Nonlinear Dynamical Control Systems

Henk Nijrneijer

Arjan van der Schaft

Nonlinear Dynamical Control Systems With 32 Illustrations

Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong

Henk Nijmeijer Arjan van der Schaft Department of Applied Mathematics University of Twente P.O. Box 217 7500 AE Enschcde The Netherlands

Library of Congress Cmaloging-in-Publication Datu Nijmeijer. H. (Henk), 1955Nonlinear dynamical control systems I Henk Nijmeijer, Arjan van der Schafl.

p.

ern.

ISBN 0-387-97234-X I. Conlrollheory. 2. Nonlinear theories. 3. Geometry, Differential. 1. Schafl. A. J. van der. II. Title. QA402.3.N55 1990 629.8'3l2-dc20

89-26360

Printed on acid-free paper

© 1990 Springer-Verlag New York Inc. All rights reserved. This work may not be lranslaled or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fiflh Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in conneclion with any form of informal ion storage and retrieval, electronic adaptation, computer soflware, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The usc of general descriptive names, trade names, trademarks, etc., in this publication, even if the former arc not especially identified, is not 10 be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Camera-ready copy supplied by the authors using ChiWriler. Printed and bound by R.R. Donnelley & Sons. Harrisonburg, Virginia. Printed in the United States of America. 987654321 ISBN 0-387-97234-X Springer-Verlag New York Berlin Heidelberg ISBN 3-540-97234-X Springer-Verlag Berlin Heidelberg New York

Preface This textbook on the differential geometric approach to nonlinear control grew out of a set of lecture notes, which were prepared for a course on given by us for the first time during the fall

nonlinear system theory, semester of 1988.

The

audience

consisted mostly

of

graduate

students

I

taking part in the Dutch national Graduate Program on Systems and Control.

The

aim of

this

course

is

to

give

a

general

introduction

to

modern

nonlinear control theory (with an emphasis on the differential geometric

approach),

as

well

as

to

provide

students

specializing

in

nonlinear

control theory with a firm starting point for doing research in this area.

One of our primary objectives was to give a self-contained treatment of all the topics to be included. Since the literature on nonlinear geometric control theory is rapidly expanding this forced us to limit ourselves in the choice of topics. The task of selecting topics was further aggravated by the continual shift in emphasis

in the nonlinear control literature

over the last years, Therefore, we decided to concentrate on some rather solid and clear-cut achievements of modern nonlinear control, which can be expected to be of remaining interest in the near future. Needless to say, there

is also a personal bias

Furthermore,

in the

topics we have finally

selected,

it was impossible not to be influenced by the trendsetting

book "Nonlinear Control Systems:

an Introduction", written by A.

Isidori

in 1985 (Lecture Notes in Control and Information Sciences, 72, Springer). A second main goal was to illustrate the theory presented with examples stemming from various fields of application. As a result, Chapter 1 starts with a

discussion of some characteristic examples

systems,

which

will

serve

as

illustration

of nonlinear control

throughout

the

subsequent

chapters, besides several other examples. Thirdly,

we decided to include a rather extensive and self-contained

treatment

of

the

necessary

geometry.

Especially

the

mathematical

required

background

theory

on

on

Lie

differential

brackets,

(co-)

distributions and Frobenius' Theorem is covered in detail. However, rudimentary

knowledge

(manifolds,

tangent

reading of the book. with

the

basic

about

the

space,

vectorfields)

Furthermore,

concepts

fundamentals

of

will

of

differential

greatly

some

geometry

facilitate

the

the reader is supposed to be familiar

linear

system

theory;

especially

some

acquaintance with linear geometric control theory will be very helpful. 110dern

nonlinear

control

geometric

approach,

has

theory,

emerged

in

during

particular the

the

seventies

differential in

a

rather

successful

attempt

formulation

of

to

deal

nonlinear

with

basic

control

questions

systems,

controllability and observability,

and

in

the

state

including

the

problems

of

theory.

It

(minimal)

was also motivated by optimal control theory,

realization

space

in particular the Maximum

Principle and its relation with controllability issues. The theory gained strong impetus at the end of the seventies and beginning of the eighties by the introduction of several new concepts, most of them having as their crucial part nonlinear feedback.

Let us illustrate this with two papers,

which can be seen as benchmarks in this development. First,

there is the

paper by Brockett on "Feedback invariants for nonlinear systems" VIIth !FAC World Congress, with

the

control

basic

question

system

can

be

Helsinki.

to

what

changed

pp.

1115-1120,

extent

by

the

(stacie

1978),

structure state)

(Proc.

which deals

of

a

nonlinear

feedback.

A direct

outgrowth of this paper has been the theory on feedback lineariza tion of nonlinear control systems. via

feedback:

a

differential

GaTi-Giorgi &. l1onaco 1981)

the

Secondly,

in the paper "Nonlinear decoupling

geometric

concept of

a

taken by Hirschorn

by

Control,

controlled invariant

various sorts of decoupling problems was

approach"

(IEEE Trans. Automat.

Isidori.

AC-26,

distribution

(independently,

SIMI J.

is

341-345, used

for

a similar approach

( .. (A, B)-invariant distributions

decoupling of nonlinear systems",

Krener,

pp.

and

Contr. Optimiz.

disturbance

19,

pp.

1-19,

1981»). It is worth mentioning that the concept of a controlled invariant distribution is a nonlinear generalization of the concept of a controlled invariant subspace. linear

which is

geometric control

"Linear

Multivariable

the cornerstone in what is usually called

theory

Control",

edition 1985). In fact,

(see

the

trendsetting book of Wonham,

Springer,

first

edition

1974,

third

a substantial part of the research on nonlinear

control theory in the eight:ies has been involved with the "translation" to ehe nonlinear domain of solutions of various feedback synthesis problems obtained in linear geometric control theory. Connected with the concept of (controlled) invariant distributions,

the above mentioned IEEE paper also

stressed the usefulness of special choices of state space coordinates, in which

the

various

system

kinds

of

struet:ure

becomes

more

nonlinear

normal

forms,

transparant.

algorithm such as the nonlinear D"-algorithm, the

dynamic

extension

algorithm,

has

usually

The

search

connected

to

for some

the Hirsehorn algorithm or

been another

major

trend

in

the

eighties. At

this moment

it is

difficult

to

say what will be

trends in nonlinear cantrol theory in the near future. feedback

stabilization

problem,

which

has

recently

the

prevailing

Without doubt the obtained

a

strong

vli

renewed

interest,

will be

a

fruitful

area.

Also

adaptive

control

of

nonlinear systems, or, more modestly, the search for adaptive versions of current nonlinear control schemes is likely going to be very important, as well as digital implementation (discretization) of (continuous-time based) control strategies. l1oreover I it seems that nonlinear control theory is at

a point in its development where more attention should be paid to the

special (physical) structure of some classes of nonlinear control systems, notably in connection with classical notions of passivity,

symmetry,

and

notions

stemming

from

bifurcation

stability and

theory

and

dynamical

systems. The contents of the book are organized as follows: Chapter 1 starts with an exposition of four examples of nonlinear control systems, rest

which will be used as illustration for

of

the

A few

book.

generalities

the theory through the

concerning

the

definition

of

nonlinear control systems in state space form are briefly discussed,

and

some typical phenomena occurring in nonlinear differential (or difference) equations are touched upon, in order to put the study of nonlinear control systems

also

into

the

perspective

of

nonlinear

dynamics.

2

Chapter

provides the necessary differential geometric background for the rest of the

boole.

Section

2.1

deals

while

in

Section

geometry,

with

some

2.2

fundamentals

of

vectorfields,

differential brackets,

Lie

(co-)distributions and Frobenius' Theorem are treated in some detail. For the reader's convenience we have included a quick survey of Section 2.1, as well as a short summary of Section 2.2 containing a list of useful properties and identities. and

observability

conditions

that

are

In Chapter 3 some aspects of controllability

treated

generalize

with

the

an

well-known

emphasis

on

Kalman

rank

controllability and observability of linear systems,

nonlinear conditions

rank for

and on the role of

invariant distributions in obtaining local decompositions similar to the linear

Kalman

input-output

decompositions.

representations

Chapter of

4

is

nonlinear

concerned

control

with

systems,

various and

thus

provides a link with a more input-output oriented approach to nonlinear control

systems,

without

actually

going

into

this.

Conditions

for

invariance of an output under a particular input, which will be crucial for

the

theory

of

analytic as well

decoupling

as

in

later

in the smooth case.

chapters,

are

derived

in

In Chapter 5 we discuss

the some

problems concerning the transformation of nonlinear systems into simpler forms,

using state-space

contains

the

full

and feedback

transformations,

solution of the local

feedback

while

Chapter

6

linearization problem

viii

(using static state feedback). In Chapter 7 the fundamental notion of a controlled invariant discribution is introduced, and applied to the local disturbance decoupling problem. Chapters Band 9 are concerned with the input-output

decoupling

problem;

an

using

analytic,

respectively

a

In Chapter 10 some aspects of the local feedback

geometric approach.

stabilization problem are treated. Chapter 11 deals with the notion of a controlled invariant submanifold and its applications to stabilization, interconnected systems and inverse systems. In Chnpter 12 a specific class of nonlinear control systems, roughly speaking mechanical control systems, is treated in some detail. Finally, in Chapters 13 and 14 a part of the theory developed

in

continuous-time

systems

the

preceding chapters

x=

f(x,u),

y

=

is

generalized

h(x,u).

to

general

respectively

to

discrete-time systems. At the end of every chapter we have added bibliographical notes about the main sources we have used, as well as some (very partial) historical information. related

,,,e

Furthermore

work

and

have occasionally added some references

further

developments.

We

like

references are by no means meant to be complete selected.

and we sincerely apologize

to

I

to

stress

that

to the

or are even carefully

those authors whose important

contributions were inadvertently not included in the references. As

already mentioned before,

included

in

the

present

many topics

book.

Notable

of interest could not be

omissions

are

in

particular

realization theory, conditions for local controllability, observer design, left-

and

right-invertibility,

linearization by feedback. methods theory.

global

and high-gain feedback,

algebraic methods.

and,

(We also like

global

issues

stabilization, sliding mode

last but not least, to refer

Isidori's uNonlinear Control

to

Systems~

in

decoupling

and

singular

perturbation

techniques,

differential

nonl inear opcimal contra I

the very recent second edition of (Springer, 1989) for a coverage of

some additional topics.) Acknowledgements The present book forms an account of some of our views on nonlinear control theory.

which have been formed

in contacts with many people from

the

nonlinear control community. and we like to thank them all for sCimulating conversations and creating an enjoyable atmosphere at various meetings. In particular Grizzle,

we

like

to

express

our

gratitude

to

Peter

Grouch,

Jessy

Riccardo Marino. {Htold Respondek and Hans Schumacher for

the

very pleasant and fruitful cooperation we have had on some joint research endeavors.

We

thank

the

graduate

students

attending

the

course

on

nonlinear system theory of the Graduate Program on Systems and Control in the

fall

semester of 1988,

for

serving as

an excellent

and

responsive

audience for a first "try-out" for parts of this book. Special thanks go to our Ph.D. students Harry Berghuis, Antonio Campos Ruiz, Henri lIuijberts and Leo van der

i~egen

for their assistance in correcting and proof reading

the present manuscript. errors

and omissions

Of course,

in

the responsibility for all

the book remains

ours.

We

like

to

remaining thank Dirk

Aeyels and Hans Schumacher for very helpful comments on parts of the text. We are very much indebted to our former supervisor Jan C. Willems for the many inspiring discussions we have had throughout the past decade. Over

the years

the

Systems

and Control

Group

of

the

Department

of

Applied Hathematics of the University of Twente has offered us excellent surroundings for our research and teaching activities. It is a pleasure to thank all our colleagues for creating this pleasant working atmosphere. Special

thanks

go

to

our

secretary Harja

Langkamp

for

assistance throughout the years. We are most grateful

her

invaluable

to Anja Broeksma,

Harjo Quekel, Jeane Slag-Vije and Harja Langkamp for their skilful typing of

the

manuscript.

We

thank

them

for

remaining

cheerful

and

patient,

despite the length of the manuscript. Also we thank Hr. H.H. van der Hey for his contribution in preparing the figures. Sontag for his publishing recommendation, Verlag

office

in

New

York

for

the

Finally we

thank Eduardo

and the staff at the Springer

pleasant

cooperation

during

the

preparation of this book.

Enschede, October 1989,

Henle Nijmeijer Arjan van der Schaft

Contents

1

Introduction

1

Notes and References

2 Manifolds, vectorfields, Lie Brackets, Distributions 2.0 Survey of section 2.1 2.1 Manifolds, Coordinate Transformations, Tangent Space 2.1.1 Differentiability, Manifolds, Submanifolds 2.1.2

Tangent vectors, Tangent space, Tangent Mappings

Vectorfields, Lie Brackets, Distributions, Frobenius' Theorem, Differential One-Forms 2.2.1 Vectorfields, Lie Brackets, Lie Algebras Distributions, Frobenius' Theorem 2.2.2 2.2.3 cotangent Bundle, Differential One-Forms, Co-distributions 2.3 Summary of Section 2.2 Notes and References Exercises

20

23 24

29 29 37

2.2

3 controllability Bnd Observability, Local Decompositions 3.1 Controllability 3.2 Observability 3.3 Invariant Distributions~ Local Decompositions Notes and References Exercises 4

5

43 43

55 61 67

69

70 73 73

93 101 111 113

Input-output Representations

117

4.1 wiener-Volterra and Fliess Series Expansion 4.2 External Differential Representations 4.3 Output Invariance Notes and References Exercises

118

state Space TransformatioD and Feedback

148

state space Transformations and Equivalence to Linear systems 5.2 Static and Dynamic Feedback Notes and References Exercises

125 135 143

145

5.1

148 165

172 173

6 Feedback Linearization of Nonlinear Systems 6.1 Geometric conditions for Feedback Linearization 6.2 computational Aspects of Feedback Linearization Notes and References Exercises 7 Controlled Invariant Distribution and the Disturbance Decoupling Problem 7.1 Controlled Invariant Distributions 7.2 The Disturbance Decoupling Problem Notes and References Exercises

B The Input-Output Decoupling Problem 8.1 static state Feedback Input-output Decoupling for Analytic systems 8.2 Dynamic state Feedback Input-Output Decoupling Notes and References Exercises 9 The Input-Output Decoupling Problem. Geometric Considerations 9.1 The Block Input-output Decoupling Problem for Smooth Nonlinear Systems 9.2 The Formal structure at Infinity and Input-Output Decoupling Notes and References Exercises 10 Local Stability and Stabilization of Nonlinear Systems

176 17B

194 205

207

211

211 219 237 239

242 242 255

270 271

274 274 286

294 296

299

10.1 Local Stability and Local Stabilization via

Linearization 10.2 Local Stabilization using Lyapunov's Direct Method

299 303

10.3 Local stabilization via center Manifold Theory

310

Notes and References Exercises

319 321

11 Controlled Invariant Submanifolds and

Nonlinear Zero Dynamics 11.1 Locally Controlled Invariant Submanifolds 11.2 Constrained Dynamics and Zero Dynamics 11.3 Interconnection of systems and Inverse systems

323 323

331 337

Notes and References

344

Exercises

346

12Mechanical Nonlinear Control Systems 12.1 Definition of a Hamiltonian Control System

349 355

12.2 controllability and Observabi1ity; Local Decompositions

363

12.3 stabilization of Hamiltonian control systems

369

12.4 Constrained Hamiltonian Dynamics

376

12.5 conservation Laws and Reduction of Order

3S5

Notes and References

392

Exercises

39.

13 Controlled Invariance and Decoupling for General Nonlinear Systems

400

13.1 Locally controlled Invariant Distributions

400

13.2 Disturbance Decoupling

414

13.3 Input-Output Decoupling

416

13.4 Locally controlled Invariant Submanifolds

422

13.5 Control systems Defined on Fiber Bundles

426

Notes and References

431

Exercises

433

14 Discrete-Time Nonlinear Control Systems

437

14.1 Feedback Linearization of Discrete-Time Nonlinear Systems

43S

14.2 Controlled Invariant Distributions and the Disturbance Decoupling Problem in Discrete-Time

445

14.3 Input-Output Decoupling in Discrete-Time

451

Notes and References

45S

Exercises

461

Subject Index

463

1 Introduction This book is concerned with nonlinear control systems described by either (ordinary) differential equations or difference equations with an emphasis the systems under consideration

That is,

on the first class of systems. are of the following type

or

xee)

f{x(t) ,u(t»,

y( t)

h(x{t)

{ {

(1.1)

,u(t»,

x(/c+l)

f(x{lt) ,u(lc»,

y(k)

h(x(k),u(k)),

(1. 2)

where x denotes the state of the system, u the control and y the output of

the system. Before we will discuss in some depth the general definitions and assumptions on the systems (1.1) or 0.2) we focus on four examples of control

systems which

fit

into

(1.1)

or

(1.2),

and which

motivation for considering nonlinear control systems. questions

will not yet be

chapters. As one will see, scientific

disciplines

addressed,

but

are

serve

as

a

Particular control

deferred

to

the

later

the examples are taken from rather different

such

as

robotics,

aeronautics,

economics

and

biology.

Example 1.1 (Robot-Arm Control) robot manipulator

Consider a

(or double pendulum)

frictionless,

with control

rigid two-link

torques

ul

and

Uz

applied at the joints.

0,

Fig. 1.1. Two-link rohO! manipulator.

The dynamics of such a robot-arm may be obtained via the Euler-Lagrange

2

formalism.

Let 0

=

«(Jl'{}:/.)

and

0

=

(Ol'O,})

and define

the Lagrangian

function

L(O.O) ~ T(O,O) - V(O)

(1. 3)

where T(o,i!) is the kinetic energy and V(O) the potential energy. For the above

configuration with

rigid massless

links

the

computed as the sum of the kinetic energies Tl and respectively •

nl

kinetic

T2

energy

is

of the masses m1

,

z . This yields

1

Z • 2.

T 1 ({}) = ;111111 {} 1 ,

T 2 (O,O) ~ :'m2(,e~ O~ + 1;(0 1 + 2.

82 )2 +

2.2)12 (cos O2

)°

1

(0 1

,,,

O2 ) ) ,

and silnilarly the potentLed energy V is the sum of the potential energi.es \'1

and

V z of

the two masses;

Vz (0)

Therefore,

Now the celebrated Euler-Lagrange equations are

i = 1,2,

(1. 5)

which yields in this ca.se the vector equation N(O)O'

N(O) =

+

[ m,

C(O,8)

'i

+ k(G) ... u

(l. 6)

+ m2i~ + mz 2; + 21112 1 1 £z cos

m21; +

/112

11. 22 cos {)z

02

mz 1;

-I- tll z l\

I112i;

12 cos

0,

l'

(1.7.a) C(Ofi!) ~

[ -ro,',',

(sin 8 2

)

mZ1 1 £2 (sin O2

)

O2 (20 1 • 2.

°1

+

1

i,)

(l.7.b)

3

k( 0) " -

[

(l.7.c)

In (1.6)

the

term le( 0)

the gravitational force

represents

and the

term

CCO,B) reflects the centripetal and Coriolls forces. Note that the matrix tI(O)

is

in (1.6) has as determinant mlm2i~f~ + rn:f~l'~ - m;.I!;l!; coszO z ' which

positive

for

all

O.

Therefore

(1.6)

is

equivalent

to

the

vector

equation (1.8)

Equation

(1.8)

manipulator.

describes

the

dynamical

behavior

of

a

It clearly constitutes a nonlinear control

state space (° 1

,0 1

,°

2

,0 2 )

E

51

x!R

X S1

x!R

£'t

TSI X TSl,

two-link

robot

system with as

Often the purpose

of controlling a robot arm is that of using the end effector for doing some prescribed task.

Though we did not

which is more difficult to model

incorporate

in the model,

the

robot hand

it is clear that the

interesting outputs of the model would be the Cartesian coordinates of the end point rather than the angles B1 and 02 between the separate links.

0,

\

0,

\ Xz

Fig. 1.2. End point of two-link robot nrm.

Denoting the Cartesian coordinates of the endpoint as Yl and Yz we obtain the output functions

(1. 9)

This is what is called the direct kinemacics for the robot arm. Of course, in practice the more important question is how to determine the angles 01 and 02 when the end position (Yl'YZ) is given (possibly as a

function

of

time). This is the so called inverse kinematics problem for the robot arm.

Computing the Jacobian of the right-hand side of (l.9) we obtain 1'1

01 +

COS

1'2

~

(01 + 02)

COS

12 (1.10)

-11 sin 01 - 12 sin (01 + Oz)!-R z

[

and thus

(1.11)

Hence

for



(° 1

with

02

kif.

".,

Ie

7l,

E

we

see

that

2 and so we lIIay apply at these points the inverse function

theorem, yielding 01 and 02 as a nonli.near function of (Yl'YZ)' We conclude thi.s discussion on robot arm control with the remark that the

approach

given

configura t:ions.

manipulat:or ...dth Euler-Lagrange dynamical

here

may

be

extended

to

various

more

complicated

For eNample one can equally well handle an m-link robot control

torques

formalism.

equations

Of

well

85

applied

course

as

in

at

the the

each

joint

analysis

direct

and

in

using

the

obtaining

by

the

inverse

kinematics

becomes much Illore involved. The study of this kind of nonlinear control

o

systems needs further invest:igation.

Example 1. 2 (Spacecraft Attitude Control) dynamics

describing

the

exchange actuators.

spacecraft

In this example we study the

attitude

with

gas

jet or

momentum

The equations describing the attitude control of a

spacecraft are basically those of a rotating rigid body with extra terms giving the effect of the control torques. Therefore one may separate the equations into kinematic equations relating the angular position wi th the angular velocity and dynamic equations describing the evolution of angular velqcity (or, equivalently, angular momentulII), The kinematic equations can be represented as follows, The angular position is described by a rotation matrix R. R transforms an inertially fixed set of orthonormal axes.

e1

•

e2

j

e3

into a

orientation as e 1

•

set of orthonormal axes e2

•

e3

),

1'1'

Je3

=1'1

for i - 1.2,3.

1':3

(with

say

the same

which are fixed in the spacecraft and have as

origin the center of mass of the spacecraft, thus

R e1

r2,'

ez

e1 Fig. } .3. Angulnr p(l~ition.

5

The evolution of R may now be expressed as

R(t) - - R(t) S(w(t» were w( t)

is

(1.12)

angular velocity of

the

the

spacecraft

at

(with

t

time

respect to the axes in the spacecraft) and Sew) is a 3x3-matrix defined by

w,

0

Sew)

with

- [ w,

=

W

-w,

three

angles

follows. if> ,

(),

about the axes r

~1

-w, An

(wI' Wz 'W 3 ).

obtained as

-w, ]

0

The

a1 ternat! va

r z and r 3

1 ,

[R3

(local)

description

angular position may be

which

JjJ,

i-th basis vector in

,

represent

consecutive

(1.12)

is

locally by

clockwise

respectively. Setting r 1

,

of

described

rotations

to be the standard

we obtain the kinematic equations as follows.

o sing, cos¢

1

0

o

o

cos¢>

sinq,

o

-sin¢>

cos¢

][

o

cosO

o

1

sinO

o

-sinO

o cosO

Therefore, sin . tanO

cosO : tane -SIn¢>

cos¢

sinq,(cosO)-l

Clearly, -~/2


0,

X

z > 0, I > D. This implies some additional

constraints on the parameters in (1.35). In particular it follows from the existence of such an equilibrium point that the right hand side of (1.35) vanishes for suitably selected controls u~,

u~. Therefore it follows that

in (x~ ,x~ ,1°) one has PI (5) ~ fI-z (S, I) and so it is necessary that /

~

p1,

and 1° is determined as

, ,

I" ~ (to.. - 1) K, > 0 . It

i,

"

reasonable

to

exceeds this value I"

(1.36)

assume

that

the

inhibitor

feed

concentration

I!

'0

u, It

1°,

2::

~

u,

or equivalently Uz

, ,

- UIKI(~ - 1)

2::

(1.37)

0,

"

which puts an extra constraint on the inputs of the system (1.35). Often

one imposes an additional constraint on the inputs u 1 and prevent assume

that that

the species in

a

first

2 will wash out, analysis

of

the

but

it

model

is

U

z in order to

not necessary

(1.35).

Altogether

to we

conclude that the model description of the mixed-culture bioreactor leads to a complex nonlinear model with state space [Ri'x [R+x lR+ and controls u 1 and U z satisfying the constraint equation (1.37). o The above examples clearly exhibit the structure of a nonlinear control system, which in continuous time is of the form (1.1) or in discrete-time of the form (1. 2). Clearly, control systems as described by either (1.1) or (1.2) are much more general than their

standard

linear

counterparts,

i.e. in continous time

+ Bu

x

Ax

y

CX + Du

(1. 38)

or, in discrete time

x(k+l)

A.;::(Jc)

+ Bu(k)

y(k)

Cx(k)

+ DuCic)

(1. 39)

where the matrices A, B, C and D are properly dimensioned. A large part of the

control

literature

is

devoted

to

such

linear

systems

and

many

structural properties and problems have been satisfactorily dealt with in the literature. Our emphasis will be on the study of similar aspects for

12

the nonlinear systems (1.1) respectively (1.2). We next discuss some basic assumptions

for

(especially

continuous

time)

nonlinear

systems.

A

continuous time nonlinear control system is usually given by equations

x(t) ~ £(x(t),u(t», (1.1)

y(t)

where x input:

h(x{t),u(t».

=

u E U

E (Rn,

c

/Rm

and

(control)

and y

the

E

output

IJIP denote

of

the

respectively the state, system.

£ : ~n

X /JIm ~ IJIn is assumed to be a smooth mapping.

means

dX>,

The

"system

the map"

In this context smooth

though many results which will be given in the next chapters

hold under weaker conditions (in mllny circumstances £ only needs to be sufficiently many times continuously differentiable with respect to x and u). Sometimes it will be useful to strengthen the smoothness condition and to require that £ is (real) analytiC. Similarly we assume the output map h :

ffin x (Rm ~ IR P

to be smooth or analytic. So (1.1) is a shorthand notation

for

~ hI (Xl (t)

Y1(t)

I

,xn (t) • u l (t) •.... ,urn (t» ,

••••

(1.4Gb)

{

Yp (t)

Together with (1.4Ga/b) we have to specify a class of admissible controls ~ for the system. Of cou~se U :

~+

-

U.

[O,m)

Here

the input functions we consider are functions

m+

(or ~)

denotes

the

time

axis.

A

main

requirement for u is that '11 is closed under concatena.tion, i.e. when u1 ( ' ) and

z (.) bath belong to '11 then for any t also

U

u(·)

E

'11, where u{·) is

defined as

u(t) ""

{

< t,

Ul

(t)

r:

Uz

(t:) .

t 2: r:.

(1.41)

_.... ....

__

/' ,/

Fig. 1.5. ConclI\cnulion ofu, (.) lind uz(·).

t

13

One possible and in many cases

acceptable

for 'U

choice

is

the

set of

piecewise continuous from the right functions on !J?m, which is obviously closed under

Throughout we will assume

concatenation.

that 'IJ

at

least

contains this set of piecewise continuous from the right functions. Next we have to make sure that solutions for

(1.40)

exist,

at least

locally. That is, consider for a given admissible control u(·) E ~! and an

arbitrary initial state Xo Em", the differential equation x(t)

=

f(x(t),u(t», (1.42)

If uC')

is

a

piecewise constant input function

small,

there exists a unique solution x(c)

unique

solutions exist

for

more

general

then for

t

sufficiently

of (1.42). To guarantee that

inputs

(for

instance

piecewise

continuous controls) we impose what is called a local Lipschitz condition on f. That is, there is a neighborhood N of Xo

in IR" such that for each

input u(·} E 'U we have (1.43)

for all x, constants

zEN and all t

I) .11

and

denotes

E (to-£,tO+f), the

usual

solution of (1.42), will be denoted as xet,to,xo,u). the

corresponding output

y(t,to,xo,u).

Note

that

function given by

(1.40b)

once

is

x(t,to,xo'u)

>

0 are

unique

local

where K> 0 and f

Euclidean norm.

The

In the same manner will be written as

determined

y(t,to,xo,u)

follows directly from (l.40b). The above conditions only guarantee the existence of x(t,to'x o ,u) for

!

Jt-t o

sufficiently small.

constant

input

x( t, to ,xo ,u)

function

For

the

u(.)

linear system

yields

a

(1.38)

globally

each piecewise

defined

solution

and thus the piecewise constant inputs form a well defined

class of admissible controls. We will not enter here the difficult problem under

which

extra

conditions

(1.42) are defined for all t. all

constant

input functions

the

solutions

of

the

nonlinear

equation

Even when (1.42) has global solutions for u,

it may happen

that no

global solution

exists when allowing for piecewise constant controls. This is illustrated in the following example.

Example 1,5 Consider on [R2 the system Xl

(l+x;)u,

x,

(l+x~)(l-u)

(1. 44)

14

Take (x1(0), x2(O»

- (0,0). For constant inputs solutions of (1.44) are

defined for all t. Now we construct a piecewise constant control u(.) for which the solution of (1.42) blows up in finite time. Let b_ 1

-

0,

80 -

1

and

l+n

Let lim an ... T


a

if there exists a solution x( t)

of

(1.50) with x(t) = x(t+T) for all t. and T is the smallest real number for which this holds true. The linear differential equation (1.51) possesses a periodic solution if and only if the matrix A has a pair of (conjugate) purely imaginary eigenvalues. If this is the case the system (1.51) has an infinite number of periodic orbits of the same period,

all lying in a

linear subspace of chs state space. In contrast with the situation for the linear dynamics (1.51) the nonlinear system (1.50) may possess a unique or a finite number of periodic orbits with possibly different periods. The following example forms a simple illustration of this. 2

Example 1.B Consider on m the dynamics d

dt

[ x, + xI(l

[::l

+ x 2 (l

-Xl

-

;: Xl

l -

Xl

- x:)1 -

(1.52)

Xl)

The system (1.52) has an equilibrium point at the origin. Moreover an easy computation shows that the circle x~ + the

system

(1.52).

In

fact.

x; -

(Xl (t),

1 forms a periodic solution of

x 2 (t»

=

(cos c, -sin t)

solution of (1.52) with initial condition (x1(0). xz(O» has period T Partly qualitDtive

as

a

substantially.

the

o

2~.

consequence

behavior

system (1.51)

is

- (1,0) and which

of

Assuming

the

of

the

systems

the system

forementioned (1.50)

(1.50)

automatically is complete)

and

to be

differences

the

(1.51)

can

differ

complete

(the

linear

the study of the qualitative

behavior of (1.50) refers to the "behavior in the large" of (1.50). Le. what happens with solutions x(t) of (1.50) when

t

goes to infinity? The

next example shows that. contrary to a linear system, a periodic orbit of (1.50) may exhibit attracting properties.

19

Example 1.9

(See Example 1.8.)

phase portrait of (1.52)

Consider again the dynamics

starting inside the circle x~ + x:

x~ + x~

while

1,

=

solutions

towards this circle.

(1,52).

The

is such that any nontrivial solution of (1.52) =

1, spirals towards the periodic orbit

starting

x~ + x~

outside

So we may conclude that,

=

1

also

spiral

except for the equilibrium

(0,0), all solutions of (1.52) tend towards the set

xi

+ x~

=

1.

-, "

o

Fig. 1.9. I'hasc portrait of (152).

The situation as described in Example 1.9 is quite common for planar

nonlinear

differential

Furthermore

(1.50).

equations

for

higher

dimensional systems a lot more complications can arise. In particular, the positive limit set of (1.50), x(t)

of

chaotic

(1.50)

\olhen

structure.

t

tends

Although

i.e. to the

the set of limit points of solutions

infinity, study

of

may have a very iolild or even the

qualitative

behavior

of

nonlinear systems is beyond the scope of this text, we will come back to some

aspects

of

this,

in

particular

those

concerning

stability

and

stabilization, later on in Chapter 10. Finally we will. briefly discuss discrete-time nonlinear systems given

as x(k+l)

=

f(x(!c) ,u(k», (1. 2)

y(k) - h(x(k),u(k)), where as before x, the

output.

u and y denote respectively the state, the input and m x E [Rn, U E IK and y E [RP, (1.2) is a shorthand

Assuming

writing for x1(1(11) '": i 1 (x 1 (k), .... ,xn (k),

ul(k), .... ,u m (lc»,

(1.53a) {

xn(k+1) - in(x1(k), .... ,xn(k), u 1 ([(), ..

. ,um(k»,

20

h 1 (Xl (k) , • . . • • X" (k). u 1 (k) • • . . .

Yl(k)

{

I

(1

then X (f) equals direCt») p dt

Let now (U,rp)

[fi",

we

obtain

a

=

in

the

curve

on

tangent with

N

IL~O mn

in any point a E

[R"

ll

(U,x1, ... ,xll

basis

a

The natural basis for T"IR is n lR .... U? are the natural coordinate

itself.

where as before r t :

n

then

mil

with

of f

derivative (-E,f)

cangent space TatRn to the smooth manifold

The can

is

Said

for

TpN

)

be a coordinate chart in

the

following

way.

Let

F: Nl ... Nz be any smooth map. Then the tangent map of F at a point p E tIl is the linear map

Let Xp E TptIl'

defined as follows.

and f

a

smooth function on tI z about

F(p). Then

(2.12) It follows linear tp:

.

U C

that

map,

-,

rp" p

•

_a_I aX i

then

a

diffeomorphism,

P E tIl.

In

-, a 1

p

=

then F"p

particular, and so

is

any

rp"p:

a

non-singular

coordinate

TptI .... T

rp(p)

map

[~n has an

Define rp~

p

ar i

rp(p)'

a 1 , ... '-a' a1 {-a' Xl

is

all

rp(U} C [fin is a diffeomorphism,

}1 ....

~nverse

if F

for

p

smooth f around p

Xn

p

i

E ~,

} is a basis for Tpil.

(2.13)

By definition we have for any

28

a

8

-a' I (f) Xi P

-(fOlp

-1

ar i

I

)

IP{p)

(2.14)

af a-r(x

=

!

with f

(p), ...

l

,xn (p»)

the local representative of

simply writ~ rentiating

at -a' (p).

t.

"i

the

local

a-

that -a~ (p)

f

representative

a4-1

Instead of

and we conclude

(f) we will usually

Xi

p

is obtained by diffe-

xl

f

of

with

respect

to

its

i-th

argument. Let now coordinate

and

(U ,I{)

(ll, 1M

be overlapping coordinate charts yielding a Sex)

transformation..

S ~ I/IOrp-l,

with

Lat

with

E Tp}!,

Xp

P E U n V, be expressed in the basis corresponding to (U,I{)

as

a

(2 15)

and in the basis for Tpi'! corresponding to (V,I/I) as

(2.16) p

then

the

coefficients!l'

(a l " , . ,O:n)1

variant1y related as (with [J ~

In

(P11'"

,fin

are

)1

contra-

the Jacobian matrix of S)

8S

ax (x (p) ) 0:

gener.al

(V'Zl.' ..

~~

fi

and

let

(

F: 1'1]

->

1'1 z

be

a

smooth

map.

,2n) be coordinate charts about p,

a

(Zj

of)

Ip

and

let

(V,x l

,· .•

2 . 17 ) ,xn

),

resp. F(p), then

8( z j oF) =

(2.18)

-::---(

So in coordinate bases for T p t1 1 and TF

(p)

Hz the tangent lIIap F"l' equals the.

Jacobian matrix of F expressed in these local coordinates.

Tangent bundle The tangent bundle TH of N is defined as manifold

with

(Xl' ... 'XII ,Vi

I'

dimension ••

,v n )

coordinates about p.

defined

2n as

TpH.

U p

It is itself a smooth

E H

with follows.

natural

local

Let

(Xl' ...• xn)

Then the coordinate values of Xp

x 1 (p) and

Vi

=

(Xp

coordinates

~

L. 1=1 )

=

0:

be i

local

aI ax, P

are

1

0: 1

'

i

En.

29

2.1 Manifolds, Coordinate Transformations, Tangent Space 2.1.1 Differentiability, Manifolds, Submanifolds

Let f

be a function from an open set A C IR

positive

integer.

differentiable) orders

:$

if

The

function

it

possesses

r!

Jc on A. If f is

is

f

n

into !R, and let k > 0 be a

C-

called

continuous

for all Ie then f

(Ie

partial is c

UJ

times

continuously

derivatives

or smooth. If f

of

all

is real

analytic (expandable in a power series in its arguments about each point W

of A) then f is called C For i E n let r

,

W

(Of course, f being C

implies that f is em.)

be the natural slot or coordinate function on IJ?n

i

(2.19) A map f

from an open set A c

mn

into lR" is C..

bijective and P as well as F-

1

P

called a

is

homeomorphism

if F

is

are continuous. Then HI and Hz are called

homeomorphic.

Definition 2.1

A Hausdorff

copological space H with countable basis is

called a topological nranitold ot dimension n it for any p E H there exists a homeomorphism 'P from some neighborhood V ot p onto an open set ot IRn.

Let 1-1 be a topological manifold of dimension n. A pair (V,rp) with V an open set of N,

and 'P a homeomorphism from V onto an open set of ~!I,

is

called a coordinate chart or coordinate neighbor/lOod. The functions (2.21)

are called local coordinate functions and the values Xl (p) •... ,xn(p) of a

p E V

point

are

called

the

local

coordinates

of

p.

The

coordinate

neighborhood (U.'P) will be also denoted by (V,x l •... IXn ) ' Let f: H ~ ~ be n a map. Then i yields a function f: rp(U) C IR .... IR defined as f - f o rp-1,

eV

that is for p f(p)

(2.22)

The function

f is called the local representative of f.

It is customary to omit the caret , and to use the same letter as defined on H and for i , instead

of i(Xl(P), ...• xn(P)]

usually

we

(Xl

I'"

delete

the

"f" for f

its expression in local coordinates. we

write

dependence

on

f(xl(p) •... ,xn(P»).

P.

and

write

f(x l

Hence

Furthermore

•...

,xn

)

where

.xn) are the local coordinates of some (unspecified) point p E V.

Let now (U ,tp) and (V. t/J) be two coordinate charts on tJ which overlap I i. e.

V Ii V

;o!

0.

Let

coordinate functions.

Xl

r

1

0

rp

and

z!

We consider the map

=

r10t/J

be

the

corresponding

(actually homeomorphism),

cf.

Fig. 2.2, S :-

-1

t/J0'P : rp(V nV)

t/J(U n V).

Ii

n

[fin

IR

(2.23)

n

This map is called the coordinate transformation from local coordinates (Xl' ..• ,Xn)

to

local coordinates

coordinates for ~n we write

(Zl"'" Zn)

on V n V _ In

the

natural

31

i E n

(2,24)

and conversely, denoting the i-th component of 5-

1

as S~l

"" ljJoTjJ-l

(2,25)

i E n

Any map f: n ..... IR can be expressed on Un If as £(X ll ... ,Xn ) alternatively as f(z1"" .Zn) (_ frn/,'l).

I

Example 2.2 Let n-IR\{(x1,X Z ) rp -

xl ;::O,X;:: -OJ,

Id with coordinate functions Xl' X z and define

(_ farp-I) , or

and let u - V - t t .

Let

T/J by

(2.26)

The coordinate functions

Z2

ZI'

corresponding to

T/J

are related with

Xl'

x2

by the map 5'1 - ipoTjJ'l given as

(2.27)

Thus (X 1 .X2 ) are Cartesian coordinates, and (Z1 ,2'2) are polar coordinates. The

function

f: H ..... ~ expressed in coordinates

(X l ,X 2 )

x~ + x:

as

expressed in coordinates (zl,z2) as z~. Definition 2.3

Two

compatible

U n V - ",

if

transformation

coordinate

S '" ffJol/!

-1

o

charts

in

or

and

called

are

case

the

is

Un V

inverse

~~,

if

the

coordinate

m

C -

coordinate

transformation

5- 1 - I/Jorp-1 are both Cm

Definition 2.4

A C"" -atlas on a

D - {Ua'~o)oEA

of

property

that

u

pairwise U

~

H.

D

topological manifold N is a collection

C""-compatible is

called

coordinate

a

maximal

charts

C""-atlas

with or

the

smooth

aEA a

differentiable

structure

if

any

coordinate

chart

(V, V')

I.. hich

is

C""-

compatible with every (Ua,rpa) E :D is also in D.

Definition 2.5

A smooth manifold is a topological manifold endm.. ed with a

smooth differentiable structure.

Remark

w We can also define C _ or cf-compatibility by requiring that Sand W

S~l in Definition 2.3 are C , respectively cf. Replacing C""-compatibility by

CW-,

respectively cf-compatibility

results in the notion of a

CW-

Definitions 2.4

and

(real analytic), respectively

in

cf-,

2.5

then

manifold.

32

Example 2.6

Talce

(U _

id).

~n,

~ _

Example 2.7

n

If '" IR ,

wi th

atlas

consisting

= H.

Example 2.B

Let H

Identify

the

in

the

single

chart

o

Let f1 be any open set of IR

single chart (U

of

~

n

wit.h atlas consisting of the

o

- id restricted to U).

Gl(n) be the set of all nXn invertible real matrices.

obvious

wayan nxn

matrix

with

determinant then becomes a continuous function on

a

~

0

point

in

~

nZ

The

2

and Gl(n) receives nZ

by Example 2.7 a smooth manifold structure as the open set of!R

where

o

the determinant function does not vanish.

Example 2.9

An open set P of a

smooth manifold H is

itself a

smooth

manifold. P endowed with the subset topology is a topological manifold. (A set V C P is open in the subset topology on P if i' = P n U for some open set U on 1-1.) For any coordinate chart (U ,cp) on

u.

U' - N n

tp'

tp

restricted to N,

rt let now

(U' ,IP')

with

be a coordinate chart for N.

This

defines a C~-atlas for N.

Example 2.10 smooth

The product

manifold

"'1

taking

x 1-12

of

two

coordinate

smooth manifolds

is

(U 1 x Uz • (tp1

charts

coordinate charts for Hl • resp. Hz·

(Uz,tpz)

(U1.IP1)'

by

o itself a

,rpz»)

with

(Obviously,

(IP1'~2)

u 2 E Uz .)

0

is defined as ('Pl ,'Pz)(u 1 ,u z ) - (IP1(u1),'PZ(u Z »)' u 1 E U1

•

The most important tools in the sequel will be the inverse function

theorem and the implicic function theorem, known from calculus. which we state next for completeness (without proof).

Theorem 2.11

f : U .... IR

n

(Inverse

: V

-+

f(V)

: f(V)

-+

Is

f : rv

-+

111

Let

U

IR

C

n

:!

be

open.

and

let

is non-singular at

chen t:here exists a neighborhood V C U of p such that a

diffeomorphism

(1. e.,

f : V .... f(V)

Is

bijective and

V is smooth). (Implicit Function Theorem) Let W C IlI

Theorem 2.12 m

Theorem)

be a smooth map. If the Jacobian matrix

some point p E U, f

Function

be

a

smooch

map.

satisfying

(p,q) E rv. I f the Jacobian mBcrix

of

f(x.y)'

n

x IR

f(p, q) '" 0 x

E [lin,

y

m

be open. and let for

E (Rm.

some

paine:

at e:he point

(q,p) Is such that the submatrix composed of its last m columns. i.e. af ay(P.q) Is non-singular, then there exists a neighborhood U c rv of (p,q)

33

and a neighborhood V that g(p)

C [fin

of

p, and a unique smooth map g : iT'"

[Rm

such

q and

=

I

I(x,y) E u

f(x,y) - OJ - I(x,g(x))

(In particular f(x,g(x»

I

x E VJ

0 for all x E F.)

-

Many smooth manifolds are generated in the following way.

Theorem 2.13

open part of

Let £1""

m:::; n,

,fm'

be smooth

funccions on [Rn

Define

[Rn).

(2.28)

fmCx) = OJ and assume

(or on an

chat: N

0. Suppose that the rank of the Jacobian

)'!

lTIiltrix

of

f-(f1.···,fm)T

,

af -a-(x) XII

afex)

(2.29)

ax

af

af

ax (x)

ax~(x)

m

"

is m at each x E N. Then N is a manifold of dimension n - m.

Proof

Let

XO

By permuting the coordinates Xl""

EN.

,Xn

we may assume

that the matrix

af , ax!

af , aXm (2.30)

af

af

aX l

aXm

is non-singular at

XO.

neighborhood V c

of xo,

smooth map g: r;

[Rn

By

the Implicit Function Theorem there exists a neighborhood rV' c

~m such that

[Rn-m

of (x~+1""

N n V equals

{ {gl (Xm+ 1 ' . . . 'Xn ) , . . . ,gm (Xm+ 1 ' . . . ,Xn ) ,Xm + 1 ' . . . ,Xn )

Now define a coordinate chart (U,~) on N about U = N n V,

a

,X,~) and a

XO

l (Xm+ 1 ' . . . ,Xn )

E W}.

by setting (2.31)

34

Doing this

any

f01"

XO E

Nt N becomes a smooth manifold of dimension n - m.

o Example 2.14 I (Xl' X z )

S1 _

The circle S1 is a

Ix~

Example 2.15

+ x: -

since

1 - O}.

0

Consider the group O(n) of orthogonal nXn-matrices (i. e. for

D(n) we have ATA

A E

smooth manifold of dimension 1,

This group can be given the structure of a

In)

smooth manifold in the following way. Define the map f by Example 2.8 can be identified with an open set of W,n

from G1 (n) (which 2

to the space of

)

symmetric nY..n-matrices (which in the obvious way can be identified with 1

-n(n+l) (R2 )

(2.32) Then O(n) ~ IA

I f(A)

E Gl(n)

In

1. I t can be checked (see Exercise 2.3) n

that

the

rank. of

the

Jacobian

of f

(seen

as

a

mapping

Z

from lR

to

1

-n(n+l)

ffi2

)

in A

E

D(n) is ~n(n+l). Therefore O(n) is a smooth 2

manifold

of

2.n (n+1) - !:.n(n-l). Actually O(n) consists two of z '}. disconnected parts: the elements in D(n) with determinant +1, denoted by dimension

112 -

o

SO(n) , and the elements with determinant -1.

