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


rank J(Ol ,(}z)

~

point

,°7.>

(° 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

<

this

e<

~/2,

description

is

cosr/J{cosO)

only

]

[w'Wz ]

~1

(1.13)

W3

locally

valid

in

the

but it serves to shoW that the equations (1.12) evolve on

a three dimensional space (which in fact is the Lie group 50(3) real

region

orthogonal

matrices

obviously depend on how

with the

determinant

spacecraft

is

1).

The

dynamic

controlled.

We

of 3x3

equations

consider

two

typical situations.

I. Gas Jet Actuators Let J be the inertia matrix of the spacecraft, h the angular momentum of b1

the •

spacecraft with

b z ,'" bm

magnitude

II· I

the

IIb111ul

denotes

axes

respect about

to

which

the the

inertial

axes

corresponding

81 ,

82

control

,

e3

•

torque

and of

is applied by means of opposing pairs of gas jets. Here

the standard Euclidean norm on 1J?3.

Using a momentum balance

6

about

the

center of mass

one obtains

the

dynamic equations

for

the

controlled spacecraft as

(1. 14)

II. Homentum Wheel Actuators We assume thae we have m wheels with the i-th wheel spinning about an axis hi' which is fixed in the spacecraft, such that the center of mass of

-Ilb i /lui

the i-th wheel lies on the axis hI and a torque i-th wheel

about

the

axis

bi

by

a

motor

Consequently an equal and opposite torque

fixed

/lb i Ilu i

in

is applied to the the

spacecraft.

is exerted by the wheel

on the spacecraft. Then, a more complicated momentum balance yields m

1 1i i

a

(w+V 1 ) + J*w

&

Rh.

h - 0,

(1. 15)

1

(1.16) where J* is the inertia matrix of the spacecraft without wheels, 1i is the inertia matrix of the i-th wheel, 11 is the total constant momentum of the system, hi is the angular momentum of the i-th wheel both measured with respect to the inertial frame e 1

,

e2

eJ

,

,

and

vi

is the angular velocity

of the i-th wheel relative to the axes r 1 • r z • r 3 • Assume that hi is a principal axis for wheel i and assume the i-th wheel is symmetric about bi

.

Then Ji

Ji +

-

Os. -Ji

},

where J 1

..

bi

bJ]dllbl !12

moment of the i-th wheel about the axis bi

•

and]1 is the inertia-

Clearly

1J. -Ji is a positive

semi-definite matrix so we may define a positive definite matrix J via m

J - J*

1 (1

+

m i

-J1 ). Let v -

lJi(w+V i

),

then (1.15) reduces to,

t"l

Jw + v - Rh,

(1.17)

and from (1.16) we obtain (LIB)

Differentiating (1.17) and substitution of (1.12) and (l.lB) yield the follOWing closed set of equations describing the control system

k {

CD

-RS(w) ,

J~ - -RS(w)h h .. O.

(1.19)

7

Both spacecraft attitude control models (1.14), respectively (1.19), show that

the

state

dynamics

space J

SO(3)x m

of here

-

are

typically nonlinear

resp.

(1.14)

matrices with determinant 1 appear (where R -

(r jk

)

(1.19)

denotes

50(3)

the

Lie

for

two

equals group

reasons,

the of

namely

Cartesian

3x3

real

the

product

orthogonal

and in both models nonlinear terms wirjk

with j ,ic ". 1, 2 I 3}. Both phenomena are essential

in a further analysis of the controlled spacecraft. Next consider again the model with gas jet actuators. It is easily seen that

forms

(I 3 ,0,0)

(R,w,u)

an

equilibrium

for

the

system

(1.14).

Linearizing the dynamiCS (1.14) around (I 3 ,0,0) yields

R - a (1.20)

Obviously, essential stability,

this

linearized

features

of

the

model

(1.20)

original

controllability,

etc.

does

model

This

not

reveal

(1.14),

shows,

that

like

any for

for

of

the

instance a

better

understanding of the controlled spacecraft, one has to develop a nonlinear analysis rather than just studying the linearization of such a model.

0

Example 1.3 (Control of a Closed Economy) The following equations describe the evolution of a closed economy in discrete time. Y(k+1)

Y(k) + a(C(Y(k))+I(Y(k) , R(k), K(k))+P(k)"G(k)'Y(kl)

(1. 21)

R(k+1) - R(k)

+ p(L(Y(k), R(kl)-P(k) "H(k))

(1.22)

K(k+1) - K(k)

+

(1.23)

I( Y(k), R(k), K(k))

F(N(k) , K(k))

(1.24)

N(k) - H(fI(k) , P(le))

(1.25)

Y(k)

In this model the quantities have the following interpretation: Y

real output

C

real private consumption

I

real private net investment

R

nominal interest rate

K

real capital stock

P

price level

G

nominal government spending

8

L

real money demand

H

nominal money stock

N

labour demand

W

nominal wage rate

a and

p are positive constants.

Equation

(1. 21)

is a dynamic

IS

(Investments-Savings)

equation and

(1.22) is a dynamic LH (Loan-Money) equation. The capital accumulation is described via

the

dynamic

Keynesian

equation

(1.23).

Equation

represents a macro-economic production function and 0.25)

(1.24)

defines the

labour demand as a function of the real wage rate. The equations (1.21-25) typically describe a dynamic economic system. To bring it into the form of a control system we have to distinguish control variables and to-be-controlled variables ("outputs"), One way to do so is as follows. Interprete G and H as the "controls" of the system (which in an economic context are labeled as instruments or instrument variables), W as a known exogenous variable (so a prescribed known control function)

and the Teal output Y and the price level P as

the

target

variables (the to-he-controlled variables), To bring the model (1.21-25) into a state space form, one rewrites the equations 0.24) and (1.25). Suppose

(Y,R,K,W,G,N,N)

is

a

particular

steady-state

solution

of

(1.21-25). Then the relation (1. 26)

N"" Her",p)

holds at the steady state (N.W,P) and provided all - -

(1. 27)

8pUv,P) '" 0 ,

we may locally apply the Implicit Function Theorem yielding locally P as a function of Nand W. say P ... Jj (fv •N),

P - H(W,N).

which satisfies Y .. F(N,K)

which holds at N

with

(1. 28)

Similarly, the relation (1. 29)

I

(Y,N,K)

may locally be transformed into

F(Y,K).

N - F(Y,K),

(1. 30)

provided that

9

aF(N- K') " 0

aN' Assuming

that

(1. 31)

.

(1.27)

and

(1.31)

hold,

we

find

the

[Dr

second

target

variable P(k) - HW(k) ,N(k»

Altogether we have obtained Y(k+1)

locally -

fl (Y(k) ,R{le) ,K(Jc)

R(k+l) {

(1.32)

- H("(k) ,F("(k) ,K(k»).

a model of the following form

,rICk) ,G(k»,

- f, (Y(k) ,R(k) ,K(k) ,"(k) ,1I(k»

(1. 33)

,

K(k+l) - f J (Y(k) ,R(k) ,K(k»

Q,(k) { Q2

- Y(k), (1.34)

- P(k) - i(W(k),f(Y(k},K(k»),

(Ie)

where Q1 and Qz denote the target variables and the functions f3

follow from (1.21-23)

and (1.32).

f1'

fz and

Therefore the model of the closed

economy as described here is a set of difference equations on the state space (Y,R,K) together with output equations given by (1.34). Note that in this

3

the state space may not be m

example

but rather some nontrivial

region in 1R3. As is clear [rom the definition of the functions fJ'

(see (1.21-23)

f1'

fz and

the dynamics (1.33) are typically nonlinear, which can

not be avoided even by assuming a simple structure on the functions C, L and I.

Although almost always in the economic literature,

when dealing

with a model of this type, one directly starts with the linearized version of the model described by analysis

incorporating

(1.33)

the

and

(1.34),

nonlinearities

is

it

seems

that a

necessary

for

a

further closer

o

study.

Example 1.4 (A Model of n Mixed-Culture Bioreactor) Let us study a model of the dynamics of a culture of two cells trains that are differentiated by their sensitivity to an external growth-inhibiting agent. based on

a

description of

micro-organisms mixed-culture

inhibitor

al tered

by

bioreactor

resistant

the

cells

unstable

fermentations

recombinant-DNA

we

distinguish

and

the

two

inhibitor

techniques. cell types ,

sensitive

cell-densities will be denoted as Xl' respectively xz' and I

The model

that

is

occur with In

such

namely

cells.

a the

Their

In addition, let S

represent the concentration of rate-limiting substrate and inhibitor

in the fermentation medium. The interactions of the two cell populations are illustrated in the following diagram.

10

Substrate

/

,

X

Xl

de-.ctivat~

z

/mibition

:I

Inhibitor

Fig. 1.4. Dhl!lmm of IWI) cell populations.

We consider a continuous mixed-culture chemostat of fixed volume with constant

inlet

parameters

in

substrate the

model,

concentration of the

Sf.

concentration namely

the

inhibiror It:.

There

dilution

rate

are

two

D and

control

the

After a certain residence

Ii

inlet

time

the

model takes the following form (using material balances of the chemos tat) Pl(S)X 1

(1.35)

Pz (S. I)x2

-px1I where

/S

the growth rate of species 1 ,

"" m'

Jll(S)

K :.::.L-

the growth rate of species 2 •

K1+I'

111, I?,

K. Kr

are specific constants describing the growth rates and p a

cons tant reflecting the

rate proportional to

Xl

I

wi th which

inhibi tor-

resistant species deactivate the inhibitor.

For

D.

the dilution rate ,

Uz =

DIt:

the total inhibitor addition rate ,

If.

the inlet inhibitor concentration

S£

the (constant) inlet substrate concentration

Yt

the yield of species 1

Yz

the yield of species 2

the

analysis

above

model

(1.35)

(which is beyond

one the

can

work

scope of

.

out

a

complete

this book).

A few

steady-state interesting

things about the model can be immediately stated. It seems reasonable to impose the condition that (1. 35) has an equilibrium point (x~ .x~ .1°)

in

11

the positive orthant

xl

> 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 <

2

1+(1+n)

u(· )

and define

2

on the interval [O,T] by

n..;.:o

-{:

u(t)

,

an :S. t

<

bn

bn :S. t

<

an + 1

Then the solution x(t.O,O,u)

is well defined for all

t

[O,T)

E

x(T,O,O,u) does not exist.

but 0

Solutions of (1. 42) which are defined for all t are called complete. From the above example we may conclude that further

restrictions

on

the

admissible controls have to be imposed in order to guarantee completeness or one has to be content at first instance with local small time solutions of (1.42). Another interesting phenomenon ts that the setting as presented so far does not directly cover the Examples 1.1, 1.2 and 1.4. The essential observation is that the state space and/or the input space and output space in these examples are not necessarily Euclidean spaces but rather manifolds (see Chapter 2).

,°

For instance the state space of Example 1.1

consists of (8 1

,0 1 ,0 2

and 01 and O2

the corresponding angular velocities. Clearly 0 1 and 02

belong to

(-1I",7I'J

2 ),

with

01

rather than IR,

and 82 the angles defined in figure 1.1 and a point 0 + Ic·271'. k

E

I,

will be

identified with 8. However in understanding the solutions of differential equations on such a manifold no difficulties arise because one can equally well consider the controlled differential equation in (1.1) on an open neighborhood of ~n and thus interprete the solutions of such differential equations as a solution defined on a neighborhood in (Rn. This is in fact the

process

of using

coordinace

charts

for

a

manifold,

as

will

be

extensively dealt with in Chapter 2. When a solution of the differential equation tends

to leave

the neighborhood under consideration.

another

neighborhood may be taken on which again (1.42) is considered. A very simple example may illustrate this. Example 51 -

1.6

(A

system

on

51)

Consider

the

I-dimensional

[(X1 'X2 )/X;+x; - 1) with unit tangent vector at a point

(Xl

sphere

,x2 )

E 8

1

15

N,

Fig. 1.6. The sphereS l ,

Consider on 51 the control system d

(1.45)

dt

Because 51 is a I-dimensional manifold this control system can also he

described in a local fashion

85

in (1.40). As neighborhoods we take NI and

Nz • see figure 1.6. and the control system reads as

o-

(1. 46)

U,

with the constraint that 8{t) belongs to Nt or Nz . When a solution leaves Nt

one

continues

to

consider

the

differential

equation

on Nz

and

50

o

forth.

There is a

particular class of continuous

will often consider in this hook.

That are

time nonlinear systems we the input-linear or affine

systems which are described as follows m

x(t) - f(x(t)) +

I

(1.47)

,.,g, (x(t))u, (t),

together with some output equation only depending on the state. In (1.47) we

assume

f,

gl""

,gm

to

be

smooth

mappings

from

~n

into

distinctive feature of these systems is that the control u appears linearly (or better, affine) in the

(u 1

mn. , •••

The ,ull!)

differential equation (1.47).

This type of control system is often encountered in applications, see for instance the examples at the beginning of this chapter. We

remark

that

everything

which

time-invariant systems of the form

has

(1.1)

extended to time-varying nonlinear systems

been

stated

in principle

so

far

for

can directly be

16

x(c) - f(x(t),u(t),t), (1. 48)

y(t) - h(x(r),u(t),t). The trick is to extend the state space of (1.48) with the time-variable t, namely to (1.48) we add the equation

i:. ... 1.

0.49)

Then (1.48) toget:her with (l.lJ9) forms a system of the form (1.1). Let us end the discussion of defining continuous time nonlinear systems with some comments. Considering the controlled differential equation in (1.1)

we

basically

deal

with

a

syst:em

described

by

t:he

following

commutative diagram

Fig- 1.1. The control system x= f(x,u) on eRn.

where (ld,f)(x,u) - (x,f(x,u», this

can be seen as

~(x.u) -

x and

~l(X,Z)

the local description of a

-

x. Mathematically

control system on a

manifold, while a global description is as follows:

x

Fig. l.8. The control systcm = f(x,u) on M.

where H denotes the state space manifold, fibers

11'

-\x), x E H.

tangent space of J-l (TH

~ U

: B

~

rr a fiber bundle whose

TxN , where T:r.B is the tangent space at

consist:ing of all velocity vectors at

TN

~

denote the state dependent input spaces. TH the

X

x in

and u stands for union over all

H the canonical projection of TN on H, and F :

H X

TH represents the dynamics of the systems, i.e. for any point (x,u) in B, in H)

I

and

11'1

-+

B ....

f(x,u), where F(x,u) - (x,f(x,u», is the velocity vector at the point x E fl. Note that locally (i.e. using local coordinates for the manifolds) this

representation is precisely as given above. The mathematical description given by

the

commutative

diagram

in

figure

1.8 has

some

interesting

17

advantages; control

in particular when studying global questions for a nonlinear

system.

f'Ioreover

there

are

examples

which

can

be

described

correctly in a global manner only by using this framework.

Example 1.7 (A system on TSz) Consider a spherical pendulum with a gas jet control which is always directed in the tangent space. We suppose that the magnitude and direction of the jet is

completely

adjustable

within

the

z

tangent plane. In this situation the state space is TS , the tangent space of the 2-sphere 52, plane

p

at

to

i.e. TS

the

z _

sphere

U T p S2, 52,

the union of all T p S2,

Let

TS Z

11"

---7

52

be

the tangent

the

canonical

projection, then B is a fiber bundle over TS'l where the fibers are defined

as

follows.

In each point x

E TS'l

the fiber ahove x

1

equals

rr- (rr(x».

Notice that in this way the manifold B locally is diffeomorphic to TS'lx but B itself is

not

diffeomorphic

to TS'lx

Observe

ill'l.

that B

=

(R'l

TS2X [R'l

would imply that the control system could be written as a smooth system

x-

fex) + gl (x)u 1 + g2 (x)u'l' however gl (as well as g'l) has to vanish at some point x ("you cannot comb the hairs on a sphere"). This illustrates

that

the

state-manifold

and

input-manifold

not

appear

as

the

usual

o

Cartesian product.

In many

cases,

however,

the

bundle, i.e. equals a product

fiber

bundle

1r

B

:

~}l

is

a

trivial

x U for some input space U. In this case

}l

an alternative but equivalent global description of the continuous· time nonlinear

control

system

(1.1)

is

provided

by

defined) vectorfields on the state space manifold inputs

u E

throughout

U.

In

the

subsequent chapters.

fact,

this

will

be

the

a }l,

setting

family

of

(globally

parametrized by the that

will

be

used

Only in Chapter 13 we will give a

further discussion on the global setting as depicted in figure 1.8 for a general bundle

1r

:

B

-+

H.

So far we have discussed various aspects of nonlinear systems described by (l.I). Let us next briefly concentrate on the dynamical behavior of the dynamics

(1.1)

in case the input u is identically zero

(or equals some

interesting constant reference value). The dynamics then reads as x

f(x,D) =: [ex)

(1.50)

which in case of a linear system (1.38) yields the linear dynamics

x

Ax .

(1. 51)

There are several features in which the nonlinear dynamics (1.50) and the

18

linear

ones

equilibrium

(1.51) points

mny of

differ.

(1.50)

A first

and

distinction

(1.51).

A point

occurs is

Xo

in

called

the an

equilibrium point of (1.50) 1f [(xo) - 0, which is equivalent to the fact

x{t)

that

~ Xo

is

a

solution

of

the

differential

equation

(1.50).

Obviously. the set of equilibrium points of the linear system (1.51) form a linear subspace

of the

state space,

whereas

the system

(1. 50)

may

possess several isolated equilibrium points. As an example, one could take the I-dimensional system

(1.50)

with [(x) - x(l-x),

having equilibrium

points at x - 0 and x - 1. Besides the difference in structure of the set of equilibrium points of (1.50) and (1.51) a similar difference appears in the periodic orbits of the systems. The system (1.50) is said to have a periodic solution of a period 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<:) ) ,

Urn

(1. 53b)

Yp (Ie)

Again we assume f

:

[Rnx

IR

m

IR

-+

n

and h : !J?"x IR

m

....

IR P to be smooth,

though

this is in what follows not very essential. Together with (1. 53a, b) we have to specify a class of admissible controls 11,

which for a discrete

time system may be any set of time functions u ; "l. ....

111

m

or u : 1+ ....

IR

m

that is closed under concatenation. The important observation is that (in contrast with continuous time systems) the difference equation x(l,H) .. f(x(1c) ,u(1c» , (1. 54)

xO'

x(k o )

admits a well defined (forward) solution x(k), k 2: Ico • for any admissible E 'U and any arbitrary initial state Xo E (Rn. As before such a

control u(·)

solution will usually be denoted as x(k,ko'xo'u) (and similarly the output will be wri tten as y(k. ko .xo . u». of

(1. 5 t l),

discuss

solutions

systems

straightforwardly

output

spaces

(e.g.

No

extends

manifolds),

to

needed to

Lipschitz condition is

of course

the

more

We

setting general

refer

to

for

discrete

state-,

Example

time

input-

1.3

and

for

an

illustration. We conclude this chapter with the remark that discrete-time nonlinear dynamics exhibit similar phenomena as their continuous-time counterparts such as,

for

instance.

isolated periodic orhits and "strange"

positive

limit sets.

Notes and References

We have discussed some

examples of typically nonlinear control systems

arising from different sources. These examples have been chosen in order to provide motivation for

the development

of a

nonlinear systems in the next chapters. Example robot-arm configurations. well

as

an

account

of

For a

further

more

structural analysis

discussion on this example,

advanced

of

1.1 is one of the simplest

manipulators

we

refer

as to

[AS), [era]. {Pal. A detailed exposition of Example 1.2 has been given in [Ar2]. [CB], [Crl], [Gr2l. [NvdS}, [SGl. tak.en from [UK]. see also [Nij

J.

The

economic

Example

1.3

has

been

Example 1.4 on a mixed-culture bioreactor

is extensively discussed in (01) .IHK]. Example 1.5 may be found in [Su1. For general

information on existence and uniqueness

of differential

equations we refer to textbooks as [Ad], [eLl I [HSj. A further discussion

21

about the possibly complicated behavior of dynamical systems may be found in for instance [Ar2], [GHJ, [HS). For more background on the formulation of

nonlinear

control

systems

on

manifolds

(including

systems

modelled

on

fiber bundles) we refer to e.g. [Suj,[Brj,[Loj,[Wij,[vdSj.

[Arl]

V. I.

Arnold,

Ordinary

differential

equations,

HIT

Press,

Cam-

bridge (HA) , 1980.

[Ar2 J

V. I.

Arnold,

Methodes

mathematiques

de

la mecanique

classique,

Editions HIR, Hascall, 1976.

[AS)

[Br]

H. Asada, J,J.E. Slotine, Robot analysis and control, John Wiley & Sons, New York, 1986. R.W. Brockett, "Global descriptions of nonlinear control problems; vector bundles

[CL] {Cra) [CB]

[Cr1] [Cr2]

[GH]

[HK]

[liS) [La] {Nij]

[NvdS]

[OlJ

[Pal [vdS]

[Su] [SG]

and

nonlinear control

theory",

Notes

for

a

CBBS

conference, manuscript, 1980. E.A. Coddington, H. Levinson, Theory of ordinary differential equations, Hc Graw-Hill, New York, 1955. J.J. Craig, Introduction to robotics, mechanics and control, Addison Wesley, Reading, 1986. P.E. Crouch, B. Bonnard, "An appraisal of linear analytic system theory with applications to attitude control", ESA ESTEC Contract report 1980. P.E. Crouch, "Application of linear analytic systems theory to attitude control", ESA ESTEC report 1981. P.E. Crouch, "Spacecraft attitude control and stabilization: applications of geometric control theory to rigid body models", IEEE Trans. Aut. Contr. AC-29, pp. 321-331, 1984. J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of vectorfields, Springer Verlag, Berlin, 1983. K.A. Hoo, J.C. Kantor, "Global linearization and control of a mixed-culture bioreactor with competition and external inhibition", Hath. Biosel. 82, pp. 43-62, 1986. H.W. Hirsch, S. Smale, Differential equations, dynamical systems, and linear algebra, Academic Press, New York, 1974. C. Lobry, "Controlabilite des systemes non lineaires", SIAN J. Contr. 8, p.p. 573-605, 1970. H. Nijmeijer, "On dynamiC decoupling and dynamic path controllability in economic systems", Journ. of Economic Dynamics and Control, 13, pp. 21-39, 1989. H. Nijmeijer, A.J. van der Schaft, "Controlled invariance for nonlinear systems: two worked examples", IEEE Trans. Aut. Contr., AC-29, pp. 361-364, 198!!. D.F. Ollis, "Competition between two species when only one has antibiotic resistance: Chemostat analysis", paper presented at AIChE meeting, San Francisco, 1984. R. P. Paul, Robot manipulators: mathematics, programming and control, lHT Press, Cambridge (HA), 1981. A.J. van der Schaft, System theoretic descriptions of physical systems, CWI Tract 3, Centrum voor Wiskunde en Informatica, Amsterdam, 1984. H.1. Sussrnann, "Existence and uniqueness of minimal realizations of nonlinear systems", Hath. Systems Theory 10, pp. 263-284, 1977. J. L. Synge, B.A. Criffiths, Principles of mechanics, lkGraw-Hill, New York, 1959.

22

[Wi] [WK]

J .C.

Willems, "System theoretic models for the analysis of physical systems", Ricerche di Automatica 10, pp. 71-106, 1979. H.W. Wohltmann. W. Kromer, "Sufficient conditions for dynamic path controllability of economic systems", Journ. of Economic Dynamics and Gontrol 7, pp. 315-330, 1984.

2 Manifolds, Vectorfields, Lie Brackets, Distributions

In the previous chapter we have seen that many nonlinear control systems x

f(x,u)

y

h(x,u)

(2.1)

are properly defined on a state space which is not equal space 1J?1l, but instead is a curved n-dimensional subset of !R called a manifold.

As a consequence,

the equations

(2.1)

to Euclidean m for some m,

usually do not

describe the system everywhere, but only on a part of the state space, and

for

a

different

representation

part

of

of

the

the

state

system

in

space

we

equations

generally

like

need

(2.1).

In

another

geometric

language we say that (2.1) is a local coordinate expression of the system we wish to describe, and that in order to cover the whole system more than one coordinate expression is needed. For patching together these different expressions the notion of coordinate transformation is instrumental. In the first part of this chapter (Section 2.1)

we will

develop

the

mathematical machinery to do this in a proper way. The approach taken here enables us

to define a nonlinear control system on a curved state space

independently

of

any

choice

of

local

This

coordinates.

so-called

coordinate-free viewpoint is often illuminating, and can provide shortcuts in

calculations

which

may

be

very

tedious

in

local

coordinates.

The

material covered in Section 2.1 is not easy to grasp at first reading. On the other hand,

in order

to be

able

to

read

Section 2.2

and

the next

chapters it is not really necessary to understand this material in full detail.

In fact for most of the chapters to follow a rough understanding

of the main ideas will be sufficient. Therefore we give before Section 2.1 a rough and intuitive survey of the material covered. One is advised first to read

this

survey and

to pass

swiftly over Section 2.1,

and

then

to

re-read Section 2.1 at later occasions. For

the

different

second part of path.

Since

the

this

chapter

material

(Section

is

much

2.2)

more

at

we the

will

follow

heart

of

a

the

contents of this book we will first go through it in reasonable detail, and give a short summary of the material afterwards (Section 2.3). In Section 2.2 we will dwell more specifically upon the properties of sets

of

manifold,

ordinary which

differential in

a

equations

coordinate-free

(without approach

inputs) are

defined described

on

a

a,

24

We show how under certain condi tlons

vee torfi elds .

there exist proper

choices of local coordinaces in which the vectorfield or the collection of vec torfields take an easy form, which is very much amenable for further analysis. These tools will prove to be instrumental for much of the theory developed in the chapters that will follow. 2.0

Survey of Section 2.1

Manifold

Consider a topological space tJ (Le., called open).

we know what subsets U c N are

Suppose that for any p E H there exists an open set U

containing P, and a bijection ~ mapping U onto some open subset of ~n for some

fixed

r l (a 1

•••• • a n )

n.

On 81 ,

-

IR i

n

we n

E

.~

have

(1, ... ,nl.

E~,

coordinate functions xl' i

the

natural

coordinate

functions

By composition with tp we obtain

on U by letting

i En.

(2.2)

In this way the grid defined on tp(U) c ~n by the coordinate functions r i i

E~,

I

transforms into a grid on U C H

Fig. 2.1. A coordinate chllrl.

The open set U together with the map tp is called a coordinate chart and is also

denoted

(U,x 1

as

1"-

,xn ).

On mn

there

is

a

natural

notion

of

differentiability, and we would like to transfer this nocion to rt, so that we can talk about differentiable functions on N. In order to

do

have to impose extra conditions on the coordinate charts (U,IP) First

of all

we

require

homeomorphism (~ and

IP-

1

that

with U n V

7'1

any

chart

(U.~)

the

map

we

above.

IP

is

a

are continuous). Secondly we require that all

charts are (C""-}compatible. (V,1/!)

for

this as

IZI the map

This means

that for any cwo charts

(U,IP).

25

(2.3) m is smooth (C ), i.e. derivatives up to arbitrary order exist.


Fig. 2.2. Coordin
Let

2i

Zl, •• ,,2 n

=

r i 01/>,

be

1 E:!.

the coordinate functions

on V corresponding to

i.e.

1/1,

The map S is called a coordinate transformation

coordinates x = (SI""'Xn ) to Z = (Zl"",ZIl)}'

(from

in fact we can write

(2.4)

i En. Finally,

N

together with its

(compatible)

coordinate charts is called a

smooth manifold.

Maps and functions Let NI F:

ttl ...

and Nz be smooth manifolds of dimension 11 1 , respectively Hz

is called a smooth map,

briefly map,

/1 2

if for any pEN!

Then

'

there

exist coordinate charts (U,
such that the map (2.5) is a smooth mOlp. We call F the expression of F in

local

coordinates,

or

local representative of F. Usually we omit the caret, and write F in the local coordinates

(Xl""

'X"l)' (Zl""

,zn2)

corresponding

to

!p,

resp.

V',

as

F(x) Note

that

(Fl (Xl""

(xl""

,Xn 1)

,x n1 ),··· ,F"2(XI""

here

is

'~l»)i

i.nterpreted as

(2.6)

a

point in IR

'"

i.e.

the

26

local coordinates of some (not specified) point p E HI . The rank of the map F at a point p

is defined as

Jacobian matrix of the map (2.6) from ~nl to is bij ective, and F and FAccordingly, a map [: H

-+

1

the

rank of the

calculated in ~(p). If F

nz

m

are smooth then F is called

diffeomorphism.

II

IR is called a smooth function, briefly function,

if for any pEN there exists a coordinate chart (U,ip) about p such that [

ip(V) c mn

fOlp-l:

=

-+

IR

(2.7)

is R smooth function. Usually We omit the caret in the local coordinate expression [

and denote

it as

[(Xl""

Xl I .••• Xn

where

IXn) I

are

local

coordinates.

Submanifolds Let l1 be a smooth manifold of dimension n. A non-empty open sel: V cHis itself a smooth manifold of dimension n with coordinate charts obtained by

restricting

the

coordinate

submanifold of H. III

< n

if

(V,Al

I'"

P

for ,xn

for

N to

each

PEP

there

exists

(q E V

I

Xl

(q)

= Xi

(p), i

=

is

adapt:ed

an

called

open

an

coordinate

chart

(2.8)

the open sets on P are of the

P n V, with V open in H), and coordinate charts (P n U,

(U,ip) coordinate chart for H satisfying (2.8) and to P,

V

submanifold of dimension

lIl+l, •.. ,nl

If we take the subset topology on P (i. e. form

V.

for N about p such that

)

n V

charts

A subset P CHis called a

~Ip),

with

~Ip the restriction of ~

then P is itself a manifold. Notice that if

Xl

I

•••

IXn

are adapted

local coordinates for M then Xl ""'Xm are local coordinates for P. An F:

I'll

important fJ z

be

a

theorem

concerning

smooth map with

submanifolds

dim

N1

=

nl

I

i

is =

the

1,2.

following. Let Pz

E

Let

lIz

and

1

suppose tha t F- (pz) is nonempty, and suppose furthermore that the rank of F in every point of F-1(pz) equals the dimension of Hz. Then

(2.9)

Tangent space Let 1'1 be a smooth manifold of dimension n.

peN

is geometrically clear:

The tangent space l'p N in

27

/' p

Fig. 2.3. TnngcllI space T p l>l

define T/l as

Formally we

ill

P E M.

of derivations;

the linear space

Xp E TpN acts on functions f

an element

defined in a neighbourhood of p, i.e.

(2.10) Geometrically Xp (f) direction Xp'

c(O)

p and c' (0)

=

be

identified

a

a

Tn

"

functions for p,

about

Xp

=

{-a I ""'-a-/ }, r 1

the

directional

otherwise, '

let

c:

H be

->

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
rf<1, cf)

if each of

its slot functions

(2.20) is C
w C ,

C-).

although

large enough.

Henceforth we will mainly work in the smooth (c''')

everything can be

adapted

to

the

ck_setting,

with

k

Sometimes we will make some additional remarks about the

cf'-case. Let us now define the notion of a smooth manifold. The basic idea is

that a smooth manifold is a set which locally can be identified with together with First

we

need

the the

,0

intrinsic notion of differentiability defined on

mO.

particular

the

notion

of

a

topological

space,

in

technical notion of a Hausdorff topological space with countable basis.

Digression

Let H be a set. A topological structure or topology on H is a

collection T of subsets of 11 satisfying (i)

the union of any number of subsets in T belongs again to T.

(ii)

the intersection of any finite number of subsets in T belongs to T.

(iii) 11 and the empty set belong to T. The elements of T are called open sets of fl,

and complements of these

closed

topology

sets.

The

set

H

together

with

the

topological space. A basis for a topology T on

n

T

is

called

a

is a collection BT c T of

open sets such that every open set can be written as a union of elements of BT

(Example: a basis for the usual topology on R are the open finite

intervals (a, b), a, b E !R.) A neighborhood of a point pEn is any open set

30

which contains p. A topological space 1-1 is Hausdorff i f any two different points Pl and Pz have disjoint neighborhoods. A mapping F: H] ... H2 between two topological spaces is called continuous if P-l(D,-) is an open set of Hl

for

any

open

set 02,

of

f/;>..

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\
~

n

is a diffeomorphism.

n, and that f 1

Then

there

, •••

exists

,f", k::;;

a

0

11,

are

neighborhood U

such that (U,fl, ... ,fk,Xktl'''''X,,) is

a coordinate chart.

Proof

Take

permuting

an

xl' .••

arbitrary

coordinate

chart

(U ,Xl""

,X" we can ensure that fl , ... ,fk

,X"tl"

,X,,) ..

independent functions about p. Then use Proposition 2.17,

about

p.

By

,xn are

o

36

The next important notion is that of a submanifold. As remarked before (Example 2.9), an open set V cHis itself a smooth manifold of dimension n (- dim H), with coordinate charts obtained by restricting the coordinate

charts for N to 1'. V is called an open subl/l8.l1ifold of N. A

suhset

P

c

fI is called a submanifold of dimension m

pEP there exists a coordinate chart

< n if for each

(U,~)

such that P n U

=

{q E U

I

Xi (q)

=

X I

(p).

i

~

m,.·l ....• n }

(2.35)

.

(V'X l , ... ,xn ) is called an adapted coordinate chart (adapted to P). If we take the subset topology on P (i.e., the open sets on P are of the form P

n V, with V open in fI), chen P becomes a manifold as follows. Let

(V.~)

be an adapted coordinate chart, then we let (P n U.~'P) to be a coordinate char.t for

Notice thnt by (2.32) the coordinates for P are

P.

An important theorem concerning submanifolds,

Xl' . . • ,Xm'

which generalizes Theorem

2.13, is the following.

The.orem 2.19

Let F ; 1'11

-t

Hz be a smooth IIIap I"itll dim Nl "" n t

•

i .. 1,2.

1

Let Pz E Nz and suppose that F- (pz) is non-empty. Assume that the rank of F in every point of F-I(pz) equal.s r:he dimension of Hz. Then 1

F- (pz) is a submanliold of Nl

Proof

Let

(U,z!"",zn

Pl E p-1(pZ)'

Since

,

(2.36)

of dimens ion n l

z ) be a coordinate neighborhood of Pz EN2 is regular it follows

F: p-l(pz) .... Nz

Proposition 2, IB that the collection of functions

•

Let from

Xi

forms pnrt of a coordinate system for HI about PI' which can be completed to a coordinate system Pl'

(Xl""'XnZ'~2+1'''''Xnl)

on a neighborhood V of

Then 1, . .

and so

,n1}'

(2.37) o

is a submanifold.

I t follows from Theorem 2.19 ths t Theorem 2.13 can be sharpened in the following way: N given by (2.28) is not only itself a smooth manifold, but also a submani.fold of IRn. Actually this situation is very general: it can be proved (but is outside the scope of these notes) that for an arbitrary m

smooth manifold N, there exists some Euclidian space R P

C {Rm,

and a 5ubmanifold

such that 1'1 and P are diffeomorphic. Hence every abstract smooth

37

manifold can be identified with a submanifold of some ~m.

Finally, let F: HI

Hz be an injective immersion. One may expect that

the image of Nl under F is a submanifold of Hz. This is not exactly true.

If fact only the following can be proved. lc

Proposition 2.20

Let F : HI ... Hz

be an immersion and let PI E HI'

there exist coordinate charts (U,x l ,··· ,XII!) about PI and (V,zl about pz - FCPl) such that F In these local coordinates equals

J'"

Then ,zn ) z

(2.38) For the proof of this proposition (which is a special case of the Rank

Theorem for smooth maps) we refer to Exercise 2.5, It follows from Proposition 2.20 that if F is an immersion then for any PI E HI

there

exists

a

neighborhood U of PI

such

that FeNl n U)

is

a

submanifold of N2 . However in general F(H l ) will not be a subrnanifold, Instead FUJl ) is called an immersed submanifold. Things that can go wrong are of the following nature.

Fig. 2.4. F(M!) is ,m immersed submllnifold.

The problems

are caused by the

fact

that

the

topology of an

immersed

submanifold FeH l ) may properly contain the topology on F(H l ) as a subset of }fz, i.e. there may exist sets F(U), with U open in Hl , which cannot be written as V n F(H l

)

for some open set V in Hz.

If these topologies are

equal, then we are in the situation of a normal submanifold,

2.1.2 Tangent Vectors, Tangent Space, Tangent Mappings

Let H be a smooth manifold of dimension n, If H is a submanifold of some ~m,

then geometrically the idea of tangent vector and tangent space at a

point p E H is clear,

38

Fig. 2.5. Tnngcnt space nt p E M.

We first proceed however in a much more formal way. Later on we will see how everything fits together. Denote by CU)(p) the set of smooth functions defined on a neighborhood of p. Definition 2.21

r/e

define the tangent space TpM

Co

11 at p to be

the

linear space of mappings Xp: C~(p) ~ ~ satisfying for all f,g E C~(p)

a.p

E~.

(linearity) (2.39)

(produce rule) Idrh the vector space operarions i.n Tpl1 defined by

(2.40) a E III .

A tangent vector N at p is any Xp E TpM. Remark

The

geometric

differentiated in p

meaning

E M

of

Xp(f)

is

that

the

function

f

is

along a curve c: (-f. t:) ... N with c(O) - p and

c'(O) - Xp' i.e. Xp(f) - (foc)'(O). This will be made clear later on. Let now F: 11

~

N be smooth. Then we define a map (called the tangent map

of F at p) Fwp: TpM'"

TF(p)N

as follows. Let Xp E TpM. For any f E CW(F(p») set

(2.41) It is easily checked that F.. pXp is indeed an element of

TF(p)N:

part (i)

of Definition 2.21 is trivally satisfied while for (ii) we compute

39

Xp(foF).g(F(p)) + f(F(p)) .X,(goF) (2.42)

- F.,X,(f).g(F(p)) + f(F(p)).F'pX,(g) Furthermore F"p is easily seen to be a linear map.

Proposition 2,22

The following properties of tangent maps are immediate.

(a)

If H

GoF is a composition of smooth maps, ellen

(b)

Let id: H .... N be the identity mapping.

=

H~p

=

G~F{p)oF~p'

then id .. p : TpN -. TpH is the

identicy matrix I.

For a diffeomorphism F : N .... N we have

(c)

Let H be a smooth manifold, and let

(U,~)

be a coordinate chart about

p E H. Recalling that any open set of a smooth manifold is itself a smooth

manifold

of

the

same

dimension,

we

have

~:

that

U .... ~(U) c

mn

is

a

diffeomorphism. Therefore by Proposition 2.22 (c)

~.,' TpH .... T1p{p) IR"

(2.44)

is a linear bijection. Hence the study of tangent spaces to an arbitrary manifold can be reduced to tangent spaces of mn. So let us first consider the tangent space Tamn for a E ffin. Geometrically it is clear that TQffin for any a E mn can be Define elements E la

af

El ~ (f)

i.e.,

E1a

=

identified with , ...

-a r, (a)

[fin.

Formally we

proceed as

follows.

,EnQ in TQlJ?n by letting

(2.45)

i E !,!,

equals the directional derivatives n

a; I ' '

i En.

It is easily

0

checked that El a' i E !,! are in TQm . In particular we have

(2.46)

i , j E !,! ,

for r 1

, ••• ,

rn the natural coordinate functions on mn. Since r 1

independent it follows that E10

' •••

,Ena

, ... ,

rn are

are independent vectors in Tamn.

We have to prove that {El~' ... ,Ena} is a basis for T~lRn. For this we need the following simple lemma.

Lemma 2.23

Let f E Cm(a) , with f(a) -

about the origin

o.

Then for

(ZIt . . .

'Zn) in a sphere

40

n

f(a1+z1,···,an+Z n )'"

Izigi(al+zl.·· .• an+zn)

J

1-1

for certain functions gl ,.,. Proof

Define h(t,z)

~

satisfying

,gn

f(a+tz) , 0

~ t

~

gi (8) -

at

-a--(a). r

i

1. Then

(2.48)

o By this lemma we can write for any f E C~(a) (2.49) Therefore for an arbitrary element En E TD~n we have by the product rule n

Ea(f) ~

I

n

En(ri)gt(a) +

I

(ri(a)-ai)Ea(gt) (2.50)

n

n

IEn(ri)Eia(f) ... : iM 1

Hence indeed

I CiiEiO(f) i- 1

(Ell!""

,Enn J

is a basis of T"IR". Since !p~p: TpH -. T

!P(pJ

IR

n

is

a bijection it follows that we can define a natural basis for TpH (natural with respect to the coordinate chart (U,!p)

a

-Xl 8

= (U.x 1 " " .xn »

a

IP ·····-a I ' Xn P

by letting (2.51)

lEn .

With this definition we have for any f E Cm(p). p E H •

A

- E

(2.52)

af

(f) ... -8--(xl (p). .. , IXn (p») i!p(p) ri

where f is the local representative of f. Hence

,

a! I (f) t

is just the i-th

P

partial derivative of f expressed in local coordinates

Xl , ••.• X n .

of the cumbersome notation ---a~ I (f) we will usually simply write Xi

p

Instead

a '· /p (f) a'~i

BE

-a' (p), );

-

f

(2.53)

C"'(p)

E

1

Let now Xp be an arbitrary element of TpN. It follows that we can write

x, -

(2.54)

°1 "

for certain constants

. ,0" E [R.

Then for f E Cffi(p)

(2.55)

and by (2.52)

(2.56)

(X1(p),···,x,,(p»).

is just the dh-ectional derivative of the function f

Hence Xp (f)

local

f)

representative

Notice that

in

the

direction

of

the

vector

,0:,,) T is the vector representation of XI'

(0'1""

-aa / , .. "-a'/ Xl P Xn p E

_ _

a_/

ari

iip(p)

Let

now

a /

_1

P

=

i

and

(U,'P)

coordinates JZ

i E :!:!. for T


,X"'

xl""

1/J" pE lV>(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
Xi.

for

.pXp )'

TH

are

They

are

also

denoted

called

as

natural

because of the following. Let r be the natural coordinate function on For an

Xp

E TpN expressed in the basis

~I ..... f-I Xl

P

Xn

I a:

as Xp = aJ. P i "1

~,

I 1

P

we have

(2.62) and so 01)

XlftpXp

~ 01

aar'

we therefore obtain Finally,

mapping

By the identification TX1 (p)!R ~ ~ (i.e. Vi

(X p

)

= X 1ftp X p

-

01

aar

with

°1 ,

let F: til .... i'f2 be a smooth map,

then we define the tangent

43

as the union of all tangent mappings F"p: TpNl .... TF (p )N;>. for p E Hl

x

z

,Xn1 ) be local coordinates for N1 , and the natural coordinates (x,v) = (Xl""

=

(Xl""

then

in

respectively

=

(Z1""

,Xn1,V 1 , · · ,

•

Let

,zn ) for 1'1 2 , z ,vn1 ) for HI

(z ,Iv)

F .. is given as (2.63)

2,2

Vectorfields,

Distributions,

Lie Brackets,

Frobenius'

Theorem,

Differential One-Forms 2.2.1 Vectorfields, Lie Brackets, Lie Algebras

Definition 2.25

A smooch vect;ol"field X on a smooch manifold H is defined

as a map X: N .... TN

satisfying 1fOX =

(Idth

11"

(2.65)

identity on H

the nacural projection from TN all 1'1).

Replacing "smooth"

throughout by

the vectorfield is called ~, adjective

functions,

"smooth".

Thus

manifolds,

W

C

resp.

resp.

,

c'.

throughollt

vectorfields

c!,

in the above definition,

In the sequel

T>'e

are

!>'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
(2.99)

An example of a subalgebra of this Lie algebra is the space of skewsymmetric matrices. Now let us consider the linear space of C~ vectorfields on H, denoted by

V'" UI).

and

take

as

bracket

operation

the

Lie

bracket

of

two

vectorfields defined above. Properties (i1) and (iii) in Definition 2.28 follow from Proposition 2.27 (b), respectively (c), while property (i) is trivially satisfied. Hence V"'(H) together with the Lie braclcet is a Lie algebra, in fact an infinite-dimensional Lie algebra. A crucial property of the Lie bracket is Proposition 2.30

Let F: N

->

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<

0 then y!!(xt(p»)

is

a

measure

understanding

p<

for the

Xt,(ys(p») for some

this

difference.

controllability

t

and s, and so This

will

properties

of

be a

nonlinear control system, as dealt with in Chapter 3. Exa.mple 2.35 Consider the folloWing simplified model of maneuvering an automobile,

___

___ A3_'_

~

I

Fig. 2.S. The front axis of a CilL

53

i.e,

the

middle

of

the

axis

linking

the

front

wheels

has

position

(Xl ,Xz ) E !Rz, while the rotation of this axis is given by the angle .\:3' The

xl

configuration

manifold

is

thus

[Rz x 51

with

local

coordinates

Consider the two vectorfields

'X 2 ,x).

(rolling) (2.108) y -

a ax,

(rotation)

.

The Lie bracket IX,)'] is computed as

sin x,

[[

cos x, 0

H~ ]l --[ ~

0

cos

0

sin x J

XJ

0

0

] [~]

-c~s

[

x, ]

51n x J

(2.109)

0

and thus the vectorfields X. Y do not commute. This also follows from the

following computation. Start in x(O) time h

yields

the

position x(h) =

Xo

=

Then rotation during time h yields x(2h)

(x 10 ,x20

+ h

(XI0 =

(x 10 + h

Rolling during

,X30)1,

sinxJO'x ZO

+ h

sinx30

x 30 + h)T, Rolling back during time h results in x(3h)

=

C05X 30 ,X Jo

)T.

,x20 + h cosxJo '

(KID

+ h

5io."30 -

h sin(x JO + h), x 20 + h cos x 30 - h cos(X30 + h), x 30 + h)T. Finally rota-

ting back during time h gives the end position

x(4h)

-

+ h sin x 30 - h sin(xJo + h) .'(20 + h cos x" - h cos (x 3 0 + 11)

[ x"

-"30

]-

r-hox-hoyi'oXh(xo} . (2.110)

Noting that sin(x3o + h)

=

sin."3o + h cosx30 + h.o.t. (higher order terms),

cos(x30 + h) - cosx30 + h sinxJO + h.o. t., we obtain

.. Xo + hZ[X,l'] (x o ) + h.o.t. (2.111)

o In Theorem 2.26 we have already shown that if X(p} ,... 0 coordinate chart (U ,xl"" another vectorfield,

,xn ) such that on U we have X

then there =

a -a x, .

is a

If r

is

linearly independent from X in a neighborhood of p,

then we may expect to find a coordinate chart such that

x _ _a_ aX l

y _

a

(2.112)

ax z

a

a

However it is immediate that [axi 'ax2l - 0 and hence by Proposition 2.30 a

54

necessary condition for the existence of such a coordinate chart is that [XtY} - D. The

theorem shows that this condition is also sufficient.

nex~

Theorem 2.36

Let Xl' .•. ,Xk

neighborhood

of p,

be

linearly

sat:isfying

coordinate churc (U,x 1

, •••

,xn)

independenr:

[Xi .XJ ] - D.

i.j

vector£ ields

E!:.,

chen

Denote Zi -

of Theorem 2.26

around p such that

i

can

~(p)

a a

(2.113)

~~Xi'

we

in is

around p such that on U

i E Ie

Proof

there

E!:., take

Z1 vectorfie1ds on a

coordinate

~{v}. (v,~)

chart:

As in the proof os

(V'zl •...• zn)

0, and

-

i E k

Define the map T: ~n ~ ~n, defined in a neighborhood of 0, by (2.114)

We compute that 1 .. l, .... k ,

(2.115) i - k+1 •... ,n

t

so tha t by the inverse func tion theorem T is a diffeomorphism around 0, which can be used as a coordinate transformation. Moreover precisely as in Theorem 2.26 we have that

(2.116) On the other hand, since [Xi,Xj J = 0, 1.j the order of integration in any way,

E!:..

i. e..

we can by Lemma 2.34 change for any 1

E

k we can firs t

integrate in (2.114) along Z1

(2.1l7) so that by the same argument

a

Tw -a r i

=

2i

i E

k

Hence T- 1 is the required coordinate transformation; map is given by T-lo~.

(2.118)

the new coordinate [J

55

r, Z,(O) IL.--"C-""

Fig. 2.9. The new coordinates conslructed in Theorem 2.36 for n = k = 2.

2.2.2 Distributions, Frobenius' Theorem Definition 2.37

A distribution D on a manifold N is a map

l .. hleh

assigns

to each PEN a linear subspace D(p) of the Cangent space TpN. D 1-1111 be called a

smooth

distribution

if

Bround any point

these

subspaces

are

spanned by a set of smooth vectorfields, I.e. for each p E 11 chere exists a neighborhood U of p and a set of smooth vectorfields Xi' i E I, ldth I

some (possibly infinite) index set, such that

D(q) In

the

=

span{Xi(q); i Ell,

sequel

distribution

q E U .

will

al1.rays

(2.119)

mean

smooth

distribution,

A

vectorfield X is said to belong to (or is in) the distribution V (denoted X E D) if X(p} E D(p) for any p E H. The dimension of a distribution D at p E fJ is

the dimension of the

subspace D(p).

A distribution

is

called

constant dimensional if the dimension of D(p) does not depend on the point p E fJ.

Let D be a constant dimensional distribution of dimension k.

Lemma 2.38

Then around any p E fJ there exist 1c independent vectorfields Xl""

,Xk

such that

(2.120)

q near p. Proof

Since dim DCp)

~

k there exist k vectorfields from the index set I

in (2.119), for simplicity denoted as Xl"" D(p)

~

,Xk

,

such that

span[X 1 (p)"",XI;(P»)

Hence Xl (p), ... ,XI; (p)

(2.121)

are independent elements of Tptl.

By continuity it

follows that for q close to p the vectors Xl (q), ... ,XI; (q) in T,/! are also independent, and hence, since dim D(q)

The vectorfields Xl""

'XI;

=

k, span D(q).

above are called the local

o generators of D,

56

since

every

vectorfield

can

XED

be

written

around

p

as

It

X(q} = Lo!(q)X! (q) for some smooth functions

°1 ,

i

k.

E

i-I

Definition 2.39

A

distribution

D is

called

involucive

if

[X,Y] E D

whenever X and Yare vectorfields in D. Remark

By Proposition 2. 27(a) it follows that a distribution D given as

in (2.119) is involutive if and only if [Xi.Xj

]

ED on

for i.j E I. In

U

particular if D is locally given as in (2.120) then we only have to check that [Xl ,XjJ ED for l.j E k. or said otherwise, [Xi ,XJ i

k

L c ij X1

form

]

has to be of the

£

for some functions c i j

•

.I!~l

Definition 2.40

A submanifold

P of H is

an

integral

manifold of a

distribution D on l1 if

(2.122)

for every q E P. (Recall that since PeN we have TqP

C

Tr/I, for all q

E

P.)

We have Proposition 2.41

Let D be a distribution on N sud} that through each

point af M there passes an integral manifold of D. Then D is involutive. Proof

Let X, Y E D and p ED. Let P he an integral manifold of D through

p. Then for every pEP

Since P is a submanifold around any pEP there is a coordinate chart (U ,Xl' ..•• xu)

for H such that

Un P

(2.123)

Writing out X and Y in the basis ~ __a__ it follows that the last n-k aX ••••• aXn l

components of X and r in pEP are zero. Hence by (2. 94) components of [X,Y] are also zero, and so [X,Y](p) pEP.

E

the las t n-Ic'

TpP = D(p) for any

o

We say that a distribution D on H is integrable if through any point of H there passes an integral manifold of D. In Proposition 2.41 we saw that

57

involutivity of D is a necessary condition for

theorem

shows

that

for

constant

dimensional

integrability;

distributions

the next

it

is

also

sufficient.

Theorem 2.42

(Frobenius' Theorem) Let D be an involutive distribution of

constant dimension k on H. Then for any pEN there is a coordinate chart

(2.124) ~(U)

such

(-E,/;) x

=

thae for

each

... x (-E,E) 8k

+1

,an'

, •••

> 0 ,

I;

smaller in absolute value

than

the

€,

submanifold

I

{q E U

xk+l(Q)

d k + 1J · · · , xn (q)

=

=

(2.125)

anI

is an integral manifold of D. Noreover every integral manifold is of this

form. Actually we usually need the following equivalent form of Theorem 2.42.

Corollary 2.43

(Frobenius)

Let D be an involutive constant: dimensional

distribution on N. Then around any pEN there exists a coordinate chart (U ,xl' ... ,xn ) such that D(q) ~ span

a

I-a I Xl

q

a I }, , ... '-a. XI: q

(2.126)

q E U .

D as in (2.126) is called a flat distribution, and usually Ive lvill simply write (2.126) as D

=

a

span (aX ' ... l

Proof

(of

Theorem

2.l12

and

(U' ,'P') about p, with 'P' (p)

~

a

'aXI: ) .

Corollary 2.[13)

Take

a

coordinate

chart

O. Mapping everything onto 'P' (U) using 'P' we

may as well assume that we are in [Rn with p

=

O.

Moreover we can assume

that D(O) C TolRn is spanned by

(2.127) Let

1r: IR

1r~o: TolRn

n ->

continuity

...

mk

TolR

be k

7t"q:

is Tq[Rn

the an .... T

projection isomorphism 7t(ql

onto when

the

k

first

restricted

to

factors. D(O) C To!Rn.

Then By

IRk is an isomorphism when restricted to D(q)

for q close to O. So for q near 0 we can choose unique vectors

58

such that

a

k .

(2.l2B)

It follows that che vectorfields Xi

(defined on a neighborhood of 0 E ~n)

1fhQXl

and

a -a--

art

(q) ..

(on

k

~

i

11I"(q)

E

) are 1f-related. By Proposition 2.30 we get

l:"l

o . By involutivic:y of D we have

(2.129)

[Xi ,Xj

1q E

and since

D(q).

Theorem 2.36 we can choose a

V C V'

E~.

Hence by

local coordinate chart (V ,xl' ...• x,,).

Xi

E~.

i Corollary

is

2./,3

(2.130)

on V . proved.

Integral

manifolds

D

of

in

these

o

coordinates clearly are given as in (2.121).

The

totality

i ." k+l •... ,n, each

with

such that

1

Hence

is one-one

1fnq

when restricted to D(q) we therefore have [Xi ,Xj] - 0, 1.J

of

(2.125)

submanifold

Frobenius'

(2.125)

submanifolds

parametrized

at.

by

la t I

is called a foliation of the open submanifold V

theorem

is

says

called

that

leaf

a

an

of

this

invo1utive

N,

foliation.

constant

dis tribution on 1-1 locally generates a foliation of N,

C

<

f,

and

Hence

dimensional

whose leaves are

integral manifolds of the distribution.

Remark integral

It is also possible manifolds

of D.

to prove

if we

that H as a \.,hole is

generaliz.e

the

manifold by allOWing for immersed submanifolds,

definition see the

foliated by of

integral

text above Fig.

2.. L, •

Example 2.44

Consider

on

H = {(x 1 ,XZ ,x3 )T

E ~J

Xi

> 0, i - 1,2,3)

the

distribution D(x) - span(X 1 (x) ,X 2 (x»), where (2.131) (These

are

the

input

vector fields

in

the

model

of

a

mixed-culture

bioreactor as treated in Example 1.4.) Since [X 1 .X2 ] = X2 it follows that D is involutive, as well as (note the definition of H) constant dimensional. In order to apply Theorem 2.42 we consider the set of partial differential equations

59

(2.132)

~(Xl ,Xl

in

Denote

,x). A possible solution is

Z1:= rp(x) ,

2 2 ;= X2 'ZJ;=

set of new coordinates for N, the that

form the

,

[(2 1 ,2"2' ZJ)

choice

of

)

2"1

xJ

then it is checked that

,

21 'Zz ,Z3

are a

in which the integral manifolds of D are of

~ constant!,

coordinates

solves

is

by

no

(2.132),

a

D -

while

span {Bz

means

unique.

and

thus

z

a

' az )' Note 3 In particular

we

can

take

o The classical version of the Frobenius' Theorem is at first sight quite different from Theorem 2.42.

(Classical Frobenius' Theorem) Consider the set of parCial

Theorem 2.45

differential equations

ak arCr,C) ~ b(r,k(r,c») with r E

k

[R1tl,

E ~" and

t

n ; !J?m X IP. ... fin.

b;

(2.134) IR m x [Rn ... N(n,m)

(nxm-matrices)

in the llnlcnol,rn

Then locally there exists a solution k i f and only if

the matrix component functions b

w

(r,s) , s E ~"

~

i

E

0, i E

~,

~,

o E

~,~

~,

satisfy

E m

(2.135)

Furthermore we can ensure that the solution k(r,c) satisfies

ak

rank at (r,t)

=

(2.136)

n.

The connection between Theorem 2.45 and Theorem 2.42 is as follows. Define the vectorfields

a zo - aT o

+

(2.137)

oEm

and the constant dimensional distribution D spanned by 2 1 "" ,2m' It is that D is involutive if and only if

easily checked (see Exercise 2 .12) (2.135) is satisfied;

in fact (2.135)

implies that [21 ,2j ) = 0,

i,j E m.

Hence by Theorem 3.36 we can find local coordinates for ~m x ~n in which D

60

is a flat distribution. By the special form of the vectorfields Za in (2.137) it follows that we may leave the coordinates r 1 while t

coordinates

l , .•.

,tn

s1 •••••

can

sn

be

transformed

••.••

to

unchanged.

rm

new

coordinates

depending on rand s in such a way that in the coordinates (r.t)

the distribution D is given as span I~ , a E mI. Denote vrQ -

t

=

her,s). and

define the inverse map k(r,t) satisfying (2.136) and

h(r,k(r,t»

- t .

(2.138)

Differentation of (2.138) with respect to ra n

ah

L

F(r,s) + a

where s

~

all ---a

'

a

E

::!. yields

8k j

(r,s) ---ar (r,t) ~ 0,

Sj

(2.139)

Q

k(r.t), and thus

Z~(h(r.s») = ~(r s) + ... as' a n

L j =1

ah

b jet (r,s) -a-(r,s) 5 j

aJr j

-a-(r,s) -ar (r,t) Sj

8Jc-1

m

L

(2.140)

-I-

et

ah

By non-singularity of the nxn-matrix Bs(r,s) this immediately yields that k(r,t) is a solution of (2.134).

Finally for later convenience, we define the sum and intersection of distributions. Let Dl and Dz be two smooth distributions on N. Then their sum Dl ... Dz is defined as the smooth distribution given in any q E H as (D1+Dz)(q) ~ spanlX 1 (q) + XZ(q)!X 1 smooth vectorfield in Dz and Xz smooth vectorfield in D'll. The intersection D1

n Dz is the smooth distribution given in

q

E

(2.141)

N as

spanIX(q)!X smooth vectorfield contained in D! and Dz )

(2.142)

Note that for two involutive smooth distributions the intersection Dl n Dz is again involutive. It follows that for any smooth distribution D we can define the smallest involutive smooth distribution containing D (because if D! and Dz are involutive smooth distributions containing D, then so is D! n Dz ), This distribution is called the Involutive closure of D. and is denoted by D:

D-

smallest involutive smooth distribution containing D .

(2.143)

61

2.2.3 Cotangent Bundle, Differential One-Forms, Codistributions Let H be a manifold, and let rpM be its tangent space in a point p. Since TpM is a linear space we can consider the dual space of TptJ,

denoted T~H,

called cotangent space of H in p,

of a

space V is called

the set of all linear functions

cotangent

vectors.

r;H

a 1 I aXl P , ... '-a Xu p

Any

Xl ' . . •

,x"

linear

Elements of r;H are

be

a

basis

for

TpM

on fl, then we denote the dual

by dx1Ip •... ,dxnl p ' By definition

a

I'(-a. x, 1' ) - '"

dX,

v.

on

I--a 1

Let

corresponding to local coordinates

basis of

(Recall that the dual V,.,

cotangent

coefficients

vector

0i>

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<

A(x) P(x) _

with 2,5

~(x)

X E V

o


a (p-r)xr matrix, depending smoothly on x.

Prove Proposition 2.20 in the following way: (a) Take

arbitrary

coordinate

(V,Zl"",Znz) about Pz F{x} Use

=

=

charts

{U,x I

, ...

,xnt }

Pl

about

and

F(Pl)' and write F in these coordinates as

{F1{X}, ... ,Fnz(X}}T

Proposition

permutation

2.17

in

the

of

order

to

coordinates

show

that,

Zl , ...

possibly

,zn z

the

after

~

F l , ... 'X nl - Fnl serve as new local coordinates system for about Pl' Prove that in the coordinates X and F talces the form

Xl

a

functions H

z

(al, .. ·,Bn1 )

H

(a1, ... ,B nl , \f!nl+l(a), ... ,\f!n/a»

(b) Define the following coordinate transformation on N

Show that in the coordinates (x,z) F takes the form (2.38). 2.6

Let X,Y

be

function f

vectorfields : 1'1

--+

on

a

manifold H.

Define

for

any

smooth

lR the object [[X,Yllp(f), p E 1'1, as satisfying

[[X,Y[[,(f) - X,(Y(f».

Show that in general [[X,Yjj is not a vectorfield.

2.7

Consider the vector fields

x-

a

Xl aX

l

a

+ Xz ax

z

Show that [X, 1'] - O. Interpret this geometrically.

m\

I

{(xl ,X z ) Xl ~ O,Xz - 0 J the coordinate transformation (see Example 2.2) Xl = r cos ~ , X z = r sin ~, and compute X and Y in the new coordinates (r,~). Check the vanishing of the Lie bracket in

Define on

the new coordinates. 2.8

Apply Theorem 2.36 to the vector fields X,i' as given in Exercise 2.7.

72

2.9

Show that the linear space of nxn-matrices with bracket operation

[A,B] ... AB

BA

A,B nxn-matrices .

is indeed a Lie algebra.

2.10 Define the 2nx2n-matrix J

"" [On - In] .

In

On (On is nxn zero matrix, In is nxn identity matrix). A 2nx2n-matrix A is called Hamiltonian if ATJ + JA - 0 . Show that the linear space of 2nx2n Hamiltonian matrices is a Lie suba1gebra of the Lie algebra. of

all 2nx2n matrices. 2.11 Consider on ~3 the vectorfields

a

a

a

- Xl 8X3 and Xz(x) = 8x 1 z 2 Let x D - (O,O,O)!. Compute that X;hQX;hOX~oX~O(XD) - h (X I ,X Z](xo )' X1{x) - 8x

2.12 Show that the distribution D spanned by the vectorfields Za' a e !E. as defined in (2.137) is involutive if and only if (2.135) holds. 2.13 Verify the local coordinate expression for the Lie derivative of a one-form as given in (2.167), (Hint: First verify the expression for the vectorfields ---ad. 1 E ~.) Xl

2.14 Let F : fl ... N and let P be a constant dimensional

distribution on N.

Prove that F"P (see (2.181»

codistribution

H.

