gl admits the basis {dx,dyl} and the space g2 admits the basis {dx,d31,d~)l,d~2}. The unicycle has one zero at infinity of order 1 and one zero at infinity of order 2. Both the essential orders equal 2.

5.3

Invertibility

System inversion is a long standing problem. It is appealing since, whenever an inverse system exists, it may give the control law that generates any desired output trajectory. Definition 5.3.1 Right inverse The system

{ ~ = u =

F(z,y,9,..,~(~)) H(z,y,y,...,y(V))

(5.10)

is a right inverse system for system (5. I) if the output y(t) of (5.1) equals the input y(t) of (5.10) whenever the input u(t) of (5.1) is chosen as the output of

(5.1o). The definition of a left inverse is obtained when the role of (5.1) and (5.10) are exchanged.

5.3. INVERTIBILITY

57

D e f i n i t i o n 5.3.2 L e f t i n v e r s e The system

{ ~ = F(z,y,i~,...,y(~)) u

=

(5.11)

H(z,y,3,...,y(~))

is a left inverse system for system (5.1) if the output u(t) of (5.11) equals the input u(t) of (5.1) whenever the input y(t) of (5.11) is chosen as the output of (5.1). D e f i n i t i o n 5.3.3 System (1.1) is said to be left- (resp.

right-) invertible if

there exists a left (resp. right) inverse system. P r o p o s i t i o n 5.3.4 [45] System (1.1) is left- (resp. right-) invertible if and

only if its rank p equals m (resp. p). E x a m p l e 5.3.5 Example 5.2.2 continued. From (5.9), a left inverse system of the unicycle is ,)

[#2

=

(tan ~)/h]~os

-

2 ,7

Yl ul

=

yl/cos

u2

=

[#2 - (tan .T})#IJcos 2 Yl

Using (5.8), one gets x3 = arctan(y2/yl) and thus, a state free inverse system exists: ~1

=

I

v~+y~

yl #2 - y2#1

E x a m p l e 5.3.6 Consider a linear system whose transfer function equals (s + 1)/s2: Xl

--

X2

&2

=

u

y

:

xl -4-x2

58

C H A P T E R 5. S Y S T E M S S T R U C T U R E

The inversion algorithm yields ~) = x~ + u and thus, a "reduced" inverse [148] is =

u

=

- 7 / + ~)

This one dimensional reduced order inverse system embodies the presence of one transmission zero.

5.4

Zero dynamics

It is shown in [80] that there exist three different notions which generalize, in the nonlinear context, the concept of transmission zeros of a linear time invariant system. One of them, namely the notion of output zeroing, provides a tool for studying local stabilization problems by means of static state feedback for nonlinear systems (see [10]). On the other hand, the notion of zero dynamics, defined below in accordance with what mentioned in Example 5.3.6, plays a crucial role in dynamic feedback control problems, particularly in the dynamic feedback linearization problem (see Section 9.2). D e f i n i t i o n 5.4.1 Consider systems (5.1) and assume that p = m. The zero dynamics of system (5. i ) is defined by the dynamics of a reduced inverse. The structure algorithm immediately yields the following. P r o p o s i t i o n 5.4.2 I f the system is invertible, then the dimension of the zero dynamics equals n E i 2=m = I hi,, where the ni' denote the orders of the zeros at infinity. -

Chapter 6

System Transformations In dealing with nonlinear systems of the form (3.1), one is interested in considering a larger class of transformations than the one used in the linear case. Adopting a differential geometric point of view, diffeomorphisms and (nonlinear) static state feedbacks provide a natural choice in many interesting situations. In the differential algebraic approach a more general class of transformations, involving not only the state and the output, but also a finite number of its derivatives, has proved to be appropriate. For this class of transformation it is possible to parallel, up to a certain extent, the procedure that in the linear case leads to the Morse canonical form. In order to describe this, we introduce in this Chapter the notions of generalized state-space transformation and of regular generalized state feedback. Regular generalized state feedbacks are a special case of quasi-static feedbacks, which have been introduced and studied in [25, 27].

6.1

Generalized State

The notion of generalized state has been introduced in connection with dynamic representations which involve a finite number of derivatives of the input and arise as solution to the realization problem in a differential algebraic context [48]. It also appears as a natural tool for deriving a canonical splitting

CHAPTER 6. S Y S T E M TRANSFORMATIONS

60

decomposition of nonlinear systems [117]. Given system (3.1), ~ is a generalized state if there exists an integer k, functions r and r such that one has locally:

r r r162 r162

u, K . . . , u(k)) ~, ~,..., ~(k))

u , u . . . . ,u(k)),u,~,...,u(k)) ~, ~,.. ,, ~(k)), ~, ~ , . . . , ~(k))

This is formalized in the following definition. D e f i n i t i o n 6.1.1 A generalized state-space transformation is a map

T : (x, u, i~,..., u(k),...) ~ (~, u, iL,..., u(k),...) such that dim~ = dimx and for some k it holds span{dr} span{dx}

C C

span{dz,du,di~,...,du (k)} span{df,du, d u , . . . , d u (k}}

Letting X = span{dz} a n d / / = span{du, d/L,..., du(~),...}, a generalized state-space transformation gives rise to an isomorphism r : E = X @ / / - + ,~ = W $ / / s a t i s f y i n g the conditions

i) ~ ( x ) ~ u = x 9 u ii) r(X) is a closed subspace of s In particular, this means that there exists n elements ~1, ~2,. 9 ~n E JC such that r(X) = span{d~l , d ~ , . . . , d ~ n } and 0(~l,~2,...,~n)/cg(xl,x2,...,zn) is generically nonsingular. E x a m p l e 6.1.2 Consider again the unicycle in Example 3.7.2. The relation

~2

~

~3 =

X2

(sinx3)ul

6.2. QUASI-STATIC STATE FEEDBACK

61

defines a generalized state-space transformation in the sense of Definition 6.1.1, whose inverse is described by:

6.2

Regular

X2

~

~2

xa

=

arcsin((a/ul).

Generalized

Quasi-Static

State

State

Feedback

and

Feedback

In this section, we aim to generalize the notion of unimodular dynamic feedbacks, known for linear systems. Such a notion has been introduced for the first time for nonlinear systems in [117]. D e f i n i t i o n 6.2.1 A regular generalized state feedback is defined by

u = ~(z, v, i4..., v (k)) such that there exists a function r which yields

u = ~,(x, r r

r

** v = r

~, ~,..., ~,(k))

Definition 6.2.1 implies that the feedback ~ is invertible. It defines a feedback group of transformations. A regular generalized state feedback is a map

F : (x, u, u,..., u(k),...) _+ (~, v, ~,..., v(k),...) such that for some k it holds span{dr}

C_ span{dx,du, d / L , . . . , d u (k)}

span{du}

C_ span{dx,dv,di~,...,dv (k)}

As a consequence, a regular generalized state feedback in the sense of Definition 6.2.1 is a special case of quasi-static state feedback as defined in [25, 27]. In the sequel the term quasi-static state feedback will be preferred and used to denote the feedback transformations we consider.

62

CHAPTER 6. SYSTEM TRANSFORMATIONS A quasi-static state feedback gives rise to an isomorphism

~:g=XOU~g=X@U such that

ii) ~(U) is a closed subspace of g This implies, in particular, that O(vx, v2,..., Vm)/O(Ut,U2,..., Urn) is generically nonsingular. E x a m p l e 6.2.2 ul u2

= =

vl v2 + / q

(6.1)

is a quasi-static state feedback since it is invertible in the sense that it is equivalent to Vl

~

U1

V2

~--

U7 -- fil

E x a m p l e 6.2.3 [26] Consider system & = u, the set of equations { u

Y

=----

X

does not define a (regular generalized state) feedback in the sense of Definition 6.2.1. However, the first single equation u = b is a feedback, but it can not be solved in v; thus, it does not define a regular generalized (or quasi static) state feedback either.

6.3

Output Injection

Concerning o u t p u t i n j e c t i o n , it could be reasonable, at this point, in order to generalize the notion used in the linear framework, to start by considering additive assignments of the form x ~+ x + f(y,..., y(7)), involving nonlinear functions of y and its derivatives. Actually, the transformations of this class may have undesired properties, since they could change the observability of the system, and the introduction of some restrictive conditions is therefore required. However, we will not undertake now the task of developing an appropriate

6.4. C A N O N I C A L F O R M

63

generalization of the notion of output injection, since for carrying on, up to some extent, a construction similar to that of the Morse canonical form we do not need to allow, for a given system, alL possible transformations induced by an assignments of the form z ~ z + f ( y , . . . , y(~)). It will be sufficient, in fact, to use only very particular output injections of the above kind for proving the following result of [117]. A deeper discussion about the generalization of the notion of output injection will be the object of Section 6.6.

6.4

Canonical

Form

With the terminology introduced above, we can now state the following result. T h e o r e m 6.4.1 Given the system = =

y

f(x) +a(x)u h(x)

(6.2)

where the components o f f ( . ) , g(.) and h(.) are meromorphic functions of x on an open subset l) C_ IRn such that generically rank g(x) = m and rank ~ = p , there exist a generalized state space transformation, a generalized state feedback and an output injection that transform the system into the following form [117], [118]

{

r = r =

Ar f(r162

y

Cr

=

(6.3)

Remark that ~' is linear from an input/output point of view and that the new input v is obtained in terms of z, u and of a finite number of derivatives of u. Proof: The proof of the statement is constructive and goes as follows.We start by constructing a suitable set of variables ((, v) by means of the following procedure: S t e p 0 define (o := Y

CHAPTER 6. SYSTEM TRANSFORMATIONS

64

S t e p k choose a subset of (r

vk) of the components of (k-1 such that

i) span{dx, d~),..., dy (~) } = span{dx, d v l , . . . , a,v 1( k - l ) ,,av2,..., dv~ k-2) ,... ,dvk}; ii) {dr d~l, d v l , . . . , dv[ k-l), d~k, dvk} is a basis for span{dx, d y , . . , dy (k)}. The procedure stops when (Ck+l, vk+l) is empty, namely when 9

span{dx,dy,..., dy(~+~)} =

span{am, d v l , . . . , dv~k),..., dvk, d~)k} and

9 {d~'0, d~l, d r 1 , . . . , dv~k), dCk, dvk, d~)k} is a basis for span{dx, d~),..., dy (k+l) } A practical way of implementing the procedure described above consists essentially in applying to the system the structure algorithm. Namely, at Step I, write ~0 = y = (Oh/Oz)(f(m) + g(x)u) =

y~2(x, u)

v13(x, u)

where

9 Oy~(z, u)/Ou is full row rank and rank Oyez(x, u)/Ou = rank Oy/Ou 9 O(y, y12)/O:e = is full row rank and rank O(y, Yl2)/Ox = rank O(y, Y12, Yl3)/Ox Then, define Vl

:=

Yll (X, U)

r

:=

y12(x,u).

(6.4) (6.5)

It is easy to check that (~1, vl) verify conditions i), ii) above. At Step k, after reordering if necessary, write

(k-l)

~k-1 ~-~ Y(k-1)2(X,U, V l , . . . , V 1

=

where

,V2,-..,

v~~-2)

yk2(x,u,...,~k-1) yka(x, u , . . . , ~k-1)

,..-,

vk-1, ~k-1)

6.4. C A N O N I C A L

65

FORM

9 Oy~l/Ou is full row rank and

rank 0(y11,..-, y(k-1)l, ykl)/Ou = rank 0(y11,..., Y(k-1)1, i/(k-1)2)/Ou 9 cOyk~/COx is full row rank and

rank O(y, Y I 2 , . . . , yl,2)/Ox = rank O(y, Y12,..., yk2, yk3)/Ox Then, define vk := Ykl(X, u , . . . , ~)k-1) Ck := yk~(x, u, . . ., i~k-1)

(6.6) (6.7)

It is easy to check that (~k, vk) verify conditions i), ii) above. The set of variables (~0,.--,~v, v t , . . . , v v ) obtained when the procedure stops can be completed by adding block variables ~+~ = v~+~(u) ~'~+1 := ~ + l ( x )

(6.8)

(6.9)

in such a way that the transformation (~0,...,~v+l, v l , . . . , v~),,...,v~,,i~,,v~,+l) - r

(~'))

(6.10)

defined by the equations (6.4), (6.5), (6.6), (6.7), (6.8), (6.9) is a diffeomorphism, linear in u, on some open subset of/R ('~+m(v+l)). Solving for u, it yields a regular generalized state feedback described locally by u = a(x, v , . . . , v(k)),

(6.11)

that satisfies the conditions span{dr} span{du}

C_ span{dx, du, d u , . . . , d u (k))

C_ span{dz,dv, d~,...,dv(k)).

In order to display the effect of the transformations (6.10) and (6.11) on the original system, let us remark that, when the procedure stops after v steps

86

C H A P T E R 6. S Y S T E M T R A N S F O R M A T I O N S

(that is when Y(v+l)l and Y(~+1)2 are empty), one has a partition of the vector (0 , or equivalently of y, into 2u + 1 blocks (Oh, for 1 _< h < 2u + 1, whose dimensions correspond, for 1 < h < v, to those of the blocks Vh, or equivalently Yhl, and, for u + 1 < h < 2v + 1, to those of the blocks Y(2v+2-h)3 respectively (note, in fact, that we have from the above procedure v blocks ykl and u + 1 blocks yk3, some of which may obviously be empty) 9 Then, expressing the system in the new variables, we get

~Oh

"-

(nhh

~

Yh

Yh

=

~0h for l < h < v

,hh

Yh

~lh

= :

(Oh for u + 1 < h < 2u + 1

=

Now, recall.ing how the variables ~0,..., ~', have been defined in (6.5), (6.7) the relations ~nhh = fh((O,...,(~,) can easily be modified by the particular output injection defined by the assignment

6.4. CANONICAL FORM with l < i < v , l _ < j _ < n h

~1!