Consider now two smooth manifolds A map F: Hl .... Hz

ively n;>..

such

coordinates

that

smooth

(U,cp) of H]

coordinate neighborhC'ods about F(p),

is

Nl

the

Xl' ... 'Xn l'

D.nd Hz of dimension n 1

respect-

I

for

each p E HI

there

exist

about p,

respectively

([T,'\b)

of H2

if

expression of F respectively

zl

in

the

I ' ••

'Zn 2'

corresponding i. e.

the

local local

representative F - '\boFo~-l, or

(2.33)

1 E ~z •

is a smooth map from ~(U) C ~nl to '\b(V) c ~nz.

F is called a diffeomorphism if F is bijective and both F and F- 1 are smooth.

Then

dim Nl "" dim 112

"'1 ,

and Note

liz

are

called

that if (U,tp)

diffeomorphic.

in

particular

is a coordinate chart on a

manifold H then according to this definition rp: U ....

mn

smooth

is a smooth map

I

and cp: U .... cp(U) is a diffeomorphism. The ranic of a smooth map F: rank of the Jacobian matrix is

easily seen

that

this

Nl ....

~~(p) rank

H;? at a point p

of is

E 1-11

is defined as the

F expressed in local coordinates (it independent

of

the

choice

of local

35

coordinates),

and is denoted as

rank p (F).

If rank p (F)

=

dim Ht

for all

dim Hz

for all

P E HI'

then F is called an immersion,

pEN),

then F is called a submersion or a regular map. A set of smooth

functions £)""

,f);

and i f rankp(F)

=

defined on some neighborhood U of p in N is called

independent if the map f ~ (f1l ... ,fk)T: U ... IRk is such that rank p (£) - k

for all p

E

1'1.

Proposition 2.16

Let

Let Nand N be smooth manifolds,

F : N -. H satisfy

rank p (F)

~

for

n

some

neighborhood U of p such that F : U ... F(U) i f ranI",

F

~

both of dimension n.

Then

pEN.

there

is a diffeomorphism.

n for every pEN and F is bijective,

then F : N

is

a

Nareaver -+

N is a

diffeomorphism. Proof

Choose

coordinate

charts

about p

(U,r{)

and

(V,lfr)

about FCp).

By

definition of rank p (F) the rank of the Jacobian matrix of (2.34) is n in tp(p). Hence by the Inverse Function Theorem (Theorem 2.11) there exists a neighborhood Let now V

=

r{

c rp(U) such that F:

r{ ...

Ferl) is a diffeomorphism.

tp~l(r{), then F: V .... F(V) is a diffeomorphism.

o

The following two consequences of the Inverse Function Theorem will be often used in the sequel.

Proposition 2.17 independent

Suppose

functions

chac

dim H

about p E 11.

~

n,

Then

and

chere

Chat

exists

f1 ' ... ,fn a

are

neighborhood U

about p suclJ that (U,f 1 , ... ,fn) is a coordillaCe chart:.

Proof

By the Inverse Function Theorem there exists a neighborhood U about

p such that f -

Proposition 2.18 independent

(f 1

, •••

,fn)T: U ... feU) c lR

Suppose that dim H

funccions

about p, and functions

about pEN. x\(p)

transformation

(1',1;,)

z

=

basis

with

local

mil, be

coordinate

respectively

ifJo!p

IO'i ax. II"

1"'1

f3 i

the ip(p)

charts

,z".

2 1 ""

about

~I

Let

JXi

·1

(x) "" S(x}.

Let

XI' E Tf,f!

n a n

and

,On)T.

in the basis

p. -

. 1 If'* pEllf'(p)

and

P

be the corresponding bases of T pH. Denote the coordinate as

these bases as

(or its

(0 1 ""

respectively

1

I f3i az:-I .

i~l

be

represented

The

coefficients

a

in 0:

1

11'

are related as

P,

~

I j"l

or, with

0-

~ [x(p»o]

(2.57)

aX j

(0 1 ' . . . ,0n)T

and

fJ

as

p - ax [X(p»)o (Classically

it

is

(2.58) said

that

under

a

coordinate

transformation

S

the

coefficients of a tangent vector transform in a contravariant fashion). The above can be conveniently used to derive a standard formula which gives the matrix representation of the tangent map coordinates. (If ,2 1

Namely

let

and

F: HI .... Hz,

F~ p

relative to local

and

let

,zn ) be coordinate charts about p E H 1 , respectively about z F{p) E Hz, resulting in the basis -aa i E ~1 respectively -aa / ' , ...

Xi

IP '

Zj

f(p)

j

E !!z' Then

(2.59) where Fj and

Zj

-

F. Hence F. p in bases for

of is the j-th component of

TpN~

F

is given by the Jacobian matrix of the local representative

TF(p)N Z

of F.

rank(F~p)

Corollary 2.24

-

rank(~~

(X(P))]'

The r:angent bundle of a smooth manifold N is simply defined as

(2.60) It follows that there is a natural projecti.on 1f: TN ... If taking a tangent vector Xp E TpN C TN to pEN. TN can be given a smooth manifold structure as follows.

that 11R" ~

First notice

TnlR

U

o

s:

aElll" in

this way TIR!I eVidently is a

chart (U,~)

=

(U,x l

, . • . • xn)

!!in s:

U

IR"

X

IR

n

1R

2n

and

I

"eRn

smooth manifold.

Then for a coordinate

on H, we define a coordinate chart (V,~) on TN

by setting

a

= -;r-l (U)

(2.61)

with for

Xi"'PXp

TN.

E T1.

The

i

(

PI IR

~ IR

resulting

XlI' .. ,Xn • V 1 , ..•• vn

(Xl:

local

(with

Vi

U elf'" IR) l i E ! ! . This defines a C'1ll

we

assume

smooth

(CcIJ) ,

will drop

that

all

unless

the

maps, stated

explicity otherwise. It follows that a vectorfield is a map which assigns to every pEN a tangent

vector

Xp E TpN

in

a

smooth

way,

as

illustrated

figure.

Fig. 2.fi. Geometrical picture of a vectorlkld.

in

the

next

44

Let (U,'p) - (U,x1, ... ,xn ) be a local coordinate chart for N, inducing the 1

natural local coordinate chart

(1r- (U).rp.) -

TN. Then the local representative of X; t1

->

(U.xl ..... xn'vll .... vn)

for

TN is the map

(2.66) which, because of (2.65). can be written as

(2.67) for some functions Xi(xt, ...• xn ):,. 1 En.

In fact as

follows

from

the

preceding section these functions Xi (X) are given by the formula

X(p)

(2.68)

Equivalently, the local representative of X is given by the vectorfield X rp(U) C ~n

on

given as

(2.69) Notice also that if we write

P for functions

Xl""

.Xn; U C f1

H

E

-t

(2.70)

I

m, then

it follows that Xl

is the local representative of these functions

Xi

Xl.'

It is customary (but at first reading a little confusing!) to omit all the carets.

Furthermore often

-aX

we

a

•

i

n

I

Hence

usually

write

-ao r

in (2.69) will be simply replaced by

i

X

in

local

coordinates

Xl""

IXn

as

a

Xi (X)7fX' or as the vector

im 1

i

_

_ [ Xl

•

(Xl : .•• ,X,,)

X(x 1 . . . ·.xn ) -

. Xn (Xl

I ' ••

1

(2.71)

. IXn)

where Xi lire of course the functions Xi from (2.67)-(2.69). Let now a; (a,b)

;, ( C)

: = a"

t

,

-t

H be a smooth curve in N. For

(Da C IL JETO(tj N

C E

(a,b)

we define (2.72)

(with t the natural coordinate on (a,b) c R). We say that a is an integral

45

curve of a given vectorfield X on H if ~(t) - X(a(t)),

In

this

coordinates

local

(2.73)

Vt E (a,b).

just

means

that

a( t) -

(al(t), ... ,anCt») is a solution of the set of differential equations

{~l(t) - X,(a,(t)' .... an(t)) te(a,b)

~n(t} ~ Xn(odt), ...

(2.74)

,

,on(t»)

with Xl as in (2.71). So, to a vectorfield X given in local coordinates as in

(2,71)

we

associate

in

a

one-to-one

way

the

set

of

differential

equations

(2.75)

also abbreviated as x - X(x)

(2.76)

,

(Xl"" ,Xn ) is the vector of local coordinates for H. (Note the slight abuse of notation, since x on the left-hand side is actually a

where x."

column-vector in [FIn.)

By

the

existence

and

uniqueness

theorem

for

smooth

differential

equations it follows that for any p E H there exists an interval (a ,b) of maximal length containing with 0(0)

=

a

and a unique integral curve o(t),

t

E (a ,b)

p. If for every p we have (a,b) - (_ro,ro), and so solutions are

defined for all time t

the vectorfield X is called complete.

Note that

vectorfields on compact manifolds are always complete, the only thing that can go wrong in general is that in finite time solutions tend to infinity (or to the boundary of the manifold, which itself does not belong to the manifold).

In any

interval (a,b) up(C)

case,

for

every bounded

set U C H,

there

exists

an

containing 0 such that for any p E U the integral curves

with op(O) - p

are defined for

all

t

E (a,b).

This

allows

us

to

define on U a set of maps (time t-integral or flow) cE{a,b),

(2.77)

by letting xt(p) be the solution of the differential equation (2.75) time t with initial condition at time 0 the point p.

i.e. l"(p) -

0p

for (c).

46

It follows from the theory of differential equations that the maps Xl are smooth.

By definition a vectorfield X defines in any p E H a tangent vector

X(p). For f: M

-t

lR

this yields in any p

E

M the directional derivative

X(p)(f). Hence by varying p we obtain a smooth function X(f) defined as X(f)(p)

;=

X(p)(f)

(2.78)

The function X(f): l'l ..... IR will be called the total derivative of f along the vectorfield X, or the Lie derivative of f along X and is also denoted as

Lxf.

Notice that if X is expressed in local coordinates as the vector

(Xl (x), , .. ,Xn(x»)T then we have af L ax(x 1 (p), ... ,xn (p) )Xi (xl (p), ... ,Xn (p»). II

Lxf(p) = X(f) (p) -

1"'1

(2.79)

1

Furthermore we have

X(f)(p) _ lim f(Xh(p») - f(p)

(2.80)

h

It is now clear how we can give a global, coordinate-free definition of a smooth nonlinear control system, given in local coordinates as

(2.Bl)

x - f(x,u).

Indeed let H be the state space of the control system and let U be the input space, then the system is given by a smooth map (the system map) f: f1 x U

-+

(2.82)

TM

with the requirement that

'/tot

equals the natural projection of N x U onto

N. -Wi th the same abuse of notation

IlS

above, f

is represented in local

coordinates x for H, natural local coordinates (x, v) for TN, and local coordinates u for U as

f(x,u) - (x,f(x,u»)

(2.83)

and so we recover the local coordinate expression x Remark

In

Chapte.r 1

it

was

indicated

that

in

f(x,u). some

cllses

the

above

definition is still not general enough. The problem is in writing the producr: H x U;

this implies that the input space is globally independent

of the state of the system. In order to deal with situations where this is not the case we have to replace N x U by a fiber bundle above the base

1,7

space 1'1, with fibers diffeomorphic to U. As a result we have only locally a product M x U. This is discussed in Section 13.5. In case the system map f: H x U

TN is affine in the u-variables we write

-+

(with the addition and multiplication defined in the linear space TxH)

L gj (x)u

f(x,u) = fex) +

ja

(2.B4)

j

1

for some functions f ,gl , ... ,gm: N

TN satisfying 1rof

-+

=

identity on

lTogj -

1'1, which hence are vectorfields on H.

Now let us return to the study of vectorfields. Since tangent vectors transform under a

(2.58»

(see

coordinate

transformation

also vectorfields

do.

In

in a

fact

coordinates (U,rp)'" (U,x1 •... ,xu ) as X ~

[ Zo (S(xl) diffeomorphism,

Sex) we have

=

i

(2.B5)

:

xo (x)

For convenience we will

vectorfield l' on

then with Z

[X,(x)]

as = ax(x)

:

introduce a new notation.

and let X be a vectorfield on N. N

Let F: N ... N be a

Then we can define a

by letting

(2.B6)

for any pEN

F is not a diffeomorphism then y is a well-defined vectorfield on N if

If and

only

F(Pl)

=

if

F

and

X

are

such

that

F~ p lXP 1 =

F" p zXp z'

whenever

F(pz). We will abbreviate (2.86) as

(2.B7)

Y .., F"X . If

local

1

a

L 2i (z)az-' 1.,1

Z,(S(X))]

fashion

in

IXi(X)~ and in local coordii'"1

(V, Zl ' . . . 'Zn) as X..,

given

a

o

o

nates (V, 1/J) -

contravariant

X be

let

F~X =

Y for vectorfields X on Hand Y on N (F not necessarily being a

diffeomorphism)

then we

say

that X and l' are F-related.

(2.86) we have for any function g: N ... F.X(g) -

Note

that by

m (2.BB)

(X(goFl)

Hence the Lie derivative of g along F*X in a point pEN is computed by taking

the

Lie

derivative

of

the

function

that we

are

now using

goF: M ... !I?

in

the

point

F-l(p) EN. We

warn

the

reader

the

notation

F~

in

two,

48

slightly different, ways: (a) as n map

F.: TH

TN, and (b) as a map from

~

vectorfields on H to vectorfields on N. The following theorem shows that outside equilibria vectorfie1ds can be given a very simple form.

~

Theorem 2.26 (Flow-box Theorem) Let X be a vectorfield on H with X{p) Then t:here exists a coordinate chart:

(U ,xl' •..• "n)

around p

SUdl

O.

that (2.89)

on U .

Geometrically this means that around p the integral curves of X Bre of the form x 1 (q)

~

constant, i - 2 •... ,n.

Proof Let (V,..p) ..p(V)

~

bounded such

(V,Zl' ••.• zn) be a coordinate chart with ..p(p) ... 0 and

a-I .

that ..p"pX" -8

Define

r1 a

T: !FIn .... !FIn

locally around

0

the

map

(2.90) (i. e.

the eime-a

(Zl , ••.• zn)'

1

ineegral of the vectorfield ..p .. X in local coordinates

I t is easily checked that

(2.91) and

that

the

T~o

equals

the

identity

coordinate transformation. Hence S :- T-

1

matrix,

implying

that

T

is

a

is the desired coordinate trans-

o

formation. For X and Y any vectorfie1d,

two

denoted as

(smooth)

lX, Y]

vectorfields

on H,

we

define

a

new

and called the Lie braclcet of X and Y by

setting

(2.92) In order

that

lX,YJ p

E

TpH

we

have

to

check conditions

(i).

(ii)

of

Definition 2.21. Condition (i) is trivial, while (ii) follows from [X,Ylp(fg) - Xp(Y(fg») - l'p(X(fg») - Xp(Y(f).g + f.Y(g)} - Yp{X(f)'g + f'X(g)} - Xp[Y(f»)g(p) + Yp(f)Xp(g) ... Xp(f)l'p(g) + f(p)Xp(Y(g») + - Yp(X(f»)g(p) - Xp(f)Yp(g)

Yp{f)Xp(g) - f(p)Yp(X(g»)

'" (X,Ylp(i)'g(p) + f(p)·[X,Ylp(g).

(2.93)

49

If X and Yare given in local X(x)

=

(X1(x), ... ,Xn(X»)T,

coordinates

respectively

(Xl""

,X,,)

as

the vectors

1"(x) = (l'l(X), ... ,l'nCx»)T,

then

[X,YJ[x(p») is given as the vector

aY

ra)X(p))

1X[x(p))

-

[ax a)X(p)) 1Y[x(p)),

as follows from computing (2.92) Indeed let

for the coordinate functions

" a x, - LX,[x(q))a:;;:-1 161

1

(2.94)

for

and

Xl""

q

,XIl •

in

the

q

coordinate chart, then for j E .::.

(2.95)

y,l [x(p)) and therefore

ax,

- -aX

1

Yi

1axa

j

(2.96)

'

It immediately follows from (2.%) that !X,l'J p depends in a smooth wayan

p, so that indeed [X,i'1 is a smooch vectorfield. The following properties of the Lie bracket follow immediately from the

definition. Proposition 2,27

For any vectorfields X.

Z and functions f,

Y,

g on a

manifold M (8)

[fX,gl'] - fg[X,i'J + f·X{g)·l' - g·fef)-..\' ,

(b)

[X,YI - -[Y,XI,

(c)

[[X,Y],Z] + [[r,Zl,X] .,. {[Z,X),l'j

(2.97) 0

=

(Jacobi-identity).

Before going on we give the general definition of a Lie algebra.

Definition 2,28

A vecCor space V (over

[~)

is a Lie algebra i f in addiCion

to the linear scructure there is a binary operation V x I'

~.

V. dcnoced by

[ , ], satisfying (1)

[olV 1

+

02V2,I"J

=

0I[V!,t.]

+

02[V 2 ,I"J,

VV 1 ,V2 .1" E V, VOl ,02

-[h',V], ItV,lv E V (anti-synulletry) ,

(11)

[V,lv]

(iii)

[V,[I",ZJ] + [I",[Z,VJJ "'" [z,[v,I"lJ

=

=

(bilincarity)

ElR,

0, VV,h',Z E V

(2.98) (Jacobi-identity)

50

A subalgebra of a Lie algebra (V,[ , that [v' ,w'] E V' tor all

Remade 2.29

Vi

,t'"

E

J) is a linear subspace V'

C V such

V'.

The most well-known example of a Lie algebra is the linear

space of nXn matrices with bracket operation [A,B]

= AB - BA

A,B

m

the

following

N. and suppose that F"Xi

-

Yi

•

i - 1.2,

tor

vectortields Xl,X Z and Y1 'Y 2 on N respectively N. Then (2.100) Proof

By (2.BB) we have for any function g: N .... IR i

-

(2.101)

1.2 .

Therefore

(2.102)

by

(2.101)

with g

replaced by Yz (g),

respectively

i\ (g). By another

application of (2.101) this equals

and hence by (2.101) with Xi

rr1,YZ ]. we have Fn[X 1 ,X2

] -

and

Yl

[Y 1 'Y2]'

replaced by [Xl ,X2

].

respectively

0

In order to develop an interprecacion of lX. YJ we first prove some lemmas. The first is immediate.

51

Let F: N

Lemma 2,31

with £101V' xt. FoXtoF- 1 .

Proof

and X a vectorfield on 1-1

N be a diffeomorphism,

-+

the f101" yt of the vectorfleld Y - F~X on N equals

Then

1 [X(gOFl)'[F- (q)) -

(F.X)(g)(q) -

(2.104)

lim h~'

lim

h1 [g[ FoX h of - 1 (q))

- g(q)

1

o

.

h~'

This lemma just expresses that if F"K - Y, then F maps the integral curves of X onto the integral curves of Y.

F"X

Corollary 2.32

Secondly

we

need

X

=

the

for

F: H'" 1'1, i f and only i f XtoF

following

derivational

=

FoX

t

for all

interpretation of

the

t,

Lie

bracket. Theorem 2.33

For any vectorflelds X and Y on N,

h->O

Proof

(2.105)

, P E ,1.

[X,Yj(p) - lim !c[(X:hl') (p) - Y(p)] h

Write out the right-hand side of (2.105) in local coordinates, and

o

check equality with (2.9l,).

We see that [X,Y] can be interpreted in some sense as the "derivative" of the vectorfield Y along X. derivative

of

along

l'

It is therefore also denoted as

X.

The

following

lemma

is

Lx 1',

crucial

the Lie in

the

interpretation of [X,Y].

Lemma 2.34

Let X and l' be vectorfields, lV'ith

if and only if Xtoyll

Proof

=

r"'oxt, for all 5,

t

£1010/5

xt,yt. Then [X,Y) ... 0

for to/hieh xt and I'll are defined.

ylloxt _ XtoY'" for all s i f and only if x:r _ Y by Corollary 2.32.

If this

is

true

for

all

t

then by Theorem 2.33

[X, r] - O.

Conversely

assume that [X, r] - 0, so that

0- lim H(X;'l')(q) - Y(q)]

for all q .

(2.106)

h~'

Given PEN, consider the curve e: (-f,d -. TpN given by c(t)

(X~l')(p).

52

Then c' (t) - lim

I:

[c(t-h) -

c(t)]

h-tO

." lim ~ [(X;-hy)p - (X;Y}p] b-+O

(2.107)

0,

by (2.106) for q _ X-t(p)

Consequently c( t) ~ c(O), so that X;Y - Y for all t. As remarked before

o

o

It follows that [X,f]

if and only i f the flows of X and Y commute,

i.e.

X'(P( --....) y'[X'(p)) p • -.....,..r. yll(p)

Fig. 2.7. Commuting vcctorlicldsX amI Y.

Hence i f [X,i')(p) the

Lie

bracket

instrumental

in

p


i,j E ~

0p

E r;H

can

be

written

I Cl:idX II' ,.,

as

for

i

and is also denoted as a row-vector

some

(O:l""'ctn ).

A function f: H ~ ~ defines in every point p an element of T;H, denoted as dfp or df(p) , by the formula

d£(p)(Xp)

~

X,(£),

(2.145)

We call df(p) the differential of f at p. If we interpret dX i Ip in (2.144) as the differential of the coordinate function The differential df(p)

are consistent.

then (2.144) and (2.145)

Xi

in the basis dx1I p ""

,dx n I p'

is

given as

af

at

(2.146)

df(p) ... -ax (p)dx 1 I p + ", + -a' (p)dxn Ip . 1

Xn

(In order to check (2.146) compute df(p)(-aa 1 Xj

Let

Z1""

,zn

be

Z "" S(x). Let 0p E

another

r;H

set

"

j

En.) -

coordinates

around

p,

with

"

(2.147)

L.8idZtlp iml

i~l

then the coefficients

a, - Ij

local

),

be represented as

LOldxilp, and as 0p -

0p'"

of

P

0i

and .8 1 are related by the formula

as, (2.148)

ax-(x(p»)fi, ,

or, with U:- (ul,·,·,u n ) and f3: - (.8 1 , ... ,.8n ), a -

(One

as

fi a)x(p») .

says

that

cotangent

vectors

transform

in

a

covariant

fashion,

62

contrary to tangent vectors which transform in a contravariant way.) For F:

ttl ....

Hz we have defined the tangent map

F"p

: TpHl

-+

TF(p)H Z

(2.150)

f: Hz ... IR •

The adjoint map of F. p will be denoted by

by

F;.

Thus dually to F. p we have

(2.151)

(2.152) In local coordinates

F;

map

is

given

Xl • . . . 'Xn 1

by

representative

of P,

row-vector then

F;OF(p)

the in

the

for

ttl'

Jacobian sense

and

Zl' .••• zn

matrix that

if

aF

2.

for 1'1 2 the linear

ax(X(p») is

0F(p)

of

the

local

expressed

as

a

is given by the row-vector (compare with (2.149» (2.153)

The cotangent bundle of a manifold H is defined as (2.154) There 0p

is

the natural projection

E T;H C T" H

bundle Xl •.••

Can

to P

be

given

11':

T· If .... 1'1

taking a cotangent vector

As in the case of a tangent bundle, the cotangent

E H.

a

manifold

structure.

Given

,xn on H we obtain natural local coordinates for

local

coordinates

r" n by letting a

n

cotangent vector op -

LOtdXi!p correspond to the coordinate values 1m 1

Now we define the dual object of a vectorfield. Definition 2.46

A smooth differential

one-form o.

briefly smooth one-

form, on a smooth manifold H is defined as a smooth map 0:

H ... T"N ,

satisfying (tdth 11'00

=

(2.155) 11'

t:be natural projection T" H ... 1-1)

identity on H .

Replacing "smooth" throughout by r:fA

(2.156) I

resp.

C!.

the one-form is called r:fA

I

63

rI.

resp.

In

the

sequel differential

one-form will

always mean

smooth

differential one-form. Hence a one-form a is a map which assigns to each pEN a cotangent vector u{p) E T;N.

Let (U,x l , .. , ,xn ) - (U,f{J) be a local coordinate chart for H about p, resulting in the basis dxIl p I ' " ,dxn Ip for r;H, then we can write

a(p) -

LUi(P)dxil

,-,

certain

for

ai

: - 0i0qJ-l:

(2.157)

p

m be

omitting the carets,

i E n,

functions

smooth

qJ(U) ~

the

local

representatives

of

we write a in local coordinates as

0t,

Letting i

E~,

and

the row-vector

(compare with (2.71»

(2.158)

a(x1,···,Xn ) ... (ol(x1.···,xn).···'on(xt'···,x,,») ,

or, abusing notation by writing

dX1

for dri l i E

~

(the natural basis for

T~ IRn) as !PIp}

I"

a -

(2.159)

a 1 (x)dx i

"1 Since

one-forms

are

the

dual

objects

of vectorfields,

natural way upon vector fields (with a a one-form and

a(X)(p) - a(p)(X(p»)

they

act

in

a

X a vectorfield) (2.160)

E •

Hence a(X) is a smooth function on ilo Any function f

defines a one-form,

denoted as df, by letting df(p) be defined as in (2,145). Notice that we have the equality

(2.161)

df(X) - X(f) - Lxf

Not everyone-form can be written as df for a certain function f.

In

fact it follows from (2.146) that

df-~dx + ... +8£ aX l

and hence, 0-

,.,0 1 (x)dX1 aa, -

aa j ax,

(2.162)

aXndxn

since

I"

aXj

1

a necessary condi tion for

a

one-form

to be of the form df is that

i , j En.

(2.163)

64

Conversely one can prove

local

that condition

function t

existence of a

such

is sufficient for

(2.163)

that

O'i

8f

8x-'

-

One-forms

df

the

are

.I.

called exact, and one-forms satisfying (2.163) are called closed. Finally let

F:

HI ~ Hz be a smooth map,

Hz, we define a one-form at on

then for O'z being a one-form on

HI' denoted as

0'1

=

F*a z ' by letting (2.164)

Notice that F'" 0z hand recall

is always a well-defined one-form on HI'

that if X is a vectorfield on Hl

is

always well-defined.)

one-forms on Hz

It is

=

df(P)(PnpXp)

that p*

easily checked

maps

then exact

in fact for any Xp E TpH we

into exact one-forms on HI'

have (P"'(df})(p)(X p )

On the other need not be a

(Of course if F is a diffeomorphism,

well-defined vectorfield on lIz. FnX

then p",X l

Xp(foF) and so

=

(2.165) Since closed one-forms are locally of the form df it follows exact one-forms on Hz are mapped by

F~

that also

onto exact one-forms on Nl

.

One may also define the notion of the Lie derivative of a one-form a along a vectorfield X. In fact we define

as the one-form

LxO'

(2.166) a

If

is

given

in

local

(ol(X)' ..•• on(x»). and

coordinates

-

df,

row

Lxa

vector

is given as the row-vector aX l

aXI

aXn

aO'n

aXn

aXn

aXn

aX l

aXn

Ban

aX I

ax!

.... (01' ... ,an) aO I

0'

the

8X 1

80 1

(Xl.··· .Xn)

where everything is

as

X as the column vector (X 1 (x} •.••• Xn (X»)T, then it

may be checked (see Exercise 2.13) that

Lxo=

Xl ••.. ,Xn

taken in x ... (Xl' ...

,Xn )

E

(2.167)

IRn.

If a

is exact,

1. e.

then (2.166) reduces to

Lxdf - lim H(~)~df - dt] == d(lim[fo~ - fJ) "" d(Lxf) h~O

(2.168)

b~D

We thus see that the Lie derivative of a one-form is the generalization of the Lie derivative of a function. Finally

we

vectorfields.

give a

the

following

one-form),

which

interesting can

be

"product"

verified

formula

using

the

(X,Y local

65

coordinate expressions (2.96) and (2.167) Lx(a(Y)) -

For

df this reduces to Lx(Y(f»)

=

0

(2.169)

(Lxa)O') + a(LxY)

X(Y(f») - Y(X(f») -

[X,YI(f)

=

Y{X(£») + df(L;.:r) , or (2.170)

,

which is just the definition of the Lie bracket [X,?].

The dual object of a distribution is a codistribution. A codistribution P on a manifold N is defined as a map which assigns

subspace

pep)

of

the

cotangent

T;n.

space

to any pEN a linear

P

is

called

smooth

a

codistribution if around any point p there exists a neighborhood U of p i E I,

and a set of smooth one-forms 0i'

with I

some (possible infinite)

index set, such that (compare (2.119» P(q) = span{ol(q); i

E 1),

q

(2.171)

U .

E

In the sequel codistribution will always mean smooth codistribution. one-form

is

0

said

to

belong

to

the

codistribution

P

(0

E P)

A if

o(p) E pep) for any pEN. The dimension of P at pEN is the dimension of

the subspace pep). A codistribution P is called the dimension of pep) (compare Lemma 2.38)

..e,

constant

does not depend on pEN.

It

dimensional

if

immediately follows

that if P is a codistribution of constant dimension

then around any p there exist

independent one-forms

.R.

0 1 ""

'Of

(called

local generators of P) such that

(2.172)

q near p .

Finally

for

any

codistribution

P

we

define

ker P

as

the

smooth

distribution (ker P) (q)

=

span[X(q) IX vectorfield such that o(X)

0, VA E P)

(2.173) Conversely for any distribution D we define its smooth annihilator ann D as the smooth codistribution (ann D)(q) It follows ker D

are

that

=

span{o(q)io one-form such that veX) if D and P are

constant

P C ann(ker P),

=

constant dimensional

dimensional.

By

definition

0,

vx

E DJ

(2.l74)

then ann D,

D C ker(ann D)

resp. and

but in general equality does not hold. However if D and P

are constant dimensional then it follows from Lemma 2.38,

(2.172),

and a

66

dimensionality D

=

argument that equality does hold, i.e.

(2.175)

ker(ann D), respectively P = ann(ker P) .

For convenience we call a codistribution P inlJolutilJe if ker P is an involutive distribution.

If P

is

locally generated by exact one-forms,

i.e.

(2.176)

q near p ,

then ker P is always involutive.

let Xl,X Z

Indeed.

E

ker P,

then by the

0

(2.177)

definition of the Lie bracket dfi(q)(rXl,XZ](q») ~ ([X1,X;d(f1»)cq) -

(f i

(X 1 (X 2 (f i ))(q) -

0,

i E

dimensional

codistribution

such

Frobenius'

Theorem

since

Xz(f!) -

X1

xl" .. ,xn such that P -

q

ann D,

e H.

-

t.

(Corollary D(q) -

Conversely,

let

P

that

D ... ker P

is

2.43)

there

exist

span{~1 •...• ~I ). UAl q ux!: q

be

a

constant-

involutive. local

Then

by

coordinates

Since in view of (2.175)

it immediately follows that P(q) - span /dxkT1(q) •... ,dxn(q)J.

or abbreviated

P - span \ dXk + 1 in

As

)

(Xz(xtCf,l»)(q) -

smooth

the

case

••••• dXn

J

(2 .178)

of distributions

codistributions

PI

and P 2

(cf.(2.41»

be

the

we

smooth

let

the

sum of two

codistribution P l + P z

defined in every q E M as

(P 1 +P2 )(q) - span\O'l(q) +

172

(q)/O'I smooth one-form in PI' (2.179) 172

The intersection Pi

n P2

smooth one-form in P2 1.

is the smooth codistribution defined in any q

e H

as (compare (2.142» (P l

n Pz)(q)

~ span{a(q)la smooth one-form contained in PI

and P z ) (2.180)

Finally let P : H

~

N, and let

P be

N. Then we define

a codistribution on

the codistribution p.p on N as

F*P(q) - span \ (P"O') (q)

I

17

one-form in PI

t

q

E

}1

•

(2.1Bl)

67

2.3 Summary of Section 2.2 1.

In

local

coordinates

B I" Xi(X)-a x,

x -

(xl

I

•••

or as a vector X(x)

IXn)

~

a

vectorfield

X

is

given

as

(Xl(Xl, ... ,Xn), ... ,Xn(Xl, ... ,xn»)T,

and corresponds to the set of differential equations

x

abbreviated as

2.

= Xex).

The Lie derivative

Lxi ".

of a function f

XCi)

along the vectorfield X

equals in local coordinates af I -a x,-(x)X, (x)

L,f(x) •

lim

f(X"(x) )-f(x) b'

1>-+0

with xt: t1 ... 11 the tirne-t integral (flow) of X.

3.

such that

4.

then there exist local coordinates

Let X{p) '" 0,

Let

X -

a ax)

or in vector notation X ,

X-(Xllo.oIXn )!

local

coordinates x

and the

r-O'l""'Yn)T Lie

bracket

Xl""

'Xn

around p

(I 0 " . 0)1,

=

be

[X, 1'] -

two

Lx Y

is

vectorfields. the

In

vectorfield

given by the vector [X.") (x)

5.

ax

BY

--(x)X(x) - ax{X)Y(x). ax

Vectorfields on H with the Lie bracket form the Lie algebra V""(l·1); that

is:

[X,l']

is

bilinear,

anti-symmetric,

and Jacobi's

identity

holds

[[X,l'j,z] + [IY,Zj.X] + [IZ.Xj.Y] - o. 6,

Suppose that FnXi

=

1'i' i

respectively N, with F: H

7.

~ -t

1,2, for vectorfields X1,X Z and r1,l'z on H Then

N.

[X,Y)(p) - lim H(X:"Y)(p) - l'(p)]. "40

8.

[X, 1"] -

0 if and only if xtoY~ "" r~oxt for all s, c.

68

9.

Let Xl" _ . ,Xk be linearly inde.pendent vectorfields with [Xi ,Xj i , j E Ie,

Xi

then

a -';. a;:-

~

E

there.

exist:

local

coordinates

Xl' •••

,xn

1-

such

0,

that

k.

1

10. A distribution D is given in any q E H as

for some vectorfields Xi and index set I.

D is involutive if, whenever x,r E D. also [X,Y) E D. 11. Let D be an involutive distribution of constant dimension 1c I

there

exist

local

a

a

l

k

D ... spanl aX •..•• 8x

12. For

any

function

such

coordinates

then that

J. on

f

H the

exact

one-form

df

is

defined

as

df(p) (Xp ) n

13. In local coordinates a one-form

U

is given

L

0l

(x)dx i

or as a row

1=1

vector a(x)

(ul(X l , ...

af

df is given as

(--a

Xl

,xn), ... ,an(x i

, ...

,X n »), and an exact one-form

af

(x) ..... ---a. (x»). XII

14. For any vectorfield X and one-form a we have

o(X)(p)

=

o(p)(X(p») E

~ n

and in local coordinates a(X)(x) =

Lo!(x)Xi(x).

!~l

15. For

any

map F: M ... N and anyone-form a

(F~o) (p) (Xp

on N we

define F*o by

)

16. F~(df) ~ d(foF). 17. The Lie derivative of a one-form a along a vectorfield X is defined as lim ~[(~)·o -

Lxa

0]

and equals in local coordinates the row vector

11",,0

aot [ (ax (x) )X(x)

18. Lx dt

...

dLx f

.

J

T

+ a(x)8X(x)

ax

69

19. For anyone-form

and vectorfields X,Y we have

0,

L,(a(Y))~ (L,a)(y)

+ a([X,Y]).

20. A codistribution P is given in any q E H as

peg) - span{Ot(q)I01' for some one-forms

01

i

E I).

and index set I.

21. Let D a distribution, and P a codistribution. Then

(ann D)(q) - span{o(q)lo one-form such that o(X) - 0 for all X E OJ, (Iter P) (q) ~ spanIX(q)

IX

vectorfield such that veX) - 0 for all

0

E PI.

P is called involutive if ker P is involutive.

22. D c ker(ann D),

P c ann(ker P),

constant dimension.

there

exist

and equality holds if D and P have

If P is involutive and constant dimensional then

local

such

coordinates

that

span{dxk + 1 , .. ,dxn l.

P -

Notes and References The material treated in this chapter is quite standard, and is adequately covered 1n many textbooks, such as {Bo 1, {Sp 1, [Wa 1, (see also (AM]), and

we have made extensively use of these sources. For more details on immersed submanifolds we refer to {Bo], definition of tangent space as given here common one;

see

however

[AM]

for

an

(Definition 2.21)

alternative

[Sp]. The

is the most

definition

(see

also

[BJ]). The proof of Frobenius' Theorem (Theorem 2.42) given here is taken from {So]. A more constructive proof can be found e.g. in [Bo], a global version of Frobenius'

[Bo]. An important extension of Frobenius'

theorem is the Hermann-Nagano

theorem for analytic distributions with no constant dimension, rNa].

tWa]. For

theorem and global foliations we refer to

This was further generalized in [Su].

see [He1,

For more details concerning

properties of distributions and codistributions we refer to [Is]. We only treated here differential one-forms. forms,

For general differential

the d-operator and Lie derivatives of differential forms we refer

to [AM],

[Bo],

[Sp],

[Wa}.

Furthermore,

involutivity of codistributions

can be defined independently of distributions, and Frobenius'

theorem can

70

be equivalently stated for involutive codistributions using differential forms, see for instance [S], [S).

[AM]

R.A. Abraham. J.E. Marsden. Foundations of Mechanics, Benjamin/ Cummings, Reading, 197B. [Bo] W.A. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic, New York, 1975. [BJ] T. Brocker. K. Janich, Einfuhrung in die Differentialtopologie, Springer, Berlin, 1973. [He] R. Hermann, "The differential geometry of foliations", J. Math. and Mech. 11, pp. 302-306, 1962. [Is] A. Isidori, Nonlinear Control Systems: An Introduction, Lect. Notes Contr. Inf. Sci. 72, Springer, Berlin, 19B5. rNa] T. Nagano, "Linear differential systems with Singularities and applications to transitive Lie algebras U , J. Math. Soc. Japan, 18, pp. 39B-404. 1966. [Sp 1 M. Spivak, A comprehensive introduction to differential geometry, Vol It Publish or Perish. Boston, 1970. [Sul H. Sussmann, "Orbits of families of vectorfields and integrability of distributions". Trans. Amer. Math. Soc., lBO, pp. 171-188, 1973. [Wal F.W. Warner, Foundations of differentiable manifolds and Lie groups, Scott. Foresman, Glenview, 1970.

Exercises 2.1

Consider the topological space lR with coordinate charts (IF: .IP) and (IF:

.w)

with IP(x) - x, W(x) .. xl. Show that these charts Bre not

compatible.

the

On

other

structure defined by the

hand atlas

show

c"D_

that lR

with

differentiable

and III

with

differentiable

(111 ,(fI),

structure defined by the atlas (1R,l/J) are diffeomorphic. 2.2

Show that f I R ... IR given by

f(x) = 0

x

~

0 •

exp(-~), x > 0

f(x)

x

is C~ but not analytic. 2.3

Prove that the rank of the map f(A) ... ALA in Example 2.15 is indeed

!n(n + 1) in points of O(n). 2 2.4

With the aid of the implicit function theorem prove the following (a) Let A(x) be a pxm-matrix, and hex) a p-vector. with

X

in some

open set U of IRn. Suppose that for some Xo E U rank A(xo ) ... p. Then Q

:

there

V ... IR

m

exists

a neighborhood V

C U

of

Xo

and a

smooth map

such that

A(x) a(x) - b(x)

,x E V

(b) Let A(x) be a pxm-matrix, and b(x) a p-vector. with x in some

71

neighborhood U of a point Xo E mn. Suppose that ~

rank A(x)

r ,

for every x E U. Then there exists a neighborhood V C U of Xo and a

smooth map

0

V ~

:

mm

such that

[~(X)]r

A(x) n(x) - b(x) -

~(x),

for some (p - r)-vector ee) Let A(x)

, x E V ,

depending smoothly on x.

be a pxm-matrix,

with x

in some neighborhood U of a

point Xo E ffin. Suppose that rank A(x) ~ r for every x E U. Then there exists

a

neighborhood V C U of Xo

and

a

smooth map

fJ :

V -. G.2{m)

(with Gl(m) the invertible mxm-matrices) such that

o [ 1
D.

(3.14)

x(t,a,x o ,u) sacisfies

,82 1(xo)

3

(3.15)

+ O(h ).

By Taylor expansion we have

x(h)

,

xeO) + hireO) + ~h2X(O)

=

+

Similarly (only writing terms up to order hZ) 1 7.

x(2h) - x(h) + hg, (x(h)) + ,h

8gz

;,x(x(h) )g, (x(h») + ... -

+ ...

ag, where we have used the fact that 82 (xO+hx 1 ) - 82 (x o ) + h ax-(xo ) + .... Next we compute (to keep notation down: all functions are evaluated at xo.

unless stated otherwise)

7. 1 =

Xo +

h(gl+gZ) + h

Bg l

ag l

[z axgl

i

ag l + axgl +

ag

z %axgz)

ag l

- h(gi + 11 ax-(gl+gZ») + h2 axgl +

- Xo

ag z ag l ag z + hg z + h2(ax-g1 - axgz + ~ axgz) +

Finally we obtain

-

78

1

x(4h) ... x(3h) - hg'l (xa+hg z +·.) +"2h

~

'l Bgz ag-{xo+,' )g2 (xo+") + ... -

Bg2 Bgl + hgz + hZ(ax-g1 - ax-gz +

Xc

Bg 2

- h(gz + h

Example .3.5 (continued)

-

Bg'}.

ax-gz } + ;h2 axg'}.

ag z 2 + h (ax g1

- Xo

og'l

i Bx g2)

dg l

-

axg'l )

+ D(h

+

J

o

).

In Example 2.35 we have directly computed (cf.

2.111) that for gl and g'l as given in (3.11)

o Formula

(3.15) implies that, at least approximately, we can steer the

system (3.12) from

Xo

into the direction given by the vector [gl,gZJ(xo );

in particular if [g1.g2](X O ) G(x o )'

However I

G(x o ) we can steer into a direction outside

~

this is not the end of the story.

By choosing more and

more elaborate switchings for the inputs it is also possible to move in and gz,

directions given by the higher-order brackets of g1 such as [g2' [g1 ,g2

J]

J

i. e.

terms

[rg 1 ,g'l 1, [g'l' [gl .gz 1]. etc, (In fact already in the

expansion (3.15) these higher-order brackets are present in the remainder D(h

3

) ,)

In case

a

drift

term

t

is

present

in

we

(3.1)

consider the Lie brackets involving the vectorfield

also have

to

t. For instance the

system

x - t(x) + ug(x) can be

regarded as a

(3.16 ) special case of

(3,12)

with u 1

cannot go back and forth along the vectorfield

t,

-

1.

Although we

by making switchings

only in u the evolution of x still can be steered in directions involving the brackets of

t

and g. (Although generally we can now only steer along

the positive or negative directions of these brackets.) Motivated by the foregoing discussion we give

Definition 3.7 algebra

f;'

is

Consider the

the nonlinear system

smallest:

subalgebra

af

(3.1).

Val (H)

vectorfields on N, ct. Chapt:er 2) chac concains t,gl""

Remark

The

(the

accessibility

Lie

algebra

at

,gm'

The smallest subalgebra of VW(H) containing a set of vectorfields

79

is well-defined,

since

the

intersection of

two

subalgebras

is again a

subalgebra.

e

The following characterization of Proposition 3. a

is sometimes useful.

e is a linear combination of repeated

Every element of

Lie brackets of the form [X"

[X,_"

where Xi' i E

Proof

Denote

(3.17)

[ ... , [X, ,X, J ... J J J ~,

is in the set (f,gl, ... ,gmJ and k - 0,1,2, . . . .

the

linear

subspace

of V'" (H)

spanned by

the

expressions

(3.17) by r, By definition of e we have r c e. In order to show r - e we only have to prove that r is a subalgebra. Let the length of an expression (3.17) be the number of Lie brackets in it, i.e, k. Consider two arbitrary

expressions (3.17) of length j, resp. 1 Z - [Z"[Z,_,,[ .. ·,[Z,,Z,J···JJJ

(3.18)

By induction we will prove that [X,Y] E r for any j and 1. For assume that [2,YJ E r

for

all Y,

.2 arbitrary,

and

dearly true for k ... 1.) Now take j

for

all Z with j :S Ie.

(This

is

- k+1 in (3.17). Then by the Jacobi-

identity (Proposition 2.27 (c»

(3.19)

[Z.YJ - -[Z',[Z"YJJ + [Zj,[Z',YJI. with Zl -

[ZJ-I,[···[Zz,Zll···jj.

Since the length Z1 equals j-1 ... k,

it

follows by the ind,uction assumption that the first term on the right-hand side is in X, and [ZI,Y] E X, so that also the second term is in r.

0

Now let us define the accessibility distribution C as the distribution generated by the accessibility algebra C(x) -

Since

e

is

Furthermore,

~:

span(X(x) IX vectorfield in ~J, a subalgebra, let RV(xo ,T)

it immediately follows be

the

reachable

(3.20)

x E 11.

set

that C is from Xo

at

involutlve. time T> 0,

following trajectories which remain for t:S T in the neighborhood V of xo.

i.e.

80

RV(xo,T)

Ix E

=

HI

there exiscs an admissible input u: [O,T) ~ U such

that

the

x(t)

E V,

evolution 0

of

(3.1)

xeO) -

for

satisfies

Xo

and x(T) - xl,

S t S T,

(3.21)

Bnd denote

R~ (xc)

V

(3.22)

U R (xo ,T) •

-

TST

We have the following basic theorem

Consider the system (3.1). Assume that

Theorem 3.9

dim C(xo ) - n.

(3,23)

Then for any neighborhood V of

and T > 0 t:he set R; (x o ) concalns a

Xo

non-empty open sec of H. By continuity there exists a neighborhood fv

Proof

t' of

c

Xo

such that

dim C(x) - n, for any x E W. We construct a sequence of submanifolds Nj in

W, dim Nj

-

j. j

in the following way. Let' be the set of associated

E~,

vectorfie1ds of the system (3.1), cf. (3.2). For j - 1 choose Xl E' such X1(X o ) ~

that

O.