(dz k + 1

, .••

on

(Hint:

First

write

,dzm 1 in sui table local coordinates

involutive co

is an involutive (cf.

for N.

(2.17B»

P-

Then compute

P"P. )

2 .15 Let V - IR

r

be a Lie algebra. For any basis (vl"'" v r

exist constants C~j' i.j,1c e

E.

}

of V there

called the structure constants, such

that i.j

e

r'.

Show that (i)

Ie Ct j

Ie

-

-cJ i

l.j,k E ::.

'

r

(ii)

Ic'kcm

k-I lj IcE

+ckc

1i

m kj

+c"'cm) ..

ji I:i

o,

i,j,me::..

Conversely, any set of constants satisfying (i), (ii), defines a Lie bracket.

3 Controllability and Observability, Local Decompositions

In the first two sections of this chapter we will give some basic concepts

and

results

in

the

study

of

controllability

and

observability

for

nonlinear systems. Roughly speaking we will restrict ourselves to what can be seen as the nonlinear generalizations of the Kalman rank conditions for controllability and observability of linear systems.

The reason for

this

is that in the following chapters we will not need so much the notions of nonlinear

controllability

and

per

observability

5e,

but

only

the

"structural p-z;operties" as expressed by these nonlinear "controllability" and

"observability"

rank conditions

that will be

obtained.

In

the last

section of this chapter we will show how the geometric interpretation of reachable

and

subspaces

enjoying

generalized

unobservable

to

the In

distributions.

some

subspaces

maximality

nonlinear this

way

case, we

for or

linear

using

make

systems

minimality the

notion

contact

as

invariant

properties

with

of the

can

be

invariant nonlinear

generalization of linear geometric control theory as dealt with in later chapters, where this last notion plays a fundamental role.

3.1

Controllability

In this chapter we consider smooth affine nonlinear control systems m

x

+

f(x)

L gj (x)u j

(3.1)

,

j"

where x

=

(Xl' . . . ,Xn

) are local coordinates for a smooth manifold N (the

state space manifold),

and £,gl' ... ,gm are smooth vectorfields on N (see

Chapter 2). £ is called the drift vectorfield, vectorfields. Given any point Xo E

}J

the

reached

set

of

points

which

can be

and gj' j

E~,

the input

one may wonder what one can say about

suitable choice of the input fUnctions u j

from Xo (.),

j

in

E m.

finite

time

by

a

This is the general

controllability (or reachability) problem. Throughout we will make the following assumptions concerning the input space U and the class of admissible controls

Assumption 3.1

(a)

~

(see Chapter 1).

The input space U is such that the set of associated

vectorfields of the system (3.1)

74 m

(f

Lg

+

j Uj

I (u 1 ' ••• ,Urn)

E

(3.2)

UI

.i~l

contains the vectorfields [.gl'··· .gm' (b) 'U consists of the piecewise constant functions which are piecewise continuous from the right. Remark

It follows from standard results on the continuity of solutions of

differential

equations

that

if we

a more

approximar::e

general

control

function u(·) by piecewise constant functions in some suitable sense, then the

solutions

of

(3.1)

for

these

piecewise

constant

functions

will

approximate the solution of (3.1) for u(.) (see also the references at the end of this chapter). In this sense many properties of systems with quite general

control

functions

can be

established by

considering only

the

piecewise constant case. Recall (see Chapter 1) that the unique solution of (3.1) at time for a particular

xeD)

t!"O

(control)

function

and initial

u(·)

t

~

0

condition

is denoted as x(t.O,xo IU), or simply as x(t). Then we state

-.\:o

Definition 3.2 any

input

The nonlinea.r system (3.1) is called controllable if for

polnes

xl.X l

in f1 r:here exists

a

finite tIme T and an admissible

cancrol function u: [O,Tl ... U such thaT: x(T,O,x 1 ,u) - xz'

One is especially interested in controllability because for linear systems

x

=

Ax +

x E 1R11. U E

Bu.

[Rm.

(3.3)

one knows that controllability is equivalenr: with the (easily verifiable) algebraic condition (3.4)

(Kalman rank condiclon for controllability), Furthermore in linear systems

theory controllability is crucial,

not only because of the concept of

controllability per se,

or maybe especially.

importance

for

e. g.

but also.

stabilizability,

(optimal)

because of its

control

design

and

realization theory. The simplest Dpproach to study controllability of the nonlinear system

(3.1) is to consider its linearization. In fact Proposition 3.3

Consider

the

nonlinear

We

system

have (3.1),

and

lee

xa

E

H

75

sa.tisfy £(xo ) = O. Furthermore let U contain a neighborhood V of u Suppose that:: c1Je linearization of (3.1) in Xo and u = 0

Z E IR

Is

O.

(3.5)

, V E [Rrn,

a. controllable linear system. Then for every T > 0 and ( > 0 the set of

points which can be reached from Xo

ue·):

functions

of

n

=

[O,T]

->

V,

satisfying

in time T using admissible control

Ilu(c)11

<

contains a neighborhood

f,

xO'

Proof

Since

linearization (3,5) is controllable there exist input n ,v (.) defined on [O,T] steering the origin z = 0 in n time T to independent points Z1, . . . , zn E !R . Furthermore we can take these

functions v

the

1

(.), •..

input functions to be piecewise constant, and so small that

satisfies Ilu(t,OII <

€

for 0 ~ t::; T and

!€il

the solution of (3.1) for the input (3.6) taking

E

E~. Denote by x(c,';)

< 1, i

initiating at x{O,O

small enough x(c,t,) will exist for all 0

$

t $

~

xo'

(By

T.) Consider now

the map

t,

H

t, near O.

x(T,O,

(3.7)

We shall show that the matrix (3. B)

is non-singular at t

=

T, and then the conclusion follows from the Inverse

Function Theorem applied to the map (3.7) (see Proposition 2.16). Since

a

atx(t,O

f(x(t,O) + Igj(x(t.O)uj(t.O.

(3.9)

j~l

we can differentiate (3.9) with respect to t, in €

~

0, so as to obtain

(3.10)

Z(O) - 0, where

af

A ~ ax(x o )

and

B ~

(gl (xo )

j ."

igm(xo»).

By

follows that the columns of Z(T) are independent.

Remark 3.4

definition

of

v

,

it

o

(see Exercise 3.2) Similar statements can be derived for the

points which can be steered in small noting that controllability of (3.5)

time

to Xo

using small controls,

is equivalent to controllability of

76

In

the linearization of the time-reversed system x - -f(x) -

Lgj(X)U j

,

j-l

However the linearization approach is often not satisfactory. Already in Chapter 1 we have seen that by linearizing a nonlinear system we may loose much of the structure of

the system,

see e. g.

Example L 2.

In

particular a nonlinear system can be controllable while its linearization is not: Exnmple 3.5

Consider the simplified model of maneuvering a car, as dealt

with in Example 2.35. If the speed of rolling and rotation can be directly controlled then we obtain the nonlinear system

(3.11) Clearly, its linearization in any point x E ~J is uncontrollable. However, as we will see,

this model of driving a car is controllable (in accordance

o

with most people's experience of driving a carl).

So let us consider again the nonlinear system (3.1). We ask ourselves in which directions we can steer from a given point xo. points which can be reached from

Xo

Le. which are the

in arbitrarily small time. First we

consider the situation where the drift vectorfield f

is absent, and for

simplicity we take only two input vectorfields while H is equal to IRn. Le.

(3.12) Clearly from any point

Xo

E

IR

n

we may steer directly in all directions

contained in the subspace of the tangent space TxOlRfi ~

mn

given by (3.13)

by using constant inputs. Can we steer, maybe in an indirect way, in other directions as well? The answer is yes and the key idea for achieving this is to srvitch be tween the vectorfields gl and gz by choos ing appropriate piecewise constant input functions u l and u z . Recall from Chapter 2 (Lemma 2.3 LI) that in case (gl .gz 1 .,. 0 then the integral flows of gl.gz commute. i.e. g~og; = g;og; for all c, s, and thus we cannot steer into a direction outside G(xo ) by SWitching between gl and gz. However. if {gl ,gzl (xo ) a: G(xo) we can steer into a direction outside

77

G(xo ) as follows from Proposition 3,6

function uCt)

=

Consider

the

nonlinear

C E

[h,2h)

(-1,0)

C E

[2h,3h)

(0,-1)

t

E

[3h,4h)

x(4h) ... Xo

system

(3.12)

the

control

[O,h)

E

(0,1)

Then the solution xCt)

Proof

c

l""'

u(c) _

for

(u1(t),UZ(t») given as

+ h2{gl

=

h > 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

<

i E

Ei

i.

,

i E

1.

Hence

by

sufficiently small

is a subrnanifold Nj C fl of dimension j. Finally we conclude that Nn is the

desired open set contained in R~(xo)'

0

Motivated by this we give Definition 3.10

The system (3.1) is locally accessible from Xo i f R;(xo )

contains a non-empty open set of !-1 for all neighborhoods V of Xo and all T> 0,

If

this holds for any Xo E it

chen

the system is called locally

accessible.

Corollary 3.11

If dim C(x)

=

n for all x E H then the system is locally

accessible.

We call (3.23) the accessibility rank condition at xo' If (3.23) holds for any x E t1 then we say that the system satisfies condition.

the accessibility rank

(The relation with the controllability rank condition (3.4) for

linear systems will be explained soon.) What can we say if dim C(xo ) < n for some xo? In case the distribution

C is constant dimensional about Xo we have Proposition 3.12

Suppose

that C has constant

dimension

k

less

tlJan

n

about xo' By Frobenius' Theorem (Theorem 2.42) ho'e can find a neighborhood W of

Xo

and local coordinates

such that the submanifold

Xl , . . . ,xn

(3.28) is an integral manifold of C.

Then for any neighborhood V C

r{ of

Xo and

for all T> 0, i;(xo ) is contained in 5 xo ' Furthermore R;{xo ) contains a non-empty open set of tlJe integral manifold 5 xo ' Hence the system restricted to 5 xo is locally accessible.

Proof

Since

f{x)

+

I

gj (x)u j

E C{x)

for

any

(u l

""

,u m) E U and x E 11

j=l

the system (3.1) for xeD) - Xo can be restricted (for sufficiently small time)

to 5 xo ' where restricted system.

dim 5 xo = dim C(x o )'

Now apply Theorem 3.9

to

this 0

82

Corollary 3.13

If

the

system

(3.1)

is

locally

accessible

then

dim C(x) - n for x in an open and dense subset of H. Proof

First. for any Xo such that dim C(xo } - n there exists a neighbor-

hood of

Xo

such that on this neighborhood dim C(x}

for which dim C(x)

=

n. Hence the set of x

=

is always open (but possibly empty). Now suppose

n

there is an open set V ...

!2)

there is also an open set V

of N where dim C(x} < -;Ii

¢ with dim

for all x

11

V. Then

E

C(x} - k < n for all x

E

V. Now

use Proposition 3.12 for the system restricted to V, Then it follows that the system is not locally accessible, which contradicts the assumption. Hence the set of x for which dim C(x) - n is dense. Usually

the property of local

accessibility

0

is

far

from

controll-

ability. as shown by the following example. Example 3.14

Consider the system on ~2

Xz

~

u.

The accessibility algebra

(3.29)

e

is spanned by the vectorfields f ""

a!z'

a!l

x;

_a_

aX l •

a!l'

g and their Lie brackets If,g] = -2Xl [[f,g),g] - 2 Clearly dim C(x) .. 2 everywhere, and so the system is locally accessible. I

However since x~ ~ 0 the xl-coordinate is always non-decreasing, Hence the reachable sets look like in Fig. 3.1, and the system is not controllable.

o

Fig. 3. j. ReaChable set rrom ,.0.

However in case the drift term f in (3.1) is absent the accessibility rank condition does imply controllability. Proposition 3.15 any X E

~

also

-x

(a) If dim C(x o )

Suppose f - 0 in (3.1), and let E =

~.

~

be symmetric, i.e. for

Then

n then R~ (x Q )

contains a neighborhood of Xo

for all

83

neighborhoods V of Xu and T > 0, (b) If

~

dim C(x)

for

n

all

x E Nand

H is

connected,

then

(3.1)

is

controllable. Proof

(a) Go back to the proof of Theorem 3.9, and consider the map

(3.30) with Xi (1t

E~,

which

of

< 51 < (1' i E

~,

the

image

is

Nn .

Now

let

(s1 •...

,sn)

satisfy

and consider the map

(3.31) Since (-Xi)

of

n

~

i

~

-

Xi

~

i

it follows that the image of this map is an open set

containing x o • and the result follows from symmetry of',

(b) (see Figure 3.2) Let R{x o ) :=

ul\xo,r), i.e. the reachable set from PO

xO' By (n) R(xo ) is open. Now suppose that R(xo ) is strictly contained in M.

Take

a

point z

on

the

boundary

of R(x o )'

By

(n),

R(z)

contains

a

neighborhood of z, Bnd hence intersects non-trivially with R(xo )' Hence z

o

can not be a boundary point of R(xo ), which is a contradiction.

Fig. 3.2. Illustrating the proof of Proposition 3. !5(b).

Remark 3,16 t(x)

It can be easily seen that Proposition 3.15 also holds if

E span(gl (x), ... ,gm (x») for all x E ff.

Example 3.5 (continued)

a + cos 1

[sin X3 aX

Since X3

a a axz 'ax J 3

(cr. (2.109», we have dim C(x)

3.15

R~(x)

contains

a

=

=

a a + sin x J ax z 1

-cos X3 aX

3 for every x E

neighborhood

of

x

[R3.

for

Hence by Proposition every

x

(and

every

neighborhood V of x and all T), and the system is controllable, as alluded to before.

o

Now let us apply the theory developed above to a linear system (3.3), written as 111

=

X

A..-x-

+

Lb i

(3.32)

U1 ,

1"'1

where b i

•...•

bm are the colwllns of the matrix B. First let us compute the

accessiblility algebra

e

in chis case.

Clearly the Lie brackets of the

constant input vectorfields given by the vectors b l

,bn are all zero:

••. .

(3.33)

i , j E ~.

The Lie bracket of the drift vectorfield Ax with an input vectorfield b i yields the constant vectorfield

(3.34) The Lie brackets of Ab 1 with Ab j

or b j are zero, while

(3.35) Continuing in vectorfields

this way we conclude bi

Ab i

,

•

e

that

i E~.

2

A b!,

is

spanned by all constant

l:ogether with

the

linear

drift

vectorfield Ax. Therefore by Cayley-Hamilton

(3.36) and

Im(B~AB1'" ~An-lB) + span(Ax).

C(x) -

(3.37)

We see that the accessibility rank condition (3.23) at with

the Kalman rank condition for

controllability

Xo

,..

(3.4).

0 coincides Hence

if we

would not have known anything special about linear systems, then at least it follows from Theorem 3.9 that a linear system which satisfies the rank condition systems

(3. t l) theory

is

locally accessible.

that

(3.4)

is

(Of course we know from

equivalent

stronger equivalence apparently is due

to

with

conrrollabllity.

the linear structure.

linear This Notice

that Proposition 3.15 does not really apply to linear systems; in fact: the extra directions in which we can steer outside 1m B are precisely due to Lie brackets with the drift term Ax.)

Remark 3.17 (3.5)

in Xo

Consider with

the

nonlinear

f(x o ) '" O.

Denote

system A ...

easily verified (see Exercise 3.4) that

(3.1)

~;(xo)

and and

its bj

-

linearization gj (xc).

It

is

85

k-times f

(-l)'A'b j

-

(3.38)

[f,[f,[ ... [f,g,[ ... []](x o )

It thus follows from Proposition 3.3 that if the suhspaces of TxoN spanned by all repeated Lie brackets of the form given in the right-hand side of

right-hand

of

Xo

for

all

side

of

(3,38)

brackets appearing in e,

stronger

then R;(xo ) contains a

E m and k = 0,1, ... , has dimension n,

(3.38) for j

neighborhood

rank

belong

cf.

conditions

T > O.

Notice to

(3.17),

than

the

a

that

very

the

brackets

in

special

subclass

of all

the

This has motivated the search for

one

given

in

(3.23)

stronger types of controllability than local accessibility;

guaranteeing we refer to

the references cited at the end of this chapter. Notice

that

the

term

span/Ax)

in

(3.35)

is

not

present

in

the

controllability rank condition (3.4) for linear systems. Furthermore for a linear system we know that not only the sets set but even the sets RV(xo ,T) for T> 0,

R; (xo )

i.e.

contain a non-empty

the points that we reach

exactly in time T with trajectories contained in V.

This motivates

the

following definitions.

Definition 3.18

Consider a nonlinear system (3.1). The system is said to

be locally strongly accessible from set RV(xo ,T)

Xo

if for any neighborhood V of

contains a non-empty open set for any T>

Xo

the

° sufficiently

small. Definition 3.19

Let

e

eo.

for all X E Co(x)

eo

=

eo

as

and satisfies [f,XJ E

r;'o

be the accessibility algebra of (3.1). Define

the smallest subalgebra I.hieh contains g1""

,gm

Define the corresponding involutive distribution

span{X(x)

IX

vectorfield in

r;'o)'

and Co are called the strong accessibility algebra, respectively strong

accessibility distribution.

Remark

It can be immediately checked that for a linear system (3.32)

(3.39)

86

Analogously of

to

Proposition 3.8 we

give

the

following

characterization

eo. Every element of

Proposition 3.20

eo

is a linear combInation of repeaeed

Lie brackets of the form E~.

j

Proof

(3.40)

k - 0.1, ... ,

o

See the proof of Proposition 3.8.

We have the following extension of Theorem 3.9. Theorem 3.21 dim

Consider the system (3.1). Suppose chat

Co (x o ) -

(3.41)

n,

then the system is locally strongly accessible from xo' Proof

The proof can be reduced to the proof of Theorem 3.9 by making use

of the following tric.k. Augment the state space equations (3.1) by the equation

t ...

I. r:: being the time variable, so that we have the augmented

system

- {X 1:. c:

defined t(x,c) ""

=

m

f(x) +

1: Sj (x)u

j

(3.42)

j-l

1

on

R-

H x ~

f(X)~x + ~t

with

state

x = (x,t),

drift

and input vectorfields gj(x,t) -

vectorfie1d

gJ(X)~x'

From the

form of the vectorfields f and gj' j E~, it immediately follows that the control algebra

C of

the augmented system satisfies for any to (3.43)

By 0.41) and 0.43) we have dim C(xo ,0) - n+l, and hence the augmented system is

locally accessible

neighborhood of Xo E

from

(xo ,0).

the reachable set

Hence

R~(XD'O»).

for

any T > 0 and V

with V- VX (-c:,T+,,) ,

> 0, contains a non-empty open set of H x IR. Hence,

there exis ts a

non-empty open set We H, and an interval (a,b), 0 < a < b S T, such that

87

Tr

conclude that We RV(x D IT). Let X E '[},

then the mapping x H X - (x) maps

U f{ onto an open set W which is contained in R (xo ,T) for some neighborhood

By choosing T small enough the intersection of

U of xo'

will

contain a

non-empty open

set

of N.

Hence

the

W with

system

RV(xo IT)

is

locally

o

strongly accessible.

(3.41)

the

strong accessibility ranle condition at xo'

dim Co (x o ) < n,

but

Co

We call

has

constant

dimension

around x o '

we

In case

have

the

following analogue of Proposition 3.12. Proposition 3.22

Suppose that Co has constant dimension k < n around

By Frobenlus' Theorem there is a coordinate chart (U,x l such

I

JSj

that

<

f,

the submanifolds S -

j

=

k+l, ... ,no

U!xk+1(q) -

(q E

nOl,r tTo/O

(1) If f(xo)

for all

E

In

, •••

0, j

-

xO'

,xn ) around Xo

,xn (q) - an]

for

and the integral

k+l •... ,no

=

possibilities:

Co (x o ).

T> 0.

+1

are integral manifolds of Co

manifold SXo through Xo is given bya j There are

8k

, ...

then f(q)

this

case

E

Co (q)

the

for all q

E

system restricted

u SXo and R1 (xo ) c SXo to

SXo

is

locally

strongly accessible. (ii) If f(x o ) f/: Co(xo ),

UC

then by continuity f(q) f/: Co(q)

U neighborhood of xo'

and dim C(q)

this case we can adapt tlIe coordinates X);+l""

,xn

for all

dim Co (q) + I for all q E

=

X); + 1 , ••• 'Xn

on

U

q E

V,

U.

In

to coordinates

in such a \,ray that as above

and if we let

(3.44)

xn(q) =0)

is

contained

in S!

o contains a non-empty open seC of S1

'0

Proof

From

the

definition

of

Co

for any T > for any T >

it

a and moreover RU (xo ,T) a sufficiently small.

immediately

follows

that

for

any

vectorfield X contained in Co we have [f,Xj E Co. Since in the above local

a

,

coordinates Co - span(ax""

a 'ax} ,

the local coordinate expression for f

takes the form (see also Proposition 3.42)

88

(3.45) fk + 1 (X\c + 1 , ••.• Xn )

(i)

If f(x o ) E Co (xo )

f(q) E Co(q)

for

res tric ted to

Sx [I'

then

immediately

it

q E Sx o '

all

Now

follows

apply Theorem

from

3.21

(3. l ls)

to

the

that system

(ii) Since C(q) - Co(q) + span\f(q)} the equality dim Ceq) - dim Co(q) + 1 for all qED immediately follows. By (3.45) we can define a vectorfield (3.46)

living

on

an

open

part

assumption :teO) ~ 0,

S; o for

contained in that

U

f-

with

coordinates

(xle + 1

••••

,xn

).

By

Then

follows

it

that

is

aXk + 1

T > 0, and by the proof of Theorem 3.21 it follows

contains

R (xo • T)

n k -

R

and hence by Theorem 2.26 there exist coordinates

that

such

XIo;+1 ••• • ,Xn

of

a

non-empty

open

set

for

T >0

any

o

sufficiently small.

Finally we give the following corollary; its proof parallels the proof of Corollary 3.13. Corollary 3.23

If the syst:em (3.1) is locillly scrcmgly ilccessible. then

dim Co (x) - n for x in iln open and dense subset of ft. Exnmple 3.24 actuators

Consider

the

equations

of

a

spacecraft

with

gas

jet

(Example 1.2). We only consider the equations describing the

dynamics of the

angular velocities

w 1 ,w2 ,w:]

(called Euler equations).

Since the inertia matrix J is positive definite we can diagonalize it as diag(a1,a Z ,a 3

)

to obtain the equations 3

al w 1 -

w2 w J(a Z -a 3 )

+

L b;llj jut j

a 2w2 -

w 1 w3(a3- a 1)

+

L b~uj jUl 3

B J W3 -

W2 W1 (8 1 -a 2 )

+

L b~Uj j-l

(3.47)

89

where

bi

=

(b:,b:,bi)T,

i

1,2,3,

=

arc

vectors

in

[113.

We

distinguish

between three cases.

I.

bl

,

are

bz , b J

independent.

Clearly

in

this

case

the

system

is

controllable. II. dim span{b 1 ,b z ,b J ) ~ 2. Without loss of generality we may assume that b 3 - 0, so that in fact we only have two inputs u 1 ,u 2 _ First consider the simple case b l - (100)1, b z = (0 1 0)7, so that the torques are around the first two principal axes. Rewrite the system as

0,

g2.(W) = (0

0)1. Compute

ig, .tl (w)

ig, .tl (w)

[

~

[

A~WJ

Al w J

A1w Z

0

Azw 1

AJw Z

AJw 1

A~WJ A 3 wZ

0

A1w Z

0

AZw l

0

WI

Q1AZW J

°lA 3 wZ

Al W3

AJ

1[~'l [

0

1

(3.lI9)

°2 Al WJ

1[;'1 [

0

°Z A 3 W l

1

On the other hand

[g" ig, ,fl] (w)

~ ~ [

which also equals [gl' [g2'

o

o

o

ulA;>.

(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 <1>: 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.,<;.

Now let

E \', As in

Xl

the proof of Theorem 3.32 it follows that (3.81)

i E ~,

and so

Xl

For

I

E SXo n V. Therefore (x xlvx o ) C 5 xo n V. converse inclusion, we note that by

the

JI1 •• ",JIk we have for q E V,

dLfH i (q)

where

_k

+ i (q)

the

of 0

E span(dH l {q), ... ,dHk (q»),

Since H1, ... ,Hk are the coordinates dfn

definition

span(dxn _ k + 1

E

subscript

n-k+i

(3.82)

Xn-ktl""'xn

, ..•

, this yields

,dxn 1 (q),

(3.83)

denotes

vectorfield. Hence f and gj' j

the

(n-k+i)-th

component

of

(3.84) (gj )n-l:+l (xn -l:+1""

Furthermore Xn-k+l""

since

,X"'

i E

the vectorfields f

xo '

the

E!E, are of the form

f(x) -

the time c -

U E 'II

and

E.. j E!E,

i E

respectively

Hi (X(t,O,x o ,u»),

hi EO,

follows

it

E.. Now let Xo and gj'

j E

E

SXo n V.

that

hi

,Xn)

only

depends

on

It follows from the form of

E!, as displayed in

(3.84)

that for any

integrals x(c,O,xo,u) and x(c,O,xo'u),

starting from

xo ' i E~,

will

satisfy

(for

small)

C

and thus H(x(c,O,x o ,u»)

lli(x(c,O,xo,u»)

= H(X(t,O,x o

,u»)

for all

HEO.

0

Corollary 3.35

Assume thac che system (3.69) is locally observable. Then

dim dO(x) ... n for x in an open and dense subset of N.

Proof

(Compare Corollary 3.13).

dim dO(x)

n is open.

dim dO(x) < n.

First,

the

set of points

x

for which

Assume that there exists a non-empty set V where

By making V smaller we may assume that dim dO(x)

=

k < n

98

for x E V. Now use Proposition 3.34 to conclude that on V the system is not locally observable.

0

In general local observability does not imply observability,

as

is

illustrated by EXllmple 3.36

Consider the nonlinear system

x

u,

x

E IR.

Yl

sin x,

Y2

(3.85)

The

observation

Clearly Hence

cos x.

space 0

consist:s

of

the.

two

dim span(cos x dX,sin x dx)

dim dCJ(x) the

~

system

is

locally

observable since points XIl

and

observable. Xl

with

functions is

However

XO-x 1

one

sin x. cos x.

for

the

all

system

x

E IR.

is

not

a multiple of 2'11" are not

o

distinguishable.

Now let: us consider a linear system, written in accordance with (3.69) as

x

- A.x

-I-

Lb

j

Uj

•

x

n

E IR ,

jml

(3.B6) i E E,

Yi where c i

r

i E E, are the rows of the observation matrix C. Let us compute

the observation space O. First (3.87) Hence

the

Lie

vectorfield b j

derivative

of

a

linear

function

along

the

constant

yields a constant function, which will not contribute to

the dimension of the observability co-distribution dO. Furthermore (3. SB)

Again

~j(ClA.~)

is constant, so we continue with (3.89)

In general we have (3.90)

99

Hence by Cayley-Hamilton

0- span{cix,ctAx, ... ,clAn-lx, i E e.l + constant functions.

(3.91)

It follows that the system satisfies the observability rank condition in

an arbitrary point Xo if and only if C CA :

rank [

(3.92)

- n.

CAn - 1

(Recall that d(c1x)

~

ci

'

d(ciA.'\':) - ciA, etc.). This is exactly the Kalman

rank condition for observability.

of course we know from linear systems

theory that (3.92) not only implies local observability but even (global) observability.

Corollary 3.35 suggests observable,

that,

codistribution

the

even if a nonlinear system is

dD

may

have

singular

points

locally

q with

dim dO(q) < n. Indeed this may easily happen, as shown by the following Example 3,7 x = 0,

Consider the system

,

y - x ,

(3.93)

X E lI1,

which is clearly observable. However 0 consists of the single function x and so dim dO(O)

~

J

,

o

O.

On the other hand the following proposition shows that in case the system is locally accessible and is analytic the codistribution dO always has constant dimension. PropOSition 3.38

Let (3.69) be a locally accessible and analytic system.

Furthermore, suppose that N is connected. Then the codistribution dO is constant-dimensional. In particular,

(3.69) is locally observable if and

only if i t satisfies the observability rank condition.

Proof

As

and

in H we can find Xl' ... ,Xn E '{} and

Xl

in the proof of Proposition 3.15 (b) it follows that for any Xc t 1 , .. "

tn E IR such that (3.94)

100

(3.95) (with ~1I inductively dofined by

Li" = ",

and ~11 ~

Lx (~.111),

k ~ 1). By

analyticity we therefore have the Taylor expansion k

\'

t

Ie!

L.

k

)

(3.96)

LX"(x ,

and so (3.97) t

It follows that (since (X .):

-0

Trt N .... Tx" N)

:

Xl

0

(3.98) and (X

-t

so

dim dO(x 1 )

:$

..

)xldD(xo) C dO (Xl ) ,

required equality.

dim dO(xo ). and

so

In

the

dim dD(x l

)

same ~

manner

dim dD(xo ).

we

obtain

yielding

the

The last statement follows from Corollaries 3.33 and

3.35

0

Remnrk 3.39

In a similar way the following refinement of Corollary 3.13

can be proved: if the system (3.1) is analytic and lives on a connected state space r1 then local accessibility implies that dim C(x) - n for all

x

E fl.

Example 3.40

Consider

the

two-link

Example 1.1. Assume for simplicity that

rigid illl

=

robot

ill:?

=

I,

manipulator

1\ ..

.22 ...

from

1. Take as

1

x = (0 1 , O2 .0 1 , ( 2 ) E T(SlXS ). (Note that 0 1 ,8 2 are angles, and so both are properly defined on the unit circle Sl.) Then the state

the

vector

system is given as

III

d dt

°2

81 8'2.

01 O2

+ 1

1

-H- (O)C(0,i)-H- (0)k(0)

0

0

0

0

H-1(O)

U1

[ ].

(3.99)

U2

f Suppose that the output is given as the angle of the first joint, i.e.

(3.100) We will prove that the system is locally observable.

For the computation

101

of

the

observation

space

0

we

note

that

Lfh ""

81

and

L!+lluOl

=

(;"1'

Computing

-

[

3+2cos8 z

1 + cosO z

l+cosO z

-1 _ ]

1

-----'1~

-l-cosO z

2 1+5in 0 2

3+2cDSO Z

(3.101) we obtain

L:h

= 1+5i0

2

02

5in0 2 "02(20 1+0 2 ) + (l+cosOz)sinOz·iJ~J

1 (3.102) + ---";-(-2gsinBl - gsin(Ol+8 Z ) + (l+cosOz)gsin(Ol+0Z)j. 1+5i0

2

02

It follows that L;h depends in a nontrivial wayan the angle 8 2 in almost all points (0 1 .8 2 ,0 1 ,0 2 ), Since LfG;>. - Dz it follows that in these points L;h - L f (L:h) will depend non-trivially on Oz. Hence for almost all points

we

have

proved

everywhere,

that

and

Proposition 3.38.

so

dim dD = il.

the

Clearly

In

system the

order

is

system

is

to

prove

locally analytic,

that

dim dO = II

observable, and

we

N = T(SlXSl)

use is

connected. To show local accessibility we compute

(3.103) 1 with (H- (8)jl

1 the i-th column of H- (8), i

=

1,2, and where

*

denote sOllie

unspecified functions of the state. Therefore the dilllension of the span of

gl' gz, [f,gl]'

[f,gzl equals

(3.104)

1

and clearly is 4 since H- (8) has rank 2. Therefore by Corollary 3.11 the system is locally accessible. (In fact, the system (3.92) is controllable, since

it is

feedback linearizable to a controllable linear system,

o

Chapters 5.6.

3.3

cf.

Invariant Distributions; Local Decompositions

Lot us first consider the linear system x -Ax+ Bu,

x E {Rn, u E [Rill,

(3.105) Y

=

ex,

y EmF.

102

A subspace V

C

II1

n

is called A-invariant if A1' c V. If we choose a basis

e t •...• en for IR n such tha t V - span {e 1

' •••• ek}

then in this bas is A takes

the form

A

with

(3.106)

=

All

a

kxk-matrix.

coordinate functions on

Let

mn.

(Xl I'"

IX,,)

be

the

corresponding

linear

For convenience we write (3.107)

Then the system dynamics (3.105) can be written as

(3,108) Now let us consider the foliation of

mn

given by the affine subspaces (3.109)

or

in the above coordinates

I

Fy

'"

I x - (xl. Xl)

IXl

...

constant}.

Consider

two po ints q 1 and ql on a same leaf. i . e. ql -ql!. E V, or equivalently 2 2 X (ql) - X (q2)' Then it follows from (3.108) that for any U E U the solutions x(t,O,ql'U) and x(t,O,qz,u) at every time instant t

~

0 will be

again on a same leaf (depending on t).

Flg. 3.3. Invariant folitJIion. n

In fact the system dynamics (3.108) project to the dynamics on II1 /1' given by

x.z -

A22x2 + B2 u. We say that the foliation Fy is invariant under the

system dynamics

x - Ax + Bu.

only if V is A-invariant.

and we conclude that this is the case if and

103

Let us now generalize this idea to a nonlinear system (3.1). We only consider foliations F whose leaves are smooth submanifolds immersed submanifolds,

see Chapter 2).

(or at least

Moreover we will assume

that all

leaves have the same dimension. Then by considering the tangent spaces of the

leaves

in

any

p E H

point

we

obtain

a

constant

dimensional

distribution D

tangent space at p of the leaf of F through p

V(p) -

which is

involutive

foliation

F

we

(see Proposition 2.41).

will

work

with

its

(3.110)

Instead of working with

associated

constant

the

dimensional

involutive distribution. First, for a general distribution we define Definition 3,41

dynamics

x-

A

(smooth)

distribution

D on

H is

invariant

for

the

£(x), f being a vectorfield on N, if

[f,X] E D,

for any vectorfield XED,

(3.111)

or succinctly, [f,D] cD. Analogously

to

interpretation

the of

linear

case

invariance

in

we

have

case

D

is

the

following

invo1utive

geometric

and

constant

dimensional.

Proposition 3.42

Let D be a constant dimensional involutive distribution

on N,

t"hich

invariant

take

local

a

is

coordinates

D - spant-a-,··· Xl

a

'-a ). xl;.

x

for

x-

Denote

correspondingly write f _ (fl

x

,

Ii!).

f(x).

=

By Frobenius' ,xn )

(Xl""

on

vc

Theorem Ive

N

can that

such

,xn ), and Then in such local coordinates f(x}

..

(Xl

I

•••

,XI;.) ,

x? -

(XI; t I ' . . .

x-

on V takes the form

(3.112)

Hence

if

ql,qZ E V

2

x [qz(t»)

for all

are

t <::: 0,

such

that

with qi(t)

X

2

2

(ql) '" x (qz)

then

denoting the solution

X

2

(ql(t»)

l .. ithin

=

l' of

(3.112) for initial condition qi (0) '" qi' i - 1,2.

Proof

a

a

Since D - span{a:-"" '8)' and Xl XI;.

a

,

[f,ax]

-I" j-'

af j

a

aX i aXj

(3.113)

104

fj denoting the j-th component of f. we obtain from (3.111)

with

i which implies

!:.

E

(3.112).

f

(3.114)

j - Jc+ I, ... ,n,

The last statement immediately follows

from

Remark 3.43

In the linear case we associate wi th a

distribution Dv on IRn. In fact if V "" spanl e 1

x1' ...• x n

then

, • . . ,ex}'

for

the

Ij1n

the

o

particular form of (3.112).

in

a 1 • . . . ,Bn

the

distribution

subspace V

lilt> the

is a basis for W such that

corresponding Dy

C

n

equals

linear

coordinates

5panla~1""'a!kl.

It

is

immediately verified that [Ax .Dv] C Dy i f and only i f AV c 1'. Next we define invariance for nonlinear systems (3.1).

Definition 3.44

A smooth

distribution

D on

H is

invariant

for

the

nonlinear system (3.1) it

[f.D] c D.

(3.11sa)

[gj ,D] C D,

e !E.

j

(3. l1Sb)

(or, equivalently, it D is invariant for every vectorfield in

~;

the set

of associated vectorfields (ct. (3.2»). As in Proposition 3.42 it follows that if D is constant dimensional and involutive I -

(X

1

of D

,x2 ).

and

if

we

that D _

choose

x -

coordinates

span(~l :- span{ axa

(xl'"

a.. ' ' ...• ax

'Xk ,Xk + 1 ' .• ,Xn )

i nva r'l.anc e I 1 axl .. is equivalent with the following local representation of (3.1):

such

2

2: gJ (xl, XZ) Uj •

j

jl' -

r(x

2

m

)

lO

l

g;

2:

+

th en

.m

+

xl _ [1 (xl, x )

l

(3.116) (X2)U j

,

j"l

and so the dynamics of x

2

are not influenced by

xl

(as in the linear case,

cf. (3.108».

Remark

At first sight there is a difference with the linear case, since

in that case, contrary to Definition 3.44, we did not impose any condition on the input matrix B. This is however explained by the fact that b j columns

of B,

are

in linear coordinates

for IR

fi

,

the

constant vectorfields,

105

while also the distribution DW associated with the subspace V is spanned

by constant

vectorfields.

Since

the

Lie

bracket

of

any

two

constant

vectorfields is zero, it follows from Proposition 2.27 (n) that condition

(3.115b) is automatically satisfied in the linear case.

We can also define invariance for the dual object of a distribution; a codistrlbution,

(2.171).

cf.

respect to a vectorfield f

We

call a

codistribution P invariant with

if

for all one-forms

0

E P.

(3.117)

Condition (3.117) will be abbreviated as LfP C P. A codistribution Pan H is invariant for the system (3.1)

Definition 3.45

if (3.118a) j

E

!E.

(3.11Bb)

The next proposition explains the relation with Definition 3,tl l l

•

Proposition 3,46 (a) Let tbe codistribution P be invariant for (3.1), then the distribution ker P is invariant for (3.1).

(b) Let the distribution D be invariant for (3.1), then the codistribution ann

D is invariant for (3.1).

(c) Let D (P) be a constant dimensional (co-)distribution.

Then D (P)

is

invariant for (3.1) if and only if ann D (leer P) is invariant for (3.1).

The basic formula to be used is (2.169), with f

Proof and

U

L,(u(X)) (a) Let

u E P

(Lfu)(X) ,.. 0.

(L,u)(X) + u([f,X])

and

X E leer P.

Hence u([f,X)

for f replaced by gj' j (b) Let

XED

and

=

(3.119)

Then

u(X) = 0,

0 and thus

and

o([f,X]) - O. Hence (Lro)(X)

~

Then

u(X) ~ 0,

and

invariant,

Lra E P

also

The same holds

since

[f,X) ED

also

0, and thus LeO E ann D. The same is true

~.

c. In case D has constant dimension, we have (cf. if D is

since

[f,Xl E Iter P.

E m.

u E ann D,

for f replaced by gj' j E

Now

and X vectorfields

a one-form:

then by

(b)

ann D

is

(2.175) ker(ann D) - D. invariant.

If

ann D

is

106

invariant.

then by (
holds

a

for

constant

dimensional

is invariant. A similar reasoning

codistribution P since

(cf.

(2.175))

o

ann(kcr P) - P.

For

a

linear

system

characterization of

there

(3.105)

:n - Im(B!ABj··. ji'-lB),

is

an

appealing

geometric

the reachable subspace, and of

n~l

N ~

11

ker

i 1 cA - •

the

unobservable

subspace.

:n

Indeed,

isO

is

the

minimal

A-invariant: subspace that: contains 1m D, and N is the maximal A-invariant subspace contained in ker C.

For nonlinear systems we will now give a

similar characterization of the strong accessibility distribution Co and the observability distribution leer dO (or the codistribution dO).

Proposition 3.47

characterizing

Consider the nonlinear system (3.1) '''ith distribution Co local

strong

accessibility,

nnd

tile

codistribution

dO

characterizing local observability. (a) Co

is

the

smallest::

distribution

chat

is

invariant

tor

(3.1)

and

that 1s invariant tor

(3.1)

and

contains the distribution G(q) :- span(gl (q) •. ..• gm (q)}. (b) dO

is

the smallest codistribution

contains the cod1stributiof1 dh(q) :- span(dh 1 (q) • ... ,dhp (q) J. (c) ker dO is the largest distribution that is invariant tor (3.1) and Is contained in the distribution ker dh.

Proof

(a) By definition Co is invariant for (3.1) and contains C. Now let

D be a smooth distribution, which is invariant for (3.1) and contains G. Then gj E D,

J

E~.

and by invariance (3.120)

for Xi E (t,gil" .,gm l , i E k. Hence (see Proposition 3.20) XED for any X E Co

and so Co (q) c D{ q)

for any q EN.

Therefore Co

is

the smallest

distribution that is invariant and contains C. (b) By definition dO codistribution

tha~

is

invariant

(3.1)

and

dh.

contains

Let P

be a

is invariant for (3.1) and contains dh. We prove that

dO{q) c P(q) for any q

E N,

Indeed,

since dh j

E

p.

J

E

E,.

it: follows by

invariance of P that (3.121)

for Xi E (f ,gl , ... ,gm). i E ~ (since Ly, i dH j

-

dLy, 1Hj. cf.

(2.161». Hence

(see Definition 3.29) dH(q) E P(q) for any H E O. (c) By

Proposition 3.46

(a)

the

distribution

ker dO

is

invariant

for

107

(3.1).

Furthermore ker dD(q) C ker dlI(q)

for any q E N.

Now let D be a

distribution that is invariant for (3.1) and contained in ker dh. Then

(3.122)

ann D :J ann(ker dh) :J dh,

and by Proposition 3.46 (b) ann D is invariant for (3.1). Hence ann D is a codistribution that is

invariant for

and contains dh.

(3.1)

By part

(b)

this yields

(3.123)

ann D :J dO,

and therefore D c ker(ann D) c leer dO.

Remark 3.48

Part (a) also holds for the distribution C if we replace the

distribution G(q) by the distribution span{f(q) ,g1 (q), ... ,gm (q»), q E N. In case Co

and dO are constant dimensIonal,

Frobenius'

Theorem we

nonlinear

system.

well-known

obtain

These

the

decompositions

decompositions

of

a

then by an application of

following

linear

local

are

decompositions

quite

system

of a

analogous

to

the

corresponding

to

its

reachable subspace n and its unobservable subspace N.

Theorem 3.49

Consider the nonlinear syscem (3.1).

(8) Let Co have consCant dimension. By Frobenius' Theorem (Corollary 2.43)

we can find

t118t Co

=

;/ =

local

coordinates x

m

fl(xl, ..l) +

I

j-' ·2 X -

If l"e

=

(Xl, Xl)

=

(Xl' ••• ' XI; 'XI; -+ 1 , •••

,xn )

such

span(....£....). In these coordinates the system takes the form axl

_l

I

j

(3.125a)

,

l

(3.12Sb)

(x ).

regard

(everything

g; (./ ,xl)u

(3.1258)

locally),

as

a

then

system l"ith for

any

value

state of

Xl

Xl

parametrized the

syscem

by

xl

(3.1258)

satisfies the strong accessibility rank condition. (b) Let dO have constant dimension. local

coordinates

ker dO

span(~J. ax2

, ,

X =

By Frobenius'

Theorem T"e can find

(x ,X ) '" (Xl' ••. ' X.I' ,X.1'-+1' ••• 'Xn

)

such

In these coordinates the system takes the form

that

108

l f1(X )

xl _

L g; (Xl)U j m

+

•

e.,

1 E

(3.126£1)

j=l m

1 2 ;.2 _ r(X ,X ) +

1 2 g;(X .X )U j

I

(3.126b)

•

j=1

If we regard (3.126ll) as a system with state xl (locally defined),

then

(3.126a) satisfies the observabiliry rank condition. Proof

(a) The

form

a

gj E span(-l}'

j

Proposition 3.20

it

ax

vectorfields

have it

Xl

from

follows

From

follows

eo

in

dim Co = k = dim

(3.125) m.

E

that no

(3.116)

in

the

above

the

fact

that in

that

local

in

components

follows

and

characterizat:ion

the

coordinates

directions.

(3.125a)

satisfies

all

Since

the

strong

accessibility rank condition for each KZ ' (b) The

(3.126)

form

ker dhl

~

D does

not

follows

a

e.

spanlaxz}' i E depend

on

X2.

(3.116)

from

and

the

fact

that

It follows from (3.126) that every function in Since

dim dO =.£

dim ,,}

this

implies

that

o

(3.126a) satisfies the observability rank condition.

Remark

The

dynamics

(3.125a)

will

be

called

the

strongly

accessible

dynamIcs of the system. Similarly the dynamics (3.l26b) will be called the I1nobservable dynamics of the system.

For a linear system (3.105) it is well-known that one can combine the above

controllabilit:y

decomposition

and

observability

decomposition

(usually called the Kalman decomposition),

into

single

B

where At

B,

C

take the form

o

All

[

AZl

A22

o

o o

o

(3.127)

in coordinates x = (xl.X2,x3,X4)

with 1/ = spanlx1,xzl

and N = span[xz,x~).

In order to generalize this to the nonlinear case. we need the following extension of Frobenius' Theorem (Theorem 2,[12):

Dz be involucive constant dimensional distributions on a manifold H. sl1ch that also Dl + Dz is involutive and Proposition 3.50

Let

Dl

and

constant dimensional. Then about any pEN there exisr local coordinates

1

J

4

x ... (x .x?,X ,x ) such t/Jilt

109

a

a

D, - spanl-,'-,I. ax ax (3.128)

D, - span{_a_,~l. z 3 ax ax

Recall the proof of Frobenius' Theorem. We may assume that we are

Proof in!R

n

with p ... D. Moreover we may assume that

a

D, (01

a ar

spanl-I Z

D z (0) -

r -

with

a ,-1 a .-1 ar

spanl-I art 0 ar Z

(rl,r2,r3,r~)

accomplished

a

by

0

3

I. 0

(3.1291 I. 0

natural

linear

coordinates

for {Rn.

transformation

of

(This

!/In.)

can

Let

be

always

dim ri - ki'

i - 1,2,3. Let

11",,: lI"b: 7rc :

Since

~n

~

IR k1 +k Z

be the projection onto span(r

,/') ,

mn

[Rk z +k3

be the projection onto span[r ,r

•n

~k,

z be the projection onto span{r ) .

DI ,D2

Z

and

D, + D,

restricted

Then lI"e·O D1 (0) nDz(O).

to

constant

have

constant dimension.

Dl (q) n

1

:

3

)

(3.130)

I

dimension,

TolRn

-t ToIRkZ

By

continuity

is

also

D, n D,

when

isomorphism

an

is

1T c ~ q

has

one-one

on

D2 (q) for q close to O. Hence near 0 we can find unique vectors

Z1 (q), .. "Zk/q) E Dl (q) n Dz (q) such that

(3.131)

Now

consider

11"".

when restricted

By

definition

Dl (0).

to

It

vectors Xl (q)",. ,Xkl + k2 (q) E

a - hi i

11"" (q)

.

1fa~O:

follows

Dl (q)

is

TorRn ... TolRkl+k2

that

near

0

we

find

unique

(3.132)

i - l, ... ,kl +k2 .

a

ar2

i - l , ... ,k2 1l"b

can

isomorphism

such that

Furthermore since 1l"a~q coincides with 1I"c*q on span(-I

In the same way by using

an

•

we obtain unique vectors

1 i t follows that q

(3.133)

110

such that

a . . a?j 1

, 1fb (qJ

(3.134)

1fb~q coincides with

Since

1fcnq

on

span(~1 ar2

J it

follows that

q

(3.135)

1 ... 1, ... , le2, .

In the same way as in the proof of Frobenius' Theorem it: follows that (3.136)

i ,j - 1, ... ,kl + Jc 2 + k::J.

and hence by Theorem 2.36 we can find local coordinates

o

such that (3.128) holds. Theorem 3.51

Consider

the

nonlinear

(3.1).

system

Assume

chat

che

distributions Go. ker dO and Co + leer dO all have const::anc dimension. Then f"e

can find local coordinat.es x ""

(Xl

4

,x'/. • x:! .x

)

such that the syst.em takes

t.he form 1 3 fl(X ,X )

·1

X

m

I

+

gt (Xl .xJ)U

(3.137a)

j

jal

•Z = X

f2

III

3 C..?, i' ,x .:/) +

L g~ (Xl.X2 ,x3 ,X4)U j

(3.137b)

j~l

J x• 3 _ f3 (x )

•4

x

=

£4 (x 3 ,X~)

Yi _ h 1 !.;rit.h

Proof

Co =

(3.137c) (3.137d)

3 1 (X ,X ).

B

spanl--

i

E

E.

~) and leer dD

axl ' ax2

(3. 137e)

=

span{~ ~I. i ax:! ' ax ,

By Proposition 3.31 ker dO is invariant for the vectorfields in

eo.

111

Hence ker dO

+

is an

Co

3,50 and interchange x

J

involutive

with

X4.

distribution.

Now apply Proposition

Then as in Theorem 3.49 the form (3.137)

o

follows. Notice that the input-output behavior (cf. local form

involving

(3.137)

only

is

the

Chapter 4)

completely described by

, , x ,x ,

states

together

of the system in

the dynamics

with

(3.137a,c),

output

the

equations

(3.137e), Moreover, if the xJ_part of the initial state is an equilibrium for

(3.137c)

then

x

J

remains

constant,

and

thus

also

the

subsystem

(3.137c) does not contribute to the input-output behavior.

Notes and References The

idea

of

using

reachability

can

subsequently in [Lol.

[HH),

Lie

he the

[Ell.

brackets

traced

in

back

to

the feh].

context of nonlinear

[SJI.

[Sull.

[Kril.

study It

of was

control

[HKI

accessibility further

theory

and

[Su2].

or

developed

in e.g. [Su3].

[Hel.

We have

confined ourselves to a purely local treatment; for more global results we refer to especially [SJ), on [SJ),

[Su1.2].

[HK]. Our exposition is largely based

[HKj, and on the lecture notes [Cd),

(Is]. For more information

concerning the remark after Assumption 3.1, we refer to [Su2 J. The proof of Proposition 3.3 is taken from [U1]. More general expressions than the one

given

in

proposition

3.6

can

Theorem 3.9 given here is due to

be

[Kr1].

found

in

[Hel].

The

proof

of

More information concerning the

issues raised in Remarlt 3.17 can be found in e. g.

[Her],

[Su3 J,

[Su4],

[Stl. Example 3.24 is taken from {Crl ,Cr31, and the final conditions for the two-input and one-input case can be found in [Ba]. Accessibility of bilinear systems was first studied in [Brl. The

study

of

observability

using

the

observation

space

as

in

Section 3.2 can be found in [HKJ, see also [So] for some modifications. In [Cr2j it is shown that an analytic system is observable if and only if the observation space distinguishes points in H. The relevance of the notion of invariant distributions in control theory, together with the resulting decomposition [IKGM].

The

(3.116), relation

was with

invariant distributions Co

brought

forward

independently

controllability (or C)

and

in

observability

and ker dO is due to

[IKGMJ.

[Hi],

and

via

the

Previous

work on local decompositions based on the accessibility algebra appeared in [Kr2],

see for later work in this direction

controllability

and

observability

with

[ReJ.

invariant

particularly stressed and elaborated in [Is].

The connection of distributions

was

112

In this chapter we have confined ourselves to affine nonlinear systems, However, Theorem 3.9, Corollary 3.11, Proposition 3.12, Corollary 3.13 and Proposition 3.15 immediately carryover to general systems

x~

f(x,u) (see

[HK)). For the extension of Theorem 3.21 to the general nonlinear case we refer to [SJ], 3.34,

Corollary 3.35

systems to

(vdSl,2]. Also Theorem 3.32, Corollary 3.33, Proposition

x

=

[vdSl, 21,

treated.

and

Proposition 3.38

immediately

carry

over

to

f(x,u), y - h(x}. For systems X - f(x,u), y = h(x,u) we refer where also

the relation with invariant distributions was

"'or more information concerning general nonlinear sys terns we

refer to Chapter 13. Finally, in Chapter 12 some specializations of the theory

of

nonlinear

controllability

and

observability

to

mechanical

nonlinear control systems will be given. {Ba]

J. Baillieul, "Controllability and observability of polynomial dynamical systems", Nonlinear Anal., Theory, Meth. and Appl. 5, pp. 543-552, 1981. [BrJ R.W. Brockett, "System theory on group manifolds and coset spaces", SIAM J. Contr. 10, pp. 265-284, 1972. W. L. Chow. Uber Systemen von Unearen partiellen Differential[Ch] gleichungen erster Ordnung, Math. Ann. 117, pp. 98-105, 1939. [Crl] P.E. Crouch, Lecture Notes on Geometric Non Linear Systems Theory, University of Warwick, Control Theory Centre, 19B1. [Cr2] P.E. Crouch, "Dynamical realizations of finite Volterra series", SIAM J. Contr. 19. pp 177-202. 1981. [Cr3) P.E. Crouch, "Spacecraft attitude control and stabilization", IEEE Trans. Aut. Contr. AC-29, pp 321-331, 1984. [El] D.L. Elliott, "A consequence of controllability", J. Diff. Eqns. la, pp 364-370, 1970. [GB] J.P. Gauthier, G. Barnard, "Observability for any u{t) of a class of nonlinear systems", IEEE Trans. Autom. Contr. AC-26 , pp. 922-926, 1981. [He] R. Hermann, "On the accessibility problem in control theory" in Int. Symp. on Nonlinear Differential Equations and Nonlinear Mechanics (eds. J.P. La Salle, S. Lefschetz), pp.325-332, Academic, New York, 1963. [Hel} S. Helgason. Differential geometry and symmetric spaces, Academic, New York, 1962. [Her] H. Hermes, "Control systems which generate decomposable Lie Algebras", J. DiEf. Eqns. 44, pp. 166-187, 1982. [HH] G.W. Haynes, H. Hermes, "Nonlinear controllability via Lie theory", SIN1 J. Contr. 8, pp. 450-460, 1970. [Hi] R.M. Hirschorn, "(A,E)-invariant distributions and disturbance decoupling", SIAM J. Contr. 19, pp. 1-19, 1981. [HKJ R. Hermann, A.J. Krener. "Nonlinear controllability and observabilicy". IEEE Trans. Aut. Gontr. AC-22, pp. 728-740, 1977. (Is] A. Isidori, Nonlinear Control Systems: An Introduction, Lect. Notes Contr. Inf. Sci. 72, Springer, Berlin, 1985. [IKGM) A. Isidori, A.J. Krener, C. Gari Giorgi, S. Monaco, "Nonlinear decoupling via feedback: a differential geometric approach", IEEE Trans. Aut. Contr. AC-26 , pp. 331-345, 19B1. [Krl] A.J. Krener, "A generalization of Chow's theorem and the Bang-bang theorem to nonlinear control systems", SIAM J. Contr. 12, pp. 4352, 1974. f

113

[Kr2]

A.J, Krener,

"A decomposition theory for differentiable systems",

SIAM J, Contr. 15, pp. 289-297, 1977 (Kr3]

[1M]

A.J. Krener, "(Ad f,g), (ad f,g) and locally (ad f,g) invariant and controllability distributions", SIAM J. Contr. Optimiz. 23, pp. 523-549, 1985. E,n. Lee, L. Markus, Foundations of Optimal Control Theory, Wiley,

New York, 1967. [La]

C. Lobry,

rNe]

Contr. 8, pp. 573-605, 1979. E. Nelson, Tensor analysis, Princeton University Press, Princeton. 1967.

"Controlabilite

[ReJ

W. Respondek,

"On

des

systemes

decomposition

of

non

lineaires",

nonlinear

control

SIAM J.

systems",

Syst. & Contr. Lett. 1, pp. 301-308, 1982. [So]

E. Sonntag, "A concept of local observability", Systems Control Lett. 5, pp. 41-47, 1985. [StJ G. Stefani, "On the local controllability of a scalar input-control system", in Theory and Applications of Nonlinear Control Systems (eds. C.l. Byrnes, A. Lindquist), pp. 167-182, North-Holland, Amsterdam, 1986. [Su1] H.J. SUssmann, "Orbits of families of vectorfields and integrability of distributions", Trans. American Math. Soc. 180, pp. 171-188, 1973. [Su2] H.J. Sussmann, "Existence and uniqueness of minimal realizations of nonlinear systems", Math. Syst. Theory 10, pp. 263-284, 1977. [Su3] H.J. Sussmann, "Lie brackets, real analyticity and geometric control" , in Differential Geometric Control Theory, (eds. R.W. Brockett, R.S. Millman, H.J. Sussmann), pp. 1-116, Birkhauser, Boston, 1983. [Su4] H.J. Sussmann, "A general theorem on local controllability", SIMI J. Contr. Optimiz. 25, pp. 158-194, 1987. [SJJ H.J. Sussmann, V. Jurdjevic, "Controllability of nonlinear systems", J. DHf. Eqns. 12, pp. 95-116, 1972. [vdS1] A.J. van der Schaft, "Observability and controllability for smooth nonlinear systems", SIAM J. Contr. Optimiz. 20, pp. 338-35l" 1982. [vdS2] A.J. van der Schaft, System theoretic descriptions of physical systems, CWI Tract 3, CWI, Amsterdam, 1984.

Exercises

3.1

Consider the

following more sophisticated model of a

with Example 2.35).

--x

Fig.

J..t. Model of a cur.

car

(compare

114

The configuration space

1

is N - [R2xS xS

1

parametrized by

(x ,Y ,fIJ. 0) •

where (x,y) are the Cartesian coordinates of the center of the front axis, the angle rp measures the direction in which the car is headed. and 0 is

the angle made by the front wheels wi th

the cal.".

(More

realistically we take -Ornax < 0 < 0rnox') There are two input vectorfields. called Steer and Drive. Clearly, Steer - ~Ol while after some analysis we see that, in the appropriate units, Drive

~

cos(rp+O):x +

Sin(rp+O)~y +

sin

O!rp'

(11)

Prove that

(b)

iJ -sin(~~O)aa x + COS(rp+O)aiJ y + cos 8 --a -: Wriggle. rp iJ a Define Slide ;- -sin rp ax + cos rp ay' Prove that ISteer,WriggleJ

[Steer,DriveJ

~

-Drive,

and

=

rWriggle,DriveJ

~

Slide.

Compute

furthermore

the

bracket of Slide with Steer, Drive and Wriggle. Compute the dimension of the distribution C spanned bye, and

(c)

show that the system is locally strongly accessible and controllable. 3.2

(see Remark 3.4). Show under the assumptions of Proposition 3.3 that the set of points which can be steered to Xo

Ilu(t)11

admissible control satisfying


!

in time T > 0 using

> O. contains

B

neighbor-

hood of x[]. Prove condition (3.53).

3.3

([Sa)

3.4

(see

also Remark 3.17)

f{x o )

~

(1)

z .. Az

+ Bv,

Y - Cz, be

its

Consider

a

nonlinear

system

(3.69),

with

O. Let

linearized

Z E

IR". v

yE

IR P ,

system

in

m

E IR ,

xa.

u

=

0,

Le.,

B (a)

Prove equation (3.38).

(b)

Show by using (ll)

that i f (1)

is controllable

V

then R (xo ,T)

contains a non-empty open set for any neighborhood V of xo. and any small T> O. (This is a weaker version of Proposition 3.3.) (c)

Prove that c i Ak = dL;lli (xI)

,

Ie

1,2, . . . ,

i E

E.

and show that if (1) is observable then (3.69) is locally observable at xa. 3.5

Consider a linear time-varying system (la) x(t) - A(t)x(t) + B(t)u(t), (lb) yet) - C(t)x(t), y

E ~P,

where A(t), B(t), C(c) are matrices of appropriate dimensions. whose elements are smooth functions of time t.

115

(a)

Rewrite (la) as a nonlinear time-invariant system x.~

(2)

=

A{t)x + B(c)u

{ t - 1

with augmented state (x,t). Define for k - 1,2, ... the matrices (3)

Dk (t) -

(B(t):

(ACt) -

(A(t) - :t)B(t)

~t)k-1B(t)1,

Use the theory of local (strong) accessibility for nonlinear systems in order to prove that if for some k

(4)

rank Dk(c) -

."

then R

(XOlt)

T> O.

(N.B.

for all

fl,

t

> 0,

contains a non-empty open set of ~n for all Xo and all Using

."

implies that R

linear

(~o It)

n

m,

=

arguments

one

can

even

prove

that

(4)

for all x o ' T> 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\<.)ds\<.".ds 1 (4.28)

k=l 0 0

!dth

as in (4.27), is uniformly absolutely convergent

t"k(t,sl .... ,sk)

on

[0 ,T).

Remark For a linear single-input single-output system x - Ax + bu.)r - ex we note that w1(c,s,x) as in (4.21) is given by (4.29) and hence does not depend on x. Therefore by (4.24) and (II. 27) we have 10'1

(t,s) -

ceA
and

to'!

,s!) = 0 for i > 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<; more equations not involving

=

x than

11);+1

and is a system

'

(4.64).

constant

rank

assumption

then

any Ie '" 1,2,.

we

call

(11.66)

is

(x, u,y) a

regular point for the algorithm.

It is clear that for a regular point the algorithm terminates after a

finite

number

of

steps, Je"

Moreover, necessarily

denoted

Jc~,

by

::s n. Denote n

in

the

n "

=

sense

that

Pk"+l =

D.

then after performing the

k +1 '

algorithm we end up with a system Xl

-

ft(x,u)

Pt(x,u, .. ,u

=

i ... 1, ... ,n

0 ,

(k~ -1)

,Y,"'Y

de" l)

o

i

Let us first consider the case n ~ O.

=

n+l, .. ,n+p

(4.8la) (4.81b)

Recall that at the k"-th step we

assumed that

rank

[::~ 1

=

i~nk "+1, ..

J

,n+p

Sk~

around

,

(x,ii,y)

(4.82)

j~l"",,n

Comparing the recursive definitions of

n"

and Pk

ic=1,2, .. Ie-O,l,.

w,

immediately

because

equations PI'"

rank

obtain

n" -

n-sk _ 1

we assume n '" 0, n = 5,,'"

[

,

Ie <:!: 1.

Hence

and,

Therefore by (l1.82) we can reorder the

,P,,+p of (4.81) in such a way that

ap, 1 ax J

.l~l • .

'" ,n

n (= S,,*), around

(x,ii,y)

(4.84)

132

where

the equations P!, .. 'P n

taken from the set P

are

by the Implicit Function Theorem, we can solve.

Uk "+1

, ..• P n + p

' Then __ _

locally around (x. u ,y) ,

frolll

. dc n _ '2. ) (/c 1 ) Pi (X,U,lI, .. ,u ,y,}" .. ,Y -) for the variables Xl'"

,X n '

!2.

(4,85)

),

i E n.

(4.86)

of

(11.B6)

in

remaining

equations

0,

i E

i,e.