=

67 andv+l
~Oh

=

~nh h Yh

=

Vh

=

~0a for

~Oh

:

~lh

~nhh Yh

= =

0 (oh for

In this way we have

(lh

l
(6.12)

v+l
]((, v,..., that, denoting (q'0,..., ~., r

by (~, ~), gives finally the desired form

{ ~ = A(+Bv = f((,r .... , v (v)) y=C( | We can now remark the following facts about the representation (6.12). The subsystems described by the first two blocks of (6.12), that represent the observable part of E", are invariant with respect to generalized state-space transformations and regular generalized state feedbacks. The first one contains the information about the algebraic structure at infinity of E, which corresponds to that contained in the list 11 of the Morse canonical form. Namely, for each h, 1 < h < u, there are as many zeros at infinity of order nh + 1 as dim (0h. In particular, with the notations of Section 5.1, we have nh -b 1 = crh -- eh-1. The list { n v + l , . . . , n 2 , + l } obtained by the second block coincides, if E is linear, with the Morse list/3. In order to decouple the last block, as done in the linear case, we should now make use of some sort of generalized output injection. Actually, as said previously, while the generalization of the notion of feedback we employed

CHAPTER 6. SYSTEM TRANSFORMATIONS

68

came in a quite natural way, the situation is much more involved for output injection. We will come back on this point in a next Section, after discussing some examples.

6.5 6.5.1

Some examples Example

The following system has been considered in [77] : ,~1

----

721

x2

=

x4 + u2

it3

=

x3721 + u 2

Yl Y2

-

-

=

xl X2 -- X3

Applying the the procedure described in Section 6.4, we obtain, at the various steps :

Step 0

~"01 :-- ~gl

Step 1

~)=

x4-x3ul

Step 2

~) --

?23 -- X37~l -- X3 u 2 -- U2Ul

~02 :--" X2 -- X3

Q

721

)

Yl : ~

721

(12:=

x4-xaul

V2 :----- ?23 -- X3~1 -- X3 722 -- U2721

The procedure stops, and we can complete the transformation by choosing (3 : - x4 and v3 := u3 so that ((, vl, v2, v3, t l ) = r ul, u2, u3, ul) is a diffeomorphism.

6.5. SOME EXAMPLES

69

As a result, the system takes the form

6.5.2

~01

-"

Vl

r

=

r

r

--

V2

~3

--

Y3

Yl

=

r

y2

=

r

xl

=

ul

5:2

=

z 4 + u2

5:3

=

u2

Yl

--

Zl

Y2

--

x2

Example

Let us consider the following system

_--

- -

X3

The system is decoupable by a regular static state feedback. Then, the transformation we get by applying the procedure described in Section 6.4 reduces to a usual state space transformation and a regular state feedback. At the various steps we obtain what follows :

StepO

Stepl

Step 2

r

=

xl

~02

=

X2 - - X3

Yl y2

=ul= = ~r4 :

~)1 = ~)2 :

yll(x,u)=vl y12(z, u) = i12

ul x3-/-u2:v2

The procedure stops, and we can complete the transformation by defining r

:'-- X3.

CHAPTER 6. SYSTEM TRANSFORMATIONS

70

The canonical representation now takes the form ~01

~-

Vl

(02 = r (12

6.6

---- Y2

(3

=

Yl y2

= (ol = (o2

v2/r

G e n e r a l i z i n g t h e n o t i o n of o u t p u t i n j e c t i o n

In order to obtain a complete analogy with the Morse canonical form in the nonlinear setting we are working in, we would now decouple (~+1 in (6.12) from ((0,..., (~) and ( v l , . . . , v~) and then, possibly, we would split the corresponding block into a controllable and a noncontrollable part. In decoupling, we would use only transformations that may viewed as a generalization of the notion of output injection, followed possibly by changes of coordinates. In the nonlinear framework, linear output injections have been used in [75]. An additive nonlinear output injection, similar to the one used in Section 3 has been employed in [95] for linearizing a nonlinear system, as well as in [64], [65] for transforming a nonlinear system into a bilinear one. In general, however, the problem of defining a quite general notion of output injection in a nonlinear framework has not received much attention in the literature. Here, we consider a class of output injections, which are not necessarily additive and whose definition is consistent with that of quasi-static state feedback formalized in [27]. The basic idea is that of considering transformations that modify the dynamics of a system of the form (3.1) by the assignment ~

t~(&,y,y,..., y(r)).

(6.13)

In other terms, this amounts to transform the system

E = ~ ~ = f(x) +g(x)u L y= h(x)

(6.14)

into

= { 5Cy == h(x).tg(f(x)+g(x)u, y, ~l,...,Y(r)), r E SV

(6.15)

6.6. GENERALIZING THE NOTION OF OUTPUT INJECTION

71

In the following, in order to make distinction between the derivatives of y along the trajectories of (6.14) and those along the trajectories of (6.15), we will denote the latter ones by y[~], whereas the notation y(k) will stand for the first ones. Clearly, some restrictions must limit the choice of the function O(f(x) + g(z)u, y, ~),..., y(r)), in order to avoid pathological situations. In particular, we would prevent the possibility that an output injection can change the observability properties of the system. Let us illustrate this point by considering the following example. Let us apply to the system xl x2 y

=

0 0

=

XlX 2

=

which is obviously not fully observable. Apply the linear, additive output injection defined by ~(x, y) = (zl + Y, 0) t. The resulting system is: ~I

~--- X l X 2

~2

=

0

y

:

XlX 2

which turns out to be completely observable (in the sense of [40]) since zl = y~/y[1] and z~ = yll]/y. Before going further, it is therefore useful to state formally, following [148], the notion of observability we will consider in defining the notion of output injection. Recall a few basic definitions from Chapter 4. D e f i n i t i o n 6.6.1 Given a system of the form (6.15), let X and gt denote, as usual, span{dx} and span{du(h),k > 0} respectively, and let Y denote span{dy (k), k > 0}. Then, the observability space of the system is defined as :

o=xn(y+u). Definition 6.6.2 A system of the form (6.14) is said to be observable if dim(X f3 (y +U)) = n, i.e. /fspan{dz} C span{dy(h),du (k), h >_ 0, k _> 0}. Now, given a system of the form (6.14) and a map of the form (6.13), we can consider the vector spaces 7-/k = span{dy, d y , . . . ,dy (k), du,d/L,...} and

72

CHAPTER 6. S Y S T E M TRANSFORMATIONS

7tk = span{dy, dy[1],..., dyM, du, du,...}. With these notations it is possible to state, as in [109], the following definition of o u t p u t injection. Definition 6.6.3 The map O ( x , y , y , . . . , y (r)) (r e ~W) that transforms the system (6.t4) into the system (6.15) is an output injection if the following conditions hold:

* /tk C Hk+r-1 9 Hk C/qk+~-i 9 9/9(x, y, !),..., Y(r))/C3x is generically invertible. The following Proposition, proved in [109], states that the output injection defined above behaves well with respect to the observability. P r o p o s i t i o n 6.6.4 Given a system of the form (6.14} and a map 0 of the form (6.13), let 0 denote the observability space of (6.14) and let 0 = Z 0 (3; + H), where y = span{dy [k], k >_ 0}, denote the observability space of (6.15). lf O is an output injection, then 0 = (P. Let us now go back to the system E" described by (6.12) at the end of Section 6.4. Denoting the block variables (ff0h. . . . , finch) for 0 < h < u, that is the variables of the first block in (6.12), by ~1, the block variables (ff0h, 99 ~nhh) for u + 1 < h < 2 u + 1, that is the variables of the second block in (6.12), by ~2 and the block variable r by r and analogously denoting the block variables (vl,...,vv) by wl and v,+l by w2, we can rewrite (6.12) as { ~1 ---- f1(~l,wl)

r

= =

Let us focus on the subsystem r = f3(r r r w , , . . . , w~~), w~))

(6.16)

corresponding to the unobservable variable ~3. The problem of decoupling ~3 from ~1, ~2 and wl by means of an output injection does not seem to be

6.6. GENERALIZING THE NOTION OF OUTPUT INJECTION

73

tractable in general. So, we limit our attention to the situations in which f3 has particular properties. Namely, we assume that, for each component f3i of f3, there exists a function Xi(r w2) such that:

/ ~ = F~(x~(r

w2), r r w l , . . . ,

w~")).

In other terms, we assume that there exists a s e p a r a t i o n f u n c t i o n of the observable variables and the unobservable ones. Then, we have the following two cases: either f3i does not depend on Ca and on w2, and we chose Xi(r w2) = 0, or (OFi/OXi) ~ O. Now, by the construction carried on in the proof of Proposition 6.6.4, ~a, ~2 and w~k) can be expressed in terms of y and its derivatives. Hence we can define, in the case in which f3i does not depend on r and on v2, o,(43,, v, 0 , . . . ,

In the case in which to get

r

= r

- F,(r

r

w l , . . ., , ~ , 1 ) .

OFi/Oxi r O, we can apply the implicit function theorem Xi : a i ( f 3 i ,

r

r

w l , . . . , w l t~))

and we define oi((~,

u, i~, . . . , C r ) ) = a , ( 4 ~ , , r

r

Wl, . . . , w[")).

The map 8 = (01,..., Oi,...) defines an output injection according to definition 6.6.3 which transforms (6.16) into

~3 = x((3, w2), where X = (X1,..., Xi,.. ,), achieving the desired decoupling and yielding a maximal loss of accessibility. Obviously, the separability condition seen above is only sufficient for assuring the possibility of decoupling (a by means of an output injection. In addition, the fact that such condition is not feedback invariant, as pointed out in [123], shows that a complete characterization of existence of an output injection with the desired property is still far from being obtained.

74

C H A P T E R 6. S Y S T E M T R A N S F O R M A T I O N S

6.7

Other

Examples

E x a m p l e 1. Let us consider the following system :

~'2 = V =

v2 G

(6.17)

The right hand side of (6.17) satisfies:

r

6 + ~2- Vl + r = F(x(r

v~), r

we have

F = ~(r

x-

OF

v~)r + r and N-X -- (2

fl - r ~2

~,(f~, v,) -

fl - y2 y

Let us transform this system into:

r Y

: :

v2 G

(6.18)

According to the previous notations, we search an output injection r Y, ~),...), which verifies definition 6.6.3 and transforms (6.17) into (6.18). We can take:

r

V) =

fl

- -

y

y2

6.7. OTHER EXAMPLES

75

So, we obtain the system :

41 =

~+vl

~2 Y

V2 G

~ =

Example 2. Consider now a case in which the non observable block cannot be split, system:

41 = 42 Y

= =

v2 G

(6.19)

Assume, in fact, that the right hand side fl of (6.19) can be written as fl = F(X((1, vl), ~2). Compute the partial derivatives i)

ii)

OF OX

fl

~vl

= I = -a--Xv---1 O

kTv

Now i) implies ~

j=~

# 0 while ii) and iii)

= 0 giving a contradiction.

There does not exist a "separation" function F of the observable variables and the unobservable ones. We can not decompose this system with an output injection.

Chapter 7

Input/Output Linearization The first problem we consider consists in linearizing by feedback the relation between input and output variables of a given nonlinear system. Input/output linearization is a basic technique in dealing with nonlinear control problems. 7.1

Problem

Statement

I n p u t / o u t p u t linearization problem Given the system E= ~ ~ = f(x)+g(x)u l u = h(x) where the state x E/R '~, the input u E/~rn, the output y G ~v and the entries of f, g, h are meromorphic functions, find, if possible, a regular static state feedback u = (~(x) + and a state transformation ~ = ~(x) such that, in the new variables, the compensated system is given by

fl(z)v

{ ~1 = AI~:+Blv: ~2 y

= =

f2(~:,~2)+a2(~:,~2)v , CI~ I

(7.:)

where vl is a subvector of v, the pair (AI, BI) is controllable and the pair (C1, A:) is observable.

CHAPTER 7. INPUT/OUTPUT LINEARIZATION

80

7.2

Single output case

The single output case of the input-output feedback linearization is certainly the most basic, and elementary, control scheme which lies in the heart of many of the various nonlinear control architectures developed below. It consists in cancelling, by feedback, the nonlinear terms appearing in the r-th time derivative of the output, r being its relative degree. In robotics this control scheme is well known under the computed torque denomination. An easy necessary and sufficient condition for the solvability of the problem can be stated in the following way. T h e o r e m 7.2.1 Assume p = 1, then the static state feedback input-output

linearization problem for Z is solvable if and only if its relative degree is finite. Proof: Sufficiency: Let r be the relative degree of the output and let hi(x) : - y(x), ..., h~_l(X) : y(r-1)(x) and vl := y(r)(x, u). One proves by contradiction that

rank O(h(x)'hl(x)''''h~-l(x)) = r

(7.2)

0x Assume that (7.2) is not satisfied, then without loss of generality, assume that dhr-1 E span~:{dh, d h l , . . . , dhr-2}. From the implicit function theorem, there locally exists r such that hr-l(X) = r The latter yields that dy(k) E span~:(dh, d h l , . . . , dh~-2} for any k > 0. Let (~11,..., ~1~) :- (h(x), hi(x),..., hr_l(x)) which can be completed to define a state transformation. In a similar vein, vl can be completed by (v2,. 9 vm) so that O(vl,..., vm) is invertible. The result follows.

Necessity:

Ou

Any controllable and observable linear system has finite relative degree. E x a m p l e 7.2.2 Let I xl x2 y

= = --

x2 u x1

|

7.3. MULTI O U T P U T CASE

81

Follow the above procedure and define ~11 = xl, ~12 = x~ and v = 2x2u. Note that the transformation ~(x)is meromorphic, whereas the inverse transformation ~-1 is not. The linearizing state feedback is u = v/2x~ and the linear closed loop system has transfer function 1/s 2.

7.3

M u l t i o u t p u t case

The above elementary solution can be easily generalized to multi output systems. The resulting condition becomes a sufficient condition. T h e o r e m 7.3.1 The input-output linearization problem for ~ is solvable if rank O(Y~r~)' Y(2~2)" Ou " "' Y(~)) = p

(7.3)

where ri denotes the relative degree of the output function hi, for i : 1 , . . . , p. Proof." Set vi : y~"')(x,u) for i : 1 , . . . , p . From (7.3), one can define v v + l , . . . , v m so that Ov/Ou is invertible. The result follows. | The matrix v, . . . . . . . .Ou . ,~v ~ appearing in condition (7.3) is usually called the decoupling matrix of E. Condition (7.3) is clearly not necessary. This is seen on the following linear example. E x a m p l e 7.3.2 Given the system

k2

---

~3 : Yl : Y2 :

X3 --[- U 1

(7.4)

u2 Xl x2

an easy computation shows that r a n k ~

= 1.

A necessary and sufficient condition for input-output linearization can be found in [82]. It may be stated as follows.