Then

sufficiently small

£1

by

the

Flow-box

Theorem

(Theorem 2.26)

for

> 0 (3.24)

is a submBnifold of H of dimension 1, contained in W. Let us now construct Nj for j > 1 by induction. Assume that we have constructed a submanifold Nj

_I

C

rl of dimension j-l defined as (3.25) j-l

where Xi' i E

H.

are vectorfields in "

and

L at

is arbitrarily small.

i~l

If j-l < n then we can find Xj

E ~

and q

E Nj~l

such that (3.26)

For q

E

if Nj

this l'

was

not

possible

then

X(q) E TqN j

_1

for

any

X

E'

and

However. in view of Proposicion 2.41, this would mean thac this

holds for any X

E

C, so thac dim C(q) < n for every q

E

Nj

-

1 C

W, which is

in contradiction with the definition of f.r. It also follows that we may

81

take q in (3.26) arbitrarily close to xo' Therefore the map

(3.27) has

rank

equal

to

j

on

some

set

a!:

0t

Proposition 2.20 the image of this map for

< (1'

ti


.

(3,50)

o fl]

(w). Hence the vectors

(3,51)

are

in Co (0), and thus if AJ .,. 0, or equivalently 8 1 .,. 8 2 , 3, and the system is locally strongly accessible from w = o.

contained

dim CoCO)

=

Furthermore

the

condition

8 1

,e. il2

is

accessibility, since if we would take

8 1

also =

il2

necessary

for

local

strong

in (3.'IB) then we obtain

90

(3.52)

which is clearly not accessible since

w3

is constant. Therefore. (3.48) is

locally st:rongly accessible it and only it

ill .-I il 2 •

For the general location of gas jet actuators the computations become more involved. Without proof we give the result

(3.48) is locally strongly accessible 1

dim span(b 1 .b2 .S(w)J- w;

wE

~

span(b l ,b 2

))

(3.53)

~ 3

III. dim spanlb 1 Ibz ,b J I - 1. Without: loss of generality we may assume that b2, - b J

0, so that in fact we have only one input u. For simplicity we

""

only consider the case

a2

81

so that the system becomes

•

(3.54)

with A '"

g -

(0

(il l

-a 3 )a~l. Computing the algebra

eo

for

t -

A(W2 w3

-WI w3

0) T and

P 1)T yields pW'J + W 2 1 -aw) ~ w1 1

rf,g) ~ -A

1

[

1 - -2A1 [-~ 1'

(3.55)

g, : - [g, [f, g J

Now g. gz, g3 span

m3

for all W E ~3 if and only if

ArfJ (3.56)

-kyO!

o There fore.

if r ... 0 and A ... O.

and no t both

0:

and fJ are zero.

then the

system is locally strongly accessible. These conditions are also necessary as can be checked as follows. If A - 0 then the system (3.54) is clearly not accessible.

If -y - 0 then

accessible. Finally i f a - fJ

~

w3

is constant,

0 then

and so the system is not

91

(3.57) Bnd

so

is

w

constrained

to

in

lie

the

,

,

z

surface - wI + -

constant.

Wz =

o

Hence the system is not accessible.

Let

us

finally

study

controllability

for

a

particular

class

of

nonlinear systems, namely the bilinear systems

I

x - Ax +

CBjx)u j

x E

,

(3.58)

[fin,

j"

where A,B 1

, ••.

from

origin

the

,Bm are

matrices. First observe that the reachable set

l1Xn

contains

only

the

origin.

\mat

can

we

say

about

the

reachable sets from other points? Let us first compute the accessibility r;.

algebra

The

bracket

vectorfield Bjx yields

of

by

the

the

drift

coordinate

vectorfield expression

and

A,.,.

(2.94)

an of

input

the

Lie

bracket (3.59)

where [A,Bj

1=

ABj - ABj

is now the commutator of the matrices A and Bj

(eL 2.99), Taking the Lie bracket of this linear vectorfield with say A....:yields (3 .60)

IAx,-IA,B j Ix) " lA, IA,Bj) )x. Continuing in this way we obtain

Consider

Proposition 3.25

the

bilinear

system

(3.58).

Let

A(

be

the

smallest subalgebra in Gl(n) (the Lie algebra of nXn matrices Idth bracket {A,B) = AB - BA)

containing the matrices A,B!, ... ,Bm'

Then

the accessi-

bility algebra fi' is given as

e= Since

(all linear vectorfields on [R" of the form tlx, Idth tI E AI).

is contained in Gl(n)

A(

finite-dimensional

Lie

~

algebra.

[R"z

it follows

Furthermore,

as

(3.61)

that AI and hence fi' is a in

Proposition

3.8

it

follows that every element of AI can be written as a linear combination of elements of the form

[Dj;' with D1

,

[Dk~l' ( ...

i

E~,

[D z ,D 1 J ... J J J,

in the set (A,B1, ... ,B m ),

(3.62)

92

eo

The analysis of the subalgebra

is completely similar. In fact let Ala

be the ideal in AI generated by the matrices Bl , ..• ,Bm' then vectorfields in eo are of the form fix with H Example 3.26

Consider

again

E Mo.

the

spacecraft

(Example 1. 2);

example

in

particular the equations describing the orientation of the rigid body

R(t) ~ -R(t)S(w(r»)

(3.63)

where R(t) - (r 1 (t),r2(t),r 3 (t)]

E

50(3) and r l (t) describes the direction

of the i-th axis of the spacecraft (with respect to an inertial frame). Let N(r:) :- R-1(t) - RT(r:). Le. the columns of N(t) describe the position of the axes of the inertial frame wi th respec t to the moving frame given by the axes of the spacecraft. Since R{t)N(t) - I we obtain

o-

R(t)N(c) + R(c)N(t)

-R(t)S{w(t»)N(t) + R(t)N(t),

and hence (3.64)

N(t) - S(w(t»)N(t). Now let us consider the

x(t)

[-w, ~t) w2 (t)

W3

time~evolution

(t)

0

of a single column of N(t}, Le.

-w, (t) 1 Wl

~t)

x

x(t),

E [R:l,

(3.65)

-w 1 (t)

and let us assume that we can control the angular velocities {w.r.t. the axes in the spacecraft} directly. Hence

ui

WI ,wl

,w 3

wI' i - 1,2,3 are

-

controls, and we obtain the bilinear system

x- [ ~

o

1

a o

o

-1

(3.66)

One computes 0.67)

(3.68)

a so

that dim Co (0) = 0

I

and dim

Co (x) -

2 for

all x

~

O.

By

Proposition

93

3,12

follows

it

the

that

reachable

sets

from

~

Xo

contain

0

two-dimensional submanifolds of ffi3. In fact, it is easily seen that Rr(xo ) is contained in the sphere in [R3 with radius r ~ Ilxo II, and by Proposition 3.15 is equal to this sphere. Of course, this expresses the fact that the

columns of Net) E SO(3) are vectors of unit length. inputs, say u 3

~

0,

then it follows from (3.67)

If we have only two

that the controllability

o

properties of (3.66) remain unchanged. 3.2

Observability

Let us consider the same smooth affine control system (3.1) as before, but

now together with an output map

x - f(x)

+

I

,-I

gj (X)U j

U"" (u

,

, ... ,U m ) E U C [p.m,

1

(3.69) i E

where h .. (h 1

, ...

E.

,hp)T: N -. Y - lR P is the smooth output map of the system.

The notion of observability we will deal with for these systems is defined as follows. Recall that y(t,O,x o ,u) = h(x(t,O,xo ,u») denotes the output of ~

(3.69) for u(·) and initial state x(O)

states

xo'

Definition 3.27

Two

(denoted x l Ix2 )

for (3.69) if for every admissible input function u the

output function t

H y(t,0,x1 ,u), and the output function

x(O)

=

for

initial

Xl'

state x(O) - x z

EN are

X l ,X2

t

said

to

be

indistinguishable

0, of the system for initial state

==-

t

H

y(t,O,x z ,u),

, are identical

on

0, of the system

t ==-

their

common

definition. The system is called observable if xlIx z implies Notice

Xl

domain

of

.. X2'

that this definition of observability does not imply that every

input function distinguishes points of N.

However,

if the output is the

sum of a function of the initial state and a function of the input (as it is

for

linear

systems)

then

it

is

easily

seen

that

if

some

input

distinguishes between two initial states then every input will do. Since our aim is to replace the Kalman rank condition for observability of

linear

systems

by

a

nonlinear

observability

rank

condition

(which

inherently will be a local condition), we localize Definition 3.27 in the following way. Let V C l'l be an open set containing say that

Xl

and X z are V -

indistinguishable,

every admissible constant control u:

[O,T]

~

Xl

as well as x z . We

denoted as

Xl IVx z , if for U, T> 0 arbitrary, with the

94

property that the solutions x(t,O,x1 ,u), and x(t.O,xz,u) both remain in V for t

~

T. the output functions y(c,O,xl ,u), respectively y(t,O,xz.u) are t ~ T on their common domain of definition.

the same for

Definition 3.28 there exists

The system (3.69) is called locally observable at Xo if neighborhood r.; of

a

such that for every neighborhood V C W

Xo

of Xu the relation XOIVXl implies that Xl - xo' If che system is locally observable at e;Jcll

then it is called locally observable.

Xo

Roughly speaking a system is locally observable if every state Xo can be distinguished from its neighbors by using system trajectories remaining close to xo. Recall that for studying local accessibility the accessibility algebra of

the

system

was

shown

to

be

essential.

Analogously,

for

local

observability the observation space will prove to be instrumental. Definition 3.29 space 0

of

containing h l

Consider is

(3.69)

linear

space

(over

(3.69). ~)

of

The

observation on H

functions

and all repeated Lie derivatives

•...• h p '

j E~.

tlle nonlinear system

the

E

e,

k - 1,2 ....

(3.70)

in the set (t,gl, ... ,gm)'

The following propositions give equivalent characterizations of O. Proposition 3.30

0 is also given as the linear space of functions on H

containing hI'" .,hp

•

and all repeated Lie derivatives j

with Z1' i E

~,

E

e.

k - 1,2 ....

(3.71)

of the form

t(x)

+

I

(3.72)

gj (X)ut •

jml

for some point u

i -

(u:, ... ,u;) E U. L.e., Z1 E'.

Proof We use the facts thut Lx1+x Z" - Lx/I + Lx l ", and Lx("l + Hz) .. Lx"l + Lx"z for any vectorfields X ,Xl ,Xz and functions ", HI .Hz • Since Zi is a linear combination of the vector fields f.g l

•...

follows

in D.

that

expressions

(3.71)

are

contained

,gm'

it immediately Conversely,

all

95

vectorfields f ,gl ' ... ,gm can be written as linear combinations of Zi' In

fact f

Remark

=

Zi for u

i

o

0, and gj =

=

Proposition

3.30

yields

the

following

interpretation

of

the

observation space 0; it contains the output functions and all derivatives of the output functions along the system trajectories. In particular, for an

autonomous

Yj - h j (x)

Yj -

system

together

(i.e.

with

The

vectorfields algebra

Xi'

i

all

inputs)

repeated

is

0

time

constructed

by

taking

Yj ... Lchj ex),

derivatives

E·

LfLchjex}, ... , j E

Proposition 3.31

no

definition

E!5.,

in

of 0

(3.70)

is not changed i f to

belong

the

to

1"8

allow

the

accessibility

e. Let X1 ,XZ be vectorfields. Then by definition of the Lie bracket

Proof

(3.73) Hence if Xl and Xz are in the set {f,gl, ... ,gm) then L!X .x )h i , i E E, 1 2 belongs to 0, and similarly if hi is replaced by any function (3.70). 0

The

observation

space

0

defines

the

observability

codistribution.

denoted as dO, by setting dO(q) -

Since

dO

span(dH(q)

is

I

generated

HE OJ,

by

(3.74)

q E 11.

exact

one-forms

it

follows

that

the

codistribution dO is involutive (see (2.166». The main theorem concerning local observability reads as follows.

Theorem 3.32

Consider the system (3.69) with dim 11

~

n. Assume that

(3.75)

dim dD(x o ) ... n,

then the system is locally observable at xO. Proof

Since dim D(xo)

= n

there exist n functions HI ' ...• Hn E 0 such that n as

dHI(x O) •... ,dHn(xo ) are linearly independent. Define the map : 11 -+!II

~(x) - [H , (x), ... ,H"(x)j'. It follows

that

the Jacobian matrix

(3.76) of

in

Xo

is

non-singular,

and

96

therefore by Proposition 2.16 there exists a neighborhood W of that I: W ~

xo. and suppose that xolvXl for some Xl E V. Then for any i E Bnd for small

such

Xo

is a diffeomorphism. Now let V C W be a neighborhood of

~(W)

tl , ... ,tic

e and

k ~ 0,

we have (3.77)

with Zl' i E!;'. of the form (3.72). Differentiating of both sides with respect

to

O. t lc -

tit -

(in

l

this

at

order)

respectively

O, ... ,t} - 0 yields

-

(3.78) for all Zj' j E!;,. of the form (3.72). By Proposition 3.30 it follows that Jf(x o ) '" H(x 1 ) for all ]( E O. In particular IIi (xo ) - Hi (Xl)' i E:!, and by injectivity of 1 on rl this yields Xo ... Xl' o

We call (3.75)

the observability rank condition. The system is said to

satisfy the observability rank condition if (3.75) holds for any Corollary 3.33

Assume

that

satisfies

(3.69)

tlle

Xo E

observability

H. rank

condition. then it is locally observable. What

can

codistribution

be

said has

dO

about

the

constant

case

dim dO(x o ) < n?

dimension

around

we

Xo

In have

case

the

(compare

Proposition 3.12): Proposition 3.34 By

Frobenius I

(U,x 1

•••.

,xn

)

Suppose that dO l1as constant dimension k < n around xo'

Theorem

around

KO

(Theorem 2.42)

we

can

find

l1.

coordinate

chart

such that the submanifold

(3.79) is an integral manifold through

Xo

of ehe involutive distribution ker dO.

There exists a neighborllood rl C U of VerI" of Xo

l~e

Xo

such that for any neighborhood

have (3.80)

Proof Ill"

As

in

the

proof

.. ,H", E

0

such

that

of

Theorem

3.32

dH 1 (x o ). ... ,dHk (x o )

there are

exist linearly

Ie

functions

independent.

Therefore by Proposition 2.18 we can take HI •••. ,Uk as partial coordinates

97

on a

neighborhood r.,r c U

manifold

of

ker dO

X Il - kt1 ' •••

,xn

in (3.79)

of xo'

we

may

definition of 5 xo as assume that the

By

as

an

well

restricted to

r"

to HI""

are equal

V c r.,r be a neighborhood of xo' Suppose that

XOIVXl

for some

integral

coordinates ,1l., 0.)

(b)

Can we always take k in (4) to be less or equal than n?

(c)

Derive a condition similar to (4) for some sort of observability

for the system (1), using the augmented system (2).

3,6

Consider a nonlinear system

x - f{x,u) (1)

h(x)

y We

say

that

the

system

(1)

uniformly locally observable at

is

XO

if there exists a neighborhood rv' of XO such that for every o neighborhood Vcrv' of x the following holds. Let with ([GB])

Then

Xl ;
for

every

piecewise

constant

input

m

u: [O,T] .... IR , with T> 0 arbitrary, such that 1 x(t,x ,u) remain in V for t E [O,T]. we have that y(t,xo,u)

1

Prove that (1)

and

for some t E [O,T].

y(t,x ,u),

;
function

x(t,xo,u)

is uniformly locally observable at

if in suitable

XO

local coordinates it is of the form y

= h(x!),

Xl -

(2 )

with

f 1 (X 1 ,X Z 'U) i - 2, ... ,n-l,

ah (x»)"! 0, -a •... 1

i - 1,2, ... ,n-l.

3.7

Consider, (3.99)

see

with

91

aft aX + (x,u) i 1

Example single

=

92,

=

)"!

3.110,

output

0 and u l =

0

for

the

every

two-link

(3.100).

rigid

and

every

robot

Linearize

0 and

Uz =

u E [Rm

manipulator

the

show that

x E 1/,

system

the

system is observable for g ,., 0, while it is not observable for g On the other hand, show that dim dO 3.8

({So]) The system (3.69)

=

4 even in case g

is called L-observable at Xo

=

in

linearized "*

O.

O.

i f for every

116

neighborhood rV' of Xo

there exists a neighborhood V C W of xa such

that the relation XOlwXl for xl E V implies that

Xo.

xl

(n) Show that if (3.69) is locally observable at xa then it is also L-observable at xo. (b) (compare Corollary 3.35) Show that if the system is L-observable at every x

then dim dO(x)

E lJ

=

n

for every x in an open and dense

subset of H. (c) Assume that the system satisfies the accessibility rank condition and that H is connected. ShoW that if the system is L-observable at every x in an open and dense subset of M, then it is L-observable at every x E H. (d)

(compare

Proposition

3.38)

Assume

accessibili ty rank condi tion and

the

system

satisfies

tha t H i s connected.

system is L-observable at: every x

E H

Flo

the

Prove:

the

the observability rank

condition holds on an open and dense subset of N. 3.9

([HK]) Prove Remark 3.39. (Hint: use Theorem 2.33.)

3.10 Prove Remark 3.48. 3.11 Let the accessibility distribution C for the nonlinear syst:em (3.69) have constant dimension. Show, analogously to Theorem 3.49, that in Z

local coordinates (x1,x )

such that C = spanl-a-J axl

the system takes

the form (compare (3.126»

;,? -

1 fl(x ,X:'.)

rn

L

+

g;

(x1,xZ)u j

•2 X ..

,

O.

jal

Assume

that

also

the

distributions

ker dO

and

C + Iter dO

have

Z

3 constant dimension. Then show that in local coordinates (x1,x ,x ,x")

with C ~ spanl~,~1 and ker dD '" spanl~,~J the syst:em takes axl ax:'. ax 2 ax 4 J the form (3.137) with (3.137c,d) replaced by ~ 0 and x~ O.

'"

x

3-.12 (a) Show that if a distribut:ion D is invariant under the nonlinear

system

(3.1),

then

(Z,D] C V

for

every

vectorfield

Z E

~

(the

accessibility algebra). (b) Let D be an involutive const:ant dimensional distribution on with i E

natural

coordinates

!.!. Prove that D ~

(Xl ,. -. ,Xn

).

Suppose

DV for some subspace V c IR

n

that

a [ax!

[JIn

,V] c D,

(cf. Remark 3.43).

(c) ([Kr3]) Let D be an involutive constant dimensional distribution for a controllable linear system on IRn. subspace

V C

IRn.

Show that D - Dv for some

4 Input-Output Representations \~e

In this chapter

x

l:

lex) +

consider, as before, smooth nonlinear systems

gj (x)u j

,

u

E,

y

j '" 1

y,

h j (X),

where x and

j

tt gl"" ,gm

functions.

(u 1

E U C [Rm,

' ••• ,U m )

(4.1)

(YI""

,Yp

E

)

m" ,

,An) are local coordinates for the state space manifold H

(Xl""

=

E

-

are

Our

smooth

aim

in

this

vectorfields, chapter

is

while

to

hI""

derive

relating the inputs u directly to the outputs y.

are

,hp

e}:plicit

smooth

espressions

Said otherwise, we want

to eliminate in (4.1) the states x and obtain a direct connection between the

inputs

and

outputs.

Apart

from

the

interest

per

this

S8,

is

potentially useful for various control purposes. Indeed, in linear systems theory it is well-known that some control or synthesis problems are easier formulated

and/or solved for

other problems

systems

given in state

may be more naturally stated

space

and/or solved

form, for

while systems

described in input-output form. We

recall

that

for

linear

systems

there

are

several

ways

of

representing a system

x

~

Ax + Bu,

u E [Rm,x E ffi"

~

Cx,

y E I!,'P,

(4.2)

Y

in input-output form. of primary importance is the impulse response macrix representation

yet) '"

with G(T) t ~

J

G(c-s)u(s)ds

CeATB

+ CeAtxo,

c

~

the impulse response matrix, yielding the output y(c),

0, as the sum of a convolution integral of the input u(t),

11 term depending on the initial state x(O) (4,3) for

yes)

Xo

=

=

(4,3)

0 ,

=

t

~

0, and

xo' Laplace transformation of

0 yields (with /\ denoting the Laplace transform).

G(s)u(s)

where the rational matrix G(s)

(4.4)

=

C(Is_A)-lB is called the Cr,:ms[er m,-"1trix,

It is well-known that G(s) can be alternatively written as a polynomial

118

fraction

6(s) - D-1(s)N(s)

(4.5)

where D(s) is a pxp polynomial matrix satisfying det D(s)

~

0, and N(s) is

a pxm polynomial matrix. (The matrices D(s) and N(s) can be also directly obtained from the matrices A,B,C in (4.2).) Inverse Laplace transformation then yields the set of higher-order differential equations in the inputs and outputs

(4.6) which is called an external diffe.rencial representation of the linear system (4.2). In

the

first

section

of

this

chapter

we

study

nonlinear

a

generalization of the impulse response matrix representation (4.3), known as the fHener-Volterra series representation of (4.1). make

contact

with

a

functional expansion.

different

series

expansion

Briefly we also

called

the

Fliess

In the second section we deal with the nonlinear

of the external differential representation (4.6). We will

generali~ation

also show how under constant rank assumptions the state

X

(in fact the

observable part of the state) can be expressed in the inputs and outputs and their higher-order time-derivatives. Finally, in the third section we will study the particular problem when a given input component does not affect

a

certain

instrumental

to

output the

component.

solution

of

The

conditions

the

disturbance

obtained

will

decoupling

be and

input-output decoupling problem as treated in Chapters 7, 8, 9 and 13. In the present chapter we will not deal with the inverse problem of input-output representations; that is, how to obtain from an input-output representation (e.g. Wiener-Volterra or Fliess series representation, or external differential representation) a state space model (4.1). For this important realization problem we refer to the literature cited at the end of this chapter. 4.1 Wiener-Volterra and Fliess Series Expansion For Simplicity we first take the input and output in (4.1) to be scalar valued

x

[(x) +

g(x)l1,

x(O) - xo

'

u

e

U

c

[R

r

(4.7) y - h(x),

yelll

119

;.;ohare X ... (Xl""

,Xn )

are local coordinates for H,

f: II?n ... {Rn, g: [Rn .... IR

n

are the local coordinate expressions of two smooth vectorfields on N. and

h: mn ... II? is the local coordinate expression of a smooth function on H. We assume throughout that U is an open subset of First we fix

neighborhood. solution

of

the initial state Xo

:= x(t,s,x,u)

conveniently denote by 'Yo (t,s,x)

Let us (4.7)

containing the origin.

II?

contained in the above coordinate

at

time

t

for

initial

state

xes) - x

and

the

input

u ". 0, i.e,

(4.8) -ro(s,s,X)

Since

x-

f(x)

""(o(t,StX)

=

x .

is time-invariant we clearly have =

""fa (t-s,O,x) .

(4.9)

For any given input function u(t), say piecewise continuous, we denote by 1 u (t,s,x) :- x(ttS,Xtu) the resulting solution of (l1.7) for xes) - x, i.e.

(4.10) "fu (s,s,x) = X

•

In order to compare h(7o(t,O,x o )) and h(7 u (t,O,xo »), i.e. the outputs for zero input and input function u, we introduce for fixed

°

the curve

t

~ s ~ t

(4.11)

Note that (4.12)

implying the basic relation - h(p(t)) - h(p(O)) -

, d ds 1>(p(s))ds.

J

,

(4.13)

Proposition 4.1 Let pes) be as defined in (4.11), chen d ds h(p(s)) - u(s) (

ahi1,(t,s,x)) ax

g(x))l x =1'u(s,O,x )· o

(4.14)

Proof By (4.9) and the chain rule (4.15)

120

The second term of the right-hand side of (4.15) equals, by (4.10),

ahb o (t-s,O,x) 8x

(i(x)

+

u(s)g(x)}

Ix. _ 1u (s, 0 ,Xo )

(4.16)

The first term on the right-hand side of (4.15) equals, by (4.8), (4.17) Furthermore, for the first term in (4.16) we observe that by (4.8) 8il("'fo(t-s,O,Z)

fez) -

Bz ah~x) ax

I _

01'o(t-s,O,z)

•

x - "'fa (t-S,O,Z)

az

(4.1S)

fez) .

We claim that since 1'0 is the flow of f

8-yo(t-s,O,Z) (4.19)

az Indeed

"'fo(r,O,1'o(t-s,O,z») = 1'o(c-s,O,"'fo(r,O,z»)

and

so,

by

Corollary

2.32, we have 'Yo(t-s,O,z).f(z) ~ fho(t-s,O,z») which is exactly (4.19). (Note

that "'fo(t-s,O,z) = ft-s(e).)

Therefore

for

z - "'fU(S,O,XO )

(4.18)

equals (4.20) and hence equals minus (4.17).

Taking everything together

we

see that

o

(4.15) equals the right-hand side of (4.14). Let us denote

w1 (t,s,x)

aJl h 0 ( c: , S

I

(4.21)

x) )

- ---:a"-x~-- g(x)

.

Then by (4.13) and (4.1LI) it follows that t

hh.,(t,O,x o »)

-

{vo(t)

+

J

u(s)W1(t,s,'Yu(s,O,xo»)ds .

Repeating the same procedure as above for

h(·)

(4.22)

rep1nced by WI (t:, s,') we

obtain s

h1dt,S,I',,(s,O,xo») ~ \"t(t,s) +

J

u(r)w2,(t,s,r,"1'u(r,O,x o »)dr,

(4.23)

121

where we denote

W2 (c,s,r,x)

=

(4.24)

a~1(C,S'70(s,r,x») -~~~a;:X:-~~~- g(x) .

Substition in (4.22) gives

, h('Yu(c,O,x o »)

J

...

wo(t)

+

I I wz (c,s,r,7 u {r,O,x o )ju(s)u(r)drds

.

+

\"t(t,s)u(s)ds

, ,,

(4.25)

After r repetitions of this process we obtain the functional expansion of yet) ... h(-ru(t,O,x o ») given by

,

yet) ~ ,

1>'0

(e)

+

J WI (t,s)u(s)ds

+

'1

JI

,,

WZ (t,sl,5Z)U(Sl)U{sz)ds z ds l + ...... +

(4.26) ,

'1

51: -1

J

II ,, t.

"1

"

J, J,

W r + 1 {C,Sl,.·,Sr+l,7 u (Sr+l,Q,X o )j

Jr

where for i

+ .... +

Iv\:(t'Sl,··,Sk}U(Sl)U(SZ)··u(s,)ds k .. ds 1

U(Sl)··U(Sr+l)ds r + 1 ··ds l

2,3, ... , and w1(t,s,x) as given in (4.21)

g(x) ,

(4.27)

We call

yet),

(4.26)

the

r-ch

and Wk(C,Sl""'S);)

order [vlener-Volterra as

given

in

(4.27)

functional the

k-th

expansion

order

of

Volterra

kernel. Notice that the kernels and therefore the expansion depend on the initial state xo'

For analytic systems we can let r a convergent Wiener-Volterra series.

in (4.26) tend to infinity to obtain (For the

(simple) proof we refer to

the literature cited at the end of the chapter.)

122

Theorem 4.2 Let f,g. and 11 in (4.7) be analytic in a neighborhood of xo'

Then there exists T> 0 such chat for each input function u(t) on [O,T] satisfying lu{t)1 < 1 ehe Wiener-Volterra series ~1

t

~k - 1

L I I .. ,J

yet) - wo{t) +

W!(t,sl"",sk)u(Sl)···U(s\ 1.

(t,sl""

and the functional

expansion reduces to

y( t) -

ce

At

t

Xo

I

-I-

ceA(t-slbu(s)ds

(4.30)

which is jus t the impulse response representation (/,.3).

We note that we can rephrase the definition of the Volterra kernels t.'1; (t, si • . . • • St;)

Let

l':

as given in (4.27) in the following coordinate free way.

11 .., 11 be the flow of the vectorfield

t (Le.

i'(x) -

1'0 (t,O,x»).

Then it can checked (see Exercise 11.1) that the kernels are also given as

(4,31) Lg { ... (L& (L (hoi t - 51

s

')

of lsl -S" ') ot(S2- -53»

0 .•. o£(llk - 1 -~k » oisk } (x

o) ,

k - 1,2, ...

The

Wiener-Volterra

functional

expansion

for

the

mulci-inpur

mulri-output case is completely similar. If (4.1) is an analytic system then there exists T> 0 such that if lui(e)1 < I, for

any

output

component

Yj ,

j

e.,

E

we

have

t E

a

[O,T], i E

uniformly

E.

then

absolutely

convergent expansion Yj (t)

-

l"~ (r:) +

L

k-l

t

L 1 1 , . . i k .. 1

III

JJ 0

0

"k

J

(lj.32)

123

\"t ,. ..

for certain Volterra kernels

kernels are given [or j E

j

{"i

l

"

1.. , (t, 51 , ' •• ,51;)'

In fact these Volterra

E as

,5):)

'ik(t,Sl'"

1,2 ...

(4.33)

Example 4.3 Consider a bilinear system (cf. Chapter 3)

x

=

I

Ay + j

~

(Bjx)u j

n

XElR , UE!p'

,

m

xeO) =x o

,

'

1

j

E. '

E

with A,B l , ... ,Ern nXn matrices. In this case the flow ft of the drift vectorfield is explicitly given as [LCX) = eAtx . Therefore the Volterra

kernels up to order two are given as

and similar expressions hold for the higher-order kernels. Let us

now briefly

indicate how

0

analytic systems

for

(4.1)

we

can

deduce from the Wiener-Volterra functional expansion, as described above, another which

functional

expansion,

called

the

Fliess

reveals

more

clearly

the

underlying

simplicity we

first

consider

the

single-input

functional

algebraic

expansion,

structure.

single-output

case,

For and

start from the Wiener-Volterra series (ll.28) with its kernels given as in (4.31),

Since

h'j (t,sl""

and 11

f,g

are

assumed

to

be

analytic

all

the

kernels

,Sj) are analytic functions of their arguments, The key idea is

now to expand these kernels as Taylor series the variables t-s l

,SI-s2""

Wj(t,sl"",Sj)

=

,Sj_l-

Sj

I kO ,k 1 ,·"

,Sj'

not

t,sl""

,sJ'

but in

i.e,

C j ,kok1' .k j '

,kj"'O

in

(4.36)

124

for certain coefficients

Cj.kOkl"

.k

j

depending on

Therefore by (4.2B)

xO'

we now have to consider the iterated integrals t

~1

l>j-l

f f ... J o

(C-s1}k n

(Sj_l-Si)k j

-

1

~ U(Sl)"

s/(j u(s.l) ~j! dS j

.. .

ds 1

(Q.37)

0

which. however. have the following appealing structure. Define t

~o

(t)

, e1(t)

t

f

=

u(s)ds

(4.38)

I

and inductively let i - 0,1 , t

I

(4.39)

to

d~lk" .de ro

II

I

=

I

d€!k(s)

o

Il

d€l k_1 ' ,.d~!o

where i o •... ,lk are 0 or 1. An easy computation shows that with these definitions (4.37) equals t

I where

(de o )11:0 del'" (d~o

(deo)k

cj,kok ... k 1 j

stands in

)k j -1

for

(4.36)

(4.40)

dEl (de o }kj

times

lc

can be

Furthermore

d€o.

directly

the

identified with

coefficients the

following

expressions (4.41)

C j • II. ok l' . , k j

(One way to obtain this identity is to use Taylor expansions of (4.3l) in the variables

t-s 1 ,51-52.'"

'Sk-1-Sk ,Sk;

for

details

we

refer

literature cited at the end of this chapter.) Combining (4.28),

to

the

(4.36),

(4.40) and (4.41) we arrive at the Fliess functional expansion yet) - h(xo ) +

m

L k ~0

t

L

LIl!o ••. L

1 0 •...• 1k wO

sik

h(xo )

I

d~ik···d€!o

(4.42)

with go :- f. Similarly for the multi-input multi-output case we have for j

E

E (compare

(4.32» t.

m

Yj (t) - hj(xo ) +

L kcO

L to·

Lg 10

•• • L E i

,fk "0

k

h j (xo )

f

d€i " .de io k

(4.43) with go

'- f,

L

and

f

t

de i ... €i(t:)

-J

u i (s)ds,

i E m

125

4.2. External Differential Representations

In this

section we

assumptions,

shall

give

an

algorithm which,

converts a system in state space form

under (L! .1)

constant

rank

into a set of

higher-order differential equations in the inputs and outputs: •

Rt(u,u, ... ,u

(k)·

,X,y, ... ,y

(k)

)

0,

=

i

E

(4.44)

E '

where u{jl and / j ) denote the j-th drne derivative of the input function

u, respectively output function y. Let us first introduce the notion of a prolongacioll of a higher-order differential equation. Consider a higher-order differential equation

(4.45) in the variables

equation

in

E IRq,

h'

We will

interpret

also

(il.lIS)

indeterminates w ,I~, ... ,w{k)

the

The

as

an algebraic

prolonged equation or

prolongation of (4.115) is defined as

P(t.',W, ... ,1,,(/0

t,,(1C't'l»

~~

:=

I"

where, for notational simplicity,

+ ap \{ + .. + al;'

ap

ap

al" (10

(4.46)

w(j+1):=

al,,(j)

The relation of (i1.lI6) with (4.115) is as follows. Let wet), C E (a,b), be a smooth solution curve of (i1.45), i.e. cECa,b),

(4.47)

then clearly for t E (a,b)

ap .

at"

Idt)

ap + - i;'Ct) + ... + al;'

and so w( c), t: E (a, b), is also a solution curve of the prolonged equation (4.46). Furthermore we note that

ap at/ k + 1 )

ap =

al,,(k)'

Now consider the nonlinear system (4.1), rewritten in implicit form as Pi{X,x,u)"" Xi - ft(x,u)

=

0 , i"" 1, ... ,11 ,

(4.50a)

126

(4.50b)

Pi (X,y) m

Lgj (x)u j

with f(x,u) := f(x) +

Remember that our aim is to eliminate x

•

j "1

and its derivatives in the equations (4.50). Roughly speaking, this will be achieved by successively differentiating the output equations along the system, and to solve from this set of equations for the state variables x. Mathematically this will be formalized by first prolonging the output equations

(4.50b),

to

substitute

xt-fl(x,u) - 0 in (4.50a),

these

for

some

of

the

n

equations

and to replace x in these prolonged equations

by t(x,u). After doing this we obtain a system of equations of the same form as (4.50) where now, however, the number of equations involving

x has

been reduced. Then the same procedure is repeated. Formally t

we

have

the

following

algorithm.

Let

(x(t),u(t),y(t».

E (-f,f), be a smooth solution curve of (4.50). This yields a solution

point

.

.

.

(x(D) ,x(D),. ,xtn)(O) ,u(O) ,fi(O),. ,u{nI(D)

,yeO) ,y(O),. ,y!nJ(O» (4.51)

of (4.50), regarded as a set of algebraic equations in the indeterminates x,x, ... ,x

(n)

,U,U, ••• ,ll

en)

,y,y, ... ,Y

In)

Algorithm 4.4 (External representation nlgorithm) Step 1 Assume that

aPi

rank

ax.

[

Denote Pl -=

(x,y)

J

51

1

~ S1' around

(x,u,y)

(4.52)

~un+l

..... n+p J-1, . . ,n

-so. with So - O.

If Pl - 0 the algorithm terminates.

If

Pl > 0 we proceed as follows. By (4.52) it follows that we can reorder the

equations P 1 " " ' Pn in (4.50), and separately the equations

Pn+1' .••• P n + p ,

in such a way that

rank

- n, around

(x,u,y)

(4.53)

127

Furthermore we re-order the variables

Xl""

.. ,Pn • so that still PiCX,K,U) ~

for Pl"

the prolonged equations

in the same way as we did

IXn

Xl -

i E~. Now consider

fi(x,u),

and replace (4.50) by the following

pn+1"",Pn+Pl

set of equations

Lemma

-

0

i

- 1, ... ,n-PI

(4.54a)

Pi (x,f(x,u),u,y,y)

-0

i

-

(4.54b)

Pi (X,y)

-

i

- n+1"

Pt(X,X,u)

-

xi

4.5 Around

- ft(x,u)

(x ,il ,y.)

tile

0

set of

n+1, ... ,n+Pl

(4.54c)

.. ,n+p

smooth

soluLion

curves

of

(4.54)

equals that of (4.50), Proof Clearly i f (x(t),u(c),u(t» is

also

a

solution

Xi - i i (x,u) - 0,

fi(x,u) for

Xi'

i

curve

E~,

then i t

is a solution curve of (4.S0),

of

the

prolonged

equations,

and

since

we may substitute in these prolonged equations

so as to obtain (4.S4b). For the converse we observe that

by (4.59)

rank (

Now let

(4.55)

] i-n+1, .. ,n+P1 lc ..n~p1 +1, •. ,n

(x(t),u(t),y(t»

be a solution curve of (4.54) around (x,il,Y).

Then it is a solution curve of (4.54c), and therefore of the prolonged equations

.. Pn + P1

Pn+1 -

-

O. Hence (x(t),u(t),y(t»

~

Since (x(t),u(t),y(t» n -PI

I

, -1

aP i 8xj

fj

(X,u)

,

aP i

"1

-a Y.

I

+

Ys"

satisfies

0, i ... n+l,.,n+Pl·

(4.56)

satisfies (4.54a) it follows that

+

I" k-n- P l+ 1

aP i

p

ax, x, +s I"1

aP i 8Yll Y s

-0

i

- n+l, .. ,n+Pl' (4.57)

On the other hand (x(t),u(t),y(t»

aP i -

ax,

satisfies (4.54b):

fj (x,u) + i

Comparing with (4.57) we see that

~

n+l, .. ,n+Pl

' (4.58)

128

L

Xk -

BPJ, ax!: fk (X,U)

n+1 •.. ,n+Pl . (4.59)

,

k-n-Pl+l

By (4.55) it now follows that

k = n-Pl+l, ... ,n , and hence (x(t),u(t),y(t»

is around

Cx,u,y)

(4.60) a solution curve of (4.50). 0

We rename equations (4.54b) by setting

Denote nl:- n, and n z :- n 1 -PI then (4.54) is rewritten as 1=1, .. ,n 2

Pi(x,u,y,y) As

II

(4.62a)

,

- 0 , i - n 2 +1, . . ,n+p

result of the first step of the algorithm we have transformed

(4.50) into (11.62). Notice that (l1.62) is of the same form as (4.50), but

x has

the number of equations involving

decreased by PI' Notice also that

(4.62) satisfies

= n, around (x,u,y)

rank

(4.63)

Step It of the algorithm Consider a system of equations

•

Pi (X,U,U, ..

(1c-2)' ,ll

,y,y •.. ,y

{/C-11

)

0, 1

=

n k +l, .. ,n+p

t

(4.64b)

for which

rank

Now assume that

o

n, around

(x,u,y)

(4.65)

129

rank [

1

ap,

'-n,'1..

aX j

-

5

k

around

,

(x,u,y)

(4,66)

,mp

jml, ... ,n

Denote Pk:=

5

k -

5);-1'

o

If PI;

the algorithm terminates.

If Pi:

> 0 we

proceed as follows. By (4.49)

(4,67)

Furthermore in the (lc-l)-th step we have assumed that

rank

ap, [ aX j

1

~

j~l,

rank

.. ,n+p

~mnk_lt1,

... ,n

(4,68) Now consider the prolongations of the equations obtained in the (k-l)-th

step . .

Pi (x,x,u, ..

By

(4,68)

(k-l) ,ll

there

,y, .. ,y exist

(k)

) -

0 ,

.

.1

=

functions

(4,69)

"1:;+1, .. ,nl;_1 ail~(x,u,

...

dC-'ll ,ll

,y, ... ,y

(k-l)

),

i - nk+l, ... ,n);_I' .2 - nl:_ 1 +1, .. ,n+p, such that if we define the following

modifications of the equations (4.69) . (k-l) de)" dC-I) do Si(x,x,u, .. ,u ,y, .. ,y ):- Pl(X,X,U, .. ,u ,y, .. , Y )

mp \' L.

0ii(X'u, .. ,u

(/C-2)

k

,Y,··,Y

(-I)"

) Pi(x,x,u, .. ,u

dC-I),

,y, .

.e .. nlo;_l +1 i

=

+

(4,70)

,ydo) ,

0",+1, •• ,010;_1'

then

aS i - 0 , i=Ok+1, .. ,nk_1,

aXj

(4,71)

j = nk +1, .. ,n.

By (4.66) we can now reorder SnJo;+l"',Snk_l in such a way that ... Plo;' around (x,Li,y)

Then by (4.65) we can permute the equations PI""

(4,72)

,PnJo; in such a way that

130

n~,

rank

Furthermore we

permute

around

(x,u,y).

the variables xl"" 'Xnk in

(4.73)

the

same

way.

Now

consider instead of (4.64) the set of equations

x -f .. (x,u)

Pl (x,x,u)

5 i (x,f(x.u) ,u, .. ,u

Pi (X,u, .. ,u

(k-2)

dC-I)

,y, .. ,y

(x,u,y)

Lemma 4.6 Around

(4.74a)

- 0,

1

,y ... ,Y c/C-l)

do

) = o.

- nk+l, .. ,n+p

) - 0,

tile

set

of smooch

solution

curves

(4.74c) of

(4.74)

equals that of (4.64).

Proof Clearly any solution curve of (4.64) is a solution curve of (4.74) (compare Lemma 4.5). Let (x(c).u(t),y(t»

be a smooth solution curve of

(4.7 /,). We only have to prove that (x(t),u(t),y(t»

is a solution curve of

(4.75)

0,

Clearly (x(t),u(t),y(t» •

51 (x,X,u, .. ,u

(k-l)

is also a solution curve of

,y ... ,y

Using the fact that 51

(k)

)

o.

is linear in x,

(4.76)

1

and (x(t).u(t),y(t»

is also a

solution curve of (4.74b), it follows that

nasi.

L -.jml

Xj

nasi

=

aXJ

L

j-l

Furthermore since

aXj

1

fj(x,u)

(x(t),u(t),y(t»

satisfies

(4.77)

(4, 74a)

and

(ll. 71)

holds,

(4.77) reduces to

as! - . - Xj

-

(4.78)

j-nk-Pk+18xj

and (4.75) follows by (4.72) and (4.73). We rename equations (4.74b) by setting

o

131

Pi(x,u, .. ,u

(k-1)

,y, .. ,y

do

):= Si+p;:(x,f(x,u),tl, ..

d{+

1)

,t/

de) ,y, .. , y ) , (4.79)

Denote

fl\:+

1:

n k -PI;' then (if. 74) is rewritten as

=

(4.80a)

i = 1, .. ,n;'+1 Pt(x,tJ, .. ,u

(k-ll

(k)

,y, .. , y )

a

=

i

Clearly (4.80) satisfies (4.65) with

in

If

satisfied

the

above

algorithm

(x, U,y)

around

for

the

n);+1+1, .. ,11+p

replaced by

11k

having PI
e

LX

1

LX

2

... LX

r

(4.114)

h(x) '" 0 ,

then we say that the input v instanta.neously affects the output y. Indeed the proof of Proposicion 4.14 shows that if we differentiate in this case y(t) (r+1)-times with respect to time, along a suitably chosen trajectory

of

the

system,

then

this

(r+l)-th

order

derivative

will

depend

non-trivially on v. One may expect that condition (4.104)

is also sufficient for output

invariance, so thllt if v does not instanta.neously affect y then v does not affect y at all. However for c"'-systems this is, unfortunately, generally not true, as shown by the following example. Example 4.16 Consider the following single-input single-output system on

x = f(x)

+ e(x)v ,

X E

~

ill (4.115)

y '" hex) ,

where the smooth vectorfield e(x) and the smooth function hex) are of the form

e(x) - I, x

~

-1

e(x)

~

0

~

0, x

o

-1

(4.1l6) hex) ~ 0, x ~ 0 h:

a

1

{ h'(x) > 0, 0 < x

~

1

Fig. 4.1. Condition (4.104) is not sufficient for output invnrirmcc.

Furthermore we let f(x) L L

r hex) .,. 0

e f

for all r

~

E

1. Condition (4.104) now amounts to (4.117:

0, which is obviously satisfied by the definition of e and h

139

v

Hence affee t

does not

instantaneously

affect

the output y in the following,

output y.

the

However v

does

indirec t, way. By (4.13) and (l,. 14)

we have for any input function v

h(t,xo'v) - h(t,xo'O)

,

=

f

v(s)

[ ah(ft-s(x»

J

e(x)

ax

ds

x

"

=

1'v(s,O,xu ) (4.118)

where "'tv (s, 0 ,xo ) is the solution at time s of the differential equation X - f(x) + e(x)v. By definition of f the term between brackets on the

right-hand side of (4. US) equals ah(x+t-s) e(x)

(4.119)

ax

By

the

definition

satisfying 1 > x +

of t

-

hand

e

it

follows

that

for

x < a and

t

-

s

s > 0 the expression (4.119) is not equal to zero.

Therefore if we take v(s) in (4,118) equal to 1 then for some Xo and t we

will have h(c,xo,l)

h(C,xo'O), and so v does affect the output y.

¢

0

Notice that in the above example the vectorfield e(x) and the function

h(x) are smooth (C"')

but not analytic.

In fact

for

an analytic system

condition (4.104) does imply output invariance:

then y is

Proposition 4.17 Suppose that the system (4.101) is analytic,

invariant under v if and only if (4.104) is satisfied.

Proof The

"only if"

direction has been proved in Proposition 4.14.

"if" direction is proved as

follows.

Let

be

Xo

the

arbitrary piecewise constant input sequence u(t)

initial state.