. d," - 2 ) ' de" V'1 (u,U, .. ,U ,J',Y, .. ,y Substitution

=

the

-I)

Pn+II"'Pn + p

yields

equations of the form

(4.81)

i E E"

\.Jhich finally constitute the external differential representation of the nonlinear system (4.1). Summarizing

Theorem 4.1 Consider the smoot11 nonlinear system (4.1), a.nd let El

regular po1.nt for the Algorithm 4.4. Suppose chat

transformed around

(x.li,y) into the equat:ions

n=

li, y) be

O. Tllen (4.1) is

(4.86) togetller {"ith (4.87).

The firsr: equati.ons (4.86) express ehe st:aCe x as function of u a.nd y and eheir deriv<1tlves, up to order k*-l:$ n-l. while t:ha la.st: equations (4,87) form the external differential represencation of che syscem.

Example 4.8 Consider the bilinear system 0

Xl

-

UAr;

='

P1 (x,x,u)

(4.88a)

0

x2

-

UXl

=: P2 (x,x,u)

(4.88b)

X3

-

x ll

-'

PJ(x,x,u)

(~,

P (x,Y) "

(4.BBd)

0

~

-

uX 1

='

BBc)

ap~

Clearly rank

ax

1 everywhere. and

i\

=

of the algorithm we c:ransform (4.88) into

o

Y

xJ

•

Hence at the first step

(~.B8a,b,d)

together with

(4.B9)

Y

Then -1

o

: J-

(4.90)

2 •

133

and

P3

=

Y-

x2

-

uX l

UX l •

-

'system into (4.88a,d),

Hence at the

second step we

transform the

(4,89), together with (h.91)

For the third step we notice that

ap,

,

-u-u

-u

ax

-u

-1

0

ax

0

0

-1

ax

0

BP,

ap,

which has rank 3 for u + u - u

as;; =u+u-u

so that

,

3

...

O. Excluding this point we define

Outside u + u - u

3 =

0 we

transform the system

aX l finally into

o

y

=

(4.94 )

From the

u

+ U - u3 xl

=

last ...

three equations we

D.

can solve

for x 1

,X

Z

and x 3

,

provided

Indeed

1 --=---', (5' u + u -

u'y)

u

(4.95) X

z

=

1 --=---, , Cuy u

+

u -

+

uy -

uy) ,

xJ

=

y .

u

Substitution of (f1.9S) in the first equation of (4.94) gives the external

differential representation (4.96)

o We shall now show how the property of n being equal to 0 immedi.ately relates to the local observability of (4.1).

In fact we will show that

n

134

is the dimension of the unobservable dynamics of the system (see Remark after Theorem 3.49).

Lemma 4.9 The subspace of ~n given by

L-

(/.j.97)

n+l •..• n+p

j

does noc depend on Proof By

,u

U ...

definition

(k"

-1)

l, ...

,n

tic" )

,Y •.. ,y

k~,

of

~

p

~

~

Ir. +1

... O.

Hence

the

subspace

(4.97) is also given as 8PL

Iter

It

[

8x , (x,U, .. ,u

(Jc" -2)

, y, .. ,y

follows

-ll

)

1

that

the

subspace

(4.9B)

thereby

de" -1)

will

(4.98)

not

if

change

we

any function obtained by prolongation of

k

and

(/.j.98)

i ... n\.:,,+l •.. ,n+p j = 1, .• ,n

i ~ n ,,+l, .. ,n+p, and substitution of

y, .. ,y

w

J

Pi ,1 = nt.,,+l •.. ,n+p,

(i1.97)

(k

x ~ f(x,u)

depend

does

on

to

herein. Now suppose that

some

variable

u, .. ,u

(k~ -2)

Let r ~ k*-l be the highest derivative of u or y such that

(4.98) depends on u and substitute

(r)

or y

(r)

Then prolong the equations Pnk"+l'" ,Pn +p

and join these equations to Pn\c"'+l'" 'Pn+l' in cr that (4. 98) depends non-trivially on u +1) or y(r+ll

f{x,u),

X -

(4.98). I t follows

which is a contradiction with the definition of r.

It follows

add

some Pi'

from Lemma 4.9 that (4. 97)

0

defines a distribution D on the

coordinate chart of N, the state space manifold, that we are working in. Furthermore, since D is given as the intersection of the kernels of exact one-forms,

the distribution D is Involutive.

(x,u,y)

that

is

dimension around

a

regular

(x,u,y).

point

for

Moreover,

Algorithm

4.4,

since we assume D has

constant

Hence by Frobenius' Theorem (Theorem 2.42) we

can choose local coordinates (Xl, Xl), with dim Xl - n, such that D - span

,.J!.-).

It follows that in such coordinates (4.81) takes the form

axl

'1 X

Pi (x

(4.9911) 2 U, ••

,u

(k~

-I

)

, y, ..

,y

(k" )

) - a .

i

~ n+l, .. ,n+p

(4.99b)

We immediately see that D is invariant (Definition 3./.j4) for (4.1). Since

135

the equations Yi - h 1 (x) in the case i ...

0\:*+1, . . .

y, .. ,Y

nn

P

(k* -

I ' ••

system.

=

0, i E E, are contained in (4.99b) it follows

i E E., only depends on

that hi (x),

Ii -

and so D is contained in ker dll. As 51:"

variables

the

for

,n+p,

1)

Xl,

0 we can solve from

of the equations P l

(= n-n)

, x

functions

as

of

0,

'"

(k"

-2)

t i " , ,ll

Z Substitution of X in the remaining p equations of the set

,Pn+p

yields

Summarizing,

the

we

external

have

differential

obtained

the

representation

following

of

the

generalization of

Theorem 4.7.

(x,u,y) be a (x,u,y)

Theorem 4.10 Consider the system (4.1), and let

for Algorithm 4.4. Then (4.1) is transformed around

regular point Into a three-

fold set of equations ·1

x

x

,

fl(X1,X'l,u)

1/t{u, .. ,u

Ri {u,u, .. ,u Here

(4.100s)

dim x

de" -2)

(1c" -1)

y, .. ,y

(k" - 1) )

y,y, .. ,y

(1c" )

)

-0

1

,

dim

x

i

E

E

(4.100a)

- n

(4.100b)

n-n

(i1.lOOc)

are the dynamics of the unobservable part of the system,

(4,100b) expresses the observable part of the state as function of

u

and y

and derivatives, and (4, 100c) is the external differential representation of the system.

Proof The

only

thing

left

to

be

shown

is

that

~

D

span

I_B_I

is

the

axl largest distribution contained in ker dh However since x

2

derivatives it follows that x knowledge of (any) definition of

that

is

invariant

for

(4,1).

is expressed in (4.l00b) as function of u and y and their 2

locally can be determined on the basis of

input function and resulting output function.

local

observability

(Definition

3.28)

this

implies

By the that,

indeed, D - leer dD, with the D the observation space (Definition 3.29).

0

4.3 Output Invariance

In this section we study the problem when in a nonlinear system (4.1) certain

input

component

uj

does

not

influence

a

particular

a

output

component Yl' For linear systems we know that this amounts to the transfer function (or impulse response function)

from u j

to Yi being zero,

look for a nonlinear generalization of this condition.

and we

In the linear case

136

there

also

existence

exists of

an

corresponding to

Yi

an

equivalent

invariant U J'

geomecri.c

subspace

condition

cont;lining

the

involving

input

the

vector

bJ

and contained in the kernel of the output function

The nonlinear generalization of this last condition given here

cix.

will be crucial for the deYelopments in Chapters 7 and 9. For simplicity of notation we consider instead of (4.1)

the

smooth

system f(x) +

x

L gj (x)u

j

+ e(x)v,

jwl

(t,.lOl)

y "" hex) •

with u 1

•..

,urn and v the scalar inputs, and y a scalar output. and we want

to deduce conditions which ensure that the input v does not influence the output y for any initial state xl) and any input functions u 1 Chapter 3 we restrict ourselves

throughout

piecewise continuous from the right) input functions u l Definition 'f.11 tIl •.. ) urn'

Consider

the

system

\' piece\"ise constant

, ••

,urn' As in

to piecel>'ise constant

(I; 101)

!llitlJ

, ..

the

(and

,um,v.

input

functions

say dwt the output y is not affected by

r,le

(or invarillnt under) tile inpuc v if for every initial st:ate x n • for every set of inpu t functions Ul . . . . urn and for all t ~ 0

(4.102) for every pair of fUllctions v

Remark 4.12 general

Suppose

input

functions

it

that

functions readily

l

2 •v •

(4.102)

holds.

If

u 1 (t) . . . . . um (t).v(t)

follows

from

standard

we

now by

approximate

piecewise

results

on

more

constant

differential

equations that (11.102) also holds for these more general input functions (SBB

also the corresponding remark after Definition 3.1).

Remark 4.13 for all

Xc

We may, equivalently, replace (4.102) by the requirement that

,u i

, ..

,urn and all t

~

y( t ,xc

0 f

til ' •.

,urn' 0) ,

(4.103)

for every function v.

We deduce the follOWing necessary conditions for output invariance. Proposition

I,

.14

Consider c1Je system (4.101) and suppose chat: the output

137

Y is invariant under v. Then for all r

,Xr in the set

Xl'"

(f

~

0 and any choice of vectorfields

. ,gm) we have

,81"

for all x

Rema.rk 4.15

Notice

that

condition

(4.104)

is

not

(4.104)

changed

if

we

let

X1""X r belong to the extended set {f,gll .. ,gm,e}. Proof

By Proposi tion 3.22,

(4.104) is equivalent wi th the requirement (i•. IDS)

for all r

~

0 and any choice of vectorfields Z1'"

,Zr

of the form

(see

(3.65» Zl - f +

I

gj (x)ut

' i

(4.106)

E.

E

j .,

for some point u

i '"

by (4.103) for small

(u~"" u!) T

E U,

Now let y be invariant under v. Then

and all x

t 1 , .. ,t):;

t,

tk o

h (Zk

where Z1'"

,Zk are of the form (4.106), and

21 " , ,2k

o

Z, (x»)

are given as

i E ~ •

(i.e.

the left-hand side of (4.107)

(4.10B)

is the output for v - 0,

right-hand side is the output for v - 1). (4.107)

with

respect

while the

Differentiating both sides of

tk,t):_1, .. , t 1

at

respectively

t):

-

0,

0, ... ,t 1 - 0, yields (compare (3.71»

tk- 1 -

L

Z ):

h(x) - L-

for all Ie
to

(4.107)

Z

, h(x) Z,

Since L

h(x)

L-

Z1

=

L-

Z,

Z2

(4.109)

1, then

, hex) . = L hex) + Le hex), Z,

- L

h(x)

(4.110)

Z

this implies (4.111)

Le h(x) = 0

i.e. (4.105) for r = D, In general we obtain

LZ h(x) ):

=

L-

Zl

L-

Z2

- L- L Z, Z,

L-

z,

h(x) ...

(4.112)

DB

and using (4.108) this yields L

e

(4.113)

L ... L h( ..... ) "" 0 , Z2 ZIe

o

for all Z2'-' ,Zk of the form (4.106).

If condition (4.104) is not satisfied, i.e. if there exists some choice of vectorfields Xl'" ,X r in the set [f,gl'" ,gm l such that for some x L

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<.t1""Xn

(4.101)

in

) such these

coordinates takes the form (see (3.116» 1 = £1(X ,X.2)

Xl

I

+ j

2 ; / .. £2(X )

I

+

g

gtex1,x'2.)uj + e1(xl ,xz)v , 1

(i1.133a)

g~(X2)Uj

j -,

Furthermore by (iii) the output equation equals (4.133b) Now it is clear from (i1.l33) that for solutions remaining in this coordi-

nate chart the input v does not affect the output y for any choice of the input functions u 1

, •• ,

urn' Since we can take such a coordinate chart around

any point x this implies that y is invariant under v. In Proposition 4.20 we

have

shown

that

ker

dO

satisfies

(i),

(ii),

if

(iii)

(4.104)

is

satisfied. Since ker dO is assumed to have constant dimension (and is by definition involutive) this shows that (4.104) implies that y is invariant under v. The converse has already been proved in Proposition 4.14.

0

Notes and References

Since the fifties, Wiener-Volterra series have been used quite extensively in the study of nonlinear systems; we refer to the booles [Ru] and [Sche] for

a

modern

account.

The

Wiener-Volterra

functional

expansion

for

a

nonlinear system in state space form as given in Section 4.1 is largely taken from [LKJ. This paper on its turn generalises the results in fBr] and

[Gi],

and

the

Wiener-Volterra

systems as given by in {BDK].

functional

expansions

for

bilinear

For realization theory of Wiener-Volterra

series we refer to [Br] and [er]. The papers,

Fliess

functional

e.g.

[Fll,F13,Fl4]

expansion has

been

and the survey

developed

[FLL] ,

in

and uses,

a

series

of

among other

144

things. the

the theory of formal power series in automata theory f Schul. and

theory of

iterated

integrals,

cf.

feh].

For realization

theory of

Fliess functional expansions we refer to [F13] and the detailed exposition in [Is]. For the proof of Theorem 4.2 we refer e.g. to [Is},fF14], and for the relation with the Volterra-Wiener expansion (4. 36), (4.41) we refer to [F12l.[Is). Functional expansions for. discrete-time nonlinear systems have been studied by [So] and [NC). An alternative

approach

to

realization

theory

starting wit.h

general

input-output maps has been developed by [Ja],[Su]. The material of Section 4.2 has been taken almost verbatim from [vdSlli for related treatments we refer to [CMPJ,[Gl] and [Re]. The importance of higher-order differential representations for nonlinear systems has been pointed

out

e.g.

in

[Wil, [F15]

and

[vdS2].

For

realization

theory

of

external differential represent.ations we refer to [vdS2],fGl). Section 4.3 is largely based on [Is], which extends the results of [11<011] and [H J. The fact that conditions (4.10 11) are Hot sufficient for output invariance in the smooth case

(see Example 4.16)

seems not

to have been pointed out

explicitly before.

[B~J

[fiDK]

[Cr1 fCh] [CMP]

. [Fll J

[Fl2]

[FI3] [FI4] [F15}

[FLL]

R.W. Brackett, "Volterra series and geometric control theory". Automatica 12, pp. 167-176, 1976. C. Bruni, G. di Pillo, G. Koch, "On the mathematical models of bilinear systems", Ricerche di Automatica 2. pp. 11-26, 1971. P.E. Crouch, "Dynamical realizations of finite Volterra series". SIN1 J. Contr. Optimiz. 19, pp. 177-202, 1981. K.T. Chen, "Iterated path integrals, Bull. Amer. Hath. Soc. 83, pp. 831-879, 1977. G. Conte, C.M. Hoog, A. Perdon, "Un th~or~me sur 18 r~presentation entree-sortie dlun systeme nonlineaire". C.R. Acad. Sci. Paris, Ser. I, 307. pp. 363-366, 1988 . ~L Fliess, "Sur la realisation des systemes dynamique bilin~airesn, C.R. Acad. Sci. Paris, A-277, pp. 923-926, 1973. rI. Fliess , "A note on Volterra series expansions for nonlinear differential systems", IEEE Trans. Aut. Contr. 25, pp. 116-117, 1980. H. Fliess, "Fonctionelles causales non linea ires et indetermi.nees non commutatives", Bull. Soc. Hath. France 109, pp.3- l ,O, 1981. M. Flioss, "R~alisation des syst6mes non lin~aires, alg~bres de Lie filtrdes transitives et series gendratrices non commutatives", Invent. Math. 71, pp. 521-537, 1983. M. niess, "Automatique et corps diff~rentiels", Forum ~lath. I. pp. 227-238, 1989. H. Flless, H. Lamnabhi, F. Lamnabhi-Lagarrigue. "An algebraic approach to nonlinear functional expansions" IEEE Trans. eire. Syst. CAS- 30, pp. 554-570. 1983. S. T. Glad, "Nonlinear stat.e space and input-output descriptions using eli fferent:ial polynomials" in New Trends in Nonlinear Control Theory (ads. J. Descusse, H. Fliess, A. Isidori. D. Leborgne). Lect. Notes Contr. lnf. Sci. 122, pp. 182-190, 1989. I

[G1 J

R.M. Hirschorn, "(A,B)-invariant distributions anel disturbance decoupling of nonlinear systems", SIAN J. Contr. Optimiz. 19, pp.

[HiJ

1-19, 1981. A. Isidori, Nonlinear Control Systems: An Introduction, Lect. Notes

fIs]

Contr. lnf. Sci. 72, Springer, Berlin, 1985. [IRGM] A. Isidori, A.J, Krener, C. Gari Giorgi, S. l1onaco, "Nonlinear decDupling via feedback: a differential geometric approach", IEEE

Trans. Auto, Contr. AC-26, pp 331-345, 19B1. B. Jalcubczyk, "Existence and uniqueness of realizations of nonlinear systems", SIAH J. Contr. Optimiz. 18, pp. 455-471, 1980. C.Lesiak, A.J. Krener, "The existence and uniqueness of Volterra series [or nonlinear systems", IEEE Trans. Aut. Contr. AC-23 , pp.

fJaJ [LK]

1090-1095, 1978. [NC]

fRu] [Re]

[Sche] [SchuJ [So] [Su] [vdSl]

[vdS2j

[Wi]

D. Normand-Gyrot, Theorie et pratique des systemes nonlin~aires en temps discret, These d' Etat, Universite de Paris sud, 1983. W.J. Rugh, Nonlinear System Theory: the Volterra-Wiener approach, Johns Hopkins Press, Baltimore, 1981. W. Respondelc, "From state-space representation to differential equations in inputs and outputs", paper presented at the IFAC Symposium on Nonlinear Control Systems Design, Capri, Italy, 1989. H. Schetzen, The Volterra and Wiener theories of nonlinear systems, Wiley, New York, 1980. H.P. Schutzenberger, "On the definition of a family of automata", Inform. Control, 4, pp. 245-270, 1961. E. Sontag, Polynomial response maps, Lect. Notcs Contr. Inf. Sci. 13, Springer, Berlin, 1979. H.J. Sussmann, "Existence and uniqueness of minimal realizations of nonlinear systems", Math. Syst. Theory 10, pp. 263-284, 1977. A.J. van der Schaft, "Representing a nonlinear state space system as a set of higher-order differential equations in the inputs and outputs", Systems Control Lett. 812, pp. 151-160, 1989. A.J. van der Schaft, "On realization of nonlinear systems described by higher-order differential equations", Math. Syst. Theory 19, pp. 239-275, 1987; Correction 20, pp. 305-306, 1987. J.C. Willems, "System theoretic models for the analysis of physical systems", Ricerche di Automatica 10, pp. 71-106, 1979.

Exercises

4.1

Verify that the It-th order Volterra kernel for a single-input singleoutput system is given as in (4.31).

4.2

Consider the nonlinear system (il. 7). and the vectorfields P t

=

([t)"g.

Define the functions 0t

=

hoft,

Show that the It-th order Volterra

kernel is alternatively given as ([Is]) Wk(C,Sl"",Sk) -

(Lp

Lp 5k

4.3

Consider order

the

nonlinear

Volterra

Sk-1

system

kernel

... L p

51

0t.)(xo )

(4.7).

Denote

for

to

initial

corresponding

h'k(C,sl, ... ,sk'XO ) ' Show that for any r
°

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\<.xo ). The Wiener-Volterra expansion

said to have

~

if l... p+j(t,s!I .... sp+j,xo) -0 for

length p

l.

(n) Prove

that

the

Wiener-Volterra

,...p(t,sl' .... Sp,x o ) does not depend on

Cb) Suppose

the

system

satisfies

expansion

has

length

if

p

xO'

the

strong

accessibility

rank

condition. Prove that if the Wiener-Volterra expansion at some Xo has length P. then \"'p(t,sl'''''sp,xo ) does not: depend on Xo ([Cr]). 4.5

Consider the nonlinear system (4 7). (n) Show that: the first-order Volterra kernel can be written as "'l(t.S,Ao)

1:1

L

=

ko

ski

L{ LgLc h(xo

lel!

ko·klaO

(b) Prove that

' ' 1 (t, S ,xo )

is independent of Xo and only dependent on

c-s if and only if LI!L~h(x) is independent of x for all Jc ~ O.

(1)

(c) Show that the output of (l1.7) for x(o) ...., AO can be written as t

yet) - Q(r.x o ) +

I

g(r-s)u(s)ds

{I

for some functions Q and geT) (i.e., the input-output response of the system is linear) if and only if condition (1) is satisfied. 4.6

Consider the bilinear system

Y

X3

-

(n) Use

=

Xl'

Xl'

Algorithm

4.4

to

determine

an

external

differential

representation. (b) Show that

the mapping (u(O) ,it(O) ,u(O) .y(O) ,j(O) .y(O»

is

1---4 Xo

not injective for those input functions that satisfy it .. -1 ... u2.. 4.7

Show that the integers P

1

""

P 1 ~ P z ~ ... ~ P k • > 0 • P l

,p

k

defined in Algorithm 4.4 satisfy

"

+ P z + ... + p)/ "" n - n.

Define kl as the cardinality of the set {Pj IPj ~ i I. i kl ~ k2 2:: ••• ;::0: kp ~ 0

k!

+ k z + ... +

(The integers lei are called observabili ty indices 4.8

E

E.. Show that

kp = n -n I

cf. [vdSl)).

Consider the Hamiltonian system (see also Chapter 12)

an

qi ~ ap~

an

, Pi -

oqj.

+ u i • Yi

where JI(q ,p) - !:../G(q)p + V(q). 2.

matrix for every q

e

1ft.

=

qi' i E

with G(q)

!2 ' a positive definite n

X

n

147

(n) Use

Algorithm

obtain

to

4.4

external

an

differential

representation, (b) Show that the external differential system given by the

Euler-

Lagrange equations d dt

where

[~l ay, L(y,y)

aL ay,

u,

- zY G 1

• T -1

i E

!!,

•

"

(y)y - V(y),

differential system obtained in

(a)

equivalent (i.e.

to

the

external

the set of input-output

trajectories is the same). 4.9

Prove Remark 4.13.

4.10 (see Remark 4.21) Consider a distribution D, 2.119) D(q) .,. spanlXi (q)

V,

; i E 1)

,

locally given as

(cf.

q E V, V open set in N. Show that

the involutive closure of D (cf.

2.1-43)

I

on V is

given as the

distribution spanned by all vectorfields of the form IX"

[Xi _"

where Xj

, j

I· .. , IX, ,X,[ .. ·111, E~,

is in the set {Xi' i

E

IJ and Ie E IN (compare with

Proposition 3.5), Fill in the details of Remark 4.21.

5 State Space Transformation and Feedback

This chapter deals with some preliminairies which are basic to controller and observer design for nonlinear systems.

In particular we discuss the

possibility of linearizing a system by state space transformations and we introduce various types of nonlinear feedback.

5.1 State Space Transformations and Equivalence to Linear Systems Consider a continuous time smooth affine nonlinear control system

x

L gi(X)U 1 ,

f(x) + i

~

(5.1)

1

where x = f,gl""

(Xl"" ,Xn ) are local coordinates for a smooth manifold Hand ,gm are smooth vectorfields on H. Together with the dynamics (5.1)

we consider an output equation y

(5.2)

h(x) ,

where 11: N .... p-dimensional

(RP

is

a

output

smoo th space

mapping and

Y

=

from

the

(Yl •... ,Yp

s tate )

is

space N in to the

output

of

the

the

system. OHthout any problem the (local) results that will be obtained can be extended to more general output functions h: N ... N,

by

interpreting

(5.2) as a local description of the output map.) There is a particular class

of systems having a

structure as

in

(5.1)

and

(5.2)

for whicr

various design problems are relatively easy to handle, namely the class of linear systems, i.e. systems having linear dynamics:

x = AX +

Bu,

with A an n x n matrix and B an n X m matrix and with a linear output map Ci.;:,

for a p x n matrix C. We may ask ourselves the question when the nonlinea: system (5.1,2) is in some way eqUivalent to a linear system (5.3,4).

T~

make this explicit we have to specify what is meant: by the equivalence 0: two systems. We will do first this for the notion of equivalence unde'

sc:ate space transformacions (see Chapter 6 for the extension to feedbac: transformations).

Since (5.1)

(and 5.2») describe a nonlinear system i'

149

local

x

coordinates

we

on

will

consider

local

state

space

transformations. Recall, see Chapter 2, that a coordinate transformation z = Sex) from x-coordinates to z-coordinates around a point Xo E diffeomorphism

S: V ..... SeV) c 1R"

for

some

x(O)

t' c

neighborhood

Consider now the dynamics (5.1) wi th some ini tial

5

mil

of

(5.5)

xo '

=

xo'

tate

and let S: V ..... SeV) be a diffeomorphism on a neighborhood V of

AD.

In the

new coordinates (5.6)

z=5(x),

we have that

.

z

as

.

(5.7)

ax(X)X'

=

and therefore

(5.8) S:

Because

V~'

SeV)

is

a

diffeomorphism,

the

mapping

S

has

a

smooth

inverse S-1: seV) -, V with (see (5.6» x

=

s"' (z).

(5.9)

Substituting (5.9) into (5.8) yields

. as -1 -1 Z - axl5 (z»)£(5 (z») +

m a5( -1 -1 I ax 5 (z»)g'(5 (z»)u"

'" ,

(5.10)

or briefly, see Chapter 2 (equation (2.85»

z -

(5J)

(z)

,.I , (5.g,) (z)u"

+

(5.il11)

which is again a system of the form (5.1) but TIm" described in the new

coordinates z and with initial state z(O)

=

S(x o )'

(5.11b)

Obviously S maps trajectories xCt,D,x o ,u) contained in V into trajectories z(t,O,S(xo),u)

in S(F) of (5.10), Let us see the effect of such a local

state space transformation in

11

simple example.

150

Example 5.1 Consider on ~+ X IR+ the system

(5.12) X2

= - xzln xl + xzu

and let Xo - (1,1)1. Introduce the coordinate transformation s: m+ x III

lit ...

x IR via

(5.13) Then in the new coordinates

the system (5.12)

(Zl'ZZ)

takes the linear

form (5.14) Z2

-

zl + u,

-

while (zl{O},Z2(O») - (Sl{l,l),Sz(l,l») - (0,0). Observe that in this case the transformation (5.13) is even globally defined on ~+ x IR+.

0

The above example shows the great effect a coordinate change can have on

1I.

nonlinear control system. Apparently the system (5.12) is intrinsically

1I.

linear system. In what follows we will assume that the point

KO

is such that f(xo) - 0

and S(xo ) = O. Without these assumptions all results that are given still hold under the modification that in the linear dynamics and output equation y

= Cz

+

constant

vectors

v

and

~

Problem 5.2 a

added,

i.e.

z - Az + Bu + v,

(Coordinate transformation into a linear system) Consider the

nqnlinear system (5.1) around a point exist

are

We now formulate our first equivalence problem.

~.

coordinate

transformation

XD

\ ..

ith f(x a ) -

z '" S(x)

with

o.

rihsn does there

S(x D )

-

0,

which

transforms the nonlinear dynamics (5.1) into linear dynamics, i.e. (5.lla) is linear? The following theorem (partially) answers this question by exploiting differential geometric tools as Lie-brackets of vectorfields. involutive distributions and so on. The answer is partial in that it only decides when the nonlinear dynamics is eqUivalent to a controllable linear system. For any two vectorfields f

and g we will define

the

repeated Lie

bracket ad~g, k - 0,1,2, ... , inductively as ad~g ~ (f.ad~-lgl. k ~ 1, with

ad~g - g.

151

Consider

Theorem 5.3

point xo.

i.e.

the nonlinear system

[(xc)

=

O.

(5.1)

around

There exists a coordinate

an

equilibrium

transformation of

(5.1) into a controllable linear syscem i f and only i f the follOTdng two

conditions hold in a neighborhood V of xo'

dim(spanlad;g,(x), i E~, j - 0, .. ,n - 1)

(i)

(5.15)

- n, Vx E V,

(5.16) Proof First suppose that (5.1) with [(xo) - 0 is transformed via z S(Xo)

z

=

where b l

(5.15)

s:

=

Sex),

0 into a controllable linear system

=

Az + Bu - Az +

,hm

, ..•

and

V .... S(V)

,.L,

(5.17)

bill i ,

are the columns of the matrix B.

(5.16)

are

satisfied.

Observe

that

and arbitrary smooth vectorfields £1

We have

[or

a

and £2

to show that

diffeomorphism

on \l we have the

identity (see Proposition 2.30)

(5.18) In the present situation we have (see (5.11) and (S.17))

5"f(z)

=

Az,

(5.19a)

(5.19b) So for i E

~,

we have

(5.20) and similarly for i E

~

and £ - 0,1, ..

(5.21) Now, from the fact that (5.17) is controllable we know that dim(span(B,AB, ... ,An-1S)

-

n,

(5.22)

and so using (5.19-21) we may conclude that dirn(span(adigi(x), i E~, j .. 0, ... ,n-1) Note

that S"

maps

the

- n, Vx E V.

distribution span(ad~gi (x), i E ~, j

(5.23) - 0, ... ,n-1)

152

into the flat distribution span(B,AB, ... An-In).

To see that

(5.16) hold,

observe that k.£ Ie £ 5" [ad£gi ,ad[gj] (z) = [5.. (ad[gi) ,S~ (adrgj

~ (_l)ktl[A k b i ,A i b j

]

)

1(z)

~

o.

(5.24)

Because S.. is a linear isomorphism for all points x E V we see that (S.16) holds.

To prove the converse we assume that

neighborhood

(i

on some neighborhood V of

S: V .... S(V)

(5.15)

an (5.16) hold on a

of Xo and we have to show the exi.stence of a diffeomorphism which transforms (5.1)

Xo

into the

controllable system (5.17). Consider the set of vectorfields

!ad~gi

E~, Ie

i

(x),

(5.25)

0, ... ,n-l).

From (5.15) an (5.16) we conclude that in a neighborhood can select n vectorfields from (5.25), say Xl" dim(span(X1(x), ... ,Xn(x»))

=

a

(5.26a) (5.26b)

(7

a coordinate

transformation S

which is such that S(xo)

=

0 and

(5.27)

i E n.

m. First

Next we compute S.f and S.gi' i

a

of Xo we

V.

Thus we may apply Theorem 2.36 and find

eVe

rr

c

Vx E V,

H,

[Xi 'Xj lex) = 0, i , j E ~. Vx E

defined on a nei.ghborhood V

V

. ,Xn • which satisfy

(z)

[S*gi ,S*Xj}(z)

observe

that,

S.[gi ,Xj](z) = 0, i E

=

using ~I j

(5.16),

E n. (5.28)

Therefore S"gl (z) = b i

,

for some constant vector b i • 1 E m. Furthermore we

have

a

U a - ,S~fl, zi

(5.16)

(use

coordinates. n x

11

a!,J

again),

)(z)

=

Bnd

S.[(Xi,f],Xj](z)

so

Because f(x o )

matrix A.

It

is

the

0, for all i , j E!!,

=

S~f

vectorfield

= 0 we conclude

straightforward

to

is

linear

that Swf(z) see

that

the

(5.29) in

the

z

= Az for some so

constructed

o

linear system (5.17) is indeed controllable.

Example 5.4 Consider the nonlinear system -

2XZX3 +

x:l

o

x;

4X2 X

+

1 r

J

- 2X 3

1

1

u .

(5.30)

153

A straightforward computation shows that

which is 3-dimensional for all x E 1R3. Therefore the condition (5.15)

satisfied.

I t is readily verified that also

(5.16)

is

is fullfilled and so

Theorem 5,3 applies to the system (5.30). In order to find the linearizing coordinate transformation we need to find a mapping S: V c [f/3 .... S(V) C

a

satisfying S(O) =

and the condition

(5.27).

An easy

(RJ,

inspection shows

that

(5.31)

does the job, and thus we may take as the

(globally defined)

coordinate

change

z

=

Sex)

=

x~

X,

+

Xz

+ xJ

[

]

(5.32)

x,

yielding the linear system in z-coordinates

(5.33)

o The importance of Theorem 5.3 lies in the fact

system satisfying (5.15) and (5.16) we may use system into a linear one -

Remark

5.5

Notice

that

that for a nonlinear

after transforming the

linear control design.

the

condition

(5.15)

alone

implies

that

the

nonlinear system (5.1) is strongly accessible in a neighborhood of xo. Of course

it

is

not

equivalent

,

,

to

strong

brackets of the form (adrgi,adrgjJ(x) Furthermore,

in order to check (5.15),

are

accessibility

because

no

Lie-

involved in condition (5.15).

i t suffices

to compute

(5.15) at

KO. When the condition (5.15) is satisfied at x o ' then it is by continuity also satisfied in a neighborhood of xo.

In order to verify (5.16) in principle one needs to compute an infinite number of Lie brackets. That this is not the case, provided (5.15) holds, can be seen after a bit more analysis of both (5.15) and (5.16). Namely,

154

if (5.15) and (5.16) are satisfied and if dim(span{gl •... 'gm) after a reordering of the vectorfields gl •...• gm'

- m, then,

there exist integers

m

1 ~!

~l ~ ~2 ~ ••• ~ ~m ~ 1 with l

-

n such that the set of vectorfie1ds

1

a

(5.34) satisfies (5.15). (In fact, the vectorfields given in (5.34) form a basis for the Lie algebra generated by the vectorfields lad~g!, i E~. k ~ OJ, that

is.

each

ad~gl

vectorfield

vectorfields of (5.30).)

(~l, ...

If

is

a

linear

is

,K:m)

combination

of

the smallest m-tuple

the (with

respect to the lexicographic ordering) with the property that the set of vectorfields (5.30) is n-dimensional, then these integers are called the concrollability indices of the corresponding linear system (5.17). Now it

is straightforward to verify that (5.15) and (5.16) can be restated as follows. Corolla:ry 5.6

Consider

assume dim(span(gl"" of (5.1)

the

,gm)

nonlinear

system

(5.1)

t
f(xo) ... 0

and

- m. There exists a coordinate transformation

into a controllable linear system 1f and only i f there exist m

controllability

indices

11:1

~

le

z

~ ..... ~

I\;m

1

~ 1,

~!

=

n such r.:hat the

folloldng tt
(i)

dim(span(ad~gi(xo)' j

0 •...• ~i-l, i

E ~})

(5.35)

= n,

As an illustration we show what conditions are needed for a singleinput nonlinear system, i.e. m - 1. In that case (5.17) is a single-input linear system which has one controllability index, namely

~

= n,

and so

(5.35) yields (5.37)

and (5.36) reduces to k+l ~ O, ... ,2n-l. which by using

the

Jacobi-identity

(see

Vx

E

V.

Proposition

(5.38) 2.27(c)

can be

simplified to

(g.ad~g)(x) "" 0

Ie

1.3,5, .... 2n-l,

Vx E

V.

(5.39)

155

Re.mark 5.7

By itself the foregoing results are of a

local nature.

The

global equivalence problem can he addressed by requiring that the state

space transformation S is defined on the whole state space H. An ohvious requirement

for

the

diffeomorphic to !lin,

whole state space.

solvability

of

the

and that conditions

global (5.15)

problem and

is

that

t1

is

(5.16) hold on the

However one has to impose further conditions on the

vectorfields f,gl""

,gin

for

the

solvability of

the

global

equivalence

problem (see the references at the end of this chapter). Problem 5.3 only addresses the question whether the nonlinear dynamics

(5.1) are equivalent via a state space transformation to the controllable linear dynamics (5.17), and so the output equation (5.2) is not taken into account. Of course, when we want to control the system on the basis of the outputs, we have to calculate the effect of a coordinate change on the output equation as well, Let z - Sex) be some coordinate transformation around x o , then besides the new dynamiCS z - (5.f)(z) +

,I., (5.g,)(z)u"

(5.11)

we obtain the new output equation

(5.40) The equations (5.11) and (5.40) are again a system of the form (5.1,2), now described in the new coordinates z. The obvious extension of Problem 5.3 can now be formulated. Problem 5.8

(Coordinate transformation into a linear system with outputs)

Consider the nonlinear system (5.1,2) around a point Xo with f(x o ) ~ 0 and h(xo ) = O. r{hen does there exist a coordinate transformation z - Sex) with 0, l.rhich transforms the nonlinear syst;em (5.1) ,.rith outputs (5.2) into a linear system with outputs, i.e. (5.11) and (5.40) are both linear?

Before we can solve the above problem we recall the following facts about Lie derivatives of functions. Given a smooth vectorfield X on &n and a smooth function h on ~n the Lie derivative of h with respect to X is the function Lxh(x)

=

X(h) (x) -

~~(x) ,X{x).

Similarly the functions r;h are

defined as follows. By convention we set ~h(x) _ hex) and inductively for k 2=: 1, r;h{x) - Lx(~-lh)(x). Analogously for a smooth mapping h: [fin ... IR P

we

define

h - (h l

k LXh

, . . . , h p ) T.

componentwise,

i.e.

k LXh -

k

k

(LX hI , ... ,LXhp)

T

where

Returning to the nonlinear system (5.1,2) we introduce

156

(in the local coordinates x) the mapping

1

hex)

ri'{x) _

Lr/(X)

(5.41)

[ L~-lh(x) which maps IR

n

into

Jc"'" 1,2,. .. . The following theorem answers

for

[Rkp.

Problem 5. B in case that the obtained linear system in the z-coordinates is minimal (i.e. controllable and observable).

Theorem 5.9 Consider the nonlinear system (5.1,2) around a point f(xo) -

a

with

Xo

and h(xo ) - O. There exists a coordinate transformation of the

system (5.1.2) into a minimal linear syst:em if and only if the following three conditions hold on a neighborhood

ii

of Xo .

(1)

dim(spanladigl (x).) - D•.. ,n-l, i E ~l)

(11)

rank rl,-l(x) - n,

Vx E

=

(5.lS)

n, "ix E 0,

ii,

(5.42)

V. )

Vx E

E

E.o lc ~ 2 and

,X): E

Xl""

(f,g, ... ,gm)' with at least two Xi's different from f.

Proof

For

simplicity we

first

assume

p

-

1.

First

we

(5.43)

show

that

the

conditions of Theorem 5.3 are satisfied, so we have to prove (5.16). From (5.42)

we

see

coordinates

that

the

h ,Lrh, ... ,L~-lh

functions

on a neighborhood of xo.

form

a

So we may define

set the

of

local

coordinate

neighborhood V c V of Xo as Sex) ." rl'-l (x) with rl'-l(x o ) - 0 because h(xo ) ~ a and f{x o ) - O. By using (5.43) now

transformation S on a

S(xo ) -

for fixed k,t and 1.) E

~

Considering !J ••••• L~-lh holds.

Therefore

transformation

we

z ~

we observe that

as may

Sex)

""

the

coordinate

apply

Theorem

functions 5./1

and

we see conclude

rl'-l(x) makes the dynamics

(5.1)

that

(5.16)

that

the

linear.

It

remains to show that h is also linear with respect to the z-coordinates.

As in the proof of Theorem 5.3 we may select n independent vectorfields Xl""

,Xn

from the set lad~gi' ) - 0, ... ,n-l, i

E

~I which are such that

[Xl. ,Xj J .,. 0 for i.) E n. Now /1 is linear in the above coordinates if for all i,) E But (5.45)

follows

~

.

(5.45) k

immediately from (5.43) and the fact that Xi = adrgq

157

and

,

for

Xj = adfg r

conditions

q,r,lc

certain

(5.15),

(5.42)

an

.e.

and

(5.43)

far

As

concerns,

as

we

of

necessity

notice

that

the is

it

straightforward to check that for a minimal linear system these conditions hold.

Finally we note

(5.16)

may be used,

functions

hI""

that for p > 1 a

by selecting a

,hp ,Lih l

, ...

similar procedure for

set of local coordinates That

,Lchp, ....

we

can

do

so

deriving

from

the

follows

of

o

course from (5 .l12).

Remark

5.10

As

mentioned

strong

accessibility

in

of

Remark

the

5.5,

system

the

(5.1)

condition

in

a

(S.lS)

implies

neighborhood

of

xo'

Similarly (5.42) implies that the system (5.1,2) is locally observable in a neighborhood of xo'

In order that the conditions

(5.15) an (5.42)

hold

i t suffices to check that (5.15) and (5.'!2) are satisfied at the point x o '

compare wi th Remark 5.5.

At

a

first

glance

condition

infinite number of functions.

(5.43)

involves

the

computation

Provided that (5.42) holds,

of

an

this is not the

case. The argument to see this is completely analogous to the one given in Corollary 5.6. From (5.42) we see that after a possible reordering of the

,

outputs Yl ' ... ,Y p there exists integers PI 2=: Pz 2=: ••• 2=: Pp 2=: 1 with

I

Pi

=

n

such that dim(span(Lidhi(x), j

n, Vx E P ,

O,···,pi-1, 1 Eel)

=

or, equivalently, the n functions (Lih!, j of local coordinates around xo.

If

=

(5.46)

O, ... ,llj-l, 1 Eel form a set

(PI' ... ,p p )

is

the

smallest p-tuple

with respect to the lexicographic ordering with the property that (S.lI6) holds,

these

integers

are

called

observabillty

the

indices

corresponding linear system. One can directly verify that and fi: 1 O!

(5.43)

re z

O!

can be •••

O!

reformulated as:

rem ;::: 1, with

There

I

K j

I

IJ i ~ n, such that

=

11,

(5.15),

exist controllability observability

and

of

the

(5.42) indices indices

'", , ;::: Pp ;::: 1, with

dim(span(adigi(x O ) ' j

=

0, ... ,1\:1-1, i E ~)

dim(span\Lidht(x o ), j

=

O, ... ,Pl-l, i Eel)

So we have obtained the following result.

,

(5.35)

n ,

(5.47)

n

=

158

Corollary 5.11

Consider the nonlinear system (5.1,2) around

Xo

satisfying

f(x o ) = 0 and h(xo) ~ O. There exists a coordinate transformation of the system (5.1,2) into a minimal linear system i f and only if there exist m

controllability indices

II:}

~

It"

L 1\:.1


~

m

and observabi-

n,

p

lity indices PI ~ Pz ~ ... ~ Pp

2:

L Pi

1.

such that (5.35),

.. n,

(5.47)

ial

and (5.48) are satisfied.

It

is

conditions

straightforward (5.35),

(5.47)

to and

single-output

case,

1. e.

single-output

linear

system

observability index

~

see

what

(5.48)

the amount

to

The

m-p-l. has

necessary in

and the

reSUlting

controllability

index

sufficient single-input single-input IC

-

nand

- n and thus (5.35) yields (5.37)

whereas (5.47) and (5.48) may be reduced to (see also (5.38»

(g,ad~gJ(x) = 0 , k - 1,3,5, ... ,2n-l,

Vx E V,

(5.39)

and

LgL~dil(x) - 0 , j - 0, ... ,n-l, Vx

E

V.

(S.49)

Problem 5.8 deals with the question when a

nonlinear system is

in

essence a transformed minimal linear system, thereby allowing for linear controller and observer design. On the other hand it does not say anything about a nonlinear system without inputs. However coordinate transformations can also be useful in this case. Consider x = t(x),

(S.50a)

around a poine x o , together with an output equation y ... h(x). As

before we

(5.50b) see

that a

coordinate

change

z

Sex)

results

in

the

equations z = (S .. £) (z)

I

Y _ h{S-l(z»,

(5.51) (5.40)

and obviously we arB interested in the question when (5.51) and (5.40) arB linear in z. We will call (5.50) an autonomous system with outputs.

159

Problem 5.12

(Coordinate

transformation

of

an

autonomous

system

with

outputs into a linear system) Consider the lIutonomous system I"lth outputs (5.50) around a point Xo with [(xo) -

~

a and h(xo )

rYhen does there

O.

exist a coordinate transformation z = Sex) with S(xo ) = 0 \.,hleh transforms (5.50) into a linear system with outputs, i.e. (5.51) and (5,40) are

linear? The following theorem answers Problem 5.12 in case the resulting linear

system is observable. Theorem 5.13 Consider the autonomous system lY'ich DutpuCS (550) 1,rich [(xo) =

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!<1te fced\l<Jck.

x

confusion arises,

166

So the actual state at time t. x(t), yielding the input at time

t

is fed back via the function

Of

as u(t) = o(x(t». One of the applications of

strict feedback is that of stabilizing the nonlinear system (5.1) around an equilibrium point Xc • Example 5.20

(see Example 1.1) Consider again the dynamics of a rigid

two-link robot manipulator with control torques u 1 and joints. Introducing the state spuce coordinates 0 -

Uz

(0 1 , O2

applied at the ),

iJ -

(f;1' 8z )

we obtain the dynamics (see (1.8» (5.83)

1'

where u - (u! ,u z ) and the matrices H(O), C(OrO) and k(O) are as in Example 1.8. It is easily verified that each point (0 0 .0) - (0 10 ,0 20 ,0,0) appears as an equilibrium point for (5.83) when setting u equal to (5.84) On the other hand let 1(0,0) -

0 1 (0,0),1. 2 (0,0»

be an arbitrary vector

linearly depending on (0-0 0 ,0). Then the strict nonlinear state feedback U -

H(O)l(O,O) + C(O,O) + keD),

(5.85)

yields the closed loop system

(5.86) Note that (S.85) evaluated at (0 0 .0) coincides with (5.84). Because the 2-vector 1(0,0) in (5.86) is arbitrarily linearly depending on 8 an may choose 1(0.8) such that (5,86) becomes asymptotically stable. A second important

type of state feedback

° we 0

is what will be called

regular static state feedback.

Definition 5.21 A regular static state feedback for the nonlinear dynamics (5.1) is defined as a relation U

=

l,there u -

o(x)

+ P(x)v,

(u 1 " ' "

the property that

urn)'

(5.87) Q:

tt ...

[Rm

and {3: H ... ~mXm are smooch mappings with

the mxm mat:rix P(x)

is nonsingular

for

v - (vI •...• vm ) represents a nel>' vector of cont:rol variables.

all

x

Bnd

167

Schematically regular static state feedback can be given as follows,

v

u.

P(x)

x

~ = lex) + g(x)u

a(x)

Fig. 5.2. Regular slmic Slate fccdhnck.

Application of a feedback (S.B7) to the system (5.1) yields the controlled

dynamics x = [(x) +

I

gi

(X)0:1 {X} +

which is

~ [I gl (x).B

jBl

i~l

(X)]V j

ij

,

(5.88)

the newly defined

inputs

i~l

again an affine control system with

(v1, ... ,vm). ~"e obtain the strict static state feedback of Definition 5.19

by setting v (linear)

0, Note that for a linear system x

state

feedback

takes

the

form u

=

Fx,

state feedback is of the form u = Fx + Gv with

~

Ax + Bu strict static

whereas

IGI '"

feedback may enable us to meet certain design goals, time

we

keep

as

much

control

on

the

system

as

linear

D.

regular

Regular state

where at the same

before

applying

the

feedback. Because P(x) is nonsingular for all x we have that

span[gl (x}, .. ,grn (x) I - span(

I

gi (x)P t

,. ,

1

(x),.

L g,(x)P'm(x»),

(5.89)

1 '" 1

which shows that the input distributions of (5.1) and (5.88) are the same.

Remark 5,22

It is a straightforward exercise to show that regular static

state feedback does not change the (strong) accessibility properties of a system.

Regular static state feedback, or for short, when no confusion arises, regular feedback, may be used in various design problems as will be shown in the next chapters. Here we will give a fairly simple illustration.

Example 5.23 Consider the two-dimensional single-input system

(5.90)

which models simple physical systems as for instance a pendulum or a cart

168

rolling i.n one direction. Suppose g(X l ,Xl)

~

0 for all (Xl ,X:!). Then we

may introduce the regular static state feedback 1

(5.91)

yielding the system (5.92) Xz

v,

which is simply a controllable linear system, that may be used for further control design.

0

An iUlportant variation

of strict static state feedback and regular

static state feedback is when only use is made of the outputs (5.2) of the system.

Definition 5.24

A stricc static output feedback for tlu.? nonlinear system

(5.1) l..rich outputs (5,2) is defined as a relation

u = ti(y) ,

(5.93)

function.

Definition 5.25 A regular statlc output feedback for the system (5.1) lvitll outputs (5.2) is defined as a relation u -

lI'here

ti(y) + ~(y)v.

u = (u1, ... ,u m),

(5.94)

Y = (Yl' ... .yp),

-

0:

p

m ...

1)(

m

and

p: IR p

-

-t

III

mXm

are

smooth mappings tdtll the property that P(y) is nonsingular for all y and v

(vI' •.. ,Vm ) represents a vector of new concrol variables.

Because y

h(x)

we observe that the output feedbacks (5. 93) and (5.94)

are special cases of the state feedbacks (5,82) and (5.87),
Example 5.26 (see Example 5.23) Consider again the system

(5.90)

169

together with the output (5,95)

Suppose that g(x 1 ,Xz)

~

0 for all (x1,XZ ), In analogy with Example 5.23 we

may try to apply a regular static output feedback to (5.90) which makes

the

overall

system

nonlinearities

linear.

Clearly

this

and g(x1 ,XZ )

f(xl,x Z )

is

possible

depend on xl

only.

only If so,

if

the

we

can

apply the regular output feedback u = _ fCy) + 1 g(y) g(y) v,

(5 96) '

again yielding the linear system (5.92).

0

The last type of feedback we are going to discuss is in contrast with

the previous ones, dynamic. Definition 5.27 A dynamic state feedback for the system (5.1) is defined

as a relation

1(Z,X) + 6(z,x)v, (5,97)

a{z,x) + P(z,x)v,

~

t"bere 0:

Z =

(2 1 , . . .

II?'I x H .... IR

v ... (vl

"'"

m

,z'l) E IRq,

and

p:

6: ~q x H ~ ~qXm

and

IRq x N

-t

IR

mXm

are

smooch

mappings,

and

v m) represent:s a nel'-' input: vect:or.

Dynamic state

feedbac1c can be viewed as

the composition of

the

system

(S.l) with the system

l(z,i) + 5(z,i)v, (5,98)

o(z,i) + fi(z,i)v, with the

interconnections

x=

x

and

u=

u.

Sometimes

the system

(5.98)

itself is called a compensator and schematically we may represent dynamic state feedback as follows:

v

X

z ~ l(Z,X) + 5(Z"'=}Vf-''--'''-''-->-j x U = a(z,x) + fi(z,x)v

Fig. 5.3. Dyn
=

f{x)

~.

g{x)u

170

Obviously dynamic state feedback includes regular static state feedback

Z ""

(taleB

0 in (5.9B». It also includes the idea of "adding integrators"

to the system (5.1). For, consider (5.1) and suppose we set i E m.

with

E~,

i

Vi'

(5.99)

the new inputs. The same can be achieved by introducing

the compensator i E ~,

(5.100) ~.

i E

find adding this to the system (5.1), so as to obtain III

Xi ...

£(x)

+

Lgi (X)Zi •

(5.101)

J."1

{

Zi

= Vi

•

i

E

~.

A mathematical example will illustrate the richness of dynamic state feedback. Example 5.28 Consider on ~3 the system

:: = :~lUl x:J ...

(

U

+

X3

(5.l02)

z

Yl ... Xl Ya

xa

Because Yl = u 1 and Yz - eX1 u 1 +

X3

we see that the control u 1 influences

both outputs in (5.102). On the other hand. applying the dynamic state feedback

(5.103)

to the system (5.102) yields:

e

from which we deduce control

VI

that

in

the

X

1Z

2

-

e

X

I VI

+

compensated system

only influences the output Yl and the control

the other output Y2'

Va

... V 2 '

(5.102,103) Vz

the

only affects

o

171

In the following chapters it will be shown that dynamic state feedback

may be used for various design purposes, In a similar way as has been done for static feedback we may introduce dynamic output feedback,

Definition 5.29

A dynamic

output

feedback

for

the

system

is

(5.1,2)

defined as a relation

z ~ l{z.y) + 5(z,y}v, (5.104)

{

u(z.y) + ~(z,y)v,

u -

where

z -

ii:IR'IxIRP .... lRrn v -

and

(Zl •••• ,Zq) E [fIq,

7:

p:[R'Ix

and

IRq x lR P

are

-+

5:

IRq,

~q

x

ffiP ~ ~qXm

mappings,

smooth

and

(v 1 • . . . , v m) represents a nel" input vector.

As in the static case, dynamic output feedbacks form a subclass of dynamic state feedbacks.

We conclude this chapter with some comments on nonlinear systems that

are not necessarily affine. Remark 5.30

Consider the nonlinear system on a manifold

H

of the form

x .. f(x,u),

(5.105) y - h(x,u).

Various

questions

and

concepts

introduced

so

far

for

affine

control

systems can be generalized to (5.105). Since many of the previous notions may be repeated (and are not changed) we will briefly concentrate on one of

the

differences.

A further

Regular static state

discussion

feedback for

the

is

system

contained (5.105)

in Chapter is

13.

defined as

a

relation of the form U

where

-

U

property

(5.106)

o:(x,v) ""

(u 1

, ••• ,

that

u m),

o:(x,'):

and 0:: H x mm . . . mm is a smooth mapping with the mm ... mm is a diffeomorphism for all x E Hand

v - (v1, ... ,vm) denotes the new input. The difference with Definition 5.21 is

that

the

relation

(5.106)

is not necessarily an x-dependent affine

correspondence between u and v. The feedback (5.106) yields the feedback modified system

x - f(x,o:(x,v»).

(5.107)

Often we will be concerned with a feedback (5,106) which is only locally

172

defined in a neighborhood of a point (xo,u a ). Regularity of the feedback u - a(x,v) is then locally guaranteed by the Implicit Function Theorem if

:~(x,v) is nonsingular. Henceforth we will refer to a mapping ~~(x.v) nonsingular for all (x,v), as a regular static state

the matrix with

0,

feedback. though

In the foregoing we assumed the controls u

one

can

also

work

with

more

general

to belong to IR

input

manifolds

U

m ,

not

necessarily being an Euclidean space. In that case a regular static state feedback a is defined as a mapping a : H xU'" U with the requirement that

:~(X,V)

is nonsingular for all (x,v).

Similarly

one

can

introduce

dynamic

state

feedback

for

the

system

(5.105) as a smooth relation of the not necessarLly affine form

{

Z-

-y(z,x,v),

u -

a(z,x,v),

(5.10S)

which differs from Definition 5.27 by the fact that the relation (S.lOS) is not necessarily an x-dependent affine correspondence between u and v.

Notes and References

In

linear

system

theory

the

study

of

state

space

transformations has attracted a lot of attention. relatively

simple

and

useful

canonical

forms,

and

feedback

As a result,

like

for

rherein.

In

the

area

of

nonlinear

transformations were first studied by Krener

control,

state

[Krl] , see also

the

[Kal and

observability canonical form (5.59,54). have appeared, see e.g. references

several

ins tance

space

[ReI], [Su]

where some corrections on that paper are given. Theorem 4.4 can be found in [Krll.

The extension including output functions

(Theorem 4.13)

comes

from [Nijl) (see also [Nij2),[RelJ,[Re2]). The study of linearizable error dynamics was initiated in (1<.IJ and continued in [KR], see also [BZ1. The result

in the single-input single-output case comes

multivariable transforming a

case

is

treated

nonlinear

in

[KR] ,

system via

see

state

also

feedback

specific applications in [Brl), (FrJ, [HC], [PoJ, [SR).

from

[K1J

and

the

lXGj.

The

idea

of

has

been used for

A general philosophy

on state feedbacks may be found in [Br2J, [Wi). see also [Ba).

[Ba1 [Srl}

S.P. Banks, Mathematical Theories of Nonlinear Systems, Prentice Hall, Hertfordshire, 1988. R.W. Brockert, "Feedback invariants for nonlinear systems", Preprints 6th 1FAC Congress. Helsinki, pp. 1115-1120, 1978.

173

[Br2[

R. W, Brockett, "Glohal descriptions of nonlinear control problems; vector bundles and nonlinear control theory", Notes for a GEMS conference, manuscript, 1980. [B2J D. BestIe, H. Zeitz, "Canonical form observer design for nonlinear time-variable systems", Int. J. Control, 38, pp. 419-431, 1983. (FKJ M. Fliess, I. Kupka, "A finiteness criterion for nonlinear inputoutput differential systems", SIAM J. Gontr. Optimiz. 21, pp. 721728, 19B3. [FrJ E. Freund, "The structure of decoupled nonlinear systems", Int. J. Contr. 21, pp. lI43-450, 1975. [Kal T. Kailath, Linear Systems, Prentice Hall, Englewood Cliffs, N.J., 1980. [Kr) A.J. Krener, "On the equivalence of control systems and linearization of nonlinear systems", SIAM J. Contr. Optimiz. 11, pp. 670-676, 1973. [KI] A.J. Krener, A. Isidori, "Linearization by output injection and nonlinear observers", Systems Control Lett. 3, pp. 47-52, 1983. [KR] A.J. Krener, \.]. Respondek, "Nonlinear observers with linearizable elTor dynamics", SIMi J. Contr. Optimiz. 23, pp. 197-216, 1985. [HC] G. Heyer, L. Cicolani, "A formal structure for advanced automatic fl ight-control systems" , NASA Technical Note TND-7940, Ames Research Center, Hoffett Field (Ca), 1975. [Nijl] H. Nijmeijer, "State-space equivalence of an affine nonlinear system with outputs to a minimal linear system", Int. J. Contr. 39, pp. 919-922, 1984. [Nij2J H. Nijmeijer, "Observability of a class of nonlinear systems: a geometric approach", Ricerche di Automatica 12, pp. 1-19, 1981. [Po] W.A. Porter, "Decoupling and inverses for time-varying linear systems", IEEE Trans. Aut. Contr. 14, pp. 378-380, 1969. [ReI] W. Respondek, "Geometric methods in linearization of control systems", in Mathematical Control Theory (eds. C. Olech, B. Jakubczyk, J. Zabczyk), Banach Center Publications, 14, pp. 1153-467, 1985. [Re2] W. Respondek, "Linearization, feedback and Lie brackets", in Geometric theory of nonlinear control systems, (eds. B. Jakubczyk, W. Respondek, K. Tchon), Technical University of \.]roc1aw, Poland, pp. 131-166, 1985. [SR] s.N. Singh, W.J. Rugh, "Decoupling in a class of nonlinear systems by state variable feedback", J. Dynamic Systems, Heasurement and Control, pp. 323-329, 1972. [SuI H.J. Sussmann, "Lie brackets, real analyticity and geometric control", in Differential geometric control theory (eds. R.W. Brocke tt, R. S. Hillman, H. J. Sussmann), Birkhfiuser, Bos ton, pp. 1-116, 1983. [Wi) J.C. Willems, "System theoretic models for the analysis of physical systems", Ricerche di Automatica 10, pp. 71-106, 1979. {XG] Xiao-Hua Xia, Wei-Bin Gao, "Nonlinear observer design by observer canonical forms", Int. J. Contr. 47, pp. 1081-1100, 1988.

Exercises

5.1

Prove Corollary 5.6.

5.2

Prove Remark 5.22.

5.3

Consider Example 1.1. Construct

il

regular static state feedback such

that the closed loop system is linear.

5.4

Show that Problem 5.14 is solvable for the system

xz

=

1 2.

1 2

-x 4 1 +

- t;X z

-Xl

Xl

2.

J

+

Xl

+

;XZ

1

'iXIX2.'

:> 2.

:>

J;Xl

"2

X1X2

'

1

Y

5.5

2-"1'

Consider on fRn the smooth dynamics x - f(x) with f(O) - 0 and the smooth

output

y

function

where

h(x),

h :

m"

~

m

and

O.

h(O)

Suppose the system satisfies the observabi1ity rank condition about X ~

Sex)

O. Define a coordinate transformation z = Sex) around x - 0 via =

(h(x),Lih(x)r.",L~-lh(x».

(n) Determine

the

system

in

the

new

z

coordinates

i.e.

= Sex),

= (S~f)(z) and h(z) ~ h(S-1(Z».

fez)

compute

(b) Assume n

2, and consider the system in the above z-coordinates.

Det:ermine under which conditions on ] and h only the system can be transformed

via

a

z = Az + P(y), y

new

z~

transformation

into

5(z)

the

form

Cz.

(c) Express the conditions found in (b) in conditions on the original

dynamics and output function. (d) Repeat

part

5.6

Consider X

on

h

mn ••

case

Have

n - 3.

input

single

o

hex). Let f(O)

~in defined

as Sex)

any

idea

what

the

output

nonlinear

system

and h(D) - 0 and assume that

(h(x) ,Lrh(x) , ' ..• L~-lh(x»

=

0, Suppose that LsL~h(x)

rank n around x

you

are for arbitrary n?

single

a

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

the map S :

in

(b)

conditions on f and

0 for Jc

=

=

has

0,1, ' .. ,n - 2.

(a) Determine the system with respect to the new local coordinates z

S(x).

(b) Suppose additionally that LgL~-lh{O) '" 0, Prove that there exists a regular static state feedback u = o(z) + {J(z)v defined around X

0

such that the closed loop system is linear. (See also Chapter 6 about feedback linearizability,)

5.7

Consider on ~ the system ~

x = u, y ~ eX +

(a) Show that the observation space (cf. Section 3.2) of the system E is given as (b) Define Z2U,

22

=

+

(e.'"

e'lx • •/;

+

(zl' Z2) = Sex)

(32 2

t:r.ajectori,es

2z1

-

of :s

)u, y are

2e =

2x

}.

(e'" + e z1'

Zx

2:.:

7.

• e' + 2e ).

and conclude

input-output

Show that

that the

trajectories

of

-

~

:

• Z1

-

input-output the

bilinear

system ~, 5. a

([ FKJ) Let

~

:

X

=

f(x) + g(x)u. y

output nonlinear system on

[Rn.

=

be a single input single-

h(x)

and ~ :

x

=

A.i(

+ (Bx)u ... bu,

Y ""

Cx a

175

single-input single-output bilinear system on ~p. The system Z can be immersed S : lR"

-+

into

the

bilinear

~

system

if

there

exists

a

mapping

IR P such that each input-output trajectory y(t,O,X o ,u)

coincides

with

bilinear system

the

E.

input-output

trajectory

y(t,O,S(x o ) ,u)

of

of :E

the

Prove that a necessary and sufficient condition

that :E can be immersed into a bilinear system ~ is that the observa-

tion space C of E is finite dimensional (see also Exercise 5.7). 5,9

Consider

the

nonlinear

the compensator

Zi

...

vi'

system U1

-

:E :

Zi'

x-

i E

f(x)

+

" gi (x)u L

,- ,

.

i

Introduce

E!, for the system :E. Show that

the system :E satisfies the strong accessibility rank condition if and

only if the precompensated system satisfies the strong accessibility rank condition. 5.10 Consider the nonlinear system with outputs L: : Y '" h(x).

Introduce

the

compensator

zi

=

Vi'

x Ui

=

f(x) = Zi'

+ i

L gi (X)lli'

'-1 E m.

Show

that :E satisfies the observability rank condition i f and only if the pre compensated system satisfies the observability rank condition.