CHAPTER 7. I N P U T / O U T P U T LINEARIZATION

82

T h e o r e m 7.3.3 Assume that the system (1.1) is right invertible. Then, the input-output linearization problem is solvable if and only if

rankx:[O(u--~

i u ( , - 1 ) ) j = rank/R [ 0 ( ~ , ~ ,

:u(--1))J

Proof." Necessity: Condition (7.5) is clearly satisfied for the closed loop system. From the chain rule, it is also invariant under invertible static state feedback9 Sufficiency: Let us rewrite the structure algorithm in Section 59 when condition (7.5) is fulfilled. At each step k, one gets from (7.5)

. ~s where L denotes a real valued matrix of suitable dimension. Rather than computing the derivative of 9(k) one differentiates ak(x) at step k + 1. By this procedure, one computes a regular static state feedback as in the proof of Theorem 7.3.1, which linearizes the input-output relation. E x a m p l e 7.3.4 ~I

~

X3Ul

x2

=

x4+2x3ul

X3

=

X4

:

X3 U2

Yl Y2

= =

(7.6)

Xl x2

From the step 1 of the structure algorithm, set vl := x3ul and Y2 - x4 + 2yl. Differentiate x4 instead of Y2 and set v2 := u2. Consequently, the linearizing static state feedback is computed as ul = vl/x3 and u2 = v2.

Chapter 8

Noninteracting Control In section 7.3, the feedback which solves the input-output linearization problem also achieves noninteracting control, in the sense that for the closed-loop system the output component Yi is affected only by the input component vi, for i = 1,..., p. A similar decoupled form appears in the canonical form derived in Section 6.4. This is formalized and completed in this Chapter. The noninteracting control problem is a fundamental control problem whose solution allows one to tackle multivariable control and design problems using SISO techniques. Further technological motivation for trying to achieve noninteraction is that human supervision of complex systems, like industrial plants, advanced vehicles and so on, is greatly simplified if different components of the output behavior are controlled separately.

8.1

Noninteracting control via regular static state feedback

Problem statement Given the system

E = ( ~: = [ y =

f ( x ) +g(x)u h(x)

(8.1)

84

C H A P T E R 8. N O N I N T E R A C T I N G C O N T R O L

where the state x E iT/n, the input u E h
i) dy} a) E s p a n ~ { d x , d v , , . . . , d v } k ) } , k

> 0

(8.2)

ii) dy} n) g span~:{dx}

(8.3)

Condition (8.2) represents the noninteraction constraint and condition (8.3) ensures the output controllability in the closed-loop system. The solution to this problem is given by a statement similar to that of Theorem 7.3.1, but here the condition becomes necessary and sufficient. T h e o r e m 8.1.1 The noninteracting control problem is solvable via regular static state feedback if and only if the following two equivalent conditions are satisfied 9-qt~(rl) 9 (r2) y(prp) rankV~Ul ,82 , - . . , ) Ou = p 9 the relative degrees of the p scalar output functions are all finite and the list of relative degrees equals the list of orders of zeros at infinity. Proof: The sufficiency follows from the proof of Theorem 7.3.1. For the necessity, note that (8.3) implies that all the relative degrees are finite and condition (8.2) yields the independence of the inputs in y~r,). I

8.2

N o n i n t e r a c t i n g control via quasi-static state feedback

Problem statement

Given the system E (8.1), find, if possible, a quasi-static state feedback u = a(x, v, 9,...) such that, for every i = 1 , . . . , r n

8.2. QUASI-STATIC NONINTERACTING CONTROL

85

i) dy} k) e spanjc{dx,dvi,...,dv}k)}, k _> 0

(8.4)

dy}") ~ span~c{dx }

(8.5)

ii)

Recall that a quasi-static state feedback is invertible by Definition 6.2.1. T h e o r e m 8.2.1 The noninteracting control problem is solvable via quasi-static

state feedback if and only if the system is right-invertible. Proof: Necessity follows from condition (8.5); the decoupled system is necessarily rightinvertible. The sufficiency follows from the construction of the canonical form (6.12). II E x a m p l e 8.2.2 Consider the unicycle in Example 3.7.2 with the two outputs Yl = Xl and y2 = x2. The condition in Theorem 8.1.1 is not satisfied since rank0(yl, fl2)/Ou = 1. The system is right invertible however and it can be decoupled by means of a quasi-static state feedback. Such a solution is obtained following the procedure described in Section 6.4, derived from the inversion algorithm. One gets tt 1 ~ V l / C O S X 3 u 2 ----- c o s 2

xa(v2 - -

731 t a n x 3 )

The closed-loop system reduces to two decoupled linear systems, a first order system yl = vl and a second order one/)2 = v2. Quasi-static state feedbacks are viewed herein as a mathematical tool which describes standard decoupling dynamic compensators acting on extended state spaces. The main benefit in using quasi-static state feedbacks comes from the fact that they define a group of transformations, whereas the class of regular dynamic compensators does not.

CHAPTER 8. NONINTERACTING CONTROL

86

8.3

Noninteracting state feedback

control

via regular

dynamic

Problem statement Given the system (8.1), find if possible, a regular dynamic compensator

{~

= F(x, z) + C(x, z)v z e ~ u = .(~, z) + ~(., z)v

(8.6)

such that, for every i = 1 , . . . , m

i) dy}k) e spanr.{dx,dz, d v i , . . . , d v } k ) } , k >_ 0

(8.7)

ii) dy} K) g span~c{dx,dz } for some K e

(8.8)

T h e o r e m 8.3.1 The noninteracting control problem is solvable via regular dynamic static state feedback if and only if the system is right-invertible.

Proof: The proof mimics the proof of Theorem 8.2.1. Necessity is obvious, and sufficiency may be proved by deriving a standard dynamic compensator, a so-called Singh compensator, from the inversion algorithm. It yields y}~") - vi for any i = 1,...,p. These statements follow from the consideration of the inversion equations (5.6), (~',) (n,~ where the derivatives of the output range from Yi to Yi . If nie ~ n i, set zil = y~n"), 9 9 zi,,~,o_n, = y}r~,,-1) and solve in u the equations

yl n~") --= vi,

8.3. DYNAMIC NONINTERACTING CONTROL

whose solution is u = o~(x, z) + ~(x, z)v. Finally, the dynamic compensator which solves the problem reads

zi,n,~-n' u

:-

vi, for i-:- 1 , . . . , p

=

Example 8.3.2 Example 8.2.2 continued. A standard dynamic decoupling compensator is

Ul

U2

,--

Z/ COSX3 CO82 X3 V 2 - - V 1 Z

tan x3

87

Chapter 9

Input/State Linearization The solution of the input-output linearization problem fully linearizes the state space equations in the case of a single output system whose relative degree equals n. The same holds true in the case of multi output system when the decoupling matrix is square invertible, i.e. condition (7.3) is satisfied, and the sum of the relative degrees equals n. The input/state linearization problem which is considered in this chapter, consist in searching for output functions that fulfill the above conditions. This issue is of major importance whenever the input/output linearization yields a closed loop system which contains an unobservable subsystem that may be unstable. Presence of such an unstable internal dynamics disqualifies the input-output linearization scheme. A solution to the input-state linearization problem allows one to master internal stability, although the given output remains nonlinearly related to the input. In this chapter, we first consider the input/state linearization problem by regular static state feedback. When such a static state feedback does not exist which fully linearizes the system, then it is interesting to seek a solution within the class of dynamic state feedbacks. In this case, the problem consists in looking for so-called linearizing outputs, i.e. outputs such that the sum of the orders of the zeros at infinity equals n.

90

9.1 9.1.1

C H A P T E R 9. I N P U T / S T A T E L I N E A R I Z A T I O N

Static Problem

state

feedback

linearization

Statement

Given the system = f ( x ) + g(x)u,

(9.1)

where the state x E ~/n, the input u E ~ m and the entries of f, g are meromorphic functions, find, if possible, a static state feedback u = a(x) + j3(x)v

(9.2)

and a state transformation

z= where a, fl, (I) are meromorphic functions, rankfi = m and q) is a local diffeomorphism from ~'~'~ to/R '~ at almost any point of/R '~, such that the closed-loop system reads = Az + By where the pair (A, B) is controllable. It is not restrictive to solve the problem by looking for the pair (A, B) in Brunovsky canonical form. Solving the input-state linearization problem consists in searching for the largest independent Brunovsky blocks, or for the largest independent strings of integrators, or equivalently for some functions in/(; which have the largest relative degree and which are controlled by independent inputs. These functions are candidates for output functions in the input-output linearization problem. The input-state linearization problem is then solvable by regular static state feedback if and only if the sum of the associated relative degrees is n, the dimension of the state space. These ideas are formalized in the following section. 9.1.2

Solution

to the input/state

linearization

problem

T h e o r e m 9.1.1 There exists a static state feedback which solves the inputstate linearization problem for ~ if and only if (i) 7-l~ = 0 and

9.1. S T A T I C S T A T E F E E D B A C K L I N E A R I Z A T I O N

91

(ii) 7-le is closed for any k > 1 Condition (i) is an accessibility condition which is obviously necessary since has to be transformed into a linear controllable system. Condition (ii) is an integrability condition which implies that the controllability indices of E are the controllability indices of the resulting closed-loop linear system.

Proof of Theorem 0.1.1: The conditions of Theorem 9.1.i are clearly necessary. For sufficiency, let k* = max{k >_ 017~k ~ 0} and 7~(kk.") := span~c{&,0(e')ld~ E 7~e*}. Let de. := dimT/k, and { d ~ l , . . . , d ~

k. } be a basis of 7~k.. From the defi+ spanK{dx} nition of k*, the dimension of span~c{d~e*)'"" .,df~i~ )irk*. span K{dz} '" - equals de* which means that the de. strings of integrators are controllable by independent inputs, otherwise 7/e.+1 would be non zero. More generally, let de := dim

7/k . Since ~ e D 7~k+1 +7~k+1, a basis 7/e+1 + 7~e+1 for "He is obtained by taking a basis of 7s its derivatives and completing it with de vectors {d~odk. +...+d~+1+1, . . ., d~a~, +...+de } " Consequently, a basis is obtained for 7~1 = span~c{dz } as {d~} j), i = 1 , . . . , de. + 9.. + d l , j = 0 , . . . , ri - t} where ri denotes the relative degree of ~'i- Inputoutput linearization of the outputs {~i,i = 1 , . . . , de. + ... + dl} fully linearizes the state equation in the coordinates {~}J),i = 1 , . . . , de* + . . . + dl, j = O , . . . , r i - 1}. II When the input-state linearization problem has no solution, it is interesting to investigate a more general problem, first stated and solved by Marino [103]. Actually, one may try to find the largest linearizable subsystem of a given system ~. The closed loop system obtained by linearizing such a subsystem will read { ~:1 = Z l Z l + B l V l ( ) ~2 = f2(z) +g2(z) vl (9.3) ?)2

where dim zl + d i m z2 = n and dim vl + d i m v2 = m. Necessarily, the dimension of the largest Brunovsky block of the pair (A1, B1) is less than or equal to k*.

92

CHAPTER

9. I N P U T / S T A T E

LINEARIZATION

This follows from 7/k-+1 = 7/00 = 0. The largest closed subspace 7/k. contained in 7/k. is uniquely defined and the maximal number p~. of k*-dimensional Brunovsky blocks of the pair (A1, Bx) equals ~(k') ,~k" + p~. = dim X 9

X

(9.4)

More generally, let 7/k be the largest closed subspace contained in 7/k and define + x P~' = dim'(k+a(~t -(9.5) i~k+ 1

~ X

and n~ = card{p~ > i}. The integer p~ represents the largest number of k-dimensional Brunovsky b~ocks of the pair (At, Bt) which are controllable independently from the blocks whose dimension is larger than k. T h e o r e m 9.1.2 S y s t e m (9.1) can be partially linearized via static state feedback, with controllability indices (of the linear subsystem) nl >_ .. 9 > nra if and only if n i ~ n i9 , f o r i = l , . , . , m.

E x a m p l e 9.1.3 Consider

/

51 52 54

X3

Xn-1

Xn

1

X4

0

0 0 0)

. ( u2 ul

0 1

then, compute 7i2 = spanjc{x3dxl - d x 2 , . . . , x n d x l - dxn-1} and more generally, for 2 < k < n - 1, "Ilk -- s p a n l c { x 3 d x l - dx2,..., x n _ ~ d x l - dxr~-a+l},

(9.6)

9.1. STATIC S T A T E FEEDBACK LINEARIZATION

93

~/,~ = 7/oo = 0 Consequently, p*1 = 2, P2* = 0, . .. and, n *1 integrability conditions (ii) in T h e o r e m 9.1.1 yields t h a t the largest feedback linearizable consists of two one dimensional subsystems. Note t h a t system (9.6) is accessible and t h a t state has relative degree 1 only.

= 1, n~ = 1, n 3* = 0, . . . T h e are not satisfied. T h e o r e m 9.1.2 s u b s y s t e m has dimension 2 and any non constant function of the

T h e o r e m 9.1.1 is a special case of T h e o r e m 9.1.2 and is equivalent to Corollary

9 . 1 . 4 System (9.1) can be fully linearized via static state feedback

if and only if i=m

Z "n i - - n . i=1 Proof of Theorem 9.1.2 T h e necessity is obvious 9 T h e sufficiency is straightforward and follows the lines of the proof of T h e o r e m 9.1.1. Let { d Z l l , . . ., dzpk,,1 } E ~P~" '~k* be such t h a t

O,tzll(k') ,..

9

,z P(k.k *"),1 )

Let {dzp~,.+l,1,.. 9, d z v b _ l , 1 } E , ~~kp. b_ -l l - p b _(k.)