(u 1 (t), .. ,Um(t»T

=

The An can

be written as

(4.120)

Consider two time instants s,t satisfying 0

~

s

~

t

~

t1

+ ... +

t

r . Then we

can write

s t

for

- t, - t,

some

+ ... +

tp _ 1

+ (tp-Tp)

('.121)

+ ... + t _ f

integers

1

+ '1

p,l

satisfying

O!:-=. p !:-=. f !:-=. r,

and

some

and

140

In

L

Denoting Z1 - f +

gj (x)ul '

with

jal

in (4.120),

then the solution 'Yo(t.s.x) of

(4.101) for v - 0 and xes} - x. and u(t) defined by (4.120), is given as tp + 1

o .••

0

Z

(4.122)

(x) .

p+l

Let us

denote

the solution of

(4.101)

for

the

above

input

sequence

u(t) = (u1(t),,,,um(t»T and for arbitrary input vet) by "I v (c::,s,x). Completely similar to Proposition 4.1 we have the following relation between the output y(t,xo,u1 •..• um.O) - h("Io(t,O,xa » of (4.101) for v - 0 and the output y(t.XO'u 1

,um,v) - h(lv(t,O,xo »

, ••

of

(4.101)

for

arbitrary

v,

namely (4.123) t

fo

[

8h('Yo(t.s,x»

ax

v(s)

e(x)

J

Ix

- 'Yu(s,o.xo )

ds

Now we will prove that the expression between brackets 8h('Yo (t,s,x»

e(x)

ax is zero for all


t

s

(4.124)
0 and all x.

By (4.121) we have that (4.124)

equals (4.125)

Since (4.101) is assumed to be analytic we can write t

£-1

Z1_1

(we take

t1

0

+ ... +

'"

tr

(4.126)

0

small enough so that (4.126)

and all following

Taylor expansions converge). In the same way we can expand for any i

(4.127) Tp

o •••

0

Zp (x».

Continuing

(4.105),

in

we

way,

and

recalling

immediately

this

see

that

(4.1011)

that

(4.125)

equals

is

equivalent with

zero,

and

hence

the

o

right-hand side of (4.123) is zero.

Remark 4.18 Notice that actually we do not need analyticity of the vector-

e.

field

This

observation

can

be

useful

e.g.

for

the

disturbance

decoupling problem (where e will denote the disturbance vectorfield,

cf.

Chapter 7).

Remark

4.19 In

functional

fact,

4.17

Proposition If we let go

expansion.

f

=

also

follows

and gm'l

=

e

from

the

Fliess

(4.43)

then by

the

Fliess functional expansion of yet) equals m+1

I

y(t)

I

J..~o

1 0 ",

(4.128) .1k~O

, The iterated integral the

indices

expressions 10 , ••• ,

i

k

J

d~ik ... d~io depends on v (= u m+ 1 )

L

g,o

.. L

gi k

equal

is

io ' . .. ,i k

}l(Xo

)

to

zero

are

m

+ 1.

However

\~henever

only if one of by

one

of

(l1.104)

the

indices

the

equals m + 1. Hence any iterated integral in (11.128) involving v

is premultiplied by zero, and therefore yet) does not depend on v.

Conditions ({I.lOL,), which are necessary, sufficient,

and in the analytic case also

conditions for output invariance can be given the

following

geometric interpretation.

Proposition 4.20

Consider

the

nOlllinear

system

(4.101).

Conditions

(4.104) are satisfied if and only if chere exists a distribution D Idth the following properties ( i)

D is invariant for

(E)

e E D,

x

=

L

f(x) + j

~

gj(x)l1 J

(see Definition 3.114),

1

(iii) DC ker dll.

Proof

x'"

(only

if)

Consider

the

unobservability

distribution

ker dO

for

dO

is

m

f(x) ...

I

gj (X)U j

,

y

=

h(x).

By

Proposition

3.LI7(c)

invariant and is contained in ker dll. By (4.l0ll) Lf./I(x)

=

ker

0 for all H E 0,

and hence e E ker dO. (if) Let D satisfy (i), D.

(U),

(i.ii).

Let X be a vectorfield contained in

By (i) and (iii.) {f,XI ED C ker dh. lienee

o.

(4.129)

Since XED c ker dh this yields LX

X

E leer

dL

f

h - 0, and so

h.

(4.130)

In the same way X E ker dL

h, j E~. Continuing in this way we have for gj any XED that X E Iter dO. In particular we may take X equal to e E D. Then e E leer dD, or equivalently Le H(x)

0 for all HE D. Hence (4.104)

=

o

is satisfied.

Remark 4.21 We may assume without loss of generality that the distribution

D in Proposition 4.20 is (iii) (i),

involut::ive.

and (iii). The fact that

(ii)

Indeed if D satisfies

D, the involutive closure of

then

D

D (see

satisfies (ii)

satisfies (iii) because ker dl1 is involutive.

Example 4.22

ED.

(11),

(i),

also satisfies

is trivial, while

To prove that

D

D satisfies

2.27(c»: let X1,X Z ED. then

(i) we use the Jacobi-identity (Proposition

similarly [X1,tf.X2,]]

(2.l43))

replaced by gj' j Em.

The same holds for f

Consider the linear system

x = Ax + Du + ev,

x E IRn.

U

m E 111 •

V

E III ,

(4.131)

ex,

y

Y E IR.

Then ker dO =

n~l

leer cAi -: N, and condition (4.104) amounts to

l .. (j

(4.132)

i .. 0.1 •... ,

and is satisfied if and only if e E N. Since a linear system is trivially

o

analytic this is equivalent with output invariance.

Up

to

sufficient adding

a

now

we

have

condition

constant

not

for

rank

yet

oucput

obtained

a

invariance

assumption

the

constructive of

above

m

C

necessary

systems.

geometric

However

and by

interpretation

together with Frobenius' Theorem yields (compare with Proposition 4.20 and Remark 4.21):

Proposition 4.23

Consider the smooth nonlinear system (4.101). The output

y 1s invariant under v if chere exists an Invo1ut1ve distribution D of constant dimension !oJlth cile properties

143

(i)

D is invariant for

(ii)

e ED,

f(x) +

x -

(iii) Dc ker dh.

In particular i f leer dO ha.s constant dimension then y is invariant under v i f and only If (4.104) holds.

Proof Let D be a distribution as above. By Frobenius' Theorem (Corollary 1

2.43) we can find local coordinates (x ,x'2.) = (x 1 that

D = span

By

{....£....).

ax'

and

(i)

the

(ii)

•••

,xl:'

system

Xl °

clarity state

1>'k(C,sl"",sk'XO ) ~ h'k(r::-r'Sl-r"",sk-r,£T(x o » In particular, if £(x o ) l>'k(C,Sl"",sk'X O )

=

0 we obtain scationary lwrnels:

"" Wk(t-T,Sl-T"",sk-T,xO),

Vr:?: O.

the Xo

k-th as

146

Verify this for the Volterra kernels of a bilinear system with

Xo

-

0

(Example 4.3). 11.4

Denote as in (4.3) initial state at

is

Xo

every j

AO

the k-th order Volterra kernel corresponding to

as t...k(t,s! ..... s\
0 and h(xo ) - O. There exists a coordinate transformation of (5.50) into

an

(autonomous)

observable

linear

system

if

and

only

if

there

exist

p

observability indices PI ~ /12 :! ... ~ Pp == 1 with follolving

tl>'O

I f!i ,.,

if of xo'

conditions are satisfied on a neighborhood

(i)

dim(span(L~dhi(x), j

(ii)

L, dh k (x)

- 0, ... ,f!i-l,

i

E: I) - n,

E

n such that the

=

'tJx E

if,

(5.46)

I-'i -1

Pk

P

- I

, -,

I

k C ij

Lldh i (x),

j -,

k Vx E ii, k E p, for some constants c i j E

(5.52)

~.

Proof First suppose (5.46) and (5.52) hold. As in the proof of Theorem 5.9 we may introduce a coordinate transformation

S

around

Xo

by setting

f!1 -1

hI (x), ... ,

Sex) = (hI (x) ,Lrhl (x), .. "Lr

(5.53) Clearly, because f(x o )

=

0 and h(xo )

=

0, we have S{x o )

is immediate that with respect tb the new coordinates z

~

0, Moreover it

= Sex)

the output

map (5.40) is linear, namely

z

i f!1 +1

y-

(5.54)

Z

!Jl1+f!Z+1

It remains to show that the vectorfield f

is linear with respect to the

z-coordinates. Let us compute Sfif. From (5.53) we have

160

Pp -1 (Zl""

,Zn) =

(hl (X). ... ,Lr

(5.55)

IIp(x)).

Therefore we have all 1

d dt(h 1 (X»)

.

aX (X)X =

8h

ax1 (x)f(x)

(5.56)

Zz,

and similarly (5.57)

Now, using (5.52) we obtain

(5.58) and clearly the right-hand side of (5.58) is a linear combination of the coordinates (Zl""

,zn)' The equations (5.56-58) show the linearity of the

first P1 components of the vector field Snf. In a similar way as above one may proceed to show the linearity of all components of S" f. coordinates S .. f

In the new

takes the form

Pl (

0

",I

o

0

*

*

*0

0

-*

*

1

o

0

0

'1 +

,'r

*'

-- ...._._...*

-I,-

z

Z

0

0

*

(5.59)

0

*.,,"

.-----~.---,~""''',,')~

0

o

o

o

I", *--

4-

....•

\

Using (5.54) and (5.59) it is immediate that the resulting linear system is

observable.

In

fact

(5.59),

(5.54)

form

a

linear

system

in

observability canonical form (without input:s). As far as the necessity the conditions (5.46) and (5.52) concerns, we note

that a

linear

the of

system

which is observable always can be put via a linear change of coordinates

161

into the observability canonical form (5.59) and (5.54). The linear system

o

(5.59) satisfies (5.46) and (5.52), which proves their necessity.

It

is

meeting

emphasized

the

observer

that

requirements

techniques.

observer,

we

first

For

for

the

autonomous

of Theorem instance,

transform

5.13

in

the

order

system

nonlinear

we

can to

into

use

systems

(5.50)

standard

linear

construct

the

a

Luenberger

corresponding

linear

system (5.59,54), which for simplicity will be written as

z - Az

(5.60a)

y = Cz

(5.60b)

and an observer for the state z of (5.60) is designed as the system

z - (A - KC)z + Ky ,

(5.61)

where K is chosen so that A-KG has all its eigenvalues in the open left

half plane. In that case the error e

z satisfies

= Z -

e - (A - KC)e , and thus e (t)

(5.62)

This shows that

converges to zero when t

x :_

S-I(2:)

converges to the state x of the system (5.50). We notice that the above construction of an observer for the linear system (5.60) can be extended to a system which contains nonlinearities depending on the observations. Specifically consider the system z

~

Az + P(y)

(5.63a)

,

y .,. Cz

where

(5.63b)

and

A

C

are

again

as

defined

in

(5,59)

and

(5.54)

and

(PI (y), ... 'P n (y»! is some smooth vectorfield depending on y, this case we replace the observer (5.61) by

P(y)

z

~

(A - KC)z + Ky + P(y)

In

(5.64)

with again K chosen such that A - KC has all its eigenvalues in the open left half plane. shows that

x~

Then the error e ,..

S-I(Z),

with

z

Z

-

z

also satisfies

(5.62),

which

given by (5.64), yields an observer for the

state x of the original nonlinear system. Motivated by this, we define

Problem 5.14

(Coordinate

transformation

of

an

autonomous

system

with

outputs into linearizable error dynamics) Consider the nonlinear system

(5.50) around a point Xo with [(xo) .,. 0 and h(xo )

=

o.

r.,Then does there

162

exist a coordinate transformation z

=

Sex) with S(xo )

=

0 which transforms

(5.50) into the form (5.63)? We will address here Problem 5.14 only for the single-output case, i.e, p

= 1.

The general case when p > 1 is in fact analogous, but needs much

more analysis. Before we can state the solution of Problem 5.14 for p - 1 we need the following result, which reformulates the conditions of Theorem 5.13.

Proposition 5.15 f(x c )

Consider the single output nonlinear system (5.50) with

0 and h(xo ) -

-

O.

There exists a. coordinate transforma.tion of

(5.50) into an observable linear system if and only if tile follm.;ing two conditions hold on a neighborllood V of xo'

(i)

dim(span{dh(x),Lrdh(x), ... ,L~-ldh(x)J) - n. Vx

(ii) the vectorfield g defined on

-r'

L&L~h(x)

I,

satisfies

[g,ad~gl(x) - 0,

E

V,

(5.65)

V via

j

- 0, ... ,n-2,

j

~

'IIx

E

V,

'IIx

E

V,

(5.66)

n-1,

k - 1,3.5 •.. . ,2n-1.

'IIx E

V.

(5.67)

Proof First suppose that (5.65) is satisfied and that the vectorfie1d g that is uniquely defined via (5.66) satisfies (5.67). Then using (5.65) we obtain that dim(span{g(x).ad£g(x) •... ,ad~-lg(x)l) - n. and using

the Jacobi-identity

(see

'IIx E

V,

Proposition 2.27)

(5.68) equation

(5.67)

yields

.e

k

= 0,

[adrg,adrg}(x)

k+i - 0, ... ,2n,

'IIx

E

V.

(5.69)

Applying Theorem 2.36 we can find a coordinate transformation z - S(x) with S(xo ) - 0 such that j

j

S" ( (-1) adf.g

)

=

a , j - 0, ... ,n-l. -8--

(5.70)

Zn-j

It is immediate from (5.66) and (5.70) that (5.71)

Now we compute S"f. For j - 0, ... ,n-2 we have

163

(5.72)

which yields that

~

S.f(z)

[a,(z,) ] [:0: ] +

(5.73)

on (zl)

for smooth functions

ol""jOn'

,

a a, (z, ) --, az,

~

0,

i E

From (5.69) it follows that (5.74)

:!2

and so indeed S"f is a linear vectorfield in the z coordinates.

On the

other hand suppose the system (5.50) is transformed via z = Sex) into the

linear

EyS

tern

z - Az,

(5.75a)

y - Cz,

(5.75b)

Define the n-vector b by j=O,1, ... ,n-2,

(5.76) j

and let g(x) ..

...

(S:lb)

0-1,

(x),

then it is straightforward to verify that this

o

vectorfield satisfies (5.66) and (5.67),

We are now able to solve Problem 5.14 when p = 1. Theorem 5.16 £(xo) -

0

Consider

and

the

h(xo ) - O.

z - Sex), with S{x o )

=

a

single

output

There

exists

nonlinear a

system

coordinate

(5.50)

l"ith

transformation

l"hieh transforms (5.50) into a system of the form

z .,. Az + P(y)'

(5.63a)

y - Cz,

(S.63b)

with (C,A) observable if and only if the follO!-ling tl>'O conditions hold on

a neighborhood (i)

V of

xo.

dirn(5pan{dh(x).Lfdh(x), ... ,L~-ldh(x)J)'" n,

'Ix E

V,

(5.65)

(ii) the vectorfield g defined as in (5.66) satisfies

[g,ad~gl(x) - 0,

k - 1,3,5, ... ,2n-3,

'Ix E

V.

(5.77)

Proof Suppose that the conditions (i) and (ii) are satisfied. As in the proof of Proposition 5.15 we see that the vectorfield g defined in (5.66)

164

helps us to define a coordinate transformation z •

n-1

d1m( span{g(x) ,ad{g(x) • ... ,ad!

Ie + l'

0,

g(x) I)

~

Sex). Namely we have

"Ix E

0,

0.1 .... ,2n-2,

V,

(5.68)

Vx E V,

(5.78)

and so I see Theorem 2.36, we can define the transformation z - S(x). with

S(xo ) = 0 and j

(5.70)

= 0, ... ,n-l.

It is obvious that in the new coordinates

}' =

(5.71)

Zl'

while 0

0

[ a,(z,)

z

z 0 (compare

(5.79)

+

0

an (zl )

5.15),

Proposition

which

a

is

system

of

the

form

(5.63).

Conversely, when a state space transformation z ... Sex) exists which brings

(5.50) into the form (5.63) with (C,A) observable, we have to establish (5.65) and (5.77). That (5.65) is satisfied follows from the fact that the

pair (C,A)

is observable and the fact that the system (5.63)

observable. Namely using the notation /i(z) n - 1

span(dCz, dL/iCz •...• dL/i

Az -I-

is locally

P(y) we have

Cz) - span(C,eA, ... rCA

n-l

(5.BO)

n.

)

Therefore (5.65) holds true. Define the n-vector b via ~

j

0, I, ... ,n-2,

(5.76) n-l,

j

and let g be the vectorfield defined by

g(x) = Clearly

using

requirements

(s.Bl)

(x) •

(5.76)

(5.66)

we

and

see a

that

direct

this

veccorfield

computation

shows

satisfies

(5.77)

that

satisfied.

Remark 5.17

the is 0

For a nonlinear system satisfying the conditions of Theorem

5.16 we obtain a system described by the equations (5.71) and (5.79). EVen in case

that

the functions

description differs

from

at, ... ,on

in

the one given

(5.79)

are

linear

in Theorem 5.13,

in

see

Zl

this

equations

165

(5.S1,)

and

(5.59).

This

is

the

difference

between

the

obseTvability

canonical form (5.59,54) and the observer canonical form (5.79,71), which in the linear case are isomorphic,

but not necessarily in the nonlinear

case. Remark 5.18

As

explained before,

in

the

observer

design

for

a

system

satisfying the conditions of Theorem 5.13 or Theorem 5.16 it is essential

to introduce output injection of the

form Ky or Ky + P(y),

see

(5.61)

respectively (5.64). For linear systems the concepts of state feedback and output injection are dual. Without formalizing here the nonlinear concept

of state

feedback and output

injection,

we remark that

in general

for

nonlinear systems such a duality is not immediate.

5.2 Static and Dynamic Feedback

So far we have discussed various versions of the question when a nonlinear system

is

(almost)

equivalent under linear

system.

a

The

change

of

state

state

space

space

coordinates

transformation

is

to

an

only

an

intermediate step in the controller and observer design. As will be clear, most nonlinear systems are not equivalent via a state space transformation to a linear one and thus the forementioned techniques will not be of much help to us.

In the next chapters we will discuss various other I.;ays of

changing nonlinear control systems. The cornerstone in this is the notion of feedback.

We will

discuss

in

this

section

some

different

types

of

feedback. Definition 5.19

A strict static state feedback for the nonlinear dynamics

(5.1) is defined as a map (5.B2)

u=a(x),

where u

Strict

=

(u 1 , ••• , urn) T and D:: N -. [p.rn is a smooth function.

static

state

feedback,

or

for

short,

when no

strict feedback, can be represented as follows:

u

x

f(x) + g(x)u

Fig. 5.1. Strict static s!-j x U = a(z,x) + fi(z,x)v

Fig. 5.3. Dyn,-3

and thus

the

(6.34)

,ZI\:-2 ,ZIl:-1 ,ZK

introduced

new

coordinates

1

~ S(x , .. ,x/{"'), defined inductively as

"z/{",)

k=/{",-l, .. ,l,

Since £(x o )

0 it follows that 5(0)

=

=

O.

In the new coordinates z

Sex)

the vectorfield £ is given as

fez)

k

= -r;~

(6.46)

11:-1, .. ,1,

2

Z

-/(",- 1 Z

and thus is a linear vectorfield,

except for the first part £l(Z),

£1 denotes some nonlinear function of z. However,

DI

=

a az 1

span{-)

o(S(x»

there

... f3(S(x»

£1(Z)

gj (z)

+

,L.,8

a

regular

feedback

u

=

o(z)

+ f3(z)v

v such that

1 (Z)(l1 (z)

L g, (x) , ·1

exists

where

since spanlgl,·· ,gm)

f3 i

j

~

(x)

0,

~

(6.47a)

{ ej

0,

,

j

~

1, .. ,ro l

,

j

=

ro l +1, ..

,Ill.

(6.l17b)

186

Clearly the resulting closed loop system in the coordinates z is linear,

o

and moreover it is controllable.

It is easily seen that if f(x u )

Remark 6.6

~

0, then the same conditions

as above guarantee that the nonlinear system (6.5) can be transformed into an

"almost

linear"

z ..

system

Az l' Bv

+ f(xu )'

Moreover

if

e

apan{gl (x o ), .. ,gm (xo )} then by an additional feedback the constant drift term f(x o ) can be also removed.

f(x o )

Finally we note

that we

can reformulate Theorem

6.3

in

the

following

slightly different way. Corollary 6.7

Tile

nonlinear

distributions D1

system

(6.5)

with

f(x a ) - 0

is

feedba.ck

Xo

to a controllable linear syscem if and only if the

••• • Dn

defined in (6.13) are all involutive and constant

linearizable around

dimensional in a neighborhood of xu. and Dn (xo ) "" T 1-1, Xu The "if" part follows from the proof of the /tif" part: of Theorem

Proof

6.3 and the observation therein that Co

=

from

6.3

the

"only

if"

part

of

Theorem

Dn' The "only if"

plus

the

fact

part follows tha t

for

controllable linear system Dn _ In,

a

o

Let us study the reSUlting feedback linearized system defined by (6.46)

(6.47)

and Gi

=

Pi I-Pi'

in i

Z ... Az

given as

some

more

2,., ,1>.,

detail. Po =

m.

Define Then

the

for

simplicity

resulting

of

linear

notation system

is

+ Bv where

oP1Xn a

A -

I

o

B

o

By a

o

permutation of the coordinates

following form,

known as

controllability indices 11:.1 : -

Thus

leI

+

z the system can be put into the

the BruIlovsky norm"l form.

~l'

i E

~I

+ .. , +

I':m

-

n.

Indeed,

define

the

as

number of integers in the set {Pl'" K'.2,

(6.48)

Notice also

, Pie)

that

11:1

which are 2: i.

(6.49)

=

simple

K..

Then by a

187

permutation (6.48) takes the form (assume for simplicity that nil

~

~

Pl

m)

(6.50a)

with 0 1

(6.50b) [ The

0,,·

proof of Theorem 6.3

("if"-part)

yields

some

useful

information

about the structure of any feedback linearizable system. Indeed i t follows

that a locally strongly accessible system (6.5)

is feedback linearizable

around Xo

if and only if there exists a coordinate system x

around Xo

such that span (&1 (x), .. ,gm (x»)

=

=

(Xl, .. ,xx,)

l_a_ 1 , and f satisfies

span

ax! (6.34) as well as (6,39). Therefore a locally strongly accessible system is feedback linearizable if and only if i t has the following flow diagram

structure,

implied

equations

by

span (g1 (x), .. ,gm (x») - span

(6.34)

and

the

condition

{--"-I ,'

ax

(6.51)

and moreover central f'l, .. ,fl':.,

(6.39)

string of

is satisfied. integrators,

Notice

that

in

interlaced with

(6.51)

the

u enters

nonlinear

the

mappings

only at the beginning and that there are only "backward feedback

loops". The system is brought into linear form by successively redefining XK.-l, . . ,Xl,

thereby successively eliminating the feedback loops in (6.51)

from the right, the

feedback

and transforming

loops

influencing

if., .. ,f'l f1

and g

into linear mappings. are

removed

by

Finally

static

state

feedback.

Example 6.8

(see also

Example 5,20)

Consider

the

two-link rigid robot

manipulator from Example 1,1 written in Euler-Lagrange form as

H(O)~ +

Ceo,i)

+

kee)

- u.

In this case a linearizing feedback is immediate, namely

(6.52)

188

c(o,b) + k(O) + H(O)v,

u -

2

v E i

with

H(O)~

the

new

input.

{6.S3} Substitution

of

(6.53)

in

(6.52)

yields

H(8)v. or equivalently, since det HCO) ~ 0,

'd = v, which corresponds to the linear controllable dynamics (6.55) Notice,

however,

SlxS

1

that

(6.55)

is

nor.;

;]

global

linear

system

since 0

;e. 1i(2.

Exnmple 6.9

Consider the controlled Euler equations from Example 1.2

(6.56) In case the vectors hl'

bz ' b3

ilre independent the system is trivially

feedback linearizable; simply set (6.57)

with v If

E

rr~J the new control vector. and solve for u ~ (u 1 , u2. ,u 3 ) (b1b2h3)

ranle

assume

without

distribution Dl

2 we

=

loss

of

effectively have

generality

as defined in (6.13)

that

which is clearly involutive

computation

of

distribution Dz

controls,

b 3 = O.

In

•

and

this

we

case

may

the

is given by the flat distribution

span{b I ,h z I

the

two

is

and of constant dimension. more

involved,

and

we

The

restrict

ourselves to the simplified situation (see Example 3.24, (3.48»

(6.S8)

with

J

=

(a 1 -a z ) a; 1,

diag(8 1 ,a 2 ,a 3 01

)

and

Al

=

(8 2 -a 3 )a;1,

Az = (8 3 -a 1 )a;1,

Aa '"

= a~ I, O'z ~ a; 1. In Example 3.24 it has been computed that (6.59)

and so the distribution Dz equals

(6.60)

189

It follows Hence if

that Dz (w)

8 1

... 8 2

everywhere,

if and only if A3 .... 0 and w 1

T [RJ

=

w

...

0 or

Wz

...

O.

then by Theorem 6.2 the system is feedback linearizable

except

for

the

w~

line

w~

=

linearization is performed as follows. Set

=

D.

23:=

Outside

w3

this

line

the

and

,

(6.61)

If w~ ...

a

then we set

£:1

w 1 and if w~ ... 0 then we set

=

w2

21

•

In the

first case we obtain in the new coordinates the equations

21

=

Al

2223

A3

21

+

0 1 til

A, 22

AJ A Z Z ;Z3

z,

z,

since

(6.61)

by

(6.62)

+

n, +

z:

A,

we

, Z2 ZJ

have

02 A3 21 tl z

z,

w,

A3

linearized by

can be

+

z

wl

setting

A3

""2

z, u,

A3

(6.62)

, two

first

The

z1

the

equations

of

right-hand side of the first

equation equal to vi' and the right-hand side of the second equation equal to v 2

,

with v

solved for For

til

the

=

and

(V 1 ,V2 )

the new input vector.

in all points for which

Uz

one-input

case

we

only

21

consider

Since

A3

WI

O.

=

che

,.
for Clny vf'ctorfi.elds X,Y and function ([J

on H.

h(~

Lf'!t "

such

~

thAt k+1

0-2.

() and

Then certi'd nly

thus

o

(fi.B9)

Since k+1

n-2 thH second term on the right-hand side is zero

alp-o

O. Hence we have proved (b) for j

)+1. Let It he

511Ch

so that

1. Now suppose that

< 11-2. Then we prove that (b) also holds for

holds for a certain j

(h)

I

thilt k+j+l

~

0-2. Then by the induction assumption (6.90)

Hf'nce

+

n-l-r.



+

(6.94)

197

Hence

n-k



= -

O.

=

such

independent it

that

and

then follows

i E P , -/C

and

the

functions

annihilate

DI\._z'

As

"'" 0,

moreover

the

func tions

f{Ji'

i E EK:' are independent. Continuing in this way, we

end up with n independent functions f{Ji ' Lr!Pi •

1

0,

Xz

linearized

> 0, 1> 0, in the sense of Remark 6.6, i.e. the

system will

Linearizing in the point (x~ ,x~ ,1

contain 0 )

an

extra

constant

drift

term.

this extra drift term can be removed

by first subtracting from the controls ul,u Z the constant terms u~, resp.

u~.

First we observe

that the distribution D,

constant dimensional and involutive since

~

span{[

=~:

], [ ~ ]}

is

202

(6.115)

Furthermore J.ll (S)X 1

J.lZ~S'I)X2

j, [)

[ ", (S ,.1)X,

[

[

-px} I

KI+I

]

(6.116)

'

-px]

and we conclude that dim Dz(x,I) = 3 since

(6.117)

Therefore.

by Theorem 6.3,

the system is feedback linearizable in any

point in the sense of Remark 6.6. The feedback linearization is performed as follows. First we have to find a function rp, with drp -

~

yi!

0, such that

0, or equivalently (6.11Sa) (6.118b)

From (6.11Sb) it follows that rp only depends on the cell-densities

Xl

and

xz' A possible solution to (6.118a) is then (6.119) By a simple calculation (6.120) The remaining new coordinate zJ only has to satisfy the requirement that are independent; we simply take

z1 ,z2,23

Z3:- Xl'

In these new coordinates

the system is described as Zl

=

22 ,

+

2Z

~ L'Z1

23

... Lr 2 3 + u1Lg

U 1 Ls 122

+ u 2 L& Zz z. -I-

(6.121)

UZLSzZ3 .

It can be immediately checked that the matrix A

,2,

L"'2 Zz

Lg1Z J

LS223

_ [ L,

1'

(6.122)

203

is non-singular, so that the linearizing feedback is given as

-A

-,

+ A

(everything depending on

-, [ v, 1

Xl ,Xz

,I). We note that the choice of coordinates

is by no means unique. Certainly we CQuid also talee for example

(6.123)

,

",

rp

is not unique, and instead of (6.119)

Z3

arctan

=

o

=: 2 1 ,

(X 2 /X 1 )

For completeness we will finally give the extension of Corollaries 6.17 aod 6.1B to the case of a general single-input system x - f(x,u),

Corollary 6.21

(6.124)

U Ell?

Consider the distributions VI ,D;.>."" ,D n + 1 for the extended

system (ct. Definition 6.11) of (6.126): x

~

f(x,u),

U

=

1>',

(6.125)

with state space coordinates (x,u) E IR

n

and control IV'.

+1

Theil

(6.124)

is

feedback linearizable around (xo ,u o ) to a controllable linear syscem i f and only if dim Dn + 1 (xO 'u O )

=

(6.126a)

n+1,

(6.12Gb)

Dn is invo1utive around (xo ,u o )'

The feedback linearization is performed by constructing a function rp(x) l-lith rp{x o )

=

0,

such

that ""

° for

all

vectorfie1ds x E Dn ,

and

defining the state space transformation z "" Sex) as

i

E

(6.127)

!::'

and solving locally u as a function

U

""

o:{x, v) from

v ... L~ !p(x,u).

(6.128)

The resulting linear system is again given by (6.105).

Remark 6.22

Using Lemma 6.15 it is clear that the functions L~-l!p,

do not depend on u, wh'l 1 e As

is well-known,

guarantees

the

aau Lr!p(xo,u " o)

~

i E~,

O.

the controllability of a linear system x "" A.>: + Bu

existence

of

a

linear

feedback

u

=

Kx

such

that

the

204

characteristic polynomial of the 'closed loop matrix A + BK is equal to any desired monic polynomial of the same degree. It is therefore clear that if a

nonlinear

system the

z-

system

Az

feedback

nonlinear

is

feedback

linearizable

to

a

controllable

linear

+ Bv then by an extra linear feedback u ~ Kz the poles of linearized

system

can

system can be

not

only

be

arbitrarily

transformed

to

assigned.

Hence

a

system

linear

the in

(6.50», bue also to a linear system with an

nrunovsky normal form (cf.

(of degree n).

arbitrary characteristic polynomial

Let us study this

in

morc detail for a single-input affine system (6.BS), which we assume to be feedback

linearizable

to

(6.105).

As

explained

in

Corollary

transformation to (6.105) is performed by taking coordinates in (6.106a) feedback

I

(Zl'"

thus defining a linear transformation z = S(x),

u - o(x} + {J(x)v

as

given

in

(6.l06b).

It

the

,zn) as

and by

clear

is

6.18

that

the the

modified feedback -(L8L~-11P(x»-1(L~1P + r u -

u

1

L~-lrp + .. + I'otp)(x) + (LgL~-ltp(x»-lv (6.129)

results in the linear system

(6.130)

with

characteristic

polynomial

Further-

more it is clear that we can always choose tp in such a way that (cL Lemma

6.15) (-1)

n-1

n-1

""

(6.131)

1.

Then (6.129) for v = 0 reduces to the static state feedback (6.132)

u

Remark 6.23

Assume that the nonlinear system (6.85) is already in linear

form x - Ax + bu, and let us see what the feedback (6.132) amounts to in this case.

First, by (6.97) and (6.131)

the function 'P will be a linear

function rp{x) = k:..:, with the (lxll)-vector Ie satisfying

o 1).

(6.133)

Secondly,

the new system of coordinates z

given

21

X

as

~ fcAj-1x.

i E

11.

Said

(Zll",zn)

otherwise,

A., ... bu the linear basis cransformacion z -

as

if

Sx given as

in (6.106a) is We

apply

co

205

kAk

s-

[

]

IeA n - l

together wi th the feedback (cf. (6.20) )

(6.135)

where

then

transformed into

(6.130).

Actually,

system

the

x

~

+ bu

A."{

-kr(A)x,

the feedback expression u

assigning the characteristic polynomial rCA) to the closed loop matrix, is well-lmown in linear systems theory (Ackermann'

formula). Furthermore, it

5

can be seen that the i-th column of the inverse matrix 5n-1

A

n-1-1

b + Pn-l A

with p(..\)

1

is given as

(6.136)

b +

the characteristic polynomial of Ii,

and it is well-known that

these columns form a basis in which the system is in conn'oller canon,ical form 0 1

(6.137)

2

[

-Po

Notes and References

The problem of feedback linearization was first posed and treated in [Brl for

the

restricted

class

of

feedback

transfonn,ltions

Previous \·.rork in this area can be found e.g. sufficient conditions

for

[vdS) , see

to

the

also

general nonlinear case [Su].

For additional

[HSH1],

=

o(x) + v.

obtained in

[JR],

see also iSu]. The

(Theorem 6.12)

results we

u

The necessary and

feedback linearization were

and in a slightly more elaborate way in [HS]. extension

in [KoJ.

refer

can be to

found

[ReI],

in

{l'lBE],

{HSH] and the survey [Gl]. The problem of partially linearizing the system was addressed in {fKJ,

[lOR).

[HBEl. while the L:ll:gest feedback subsystem

was identified in {Ha2]. see also state

feedback

has

(Rel],

been dealt with

[Re3].

Linearization by dynamic

[GUIl,2),

[lsJ

(see also

{RuJ,

(CIRT]),

{BoJ,

of

[Di-lE]. Feedback linearization of the input-output map of the system [IR],

[I-lSN2],

problem

rCTI],

in

e.g.

The

feedb,1ck

treated e.g.

studied in

[II>1LJ.

global

was

linearization was

in

while

[Re2),

feedback

206

linearization of systems with outputs was studied in [CIRT]. A different

approach to Hnearization by feedback was taken in e.g. [RC], Finally,

the

problem

of

approximate

feedback

[C~1RJ,

linearization

has

[WR]. been

addressed in [KrJ. The non-genericity of feedback linearizable nonlinear syscems,

for n (= dim H)

II!

(= dim U)

not

(00

small.

has been shown in

(Tc).

[go J [8r] [CIRT)

[GIl

[ClJvll J

W. M. Boothby, "Some comments on global linearization of nonlinear systems". 5yst. Control Lett., 4, pp. 143-147, 1984. R.W. Brockett, "Feedback invariants for nonlinear systems", Proc. VIIth TrAC World Congress, Helsinki, pp. 1115-1120, 1978. O. Cheng, A. Isidori, W. Respondek, T.J. Tarn. "Exact linearization of nonlinear systems \"ith outputs", Hath. Systems Theory, 21, pp. 63-83. 1988. D. Glaude, "Everything you always wanced to know about linearization", i.n Algebraic and Geometric Methods in Nonlinear Control Theory (eds. H. Fliess, H. Haze,,,inkel), Reidel, Dordrecht, pp. 181-226. 1986. B. Charlet. J. Levine, R. Harino, "Two sufficient condi tlons for dynamic feedback linearization of nonlinear systems", in Analysis and Optimization of Systems (eds. A. Bensoussan. J.1. Lions), Leet. Notes Gontr. Inf. Sci., Ill, Springer, Berlin, pp. 181-192,

1988.

leD-12l [GHRJ

[CTI)

[DUE]

[HSJ

[HSH1]

B. Chariet,

J. Levine, R. Harino, "On dynamic feedback linearizat:ion", Systems Goncrol Lett:. , 13. pp. 143-151, 1989. C. Champetier. P. Houyon, C. Reboulet, "Pseudo-linearization of multi-input nonlinear syscems" , Proe. 23rd IEEE Conf. on Decision and Control, Las Vegas, pp. 96-97, 1984. D. Cheng, T.J. Tarn, A. lsidori, "Global feedback linearization of nonlinear systems", Proc. 2Jrd IEEE Gonf. on Decision and Control, Las Vegas, pp. 74-83, 1984. W. Dayavlansa, {.J.t-!. Boothby, D.L. Elliott, "Global state and feedback equivalence of nonlinear systems", Systems Control Lett., 6, pp. 229-234, 1985. L.R. Hunt, R. Su, "Linear equivalents of nonlinear time-varying systems", Proc. Int. Symposium on Hath. Theory of Networks and Systems, Santa Monica, pp. 119-123. 1981. L.R. Hunt, R. Suo G. Heyer, "Design for multi-input nonlinear systems", in Differential Geometric Control Theory (eds. R.W. Brockett, R.S. Hillman, H.J. Sussmann). Birkhiiuser, Boston,

pp. 26B-298, 1983. rnSH2}

1.R. llunt, R. Su, G. I'leyer, "Global transformations of nonlinear systems". IEEE Trans. Automat. Contr. AC-28, PP 24-31, 19B3. A. Isidori, A.J. Kraner, "On feedbi1ck equivalence of nonlinear systems", Systems Control Lett., 2. pp. 118-121, 1982. A. Isidori, C. Hoog, A. de Luca, "A sufficient condition for full linearizability via dynamic state-feedback", 25th IEEE Gonf. Decision and Control, Athens, pp. 203-208, 1986. A. raidori, A. Ruberti, "On the synthesis of linear input-output responses for nonlinear systems", Systems Control Lett., 4, pp. 17-22, 1984. A. Isidori. "The matching of a prescribed linear input-output behavior in a nonlinear sys tern", IEEE Trans. Automat. Contr.. AC-30, pp. 258-265. 1985. I

[lKJ [lHLJ

[IR]

llsl}

207

{I'2J

A. Isidori, Nonlinear Control Systems: An Introduction, Notes Contr. Inf. Sci., 72, Springer, Berlin, 1985.

{KIRJ

A,J. Kraner, A. lsidori, W. Respondek, "Partial and robust linearization by feedback", Proe. 22nd IEEE Conf. Decision and Control,

{KoJ

W. Korobov, "Controllability, stability of some nonlinear systems", Differencialnyje Uravnienje, 9, pp. {166-469, 1973. A.J. Krener, "Approximate linearization by state feedback and coordinate change", Systems Control Lett., 5, pp. 181-185, 1984. R. Harino, "Stabilization and feedback equivalence to linear coupled oscillators", Int. J. Control, 39, pp. 487-496, 198',. R. Barino, "On the largest feedback linearizable subsystem", Systems Control Lett., 6, pp. %5-351, 1986. R. Marino, W.H. Boothby, D.L. Elliott, "Geometric properties of linearizable control systems", Math. Systems 18, Theory, pp. 97-123, 1985. C. Reboulet, C. Champetier, "A new method for linearization nonlinear sys tems: the pseudo-lineariza tion", Int. J. Control, liD, pp. 631-638, 1981l. W. RespondeJt, "Geometric methods in linearization of control systems", in Mathematical Control Theory (eds. Cz. Olech, B. Jalcubczyk, J. Zabczyk), Banach Center Publications, Polish Scientific Publishers, Warsaw, pp. 453-467, 1985. 1,]. Respondek, "Global aspects of linearization, equivalence to polynomial forms and decomposition of nonlinear systems", in Algebraic and Geometric Methods in Nonlinear Control Theory (eds. M. Fliess, H. Hazewinkel), Reidel, Dordrecht, pp. 257-2811, 1986. W. Respondek, "Partial linearizations, decompositions and fibre linear systems", in Theory and Applications of Nonlinear Control Systems (cds. C.I. Byrnes, A. Lindquist), North-Holland, Amsterdam, pp. 137-1511, 1986. W.J. Rugh, "An input-output characterization for linearization by feedback", Systems Control Lett., 4, pp. 227-229, 198f,. R. Su, "On the linear equivalents of nonlinear systems", Systems Control Lett., 2, pp. 48-52, 1982. A.J. van der Schaft, "Linearization and input-output decoupling for general nonlinear systems", Systems Control Lett., 5, pp. 27-33, 1984. K. Tchon, "On some applications of transversality to system theory", Systems Control Lett., 4, pp. 1119-156, 1984. J. Wang, W.J. Rugh, "Feedback linearization families for nonlinear systems, IEEE Trans. Automat. Contr., AC-32, pp. 935-940, 1987. H. Zribi, J. Chiasson, "Exact linearization control of a PH stepper motor" , Proc. Ameri.can Control Conference, 1989, Pittsburgh, 1989.

Leet.

San Antonio, pp. 126-130, 1983.

{KrJ {MalJ {Ma2)

{MBE)

{RG) {R")

{Re2)

{R,3J

{Ru) {Su)

IvdS]

{Te) {WR) {ZG)

Exercises

6.1

(see also Remark 6.2)

f(x o ) x

=

=

0.

Denote

A'( -I- Bu, with A

(a)

its =

Consider the nonlinear system (6.5) with linearization

af

8x(x O ) and B

=

around

(gl(XO)~

system is feedback linearizable around x(). system

can

be

transformations)

also

transformed

to the

(using

linear systelll

Z

. !gm

(xo»)·

u

=

a

space

Az -I- Bv,

by

Suppose the

Show that around Xo

state 0'

and

Xo

and

the

feedback

with A and B as

208

above.

(In applications this may be a more sensible thing to do than

to transform the system into Brunovsky normal form (6.50).) Consider a

(b)

linearizable

x = f(x,u),

system

(xo ,uo)'

around

that

6.2

af

au (-"0

I

Consider

system

cnn

feedback

be

also

af

with A"" a./-"o,u o ),

lIo) .

the

single-input

2n-dimensional manifold fl,

nonlinear

system

f(x) ... g(x)u

x

on

a

with f(x o ) - 0 and satisfying the strong

accessibility rank condition in xo' can be

is

which

0,

the

z - Az + Bv,

transformed into the linear system B ~

=

f(x o ,uo)

Show

transformed using state

Show that

space

transformations into a system of

the system around Xo

transformations

and

feedback

coupled linear oscillators with

11

unit masses

Z

l ].. [n

~

0

I

nXn

"XU

v

-K

0

/;:1

leI:!

0

len

kz

kZ3

0

1c 2J

k3

.

nXn

\1here

K

0

0 len·

0

1 •n

/ell

len - 1. n

if and only if the system is feedback linearizable around xa ([Mal J).

6.3

Prove

that

the

D.k

distributions

defined

in

(6.19)

are

feedback

invariant.

6.4

(IZC])

Consider

the

following

nonlinear

system

(a

model

of

a

permanent magnet stepper motor) Xl

= -K 1 x 1 + KZ x 3

x2

-K 1 -"2

x J = -K 3 x 1 (Here

XI'X;!

+

KZxJ

sin(K!ix~)

sin(K5.'i:~)

u1

+ K3 x Z cos(Ksx,) -

denote currents,

motor position, J

+

cos(K5x~)'" u z

xJ

denotes

is the Totor inertia I

K~X3

-I- K6sin(4K5x~)

the rotor speed I and

1'1.

x4

- rr./J

is the

is the load torque,

which is assumed to be measurable.) (n)

Verify the condit:ions for feedback linearizability of the system

0, and

(in the sense of Remark 6.6) in the point Xl compute the controllability indi CBS. (b)

Show

that

the

coordinate

transformation

linearizing transformation is given as Zl 22

~

x 4 /K."J x31 KJ

involved

in

the

209

Z3

=

-Xl sinCK"x") + X2 cos(K 5 x ,,) - K4X3/KJ (KrJK J )sin(L!K 5 x,,) -

Z"

=

+

cos(K5x~)

Xl

f

L

/(JK 3

·f·

)

sin(K 5 x,.)

X2

and compute the corresponding lillearizing feedback u

6.5

=

+ {J(x)v.

u(;;:)

Consider the following feedback lillearizable system (motivated by the system (6.112) considered in EXalliple 6. LO)

> 0, x 2 > 0, x J > 0.

wi th

Xl

(8)

Show

for

this

the

is

coordinate

transformation system.

Sbow

involved

that

the

in

a

linearizing

resulcing

tr,msforJUation

closed-luop

system

is

a

global linear system on Ip:l

also part of a linearizing trausfonnation.

ShOI-1,

however,

that

the

resulting closed-loop system is not a global linear syst(.lm.

6.6

(see Remark 6.23) Show, i-th column of

the

using the Cayley-Hamilton theorem,

inverse

of

the llIatrix S

defined

in

that the

(6.13L!)

is

given as in (6.136). Furthermore, show that the columns form a basis in which the system is in the fonn (6.137).

6.7

[Br] Consider the single-input system (6.H5). Show that tile system is feedback linearizable around Xo to a controllable linear system using the

restricted class of feedbacks

II

u(x)

=

-I-

v

(i.e.

{3(x)

=

1),

if

and only i f

6.8

(i)

dim D,(x o ) - n

(ii)

[adig,ad~gJ (x)

E

Dk

(x),

for every

i :s j :s k and k E Jl-l,

Consider the nonlinear system

x2

=

Xl X 2

e>'l u1

X3

-1- x3

about the equilibrium Xl ~ 0, '"'z

=

1, .\:]

=

0, x"

(a)

Verify the conditions for feedback lineal'iz..J.bilicy.

(b)

Compute

the

linearization using

that the syscem is in "decouplcd form".

6.9

° :s

Consider the Hamiltonian cOlltIol SYbtem

all

ql

=

api (q,p)

Pi

=

-

aH

aq;(q,p)

i E II

+

Ui

Corollary

6.18,

and

t.he

fdct

210

H(q,P) - !pTG(q)p + V(q)

where G( q)

for

2

some

positive

definite

matrix

and dV( qo) - O. Check feedback linearizability about the point

I

(qo ,0).