6 Feedback Linearization of Nonlinear Systems

In

the

previous

z

transformations

x

chapter =

we

have

seen

that

by

applying

a nonlinear system given

Sex)

state

spnce

in local coordinates

(Xl •...• Xn ) as

=

x

f(x)

I

+

gj (X)U j

,

(6.101)

e,

(6.lb)

j=l

i E

transforms into a system which may look rather different from the system

in the original coordinates. In fact. in Theorem 5.3 and Corollary 5.6 we have

given

necessary

and

sufficient

conditions

under

a

which

locally

strongly accessible nonlinear system (6.la), which may be highly nonlinear in the original coordinates, can be transformed into a linear system

z

(6.2a)

Az "" Bll

,.,hcl.""e (5,,£) (z)

Az and (S"gj) (z) ... b j

t

the j-th column of the matrix

n.

Furthermore in Theorem 5.9 conditions have been given which ensure that also the output equations are transformed into linear equations

y

(6.2b)

Cz

~"ith 11i(S-1(Z)) = c,z and c i

ehe 1-th row of the matrix C. Although the

conditions as given in these theorems in principle only ensure the local ex:i.stence of such a linearizing coordinate transformation z

S(x)

still

these theorems are potentially very useful for control purposes since in this way the control of a seemingly highly nonlinear control system may be reduced to the control of a linear control system, for which we have many tools at our disposal. IntUitively it is clear that the set of nonlinear systems (6.1) which can be transformed (locally) into a linear system forms a very thin subset of the set of all nonlinear systems. On the other hand in the preceding chapter

we

u - a(x) +

x

have ~(x)v

[(x) +

also

seen

that

application

of

nonlinear

a

feedback

to (6.1a) results in the system (5ee(5.88»

I j''l

gj (x)a j (x)

+

I (I .le 1

Lu 1

gt (x)f3 1 j (x»)

Vj

t

(6.3)

177

which may he quite different from the original system (6.1a), Therefore, a logical

next

tep

5

is

to

ioves tiga to

when

the

dynamics

(6. la)

can

be

transformed by a (local) state space transformation z - Sex) and a regular static state feedback u

is

linearizat::ion linearizable

problem.

the

et(x) + {3(x)v into a linear system (6.28) where u

input

split

sys tem

two

(in sui table

dynamics

in principle

then

a

parts: local

imposed linear control strategy for

is

This

controlled

sense

into

v.

vector

the

If

the above

in

system can be renders

=

the new

replaced by

nonlinear

coordina tes)

called

the

feedback

is

feedback

(6.2a)

the

control

feedback linear,

of

loop and

the obtained linear system.

added the words" in principle", because the nonlinear feedback,

the

which

a

super

He have

required

to make the system linear, may be very complicated, and very sensitive to parameter uncertainties. are bounded, new

inputs

say if U v1

Nevertheless,

Furthermore,

, ...

,vm

may

it

is

clear

if the original controls u 1

, ... ,

um

,um)1 Iuil::;: 1), then the bounds for the

{(u1, ...

become

complicated

that

feedback

functions

of

linearizability

the is

state. a

very

enjoyable property for a nonlinear control system.

The

feedback

linearization

problem

is

also

important

from

a

mathematical and system theoretic point of view,

since it forms a first

sy~tems

(under state space and

in the classification of nonlinear

step

feedback transformations) by singling out the essentially linear systems. Finally we note that the problem of linearization of (6.la) by state space transformations, transformations,

or

by

can be

space

state seen

as

the

and

transformations

generalization

of

the

feedback problem

of

linearizing a single vectorfield x '" fex),

(6.4)

by means of coordinate transformations z equilibritun point of f.

Sex)

=

in a neighborhood of an

This problem has a long history in mathematics,

starting at least with the work of Poincare,

and is known to be hard in

general. Surprisingly, we shall see that in a sense the addition of inputs makes

the

problem

much

easier,

since

by

assuming

local

strong

accessibility we add a lot of extra structure to the system. This chapter is organized as follows.

In the first section we shall

give the geometric conditions for feedback linearizability of (6 .la). We will also show how these conditions nicely generalize to the solution of the

problem of feedback

x - f(x,u),

using

linearizability of general

feedback

U

=

(l(x,v).

In

the

controlled dynamics

second

section we

will

approach the problem from a more computational point of view. We will show

178

that the geometric conditions as deduced in the first section contain a lot of redundancy I

and moreover that often there is a simple recipe to

construct the required state space and feedback transformation. This will become especially clear in the single-input case,

6,1

Geometric Conditions for Feedback Linearization

Consider a nonlinear affine system (without outputs) In

X-

L gj(x)U j

f(x) +

(6,5)

jm1

where x "" (Xl'"

are local coordinates for a smooth manifold Hand

IXn)

f ,gl , .. ,gm are smooth vector fields on M. For simplicity we first throughouc take U '"

m 111

•

Let Xo be an equilibrium poim: for f, i.e. f(xo) '" O. The

Definition 6.1

system (6.5) is feedbaclt linearizable around (i)

il

trllnsformation S: 0 c Ill

coordlnace

n

Xo

....

if there exists

IR"

defined

on a neighbor-

hood 0 of x o ' {
Idt:h a(x) an mxl

vector satisfyIng a(xo ) - 0 and (J(x) an mxm invertible matrix, both defined on 0, such chat the feedback transformed system

X-

m

Lgj (x)v

f(x) +

(6.6)

j ,

j-l

Idth (see (5.88),(6.3)

L

f(x) = f(x) +

1- 1

(6.7)

m

L

gi (x)fJ l

j

(x) ,

J

E ~I

1-1

transforms under z

~

Sex) into the linear system

z - Az + Bv

(6.8)

j

Remark 6.2 with

the

The

(6.9)

E Ill.

above notion of linearizability should not be confused

classical

notlon

of linearizing

the

system

(6.6)

around

the

179

equilibrium point Xo and v

O.

=

case one wri tes

In this las t

(6.6) in a

Taylor expansion

at

ax(xo ) (x-x o ) +

x

m

I gj (Xu )Vj

(6.10)

+ higher-order terms in x,

jml

and

the

linearized system

defined by

is

omitting

all

the

higher order

terms, yielding the linear system (6.11)

The relation of both notions of linearization is as follows. Assume (G.9) holds,

then the matrices A and B in (G.9) are related to the matrices in

(6.11) by A

=

8I

P ax(X o ) P

as

where P = 8x{x o ).

sense amounts

-1

E

j

,

Thus we

conclude

that

(6.12)

~,

linearization in

to neglecting the higher-order terms

the

classical

in (6,10),

while in

feedback linearization the higher-order terms of (6.10) are eliminated by

state space transformatioos (see also Exercise 6.1),

With every nonlinear system (6.5) distributions

Dl C Dz C DJ C

inductively ad~gj ... [f ,ad~-lgJ

as

1,

k

=

we associate a nested sequence of follows.

ad~gJ

Denote

=

gj'

and

1,2, .. , with [f ,gj the Lie bracket of

vectorfields f and g. Then define Dk(x)

=

span{ad: gl(x), .. ,ad;gm(x)lr

=

D,l, .. ,k-l), Ie ... 1,2, .. (6.13)

The main theorem of this section reads as follows

Theorem 6,3

that

the

Consider che nonlinear system (6.5) \.,ith f{x o )

strong

accessibility

rank

condition

in

(cf.

Xo

D.

=

Assume

(3.41))

is

satisfied. Then the system is feedback linearizable around Xo if and only if the distributions D1 constant

dimensional

in

, ..

a

,Dn

defined in

neighborhood

(6.13)

of xo'

are all

invo1utive and

Noreover

the

resulting

linear system (6.9) is controllable. First we prove the "only if" direction.

Proof of Theorem 6.3 ("only if") linearizable

around

xo '

i,e,

Suppose (6.9)

tha t

holds

for

the

sys tern

some

u - a:(x) + {3(x)v and local coordinate transformation z

is

feedback

regular

feedback

~

S(x) around xo'

180

For the linear syscern (6.8)

the distributions D1

, •• • Dn

as in

(6.13) are

the following flat distributions (superscripts A,B indicate the dependence on the vectorfields Az and colwnns of B), defined for z around 0:

kEn.

(6.14)

Clearly the distributions D:,B, ••. ,D~·B are all involutive and of constant: dimension.

For

the

feedback

transformed system

(6.6)

the

corresponding

distributions are given as (x in a neighborhood of xo):

(6.15)

D~·g •.. ,Dr:'/!

Also

are

constant

dimensional.

and

by

invo1utive because of the involutivity of D~·B, .. ,D~·n.

D;'Il, k E

E!;

We will now show

D:' s defined in (6.13) for system (6.5) equal

Dr.

that the distributions

2.30

Proposit:ion

in ather words they are feedback inva.riant:. First, for k ... 1

we have

!.gl(x) •. ,gm(x)l

D:'&(X) .. span

since the matrix P(x) D;'s(x)

is invertible.

ad~ gj

D:~ ~

E

L

Now assume that for some k we have

then

I

m

ad~ gj .. [£ +

(6.16)

then we will prove that D:~~(x) ~ DH1(x).

Di;.(x) for x near xo.

Indeed take

= span(gl(x),.,gm(x») - D 1 (x),

ad~-l gj]-

grol1

1=1

(6.17)

k-1( Il ad[ gj E Dk '

Now DI:+1'

Furthermore

Eo11m
gj

EDlin

Dk

=

since }:-1_

Igl,ad e and

and

•

I

gj

Dk - D;"s

is

invo1ucive

k-l_

and ad!: gj

E DII

D:'~~(x)

thus

1<;-1-

rE, ad! gj) E Dk + I

thus

cDk+1(x).

(01

and

by

definition

of

gl E Dl C Dk.

it

)gi E Dk . Therefore by (6.17)

For the converse

inclusion we

observe that m

I 1

ad;~l

giOi'

gj

1~

(6.18)

~l

(f.i2d~·lgj

m

J -

I

1 "1

rn

0i [g1

,ad~-lgj 1 + I ad;-lgj (a i )gl E D:~~. ial

0

181

The above proof shows that in general the distributions Dl ,D z ,'"

Remark

are not feedback invariant, In fact

k = 1,2, ...

i. e.

in general we do not have

the feedback invariance of Dl ,V2 , ...

V:'

g

...

in the

D:'

g

above

proof follows from the assumption of involutiveness. Therefore for systems which are not feedback linearizable these distributions are not the right

tools [Dr studying equivalence of the system under coordinate and feedback transformations. Instead we may consider the following distributions span (g1 (x), .. ,gm (x))

(~Dl ex))

(6.19) lIk(x)

=

[f,llk_1J(X)

L [gj,l'Ik_1]e x ),

+

k

2,3, ..

~

j" 1

o

which are always feedback invariant (see Exercise 6.3).

The key mathematical tool in the proof of the "if" part of Theorem 6.3

is

the

following

generalization

of

Frobenius'

Theorem

(Theorem

2.42)

(compare also Proposition 3.50):

Lemma 6.4

Let Dl C Dz C D3 C

C

constant dimensional distributions

be a

DIl

011

N,

Idth

sequence

of involutive

dimensi?ms ml

::-::::

mz : -: :

and niH'

Then around any point xQ E N there exist local coordinates

(6.20) such that

(a span axl'"

Proof

. ,_a_) , axk

k

=

(6.21)

l,2, ... ,N.

First consider the distribution Du'

By Frobenius' Theorem (Theorem

2.lI2) there exists a partial set of coordinate functions

on a

neighborhood U of x o '

such

that

the

x

=

(x + 1 mu

integral manifolds

.. ,xn )

of Vu

are

given as

lq for

E Ulx(q)

(YmJl-l" 1

(6.22)

c),

constant vectors

Frobenius'

DH -

=

Theorem

c E [R,,-mu. there

Next

exist

consider

partial

.. ,Yn) on a neighborhood of Xo

the

distribution DU _ l '

coordinate

By

functions

such that the integral manifolds of

are locally given as (6.23)

182

for cons tant vec tors d integral

manifold

manifold

(6.22),

manifolds

(6.23)

fortiori for X

ll

the

of Dll DlI -

C

I

m

Xo

to

contained

are

X

that LX

DII ). Define Pu:=

we can permute the coordinate functions 11-1

is

functions (Recall

1 •

Since D,,_ 1 C DII it follows tha t every

ll-l,

close

(6.23)

i.e.

E

Il?n-

- 1 E

mil

x- 0 -

in

some

integral

on

the

integral

any X

E

constant for

DII

that

as

yll-l

!l -

Y

and a

•

then it follows

mil-I'

(X ,y),

(6.24)

PH

and dim y - n-m/l suc:h that the functions xU and x are

wich dim

independent. Then the integral manifolds (6.23) are also given as (6.25)

e"

for constant vectors

CE

E [R1l.

Next consider DII -

IRn-mJ/,

Theorem there again exist partial coordinate functions integral manifolds of

z '

By Frobenius'

yll-Z

such that the

are given as the sets where all these functions

DJ/-z

are constant. However since DI/

2

C

DU -

all the functions (xu,x) are also

1 '

cons tant on these integral manifolds. 1 t follows that we can permute as

l-z

with

(xll-1.ylJ-l)

integral manifolds of DII -

dim(xll-l):=

PII-l

'" m/i-l-mll-

the

are also given as

2

II C

(6.26)

,

we obtain local coordinates x _

Continuing in this way,

yll-Z

that

such

Z

/I -

, ..• x .x)

(Xl

such that for every k = 1 •..• N the integral manifolds of Dk are given as II

c ,x(q)

for P'r.:=

constant nil'.

Ink

-

=

-

c

l.

(6.27) C E (Rn-mu

vect:ors k = N.N-i •... ,

l'

with mo

O.

By Corollary

2.43

where this

is

equivalent to (6.21),

Remarlt 6.5

(6.27).

o

Notice that the description of the integral manifolds of DI: in

and

therefore

of

the

distributions

Dk

in

(6.21),

invariant under any coordinate transformation z = S(x),

z -

k

E~,

(Zl ••••

is

z".Z)

I

of the "triangular" form

Z Z

z

II

sf! (XII .x)

11- 1

S'/-l (x"

Ir.

S....

1

(x k

(6.28) 1

.x" .x)

- 1 • • • • X II

,x),

k '" N

2, ... ,2

lB3

Consider the distributions Dl C D2 C D3

Proof of Theorem 6,3 "if"

... as

defined in (6.13), Since they all have constant dimension and dim Dk for all k, it follows that there exists a least integer D

K.

'"

Clearly,

1f,:!O

n

(6.29)

D

r.

=:.;

such that

n.

We claim that VIC is invariant under the system dynamics

(6,5), see Definition 3.44. Indeed DIf, is invariant under f by (6.29) and

invariant

under

involutive, Die

Co

x

=

E~,

Since gj E VIC'

and since (6,5)

n = dim fl.

Die

j

gj.

(Xl, .. ,XIC)

j

gj E Dl C D

since E~,

K

it

by

follows

is strongly accessible

Thus by Lemma 6.4 o such that x(x ) = 0, and

k

=

there

,

j

E

and

!:!'

K

Proposition

3.47

it thus follows

exis t

is

D

local

that

that dim

coordinates

(6.30)

1, ... ,K,

where (6.31) with

=

/D);

dim Dk , and

/D

o := O. By definition of VI: it follows that

(6.32)

k=l, . . ,1C-1. Using (6.30) this yields for any k E span

=

1, ... ,,,,-1

1-"-l '

ax

.. ,

_'_1 k+l

Writing [ corresponding to (xl, .. ,xl\,) as [

j = 1,2, ... ,i-2,

- 0,

i

j

ax

=

=

E k.

([1, .. , ftC)

3, ...

(6.33)

this yields

(6.34)

,11:.

Thus the vectorfie1d [ in such coordinates is of the form

f'

f

-

f' f'

, .. ,.,"t") , (x , .. . ,xl'.) , (x , .. . ,xl'.) (x , .

X

Hon'over, the [arm

1'.- 1

(6.35)

I'.

,x)

since gj E D 1 • j

1 .... m,

the vectorfields gj'

j

E!E.

are of

184

1

gj (X gj

1

O...

Ie

,X )

(6.36)

:

~

[

6

Furthermore. by definition of Dk

(6.37) Togehter with (6 34) this implies that k 1

dim(x +

rank

)

~ dim Dk + 1

dim Dk

-

(6.38)

•

I'lence for any 1.

rank

(6.39)

x near xo'

Since dim(x

l 1 - )

Pi

1

it immediately follows

that

Pl

~

P2

~

••

~

PI{.

> O.

Now we are going to define a coordinate transformation z = Sex) with S(O)

= 0 such that in the z-coordinates the system can be made linear by

feedback.

Furthermore this coordinate trans formation will be of the form

(6.28), i.e. (recall that dim DIC - dim H)

z

I\.

SIC-1

Z

i

S (x

Z

Z

(6.40)

Sl\. (:/")

1\.-1

1

i 1

Sl (xli' .. ,XK)

and thus

span(~,

Dk .,.

First we set zK.

..

I~)'

It E

. For the definition of

(6.U)

/c •

zl\.-l

we observe that by (6.39)

rank Now set

(6.43)

1' -le-l

are

where."C

_ IC-1

that x

and

coordinate

PK.-l-Ple

functions chosen from the set x

nre independent functions of

transformation

of

the

farm

IC 1 x -

(6.40),

IC 1 -

in such a way

Clearly this defines a so

that

in

the

new

185

1 X, ••

coordinates

,x

~~Z

"'-1 ,Z

If,

,2

the

distributions

are

still

given

in

as

(6.tll), Therefore (6.34) and (6.39) still hold i f we replace x'" and /~-l

z"',

by

respectively

(Although the component functions

ZIl:-1

£1, ••.

,1/'"

in

the new coordinates are not the same as in the old coordinates!) Thus by (6.39) for i - ",-1 in these new coordinates we have

rank

~ Ptr.-l

dim(z

=

<-1

(6.43)

).

Then let < ] f <-1 (x <-, ,Z <-1 ,2)

<-, z

=

_/1:-2 X

[

'

where i{·-2 are PIC-Z-P"" functions that

and

£"'-1

are

coordinate

transformation (6.44)

the set x

chosen from

independent

functions

is of the

In

this

way

we

continue

till

we

have

in such a way Clearly

of

form (6.lI0),

and (6.39) still hold in the new coordinates xl, ..

(Zl"

1\.-2

,x/>,-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

,.<

...

0 'this can be

simplified

sicuacion

treated in Example 3. 2l! (see equa cion (3.54)

(6.63)

with

A

and

thus

=

Clearly

(8 1 -a 3 )a;1.

is

trivially

the

3.24 it is computed that [f,g]

D, (w) -

'pan {

[

distribution Dl

equals

span(o,fl,1)T),

involutive and of constant dimension. =

In E:{ample

-A(j3w J + wz ,,(,-aw J -w 1 1,O)T so that

~ l' [-~~:~~;~ 1} A

Clearly D2 does not have constant dimension (take

W

=

O!).

Furthermore in

general D2 is not involutive, since

(6.65)

for general values of A,cr,j3,,,(, linearizable.

and so the system (6,63)

is noC feedback

o

190

We sh
nonlinear systems x

to gener
For completeness we give (compare Definition

f(x,lI).

6.1) . Consider a smoor:h nonlinear sysc:em

Definition 6.10

x ""

(6.66)

f(:':,u)

wit11 equilibrium (xc ,u o ). i.e. f(xo ,u o ) = O. Tllroughouc '
(1)

Xo I"ith S(x o ) (i1) a regular o(xo ,0) -

=

n

mn

-l'

defined on a neighborhood 0 of

0,

scacic state feedback u = o(x,v) U

(see

(5.106)

satisfying

o and defined on a neighborhood 0 x V of (;"0.0), V CU, non-singular on 0 x V such chat: for suitable constant

:~(x, v)

lolith

matrices A and D - A Sex) + Bv,

(S) f(x,o(x,v» .. x

The

conditions

for

feedback

notion of the extended

SYSCDIll

x

linearizability

E 0,

in

v

(6.67)

V

case

this

rely

on

the

of a general nonlinear system (6.6) (compare

with Definition 5.27 and (5.101».

Definition 6.11

The excended system of

is

(6.66)

tlu;

affine nonlinear

system x - f(x,u) U

(6.68)

I>'

Ivlch state (x,lI) E H x U and input

Theorem 6.12

Consider

ehe

m

I>' E 111 ,

nonlinear

system

(6.66)

Idth

O.

f(xo, u ll )

Suppose chat t11e excended sysr:em (6.68) satisfies tlle scrong accessibility rank condiclon in (xo,u o )' Then tlle nonlinear system (6.66) is feedbacl, 1inearizable around (xa,u D ) if and only if the extended system (6.68) is feedback

linearizable around

(x o , u o ),

i. e.

satisfies

the

state

the conditions of

Theorem 6.3.

Proof (Only if) x "" S"1 (z)

space

One

can

together with

transformation

interpret the feedback

(x,u) .... (z,v)

for

II =

(l(x, v)

the

space for

transformation

(6.66),

extended system.

coordinates (z,v) the extended system has the form

as a In

state

the

new

191

z = Az

v

+

Bv

(6.69<1)

(6.69b)

G(Z,V,h')

for some function G. By the regularity of the feedback u locally solve from the relation u

i.e, v

=

P(x,u) with a(x,p(x,u»

=

a(x,v)

=

a(x,v) we can

for v as function of x and

tI,

u. It follows that (6.69b) is given as

=

(6. 6gb') a{J

+ au (5

-1

(z) ,0:(5

-1

(z), v) )1..

Setting the right-hand side of (6.69b')

equal to a new control vector

10'

defines the required linearizing feedback for the extended system. (if)

be

Let Dl C Dz C

the

sequence of

distributions

as

defined

in

(6.13) for the extended system. By assumption they are all involutive and of constant dimension.

Hence by Lemma 6,L1 and local strong accessibility

q ~ (ql, .. ,q"') around (xo ,u o ) E N x U such

there exist local coordinates that

Dk = span

Since

lc=2, .. ,N.

a

a BQm +1'" 1

(...E...., .. ,....£....], aql aq'"

' BQm 2'

Dl

=

with

a

span

(au 1

, q

and u

Vi;

=

1/11:

a

'"

'aum )

(x, u)

coordinates

and

it

Now

In coordinates

for N around x o ,

dim

dim

follows

consider

for N xU,

where x

for U around u o ,

we

the

can

take

vectorfields

are coordinates they

are

of

the

general form

(6.70)

a

a

By definition [aqj 'au J

=

0,

1

j

m1+l, .. ,mz ,

iE~,andthus

(6.71)

Therefore Aj (x,u) and Bj (x,u) do not depend on u.

Hence we may write in

the (x, u)-coordinates m1+l, .. ,mzl. In the same way Dk for any k

(6.72)

2, .. ,N is of the form

(6.73) Define the nested sequence of distributions E], .. ,EU-1 on N

192

k = 2, .. ,N.

Since

the

Di , .. ,Do

distributions

dimension it immediately follows

are

all

involutive

and

that the distributions E 1

also have these properties. (Note that Ek

of ,

is the projection of Dk on

there exist local coordinates x _ (Xl, .. • x

Hence by Lemma 6.4

constant

,Ell_Ion

•

U 1 - )

}1

n.)

around

for H such that

Xo

Ie

E

(6.75)

N-l ,

implying that

) + spanl~, ... ,_8_ l . axl a.\Cr.-l

span

(6.76)

Ie E N.

In the same way as in the proof of Theorem 6.3 (if-purt) it now follows that in these coordinates f(x,u) has the form

i

(u

£2

Il 1 (xl • ••..• X - )

Il 1

•.. ,X - )

£(x,u) =

(6.77) 11- 2. X

0-1 ,X

)

and furthermore

a? ax k-1

rank - -

where

we

0

x:

denote

l1 1 x - , •• ,Xl. U

u.

to coordinates

of Theorem 6.3 (if-part). is

, .. ,zll-l)

Furthermore U

-

the

the

,

lc E N-l

dim(.l) ,

We

(6.78)

modify

Zll-l, .•• zl,

coordinates

the

successively

v in the same way as

in the proof

The thus defined transformation (xl, .. ,X"-I)

required

transformation

state

from

space

u

to

v

transformation

is

the

H

z - S(x).

required

feedback

Indeed, recall that the transformation from u to v i,s defined

o(x, v).

by setting (see (6.45» 1

11- ~

1

f(u,x, .. , x )

v - [

(6.79)

ii

where ii are m - dim(xl) functions from the set (u l way

that

Implicit (X01

ii

and

fl (x.

Function

u)

are

Theorem

we

independent can

solve

'"

functions u

from

,um ) , taken in such a of

u.

(6. 79)

Hence locally

,u o ), yielding an explicit regular static state feedbaclc

by

the

around

193

U

(6.80)

(J(X,V).

=

o Remark

Notice

that

the

adaptation

of

the

state

space

coordinates

performed in every step is of the same nature as the final adaptation of the

u-coordinates

defining

the

required

feedback

transformation.

Indeed

consider (6.45), then by the Implicit Function Theorem we can solve for /. as tunc tion of

2k

k

.

and zkil .. ,z , 1. e.

(6.81) This can be interpreted as "feedback" for a fictitious lower dimensional system with old input u = xl<, new input v

Example 6.13 The

Consider

the dynamics of a

forces which act on the rocket are

=

zk and state (zk+1, ... ,zk).

rocket

outside

the

atmosphere.

the gravitational force

and the

force as delivered by the rocket motor.

Fig. 6.1. Rockel ou1side the atmosphere.

The control variable is the angle a: expressing the direction of the force as

delivered

X,,- = 0,

x3 -

by

i-,

the x4

=

rocket

iJ,

motor.

Take

state

thus x E T(IR+ X 51).

space

variables

Xl

r,

-

Then the dynamics are given

a,

x,

x,

x,

x,

x,

_gR'l/xi +

x4

- 2X 3 X 4/Xl

T

- co,

m

+

T

u

+

,

sin u

1l1X1

g

(with m the mass of the rocket; radius

of

the

linearizable extended

earth).

(around

system,

(6.82)

X 1X4

i.e.

an

In

order

to

arbitrary (6.H2)

extended system is of the form

the gravitational constant and R the check

together

xa

if

equilibrium

= f{xc)

with

the

system

point) the

is

we

equation

+ g(xo)w, where x~

feedback

consider

=

u

=

I;'.

the The

(x, u) and

o o

x4 2

_gR /x; + XIX: + T

f(x,u)

,g(x,u)-

cos u

m

o

(6.B3)

o o

1

Hence

o o ~ sin u

[f ,g)(x, u} -

m T

(6.84)

cos u

o It follows that Dz(x,u) - spanlg(x,u}.[f,g](x,u}} is not involutive, since

o

o cos u

[g, [f,g]] (x,u)

e

(6.85)

Dz (x,u).

sin u

o Therefore by Theorem 6.12 the system is not: feedback linellrizable. 6.2

0

Computational Aspects of Feedback Linearization

Consider a nonlinear system (6.5) with f(x c ) - 0 and satisfying the strong accessibility rank condition in x D ' for which the. distributions DI , •• ,D" are all involutive and of constant dimension. Then Theorem 6.3 gives us the following recipe to construct a state space transformation z - Sex) and a feedbaclc u ... o(x) + P(x)v which will transform the system into a linear one: (a) Construct Dk

(b) Adapt

as

in

6.4

Lemma

a

spanl

axl .. 'ax"')' successively the

coordinates

Ie - 1, ...

,~

x ~ (xl, ... ,x~)

such

that

.

coordinates

X

~-l

, ••

,x

1

to

z

"-1

, .. ,z

1

as

in

(6.45),using the vectorfield f. (c) Apply the feedback (6.47).

In this section we shall show

that in general

this

is not

the most

efficient way to perform feedback lineariza tion. The crucial observation is that the sequence

DI

C

D2,

C

•• C D~

is not just an arbitrary nested

sequence of distributions as in Lemma 6.4, but instead is constructed in a very special way. coordinates xl, ..

As a result,

we do not have

to construct all

the

as in (a), but only a (possibly small) part of them,

195

and

still

step

(b) will <

I

provide

z , .. ,z . Also

coordinates

this

us

with

a

imply

will

complete

redundancy in requiring that all distributions Di

set

there

that

are

of is

required a

lot

of

involutive and of

constant dimension. These considerations become most clear in the single-input case

(6.85)

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

in which case the distributions Di are simply given as Di (x)

=

(6.86)

span(g(x} ,ad!g(x), . . ,ad:-1g(x)},

:$

dim D1 (x)+1, DIl(xo ) is equal to T:,:/J if and

i E n.

(6.B7)

and thus, since dim Di+1(X)

only if

The

main

technical

proposition

convenience we denote L,;rp

~ drp(X)

reads

follows.

85

as ,

For

notational

for any vectorfield X and

function rp.

Proposition 6.14

Consider the single-input system (6.85).

exists a function

~

Suppose chez-e

such that

(6.88)

Ie - 0,1, .. ,n-2

(or,

equivalently,

- 0

statement (a) (x o ) ,...

for

X E Di'

i

~

1, .. ,n-l).

Then

the

°

implies the follol>'ing two statements (b) the functions rp, L!~, .. ,Lrl~ are independent around xo' (e) the veetorfields g, [f,g), .. ,ad~-lg are independent around xo' Conversely, if g(x o ) ,... 0 then (b) implies (a), and if drp(xo) ,... 0 then (e) implies (a).

Remark

Note that conditions (6.BB) and (xo) ,... 0 are in a sense

dual to the conditions for linearizable error dynamics, cf. Chapter 5; in this last case rp is known, while g has to be found.

In order to prove Proposition 6.14 we state the following auxiliary lemma.

Lemma 6,15

Let

f

and g

be

vectorfields

following statements are equivalent

and

~

a

function

on

N.

The

196

k - 0,1,2, .. ,0-2,

(a)

0,

(b)

i+j

0,1,2, .. ,11-2,

=

k = 0,1,2, .. ,n-2.

(c)

Furti1el.-m(Jt"c If arm of chn a/love scatpments hoLd,

(d)

~(-1/ ,


Pr('lof

C] Pflrly (b)

imply (b).

Af';!':;l1l!ll:!

ilnpli.es

(,,)

(c).

<Jnd

thllt (1.1) hnlr:l5.

tl1en

k = 0,1, ... ,0-1.

we prove thAt both (A) and (c) Ch.:;pt~r

From

2,

cf.

(2.17()), we recall

+ 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

+
o

(6.91)

. so

that:

fo110W5.

thHn

Wf!

inc1eed

(b)

holds

j+l.

for

Similnrly

(b)

follows

from

(c)

as

Suppose (b) holds for a certain i < 0-2 (i t does hold for I

0),

such

that

prnve

it

also

holds

for

HI.

Indeed

let

Ie

be

n-2. Then by the induction assumption

k+j+l

o

(6.92)

lip-nee

.

o

(6.93) so thAt indeed (b) holds for 1+1. Finally (d) will be proved by induction to Jr. For k=O the identity is trivial. Let (d) be satisfied for k-l. Then eli

fff'n~nti.at:e

Yo -

the fllnction
n- 1

1

tp,

which is zero by (b). with

ad f

n::spf'ct: to f: 0

n-l-k

ad e

g>

n-l-r.


g>

+

(6.94)

197

Hence

n-k


ad f

\:-1

g>

= -
'"

rp,

-(-1)

k-1

n-1


that

g>,

o

(d) holds for k.

Proof of Proposition 6.14

Consider

the

following

product

of

two

IlX/J

matrices

(6.95)

1



(Xo)


1 tp,

ad~-lg>

(6.96)

Now suppose

(6,96)

that

side of (6.95) suppose

Also,

(a)

holds,

is nonsingular,

(c)

are nonsingular,

(b)

holds

follows that

(b)

then by

and g(x o ) '" D.

0, Ie

of

Lemma

6.1S(d)

the

two nxn matrices on the

so that

and drp(xo) ... a

holds

suppose

then because

and hence the

and (c)

(6.88)

By

(6.88)

follow.

necessarily

and

Lemma

matrb:

left-hand

Conversely, (a)

holds.

G.1SCb)

it

0,1,2, ... ,n-2. Since g(x o ) '" 0 necessarily (xo) ... 0 and thus by Lemma 6.15(d) (a) follows. 0

We now obtain

the

=

following

=

refined versions

of

Corollary

6.7

in

the

single-input case.

Consider the Single-input sysc:em (6.85). Then the system

Proposition 6.16

is feedback linearizable around Xo to a controllable linear system if and only if there exists a function (p such c:hat

(x)

=

0,

Ie

=

0,1, .. ,n-2, x in a neighborhood of x O

'

(6.97a)

(6,97b) Furthermore for

sllch

a rp the distributions

DI;

are given as

i=1, .. ,n-1.

(6.98)

19B

The single-input system (6.85)

Corollary 6.17

around

is feedbr'1Ck linearizable

to a cOlltroll<1ble linear system if and only if

Xo

(6.99a)

Dn

(6.99b)

is involutive around xo'

1

If

Proof of Proposition 6.16 and Corollary 6.17 Proposition 6.14 Lemma

6 . 15 (b)

(drp,L,'P, ..

,dL~-l-i!p}.

By

dim Di (dtp, ••

D1 , •. ,Dn

Then

and

i,

by

are all

from

by so

the that

and

independency

then

by

the

linear

system.

system

feedback

is

Conversely

if

the

that

dim span

Hence

invo1utive and constant dimens ional 6.7

Di C leer span

it follows

(6.86) of

follows.

(6.9B)

that

(6.97a)

Proposition 6.l4(c)

thus

Corollary

controllable

follows

it By

n-k,

,dL;-l-ktp}

holds

(6.97)

ehe functions tp,L{!fJ, ..• L~-l'P are independent around xo'

the

distributions

and Dn (xo ) linearizable

system

is

to

a

feedback

linearizable to a controllable linear system then by Corollary 6.7 (6.99) is satisfied. (Theor.em

Finally if (6.99)

2.42)

applied

(Xl •.•• ' x l1

coordinates

to

is satisfied then by Frobenius'

the

distribution

Dn - 1

there

Theorem

exist

local

such tha t the integral manifolds of Dn ~ 1 (which by

)

(6.99a) and (6.87) have dimension n-l) are given as

constant}

{q near Xo IXn(q)

(6.100)

I

~I'

or equivalently, (6.97a) holds for tp (6.97b) holds for tp

''{n'

We

(6.99)

conclude

that

Moreover by (6.99a) necessarily

o the

implies

involutivity

and

constant

dimensionality of all the distributions D,;;, k-I,2, .. ,n. Horeover if we have found a function rp with d!p(x[]) .,.. 0 and (x) - 0 for all X E Dn - 1 then by (6.9B) the "rectifying" local coordinates as in Lemma 6.4 for the sequence

of

L~-1!p,L~-2tp,

•••

distributions I

D1 c D2 C •.. c Dn

- 1

are

directly

as

given

Indeed if we define the coordinates z

Lftp,'P.

,zn)

(ZI , ••

by

(6.101)

i

then it immediately follows from (6.98) that

i

E

(6.102)

11.

(Notice that for notational convenience we have reversed the ordering of Zl •••

I

zn

in comparison with the ordering of the coordinates

X

1

•••

1I

,x

or

199

1 2 , ..

,2

K

2 1 " , ,Zu

of Lemma 6 are

respectively Theorem 6.3.) Horeover the coordinates

,l"

already

1055 of generality,

the

11110.8t'1.:::111g coordinates.

that rp(x o )

0,

=

If we

implying that 2 1 (X O)

assume =

without

L~'\p(xo)

0,

i E n. In fact, for any i E 11-1 (6.103a)

Zi+l'

since L8L~-1
By

(6.97b)

=

(X)

and Lemma

6.1S(d)

0 by Lemma G.ISCb), and

=

LgL~'-\p(x) '" 0 in a neighborhood of Xo

so

that we can define the regular static state feedback

(6.104)

resulting in the linear system

(6.105)

Summarizing, we have obtained

Suppose the single-input system (6.85)

Corollary 6.18

Then

locally around

tp(x o ) =

o.

Xo

satisfies

chere exists a function rp satifying

By defining the state space transformation z

=

(6.99),

(6.97)

and

Sex) around Xu

as (6.106£1)

i E ::.' and t:he regular st:at:ic st:at:e feedback u

=

a(x)

+

{3(x)v as

(G. 10Gb)

the syst:em around

Consequently

the

Xu

is t:ransformed int:o t:he linear syst:em (6.105).

only

possibly

difficult

step

in

linearization lies in the computation of the function and ...

d<.p(xu);"'- 0

,ad~-lg(xo)

6.14), i.e.

are

(and

therefore.

independent,

assuming

also

that

satisfying

performing <.p

feedback

satisfying (6.97£1)

g(x u ),

(6.97b)

adfg(xU )"

by

Proposition

in finding a non-trivial solution of the n-dimensional set of

partial differential equations

200

Lx if{J(X)

-1

0,

=

1

g,

E

n.

(6.107)

The treatment of the multL-input case follows the same lines as in the single-input case, and will be only sketched here, leaving the details to the reader. Suppose the distributions D1 such that dim DtC and let Pk tive

dim H)

constant

dimensional

there

exist

of Lemma 6.15 it then follows
by

such that
independent functions ifJj

i E etC'

,D"_l(,..rith

I\.

the least integer

dim Dk _ l ' Ie E ~, wi.th Do = O. Since D1\.-1 is involu-

dim Dk

and

, .•

are all involutive and of constant dimension,

independent.

However,

and

that

the

this

point

at

Theorem

PIC

f!..K.' As in the proof

E

i E f!..tC'

0,

=

it follows

Frobenius' i

0,

Furthermore as in

functions there

!Pi'

is

L!!Pi

I

maj or

a

difference with the single-input case, in general r:he codimension of D",_z' which

P/C +

is

larger

is

p "-1'

than

the

number

which is 2p/C (since Po ~ Pl ~ ••. ?::PK.'

i E f!..",'

of

functions

L r !Pl'

!Pt.

see the proof of Theorem

6.3). Therefore we have to use the assumption that D"'-2. is involutive and cons cant dimensional to construct by an application of Frobenlus' Theorem functions

P"'-l-P",

i

Lemmo

6.15

i E P

-/C-l

•

and

i E f!..1\.'

are

Proposition

6.14

and
/C-:!

i E e.1\.-l' L;!Pi'

LrlPi'

px:+1, ... ,P"'-l'

i

ifJl'

and LrifJi'

e. - ' tc I

>

O.

=

such

independent it

that

and

then follows

i E P , -/C

and

the

functions

annihilate

DI\._z'

As


in

that "'" 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


i E

fl·

i

£'2.'

i E

etc'

(6.108)

which form the required coordinate system in which the system can be made linear

by

u

+ P(x)v of (recall the definition of the cDntrollability indices

a(x)

feedback.

This

feedback

is

given

as

the

solution

in (6. 1,9»

K.i

Vj

=

Lr
where we have supposed for simplicity that

P1

=

E

!E,

(6.109)

m (i.e. the vectorfields

gl •.. ,gm are independent around x o )' As in the proof of Lemma 6.15 (d) the

m

X

m matri.x -l!pJ )

(X)] I"~,••..• m j=l •..• m

(6.110)

201

is nonsingular around KO ' and so (6.109) can be solved for u yielding the feedback (compare wi th (6.104»

where b(x)

=

Remark 6,19

(L~l~l' .. ,L;m~m)T(x), and v is the new input.

It is clear that like in the single-input case the assumption

of involutivity aod constant dimensionality of all the distributions Di in Theorem 6.3 can be somewhat relaxed. In particular if for a certain j Pj

then the involutivity and constant dimensionality of D j

... Pj+l

the same properties for the distribution Dj

Example 6.20

<

1':.,

implies

-1'

Consider the model of a mixed-culture bioreactor

treated

85

in Example 1.4 P, (S)x, [

/12 (5 ,I)xz

]

+

(6.112)

-px1I where

}ll

(S) and liZ (5,1) depend on Xl and x z ' resp. Xl ,x... ,I, through

As discussed in Example 1.4 we assume that there exists a point (x~ ,x~ ,1 in the positive orthant for which PI (8)

=

IlZ (S, I) "" p.

Then (x~ ,x~

,l)

0

)

is

an equilibrium point of (6.112) for the constant inputs

"

u,

(6.114a)

We will now show that (6.112) can be feedback linearized around any point (Xl

,xz ,I),

feedback

Xl

> 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,<.- vectorfield

234

(7.89b) As this vectorfield does not belong to the distribution spanned by the input-vectorfields gl'

g:z

and Xl

we

observe

that

the

one-dimensional

distribution span(X 1 J does not satisfy (7 .15a). However letting (7.90)

D = span/Xl 'X 2 I ,

it is rather easily seen that this distribution is involutive and fulfills the conditions (7,15a,b). but is not of constant dimension. One may only verify that D is not the maximal locally controlled invariant distribution in u

kar drJ

•

Nevertheless

we

will

show

that

there

exists

a

feedback

= a{r,w) + p(r,w)v which makes the distribution D invariant (note that

Theorem 7.5 only applies around points where D is constant dimensional). A straightforward computation shows that (7.91) produces new input vectorfields

that leave D invariant, W1

w2

""

though {J(w)

is Singular at points

(r ,w) where

O.

Next we will determine a 2xl-function a(r ,w) - (a 1 (r ,w) ,a l (r ,w») t

such

that f(r,w) - f(r,w) + gl(r,w)Ql(r,w) + gz(r,w)a 2 {r,w) leaves D invariant. This yields the following set of partial differential equations for a(r,w)

w:zXl(al(r.w») + w1X1(a:z(r.w») ~ -(b 1 -l)w 2 { -w1X1(al(r,w») +

WZ X1

(7.93a)

(o2(r,w») - -(b2 +l)w1

and

wlXZ(a1(r,w») + w1 Xl (a2(r,w») { -w1XZ[a1(r,w») + w2XZ(02(r,w))

=

(b1+bz)w1wJ

(7.93b)

-(b 1 +b 2 )wZw 3

Similar as in the proof of Theorem 7.5 we find the (non-unique) solution

a1(r,w) {

=

Z ( (1-b 1 )w2 w3

+ (1+b l

2 ) 2.2-1 )W 1 W 3 • (W 1 +WZ )

(7.94)

0z(r.w) - -(bl+bz)wlWZW3)(W;+w:)-1

Or. with respect to the original vectorfields g1 and 8z we have, using (7.92), [(r,w) = f(r,w) + glo1(r,w) + gzo2(r,w), with

235

: ' (r,w) {

02

(7.95)

(r ,W) -

from which it follows that the everywhere defined static state feedback

(7.96)

[ :: 1

leaves the distribution D invariant. As noted before, the feedback (7.96)

is not regular at points where

WI

O.

Wz ~

=

So far we have solved the DDP for the derived system (7,88n,b). We now consider the problem for the complete spacecraft model (7.Bll) with outputs

(7.87). The solution is delivered via the following coup de grace. Instead of considering the first column r'" (r 1 ,rZ ,r3 )T in (7.88a,b) we could also

have used the other two columns

with outputs Y2

=

53

5

=

(5 1 ,5 2 ,5 J )T

respectively YJ

Posing for

tJ .

=

and t -

(t1,tz,tJ)T of RT

these systems

same Disturbance Decoupling Problem gives the same feedback (7.96)

the as a

possible solution, because (7.96)

is only depending upon (w 1

not on r (or sand t)! Therefore,

(7.96) is a decoupling feedback for the

,WZ

,w 3

)T

and

system (7.84) which decouples the last row of RT from the disturbances. The system in decoupled form reads as

o Although the preceding example does not completely match with the general theory developed so far (as the matrix {3(x) is not invertible everywhere), it does if we restrict ourselves to an open dense submanifold of the state space

H.

Such

difficulties

are

common

in

the

treatment

of

nonlinear

control problems and cannot be avoided a priori. In have

the

formulation of the

required

that

a

(local)

(locally

Disturbance Decoupling Problem we

defined)

regula.r

static

state

feedback

exists that solves the problem, i.e. in the decoupling feedback u - a(x) + {3(x)v

(7.3)

we impose the condition that the mxm-matrix {3(x) domain of definition.

This

requirement guarantees

control on the system as before

is nonsingular on its that we keep as much

(but the outputs are isolated from the

236

dis turbances), purposes.

and so we

Of course,

can use

the new controls

v

for

other design

when no further design obj ectives are

imposed,

we

could be content with a solution of the form (7.3) where the matrix {3(x) is not necessarily nonsingular. The most extreme situati.on appears when we have no further access to the system, i.e. {3(x) = 0, and thus one tries to solve

the

(local)

DDP

via

strict

static

state

feedback

u

= a:(x).

At

present no complete solution of the (local) DDP is Icnown when allowing for strict static state feedback. It is clear that in practice one would add further objectives (local)

DDP.

asymptotic v

~

The IIIOSt

stability

logical

of

the

additional

disturbance

requirement would be decoupled

to the that of

system when

setting

in

achieve

O. We will return to this aspect in Chapter 10. So

far,

we used

regular static

scate

feedback

order

to

disturbance decoupling. It is natural - see also the discussion at the end of Chapter 5 different dynamic

to study che (local) Disturbance Decoupling Problem using

control

schemes,

state or output

such

as

feedback.

regular

static

For instance

output

feedback

or

in allowing for regular

static output feedback one would like to find a control law U

=

o(z) + ~(z)v,

(7.98)

where (7.99)

z - Ic(x)

denotes system

another set of measurements A.X' + 811 + Ed,

x

y

subspace 'V, satisfying 1m E

Cx,

c

'If

i. e,

c

z

made =

Kx,

on

For to

a

linear

finding

a

ker C, which is controlled invariant and

invariant;

requirement,

involving controlled invariance appears

system, amounts

A(V n leer K) C lr.

conditioned

for a distribution D,

the

this

in

the

A

somewhat

similar

and conditioned invariance

nonlinear

situation.

We will not

pursue this further here (see the references at the end of this chapter), There

is

a

slight modification of

the

DDP

involving regular

static

state feedback that can be solved completely analogous to the DDP, This is the so-called Modified Disturbance Decoupling Problem.

Problem 7.23

Modified Disturbance Decoupling Problem (MDDP) Consider the

nonlinear dynamics (7.43) with outputs (7.44). Under '''hieh conditions can lV'e

find u ~

a

regular static state feedback

a{x) + P(x)v +

1(x)d

(7.100)

237

with

!l

and fJ as in (7.3) and

-rex)

an mXJ!-matrix, such that in the closed-

loop dynamics the disturbances do not influence the outputs? The difference between the DOP and the HDDP is that in (7.100) we allow

for feeding forward the disturbances d in the control law. Obviously this requires knowledge of the disturbances d. Similar to Theorem 7.14 we give

the local solution of the MDDP, i.e. for each arbitrary point Xo in H the control law (7.100) is defined on a neighborhood of xo'

Theorem 7.24

Consider the nonlinear system (7.43,44).

Suppose that the

distributions n", D~ n G and G are constant dimensional.

Then the local

Modified Disturbance Decoupling Problem is solvable i f and only i f

(7.101) As the proof of this result parallels that of Theorem 7.14 we will leave it for the reader. The notion of controlled invariance for affine nonlinear systems (7.1) can also be extended to general nonlinear systems, x

~

f(x,u). A discussion of this,

locally described as

together with a study of the Disturbance

Decoupling Problem for such systems, will be given in Chapter 13.

Notes and References

In linear system theory the notion of controlled invariant subspaces dates back to the end of the sixties, see [BMJ, use

[\>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

-<-

IR P ,

m

p:-

L (Pi +1),

as

Sex) ~ (hI (x) ,Lfh! (x), .. ,Lfih 1 (x) ,h z (x), .. , .. ,hm (x), .. ,Lfmhm(x)).

(8.37) Then we have Consider the smooth square system

Proposition 8 .11

(8.1,2)

and suppose

(8.25) holds. Then

rank Sex) - p =

L (Pi +1) ,

(8.38)

for all x E N.

'-1 Proof

Assume (8.38) is untrue at some point Xo. So the one-forms

(8.39) are

linearly

dependent.

numbers elk' i E

!E,

This

is

equivalent

k = 0, ... ,Pi' such that

to

the

existence

of

real

252

o. Using the definition of the characteristic numbers, we find m

I

c

i~l iPi

where that

a

(8.41)

(x), ij

is the (i,j)-entry of the matrix A(x). From (8.40) i t follows

clj.j(X)

this expression should vanish at

the point xo'

However

this would

imply that the rows of the macTix A(x) at Xo are not linearly independent,

c

SO we conclude c

rnP m

IPl

= 0, and (8.40) reduces to (8.42)

Then

P-1

1.11

LSiL{{I l<

i~l

t

CnL~lJi(X»)

(8.43)

~o

B.7 (see 8.20a). By the same reasoning as

where we have used Definition before

we

conclude

that

the

lilst

expression

in

(8.43)

equals

m

I

a.

c

i-1 iPi -1

(x).

Now (8.40) and the i.ndependence of the rows of A(x) at Xo

ij

yield that c

-1

=

C

1P1

= mPm 1

O.

A repetition of this argument shows

that all cik's in (8.40) equal zero and thus (8.38) holds true.

0

FLom Proposition B.ll it follows that the decoupled system (8.1,32,2) admits a local "normal form". Namely, define for i Em

and

let

.2

be

(n-p)

supplementary

coordinate

functions

such

that

(i,zl, ... ,zm) _ Sex) forms a local coordinate system. Then with respect to these new coordinates the decoupled system (8.1,32,2) reads as

!!!,

E

i

rn

z

-

1 f(z,z

... Igi(Z,z1, ...

,Zffi)Vi.'

(8.45)

im 1

where the pairs

(Ai ,b i

),

i

E!!!. are in Brunovslty canonical form (6.50).

253

(8.45) holds equally well for smooth

Notice also that the "normal form"

systems satisfying the rank condition (B,25).

Remark 8.12

The

regular static state feedback

(8.32),

suggested in

proof of Theorem 8.9 for achieving input-output decoupling, only solution of the

decoupling problem.

regular static state

For

instance,

see

feedback strong

(8.45),

for

the

the

is not

the

input-output

controls

Vi

in

(8.32) we may introduce additional feedbacks of the form

i E

which keep

more

the system in a decoupled form.

decoupling

surprising

linear,

is

see

feedbacks that

than

(8.32)

(8.45).

Of

those

makes

course

the

the

In general

suggested

there exist even

above.

decoupled

complete

(8,1,32) is still nonlinear, except in case

~,

What

might

input-output

behavior

feedback modified dynamics

L (pi+l)

- n, see the "normal

o

form" (8.45).

We

be

emphasize

that we

require

in Theorem 8.9

that

the

characteristic

numbers of the square system (8,1,2) are finite. If this is not the case, for some i, controls u I

, ...

,urn'

then the output Yi is not influenced at all by the

and this will not be altered by applying a

regular

static state feedback, cf. Proposition 8.B. In the situation where result

on

the

local

decoupling problem; there exis t such that

(8.25)

regular i.e.

is not met for all x E N we obtain a

static

given a

state

feedback

point x o ,

strong

under what

input-output

conditions

does

a regular feedback (8.13) defined in a neighborhood V of Xo

the system is strongly input-output decoupled (see Definition

8.3) ?

Theorem B.13

Consider

characteristic numbers

the PI"'"

square

system

(8.1,2)

with

finite

Pm' The regular static state feedba.ck strong

input-output decoupling problem is locally solvable around a point Xo in N (i.e, a. decoupling feedback is only defined in a neighborhood V of x o ) if and only if

rank A(xo ) Proof

The

(8.46)

u/,

essential

observation

is

that

when

(8.46)

holds

then

m for all X in a neighborhood

rank A(x)

\l

of xo' By replacing N by V the

o

proof follows from Theorem 8.9. For obvious

reasons we will henceforth

refer

to

the matrix A(x),

defined in (B.2 l ;). as the decoupling matrix of a square system (8.1,2). Example 8.14

Consider

the

robot-arm

introduced

configuration

in

Example 1.1. The dynamics are given as (see equation (1.6» '1

X

=

X

l

,

(8.47)

where

Xl

matrices

=

(Xl.X z

l

H(x ) ,

(8 1 .0 2 ), u = (u 1 ,u z ) and the ) = (Ol,(JZ)' ~l = (x J ,x4 ) l l 2 C(x ,X ) and k(x ) are defined as in (1.7a,b,c). As

outputs for the system (8.47) we take the Cartesian coordinates of the endpoint, i.e. see (1.9),

(8.48) A direct computation yields that the characteristic numbers of the square system

(8.47,/18)

equal PI

=

Pz

= 1.

and

the

decoupling matrix of the

system takes the form ilC~SXl + iZcos(x1+x'l) [ -11S1~Xl

izsin(x1+xZ )

,£zcos(X1 +XZ ) -l z sin(x1 +x 2

1

1-1

H(x)

.

(8.49)

)

Note that the rank of A(xl,x Z) is depending on xl. We have (8.50)

(8.51)

So we conclude from Theorem 8.12 that the robot-arm is locally strongly input-output decouplable via a regular static state feedback around each point: satisfying (8.51). Note that the points we have to exclude are given as

X

z = 0 or x 2

~

rr and it is physically clear that at those points we can

not have input-output decoupling.

For instance x 2 = 0 corresponds to a

stretched position of the robot-arm and we may not allow that the endpoint (Yl'Y2) reaches outside the working space of the robot arm:

255

Around each point

(x~ ,x~) satisfying (8,51) we may use the control law

(8.32) proposed in Theorem 8,9, As we have in the present situation that

,

L (Pi +1)

=

l,

equals

the

dimension of

the

state space,

we may

introduce

i-i

the coordinate transformation (8.44) around (x~,x~), i.e, flSi~Xl+£2sin(xl+x2)

£ 1 casx l

+£2

cos (Xl +X2

)

(8.52)

x311COSXl+(XJ+X4)£ZCOS(Xl+XZ)

yielding the modified dynamics Z11

~

Z 12

2"

V,

Z"

Z" v,

Z22

(8.53.1)

with the transformed outputs and

Yz

=

(8.53b)

Z2'

o 8.2

Dynamic State Feedback Input-Output Decoupling

So far we have presented a rather complete description of square analytic systems

that

can

be

(locally)

regular static state feedback.

strongly

input-output

decoupled

via

a

The essential requirement for decoupling

turns out to be the nonsingularity of the decoupling matrix.

For smooth

nonlinear systems the nonsingularity of the decoupling matrix is again a sufficient condition for

the solvability of the input-output decoupling

problem (see Exercise 8.1). So square systems having a decoupling matrix of rank smaller than m on the whole state space are certainly not inputoutput decouplable via a regular static state feedback. look at more general control loops

for

This suggests to

the analytic nonlinear dynamics

(8.1) in order to achieve (locally) input-output decoupling. In particular we

are

going

to

study

if

allowing for dynamic state

we

can

feedback.

achieve

input-output

Recall,

dynamic state feedback is defined as a relation

-y{Z,x) + S{z,x)v O(Z,x) +

~(z,x)v

decoupling

see Definition 5.27,

by

that

256

where z =

,Zq) E IRq,

(ZI'"

and f3: IRll x H .... IR new input.

mXm

1': IRq

x}f .... IRq,

£: 1J?'l x H ....

are smooth mappings, and v ""

a: Ii/'l x}f'" IR

IJ?'1Xrn.

(VI ••••• Vm )

represents a

see that dynamic state feedback may be of use

To

m

in

the

input-output decoupling problem we consider the following example. Example B.15

Example 5.2B)

(see

Consider

on

the

square

analytic

system

(8.55a)

(8.55b) For the system (B. 55) we compute Pl - Pi'. = D. and the decoupling matrix of the system is

A(x) '" which

has

[

1 e

0 XI

1'

0

rank 1

decouplable via

a

(B.56)

everywhere.

So

(B.S5)

is

not

regular static-state feedbaclc:.

locally

input-output

let

add to the

Now

us

system (8.55) the dynamic state feedback

(8.57)

we obtain as dynamics

r'~ x

Again

we

- z x

-

e"

-

vi'.

-

VI'

1Z

+

X3

(8.58)

compute

(8.SB.55b). We find

the

PI -

characteristic

numbers

of

the

modified

system

P;>. - 1 and the decoupling matrix of this system

has the form (8.59)

257

which has

rank 2 at

each

according to Theorem B. 9 state

feedback

of

point

(x,z) E

ml'.

So

the

input-output decouplable

the

modified

via

a

v = o(x,z) + {3{x,z)v.

form

system

is

regular static Actually

a

straightforward calculation shows that the regular feedback

(8.60) decouples the system (8.58). In fact the dynamic state feedback consisting

of the cascade of (8.57) and (8,60) equals the one given in Example 5.28,

o Hotivated by the foregoing we formulate the following problem.

Problem 8.16

(Dynamic

Consider

problem)

state

the

feedback

square

strong

analytic

input-output

system

decoupling

Under

(8.1,2).

which

conditions does there exist a dynamic state feedback (8.54) such that the

feedback

modified

dynamics

(8.1,54)

lvith

outputs

is

(8.2)

strongly

input-output decoupled?

Before going to address Problem 8.16 for nonlinear systems we briefly discuss it for linear systems.

Example 8.17

X-

Consider the square linear system

A, + Eu

(8.61)

{ Y

~

Cx

with x E [Rn,

u,y E

mm

and A,

Band C matrices

of appropriate size.

The

linear dynamic state feedback input-output decoup1ing may be formulated as follows, When does there exist a linear dynamic state feedback

pz + Qx + Rv (8.62)

Kz+Lx+Nv such

that

Because

a

the

closed-loop

linear

system

system is

(8.61,62)

input-output

is

input-output

decoupled

if

and

decoup1ed.

only

if

its

transfer matrix is an invertible diagonal matrix, one may check that the necessary dynamic

and state

sufficient

conditions

feedback

input-output

for

the

solvability

decoupUng

problem

of is

the

linear

that

the

transfer matrix 1/(5) of the system (8.fi1),

(8.63)

258

is nonsingular. i. e.

the determinant of H(s) is a nontrivial function in

s. For the proof of this we refer to the references given at the end of

o

the chapter.

We now return to the nonlinear Problem 8.16. In what follows we will address this problem in a local fashion. That is. we will give conditions which assure the existence of a pre.compensator of the form that

the

overall

decoupled

system

around a

(8.1,54)

point

(x o , 2 0

with

in 1-1 x IRq.

)

(B.2)

is

In order

to

outputs

such

(S. Sil)

input-output obtain

these

conditions the following algorithm is essential. Algorithm B.18

(Dynamic Extension Algorithm) Consider the analytic square

sys tem (8.1,2). Step 1

Compu te

the

characteristic

(8.1,2). Then by Definition 8.7 we

YiP~+l)l

[ y~

1

numbers

ob~ain

1

the

for

PI ' .••• Pm

system

the following vector equation

E1(x) + nl(x)u

=

(B.64)

Prn+ll l

for an analytic lII-vector E\x) and an analytic lIIxm-matrix n (x). In fact, the exact structure of

these matrices

£l(x)

and D1(x) 1

and in particular the matrix D (x)

equation (8.30),

is

as

given

in

coincides with the

decoupling matrix A(x) given in (B.2LI). Let

(8.65) By the analyticity of the system (8.1,2) r 1 (x) is constant on an open and dense submanifold Nl of H. say rl.{x) - r 1 for x in Hl neighborhood contained in 1-1 1 that the first r

1

•

Assume we work on a

Reorder the output functions hl , ... ,hm such

,

rows of the matrix Dl are linearly independent.

that this may require that we shrink the neighborhood in 111 output -h P

1 1

map

as

h

=

(}11: 1+ 1 ' ••• , h) m'

-

1 Pl""

1

11

Deno t e -1

,P r1 ) and P

where

(hl.111).

in

t he

1

(P r1

T

l""

y

as

Note

Write the and

hl ... (hl1 ... ,hrl)

carr-espon d'l.ng way 1 .P m ).

,

(yl.y-l)

an d

We can choose (see Exercise 2.4)

an analytic m-vectar a:\x) and an analytic invertible mxm-matrix /31(X) on a

(possibly smaller)

neighborhood in fl l

•

such

that after applying

the

s tate feedback U

with v

-

1

ol(x)

+

'" (VI""

/31(x)

,vr1 )

[~~ J -1

and v

(8.66) -

(v r1 ""

.vm )

the corresponding partition-

259

iog of the new input v, we arrive at

1

(y') (::.')

(B.67)

(yl) (p +1)

In

(8.67)

is

.\l(x)

an

analytic

(m-r1)xr1-matrix and Irl

the

(m-r 1 )-vector

and

r1xr1-identity matrix.

}/(x)

an

analytic

Define

the

modified

vectorfields fl(X)

=

rex)

+ Igj(x)ul(x),

(8.68a)

jDl

g:<x)

~

,.Igj(x)pll(X), ,

i

E:E.

(B.6Bb)

and consider the modified dynamics

I g:

x = fl(X} +

(x)u i .

(B.69)

i~l

Note that in order to simplify notation we have renamed in (8,69) the new

controls

as

applying

static

Ul "'"

What has been done so

Urn'

state

feedback

input-output channels. local

static

state

to

achieve

In particular when r 1

feedback strong

far

is nothing else

decoupling -

input-output

m we

of

the

have

than

first

repeated

decoupling as

given

r1 the in

Theorem 8.13.

Step 2

In this step we are only concerned about the outputs? = J?(x)

and we want to examine their dependency on the inputs (The

inputs

u

,

y'

=

(u 1 , ... ,u"1)

one

by

one

ill ~

(U r1 + 1 ,· •• ,urn)'

control

the

outputs

see (8.67).) In order to do so, we differentiate these

outputs with respect to (8.69) to see when

ill

appears for the first time.

Let for i - r1+l, ... ,m, p~ be the smallest integer such that the (p~+l)-th time derivative of Yi

explicitly depends on

ill.

Observe that such a time 1 and their time

derivative possibly also depends on the components of u derivatives.

is

Thus

the

characteristic

number

of

the

channel of the system (8.69) with respect to the inputs where

the inputs

(u l

'."

,u"1)

and their time

derivatives

i-th

output

(U"1+ 1 " " ' um),

are viewed as

parameters. We have

(B.70)

260

for

an

analytic

(m-r l )-vector

2

matrix V (x .l?),

VI

whel:e

time-derivatives u:

jJ

,

E2(.'{ ,Ul)

consists

of

i - 1, .. ,r},

J ~

that the highest derivative of u at most of order derivatives

0,

an

analytic

(8.70)

as

u

1

and

their

which occur in (8.70). We note

appearing in

parameters

(m-r 1 )x(m-rl)-

of

components

Vi, and

thus in (8.70), is

To see this we simply consider u

11-1.

in

1

and all

and

we

1

and their time

observe

for

that

the

parametrized system each of the characteristic numbers p~ is smaller than n-l

(cf.

Proposition

components

of

immediately bound of Ilr l ,

u

1

are

follows

the

B .11 with at

most

the n-l

times

2

that

when interpreting u

1

output Yi "" hi (x».

differentiated.