,

-: Pk*"

Ou

such t h a t

_(k.-,)

(,'-1),

'''''Zp* k.,l'zpi.+l,l'''''zp*k,

1,1) - - Pk"--1"

This procedure can be repeated by induction. Rename z12 := 41t, zla := z 11 (2) , ' ' ' , Z l k *

.(k'-l) : = ~11 (9.7)

Zp~2=Zp~l

CHAPTER 9. INPUT/STATE LINEARIZATION

94

and solve in u the equation

u)

Yl

(9.8) z (1) ~x u)

Vrn

Pl lk

From the proof of Theorem 9.1.2, one has the following C o r o l l a r y 9.1.5 For k = 1 , . . . , k*

p~ =

dimT/k + ~ k + l 7~k+1

(9.9)

Note that m = p~ >_ p~ >__... _> p~. is the list of dual controllability indices as they have been defined in Section 3.7. Equations (9.2) and (9.7) yield the Brunovsky form Zll

----

Z12

Zl,k*-I

"-"

Zlk*

Zlk*

:

Yl

Zpl,1

--

Vm

(9.10)

The number of controllable blocks of dimension k in (9.10) equals (p~ -P~,+I). II E x a m p l e 9.1.6 Consider the unicycle in Example 3.7.2 with the slight modification in the inputs: let the velocity Ul be a fourth state and let the acceleration /q be a new input, denoted Vl. The angular velocity u2 remains the second controlled input and is denoted vs. The system's description then becomes,

=

(x4cosx3)(00) x4 sin x3 0 0

+

0 0 0 1 1 0

Vl v2

(9.11)

9.2. DYNAMIC STATE FEEDBACK LINEARIZATION

95

Compute, 7/2 = spantc{dxl,dx~}, 7/3 = 0 and 7/~2) : spantc{d(vl c o s x 3 _ v2x4 sin x3), d(vl sin x3 + v2x4 cos x3)}. System (9.11) is fully linearizable by regular static state feedback obtained by solving the following equations in v: V 1 COS X 3 - - V 2 X 4

Vl sin

sin xa

=

X 3 -~- V 2 X 4 COS X 3 :

Wl

W2

which yield V 1 ~- W 1 COS X 3 - ~ W 2

sin X 3

v~ = (w2 cos x3 - wl sin x3)/x4. The linearizing state coordinates are zl - xl, z2 = x4cosx3, z3 = x2 and z4 -- x4 sin z3. This solution is equivalently obtained when considering xl and x2 as outputs and applying the input-output linearization technique.

9.2

Dynamic

9.2.1

Problem

state feedback

linearization

statement

Given the system E of the form (9.1)

-- f(x) + g(x)u,

(9.12)

find, if possible, a dynamic state feedback

( ~ =

u =

M(x,~) + g(x,~)v

(9.13)

where ~ E / R q for some integer q and a state transformation z = (I)(x, ~) where (I) is a local diffeomorphism from ~n+q to ~n+q at almost any point of IR n+q, such that the closed-loop system reads

= Az + By where the pair (A, B) is controllable.

CHAPTER 9. INPUT~STATE LINEARIZATION

96

It is not restrictive to solve the problem by looking for the pair (A, B) in Brunovsky canonical form. Solving the dynamic input-state linearization problem consists in searching for some functions in K: which have the largest structure at infinity. These functions are candidates for output functions in the input-output linearization problem. The input-state linearization problem is then solvable by regular dynamic state feedback if and only if the sum of the associated orders of zeros at infinity is n, the dimension of the state space. Equivalently, these linearizing output functions define a system without zero dynamics, in the sense of Section 5.4. The result was first stated in [81]. These ideas are formalized in the following section. 9.2.2

Solution to the dynamic

state feedback linearization

A solution to the dynamic state feedback liaearization problem may be derived by inspection on the 7/k spaces. Since the closed loop system has to be fully controllable, a necessary condition is 7ioo = 0. Then a canonical basis for the 7/k's can be constructed as follows. Let {wk. } be a basis of 7/k.: 7/k" = span,:{wk. } Let {wk*-i } be such that {wk., &~o,wk.-1 } is a basis of 7/k.-1: 7/k'- I = span/c {Wk., (zk. , Wk. } More generally, let {wk} be such that

for k = 1, ..., k*. A sufficient condition for dynamic state feedback linearization then follows. T h e o r e m 9.2.1 If {oal, ...,wk.) is integrable, then the dynamic state feedback

linearization problem is solvable. E x a m p l e 9.2.2 Consider the unicycle in Example 3.7.2. From the computa-

tion 7/1 7/2 7/~

-- spanjc{dx) --- spanpc{(sinxa)dxl - (cosx3)dx2} = 0.

9.2. D Y N A M I C S T A T E F E E D B A C K L I N E A R I Z A T I O N

97

one has k* = 2, w2 = (sin xa)dxl - (cos xa)dx2. Since d~2 = us cos xadxl + us sin xadx2 - uldxa, one may pick ~o1 = dxl, so that 741 74~

-_-

span~:{u~2,~J~,~l} spanlc{~z2}.

Finally, span~c{w2, wl} is integrable and equals span~:{dyl, dy2} where Yl = x l and y2 = x2 is a set of linearizing outputs. Proof of Theorem 9.2.1: Let {dyl, ...,dy,n} be a basis for span~:{wl, ...,wk'}. By construction, the sum of the orders of zeros at infinity of {Wl, ..., wk. } equals n. This sum can not decrease by a change of basis, so it equals also the sum of the orders of zeros at infinity of {dyl, ..., dym}. Consequently, {dyl, ..., dym} defines a set of output functions without any zero dynamics and it embodies a solution of the dynamic feedback linearization problem. |

Chapter 10

Disturbance Decoupling The disturbance decoupling problem is a basic problem in geometric linear systems theory [144] as well as in nonlinear systems theory [78, 112]. The solution of the problem by means of an invertible state feedback is well established by standard geometric tools in [78, 112] and it will not be considered here. We will instead concentrate on a more general situation, looking for a quasi-static state feedback that achieves the decoupling of the disturbance from the output. In the single output case the disturbance decoupling problems via static or quasi-static state feedback are equivalent. The idea is to use feedback to cancel those states which affect the output and which are affected by the disturbance. The solution strategy is clarified on the following linear academic example. E x a m p l e 10.0.1 Xl

-~

x2 -[- U

y

=

xl

(lO.1)

where u is the control and w the disturbance. Through x2, the disturbance w affects the output y: i) = w + i~.

100

C H A P T E R 10. D I S T U R B A N C E DECOUPLING

The invertible (static) state feedback u = -x2 + v renders x2 unobservable and decouples the disturbance w from the output y in the closed-loop system: y(~) = v(k-1),k >_ 1. The general solution to the problem achieves the same goal, it renders unobservable (under feedback) the largest subspace of the state space and the disturbance is rejected from the output if the disturbance only affects the largest possible subspace. This scheme is displayed in Figure 10.

,

U nobservabte subsystem W

Figure 10.1: Disturbance decoupled system.

10.1

Problem Statement

Given the system

l

y

=

h(x)

Y

10.2. SOLUTION TO DISTURBANCE DECOUPLING

101

where the state x E /Rn, the input u E JRm, the output y E /Rp, the disturbance w E /Rq and the entries of f, g, p, h are meromorphic functions. The Disturbance Decoupling Problem (DDP) is defined as follows: find, if possible, a quasi-static state feedback

u = (~(x, v, iJ,

., v(O)

such that dy (0 E span~c{dx,dv, d~), 9.} for any i E W

(10.3)

Analogously, the Dynamic Disturbance Decoupling Problem (DDDP) for the same system consists in finding, if possible, a regular dynamic compensator { ~=F(x,z)+G(x,z)v u = ~(~, z) + ~(x, z)v

ze~

such that dy (i) E spanlc{dx,dz, dv,di;,...} for any i E W

10.2

(10.4)

Solution to disturbance decoupling: quasi static feedbacks and dynamic compensators

The key tool solving the described disturbance decoupling problems is provided by the smallest subspace which is observable under any quasi-static state feedback. Recall from Chapter 4 that the observable space of system (10.2) with w - 0 is given by A' n (y + U). The subspace X n Y of the observable space is isomorphic to the smallest observable subspace, under any quasi-static state feedback and it is instrumental in solving the disturbance decoupling problem. This is shown in the next Theorem. T h e o r e m 10.2.1 The disturbance decoupling problem is solvable via quasistatic state feedback if and only if p(x) is orthogonal to the subspace X M Y. One equivalently solves the DDDP as follows. T h e o r e m 10.2.2 The dynamic disturbance decoupling problem is solvable if and only if p(x) is orthogonal to the subspacc X M Y.

CHAPTER 10. DISTURBANCE DECOUPLING

102

Before proving Theorems 10.2.1 and 10.2.2, let us give an illustrating example, taken from [73].

Example 10.2.3 ~:i

:

X2ttl

X3

=

X~ "t- X4 § X4Ul

~4

=

9~5

=

Yl Y2

--=

u2 xlul

(10.5)

+ w

Xl x3

-~

From the structure algorithm, one computes X r3 y = span~c{dxl, dx3, (1 - x4~)t ,, --~-~o~ +

/~

(1 + ~)d~}

0

Since

p(x)

=

0

, the condition in Theorem 10.2.1 is satisfied. Solving in

0 1

u the equations

91

=

vi

Y2

~

?92

(10.6)

we can construct a quasi static state feedback which solves the disturbance decoupling problem U1

~

Vl/X 2

tt2

--

X2 ~

(

)

V 2 - - X5-t-

It is now possible to check that, as expected, the output of the compensated system is not affected by w, since ~)l(t) = vl(t) and ~2(t) = v2(l). A dynamic compensator is equivalently derived by setting II 1

~

W1

~

Z/X 2

=

~

( X2

x2

,]

10.2. SOLUTION TO DISTURBANCE DECOUPLING

103

This yields ~l(t) = wl(t) and ~2(t) = w2(t).

Proof of Theorem 10.2.1 The necessity follows from the fact that the space X n y does not depend on the feedback, and for the closed-loop system, p(x) • (X n Y). Conversely, ifp(z) A_(X n y ) , equations (5.6) in the inversion algorithm do not depend on the disturbance input. Standard quasi-static state feedback which solves the noninteracting control problem for a right invertible subsystem (see Section 8.2) will solve the disturbance decoupling problem. I The proof of Theorem 10.2.2 follows similar arguments from Section 8.3.

Chapter 11

Model Matching In the nonlinear framework, the model matching problem was considered in [38] and, in the case of a linear model, in [34], [76]. Some further contributions are in [115]. The formulation of the Model Matching Problem we give in the following differs slightly, from that of [38]. However, our approach provides a condition for the solution of the problem which is at the same time necessary and sufficient whereas the conditions given in [38] are either necessary or sufficient.

11.1

A special form of the inversion algorithm

Given the system E we may consider its input u as divided into two subsets u = (v, w), where v is viewed as a set of controls and w as a set of parameters. In this case, we apply to E the following algorithm, denoted by Singh,. S T E P 1 : Calculate

Oh

!) = ~ [f(x) + gv (x)v + gw (x)w =: fl (x, w) + gl (x)v and set Gl(x):= gl(x) and sl : - r a n k G l ( x ) . Permute, if necessary, the rows of the output so that the first sl rows of G1 (x) are linearly independent and decompose 1) as

106

C H A P T E R 11. M O D E L M A T C H I N G

(11.1) Yl

'

where dim .Vl = sl =: Pl, 9 Then, eliminating v in the last rows, write

(~)1) : ( ]I(X;(I;!W'9~~I(x)v,~I))

(11.2)

and set d l ( x ) =: ~l(x). S T E P k-t-1 : Suppose that from steps 1 through k one has ~

=

]~(~,w)+~l(~)~

~1 ,..., ~k-1, ~)k- 1)+ ,~k ---- ]k(x, w,..., w(k-1), ~)1,..., ~(k-1) +.~k (;r, w,..., w(k-2), 91,..., t2~k-~),..., .Ok-1)v 9k

~)k(X,W,..., W(k-l), Yl,''', u

=

,..-, ,Yk)

where

Ok has full rank sk. Then Yk

-" .fk+l(X, W ,

. . . , W (k),

- t - g k T l (;r,, w ,

DefineGk+l:=(

) dk 9k+l

~k+l

=

8k

" " " , Ylk),

" " ", Y k ,

Yk)-t-

yi , . . . , ~[ k-1),

. . . , ,~k ) ' v

'

and 8k+1 :-~-rankGk+l(x). Decompose ~)k as

Yk = where dim~k+i = 8k..F1 - write

Yl,

. . . , W ( k-1),

9k+1

: : P(k+i)v" Then, eliminating v in the last rows,

L+l(x,w,...,w(k/,~l,...,~),...,~k,bk)+

+~k+l(x, w,..., w( k-l/, ~1,..., 9~k-1),..., ~k)v

11.1. A SPECIAL FORM OF THE INVERSION ALGORITHM

107

and set Gk+l

:~

"

9

Ok The above algorithm performs the inversion of E, viewed as a system depending on the parameter w, with respect to the input v when p~ := s,~ equals the dimension of v. When w is empty Singh~ reduces to the usual Singh's inversion algorithm. The indices cri, si and pi contain essentially the same information and each of them could be used in the following. We choose to state the next results in terms of the pi, which have a direct interpretation as numbers of zeros at infinity of order i (see [106]), although the other indices are often used in proofs and calculations. L e m m a 11.1.1 Let the systems

T = {

yT

= =

+ g(x)u +

(11.3)

and G-- { ~yG -: fG(Z)hG( z)+ ga(z)v

(11.4)

with outputs of the same dimension, be Tven and let (GT) denote the composite system

{x

= f(x)+g(~)u

(GT) :

i = fc(z) + gG(z)v (11.5) YGT : h(x) - he(z) + h'(x)u Then we have piv(GT) -- pi(G) for all i and, in particular, pv(GT) = p(G). Proof. Let /C' denote the field of meromorphic functions in the variables x, z, v,..., v (N-a) and the parameters u,..., u (N), where N = dim x + dim z. We denote by gift the vector space spanned over/C' by {d3:, dz, d~laT, 9 . . , O-Y G T(0 ~. , Remark that to consider u , . . . , u (Iv) as parameters instead of variables means that the differential d(.) is given by N-1

d(.) : (O(.)/Ox)dx + (O(.)/Oz) dz + E i=0

(O(')/Ov(i))dv(i)"

108

CHAPTER

11.

MODEL

MATCHING

Following the proof given in ([37], Thm. 2.3) one can show that p I v ( G T ) = dim,:, s T. From this, since d (YGT J)

~

dy(~)

dy(~ )

=

~(~, u,..., u(J))d~- dCg )

with Cj E ]~l, it follows piv(GT)

=

dim span,:,{dx, dz, d~IG,...,dy~ )} - d i m span,:, (dx, dz, d~)a,..., dY(~-l) },

and hence piv ( G T ) = dim span,:, {dz, d~)c,..., dy~ ) } - dim span,:, {dz, d y a , . . . , dy(~- 1)}.

Now, let ( w l , . - . , , wr,} C {dz, d~)G,..., dy(~) } be a basis over K' of s an element of { W l , . . . , w r , } . We write

be

Cv = E ' ~ j ( x , z , v, . . ., v ( N - 1 ) , u, . . ., u ( N ) ) w j

with 7j E K r and, computing the derivatives with respect to x, u , . . . , U(N), we get C~W O"[j Wj 5-;

=

E-~

=

o

a--~

=

Z-57

=

o

-

~ o-;-~

=

o

C~tV Ou(NI

O"/j Wj

Therefore O~/j/3x = 0 and c%/j/Ou = . . . = a"/j/Ou Uv) = 0 for all j, or, equivalently, ~j = "/j (z, v , . . . , v( N - 1)). This says that {w l , . . . , wr, } is a set of generators over the field K of meromorphic functions in the variables z, v , . . . , v (N-l) of span*:{dz,dyG,...,dy(~ )} = g2- Moreover, since K C /C', dim*: s = dim*: giafor all i and the result follows. |

11.2. MODEL MATCHING PROBLEM

11.2

109

M o d e l Matching P r o b l e m

Let us now state the Model Matching Problem (M.M.P.).