6.10 Consider the nonlinear system (6.5) with [(xI]) - 0, the strong accessibility rank condition,

and satisfying

together with its extended

system III

X

-

Lgj (x)u

f(x) + j

u

j

,

U

l

U ... I...

(n)

Prove,

as a direct consequence of Theorem 6.12,

that (6.5)

is

feedback 1inearizab1e around Xo if and only if the extended system is feedback. 1inearizable around (xo ,0). (b)

[eLMl] Cons ider the sy stem

Show that this system is not feedback linearizab1e around O. Consider t:he part.ial extended system

with state

(Xl IX ... 'Xl 'X 4 IU1) and inputs (1.1'1 ,u z ). Show that this system is feedback linearizable, and compute t:he linearizing transformation.

7

Controlled Invariant Distribution and the Disturbance Decoupling Problem

In this chapter, Section 7.1, we will introduce and discuss the concept of

controlled

invariance

for

nonlinear

systems.

Controlled

invariant

distributions play a crucial role in various synthesis problems like for instance

the

decoupling

disturbance

problem.

decoupling

A detailed

problem

account

of

the

and

the

input-output

disturbance

decoupling

problem together with some worked examples will be given in Section 7.2.

Later, in Chapter 9, we will exploit controlled invariant distributions in the input-output decoupling problem.

7.1 Controlled Invariant Distributions

Consider the smooth nonlinear control system

x ~ f(x)

+

L gi (x)u ,.,

(7.1)

1

where x

=

f,gl""

,grn are smooth vectorfields. Recall,

,X n )

(Xl""

are local coordinates for a smooth manifold M and see Definition 3.31,

that a

smooth distribution D is called invariant for the system (7.1) if

[f,D] c D,

(7.2a)

(gl,D] cD, i

(7.2b)

E III.

Such invariant distributions playa central role in the output invariance of a nonlinear system, invariant

cr.

distribution for

Section 4.3. the

system

We generalize (7.l)

the notion of an

by allowing

for

a

regular

static state feedbaclc, i.e. u - o(x) +

where a: N

->

IR

rn

p(x)v and

p:

(7.3) N ....

for all:;: in N, and where v

mrnXrn =

are smooth mappings with P(x) nonsingular

(vI""

,vrn) denotes the new inputs. Applying

(7,3) to (7.1) yields the feedback modified system m

X

where

=

"lex)

-I-

L gi (x)vi

(7.4)

212 m

f(x) '" f(x) +

L gl (X)lli (x) ,

(7.5a)

1=1 m

L gj (x)f1 j

1

(X) ,

i E

m.

(7.Sb)

j=l

We now define Definition 7,1

A

smooch

discribution

D

011

N

is

controlled

called

invariant for che dynamics (7.1) if tl1ere exists a regular scatie state feedback (7.3) such that D is invarianr:: for the feedback modified system (7.4), i.e.

[f,D] c D,

(7.6a) i E m.

(7.6b)

As we will see in the next section, this generalization of an invariant distribution will be instrumental in the solution of various synthesis problems. At this point we observe that it may be difficult to check if a given distribution D is controlled invariant,

because this requires to

test if there does exist some feedback (7.3) which makes D invariant. Before establishing convenient criteria on the distribution D and the original dynamics (7.1) which guarantee that D is controlled invariant:, we will briefly discuss controlled invariance for a linear system. Example 7.2

Consider the linear system

x = A;: ... Bu.

with x

E

!FIn, u

E

(7 . 7)

tRill and A and B matrices of appropriate size.

with Section 3.3,

a subspace 'IT

sometimes

(A,B)-invariant)

u

called

Fx + Cv,

Icl

#

C [Rn

In analogy

is called controlled invariant (or if

there

exists

a

linear

feedback

0, which makes 'IT invariant, thus

(A+BF)V c V.

(7.S)

A standard simple result from geometric linear system t:heory states that such a feedback matrix F exists if and only if A1! C 'If

+ lm B.

(7.9)

As in Section 3.3 We can put this in a more differential geometric setting

213

by identifying the subspace Let {vi"" by

the

V

with its corresponding flat distribution

DV'

form a basis for V, then DV is the distribution generated

,vr }

constant

vectorfields

,vr .

v11 . . .

The

condition

then

(7.8)

translates into [(A+BF)x, v.2 1 E DV(x) ,

which

is

the

"linear"

(7.10)

counterpart

of

(7 .6a).

Denoting

(BG)i

as

constant vectorfield formed by the i-th column of the matrix BG, i E

the

~,

we

also obtain that

.e which yields

the

E :::. i

counterpart of

E

!!!. x

(7. 6b).

"

(7.11)

E !R ,

Since

the

condition

(7.11)

is

automatically satisfied for a linear system, we obtain as a necessary and

sufficient condition for the controlled invariance of DV that, see (7.9),

"

(7.12)

iEE:,xE!R,

o

where DIm 8 is the flat distribution corresponding to 1m B.

We next turn our attention to the question under which conditions a smooth distribution V is controlled invariant for (7 .1).

First we

identify

a

seC

of necessary

the nonlinear system

conditions

on D and

the

vectorfields f,gl""

,gm that should hold when D is controlled invariant

under

(7.3).

the

feedback

Because

gi (x) -

I gj

(x)f1;:(x) ,

i

E~,

(see

7.Sb), we obtain from (7.6b) that for any vectorfield XED

~

[g, (x) ,X(x) [

-j-'rgj

[

Iii (x)P;;

j-'

j

(x)Lxf1;: (x) E Vex)

+

I L.

(x) ,X(x) J

G(x) ,

j-' i

j

(x) ,X(x)

JP;; (x)

-

(7.13a)

E ~,

where G(x) is the distribution generated by the input vectorfields: (7.14)

Similarly (7.6a) yields, using (7.l3a) m

[f(x) ,X(x) J -

(l(x) ,X(x) 1

[f(x) +

- I

I

m

g, (x)., (x) ,X(x) J - [

I

g, (x)., (x) ,X(x) J -

m

(gi (x) ,X (x) jUi (x) +

I

gl (x)LxU 1 (x) E Vex) + G(x). (7.13b)

21Ll

Summarizing I

we have obtained the following necessary conditions for

the controlled invariance of the distribution D

(7.15a)

(f,D) c D + G. [gi.D] cD + G, Assuming

some

(7.1Sb)

i E m.

regularity

conditions

we

will

see

that

the

conditions

(7.15a,b) are also sufficient for local controlled invariance.

A smooth distribution D on N is called locally controlled

Definition 7.3

invariant for the dynamics (7.1) if for each poinc Xo E H there exists a

neighborhood V of V sucll

thae

Xo

and a regular scatic state feedback (7.3) defined on

the feedback modified dY1lClIIJics

(7.4)

defined on V satisfy

(7.68,b) on V.

Remark 7.4

Notice that a locally controlled invariant distribution is in

general not controlled invariant. The point is that the locally defined feedbacks

of

Defini tion

7.3

need

not:

patch

together

into

a

globally

defined SlIIooth feedback which makes the distribution invariant.

As announced, for

local

the following theorem shows the sufficiency of (7. 15a. b)

controlled

invariance,

provided

some

constant

dimension

conditions are met.

Theorem 7.5

the

Consider the smooth nonlinear system (7.1) and assume that

G ha.s

distribution

discribution

of

constant

COlJscant

dimension.

dimension

dimension. Then tile distribution

and

Let

assume

D be D

n G

l1n has

involutivB constant

D is locillly controlled invariant if and

only if

[f.D) c D

-I-

G,

(7.15a)

i E m.

Proof

(7.1Sb)

As we already have shown the necessity of (7.15a,b), we only have

to prove the sufficiency part of the theorem. Let Xo E H. We have to show the existence of a (7. 6a, b)

feedback

hold true on V for

(7.3)

in a neighborhood V of

the modified dynamics

(7 4).

X'D

such

Consider

that the

constant dimensional distributions D, G and D + G ilnd assume first that D n G - O. Let dim D

k and dim G = m. In a neighborhood Vi of

choose r = n-m-k vectorfields Xl""

,Xr such that

Xo

we may

215

+

,X r J) .. dim D + dim G

dim(D+G+span[X 1 ""

dim(span{X 1 , ... ,X t 1) .. n (7.16 )

by Corollary 2.43

Also,

(Frobenius),

we can find local coordinates on a

neighborhood Vz of xo. again denoted as (Xl , ... ,Xn ) such that

D

a

a

Xl

'''1;

(7.17)

span{---a _ , ... '---a' I

=

and thus D + G + span/XI"",X r } So we

find

a

a

the distribution G + spanlX I

that

(7.18)

span{aXl ""'axnJ.

=

, ...

,X r )

is spanned by

n-/;::

vectorfields

a

z, (x)

BX);+l

+

(7.19)

[ Zn_k

+

ex)

a

n _);

k

I

~i

(X)ax

1"'1

1

In the sequel we will use G{x) to denote the distribution of input vectorfields (7.14) as well as the IlXm-matrix formed by the input vectorfields: G(x)

(&1 (x), ... ,gm(x)j. Define the nx(n-k)-matrices BCx) and Z(x) by

=

B(x)-(g, (x), ... ,gm (x) ,X, (x), .. . ,X" (x) J {

(7.20) Zex)=[Zl

Letting G(x)

ex),.

and Z(x)

matrix which

are

skipping

first

n Vz

V '" V1

aG

aX i

for

the

. ... ,Zn-k the

(n-k) xm-matrix,

obtained from

Ie rows,

we

the

respectively

matrix G(x),

obtain from

(n-k)x(n-lc)-

respectively Z(x),

(7.lSb)

on

the

by

neighborhood

of Xo that

i

(x)

some

be

ex»)

mxm-matrices

E

£5.,

(7.21)

Kl (x), ... ,Kk ex).

Because

1m Bex) + DCx)

have that

i

where B(x) first

k

is

rows

the

E

£5.,

(7.22)

(n-k)x(n-k)-matrix obtained from B(x)

and Kl (x), ...

,Kk

(x)

are

by deleting the

some suitably chosen

(n-k)x{n-Ic)-

matrices. From the special form of the matrix B(x) we conclude from (7.21) and (7.22) that

216

1 E k.

(7.23)

As (gl(x), ... ,gm(x),X 1 (x), ... ,Xr(x») and /Zl(x) •... 'Zn-k(x») both span the same

distribution

G + span{X 1

, •••

,Xr

) ,

there

exists

a

nonsingular

(n-k}x(n-k)-matrix H(x) such that B(x) -

(7.24)

Z(x)H(x).

Partitioning the matrix M as 2

H(x) -

[

Hi(x) H (x) ..................... ;"'.,. . ".................. M3 (x)

1 1m ,

(7.25 )

'1 (x)

then we may assume, without loss of generality, that the mxm-matrix H1(x) is nonsingular. For if this is not the case, then a permutation of columns of the matrix Z(x) will produce a nonsingular mXffl-matrix in the upper left corner of H(x). From (7.24) we obtain, using (7.20) and (7.19), that

aB

-

aN

-(x) - Z(x)-(x)

ax!'

8xt

(7.26 )

i E !E"

and so, from (7.22) and again (7.24), we derive B(x)K L (x)

=

-

B(x) ( H(x) )

-1

8H "iix.'"(x).

i

(7.27)

E ~,

1

which implies i E

!;;.

(7.28)

since the matrix B(x) has full column rank. Using (7.25) and (7.23) we conclude that the nonsingular 1!!Xm-matrix flex) satisfies (7.29)

i E k. Define now the nonsingular mXm-matrix P(x) as

(7.30)

then this matrix

/3(x)

yields the desired change of input

Vee: torfields.

Namely using the identity i E ~,

(7.31)

217

we obtain -aa (e(x)p(x)) ~ - a a _ (G(x) (II' (x) )-') -

x,

Xi

ae_ (x) (II , (x)) -, + a (' )-' -a G(x)-a' x, n (x)

=

Xi

5..

i E

and thus [G,8,D] cD.

So

far

we

(7.32)

have

distribution.

assumed

In

case

that

the

the

distribution

constant

D n G

dimensional

equals

distribution

the

zero

D n G

has

positive dimension, say q, we first construct an mxm transformation-matrix

PCx)

such

(7.5b)

g1 •... ,gq

for

that have

we

the

D n G

gl""

transformed vectorfields span[gl""

=

satisfy (7.6b),

and for

Obviously

,gq)'

,gm

via

defined

the

vectorfields

the other vectorfields g'l+l •... ,gm

may use a similar procedure as given in case D n G

=

we

O.

Finally we have to show the existence of an w-vector a(x) such that the vectorfield

have

f(x)

obtained

fulfilled,

f(x) - [(x) +

are

the

we

I

f{x) +

=

as

well

I gi (x);;i (x)

related via f3(x)a(x)

construct

satisfies =

satisfies

/lJxIIJ~matrb:

nonsingular

may

(l(x).)

(7. 6a).

In

the

above

... ,;): (x) and vectorfields DI (x), ... ,D): (x)

where Clex)

-

+ Di(x),

yields

i

such

As

we

that

the

coordinates existence

already

(7.6b) such

;;(x)

(The vectors (l(x)

;1 (x),

G (x)oi (x)

(7.15)

(7. 6a).

m~vector

via

,

equation

{3(x)

an

given

~ex) aX i

(7.17),

gi (X)Q i (x)

and ;;(x)

where

of

is that

D is

m~vectors

in D such that

(7.33)

E ~,

[gl (x), ... ,g~(x)l. Skipping again the first Ie rows in (7.33)

yields an equation of the form

i

E

(7.34 )

k.

C1 (x) satisfies (see (7.32»

As the (n-Jc)xm-manix

i

(7.35)

E ~,

and for all i , j E we obtain from (7.34) that

~,

(7.36)

218

(7.37)

i , j E ~.

However,

this is a well-known set of integrability conditions.

In fact,

define the m-vector o(x) by Xl_

o(x) -

I

_

_

1l'1(Xl,O, ... ,O,Xk+l"",xn)dxl

+

o

Xz_

+

I

_

_

Il'Z(xl.X2.0 •...• 0.Xk+l.···.xn}dxz

+

D

Xx_

+ ... +

f

_

_

(7.38)

,Xk-l'Xk 'Xk + 1 ' · · · ,Xn)dXli;'

Il'li; (XII'"

o

then it follows from (7.38) that this vector satisfies

k.

i E

(7.39)

o(x-) is the required feedback since it can be checked that the vectorfield m

f(x) ~ f(x) +

L gi (x)01 (x)

satisfies

indeed

(7. 6b).

This

completes

proof.

0

Rernarlc 7.6 set

the

of

The underlying result of the if part of the proof is that the

partial

differential

equations

(7.29)

has

a

locally

defined

solution NI(x). The necessary and sufficient condition for the existence of such a solution is that the matrices Ki(x). i

E~.

satisfy

(7.40)

i,j E ~.

These

equations

(Compare Chapter

are

this with 2,

i.e.

called

the

integrability

the classical version of

Corollary

2.45.)

The

assuming that a solution Hl(x) of (7.29)

conditions

the

necessity

Frobenius' of

(7.40)

exists. Then (7.40)

for

(7.29).

Theorem of follows

by

follows by

the fact that

i,j E ~.

(7.41)

On the other hand one obtains (7,i.O) by using (7.22) and

i.j E ~.

(7.42)

So the remaining thing t.o be shown is that che integrability c:onditions

(7.40) are indeed sufficient for the existence of a solution of (7.29).

219

Theorem 7.5 gives a "geometric" proof of this. The proof of Theorem 7.5 reveals only

guarantee

the

local

distribution invariant.

that

existence

the conditions

of

a

feedback

(7.15a,b)

which

will

makes

the

One needs further assumptions on the manifold N

and the distribution D in order that a regular feedback on /1 exists, which renders

D invariant.

local versus

global

We

shall not

pursue

the

mathematical

controlled invariance here,

but

problems

confine us

to

of

the

local solutions as obtained in Theorem 7.5, 7,2 The Disturbance Decoupling Problem

In this section we study in detail the Disturbance Decoupling Problem for nonlinear control systems.

Instrumental in the

(local)

solution of this

problem will be the notion of controlled invariance as introduced in the previous

section.

As

announced

in Chapter 4

an

essential

role

in

the

solution is played by the concept of output invariance, cf. Section 4.3. Consider the nonlinear dynamics

x = f(x)

+

I

,.,

,

I

gi (X)U i +

(7.43)

8 t (x)d i

where f,gl, ... ,gm and (u1, ... ,um ) are as in Section 7.1, while e , ... ,e 1 l are smooth vectorfields on Nand d - (d , .•• ,d ) is an arbitrary unknown l 1 time-function. The elements of the vector d can be interpreted as disturbances or unknown inputs acting on the system.

Together with the

dynamics (7.43), we consider the outputs y - hex)

where h: H ....

(7.44) (RP

is a smooth map.

From Proposition 4.16 we know that the

disturbances d do not affect the outputs y dimensional

involutive

distribution

D

on

if there exists H

with

the

a constant

following

three

properties (i)

If,Dj CD,

(7.2a)

Ig, ,Dj C D, (ii)

ej

(iii)

D c leer dh.

ED,

j

i E

E

(7.2b)

~,

!,

(7.45) (7.46)

Obviously, these conditions for output invariance are usually not met and thus

the

disturbances

d

do

Disturbance Decoupling Problem.

influence

the

output.

This

leads

to

the

220

Problem 7.7 system

Disturbance Oecoupling Problem (OOP) Consider the nonlinear Under Ivhich condltions call !;'e find a regular static

(7.43,44).

state feedback (7.J) such thtlt in the feedback modified dj'namics 1

m

f(x) +

x

I gl (X)V I

I

+

C j (x)d i

the disturbances d do not influence tile outputs (7.44)7 Completely analogous to section 1.1.3 we obtain the following result.

Proposition 7.B

TIle Disturbilnce Decoupling Problem is solvable for the

smooth system (7.43,44) if there exists a consCant dimensional involutive distribution

D Ivhich

is

controlled

invariant

ilnd which

saeisfies

the

condition p

n leer dh j = ker dh.

(7.48)

1

In

case

tile

syscem

condition for

(7.43,44)

is

analytic

a

necessary

and

sufficient

the solvability of ehe Disturbance Decoupling Problem is

that there exists an analytic involutive controlled invariant distribution

D satisfying (7.48). Proof

The

first

Proposition

4.16,

part

of

whereas

the the

statement second

follows

result

is

immediately a

from

consequence

Proposition 4.14.

of 0

Proposition 7.8

completely solves

the

DDP

for

analytic systems

and

provides a sufficient condition for its solvability in case the system is smooth. However, in both cases this result is by itself not very useful as it may be

difficult

to

checl~

if

there

exists

a

controlled

invariant

distribution satisfying (7 .4B). To circumvent thi.s difficulty I we approach the

problem

in

a

slightly

different manner.

We

first

search

for

the

maximal controlled invariant distribution D* in ker dh - provided such an W

object does exist - and then we check whether D contains the disturbance vectorfields

The

following

example

shows

that

this

approach

indeed works for the linear DDP.

Example 7.9 =

Consider the linear system Ax + Bu + Ed

(7.49)

Cx

221

with x E size,

~n,

In

linear

U E mm,

the

y E mP, dE mi, A, B, C and E matrices of appropriate

linear Disturbance

state

feedback

Decoupling Problem one

u - Fx + Gv,

IGI"

0,

such

that

dynamics the disturbances d do not affect the output.

searches in

the

for

a

modified

The solvability of

the linear DDP is known to be equivalent to (see the references cited at

the end of this chapter) the existence of a controlled invariant subspace

V which satisfies

1m E eVe leer C which

is

the

linear

(7.50 ) counterpart

of

the

condition

(7.48)

stated

in

Proposition 7,8. On the other hand, given the subspace leer C, there exists

a

unique maximal

(7.49)

controlled invariant subspace V~

contained in leer C,

i.e.

for

the

dynamics

of

V" is controlled invariant and contains

any other controlled invariant subspace contained in ker C. Therefore, it immediately follows that the linear OOP is solvable for (7.49) if and only if 1m E C V·, with

V·

the

(7.51)

maximal

controlled

invariant

subspace

in

ker C.

Provided

(7.51) holds, a feedback u = Fx + Gv which solves the linear OOP is given by an mXn-matrix F such that (A+BF)V"

c

V"

and an arbitrary nonsingular

o

mxm-matrix G.

In the sequel we will closely mimic the solution of the linear

oor

as

sketched in Example 7.9. The next observations show that similarly to the linear case,

there exists a largest involutive distribution D" contained

in ker dh which satisfies (7.lSa,b).

Proposition 7.10 (7.1Sa,b).

Then

Let D be a distribution contained in ker dh satisfying also

D,

the

invo1utive

closure of D,

see

(2.132),

is

contained in ker dlI and satisfies (7.1Sa,b). Proof

As ker dh is an involutive distribution, we immediately have that

OeD c ker dh. Now let Xl and Xz be smooth vector fields in D. Then by the Jacobi-identity

[f,[X"X,I] ~ -[X,,[X,.£I] - [X,,[f,X,]] E jj + G, and also

[g" [X, ,X, I] - -[X" [X, ,g, I] - [X" [g, ,X, I] E jj + G, i Em.

222

Repeating

this

argument for

in D

iterated Lie brackets of vectorfields

o

yields the desired conclusion.

Let D1

Proposition 7.11

and D2

be

distributions

in

leer dh

satisfying

(7.15a,b). Then the distribution D1 + Dz is contained in leer dll and also satisfies (7.15a,b).

Proof

This follows immediately by observing that a smooth vectorfield X

in D1 + D2 may be decomposed (locally) as the sum X

o

and Xz E D2 and then writing. out [f ,Xl and [gi ,X], i E m. Because

the

zero-distribution

trivially

is

contained

leer db

in

and

satisfies (7.l5a,b) we have as a result:

There exists a unique involucive distribution

Corollary 7.12

in ker dll

that sacisfies O.ISel,b) and t..hich contains all distribut:ions in leer dh sacisfying to

(7.l5a,b),

I'

This

distribution

Ivill

be

denoted

as

..

D (f,g; n leer dh j

)

or, Ivhen no confusion arises, as D .

j~l

Using

the

foregoing analysis we

can effectively solve

the

DDP

in a

local way, That is, we will solve

Problem 7.13 (Local Disturbance Decoupling Problem) Consider the nonlinear

system (7,43,44). Under Ivhicl1 conditions can !ve find for each point Xc E H a regular static state feedback (7.3) defined on a neighborhood V of

Xc

such that in the modified dynamics (7.47) defined on V the disturbances d do not influe1lce ehe outpucs?

Using Corollary 7.12 we obtain a solution of Problem 7.13 in case that the

"

~

distributions D , D n G and G are

Theorem 7.14

cons~ant

dimensional.

Consider tile nonlinear system

distributions D",

D~

(j

(7.43,44).

Suppose that the

G and G are constant dimensional.

Then the Local

Disturbance Decoupling Problem is solvable if and only if span/e l

•...

,eil CD",

(7.52)

The effectiveness of Theorem 7.14 lies in the fact that there exists an algorithm which computes DR in regular cases. Consider the algorithm (the

223

D~ -algorithm): =

TN

=

ker dlJ nIX E V(N)I [f,X] E

vi!

+ G, [gi ,Xl E nil + G, i E!:!!l

(7.53) where V(N)

=

V"'Ul)

denotes

n.

the set of smooth vectorfields on

Suppose

the following holds. Assumption 7.15

For all

~ 0 the distributions

}l

nil and nil n G as well as

nP

the distribution G have constant dimension on N (or equivalently

Il

~

+ G,

0 has constant dimension).

Proposition 7.16

Consider

the

algorichm

(7.53)

under

Assumpt:ion 7.15.

Then (7.54)

( i)

(E)

nil is involutive for p ~ 0,

(7.55)

(iii)

n" _ nn,

(7.56)

(iv)

If

is

Dc ker dh

a

distribution

meet:ing

the

requirements

of

Theorem 7.5 then D c D~. Proof (i) Clearly nO :J

nWZ

ker dh n

=

c ker dh nIX

nl.

Now suppose nil :J nJi.-!l, then

IX E V(N)

I [f,X]

E V(N)I [f,X]

E nlHl

Ed! + G,

+ G,

[g1'X) E

nPtl

[g1 ,Xl E n P + G,

+ G, i E~)

i E~)

nlHl

=

which proves (i). (U) Clearly

DO

is

involutive.

Xl,X Z Ev'Hl. This implies i E!E'

the

as well Xl,X Z E leer dh.

Jacobi

identity

one

[gi' [Xl ,Xz]] E [VWl ,DP+G] , {D P+1,D P+G] c

vP +

(iii) From (i)

Next

involutive

suppose

[f,X,,] EDP+G and

Then

finds i E m.

and

[gi'X k ] EDP+G,

let

k=1,2,

[Xl ,Xz ] E leer dh and moreover using that [f, [Xl ,Xz]] E [DWl,DP+G] and

As

is

Dll

involutive

we

have

G, which proves the assertion.

and (ii) we conclude that the distributions

(oil)

form a

decreasing sequence of involutive distributions which by Assumption 7.15 are of constant dimension.

The only thing we need to prove is that the

sequence stabilizes, i.e. if for some p, DIHI Jc=2,3, ...

nlHl

=

DP

But

implies

=

oil,

this follows directly from the z DP+ = DP+1. As long as we have

then DIHk

= DP

algorithm strict

for all

(7.53)

as

inclusion

in

(7.53) the dimension of the distributions Dll decreases with at least 1 in

224

each step of the algorithm, from which we may conclude that the algorithm will terminate in at most n steps. (iv) Assume D

C

ker dh is involutive. has constant dimension as well a.s

D n G and G and satisfies [f,D] c D + G, [gi tD] C D + G, i E m. Obviously we have Deiter dh - D1. Now assume D

C DIi ,

then

D - ker dh n (X E V(H)I [f,X] ED + Gt [gi'X] ED + G, i Em} c ker dh n IX E V(H) I [f,X] E Dli +

G,

[gt ,Xl E DP + G, i E~)

_ DP+!.

Therefore D C Dli for all 11, and so D c Dn Note

that

the

algorithm

(7.53)

under

_

o

Dft. the

Assumption 7.15

precisely

produces the maximal distribution in ker dh meeting the requirements of Theorem 7.5, and thus in order to find a local solution to the DDP we only need to verify the hypothesis (7.52) of Theorem 7.14 for it. As we will see later the algorithm (7.53) is very much inspired by a corresponding algorithm for computing the maximal controlled invariant subspace for a linear system. For computational reasons we also give a dual version of it, which in some cases is somewhat easier to handle. With the smooth distribution ann

G

which annihilates

G

G,

we define, see Chapter 2 t i.e. for x E

the co-distribution

H

ann G(x) - (w(x)1 w is smooth one-form on H with w(X) - 0 for all

(7.57)

X E G) •

Consider the algorithm

(7.58)

In analogy with the Assumption 7.15 for the algorithm (7.53) we assume The co-distriburion ann G and the co-distributions p~ and

Assumption 7.17 pli nann G,

~

2:.

O.

have

constant

dimension

on

N

(or

equivalently

p~ + ann G has constant dimension).

Under the constant dimension hypothesis the algorithms (7.53) and 7.58) are dual. Precisely:

225

Consider

Proposition 7.18

the

algorithms

and

(7.53)

(7.58)

the

under

Assumption 7.15 respectivelJ' 7.17. Then

nI'

~ Icer pi',

(7.59)

or, equivalently ann DI1

Proof

pJi,

=

j1

(7.60)

:.>: O.

The claim is obviously true for /,

assertion for /,

=

Let X E Icer p2

then we have

following

to

prove X E DZ.

1. Let us show the

=

have,

see

all

for

an~

E pI n

W

(7.58),

X E ker p'

(i)

(iii) X E ker Lf',;(pl n ann G) ,

and

- (Lgiw) eX) - 0

\Je

X:

for

properties

three

ker L[ (pi n ann G) (Lrw) eX)

0 and /,

=

2. The proof for arbitrary /' is completely analog.ous. the

(U) X E

i E m.

So

E m. Now, using the

G and i

properties of Lie-derivatives for one-forms, see equation (2.169), we have (L,w)(X) - L,(w(X)] - w([f,X])

and similarly (LSiw)(X)

As

w(X) = 0

L"'i(w(X») -

=

we

w(lt,X)

obtain

[f ,xl, [gi ,Xl E ker(pl n ann G), assumptions may

kerCpl n ann G)

=

i

=

E

m.

W([gi ,X)) Ill.

a

=

Now under

i E

for the

11!.

constant

D' +

Therefore dimension

.no

G,

{gi ,Xl E VI + G,

and

This shows Iter pZ C DZ.

nl.

E

ker pl + ker(ann G)

=

{f,Xl E Dl + G

conclude

X E Icer pI

i

w([gi ,Xl),

thus

i E!::!,

we

where

In a similar way one shows that

o From the above proposition we conclude that under the Assumption 7.17 the maximal locally controlled invariant distribution is also given as D~ = ker p* = ker pll.

Note

that

in

particular

contains span(dh 1

,

L, (P" n G) C p", the

minimal

(7.61)

, ...

,dh p

i E

11!.

-

)

p*

is

and

for

an

involutive

which

we

codistribution

have

Lr CP" n G) c p"

Moreover by the duality between

codistribution

having

these

properties.

n"

that and

and p", p* is

Observe

that

the

Assumption 7.17 about constant dimensions is not really needed for having convergence

of

the

sequence

limiting codistribution p",

of codistributions yielding D"

(pJJ)

as ker p".

main result on the solution of the local DDP,

i.e.

/I:!D

in

(7.58)

If we return to

to

a

the

Theorem 7.ll!, we see

226

that in order to solve this problem. we need to do three things. First we compute D" via the algorithm (7.53)

or the dual algori thm

(7.58)

and

suppose the Assumption 7.15 (or Assumption 7.17) holds. Then, one has to check i f the condition (7.52) is fulfilled. I f not, Problem 7.13 is not solvable;

if

(7.52)

is

true

then one solves

for

the desired

(local)

regular feedback by using Theorem 7.S. Like we have seen this involves the solution of a set of partial differential equations. However, we will now show that this is nor necessary. In fact we will give an effective way of determining

the

codistributions

p~,

~ ~ 0,

from

the

algorithm

(7.S8)

provided Assumption 7.17 holds, and at the same time we obtain a local feedback which renders the limiting codis tribution p" invariant. Because Df< _ ker p", see Proposition 7.18, this feedback makes D" invariant.

Algorithm 7 .19

(Computing

P~.

~ ~ 0J

locally,

provided

Assumption 7.17

holds).

Step 0

Suppose the dimension of pi

span ( dh 1

' •••

I

dh p J equals Pl'

Then

after a possible permutation on the outputs we have around p Step 1

1

=

Xo

span(dhl.···,dh pl )·

Define the P1xm-matrix A1 {x) and the PI-vector Rl(x) via (7.62a) (7.62b)

Because pl II ann

G

has constant dimension.

constant rank. say r 1 we may

assume

•

the matrix Al (x) has

After a possible permutation on the outputs

that the first r 1

rows

of A1(x)

are

linearly

independent. Then (see Exercise 2.4) We may select an m-Vector alex) and a nonsingular

where

1P1 (x)

m~n-matrix

P1(x) such that

is a (Pl-r 1 )-vector and rPl (x) a (Pl-r1 )xrl matrix.

Denote the differentials of the ent:ries of IPI and tPl as depl and dtPl' Then we have (7.64)

227

Before proving (7.64) we continue the computation of the pP's. By assumption

, P -

has

pZ

span{dIJ 1 , ... ,dh p1 " " ~1

the entries of Step 2

fixed

dimension, ,dhpzJ

say

and

Pz,

we

may

set

for well chosen differentials of

~1'

and

Repeat step 1 with the functions h1 •... ,h pz . This yields a matrix

Az(x) of rank r z • a vector Bz(x) and new feedback functions az(x) and fJz(x)

such

differentials

that equations

of

the

entries

of of

the the

form

(7.63a,b)

matrices

hold. and

'Pz (x)

The

1fz (x)

appearing in the modified equations (7.63a,b) enable us to compute

analogously to (7.64) as

p3

In a

p3 _

span(dh 1

completely similar way the next steps

,.,.

,dhpz,d'Pz ,d1jJz l.

are executed.

Clearly,

see

Propositions 7.16 and 7.18, we are done in at most n steps (more precisely this will be in at most n-P1 +1 steps).

So going through the above steps

enables us to compute the P~'s. Moreover, one straightforwardly shows that the inductively defined feedback u invariant. Here

a""

and

t/

+ t/(x)v makes p" and thus n""

a""(x)

=

are the matrices determined in the last step. It

remains to prove (7.64).

Proof of (7.64)

Define the locally defined regular static state feedback m

U =

Oil

It

is

ex) + a

Pl

straightforward

produces

the

applied.

So

po;. _ pI

same in

list

exercise

to

show

of codistributions

particular

I

+ L_ Cp1 n ann G) +

a

the a

,.,

.

algorithm

(7.58)

regular

feedback

of

we

neighborhood

(pl n ann G).

L_

in1

f

in

that when

I gi (X)V i

+

(x)v and write the modified system as x = [(x)

Xo

Inspection

of

is

have

(7.63b)

IIi

yields that

dh i ex) rI ann G(x),

for 1 - 1, ... ,r 1

(7.65a)

,

as well as

dh i (x) So

the

",

,.I , (1/1 1 Cx) 1

one-forms

ik

dhJo;

e7.65b)

E

(7.65b)

ann G(x),

exactly

span

the

Therefore po;. consists of the one-forms in pl one forms

",

E

m. Now

, ..•

p1 nann

G.

,dh p1 , plus the

and L_ (dhi-l: (V'l)il;dh k ), 1 ~ r 1 +l, ... ,Pl' 1\ j

J

codistribution i.e. dh 1

k" 1

.

228

r 1

L_(dhl[

L

r1

(¢1)lk dh k)

L_dh j

=

ktt 1

f

L

-

(L_(¢1)1I: dh \o:

K~ 1

+ (¢1)n L _dll k) r

£

"1

L L_ (¢1 ) 1k dhr,

L_ dh 1

k~ 1

f

,

f

and similarly

for 1 - r1+l •... ,PI' j E m. This because L_dh t

-

£

dL_ll k f

d(zero function) - 0, k - 1, ... ,r l

•

respectively 0,

Ie

1, ... ,r 1

j E m.

,

Therefore. we find p2. ~ splln{dl1 1 .... ,dh p1 ' + spanlL_dhkl Ie - r1+l, ...• ptl + f

As L_dh k f

~

dL_h k we find, see (7.62a,b) r

(7.64)

o Although the above computations are generally quite complicated, there is a large class of systems for which these computations are not involved that much. This is in particular true, as

We

will see later, for single

input single-output systems and for the static state feedback input-output decouplable systems that will be treated in Chapter 8. Let us next investigate how the

local DDP works

out for a

linear

system. and afterwards treat some typical nonlinear examples. Example 7.20

x

Consider as in Example 7.9 the linear system

'" A..'I{ + Bu + Ed, (7 . 49)

Y - ex. Let:

Xu

be an arbitrary point in

IR

n

and let us try to solve the DDP in a

229

neighborhood of xo' restrict

ourselves

Note a

Because we want to

that

priori

in

to

contrast

regular

with

linear

apply Theorem 7.14 we

Example static

7,9

we

state

first have

do

not

feedbacks.

to determine

the

maximal locally controlled invariant distribution V* contained in the flat distribution ker C. For the system (7.49) i t is relatively easy to apply

the algorithm (7.58) or Algorithm 7.19. Let

(b 1

"

..

,bm )

and

(e l , ... ,e p

denote

)

the

columns

and rows

of

the

matrices Band C. We may interprete the bi's as constant vectorfields on Ulo and the

7.19

C

dimensional pI

as constant one-forms on

' 5

j

find

we

pO =

0

and

pl _

codistribution.

n ann(span(bl, ... ,bm )

span! c 1

Before

According to

[R0.

,c p I,

' ...

which p2

computing

the Algorithm

is

we

a

constant

observe

that

equals the codistribution ann{ker C + span(b 1 , ..

... ,bm ) , which is again a constant dimensional codistribution generated by

a

set

of

constant

one-forms

constant

one-forms as

row

span{el, ... ,e!;,) C span(c 1 , . . . ,cp

(in

the

x-coordinates).

vectors

Denote

these

note

that

and

Then

)'

m

p2 _ span{c1, ... ,c p ) + LAXSpan(el, ... ,ekl +

ILt,iSpanlel"",Ck)' 1-1

(7.66) In order to compute the last two terms of the right-hand side of (7.66) we have to determine L Ax e

1

and ~icl' Using (2.167) we obtain

1 E Ie

(7.670)

lE~,iEU/.

(7.67b)

k

Let w(x)

=

I w (x)c 1 be an arbitrary one-form in spanIel I'" 1

le k ). For an

'-1

arbitrary vectorfie1d X(x) we have, see (2,167), k

I (L,(w,(x))c 1

L,.:w(x) -

(7.68)

+

1-1

Therefore, span{e l , ...

(7.67a,b)

using

,ek I

and

(7.68).

the

and

fact

that

c spanlc l , ... ,c p )' we find

Thus p2 is again a codistribution generated by a set of constant one-forms and

is

therefore

explicitly

on

of

the

constant input

distribution D2 = ker p2 is

dimension.

vectorfields given as

Note bI

that

, ...

the flae

,b m.

p2

does

The

not

depend

corresponding

distribution generated by

230

-1

(Here A r,r

the linear subspace 1'z - ker C n £l(ker C + span[b 1 , ... Ibm))' is defined as the linear subspace [z E ~nIAZ E W).)

The next steps in che Algorithm 7.19 proceed in a similar way. Proposition

7.18

computation

as

obtain

we

shows

above

nlJ •

distributions

the

that

A

nlJ

distributions

the

Using

straightforward are

fl.at

distributions which are generated by the linear subspaces

(7.70) The algorithm (7.70)

is exactly the linear algorithm for

computing the

maximal controlled invariant subspace of the system (7.49)

in the kernel

of C. So the maximal locally controlled invariant distribuc:ion system

(7.49)

in

t:he

corresponding co V"

distribution

ker C

equals

the

flat

n" of the

distri.bution

the maximal controlled invariant subspace of (7.49)

I

in the linear subspace kerC.

Obviously

n" and D"nspan(b11 ...• brn } are

constant dimensional. The next step in solving the local DDP for (7.49) is to test (7.52), i.e. (7.71)

where e l

, ...

,e,2 are the columns of the matrix

E. Observe that (7.71) is an

inclusion between distributions, which parallels the subspace inclusion (7.72)

Equation (7.72) expresses the standard necessary and sufficient condition for the linear DDP. Now, when (7.71) is fulfilled, we know by Theorem 7.14 that around Xo

a solution of the local DDP exists.

solution one may

resort on Theorem 7.5

or

on

To find

Algorithm 7.19. However, as (7.71) and (7.72) are equivalent, much

easier

in

this

case.

Namely,

take

a

an actual

the computations

regular

linear

in

the

things are

static

state

feedback u ... Fx .... 1m v, which solves the linear DDP. Thus the matrix F is determined such that: (A+nF)V~

c

VW. Then this same feedback of course also

solves the local (nonlinear) DDP. So we gain nothing in trying to solve the DDP for the system (7 .llg) by allowing for nonlinear feedbacks! Another by-product of the equivalence of (7.71) and (7.72) is that we indeed find a feedback defined on eha whole state space

I

which was not guaranteed by

o

Theorem 7.14 (or Theorem 7.5).

Next

we

nonlinear

discuss system.

the In

D" -algori thm

this

case

it

for is

a

single-input

straightforward

single-output to

develop

an

231

formula

explicit

for

the

invariant

controlled

locally

ma:dmal

distribution.

Theorem 7.21 Consider the single-input single-output: nonlinear system on N x -

f(x) +

y

hex).

g(x)u,

(7.73) Let p be the smallest nonnegative integer such that the function Ls L~h is

not identically zero. Assume that

< '" and that

p

(7.74 )

for all x EN.

Then

v* Proof

=

ker(span{dh,dLrh, ... ,dL~h)).

We

compute

the

pi!, 5 ,

(7.75)

by

11
us ing

the

expression

(7.64).

Computing the lxi-matrices Al (x) and B} (x) from (7.62a,b) yields

A, (x) (7.76)

{ B} (x)

In case p > 0,

satisfying !PI (x)

=

A} (x)

(7.63a,b)

a

=

for all x and we may choose

as

0l(X) = 0

and

~

PtCx)

1

for

01

and PI (x)

(x)

all

x

and

so

Lfh(x), which yields

(7.77) In case p = 0, the function Al (x) coincides with the nonvanishing [unction given O'l(x) and

in =

{7. 74}.

A

solution

-(Lllh(x}r1Lrh(x)

~l(X}

of

(7.63a,b)

and fJ1{x)

in

(Lgh(x}r

=

1

this ,

case

and no

is

given

functions

by

!Pl(x)

appear on the right-hand side of (7.63a,b). So

pZ ~ pl

=

span{dh),

(7. 78)

and thus, see Propositions 7.18 and 7.16, p~ = span{dh],

which is precisely

(7.79) (7. 75)

for p

=

O.

For p

> 0, one iterates the above

computations starting from (7. 77), until pPil is reached. Clearly

p~ from

=

pP+l = span{dh,dLrh, ... ,dL~lJ)

which

(7.75)

readily

follows.

Using

(7.80) Algorithm

7.19,

the

feedback

232

u .. a"(x) + (3"'(x)v,

with

Q"'(x)

lex) -

and

(LgL~h(X»)-I, leaves D~ invariant.

o

So far we have developed the theory on the local DDP in the regular case. i. e. we have assumed throughout that the distributions D*. D* n G and G are constant dimensional. The following example illustrates that in some circumstances this is not needed. Moreover it shows a method how One can heuristically obtain a decoupling control law which not necessarily leaves the maximal locally controlled invariant distribution D* invariant. Example 7.22

In Example 1.2 we have seen that the equations for a gas jet

controlled spacecraft are given by (see (1.14»

it ~ {

-RS(w)

m

(7.81)

L btu!

J(" - S(w)Jw +

1"'1

where the orthogonal matrix R( t} denotes the position of the spacecraft with respect to a fixed set of orthonormal axes, w- (Wl,w2 ,WJ )T is the angular velocity with respect to the axes. S(w) is a skew-symmetric matrix

(7.82)

The positive definite matrix J. the inertia matrix, will be assumed to be diagonal

[",

J -

:

0

0

82

0

0

a3

j,

8i

> 0, i - 1,2,3,

(7.83)

which means that the eigenvectors of J, the principal axes, coincide with the columns of the matrix R. We assume that there are 3 controls on the system,

one of them being unknown

(a disturbance).

acting as

torques

around the principal axes. Therefore we henceforth consider the system

[al~11 a2~2

s3 wl

[0

-W 3

w2

J

w

0

-WI

-W21l~alWll WI

B Z W2

0

a 3 W3

(7.84)

+

[ 01 1

U1 +

0

[ 10 0

where the first equation of (7.84) follows from (7.81) and the fact that

RTR - 1 3

,

so

:t(RT)

=

_RTkR

T

,

see

also

Example 3.5.

Together

with

the

233

dynamics (7.Bl!) we consider the output function y

=

last row of RT

(7.85)

last column of R.

=

Let us write

(7.86)

R' - [ :: r,

s,

then

(7.87)

Note that r; + s~ + Yl

and Yz

the

c;

=

1 and so the output map (7.87) has rank 2; given

third' output

is

except

for

+ or -

a

sign

completely

specified. We want to solve the (local) Disturbance Decoupling Problem for the system (7.84,87). We solve the problem first by considering only the first

column

of

the

matrix

RT

and

the

output Yl

first

That

r).

=

is

consider the derived system

d dt

r,

w3

r

Z

- wZ r

3

0

0

0

r,

-"'3

r

l

+ w1 r

3

0

0

0

0

0

r,

wZr 1 -

w, w, w,

wir l

b l "'2 w3

+

a,-, u,

+

0

b" WI WJ

0

a,-,

b 3 WI W 2

0

0

0

u,

+

0

d,

(7.88a)

0

a,-, (7.8Sb)

where b i

-

a~1(a2-a3)'

b1 - a~~(a3-al)

and b 3

Proposition 7.8 and Theorem 7.14 we need

=

a~1(al-a2)' According to

to find a

controlled invariant

distribution D which contains the disturbance vectorfield {D,O,O,O,O,a;l}! and which is contained in the distribution leer dr J

.

In what follows we

search for an involutive distribution D which is contained in Iter dr J

and

which satisfies (7.lSa,b), but is not required to have constant dimension. Nevertheless - compare Theorem 7.5 where constant dimension of D is needed - we show that the distribution D is controlled invariant. Let (7.89a) Clearly we need to have Xl E D. vectorfield f

Computing the Lie bracket of the drift

in (7.88a) with Xl yields ll,111]. A modern account of the

of controlled and conditioned invariant subspaces and

linear synthesis problems is given in [Wo].

their use

in

The nonlinear generalization

of the notion of controlled invariance together with their applicability in various nonlinear synthesis problems has been initiated by Hirschorn in

[Hi] and !sidori et a1. characterization

of

a

in [IKGtU], controlled

Theorem 7.5 can be found in [Hi],

see also invariant

{IKGH2],

a modification of the one given in [Nij].

[11\>1]

and [lsI),

distribution

as

{Is2]. The given

in

{Nij]. The proof given here is A relaxation on the constant

dimension assumptions of Theorem 7.5 is discussed in [CT].

The algori thm

(7.58) for computing the maximal locally controlled invariant distribution has

been

given

in

[IKGMI]

and

its

dual

(7.53)

comes

from

[Nij].

The

Algorithm 7.19 is due to Krener [Kr2]. The difference between locally and globally

controlled

invariance

has

been

Example 7.22 has been taken from [NvdS3].

studied

in

[Krl]

and

[BKJ.