1

maxlPrl - P rl " "

time derivatives of u 1

single

2

uppearing in

i3nd the time derivatives

(u )

(Actually f

it

better

upper

Sop 1 : ~ dim

ff-. ) 1

so

a

is

,Pm

and

(j)

U1 ~

as independent

variables. Let -I

2

-1

(8.71)

rank D (x. U ) ,

rz(x,U)

then r 2 (. , .) is constant on an open and dense submanifold 1-12 say r 2 (x,U

1

)

r 2 for (x,D1) E fJ 2

=

•

of N x fR}Jl,

Let (8.72)

Assume we are wod.:ing on an open neighborhood in Hz step 1 has been pel:formcd on the projection of 1'1 2 output functions

71 1

which is such that

into Ht .

linearly independent on this neighborhood and write 11 -2

h

=

2

P

(h q 2+ 1 ' ••.• lim ) • 2

2.

(Pr1'1.··· ,P qz )'

l(x ,U

1

)

(l

2.

=

(P qZ+1""

and

an

write

we

Accordingly -2

r:/ (x ,[;1)

(m-r1 )-vector

Reorder the

such that the first r 2 rows of the matrix V

2

,Pm)'

2

~

y1

(h r

2

(x,i?)

are

1+ 1 , ••• ,hq2,) ,

(l.y2)

and

Then we can choose an analytic

invertible

analytic

(1Il-r 1 )x(m-rl }-matrix

on the above neighborhood such that after applying the control

law (8.73) we obtain

(8.74)

In (8.73) v (8.74)

2

is a r 2 -control vactor tind ,;2 a (m-qz)-control vector and in

)..2(x,U

1

)

(m-q2)x~-matrix.

is

an

analytic

(I/l-Q2}-vector

and

l(x,il)

an

Define the modified parametrized vectorfields

analytic

261

rn

I

+

fZ(x,i/) "'" flex)

2.

1

-1

(8.75.)

gj (X)Oj (X,V ).

i

=

1, ... ,r 1

(8.75b)

,

I

g:Cx,U) =

(8.7Sc)

j ~rl +1

and consider the dynamics

x .. ['lex,VI)

rn

+ Ig:(x,ii1)U i

(8.76)

.

'-1

1

In (8,76) we have renamed the controls (u ,v2,;2) as U, and we have a new

split~ing for the newly defined control variables u. Namely u _ and

and

their

time

-, U

derivatives

parameters. Alternatively,

-

(uqz+1""'u m), t

occuring

in

V

the

controls

will

be

(U

u1

2

,;:?) ,

,···, Ur 1

considered

as

as we will see later in the proof of Theorem

B.19 we can interpret them as additional state variables and new controls for a suitably defined precompensated system of (8.1). Note that for the modified

system

(8.75)

the

first

qz - r1+r Z

input-output

channels

are

decoupled, see (B.67) and (8.74),

From the foregoing reasoning the general step is easily established,

Step .2+1

Assume we have defined a sequence of integers r

1 ,. , .

,r.2 and let

(8.77)

We have a block partitioning of II as h

=

1 .2 -.2 (ll , ... ,h ,h ) and the parame-

trized dynamics 1.

-.2-1

x~f(x,U

~ 1 -.2-1 )+t..gt(x,U )U i

(8.78)

"1 where the controls output

channels.

one by one influence the firs t Similarly

to

the

dependency of the remaining outputs

-,

yl

second

step

we

examine

now

q 1!

the

_ (h'(x»

the remaining inputs U ~ (U q p+l"" ,urn)' So we differentiate these . .2+1 outputs with respect to (B.78) until 11.2 appears. Let for ~ > Ql' Pi be the

smallest

integer

such

that

the

,.,

(Pt

+l)-th

time

derivative

of Yi

explicitly depends on 111. Analogously to (B.70) we obtain the equation

262 Jl.; 1

(P q li 1 i1)

Yq1 +1 (8.79) 2+1

(Pm

t 1 )

Ym for an analytic

(m-qi)-vector E

-£

1+1

and an analytic (m-qi!)x(m-ql)i matrix Di!+1( x,U i!) where U£ consists of u = (u 1 ,u q1 ) together with suitable time derivatives of the components u 1 " ' " u q i!' which appear when (x,U)

I

I

•••

differentiating the outputs ;;1 with respect to (B. 78). By convention V1 - 1 • Note that, as in step 2. we have that J11:~ dim Vl satisfies

Ui! :J J1£ S

Let

(n-l)q£.

-J!

(B.BO)

(x,U )

is

constant

on

an

open

-£

J!

say r 1+1 (x,U ) = r 1+1' for (x , U

and

)

E l-l1+1'

dense

suhmanifold

Let

1+1

ql+l

=

ql!+r£tl'" i

Er i o

(S.Bl)

·

1

Assume we are working in a neighborhood in Nl+1 I

for which the projection

of it on M is contained in the projection of H£ on H and which is such -£. 1+1 that after a relabeling of the outputs y the flrs t r 1 rows of D a r e £ +1 linearly independent on this neighborhood. Wri te 11 +1 (11'1 £+ l ' •••• h'1. J! ). -hi'+! _ (h I h-£ _ (J/H .-Jl£+l) -11_1 'lin'" 1m)· then -, and partition the vector Y and

accordingly, 1+1

i+1

1'1

1. e. £+1

«Pq1+1 •... ,P'1.2+1).(Pql'+1~1 •... 'Pm l

a£+l(x.V )

and

an

invertible

;;1 = (/+1.;;1+1)

».

Choose

an

(/+1, pl'+l) ..

and

analytic

(m-qi)x(m-q.Q)-rnatrix

(m-q£)-vector

;/+l(x,U 1 )

on

the

forementioned neighborhood such that by applying the control law

-£

a 2+1 (x.Vl!) + "l+l(X,U 1 )

u

[

v

2~ 1

-i+1

v

(8.B2)

1

we arrive at

[ (y

1+1 (p ) i+> )

0

+

-£+1 -£+1 (p (y )

In

8 ( .82)

v

)

£+1.

15

~.£+l(X.Vjl) an

0

[ I""

1

J1

1+1

-.£+1

ri-l1-vector and v

a

[ -

(x,U)

0

:'., 1+>

(m-q£+l)-control

1

(B.83)

. vector and

the matrices appearing in the ri~lt-hand side of (B.83) are analytic in x

263

-f and U .

In fact we have applied a feedback parametrized by -f

inputs

without

u ,

changing

the

u

inputs

l

Define

the

Vi

on the modified

parametrized vectorfields

(8.84a) 1+1

gi

-1

(X,U)

1!

-.£-1

gi(X,U

=

i

)

=

(8.B4b)

1, ... ,Q1"

and consider the dynamics •

X

f

=

-1'

1'-+1

_£

1'+1

m

+ Lgj

(X,U)

,. ,

(X,V)U i

(B.B5)

•

1

where we have again renamed the controls (u ,v

£+1 -1+1

,v

) as u. Notice that

the sequence of qi 's as defined in (8.81) is increasing and bounded by m, i. e.

:s q.i! As

ql = qit1

outputs

:y1'

for

some

:S

:S qitl :S

.e implies r itl

=

(B.B6)

m.

0,

and

therefore

are independent from the remaining inputs

the sequence (8.86) stabilizes. That is,

1

u

the

remaining

we conclude that

there exists a finite integer k

such that :S

This limiting value will be denoted as q , so q

(B.B7)

m.

is defined as

(8.B8) The

Hi

integer n

H2

q

n ... n

is

HI<'

well where

defined Hi

is

on the

an

open

and

projection

dense of

the

obtained in the i-th step, on the manifold M. Therefore q

submanifold manifold

of Hi'

is well defined

on an open and dense submanifold of the state space manifold M. Henceforth q~ will be referred to as the rank of the square analytic system (8.1,2).

o Remark

One can readily verify that for a linear system the above defined

rank coincides with Example 8.17.

the rank of the corresponding transfer matrix,

see

264

We are now able to give a solution of the local dynamic state feedback input-output decoupling problem. Theorem 8,19 tlvO

Consider the square analytic system (8.1,2). The following

condlt:.ions are equivalent:..

(i)

The dynamic state feedback strong input-output decoupling problem is locally solvable on an open and dense submanifold of N.

(1l) The

rank q

-

of

system (8.1,2) satisfies (8.89)

m.

(ii) ~ (i).

Proof

the

Suppose (8.89) holds on an open and dense subrnanifold

of H. That is. the successive application of the Algorithm B.18 yields see (8.67.74,83) - a parametrized input-output decoupled system on an open and dense submanifold of H x the

sequence

input-output

(8.87)

{R/J

n

where

,

stabilizes.

decoupling

can

be

It

p."

equals

remains

achieved by

and k is such that

p.1!

to

be

shown

applying

that

dynamic

a

this state

feedback of the form (B.54). To see this we carefully study the steps in the algorithm. At step 1 we apply the regular static state feedback (8.66) yielding

input-output

decoupling

of

the

first

ql -

input-output

rl

channels. Clearly the feedback (B.66) is of the form (B.54). In the second step we proceed as follows. Let the inputs u

1

VI

denote the highest time derivntive of Z (U • til)

appearing in VI. Note that

denote the new input of

the feedback modified system (8.69). Introduce the precompensator (8.90a)

i ~ 1, ... , ql '

(a.90b)

1 "" 1, ...• ql .

Construct the composition of (8.66) with (B.90a,b) via 1

ul Because

a:(x,V

-

VI

1

=

(8.91a)

1 •... , ql •

m

)

+

r

(B.91b)

is the highest time derivative of the inputs u

it follows from (8. 90a) that all the time derivatives can be

expressed

(u1 •...• um )

as

Z;j'

Therefore

the

ui

j

)

1

""

relation between

and the newly defined inputs (h'l •...• Wm )

(u l

,···.

uq1 )

appearing in the

iP

inputs

in (8.90b,91b)

is

precisely in the form of a dynnmic state feedback (8.54). Notice that the two feedbacks (B 66) together with (8.90,91) indeed form a dynamic state feedback of the form (8.S fl) with as new inputs (w l

•.•• ,Ivm ) .

From (8.67)

265

and

(8.74) we see that we have obtained input-output decoupling between

the first

Il

+r 2

q;.>.

=

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

<

zi j + 1

1

~

j

"'i

i

~

1,

i

"j

1,

~

(8.92a)

,Ql '

(8.92b)

,qi.'

Compose the first J! feedbacks with (8.92a,b) via the linking maps i -J! (x,V)

i+l Qi

=

(8.93a)

1, . . , q}!,

+

(8.93b)

From the definition of

i/'

all

u i (j)appearing

s

time

derivatives

1, ... ,.1',

=

i

=

and the preceding l' feedbacks

j

1, ... ,qn'

~

1, ...

Vi

in ,v~.

can

Observe

be

it follows

that

in

expressed

that the composition of

the previous J! feedbacks with (8.91,92) is again a dynamic state feedback.

Now,

because

that

after

holds and q"

(8.89) applying

input-output behavior channels.

a

sequence

indeed

We emphasize that

qk

of

Ie

consists

for Ie sufficiently large, feedbacks

of

the control

q" '"

laws

of

III

the

above

decoupled

of the

form

only valid if the matrices D£tl(X,ij.£) have constant rank,

we see

type

the

input-output (8.92,93)

As

are

this is the

case on an open and dense submanifold by the analyticity of the system, the

above

(8.54) (Xo

,z; j

procedure

around (0), ' ..

Therefore

we

an

,z~ j (0» have

yields

the

existence

open

and

\~hich

achieves

shown

that

output decoupling problem

is

the

of

strong

dynamic

a

dynamic

,et

dense

of

state

feedback

initial

points

input-output

state

feedback

decoupling.

strong

input-

locally solvable around an open and dense

submanifold of initial states Xo in}l. (i) ;} (ii)

Assume there exists a compensator -Y(Z,X) + 05(z,x)v (8. " )

~

a(z,x) + P(z,x)v

266

with

Z

m\

E

which achieves strong input-output decoupling of the overall

system (1l.l,54.2) around a point (xo,zo} EN x

IRq,

By Definit.ion 8.3 it

follows that the precompensated system has finite characteristic numbers 01 •• ' , • am

de fined in a neighborhood of (xo • Zo)' and by Theorems 8.13 and

8.9 we may equally well assume that the input-output behavior is locally given as i

E

(8.94)

m.

Observe that: (8. 9 il) implies the following local reproducibility property. Given an arbitrary set of analytic functions find

controls

Vi

i E!E,

(t),

such

that

these cont.rols produces as output y(c) -

the

~i(t),

system

E~,

i

one is able to

(8.1,54)

feeded with

on a possible

(yl(C) •...• ym(c)}

small time inter.val, such that

(B.95)

i E ~,

for any fixed set of (8.1,2)

cont.rols

possesses

u (t) • l

oi

the

i E

~.

with a

~

i E m. Therefore the original system

ai '

same

reproducib ili ty

such

that

(8.95)

property,

holds

for

Le.

small

t,

there

exist

when

these

controls are applied. This follows from computing the controls ui(t) from (8.54) with inputs v 1 (t) .... •

vnl (t:).

To prove that. (8.89) is necessary for

input-output decoupling we show that if q"<m then

t.he system

(8.1,2)

does not possess the above reproducibility propert.y, In order to see this we follow t.he decoupling procedure based on the algorithm. which according to the proof (ii)

~

(i) yields as inpuc-out.put behavior

(8.96)

i - I , ... ,q", for suitable chosen 0t'

i ~ 1, ... • q",

~

and the outputs Yi'

do not depend upon the remaining inputs Wi'

i ...

i - q +1, ...• m

q·'+l, ...• m, Therefore the

forementioned reproducibility property is violated.

o

Remark B.2D

The above algorithm can also be applied to smooth systems as 2 long as the constant rank hypothesis of the mat.rices D (x,U'£-1) are met. In the same manner as in Theorem 8.19 it follows in this case that, when the number q" defined in (8.89) equals m on a neighborhood of a point x o ' then the Problem 8.16 is locally solvable about xo' Notice that the rank

of an analytic system is an intrinsic number associated with the system, and which is independent of the particular feedbacks chosen in Algorithm 9.18. Clearly, this follows from the implicit. characterization of q~ given in the second part of the proof of Theorem B.19. as being the number of "reproducible outputs".

267

Example 8.21

A simplified model of a voltage fed induction motor can be

described as

-(0+,8)

w

d

-(JoL~

dt

-1

+

-wa

-1

0

0 o

DL;

-w

-(o:+fJ)

-ooLs

-1 L~

wa

1 -1 L~

-1

-1

L~

fJL~l

0

0

0

0

X, x, x, x,

+

0 -1

-1

0

a L,

1

0

0

1

",

[ ", 1

(8.97)

where

Xl and X z are the components of the stator current and X3 and x 4 are the corresponding flux components all measured with respect to a fixed

reference

frame,

0;

R" 0 -I L : 1,

_

fJ _ Rru -I L : 1 and a _ l-llL: lL~ 1 .

Here

parameters R" and Rr denote the stator and rotor resistances, L"

the

and L t

the stator respectively rotor self-inductance and H is mutual inductance. The mechanical speed w of the motor is assumed to be constant. In order to

have

a

voltage-frequency

control

(U l ,U2 )

voltage input vector

scheme

for

the

induction

motor

(B.98)

[ :: 1 [ : ::: : 1 where

e

the

is expressed as

is the angular position, V the amplitude and wa the voltage supply

frequency, which shows that, more realistically, we should take

«',w

a ) as the input of the induction motor. This can be achieved by adding to (8.96)

the state variable x5 = 0, which yields as dynamics

X, d dt

-(0+,8)

x,

-(0+{3)

w

XJ

-oaL~

X~

0

x,

-cIOL~

0 - 1

cos Xs

0

a -lL~lsin Xs

0

CO, Xs

0

sin Xs

0

0

1

a

L~

+

where (u1.U Z )

=

-wa

0

0 -1

,8L~l

-w

(V,w a ).

- 1

wa

-1 L~

- 1

-1 L~

{3L~

1

0

Xl

0

x,

0

0

0

x,

0

0

0

X~

0

0

0

x,

[ ",", 1'

+

(8.99)

Together with the dynamics (8.99) we consider the

268

stator flux and stator torque as outputs. So we have

(8.100)

Let us investigate if the square analytic system (B.99,100) is statically or dynamically input output decouplable. It is straightforward to verify that the characteristic numbers

pi

p:

ilnd

of (B.99,100) both equal zero,

and the first step of Algorithm 8.18 yields

[

~l 1 [2XJ'~J Yz

X 2 ';::I

-2oaLgXlX3 [

WX I X 3

'r {o+{J)X 1 X4 -

2xJ cos [

(Xl +0

X5

IX::!)

...

2x" sin

sin

+

X5

WX2X~

+

(o+{J)X 2 X 3

-

wa-1L:1x; - wo

"5

(X2

1 0-lL: X4)

-

cos

x5

:][:J

(B.IOl)

which will be abbreviated as

(8.102)

: ][ :: 1 Because the 2xZ matrix Dl(x) that the system (8.99,100)

=

(d: j (x»

has at most rank 1 we conclude

can not be locally decoup1ed by applying a

regular static state feedback, cf. Theorem B.13. On the other hand, around initial points where either d;\(x) or d;I(X) is nonvanishing we find that the rank of the matrix D1(x)

r1

1. To complete step 1 of Algorithm

B.18 we will assume that d:I(x) _ 0, i.e. we are working in a neighborhood of an initial point

Xo

with d~l(XO)

¢

O. Define around

Xo

a reguLar static

state feedback via 1

(x) + (d11(x»

-1_

u1

where (U I ,L/ Z > denotes the new contra!. Applying (8.103) around system yields as input-output behavior

(8.103)

Xo

to the

269

In the second step we consider the modified dynamics (8.99,103) and verify when the output Yz is going to depend upon

uz .

After a straightforward but

tedious computation we find

(B.105) for

a

certain

function

2

e (x,li I ,U I

According

).

to

the

second

step

of

the algorithm we conclude that p~ - 1 and the "second decoupling matrix"

is given as the lxl-matrix

ai'

1 d Z (X,U I )'" -d:1 21 (x)e 1 (X)-a_ d ll (x)

1-1

Xs

1 ai' + dZ1(x)ax d I1 (X) I"'" ul •

which is nonvanishing for an open and dense set of points (x,u I ~

)

E [R5 X IR.

r 1 +r z = 2. Therefore the model of the induction motor is dynamically decouplable around points where both d;l (x) and dZ(x,u) are

Thus r z

1 and qz

(B.106)

5

nonvanishing.

=

In order

to

find a

decoupling control

law we

follow

the

procedure of Theorem B.19. First we note that the highest time derivative

u1 in the expression (B.I05) is order integrator for u1 via of

u1 '

thus ~l

=

1, and we define a first

(B.I07a) and the second input

Uz is modified as (B.107b)

Thus a dynamic feedback which decouples the system (B.99,100)

locally is

given as, see (B.I03) and (B.I07)

(B. lOB)

Notice

that

applying

the control

law

(B.10B)

locally yields

as

input-

output behavior for the induction motor

(8.109) provided that d~l (x) and dZ(X,Zl) are nonvanishing.

o

270

The input-output decaupling problem has received a lot of attention over the

last twa or

three decades.

The here presented solutions have been

given for linear systems in [FW] [Wa],

and in

using static state feedback

where dynamic state feedbacks have been allowed.

Example 8.10 in

fact summarizes the result of [FW), whereas the remark folla';ling Algorithm B .1S

refers

to

[Wa).

problem may be found

Further aspects in

[er].

of

the

A geometric

lienar dynamic theory

on

the

decoupling

input-output

decoupling problem for not necessarily square linear systems is given in tWo J. A survey of results has been given in [I1W2].

[WH),

[HWl].

In [Po J

the decoupling problem for time-varying square linear systems is treated and probably forllls

the first approach to generalizing the problem in a

nonlinear context. The result on seatic feedback input-output dc-coupling as is given in Theorem 8.9 is based on [SRJ,

[Fr).

lSi},

see also [Is],

[IKGM). The result of Proposition 8.11 is borrowed from (Is]. A different approach

to

the

nonlinear decoupling

problem

is

presented

[Gl J.

in

A

nonlinear differential geometric treatment of the problem for smooth not necessarily square dynamic

feedback

systems will be input-output

discussed

decoupling

In

problem

the for

next chapter. square

The

nonlinear

systems has been studied first in [DH). The Algorithm B.1B given here has been taken from INRII. [NR2]. The limiting value q" appearing in Algorithm 8.18

is

called

introduced in nonlinear

the

[FH],

systems

is

rank

[F12]

of

the

system

in

analogy

with

terminology

where a differential algebraic approach for

given,

and

is

in

agreement

with

the

linear

terminology. In fact the rank as introduced here equals the rank as given in [FllJ,

[F121, as has recently been demonstrated in [DGl1]. The Example

8.20 on dynamic decoupling comes from [LU).

[C1] {Cr]

[DCM] [Dt1]

IFW]

lFill

D. Claude, "Decoupling of nonlinear systems", Syst:. Contr. Lett:. 1, pp. 242-248, 1982. H. Cremer, "A precompensator of minimal order for decoupling a linear multivariable system", Int. J. Gontr. 14, pp. 10B9-1103. 1971. M. Di Benedetto, J.W. Grizzle, C.H. 1100g, "Rank invariants of nonlinear systems", SIAH J. Contr. Optimiz. 27, 1989. J. Descusse. C. II. Hoog, "Decoupling wi th dynamic compensation for strong invertible affine nonlinear systems". Int. J. Contr. LI2, pp. 1387-1398, 1985. P. L. Falb, W. A. Wo1ovich, "Decoup1ing in the design and synthesis of lnultivariable control systems", IEEE Trans. Aut. Contr. AC-12, pp. 651-659, 1967. H. Fliess, "A note on the invertibility of nonlinear input-output differential systems", Syst. Contr. Lett. 8, pp. 1117-151, 1986.

271

[Fl2]

[Fr] [GNJ

{HvdSj

[Is] [IKGMJ

[LUJ

(MBE]

M. Fliess, liVers une nouvelle theorie du bouclage dynamique sur la sortie des systemes nonlineaires", in Analysis and Optimization of Systems. (ads. A. Bensoussan, J.L. Lions), Lect. Notes Contr. lnf. Sci. 83, Springer, Berlin, pp. 293-299, 1986. E. Freund, "The structure of decoupled nonlinear systems", Int. J. Gontr. 21, pp. 443-450, 1975. L.C,J.H. Gras, H. Nijmeijer, "Decoupling in nonlinear systems: from linearity to nonlinearity", lEE Proceedings 136 Pt.D., pp. 53-62, 1989. H ,J. C. Huijberts, A.J. van der Schaft, "Input-output decoupling with stability for Hamiltonian systems", preprint 1987, to appear in Hath. Control, Signals, Systems, 1989. A. Isidori, Nonlinear control systems: an introduction, Lecture Notes Contr. Inf. Sci. 72, Springer, Berlin, 1985. A. Isidori, A.J. Krener, C. Gori-Giorgi, S. Monaco, "Nonlinear decoupllng via feedback: a differential geometric approach", IEEE Trans. Aut. Contr. AC-26, pp. 331-345, 1981. A. de LUca, G. U1ivi, "Dynamic decoupling in voltage frequency controlled induction motors", in Analysis and Optimization of Systems (eds. A. Bensoussan, J .L. Lions), Lect. Notes Contr. In£. Sci. Ill, pp. 127-137, 1988. R. Harino, W.H. Boothby, D.L. Elliott, "Geometric properties of 1inearizab1e control systems", Hath. Systems Theory 18, pp.

97-123, 1985.

[MW1[

A.S. Morse, W.H. Wonham, "Decoupling and pole assignment by dynamic compensation", SIAH J. Contr. Optimiz. 8, pp. 317-337, 1970. A.S. Horse, W.H. Wonham, "Status of noninteracting control", IEEE Trans. Aut. Contr. AC-16, pp. 568-580, 1971. H. Nijmeijer, W. Respondek, "Decoupling via dynamic compensation for nonlinear control systems", Proc. 25th. CDC, Athens, pp. 192-197, 1986. H. Nijmeijer, W. Respondek, "Dynamic input-output decoupling of nonlinear control systems", IEEE Trans. Aut. Gontr., AG-33, pp, 1065-1070, 1988 W.A. Porter, "Decoupling of and inverses for time-varying linear systems", IEEE Trans. Aut. Contr. AG-14, pp. 378-380, 1969. S.N. Singh, \.,T,J. Rugh, "Decoupling in a class of nonlinear systems by state variable feedback", J. Dynamic Systems, l-leasurements Gontr., pp. 323-329, 1972. P.l(. Sinha, "State feedback decoupling of nonlinear systems", IEEE Trans. Aut. Contr. AC-22, pp. '+87-489, 1977. S.H. Wang, "Design of precompensator for decoupling problem", Electronics Letters 6, pp. 739-741, 1970. tJ.H. Wonham, Linear multivarillble control: a geometric approach, Springer, Berlin, 1979. W.H. Wonham, A.S. Horse, "Decoupling a pole assignment in linear multivariable systems: a geometric appraoch", SIAH J. Contr. Optimiz. 8, psop. 1-18, 1970.

[HW2] [NR1] [NR2]

[PD]

[SRI lSi] tWa)

tWo] [WH]

Exercises

B,l

Consider the square smooch system (8.1,2)

and let

P l " " , Pm

be the

characteristic numbers defined as in Definition 8,7. (a)

Assume the decoupling matrix A(x) (see (8.24)) is nonsingular at

a point Xo EN.

Prove that the regular static state feedback input-

272

output

decoupling

is

locally

solvable

around Xo

for

the

system

(S.1,2). (b)

Assume the system satisfies (8. l l) and the subset Ho of (B.IO)

coincides with H.

Show that the system is not necessarlly input-

output decoup1ed (in che sense of Definition 8.1). Hint: see Example 4.16. 8.2

([GN}) Consider the square analytic system (B.1,2) about an equilibrium point Xo i

E~,

(a)

of

Yl ~ cj.x,

Let x - Ai + Du,

the veccorfield f.

be the linearization of the system (8.1,2) around xc'

Prove that if Problem 8.6 is locally solvable about Xo for the

system (8.1,2) then the (linear) input-output decoupling problem is solvable for the linearized system. (b)

Show that the converse of (11) is true in case (81.2) and its

linearization have the same characteristic numbers. 8.3

«MBE], [HvdS]) Consider an input-output decoupled system of the form

This

(8.1.32.2).

(8. lIS).

Assume

involutive.

decoupled the

Prove

that Z

fO~,zl .... ,zm),

Z

system

admits

distribution

Le.

G(x) -

there

i

as

a

local

exist

defined

in

VI

I

••

form" is

local

coordinates

(8.44)

such

z

the time-derivative of

depending upon the inputs

"normal

spanlg1 (x) •... ,grn (x) I

that

is not implicitly

,vm • Conversely show that when in the

,

"normal form" (8.45) the equations for z are independent of the new inputs v 1 • " " vrn then the distribution G is involutive. 8.4

Consider

the

square

analytic

system

(8.1,2)

with

characteristic

nUJllbers Pl •... ' Pm' Assume that rank A(xO ) ~ m. Let D" be the largest locally controlled invariant distribution in ker dh (see Chapter 7). "

Show that D (x) ...

m

()

Pi

j

ker dL,h l (x) on a neighborhood of xo'

()

ivl J"'O

8.5

Consider

x ...

~:

as

in

Chapter 7

f(x) +

Lgl (x)u

an

analytic

with

system

i

+

luI

L el (x)d l

'

Yi - hi (x). -

the strong input-output decoupling problem for

•••

~o

d..2 - O.

-

Xo

l:

Q

Suppose

is solvable and the

local disturbance decoupling problem for :E is solvable. locally about every point

Let

1 E m.

1 m!

denote the same system with disturbances d 1

U

disturbances

1

m

Prove that

there exists a regular state feedback

et(x) + (J(x)v such that the closed loop system

Ls

input-output

decoupled as well as disturbance decoupled. 8.6

Consider the square smooth system (8.1,2) about an equilibrium point Xo

of f. AssUJIle the decDupling matrix A(x) is nonsingular at xo' Show

that the system is feedback linearizable about Xo

(see Definition

273

m

6.1) i f

I

(p,+1) ~ n.

i"1

8.7

Consider the square analytic system (8.1,2). 2i -

Vi' Ul

Define the compensator

Zi' i E ~. Prove that the regular static state feedback

=

input-output decoupling problem for the system (8.1,2) is solvable if and only if it is solvable for the precompensaterl system. 8,8

Consider

the

square

analytic

system

(8.1,2)

and

compensator of the form (8,54) and assume P(z,x)

=

a

an

analytic

for all (z,x).

Assume that for each X E H the compensator has rank m. Prove that the

rank of

the

system

(8.1,2)

equals

the

rank

of

the

precompensated

system (8.1,54,2). B,9

Consider the analytic system (8.1,2)

and assume that the number of

inputs m is larger than the number of outputs p. Define the pxm-decoupling matrix A(x) as in (8.24).

(a) there u

=

exists

locally

about

a

KO

a(x) + P(x) such that the inputs

outputs Yl""

regular

static

Yp + l " ' "

Yrn

Prove that

state

feedback

do not influence the

,Y p while the closed loop system with inputs VI""

is strongly input-output decoupled, if and only if rank A(x o ) (b)

Formulate

in

a

similar

way

the

dynamic

strong

=

,vp

m.

input-output

decoupling problem and show that it is locally solvable on an open and dense subset of H if the ranlc of the system (8.1,2) equals p.

a .10

(see

Exercise

7.7)

x3 "" x 4 u l + x 2

,

there

a

exists

achieves

strong

x4

Consider = u2'

dynamic

Xj

-

on Xl

the

[Rj Ul

+ d

compensator

input-output

decoupling, provided that xju

J

sys tern YI = Xl'

for

decoupling ~

O.

Xl

this as

Xz

= Xz ul ,

Y2 "" x z .

as

Xj ,

Show that

system which well

=

locally

disturbance

9 The Input-Output Decoupling Problem: Geometric Considerations In

the

previous

chapter

we

have

given

an

analytic

approach

to

the

input-output decoupling problem for analytic systems. This resulted in a rather complete description of square analytic systems (1. e. systems with an equal number of scalar controls and outputs)

that are decouplable

either by static or dynamic state feedback. The study presented in Chapter B is based on purely analytic considerations such as the determination of the decoupling matrix and computation of its rank. On the other hand, as we will see, for the static state feedback input-output decoupling problem a

much

more

distributions,

geometric

approach ,

is possible.

The key

involving is

controlled

invariant

that for analytic systems

noninteracting condition for an input u i

the

not to affect an output Yj' see

(8.4), is equivalent to the geometric condition derived in Chapter 4 (see Proposition

il.

20). Furthermore, also for smooth systems such a geometric

characterization (Proposition 4.23) will prove to be a good starting point in the study of the input-output decoupling problem. It turns out that the differential geometric approach based on Proposition 4.23 will also enable us

to

treat

the

block

input-output

decoupUng

problem

(for

smooth

systems) . 9.1 The Bloclt Input-Output Decoupling Problem for Smooth Nonlinear Systems Consider a smooth affine nonlinear system

L gt (x)u i

x - f{x) +

(9.1)

i OIl

together with the partitioned output blocks

(9.2)

where x gl""

,gm

(Xl' ... ,Xn )

are local coordinates for a smooth manifold H, f,

are smooth vectorfields on Nand 11 1 : H

smoo th mappings.

\~e

assume throughout that the

equals the number of scalar controls, thus

-!o

IRPi,

number

PI > 0, i E £,. are

of

output

blocks

275

p - m .

(9.3)

Clearly when all Pl' 5 are identically one,

then we are dealing with the

square systems of Chapter 8, but when one of the Pi's is larger than one

the

analysis

given

50

far

is

no

longer

adequate.

The

differential

geometric formulation of the static state feedback input-output decoupling

problem also differs from the one given in the previous chapter in that we allow for smooth (not necessarily analytic)

systems.

Throughout we will

make the following assumption. Assumption 9.1 The system (9.1)

satisfies tile serong accessibility rank

condition at each point Xo E H. Therefore

(see Definition 3.19 and Theorem 3.21)

corresponding

to

vectorfields gl"" all X E Co

-

the

smallest

subalgebra

of

the distribution Co which

V(m

,gm and which is invariant under f

-

contains

the

so [f,X} E Co for

has dimension n at each point Xo in I'l.

In accordance with Definition 8.1 we will say that the system (9.1-3) is

input-output decoupled if after a possible relabeling of the inputs

(u 1

, ...

,urn), the i-th control u 1 does not influence the outputs Yj' j

Denoting hl ..

,h iPt )' i E:!,!,

(llil""

this yields

'" i.

the necessary condition

that (see Chapter 4, Proposition 4.14)

(9.4)

J!EEj,XEI'l.

We note that at a first glance no condition on the interaction of the i-th input

on

the

satisfying

i-th

output

Assumption

block

9.1

which

has

been

fulfils

given.

(9.4)

However,

a

automacically

system has

the

property that the i-th control affects the i-th output block. To see this we introduce the following objects. In the algebra ~,

is

eo

we define

eOl

'

i

E

as the smallest subalgebra which contains the vectorfield gt and which such

that

{f,X] E

eOi

'

[gj ,Xl E

eOl

j

'

E:!,!,

for

all

X

E

eOi

'

Furthermore, we let Co i be the corresponding distribution COi(X)

-

span (X{x)IX vectorfield in

By definition the distributions Co ex) ~ Co

1

Co

ex) + ... + COm (X)

Now consider the distributions

and COi X

eOi ) ' '

i Em.

1 E~,

E N.

(9.5)

are related via

(9.6)

276

Pi

~

leer dh!

(9.7)

i E m.

n ker dhd

i., 1 From the noninteraction condition (9.4) it follows that COj c ker dh i

.V

i.j E !E, j

...

i, xEN

(9.8)

and so when Assumption 9.1 holds we conclude that for 1 E m 11i ~ (Trt)

1m hi" where hi~ : TH

-7 TIRPi

=

...

h1n (Co ) - hi n (Co 1 +

+

=

Co 1 )

h!"(C Oi

(9.9)

)

is the tangent mapping of hi' This immediately shows

that the i-th control does influence the i-th output block. Namely if we introduce in analogy with the set RV(xo.T) given in (3.21) the subset of reachable

points

in

the

i-th

output

(= ~Pi),

space

i

E

~.

for

a

neighborhood tf of Xo as IYi

E

~Pi IThere

exists [O.T]

of (9.1) for x(O)

~

and hi(x(T»

piecewise

a

u ~ (u1 ••.• ,um)T:

constant

satisfies x(t) E V. 0

Xo

Yl.J •

-

input

[Rm such that the evolution ~

t

~

T.

(9.10)

then i t follows from (9.9) that hi (RV(x o .1) has nonempty interior in 1m hi for any neighborhood V of Xo and any T > O. This is what we call output controllability. Horeover by manipulating only the i-th input function ui

we can steer in an open set of lrn hi' no matter how the other inputs u j j

•

... i, are chosen (see again (9.9». For a square system (a system with m

scalar outputs) Assumption strongly

9.1

the

noninteraction

almost,

input-output

condition

but not entirely, decoupled

in

the

(9.4)

implies sense

of

together that

the

Definition

with system B.2,

the is as

illiustrated by the next example Example 9.2

(=

Example 8.4)

Consider the square system

Xl Xz

.:\:3

(9.11)

1

Yl Y::

Clearly the system (9.11) satisfies the noninteracting condition (9.4). Moreover one straightforwardly checks that accessibility assumption at each poi.nt di.'stributions

COl

and CO2 we obtain

(9.ll)

(Xl ,X z ,Xl)

satisfies the strong E

J

IP. •

Computing the

277

a x,

(9.12)

span(-a- I

C O2 -

and so at each point in ~3 we have (9.13)

and we may conclude that the i-th control instantaneously affects the i-th

output. However, see Example 8.3, the system (9.11) does not globally meet

o

the requirements of Definition B. 2.

The following example illustrates the output controllability condition for block outputs.

Example 9.3 (see Example 9.2) Consider the strongly accessible system

As

,

x, x,

- x, - u,

xJ

u,

in

the previous

system

- x,

y,

(X

y,

(9.14)

1

example

is

,X )T 2

(9.14)

the noninteracting condition

immediate.

Using

(9.12)

we

see

(9.4)

that

the

for

the

output

controllability conditions are met. In particular one can u,e the control to

u,

, y,

-

the

steer

0

output

y,

-

(x~,x~)

into

an

open

(x: ,x~) in IRz.

,et

of

points 0

In what follows we will discuss the possibility of (locally) achieving the noninteraction condition (9.4)

for a system (9.1-3)

which satisfies

Assumption 9.1, by applying regular static state feedback. As in Chapter 8 this means that we search for a regular feedback u

with

et(x) + fJ(x)v

~

N

-Jo

!R

all x and v

=

0:

m

,

(9.15)

fJ: N -,

[RmXm

(v 1 , . . . , v m)

smooth mappings and with fJ(x) nonsingular for

representing the new controls. such that in the

feedback modified dynamics

x

~

f(x) +

with

f(x)

.. ,

I gj. (x)v i I

- f(x) + j

gi(X)

- I

,. ,

gj

~

(x)fJ j

(9.16)

gj (X)Oj (x)

(9.17a)

1

i

(x).

i E

!!!.

(9.17b)

278

we have that the i-th new input v t does not affect the outputs

Yj.

j

~

i.

Though in this formulation of the input-output decoupling problem only the noninteracting condi tions

are

required.

we

emphasize

the

role

of the

standing Assumption 9.1. First we derive

Lemma

9.4

The

(9.1)

system

condition at each point x ll

satisfies

the

strong

accessibility

rank

in H if and only if the feedback modified

syscem (9.16.l7a,b) satisfies tlle strong accessibility rank condition at each point xa in If.

Proof The result follows in n direct manner from the representation of

o

vectorfields in Co as is given in Proposition 3.14.

The above result implies that whenever we can nchieve noninteraction by applying

regular

a

nonlinear system,

static

state

feedback

to

a

strongly

accessible

then the resulting feedback modified system possesses

the output controllability property, and thus the neW input v l may be used to steer the output Yl in sOll1e neighborhood with nonempty interior. As a motivation to the local solution of the nonlinear input-output decoupling systems.

problem

we

first

recall

the

following

result

for

linear

(The proof may be found in the literature cited at the end of

this chapter.)

Theorem 9.5 Consider a controllable linear system with m inputs x - Ax + Du

(9.1S) i

E

m.

There exists a regular linear state feedback

u = Fx + Gv

det G

~

0 ,

(9.19)

'''hieh achieves noninteraccing betl-leen 1m B - 1m B n V;

,,,here

V;

is

Vi

ilnd Yj' j

-,.I

i, if and only if

+ ... + 1m B n V~ •

the maximal

conr:.ained in the subspace

conr:.rolled

n

ker Cj

,

invariant

(9.20)

subspace

of

the

system

i E m.

J""i

The essence of condition

(9.20)

may be explained as

follows.

In

the

decoupling problem one senrches for a feedback (9.19) which renders the noninteracting between

Vi

and Yj' j

-,.I

i. In case such a feedback exists we

279

observe that the subs paces (A+BF) (BG)

1Ii = Im( eBG) i

(A+BF)n-l(BG)l) ' i E~,

i

(9.21)

satisfy

(9.22)

Clearly

each

of

the

subspaces

invariant (actually the 1Ii

1Ii

defined

in

(9.21)

are

controlled

are controllability suhspaces for the system

'5

(9.18», so they satisfy

!E.

(9.23)

and also (9.22). The subspaces 11 j

which satisfy (9.22) and (9.23) are also

A1Ii

+

C .111

1m B

i.e.

compatible,

i

E

there exists

a feedback matrix F which makes

the .1I 1 's

simultaneously invariant: i

(9.24)

E m.

Furthermore we have that Im(BG)i - 1m B n V:, i E!E. which implies (9.20). Conversely condition (9.20) implies the existence of suhspaces 111 , i E

~,

satisfying (9.22), (9.23) and (9.24). So there exists a compatible family of subspaces .11 1

,:TIm which determine the feedback described in (9.19).

""

In fact the matrix F is chosen as in (9.24) and the matrix G is computed via Im(BG) 1

=

n

1m B

'If;,

i E

m.

Next we return to the nonlinear decoup1ing problem.

Define for i E m

the distribution

D~~ as

=

the

D"(f , g', j""1 n ker dh,), ~

maximal

contained in

/;1i

locally Iter dh j

•

(9.25)

controlled

invariant

distribution

of

(9.1)

We make the following assumption.

Assumption 9.6 (i) The dist:ribuCions G:= span(gl, ... ,grnJ, D; and D; n G,

i E (ii)

~,

have const:ant: dimension. The

distributions

Vi

involutive

closure of

(iii) The

out:puc

I

D;, i E~,

j""i

const:ant dimension. maps

hi: H - ) lR?t, i E

:!!,

are

have

(9.26)

non-trivial,

i.e.

Iter dlt i (x) ,.., 0 for all x E N.

Theorem 9.7 Consider t:lle system

9.6.

Then

(9.1-3)

the st:at:ic state feedback

locally solvable if and only if

sat:isfying AssumpCions 9.1

and

input:-output decoupling problem is

280

D; () C +

G ...

... +

D; () C

(9.27)

Proof ("only if" part) Assume u - a:(x) + P(x)v achieves locally around a point Xo

the input-output decoupling property. Clearly,

it follows that

for any i E m the distribution

:= involutive closure (span{ad~

Ci

!

gi' ad~ j 8-J,

Ie

I

g

2:;

0, j E~)

(9.28)

is invariant under f and gl'··· ,gm' while (9.29)

As

gJ,

E Ct

,

it follows that (9.30)

G

and because the distributions C t are controlled invariant we have C1 CD;.

o

Together with (9.30) this yields the desired conclusion (9.27). Before proving the converse we need a few intermediate results.

Lemma 9.B Consider chs system (9.1-3) sllcisfylng Assumptions 9.1 and 9.6.

Suppose (9.27) holds true. Then che codiscributions ann V1 linearly

independent:

at

each

x E H,

point

, •••

,ann Vm are ann Vi (x) n

i.e.

(ann Vdx) + ... + ann Vi-dx) + ann V1 + 1 (X) + ... + ann Vm(x)} ". O. m

Proof

Let

w1 E

ann VI

and

suppose

w 1 (x)

-L -

fPj

(x)w j (x)

for

smooth

We

have

D;

because

j m2

functions w1 E

ann V 1

Wj

E

ann Vj

1..)1

E

ann

D;

W1 E

Le t

and

fPz ' ••• ,rpm

C C

ann(D~ + ... +

D;) ,

Wj

E

ann

Vj ,

j

2, ... ,m.

III

as well as

WI

=

L

j

~2

D;)

ann{D~ + ... + D;_l + D;+l +... +

I

j

IPj Wj E ann po!

1. This implies that

n ann(D; + ... + D;) • Le.

ann(D; + D; + ... + D; )

D : = D; + D; + ... + D;.

then

(9.31) D:J G and

invariance of the D;'s implies chat [t,D]

C

thus

the

D and [gj ,D]

local C

controlled

D, j E~,

SO

D

is an invariant distribution for the system (9.1) which contains G, The strong

accessibtlity

ann D

O. From equation (9.31) we then obtain

=

assumption

implies

that

dim D WI

~

dim N and

thus

0 implying that the

codistributions ann V 1 and ann V2. + ... + ann Vm are linearly independent. The same argument shows that ann Vi and ann Vl + . . + ann

+ ... + ann Vm are linearly independent for any i E

nl.

1'1-1

+ ann V1 + 1

o

281

Corollary 9,9 Under the conditions of Lemma 9.B there exists ilround each xQ E N

paine i E !,!!,

coordinaces

x

=

,x n )

(Xl'"

a

is

l"here

such

i

ann Vi

tlJat

span! dx 1,

=

peJrtitioning

block

the

of

x-coordinates.

Proof By induction. The codistribution ann VI is constant dimensional and so there exist coordinates 1 Denoting the remaining span{dx ).

involutive ann VI -

that ann V z

observe

aun VI nann V z so

2

coordinates x

such

Xn )

x

of

spanldr/)

as

-, x

that

=

the

d:?

be

may

(dim x

2

=

as

chosen

new

a

set

local

partial

of

dim(ann ttl»' Repeating the above argument yields

o

the desired set of local coordinates.

The

above

corollary

In

we

dimCann V z )

for

ann Vz ), As involutivity of d 2 equals dim(ann Vz ) and 0 it follows that the rank of L

use

we

functions

the

(Xl"'"

=

components

locally can be written as

(here

functions

x

the

describes

sequel

we

will

the

structure

describe

the

of

the

.

distributions

distributions

Vi'

i E

!E.

under the assumption that (9.27) holds.

Lemma 9.10 Consider the system (9.1-3) satisfying Assumptions 9.1 and 9.6.

Then the condition (9.27) implies that

n

dim G

D:

1.

=

m

i E

(9.32)

Proof We first show that for each i E m

(9.33)

Indeed, of

suppose

Then G

(9.33) does not hold.

j

,I,.., D;.

involutive

V:

the definition of the distributions

I

of j

herewith

"'i

v;

C ker

dh i

contradicting

I

II

by Assumption 9.1. However by

it follmoJS

D; C

I

that

involutive

implying

,

the

that

assumptions.

the Now,

hi

map let

is

a

trivial

map,

V~)

and

1'1= dim(G n

k-l

G n D;) -

and in exactly

that

involutive closure

j ",

k

/c,

v; c

sum of locally controlled invariant distributions)

and contains G and therefore has dimension

1'k= dime

"'i

Note that this last distribution is locally controlled invariant

(being the

closure

I

c

1'k ~ 1

for

dime

I

G n V;), Ii: = 2, ... ,m.

Obviously 1'k <=- 0 for all

the same way as we established

k E Ill.

On

the

other

hand

it

(9.33)

it even follows

follows

from

(9.27)

2B2 m

m

L 'Yk=

that

dim(

kal

I

m. So the "Yk' s form a set of m integers C!: 1

G n D:)

i-1

which add up to m. Therefore 1k = I, k

E~,

o

establishing (9.32).

Corollary 9.11 Under the same conditions as in Lemma 9.10 it follows that

o.

G n D"

D~

[.Jhere

is

(9.34)

r:::he

n

contained in

maximal

l~er

dh i

locally

controlled

invarianr.

disr:ribut:ion

.

iD 1

Proof This follows directly from (9.32) and the fact that D~ C D~ for all

o

i E m.

Lemmo 9.12 Consider the syscenl (9.1-3) and let Dl and Dz be two involucive distributions satisfying

[f .D11

C Di

[gj ,D l

+ G

C Dl + G

]

i - 1.2

(9.35a)

i - 1.2. j Em.

(9.35b)

Assume that G - G

n Dl + G n Dz .

then their

Proof Le t

(9.36)

intersection

XED 1 n D2 .

D1

n Dz

Then

also satisfies (9.35a.b).

(9.350.)

implies

(f,X]

. .

Y1 + b l

~

Yz + bz •

b l E G and so, - Yz - b z Y2 = bi + bz for some vectorfie1ds b i" E G n Dl and b z G n Dz . Clearly 1'1 b I" - Yz + bz E Dl n Dz and so we , A similar find [f,X] - Y1 + b l 1'1 b l + b l + b l E D1 n Dz + G.

where

using

1'1

E

Dl

(9.36),

and b l .b 2 E G. This implies Y1 we

obtain

that

1'1

-

.

reasoning applies to the vectorfields gj. j

o

m.

Before giving a canonical characterization of the distributions

D:

we

need one further result. Let ker(ann VI + ... + ann Vm )

(9.37)

Lemma 9.13 Consider the system (9.1-3) satisfying the Assumptions 9.1 and

9.6. Then the condition (9,27) implies thar: m

(1)

Do -

LD;

lel

n Vi

•

(9.38)

283

(li) Do is locally controlled invariant.

We

Proof

will

(9.39)

(9.38)

establish

the

using

coordinates

local

(9.40) Obviously (9.37) and (9.40) imply that spanl_a_1 axo Let X

X E Do.

(9.41)

Since

V;

dxj(X j

by definition

},

so

dxJ(X)_O,

. I D, n

+ ... +

n;

~ TN,

of

dxJ(X j )=0

jEE!' m

i E

Conversely,

i'i .

!E.

such

that

Lemma

9,8,

we

can

write

i E:!!. Then dxJ(X) - dxj(X 1 ) + ... + dxJ(Xm ) the distribution Vi' However X E Do implies

m

,.,

see

Xi En:,

Xl + . . + Xm with

=

X

~

for X

X

,.I ,D,

E

, + ... + X

.

Therefore

there

n Vi'

X,

exist

J

Clearly

m •

XjEVj'

dx (X 1 )

-0

for

i.e.

XE

n;

n V, '

E

i ,j E

!E,

yielding X E DoIn

order

to

establish

(ii)

we

note

from

(1)

that

the

distribution

m

,.,In:

n

Vi

is

distributions

involutive and constant dimensional.

n:

and Vi' i E El.

satisfy

Lemma

9.12

Moreover each

and

of

therefore

the

each

D: n Vi' i E~, is locally controlled invariant. Now Do, being involutive,

is locally controlled invariant as

it is

the sum of locally controlled

o

invariant distributions.

We

are now prepared for giving a

distributions

D:,

canonical characterization of

the

provided (9.27) holds true.

Lemma 9.14 Consider the system (9.1-3) satisfyIng Assumptions 9.1 and 9,6,

Then the condition (9,27) implies that for I

v: -

lcer(ann Vt +" ,+ ann Vi

-

Em

1 + ann Vi + 1 + ... + ann Vm )

(9.42)

We establish (9.42) again using the local coordinates m 1 x - (XO,x , ... x ) of Corollary 9.9. In these coordinates (9,42) comes down Proof

to showing that

,pan{~ _a_I 0'

ax ax As

n:

,

i E m

C Vj ,j ... i, it immediately follows that

(9.43)

284

i E m.

show

the

converse

span{~)

inclusion

we

first

note

that

the

distribution

is locally controlled invariant, cf. Lemma 9.13.

axo

from

it immediately follows

(9.38)

that Do

controlled invariant distribution D" in

n

~loreover.

is contained in the maximal leer dh j

D:,

As D~ C

•

i E!E. We

obtoin that 1 E I1J

let

Next

X

and write

E

iJ O'! (Xl) + ... +

X ~ Xl + ... + Xm

J

Xj E D; ,

with

E m.

.~

Then dxj(X)

dXj

0,

x-

=

dX j (Xj ). Now for j .,. i we have that

yielding Xj E Do, j ,.. i.

thereby

Do + D; C

Xl + ... + .Ym

(Xm )

D:.

Therefore

we

This completes the converse

obtain

that

inclusion of

o

(9.44). We are now able to prove the remaining part of Theorem 9.7.

Proof of Theorem 9.7 (" if" part) In order to produce locally a feedback which achieves apply

a

locally

defined

feedbaclc

which

renders

follows.

the

We

first

distribution

Do

see Lemma 9.13. Using the local coordinates of Corollary 9.9

invariant. and

input-output decoupling we proceed as

x

writing

(Xl •••.

,xm)

and

x

(xo ,x)

we

obtain

the

following

decomposition •0

x

f'l

X

(9.46a)

m

un

Igz1(x)U i

+

i=l }'i

where u

~

-

1 E m

hi (xl)

(u 1

, ...•

urn) denote the. new controls.

the dynamics modulo Da

x = f2 (x)

+

I

In the sequel we deal with

i.e.

1 g2i (X)U i

(9.47)

i~l

and we will construct a regular feedback u input-output decoupUng.

a(x)

=

+

{J(x)v which achieves

An easy inspection of (9.47,46b)

the results of the preceeding lemmas,

yields,

using

that the maximal locally controlled

n

invariant distribution of (9.47) contained in j

"1

ker dh j

is given as

285

span{~)

, i

(9.48)

E m

aXl

and moreover dim(G n V;)

1, i E ~, where G(x)

=

=

span(gz 1 (x), ... ,gzm (X») .

Notice that the system modulo Do still satisfies the decoupling condition (9.27).

Applying a preliminary feedback involving only a

input vectorfields gZi(X), i &2i (x) E

i5:,

i E

Using

III.

change of the

E~,

we may also assume in the following that

the

local

invariance

controlled

of

the

distributions

i.J;

,V:} (Ez.v;]

C

V;

+

G

jenJ,iEm,

(9.49a)

c

i.J:

+

G

i E m

(9.49b)

[gZj

we find that

From (9.49a) we obtain for j E m

[gZJ'V;

+ ... +

V;_l

n; Using Theorem

7.5,

iJ;-t1

+

+ ... + D~) c

+ ... +

n;_l

+

can

find

a

we

modified input vectorfields

gZJ

5;+1

+ ... +

V;

+

(9.50)

span{gZJ)

feedback matrix

(J(x)

such

that

the

satisfy for j E m

(9.51) which in our local coordinates means that

o o j

A

completely

feedback a(x)

analogous

reasoning,

(9.52)

E m

again

using

Theorem

such that the modified drift vectorfield

7.5,

12 (x)

yields takes

a the

form ~

,

£21 (x )

(9.53)

Combining

(9.52),

system of the form

(9.53)

and

(9.46b)

we

have

constructed

a

decoupled

286

(9.54)

1 Finally, since (9.54) is still strongly accessible (cf. Assumption 9.1). (Xi) + gZi (Xi)v i

each of the subsystems ;/ and

i E ~.

the

thus

output

is strongly accessible.

controllability

of

is

immediate.

D

Remark 9.15 We directly obtain for the input-output decoupled system of Theorem 9.7 a local normal form described by. see (9.46,54),

xO

~ f1

0

m

1

m

(x ,x , ... ,x ) +

Lgl! (xO ,xi, . .. ,xm)Vi i=l

. 1 X

-

1

-If,,(X)

1 'm

X

-

1

+ 8z,(X )v,

m fzm(x ) + gZm(xm}Vm (9.55)

Yl {

Ym

Notice that this extends the normal form of square decouplable systems of Chapter 8 (see (8.45». Theorem g. 7 gives a geometric solution to the static state feedback input-output decoupling problem in case the number of output blocks equals the number of inputs (see (9.]}). For a square analytic system with scalar outputs

Yi'

i

E~,

and satisfying Assumptions 9.1 and g. 7, the geometric

decoupling condition (9.27) follows from the analytic decoupling condition (8.2S), but, on the other hand (9.27) only implies the rank condition on an open and dense subset of N (see Exercise 9.2). We finally remark that also the more general block-input block-output decoupling problem can be sc:udied geometrically.

see for instance Exercise 9.3 or the literature

cited at the end of this chapter. 9.2 The Formal Stucture at Infinity and Input-Output Decoupling In 1 in ear system theory there is often a direct way to pass over from a state space problem formulation to a problem description in the frequency

287

domain. This is due to the fact that a minimal linear state space system

xeD)

A.x ... Bu,

0 ,

=

(9.56)

- ex can be

represented by

equivalently

its

transfer

matrix C(sI

similar frequency domain description for a nonlinear system

f(x} +

X -

xeD) =

Xo

,

(9.57) {

does

y - hex)

not

exist

structural given by

(9.57)

=

the

also

Chapter

which

in

transfer matrix,

in geometric

structure G(s)

(see

information,

terms.

infinity.

at

4).

the

Nevertheless,

linear

can be

case

defined

We will illustrate

(A

linear

system

is

for

the

this

(9.56)

some

most

nonlinear

with

with

the

Consider

the

=

G(ljs) has'£ zeros

nonlinear

system

of

(9.57)

locally controlled invariant distribution

orders and

n"

11l' ••••

recall

matrix

°1 ""

n.e

that

of (9.57)

system

so called

transfer

G(sI-A) -In is said to have £ zeros at infinity of orders

if the matrix G(s)

important

conveniently

at the

,nl'

0.) maximal

in ker dh may be

computed via (see Chapter 7. (7.53»

D'

TH

{ nJ.l+l '" ker dh n IX E V(H)j[f.X] E DJ.l + G. {gl.Xj E oil + G. i E ~l (9.58)

and

the

following

constant

dimensions

assumptions hold (see Assumption 7.18).

Assumption

9.16

For

all

J.l ~ 0

t:he

dist:ribut:ions

nJ.l

and

nJ.l n G

have

constant: dimension on H.

Provided Assumption 9.16 holds we have that (see Proposition 7.16)

D~(f.g; ker dh)

=

n D .

(9.59)

From the n*-algorithm (9.58) the following structural information will be extracted. Definition 9.17 Consider t:lle nonlinear sysr:em (9.57) for

I ..hich

Assumption

9.16 holds. Ler:

(9.60)

288

tvhere Dk - D~ (f.g,· ker dh) and let: nj.( - number of

i'

(9.61)

,{hich are grea.ter than or equa.1 to IJ.

IS

Then t:he sysc::elll (9.57) is said to have pi formal orders (n Jl ).

zeros at infinity of

We observe that for a single-input single-output nonlinear system the formal structure at infinity can be expressed as follows. Let p < characteristic number

of

the

single-input

Then by definition the function Ll!L~h(x)

x

EN.

Assume it is nonzero everywhere;

single-output

system

~

be the (9.57),

is nonvanishing in some point then this

single-input single-

output system has one formal zero at infinity of order p+l. In order to see

this we

note 1

dLrh • ...• dLi- h)

that

in

this

case Dj.(

is

given

as

Di-J

lcer(span\dh ,

and for k ~ 0, ... , p-l the function Lg L~h is identically

zero, cf. Theorem 7.21. Analogously it follows that a square decouplable system with characteristic indices PI"'" Pm has

zeros at infinity of

111

orders PI +1 •...• Pm+l.

We stress that the above geometric definition is I

for a

indeed equivalent to the more usual definition of

linear system.

structure at infinity of its transfer matrix. In other words, for a linear system the formal structure at infinity of Definition 9.17 coincides with its structure at infinity (see the references). To show the importance of the formal zeros at infinity for a nonlinear system we will discuss

their role

in the static state feedback input-

output decoupling problem. Consider again the system (9.1-3) and suppose Assumptions 9,1 and 9.6 are met. We

introduce

the following regularity

assumption, which extends Assumption 9.16, Assumption 9.18 For each subset I C ~t V:.

invariant distribution in

n

leer dh j

•

the maximal locally concrolled

may be computed via tlle algorithm

JEI

VO

{

I

- TH

V'H1 - n ker dh j n IX I

E

V{H)

JEI

I

[f,X]

G + VJlI

E

I

[gi ,X]

E G

i E !::}

,,,here

for

each

Jl

~

0

t:he

distributions

dimension on H. In the sequel we will also write

Vp. I

and

I1J..! () G have 1

+ Vi-Jr •

(9.62) constant

289

=

, 1 C E!.

V"

lIl/t

and in this notation

(9.63)

v;

equals

i)

V:

the distribution

introduced before.

Also note the following identiti.es

U"

o

TN

=

(9.64)

and On 0

With the above family of controlled invariant distributions we denote the corresponding lists of orders of zeros at infinity as

- dim(G n Vr-1) -

p~

dim(G n V:)

i E

"-

- dim(G n d - dim(G n D· ) - dirn(G n D~-l) - dim(G n D;) - dim(G n VJ1--1) , - dim(G n V"), - dim(G n DWl) , - dim(G n D·,) H

p" q,I' p"(I) q"(I)

)

" - 1,2, .

11/

"" - 1,2, ... 1,2, " "-

1,2, ...

Ie m

c m

I

(9.65b)

1,2, .

i E m

(9.650)

(9.65c)

(9.65d) (9.65e)

From (9.63-65) the following identities are obvious 1 E III

IcE!. '

J1- =

(i)

rp(o)

eU) !p(I

~

peE!)

--1 !N

P

1,2, ...

P(~)

(9.66) (9.67)

1,2,.

We will need one further definition. Let of m. A function 11':

,

denote the family of subsets

is called a lveighc function if

0

(9.68a) !pCl) + rp(J) - rp(I n J), for all I,J C m

U J)

(9.68b)

Theorem 9.19 Consider the system (9.1-3) satisfying the Assumptions 9.1,

.

9.6 and 9.18. Then the follOl-ling conditions are equivalent

D:

(i)

G -

G n

(ii)

p"

- iEm I

+. .. + G n

p,I'

"

D~

(9.27)

1,2, ...

(9.69)

(iii) p~: P(~) ~ ~ is a weight function, p (iv)

=

1,2,.

(9.70)

The input-output decoupling problem is locally solvable.

For the proof of this theorem we need some preliminary results.

Lemma 9.20 Consider the system (9.1-3) satisfying Assumption 9.18. Then if for some

p ~

0 and I,J C

I1!

290

(9.71)

and

G- G

Ii

D~ + G Ii D~ I

(9.72)

J

then also (9.73)

Proof DJ,l+l

Ii

Ii

DJ,l+l

E V(ll}

Iter dh j

)

Ii

DJ,l

[f,X] E

G + DJ,l,

[g!,X] EG+

[f,X]

G -1'

[gl ,X] EG+ DJ,lJ' i E ~l

leer

Ii

n

Ii (

)

jEnllJ

IX E VO'l)

Ii IX

leer dh j

Ii

J

I

E

IX

Ii

I

DJ,l, J

I

E V(M)

[f,X] E G Ii D~ +

IX

I

E If UO

E

G Ii

I

, i E Ii

i

I'

!!!l

nilJ

Ii

+ DJ,l Ii DJ,l J

!

,

~I

[f. XJ E G + D~

1m • [g I • X]

E G

i

+ DJ,l 1m,

E !:!l

_ DP.+1

o

1m

Lemma 9.21 Consider the system (9.1-3) satisfying Assumpt:lon 9.18. Then

(9.27) implies that for all I,J C

~

.snd J,l

~

0 (9.71)

Proof Choose I,J C G :J G

Ii

~

D: + G Ii

and let us first assume that I U J - m. Then we have

D~

1.:

:J G Ii

i.EI

So, by (9.27) we obtain G - G G - G Ii D

Ji 1

+ G Ii

Ii

D: + G

n 1.: D~ lEJ

:J

1.:

G Ii D:.

lErn

D; + G Ii D~ , and thus for all

p.

~ 0,

(9.72)

DJ,l J

Induction and Lemma 9.20 then lead to the desired result (9.71). Now, for

arbitrary I,J c !!! we have

and because I U J U

~\I

-

III

we have that (9.75)

291

On the other hand

nil

c

InJ

nil n nil I

which together with

J'

(9.74) and (9.75)

yields the desired conclusion.

0

Lemma 9.22 Consider the system (9.1-3) satisfying Assumption 9.18. Then

Dr

LG n

G -

(9.76)

p 2! 0,

lEm

i f and only i f

,

Proof

(~)

.

m

11!\!VJ

-

nil

G n

m

IVJ

GnvlIJ +

=

all

c m. Then, for

)nD ll

_ G n

,

+ G n D"

r

G - G n D

(,,) Let I,J

Gnv il

G n DP

c!!!.