Problem Statement Given a model

T = { YTiC == h(x)f(z)+ g(x)u

(11.6)

and a system G as in (11.4) find a proper compensator

H = : ~ = ftl(~,z,u) ( v = hH(~,z,u) with state s p a c e / ~ q and a m a p r : ~ n _4 Rq such that, denoting by YGI-I the output of the composite system GH, one has that yT(u, x) - Y G H ( U , r z), t h a t is the difference between the output of the model, viewed as a function of u and of the initial state x, and the output of the composite system, viewed as a function of u and of a suitably defined initial states z and ~ = r does not depend on u. In order to gain a better insight into the Model Matching Problem we are considering, we now state it in a generalized form (G.M.M.P.), which includes in particular the left inversion problem. Specializing such formulation by requiring a proper compensator we get the most interesting case from the point of view of control theory.

Problem Statement (Generalized form) Given a model T as in (11.3) and a system G as in (11.4) find an integer u > 0, a possibly nonproper compensator

H=

{ ~,:

=

=

fH(~,z,u,...,u("))

h,

(11.7)

with state space ~ q and a m a p r : IR'~ -+ ~ q such that, denoting by YGH the output of the composite system GH, one has that yT(u, x) -- yGH(U, r z), that is the difference between the output of the model, viewed as a function of u and of the initial state x, and the output of the composite system, viewed as a function of u and of the initial states z and ~ = r does not depend on u(~). The M.M.P. is the special case of the G.M.M.P. for u = O.

110

CHAPTER

11. M O D E L M A T C H I N G

R e m a r k 11.2.1 In the M.M.P. the requirement that yT(U, x ) - - y a g ( u , r z) does not depend on u amounts, in the linear case, to the equality of the transfer functions of the model and of the composite systems. Prom this point of view, therefore, our formulation represents the natural extension of the one currently understood for the linear model matching problem (compare with the quoted references and with [78], [76]). We recall that a stronger formulation of the M.M.P., requiring the equality of YT and Yogi, has been considered, only for a linear model, in [34]. Remark that the problem we stated qualifies as an exact M.M.P., as opposed to an approximate or an asymptotic M.M.P. that could also be considered, see e.g. [711. Let us consider the left inversion problem in the linear framework. The solution provided by the Silverman algorithm [128] has the form (11.7), where u is the inherent integration order of the system [127], [113]. In the simple example given by T = {YT = Y and by G={~ y

= --

v z

we obtain

The difference between the outputs of the identity model T and of G H is YT -- YaH = Y -- z, the latter depends on the input y and is independent on the first derivative Y. E x a m p l e 11.2.2 Let T=

~ ic=u L yT=X

and

G=

YG ~- Z2

be the data of a M.M.P.. The pair consisting of the compensator H = {v = u / 2 z and of the empty function is a solution in the sense of Remark 11.2.1. In fact, for zo r O, we obtain yvH(U, Zo) = f~ u ( r ) d r + z~ for all input functions u(t) and for all t > 0 such that f~ u ( r ) d r + z02 > 0. Then, we have yT(u, x) -- YGH(U, z) : X -- z 2 and d(yT(u, x) -- yGH(U, Z) ) (k ) E span~: {dx, dz}.

11.2. M O D E L M A T C H I N G P R O B L E M

111

In particular, if, for example, the input is bounded by lu(t)l < M, and the initial conditions x0, z0 5~ 0, are chosen, YT -- YaH is independent from u over the time interval [0, z~/M). It may be useful to remark that v = u / 2 z is a solution of the M.M.P. in the same way, that is with the same limitations, in which it is a solution, in the sense of [79], of the disturbance decoupling problem with disturbance measurement described by ~ = u, ~ = v, y = x - z 2, where u is the disturbance and v is the control. Note that taking, for instance,

T= ~ d:=xu

l

G= I Z:

ZU YG = z,

and

YT .= X

contrarily to what happens in the linear case, the identity compensator H = {v = u does not give a solution of the M.M.P.. In fact YT -- YGH ---(xo - zo) exp (fo u ( r ) d r ) is independent from u only if the initial states of the model and of the system coincide. In this case a solution is given by the compensator

H = ~ ~ = ~u, [ v = (~/z)u,

where ((t) E/R '~

and by r = id.

A structural condition under which a compensator exists and a procedure to compute it are given in the following theorem.

T h e o r e m 11.2.3 The Generalized Model Matching Problem is solvable if

p(GT) = p(G) where ( G T ) is the composite system (11.5).

(11.8)

CHAPTER 11. MODEL MATCHING

112

Proof. Applying Singh, to (GT) we obtain

[~1(~:, z, 4, i~)

9N

J~N(X, Z, 72,..., lZ(N), ~'1,..., yI(N-1), ' ' ' , 9N-l,

YN-1)

/~N (X, Z, U,..., U(N) , 1~'1,''', yI(N-1), ' ' ' , YN-1, YN-1, f'U)

(01(z)

/

G2(x, z, u, gl) "

+

v

GN(~.,Z,'tI,...,~(N-2),91,. ..,Y(N-2) YN_l ) =

(~

0 +

O) v

(11.9) with rankG = # rows G = p, (GT) and where Yi represents a suitable subset of rows of'.YGT(i),which will be useful to denote also as Y(~)i_y(i)ci. We can choose constant values

Y1

( ~T,'I(N-2) for]P=

Y =

YN-1

]~'N- 1

0

such that the generic rank of G evaluated at Y is equal to the number of rows of G. Then, solving for v the system

=Fly+0IVY,

11.2. MODEL MATCHING PROBLEM

113

obtained by replacing Y with Y in (11.9), we get

v = ~(x, z, u,..., ~(~), ~ , . . . , 9~(~-~), Y~,..., ?~_~, z,,_~, Y~_~). Now, denoting by w0 a vector of same dimension as x and by wi a vector of dimension (N - i) 9dim Y/, we set t, = N and we construct the compensator

I (Oo= f(wo) + g(wo)u 0 1 H= (vi = 1 0 0

/ ~)

wi +

for 1 < i < N - 2,

(11.10)

0

Y/

Y -" (~(WO, Z , U , , U ( N ) , w l , . . . , w N _ 2 , Y N _ I )

and, letting r - (x, 0, . . ., 0), we claim that (H, r is a solution of the G.M.M.P.. In order to show this, let us first remark that, in (11.9) YN is actually independent from u (k) for all k. In fact, by Lemma 11.1.1 and the rank equality (11.8) it follows p(GT) = p~(GT) and we know, from ([37], Theorem 2.3), that the dx,dz,dY1 . . . . ,dY (N-l), . . . , dYN are independent over the field K. So, if OFg/Ou(k) r 0, for some k >_ O, dYg does not belong to spanpc { d x , d z , d Y 1 , . . . , d y I ( N - 1 ) , . . . , d Y N } and then p(GT) > pv(GT), contradicting the assumption. Now let us consider the composite system (GH):

= F(w) + a(w)u,

(GH) - {

= f c ( z ) + ga(z)r

Z, u , . . . , u(N)),

YG~ = h a ( z )

initialized at r = (x0, 0 , . . . , 0) and the difference Y T - - Y G H between the output of the model and that of (GH). Recalling the notation ~ = Y(T!i-- ~(a!i, by substituting the output of H to v in (11.9) and taking derivatives we get

•T.i( N - a )

9 (N-l)

-- OGH,i

~ ( N ) _ y (GNH).N T.N

Therefore

: _

Yi for 1 < i < N - 1

O.

d(yT -- yaH)(k) E span~: {dx,dz,dw, du,...,du (N-l)}

for all k.

II

CHAPTER 11. MODEL MATCHING

114

It is worthwhile to note that, although in (11.7) the compensator H is described in a very general form, the construction illustrated in the proof of Theorem 11.2.3 produces always a system whose state equations have the same form of those of the model. In particular, the derivatives of the input appear only in the output function hH(~, z, u , . . . , u(n)). A structural condition under which there exists a proper compensator H, that is one which does not depend on the derivatives of the input u, is given in the next Theorem. T h e o r e m 11.2.4 The M.M.P. is solvable with a proper compensator H of the form

H = ~ ~ = fH(~,z,u),

I

v

,f pi(GT) = pi(G)

(11.11)

for all i >_ 1. Proof. Assume that (11.11) holds, then, for all i, we have by Lemma 11.1.1 p~v(GT) = pi(GT). In particular, this implies that at the first step of the algorithm Singh~ applied to (GT) we have

71

(x, z, u) F1 (x, z, tL, ~'Zl)

0

/ V,

where actually OF1/Ou = O, since otherwise Pl (GT) would be strictly greater than Ply (GT). Repeatedly applying the same argument, we get at the last step N

?

and, hence, v = r z, ~ ' , . . . , ~(N-3)). Therefore the compensator obtained following the construction described in the proof of Theorem 11.2.3 is, in this case, proper. |

11.2. MODEL MATCHING PROBLEM

115

(oo)

E x a m p l e 11.2.5 (i) The M.M.P. concerning the model

/

1 0 0 0 0 1

x : ( o ' / +

zs0)

and the system .-~

Z4

G=

u,

0 0

+

0

1

1

1

0

1

v,

I z2 -- z3 Zl

was considered in [38]. It was shown that the geometric necessary condition given in the same paper is not verified, although the compensator

H= v=

~ r162 + zs)

0 z3

+1/(r

0 1

u

provides a solution of the problem (see [38], Example 5.4). It can be easily checked that (11.11) is verified, that is pi(GT) = pi(G). Then, applying the procedure illustrated in the proof of Theorem 11.2.3 we get the proper compensator

H~=

{

~=

v

+

(z4 -

,~ -

10

0 0

~,z~

0 1

u,

z4 - ~2 - u2)/(,~

'~ -

zs -

z4)

)

"

Clearly, by removing the unnecessary equations ~1 = ~2, ~3 = ~ 4 , and ~4 = us, we obtain an other compensator, say H", which solves the problem. Now, the change of variables ~ = z4 - ~2 transforms H" into H.

CHAPTER 11. MODEL MATCHING

116 (ii) Consider the model

01 )

~ ( 0x2) §

U,

Y T = ( )0X l

/ ( 1~)

and the system

~=

0 z3

YG =

for which

p(GT) = p(G).

1 1

v,

(z~)

G =

Z2 - - Z 3

Note that, since

f~ vl(~)d~ + zl(0) YG =

and yT1 = 0, contrarily to what happens in the linear case, it is not possible to find a compensator H such that YT -- YaH = 0 for u r 0, also when we are allowed to chose the initial condition z(0). Applying Singh~ to (GT), we get

r

=

(0)(, u - z3r

-

0)

z31)1 r

v= # +~,,

and then, in fixing constant values Y for Y, we are obliged to chose 1)1 :p 0. Taking for instance Y =

0

we get v = '

, which represents by U+z3

itself a compensator H that solves the problem. R e m a r k 11.2.6 The conditions of Theorem 11.2.3 and Theorem 11.2.4 are not necessary for the existence of solutions to the G.M.M.P. and the M.M.P.,

11,2. MODEL MATCHING PROBLEM

117

as pointed out by the following example, taken from [70]. Let

/ Ix2)(~ ~: =

x3 0 x4

T =

YT

=

1 0 0

0

1 0

0 1

+

1

ul u2

0

x4 Xl

and

I

z2

G=

0

YG =

Z3 Zl

vl v2

9

By applying Singh's Algorithm to G, we get pl(G) = p2(G) = pa(G) = 2. The same procedure applied to (GT) gives

YGT1 "~ X3 ~- 711 -- Yl YGT2 ~- X4 -- V2: YGT3 = z 2 + z 2 ( y c T 2

- x 4 -- 1),

and then O a T a = i J c T 1 + z 2 ( i I G r 2 -- x 4 ) -- ( i J G r l -- ~ a -- ~ ) ( Y G T 2

-- x 4 )

So pl(GT) = 2, p2(GT) = 3 and the sufficient conditions of 11.2.3 and tl.2.4 are not satisfied. However, the compensator

H =

0 vl = ~3 + Ul, V2 =(~;

and r = id give a solution to the M.M.P..

1

ua

CHAPTER 11. MODEL MATCHING

118 From (11.9) we get the equality

= p ( , , ~ , , , . . . , CN> ? , , . . . , ? F '-1>, 9 9 ?N- 1, Y~- ~) +dN(~, z, ~ , . . . , r

,~-~

..., ~_~1~

(11.12)

and, by differentiation of ]?N, the equalities

?(Nrl+q-N)

~_ pr~.i_q(.~,

Z, ll, . . ., ll(n+q)

yl,

. . . , ~zl(n'l'q-1)

. . ., ~ZN_l,

(11.13)

~(n+q-N+l) YN),...,~lZ(Nn+q-N)), from which it can be understood that a weaker condition for the existence of solutions to the M.M.P. is, in particular, that there exists a vector of functions

Yl(~, z). ) Y(r = ( YN(~,z) such that i) OY(k)/Ou = 0 for all k;

ii) substituting Y(x, z) and its derivatives for ~>=

9

and its derivatives in 0, the generic rank is equal to the number of rows; iii)

substituting Y(x, z) and its derivatives for ~-=

9

11.3.

L E F T FACTORIZATION

119

and its derivatives in/~g, 99-, Fn+q, F, all the coefficients of the monomials in u , . . . , u (~+q), and respectively, all the coefficients of the monomials in f i , . . . , u(n+q), are zero. Such condition is verified in the above example

11.3

Left

Factorization

It is well known [22], [63] that in the linear case (11.8) and (11.11) are necessary and sufficient conditions for solving the G.M.M.P. or the M.M.P., also when no feedback connection between the state of the system G and the precompensator H is allowed. In such formulation, the linear G.M.M.P. amounts to the problem of factoring the transfer function of the model T through a possible left factor, represented by the transfer function of G. It is natural, then, from an abstract point of view, to consider also the dual problem, which consists in factoring the transfer function of T through a possible given right factor (see [21], [22], [63], [98]). In the more general context we are considering, this leads to the following formulation for what we call Left Factorization Problem. 11.3.1

Left factorization

problem

(L.F.P.)