Other examples can be found in

238

[Cll,

[GBBl},

7.24,

has

(HG]. The modified disturbance decoupling problem, Theorem been

(C,A,B)-invariance

treated is

in

discussed

[MG] in

and

the

[IKGlH]

nonlinear and

version

[NvdS21.

of

Controlled

invariance for general nonlinear systems is studied In [NvdSl),

Another

approach

on

in studying

the

disturbance

decoupllng

problem

based

the

so-called generating series of a system can be found In (Cl].

[B1-1]

[BK)

[dBl] [Cl]

(eT] [GBB!1

G. Basile, G. Marro, "Controlled and conditioned invariant subspaces in linear systems theory", J. Optimiz. Th. Applic. 3, pp. 306-315, 1969. C.r. Byrnes, A.J. Kraner, "On the existence of globally (f,g)-invariant distributions". in Differential Geometric Control Theory, (eds. R.W. Brockett, R.S. Hillman, H.J. Sussmann), Birkhauser, Boston, pp. 209-225, 19B3. M.D. di Benedetto, A. Isiclori, "The matching of nonlinear models via dynamic state feedback", SIAH J. Contr. Optimiz. 24, pp. 1063-1075, 1986. D. Claude, "Decoupling of nonlinear systems", Syst. Gontr. Lett. 1, pp. 242-248, 1982. D. Cheng. T.J. Tarn, "New results on (f,g)-invarianc.e", Syst. Contr. Lett. 12, pp. 319-326, 1989. J.P. Gauthier, G. Bornard, S. Bacha, M. Idir, "Rejet des perturbations pour un modele non lineaire de colonne a distiller" , in QutUs at Modeles Matheml1tiques pour l'Automatique, l'Analyse de Systemes et Ie Traitement du Signal, vol. III (ed. I.D. Landau), Editions du GNRS, Paris, pp. 459-573

19B3. [Hi)

[IKGN1)

[IKGl12]

[151]

[152] (!Cr1]

[Kr2)

[HG]

R.M. Hirschorn, "(A,B)-invariant distributions and disturbance decoup1ing of nonlinear systems", SIAM J. Contr. Optimiz. 19. pp. 1-19, 19B1. A. Isidori, A.J. Krener, G. Gori-Giorgi, S. Honaeo, "Nonlinear decoup1ing via feedback: a differential geometric approach". IEEE Trans. Aut. Contr. ~C-26, pp. 331-345, 19B1. A. Isidori. A.J. Kraner, C. Gori-Giorgi. S. Monaco, "Locally (f,g)-invariant distributions", Syst. Contr. Lett. I, pp. 12-15, 1981. A. Isidorl, "Sur la theorie structurelle et la probleme de In r~jection des perturbations dans les systemes non lin~aires", in Qutlls et Modeles Mathematiques pour l'Automatique. l'Analyse de Systemes et Ie Trniteme.nt clu Signal, Vol. I (ed. 1.D. Landau) Editions du CNRS, Paris, pp. 245-294, 1981 A. lsidori, Nonlinear Control Systems: an Introduction. Lect. Notes Contr. Inf. Sci. 72, Springer, Berlin, 1985. A.J. Krener, "(f,g)-invariant distributions, connections and Pontryagin classes", Proceedings 20th IEEE Conf. Decision Control, San Diego, pp. 1322- 1325. 1981. A.J. Krener, .. (Ad f, g), (ad f. g) and locally (ad f, g) invariant and controllability distribucions", SIAN J. Gontr. Optimiz. 23. pp. 523-549, 1985. C.H. Hoog and G. G1ulllineau, "Le probleme du rejet de perturbations measurab1es dans les systemes non linciaires-applications ~ l'amarage en un seu1 point des grands petroliers", in Qutils et Modeles Mathematiques pour l'Automatique, l'Analyse de Systemes et le Traitement du Signal.

239

Vol III (ed. 1.0. Landau), Editions du CNRS, Paris, pp. 689-698,

[MW]

1983. S.H. Hikhail, W.H. Wonham, "Local decomposability and the disturbance decoupling problem in nonlinear autonomous systems",

[Nij]

H. Nijmeijer, "Controlled invariance for affine control systems"

[NvdSl]

H. Nijrneijer, A.J, van der Schafr, "Controlled invariance for nonlinear systems", IEEE Trans. Aut. Contr. AC-27 , pp. 904-914,

[NvdS2J

H. Nijrneijer, A.J. van der Schaft, "Controlled invariance by static output feedback", 5yst. Gontr. Lett. 2, pp. 39-47, 1982.

[NvdS3)

H. Nijmeijer, A.J. van der Schaft, "Controlled invariance for nonlinear systems: two worked examples", IEEE Trans. Aut. Contr. AC-29 , pp. 361-36/\, 1984. W.H. Wonham, A.S. Horse "Decoupling and pole assignment in linear multivariable systems: a geometric approach", SIAl-t J. Contr. Optimiz. 8, pp. 1-18, 1970. W.H. \.Jonham, Linear multivariable control: a geometric approach, Springer, Berlin, 1979.

Allerton Conf. Gomm. Contr. Compo 16, pp. 664-669, 1978.

Int. J. Contr. 34, pp. 824-833, 1981. 1982.

[OM]

[WoJ

Exercises

7.1

Prove

that

the

Algorithm

7.19

is

invariant

under

regular

static

state feedback. 7.2

Prove Theorem 7.24.

7.3

Compute the maximal locally controlled invariant distribution V" for

7.4

Consider a

the system (7.BBa,b) of Example 7.22.

.X

=

[(x)

smooth single output nonlinear system on a manifold N, i gi (x)u i + Lei (x)d i , y = hex). IHth this system we can

L

-I-

associate

two

systems,

and :Ed: x

=

smallest

integer

[(x)

namely,

i Lei (x)d i

-I-

such

,

Y

x

:Eu:

[(x) +

=

= h(x).

Lgi(X)U i ,

Y

hex)

=

Let p, respectively a be the

(LS1L~h(x), ... ,LsmL~h(x») ... (0, ... ,0),

that

(L01L~h(X), ... ,Lo.eL~h(x») ... (0, ... ,0). Assume that these inequalities hold for all x E N. ea) Compute (b)

D:,

Show that

V;,

respectively for :E u ' respectively :Ed' the Disturbance Oecoupling Problem for the

system is solvable if and only if D; C

D:.

original

(c) Show that the condition found under (b) is equivalent to p < o. 7,5

Consider

the

single-input

disturbance d, :E: Xo for

y

=

which

x

f(x o )

single-output

nonlinear

[(x) + g(x)u + e(x)d, y

= =

0

and

h(x o )

=

O.

Let

=

system

hex)! around a :E.e: x

=

A.:;;: +

bu

with point -I-

ed,

cx be the linearization of :E around Xo and u = D. Let p and 0 be

the integers as defined in Exercise 7.4 and assume LIlL~h(xo) ... 0 and

240

Lo L~ h (x o ) ~l

for 7.6

po!

O.

Prove tha t

the Local Dis turbance Decoupl ing Problem

is solvable.

Consider a particle of unit mass moving on the surface of a cylinder according to a potential force given by the potential function V

qz - pz

ql - Pl

av

PI - - iJql (ql,q'Z) + U

Pz

where (qI' qz. ,Pi ,pz) E Ttl (SIX !R) ,

av iJqz (ql ,q~,>

-

+ d

nnd d represent the control and

U

disturbance respectively. Let the output be given as y - qz. ea) Show that the Disturbance Decoupling Problem is solvable.

(b) Let z - q1 be the measurements on the system, Show that if the potential function V can be written as V(ql,Q2) - f(Q1) +g(QZ)qI + for smooth functions f, g and h, then there exists a regular

h(qz)

feedback depending on z only, which solves the Disturbance Decoupling Problem. 7.7

Consider x~ -

(a)

on

the

system

+ d,

u z • Xs = x1U 1

Y1 - Xl' Yz XlI> Show that D ... 0 and conclude that the Disturbance Decoupling

Problem is not solvable for this system. Introduce the dynamic compensator z

(b)

show that for the precompensated system the Disturbance Decoupling Problem is

locally solvable

(xl •.... Xs

Braund any point

,z)

with

xsz ,. O.

7.8

Let

Dl

and

Dz

be

distributions

satisfying

the

requirements

of

Theorem 7.5_ (a) Show by means of a counterexample chat D1 n

Dz is not necessarily

locally controlled invariant. (b) Assume Dl C Dz . Prove that around any point Xo there locally exists a regular state feedback which makes Dl and Dz simulr:a.neously invariant.

7.9

Prove Theorem 7.24.

7.10 Consider P:

x""

Ym

...

the

f(x)

hm (xm )

single-input

+ g(x)u. y

=

nonlinear

single-output

h(x) (plant), and N:

xm

~ fm(xm)

...

system gm(xm)um ,

(model). The local nonlinear Model 11atching Problem can

be formulated as follows ([ dBI J). Given initial points Xo and xmO find

a

precompensator

u - c(x,xc

)

F: (x.xm )

H

+

d(x,xc)umr

F(x,xm ) =

independent of um I (xc ,xlIlo).

Here

ypoQ

Xc

Q of for such

for all t

the the

form

'

Xc - a(x.xc ) + b(x,xc)um,

system

P

and

a

mapping

that l'oQ(x,F(X,xm).c) - ym(xm,t) and all (x,xm )

is

in a neighborhood of

denotes the output of the precompensated system

2'1

poQ.

The

solution a,

obtained

x• = fll(x a ) + (f'ex) .f!e x m ))',

Za:

ha (xa)

of

ga (xa)u +

gil (x a )

hex) - h m (xm).

=

this

follows.

-

local

Hatching

Yo

(iex),O)',

=

Problem

ha (x,,) ,

P a (x,,)

where

in

ker dh a .

be

system

fa (x,,)

- (O,g!CXm»)T

-

and

Prove that the local Hodel Hatching Problem

en;

+ spanlgll), where D: is the

locally controlled invariant distribution of

contained

can

augmented

the

P a (xlI)um ,

is solvable if and only if span(Pa1 maximal

Model

Define

Hint:

Relate

the

problem with

the the

system :E" Hodified

Disturbance Decoupling Problem. See for the multivariable case [dB!).

7.11 Let Dl

Theorem

and D z be two distributions satisfying the requirements of 7.5. Assume Dl n D2 = 0 and Dl + Dz is an involutive

distribution. Prove that locally Dl and D2 can simultaneously be made invariant by applying a regular static state feedback.

8 The Input-Output Decoupling Problem

In this and the next chapter we discuss various versions of the inputoutput decoupling problem for nonlinear systems. As a typical aspect of input-output decoupling is the invariance of an output on a subset of the inputs

we have to make,

J

like in Chapter 4,

some distinction between

analytic and smooth systems. In this chapter we first present a general definition of an input-output decoupled system. Next we give an approach to the static state feedback input-output decoupling problem which is most suited to square analytic systems. A geometric treatment of the static state

feedback

input-output

decoupling

problem,

applying

to

smooth

systems, will be given in Chapter 9. This last treatment will also allow us to give a solution to the bloclc input-output decoupling problem. In Section 8.2 we will treat, for square analytic systems, the dynamic state feedback input-output decoupling problem. B.l. Static State Feedback Input-Output Decoupling for Analytic Systems Consider the smooth affine nonlinear control system m

X

.,.

f(x)

L gi (x)u 1 '

+

(8.1)

i"'l

with outputs

y .. h(x),

(8.2)

where x - (Xl"" f'gl •...

are

,Xn )

local coordinates

for

,gm are smooth vectorfields on f1 and h -

a

smooth mani fold

(hI""

,hp): H

-+ [RP

}of.

is a

smooth mapping. Roughly stated the input-output decoupling problem is as follows.

Suppose

the

outputs

(B.2)

are

partitioned

into

m different

blocl;:.s, then the goal is to find - if possible - a feedback law for the system (8.1) such that each of the m output blocks is controlled by one and only one of the newly defined inputs. Depending on the way of output block partitioning systematically

and

treat

the

type

of

several versions

feedback of

the

we

allow

for,

input-output

we

will

decoupling

problem. We start our discussion with assuming that each output block is onedimensional, so we have

243

p

m,

~

(8.3)

i.e, the number of scalar outputs Yl equals the number of scalar controls u1

A system (8.1,2) satisfying (8.3) will be called a square system. We

,

say that

the square system (8.1,2)

possible

relabeling

j

~

of

the

inputs

the i-th output Yt

influences

is input-output decoupled if after a

u1

, ...

,urn'

the

input

i-th

u1

only

and does not affect the other outputs Yj'

i. More precisely, see Definition 4.11,

Definition B.1

The

nonlinear

system

(8.1-3)

is

called

input-output

decoupled if, afcer a possible relabeling of the inputs, ebe follcMing two

properties hold. (1)

For each i E

~

the output J't is invariant under the inputs u j

(Ii) The output Yi is not invariant with respect to the input u i

'

'

j

i

Em.

,.. i.

Using Proposition 4.11, we inunediately obtain as a necessary condition for input-output decoupling that, cf. Definition 8.1 (i),

Next we discuss part (ii) of Definition 8.1, To avoid complications such as

demonstrated

output

in

is analytiC,

(8,1,2)

Yi

Example

4.16

we

In that case,

assume

throughout

that

the

system

tii

on the

the effect of the control

is determined by the functions

Consider now the subset of functions of (B.5) given by

,

LstLeh t (>:), k O! 0,

X E

N.

(8.6)

Clearly when all the functions in (B.6) are identically zero, so

then also all the functions given in (B.5) are identically zero, and in no way the input u l

is going to interact with the output

Yi'

cf. Proposition

4.17. Therefore we assume (8.7) is not true, and we define for i Em the finite

nonnegative integers PI""

the function LgIL~hi

{

,

LgiLehi

""

0,

LgtLfih1 (x)

,Pm

as

the minimal integers

for which

is not identically zero. Thus Pi is determined as k '" 0,1, ... ,Pi -1, (8.8)

"°

for some x EM.

Using (8.4) and (8.8) we have for the (Pi+l)-th time derivative of yl. i E

(8.9)

11)

and so at the open subset of N

(8.10) the inputs

instantaneously do influence the output Yi' We are now able

lit

to give a formal definition of an input-output decoupled system. Definition 8.2 input-output

The square analytic system (8.1.2) is said to be st:rongly decoupled

if

(8.4)

holds

and

if

there

exist

finite

nonnegative int:egers Pl •.•• 'P m as defined in (8.8) such that: elle subset: No given by (8.10) coincides with N.

We

have

given

here

a

global

definition

of

strong

input-output

decoupling. We localize it as follows. Definition 8.3

Let

Xu

E N. The square analytic system (8.1,2) is said to

be locally st:rongly input-output decoupled around Xo neighborllood V of Xu such integers Pl""

c]U1f:

if there exists a

(8.4) holds on if and if there exist: finite

,Pm as defined in (8.8) Idtll N replaced by V such that the

subset No given in (8.10) contains V.

In the Definitions 8.2 and 8.3 we require a strong form of (instantaneous)

input-output decoupling.

In particular the

requirement

that the

subset No

coincides with N is in some cases not entirely natural. The

following

examples

assumptions on Example 8.4

no

Consider on :I

Xl

X2.

xJ

=

illustrate

the

difficulties

that

arise

when

no

are made.

Xz '

Y1

U1 •

Y2

=

m3

the square analytic system

Xl'

x3

(8.11)

•

uz ,

It is straightforward to check that (B.4) holds for the system (B,l1). Next we compute the functions appearing in (8.B). In the present situation we P2.

have -

LSlhl = 0,

LB1Lrhl(x}

=

O. The subset No is given as

3x;.

so

p)

245

Therefore

the

according

to

system

(8.11)

Definition 8.2

is

strongly

not

though

is

it

input-output

locally

around

decoupled each

point

(XOl,x02,x03) with X02 ~ O. However, the system (8.11) is globally inpucoutput decoupled in the sense of Definition 8.1. One computes that L"'lL;h1 (x) ... 6 "" 0, control

which

shows

that

the output Yl

no matter how the initial state

,

li l

(X OI

affected by

is

the

,X02 ,XOJ ) was chosen (cf.

o

Definition 4.11). Example B. 5

Consider on [Rz the square analytic system

(B.12) X

Again have

uz ,

=

z

this

Yz

sys tern

coincide

for

0

with

N

xz '

(8.12)

LSlhl ex) - Xl'

L81hl (x) -

=

L82hz ex) - 1

(Xl

,Xz) -

The

U?2.

=

satisfies

the

condi tion

yielding

(O,x z )

the

system

(8.12)

(8.4).

PI - 0,

subset

is

and

of

No

not

Furthermore Pz - D.

(8,10)

strongly

we

Because

not

does

input-output

decoupled but it is locally strongly input-output decoupled around each point

(XOl'X OZ )

Yl (t) =

Xl

(t)

=

provided

XOI;oo!

O.

In

case

X OI

0 for all t, no matter which input

-

III

0

we

will

have

(t) we choose, which

indeed shows that the output Yl is for such initial states unaffected by the control u l

0

.

In Example 8.4 as well as

in Example 8.5

the subset Ho

is

open and

dense in H, but as has been shown this is by itself not sufficient for (global)

input-output

decoupling.

We

will

see

later

that

by

using

invariant distributions the distinction between the two examples can be clarified (see Chapter 9). Note that for an analytiC system that satisfies Definition 8.1, the subset 110 is always open and dense in H and thus the system Xo

is

locally

strongly

input-output

decoupled

about

each

point

E Ho . If the square system (8.1,2) is not input-output decoupled (or locally

input-output

decoupled)

adding control

loops

we

may

try

to

alter

such that it becomes

the

system's

dynamics

input-output decoupled.

by

As a

first attempt one may try to achieve this by adding regular static state feedback. Recall that regular static state feedback has been defined as U

~

(l(x)

+

where a: H -. [Rrn,

(B .13)

[.J(x)v

/3: N

->

!R

rnXrn

are analytic mappings with /3(x)

nonsingular

246

for all

and v -

X

v m) represents a new control. By applying (8.13)

(V 1 •••••

to the dynamics (8.1) we obtain the modified dynamics m

X = £(;1{)

L gi (X)V i

+

'

where f(x)

=

f(x)

+

L gj (x)a

j

(8.15a)

(x) ,

j~l

m

L gJ (x)/3j i

g.\(x)

(B.ISb)

E Ill.

(x) ,

jal

We now formulate Problem 8.6 problem)

(Regular static state feedbaclr strong input-output decoupling

Consider

the

square

analytic

system

(8.1.2).

cOlldlcions does there exist: a regular static state feedback that

the

feedbaclc modified dynamics

(8.14)

Idth

Under

which

(8.13) such

t:he outputs

(8.2)

is

strongly input-output decoupled?

Before we are able to give a (local) solution to Problem 8.6 we need a few more things. Consider the dynamics (B.l) with the outputs (8.2). For

J

E

E we

have Yj(t) - hj(x(t» d

and so m

(t»)

=

Lrh j (x) +

L L", ihj (x)u i

.

(8.16)

1:1

Consider the function

This m-valued vector is ei ther identical zero for all x EN.

or there

exist points in N where it is different from the zero vector. In the last case we define the clJaracceriscic number Pj of the j-th output to be zero. In case (8.17) vanishes for all x in H we differentiate (8.16) once more to

obtain (B.lS)

Now consider the function (B.19}

Whenever this m-valued function Le, Lrh j (x) is nonzero for some x we define

Pj - 1 otherwise we repeat the above procedure.

247

The

Definition 8.7

characteristic

numbers

PI' ... ,P p

the

of

an;l1ytic

system (8.1,2) are the smallest nonnegative incegers such that: for j

LgL~hj(x)

=

E

E

(LgIL~hj(x), ... ,L8rnL~hj(X») ~ 0, k=O"",Pj-l, VXEH,

(8.20a)

for some x E 1'1,

(8.20b)

and when

LgL~hj(x)

=

(LgIL~hj(x),

..

,LllmL~hj(X)J

=

0 for all k ~ 0 and x E H

(B.20c) I{e set

(8.20d) Sometimes the integers PI"'" Pp are also referred to as

the relacive

orders or indices of the system (8.1,2) and they represent the "inherent number of integrations" between the inputs and the output Yj' j E p. Thus, see (8.16) and (B.18),

the (pj+l)-th time derivative of the output Yj may

depend upon the inputs u provided we are at a point x where (8.2Gb) holds. Note that the integers introduced in (8.8) are exactly the characteristic (locally) =

decoupled

0 for all i

-F-

j

system.

and k

~

This

because

in

a

decoupled

0, see also equation (8.4).

From

this observation it should not be too surprising that the characteristic numbers

play

a

key

role

in

the

input-output

decoupling

problem.

The

following proposition shows that the characteristic numbers are invariant under regular static state feedbacks.

Proposition B.B

Let

PI""

,P p

be

the

characteristic

numbers

of

the

analytic system (8.1,2) and let (8.13) be a regular static state feedback applied

to

(8.1).

Let

Pl""'P p

be

the

characteristic

numbers

of

the

feedback modified system (8.14). Theil

(8.21)

j E E,

(8.22)

If PJ < "" then

j

E

E.

x E

n.

(8.23)

248

Proof

Let u - o(x) + P(x)v be a regular static state feedback and recall

the defining equations (B.lSa,b) of the modified dynamics. Clearly (S.22) is true for k ... O. Assuming (8.22) holds for some Ie wi th 0 :S k < Pj ' we

have that

L(L~hj (x)

-

L~+lhj (x)

I

+

0i

(x)L s

iL~hj (x)

...

i"l

where the last equality follows from (8.20a). Having established (6.22) we prove (8.23). Using (B.lSb) and (B.22) we have

(If3'r.1(X)LgkL~JhJ(X)' ... , IPkm(X)Le.kL~jhJ(X») 1t=1

=

kDl

From (8.23) we now immediately conclude that (B.2l) holds true. As P(x) is nonsingular for all x in

!-J

some x in H (see (8.20b»

when Pj is finite, and thus the left-hand side

the right-hand side is a nonzero vector for

of (8.23) is a nonvanishing vector at x. On the other hand from (8.20a) and

(8.22)

We

immediately deduce

that

(L_ L~IJj (x) • ... • L_ L:hj (x») .. 0, II 1

(

!'>m

{

1c - 0 •... ,Pj-l, for all x in f1. The fact that (6.21) is true when Pj

o

immediately follows from (8.22).

We are now prepared to solve the strong input-output decoupling problem via regular static stace feedback.

Given a square system (8.1,2) with

finite characteristic numbers Pl •... 'Pm' we introduce the mxm-matrix A(x) as Le.lLf1h1 (x)

L8 Lfl hI (x) m

(B.24)

A(x) -

[L II 1LPmh !

m

(xl

L

LPmh (x)

!lm i

m

1·

We then have Theorem 8.9

Consider

c::l1e

square

ana.lyc::lc

system

(8.1,2)

,.,itll

finite

cIJaraeteriscic numbers Pl •...• Pm • The regular stacie state feedback scrong inpuc-outpuc:: decoupling problem is solvable if and only i f

249

Proof

(8.25)

for all x E N.

rank A{x) = ro,

Suppose first that the regular static state feedback strong input-

output decoupling problem is solvable. So there exists a feedback (8.13) such that the modified system (8.14,2) is strongly input-output decoupled. In particular, combining (8.4) and (8.8) we have that for j

E

~

while by Definition 8.2

(8.27)

~.

for all x EN, j E

So for the modified system (8.14,2) we compute according to (8.24) L- L~' h, (x)

l

"'1

A(x) ""

L

which is by

Bl

....

(.

L_

Sm

L~mh (x) f

a

h, (X)]

L_ L~m!Jm(x)

m

(8.27)

L~' f.

'm '

(8.28)

nonsingular mxm-matrix

for

all x

in N.

Now using

Proposition 8.8, we obtain from (8.23)

A(X) "" A(x){3(x). Since (3(x)

(8.29)

is nonsingular for all x E N we conclude that the matrix A(x)

N.

has rank m for all x in Next we assume static

state

(8.25)

is satisfied and we have to construct a regular

feedback

u

=

(l(x) + {3(x)v

that

achieves

input-output

decoupling. Recall that by Definition 8.7 of the characteristic numbers we have

the following set of equations

(y:Pi+

1l

denoting the

(Pi+l)-th

time

derivative of the i-th output function): [ y,IP, . H)

L~: +lhl

L~ lL~~ hl

(x)

(x)

(8.30)

+ ( pm+l)

Ym

L~m+lhm (x)

L g 1LPmh f m ex)

which by (8.24 ) yields [ h. IP,U) (Pm +1)

Ym

Li' "h, (X)]

LP~'!-lh ,

+ A(x)u. rn

(8.31)

(x)

As the matrix A(x) is nonsingular for all x in N we may define the regular

250

static state feedback

(8.32)

Application of this feedbac1c to the system (8.1) yields

(8.33)

which obviously shows that: the modified system is strongly input-output

o

decoupled.

Remark

B.9 also holds

It immediately follows from the proof that Theorem

true for smooth systems.

if we define strong input-output decoupling for

smooth systems in the same way as we did for analytic systems (Definition 8.2). Horeover the rank condition (8.25)

implies that the smooth system

can be input-output decoupled in the sense of Definition B.l.

as can be

readily verified (see Exercise 8.1 and also Exercise 9.2).

At

this

point

let

us

see

what

Theorem 8.9

amounts

to

for

linear

systems.

Example 8.10 A.x

X

Consider the m-input m-output linear system

+ Bu (8.34 )

y - Cx where x (8.3[1)

IR" and A, D and C are matrices of appropriate size. The system

E

is input-output decoupled when the corresponding transfer matrix

C(sI-A) -In

is

a

diagonal

invertible

matrix.

Let

us

next

see what

the

mxm-matrix A(x) of (8.24) for this system is. Denote the I-th row of C as ci

•

and similarly the i-th column of B as hi' i

(8.34)

a

function

of

the

form

E

IE. For the linear system

Lg.L~hj(x)

takes

the

form

.!.

LlliL!XcjX = CjAkb l • which is a constant for all i , j E!E. k:i:! O. Therefore.

see Definition 8.7. minimal nonnegative Pi - ""

if

ciAKn -

the

characteristic numbers

integer for which 0

for

all

Ie

2::

D.

the From

Pi'

are

row-vector this

we

defined

c i APt B ~ O. conclude

mxm-matrix A(x) of (B.24) reduces to the conscant nlxm-matrix

as

the

while

that

the

251

(8.35)

From

B.9

Theorem

~

conclude

decouplable

input-output u

we

that

via

the

system

regular

a

(8,34)

static

(lex) + f3(x)v whenever the matrix given in (8.35)

is

(strongly) feedback

state

is nonsingular.

If

(8.35) is nonsingular we obtain as a decoupling feedback, see (8.32),

(8.36)

which is a regular linear feedback. the

nonsingularity

of

Of course this is not surprising as

the matrix given in

(8.35)

is

the

necessary

and

sufficient condition for the linear system to be input-output decouplable

by regular linear feedback u

~

Fx + Gv. So we may conclude that the linear

system is decouplable via a feedback u = o(x) + {J(x)v if and only i f it is

o

via a linear feedback.

The condition (8.25) needed for the strong input-output decoupling by regular static functions

state

L~hi'

feedback has an interesting consequence about

i E!E'

k

~ 0, ... ,Pi.

Define

the

mapping

s:

N

the

-.

=

input-output channels. Observe that,

characteristic numbers of the first

number

II

ql

=

see (8.90),

the

outputs are increased with the

In the third step we proceed in a completely similar way.

vI'

We

will describe here the general (f+l)-th step. As in the algorithm denote the

inputs

of

feedbacks

(t"1 ' ... ,I>'rn)'

,

the

of

system after applying

the

form

Note

that

(8.90,91)

the

the

by

again

composition of

feedback u

(u l

=

these

(8.66)

""

,urn)

feedbacks

f

(i-I)

and

instead

is

of

of

the

desired form (8.54) . Let v J.l be the highest time-derivative of the inputs u

Vi.

appearing in

,

., ., z

2i j

~

ilJ

p

Introduce the precompensator


O.

> 0 \>]e have

(H) For all JJ

+

pJJ ( ( 1\)

l/' ( (2 •... ,m)

P'l ( (2 , ...• m \ ) I

=

pr

(9.69)

1E11I

(ii)

'*

(i) By induction we show that i f (9.69) holds, then for all JJ ?!: 0

G

~

I

G n

Dr

(9.76)

iEnJ

as well as

n D~J !Em

DIl

(9.BS)

and

nili

=

n t,~l

i

(9.86)

m

j;>
Clearly the statement is true for Jl certain Il> 0,

O.

=

Assume (9.76,85,86) hold for a

by repeated application of Lemma 9.20.

then.

(9.85)

and

(9.86) hold true for Jl + 1. Furthermore we have for all i E m

G- Gn

Dr

-I-

vr

Gn

(9.B7)

1 Next we compute dim(G n DJ..I+l + G n VJS+ ). i

I

dim(G n DJl) ~

H1

dim(G n V/

j";>
dim(G n DJJ+l) =

i

I J~

)

-

(m-2)m

-I-

j

dim(G n {lJJ+l) J.

dim(G n VJ..l+l) j

(m-2)m -

dim(G n Djl+l).

(9.88)

293

Using (9.69) we obtain the following identities m - dim{G n n")

L:

L em

=

- dim(G n V"», so j

dim(G n V~) - dim(G n n")

,

j~

(9.B9)

(m-l)m .

=

Harcover, dim(G n DJ.l+1) - dim(G n

n")

I

=

dim(G n v~+1)

- dim(G n v:),

(9.90)

jEm

which by (9.89) leads to

I

1

dim(G n V Il + ) - dim(G n

,

nP+1 )

_

(9.91)

em-I)m.

So from (9,BB) and (9.91) we conclude dim(G n nl-l+1 + G n VP+1) ;::; m that is i

i

,

,

GnnJJ+1+cnvJ.l+1_G

(9.92) E~,

Having established (9.92) for all i G -

,

n (G n nJ.l+1 + G n VI.Hl) ,Em

-I

we see that

,

DI-J.tl G n

,Em

+

G n nJ.l+l,

so

I

G -

,

nJ.l+1 G n

lEw

(9.76)

o

(9.27) follows by taking JJ sufficiently large.

and thus

It was

shown in Corollary 9.11

(9.27), we have dim(G n n") this fact

=

0,

that

given the

decoupling condition

Notice that we have not explicitly used

in the proof of Theorem 9.19.

In fact,

as noted before,

the

theory on input-output decoupling by static state feedback can be extended to the block-input block-output decoupling problem (see also Exercise 9.3) and

Theorem

dim{G n Dn)

Example

=

9.23

9.19

can

be

extended

to

that

situation,

in

which

case

0 is no longer necessarily true.

Consider

again

the

rigid

two-link

robot

manipulator

of

Example 1.1 (see also Examples 5.20 and 6.8). Its dynamics are given as

~r 1

[

(9.93)

where U" (ul,U z )' 0 = (Ol'OZ)' 0 = (Ol'OZ)' and the matrices M,e and Jc are defined as in Example 1.1. With the dynamics (9.93) we consider the

294

outputs Yl and Yz given

lIS

(9.94) It is easily seen that the dynamics Assumption 9.1

is

satisfied,

see

(9.93)

Example

is strongly accessible, 6.8.

Next

we

determine

maximal locally controlled invariant distributions in ker dh i

•

so the

i = 1,2.

using the standard algorithm

D;

D11

ker dh1

D;

Dl z

- ker dh,.

span 1_8_ •...£.... ,

(9.95a)

a02- aD,. (9.95b)

a0 1

Using (9.95) we find indeed

c ...

span(...£....) + span(...£....)

G fI

00 1

D;

+ G

fI

D; (9.96)

and so the necessary and sufficient condition (9.27) decoupling is satisfied.

Of course r

this

for input-output

is not surprising;

an easy

inspection of (9.93,94) shows that u

= C(O.B) + keD) + H(O) v

(9.97)

yields the feedback modified dynamics (see Example 6.8) (9.98) It is also straightforward to compute the formal structure at infinity for the system (9.93.94), One obtains pl = 2 - 0 = 2. p2 = 0, and in a similar way

p~

=

2

1

1, p~ ~ 1 - 1 ~ 0, p~ = 2 - 1 = 1. p~ - 1 - 1 - 0

and

thus, indeed, cf. Theorem 9.19, pl _ p~ + p~ and p2 ~ p: + p: .

0

The geometric theory for the linear static state feedback block decoupling problem has been developed

in

[BM] , [MW], [Wo], [UM].

Various

equivalent

formulations of the structure at infinity for a linear system have been given

in

structure

[Raj ,[Hal.[Mo). at

infinity

is

A geometric given

in

characterization

[Mal.

and

its

of

the

relevance

linear in

the

input-output decoupling problem follows from [DLMJ.[Di]. The differential

295

geometric theory for the nonlinear (block-)input (block)-output decoupling

problem

been

has

in

initiated

[ IKGK],

followed

and

up

in

[Nij3],[NSl],[NS2],[NSL!j, where the problem under static state feedback is

solved. Recently, been developed

in [DCM] , [HGJ ,[GDM] differenti..l algebraic methods have

for

solving

the

dynamiC

state

feedback block decoupling

problem. The nonlinear formal zeros at infinity have been introduced for a particular

class

of

nonlinear

systems

definition as given here comes form [NS2j.

in

[lsI], [152],

whereas

the

Further results on the formal

structure at infinity have been reported in

[Nij2J, [NS3], [Is3J.

Theorem

9.7 was first proved in [NS4j. The proof given in [NS4] was based upon the so called "controllability distributions"; ideas

from

[Ch]

and

[NS4], [Nij1],

and

the proof given here combines avoids

the

introduction

of

"controllability distributions". The proof of Theorem 9 .19 is taken from [NS2]. A study of the input-output decoupling problem with stability has been given in

[IG);

see also

[HG]

for a characterization of decoup1ing

feedbacks.

[BM] [Ch) [DiJ [DGM] [0111]

[GDM]

[Hal

[HG]

[151]

[Is2] [Is3]

[IG]

G. Basile, G. Marro, "A state space approach to noninteracting controls", Ricerche di Automatica 1, pp. 68-77, 1970. D. Cheng, "Design for noninteracting decomposition of nonlinear systems", IEEE Trans. Aut. Contr. AC-33 , pp. 1070-1074, 1988. J .M. Dian, "Feedback block decoupling and infinite structure of linear systems", Int. J. Contr. 37, pp. 521-533, 1983. M.D. Di Benedetto, J .W. Grizzle, C.H. Moog, "Rank invariants of nonlinear systems", SIAM J. Contr. Optimiz. 27, 1989. J. Descusse, J.F. Lafay, 1>1. Malabre, "On the structure at infinity of block-decoupab1e systems: the general case", IEEE Trans. Aut. Contr. AC-28 , pp. 1115-1118, 1983. J.W. Grizzle, M.D. Di Benedetto, C.H. Moog, "Computing the differential output rank of a nonlinear system", Proc. 26th IEEE Conf. Decision Control Los Angeles, pp. 142-145, 1987. M.L.J. Hautus, "The formal Laplace transform for smooth linear systems", in Mathematical Systems Theory, Lect. Notes Econ. Math. Syst., 131, Springer, Berlin, pp. 29-/.7, 1976. 1.J. Ha, E.G. Gilbert, "A complete characterization of decoupling control laws for a genral class of nonlinear systems", IEEE Trans. Automat. Contr., AC-31, pp. 823-830, 1986. A. Isidori, "Nonlinear feedback, structure at infinity and the input-output linearization problem", in Hathematical Theory of Networks and Systems, Lect. Notes Contr. Inf. Sci., 58, Springer, Berlin, pp. 473-493, 1983. A. Isidori, "Formal infinite zeros for nonlinear systems", Proc. 22-nd Conf. Decision Control, San Antonio, pp. 647-653, 1983. A. Isidori, "Control of nonlinear systems via dynamic state-feedback", in Algebraic and Geometric Methods in Nonlinear Control Theory (eds. M. Fliess, M. Hazewinkel), Reidel, Dordrecht, pp. 121-146, 1986. A. Isidori, J.W. Grizzle, "Fixed modes and nonlinear noninteracting control with stability", IEEE Trans. Aut. Contr. AC-33 , pp. 907-914, 1988.

296

[IKGM) A. !sidod, A.J. Krener, C. Gori-Giorgi, S. Monaco, "Nonlinear decoupling via feedback: a differential geometric approach", IEEE Trans. Aut. contr. AC-26, pp. 331-345, 1981. [Ma] M. Ha1abre. "Structure a l'infini des triplets invariants. Application iJ la poursuit:e parfaite de modele", in Analysis and Optimization of Systems, (eds. A. Bensoussan & J.L. Lions), Lect. Notes Contr. Inf. Sci. 44, Springer, Berlin, pp. 43-53, 1982. (HG] C.H. Moog, J.W. Grizzle, "0ecouplage nonl1neaire vu de l'algebre lineaire", C.R. Acad. Sci. Paris, t.307, Serle I, pp. 497-500, 1988. [Mo] A. S. Morse, "Structural invariants of linear rnultivariable systems", SIAM J. contr. Dptirniz. 11, pp. 446-465, 1973. [MW] A.S. Morse, W.M. Wonham, "Status of noninteracting control", IEEE Trans. Aut. contr. AC-16, pp. 568-581, 1971. [Nij 1] H. Nijmeijer, "Feedback decomposition of nonlinear control systems", IEEE Trans. Aut. Gontr. AG-2B, pp. 861-862, 1983. [Nij2] H. Nijmeijer, "Zeros at infinity for nonlinear systems, what are they and what are they good for?", in Geometric Theory of Nonlinear Control Systems (eds. B. Jakubczyk. W. RespondeJc, K. Tchon), Scientific Paper of the Institute of Technical Cybernetics of the Technical University of Wroc1aw, Poland, no 70, pp. 105-130, 1985. [Nij3) H. Nijmeijer. "On the input-output decoupling of nonlinear systems" in Algebraic and Geometric Methods in Nonlinear Control Theory (eds. M. Fliess, 11. Hazewinkel), Reidel, Dordrecht, pp. 101-119, 1986. [NSlj H. Nijmeijer, J.I1. Schumacher, "The regular local noninteracting control problem for nonlinear control systems", Proc. 22-nd IEEE Conf. Decision Control, San Antonio, pp. 3BB-392, 19B3. [NS21 H. Nijmeijer, J .M. Schumacher, "Zeros at infinity for affine nonlinear control syst:ems", IEEE Trans. Aut. ContI'. AG-30, pp. 566-573, 1985. INS3} H. Nijmeijer, J.H. Schumacher, "On the inherent integration structure of nonlinear systems", IHA J. Mat.h. ContI'. Inf. 2, pp. 87-107, 1985. [NS4 J H. Nijmeij er, J .M. Schumacher, "The regular local noninteracting control problem for nonlinear cont.rol syst.ems" , SIMI J. Contr. Dptimiz. 26, pp. 1232-1245, 1986. {Ro] H.H. Rosenbrock, State space and Multivariable Theory, Wiley, New York, 1970. [Wo} W.N. Wonham, Linenr Multivariablc Control: a Geometric. Approac.h, (2-nd edition), Springer, Berlin, 1979. [WI1) W.M. Wonham, A.S. Morse, "Decoupling and pole assignment in linear multivariable systems: a geometric approach". SIAM J. Contr. Optimiz. 8, pp. 1-18, 1970.

Exercises

9.1

Show that Assumption 9.1 in Theorem 9.7 may be relaxed as follows. Instead of assuming that the strong accessibility distribution Co is n-dimensional we only require that Co is constant dimensional and Im h,. - i1"(Co )' Prove Theorem 9.7 under this weaker assumption.

9.2

Assume

the

system

(9.1-3)

is

analytic

and

each

output Yi

is

1-

dimensional. Suppose Assumptions 9.1 and 9.7 are satisfied. ea) Prove that the analytic decoupling condition (8.46) implies the

297

geometric

decoupl ing

condi tian

kar spanldhj,.",dL/Jhj

(Hint

(9.27) ,

Prove

that

D: =

, j "" 1); see also Theorem 7.21)

(b) Prove that (9.27) implies that (8.46) holds on an open and dense subset of N. 9.3

Consider the nonlinear system (9,1,2) where the number of inputs m is larger

than

the

assumptions

as

decoupling is

number

of

output

Theo~'em

in

9.7

locally solvable

denotes direct sum.

blocks. that

if G

Discuss also

=

Prove

the

D~ n

under

n;

G 0 ... 0

the

n G,

where 0

that dim(G n D~)

the case

same

input-output

(block)

m - p,

=

compare Corollary 9.11 (see also [NSfl]). 9.4

Consider a system (9.1-3) satisfying all requirements of Theorem 9.7. Recall,

see

(9,62)

and

,

9.5

=

III.

,

Prove

definition of V", I c -m.

Consider a system (9.1-3) satisfying all requirements of Theorem 9.7. fRPm

assume

as hex)

(h l (x), ... ,11m (x) is a local diffeomorphism about each point

=

that the output map h: H

P

Horeover,

x E 1'1. Prove that D; = 9.6

the

(9,63),

that for all I c ~, dim(D~ n G)

Consider

a

nonlinear

n

leer dh j

system

,

i E

(9.57)

III

and

[II

----)0

1 X ... X

defined

(see also [Nij2]). assume

the

system

has

p

formal zeros at infinity of orders (n P ), cf. Definition 9.17. Suppose the system (9.57) i E m.

Prove

infinity of orders (n 9.7

21

is precompensated by the system

=

vi'

Ui

=

21 ,

that the precompensated system has pI formal zeros at Ptl

).

For a linear system the number of zeros at infinity is bounded by min(m,p).

Show by means of a counterexample

that

the number pI of

formal zeros at infinity of a nonlinear system does not necessarily satisfy pI

9.a

;$

min(m,p).

Consider the nonlinear system (9.57) and assume D" (n) Assume m be

a

=

1. Prove that necessarily also p

precompensator for

system.

Show

that

this single

the

maximal

=

D.

1. Let z = v,

U

=

z

input single output nonlinear locally

controlled

invariant

distribution of the precompensated system is also identically zero. (b) Assume Ill> 1 and suppose only the first input is integrated, i.e. we add the precompensator z system

(9.57).

locally

= VI' U I = Z, U i = Vi i Show by means of a counterexample

controlled

invariant

distribution

of

=

2, . .

that

the

to the

,III,

the maximal

pre compensated

system is not identically zero. 9.9

Cons ider

the

sys tern

(9.1-3),

satisfying

Assumption

Triangular Oecoupling Problem consists of finding

a

9.1.

The

regular

static

state feedback u = o(x) + P(x)v such that for the closed loop

system

298

output

the j

2 ••..

is

YJ

,Ill.

nat

R;

Let

be

influenced the

the

by

maximal

controls

vI • , •.• Vj -1 '

controlled

locally

invariant

m

distribution

n kar dlIi • k

in

=

0,1, ...

1.

,lI!

Prove

that

l-m-l:

suitable

regularity

Triangular

n

dim(R;

p -

an

arbitrary

~ 2. Suppose that

111

is

on

the

locally

distributions solvable

R:

the

if and only

if

Ie - 1.

G)

9.10 Consider

assumptions

Decoupling Problem

under

analytic

n"

nonlinear

system

(9.1,3)

with

O. Prove that the dynamic input-output

decoup1ing problem is locally solvable on an open and dense subset of

H. Show by means of a counterexample that

D

W

0 is not n sufficient

-

condition for dynamic input-output decoupling when

III -

P > 2.

9.11 Consider the nonlinear system (9.1,2) about an equilibrium point x o , i.e.

f(xo)

around

x[}'

(9.1,2) 1:£.

On

implies

y ex

= O. Let E : x = AX + Bu, = be the linearization i Show that in general output controllability of the system

does not imply output controllability of the linearization the

other

output

neighborhood

of

hand.

prove

that:

controllability Xo

(see

also

of

output the

controllability system

Proposition

3.3).

of

(9.1,2) Discuss

in

1:,l!

a the

input-output decoupling problem for (9.1-3) and 2:jI in light of these results.

10 Local Stability and Stabilization of Nonlinear Systems

In

this

chapter

we

will

discuss

some

aspects

of

local

stability

and

feedback stabilization of nonlinear control systems.

10.1 Local Stability and Local Stabilization via Linearization

We

first

present

some

standard

definitions

and

results

on

the

local

stability of an autonomous system. i.e. a system without inputs. Consider x~f(x),

where x

(10.1) ,Xn )

(Xl""

=

are local coordinates for a smooth manifold Nand f

is a smooth vectorfield on N.

Let

Xo

be an equilibrium poine of (10.1),

i. e.

(10.2)

In

the

sequel we will

study

the

qualitative

behavior

of

the

dynamics

(10.1) in a neighborhood of the fixed point xo' The equilibrium point Xo is said to he locally scable if for any neighborhood V of Xo there exists a

neighborhood

V

of Xo

belongs to I' for all

stable if Xo

t

V,

such that if x E ~

then

O. The equilibrium Xo

-t

two

roo

solution x(t,O,x)

is locally asymptotically

is locally stable and there exists a neighborhood Vo of Xo

such that all solutions x(t,O,x) of (10.1) with t

the

x E Vo'

converge to Xo as

In what follows we will study local asymptotic stability. There are

important

classical

ways

to

decide

about

the

local

asymptotic

stability of an equilibrium point xo' These are the so-called first and second

(or

direct)

stability of Xo

method of Lyapunov.

In

the

first

method

the

local

for the system (10.1) is related to the stability of the

linearization of (10.1) around the equilibrium point (10.2). So, consider the linear dynamics

x

=

(10.3)

tL'L,

with

at

A=ax(x o )'

Theorem 10.1

(First method of Lyapunov) The equlllbrlutIJ point xa

(10.4) of the

system (10.1) is locally a.symptotically sta.ble if the matrix A given in

300

(10.4) is asympt::ot::iclJlly stable, I.e. the mat::rix A hils all its eigenvalues

in r:he open lete half plane. The equilibrium point xI) is not st::able if at least one of clle eigenvalues of the matrix A has a positive real part.