I,J

" >

0,

G

nPIU,

n

IVJ

G n

nil r

ID\{l}

J

1

nil + G n nil + G n nil (by Lemma 9.20) J

I

G n D"

=GnnP+GnnP+GnnP

G n

0

(9.77)

D"ru,

=

G n

-L

G n

rem

- (G

n

nPI

Dr +

G

,

n DP +

nn il

m\IUJ

nlJI +

G n

nP + G n nP ~ J

+ G n nil

o

,

We are now prepared to prove Theorem 9.19. Proof (of Theorem 9.19) (i)

~

By Lemmas 9.21 and 9.22 we have for all I,J C

(iii)

G n nil + G n nil 1

~

and p

~

0 that

G n nil

=

lVJ

J

(9.77)

and so it follows using (9.64) that for all I,J C

~

and

GnvP+GnVP-GnV.u !

J

I

Therefore, for all

J

Il

o

that

(9.78)

IVJ'

(G n Vp) n (G n UP) - G n

p ~

vPIVJ

(9.79)

> 0

pll(o) ... dim(G n vJ-'*1) -

o

dim(G n V") - m - m - 0 , 0

(9.80) (9.81)

Using (9.79) we have

(9.82)

292

and from (9.78) we obtain

(9.B3) Furthermore (9.82) and (9.83) hold true i f we replace the superscript JJ by i.e. by taking JJ sufficiently large. Combining (9.BI-B3) leads to

dim(G n V*) - dim(G n Vn) + dim(G n V· I

InJ

J

)

So (9.80) and (9.BLI) readily yield that pJJ is a weight function for (iii)

~

= p~ +

J..I

> 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 <.! D.

Tv

to

and

contained

in Tv

is

the

This holds because along each trajectory of which

Lyapunov function is constant;

is

hence

thus

the

il

solution

of

(10.1),

the

trajectory cannot approach xu'

Summarizing we have obtained.

Lemma 10.6

Consider

the

affine

control

system

(10.17)

around

the

equilibrium point (10.2), Suppose there exist:s a Lyapunov-function ;e on a neighborhood Va

such

of Xo

that

(10.5)

holds

true

and l"h1ch

satlsf les

(10.22). The smooth feedback (10.18) locally asymptotically stabilizes the eqUilibrium point

Xo

(defined in (10.21)

if and only if the largest f-invariant subset in r.r equals (x o )'

The difficulty of applying Lemma 10.6 lies in the fact that we need to know the trajectories of the dynamics when u '" O. To avoid the computation of

solutions

of

we

(10.1)

will

establish

sufficient

conditions

in

geometric terms. Define the distribution D(x)

=

span{f(x), ad~gi (x), i E

:E,

Ie


OJ, x E 11 0

(10.23)

,

Then we have

Lemma 10.7

Consider the system (10.17) around the point (10.2).

Suppose

there exists a Lyapunov-function ;e on a neighborhood Vo of the equilibrium point Xo such that (10.22) and (10.5) hold. TiJen along any trajectory of the closed loop dynamics (10.19) lying in the set [v' given in (10.21)

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<'1) when u

I-lb. =

u

(10.34)

u

Clearly

(w1,wz,w J

)

=

0

is

an

equilibrium

O. An obvious choice for a Lyapunov function for

the drift vectorfield in (10.3[1) is the kinetic energy of the rigid body, i.e.

308

(10.35)

l'!(w)

w - O.

!f. is a smooth positive definite function having a unique minimum in

Compucing Lel'! yields (10.36) which shows

that w

0

is

a

stable

equilibrium point

of

(10.34)

when

u - O. Define the smooth feedback (10.37) In what follows we will

w

see whether or not

the

feedback

(10.37)

makes

0 asymptotically stable. Define the distribution

D(w}

=

spanlf(w), ad~c(w), k ~ 01,

(10.38)

\IIhich after some computations yields 1 23 w2 w3

D(w)

=

131

w:lw 1

1 12. w1 w2

(10.39) In order that we may apply Theorem 10.9 we nee.d to find subsets VIand V2

of

[R3

such that Asswnption 10.8 holds. Define.

VI - Iwl dim D(w)

3J ,

dim D(w)

< 3).

(wI

!21 and

t\

(l0.40b) U \1 2 =

3

m

•

as well as 0 E tl z

. Now

a straight-

forward analysis yields that (10.41) where

(10.42<1)

0),

(10 .42b) 0) .

309

Let W be the set defined by

(10.43) In what follows we will make the following assumption

So the control axis is not perpendicular to any of the principle axis of

the

rigid body.

To verify the Assumption 10.8 we

need

to

VZ2 and [y. Note that IJIC~ imply that V Z1 is in this case

intersection of the sets Vl1J

< 0, which

I z3 c; - 1 31 Ci

dimensional

plane.

Vzz

as

given

in

(lO.42b)

is

a

cone

compute

the

I12C~ > a and always

in

J

m,

a

two

which

degenerates into the union of two planes in case that

(10.45) Provided (IO.L!S)

is not satisfied the intersection of the plane tin

and

the nondegenerate cone 1'22 equals a finite number of lines through w = O. Taking the intersection with fv we obtain the origin w - 0 and probably some

of

these

lines

in VZ2 n V Z1 '

Let ;:;(.)

be

some

trajectory

of

the

system (10.34,37) belonging to one of these lines; obviously ;:;(.) is also trajectory of the vectorfield f. Clearly differentiating the Lyapunov d function !£ along w(.) yields dt!£(w(t» = 0, t 2:: 0, so that .I-:(w(t» is

it

constant for all t 2:: O. (wi !few)

constant).

=

equilibrium

point

of

Therefore ;:;(.) belongs

Intersection with the

dynamics

a

to an ellipsoid given by

line

(10.34,37).

yields However

that one

;:;(.)

is

an

immediately

checks that given (10.44) the only equilibrium point in r? equals w "" O. So Assumption 10.8 is satisfied i f (10.44) holds and 1 12 by

Theorem

10.9

we

conclude

asymptotically stabilizes w

=

that

in

this

case

c; -

the

1 23

c; ,. 0,

feedback

and

(10.37)

O. Next we investigate the case that (10.45)

holds. V22 reduces to the union of the planes

Iw

w

2

=

(10.46a)

0),

(lO.46b)

Iw

Taking the intersection of (10.46a,b) with the plane r? we again obtain a finite

number

(2)

of

lines

(Notice

that

we

use

here

Similarly as before we conclude that the feedback (10.37) stabilizes w = O.

(10.44)

again.)

asymptotically 0

310

10.3 Local Stabilization via Center Manifold Theory

So

far

we have

discussed

the

local

feedback

stabilization problem via

procedures based on Lyapunov/s first and second method. Another approach to this problem is based on the Center Manifold theory. theory

forms

an

extension of

the

first

method

Center manifold

of Lyapunov

in

that

it

provides a way of studying the stability of an autonolllous system for which the linearization at the equilibrium has some eigenvalues located on the. In the seque 1 we firs t brie fly describe this

imaginary axis. then we

explain how

problem.

it can be exploited in

Consider again

the

system

(10.1)

the

around

feedback

theory and

stabilization

the equilibrium point

(10.2) and let the nxn-matrix A given by (10.4) denote the linearization of (10.1)

at xo'

In order to have local asymptotic stability of

(10.1)

around the equilibrium point we conclude from Theorem 10.1 that the matrix 11 should not have eigenvalues in the open right-half plane.

Therefore the

set of eigenvalues of A, O'(A), can be written as the disjoint union

(10.47) where the eigenvalues in

0'_

lie in the open left half plane and those in

Go lie on the imaginary axis. Let f be the number of eigenvalues (counted with their multiplicity) contained in (counted

with as

eigenvalues

0_.

in

Then there are n-i eigenvalues

their

multiplicity)

above

there exists a linear coordinate

0'0'

Given

the

splitting

of

the

transformation T

such that

~-

[ 6°

TAT- 1 =

],

(10.48)

where the {n-i,n-f)-lIlatrix AO and the (i, i)-matrix A O'(Ao)

z

Tx

-

{ where of z,

. ., Z

Zl

AOz 1 +

fO(zl,zL)

- L Az

f (z

.t-

-

In

the

have as eigenvalues

transformed

coordinates

1

(10.49)

2 ,2 )

and z'l denote the first {n-.O. respectively the last l' components

and [0 and [

representing

z

0'- •

the system (10.1) takes the form

X'o

'1 2

o(A-)

respectively

.,. 0'0 •

the

are smooth appropriate dimensioned vector

second

and higher

order

terms

around

the

functions

equilibrium

O. So

0,

(10.50a)

311

[(0,0)

~ 0,

(10.50b)

dfo(O,O)

=

0,

(IO.SOc)

O.

(10.50d)

We are now prepared to formulate the so called Center Manifold Theorem.

Theorem 10.11 (10.49) (lO.1). tp:

(Genter Manifold Theorem)

around

the

point

eqUilibrium

Consider z = 0

the

of

local

the

description

autonomous

syscem

Then for each lc - 2,3, .. there exists a Ok > 0 and a c! -mapping n1 E IR - IIz111 < 51; 1 ... {Rl Idth 'P(O) = a and d'P(O) = 0, such thar: the

!

(Z1

surface (the center manifold)

z' ~ "(z'),

Ilz'll

< ',.,

(10.51)

is invariant under the dynamics (10.49).

For the proof we refer to the literature cited at the end of this chapter.

Remark 10.12

center

(10.49) do not possess a unique

(i) In general the dynamics

manifold,

but

may

have

an

infinite

number

of

such

c!'

center

manifold

invariant

manifolds. (ii)

The

(finite) (10.51» (10.LIS)

smooth

dynamics

k = 2,3,.

on k

depends

is analytic,

manifold.

In

what

(10.49)

However

the

has

a

size

of

and may shrink with

the

center

for

manifold

increasing k.

(li k

Even

each in

in case

there does not necessarily exist an analytic center follows

we

will

throughout

work

with

a

C

2

center

manifold. The dynamics on the center manifold (10.51) are given as (10.52) The

following

theorem

shows

that

the

dynamics

(10.52)

contain

all

necessary information about the local asymptotic stability or instability of z - 0 for (10.49).

Theorem 10.13 (10.51)

of

Let t:he

asympt:ot:ically respect:ively,

represent: [;he dynamics on the cent:er manifold

(10.52) system

s[;able, implies

(10.49). locally

that

Then st:able,

(Zl,Z2) = 0

is

lI'e

have: or

Z1

=

unstable

0

is

locally

for

(10.52)

locally asympt:otically stable,

locally st:able or unstable for (10.49) respectively.

312

Essentially this theorem states that the local asymptotic stability of the n-dimensional system (10.49) can be deduced from the local asymptotic stability of the reduced (n-i)-dimensional system (10.52) on the center manifold (10.51). Before one can apply this result one needs a way to determine the center manifold (10.51), and in particular the mapping

~.

This is in general not possible since it is equivalent to solving (10.49) analytically. Differentiating (10.51) with respect to c and substituting this into (10.49) yields the partial differential equation for

~

(10.53) together with the boundary conditions ~(O)

~

d~(O)

0,

(10.54a)

O.

(10.54b)

Although we

may not be able

to solve

(10.53,54)

analytically we

can

approximate the solution to any degree of accuracy. Theorem 10.14

Suppose~: ~n-l

~£ is a Cl-mapping sacistying (10.50) and

for some q > 1

(10.55) tor

zl ....

O. Then

(10.56)

For the proofs of Theorems 10.13 and 10.14 we refer to the literature. The following example demonstrates the power of center manifold theory. Example 10,15

Consider on ~2 the system

(10.57)

around the equilibrium point (0,0). Clearly this system is already in the form

(10.49).

From Theorem 10.11 we conclude that

(l0.S7)

possesses a

center manifold of the form (10.58)

313

with !p(O) ip(x 1

!p'

=

(0)

D,

=

Trying an approximate solution ;P(x1

)

of the form

~ ip2x~ + ,p3X~ + .. ' we obtain

)

(10.59)

Using Theorem 10.14 we see that (10.60) Substituting

this

into

the

dynamics

of

the

center

manifold

(10.58)

we

obtain Xl

-2x~ +

=

which has

Xl

a

=

x:

+ O(x~')

=

-x~ + O(x~),

(10.6l)

as a local asymptotically stable equilibrium point. Thus

from Theorem 10.13

\~e

may conclude that (0,0)

is locally asymptotically

o

stable for the system (10.57).

The

computations

in

Example

10.15

are

rather

straightforward.

In

general, however, when the dimension of the center manifold is larger than one,

the problem of verifying whether

or

not

the

dynamics

(10.52)

is

asymptotically stable is hard. In

the

following

i.,le

investigate

how

to

exploit

the

center manifold

theory in the local feedback stabilization problem. Starting point will be again the control system x

=

(10.7)

f(x,u)

around the equilibrium point (10.8) From Theorem 10.5 we know that the only case of interest arises when the linearization

x around

Ax

+

(10.8)

Bu

(10.10) with

A

and

B

defined

as

in

uncontrollable eigenvalues on the imaginary axis;

(10.11)

has

some

in all other cases the

stability or instability follows from this theorem. This is the case when the reachable subspace 11 of (10.10) has a dimension smaller than n, and so after a linear change of coordinates (10.10) can be written as

314

~ ~~ 1 [ t [

X

A11

Au AZZ

0

and the eigenvalues of pair (All ,B 1

)

such

u -

1[ :: 1 +

AZL

=

+

Uo

0

(10.13)

are all located on the imaginary axis. As the

is controllable, we know that there exists a linear feedback that All + BIFI

suggests to apply a feedback u u

1u,

[ D1

X

is

an

asymptotically stable

matrix.

This

a: (x) of the form (10.62)

F1

to the system (10.7). In (10.62) Xo

(x;,x;) and Q is a smooth feedback

0 and do(O,O)

O. The linearization of (10.7) with

function with 0(0,0)

the feedback (10.62) equals

(10.63)

and so we conclude that after a linear change of coordinates (10.7,62) is

a system of

the

form

(10.49),

which according

to Theorem 10.11 has

a

center manifold

(10.64) where Zl _ xl_x;, of

the

z2

x 2_x; and the superscript a: denotes the dependency

=

center manifold on

the

feedback chosen.

locally asymptotically stabilizes the equilibrium

In order that we have

Xc

(10.62)

to use

the

freedom in the feedbac.k (lO.62) in such a way that the reduced dynamics on the center manifold (10.52) is locally asymptotically stable. Notice that in (10.62) we have a complete freedom in selecting the linear map F z and a,

whereas

Fl

is only constrained by the requirement that

asymptotically stable.

Cle
dynamics asymptotically stable in this way, Taylor expansion of

All

whether or not we can malte

+ B1 F t is

the reduced

depends on the higher order:

the closed loop system dynamics.

As

stated before,

this is the most difficult: part of the center manifold theory. To see how the theory works we disc.uss a smooth control system

x~

f(x) + g(x)u

(10.65)

on ~z around the equilibrium point f(O) to

the

Flow-Box

Theorem,

see

Theorem

o

and asswne g(O)

2.26

I

there

exists

~

O. According a

coordinate

transformation in which g(x) - 8/8x2' In these coordinates (again denoted by x

(Xl

,xz»

the system takes the form

315

(l0.66)

Application of the feedback (10.67)

where 0(0,0)

0 and da(O,O) = 0, yields the closed loop dynamics

=

(10.68)

Linearizing (10.68) about (0,0) yields

"dt [X' x,

1-

,

il!..L(0 0) aX

[

*"(0,0)

'

1

(10.69)

-1

P,

First of all we require that

at, -a (0,0) - 0, Xz

(l0.70a)

because otherwise the linearization of (10.66) would be controllable and thus (Theorem 10.5) the local feedback stabilization problem is trivially

solvable. Also we ask that

at, In

case

(l0,70b)

- O.

aX (0,0) l

violated the system (10.7,67) is af, af, if -a-(0,0) < 0 and unstable if -a' (0,0)

is

(l0.70b)

asymptotically stable

x,

Xl

matter how we select the feedback (10.67). Now,

assuming all

locally

> 0, no

data to be

analytic, we may take a Taylor series expansion for the closed loop system (IO.7,G7). 0(X 1 ,X 2 )

To simplifY things we assume that the higher order terms in

(10.67)

only

depend

on

Xl'

x1X2 +

b2xI

This

yields

the

in

following

expansions, tl

-

(Xl ,X2 )

0(X1 ,X 2 )

Writing Xl ...

Xl

,+

8 2X1

,+

8 3Xl

b

1

and Xl ...

-

PIXl

(in order to obtain the Jordan form for

the linearized dynamics as in (10,49», the dynamics we obtain

..

(10.71b)

1

X2

,

(lO.7la)

, + pJx , +

P2 X l

,

x; + b3X~X2 + c 2 X l + cJxl +

and substituting (l0, 7la, b)

into

316

+ 8JX~ + b1 x1 XZ + bIPIX; + ... +

(10.72)

P2X~ - Pl(a2X~+a3i~+bliliz+blPlX~+,")'

From Theorem 10.11 we know there exists a center manifold (10.73) for

Note

(10.72).

approximation of

'f! -

that

by

definition

sse Theorem 10.14 -


we

tp' (0)

- O.

To

get

an

let (l0.74)

and substitute this in equation (10.55) yielding (2
3rp:lX~)(a2x~ + a3x~ + b1lPzxi + blrp3X~

-I-

+
'f!:Jxi -

.j-

P2X~ ... Plazxi + PlaJx~ + blP~xi

blPlX~ ...... ) of·

blrpZX~ + (10.75)

Letting q = 3 we find that (10.76) yields a second order approximation of the center manifold _ The reduced equation (10.52) then takes the form (10.77) from which

we

conclude

that a necessary condition for

the

asymptotic

stability is that (10.78)

0,

and given (10.78) the reduced equation (10.77) is locally asymptotically stable if (10.79) Therefore we conclude that (10.77) (10.78)

and

(10.79)

hold,

cf.

is locally asymptotically stable i f Theorem 10.13

Note

that

these

two

conditions can be fullfilled by selecting a suitable feedback (10.67,71b) if the coefficient b l

is

nonzero.

In

terms

of

(10 66) this is equivalent to the requirement that

the original

dynamics

317

a2 fl ax1ax (0,0) Z

" o.

(10.80)

We conclude our exposition about the use of center manifold techniques in the smooth stabilization problem with the following example.

Example 10.16

Consider

the

equations

for

the

angular velocities

of

a

rigid body with two external torques aligned with two principal axes (see Example 1.2): I~ - S(w)Iw + blu l + blu Z '

where

w

~

Sew)

(10.81)

(wl'w Z 'w 3 ) and

0 w, -w, -w, - w, 0 w, 1' -w, 0

[

I -

U'

0

0

I,

0

0

I,

(10.28)

J.

(10.82) II'

I2

and 13

denote

the principal moments of inertia.

We will

assume

1 J > 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<_l+l(X)

hpl«x) = 0).

=

Define

the

and the PI:. x 1 vector B); (x) as

m matrix AI:. (x)

(1l.14)

Assume that AI:. (x) has constant rank r):

in a neighborhood of

in N);. After a possible permutation of the functions h j assume that the first r k

Xo

we may

rows of AI< are linearly independent.

Therefore there exists on a neighborhood of Xo in NI:. a feedback u

O'k (x), with O'k (xo)

=

AI:. (x)O'(x)

0, such that

+ B (x) _ [ I:.

0

'I\; (x)

]1 r,

(ll.IS)

Assume that !Pk has constant rank 51:.+1 on a neighborhood of Xo in N);. Then locally around x o ' NUl: submanifold,

(n-Pk+l)-dimensional

S):+l

as

0) is an

where

0 and a(xo ) = 0.) Take entries of IPt. which are independent on Ni: and denote them

(Notice that 'I\;(xo )

hpi:+l""

=

0, since £(xo)

I

(x E NI:. !PI:. (x) -

=

=

,hp);-Il'

If at every step of the algorithm the two constant rank assumptions are satisfied then we call Xo a regular point for the algorithm.

Remark 11.7

As already indicated in Algorithm 11.3 it is not necessary to

assume that [(xo) may

still

O. In fact if [(xo )

exist

a

feedback

u -

~

0,

then at the k-th step there

satisfying

(x)

QI:;

(11.15)

such

that

'I\: (x o ) - O. (However in general Ok (xo) can not be taken equal to zero in this case.) If this holds for every k ... 1, ...• k". and the constant rank assumptions of Algorithm 11.6 are satified then we will still call

xG

a

regular point for Algorithm 11.6. Let us now check that the submanifolds Nk as produced by Algorithm 11.6 coincide with those as defined in Algorithm 11.3. Suppose x D is a regular poine for Algorithm 11. 6. Clearly Nl for Algorithllls 11. 3 and 11.6 are the same around x[J' Now assume that Nj

of Algorithm 11. 3 coincides with Nj

Algorithm 11.6 for j :S k.

(xl

such that Nt;

(xlx

=

1

Let x -

be local coordinates around Xo

~ 0). Indeed. let us take

ly speaking the functions hp j+l'

of

xl -

(hi""

.hpl<)'

(Strict-

.hp j+l are only defined on NJ , j :S k;

•

however they can be easily extended to independent functions on a whole neighborhood of Xo in H.) Partition accordingly f and g

(81

=

... ~gm) as

(11.16)

i,

Notice that in these coordinates Ai; =

£1. Therefore (11.15)

while Bit.

amounts to

i(0.x Now,

2

)

-I-

i(O x

2

(x

)Q

'I;

2

f(O,x ) E TxNi;

+

G(x}

2

) _

[

0.2]

(11.17)

IPk (x )

1

if and only i f f (0,X

2

)

of (11.17) this last inclusion holds i f and only i f

E Im gl(O,Xl).

['I\:~X2)J

In view

E 1m i(O,x

l

).

and since the first rk

rows of gl(O.x2) are independent this is true i f

and only if ~(X2) = 0,

thereby proving that Nk + 1 of Algorithm 11.3 coin-

cides with Nk + 1 of Algorithm 11.6. In comparison with Algorithm 11. 3, in the \c-th step of Algorithm 11. 6 the additional asswnption is made that the rank of Ai: neighborhood of Xo in Nk

,

thus enabling the construction of the feedbacks

Ie ~ 1, ... ,k", solving (11.15).

Clk(X).

obtain by

definition of

is constant on a

1/

In particular, at the kn_th step we

that fPK" - 0

on Nit." -: N"

(or

equivalently

0), and thus 0k"(x) -: aW(x) satisfies (see (11.15» (11.18) W

with A

:",

At;" , n":"" Bk~'

rank

i"-

r" ;- rl
=>

cr"(x) is a feedback

W

which renders N

invariant. Thus,

if

Xo

is a regular point for Algorithm

329

11.6, then N" is automatically locally controlled invariant (compare with Proposition 11.4 and Proposition 11.2). Finally, because the submanifolds

Nk

as

produced

Algorithm

by

submanifolds Nk

11.6

of Algorithm 11.3,

equal

lntrinsically

the

it follows

that the Nk

11.6 do not depend on the particular choice of

defined

of Algorithm

satisfying (11.15)

ok

or

the selection of the independent entries of ({\;.

Remark

for

Notice that Algorithm 11.6 is very close to Algorithm 7.19,

computing

the

maximal

locally

controlled

invariant

used

discribution

contained in ker dh. The main difference is that in Algorithm 11.6 we do

not have 11.1,2).

to compute

PI; (x), nor

the matrices

Additionally,

the

constant

rank

JjJ); (x)

(see also

assumptions

Exercises

in Algorithm

11.6

need only be made on neighborhoods of Xo contained in the submanifolds Nk . The indices sk and r k defined in Algorithm 11.6 are related as follows.

Lemma 11.8 sic 'Yk'

Let Xo be a regular point for Algorithm 11.6. yielding indices

k "" 1, ... , Ie ~

Then (11.19a) (11.19b) (11.19c)

Proof Clearly

51

::S p.

Since 'Pt

in (11.13)

is an (St-rl)-vector,

(11.19b)

immediately follows. Now consider (11.15) for k and Ie-I, i. e. Ak (x)ok (x) + Bk (x)

Ak

First,

_

1 {X)ok_l(x)

+

Bk

['P~ (x) J'

=

_

1 (X) ""

(11.20)

[!f.'k-~(X)J·

the right-hand side of (11.20) has

(11.21)

SY.

more entries than the right-

hand side of (11.21). Since rank Ak (x)-rank Ak _ 1 (x)

=

r k _ 1 -rk _ 2 it follows

that the right-hand side of (11.20) contains r k _ 1 -rk_2 more zero entries than the right-hand side of (11.21). Furthermore we can choose satisfying (11.20),

resp.

(11.21),

'Pk _ 1 also appear as entries of IPk

Ok

and 0k_1

in such a way that all the entries of (since the rows of Ak _ l' resp. Bk _ l' are

also rows of AI;' resp. Bk ). It follows that rpk (x) consists of the entries of

!f.'1;-1 (x)

additional

together entries

(11.19c) follows.

with

can

sk-(rk-rk _ 1 )

contribute

to

additional the

rank

of

entries.

fA:

on

Only

Nk ,

and

these thus

o

330

r . we immediately obtain Corollary 11.9 The indices r

..

satisfy

51: • rr;

sS\r"+rk"-l:S

In particular if p

=

m and r

then (11.22), or equivalently (11.19).

In,

holds {.rich equillicy.

Remark

11.10

0'1' . . . • ok·

entries

of

entries of

As

noticed

in

in such a way rp.". ~k

Remark 11.11

Ie

=

2, ..

the

,1/

It

condidon r

n

=

of

Lemma

11.8

we

the entries of rp." -1

follows

~-l

not appearing in

The

proof

that all

that

if

can

choose

also appear as r,

p '" m

then

the

are independent on NI:'

m is

equivalent

to

the

existence

of a

unique solution an(x) of (11.18).

Remark 11.12

The

k-th step

of Algorithm 11.6

following different: but equivalent form. back

U

can be

recast

into

the

Instead of constructing a feed-

= Q",(x} on NI: satisfying (11.15) we can construct on a neighborhood

of Xo in Nk a {PI: -rl:} X PI: matrix RI: (x) of full row rank satisfying (see Exercise 2.4) Rk (x)A k (x) -

x E

0

(11.23)

NI;'

Then locally around x o ' Nk + 1 is alternatively given as (11.24 ) Indeed, since the first rio; rows of Air. are independent, it follows that Rk is of the formR k = (~~iTk)' with Tic an invertible (Pr;-rk) x (Pk-rk) matrix (In fact we can choose Rk such that: Tk is the identity matrix.) Premultiplying (ll.lS) by Rk(x) yields

(11.25) and thus Rk (x)B k (x) = 0 for x E Nt if and only if IPk (x) Finally

let

us

give

the

following

particular

~

o.

case,

where

the

that

the

computation of N" and o' becomes especially easy. Proposition 11.13

Consider

the

system

(11.10).

Assume

331

cha.racteristic numbers PI"" ,Pp

as

defined

in Definition 8.7 are all

i1n1l:e. Furthermore assume that the p x m mat:rix (the decoupling matrix)

(Lg JL~i hi (x») .

A(x)

~gl

(11,26)

... "p

jgl, ...

,m

has rank equal Co p on the set 0, i E

If

i'

eJ

(11. 27)

is nOll-empCy (for instance i f there exists

f(x o }

0), then the functions hi •... ,Lfih i

=

and thus

N"

•

i

E

Ka

e.

\.rich 11(xo ) - 0 and

are independent on N", p

is a smooth submanlfold of dimension n -

I

Further-

(Pi +1).

j-1

more N" equals the maximal controlled invariant: output-nulling submanifold for

(11.10).

The

feedbacks

u

=

Ct"

(x),

x E N",

I...hich render N*

invariant

are given as the solutions of A(x)o:" (x) + B (x)

l"here B(x)

is

o ,

(11,28)

the p-vector with

i-eh

component L~Pi i'l)h i (x).

fact,

In

A(x) ~ A*(x) and B{x) "" B~(x).

Proof

(see

follows

as

also in

Exercise

11.3).

Proposition

8.11.

Independence It

is

of

hi""

immediately

seen

,L~ihi' that

i E E, u

=

o(x)

satisfying (11.28) renders N* as defined in (11.27) invariant, and thus N* is controlled invariant and output-nulling.

Clearly,

if the outputs y( t)

have to be kept zero then also all their time-derivatives have to be zero. Maximality y:j) have

of

N"

now

follows

L~hi' j=l, ... ,Pi' i

=

E

E,

i E E,

since

implying that the functions hi""

,L~ihi

by

definition

of

P,

to be zero on any controlled invariant output-nulling submanifold.

o

For the last equalities we refer to Exercise 11.3.

Remark 11.14

Notise that N" as obtained in Proposition 11.13 is globally

defined, not just in a neighborhood of a particular point xo' Furthermore, if

rank

A(x)'" p

everywhere,

then

the

maximal

controlled

distribution V* contained in ker dh is given as ker span {dh i

i E

e)

invariant

, ...

,dLfih t ,

(see Exercise 8.4 and Theorem 7.21), and thus N" given in (11.27)

is an integral manifold of D".

11.2 Constrained Dynamics and Zero Dynamics

In the previous section we have given algorithms to compute the maximal

332

locally controlled

output-nulling submanifold N'"

invariant

around a point Xo satisfying f(x a ) n

on N

for

(11.10)

0, h(x o ) - D. The resulting dynamics

~

is given as

2:

x - f(x) -t.

x

gj (X)t:I; (x),

E

N".

(11.29)

j=1

(Q;(x) ..... cr~(x»

where an (x)

is any solution of (11.18). Now let a(x)

be one particular solution of (11.18). Locally around Xo We can find an m x (m-r")

matrix {3(x)

a

of full column ranlt such that A'" (x){3(x) ..

(see

Exercise 2.4). Then the full solution set of (11.18) is given as

(11.30)

and thus locally around Xo the resulting dynamics on N~ is also given as iii j

~

n

gj (x)v j

x ~ I(x) + ~

(11.31)

EN,

X

1 m

where

f(x) +

I(x)

2: gj (x)a j

j E ~,

(x) , 1=1

j~l

Iii

:=

with inputs v 1

r.

III

applying

to

(lLIO)

the

, ••• •

Alternatively (11.31) is obtained by

viii'

degenerate

feedback

u

= a(x) + {3(x)v,

v

E 111

given by (11.30),

and then restricting the closed-loop system to N".

will call

the constrained (or clamped)

(11.3l)

i.e.,

(11.10),

all

motions

of

(11.10)

dynamics for

compatible with

the

m

•

We

the system

constraints

hex) = D. Next

let

us

consider

distribution Co

for

the

(Definition 3.19),

constrained

dynamics

(l1.31)

the

characterizing local strong accessi-

bility with respect to the controls v 1

"'"

.

By Proposition 3 .47 Co

is

the smallest distribution on N" that is invariant for (11.31) and contains the vectorfield

gt, ... ,gJii"

neighborhood of

';::0'

x ~

(.;? ,;?) -1

X

=

Assume that Co

has constant dimension on a

Then by Theorem 3.49 we can find local coordinates

for N" around Xo such that (ll. 31) takes the form

.x

iii

2 )

+

2: g] (xl '~?)Vj

•

(11.32)

j&l

Definition 11. 15

The dYIJamics

the system (11.10) around xo'

xl

=

r(x2)

are called the zero dynamics of

333

For a controllable and observable linear system x

Remark

=

t\.."I:

+ Bu the

eigenvalues of the zero dynamics (which in this case are linear) coincide precisely with

the

transmission zeros

of

the

transfer matrix C(Is-A) -l B

(see the references cited at the end of this chapter). Notice that the zero dynamics do not depend on the particular choice of

n(x) satisfying (11.18), and so are intrinsically defined. Furthermore if r

~ In

then

(see Remark 11.11)

the

zero

dynamics

equal

the

constrained

special case considered in

Proposition

dynamics. Finally,

let us

consider

the

11.13, with the additional requirements that p "" m and that rank A(x) everj'l"iJere i.e.

=

m

(and not only on N" and therefore on a neighborhood of N"),

the case of an input-output decouplable square system (see Theorem

8.9). Then the functions Zjj

-

L~-lhj ex), j

1, ... ,Pi + 1,

=

i E

(11.33)

~,

are everywhere independent (Proposition 8.11), and as explained in Chapter 1

8, the decoupling feedback u = A- (x) (-B(x)

+

(11.28),

into

transforms

the

system

(11.10)

v), with A(.x) and B(x) as in

the

"normal

form"

(cf.

(8. llS»

.,

z

·m

(11.34)

Z

m

l(z,zl, .. ,zm) +

Z

Lgj(z,zl, .. ,zm)v j j

with

/:=

(Zi1""Z.

HPj

), +1)

~

,

1

i E

11!, -

and

z being additional coordinates,

satisfying z(x o ) = O. Furthermore the matrix pairs (Ai ,b i ), i E:!!, are in Brunovsky canonical form (see (6.50». It immediately follows that in this cuse the zero dynamics are equal to the constrained dynamiCS and are given

"' z

=

and that From

£(z,O, .. ,0),

z can linear

respectively

(11.35)

be regarded as local coordinates for N~ around xo' control

instability,

theory of

it

the

is zero

well-known dynamics

is

that very

the

stability,

important

for

design purposes, and we may expect this to be true in the nonlinear case as well. We immediately obtain

334

Consider

Theorem 11.16

il

square

system

(11.10),

f(x o )

t"ith

~

and

0

h(xo } .. 0, and rank A(x o ) - m. Assume that its zero dynamics are locally

asymptotically

stable around

system for Proof

v

=

N~.

Xc E

Then

Q(X) +

regular static state feedback u

there

~(X)VI

exiscs

0 is locally asymptotically stable around

First apply the decoupling feedback law u

that locally around

Xo

a

decoupling

such that the closed loop

=

xO'

£\x)(-B(x)+v),

such

the closed-loop system takes the form (11.34). Then

apply additional linear feedback

!E.

i E

such

that

the

Ai:~

matrices

(11.36)

Al + bile!,

i

are

E!E.

all

asymptotically

stable. Notice that the closed-loop system with inputs VI"" input-output decoupled. Setting

=

vi

E ~,

0, i

Vm

is still

we obtain the system

Z1 'm

(11.37)

Z

(

m

I

Z

gj

(z

,Zl, .. ,zl1\)kj Zi.

jml

Clearly. the eigenvalues of the linearization of (11.37) in i E !E, are the eigenvalues of

Al , ...

I

~

Z=

0,

O,Zi

together wi th the eigenvalues of

che linearization of the zero dynamics (1l.35)

z

in

(10.49) we can write (11.35) into the form (with z

=

=

O. Accordingly to

(sl,s2))

(11.38)

AO are on the imaginary axis and the eigenvalues of A- are in the open left half plane, and fO and [ only contain second-

where the eigenvalues of

and

higher-order

10.11)

Thus

by

the

chere exists around Xc

terms.

an

lnvariant submanifold s2

Center Hanifold

Theorem =

(Theorem

!p(sl)

of N*

wi th dynamics (11.39) Since N~ is an invariant submanifold for foillows that the submanifold

s2

!pest)

the whole dynamics

(11. 37)

it

of N" is also a center manifold

for (11.37). Since the dynamics (l1.35) on N" are assumed to be locally asymptotically

stable

asymptotically

stable

about about

Xo

xo '

it

follows and

thus

that by

(11.39) an

is

locally

application

of

335

Theorem 10.13, Xo

is a locally asymptotically stable equilibrium for the

o

whole dynamics (11.37).

Remark

The above sufficient condition for

input-output decoupling with

local asymptotic stability is far from being necessary,

ct.

the references

at the end of this chapter. The reason is that decoupling feedback laws do not necessarily have to make

N"

invariant.

Theorem 11.16 immediately suggests an alternative approach to the problem

of local feedback stabilization of a nonlinear system

x

= [(x)

+

L gj (x)u j

f{x o )

,

=

(11.40)

0,

j"'

as

treated

in the preceding chapter.

Indeed suppose

that we can find

III

(dummy) output functions i E

(11.41)

!!!.

such that the system (11.40), (11.LI-1) has nonsingular decoupling matrix and

its

zero-dynamics

are

locally

asymptotically

stable.

Then

by

an

application of Theorem 11.16 the system can be made locally asymptotically stable by feedback. This idea is illustrated in the next example.

Example 11.17

Consider

the

equations

for

the

angular velocities

of

a

rigid body with two external torques aligned with two of the principal axes (see Example 10.16)

(11.42)

By a preliminary feedback the system can be transformed into

W

z

=

VI

(11.43)

Now consider the dummy output functions

(11.44)

336

In the coordinates Yl'Y2.wl the system takes the form

(11.45)

Clearly the zero dynamics of (11.45) is given as (11.46) which is asymptotically stable by the fact that I 2 :3

-

(I 2 -I 3 )/1 1 < O. Thus

for example the feedback

(11.47) asymptotically stabilizes (11.45). (Compare with the stabilizing feedback

o

obtained in Example 10.16.)

Finally. in order to make contact with Theorem 10.5, let us suppose that the linearization of (11.40) in Xc and u - 0 is not controllable. As explained in Chapter 10 (cf. Theorem 10.5), this means that there are some eigenvalues of the linearization, called the unconcrollable modes, which are invariant under feedback (namely the eigenvalues of

A22

in (10.13».

Now suppose there exist output equations (11.41) for which the decoup1ing matrix A(x) has everywhere rank m, such that after feedback the system takes the form (11.34). For simplicity (see Exercise 11.6), assume that we can choose the additional coordinates (11.34)

j

E~.

(This

is

the

case

z

if

in such a way and

only

if

that gj - 0 in

the

distribution

spnn{gl(x), ... ,gm(x)) is involutive about x o ' see Exercise 8.3.) Clearly the uncontrollable modes of the linearization of (11.40) are the same

as

the uncontrollable modes of the linearization of (11,34), i.e. of i E !E,

at

- + 1...~ al( 0 , i

- -(0, .. ,0);;

az

1-1

az

... ,O)E

i

(11.48)

Since (Ai ,bl.)' i E!!!, are controllable pairs it follows that the uncont:rollable modes of the linearizacion are necessarily modes of the linearized zero dynamics (cL (11.35»)

337

al

-

(11.49)

-(0, ... ,O)F,.

8z

In view

of

becomes

difficult

Theorem

10.5

the

when

problem

(some

of)

of

the

linearization are on the imaginary axis.

modes

also

appear

as

eigenvalues

of

local

feedback

uncontrollable

\~e

stabilization modes

of

the

conclude that these particular

the

linearization

of

the

zero

dynamics.

11.3 Interconnection of Systems and Inverse Systems

In

many

instances

systems

are

at

first

instance

given

connecr:ion of a number of (small and relatively simple)

as

the

inter-

subsystems.

EVen

if these subsystems have explicit state space models, the system resulting from

the

interconnection

differential

and

is

algebraic

a

described

relations,

priori

and

a

by

a

maj or

mixed

problem

set

is

of

the

transformation of these relations into a set of explicit differential and algebraic

relations,

i.e,

a

state

space

model

(which

is

usually more

convenient for control and simulation purposes). In the present section we will show how such a

transformation may be naturally interpreted as the

computation of some kind of constrained dynamics. Consider k affine nonlinear systems

Xi

=

f~(x~)

Ii

+ j

where

Xi =

manifold

g~ (xi)u; ,

(x~, ... ,x~,) ,

ll,

i

E!5..

i

E

!::'

(11.50)

1

~

are

local

coordinates

for

an

n l -dimensional

Suppose these systems are interconnected by an inter-

connection constraint of the form (11.51)

where

rp: til x

struct

the

fl

x ... x

system

II

resulting

IR

O

is

from

a

smooth mapping.

this

In

order

interconnection we

to

consider

conthe

produc t sys tem

Xl '" f1(./) + [1 j

~

g; (xl)u;

1

(11.52a) ~,

+

L

g~ (Xk)u~

,q

with

state

space

fl

x ... x

t/,

input

space

lR m1

x

X

~!ttJ.y.

and

(dummy)

338

output equations j

(1l.52b)

E c.

The system resulting from the interconnection is precisely given as the constrained dynamics for (11.52), which can be computed using one of the algorithms given in the first section of this chapter. Remark

If the interconnection constraint

the inputs

Ill' . . . 'Uk'

~

in (11.51) depends on any of

then the output map (1l.S2b) will depend on the

inputs and we have to take recourse to the computation of the constrained dynamics for a general nonlinear system, see Chapter 13. Example 11.1B

Consider two rigid two-link robot manipulators (see Example

1.1), mounted on a same platform. at a fixed distance d.

Fig. 11.1. Co.operating roOOt mnnipul!ltors.

The

equations

of motion of both

robot manipulators

are

described

in

Example 1.1. Now suppose that both manipulators have to co-operace; in particular. suppose that the endpoints of both manipulators have to be at the same place at every time t (for example for simultaneously grasping an object). Also suppose that the endpoints of both manipulators merely have to touch each other. in such a way that no reaction forces are present (as an idealization of the situation that the object that has to be grasped by both manipulators is easily damaged). Then we have the interconnection constraint equations (cf. (1.9))

339

It is easily seen that the product system of the two robot manipulators with

the

two

output mappings

corresponding

to

the

constraints

satisfies the assumptions of Proposition 11.13, with Pl = Pz

=

N" is non-empty and is given as the set of all points (e~, O~,

b;,

O~,

iJ~,

b~)

satisfying

(11.53)

equations

and

their

(11.53)

1.

Indeed

8~, O~,

first-order

time-

derivatives

,,' .,

,

".,, , 0,

0, cos 0' +

"

, 0,.,

(0

~ +

,

cos 0' + ..e~ (0 ~ +

.,

°

2 )

/} i) cos (e~ + O~) (11.54)

,

",

sin 0' +

"

.,

(0 ~ + o'}) cos

, 0,. ,

"

(0;

+ Iii)

,

sin (0; +

, .,

sin 0' + 12 (IJ 1 +

o~)

0; ) sin (O~ + O~)

o

and the required feedback is computed as in (11.27) .

Let us apply this same methodology to the problem of inversion of a

nonlinear control system. That is, we want to reconstruct the input functions u on the basis of the knowledge of the output functions for

and the initial state of the system. of left-inversion;

t

;?:

0

(This is usually called the problem

the dual problem of finding input functions such that

the resulting output functions for a fixed initial state are equal to some desired functions of time is called the problem of right-inversion, and is touched upon in Chapter 8, cf. the proof of Theorem 8.19.) Let us again consider the system

x

f(x) +

L

gj

j-'

y,

(x)u j

,

x(O)

=

xo• (11.55)

i E E,

where x -

(Xl""

,X n )

are

local coordinates

for

the

state

addition,

consider an am-:iliary system consisting of p

space

N.

In

parallel n-fold

integrators

(11.56)

y

- w, input

initialized

at

some

point

v

(Yo ,Yo, .. ' ,y~n-1)

and E [Rnp.

(11.55) to (11.56) by the interconnection constraint

output Let

y,

us

which

is

interconnect

340

hex) - y '"

So (X,y)

(11.57)

O.

Now apply Algorithm 11.6 to the product of systems with output equations So (x,y),

and assume

that

(11.55)

(xo ,Yo

and (l1.56)

,Yo, ... y~n-l))

is a

regular point (in the sense of Remark 11. 7), Then we obtain a descending sequence

of

subrnanifolds

u"

Nt:J N2 :J ..• :J N);· =

of

x Il?n

11

p

around

(xo 'Yo" .. y~n-l}). of the form

(l1.SB) where S,;: has

components. By the special form of (11.56) and (11.57)

5';:+1

we have the folloWing extra information concerning the mappings

At the

51:'

firs t step of the algori thm the matrix Al talces the form

i

hj(x»).

J~l,

.. ,Pl

0

PI Xp

J.

(11. 59)

1=1, .. ,m

Moreover, N x li'!n

p

,

since we have

the components of So (x,y) PI

51

=

hex)

p. It follows that r l

yare independent on rank.(Ls h J. (x»).

."

J~l .... p

i

Furthermore the matrix B1 is given as

.

1"'1, .. ,m

(1l.60)

Therefore 51 has

P-rl entries, and may be tak.en of the form

52 =

(11 61) where

rank

especially

G1 (x ,y) clear

=

S2

if we

in use

a

neighborhood

the

procedure

of

(xo .Yo ).

given

(This

in Remark

becomes

11.12.)

In

general we obtain that the mappings Sk can be taken of the form •

S,;: (x,y,y •.. ,y

where

(1';)

(11.62)

)

rank Gk (x ,y, ..

,y(k-U)

5\c+l

around

(xo ,Yo, .. ,Yo

(k-ll).

Furthermore

(cf. (11.22»

1c=3, .. ,n,

(11.63)

and for k < n the last p columns of the matrices A,;: (x,y, .. •y!k-l) responding to dependencies on the inputs in view of

(11.62)

and

the special

III')

form

(cor-

are identically zero. Finally, of

the

dynamics

(11. 56)

I

the

equation

o

(lI. 64)

341

can be always solved for

Q

on Nn , implying that k"

n. Using (11.63) we

:$

see that (11.22) holds with equality everywhere, and thus r" - p. Let us now define one additional integer, namely (11.65)

AI;"

where

follows

denotes the first m columns of At;'"

that

p.

~ r

(since

the

last

p

zero). However if k" ... n, we only obtain / Now assume that p ~ m and /

=

For lc" < n it immediately

columns

:s

of AI;"

for

Je" < n

are

r"

m. Regarding the algorithm as above as

the constrained dynamics algorithm for the system (11.55), parametrized by

y,y, ... ,y{n-1J

it follows from Corollary 11,9 (see Remark 11.10) that

as,

ax

rank

(x,y, ... ,yO:»

Thus from the PI; ~ -

So (x,y)

51

~

+ ... +

5);+1 on Nk

51;"

Jc ~ 1,2,. ,//-1.

,

(11.66)

equations defining N", i. e.

.. 0,

(11.67)

0,

we can locally solve for Pk" components of the state x,

y,y, ... ,y(k denote by

Z

-1)

and

of

the

remaining

n-Pk"

state

as functions of

components,

which

we

E IRn-Pk". Furthermore since p" - m, the equations (11.68)

have a unique solution u as

function of x and y,y, ... ,y(k)

Combining

this, we obtain equations of the form

z _ F(z,y,j, ... ,yO: -1», u

=

G(z,y,y, ...

,y(k

(11. 69a)

z(O)~zo'

».

(11. 69b)

Equations (11.69a) together with the auxiliary system (11.56) describe the dynamics on the constrained dynamics submanifold N~ c M x {Rn

p

•

The system

(11.69) is called an inverse system, since for every output function yet), t ~ 0

small,

with

(y(O),Y{O), .. ,y(n-ll(O»

close

to

(Yo,Yo, ... ,Yo(n-ll),

it reconstructs the unique input function u(t), to! 0 small, yielding this particular

output

function

for

initial

state

xeO)

=

Xo

if we

set

the

342

components

of

to

20

be

equal

to

the

corresponding components

of xo'

Summarizing we have obtained

Proposition 11.19

Consider

(xo ,Yo" .. ,Yo (n-1»

is a regular point for Algorithm 11.6 applied to the

the

(11.55)

system

{,rith

m.

p

Suppose

product system (11.55), (11.56), !,rlth output equacion (11.57). Assume that P

=

m. l,rhere p" is defined as in

(11.65). Then there exists an inverse

system (11.69) for (11.55), l,rit:h Zo determined by Xo

J

l,rhieh reconstructs

in a unique manner the input function from the observed output: function.

Example 11.20

Consider the system on ~4

(11.70)

Thus So(x,y)

and so

Al (x)

Bl (x .y)

[.~~ ~ 1 has

Y2, then A z (x .y) while Ie" = 2. Solving for u l

~J.

[;4

=

o

. Take

rank 2 for Yl .,. 0,

Yl

and u z . one obtains

(11. 71a)

From

the

equations

o

and 51

So ... 0

we

can solve

for

x l ,x2

and

x3

'

thereby leaving the equation (l1.71b)

o

and thus (II.7Ia,b) constitutes the inverse system for Yl .,. D.

In analogy with Proposition 11,13 we now state an important particular

case, where the computation of the inverse system becomes especially easy. Suppose that p everywhere.

111

j

'" 1, ...• PI +1.

coordinates Pi'

m and that the decQupling matrix A(x) of (11.55) has rank Then 1 E

by

Proposition

8.11

the

functions

Zij

~ Lt1h i

,

E!. are everywhere independent. and we can take local

(zl, .. ,zm.

z),

with

ZI

-

(Zil""'Z i

'Pi +1}

),

By

definition

of

i E ~. we have j

while

= 1 •.... Pt + I,

i

E~.

(11.72)

343

y,
Ym

(11.73)

(Pmil)

the

lI1-vector

has

BCx)

i-th

component

equal

L~Pi+l)hi (x).

to

It

follows that u is uniquely given as

u=A

-1

1

'm [ -B(z, .. ,Z,Z)

III -

(Z, .. ,2,2)

where we have expressed A(x) and B(x) in the new coordinates (Z1, .. ,zm,z).

Furthermore, the dynamics of the z-coordinates are given as (see (11.34» m

Z = f(z,zl, ... ,zm) ...

Lgj(Z,Zl, .. ,ZIll)y/Pj+lJ j

with

f

and gj' j E!E, as in (11. 34). Substituting in (11. 74a), (11. 7llb) for

Z1 the vector (Yi'"

,y:Pl l), i E!E. it immediately follows that (11.74) is

Z.

an inverse system with state N~ (see (11.3L.»

globally Yi = ..

(11.74b)

·1

defined =

yi Pi )

Remark 11,21 given above

In fact, since

z are

local coordinates for

it is concluded that in this case the inverse system is with

0, i E

=

Let us

~,

coincides

and

space

state

for

with the zero dynamics.

finally

indicate

the

connections

of

the

algorithm

for computing an inverse system with the dynamic extension

algorithm as dealt with in Chapter 8 (Algorithm 8.18). us show that p

In particular let

as defined in (11.65) coincides with q,

the rank of the

system, as defined in the dynamic extension algorithm (cf.(8.BB». The key observation

is

as

follows.

Consider

extension algori thm. We obtain (cf. , =

Therefore

we

V

can

=: u

the

first

step

of

the

dynamic

(8.67)

1

(11.75)

replace

in

the

feedback

defined

in

Step

2

of

Algorithm B.18 (11. 76) 1

the variables iJl (i.e., u

l

and its time-derivatives) by (/)(P

11)

and its

time-derivatives. The same holds at the next steps, i.e. the dependency of the feedbacks

in Algorithm 8.18 on

f/

can be regarded as dependency on

time-derivatives of the output functions. Therefore these feedbacks reduce to

the

feedbacks

as

used

in

the

above

algorithm

for

constructing

an

344

inverse system (see also Exercise 11.11 and the references). Let now Xo be in the open and dense

submanifold where

q

is well-defined.

Then it

readily follows that

.. :~

q

i'

-i" )

rank D (x,U

(11.77)

p

(cf. 8.78), where 1* is the least integer such that ql* ~ q£~+l(- q"). Notes and References The notion of zero dynamics was first identified, single-output case, in (BIll and [Mal,

in the single-input

in particular in connection with

high-gain feedback. (The relation between the maximal controlled invariant distribution and linear zero dynamics was previously stressed in

[IKGM).)

Applications

of

the

notion

of

zero

dynamics

to

(IGJ,

feedback

stabilization and adaptive control were already obtained in (BI2,3]. The general definition of constrained dynamics, and zero dynamics for r· - m, was given in [11-1]; see also [vdSl] for Hamiltonian constrained dynamics. The version of Algorithm 11.6 as given in Remark 11.12 was formulated in

[1M]; it is

a

generalization of Hirschorn's algorithm [Hi], which in turn

is based on the linear structure algorithm [SilJ. The equivalence between Algori thm 11.6 and its version in Remark 11.12 has been shown in [vdS4, 5] , see also

[BI4].

Propos i tion 11. 13 for p < m can be found

in

rvdS2] .

Algorithm 11.3, which is a coordinate-free version of Algorithm 11.6, was first formulated in

r BI4]. The definition of zero dynamics for r'" < m is

taken from !vdS3. 5]. The idea of exploi ting zero dynamics for feedback stabilization was further elaborated in [814,5,6]; Example 11.17 is taken from

[BI6).

The

observation

that

the

uncontrollable

modes

of

the

lienarization are contained in the linearization of the zero-dynamics is due to (814). The algorithm described in equations (11.57)-(11.68) is

a

version of Singh's algorithm lSi]; see [1M} for a clear exposition of this algorithm, and IdBGH) for a somewhat alternative formulation. The present treatment based on interconnections was inspired by [vdS4]. The definition of inverse system is taken from [Hi], [1M)

[1M].

Example 11.20 is taken from

The equivalence of Singh's algorithm with

the dynamic extension

algorithm has been shown in ldBGM].

[BIlJ

C Byrnes, A. Isidori, "A frequency domain philosophy for nonlinar systems, with applications to stabilization and adaptive control", Proc. 23rd IEEE Conf. Decision Control, Las Vegas pp. 1569-1573, 19BII.

345

[BI2 [

[B13]

C. Byrnes, A. Isiciori, "Asymptotic expansions, root-loci and the global stability of nonlinear feedback systems", in Algebraic and Geometric Methods in Nonlinear Control Theory (eds. H. Fliess, M. Hazewinkel), Reidel, Dordrecht, pp. 159-179, 1986. C. Byrnes, A. Isidori, "Global feedback stabilization of nonlinear systems", Proc. 24th Gonf. Decision Control, Ft. Lauderdale, pp.

1031-1037, 1985. [BI4]

C. Byrnes,

A. Isidori,

"Local

stabilization

of

minimum-phase

nonlinear systems", Systems Control Lett. 11, pp. 9-17, 1988. [BI5] [BI6]

[dB]

[dBGH] [DP] [Hi] [IKGH}

(IH]

{KI] (Ma] [Ho] [Nij]

lSi] [SilJ [vdSl]

[vdS2]

[vdS3]

[vdS4] [vdS5j

C. Byrnes,

A. lsidori,

"Attitude

stabilization

of

rigid

spacecraft", Automatica. C. Byrnes, A. Isidori, "New results and examples in nonlinear feedback stabilization", Systems Control Lett. 12, pp. LI37-4 LI2, 1989. M.D. di Benedetto, "A condition for the solvability of the nonlinear model matching problem", in New Trends in Nonlinear Theory (eds. J. Descusse, H. Fliess, A.lsidori, D. Lebargne), Lect. Notes Contr. Inf. Sci. 122, pp. 102-115, Springer, Berlin, 1989. H.D. di Benedetto, J .W. Grizzle, C.H. Haag, "Rank invariants of nonlinear systems", SIMI J. Contr. Optimiz. 27, pp. 658-672, 1989. B. D'Andrea, L. Praly, "About finite nonlinear zeros for decouplable systems", Systems Control Lett. 10, pp. 103-109, 1988. R.H. Hirschorn, "Invertibility of mu1tivariable nonlinear control systems", IEEE Trans. Automat. Contr. AC-2l\, pp. 855-865, 1979. A. Isidori, A.J. Krener, C. Gori-Giorgi, S. Honaco, "Nonlinear decoupling via feedback; a differential geometric approach", IEEE Trans. Autom. Contr, AC-21, pp. 331-3l\5, 1981. A. Isidori, C.H. Moog, "On the nonlinear equivalent of the notion of transmission zeros", in Modelling and Adaptive Control (eds. C.I. Byrnes, A. Kurzhanski), Lect. Notes Contr. Inf. Sci., 105, , pp. 146-158, 1988. A.J, Krener, A. Isidori, "Nonlinear zero distributions", Proc, 19th IEEE Conf. Decision Control, Albuquerque, pp. 665-668, 1980, R, Marino, "High-gain feedback in nonlinear control systems", Int. J, Contr. L12, pp. 1369-1385, 1985. C,H. Haag, "Nonlinear decoupling and structure at infinity", Hath. Control Systems, 1, pp. 257-26B, 198B. H, Nijmeijer, "Right-invertibility for a class of nonlinear control systems: a geometric approach", Systems Control Lett. 2, pp. 125-132, 1986. S.N. Singh, "A modified algorithm for invertibility in nonlinear systems", IEEE Trans, Automat, Contr. AC-26, pp. 595-598, 1981. L.H. Silverman, "Inversion of multivariable linear systems", IEEE Trans. Automat, Contr" AC-14, pp, 270-276, 1969. A.J. van der Schaft, "On feedback control of Hamiltonian systems" in Theory and Applications of Nonlinear Control Systems (eds, C.I, Byrnes, A, Lindquist), North-Holland, Amsterdam, pp. 273-290, 1986, A.J, van der Schaft, "On realization of nonlinear systems described by higher-order differential equations", Nath, Systems Theory 19, pp. 239-275, 1987, A.J. van der Schaft, "Realizations of higher-order nonlinear differential equations", Proc. 25th IEEE Conf. Decision Control, Athens, pp, 1569-1573, 1986, A.J, van der Schaft, "On clamped dynamics of nonlinea,· systems", Hemo nr, 634, University of Twente, 1987, A.J. van der Schaft, "On clamped dynamics of nonli"ear systems" in Analysis and Control of Nonlinear Systems (eds, C.l. Byrnes,

346

C.F. Hartin, 1988.

R.E. Saeks), North-Holland.

Amsterdam,

pp.

499-506,

Exercises 11.1

Consider suppose

the

system is

Xo

a

(11.10)

regular

with

point

distribution 6- - ker span(dh 1

•...

0

f(x o ) -

for

and

O.

and

Define

the

h(xo )

11.6.

Algorithm

,dh pk * J

Show that 6· is an invariant distribution for the vec tor field

(n)

m

W

+

Igja;. where a is any solution of (11.18). j"'l (b) Is 6* a controlled invariant distribution for (11.10)?

f

11.2

Consider the system (11.10) with f(x o ) = 0, h(x o ) - 0, that Xo

is

a

regular

point

for

Algorithm

7.19

as

and suppose well

Algorithm 11.6, yielding respectively the distribution D~,

11.3

for

Show that dim N ~ dim D*. W

ft

submanifold N

as

and the

•

Assume that the conditions of Proposition 11.13 are satisfied. Show ft

that Algorithm 11.6 yields N as given in (11.27). 11.4

Let

D be

a

distribution

which

is

invariant

for

x - f(x).

i.e.

[f,D] C D. Suppose f(xo) E D(xo ) for some xo' Let N be an integral manifold of D through xo' Prove that N is an invariant manifold for .~ - f(x). 11.5

[DP] Assume that the conditions of Proposition 11.13 are satisfied, and that f(x o ) - 0, h(xo ) = O. Show that the linearization of the zero dynamics at Xo equals the (linear) zero dynamics for the system (11.10) linearized at xo'

11.6

Show that in general

[BI4j

not

involutive}

(i.e.

also i f span(gl(x)' ...• gm(x)}

the uncontrollable modes

of

the

is

linearization of

(11.34) are modes of the linearized zero dynamics (11.49). 11.7

(see Example 11.20) Show that the constrained dynamics for (11.70)

is trivial; i.e. N~ - (O). 11.8

(Output x(O) -

cracking) xo'

satisfied.

Yo (t), (a)

C::?:

Show

Asstune

Consider the system that

the

(H.lO)

conditions

U

Proposition

11.13

are

O. chat

this

is

possible

us ing

A-1(zl, ... ,zm;Z)(_B(zl, ... ,zm,z) ...

the

control

[~~~l+l) j) CPm i1 )

if

initial state

Suppose one wants to track a desired output trajectory

(1l.74a» (1)

of

with

Ydm

(compare

with

347

j Em.

(2 )

(Here

(b)

Z are

i E~, and

Zi,

as in (11.34».

Consider instead of (1) the control law u ~ A-1(zl, ... ,zm,Z){_B(zl, ... ,zm,z)
~dl

+

(3 )

[

y
+

the

1J

(t)

+ a

1

+ 810(Ydl-Yl)

clm

+

m

ai(s) = SPi+1

+

a

'P,

SPl

8

mO

(Y -Y

+ ... +

drn

rn

l)

)

i

aiD'

E~,

Show that the following higher-order differential

equations are satisfied for e(Pi+

+

(y _y )
polynomials

are all Hurwitz.

)
+ a

dm

where

(

a lPl

+

Vi

=

Ydi

e
Yi'

~,

1 E =

0,

i E!!!,

i

lPi

and thus that e 1 (t)

-

+ aiOet(t)

-4

0, for t

-> "",

i E

!!!. Therefore, if (2) is not

satisfied we obtain asymptotic output tracking. 11.9

[dB]

Consider the nonlinear model matching problem

(I-U1P)

that was

defined in Exercise 7.10, If we instead require the existence of an initial state xeO of Q such that ypoo{x o ,XeO ,t) - Ym{XmO ,t) '" 0 for all

t,

we refer to the problem as the local strong model matching

problem (SHHP). Assume the constrained dynamics manifold augmented system (a)

~u

(cf. Exercise 7.10) exists.

Show that the local SHHP is solvable from any xoD E

only if the local l-ll1P is solvable around any (c)

of the

Show that the local SMMP is solvable from xoo:= (XO,xmO ) if and

only if xoo EN:. (b)

N:

N:

if and

xoo.

Prove (a) and (b) for a square multivariab1e plant and model.

For (b) asswne that the plant has nonsingu1ar decoup1ing matrix. 11.10 [Mo]

Consider the algorithm for computing an inverse system,

i.e.

(11.55)-(11.69). Define P~

=

p

P~

=

p -rk -1 '

k

~

2,

and n~

:= number of

p~s

that are greater than or equal to i, i
This defines an alternative formal structure at infinity; prove that for a system with nonsingular decoup1ing matrix it coincides with the formal structure at infinity defined in Chapter 9. 11.11 ([dBGH]. p

[Nij]) Prove (11,77). Furthermore, show that =

a('Y,Y,···,Y . (n» a(u,~, .. ,u
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.
(l2.6)

x

Jj the n

III

matrix

1m B

(12.7)

J'

~

0

yielding the 2n-dimensional standard state space system

(12.8)

Example 12.1

Consider the rigid frictionless two-link robot manipulator

1.1.

treated

in

ql =

qz = Oz (relative angles), and as controls u 1 and

°

1 ,

Example

with

generalized

configuration

coordinates

Uz the torques

at the joints. Altern
D

Equations (12.8) constitute a class of mechanical control systems which is often encountered in applications. However control or input variables in mechanical systems do not necessarily have to appear as external forces as in (12.ll) or (12.8). as is illustrated by the following simple example.

Example 12.2

Consider a linear mass-spring system attached to a moving

frame

Fig. 12.1. Moving linear mas.%pring system.

where the input u is the velocity of the frame. energy

.!1II(q + 2

function

U)2

depends

directly

L(q,q,u) = .!:m(q + u/ - -21kq2, 2

(cf. (12.3) with F~ = 0)

on

u,

and

yielding

In this case the kinetic so the

does

the

Lagrangian

equations

of motion

351

aL

d dt

In

aL

.

(-,-(q,q,u)) aq

general,

for

.