Problem statement Given a model T as in (11.6) and a system

H = [ ~ = fH(z) + g~(z)~

[

V = hn(z)

(11.14)

find a proper compensator

a = [ ~ = sa(~, v) [ Ya = ha(~, v) with state space/R q and a map r :/R n -4 ~ q such that, denoting by YaH the output of the cascade GH, we have yT(u, zo) --yaH(U, r z0) = 0 for any initial states z0, z0.

120

C H A P T E B 11. M O D E L M A T C H I N G

Problem statement (Generalized version) - G_L.F.P. Given a model T as in (11.3) and a system H as in (11.14) find an integer v > 0 and a possibly nonproper compensator

[

YG = h a ( ( , v, . . . , v("))

(11.15)

with state space ~q, a map r :/~" ~ ~g~qsuch that, denoting by YGH the output of the cascade GH, we have YT - (u, xo) - YGH -- (u, r

z0) = 0

(11.16)

for any initial states z0, z0. R e m a r k 11.3.1 The same considerations as in Remark 11.2.1 apply to the present situation. Therefore a solution (G, r will be one which achieves (11.16) for all initial states x0, z0 in an open and dense subset of the state spaces. The first result we have in this framework is the following Theorem. T h e o r e m 11.3.2 The G.L.F.P. is solvable only i] P where(H

T

(') H

= p(H)

(11.17)

) is the system consisting of the state and output equations of T

and H. Proof. We start by proving the Theorem under an additional technical assumption on the system H. Assume that the maximal regular controllability distribution ?~*n of H contained in kerdhH is locally well defined, i.e. that the regularity conditions of ([78], w are satisfied. Denoting by G the distribution spanned by gH(z), we assume that the following holds:

dim(G U Ti*H) = m - p(H),

(11.18)

Now, let the regular feedback u = a(z) + i3(z)w be a 'friend' of 7 ~ ) let us denote by ( T H

) the system obtained by compensating ( T H

and with

11.3.

LEFT FACTORIZATION

121

u = a(z) + ~(z)w. By (11.18), the action of the feedback u = a(z) + ~(z)w transforms H into the system H which, up to a'change of coordinates, is of the form [78] il = fi(zl) + gl(zl)wl, z2 = f2(zl,z2) + g2(zi,z2)w, v = hg(zi), where w = (tbi, t~2), wi = ( w l , . . . , Wp) and p = p(H). Hence we have Ov(k)(w, Zo) _ 0 atvi

(11.19)

for all i > p + l and for all k. Moreover, if (G, r is a solution of the G.L.F.P., we have that the output trajectory Y(w, zo, r z0) of the cascade composition between

(') /)

and the system

O= [ ~ =fG(~,, .... ,v(")) [ u = y~ - hG(~, v,..., vl~)) initialized at (x0,r implies

ou (k) (w, xo, r

z0), is identically zero. This, together with (11.18),

zo)

Owi

Owi

Owi

oy(~k)(~, xo, =o)

oy(~) (y,~ (~, zo), ~(xo))

Owi Oy~~) ( w , Zo, zo )

Owi

~wi o~a(k) (~( w , zo), r

oy~(w, =o)

Oyff

Owi

Oy~k) ( w, z o, zo) = Therefore, p

p(/)

/~

) = p(H).

0

c3wi for a l l i _ > p + l

and allk.

is not greater than p(H) and, as a consequence, p

H

=

122

CHAPTER

11. M O D E L M A T C H I N G

The general case can always be reduced to the previous one. In fact, if (11.18) does not hold, one can pick p ( H ) independent output components of H that can be decoupled with a regular dynamic state feedback [43], [33], [111]. Then, the extended system H E verifies diin(GE t.J ~ H E ) = m -- p ( H E ) . Since any solution of the G.L.F.P. concerning T and H solves also that concerning TE and H E and since the regular dynamic state feedback does not affect the systems' rank, the conclusion follows from the first part. | In general (11.17) is not sufficient for the solvability of the G.L.F.P.. However, under (11.18) and an additional technical condition, which essentially assures the possibility of expressing locally z as a function of the output and its derivatives, it is possible to get a local result. More precisely, it is possible, for any z0 in an open and dense subset of the state space, to find a neighborhood 7)o and to show the existence of a compensator G and a map r which achieve (11.16) for z E 7)o. We will say, in this case, that the L.F.P. is locally solvable. T h e o r e m 11.3.3 The G . L . F . P . is locally solvable if the following conditions hold :

i) p

H

=

p(H);

ii) dim(GE UTIHE) = m -- p ( H ) ; ill) ~'~i>0(P(U) - s~) -- n, where n = dimz, so = 0 and the si are obtained by al~plying Singh's inversion algorithm to H. Proof. We consider a friend of 7~H, u = a ( z ) + fl(z)w, as in the proof of Theorem 11.3.2 and we use the notations introduced there. By the rank equality

P

H

= p

/4

= p(H), since (11.19) holds, the input components wi,

with i >_ p + 1, do not affect the output of

/)

. By applying Singh's

inversion Algorithm t o e we get ff;t = r

v, i~, . . ., v ('~))

(11.20)

where r is a meromorphic function of its arguments and, in particular, it is defined for all z in an open dense subset of the state space. Moreover, using

11.3.

LEFT FACTORIZATION

arguments as in ([79], w

123

one can show that, by (iii), the Jacobian matrix

0

751 -- ?)I(Z,Vl)

~zz

'

~)N -- ~)N( z , v l , . . . , v l

'

~(N-l)

,''',VN))

whose elements are obtained by applying Singh's algorithm to /~, has rank n. Then, for any z0 in an open and dense subset of the state space, there exists a neighborhood 7)0 of z0 such that z = X ( v , ~ ) , . . . , v (")) for z ff 7)0. By substituting in (11.20), we get then, ~: = ~(v, ~3,..., v(V)). Now, writing the state equation of

fI

as & = f : ( x , z) 4-g:(x, z)~v: + g2(x, z)~2, ]: =

f2(z) + g 3 ( z ) w , we can consider the system

c

= =

[ ~ = f:(~,x(v,~,...,v(")))+g:(v,~,...,~(-)))~(~,~,...,~(-))), L ~c = hff) ( ~-- fG(~,v,i~,. ..,v(v)), y ~ = h(~),

and we claim that (G, ~), where ~ is the identity map, is a solution of the G.L.F.P. relatively to :Do. In fact, by inspection, one sees that the output trajectory Y ( w , Xo, ~(xo), Zo) of the system

JC = fl(X,Z) "k gl(x,z)Cvl "k g2(x,z)Cv2 = f2(z) + g3(z)w,

-----fl(~, Z) -~-gl(~, Z)?~I, y -'- h(x) - h(~) is identically zero for all w. Inverting the feedback u = a(z) +/~(z), we obtain

yr(u, xo) = y a H ( u , ~(z0), z0). E x a m p l e 11.3.4 Let the systems T=

{ d:=u,

YT -- x

and

H =

I Zl ----z2 z.2 "- u, v=z 2

CHAPTER 11. MODEL MATCHING

124 be given. T h e well as (iii) is, there is no need u = ~b(z, v, 5, i))

conditions (i), (ii) of T h e o r e m 11.3.2 are clearly verified as since p(H) = 1, d i m z = 2, sl = 0, s2 = 1. In this case to apply any feedback. By Singh's inversion a l g o r i t h m we get = (i) - 2z~)/2zl and v - z~ = 0,/J - 2zlz2 = 0. Since

Oz

~) - 2ZlZ2

=

z2

zl

has rank 2 for zl r O, we can express z as a function of v, + in the n e i g h b o r h o o d of any point for which zl r O. In particular, here we have

z2

v/2x/~

)

for Zl 0.

T h e n , the c o m p e n s a t o r s

GI

{ ~ _ i) - i~2/2v =

2V~

{ ~ and

G~ =

yc = ~

i) - i~2/2v - -

2x/~

'

,

yc =

together with the identity m a p , are local solutions of the G.L.F.P. respectively for zl > 0 and for Zl < 0. W h e n a proper c o m p e n s a t o r is sought, the necessary condition (11.17) has to be strengthened into the equality of structures at infinity, and one obtains the following result. Theorem

11.3.5

The L.F.P. is solvable with a proper compensator a = ~ d = fG(~, v), t YG = ha({, v)

only if Pi for all i >_ 1.

H

= pi(H)

(11.21)

11.3.

125

LEFT FACTORIZATION

Proof. Let K: denote the field of meromorphic functions in the variables (x, z, u , . . . , u (N- 1)) where N = dim x + dim z. By definition Pi ( T k

=

dimspanpc { d z ' d z ' d g T ' d i J " ' "

dy(~)' dv(i)}-

J

dim span~ {dx, d,, d~r, d~,..., ay~ -1~, d~(;-l~}. Denoting by/C' the field of meromorphic functions in the variables

(x, z,~, u,..., u(N- 1)), since neither T nor H depend on ( we have T ) Pi ( H

=

dim span~:, {dx, dz,d(,di/T,diJ, "" .,dy(~),dv(i)} dim spanjc, {dx, dz, d~, dyT, d ~ , . . . , dy(~-1), dr(i-i)}.

Since YaH = YT, one can substitute ayaH ~ (k) into dY(Tk) for all k in the equation above, thus pi

(T) H

=

dimspan~c,

{dx,dz,d~,dimH,di~,...,dyg~,d~C~)}-

dim span,:, {dz, dz, d(, dilaH, d~),..., dY(~H~), dv(i-1)}. Moreover, by the properness of (GH), we have also that dy(ak) E spanpc, {dx,dz,d(,d~),...,dv(k)}, thus Pi

T) H

=

dim span~c, {dz,dz,d~,d~),..,dr(i)} dim spanpc, {dx, dz, d(, d~),..., dr(i-i)}.

Let K:" denote the field of meromorphic functions in the variables ( z , u , . . . , u ( n - l ) ) where n = dimz. Since dv (k) E span~:,, {dz,du,d/~,...,du (k-l)}

C H A P T E R 11. M O D E L M A T C H I N G

all k <_ n, we get finally Pi

H

=

dim spanx:,, {dz,d/~,... ,dr(i)} dim spanlc,, {dz, d i ) , . . . , dv (i- 1)}

p;(H).

Chapter 12

Differential Algebraic Methods in Nonlinear Systems Theory 12.1

Basic notions

In this Chapter we will introduce some elementary notions concerning Differential Algebra, which find application in system and control theory. Differential Algebra spans from Algebra when algebraic structures, like rings and fields, are enriched by means of a formal way of defining derivatives (see, for a more comprehensive discussion [90, 93, 124]. From this point of view, the basic notion we need to consider in order to build up differential algebraic structures is that of derivation, or derivative operator, on a ring A. D e f i n i t i o n 12.1.1 Given a ring A, a derivation on A is a map d : A --+ A satisfying 9 d(a + b) = d(a) + d(b) for all a, b E A; 9 d(ab) = d(a)b + ad(b) for all a, b E A.

130

Differential Algebraic Methods

D e f i n i t i o n 12.1.2 A commutative ring A with unit (respectively a field K),

endowed with a derivation d is called a differential ring (respectively a differential field). Examples of differential rings or fields are easily found. Besides trivial examples (any commutative ring can be viewed as a differential ring taking d equal to the null map), such are in facts the ring of infinitely differentiable functions on N and the ring of analytic functions on an interval I, as well as the field of meromorphic functions on I, endowed with the usual derivative operator. Working with differential rings or fields, we will need to consider differential homomorphisms, that are simply homomorphisms of the underlying algebraic structures which commute with the derivative operators. If I is an ideal in a differential ring A with derivation d, we say that I is a differential ideal if a E I implies d(a) E I. Differential ideals are generally not easy to handle. In particular, it is important to remark that the differential ideal generated by a set of elements consists of all linear combinations of the generators and of their derivatives. This can make difficult to check whether a given element belongs to a given ideal or not, even if a nice set of generators of the ideal is known. In case the differential ring A is an integral domain, its derivation d can be extended to its quotient field Q in a unique way, by defining d(a/b), where a/b is any element in Q, as follows d(a/b) = (bd(a) - ad(b))/b 2. Sometimes we will be interested in adding a differential indeterminate to a differential ring A with derivation d. This amounts to extend d to the ring of polynomials A[I], where I is the infinite set of indeterminate I = {z n, n E ~W}, by letting d(x(n)) = z(n+l) for n E gr. The resulting differential ring will be denoted by A{x} and, if it is an integral domain, its quotient field, endowed with the induced derivation, will be denoted by A < x >. Given a differential ring A and a differential subring B C A, we say that A is a differential extension of B and we write A / B to denote the extension. Differential extensions of fields are defined in a similar way. From an algebraic point of view, given an extension A / B , one is interested in considering the elements of A which satisfy algebraic equations with coefficients in B, that is the elements a E A such that P(a) = 0 for some polynomial P ( z ) E B[x]. Such elements are called algebraic over B, while possible elements of A which do

12.1. BASIC N O T I O N S

131

not have such property are called transcendental over B. From a differential algebraic point of view, one can consider the elements of A which satisfy differential algebraic equations with coefficients in B, that is the elements a E A such that P(a, d(a), d 2 ( a ) , . . . , d"(a)) = 0 for some n and some polynomial P(zo, zl, . . . , z,) ~ B[zo, zl, . . . , z,~]. Such elements are called differentially algebraic over B, while possible elements of A which do not have such property are called differentially transcendental over B. In a differentially algebraic extension A / B , all the elements of A are differentially algebraic over B. An extension A / B is differentially transcendental if A contains some elements which are differentially transcendental over B. E x a m p l e 12.1.3 Let C ~ denote the ring of C ~ functions f : IR --+ 1~, en-

dowed with the usual derivative. Cc~ is, in an obvious way, a differential extension and the differentially algebraic elements of C ~ are simply called differentially algebraic functions. The C ~ function f ( x ) = exp x is differentially algebraic, since the polynomial P(zo, zl) := zo - zl is such that P ( I ( x ) , f ' ( x ) ) -= O. Similar conclusions hold for, e.g., the C ~ functions sinz, cosz. Given a differential field extension K / k and a subset L C K , we denote by k(L) the smallest differential subfield of K which contains k and L. In case K is differentially algebraic over k(L) and L is minimal with respect to this property, i.e. K is differentially transcendental over k(L') for every L' ~ L, we say that L is a differential transcendence basis of the extension K / k . If a finite set L is a differential transcendental basis of K / k , the n u m b e r of elements in L is called differential transcendence degree of K / k . To say that the differential transcendental degree of K / k is zero means that the extension is differentially algebraic.