Note

that

it is

immediate

that

the

results of

theorem 10.1 are not

changed under a coordinate transformation z = S(x) around the equilibrium point xo'

Essentially local asymptotic stability and instability can be

decided via Theorem 10.1 from the linearized dynamics (10.3) provided that the matrix A given in (10.4) has no eigenvalues wi th zero real part. An equilibrium point xI) for which the linearized dynamics has no eigenvalues with zero real part is called a hyperbolic equilibrium point. The

second

(asymptotic)

or

direct

stability

method of

of

the

Lyapunov

equilibrium

for point

deciding Xo

about

involves

the the

introduction of positive definite functions and invariant sets. A smooth function !l defined on some neighborhood V of Xo 0 and !lex)

.I!:(xo)

>

0 for all x

yA

xo'

A set

{if

is positive definite if in N is an invariant set

for (10.1) if [or all x E W the solutions x(t,O,x) of (10.1) belong to W for all t.

Theorem 10.2

(Second method

of

Lyapunov)

Consider

t::lle

dynamics

(10.1)

around cfw eqUilibrium poine (10.2). Let !£ be a positive definit::e function on some neighborhood \1 0 of xo' Then I.e have

(i)

Xg

is locally stable if (10.5)

(ll)

Xo

is locally asymptotically stable if (10.5) holds and the largest

invariant set under the dynamics (10.1) cont::ained in the set

OJ

(10.6)

rv

equals Ix o ); i.e. tile only solution x(t,O,x) srarelng in x E t ~ O. coincides td tll Xo •

Ivllich

remains in r,r for all

Note that the condition (10.5) expresses that around Xo

the function f

is not increasing along solutions x(t,O,x) of (lO.I). A positive definite function !l satisfying (l0.5) is called a Lyapuno\T funcCion for the system

(10.1). It follows in particular that Xo is locally asymptotically stable, when !J!

is

strictly decr-easing along all

solutions x(t,O,x),

x

E

V\lx o I

because in this case the set W trivially equals {xc), The main interest of Theorem 10.2 in comparison with Theorem 10.1 lies

301

in the case

fact

that Theorem 10.2 may decide about asymptotic stability in

the linearized dynamics

identically zero.

Horeover,

(10.3)

has some eigenvalues with real part

although we will not pursue

this here,

the

direct method of Lyapunov may be used in the determination of the domain of attraction of an asymptotic stable equilibrium. On the other hand,

the

drawback of the second method of Lyapunov for the study of stability of an equilibrium point x o ' is that in general there does not exist a systematic procedure for constructing Lyapunov-[unctions. An exception is formed by the class of mechanical systems where the total energy serves as a good

candidate Lyapunov-[unction (see also Chapter 12.3). The

following

interesting

result

shows

that

the

converse of Theorem

10.2(ii) is also true (see the references).

Theorem 10.3 (10.2).

Consider the dynamics

Assume

the

(10.1)

a.round

the equilibrium point

equilibrium point is locally asymptoCically stable.

Then there exists a Lyapunov-function !i!. defined on some neighborhood Vo of Xo and for which the set f{ defined in (10.6) equais (x o )'

We

emphasize

that Theorems

about the local nature of the point xo'

10.1 and 10.2 (asymptotic)

by

themselves

only

decide

stability of the equilibrium

In order to decide about the global character of an asymptotic

stable equilibrium more advanced techniques are needed.

For this we refer

to the literature given at the end of this chapter. In the sequel we will show how Theorems 10.1 and 10.2 can be exploited in

stabilization problems

for

nonlinear

control

systems.

Consider

the

control system

fCx,u) ,

x where u -

x

(Xl""

(10.7) ,Xn )

are

(u l , . . . ,urn) E U C [Rm,

field

for each u E U.

local the

coordinates

input

space,

for

and

a

fC.

smooth

,u)

a

manifold

N,

smooth vector-

We assume U to be an open part of [Rm and that £

depends smoothly on the controls u.

Let (xo ' u o )

an equilibrium point of

(10.7), so (10. B) Our concern is to see if the equilibrium (10. B) is locally asymptotically stable or can be made so by using some suitably chosen control function. In the first case we simply may check if the vectorfield f(· ,u o ) satisfies the conditions given in Theorems 10.1 and 10.2.

If not,

we will

see if

302

addition of a

strict state feedback u = cr(x)

to

the

(10.7)

system

can

improve the stability of the equilibrium (xo,u o )'

Problem 10.4 (Local feedback stabilization problem) Under which conditions does

there

exist:

a

smooth

U, {\fit}] cr(xo )

cr : Ii

x

strict

st:atic

state

feedback

u - o:(x) ,

UD , sucil that the closed loop syst:em

(l0.9)

f{x,o:{x»

has xI) as a locally asymptotically scable equili.brium7 A solution of Problem 10.1. can be obtained on the basis of Theorem

10.1 by using the

linearization of

the system

(10.7)

around

the

point

(xo,u o )' That is, we let

x = AX + Bu,

(10.10)

where

(10.11) Define :II as the reachable subspace of the linearized system (10.10).

see

also Chapter 3. So (10.12) Clearly

the

subspace :II

is

invariant under A,

A3l c :II,

i. e.

so after a

linear change of coordinates (10.10) can be rewritten as

(10.13)

where

the vectors

C:;;?, 0) T

correspond wi th

vectors

lying

in :II.

We

then

obtain

Theorem 10.5

The

feedback

stabilization problem for

admits a local solution around xn

t:he system

(10.7)

if all eigenvalues af the matrix Azz

appearing in (10.13) are in C , che open left half plane of ([;. Moreover if one of the eigenvalues of A z z has a posi t:;ive real part,

t:;llen r:here does

not:; exist: a solution to tile local feedback st:;abilizllt:ion problem.

Proof

Consider the linearized dynamics (lO.13) around (xn,u n ) and assume

all eigenvalues of

AZ2.

belong to ([; . Then a standard result from linear

303

control theory tells us that there is a linear state feedback u = the system (10.13) which asymptotically stabilizes the origin that

we

may

Taking

the

actually smooth

take

a

feedback

u

=

Uo

+

feedback only

F(x - x o )

for

x

=

depending

the

Fi

O. on

nonlinear

for

(Note

:;?)

system

(IO.7) we obtain the dynamics f(x,u o + F(x - x o

x

»,

(10.14)

of which the linearization around Xo equals

x

(10.15)

+ BF)x.

(A

By construction the linear dynamics (10.15) so by Theorem 10.1 we conclude that Xo

is asymptotically stable and

is a locally asymptotically stable

equilibrium point for (10.15). Next suppose that at least one of the eigenvalues of the matrix (10.13)

has

a

positive

feedback wi th o(xo )

x

=

=

real part.

Let u

=

a(x)

be

an

arbitrary

in

smooth

uo. Linearizing the dynamics (10.9) around Xo yields

a. o ) )-x, + Bax(x

[A

AZ2

(10.16)

which still has the same unstable eigenvalue of the matrix

A 2Z '

By Theorem

10.1 we may conclude that Xo is an unstable equilibrium point of (10.9). 0

Remark

Note

that

the

above

theorem

yields

no

definite

answer

to

the

feedback stabilization problem when some of the eigenvalues of the matrix lie on the imaginary axis (compare Theorem 10.1).

A22

10.2 Local Stabilization using Lyapunov's Direct Method In the following a stabilization result using Lyapunov's direct method is given.

It enables us to improve local stability of an equilibrium point

for an affine nonlinear system into local asymptotic stability.

Consider

the system

x ~ f(x) +

I

(10.17)

gi (X)U j

i~l

with

o. In

(10.17)

x

=

(10.2) (Xl""

,X n )

are local

coordinates around

the equilibrium

point Xo on a smooth manifold f1 and f, gl'" .gm are smooth vectorfields.

304

Suppose there exists a Lyapunov function .I': defined on some neighborhood Va of Xo for the dynamics (10.17) with u

0, so for the system

e

(10.1)

x - f(x)

we have L!~(x) ~

0 • V

X

EVa'

(10.5)

Then according to Theorem 10.2 the point xa

is locally stable for the

system (10.17) by setting u - O. In what follows we will show that under some

additional

conditions

we

are

able

to

produce

an

asymptotically

stabilizing feedback. Consider the smooth feedback u - o(x) with i E m

(10.18)

x E Vo,

yielding the closed loop behavior m

X

-

I

f(x) ...

(10.19)

81 (X)oi (x).

i-I

Clearly. Xo is also an equilibrium point for (10.19). At each point x E Vo we have, using (10.18) and (10.5), that

I (L8i~(x»2 ~ 0.

III

I

L

.I': (x)

- Lf..I':(X}

01 IIi

i

which shows by Theorem 10.2 that

Xo

IM

(10.20)

1

is locally stable for the closed loop

dynamics (l0.19). In order to study the local asymptotic stability of Xc for (10.19) we introduce the set

W - Ix E Vo

I

L[~(x)

I

i

... {x E Notice that

Xo

Val E

L!~(x) -

K

(LSi.l':(x})2 - OJ l

a, Lgl..I':(x) - 0,

i E mI.

(10.21)

rtf. Let ria be the largest invariant subset of rtf under the

dynamics (lO.19).

In case that TiD equals {x a J we conclude from Theorem

10.2 that Xo is locally an asymptotically stable equilibrium point. Now let xD(t,O,x) denote the solution of (10.19) starting at t - 0 in x EVa' Observe

that

any

trajectory xO(t,O,x)

in flo

is

n

trajeccory

of

the

dynamics (l0.1); this because the feedback (10.18) is identically zero fOl" each point in

r".

Therefore

is locally asymptotically stable for the

Xo

dynamics (10.19) i f the only trajectory of (lO.l) contained in ri is the trivial solution x( 1:)

-

-"0

,

t

~

0. Henceforth we will briefly refer to r"u

305

as

the

largest

[-invariant

subset

Tv.

in

On

the

other hand,

when

the

Lyapunov-function 2 satisfies d!f(x) .. 0,

(10.22)

also the converse is true. That is, if Xo is locally asymptotically stable for

(10.19),

then

the

trivial solution x(c), (10.19)

belonging

only trajectory of (10.1) t
the

distribution D of (10.23) has dimension strictly smaller than n.

Proof

Consider the function Lf!l on \'0'

From (10.5) it follows that this

function has a maximum at each point x E r.r. Therefore

dLr!l(x) Consider

a

=

0,

Vx E

trajectory

rl. Q

(10.24) -

x (t,O,x),

t
0,

noted

solution also satisfies (10.1). Define the functions 'Pi (t) as

before,

this

306

~!

(t) -

Obviously

Lgi~(X

these

Q

-

time

(10.25)

i E m.

(t.O.X», functions

are

identically

zero

and so

all

time

derivatives (d/dt)k~i(c) vanish. On the other hand we compute for k ~

a

(10.26)

i E m.

o

The conclusion now follows from (10.22). Lemma 10.7

leads

to

several

asymptotic stability of xa

sufficient

for

conditions

for

the closed loop dynamics

the

local

(10.19).

The

simplest situation is that where we have dim D(x o ) "" n,

(10.27)

which implies that on some neighborhood

Vo

c Vo

of

Xo

(10.28)

dim D(x) = n. Using Proposition 10.7 we

conc.lude

that

is

Xo

locally asymptotically

stable for the closed loop sys tern (10.19). Of course.

this resul t also

foliows from Theorem 10.5 because for the linearization (10.10) of the system (10.17) around xo.

the reachable subspace 11 given in (10.12) is

n-dimensional

seen

as

can

be

from

(10.27)

and

using

the

fact

that

f(x o ) ~

a and ad~gl (x o ) = (_1)KAk b1 • k ~ 0,1,2, ... (cf. (3.38». Other interesting sufficient conditions based on Proposition 10.7 may

be obtained as follows. Define the set (10.29)

Assumption 10.8 V1

U

There exist subsets V 1 and Vz of Va "rith VI n V z

eI and

=

V z = Vo such chat

(i)

Xu

(ii)

dim D(x)

E 1'2 •

n, for all x E V1

'

(iii) There exists a neighborhood

Vo in

Vo

of

Xo

such that (xu I is che

largest invariant subset of (10.1) in the set 1'z n

Wn Va'

Them we have

Theorem 10.9

Consider

the

affine

control

system

(10.17)

around

equilibrium poinr::: (10.2). Suppose chere exists a Lyapunov-funcr:::ion

!f.

the on a

307

neighborhood Vo

of

feedback

locally asymptotically stabilizes the equilibrium point

(10.18)

such

Xo

that

(10.5)

and

(10.22)

hold.

The

smooth

Xo i f Assumption 10.8 is satisfied.

Proof

Without

1055

of

generality

we

may

suppose

Clearly each nontrivial solution of (10.19)

contained

W.

in

trajectory

is

On

also

the

other

contained

hand

in

Lemma The

ITt.

Theorem 10.2 by observing that a

Va

that

equals

Vo'

lying in the set rv is also

10.7

implies

conclusion

then

trajectory of (lO.19)

that

such

follows

a

from

in rV' is also a

o

trajectory of (10.1).

We note that a simple typical special case of Theorem 10.9 is obtained

when dim D(x)

Example 10.10

n, for all x E Vo \(x o J.

=

Consider

the

equations

(10.30)

for

the

angular

velocities

of

a

rigid body with one external torque (see Example 1.2); I~ - S(w)Iw + bu

w, '

wJ) ,

0

[ w,

w,

-w,

0

w,

-wI

0

with w = (wI'

Sew)

(10.31)

-w,

1

I

~

[ I, 0 0 o I, 0 o 0 I,

(10.32)

1

I, > I, > I, > 0 denote the principal moments of inertia. Let I"

=

{

I"

~

I"

=

(Iz

- I,) I I, ,

(I, - I}) I I, , (II

(10.33)

I I, .

- 1 2)

Then (10.31) may be written as

w, w, w,

{ with

C

=

= 1 Z3 -

131

~ lIZ

W, W, W, W, W, W,

(c 1 ,C Z ,C 3 )T

+

c,

+

Cz u

+

c,

=

point of (10.3 12 > II > D. The system (10.81) can be rewritten as

w, w, w,

{

- IZJwZw J - I

J1

w3 w1 +

C

z u1

(10.83)

1 12 w 1 wZ + c J u"-

where

I"

{

.,. (l z - 13 )/1 1

(10.33)

I" - (IJ - II )/l z I" - (II - 1,,)/1 3

and

c,

-,

- I,

c,

~

-,

1J

.

Consider the feedback law

u, {

(10.85)

u,

which yields, as in (10.49), the equations

318

(10. (6)

From Theorem 10.11 we deduce that there exists a center manifold described

by (10.87)

with

We

approximate

(see

Theorem

QIW~ + Qzw: and jz(w 1 )

-

10.14)

the

center

manifold

by

jl(w 1 ) -

filW~ + fi2W~. resulting in the equations

O(W~).

(10.89a)

(2~lWl + 3Pzw~)IZJ(Qlw: + 02W~)(~lW~ + P2w~) + ~lW~ + pzw: a2w~) -

- lIZW1(Q1W; +

Ql w; -

q2w~

=

D{w~).

(lO.89b)

From (10.89u,b) we obtain

:: -~~1-_0~Z 1 ~1

-

=

0 (10.90)

ql = 0,

Pz - 112Q 1

-

qz -

o.

So the center manifold (l0.87) is approximated by

{ ~'(Wl) = ;P2 (wI)

=

+ (I 31 ql + P2)W~,

2

PI

W\

(10.91)

qlW~ + (IIZPl + qz

Substituting this in the (approximated)

dynamics on the center manifold

yields (l0.92)

or, WI -

12JPlqlw~

-I-

[I Z3 pd I 12Pl+Q2)

+

I Zl ql{I31Ql+PZ)]W: ... D(w;).

(10.93)

319

In order that wI

0 is a

=

locally asymptotically stable equilibrium of

(10.93) we need to have I 23 PIQl PI

=

0, so (10.%a)

0,

=

or (lO.9l!b)

The

system

(10.93),

and

therefore

the

system

(10.86),

is

then

locally

asymptotically stable if either +

IZ3Ql(IJlQl

P2)

(10.95a)

< 0,

or (lO.95b)

o In the feedback stabilization problem we have concentrated on the local

existence

smooth

of

stabilizing

state

feedbacks.

smoothness

This

assumption fits naturally into the context of this chapter. However, one

may

relax

this

assumption

and allow for

instance

differentiable feedback functions. Clearly, when not differentiable, x

when

0;

there

is

an extra

problem,

continuously

since

the

solutions

need not be uniquely defined for positive time.

f(x,a(x»

=

k-times

is only continuous but

Ct'

of

Horeover,

is only continuous one can no longer use a result as Theorem 10.5

for testing the stabilizability via this non-smooth feedback and, in fact, the requirement that the linearization of the system should not possess an unstable uncontrollable mode in order to be stabilizable, no longer need to be true for

one

example feedback E

and

K,

feedbac\t.

the existence of a CO stabilizing state feedback.

can -Xl

shoW

that

for

+ Ex~/3 + K(x z

although

the

the

-xi)

system

is

system

Xl

= U,

Xz =

Xz

As an

- x~,

the

is a stabilizing feedback for certain not

(locally)

stabilizable

by

a

C

l

We will not pursue the non-smooth stabilization problem here,

but instead refer to the relevant literature cited in the references.

Notes and References

The

stability

theory

for

autonomous

differential

equations

has

a

long

standing history and is today still far away from its completion. From the many textbooks on the basic results on stability we mention {LL,Ha,HSj. The first and second method of Lyapunov were originally described in fLy]. Theorem 10.3 on the local existence of a Lyapunov-function for a stable equilibrium can be found in [Mas,Mal,Ha,tHlsj, see also [Br]. The feedback stabilization problem for nonlinear control system is widely studied in

320

the control literature.

Theorem 10.5 can already be found in [LMJ.

The

feedback stabilization problem using a Lyapunov-function as in Lemma 10.6 -

Theorem 10.9

-

is

studied

in

(JQ,Sl,KT,LA];

we

have

more

or

less

followed the survey paper [Ba1. Example 10.10 is borrowed from [AS] where in a slightly different way the same result is obtained. A standard reference on center manifold theory is rCa). Center manifold theory as a tool in the (smooth) feedback stabilization problem was first studied by Aeyels, see [Ael,Ae2, Ae3] and [AS]. A survey of this approach is given in [Ba]. The application of center manifold theory for a

two dimensional control

system follows that of [Ba). Example 10.16 is essentially due to non-smooth feedback stabilization problem for a system was studied in IKa];

I AS].

The

two dimensional control

the example given at the end of this chapter

has been taken from this reference. For an approach to feedback stabilization based on the notion of zero dynamics we refer to Chapter 11. A recent survey about the feedback stabilization problem has been given in [So].

[Ael)

D. Aeyels. "Stabilisation of a class of nonlinear systems by a smooth feedback control fl. Systems Control Lett. 5, pp, 289-294,

1985, (Ae2]

[Ae3]

D. Aeyels, "Stabilisation by smooth feedback of the angular velocity of a rigid body", Systems Control Lett. 6, pp, 59-64, 1985 D. Aeyels, "Local and global stabilizability for nonlinear systems". in Theory and applications of nonlinear control systems (eds. C.l. Byrnes, A. Lindquist), North-Holland, Amsterdam, pp. 93-

105, 1986. [AS]

[Sa] [Br)

[Cal

[Ha] [HS) [JQ]

{KT]

D. Aeyels. H. Szafranski, "Comments on the stabilizability of the angular velocity of a rigid body", Systems Control Lett. 10, pp. 35-40, 198B. A. Bacciotti, "The local stabilizability problem for nonlinear systems", IHA J. Hath. Contr. Inform. 5, pp. 27-39, 1988. R. W. Brockett, "Asymptotic stability and feedback stabilization", in Differential geometric control theory (eds. R.W. Brockett, R.S. Millmann, H.J, Sussmann), Birkhauser, Boston, pp. 181-191, 1983. J. Carr, Applications of centre manifold theory, Springer, New York, 1981. W. Hahn, Stability of motion, Springer, New York, 1967. M.W. Hirsch, S. Smale, Differential equations, dynamical systems and linear algebra, Academic Press, New York, 1974. V. Jurdjevic, J.P. Quinn, "Controllability and stability", J. Diff. Equat. 28, pp 381-389, 1978. N. Kalouptsidis, J. Tsinias, "Stability improvement of nonlinear systems by feedback", IEEE Trans. Aut. Contr. AC-29 , pp. 364-367,

19B4. (Ka) (LA]

M. Kawski. "Stabilization of nonlinear systems in the plane". Systems Control Lett. 12, pp. 169-175, 1989. K. K. Lee. A. Arapostathis, "Remarks on smooth feedback stabilization of nonlinear systems", Systems Control Lett. 10, pp. L11-44 , 1988.

321

[LLl

LaSalle,

J,

S. Lefschet::,

Stability

by

Lyapunov's

direct

method

with applications, Academic Press, New York, 1961. fUll

E.B. Lee, L. Harkus, Foundations of optimal control theory, John Wiley, New York, 1967. B.A. Lyapunov, "Probleme general de 101 stabilite du mouvement", reprinted in Annals of Mathematical Studies, 17, Princeton

[Ly]

University Press, Princeton, 1949. [Hal}

I.G. Halkin, "On the question of reversibility of Lyapunov's theorem on asymptotic stability", Prikl. Hat. Heh. 18, pp. 129-138,

1954. [Has]

1.1. Hassera, "Contributions to stability theory", Ann. I·lath. 64, pp. 182-206, 1956. Erratum in Ann. Hath. 68, p. 202, 1958. [SIJ H. Slemrod, "Stabilization of bilinear control systems with applications to nonconservative problems in elasticity", SlAB J. Contr. Optimiz., 16, pp. 131-141, 197B. [So) E.D. Sontag, "Feedback stabilization of nonlinear systems", Proceedings HTNS-B9, Amsterdam, Birkhiiuser, Boston, to appear. [Will) J.L. Willems, Stability theory of dynamical systems, Nelson, London, 1970. [Wils) F.~.;r. ~.Jilson Jr., "The structure of the level surfaces of a Lyapunov functions", J. Dif£. Equat., 3, pp. 323-329, 1967. Exercises 10.1

L:

ConsIder on [Rn the smooth system Suppose

X

0

=

is

Lyapunov-function D{x)

=

locally

stable

for

f(x).

x

=

x

for

a

implies

O.

that

x

=

=

f(x)

the

that

the

=

and

let

there

be

V

a

D via

distribution

Suppose

such that L~Lx Vex) Prove

x

Define

spanlf(x) ,ad~g(x), k?: OJ.

neighborhood fV- of

fex) + g(x)u with f(O) = O.

=

e:.:is ts

a

0 for all XED and k ?:

feedback

u

locally

-LgV(x)

=

a

asymptotically stabilizes the origin (see [LA]). 10.2

Consider

x

form

a

=

smooth single-input

f(x) + g(x)u,

Lg!J(X) ,.. O. Let K

y

=

single-output

hex) = x".

Assume

system that

is an invariant set for

all:

for

n

of

the

x E IR

all

n

Ix E [Rnl hex) = OJ.

=

(a) Show that there exists a smooth function a; IfIn

that

on IR

the dynamics

x ""

->

IR

such that K

f(x) + g(x)a(x)

and show

is uniquely determined.

(b) Prove

that any point xG E IR" can be steered into K in

finite

time by a suitably chosen input u. (c) Suppose

B~

=

Ix

E

and

IRnl

distance (x,K)

:S r).

IIL,h(x)II ' ,

Prove

that

Define

> O.

there

exists

a

constant c ,.. 0 such that all points xG E B~\K are steered by either

u

=

+c or u

(d) Assume

=

-c in finite time into

f(O)

linearization

~

0

of

transfer function of

+ qo' pes)

=

and

let

about

x

L..e

K. bu,

=

O.

Let

with 11(5) = q(s)/p(s),

5" + Pn_lS,,-l +

... + Po

y

ex

11(5) ~ c(sI-A) -lb

and

q(s) ~ qn-ls

assume

pes)

be

the

be

the

"-, +

and

q(s)

322

have no common factors.

Show that qn-l - cb ~ 0 arid show that the

linearization

system

i

of

the

found

under

(a)

is

given

as

= (A-(cb) -l bcA )x restricted to the subspace kar c. Prove that the

characteristic polynomial of this matrix equals q(s). (e) Show

plane,

that when all

for

yet) - 0 t ....

CD,

of q(s)

zeros

lie

in

the

open left half

a

then there exists a neighborhood N of x sufficiently

t

where

EN,

''''0

t:he

large,

and

such that

x(t,O,xo ,u)

(ii)

to be ± c

control u is defined

....

(i) for

0

outside

K n Nand u - u(x) in K n N. 10.3

Investigate the center manifold approach for the system (10.66) in case that in the Taylor-series expansion (10.71a) condition (10.80) is not fullfLlled.

10.4

Show that in Example 10.10 the assumption needed

for

the

asymptotic

stability of

is essent:ially

(10.411)

the

closed

loop

dynamics

( 10 . 3L, , 37) .

10.5

Consider a nonlinear system x

=

(xo ,uti).

feedback

Suppose

solvable

10.6

for

the

this problem

x~

w.

U~

((Br])

Consider

is

f(x,u) about an equilibrium point Prove

system.

stabilization

f(x,u),

local

the

an

equilibrium

the

for

smooth

local

the

feedback

extended

x .. f(x,u)

system

is

system

and

let

(X,u) - (0,0)

be

the solvability of the local feedback stabilization

problem is that the mapping (x,u)

«(Ael])

~

point.

H

Prove

f(x,u)

Show that

in Example

10.16

([Ael)) Show that for the system exist

a

that

a

necessary

is onto on an open set

0 for (x,u) belonging to a neighborhood of (0,0). the

feedback

-c;lwz + Cz Uz - -c~lI12.WIW2 - c;lwJ - c;lw: ally stabilizes the origin.

10.8

problem

condition for

containing x 10.7

that

solvable

on

stabilizat:ion

linear

feedback

U

Xl

... a1x 1

-

x I XZ '

+ 8 zX z

u 1 - -c;lI31W:lWl locally asymptotic-

x2 -

U

there does not

rendering

the

origin

asymptotically stable, but the local feedback stabilization problem is solvable via a quadratic feedback function. 10.9

Consider the bilinear system and B - [;

x'"

Ax + (Bx)u on [1/2, with A -

[~ ~]

;]. Show that the matrices A and B can be diagonalized

Simultaneously. Determine all possible constant feedbacks u - c such that the closed loop system 10.10 Cons ider values

on IR

the

k

there

of

system exist

x - (A+Bc)x

xa

x + uk,

is asymptotically stable. k E IN.

continuous

Determine

feedback

for

which

u - u(x),

with

0(0) - 0, such that the closed loop system is asymptotically stable.

11 Controlled Invariant Submanifolds and Nonlinear Zero Dynamics

In Chapter 3.3 we have seen that the notion of an A-invariant subspace n (Rn for a linear set of differential equations ~ Ax, x E m , can be

x

'If c

conveniently x E N,

generalized

to

nonlinear

differential

equations

x

f(x),

=

by introducing the notion of an invariant foliation or invariant:

(constant Chapter 7 nonlinear subspace,

dimensional

and

involutive)

dlstribucion.

Subsequently

in

(and also in Chapter 9) it has been shown that an appropriate generalization at

least for

input-output decoupling,

of

the

concept

applications is

such

of as

a

controlled

disturbance

invariant

decoupling

and

that of a controlled invariant distribution.

In the present chapter we will show that the concept of a

(controlled)

invariant subspace also allows for a different nonlinear generalization, namely that of a (controlled) invariant submanifold. Furthermore, we will show that this second generalization is the appropriate tool for dealing with problems such as interconnection and inversion of nonlinear systems, and for defining the nonlinear analog of the concept of cransmission zeros of a linear system.

11.1 Locally Controlled Invariant Submanifolds

Consider a linear set of differential equations u

x-Ax, xElR . With

any

linear

Ix + fix E [flu} of

(11.1)

subspace [Rll.

V C [flu

we

can

If V is A-invariant,

associate

the

i.e. Atr c V,

foliation

F .,. 1r

then this implies

that the foliation Fy is invariant for (11.1). On the other hand AY C V is also

equivalent

xeD) E f

remain

to

the

requirement

in 'IT for all

t == O.

that While

the the

solutions first

of

(11.1)

for

interpretation of

A-invariance gives rise to the nonlinear generalization of an invariant foliation or invariant distribution,

the second interpretation leads

the notion of an invariant submanifold.

to

Indeed consider a vectorfield on

M, locally represented as

x - f{x). A submanifold N C N is called invariant for (ll.2) if

(11.2)

for all x E N.

f(x) E 1'.~N,

If N is connected then this

(11.3)

immediately implies that the solutions of

(11,2) for x(O) tn N reml').] n in N for all t:

0, (In the preceding chapter

~

we already encountered the more general notion of an invariant subset of H for (11.2), cf. 1beorBm 10.2.) Now let us consider the smooth nonlinea.r dynamics m

X

= f(x)

L

+

gj

(x) u j

(11.4)

u

•

j nl

where x

are local coordinates for some n-dimensional lIIani-

'Xo )

(Xl""

fold N. Definition 11.1 A sublllanifoid N cHis (locally) cont:rolled invariant for (11.4) U -

if there exiscs

o(x), x EN,

(l.oeally on N)

a

serlet: st:at:ic st:at:e

feedback

such chac

for all x E N,

(x) E T'1!N,

(11.5)

m

i.e .• N is invariant for

i

L

f(x) -I-

gol (x)Oj (x).

1

j

We immediately obtain (compare with Theorem 7.5) Propos ition 11. 2 Consider (11.4) and a submanifold N c H. Denoee G(.;::)

span Ig l (x) , ... ,gm (x) ) ,

and assume rhat dim(T:r.N + G(x»

(11.6)

X E N,

is constant for every

X

EN.

Then N is

locally controlled invariant for (11.4) if and only i t f(x) E T:r.N

+

for every x E N.

G(x),

Proof The "only if" direction is g(x) ~

(g1

(x)

(x)

such that N

J.

trivial.

Suppose

(lL 7) (11.7)

holds.

Locally we may choose coordinates x

01. Write accordingly

f(x)

(11.8)

Then by assumption to

fl(O,XZ) E 1m

locally find an tn-vector gt(0,X2)D(O,X2)

+ fl(O

has constant rank, while (11.7) is equivalent It

follows

2 D(O,X ),

O.

(see

Exercise

2.4)

depending smoot:hly on

that Xl,

we

such

can that 0

325

Remark If the assumption of constant dimensionality of TxN + G(x), x E Nt is not satisfied, then Proposition 11.2 is not valid anymore, as shown by the following example

(11.9)

Let N -

(X,jXl

~ OJ.

It is easily seen that

(11.7)

there does not exist a smooth feedback u = o(x),

is satisfied.

hood of any point ex: ,0), which renders N invariant. suggested by (11.9) is u - l/xz' for

Xz

F

However

defined on a neighbor-

(Indeed the feedback

0, which cannot be extended to a

smooth feedback around Xz - D.)

Now let us consider (11.4) together with output equations, i.e.

x

=

I

+

f{x)

gj (X)U j

u

,

j~l

(11.10)

y - hex) , 1 A submanifold N cHis called output-nulling if N c 11- (0),

i f the

i. e.,

output value corresponding to states in N is zero. Recall that in Chapter 7 algorithms have been given to compute, under constant rank assumptions, the maximal

locally controlled invariant

distribution contained

in

the

distribution leer dh (cf.(7.53), (7.58) and Algorithm 7.19). Similarly, we now want

to

compute

the

maximal

locally

controlled

invariant

output-

nulling submanifold for the system (11.10).

Algorithm 11.3

(Constrained

(11.10), and suppose h(xo ) neighborhood of xo'

=

dynamics

algorithm)

Consider

the

system

O. Denote G(x) as in (lL6). Let O(xo) be a

Step 0 Step k> 0 Assume that Nk is a submanifold through xo' Then define Nktl =

If

we

can

submanifold

(x E Nk jf(x) E

find

through

O(x o )

xo ,

TxNk + G(x) l.

such then

that

Xo

is

at

every

called

a

step

It

Nk ,

regular

~

point

0,

is

for

a the

algorithm.

Let

Xo

be

a

regular

point

for

Algorithm

11.3.

Then

we

obtain

a

descending sequence of submanifolds (11.11)

326

Since

Nk + 1

dim

dim N't;.

:::;

k~ :::; n such that N\c"+j component of Nle

sacisiies

siltisfying

j

it

follows Denot:e

1,2, ...

that

there

a

e1tists

the maximal connected

containing Xo by N"

ft

Proposition 11./, N"

1,2, ... ,

k

t

N\c",

=

Suppose Xo is

il

regular point for Algorithm 11.3. Then

Furtlwrmore

(11.7).

for

any output-nulling

chere exists some neighborhood O(x o )

(11.7)

submanifold N

of

Xo

such that

N n O{xo ) eN". W

Proof

Since

follows 1

from

N C 11- (0)

a

on

the

neighborhood definition

of x o ,

of

N = Nk

N"

that

Nkft,q

= Nt."'!-l

"

f

it

immediately

satisfies

(11. 7).

Let:

(11. 7) _ By induction to Jc it follows thar: N n 01:; (x o )

satisfy

C Nk for suitable neighborhoods Ot; (xo ) for all k.

0

H

Thus N to

x

is the maximal oULpuL-nu1ling submanifold through Xo wiLh respect

properLy E N"

(11.7).

If

additionally

(T7.Nw + G(x»

dim

is

constant

for

then it immediately follows from Proposition 11. 2 that N" is

the

maximal locally controlled invariant output-nulling submanifold around

Remark 11.5

For

a

system

linedr

x - Ax + Bu,

y -

ex,

xI}'

11.3

Algorithm

since the definition of VIJ +

1

simply reduces to the algorithm (7.70).

can

be rewritten as

We will now give a more constructive version of Algorithm 11.3,

which

actually is very much related to Algorithm 7.19.

Algorithm 11.6

(Constrained

dynamics

algorithm)

(11.10) and suppose that 11(x o ) - 0 and f(x ll Step

a

Assume

~

11

that

(hi""

neighborhood of Xo Nl S1'

=

h -1 (0)

Permute

is

an

the

,1lp)

in 11-] (0),

)

in

the

system

has

conSLant

rank

such

a

submanifold, way

that

hI"

a

in

51

Then locally around Xo the

(n-Pl )-dimensional outputs

Consider

= D.

where

__ ,hPl

set

PI:are

independent around xoStep 1

Define the PI x m matrix Al (x) and the Pl x 1 vector B1(x) as

(11.12) .. ,PI

327

Assume that Al (x) has constant rank r 1 in a neighborhood of Xo

in N1

•

may

assume

After a possible permutation of the output functions we that

independent. Exercise

=

0:

there 1

first

by

Then

2.4)

feedback u

the

r1

the

implicit

exists

(x), with

(11

rows

on

function

, x

!PI (x)

Assume that Then

.

has constant rank

'Ill

locally

around

on

S2

Nz

xo ,

~

the

of

entries

0

and f(xo)

such

!PI

=

linearly

theorem

(see

in NI

Ii

EN, .

a

0 we

that

(ll.13)

neighborhood of

Ix E N1jIPl (x)

=

(n-pz)-dimensional subrnanifold, with pz:= PI + because 11(xo )

are

0, such that

=

_ [0 ]1 r, NI

Al

neighborhood of Xo

Ii

(xo)

of

O}

=

first

S2

in an

(Notice that

S2'

have 'PI (x o ) = 0.)

the

Xo

is

Permute

entries

are

independent on Nl , and denote them as hp 1+ 1 ' ... ,h pz '

Step k> 0 Let Nk be a smooth (n-p); )-dimensional submanifold through x o , given

x

p):

as

Ix E N);-llh p l
where the right-hand side denotes the Jacobian matrix with respect

348

to

U.~l.,

••

,u(n~l)

of

the

vector with block components

inductively defined by (with (/0) (k+1)

y

~

ax

B\,tk)

(f(x)

+

I jnl

gj (X)U J

Y1 .... ,yen)

11(X)) )

+

k~l ay(kl L.

I~O aU(~)

(1+1)

U

1c

> 1

12 Mechanical Nonlinear Control Systems In the present chapter we focus on a special subclass of nonlinear control

systems, which can be called mechanical nonlinear control systems. Roughly speaking these are control systems whose dynamics can be described by the

Euler-Lagrangian or J/amiltonian equations of motion. It is well-known that a large class of physical systems admits, at least partially, a representation by these equations, which lie at the heart of the theoretical frame-

worle of physics.

Let us consider a mechanical system with n degrees of freedom, locally by

represented q

generalized

n

(ql"'" qn)·

=

configuration

(position)

coordinates

In classical mechanics the following equations of motion

are derived

~[~l dt . Here T(q,

q),

(12.1)

i E n.

aq,

with q

=

(ql' . . , qn) the generalized velocities, denotes the

total leinetic energy (strictly speaking kinetic co-energy) of the system, while Fi

are the forces acting on Lhe system. Usually the forces Fi

decomposed into a part which are called conservative forces, that i E

are ~,

derivable

from

a

pot:ent:ial

energy,

and a

are

i. e., forces

remaining part

F:,

consisting of dissipative and generalized ext:ernal forces: i E

with V(q)

(12.2)

~,

being the potential energy function.

function Lo (q,q)

as T(q,q) - V(q),

Defining the

Lagrangian

one arrives at the celebrated Euler-

Lagrange equations

(12.3)

i E n.

From (12.3) forces

and

a control syst:elll is obtained by disregarding dissipative

interpreting

control variables u i

.

the

external

forces

Fi"

in

(12.3)

as

input

or

Hore generally, if only some degrees of freedom can

be directly controlled, then one obtains the control system i

=

1, ...

,m, (12.4)

i=m+l, ...

with u l

, ...

,n,

,urn being the controls. Notice that (12.4)

is not yet in the

350

m

standard state space form x ~ f(x) +

Lgj(X)U j

indeed (12.4) is a set of

;

j =1

implicit

second-order

differential

equations.

However

for

mechanical

systems the kinetic enet:'gy T(q,q) is of the form .

1'T

.

T(q,q) - i q H(q)q for

some

(12.5)

positive-definite

matrix

Thus

N(q).

(12.4)

takes

in

obvious

vector notation the form

H(q)ij + C(q.~) + k(q)

av

with k i (q)

aql (q) • i

E

Bu

!2. 0, definite,

it

o.

(q,p)

follows

and by assumption V(q)

chat 1:(q,p)

is

posll:'1ve

on

V(qo)

some

(12.80)

is

positive

neighborhood

of

(qo,O). Since by (12.80) ~t 1:(q,p) = 0 it also follows that (qo,O) can not be an asymptotically stable equilibrium.

o

Now we are heading for a specialization of the stabilization result using Lyapunov' s

direct method as given in Chapter 10 (in particular Theorem

371

10.9). It follows from Theorem 12.27 that i f V(q) - V(qo)

is a positive

definite function on some neighborhood of a point qo with dV(qo) £(q,p) - Ho(q,p) - Veqa)

librium

(qo, 0).

=

0, then

is a Lyapunov function for the system with equi-

Furthermore

the

feedback proposed

in

(10.18)

takes

the

form

(12.81) = Xlli(Ho)(q,p)

= -(Hn,H i }(q,p).

Furthermore we have by (12.7gb) (12.82)

and thus the feedback is simply given as

(12.83)

i E ~.

which

physically

(Notice

that Yl

coordinate,

and

can be =

Hi (q)

thus

interpreted

as

adding

can be regarded as

Y1

as

a

damping

to

the

system.

a generalized configuration

generalized velocity.)

Indeed

with

this

choice of feedback we obtain (see (10.20» d dt req,p)

=

which expresses equals

I

(Yi)2,

,. ,

(12.18),

.

d dt Ho(q,p)

the

fact

,

(12.84)

CYi) , that

the rate of decrease

of

internal

energy

the dissipation of energy due to damping (Compare with

where a similar expression has been derived in a more general

situation.) We now come to the following specialization of Theorem 10.9. Define the codistrihution P(q,p) = span(dllo(q,p), d(ad

k

llo I1 i )(q,p),

i E

~,

lc 2: 0)

(12.85)

where we have defined inductively 02.86)

Ie - 1,2,.

Theorem 12.28

Consider the simple Hanliltonian control system (12.79) on

T~Q. Let qa satisfy dV(qo)

=

0,

and let V(q) - V(qo) be positive definite

on a neighborhood Ua of qo such dwt dV(q) ,., 0, q,., qo' q E Un' Then there

exists some neighborhood rIo of (qa ,0) stlch that £(q,p) is

positive

definite

all

r.,o

alld

d£(q ,p) .,..

a

for

all

=

}fa (q,p) - V(qo)

(q ,p) E IVa

with

372

(q,p) ,.. (qo ,0). Assume there exist: subsets = ~

and [v l

and

[vI

[v 2

of rVa

,.,rith fVl n fV2

'" fvo such rhar

U fv ...

rv . . '

( i)

( q0 ,0) E

(ii)

dim P(q,p) = 2n, V(q,p) E WI'

(iii) there exists a neighborhood Wo C fvo of (qo ,0) such that {(qo ,0) I is

rhe largest invariant subset of the dynamics q. = ~ ap , p = - £!!.o.. aq in the set f"l n

Wo

n ((q,p) E

= (llo,Hil(q,P) = 0, i E ~I

[VOIYi

Then the feedback (12.83) locally asymptotically st:abilizes the system in (qoIO).

Follows immediately from Theorem 10.9 by noting that by Lemma 12.19

Proof

dim P(q,p) = 2n if and only if dim D(q,P) - 2n where (see (10.23» D(q,P) = span(Xuo(q,p), ad~

110

XH (q,P) l i E ro, 1

-

k

~

(12.87)

01.

o A typical special case of Theorem 12.28 is obtained when

(12.88)

dim P(q,p) = 2n, for all (q,p) E Wo with q ,.. qo, 1. e .

I

when

r" . .

(q,p) E Wolq - qol. Indeed since

G(q)p,

~

02.89)

and G(q) > 0 it immediately follows that (qo,O») is the largest invariant subset contained in V.... Furthermore we mention the following simple

Corollary 12.29 ro = n.

Then

Let 11l(q)I""lIm(q)

dim P(q,p) = 2n,

for

be

all

independent (q,P).

qo'

about

Hence

ilnd

V(q) - V(qo)

if

let is

positive definite on a neighborhood Ua of qo such that dtf(q) ,.. 0 for all q E U o ,.;ith

q ... qo'

r:hen

(12.83)

locally asymptotical1y stabilizes

r:he

system in (qo ,0).

Proof

Since

11 1

""

are

Illn

independent

we

may

take

local

coordinates

qi - Hi' i E n for Q. Then in corresponding natural coordinates

(q,p) (12.90)

Since G(q) > 0 for all q it follows that dim (dll i I d(llo Remark 12.30 as

the

Illi

1. i E n I = 2n. 0

Note that the feedback 02.83) can be alternatively regarded

addition

of

a

Rayleigh's

dissipation

function

(see

(12.19))

373

~L.

y.,'

- Hi (q),

'Yi

i E

!E.

to the

system

A main asswnption in Theorem 12.28 was

On

V(q)-V(qo)'

the

other hand,

equations

for

u ... D.

the positive definiteness of

application

of

the

linear proportional

output feedback

(12.91) with

vi

the new controls, to the simple Hamiltonian control system (12.79)

is easily seen to result in another simple Hamiltonian control system

aHo qi'"

api (q,p),

i E

where

No (q,p) :=! V(q) ... V{q)

pTG(q)p

+

(12.92)

:!'

V(q), and V(q} is the oel.. potential energy

,

+ ~

(12.93)

Hence by a feedback (12.91) we have the additional possibility of shaping

the potential energy. The following lemmas will give a partial answer to

the question when it is possible to shape the potential energy in such a way that it becomes positive definite.

Lemma 12.31

Let Q be an n x n symmetric matrix and let C be a surjective

matrix. Then there exists an m x m symmetric matrix 11 such that

m x n

Q + CIRC > 0 if and only if Q restricced Co ker C is positive definite. Furthermore \"e can take II to be a diagonal matrix.

Proof

The "only if" direction is clear. Let now Q restricted to ker

C

be

positive definite. Let rl be an n x (n-m) matrix whose columns span ker C, and let V be an n x

matrix whose columns span the orthogonal complement

11/

First we prove that the n x n matrix (v:[n nm Indeed let Va; + [Ifl = 0, with a; E mm and fl E IR - • Then of Q(ker C).

a

~ [/Q(Va + [I"fl)

=

is nonsingular.

r/Q[lfJ.

Since Q restricted to leer C is positive definite this implies fJ hence () - O. It is easy to see that

a

and

374

Since

rank

V'ICTfICV -

rank

(v!rv)!CTHC(vjrv) "" rank

CTHC

follows

it

that

Q + CrHC can be made positive definite by choosing an appropriate H _ HT

o

(if necessary diagonal).

Lemma 12.32

o

Consider rhe simple Hamiltonian control sysrem (12.79) It'lth and Hj(qo) = 0, j Em. Assume that the matrix

(12.94)

is posirive definirE.! \vhen restricted to the subspace

(12.95)

niter dJij{qo) j~l

Then ehere exisrs a feedback (12.91) such that V(q} = V(q) + ~

ro

kjY~ is

I

J "1

positive definite on a neighborhood Uo of qo. and dV(q} \,rith q

;ol

Proof

Apply Lemms 12.31 to

;ol

0 for all q E Uo

qo'

Q

(qo )

J.. 1

,C = [ aH!. (q[)

J.

"

02.96)

1~,J~

,JE!2,

This yields the e:ds tence of a diagonal rna trix }{ = diag (k l that Q + CTUC > O. No,>! consider the function V(q) = V(q} +

, .. ,

/em)

~ ~ kj)'~'

such Then

j "1

(12.97)

and thus

V is

o

as required,

We conclude that if the potential energy V(q} satisfies the assumptions of Lemma 12.32 then there exists a proportional output feedback (12.91) such that I'lenee

V

as defined by (12.93) satisfies the assumptions of Theorem 12.2B. by

Theorem

12.27

the

Hamiltonian

system

for

Vi

= 0

is

stable.