.

aq(q,q,u)

a

m(g + u) -

=

mechanical

system

Lagrangian L(q,q,u) depending directly

other external forces, d dt

[~(q,q,U)l aqi

-

of

leq

n

o

0,

=

degrees

of

u we obtain,

DO

freedom

with

a

in the absence of

the equations of motion

~L

(q,q,u)

=

0,

(12.9)

1 E n.

qi

(12.4) can be regarded as a special case of (12.9) by taking

Notice tha t

in (12.9) the Lagrangian m

Lo(q,rJ) +

L(q,q,u)

LqjU j

(12.10)

.

j "1

We will call (12.9) a Lagrangian control system. Let us now pass on to the Hamiltonian formulation.

For the Lagrangian

control system (12.9) we define the generalized momenta

aL

.

p, - -.-Cq,q,u), aqi In general singular Lo

=

the

i

n x n

everywhere

(12.11)

matrix with

(for

example

implying that P

T - V),

n.

E

(i,j)-th element

if L

is

as

a'L Bqi aqj

defined

in

will

be

OOll-

(12.10)

with

(Pl""'P n ) are independent functions. One

=

now defines the Hamiltonian function H(q,p,u) as the Legendre transform of L{q,q,u),

i.e.

H{q,p,u)

" L

=

I

q

where

(12.12)

Piqi - L(q,q,u),

~l

and P

are

related

by

(12.11).

Since

by

(12.11)

the

partial

qi

are all

derivatives of the right-hand side of (12.12) with respect to

zero, one immediately concludes that H indeed does not depend on well-known

that

with

(12.11)

and

(12.12)

the

Euler-Lagrange

q.

It is

equations

(12.9) transform into the Hamiltonian equations of motion q1 =

all

~(q,p,u),

(12.13a) i

E n.

all

Pi ~ - ~(q,p,u), Indeed (12.13a)

(12.13b) follows

(12.13b)

follows from

from

(12.12).

substituting We

call

(12.11)

(12.13)

a

into

(12.9),

Hamiltonian

while

control

352

system. A main advantage of (12.13)

in comparison with

(12.9)

is

that

(12.13) immediately constitutes a control system in standard state space form,

with state variables

(in physics usually called

(q,p)

tbe phase

variables). Moreover, as we will also see later on, the variables q and p are

conjugate variables

and

the

Hamiltonian H(q,p,u)

can be

directly

related to the energy of the system. In particular, if L(q,q,u) is given as in (12.10) then it immediately follows that m

R(q.p,u) - Ro(q,p) -

r

(12.14)

qjU j

j '" 1

with Ho(q,p) the Legendre transform of Lo(q,q). V(q) with l' as in (12.5)

110 (q,p)

2'1

=

T-1

pH

(q)p

If LD is given as T(q,q)

then it follows that (since p - If(q)q)

+

(12.15)

V(q),

which is the internal energy of the system.

Example 12.3

Consider again

the

Example 1.1. Denote 01 = ql' 02 •

l' T

=

two-link rigid robot

manipulator

.

T(q,q) = zq H(q)q, with }f(q) as in (1. 7a). Take for simplicity m1

and

1\ -

-

mz "" 1

1'2 - 1. Then the generalized momenta are given as

aT - oql

Pl

from

qz. The total kinetic energy is given as

aT

Pz

-

(3 + 2 cos qz );11 + (1 + cos q2. )Ch.

= (1 + cos qz )Ch + qz·

8qz while the Hamiltonian H(q,p,u) is given as (see also Example 3.40) 1 2.

T

P /'f

~l

(q)p + V(q) -

(1 -I' sinZq2)-1(~P12 2.

q1U 1 - qzu z

~

(1 + cos qZ)PIPZ + ~(3 + 2 cos qz)p/)

z

o Example 12.4

l'

Consider the system of Example 12.2 with L - im(q

! kq2. We obtain p z

i;pZ

+ ~

m(q

+ u) and H(q,p,u) = Ho(q,p)

o

Consider Ie point masses

mi' with positions ql

in their own gravitational field corresponding to V(ql,

..

up, with Ho(q,p) ~

JcqZ being the internal energy.

Example 12.5

,l) ...

+ u) 2- -

E (RJ,

i E ~.

the potential energy

rmimj/llqi_qjll. Suppose the positions of the first £ masses i <j

353

(J! :$ k) can be controlled. We obtain a Hamiltonian control system

aH aq~

,i

=

1+1, ... ,Ie, j

~ 1,2,3,

In this chapter we mainly confine ourselves to affine Hamiltonian control systems,

i.e"

Hamiltonian control systems (12.13) with a Hamiltonian of

the form

L

ll(q,p,u) - Ho (q,p) -

Hj (q,p)u j

(12.16)

,

~1

j

where Ho (q,p) is the internal Hamiltonian (energy) and Il j (q,p), j

E ::!' are

the interaction or coupling Hamiltonians. We now come to the definition of the natural outputs of a Hamiltonian

control system 02.13), which we define as

Yj

=

-

aH ---a

j

(q,p,u),

E~.

(12.13c)

"j

In particular j

and

E~,

we

for

an affine

obtain

Hamiltonian

affine

the

(12.16)

llamlltoni.an

we have Yj

input-output

=

Hj(q,p),

system,

or

briefly Hamil conlan system

-

q,

all o ap, (q,p)

j

aHo aqi (q,p) +

p,

-

api (q,p)u j

,

~1

all j aqi (q,p)u j

I j

Yj

all j

- I" ~

E

(12.17)

!!,

,

1

j

lfj(q,p),

i

E m.

There are several reasons for adopting this definition of natural outputs. First

in

this way

duality between inputs

and

outputs

is

induced.

For

instance, if u l , ... ,urn are generalized external forces, then Yl'·· 'Ym will be

the

corresponding

generalized

configuration

coordinates.

(This

is

usually called the case of collocated sensors and actuators.) Secondly a strong type of symmetry or reciprocity between the inputs and the natural outputs results, as will become clear especially in Section 12.2. Thirdly, differentiating the internal energy lfo(q,p) obtain the energy balance

along the system (12.17) we

356

dH o

i

L aq: m

Pi J

ql +

1

n [alia aHa L 1"1 8qi

... 0 +

aHo

[aH O

n

dt~

aHo aHO] + aqi j

CHj CHo aqi apl lui

L L jol

n

L L o 1 i

D

1

8nk]

m

8110 all j

aHo aUJ

[- aq; ap; +

- k~i LUk -BPi

BHj

+api

aq; Ju

BPi

aHo

[- -aqi- +

...

j

L Uk

aH

k

JJ

U

j

k~l

(12.18) expressing that the increase of internal energy equals the total external

L Yj u

wot'le performed on the system (the time-tntegral of j

Exnmple 12.3 (continued) If.

The on

natural

the

0

outputs

ther hand,

are

m

j ) •

1

the

relative

the controls

angles

would be

horizontal and vertical forces at the endpoint of the manipulator,

the then

the natural outputs are the Cartesian coordinates of the end-point. cf.

o

(1.9).

Example 12.4 (continued) that in this case y

The natural output is

the momentum p.

Notice

P is the reaction force of the moving frame, so that

o

uy is again the instantaneous external work.

l

i E ..e, i. e . the GU i forces as experienced by the first 1 controlled point masses. This is an

Example 12.5 (continued)

Natural outputs are

o

example of a non-affine Hamiltonian system.

The adopted definition of a Lagrangian or Hamiltonian control system basically

only

covers

internally

conservative

mechanical

systems.

In

particular it follows from (12.18) that for u - 0 the internal Hamiltonian Ho(q,p) is a conserved quantity, expressing the conservation of internal

energy.

On

the

other hand,

in practice

mechanical

systems

always

do

possess inherent damping (which, however, is often difficult to quantify). Nevertheless 'the

conservative

idealization

(disregarding

dissipative

forces) is usually a natural starting point for analysis, as well as for control purposes. In fact the actual presence of urunodelled damping will in many ca'!:res only improve the characteristics of the controlled system, see also Section 12.3.

]55

Dissipation of energy can be included from the very beginning by adding to the Euler-Lagrange equations (12.3) a dissipation term in the following

way

i E

where

is

R(q)

called

(12.19)

~,

Rayleigh's dissipation

a

function.

This

will

be

further elaborated in Section 12.3,

12.1

We

Definition of a Hamiltonian Control System

now

want

(affine)

to

give

a

Hamil toni an control

intrinsic

definition

of

Hamiltonian system has

and

global,

ayB tern

the

fact

coordinate-free, (12. 17).

that

the

local coordinates

First state

Ql""

definition

we

have

to

of

an

give

an

space

for

an

affine

,qn ,Pl""

,P n ,

where

the

configuration coordinates ql"" ,qn are in some sense conjugate, Dr dual, to the momentum coordinates Pl"

Definition 12.6 functions

on

. ,P n .

Let N be an manifold and let C"" (1'1)

N.

A

Poisson

structure

on

N

is

a

be the smooth real bilinear

map

from

c"" Uf) x c"" UJ) into em (N), called the Poisson bracket and denoted as (F,G)

H

{F,G},

t"hich sacisfies for any F,G,H E (G,F)

IF,GI

c""un

=

the following properties

(sket,,-symmetry) ,

[F,[G,Hll + (G,(H,F)) + [H,(F,Gll [F,GH)

(12.20)

F,GEC""Ul),

{F,Glll + G(F,Hl

=

0

(12.218)

(Jacobi identity),

(Leibniz' rule).

(12.21b)

(12.2lc)

N together l"lth a Poisson structure is called a Poisson manifold.

Remark

It follows from (12.2la,b) and the bilinearity, that e""(N) endowed

with the Poisson bracket is a Lie algebra (see Definition 2.28). Now let us define for given FEe"" U1 ,tR) and arbitrary x E 1'1 the mapping XF (x): C""(1) ... tR as XF(x)(G)

:= (F,G)(x)

(12.22)

356

(F,GI(x)H(x) + C(x)(F,H}(x)

Then i t follows from (l2.21c) that XF(x)(GJl) and thus (see Definition 2.21)

X F (x) E T)Jl

for any x EN. Hence for any

F

we obtain a smooth vectorfield XF on N satisfying (12.23) XF is called the /{illIllltonian vecLorfield corresponding to the Hamiltonian

fUnction F.

Example 12.7 (ql

C(q.p)

with

natural

coordinates F(q,p),

as

is

easily

H{q.p) E c
aF ac

L

(F.G)(q,p) =

(It

~;21\

N

Let

'.··.qn.Pl"·· ,PII ) · Define the Poisson bracket of two functions

- aq;

verified

this

bracket

(12.21).)

satisfies

Let

then the Hamiltonian vectorfie1d Xu is given as

au

(JH

Xl! (q .p) ~

that

aPi)(q·P)

aH )i (q.p). .... --aqll

(-a-, ... '~a-' PI

Pn

(12.25)

Indeed

E n.

(12.26)

(H,Pi I = -

o

The bracket (12.34) is called the standard Poisson bracket on

Example 12.8

(see also Exercise 12.1) Consider N - 1

dinate functions x

~

(xl ,xZ,xJ

).

with natural co or-

Define for any F,G E BG

-a' X )ex) z

iJF

3

-I- Xl

the bracket

of ac aXl

aF aG -

aX

ac

- ax z ax 1 ) (x)

1

ax::,) (x)

(12.27)

It can be verified that this bracket satisfies (12.21). and thus defines a Poisson bracket on n;3. Consider the Hamiltonian 2 Xl

H(x) for

certain

becomes

~

2

+

Xz 28;,

...

constants

l X3

(12.28) a 1 ,a 2 .a J

•

Then

the

Hamiltonian

vectorfield

XH

357

Xl

~

x,

",

(R,xll

~

IH ,X,,- J

~

- a, 8 2 i:l3

",

-

X,,-X3 ,

a,

8]a 1

xJx 1

(12.29)

•

a, - ", Identifying

i = 1,2,3, we have obtained the Euler equations

with

Xi

for the dynamics of the angular velocities of a rigid body;

see Example

o

1.2 and Example 3.24.

The map F H X F is a linear map from CtrJUf) to V""(}1) , the linear space

of smooth vectorfields on H. Actually this map is a Lie algebra morphism

c""ut) ,

from

endowed with the Poisson bracket,

to

V"'un,

endowed with the

usual Lie bracket of vectorfields:

Lemma 12.9

For any F, G E

em (1)

1"8

have (12.30)

Proof

Let

H E C""un,

(F,(G,ll))

-

By (12. 2lb) and (12.23)

(G,IF,H}) =

([F,G)'H) ~ X{F,G){ll).

o

A diffeomorphism rp: N ... N for a Poisson manifold N is called a Poisson

automorphism if (Forp,Gorp)

=

(F,G)orp,

for all F,G

E

emU!),

(12.31)

and a vectorfield X on N is called an infinitesimal Poisson automorphism if the time L-integrals all

t

for which

xt

xt

of X (cf. (2.77)) are Poisson automorphisms for

is defined, i.e. for any F,G for

Lemma 12.10

(12.32)

t :! D.

Let 1'1 be a Poisson manifold.

A vecLorfield X on 1'1 is an

infinitesimal Poisson automorphism if and only if X«(F,G}) Proof

=

(X(F),G) + (F,X(G)),

First note that for any to

ro for all F,G E C (I'1).

(12.33)

358

(12,34) ( F,G)

left-hand s ide of (12.34)

vanishes.

Taking to

0 in

and thus the

I

(12.34) we obtain

(12.33). Conversely i f (12.33) is satisfied then the right-hand side of is

(12,34)

zero

for

co,

any

and

elms

(FoXL,GoXt)oX-

t

which

(F,GI,

implies (12.32).

Notice

D

that for a Hamiltonj.an vectorfield XII

XIIIF,GI -

(1f,(F,G))

=

IXu(F),GI + (F,Xu(G)I,

+- IF,IH,G))

IIH,F),G)

the Jacobi-identity yields and

thus Xu satisfies (12.33). Hence any Hamiltonian vectorfield is necessarily an infinitesimal Poisson automorphism. Let 11 be a there

exist

Poisson manifold with

locally

smooth

local coordinates

Xl""

l , j E!::,

such

functions

h i j (x), 1

,xr

.

Then

that

the

Poisson bracket is given as r

L

(F,G) (x)

1, j

\v1 J (x)

BF

BG

- a (x) -a (x). v

~l

(12.35)

""J

Xi

Indeed, since

{F,G)(x)

(XFG)(x) - dG(XF ) (x) , and

(F,G)(x) -

-

(G,FI(x)

(XGF)(x)

=

-

dF(Xr:;)(x).

the Poisson bracket (F,G)(x) only depends on dF(x) and dG(x) , and thus is of the form (12.35), Furthermore from (F,G) = -

i,J

(G,FI it follows that

(12.36)

E ~.

Also note that w'J is directly determined by

i,J E r, and from the Jacobi-identity

(Xi

I

(x j

IX):

(12.37) \1 .+ (xj

, IXk ,Xi))

+

IXk

I

(x, ,Xj II

=

0

it thus follows that

i,j,k E

Conversely,

if some functions

then by (12.35)

\"'1

J'

1,J

E ~.

~.

(12.38)

satisfy (12.36) and (12.38).

they define a Poisson bracket. We conclude that locally

any Poisson bracket is determined by a skew-symmetric matrix

359

"(x) -

(12.39)

(W'j (x») i-I, _ ... , r j-l, ...• r

with wij(x)

satisfying (12.38),

The matrix [/Cx)

is called the structure

matrix. The raille of the Poisson bracket in every x E H is simply defined as

the rank of the structure matrix [v(x).

necessarily

its

rank

is

even.

A

Poisson

Since r.,r(x)

is skew-symmetric

braclcet

said

degenerate if rank [vex) = dim N for every x E tJ.

is

to

be

no/]-

In particular for a 001l-

degenerate Poisson bracket we have for any x rank [v(x)

=

dim N "" 2n, for some

Example 12.7 (continued)

(12.40)

11.

In this case (12.41)

X E jR2n,

o

and thus the standard Poisson bracket is non-degenerate.

Here

Example 12.8 (continued)

[vex)

=

l-~,

xJ

0

Xz

-x, Xl

1

(12.42)

0

-Xl

which has rank 2 at any point

(Xl

,xz ,XJ

)

P!

o

(0,0,0).

The following theorem shows that locally every non-degenerate Poisson structure is as the standard Poisson bracket as given in Example 12.7.

Theorem 12.11

Let

N

be

a

2n-dimensional

manifold

Poisson bracket (, J. Then locally arvund any Xo E nates

(q,p) '" (ql, ... ,q",Pl""'P,,),

}l

hl'ith

non-degenerate

I"e can

called canonical

find coordi-

coordinates,

such

that [F,G) (q,p) i

Remark

BF BG BF I" (aPi ~ - ~ ~1

BG BP1JCq,P)

(12.43)

Equivalently, ql"'" q" ,P!, ... ,p" are canonical coordinates if and

only i f

(12.44)

0, i ,j En.

In the proof of Theorem 12.11 we use

Lemma 12.12

Let G1, ... ,G", }{ E caJUl) , and F E

c'D CjR ").

Then

360

k

(F (G 1 , . . . ,Gk

Proof

) • H)

L

(x)

(x), .. ,

Follows immediately from the coordinate expression (12.35).

Take any function ql wi th dql (xc) '" O.

Proof of Theorem 12.11

Poisson bracket is non-degenerate it follows that Xq1 (X O ) Flow-Box Theorem (Theorem 2.26) there exist coordinates such

Xo

(12.45)

,Gk (x» IG i ,HI (x)

a = ax"

that

Denote

J06

Since the

O. Then by the

x;,...

It

Pl'-

0

around

follows

that

1

{Pl.ql) -

xJ

••••

,X211

l(Pl) = 1.

-{ql,PI} -

O.

XCP1.Cll}

Hence

by Theorem

Furthermore by Lemma 12.9.

2.26

we

can

find

independent

[XP1,Xqll-

functions

such that (12.46)

for

i

~

3, ... ,2n.

ql.Pl.X 3 •... ,X Zn

Since

follows

it

that

form a coordinate system around xo' Then by the Jacobi-

identity (12.47) and thus by Lemma 12.12

o

(12.48)

It follows that the functions

{Xl 'Xj }.

i -

3 •...• 2n do not depend on qt

and PI' so that

(12.48) for

some

functions

Vi J

Poisson structure on 1R

Recall

2n

-

i.J -

I

z

3 •...• 2n.

which

define

a

non-degenerate

. The theorem now follows by induction to n.

0

that for any Poisson bracket the Hamiltonian vectorfields are

necessarily infinitesimal Poisson automorphisms. The converse is generally not true as illustrated by

Example 12.13

Consider

[RJ

with natural

coordinates

x

=

(Xl • x;! ,x3 )

and

Poisson bracket {F,G)(x) -

BF

(ax2

BF

- ax

BG

l

ax) Z

(x)

(12.49)

361

Then X

=

is easily seen to satisfy

-"-

ax)

simal

Poisson automorphism.

since

this

1 ~ ,-\:)

=

would

(1l,x 3

require

(12.33),

However X is

the

not

existence

of

However by (12.49) (ll,x J 1

).

a

and so

a

is

an infinite-

Hamiltonian vector field

function

H(x)

satisfying

o

0 for any If,

=

The following proposition shows that for non-degenerace Poisson brackets

the converse does hold, at least locally. Let N be a Poisson manifold I,'ith non-degenerate Poisson

Proposition 12.14 bracket.

the vectorfield X on N be

Let

an

infinitesimal

Poisson

auto-

morphism. Then locally around any Xo E N there exists a function 11 such

that X

=

XH ,

Proof

By

Theorem

(ql , ... ,qn ,PI""

nates as X

12.11

we

can

take

i~rite

,Pn) around xo'

(xi, ... ,x;,xi, ... ,x~)!.

=

canonical

coordinates

(q,p)

=

the vectorfield X in these coordi-

Since X is an infinitesimal Poisson

automorphism we obtain for any i ,j E n 0= X(6 i j

)

axr

axJ

apj +

aq;'

Similarly

axf

axf

aqj + aqi'

xi, ..

Thus the one-form represented by the row-vector (-Xi, .. ,-x~,

closed

-X~

=

1

(cL

2.163),

~ Xq aqi'

i

=

~

api'

and

hence

locally

degenerate PI" again

Poisson

!p:

n ... n

coordinates

canonical

that

for a non-degenerate Poisson bracket is

bracket

coordinates,

Also,

a coordinate

transformation S

(q ,p)

into

. ,P n ) is called a canonical

i , j E 11.

H(q,p)

is

,X;)

such

o

usually called a canonical mapping. canonical

exists

i E n.

A Poisson automorphism

mapping

there

i.e.

new

coordinates

(Q,P)

(Pi,Qjl

=

Ojj'

(Qi,Qjl

(Ql""

=

coordinate transformation

if (Q,P) =

0

=

,Qn'

are

(Qi'P j

),

362

For later use, see Section 12.4, we mention that to any non-degenerate Poisson

structure

dual,

another,

geometric

object

can

be

associated.

Namely, let (, I be a non-degenerate Poisson bracket on N, locally given by the skew-symmetric structure matrix: rl(x). Deftne then the bilinear map (12.50) by setting

(12.51) Although

W:x;

is in principle now only defined on tangent vectors of the

form XF (x) E T,/1 for F E c"'(tn it follows by linearity and ehe fact that we

can choose

2n functions

F1

, •••

independent that (12.51) defines

Wy;

,F2n

such

that XI" 1 (x) , ... 'X F 211 (x)

are

completely as a bilinear map (12.50).

Since in local coordinates the column vector Xn(x) for H E CW(N) is given as (dJl(x)wx)T, with dH(x) a row vector, it follows in view of (l2.3S) that Wx

has the matrix representation (l2.52)

By letting x vary we obtain a so-called differential two-form w, which is called a symplectic form on N.

In canonical coordinates (q,p)

equals

Wx

the constant matrix W(x) in (12.41). Furthermore note that by (12.51), and -(G,F} = -XG(F)

(F,G) ...

-dF(G)

I

the Hamiltonian vectorfield Xli

corre-

sponding to the Hamiltonian /{ is uniquely determined by the relation W"

(XII (x) ,Z) - -dH(x) (2).

for any Z E Tr.N, x E l1.

The manifold If endowed with the symplectic form

W

(12.53)

is called a symplectic

manifold. We are "now able to give a coordinate free definition of a Hamiltonian control system (12.17),

Let N be a manifold Idth non-degenerate Poisson bracket.

Definition 12.15 Let Jio ,H 1

x

' •••

,11m

E Crt> (/'/).

~ XlIo(x) -

L

Then

XUj(x)u J ,

j =1

(12.54)

x E fl. j E is

:!!.

an affine Hamiltonian

system.

input-output

system,

or briefly.

Hamiltonian

363

With the aid of Theorem 12.11 we immediately obtain Let (12.54) be a. Hamiltonian system on 1'1, Then around any

Corollary 12,16 Xu

there exist canonical coordinates ql'" ,qn ,Pl'" 'P n for l'l, such that

(12.54) takes the form (12.17).

Remark 12.17 Hamiltonian

Sometimes

it

by

system

is

useful

requiring

that

to lIo

relax

is

only

the

definition

locally

of

defined,

or

equivalently (see Proposition 12.14) that the Hamiltonian system is given

- Xo (x) - llj (x),

x

Yj

I

Xl!j(x)u j

,

~1

j

j

E

(12.54')

~,

with Xo an infinitesimal Poisson automorphism.

Remark 12.18

If the Poisson bracket in Definition 12.15 is

degenerate,

then (12.54) will be called a Poisson system.

12.2

Let

Controllability and Observability; Local Decompositions

us

consider

a

Hamiltonian

system

(12.54).

By

Definition

3.29

the

observation space 0 is the linear space of functions on N spanned by all repeated Lie derivatives

J with

Xi'

(F,ll::,

J

i E~,

in

for all FE

Proposition 12.19

the

E~,

set

emu!),

k

=

(12.55)

0,1,2,.

(Xn,-Xul"",-XurnJ.

Now by

(12.23)

LxfH j

=

and thus we immediately obtain

The observation space 0 of (12.54) is the linear space

of functions on 1-1. containing 111 •... ,Urn and all repeated Poisson brackets

(12.56)

JEE,kEIN, Idth F i

,

i

E~.

in the set (Ho,H1, ... ,Hml.

With regard to controllability of 3.20

every

element

of

the

Lie

(12.5'1),

algebra Go

we note

that by Proposition

characterizing

local

accessibility (see Theorem 3.21) is a linear combination of elements

strong

364

E~,

j

k

~

0.1, ... ,

(12.57)

where Xi' i E!5. is in the set (X!![J'-XIf1 •... '-Xllm , . Using Lemma 12.9 and Proposition 12,19 we. immediately obtain Proposition 12.20

Every

element

Hamileonian vectorfields Xr

Remark

Itlith

By Lemma 12.9 it also

of

is

Go

linear

a

combination

of

FED.

follows

that every element of Go

is

a

Hamiltonian vectorfield. Remark

Proposition 12.19 en 12.20 also hold for Poisson systems.

Since the Poisson bracket for a Hamiltonian system is non-degenerate it follows that the kernel of the map F consists precisely of

the

constant

H

Xr defined by the Poisson brack.et

functions

on N.

From Propositions

12.19, 12.20 and Theorem 3.21, Corollary 3.23, Theorem 3.32 and Corollary 3.33 the following is now immediate.

Proposition 12.21

Suppose dim dO(x o )

2n

(~dim

i'J)

for

the llamlltonian

syscem (12.54). Then the system is locally observable at x o ' and locally strongly accessible at xo' Conversely, assume that dim dO(x) = constant. Then (12.54) is locally observable if and only if it is locally strongly accessible.

Example 12.22

Consider the two-link robot manipulator from Example 12.3

(see also Example 1.1), with for simplicity Hamiltonian

H(q,p,u) = Ho(q,p} -

}{1(q,P)u 1

1111

rIIz

II z (q,p)u z

~

..1'1

is

..I'z

as

~

1.

given

The in

Example_12.3. It is easily seen that the observation space 0 corresponding to

]{o

,H 1 ,H2o satisfies dim dO(q ,p) - 4 for all points (q ,P)

I

and thus the

system is locally observable, as well as strongly accessible. If we only take one input, say u2

,

and the corresponding output y - qz.

then things become lIlore complicated. However it can be checked that in this case still dim dO(q,p)

4 everywhere, see also Exercise 12.B.

o

Let us now return to the characterization of the observation space 0 of a Hamiltonian system. as given in Proposition 12 19. Comparing this characterization with the characterization of

(;'0

as given in Proposition 3.20

and replacing Lie brackets with Poisson brackets it immediately follows from Definition 3.19 and Proposition 3.20 that 0 is alternatively given follows.


365

Proposition 12.23 is

The observation space 0 of a Hamiltonian control system ttJ

the smallest subalgebra of C (1)

contains HI'"

(under the Poisson

brackec)

~

Let us introduce some additional terminology. A collection functions on a symplectic manifold i)

ii)

'f1

is a linear subspace over

Let

which

"Hm and satisfies {lID ,F) E 0 for all FED.

F I , ...

[R

n

of smooth

will be called a funcClon space if

(12.S8a)

of C
,F~ E '5, and G:[RB .... IR be a smooth function,

then

(12.58b) Furthermore, we call 'J a function group if also the following holds. iii) Let Fl,F'l. E 'J., then IFl ,Fz } E '!I, ('!! is a subalgebra of

Given some functions Fi' i E

span IFi

;

(12. SSc)

c""CH) under the Poissonbracitet) I (index set), on H we denote by (12.59)

i E I)

the smallest function space

in Cro(H)

containing Fi'

i E I.

Recall from

Chapter 3 (Definition 3.19) that the subalgebra eo defines a distribution Co. In an analogous way the subalgebra 0 c Cro(H) now defines the function

space '}O := span O. In fact, since 0 is an algebra it immediately follows that'} is a function group (just like Co

is an involutive distribution).

Furthermore note that dO(x)

=

d'JO(x),

and in particular dim dO(x) We now want to

(12.60)

for any x E N, =

dim d'JO(x) for every x E N.

give a Kalman decomposition of a Hamiltonian system

similarly to the decomposition of a general nonlinear system as obtained in Theorem 3.51, but adapted to the Hamiltonian structure. The following theorem

is

crucial

in doing this.

First we

introduce

for

any

function

space'} its polar group '}~ as

'}~

-

(G

I

E Cro(H) [G,F)

=

0, VF E ;/).

(12.61)

It immediately follows that '}~ satisfies properties (12.58a,b,c) and thus that :f~ is also a function group.

Indeed,

let G1 ,G" E '}~ and K :

!p." -+ [R.

Then by the Jacobi-identity (12.62)

for any F E

';J,

and thus (G 1 ,G,,) E '}~. Furthermore by Lemma 12.12

366

(12.63)

Lee f1 be a manifold t"ith non-degenerate Poisson bracket,

Theorem 12.24

and let' be a function group on H. Assume that dim that dim d('J n :7"1) (x) - constant. Let dim d:1

=

d~(x)

= conscant, and

Ie, and dim d('7f n 'J1) = r.

Then locally there exist canonical coordinates (ql, ... ,qn'Pl , .. "P o ) for N such chat (12.64 )

span I ql , ... , ql' Pl , ... ,p l' P 1+1 •... ,p i+r J • Ideh 21

Proof

r

'1'

~

Since

Ie. dim

d'!J

~

Ie

we

can

locally

FI ' ... ,F'); E '!J such that '!J = span{F 1

t

Ie

find

,Fk

•••

independent

functions

Since 'J is a function group

).

i t follows that

i,j E ~,

for

some smooth functions

these

functions

define a Poisson structure { , that

IRk ... IR.

tl"ij

= r

dim

it

of

Exercise 12.2), " .. ,Pl,P i

-l'

l

""

Theorem

it follows

and

and

(12.38).

that thus

k

immediately where 21

12.11 that

immediately checked

(12.36)

on IR • Furthermore by the assumption

)[Rk

structure has constant rank 21, generalization

It is

satisfy

t,'i.1(yl •...• h)

(12.65)

to

follows k - r.

degenerate

there exist

that

this

Poisson

Then by an immediate Poisson

coordinates

brackets q~,

..

(see

,qQ,p~,

..

t . ,Pi+r on IRk sa t"1S f y1ng

(12.66)

i=l, ... ,f,

(12.67) i = 1, ... ,f + r. In view of (12.66)

the functions

qi'

i

~.

Pi'

i E~.

are pClrt:ial

canonical coordinates for N. A straightforward generalization of Theorem 12.11 q f+1

then

yields

' ••• • qn • P .Q+r+l ' •••

che

,Pn

existence ,

of

additional

and (12.64) resul ts .

canonical

coordinates

o

367

Now let us return to

function group ']0 defined by

the

the observation

space O. Recall that in Theorem 3.51 we assumed that the distributions Co' ker dO and Co + ker dO all have constant dimension. By Proposition 12.19 and

(12.60)

is

it

immediate

and

if

only

d('J

dim

O

then

Co

Furthermore it immediately follows

the definition of '?J~ (see (12.61»

if

if dim Q:}O(x) - constant

that

ker dO have constant dimension.

that Co + leer dO has constant dimension

'J~) (x) ~ constant.

n

and

from

assumptions made in the following theorem,

This

which is a

motivates

the

generalization of

Theorem 3.51.

Theorem 12.25

Consider

a

Hamiltonian

~

system

(12.54)

lo'ith

observation J.

Ie and that: dim dC10 n :1 ) (x) 0 constant - r. Then locally there exist canonical coordinates

space D. Assume that dim &O(x)

(ql'"

,qi ,qjl+l'"

=

,qi+r,qjl+r+l'"

const:ant

~

,qn ,PI' .. ,Pi ,P i + l , · · 'PjI+r,PjI+r+I"

. ,Pn)

(12.68) l"here k

=

2i + r, such that

(12.69) Furthermore Ive have

(12.70) Ivhich implies that lIQ is locally of the form

(12.71) for some smooth map f

r

-> [Rr.

Noreover Hj

E :1 , j E:!.I'

only depend 0 In the coordinates (12.68) the system talces the form

ql, q'1. ,pI

: IR

on

au l

o I I '1. -,-(q ,p ,q )

ap

all o

{., p

I

I

'1.

- , - { q ,p ,q )

aq

.,

q

(12.72a)

i

., p ~

f(q'1.) ,

[~~2(q2)r aq'

+

I

",

aH j

aq

,

(ql,pl,q'1.)u

j

,

(12.72b) p'1.

(12.72c)

368

aH;

-3

q {

(q3 ,p3 ,q2), 8p3

BH3

.J

p

aq3

(12.72d)

(l.i,l),

Yj ... Hj (ql,pl,l)

j

a

8q

the

m.

3

(12.72e)

~) and Co .. span { -al ' - a _a_I. 1' op'J iJq op iJp2

span 1 - ,

Furthermore, ker dO

Remark

E

op2

As in Theorem 3.51 we can call l',p2,pJ the unobservable part of

system,

and ql ,pl, p2

the

strongly accessible part:.

The

(12.72 a., b, e) is locally observable, while for fixed q2

sub-system

the sub-systems

(12.72a,c) as well as (12.72a) are locally strongly accessible. Proof

Application of Theorem l2.2l. yields (12.69), By definition of 0 we

have {Ho,'O} C

~O

(cf. Proposition 12.23). By the Jacobi-identity (12.21b)

we have for any F E

~O

.l

and F

E

~O

(12.73)

since

{F,lI o I E'li ' O

Thus

(H o .Fl) E

(f}~).l

For a

function

group ,

with

dim d~(x) - constant it easily follows that (:1~).l - :1 , and thus (12.70) 0

follows. Writing out (12.70) in the coordinates (12.68) yields

(12.74)

o

and (12.71) and (12,72) follow. It

follows

from the local representation (12,72)

behavior

of

the

Hamiltonian

system

behavior

of

the

lower-dimensional

is

the

system

same

that the input-oucput as

the

(12.72a,b,e),

regarded as the lOI,ler-dimensional Hamiltonian system

input-output which

can

be

369

.,

q

-

all~

.,

P

aq

1 ' (q ,p 1 ,q-) +

1

I j -,

- lfj(ql,pl,q2),

Yj

j

all j

aq E

,

(12.75)

(ql,pl,q2)U.l '

~,

with state (ql ,pI), driven by the autonomous dynamics (12. 72b).

12.3

Stabilization of Hamiltonian Control Systems

For clarity of exposition we will restrict ourselves in this section to an important but particular subclass of Hamiltonian systems,

called simple

.

Hamiltonian syst:ems. Let Q be an n-dimensional milnifold, denoting the coofiguration space, and let T Q be its cotangent bundle, denoting the phase

space or state space. On T~Q there is a naturally defined Poisson bracket, which

is

defined

q = (q1""

in

local

coordinates

as

follows.

Let

be local coordinates for Q Bround qo'

,q,,)

let

,qn ,P!, .... ,p,,) for ThQ. Now

see Chapter 2.2.3, natural coordinates (ql"" let F and G be two smooth functions on ThQ,

C/o E Q and

Then there exist,

then we define their Poisson

bracket as

aF aG aF aG I (api ~ - ~ api )(q,p).

(q,p)

[F,G)

(12.76)

In order to check that this Poisson bracket is well-defined,

i.e. does not

depend on the particular choice of natural coordinates, we let (ill"" =

q be

another set of local coordinates around qo' If

q is

,qn)

linked to q by

a coordinate transformation

q• (with

(l2.77a)

Seq)

q and

then i t follows that p

q being column vectors),

is related to P as (see (2.l49» p _ -p

(with

as(q) aq

p and

(l2.77b)

p being row vectors) . It immediately follows

from (12 .77) that

ro

for any F,G E C (T"q)

I" (a~ i"l

ap l

vC

aF

aqi

aq~

ae; ) - - . -=(q,p) ap,

I i"1

aF aG aF vG (ap, dq, - aq, 8p,)(q,p),

(12.78)

370

where

F and G are

q, p

and where the variables

I

the functions F and G expressed in the new coordinates

the variables (q,p)

(q ,p)

in the left hand side are related to

in the right hand side by (12.77). In particular,

it

follows that natural coordinates for T~Q are canonical coordinates for the above Poisson bracket on

Definition 12.26

r-Q.

A simple Hamiltonian

system on T"Q

is

a

Hamiltonian

system (12.54) wlIere Ho ,HI' ... ,Hm are of the form (in natural coordinates (q ,p) for T~Q)

Ho(q,p)

(12.79a)

wich C(q) a positive definice j

The

pTC(q)p

expression

E

n x n matrix for every q, and

syn~etric

(12.79b)

!E'

is

called

the

kinetic

energy

and

V(q)

the

potential energy. Let us Xo

-

consider a

simple Hamiltonian system with equilibrium point

(qo ,Po)' It immediately follows from

Po ~ O.

~

Furthermore qo satisfies d[!(qo)

q '" ~ op

C(q)p that necessarily

0. The following theorem is an

immediate consequence of Theorem 10.3.

Theorem 12.27

Let (qo ,0) be an equilibrium point of the simple Hamilton-

ian sysr::em given by

(12.79).

Suppose

that V(q)

V(qo)

is a positive

definite function on some neighborhood of qo. Then the system for u = 0 is stable (but not asymptotically stable). Proof

We will show that l.(q,p);- Ho (q,p) - V(qo)

for the system with u

=

is a Lyapunov function

O. Indeed (compare with 12.18)

r(q,p) Furthermore since G(q) > 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

'<1e

finally conclude

Consider tlle Hamiltonian system (12.54) on N.

is non-empty and

that rank A(x)

= m for all x Elf.

Assume that

Then

the zero

dynamics (12.108) is the Hamiltonian vectorfield on N" with respect to the

Dirac bracket.: (12.131) (with N replaced by N*, and ljJj

J

j

E::..

as given in

12,113) and Ilamilt:onian ii o • i.e. the zero dynamics is given as

(12.139) Clearly the theorem also holds if rank A(x) ~ m only for x E N

D

Remarlt

•

383

Example 12.42

Consider

the

Hamiltonian

system

on

with

[R2n

canonical

coordinates (q,p), given as

}{l(q,P) i.e.

n

independent

constants

lei '

sphere

Illqll

[q

Since

{lfo ,H l

Ilqll

for

mass-spring

2

=

L Pi qi ,.,

.-

and

" 0,

with

in IR" . The zero dynamics

- I)

(q,p)

systems

iq:-L ,., masses

and

~

unit

following form.

the

Hence A(q ,p)

_

I q: '" °

L,

i~l

0

[(q,p)1 Lq; - 1

N

spring

output on the

zero takes

- 1.

we have P,

(12.140)

unit

configuration evolves for

whose

)

=

L Pi qi

0,

0)

(the

tangent

i~l

i"1

bundle to the unit sphere) is a symplectic submanifold of [R2.n.

The feed-

back u ~ a"Cq,p) which renders N" invariant is computed as the solution of

adl~oHl(q,P)

+

A(q,p)r/(q,p)

=

4

I

q:c/(q,p)

+ 2

I p~ I 1 I P: + .:, 2

-

kiq:

o

yielding the

on

(since

zero

dynamics

is

given

as

x

i~

=

XI! (x) ,

X

1

E N",

=

o

I

a

on N",

,

kiq~·

Thus

'"1

with

II ~ Ho - a"H!,

resulting in the equations

-

I (kjq~

kiqi +

-

P~)qi'

(12.141)

i E :.:'

j"1

o

under

the

I

constraint

q:

=

1.

The

vector

20/'(ql'"

,qn)T

yielding

the

'"1

second

term

on

the

right-hand

side

of

(12.141)

is

the

normal

force

required to constrain the motion of the mass-spring systems to the unit sphere.

The

Dirac

bracket

is

given

as

(with

(12.142) For n

=

q

2 a convenient set of canonical coordinates for ( , lu* is

:= arctan

(12.143)

ql

and we can express

No

in these coordinates as (12.144)

yielding the following equivalent equations for the zero dynamics

(12.145)

q

o Exa.mple 12.43 Example

Consider

12.38,

Dirac bracket

i. e. , {

1u"

the

simple

Hamiltonian

Hfj(q,p) _ !pTG(q)p + V(q). 2

is given

as

system

as

(q)

qj ,

Hj

-

eli

in (12.131), where

treated j

E ~.

in The

is the (i ,j)-th

element of the inverse matrix of (12.122):

Gll;G

[

-I

- 1

G ll

- I ll

(12.146)

1

0

It is easily checked that the canonical coordinate functions qm+l •.. ,qn' for 1R

Pm+l , •• 'Pn

2n

when restricted to N* form a set of canonical coor-

dinates for

i'

Pm+l •..• Pn)T

and (q,p) - (ql, .. ,qm'Pl, .. ,Pm)T, then Ho(q,p) is computed as

with Poisson bracket {, III'"

(q ,P) -

Denote

(qm+ l' .. ,qn •

follows. Write , GIl

(12.147)

m x m matrix.

n

then

the

equations

GIIP

+ G12 P ~ 0,

-T

-

P Gup +

~

gljPj

...

jBl

yielding

1

~

... -1

gmjPj

o

can

j"'l;..

P - - G11 G12 P.

be 1

Expanding

rewritten

T

1 -T

as -

;: P Gp - ;: p Gup +

we thus obtain

'2

(12.lLlS)

Gi2G~{G12)(O.q) is again positive definite. The zero dynamics is simply given as

(12.149)

p

q -

More concretely. consider the simple Hamiltonian system as treated in Example

12.3

(two-link rigid robot manipulator

lengths), Consider first the output y -

ql

with

and set u 2

-

unit

masses

O. Then

and

HO(q2,P2)

is given as (substitute ql = 0 and PI = (I + cos qz)pz in Ho) 1

2

(12.150)

2 P2 - g cos q2 - 2g

resulting in the zero dynamics q2 - -

ap2 •

the zero dynamics for

ul

-

P2

. On the other hand,

0 and holding the output Yz equal to zero is

governed by the Hamiltonian

(12.151)

385

Physically the zero dynamics are obvious in both cases, as is illustrated

by the following figures.

I q, I

I

P / /

..

I

/ q,

I /

/

I

•~(12.151)

/

~(12.150)

Fig. 12.2. Zcro dyrmmics for the two-link manipul'IlOr.

o Using Theorem 12.41 we obtain the following analog of Theorem 11.16.

Consider

Proposition 12.4/, Xl!o(x o )

=

0 and Ifj (xn)

Suppose that Ho

minimum in xn feedback u

=

E

(i.e.

N"

the

0, j

=

E

Hamiltonian

system

(12.54)

:E' such that: rank A(x)

the restriction of Ho

Co

N")

= m

on

N,

with

for all x E N.

has a strict local

Then there exists a decoupling regular static state

o:(x) + P(x)v such

that: the closed-loop system for

v

=

0 is

locally stable around xnProof

First we note that since the zero-dynamics (12.139) is Hamiltonian

with Hamiltonian function

No,

the equilibrium

Xo

is locally stable if

Ho

has a strict local minimum in xo' Then we apply Theorem 11.16, with local

o

asymptotic stability replaced by local stability.

Remark

Since an equilibrium of a Hamiltonian vectorfield is never locally

asymptotically stable (see Exercise 12.10), but at most stable, it follows that for a Hamiltonian system we can never obtain decoupling with local asymptotic stability if dim N* > 0 and if we

take decoupling feedbacks

which render N~ invariant.

12.5

Conservation Laws and Reduction of Order

Let us first consider the Hamiltonian system (12.54) for u = 0, Hamiltonian vector field

i.e.

the

386

(12.152) on a 2n-dimensional symplectic manifold

H. A

H ....

function F :

is called

IF.

= (No ,Fl 0 clearly reduces

a conserved quantity (or first integral) for (12.152) i f XHo(F)

O. The existence of a conserved quantity F with dF(x) the solution of (12.152) for x(O)

».

sional level set P-1(P(X Il

~

Xo to the solution of a (2n-l)-dimen-

The Hamiltonian structure, however, allows us

to reduce the order by two, in the folloWing sense. Let F be a conserved quancity for

(12.152)

with

dF(x o ) "" O. Denote P 1

:-

F.

By

the

Flow-Bor:

Theorem (Theorem 2.26) we can find, locally about xo' a function Q1 wi th dQl(x o ) "" 0, such that near Xo

(12.153)

1.

Then. as in the proof of Theorem 12.11. we can find additional coordinate

(Q,P) - (Qz'"

functions

such

,Pn )!

,QII 'P'l'"

that

Ql'"

are

,Qn ,P l ,·· 'P n

canonical coordinates about x o - Since 0 = (Ho ,PI) = ~ the Hamiltonian lin only depends on P 1 ,Q,P, and hence in such coordinates the differential equations (12.152) cake the form 8110

Q1 - aP I i\

=

(12.l54a)

0

tIt

(Q,P,P 1 )

is clear

(12.154b) aHo

(Q,P,P l

8Ji

)

(12.l54c)

aHo

(Q,P,P 1 )

ali

that by solving

(2n-2)-dimensiona1 set of Hamiltonian

PI (xo ) one immediately obtains the solution

equations (12.lS4c) for P l of the full

the

system (12.154);

in fact Q1

is

Q,F and the

obtained from

constant PI by a simple integration of (12.154a).

Remark 12.45

The

knowledge

of

conserved

quantities

also

leads

to ~r

a

sharpening of the conditions of Theorem 12.28. Indeed, let FJ..

i E

conserved quantities

condition

for

XHo'

Then

dim P(q,p) - 2n for (q,p)

E VI

can be replaced by the (weaker) condition

dim (P(q.p) + span(dFi(q,p)li

E ~))

it

can be seen

that

the

be

~ 2n for (q,p) E VI'

Let us now consider the Hamiltonian system (12.54) for non-zero inputs u.

Analogously,

we call F : N .... IR

a

conserved quantity for

(12.5 l l)

if

387

m

I

uto -

lljU j ,F)

0, for all u, or equivalently

=

jKl

(H j ,F)

Note

that

the

the

above

~

of

a

conserved

that [G,F) - 0 for all G E 0,

system

dim dO(x)

definition

quantity

In fact by applying the Jacobi-identity

restrictive. from (12.1S5) of

(12.155)

j=O,l, ... ,m.

0,

=

(12.54)

Proposition

(cr.

is

quite

(l2.21b) we obtain

with 0 the observation space 12.19).

particular,

In

if

dim H for all x E l-1 (and thus the system is locally observable

as well as strongly accessible, cf. Proposition 12.21) then F is necessarily a constant function, and thus a trivial conserved quantity. A more general concept is that of a conservation law. Let F:N ... IR, and let Fe (y, u) be a smooth function defined on [Rm x [Rm,

the space of outputs

and inputs. The pair (F,F") is called a conservation 1a\0{ for (12.54) if

(12.156)

which expresses that the time-derivative of F along the system (12.54) is a function only of the outputs and inputs. Because

X _ Ho

I

H u (F)

j~l IH j ,F}u j

{li o ,F) -

=

j

j

,

j-1

it immediately follows that FQ(y,u) is of the form V(y)

+

I

Kj (y)u j

,

for

j-1

certain smooth functions Kj

,

j E!!!, and V on IR

m ,

and thus (12.156) reduces

to {li o

,F) (x) ~ V[lIl (x), .. ,lIm (x») , (12.157)

(HJ,F}(x)

=

-

(H 1 (x),··,ll m(x»),

Kj

In particular, if V

=

j E

!E.

0 then F is a conserved quantity for Xllo. Under an

extra assumption we will show that V in (12.157) always can be made equal to zero by applying a special type of feedback.

a Proposition 12.46 Consider a conservation 1810{ (F,F ), with F''(y,u)

I

Kj (y)u j

for

,

the Hamiltonian

system

(12.54).

Let Xo E N and

=

V(y) + denote

j-1

Yo -

(111

(x o ), .. ,Hm (x o »

non-zero. feedback system

Then U

=

there

T.

Assume

exists,

a(x) + (3(x)v

that

the

vector

(K 1 (yo), .. ,Km (Yo» T is

locally about x o ' a regular static state transforming (12.54) into another Hamiltonian

38B

r,1

X = x-

I

(x) -

110

(12.158) j E ~.

Ho (x)

h'here

+ S( Hl (x) ... ,Hm (x») for a cerr-ain function S(y). and

= Ji n (x)

jE~.

Hj(X) = Rj(H1(x) .... Hm(x»).

(R 1 , ... R.n)T: lR m ... lR m

lvith

a

being

coordinate transformation about Yo' in such a t.. ay t.hat: (12.159) 1. e.. (F, li l

is a conservation law for (12.158).

)

Flow-Box Theorem

(Theorem 2. 26)

.y",)

formation (Y1'"

there exis ts

(R1(y) •..• P'm(y)]

about

a

local

Yo

such

coordinate that

t::rans-

a aYl

K(y)

Define the preliminary feedback u "'" {J(x)u as

(U 1 ' ..

1

(12.160)

Um)

m Y = (H 1 (x), ..• Hm(x»'T,

with

x

system

m =

- I

X"o(x)

j~l

R j (Hj (x)

,11", (x») •

1"

Yl'" 'Ym

coordinates such

that

V
j

E

by

transforming

x-!l j

(x)u j

m.

-

Denote

V(Yl'"

Hj

Yj

,

into j

(x) •

the

Hamiltonian

where

E ~.

expressed

V(Y1' ..• Ym)

,Ym)'

in

Next define a function (this

(Yl ••.• Ym)

,Ym)

(12.54)

is

always

Hj

(x):-

the

new

S(Yl'" .Ym)

possible).

and

introduce the feedback

=

Uj,

-

vl

1 E

•..• Ym)'

(12.161)

~,

BY1

with Yj are

~

Hj

(x) • j

canonical

111.

It is immediately checked that (Y1'"

coordinates

for

T"lR

m

IR

m

x IiI

m

and

that

,rm ,v

1 •·•

the

feedback

,v

m)

(12.160) and (12.161) transforms (12.54) into (12.158), where S(Yl, .. ,Ym) equals

S
,Ym) expr-essed in the original coordinates Yl'" '"

I

i= 1

and thus

8R j

m

U{i ,F)

=

L i;1

.Ym • Finally

8Rj

8Yi Ki ~ ~:Rj ~ - 5 1j

,

j

E

:E.

389

o

yielding (12.159), Remark 12.47

The above proof shows that

[RID

m X IR , the space of outputs and

is most naturally seen as T"mm endowed with its natural Poisson

inputs,

bracket eef.

(12.76». Furthermore the set-up naturally generalizes to a

space of outputs and inputs given as T~l',

where l' is any m-dimensional

output manifold. Motivated by the preceding proposition let us consider a conservation law e

(F,F )

Xu o.

with F"Cy,u) "" u 1

It follows

.

CQ,P) ~ CQl'"

canonical coordinates

before,

all, aQ,

since

-

IH j ,P t

)

-

0lj'

,Q n ,PI'"

the

{lfo,P!) = 0

-

,P n )!,

function

(cf.

(12. 154) )

with Pl:~ P.

H,

only

Then,

depends

as on

all j and

-

that F is a conserved quantity for

and thus we can cons truct in the same way as before

Furthermore

P, .

implies thatH j

j E~,

forj~2,

the

condition

aQ;

-

... ,m only depends on

(Q,P) and PI' while HI is of the form (12.162) For ease of notation we write Hj(Q,P,P I in

the

new canonical

coordinates

(Q,P)

):-

Hj(Q,P,p 1 ). j ~ 2,,,.,m. Then

the Hamiltonian

system

(12.54)

takes the form (compare with (12.154»

ii,

P,

~

-

aHj

aH, ap, (i),F,P,) -

I ap,

",

(12.163b)

all,

{ y,

Yj

j-' ap

aF all,

ail

-

aH j

m

I

(i),F,P,) -

(:

aH j

m

(Q,P,P I

)

I

+

j"l

aq

(Q,P,P1)u j (l2.163c)

(Q,P ,PI )U j

Q, + Ht (Q,P,Pd

Hj (Q,P,P 1 ),

(12.163d) j

Thus we have obtained the

~

2, ... ,m.

(2n-2)-dimensional Hamiltonian control system

(12.l63c), which is driven by the variable F remaining

(l2.l63a)

(Q,P,PI)u j

j~1

variable

Qt

is

obtained

from

integration of (12.163a). Pictorially we have

P l - Ju1dt. Furthermore the

Q,P,P 1

and

u

by

a

simple

390

(ft,P)

u

(12.163c) u

Pi GF

U1

~

J

(U 1 , ••• ,Urn)

U

static nonlinearity

Fig. 12.3. HamillOnian system with conservation law (F,UI).

Summarizing we have pLoven Theorem 12.48 la," (F .F exist

e

Consider a Hamiltonian system (F, u 1

)

)

Ivich

(Q .P) = (Ql' QI P 1 .~

coordinates

canonical

(12.54),

conservation

satisfying dF(xfj) ,... O. Then loca.lly a.round Xo in

there

which

the

Hamilconian syscem talces che form (12.163).

Remark 12.49

If F is a conserved quantity for (12.54) (or, equivalently,

o takes

the

form

is a conservation law) then the Hamiltonian system

(12.163)

with

(12.163b)

(12.l63d) replaced by Yj - Hj(Q,P,P I

Example 12.50

Consider

the

),

replaced by i\

O.

and with

with

canonical

j Em.

Hamil toni an

system

on

coordinates (ql .QZ,Pl,P2)' given by the internal Hamiltonian (12.l64a)

and the interaction Hamiltonian (12.l64b) This describes the motion of two particles of unit mass moving on a line, whose interaction comes from a potential VCr) depending on the distance r between

the

two

particles

(for example,

corresponds

tT(r)

to

a

linear spring between the two masses). Moreover the second particle is controlled by an external force u. It is easily checked that (F,F F(q,p) .. PI

'1'

(the

P2

total

linear

momentum)

and

F"(y,u) = u

e

)

with is

a

conservation law. Introduce new canonical coordinates F

PI

+

P2'

In chese variables the internal Hamiltonian is given as

P z = Pz'

(12.165)

391

02.166a)

while (12.166b)

form (ef. (12.163»

and the system takes the

Q,

~

P,

~

Q,

- -P,

P, - P z ,

Y

Q1 + Qz,

=

u,

(12.167)

+ 2P~. '

dV

dQ, (Q,) + u.

Notice that for almost all potential functions \l(Qz) the observation space o satisfies dim dD{Q,P) conserved quantities.

=

4, in which case there cannot exist non-trivial

On

the

other hand,

(12.164b) into H} (ql ,qz ,PI ,pz) .. qz - ql the distance between the

if we change H}

as

given

(i.e., an actuator is controlling then F

two particles),

PI + pz

=

is a conserved

quantity for the resulting Hamiltonian system.

Remark 12.51 XII (F)

Let F be

(SYl1ulletries)

a

in

0

conserved quantity

for XII'

i.e.

O. Then clearly XF (ll) = (F ,R) = -(H ,F) "" -XII (F) = O. A vector field X satisfying X(H) = 0 is called a symmetry for the Hamiltonian H. We thus =

see that F is a conserved quantity for Xu if and only if XI' is a symmetry for

(Statements

H.

-

IF,H)

Lemma

X; aXI~

0

implies

2.25

i,

"

of

this

type,

relating

symmetries

to

conserved

usually go under the heading of Noecher's theorem.

quantities,

(cf.

Lemma

equivalent

to

12 .9)

that

[XI' ,Xu]

the

fact

that

-

the

0,

which

flows

Horeover

in view of

X; ,Xl~

satisfy

XI~ aX; for all s,t for which X; ,XI~ exist. In particular this means

that for any t

the mapping

X;

maps solutions of XIl onto solutions of Xu;

therefore XI' is called a symmetry for the vectorfield Xu' This generalizes

to conservation lat... s

(12.156)

as follow·s.

Let

(F,F~)

be a conservation law for the Hamiltonian system (12.54). Then we consider the pair of vectorfie1ds

(XF'XI'~)

with Xl' the Hamiltonian vectorfie1d on

,

tJ, and X .. the Hamiltonian vectorfie1d on T*[Rrn = [Rrn X Worn (space of outputs and

inputs),

bundle.

with

respect

to

its

symplectic

structure

It can be proved (cf.

the

references

cited at

as the

a

cotangent

end of

this

t t (XF,X ,,): N x (IF: rn x ill rn ) ..... N x {IF: OJ X mm) LtI map F solutions (x(s),y{s),u(s») of the Hamiltonian system onto other solutions,

c h apter )

t Ilat

t he

mappings

392

and

thus

(XF,X

re ) can be called a symmetry for the Hamiltonian system.

Conversely it can be shown that if (XF,XI'0) is a symmetry,

then (F,F

e

)

is

a conservation law.

Notes and References

Classical references to the Euler-Lagrange and Hamilton equations are [Go] and [Wh]. For the modern geometrical treatment, using symplectic geometry, we refer to e.g.

[Ar] and tilt]; in particular in [U1] an extensive

[Al1] ,

treatment of Poisson structures can be found, see also [We]. The definition of a Hamiltonian system with inputs and outputs is due to

[Br],

and

was

further

developed

in

[vdSl,vdS3].

The

controllability and observability for Hamiltonian systems [vdSl, vdS4]. [CIl,CI2], systems

For realization [Jl,J2],

using

especially

(vdSl]

and

[CvdS].

robotics

in

by

[TA),

treatment is largely based on (vdS8J,

refer

to

The stabilization of Hamiltonian

as

context,

of

taken from

theory of Hamiltonian systems we

"PD-controllers",

in a

theory

is

12.3,

was

see

also

see also

[Ma],

first [Ko].

advocated, The

present

(TK]. The treatment

of constrained or clamped dynamics for Hamiltonian systems as given here was developed in

[vdS9,vdS10].

For a

study of Hamiltonian systems with

additional physical constraints we refer to [MB].

The treatment of Dirac

brackets is largely taken from [DL1'] , see also (MB). Theorem 12.44 can be found in (HvdS]. The use of conserved quanti ties for order reduc tion of Hamiltonian vectorfields and ultimately,

for explicitly solving a set of

Hamiltonian differential equations is very classical, (01 J.

present treatment was much influenced by

symmetries and conserved quanti ties (Noether' s

see e.g.

(WbJ.

The

For the relation between theorem) we refer to

[AM]

and [OlJ. The problem of using multiple conserved quantities or a group of symmetries

orde~

for

reduction

is

much

treatment is a vast subject, see e.g. as

well

as

symmetries,

[vdSl,vdS2,vdSS],

where

for a

more

(AM],

Hamiltonian generalization

[MW],

involved.

geometric

[MRJ. Conservation laws,

systems of

Its

were

dealt

Noether's

with

theorem

in was

obtained. The use of symetries for decomposition of systems was emphasized in

(GH1, GH2];

extended

to

Hamiltonian

in particular an

and

Abelian Poisson

in

group

(GH2) of

control

purposes were studied e.g. in [Kr),

the

decomposed

symme tries. systems (KM],

and

form

Conserved their

use

(12. 164)

was

quantities

for

for

reduction

[Sa] and [SOK}{]. For a treatment

of general Hamiltonian systems (not affine in the control variables) and applications to optimal control theory, we refer to e.g.

(vdSl].

393

{Ar]

V. 1.. Arnold, Mathematical Methods of Classical Mechanics, Springer, Berlin, 1978 (translation of the 1974 Russian edition). [AN J R . A. Abraham, J. E. l1arsden, Foundations of Mechanics (2nd

edition), Benjamin/Cummings, Reading, Hass., 1978, [Br]

R. W. Brockett, "Control theory and analytical mechanics", in Geometric Control Theory (cds. C. Hartin, R. Hermann), Vol. VII of Lie groups: History, Frontiers and Applications, Hath. Sci.

{CII]

P.E. Crouch, H. Irving, "On finite Volterra series which admit Hamiltonian realizations", Hath. Systems Theory, 17, pp. 293-318,

Press., Brookline, pp. 1-46, 1977. 1984. [CI2]

[CvdS]

(DLT]

[Go] IGB1]

[GM2)

[HvdS]

[Jal]

[Ja2] [Ko] [KrJ

IICH]

[Ul] {Ha] [MB 1

[MR) [MW] [01]

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394

[Sa J

G. Sanchez de Alvarez. Geometric Methods of Classical Mechanics applied to Control Theory, Ph.D. Thesis, Dept. Mathemat.ics, Univ. of California. Berkeley, 1986. [SOKI1] N. Sreenath, Y.G.Oh, P.S. Krislmaprasad, J.E.14arsden, "The dynamics of coupled planar rigid bodies. Part I: Reduction. equilibria & stability". Dynamics and Stability of Systems. 3, pp. 25- l I9. 1988. [vdSl] A.J. van der Schaft, System theoretic descriptions of physical systems, CWI Tract 3, CWI, Amsterdam, 1984. [vdS2] A.J. Van der Schaft, "Symmetries and conservation laws for Hamil tonlan systems with inputs and outputs: A generalization of Noether's theorem", Systems Control Lett., I, pp. 108-115, 1981. [vdS3) A.J. van der Schaft, "Hamiltonian dynamics with external forces and observations", Math. Systems Th., IS, pp. 145-168, 1982. [vdS4] A.J. van der Schaft, "Gontrollability and observability for affine nonlinear Hamiltonian systems", IEEE Trans. Autom. Contr. I AC-27 , pp. 490-492, 1982. (vdS5) A.J. van der Schaft, "Symmetries, conservation laws and time-reversibility for Hamiltonian systems with external forces", J. Bach. Phys., 2[1, pp. 2095-2101, 1983. {vdS6] A.J. van der Schaft, "Linearization of Hamiltonian and gradient systems", IMA J. Math. Control Information 1, pp. 1B5-198, 1984. [vdS7] A.J. van deL Schaft. "Controlled invariance for Hamiltonian , Math. Systems Th., 18, pp. 257-291, 1985. [vdS8] van der Schaft, "Stabilization of Hamiltonian systems", Non1. An. Th. Meth. Appl., 10, 1021-1035, 1986. rvdS9] A.J. van dar Schaft. "On control of Hamiltonian systems", in Theory and Applications of Nonlinear Control Systems (eds. C.!. Byrnes, A. Lindquist), North-Holland, Amsterdam, pp. 273-290, 1986. of motion for Hamiltonian systems {vdS10] A.J. van del' Schaft, Math. Gen., 20 pp. 3271-3277, with constraints", J. 1987. [TK] J. Tsinias, N. Ka1ouptsidis, "On stabilizability of nonlinear systems", 21st IEEE Conf. Decision Control, pp. 712-716, 1982. [We] A. Weinstein, "The local structure of Poisson manifolds", J. Differential Geom., 18, pp. 523-557, 1983. [\<1h] E.T. Whittaker, A treatise on the analytical dynamics of particles and rigid bodies, 4th edition, Cambridge University Press, Cambridge, 1959. I

Exercises

12.1

Let: v1

be

if

•..•

an

r-dimensional

Lie

(Def.

2.28),

with

basis

,v" satisfying the commutation relations i,j

with

algebra

c:

E ::,

k=l j '

i,j,lc E::"

Gonsider N - IR

r

the structure constants

(see Exercise 2.15).

with natural coordinate functions x'" (xl""

,Xr ).

(a) Show that (F,G}(x) -

I

C~jXk

F,G: N .... II? ,

i .j. k"l

defines

a

Poisson bracket

on N.

called

the Lie-Poisson bracket.