Chapter 13

State space s y s t e m s 13.1

A differential algebraic point of view

In this section, we investigate the possibility of associating to a nonlinear system E of the general form E = { f(:~'x'u)

0 h(y,x) == O.

(13.1)

a differential field Ez < u, y > which can be viewed as an extension of/~/. Our construction will particularly provide useful results when specializing, as in [44], to systems, i.e. to systems of the form

explicit

E= { 5Cy == h(x).f(x'u)

(13.2)

where the components of the vector field f and the function h are assumed to be analytic and differentially algebraic over a suitable ring. It is worth to remark that such assumption hold for a large class of dynamical systems that arise in modern control applications, like those used for modelling of mechanical systems and robotic manipulators [2]. The construction of E~ < u, y > is of importance in the study of general systems of the form described above since it embraces a relevant structural

CHAPTER 13. STATE SPACE SYSTEMS

134

information, both from the differential geometric point of view and for the differential algebraic one. Roughly speaking, /C~. < u, y > describes the differential algebraic relations existing between the inputs and the outputs of ~. Its construction represents therefore a solution, in a nonlinear setting, to the difficult problem of establishing a path from the state space, or internal, system description to an input/output, or external, system description. In particular, it provides a framework where the dynamical properties that depend on the i n p u t / o u t p u t behavior can be characterized and studied. As an illustration of this we recall (see [44, 45]) that the invertibility properties of Z can he characterized by means of the subfield K:~ < y > of/C~ < u, y >, which consists of all nonvanishing rational combinations of the outputs and their derivatives. From a technical point of view, the key point in the construction of /C~ < u , y > consists, after establishing the suitable framework, in proving that the differential ideal 27~ associated to the dynamic equation x = f(z, u) of E is prime. This is proved in Theorem 13.2.2. An elementary construction of the field/C~ < u, y > in the case in which f and h are rational functions in x and u has been described in [108] (see also [120, 62]). 13.1.1

The

basic

differential

ring

Let C denote the ring of real functions in the variables (z, u) which are analytic on an open disk 79 C ~ n + m . We remark that C is an integral domain, that is the product of two elements gl (z, u)g2(z, u) of C is the zero function on 79 only ifgl(x,u) = 0 or g2(z,n) = 0. We denote by O/Oxi for i = 1,...,n, O/Ouj for j = 1 , . . . , m the usual partial derivative operators acting on C. Starting from C we build up a differential ring in the following way. First we introduce an infinite set of indeterminates

I={x~h),u~ h), f o r i = l , . ,

...,m; h > l ,

Then we consider the ring C[I], whose elements are polynomials in the indeterminates of I with coefficients in C, and we define a derivative operator d on C[I] by means of the following rules

d(x,) =

xll ; ...d(xl hI) =

for i = 1 . . . , n ; h > 1;

(h+l) (13.3)

13.1. A D I F F E R E N T I A L A L G E B R A I C P O I N T OF V I E W

135

d(uj) = uJl); ...d(u~ h)) ='uj(h+l) for j = 1,...n; h >_ 1; d(g(x, u)) =

E (0/(9xi(g(x'u))x~I)q- Z i=l,...,n

(13.4)

(~/Ouj(g(x'U))U~ I>

j=l,...,m for g(x, u) E C

(13.5)

d(gl (x, u) + g~(x, u)) = d(gl(x, u)) + d(g2(x, u)) for

gl(X,U),g2(X, U) E C[I]

(13.6)

d(gl(x, u)g2(x, u)) = d(gl (x, u))g~(x, u) + gl(x, u)d(g~(x, u)) for gt(x, u),g2(x, u ) E C[I]

(13.7)

Remark that by means of the indeterminates xl h) and uJ h) we essentially add to the ring C all possible derivatives, with respect to d of the elements xi and uj. In the following it will be convenient to write x}~ for xi, i = 1 , . . . , n and u~~ for u j, j = 1 , . . . , m. The following Proposition is obvious. P r o p o s i t i o n 13.1.1 (C[I], d) is a differential integral domain. The differential integral domain C[I] will be the basic ring for developing the control theoretic constructions of the next sections 13.1.2

Differential

Algebraic

Analysis

Let us consider asystem Z of the form (13.1), with state x E j/~n, input u E / R m and output y E/f~ p evolving over a time interval 0 < t < to, f being an analytic vector function of dimension n with domain in ~ n + m and h being an analytic vector function of dimension p with domain in ~n+p. D e f i n i t i o n 13.1.2 Given the system ~, described by (13.1), we associate to the dynamic equation f(fc, x, u) = 0 the differential ideal I s of C[I] defined as follows /:~ = (fl(x (1), x, u), . . ., fn(X (1), X, u) ) (13.8)

where f, = fi(x, x, u) for i = 1,..., n are the components o f f ( x , u).

C H A P T E R 13. S T A T E S P A C E S Y S T E M S

136

Remark that 27~ consists of the elements g E C[I] that can be written, for V~k>~.....n~Jix /r ix(l) , x , u))(})gik where only a finite some gik E C[I], as a sum g = z..~i=l number of terms are different from zero. In case the differential ideal satisfies the key assumption of being prime, it is now possible to construct a differential field associated to E as illustrated in the following. L e m m a 13.1.3 / f the ideal 72~ is prime, then the ring (C[l],/-Zr., d'), where d'

is the derivative operator induced by d, is a differential integral domain. Denoting provisionally by/C the quotient field of C[I]/I~, we have the following definition.

Definition 13.1.4 i) The differential field associated to E is the smallest differential subfield of IC containing 1~, [uj] for j = 1 , . . . , m, [yk] for k = 1 , . . . , p , where yk(z) = hk(z) are the components of y, and it will be denoted by IC~.< u, y >;

ii)

The differential output field associated to E is the smallest differential subfield of lC~ < u , y > containing IR and [Yk] for k = 1 , . . . , p , where Yk = hk(z) are the components of y, and it will be denoted by ICE < y >.

13.2

Explicit systems

In the case of systems of the form (13.2), we give a direct proof of the integrity of C[I]/:~.. It may however be remarked that the construction of the associated field/C~ < u, y > could actually be carried on starting from the construction of the field/C considered, for systems of the form (1.1), in Chapter 1, Section 1.1. Remark, first of al,l that, in case E is of the form (13.2), 27~ consists of the elements g e C[I] which can be written, for some gik E C[I], as a sum v-~k>0 , (1) g = ?-,i_~l..... ntZi - fi(x,u))(k)gik where only a finite number of terms are different from zero. This motivates the introduction of the following lemma. L e m m a 13.2.1 Let g be an element of C[I] and assume that its monomials do

not contain xl h) for any i and for any h > O. Then g E I s if and only if g = O.

13.2. E X P L I C I T S Y S T E M S

137

Proof: Assume g satisfies the hypotheses of the lemma and suppose that it belongs to 9---,k> o , (1) the ideal 7::c. Then we can write g = 2-.,i=1..... n(Xi - fi)(k)gik, for some 9ik E C[I]. Since the x}l)'s are mutually independent, substituting fi for x} 1), i = 1 , . . . , n and dh(fi) for x}h) in both sides, the previous equality is preserved. Moreover, the first member is unchanged, since no quantity to substitute appears in it, and the second member becomes zero. So we get 9 = 0. The converse is obvious. T h e o r e m 13.2.2 Assume that ~ is of the form (13.2), then (C[I]//Z~, d'), where d I is the derivative operator induced by d, is a differential integral domain, or, equivalently, Z~ is a prime differential ideal.

Proof: Let us start by taking an element 90 of r

: g0 can be written, in more than

one way, as k>0

go =

(k+l) xi goik + goo

i=l,...,n

where the monomials constituting g00 do not contain x} h) for any i and for any h > 0. Then, if we consider the element gl of C[I] defined by k>0

gl =

(k+l) Xi

g l i k q-go0

i=l,...,,n

V'k>~...... n(xl 1)-fi)(k)goik belongs to 27~. Now, writing we have that (9o-91) = z-.,i=l gl as k>0

gl =

Xi

g l i k W glO

i=l,...,,n

where the monomials constituting g10 do not contain xl h) for any i and for any h > O, we consider the element gz of C[I] defined by k>O

g2

f k)9 ,k

=

i=l,...,,n

+

C H A P T E R 13. S T A T E SPACE S Y S T E M S

138

and we have that (gl-g2) = v-k>0 z..,i=l ......ntiX(t) i --fi)(k)glik belongs to Z=. Iterating this procedure we construct a sequence go,g1,... ,gr of elements of r such that (gi - gi+l) and, as a consequence, (go - gr) = (go - gl) + (gl - g2) + -.. + (g~-I - g~) belong to 1: and such that the monomials constituting g~ do not contain x~h) for any i and for any h > 0. In particular, if we denote by square parenthesis the equivalence classes of elements of C[I] in C[I]/I, we have that [go] = [gr]. Now, given two elements g' and g" of C[I] such that [g'][g"] = [0], let us consider two elements, say g;~ and g"q, constructed as above, such that the monomials constituting them do not contain x}h) for any i and for any h > 0 and such that [g'] = [g~],[g"] = [g"q]. Then, by I " I ,, [9pg q] = [g'P][g"q] = [g'][g"] = [0], we have that the product gvg q belongs to /:~. Hence, by Lamina 13.2.1, g~g"q = 0 and, since C[I] is an integral domain, either g~ = 0 or g"q = 0. In turns this implies that either g~ belongs to Z or g" belongs to Z~. In case E is of the form (13.2), the differential field K~ < u, y > contains in particular the input and the output functions of E modulo the identifications induced by the system equations. Remark that the construction of s < y > follows the lines of the construction of the differential field denoted by Y in ([45] sect. III). However, comparing with [45], Theorem 13.2.2 provides a rigorous proof of the possibility of carrying out such construction. If we restrict slightly our set-up, the field ]Cz < u, y > may be viewed as an input-output description of ~ and, at the same time, the properties of K~ < y > may be employed for characterizing the dynamic properties of E, More precisely, with the notation established in Section 13.1.1, let us consider the infinite set r

=

for i = 1, .., n; j = I , . . . ,

m ; h > 0, h e

and the differential subring /R[I*] of C[I], consisting of the polynomials with real coefficients in the indeterminates of I*. In the rest of this chapter, we will restrict our attention to the dynamical systems of the form (13.2) for which the following Assumption holds.

Assumption

13.2.3 The components fi(x, u), for i = 1,..., n o f f ( x , u) and hi(x), for j = 1,... ,p of h(x) are elements o f t C C[I] which are differentially algebraic over IR[I*].

13.2. EXPLICIT SYSTEMS

139

R e m a r k 13.2.4 The above condition means, in more explicit terms, that

fi(z, u) and hj(x) are solution of some algebraic differential equation with coefficients in ~:~[I*]. An elementary example of an element in the class determined by Assumption 13.2.3 is, for instance, that given, for n = m = p = 1, by the system

E = { Yk == xSin(u)

(13.9)

In fact E above verifies the Assumption 13.2.3 since f(x, u) = sin(u) and h(x) = x belong to C and are solutions, in C[I], of the algebraic differential equations u(1)d2X - u(2)dX + (u(1))3X = 0 and, respectively, d X - x (1) = O. For such system, K:c < u, y > can be described as the quotient field of the ring of polynomials in the indeterminates Ix], [u(h)], for h E SV, [sin(u)], [cos(u)] provided with the derivative operator determined by d' and by the equality d'[x] = [sin(u)]. From a general point of view, the class of dynamical systems determined by Assumption 13.2.3 includes essentially all systems of the form (13.2) currently dealt with in modern control theory (compare with [48, 125] for further comments). As a consequence of Assumption 13.2.3, if we denote by ]C~ < u > the smallest differential subfield of K containing IR and [uj], for j = 1 , . . . , m, the elements [Yk] for k = 1 , . . . , p of K~ < u, y > are differentially algebraic over K~ < u >. Therefore , the differentially algebraic extension

1 ~ < u,y >/]C~< u > defines an input-output system in the sense of [46] or a dynamics in the sense of [10]. Its construction may be viewed as a way of deriving, in a general nonlinear setting, an external description of ~ from the internal one given by (13.2). Moreover, the field 1 ~ < y > turns out to be a differential extension of ~ / w h o s e differential transcendence degree is finite.

140 13.3

C H A P T E R 13. S T A T E S P A C E S Y S T E M S

Further

remarks

Let we add some remark about the framework we have chosen for our investigation and let us discuss briefly a possible extension of our result. R e m a r k 13.3.1 The property for E of being described by equations with analytic coefficients, is considered by some authors as a standard assumption in nonlinear control theory. Moreover, such hypothesis appears to be a very natural one for developing the construction of K~ < y >. In order to see this, let us consider the nonanalytic C~-system E described by { x

=

Yl

=

y2

=

u e x p ( - 1 / x 2) 0 0

xp(-1/x

ifx 0 ifz 0

As YlY2 = 0, although yl # 0 and Y2 # 0, the construction of the field/Cz < y > cannot be carried on. R e m a r k 13.3.2 One could argue that the class of systems described by (13.2) does not represent the most natural framework for a differential algebraic approach. More general descriptions, in which the dynamic equation is in implicit form, i.e., descriptions of the form E - - { f(~'x'u)y

== h(z)0

(13.10)

appear in fact to be more appropriate. However, it is easy to remark that, in general, the differential ideal 77~ of C[I] associated to an implicit dynamic equation, that is the ideal Z~ = ( f l , . . . , f n ) (13.11) where

f, = for i = 1,..., n are the components of f(/:, z, u), is not prime. As an example, t a k e / ( z , x, u) = z2 _ u 2 : neither (zC:) - u) nor (z(:) + u) belong to Zs, but their product does and, again, the construction of the field/C~ < y > cannot be carried on..