Furthermore if also the remaining assumptions of Theorem 12.28 are met for the system with internal energy feedback

Ho -

Ho

+;-

m

L

JcjY~' then the derivative

375

(12.98)

~,

i E

will result in local llsymptotic stability.

If m

Remark 12.33

=

nand H1

, ...

,Hn are independent then the assumptions

of Lemma 12.32 are automatically met. Hence by Corollary 12.29 the system can be always made locally asymptotically stable by a feedback of the form i

E

(12.99)

n.

It is clear that Theorem 12.28 remains valid if we replace the feedback

(12.83) by the more general expression i

(12.100)

m.

E

Thus if the assumptions of Lemma 12.32 and Theorem 12.28 (for the system with

internal energy No)

are met,

then every feedback of proportional-

derivative (PD) type 1 E

ki

with

sufficiently

system. lei ,c i

'

Furthermore, iEEE,

large the

will

locally

freedom

in

the

(12.101)

~,

asymptotically choice

of

the

stabilize

the

gain parameters

can be used for ensuring a satisfactory transient behavior

(analogously to classical PD control for linear second order-systems).

Remark 12.34 in (l2.101)

110tivated by the fact that the damping terms -CiYi'

i E!!.!,

, I ci.Yt,

correspond to the Rayleigh dissipation function":

and

i~l

the terms -leiYi'

i E!!.!, correspond to the extra potential energy

we could even generalize (12.101) to the "nonlinear PD controller" i E

!!!,

(l2.102)

corresponding to the addition of a general potential energy term P(y) and Rayleigh dissipation function R(}').

Example 12.35

(see also Example 12.3). Consider the two-link rigid robot

manipulator from Example 1.1, mi

~ all

=

1.

configuration

where we

Furthermore we take qi

=

Jr,

ql ""

1f

U

z

=

O.

take

asymptotically

First we apply the linear feedback

for simplicity 1'1

Suppose one wishes stable by

=

1'l

to make

smooth

=

1, the

feedback.

376

(12.103)

It is easily seen that for k > 2g the potential energy V + ,! lc(ql - 11")2 2

has

a

unique

minimum

in

(11",11").

Since

dim

P(q,p) =

4

the

additional

derivative feedback (12.104) will

thus

result

in

global

(except

for

the

point

ql - 11",

qz = 0)

o

asymptotic stability.

12.4 In

Constrained Hamiltonian Dynamics

this

section we make a

closer study of

the

constrained and zero

dynamics, as treaced in Chapter 11, in the case of a Hamiltonian sysr:em (12.54). We confine ourselves to the case as dealt with in Proposition

11.3, i.e. we assume throughout that the m X m decoup1ing matrix (12.106) has rank equal to m on the set (12.107) Then we

know

from

Chapter

11

that

the

constrained dynamics

for

the

Hamiltonian syscem (12.54) are given as m

I

(12.108)

XII. (x)a; (x) , J

where a- (x) is the unique solution of (11.28). Moreover since p = m the constrained dynamics equals the zero dynamics, cf. (11.35). We will show that because of the Hamiltonian structure the zero dynamics (12.108) has a very special form.

First of all we note that the matrix A(x)

can be

rewritten into the following more convenient way. By (12.23) we have Ic-O,l,··,pi'

i E

~,

(12.109)

where the repeated Poisson bracket ad~ Hi is defined as in (12.86), and o is the smallest nonnegative integer such that

Pl

(12.110) Therefore

377

(12.111)

In particular it follows that [or the computation of A(x) we do not have to

go

through

the

aqua tions

I

Hamiltonian Ho -

Hj u j

of motion

(12.54);

the

knowl edge

of

the

suffices.

j"'l

A submanifold N of a manifold N with symplectic form w (defined by a non-degenerate

T"N x T,,/J

-+

Poisson

if

submanifold

[R,

the X E N,

bracket,

following

cf.

(12.51»

holds.

is

Restrict

called

the

to a bilinear form w,,: T"N x TxN

a

symplectic

bilinear ->

form

~,,(X,Y) := W,,{X,y) ,

(12.112)

Then N is called symplectic if every x EN,

i. e.

if

w,,:

IR, X E Nt i.e.

the

is a non-degenerate bilinear form for

Wx

rank of a matrix

representation of

w"

equals

dim N for every x E N. (In particular N is even-dimensional.) Theorem 12.36 N"

Consider the Hamiltonian system (12.51/)

is non-empty and Chat rank A(x)

sympleccic

submanlfold

Hi (x) ,adlloHi (x), ...

of

,adfr~/{i (x),

=

N.

Noreover,

i E m

N.

all

m for every x E N".

(lvhich

denote a.re

Assume that Then N" is a

che

independent

fUllctions on

by

Proposicion 11.3) as

l, ... ,s

(12.113)

then the s x s slcelv-symmecric matrix (12.114)

has rank s for every x E N"

Proof

First note that by the Jacobi-identity we have for any i , j E m

(12.115) By definition of Pi' cf.

(12.110), the last term is zero, and inductively

we obtain

(12.116) By permuting

the

indices 1, ... ,m we may assume

Fi.rst suppose that PI > P2 >

>

Pm'

that Pl ;;: P2 ;;:

Then i.t follows that

378

< i.

for j Hence A(x) (cf. A(x)

tion

(12.111»

is

(12.117)

is an upper t:riangular matrix. Since by assump-

non-singular

x E N" I

for

it

follows

chat

the

diagonal

elements (adft~Hi'}{jl(X), x E N~, i E!E. are all non-zero. By (12.116) this implies that

lad~i-kHl.adl~ o

..pj

as

Pt -

in

P'2.

>

(12.113) P3

}(x)

po!

that

i Em. x E Nr..

for k'" O.l ..... p,l.

0,

as in (12.113)

-

there exists another function

("fjIr,"fjIJ) (x) ... 0,

X

E

i'.

Now

suppose

that

Pm' By the same argument A(x) has the form

~1

o where

such

> ... >

H 0 I

~i

Hence for every function

(12.ll8)

the 2 x 2 submatrix ((ad~1 Hi. .Hj)). has rank 2. o i.J~l.Z

point x E N". If {ad~i Hi ,III I (x) tuation as above. If (adit 1 Hl .H I o

)

fixed

O. i ~ 1,2. chen we are in the. same si-

po!

D

Take a

(x) '" 0 then necessarily

lad~lf{l ,Hz }(x) ... 0

O. But since Pz "'" Pl ~ P this implies by (12.116) that (ad~-kHl' ad~ Hz I (x)

o

pO

0 for Ie "'" O.l .... ,p.

Hence,

there exists another function

again for any function "fjIl

V'j

0

as in (12.112)

as in (12,113) such that (1,1'1 ,"fjIj lex) ... O.

If more integers Pi are equal then we proceed in the same way by looking at the corresponding non-singular submatrix of A. Now talce x E N~ and an arbitrary X E T~N~. By definition of N" we have X(..pj)(x) - 0 for all

..pj defined in

(12.113). Therefore

(12.119) and thus the vectors X..pj (x). j

E ~,

are all elements of

0,

for all X

(12.120)

E Tl(N}

Since dim (TxN) 1. = dim TxN - dim TxN, and dim TxN

~ dim N -

s i t follows

E ~, form a basis for (TxN)l. Since for any"fjll

thac che vectors X..pj (x), j there exists "fjIj such that w,,(X/

lPl

it

(x),X. 1 (x))

follows

w,,(X,i')

for any

po!

O.

XE

=

Y)j

that

for

Thus TxN

l!Jl i ,..pj lex) ... 0,

any

(TxN)l

X E (T x N)1.

n

there: exist

symplectic submanifold.

«TxN)l)l

Y

(12.121)

there

~

exists

r

(TxN)J. n TxN =

E TxN such that

E

such

that

implying

that

(TxN)l

(0),

wx(X,Y) ..

D.

Thus N· is a

379

Finally suppose that C(x) constants a 1

is singular in some x E N*.

Then there are

,as such that

, ..•

Iai{lPi'¥'j)(X) = 0,

j

E-'

E

im 1

, I

I8 i XJjJ (,pj) ex) ... 0, j E~, and thus that

implying that

i~l

n

This would yield

n

Ia i x1jJ

{

wx

(x),X) ~

Xl/1, ex) E TxN~.

t

~

S

IaiwX(X, (x),X) 1ml 'Pi

i

iwl

8

i~l

i

=

IaiX(¢i)(X)

0,

=

1m!

for all X E TxN* and thus w" restricted to TxN~ is degenerate, which is a

o

contradiction. Remark 12.37 functions ~}

E

j

Similarly

1/Jj'

E~,

j

it can be the

shown

non-empty

is symplectic if and only

that

arbitrary independent

for

subrnanifold

:= (x E nl1jJj ex) .. 0,

N

if the corresponding C(x) is non-singular

for all x E N (see Exercise 12.16), Example 12.38

Consider

12.26) with Ho(q,p) Then

Pi = 1,

=

E~,

i

a

simple

Hamiltonian

system

on tR

2n

~ pTC(q)p + V(q), and take J{j(q)=qj' j and

ad~o}{i

L gij (q)Pj'

= j

i

E~,

(Definition =

with

l, .. ,m

$

n.

the

gij (q)

ml

(i,j)-th element of G(q). Hence A(q,p)

~

Gll(q) , where Gll(q) is the m x m

leading submatrix of C(q), and thus A(q,p) is non-singular, implying that

i'

!(q,p) E

=

!p'znl q1

-

...

~ qm -

0,

Lqlj(q)(Pj

-

......

jul

is

a

symplectic

submanifold

" gmj (q)Pj L" glj (q)Pj, ...• L j"l

as

of 1R2n.

1/'1'"

Denoting

Lgmj(q)Pj

~

0)

jal

the

functions

ql"'"

qm'

,"-'Zm it follows that

jRl

- C,,(q)],

C(q,p) _ [ 0

C" (q)

(12.122)

S(q,p)

where S is the m x m matrix with (i ,j)-th element

Sij

(q,p)

I

(12.123)

k,P,,!

o

Note that the non-singularity of A(x) for x E N" implies that A(x) nonsingular on a neighborhood in N of every point x E N", for C(x).

is

and similarly

In order to simplify the exposition we will henceforth assume

that A(X), and thus C(x), is non-singular for all x EN. Now let us consider a symplectic non-empty submanifold N given as

380

(x E H'~j (x) - 0,

N

E ~)

j

(12.124)

for some arbitrary independent functions Y'j' j (12.113»,

such

that

the

matrix

~

E

as

C(x)

(not necessarily

defined

in

non-singular everywhere. Using the restricted symplectic form

won

(12.112»,

Xp

we can define for any F

CaJ(N)

the vectorfield

BS

N

in is

(l2.1111)

(cr.

on N by

setting (see (12.53»

(X),Z) = - dF(x)(Z) , for any Z E TxN, x E N.

(12.125)

co

In particular, denoting the restriction of H E C (1) to N as can define the vectorfield XII

X«

on N.

Ii

E d:rJ(N) we

will be different from

In general

(the Hamiltonian vector field on N with Hamiltonian If) restricted to N.

In fact Lemma 12.39 N for j

Let If E ceo (N),

~

then

Xl! on N i f and only i f (ll,1/Jj J

E s. Furthermore for any 11 E Cet) Uf)

o

on

def i118

s

a

(x) = -

i

L

(l1,l/Jj }(x).

ciJ(x)

x E N,

(12.126)

j"'l

{>'id] (clj(X»)i,]E!!..'

H"(x)

Proof

The

XII (x)

E Tl(N

the inverse l1latrix of C(x) , cf.

L

H(x) -

first

I/Ji(x)a i

statement

(x),

is

(12.114), and

x E N,

easily

(12.127)

deduced

from

the

for all x E N i f and only i f Xu (if'j) ... (H,1Pj) - 0

fact

that

on N for

j E s. The second statement follows from the fact that 5

(1I",Ylt ) -

L

{H,Th.} -

{ibi,YJk}ai -

L

1/>1 (ai,l/>k l .

(12.128)

The last term on the right-hand side is zero on N. Furthermore the second term equals by definition of a j s

g

L cHcljlll,YJj I 1.j~1

implying that

L Cucij{ll,YJ

= -

(12.129)

j )

i.jal

Ill', Y'k

)

o

on

N for

j

E s. Since

li"

~

Ii it follows from the o

first statement chac

We will now show that for

any H E

ern un

the vectorfield

on N is a

3Bl

Hamiltonian vectorfield on N. First we note that the restricted symplectic

form

wdefines

a bracket on N by setting (12.130)

(F,GII! (x)

where

XF

and

Lemma 12. ',0

XG

are defined as in (12.12S).

c''' (N)

For any F ,G E

{F,Gluex} -

I

(F,G)(x) t,

(F,V.l)(X)C

ij

(x)(lfrj ,G)(X),

(12.131)

x E N

jml

where the right-hand side is computed for any smooth extensions of F and G to

a

neighborhood of x

in H.

Furthermore

, III

is a non-degenerate

Poisson bracket: on N, called the Dirac bracket, and for any F E eIDCN) the vectorfield XF is the Hamiltonian vectorfield

Oil

N with respect to F and

the Poisson bracket ( • 111' 1. e. for any G E Cro(N)

XF (G)(x)

-

(F,Gl n (x).

(12.132)

x E N.

By (12.130) and Lemma 12.39 we have for all x E N

Proof

(F,GI!!(x) = w,,(XF(x),XG(x») = w,,(XFIl.(x),XGIl.(x)) =

{F,G) (x)

(F,GI (x)

+

+ c

I'

(F,G)(x) -

{F,~,I(x)c

lk

(x)

+ c

k.E

(x)}{F,V\.)(x)(¢l'G)(x)

=

" (x){~j,G)(x).

(12.133)

i, j"l

Clearly (12, 2lc),

the

bracket

while

(,)"

as

given

the Jacobi-identity

in

(12.131)

(12.21b)

satisfies

follows

(12.21a)

and

straightforwardly.

Thus ( , III is a Poisson braclcet. By (12.130) this implies that

w is

the

symplectic form on N corresponding to { , III' and thus by (12.125) XF is the corresponding Hamiltonian vectorfield on N for any F E C~(N).

0

382

Now

let us

(12.108)

come

back

to

the

constrained dynamics

evolving on the symplectic submanifold N".

since Hj(x) =

a

or

zero

dynamics

First we note

that

for x E N* the zero dynamics can be rewritten as m

X

-I

X1!o{x)

Xa,(x)o;(x) -1Hj(x)Xa"(x)

j~l

j~l

J

j

j=l

(12,134) n

Furthermore by the definition of Pi {cL (l2 .110)) we have for x III

tHo -

I

lT J

0;.

ad~ /ll

0, k - O,l""Pj-I, i Em

J (x)

j"'l

while for x

N

(12.135)

E

N"

rn

IHo -

I

HJo;. a.d~~][i )(x) ~

j~l

IH j ,ad:~Hl}(X)O;(X) - 0,

I

I

E~,

(12.136)

j~l

by definition of o"(x). Therefore, with

~j'

I

(In fact

E ~.

(11.28)

exactly amounts

to

(12.136),)

as defined in (12.113),

m

(llo

-IHjo:'~il(X)

-0,

(12.137)

forxEN", lES,

jPl

and thus by Lemma 12.39 we have

(12.138)

since the restrictions of Ho

I

Hj

0;

h

and Ho

to N

are clearly the same.

j~l

Using Lemma 12.40

Theorem 12.41 W

N

''ith

11',

for any and

resp. Zq

1I'''q Zq

11' ~ qZq

being the nacural projection of N X U on N, resp. U (i.e.

11',

E TqUl x U), q E /'1 x U, witlI Zq E D(q), I>'e have 1I'*qZq E D(1I'(q» =

0,

for some

11ile moreover every element.of D(1I'(q»

1..

Zq

can be I>,ricten as

E D(q) ). Then D is invariant for (13.1) (cf. Def init100

13 .1) if and only if (13 .36)

406

q E H x U.

Proof The proof is based on the following formula. Let X be a vectorfield on N,

and

let XI!

be

the

unique vectorfield on N x U with

7r"Xo - X,

7rwXc = O. Then

where X(x)

(Xl (x), ... IXn(x»

is the local representative of the vector-

field X on N, and where the last vector is taken at ehe point (x,f(x,u»

E

TH. This last vector can be trivially rewritten as x(x)

0

ax ax(x)f(x,u)

[

ax ax(x)£(x,u)

[

J .

u)X(x)

In view of (13.19) and (13.21) we thus obtain .

P.. Xo

X - [f,X]

=

1

(13.37)

•

with [f,X] - [£(',u),X(·)](x) depending on (x,u) e H x U. Now let XED. Suppose that D is invariant for (13.1), i.e. for all u [f(·,u). X(·)] ED. Then clearly the right-hand side of (13.37) is in b. proving that F"Dc C

b. Conversely, let F"Xc

this

(13.37)

implies

by

that

E

D for

[f(.,u),X(')]£

Proposition 13.S(e) (f(·(·,u),X(·)]

E

any XED. Since

e fl,

and

XED

therefore

by

o

D.

Remark If D is involuti va and cons tant dimensional then (13.36) takes the fo llowing simple form.

nates x =

2 I (X , X )

By Frobenius'

Theorem we can find local coordi-

such that D = span (~l, and thus (cL

(13.34»

b ..

axl

span the

Writing accordingly f equality

0,

,U)

which

=

1£2),

implies

then (13.36) amounts to the

local

decomposition

(13.9).

Now

let

us

proceed

associate with (13.1) the x

= f(x,u)

to

local

e,.~tended

controlled

invariance.

First

let

us

system (Definition 6.11)

I

(13.38) 11 -

Iv

,

407

which

is

(Recall

an

system with

affine

that

we

assumed

to

U

state be

space

an

open

M x U and

subset

of

I" E !Rill,

inputs !Rill.)

The

drift

f(x,u)aa _ on H X U will be denoted by to' and the distribution

vectorfield

a

x

a

of input vectorfields I-a--""'-a--} for (13.38) by Go' ul urn Consider

Theorem 13.7 1-1 xu ..... TH.

the

nonlinear system

(13.1)

system map F:

h'ith

Let D be an involutlve distribut:ion of constant dimension on

H. Assume that cbe distribution

(13.39)

on N x U has constant dimension. Then D is locally controlled invariant: i f and only i f

b +

F.D" c

(13 .40)

F"G"

Remark Notice that (13.40) may be equivalently replaced by the require-

n being

ment (with 1T: t1 xU ....

the projection)

(13 .40') Proof

By Frabenius' Theorem we can find local coordinates

a

a Wrlte . x1 'a-:-J.

i sue h tlat D - span ta-:-'" Xl

!

XI:

and correspondingly f= (fl, .. , fn

)

=

x = (xl' ... ,xn ) z ( ) ' = X):;. 1 , .. 'Xn

,XI:) , X !

1

f

,

(Xl'"

=

(f 1 , .. ,£):) ,

!

_2

r

=

(f k + 1 , .. , fn ) .

Then (13.40) is equivalent to

ai

Z

u'

1m -l(X,u) c 1m au{x,u) ,

(13 .41)

for every (x, u) .

ax

Now suppose D is locally controlled invariant. Then there exists locally a feedback u

=

o:(x,v) such that (f(o,v),D]

CD, with f(x,v):= i(x,a(x,v».

Equivalently

=

which P

=

implies

ann D.

Then

ai'.

0, or,

(l3.41).

P

is

coordinates given as

ai

Z

an ax

(13. L.2)

ax1(X'u) + iit1(x,u) -l(X,V) - 0,

an (cf.

Conversely

let

(13.111)

be

satisfied.

involutive codistribution on TH, Exercise

This implies (see Exercise 2.111)

2.8

and

(13.34»

P=

in span

Denote

the

above 2

(dx ,dx

2

).

that F*P is an involutive codistribution

on H xU, locally given as

(13.43)

408

E~,

By (13.41) there exist l1l-vectors b i (x,u). i

!5.,

I E

and thus in view of (13.43), writing b i

a

ker Fhi' = span (aX

+

i

I

,,-]

(X,U)

b S!

af2

that

Gn

F"'P

O. Since leer

-

a au;; ,

Go

Notice furthermore that ker ifUdu -

(b 1l

-

satisfying

(13.44)

, ...

,bmt)T,

atL.

au duo

i E k) + leer

(cf.

(13.45)

(13.39». Let: us first assume

is an involutive distribution the Lie brackets

F"P, and thus

are contained in ker

8b s i

brj

]

...

!!,

0, i,j E

s

Em.

(13.46)

These partial differential equations are exactly the integrabIlity con-

ditions of the classical Frobenius' Theorem (Theorem 2.45), with the only difference

that

in

(13.46)

are additional parameters xk+1'" ,xn

there

Thus by Theorem 2.45 there exists locally an m-vector a(x 1 (regarded as

(xk + 1 ,

aa

••

ax;

a

function

of

,xk

(xl'"

)

and

(v 1

""

vm )

, ••

,xn ,v1

•

,vm )

' ••

parametrized

by

,xo » such that

(x,

Moreover

v)

bi(x,a(x,v»,

for

any

x

and

v

i

the

E

Ie

matrix

all

u = a(x,v)

defines

t(x,a(x,v»

it immediately follows from (13.47) that

v) -

a

at 2

regular

aX (x,o(x,v» I

static

at].

state

+ au(X,Il(X,V»

rank

has

av(X'v)

m,

and

thus

Denoting £(x,v)

feedback.

ao

aX (x,v) i

(13 .48)

af l

-a' (x,a(x,v» Xi

+

at 2 (x,a(x,v»)b -au i

(x,a(x,v») -

0

and thus D is invariant for the closed-loop system x

i

E!5. '

l(x,v).

Z

af Now let dim ker a;:J(x, u) '" m - m > O. Since for each x the distribution

af

l

leer au(x,u)du is an involutive distribution of constant dimension on

u (x),.

U we can find for each x local coordinates

af'

a span I__

ker au(x,u}du

(13.49)

au_mH

Equivalently, there locally exists a mapping u such

that

(13.49) holds.

feedback,

u

., m(x) for U such that

l

transforming

system

o(x,u), with rank ao au

- m,

o(x,u) defines a (preliminary) l(x,u):= f(x,o(x,u». It

The mapping u = the

=

into

x

follows from (13.lll) that

aI Z

_

au-,

(13.50)

C 1m -(x,u),

1m

(U I

,urn)' and thus there exist m-vectors bi (x,u) such that

""

(13.51)

By considering the distribution (see (13.45»

rn a a (ax i ... I b,;i(x,u) I aU

span

~ ~

we now have

reduced

locally a feedback the

system

x

=

(13 .52)

i E ~I

B

the problem to

u l

=

0: \x· , vI),

1 f(x,0:1(X,V )

lil,V

,l'?) ,

the

case

1

m

E IR

Go

=

0,

Thus

exists

such that D is invari •• nt for

(u_mH , ... ,u

with

there

m

).

The

total

feedback which renders D invariant is therefore given as

(13.53)

o From a geometric point of view the above theorem can be interpreted in the following denote P

manner. =

Let

D

satisfy

ann D. Then E:= ker

the

F*P

assumptions

of

Theorem

13.7,

Horeover if (13.40) holds, then E is constant dimensional and satisfies (i.e. 1T"(X,U)E(x,u)

and

is an invo1utive distribution on t1 xU.

=

DCx),

V(x,u».

Furthermore by definition of E and by Proposition 13.5

410

F"E = F" (ker F~P)

c ker P

=

D.

(13 .55)

Hence from Theorem 13.7 we obtain (compare with Proposition 13.6)

Corollary 13. B Let D be a distribution on N as in Theorem 13.7, involucive and conscant dimensional and such

chat

Go

(see

i. e.

(13.39)) has

conscant dimension. Tllen D is locally concrolled invariant if and only it chere exiscs an involutive constant dimensional distribution E on H x U, lvi th

cons Can C dimens iona I,

EnG"

sa tisfy ing

(7r: N x U -. N

being

che

natural projection) (13.56)

7r w E = D,

F.E c b. Notice that if ~e = 0 then E - ker F~P has dimension equal to dim D, and is the unique distribution satisfying (13.56). If G"

#

0 then the proof of

Theorem 13.7 shows that at least locally we can define a (non-unique) distribution E, contained in ker F"P, satisfying (13.56) and dim E = dim D. Indeed we may take E as the distribution defined in (13.52). Horeover

I

a

distribution E satisfying

(13.56)

and dim E = dim D is

directly related to a feedback u = a(x,v) which Lenders D invariant. fact, E is

necessa~ily

E = span ( and

the

a

---a' Xl

functions

+

In

of the form

L

bSi(x,u)

~~1

b 51 (x,u)

au:a ,

1 E~},

determine a(x,v)

(13.57) by

(13.47).

Conversely

if

u = a(x,v) renders D invariant then denoting aa~

b s1 (x,u):=

aX

i

(x,v) Iv _ a-1(x,u)

(13.58)

,

the dis tribution E defined by (13.57) satisfies (13.56). From a more geolIIetrical viewpoint E is determined by a(x,v) as the distribution on H x U ~lose

integral manifolds are of the form {(x,u)

= a(x,v»lv is constant).

Let us now see how Theorem 13.7 specializes to affine systems x = f(x)

L

+ J

gj (x)u j

,

(13.59)

~l

and how we recover the results of Chapter 7 (e.g. Theorem 7.5). First we note that condition (13.40), or equivalently (13.41), reduces to

(13.60)

411

where G'l.(x) denotes the matrix composed of the last n-k rows of the matrix G(x) with columns gl(x), ... ,gm(x), It is easily seen that (13,60)

is equi-

valent to

as

[f,D]ex)

c Vex) + G(x),

[gj ,D] (x)

c D(x) + G(x},

in Theorem 7.5.

(13.51) j

Furthermore,

E

!E.

condition

(13.60)

is

equivalent

to

the

existence of m-vectors b i (x,u) such that (ef. (13.44» m

8£'

ax (x) + i

ag~

I aX i j

i E

"j

k.

(13.52)

~1

It follows that in the affine case the vectors b i

are of the form

03.53) for certain m-vectors ,Ill (x) , ... ,1m (x), and m x m matrices Kl (x), ... ,Km (x). Therefore the set of p.d.e.'s (13.47) takes the form

(13.64)

i E k.

It follows that 1(X,V):= a(x,v) - o{x,O) satisfies (13 .55)

i E ~, which implies that -y{x,v) {J(x)v,

for some m x

III

can be taken to be linear in v,

Le.

(13.66)

i E le.

has

Since Denoting o{x)

=

r(x,v) -

matrix P(x) satisfying

rank

m it

follows

that

rank

P{x)

=

m

everywhere.

a(x,O) it is concluded that the feedback which renders D

invariant can be taken of the affine form u

=

o{x) +

~(x)v,

in accordance

with Theorem 7.5. Furthermore we see that o{x) satisfies (13.57)

i E ~ ,

Finally, the integrability conditions (13.46) reduce to

aKi aX

aXj -

aX

j

i

+ KiKj - KjK t

~ 0,

i,j E k (13.GB)

ali

aX

al j

-ax j

+ Kilj -Xj1!i =0,

i,j E~,

i

which are exactly the integrability conditions for

the partial differ-

ential equations (13.66) and (13.67); compare with (7.40) and (7.37). Hotivated by Corollary 13.8 we will now relate controlled invariance

a

for

(13.1)

gene1:al nonlinear system

to controlled

invariance

of

its

exeellded syst:.em (13.38).

Proposition 13.9 Consider the nonlinear system (13.1) l"leh its extended

a

a

syseem (13.38). Denote as before fo = f(x,U) ax and Go

a

span(~ •..

'ou " m

Let D be a distribution saeisfying r::ha assumpcions of Tl1eorem 13.7.

(a)

Then D 15 locally concr011ed invariant for

(13.1)

If and only if r::llere

exists an involucive constant dimensional distibutlon E on N xU, 1f n

D and

E

constant dilllensional.

EnG"

,,,hieh

Is

I"iell

locally controlled

invariant for (13.38). i.e.

[fe' E] c E + Ge

a [~.E)

c E+

•

(13.69) j E !E'

Gel

J

(b) Conversely,

011

x

N

let E be an involutive constanC dimensional dist:ribllcion

sucll

U

chac

satisfies

E

dJlIlensional. Then D:-

1r .. E

and

n

is

constant

is a I"ell-defined discribucion on N.

l"hieh is

(13 69).

E

Ge

involucive and constant dimensional. Noreover assume that tlle distribution

Go

D

for

(13.39»

(cf.

has

constant

dimension,

then

D

is

locally

controlled invariant: for (13.1).

Proof

Part

In view of Corollary 13. B we

(a).

only have

to

show

that

(13.69) is equivalent to (13.56). By Frobenius' Theorem we can find local



2 (vI, v )

v

It x U,

for

and

N x U such

E

span

ai"

o.

~

(Xl

~).

ax l '

av 1 the

is

In these coordinates F"E c

-l(x,v)

Since 1[,,£

=

D and

that we can find coordinate

that

(~

F(x,v) = (x,f(x,v»

where TN.

(~}. axl

is constant dimensional it follows

functions

It x U

for If such that D

b

2 ,vI, v )

Denote local

is

a

coordinate

correspondingly

representation

f

of

F:

is equivalent to

o.

(13.70)

ax On the other hand from (13.69) the same equations are obtained. For part: involutive

(b) and

we observe has

that by

constant

(13.69)

dimension.

the distribution E + Go

Hence

by

an

Proposition 3.50 we can find local coordinates z"" (zl,zz.zJ,./') for such

that

~ span (_8_ ~}

E C

'

az3

and

Go

=

span

I~, ~l. az3

az~

is

application

Since

of

f1)( U

G I)

-

413

(F-"" ,F-)

span

u1

system E

for

span

=

H.

urn

it

immediately

Denote

a a (-1 '-1)' ax Bv

x

and D

1

1r~E ..

=

follows

1

2

z,

x:-

span

that

is

(21,Z2) 1 3 v:~ z,

Z

z,

a I-I

a

coordinate

Z

z

v:""

4

Then

and the result follows from

ax! '

Corollary 13.8. We

0

conclude

correspondence

from

Proposition

between

locally

13.9

that

controlled

there

invariant

is

a

one-to-one

distributions

for

(13.1) and for its extended system (13.38). Also the feedbacks required to make these distributions are intimately related.

distribution D

~ span

for

(....£....). ax!

(13.1)

and

Let now u

=

D invariant,

renders

invariant for Indeed,

(13.1),

respectively for

(13.38),

let D be a locally controlled invariant

choose

l

coordinates

2

x _ (X ,X )

such

that

o:(x,v) be a regular feedback which locally

[fC· ,v),D] CD,

i.e.

for

all

v,

where

f(x,v):-

f(x,o:(x,v». Then (x,v) is a new coordinate system for N x U. Defining in these

new

coordinates

distribution E as

the

~x)

follows that (with fo:= f(x,v)

[fo'

c

E]

[-'aV j , E[

span{~), ax!

it

immediately

(13.71)

E,

j E

C E,

:E.

Hence, E is invariant with respect to the dynamics

x - f(x,v) (13.72)

W

v

with state (x, v) and inputs related to

(13.72)

by the

I{.

The extended sys tern (13.38) of (13.1)

Cextended)

state space

transformation x

~

is x,

u - a(x,v), and in the old coordinates (x,u) the system (13.72) takes the form (13.73)

x -

f(x,v)

u

au ax(x,v)f(x,v)

=

an()_

+ av x,v w

where v is such that n{x,v) that

I{

is related to

w

=

Iv

u.

Comparing (13.73) with

(13.38) we see

ao: an()_ ax(x,v)f(x,v) + av X,v w ,

with v satisfying o:(x,v) follows

=

via the feedback transformation

that

the

=

affine

(13.74) 1

u. Denoting v - a- (x,u) (abuse of notation) it feedback

extended system (13.38) is given as

which

renders

E

invariant

for

the

Ld4

ao

ax(x,a

IV'

-1

ao

(x,u»f(x,u) + av(X'o

-1

_ (X,U»III'.

(13.75)

13.2 Disturbance Decoupling results

The

on

local

controlled

invariance

section will now be used for solving

obtained

in

the

the local disturbance

previous decoupling

problem for general nonlinear systems. Consider a general nonlinear system with disturbances q

x

£(x,u,q),

)' =

h(x.u).

U E U,

q

E

(13.76a)

Q,

(13.76b)

Here, as before, x = (Xl •...• Xn

)

are local coordinates for an n-dimension-

al manifold !-I. u are coordinates for U (the input space), which is an open and q are coordinates for Q (the space of disturbances),

subset of IRm.

which is assumed to be an open subset of Ii. Everything is assumed to be smooth. Alternatively (13.76a) is given by a system map F: H x U x Q locally

represented

as

F(x,u,q) - (x,f(x,u,q».

First

we

-!;

state

TN, the

following generalization of Proposition 4.23 regarding output invariance with respect to the disturbance q. Proposition 13.2.0 Consider

the system (13.76) Ideh system map F.

The

output y is Invarianc under q if rhere exists an involutive and constant

dimensional distribution D on H such that (i)

[f(·,u.q),D] cD,

(Ii)

F"TQ c b,

(iii)

Dc ker

d~h(·,u)f

for all (u,q)

E

U x Q,

for all u.

(Here TQ denotes tIle f.-dimensional distribution on /'f x U x Q given in

a

ah

local coordinates (x,u,q) as span laq}; and dxh(x,u):- 8x(x,u)dx.) Proof By Frobenius l Theorem we can find local coordinates x = (x ff

such

that

D - span

{_o_).

Write

accordingly

f

(£1,£2),

Z ,X )

for

then

(1)

1

8)/ implies that (13.76a) is of the form

(13.77)

Furthermore by (U) fZ does not depend on q, and by (iii) h(x,u) does not depend on xl, implying that y - h(x,u) is invariant under q.

0

The local disturbance decoupling problem (cf. Problem 7.13) consists of finding a locally defined regular static state feedback, briefly feedback, u

o:(x,v) for

=

(13.76), such that in the feedback transformed system the

outpllt y is invariant under q. Following the same approach as in Chapter 7 this will be done by looking for a distribution D on N which satisfies the conditions of Proposition 13.20 with respect

to

a

feedback

transformed

system. As in Chapter 7 the notion of local controlled invariance will be crucial in doing this. Hotivated

by

the

one-to-one

invariant dis tributions

correspondence

wi th respect

to

(13.1)

of

locally

controlled

and its extended sys tern

(13.38) (cL Proposition 13.9) we consider the extended system of (13.76a) x

~

f(x,u,q)

U

=

Iv

q -

d

(13.78)

(x,u,q),

with state the

disturbances

differentiable.

and inputs (w,d). (as

q

However

(13.78)

assumptions, 7.19), for

is

an affine system,

using

i.e.

(with fe = f(x,u,q)

the

algorithms

the

inputs

u)

for

are

assumed

to

be

disturbance decoupling

we can compute under treated in Chapter 7

constant

(e.g.

rank

Algorithm

maximal

I!'...-)

[...!'...-,EJ c E + span aU j

I!'...-)

a

[aq:;-,EJ c E + span E this

c

follows as

71":

ker dh

maximal

intersection £

•

distribution £

on N x U x Q,

satisfying

aax)

c E + span

{te' EJ

with

the

conditions

the maximal controlled invariant distribution contained in ker dh

(13.78),

Denote

as

final

will be valid for arbitrary disturbance functions.)

(Proposition 13.21) Since

well the

(Notice that for defining (13.78)

au +

span

a laq 1. a

(13.79a)

+ span laq)

~,

(13.79b)

aq

j E ~,

(13.79c)

(with h(x,u,q)

:= h(x,u».

(13.79d)

au

au +

I!'...-)

span I!'...-)

distribution

a

E

as

a

£*

and

assume

that

£"

and

the

n (span (aul + span (aql) have constant dimension. Then it

in the proof of Proposition 13.9, N x U x Q ... N

the

natural

part

(b),

projection,

is

that 1':,,£" =: D", a

well-defined

distribution on N, which is involutive and constant dimensional. We obtain

Proposition 13,21 Lee E" and V" be as above.

decoupling problem is solvable if F"TQ

en".

TheIl ehe local diseurbance

= (xl,x L )

Proof By Frobenius' Theorem we can find local coordinates x N such

that D" = span

for

[~). Write accordingly f ~ (i ,£2). Regard now axl

(13.76a) as a system with inputs u and q. Then by Proposition 13.9 D- is locally controlled invariant with respect to (13. 76a). Thus (cf. there exist m-vectors bJ.(x,u,q),

iE~,

(13.44»

and .2-vectors cj(x,u,q),

iE!5.,

such that for i E k Bf'l

aX

i

Now

af

2

8f'l

(x,u,q) + au-(x,u,q)bi(x,u,q) + aq-(x,u,q)c 1 (x,u,q) =

i/

c

F"TQ

implies

that

f2

does

not

depend

on

q.

o.

(13.80)

Therefore

in (13.BO) can be taken to be independent of q (and c i

nJ-vectors b i

the

can be

taken arbitrarily, say equal to zero), thus reducing (13.BO) co af 2

af'l

ax- (x,u) + au(x,u) b j (.'[,u) = 0,

i

k.

E

(13.81)

1

As

the proof of Theorem 13.7

in

u = o:(x,v) ker

the vectors

which renders D"

bi

determine

(or E*)

the feedback

invariant.

Indeed,

if

2

au .,., af

(13.49»

(ct.

case ker W

E c ker

0 then o:(x,v) is determined as the solution of (13.47), and in 2

au . . af

dh

0 we proceed as in the proof of Theorem 13.7. Finally since

it follows

that D" satisfies

13.20 for the feedback transformed system

the

conditions

x - f(x,v,q):=

of Proposition

f(x,o:(x,v),q).

0

13.3 Input-Output Decoupling IJe will briefly show how the approach to the input-output decoupling of square

affine

systems

as

dealt

with

in

Chapter

8

can

be

readily

generalized to square general nonlinear systems x = f(x,u),

y As

~

U

E U, open subset of ~m,

(13.82)

h(x,u),

in Chapter B we will

throughout assume

that

(13.82)

is

an analytic

system, although the results will partly also hold for smooth systems as well (see Chapter 8). First we give (compare with Definition 8.7)

Definition 13.22 Consider the system (13.82). The characteristic numbers Pj

are the smallest integers

~

- I such that for j E

E

a

L~hj"" '~aa L~hjJ (x,u) urn

au

=

0,

k=O,l"",Pj' V{X,u) E N,

(13.83)

for sO/lle (x, u) E N xU.

If

~ au Lkf h;~

Remark

o

(x,u)

Notice

that

for all Jc

for

an

~

0 and x E N,

affine

system

then h'e set Pj

Definition

=

13.22

ro.

reduces

to

Definition 8.7.

Analogously to Definition 8.3 we state

Definition 13.23 Lee (x o ,u a ) EN x U.

The system (13.82)

locally strongly input-output: dccoupled around (xo ,u D )

is said to be

i f dwre exists

il

neighborhood V of (xo ,un) such thae

a

-a u,

,

Lf

hi(x,u)

=

0,

k 'e denote the fiber bundle simply by (B,H).

Notice that a fiber bundle is only locally isomorphic to a product space 01 x U. Nonlinear systems on fiber bundles are now defined as follows.

A smooth nonlinear control

13.33

Definition

(B,H,1r,U, (° 1 ) iEI)'

is

defined

by

a

map

system on

(the

system

a

fiber

map)

bundle

F: B

-+

TN

satisfying (IT M.- TN ... N being the projection)

(13.129)

The system 1>'i11 be briefly denoted by (D ,N ,F).

Let

Xo E 01

and

around x o , and U

b o E ;r-l{xo ). =

Choose

local

(ul, ... ,u rn ) around Uo

coordinates

x ~ (Xl'··· ,xn)

:=

7r"

(0)

(14.11)

440

Step i + 1 Suppose that around (xc ,u c ) Di + K is an involutive constant Then define in a neighborhood of

dimensional distribution on T(H x U). (xo ,u o )

and stop if Di + K is not involutive or constant dimensional. The effectiveness of

the

above algorithm rests

upon

the following

observation. Lemma 14.3 Let (xo,u o ) be an equilibrium point of (14.3) and assume that

f. has full rank around (xo,u o )' Let D be an involutive constant dimensional distribution on 1'1 x U such that D + K is also involuclve and constant dimensional. Then there exists a neighborhood 0 of (xo,u o ) such that f.(Dl o ) is an invo1utive constant dimensional distribution around xo' Proof Choose local coordinates on H such that Xo - O. From the fact that f*

has

full

rank

around

(xo.u o )

it

follows

that

K is

a

constant

dimensional involutive distribution around (xo.u n ). Therefore, see Theorem

2.42,

there exist local coordinates z

=

around (xo ' un) in H x U

(zl

such that

K - span(.J!..-} :2

(14.13)

I

8z

where

Z2

is

an

m-dimensional

coordinates fez) = f(z1). Moreover around

Zo

vector.

This

implies

that

in

these

~(zl) is a nonsingular (n x n)-matrix 1

az

So, using the Inverse Function Theorem, we may introduce new

local coordinates

(£(Z1). z2)

around

these coordinates the function f

(xu. u D )

in H xU.

With respect

takes the form f(zl tZ2) -

zl

to

(see also

Exercise 2.5), and thus locally £ is a projection. In the rest of the proof we will use these coordinates and drop the bar notation. Obviously. we will have that

(14.13) holds true in these coordinates.

Next,

let

X1 ••••• X1 be a basis for D. The involutivity of K + D implies that (14.14) and the constant dimensionality of K + D, and so of K n D. implies that we may apply Theorem 7.5, yielding a basis {X1 ....• X11 for D with ,X]

ex,

i

E

1

(14.15)

441

In the above coordinates

this

Xi'

implies that the vectorfields

i E

!.

have the form

(14.16)

f.(D)

Clearly,

is

then

a

distribution

spanned

by

the

vectorfields

X~(Zl). i E!. which by the constant dimensionality of K n D has constant -2

dimension. Also,

-2

the involutivity of span{X1 , ... ,Xl) immediately follows from the involutivity of D. o

Using inductively Lemma 14.3 we obtain the following result. Corollary 14.4

Locally

around

(XU'u D)

(14.12)

defines

an

involutive

constant dimensional distribution Di + 1 ,

Theorem 14.5 Consider the discrete-time nonlinear system (14.3) about the equilibrium

point

(xu ,u a )

to a

applied

to

(xu ,u o )'

The

system

(14.3)

is

linearizable

around

controllable linear system i f and only i f Algorithm 14.2

the

syst:em

gives

(14.3)

dist:ribut:ions

Do, ... ,Dn

such

that:

dim VII .., n + m. Proof First, suppose (14.3) is feedback linearizable about (xo,u o ) into a controllable linear system (14.7). In these coordinates we find that

B) , which

by

the

(14.17)

controllability

of

(14.7)

must

have

full

rank.

One

immediately calculates that

D, - span(....£} ov D1

where If

+ ... +

(14.18.)

' 1-1

a

(14.1Sb)

If

=

1m B. Hence the involutivity and constant dimensions conditions

A

If

+ span{av1 • i - 1,2, ... ,

=

of Algorithm 14.2 hold. Moreover, that dim VII

=

the controllability of (11•. 7)

implies

n + m.

In order to prove the converse, we proceed as follows.

Let locally

around (xo , U o ) i

E n

(14.19)

442

By Lemma 14.3 and Corollary 14.4, Ai is a constant dimensional involutive in H, i E n.

distribution on a neighborhood of xo Let

Ai C A1+1 •

n

with set Pi - mi - m1 - l , i E !!" + m one obtains the existence of a n such that dim A n. Lemma 6.4 applied to the

.. dim A1

mi

i E

,

mo - O. From the fact that dim Dn minimal

number

1t.:S

Clearly we have that

and

-

n

K.

sequence of distributions A1 C A2 C , •• C A" yields local coordinates x around

such that

Xo

i E "

(14.20)

J

where dim Xi - Pi' 1 E ~. With respect to the above coordinates x we write f(x,u) -

(f\x,u) •...

,r(x,u»T

accordingly.

We

investigate

next

the

particular structure of f with respect to the distributions Ai' i E ". We see (14.19)

have.

and (14.12),

that f .. Do - 6 1

which implies that in a

,

neighborhood of (xo.u o ) spant

i E

span(~1

!:!)

(14.21)

axl

j

. Id'Lng yH!. floDl

-

af x. u ' " au (

)

0 f or J. - 2 , ... , IC, an d

a 1 so rank

~ Pl'

Similarly,

A2 gives

(l4.22) a~

from which we obtain - ( x , u) - 0 for j - 3, ...

ax 1

,.JC

A repetition of the above argument, using fwDi

1

and rank -

61

a~

- 1 (x

ax

I

u) -

P2'

i E ~I yields the

,

following form for f:

l (X I ,X 2 , ... ,x",u) 2 £2 (Xl • ."1. • ••• • x") f

[(x,u) _

f3(X'2. ••••

,x")

(14.23)

Note that this is exactly the form as obtained in (6.35) or (6.77). Next 8f!

we

exploit

the

fact

that

rank --:t=t(x,u) - Pi' i

ax

E~.

in

order

to

successively change the coordinates (Xl, ... ,/'") (analogously to the proof of Theorem 6.12).

Observe first

that Pi

~ PU1

for 1

step we introduce new coordinates (Zl •.. "Z~) via zj - x -

([IC(X"-l ,x"), ;rIC-I)

where

i"-l

are

Pk-l - Pk

E r;;. j

,

In the first

for j

components

¢

,,-1 and of

X"-l

443

cn

D* = 'frnE~ We will

(14.45)

now briefly discuss

the

Disturbance Decoupling Problem in

discrete-time. Consider the system x(k+l) { y(k)

where

the

=

f(x(k), u{k), q(k»

h(x(k),u(k»

system

map

f: H x U x Q

-