13.3. FURTHER REMARKS

141

R e m a r k 13.3.3 A perhaps more general setting would have been obtained by considering dynamical systems described by systems of differential equations of the form

E= [ dz/dt

= f(z,u,du/dt,d2u/dt~,...,dku/du k) =

L

(13.12)

where d/dt represents the time derivative, in which the state evolution involves the derivatives of the input u. The basic construction of our framework can be easily extended to this situation. We briefly described how to do this, using the same notations as before for denoting the objects which correspond to those introduced in section 13.1.1. First, denoting by z = ( z l , x ~ , . . . , x n ) ; u0 = (UOl,UO~,...,uom); ul = (ull,u12,...,u1,~) . . . u k = ( u k l , u k 2 . . . . ,Uk,~) k + 2 vectors of real independent variables, one starts by considering the integral domain C of real functions of variables (z, u 0 , . . . , uk) which are analytic on an open disk 7) C / R + n + (k + 1)m. Then, one introduces the infinite set of indeterminates l={~h),u~.h), fori=l,...,n;

h>_k+ l, h E M }

and constructs the ring C[I], whose elements are polynomials in the indeterminates of I with coefficients in C. A derivative operator d on C[I] is defined by means of the rules (13.3), (13.4), (13.6), (13.7), while (13.5) is substituted by: d(urj) -- U(r+l)j d(ukj) = uJ 1) , , (h), (h+l)

for j = 1 , . . . , m ; r = 0 , . . . , k - i; (13.13)

a[uj ) = u j

for j =

1,..,m;

h_> 1.

Remark that by means of the indeterminates xl h) we essentially add to the ring C all possible derivatives, with respect to d, of the elements xi and by means of the indeterminates uJh) we add all possible derivatives, with respect to d, of the elements uoj of order greater than or equal to k, being represented by the elements urj for r = 0, ..., k, which already belong to C. Clearly, (C[I],bf d) is a differential integral domain. Moreover, the differential ideal/7~ of C[I] associated to ~ and defined as =

_ fl,...,

_ y,)

(13.14)

142

CHAPTER 13. STATE SPACE SYSTEMS

where fi

= fi(z, uo, u l , . . . , u k ) for i = 1 , . . . , n are the components of f(x, uo, u l , . . . , uk), can be shown to be prime just as done in the proof of Theorem 13.2.2. In fact, the key point in that proof is that the ideal at issue is defined by linear expressions in the zl 1), and this holds true also in the present more general situation. Then, the construction of/E~ < u, y > and of En < y > can be performed and, assuming that the components fi = fi(z, u0, u l , . . . , uk) for i = 1 , . . . , n, of f(x,uo,ul .... ,uk) and hj(z), for j = 1 , . . . , p , of h(x) are differentially algebraic over the suitable ring, the definition of output rank for systems of the form (13.12) can be given as in Definition 14.1.5.

Chapter 14

Input-output systems As mentioned at the end of Section 13.2, the construction of a differential field /(:~. < u, y > associated to the state space system E suggests to define inputoutput systems as differential field extensions (see [46]). Keeping in mind the situation represented by the extension /Es < u , y > //R, we can state the following abstract Definition.

Definition 14.0.4 An dynamical system is a finitely generated differential extension K / IR.

14.1

Inputs

states and outputs

Recalling that the field/C~ < u, y > constructed in Section 13.2 contains the input and the output functions of Y:. modulo the identification induced by the system equations, one can think of the elements of K as the possible behaviors exhibited by the system K/IR. Then, the following Definition makes sense. D e f i n i t i o n 14.1.1 Given the system E = K/IR, let u = ( U l , . . . , U m ) be a finite set of elements in K. Then, u is said to be a set of inputs for E if the differential transcendence degree of K / ~ < u > is zero. Moreover, if the differential transcendence degree of K / ~ equals m, then u is said to be a set of independent inputs.

C H A P T E R 14. I N P U T - O U T P U T S Y S T E M S

144

A set u of elements can therefore be interpreted as a set of inputs for E if the behavior of the system is described in terms of u and differential relations. E x a m p l e 14.1.2 Consider a system described by K / ~ , where K is the differential field ~ < y, u > modulo the equivalence relation generated by ~1- u = O. Since K is differentially algebraic over IR < y >, it is possible to see y as an input. Clearly, in accordance with the above Definition, one can alternatively see u as an input. A notion of state for a system E of the form K/B:~ can now be given, expressing the fact t h a t the behaviors of the system are described in terms of states, inputs and system equations. D e f i n i t i o n 14.1.3 Given the system E = K / ~ with input set u, let x = ( x t , . . . , x , ) be a finite set of elements in K . Then, x is said to be a state for E if it contains a non-differential transcendence basis of K / I R < u >. Given the system E = K / R , any set y -- (Yl,...,Yp) of elements in K can be viewed as a set of outputs. Then, given a set of inputs u and a state x, expressing the algebraic dependence of the components of k and of y over /R < u >, one gets an implicit realization of E of the form

F([c,x,u, it,...,u (v))

=

0

(14.1)

H(y,x,u,

=

0.

(14.2)

Possibly, the above equations m a y be locally solved with respect to x and y, yielding a local explicit realization of the usual form

~. =

f ( x , u , i t , . . . , u (u))

(14.3)

y

h ( x , u , it,...,u('~)).

(14.4)

=

As a consequence of Definition 14.1.3 we can now introduce the following notion of order.

Definition 14.1.4 The order of the dynamical system E -- K / I ~ with input set u is the (non-differential) transcendence degree of l i / ~ < u >.

145

14.1. INPUTS, STATES AND OUTPUTS

The notion of order given above generalizes the notion of number of poles for a linear system. Together with the order of a system, an index which characterizes basic dynamical properties is the so-called output rank, defined as follows (see [45]). D e f i n i t i o n 14.1.5 Given the system E = K / ~ with output set y, the differential output rank p(Z) of S is the differential transcendence degree of ~/$t. In case E has the state space form 13.2 and satisfies Assumption (13.2.3), its differential output rank p(E) is given by the differential transcendence degree of/Cr~ < y > / ~ , where/Cr~ < y > is the differential output field defined in 13.1.4. Then, if E is linear, p(E) turns out to coincide with the rank of the transfer matrix of E ([46, 108]). If E is a nonlinear affine system, p(E) can be computed by means of the structure algorithm as shown in [37]. E x a m p l e 14.1.6 Let us consider the system P, described by xl

=

sin(z2)

X3

:

x4

:

Xl + X2X4 U2

Yl

:

Xl

Y2

:

X4X3

(14.5)

An easy direct computation of the output rank p(P.) lows. Since the dimension of the output space is 3, dence degree of K.~ < y > cannot exceed 3. Now, let ~l = [xl], ~2 = [x2], ~a = [z4x3] of/C~ < y >. By the %1%2

]

+

-

=

can be carried on as folthe differential transcenus consider the elements equality o

we have that p(H) is less than or equal to 2. Moreover, since ul and u2 are mutually independent, so ate

~1) = [Ul] and ~(1) = [u2x3 + (xl + x2x4)x4] and we conclude that p(H) = 2.

C H A P T E R 14. I N P U T - O U T P U T S Y S T E M S

146

The notion of output rank was used in [44, 45, 46] to obtain a characterization of invertibility that holds, in our context, for affine systems which satisfy Assumption (13.2.3). More precisely, recalling that left invertibility of a dynamical system means the possibility of determining, as solution of a system of differential equations, the input which generates a given output, we can state the following Proposition. P r o p o s i t i o n 14.1.7 A system ~ of the form (13.2) satisfying Assumption

(13.2.3) with u E ~ m is left invertible if and only if p(~) = m. A similar characterization of right invertibility can be given as follows. P r o p o s i t i o n 14.1.8 A system E of the form (13.2) with y(t) E IR p is right

invertible if and only if p(E) = p.

14.2

Structure

By exchanging the role of inputs and outputs, we can derive a notion of zero dynamics for a system ~ of the form K / ~ . Assuming that y is the output of and that the differential transcendence degree of K / I R < y > is zero, the zero dynamics can be explicitly constructed, in the same way as a realization, by means of a finite set z = ( z l , . . . , z r ) of elements which contains a nondifferential transcendence basis of K/]R < y >. Then, in particular, we have the following result. P r o p o s i t i o n 14.2.1 The dimension of the zero dynamics of the system E = K/H:~ with output y and diff. tr. deg. ( K / ~ < y >) = 0 equals the non

differential transcendence degree of E = K / ~

< y >.

The dimension of the zero dynamics can be interpreted as the number of finite zeros of E. Concerning the zeros at infinity, if x denotes the sate of E and y denotes its output, we have the following definition. D e f i n i t i o n 14.2.2 The number of zeros at infinity of order 1 equals the non

differential transcendence degree, if it is finite, of ~ < x, y > over ~ < x >. The number of zeros at infinity of order k equals the non differential transcendence degree, if it is finite, of ~ < x, il,.. . ,y(k) > over ~ < z , y , . . . , y ( k - 1 ) >, fork>2.

Chapter 15

Accessibility and observability from a differentially algebraic point of view The notion of accessibility has been characterized in Section 3.4 by means of the concept of autonomous element. From a differential algebraic point of view, autonomous elements can be defined as follows. Definition 15.0.3 Given a differential field extension K / m , an element E K is said to be an autonomous element i f ( does not belong to IR and diff. tr. deg. /R < ~ > / / R -- 0.

(15.1)

Condition (15.1) parallels (3.4) in Definition 3.4.1 by saying that ~ satisfies a differential algebraic equation with coefficients in/iS, while in (3.4), F needs not to be algebraic.

148

Example 15.0.4

C H A P T E R 15. A C C E S S I B I L I T Y A N D O B S E R V A B I L I T Y Let us consider the system E given by xl

=

x2

=

z2 + u sinxe

The element ~ = z2 is autonomous in the sense of (3.4) since it is solution of the first order differential equation f( - sin X = O. It is also autonomous in the sense of (15.1), as it solves the higher order polynomial differential equation 2 4 + 2 2 - 2 2 = O.

Using the concept of autonomous element given by Definition 15.0.3,the notion of accessibility in the differential algebraic framework can be characterized, as done in Definition 3.5.1, by the absence of autonomous elements. More precisely, we have the following Definition. D e f i n i t i o n 15.0.5 A dynamical system E = K/IFI is said to satisfy the strong accessibility condition if there are no autonomous elements in K . Concerning observability, the differential algebraic framework provide a simple and quite intuitive notion. D e f i n i t i o n 15.0.6 (see [~0]) Given a differential field extension K/1F•, an element ~ E K is said to be observable with respect to the differential field k, where IF~ C k C K , if it is algebraic over k. In case of a dynamical system E = K / I ~ with input set u and output set y, we are interested in observability with respect t o / R < u, y >. Actually, this notion characterizes the property of being solution of an algebraic equation involving inputs, outputs and their derivatives. E x a m p l e 15.0.7 Consider the system E given by Xl Jc2 ~3 y

= = = --

x2 u § x3 u x~

Note that x l , z 2 , x3 do not belong to IF~ < y , u >.

However, x2 and x3 are

algebraic over ~ < y , ~ > since ~ - y = 0 and x] + 2u~3 + (u - ~ ) = 0 ~

and z3 are observable whereas xl is not.

Chapter 16

Control P r o b l e m s 16.1

Dynamic disturbance decoupling

In this Section, we consider a system with control inputs u, disturbance inputs q and outputs y. We state without proofs some results taken from [29]. The rank of the system is denoted p(y). T h e o r e m 16.1.1 The dynamic disturbance decoupling problem is solvable if and only if non diff. tr. deg. /R < y > (u,z)/IR < y > (x) = m -

p(y)

For a complete study of the problem and additional results on dynamic disturbance decoupling with measurement of the disturbance (and a finite number of its time derivatives), the reader is referred to [29].

16.2

Left factorization problem

T h e o r e m 16.2.1 The left factorization problem is solvable if and only if diff. tr. deg. 1R < y, YM > /J~ -: diff. tr. deg. ~ < y > / / R

150

C H A P T E R 16. C O N T R O L P R O B L E M S Proof." Consider the tower of fields 1R C 1~ < y > C IR < Y, YM >,

one has

diff. tr. deg. h~ < y, YM > / J ~ = = diff. tr. deg. /~ < y, YM > /~t~ < y > +diff. tr. deg. ~ < y > / / ~ . The Left Factorization problem consists in finding a post compensator driven by y and whose output equals YM, thus diff. tr. deg. ~ / < Y, YM > /IR < y >---- 0 and the result follows.

16.3

Noninteracting control

The notion of output rank has been useful in providing solutions to decoupling problems in I33]. Without entering in the details, we remark that the characterization of input-output decouplability given in [33] holds, with the same proof, in our framework. T h e o r e m 16.3.1 The three following statements are equivalent (i) the noninteracting control problem is solvable via dynamic state feedback, (ii) the noninteracting control problem is solvable via quasi-static state feedback, (iii) the system is right invertible.

A recent survey on solving control problems using filtrations of differential field extensions can be found in [31].

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Index accessibility, 29, 148 accessibility criterion, 32 accessible, 20, 29 analytic, 4, 5, 7 autonomous, 147 autonomous element, 30, 31, 147 canonical form, 63 closed, 13, 14 controllability canonical form, 34 controllability indices, 34 controllable, 28, 29, 70, 80 derivation, 129 derivative operator, 9, 10,134, 136, 137, 141 differential extension, 130 differential field, 9, 130, 136, 138 differential form, 8 differential operator, 10 differential ring, 130 differentially algebraic, 131 differentially algebraic functions, 131 differentially transcendental, 131 disturbance decoupling, 99,101,103, 149 dynamic feedback, 86

essential order, 53, 55 exact, 10, 13, 14 exterior product, 12 feedback linearization, 80, 90, 96 field extension, 131 Frobenius Theorem, 14 generalized state, 7, 60 generalized state feedback, 59 indeterminate, 8,141 indistinguishable, 39 input, 143 input-output system, 139 input/output linearization, 79 input/state linearization, 89 integrable, 14 integrating factor, 14 inversion algorithm, 105 left factorization, 119 left inverse system, 57 left invertible, 57 linearizable subsystem, 93 linearizing outputs, 90, 96, 97 meromorphic, 7, 9

166

INDEX

model matching, 105, 109, 114

unicycle, 35, 39, 42, 85, 94

noninteracting control, 83-86, 150

vector space, 9-12, 32, 52, 71, 107

observability canonical form, 41 observability criterion, 41 observability indices, 42 observable, xi, 20, 37-43, 67, 71, 75, 79, 101, 148 observable space, 40 one-form, 10, 46 order, 144 output injection, 43, 62, 72 output injection linearization, 42

wedge product, 12

Poincar6 Lemma, 13 pole, 145 quasi-static feedback, 59, 85, 99 rank, 55, 145 reachable, 28 realization, 17, 144 relative degree, 30, 31, 53 right inverse system, 56 right invertible, 57 s-form, 12 state, 144 state elimination, 20 strong accessibility, 32 structure algorithm, 105 structure at infinity, 52, 67 transcendence basis, 131 two-form, 11

zero at infinity, 55, 58, 146 zero dynamics, 58, 146

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