395

Furthermore,

n with V", the dual of V. Then show that the

identify

above Poisson bracket has the coordinate free form

(F,G)(x)

~ ~~,[dF(x)JdG(x)l>

where dF(x)

and dGCx)

are

regarded as

elements of

and

(V")" = V,

where < , > denotes the natural pairing between V and V" (b) Consider the three-dimensional Lie algebra so(3) of the rotation

we [V 3

have

,v 1

]

the

v2

=

commutation

[vl,v Z ] ~ v J

relations

[vz,vJl

•

~ vI'

Show that the resulting Poisson bracJcet on 50"(3)

,

is

given as in (12.27). 12.2

Prove

the

following generalization of Theorem 12.11.

rank 2n (2n s m). coordinates

Let N be

an

( , J having constant

m-dimensional manifold with Poisson bracket

Show that locally around any Xo E N we can find for

(q,p,z) = (ql,,,.,Qn'Pl""'Pn,ZI""'Z1')

H

(so

2n + l' = m), satisfying (Pi' qj J

5 ij ,

(Pi

0,

,2k J

(qi,qj)

(qi,zk1

=

0,

=

0,

(Pl,Pj)

{Zr'Zk}

=

i,j E !2

0,

=

i E

0,

!2, r,1e E £

(see for the non-constant rank case [01] or [We]). 12.3

Let

be

fi(x)

the

structure

matrix

of

structure on H with local coordinates x (i,j)-th element of the

a =

non-degenerate (xl""

inverse matrix r(l(x)

Poisson

Denote the

,x2n ).

i,j E 2n.

by I/J{X),

Show that the functions l / J (X) satisfy ( i)

I"ij

(x)

=

-

i ,j E 2n,

l"j\X),

jk

+ ~(x)

(ii)

=

aX1

0,

i,j,k E 2n.

Conversely, show that any matrix (2nx2n}-matrix Sex) whose elements 5 i j (x)

satisfy (i) and (ii) defines a symplectic structure on 1R2n;

i.e. -(S(x)r 12.4

Consider

a

1

is the structure matrix of 11 Poisson bracket.

Poisson

control

system

on a

manifold with

degenerate

Poisson structure. Let Co be the distribution as in Definition 3.13. Show that dim Co (x) structure system

in x,

cannot

is always

x E H.

be

less

than

the rank of

In particular ShO\o1 that a

locally

strongly

accessible.

observability? 12.5

( [vdSl J) Cons ider two Hamil tonian sys terns,

.:..:i

=

x

i ( .../ )

"0

~

X

i

J~ 1 Il j

{xi)u j

,

Xi E

ll, Yj

the

Poisson

Poisson control

What

about

local

396

where (

,

fl

and

HI

are

symplectic

respectively

)1'

manifolds

1

Poisson brackets

,)z. Assume that both observation spaces 0

satisfy dim dOi(Xi) - dim ni, for all Suppose that :8

with

Xi

E

H1,

1

= 1,2.

i

and :82. are equivalent with equivalence mapping

~: HI ~ HZ, i.e. if Xl(t) is a solution of ~1 for a particular u(t), then xZ(t) = ~[xl(t)l

is

solution

a

yielding the same output H; (x( t»)

of

j

this

j

E m.

(0) Prove that tp is a Poisson bracket isomorphism,

i. e.

~~Xlll =

X

II2

j

j

,

= 0, l, ...

u(t),

j

{F.G11D~, for any F.G: HZ ~ (b)

same

Formally

"It.

~,

E

U;,

H~O~ ~

,m.

for

L2

H~(x(t») ,

Prove

H; D~

that

=

!Fo~,Go~J1

=

m.

H~

+ cons tant,

i. e.

the

internal

energy

(modulo constants) is the same. 12.6

(0)

Show

that

a

linear

system

y ~ ex.

x = A,y + Bu.

x E Il/zn,

is a Hamiltonian system if there exists a skew-symmetric

U.y E [Rm.

invertible matrix W such that

(1)

1

+ [v' A = 0,

A[v'

(or equivalently. i f ,~e let J := _r{l, A!J + JA = 0

I

BTJ

G). Then

the system is called a linear Hamiltonian system. (b) ([vdS61,

compare also with Theorem 5.9) Consider a Hamiltonian

system (12.54)

with Xno(x o ) = 0 and lI j (x o ) = 0, j E!,!!, satisfying dim dO(x o ) = dim H = 2n. By Proposition 12.19 the observation space

o

is spanned by all functions

j E k = 0.1.2 .... F = {Fl' {Fz • { ... {Fk .llj J J ••• J} . (for k = 0 we let F - Hj ) . Define the grade of F (denoted gr(F»

(2)

the number of times that one of the functions Hj

j

,

E!,!!.

as

appears in

F. Show that we can choose coordinates z about xo. with z(x o ) = 0, such

tha t

in

[hese

coordinates

(12.54)

is

a

linear

Hamil tonian

system z = Az + Bu, Y = Cz. if and only if all elements F as in (2) wi th gr(F) = 3 are zero in a neighborhood of xo' that tv' in (1.) equals [v'(0) , with fv'(z)

Furthermore show

the structure matrix of the

Poisson bracket in the coordinates z, and that [v'(z) = [v'(0) for all z in a neighborhood of O. with gr(F)

~

(Hint: Prove that all elements F as in (2)

3 are zero in a neighborhood of Xo

if and only if all

elements F as in (2) with gr(F) = 2 are constant in a neighborhood

12.7

of x o ') (a) Let' be a function group on the symplectic manifold H. Define the distribution D';} as D,(x) = span(XF(x) involutive. A

function

group

':1

is

I

F

E ~J.

called

Prove that D, is

invariant

for

the

Hamiltonian system (12.54) if (H j ,':11 C', j - 0.1 .... ,m. Prove that

397

then

D~

is an invariant distribution for (12.54).

(b) Let' be an invariant function group with dim

~(x)

Suppose

Prove

j

'fJ n ,1

- 0.1, ... ,In,

are

are

the

constant

in sui table

Hj(ql •... ,qn'Pl'''''Pn )

functions.

canonical

~p~f!(q).

-

coordinates

can be

What

constant.

-

that

lfj'

the

form

of

said

about

the

case that 'fJ satisfies ,1 C', Dr' C ,.i.? (c) Let D be an involutive distribution of constant dimension on the

symplectic manifold N.

By

Frobenius'

theorem

independent functions K1 .... ,Kk such that Vex) ... ,dKk (x)

there =

exist

locally

ker span(dK 1 (x), ..

J. Assume that Xl"" ,X" are globally defined. Prove that

there exists a function group 'II such

that D "" D'J if and only if

span{K1, ... ,KkJ is a function group ([vdS7]). 12.8

Consider the two-link robot manipulator from Example 12.3, see also Example

Consider

12.22.

only

the

first

input

u1

U -

and

corresponding output y - ql' Compute the observation space 0, for

g

~

and

0,

then

g

for

0#-

accessibility and observability.

O.

Decide

about

Do the same for

the first

local

strong

input u -

U

z and

0

and

output y - qz. 12.9

Consider

a

lfj (x o ) - 0,

Hamiltonian j

E~,

sys tem

satisfying

with

(12.54)

dim dO{x o )

=

dim N

Xu 0 (xo )

2n.

=

=

Define

the

function spaces ;}k

span(adl~olfl' ... ,ad~oHml r " O,l, ... ,k-l),

..

(a) Prove:

1c

=

1,2, ..

the system is feedback linearizable around Xo if and only

if '1"" ,'Zu are function groups satisfying dim d:J k (x)

=

constant,

x around x o ' 1c E 2n (compare with Chapter 6). (b) Consider the simple Hamiltonian system of Example 12.38.

Show

that

that

a

necessary

condition

Gll (q) only depends on ql"

for

as div(X) (x) -

=

that

ax' (xl)

det(ax

is

a vectorfield on [fin

with natural

(xl, ... ,xn ). The divergence of X, div(X), is defined n aX i

L ax

i~ 1

(a) Show

linearizability

.. ,qm'

12.10 Let X(x) = (Xl (x), ... ,Xu (x») T be

coordinates x

feedback

the

(x). l.

maps

n - 1 for all x E tII ,

are

volume-preserving,

i. e.

i f and only i f div(X) (x) ~ 0 for all

x E [fin.

(b) Prove Conclude

that div(Xu ) (x) .. 0 for any Hamiltonian vectorfield Xu' that a Hamiltonian vectorfield cannot have locally

asymptotically stable equilibria.

39B

12.11 Consider a Hamiltonian system (12.54). Apply feedback u - o(x) + v. Show that the feedback transformed system is again Hamiltonian (with

if

respect to the sallie Poisson structure)

and only if there exists a

function P(YI ' ... •Ym) such that i

m.

E

Show chat the internal energy of the resulting Hamiltonian system is given as Un (x)

-I-

12.12 (/1) Consider

P(lIl (x) , ... ,11m (x)).

a

Hamil tonian

(12.54)

sys tern

with

non-singular

decoupling matrix. Suppose that 110 has a strict local minimum in Xo (implying that Xo

is a locally stable equilibrium for XII 0) . Suppose

E!!!. Show that

0, j

is also a locally stable equilibrium

Xo

for the zero dynamics. (b) Suppose that the conditions of Lemma 12.32 are sati.sEied. Prove that

Xo

is

(qo ,0)

=

a

locally

stable

equilibrium

for

the

zero

dynamics.

12.13 (n) Consider decoupling

Hamiltonian

a

matrix.

Define

(12.54)

system

new

Yj

outpucs

with

non-singular

= R j (Yl •... tYm ),

j

E

:E,

where

J (y)

aR.

ay:

rank [ and Yl

'"

~

= m everywhere,

o

Ym

i f and only i f YI

O.

Ym

Show that

the decoupling matrix of the resulting system is still non-singular, and that the zero-dynamics remain the same. (b) Apply (a) to the case of the two-link rigid robot-manipulator of Example 1.1 with (~l'~2) the Cartesian coordinates of the endpoint. Are there any singularities? 12.14 ([CvdS]) Consider a nonlinear system m

L: x

f(x) +

L gj (x)u

j

Yj

,

j E

!E, x E H.

With any vect::orfield X on N 've can associate a function

a( X(x»),

to IR by setting }(\x ,n) (x,p)

(xl""

pTX(x).

11£ :=

,Xu

Furthermore

1]071:,

the

,P!, ... ,Pn)

for

any

Q

E

r;n.

function

h: X

-+

It?

Jf' from r"N

In natural coordinates is

}((x,p)

define

given

hi: TON .. IR

as by

11': T"N -. N projection. Define now the following system on

T"n m

X,(X)+LXs,(X)!l

and show that Le

G

j"l

II

J

C

m LXjl(x)u;, j~l b j C

is a Hamiltonian system on rftfl with

inputs u,

U

U

399

and outputs yfi, y, called the Hamiltonian extension. Show that the observation space of :Eo is spanned by all functions }/-, with H in the observation space 0 of 1:,

and all functions;/',

with X in the strong accessibility algebra Go 3.19).

and

Prove

that

locally

strongly

observable

and

of :E (cf.

the Hamiltonian extension is

if

accessible

locally

strongly

and

only

if

accessible.

observation space of :Eo is not changed if we set u; 12.15 Consider

a

particle

in IR

J

with mass

III

in a

Definition

locally observable

2

is

locally

Show

that

0,

E m.

=

potential

the

field with

potential V, subject to an external force u. The system is given by 1

qi - ;;; Pi ,

Pi

Suppose that I'(q)

~

-

~(q) + aq,

ul

Yi

•

=

qt.

i = 1,2,3.

is invariant under rotations around the e 1 -axis.

e

Show that (P,p ) with F(q,p) = qZP3 - P2q3 and po(y,u) is

a

conservation

law.

(F

equals

the

angular

=

Yzu J - uzYJ

momentum).

Apply

Proposition 12. ',6 and subsequently Theorem 12.48.

12.16 Prove Remark 12.37 (see (DLT]).

12.17 Consider a simple Hamiltonian system (Definition 12.26). Show that the

map

(p(q,p)

=

1p: T"Q (q,-p),

r"Q,

in

local

natural

coordinates

defined

as

satisfies

1p"Xllj =-XHJ , j=O,l, ... ,m, Hjo(p=H j , jE~. Show that this implies that if (u(c),x(c),y(t») is a solution curve

of

the

curve,

system, with

reversible,

then also (Rf)(t)

[vdSl,S].)

(Ru)(t),1p(R..y)(t)],Ry(C») f(-t).

(The

system

is

is

a

solution

called

time-

13 Controlled Invariance and Decoupling for General Nonlinear Systems

In Chapters 7-11 we have confined ourselves to affine nonlinear control systems. The aim of the present chapter is to generalize the main results obtained to general smooth nonlinear dynamics x = f(x,u)

where x

=

(Xl"

U E

I

U,

(13.1)

.,Xn ) are local coordinates for the (state space) manifold

M. Throughout this chapter we will assume that the input space open subset of ~m, with local coordinates

U

From Chapter 2 we recall that (13.1)

(u l

'"

is an

U

,um ).

can be regarded as

the local

coordinate expression of a smooth map (the system map)

F: N x U -; TH.

(13.2)

satisfying the commutative diagram

F H x U ------,)) TN

~

H

(13.3)

~N

with 11": N xU .... H and TiN: TN ... H being the natural projections. Indeed, let x -

(Xl

I'

,'xn

)

be local coordinates for H, and let (x,v) -

(XI

I'

.,xn

'

v1' .. ,vn ) be corresponding natural coordinates for the tangent bundle TN,

cE.

(2.61).

Furthermore let u- (u1 .... um )

be local coordinates for U.

Then by the commutativity of (13.3) the map F is locally represented as F(x,u)

(x,f(x,u»,

(13.4)

thereby recovering (13.1), 13.1 Locally Controlled Invariant Distributions

Analogously to Chapter 3 (Definition 3.44) and Chapter 7 (Definicion 7.5) we state Definition 13,1 A distribution D on N Is Invarlam: for (13.1) i f

/jOI

for every

[f(·,u), DJ cD

E U,

t1

(13 .5)

and locally controlled invariant

for

regular static state feedback (cf.

(5.102)), briefly feedback,

u

rank au (x, u)

u(x,u),

=

=

au

(13.1)

if there

exists locally a

for all x,u,

m,

(13.6)

such that the closed-loop dynamics

x

f(x,u):= f(x,o:(x,u»

=

(13.7)

satisfies [f(.,G),DJ cD,

for every

U.

(13.8)

Remark For global controlled invariance we additionally have

to require

that u - u(x,u) is globally defined with the property that for every x the

u ~ u(x,u)

mapping

is a diffeomorphism.

If D is involutive and constant-dimensional, (Theorem Xl =

2./j2)

we

(xI, . . . ,Xk ),

a

span(-a - , ... Xl

X

can Z

a

find

It

'-a' xk I.

local

(Xk + 1 , . . . ,x n ),

=

follows

from

x

coordinates

=

for

(XI,XZ)

Ie = dim D, such that D

Theorem with

N,

span(~l:=

=

ax!

(13.5)

that

(13.1), then (13.1) locally decomposes as, cf.

Xl

then by Frobenius'

if D

is

invariant

for

(3.l09),

f1(X1,XZ,u) (13.9)

xZ=fz(xz,u),

and similarly for the closed-loop dynamics (13.7) i f D is locally controlled invariant. Our

first

goal

characterizing

is

locally

nonlinear systems.

to

obtain

controlled

In order

to do

a

generalization

invariant so we

of

Theorem

distributions

first

set up

for

some

7.5, affine

additional

mathematical apparatus. First, let h: N

~ [R

be a function (everything in the sequel is assumed

to be smooch). The complete lift (or prolongation) of h is defined as the function

h:

TN -.

[R

given by (13.10)

Notice

that

for

any

set

of

resulting natural coordinates also

given

as

(xl'"

'Xn ,Xl'"

local (Xl'"

,xn ).

coordinates ,Xn

' Vj

In

, .. ,

the

(xl""

,xn )

for

v n ) for TH, cf. sequel

we

will

fI,

(2.61),

the are

therefore

402

througllout denote the natural coordinates for rtf by (x ,x) ;=

Xl'"

.x

n ).

It is easily seen that the prolongation

iJ

(Xl'"

'Xn '

in natural coordi-

nates (x,x) is locally represented as

(13.11)

The vertical lift 1,1: TN

-+

!R of h is defined as

'lrN:

TH .... N projection.

(13.12)

and thus is locally represented as (13.13)

Secondly, let X be a vectorfie1d on H. We define the complete lift (or prolongation) of X as the vectorfield

Xon

TH satisfying (13.14)

for all h:N .... JR. This uniquely defines Lemma.

Let

13.2

11: N .... IR. TheIl

X

be

X~

O.

il

X.

as follows from

vectorfield on TN satisfying

X(h) -

0

for all

Proof In natural coordinates (x,x) we can write

(13.15)

First take h -

Xl'

from X(:X i

then it follows

Hence, using (13 .11). 0 -

X(i}) ~

(X,(x,x), ... ,X.(x,x») n'h(x)

n

I

Xi (x

,x)x

j

[7' ]-

)

0

-

a2h --ax! j

aX

that 21

=

O.

i E n.

(x), or

0

(13 .16)

XII

2

with D h(x) being the Hessian matrix of h. Since h is arbitrary (and thus Dlh(x)

is an arbitrary symmetric matrix) this implies i,j

Let now x t j

¢

O. Putting i - I in (13.17)

E ::. and thus

(13.17)

E ::.

we obtain XJ (X,X)Xl

=

-Xl (X'X)X j

,

403

(13.18)

for some function o{x,x)

(not depending on j).

(13.17) we find 2o(x,x)xi

x

obtain

o(x,x)

Xj (x,x)

Xj (x,x)

=

0,

=

or

j

= 0, i,j E~, and thus if

If X is locally represented as

X(x,;n

I i

,L.

X=

a

X i (X)

aX

i "1

X is

-

j E!2.

0,

=

E~, and thus

0 for all (x,x), j

(13.14) that

Substituting (13.18)

By

Xi

~ 0 and

continuity

Xj

this

into

~ 0 we yields

D.

0

then it readily follows from

i

locally given as

a ax,

Xi (x) - - +

~1

ax

I" i •j

.

a

~(x)Xj ~1

(13 .19)

aXl

j

The vertical lift of X is defined as the vectorfield Xi on TN satisfying

x'(h) for all h:

(X(h)j' N ..... II?

(13.20)

This uniquely defines Xl! as again follows

from Lemma

13.2. Furthermore in local coordinates

I

Xi(x,x) = i

Thirdly,

Xi (x)

a

(13 .21)

aXl

~1

let a be a differential one-form

DO

N. The complete lift (or

prolongation) of a is the differential one-form on TN defined by setting (13.22)

where 0: TN . . . !R is the function given as

(13.23)

I f a is given in local coordinates as a follows

that

~(x,x)

- I"

aO i ax. (x)x j dX i +

,

'" 1

Finally

the

vertical

life:

"

,.I ,

of a

Oi

is

- ' L.,

(x)dx i

the

oi

(x)dx i ,

then

it readily

.

differential

one-form of

on

TN

defined as

,

a ;=

i. e.

1fn

0,

in local coordinates

(13.25)

404

1

n

.

o (X,x)

L

= !

a

(13.26)

(x)dx i •

0i

1

Using the local coordinate expressions the following identities are easily verified.

Proposition 13.3 For any function h: N

IR. any vectorfields X ,Xl and Xz

-+

on N, and any differencial one-form a on 1-1, we have X(hi.)

(X{h»)

a(Xi.}

(o(X»)

1!

dli = (dh)

dh

X(il)

1:.

=

J = ([Xl ,Xz

[Xl

xi. (h) •

jl -

]) . ,

1

a(X)

(XL

0

.

R.

=

(X(h») . ,

Xi. (hl) - 0

(13.27)

(a(X»)' ,

of.(Xf)

0

(13.28)

[X;,X;]

- a

(13.29)

[Xl,Xz]i,

[Xl ,Xz ]

(13.30)

(dh/.

Prolongations of distributions and co-distributions are now defined as follows. Definition 13.'. Let the disr:ribution D on N be given a.s i

E

1-1, lvich Xi

11, P

the distrlhuCioll

D on

vectorfields on fl,

D(p) = spanlXi (p)

then the prolongaCion of D is

TN defined as

(13.31)

q E TN.

Analogously, lee the co-distribueion P on N be given as pep)

P of

II, pEN, !dth

0i

differencial one-forms

P is the co-distribucion

P on

011

E

(13.32)

Ttf.

involutive Clnd constant-dimensional we obtain

following simple local representations of band

P.

(Theorem 2.lI2) we can find local coordinates x =

a

D

span (ax! •... for TN we have

b

span

8

span{ol(p)1

H, then the prolongation

TN defined as

q In case D and Pare

I

the

By Frobenius' Theorem (Xl •..•

,xn

)

such that

lc = dim D. Then in the naturnl coordinates (x,x)

a

_8_ •...

... 'ax,_'· ~

aXl

,_a_I.

03.33)

.

ax;;

Similarly if P = span (dx 2n' .. ,dXn I, n-l

~

dim P, then

405

For general (co-)distributions D and P we obtain Proposition 13.5 Let D be a distribution, and P be a co-distribution on H. Then If D

=

ker P (see 2.173), then

(b) If P

=

ann D (see 2.174), then

(a)

b ker P. P = ann b. b

(e) It D (resp. P) has constant dimension then

(resp.

PJ

has conscant

dimension. (d) If D (resp. P) is involutive then

b

(e) If Xi E

Proof

(a)

follows

and

from

(b)

follow

(13.29),

from

Part

(e)

P)

Crespo

is involutive.

then XED.

(13.28).

Part

(c)

is

is proved as follows. Then Xi

P E H. functions

b

for a vectorfield X on 1'1,

~

L:

trivial,

while

(d)

Let Xi E b, where i for certain )

(0)'1 + PiXi

Pi on TfL By (13.27) we obtain

CIt,

for all h: N -. IR and thus by Lemma 13.2,

I

. CliX i

~ 0, Hence X

i =

lEI

and from the local expression (13.20) can be chosen independent of

X,

I

PiX!

,

'

iEI

it is easily seen that 13 1

implying that X -

I

,

i E I,

o

P1X! ED.

lEI

A key observation with regard to the use of the prolonged distribution

b

for purposes of invariance is contained in

Proposition

Consider

13.6

system map F: N x U

-+

the

TN as

nonlinear dynamics in

(13.2).

(13.1)

given

by

the

Let D be a distribution on N.

Define De as the unique distribution on N x U such chac (13.35)

l>'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

<:/

coordinates x = £ n Go

=

span

system for (fl,f2), ->

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 <=- 0, (X,U) E

v,

i;" j, i,j

E~,

(13.84)

and finite integers Pl"" ,Pm satisfying k=O,l"",pi' CX,t!) EV, IE!!!,

(13.85)

such dwt moreover the set (13.86)

contains V. From now on we will drop for simplicity throughout the adjective strongly. The

(cf.

local regular static state feedback input-output dccoLlpling problem Problem

feedback u is

=

8.6) a(x,v)

is

to

locally

input-output

problem is

approached as

(13.82)

finite.

are

find

a

locally

defined

regular

static

state

for (13.82) such that the feedback transformed system decoupled. follows.

Then

define

Analogously

Assume the

to

throughout

decoupling

Theorem that

matrix

8.13

Pl"'"

of

Pm

(13.82)

this for as

(compare with (8.25»

a A(x,u)

-

Bll;

L;Pl+ll h1 (x,Ll)

a aUm

L;Pl+ 1 )h (X,Ul 1

(13.87)

a ~

L;Pm+ll hm

(X,

Lll

a all",

L;Pm+l)hm(x,u)

418

Theorem 13.2/, Consider tile square system (13.82) Idth finite characteris-

tic numbers

Pm' Let (xo ' u o ) E f1 xU,

p} •••• ,

Then the local regular static

state feedback input-output: decoupling problem around (xo ' u o ) solvable if and only i f

is locally

(13.88)

rank A(xo,u o ) - m . Proof Let (13.88) be satisfied. Consider the equations

1l::]

) L ( PI +l)h 1 (" X. U

r!

L~Pm+l)hm(x.u)

By

the

(13.89)

Implicit Function Theorem

solution

II

of (13.89)

= a(x,v)

there

with

exists

:~(x.v)

locally

about

inverti.ble,

(x"

Since Yi

1(10)

a

(Pl+1)

-

i E~, this feedback solves the pr.oblem. For the converse

1

L r (Pi+ )h 1 (x.u).

direction we refer to the proof of the "only if" part of Theorem 8.9.

Similarly (13.88)

to

is

Chapter

also

for

B

(see

Exercise

smooch systems

8.1),

(13.82)

it

r.eadily

0

follows

that

a sufficient condition for

local input-output decoupling (see Exercise 13 ,11). For the generalization of

the geomecric

theory of input-output

decoupling

(see

Chapter 9)

to

general smooth systems (13.82) we refer to Exercise 13.5. Now suppose rank

A(xo ,uo) "" m. Consider the functions (13.90)

By

definition

Furthermore,

of

E~.

i

Pi.'

all

these

functions

do

not

depend

an immediate ex tens ion of Propos i tion 8.11 shows

on

that they

are independent functions of x. Hence we can choose additional functions about Xo such that

(Z1

f

•

,

,

f

u.

zm, z) are local coordinates for N about

Xo

z

with

z(xo ) - O. After applying the feedback resulting from solving (13.B9) for u we obtain in these coordinates,

analogously to

(B.l1S),

the following

local "normal form" (valid for analytic as well as for smooth systems) i

E m, (13.91)

where the pairs (Ai ,b i

),

i E:E. are in Brunovsky canonical form (6.50).

Example 13.25 Consider the simplified model of a voltage fed induction motor, as dealt with in Example 8.21. As in (8.98) we let the supply voltage vector be expressed as

ii, [ U z

1

1

[V

cos

V

5ln

(13.92)

0

where V is the amplitude and V

=;

u, and

,

,

the angular position. Now lot us consider

u z as inputs to the system (compare with Example 8.21) .

Clearly for this choice of inputs the system is nor affine in the inputs.

As

in

Example B.21 we consider the

flux

stator

and stator

torque

as

outputs, i,e.

(13.93)

Pz

~

0,

and

the

equations

(l3.B9)

are

given as

(compare with

(8.101» (13.94)

+ (0 + P)X 1 X 4 - (0 + P)X Z X 3 + WX Z X 4 - wo

WX1X J (-Xl

+ 0-lL~lXJ)Ulsinuz +

(X z

-

- 1

a-lL~lx4)U!COSU2

- 1 2 L~ X J

= Vz

Hence the decoupling matrix A(x,u) equals

2xJ cosu" +

2X4 5

iou z

[ (-x + o-lL~lxJ)sinU2 1

I-

(X2-0-1L~lXdcosuz (13.95)

which is a non-singular matrix in all points (x, u) where det A(x, u)

Clearly

01

is

~

(13.96)

2u, [

always

different

from

zero,

being

the

amplitude

of

the

voltage. Furthermore the term inside the large brackets is proportional to the

scalar product of the stator and rotor flux,

during

normal

mode

operation.

Bence

in view

and

thus

of Theorem

is non-zero

13. 2LI,

during

normal mode operation the system is locally input-output decoupable by a static state feedback and this feedback is locally given as the solution

o

of (13.94).

Now,

as

in

Chapter

8,

we

proceed

to

the

problem

of

input-output

decoupling of the square system (13.82) by dynamic state feedback ),(2,.'{,

v),

(13.97) a(z,x,v),

420

where

and i' are smooth mappings.

Q'

As in Chapter B we assume throughout

that (13.B2) is an analycic system, and we only solve a local version of

The key tool is

the problem.

the

following generalization of Algorithm

B.18. Algorithm 13.26 (Dynamic extension algorithm) Consider the analytic square system (13.82).

Step 1 Denote

the

ticity r] (x,u)

1

by D (x.u). Let

is constant,

Assume we work

N x mm.

output functions independent.

pi

characteristic numbers by

ling matrix (cf. (1.3.87)

on

h

o

I

P~.

and

the

1

neighborhood contained

in such

}ll •..• Illm

\Jri te

I ' ••

(x,u):- rank D (x.u).

decoup-

By analy-

on an open and dense subset B1

say r!. a

t"1

way

a

h

Reorder

,

of the

that the first rows of Dl are

-(l

(hI' ...• Ill' I) ,

in 8 1

=

pondingly y - (yl,~l). Consider the equations

(lJ r 1 - 1 '

I

,iJ rn )

•••

and

corres-

I

J'l

L ~ (Pl+l)h 1 (. x, u )

(Pl +1)

1

1

Yr1

p~

;

(13.98)

+1)

1

By the Implicit Function Theorem we can locally solve for r l u as a function a1(x,v) of x and v,

components of

so as to obtain

1

Yr

(Pr

11)

=

(13.99)

i

Vi •

Thus after a possible relabeling of the input components obtain a and

v

1

"partial"

(v 1

-

system

r1

transforms

1

Jjo(x,a (x,v

».

1

).

feedback

Leaving

the

l

=

vI

as u

l

where

11l(X,Vl),

remalnlng

inputs

vl,~I) :~ f(x,a\x,v

into

Renaming

u

l

(U l •••. , Urn)

u

~1

l

(ul . . . . 'U

unchanged. and

)

Vi)

we r1

)

the

:~

we obtain the system

1 -I

x

u ,u )

-1

-1

(13 .100)

1

11 (x,u )

Y with

•••• • v

state

inputs

-1 l/ •

outputs

-1

y,

and

where

1 U

is

regarded

as

a

set

of

parameters. Step £+1 Assume we have defined a sequence of integers t"1 •••• ,r,£. with J! 1 i -J' qJ!: I r i I and we have a block partioning of h as (h .... ,lJ,}l), with ~1

dim

r

i

I

i E ~,

and

correspondingly Y

assume We have obtained a system

~

1

(y •...

.i! -1

,Y ,Y ).

Furthermore.

421

'1 cere and

t

(13.101)

-1 -1-1 £ h (X ,U ,u),

-i

Y

h e new contro 1 5 u are correspon d'l.ng 1 y sp l'I.t as u -

V1 - 1

denotes

1 U - 1 :=

1 (u , .. , ,U.2-1)

and

regard (13.101) as a system with inputs _

ilJ!,

•

1+ 1

Denote the characterlstlc numbers of (13.101) by Pqr l"" £+1

decoupling matrix of (13.101) by D

(m-ql) x (m-q,r)

( u, 1 .. , ,u .2 ,u -1') ,

time-derivatives. We i 1 parametrized by V - and / .

suitable

1+1 ,Pm '

-1-1'

(x,U

,l.l

),

and the

Byaoa1y-

ticity the rank of this matrix is constant, say r . ' on an open and dense l .1_ .2 .2 1 such that subset B.2+1 of points (x,V ,il ). Reorder the output functions h the

first

r

q.e + r 1 +1 , h

rows

iH

1+1 =

of

(h q1 + 1

niH

...

·,h'l,e+l)'

(yl+1,yl+1) , til

accordingly;;1

are

~

independent, -1'+1 h =

and

write

(h'l1+1+1, .. ·,hm),

with and

denote

1 (U 1 + ,ljlil). Consider the equlltions

(13.102) By the Implicit Function Theorem we can locally solvo for r ft1 = q1+1 - q f components of

U-

f

as

functions

of x,

Vf

as to obtain

(13.103) Thus after a possible relabeling of the input components of ;:'/ we obtain a f+1 f+1 -f f+1 ~1+1 "partial" state feedback u = a ~(x,U , v ) . Leaving the inputs u 1 1 f unal tered the sys tern transforms into fft1 (x, V , vf-t 1, U- + ) : = 1H f1(x,Vf-l,,/+l(x,Vf,vftl),uf+1) with output function Ill'+l(x,Vl',v ):_ 1 1 1 f 1 1 Il (x,V - ,c/+1 (x,i/,v +1». Renaming again u.!'+1 '= V + we obtain the system

x

f1'fl(x,f/ ,u1t1 ,[,/+1)

-1+1 Y

Ill+1(x,Vl,ul+1) ,

(13.104)

o

As in Chapter 8 it follows that there exists some finite integer lc such that

(13.105) and the integer q~ = qk will be called the rank of the system (13.82), Analogously to Theorem 8.19 we obtain

422

Consider the square analytic system (13.82), The follol,,ring

Theorem 13.27

t!"O condit ions are equivalent: (1)

The dynamic scate feedback inpuc-output' decoupling problem is locally solvable on an open and dense subset of 1-1,

(ii) The rank q" of the system equals m.

For the proof of Theorem 13.27 we refer

the proof of Theorem 8.19;

to

details will be left to the reader.

13.4 Locally Controlled Invariant Submanifolds Analogously to Definition 11.1 we will call a submanifold N C N locally

cont:rolled locally

invariant:

around

any

feedbac1c. u - a (x),

X

for

the

Xo E

N

nonlinear a

strict

system static

(13 .1) state

if

there

feedback.

Locally

we -

lIIay

briefly

EN. such tha t

(13 .106)

x EN.

N - (Xlx l

exists

choose

coordinates

x '"

(Xl

,Xl)

for

H

such

that

0). If we write accordingly

1

f 1 (x 1 ,x 2 ,u)

f(x,u) [

(13.107)

i~(xl,X2,u)

then we obtain the following analog of Proposition 11.2.

Proposition 13.27 Consider elle nonlinear system (13.1) with N C H, locally

given as (X/Xl

af l

0). Assume that rank au(O

,u) does not depend on xZ,u.

Then N is locally controlled invariant if and only if for any x E N there exists u E U such thar:. f(x,u) E TxN. ilfl

2

~

Proof Let rank au(O ,x ,u)

s. Then by the Implicit Function Theorem the

equation £1(0 ,x'2, u) - 0 can be solved for s of the u-variables as smooth func tions of

o

X2.

Remark 13.28 Moreover it follows that we can locally define a degenerate ~

feedback u = a(x,v), v E III •

m :~

111 -

5,

ax

rank 8v

x EN,

m,

such that

for all v E ~m

Now let us consider a nonlinear system with outputs

•

(13.108)

423

x

f(x,u)

xEN,UEU,

(13.109)

y

h(x,u)

=

A submanifold N C N will be called locally controlled invariant outputnulling if there exists locally a feedback u

t{x,oCx» In

order

11(x,o:(x»

E T"N

to

compute

the

nulling submanifold for

a

=

(13 .110)

N

X E

locally

llJaximal

(13.109)

o(x), x E N, such that

=

controlled

invariant

we again take recourse

to the

outputeXTended

sysCem x ~ f(x,u) {

u

~

Iv

y

=

h(x,u)

which

is

(13.111)

an

for

Y

E [RP

system

affine

(xo ,u o ) E 11 x U, point

(X,u)ENxU,

with

state

satisfying £(xo ,u o )

Algorithm 11.6

applied

=

to

(x,u)

and

input

0 and hexo ,u o ) the

=

0,

extended system

Now

Iv.

be a

let

regular

(13.11).

Then

locally through (x o ' liD) we obtain the maximal locally controlled invariant output-nulling submanifold N::= N:* c N xU, given in the form N:

I (x,u) near (x o ,un) !h(x,u)

=

=

!PI (x,u)

'f);"_I(X,U)

=

OJ

=

(13.112) By the first assumption of regularity in Algorithm 11.6 the matrix h(x,u)

a au

(13.113)

!PI (::"u)

!P,,*-I (x,u)

has constant rank

y";=

r k " on

N:

(due to the particular form of the input

vectorfields in (13.111». An application of the implicit function theorem

.

yields = r

the existence of a regular feedback u

, such that locally around (x Q , u Q h(x,.(x,u ,u"»

rank"';"

u

I

[

'Pk~-l (x,cr(x,u' ,u"»

=

o(x,u' ,u"),

dim u"

h(x,.(x,U' ,u"»

a

]

with

)

- r

]

'_0

ali' !Pk"~l

(X,CI(X,U' ,u"»

(13.l11l) and

thus

we

can eliminate

the variables

u"

as

Furthermore, by the second regularity assumption matrix

a in

function

u"(x)

Algorithm

of x.

11.6

the

a[

1

11 (X , a (x , U' ,U II (x) »

~

ax 'PI<

h _

1

(13.115)

(x 1 a (X,

has constant rank on state-variables

U

I 1

u" (x) )

N; .

It follows that we may solve for a part of the

say x", as a function x" - lJ!(X') of the remaining state-

I

variables, denoted as x'. Thus finally we obtain a system of the form

x'

(13.116)

f' (x' • U ' )

=

where f'(x',u'):= [(x',x"(x'),u'), with [(x,u'):" f(x,a(x,u',u"(x»). set

Ix -

I

(x' ,x")

!/,(x '

X" -

»)

defines

The

a subrnanifold,

locally around x(]

denoted as N·, of H. By construction N" is a locally controlled invariant output-nulling submanifo1d for (13 .109). which is maximal in the follOWing sense.

N be

Let

submanifold

N - (xlx

1

is

xo'

locally Locally

controlled around

ah

invariant we

may

output-nulling

represent

1

au

N

as

(13.117)

2

(O,X ,u)

constant,

say

r,

Bround

(xo,u o )'

Then

locally a degenerate feedback l 1 Go. =m-r , rancGv=m, such t1at

exists 171:

Xo

~ OJ. Assume that the rank of the matrir: (cf.(13.107»

ru(D,x afl Z ,u) [

another

through

f(x ,o:(x, v»

E 1'xN

h{x,o:(x,V»

- 0 ,

I

u

(see

Remark

o.(X,V) ,

(13.28»,

there

o:(xo,O)

, x E N, for all v E ~m

(13.118)

Now one easily checks that the following submnnifold of H x U

:~

No is

a

(X,a(x,V»

locally

x EN, V E

controlled

extended system Nc

I

(13.111),

invariant and

thus,

~m) output-nulling since

No

is

submanifold maximal,

for

the

necessarily

eN:. Since x' serve as coordinates for

on N",

which will be called,

i',

(13.116) describes the dynamics

in analogy with

(11.31).

the

constrained

dynamics for (13.109). (For a definition of zero dynamics. analogously to Defini.tion 11.111

I

we refer to Exercise 13.8.)

As in t.he affine case

(cf.

Proposition 11.13) we have

the following

important. special case where the computation of N- becomes more easy.

Proposition 13.29 Consider the system (13.109). Assume chilt the characte-

425

ristic numbers PI""

,Pp as defined in Definition 13.22 are all finite.

Furthermore ilssume that ella p x m matrix A(x,U)

~ au,

[

L(Pi +l)h i (. f :;:

,U

)

1,

(13.119) 1, ... ,p 1, .... m

lU1s rank p on the set

(13.120)

(Note that by definition of Pi

depend on uf).

I f N*

wi th h (xo ,un) ~ 0 and f (xo ,un) of

dimension

controlled

feedbacks u

I"

n -

(xo,u o )

then N~ is a smooth submallifold of N

0),

=

and

(Pi +1),

invar lane =

,L~ihi' i E E. do not

elle functions hi""

is non-empty (for example i f there exists

is

equal

outpuc-nulling

to

the

submanifold

maximal for

(globally)

(13.109).

The

o:~(x) which render N~ invariant are given as the solutions

of Lf

(Pi+ 1 )

"

(13.121)

,iEE

chere exists locally around any point in N" a degenerate feedback 80 _ P , rank av = nI, such that

In fact,

U

*

ht(x,cr(x»=O,xeN

o(x,v), v E [\1m , iii := m -

=

X E

N~,

for all v E IR

and the constrained dynamics are given as

m J

i

the dynamics

E. '

E

x

=

(13.122)

f(x,o(x,v»

restricted to Ni'<

Example 13.30 Consider the model of a

rocket outside

treated in Example 6.13. Consider as output y the angle Then clearly p

=

1,

which is non-zero 0),

and

and

for

the

decoupling matrix

u around O.

a"ex)

solving

z constrained (and zero) dynamics

X

=

XI,

=

Thus,

N"

(13.121)

=

is

the

atmosphere

given as

(Xl ,Xz ,XJ 'X4)

equals

0,

as

i.e, )' = Xz' (T/nL'C 1 )cosu,

E T(IR+X Sl);

resulting

in

a

(13.123)

o The theory of interconnections of systems, as dealt with in Chapter 11 for affine systems and for

interconnections only involving the states of

the systems, can now be immediately extended to general nonlinear systems

i E k

(13.124)

426

interconnected by a interconnection constraint of the form (13.125) where rp is a smooth mapping.

Indeed we have to compute the constrained

dynamics for the general nonlinear system defined by the product: system (cf. (11.52a»

of the systems (13.l2Lf), with output mapping cp.

Example 13.31 Consider two nonlinear systems Xi - fl(x l ,u t i

),

Yj - hl(xi,u t

1,2. Suppose the second system is placed in a feedback loop for the

=

first system, resulting in the interconnection equations

(13.126)

-1. while the decoup-

Clearly. all the characteristic numbers are equal to

ling matrix I

(13.127)

81Jl ab 2 ]

au

has full rank if and only if the matrix [ I condition is usually imposed

85

z

is invertible. This

a prerequisite for "well-posedness" of the

interconnection.

The

extens ion of

the

theory of inverse

systems,

as

wi th affine

deal t

systems in Chapter 11, to general nonlinear systems (13.109) will be left to the reader (see also Exercise 13.6).

13.5 Control Systems Defined on Fiber Bundles

As

already

definition F : l!

xu ....

al:gued of

a

in

the

nonlinear

TH has

Introduction c.ontrol

(e. g.

system

some serious drawbacks.

X

Example

f(x,u)

1. 7).

the

as

system

a

The problem is

thus

that

input

state and

the

map

that in some

cases the input space U is not independent of the state. the use of the product N x U of the

global

implying is

too

restric.tive. The mathematical generalization of a product space is a fiber

bundle: Definition

13.32

(B,H.~,U,(O!

liEI)

is

called

a

fiber

bundle

if

the

iollOl..ring holds. B.N and U are smooth manifolds, respectively called the

),

total space, the base space and the standard fiber. The map if: B

-+

t1 is a

surjective submersion, and 10i)i E I is a covering family of open subsets for

t1

such

that 01

for

every

ieI

there

diffeomorphism

a

is

U, satisfying the conunutative diagram

X

---------;) 01 X U

/

(13.128)

projection

0,

The

submanifolds

1r

-1

(x), x E tl,

are

called

the

fibers

at

If

x.

no

confusion can arise I>'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)

:=
are

local

coordinates

for B around b o ,

' ... ,urn) = (X, u).

which we will

,U 1

(13.129),

is locally represented as F(x,u)

the local coordinate expression Observe

.... 0 1 X U,

x

=

=

(x,f(x,u».

(x,u)

as above,

thus recovering

we can take local

either using the diffeomorphism 1'1

or the diffeomorphism q'j

as

in view of

f(x,u).

that on any non-empty intersection 01 n OJ

coordinates

simply denote

In such coordinates the map F,

(Xl' ... 'Xn

: ;r-l(Oj) ... OJ x U.

: IT-I{Ol)

The relation is as

follows. There exists a diffeomorphism q'ij rendering the following diagram commutative 11"-1(0 1 n OJ)

/

(Oi n OJ) x U

'>"

~

~(0

(13 .131)

1 n OJ) xU

428

Let now xI] E O! n OJ 1

let

b o E n- (x o ) ,

ifo i (b o ).

and u

j

and take local coordinates x around xo'

and

take

local

coordinates

(ur, ... , u~) around IJ"'j

-

(b o ).

u

i

Furthermore

(ui ..... u~)

=

around

In such coordinates
j

is

locally represented as (13.132) Alternatively, as in (13.130) we obtain two coordinate systems (x,u (x, uj)

around

since II,! j

bo

which

E B,

are

related

as

u

j

'PI

""

(x. u!).

j

_a_

is a diffeomorphism it follows that rank

'Pi j

i )

and

Furthermore

m everywhere,

-

au!

and thus this change in input coordinates can be interpreted as locally defined feedback. More generally, we give

Definition 13.34 Let (B.H.F) be a nonlinear system. A (global) feedback is

a diffeomorphism

B

A

B

B

1f~ (A

B satisfying

-t

(13.133) /1<

lJ

is called a bundle isomorpl1ism.)

(x,v)

and

be

(x,u)

respectively

around

sets

tl>'O

local

of

b o E B and

Let Xo E 0i

around

and b o E

coordinates A(b o )

B.

as

1f-

1

(13.130),

A is

Then

Let

(xo )'

in

locally

represenr:ed as A(x,v)

~

(x,a(x,v)

u)

(13.134)

.

The feedback transformed system is denoted as (B,N,FaA). The above definitions specialize to affine nonlinear control systems m

x

=

f(x)

I

+ j

called a

gj(X)U j

follows.

A

vector bundle if U is a

diffeomorphisms (11.1 'P!j

as

fiber

bundle

(B,N,7!',U,{OlliEI)

is

~l

:

1f-

1

as in (13.132). i , j

(01) .... 01 E

linear space,

xU,

i E I,

I, are linear in u

I.e.

U - rn

m

,

and

the

are such that all the maps i

if we take u! and u

,

j

to be

linear coordinates for U. An affine control system (B.N.F) is now given by a vector bundle (B.N) and a system map

,,-lex)

frorn

C B

to

tr~l(x)

linear coordinates for U expression

F(x,u)

C TN.

IR

m

F: B

Taking

-t

TN which is affine as a map

local

coordinates

x

for

Nand

we immediately obtain the local coordinate

(x.i = f(x) +

I

gj (x}u j

).

Finally

a

(global)

feed-

j"l

back A will be defined as in Definition 13.34, with the restriction that A as

a

map

from

1<-I(x)

to

itself

is

affine,

resulting

in

the

local

429

coordinate expression (3(x) an invertible

Let

us

now

(cf.

(13.134»

A(x,v)

(x,a(x) -I- P(x)v = u),

=

with

x m matrix.

III

indicate

how

the

material

of

the

generalizes to nonlinear control systems (B ,N IF).

previous

sections

First let us consider

Proposition 13.6. We immediately encounter the problem of defining De as a distribution on B instead of on N x U. we can define, such

that

Clearly on any subset

similarly to Proportion 13.6,

pr z : 0 1

=

D

01

and

1

(Ol) C B

n!

the unique distribution

1rlnD!

=

0,

*i

:= pr z 0
on

1f-

where

does not necessarily define a distribution D" on the \"llole fiber bundle B; in fact we need that is ensured by the Suppose

n! -

D~ on

1f-

1

(Oi n OJ)'

The global definition of De

following additional requirement on the bundle

that all the maps

i,j E I,

'P1j'

(D ,N).

(13.132) do not depend on

as in

n OJ' Defining on every lI'-l(Oi) the distribution 11' as :IT~;(O),

X E 0t

easily

follows

that

Hi

=

H

j

on

1r-

1

and

(Oi

globally defined distribution H on B with 11 follows

lI'~H

that

.. TN and

that dim H

there

exists

on 11'-1(0 1 ), i E 1.

Now we define De

dim N.

=

thus

i

1I

=

it a

It

on B by

setting

(13.135) Contrary

to

Proposition

13,6,

the

generalization

of

control sys tems (B, N, F)

13.7

Theorem

(characterizing locally controlled invariant distributions)

to nonlinear

does not pose any problems.

Indeed,

Ie t D be an

involutive distribution of constant dimension on N,

Define

the vertical

distribution Ge Assuming

on B as Ge = 11';1(0),

G"

that

has

constant

Ge

and denote

dimension

we

=

obtain

controlled invariant for (B,/'I,F) i f and only i f (cf.

!Z E Ge

D

that

I

F*Z E

is

V).

locally

(13.40'»

(13.136) Hence

if

the

(x,u)

as

in

regular

system (/'1, D ,F)

(13.130),

static

x - l(x,v)

is

feedback

locally represented as x

(13.136)

then u

=

implies such

o(x, v)

the that

Let us now briefly elaborate on such local

0

D

~

f (x, u),

existence

is

invariant

wi th of

a

for

:- f(x,o(x,v»,

Take local coordinates x Xi

local

=

(Xl""

11' : B ..... IR we can regard

Xl"

,Xn ) for N. ..

feedback

transformations. N ..... IR with

Identifying Xi

,xn as coordinate functions on B. Thus ~ (u 1 "" ,u m) on B such that Such coordinates for B are called

we can take additional coordinate functions u (x,u)

fiber

is

a

coordinate system for B,

respecting;

indeed

the

fibers

of

B

are

given

as

the

sets

x = constant.

(Note

that

the

coordinates

(x,u)

in

as

(13.130)

are

automatically fiber respecting.) Now let (x, v) be another set of fiber respecting coordinates on

the same neighbourhood of B.

It immediately

follows that (x,v) is related to (x,u) by a mapping

a(x,v)

~

u

rank

aa

(13 .137)

m,

av(x,v)

and conversely, every mapping u - o(x,v) as in (13.137) defines a new set of fiber

(x,v).

respecting coordinates

conclude

We

that local regular

static state feedback (13.37) can be regarded as the cransition from one set of fiber respecting coordinates to another. Furthermore, in case (B,M) is

a

vee cor

it

bundle

is

natural

coordinates (x. u) such that u fibers

of B ~

det [J(x)

and a

I

0,

to

regular static

corresponds

to

restrict

the

to

fiber

respecting

are affine functions on the

(u1 •... ,um )

state

u

feedback

transition

from

one

£leX) -I-

set

of

{J(x)v, fiber

respecting coordinates (in the above restricted sense) to another. Let us now pass on to disturbance decoupling. First of all, what is the appropriate global description of a system with disturbances (13.76)? We will choose to model the input space U as state-dependent (as before), and the space Q of disturbances as state- and input-dependent:. Formally we let 11'

:

and ;r ;

B -. 1'1 be a bundle with standard fiber U,

B -.

B be

a fiber

bundle (with base space B!) with standard fiber Q. Definition 13.35 A nonlinear control system with disturbances (B,B,N,F) is given F :

by

B ...

fiber

bundles

(8,11)

(.8 ,B)

and

as

above,

and

a

system

map

TN saclsfying

(13.138)

Let (x, u), respectively (x, u. q), be fiber respecting coordinates for B respectively for li,

then we immediately recover from (13.138)

I

the local

coordinate expression (13.76a). Consider now a dis tribution D on N. controlled

invariant

coordinates (x, u) (x,u,q) for

B We

for

(li.B of! ,F),

if

We will say tha t there

exist

D is

fiber

locally

respecting

for B such that for all fiber respec ting coordinates

have that [f(. ,u,q) tD)

Proposition 13.36 Let (8 ,B ,fl,n be

il

C

D, for every lI,q.

nonlinear cont:rol sysc.em IIlit:h d1.stllr-

431

bances. Let D be an involutive distribution of constant dimension on H. Assume

that

dimension.

the

distribution

{Z E

1r;-l(O)

I

F~Z

E

D)

on

Then D is locally cont:rolled invariant for

Ii

llas

constant

(B,B,N,F)

i f and

only i f

0) F*{7r;-l(D»

c b + p,,(1J";-l{O» , (13.139)

Proof Use Theorem 13.7 together with the observations made in the proof of

Proposition 13.21.

0

Finally consider the system with disturbances (B,B,H,F),

together with an

output map

h : B

-+

IRF

(13.140)

It is easily seen that Proposition 13.21 immediately generalizes to this

case.

Furthermore,

all

the

results

obtained

in

Section

13.3

about

input-output decoupling, being local in nature, immediately carryover to

the case of a nonlinear control system (B,N,F) with output map (13.140). The same holds for the material on controlled invariant submanifolds and constrained dynamics as covered in Section 13.4.

Notes and References

The generalization of the notion of controlled invariant distributions to general nonlinear systems, [NvdSl].

The

including the basic Theorem 13.7,

present proof

is

also

partly based

on

is due

{vdS2],

while

to the

definitions of prolongations are largely taken from {Ylj. For a treatment of controlled invariance by output feedback for general nonlinear systems we refer to [NvdS2]. The relation between feedbacks for the system and its extended system were noticed in [vdS3];

see also

[NSj.

The treatment of

the disturbance decoupling problem for general nonlinear systems (Section 13.2)

is due

to

general nonlinear approach

[NvdS3j.

The

input-black-output decoupling problem for

systems

was

studied

present

analytic

(Theorem 13.24) and the extension to dynamic feedback

in

[vdS3];

the

(Theorem

13.27) seem not to have been stated explicitly before.

Example 13,25 is

based on [dLU]. The generalization of the notion of controlled invariant suhmanifold and of constrained dynamics to general nonlinear systems has been dealt with in [vdS4].

432

The definition of control systems on fiber bundles is due to [Brl,2J, see also

(WiJ,

and was

further

developed

in

[vdSl,2], [NvdSl,2,3];

in

particular Definition 13.35 and Proposition 13.36 can be found in lNvdS3]. Relationships between controlled invariant distributions

and integrable

connections on fiber bundles were studied in [NvdSI).

[SrI)

[Br2] [CvdS} [dLU]

(NS)

[NvdSl]

[NvdS2)

(NvdS3]

{NvdS4j [vdSl]

[vdS2] (vdS3j

(vdS4j

[Wi) [Ylj

R.W. Brockett, "Control theory and analytical mechanics", in Geometric Control Theory (eds. C. Martin, R. Hermann), Vol VII of Lie Groups: History, Frontiers and Applications, Hath Sci Press, Brookline, pp. 1-46, 1977. R.W. Brockett, "Global descriptions of nonlinear control problems; vector bundles and nonlinear control theory", manuscript 1980. P.E. Crouch, A.J. van der Schaft, Variational Hamiltonian Control Systems, Lect. Notes Contr. Inf. Sci. 101, Springer, Berlin, 19B7. A. de Luca, G. Ulivi, "Dynamical decoupling of voltage frequency controlled induction motors", in Analysis and Optimizlltion of Systems (ads. A. Bensoussan, J.L, Lions), tecto Notes Gontr. Inf. Sci. Ill, pp. 127-137, 198B. H. Nijrneijer, J.H. Schumacher, "Input-output decoupUng of nonlinear systems with an application to robotics", in Analysis and Optimization of Systems, Part II, (eds. A. Bensou5san, J.L Lions), Lect. Notes Contr. In£. Sci. 63, Springer, Berlin, pp. 39l- l tll, 1984. H. Nijrneijer, A.J. van del' Schaft, "Controlled invariance for nonlinear systems", IEEE Trans. Aut. Contr., AC-27 , pp. 904-91l1, 1982. H. Nijmeijer, A.J. van der SChl1ft, "Controlled invariance by static output feedback for nonlinear systems", System Control Lett., 2, pp. 39-l I7, 1982. H. Nijrneijer, A.J. van der Schaft, "The disturbance decoupling problem for nonlinear control systems", IEEE Trans. Aut. Contr .• AC-28 r pp. 621-623, 1983. H. Nijrneijer. A.J. van der Schaft r "Partial syrnmetrles for nonlinear systems", Math. Syst. Th. 18, pp. 79-96, 1985. A.J. van der Schaft, "Observability and controllability for smooth nonlinear systems", SIAM J. Contr. Optimiz., 20, pp. 338-354, 1982. A. J. van der Schaft, System theoretic descriptions of physical systems, CWI Tracts 3, CWI Amsterdam, 1984. A.J. van der Schaft, "Linearization and input-output decoupling for general nonlinear systems", System Control Lett., 5, pp. 27-33, 1981,. A.J. van der Schaft, "On clamped dynamics of nonlinear systems", in Analys is Ilnd Control of Nonlinear Sys tems (eds. C. I. Byrnes, C.F. Hartin, R.E. Saeks), North Holland, Amsterdam, pp. 499-506, 1988. J.C. Willems, "System theoretic models for the analysis of physical systems", Ricerche di Automaticil, 10, pp. 71-106, 1979. K. Yano, S. Ishihara, Tangent ilnd cotangent bundles, Dekker, New York, 1973.

'133

Exercises

13,1

([CvdS)

Consider an affine nonlinear system m

x

L

f(x) +

=

j

~

gj (x)u j

i E Along

every

x EN,

U E [Rm,

y

E

mP

,

1

P. '

(u(t), x(t), yet»~

solution

of

2::

we

can

define

the

linearized system, which is a linear time-varying system, If we take 2:: together with all its linearized systems we obtain the system

f(x) +

at ax(x)v +

ag j ax (x)v

L ~

j

m

+

uj

1

L j~

gj

(x)u~

1

i EO P.

i E P. with v the variational state,

u~ the variational inputs and y~ the

(2::i' is called the prolongation of 2::,)

variational outputs.

(a) Show that in a coordinate-free way 2:1' is given as

- f(x + - hi1 (xl')

xI'

p

y,

m

L

)

gj(xp)U j

+

j~

j" ,

~

1

L

gj (xl' )u.i

1

xl' E Til

i E E

i E P. with state (natural

m

(natural coordinates (x,v», input space 71R v coordinates (u,u » and output space 71Ri' (natural space

TN

coordinates (y ,yv».

eo

(b) Let

be the strong accessibility algebra and 0 the observation

space of 2:.

Show that the strong accessibility algebra fJ~ and the

observation space 01' of I: P are given as [;'1'

~

0"

+

HE 0

Furthermore

13.2

eo } +

X E

}.;

prove

dim Co

=

dim N,

dim dO

=

dim N.

Consider E~ and V"

I

that

and

that

as

X E Co

Xf

,,1

HE 0

dim C~

if

dim TN

=

dim dOl'

=

in Proposition 13.21.

and

if

dim TN

Suppose

and

only

if

only

if

that t(x,u,q)

happens to be affine, i.e. f(x,u,q)

=

L gj (x)u

f(x) + j

~

1

1

j

Lei (X)qi

+ i

~

'

1

while h(x,u) only depends on x. Show that V* is equal to the maximal

434

distribution D satisfying

1 E .2 W

show

Furthermore

that

the

F.TQ C b

condition

reduces

to

span(e1 •...• e ) c D*. Finally show how the conditions of Proposition i 13.21 reduce to the conditions of Theorem 7.14. 13.3

Consider a finite

general

nonlinear

characteristic

characteristic

y - h(x,U)

nWllbers

Are

equal

of

the

f(x,u),

e·

Pi' i E

extended

Pi + 1. i E

to

x ""

system

numbers

X-

system

e.

y - h(x.u). Show

Show

that

with

that

f(x,u).

the

the ~ - w.

decoupUng

matrix of the extended system equals the decoupling matrix A(x, u) defined in (13.87). Relate the decoupling feedbacks for ehe original system to the decoupling feedbacks of the extended system. 13.4

smooth square system (13.82).

(n) Consider a

(13.88)

condition solvability

of

is

the

sufficient

a

regular

static

Show

condition state

that for

feedback

1:he the

rank local

input-output

decoupling problem (using the obvious generalizatl,on of Definition

8.1 for an input-output decoupled system), see Illso Exercise 8.1. (b) Consider Y2

the

analytic

Verify that the system cannot be decoupled using regular

xL.'

=

static state feedback. On the other hand. show that: the CO-feedback u1 = v 1

13.5

u2 ~ (vZ -V 1 )1/3. does decouple the system.

,

([vdS3 J)

Consider

a

Yi ~ hi (x,u), where

general

111

:

N ....

smooth

mPi

,

i

nonlinear ~.

E

U

system

x - f(x, u).

Show that the block

E (Rill.

input-output decoupling problem (as treated in Chapter 9 for affine smooth nonlinear systems) extended

system,

n;" . i

1!!,

E

as

and in

can be solved as

define

Chapter

for 9.

thi.s

Let

the

follows.

system

the

same

Consider

the

distributions

constant

dimension

assumptions hold as in Chapter 9. Prove that the block input-outpue decoupling problem is solvable if and only if the block input-output decoupling problem is solvable for the extended system, if and only if (with Go

=

a

span{

a

Show that the distributions D:~ feedback

in

dimensional n +

III

the

following

distributions

1, such thae

immediately determine a decoupling

way.

Construct

E£, i E~.

on

N

X

involutive !R

m

,

all

of

constant dimension

435

and m

Show that E :-

n

is again an involutive constant dimensional

E\.

1-1

distribution on /'J X 1R1Il. of dimension n and satisfying '/t"H"E = TM. The

required decoupling feedback u

I

sets {(x,u - n(x,v)) 13,6

~

has the property that the

n(x,v)

v is constant) are integral manifolds of E.

Consider the normal form (13.91). Prove that we can choose the local

z in

coordinates

such a way that

f in (13.91) does

no t

depend on

a

v l , .. ,vml if and only if the distribution (with fo :- f(x,u)ax'

a

a

a , ... , [f, '-aa-I ) u

[ f " -u-I

span! aU •... 'au m I

a

1

1

on H x U is involutive. 13,7

Show

x

that

feedback

the

linearization

0, x E [fin, U E!R

f(xolu o ) -

= f(x,u),

rn

problem eeL

for

a

system

6.10)

Definition

can

be equivalently rephrased as follows: Find m functions hi(x,u), with hI (xo ,u o )

0, such that the decoupling matrix A(xo ,u o ) has rank m

=

m

L (Pi

and 13.8

+ 1)

=

n.

Consider the square system (13.82), and assume that rank A(x,u)

=

m

for all (x,u). Show that the inverse system is locally given by the equations

z for

fez,

=

suitable

2

,

.

, . . . ,2 ,

initial

conditions

where

2(0),

2

and

f

are

as

in

(13.91) and Zi := (Yi"" 'Yi (Pi»), i E~, together with the equation U

where (d.

(Here

a(Z",

=

Z1, •••

,2m , y~P1t1) , ... ,y~Pm+l»

a(z,zl, .. ,zm, y~Pl+l)". ,y~Pm+l»

is

the

unique

solution

of

(13.89))

h1(z,zl, ... ,zm,u)

denotes

hi(x,u)

expressed

in

the

new

coordinates z,zl, ... ,zm,u.) Generalize Exercise 11.8 to the system (13.82) .

13.9

Generalize nonlinear

Exercises 7.10 and 11.9 square

system

x

=

(tlodel Natching)

f(x,u),

y = h(x,u),

to a

general

satisfying

rank

A{x,u) = m for every (x,u).

13 .10 (Zero-dynamics for general nonlinear systems, constrained dynamics

(13.116),

where

x'

are

[vdS4]). Consider the coordinates

for

N~.

436

Compute

for

its

extended

x'

system

I' (x' IU') • U· -

strong accessibility distribution,

and show that under

constant

distribution

ranlt

dis tribution dynamics.

,

conditions ~

on N,

Co

Assuming

Theorem 3.35,

this

which

that Co

is

has

the

dimension,

the

I

suitable

projects

invariant for constant

W·

to

a

constrained show,

using

that the constrained dynamics proj ects to the lower

dimensional dynamics (the zero-dynamics)

X,2

=

f,2(X'Z)

Show that this definition is consistent with Definition 11.14.

13.11 A vectorfield X on H is called a symmetry for the vectorfie1d £ on H if XL

(Indeed,

[X,f] ... O. G

~

_

~

D

XL,

and

in view of Lemma 2.34 thus

xt

maps

solutions

this of

implies ~ - rex)

that onto

solutions of E.) Show that [X,£] ~ 0 if and only if F.X = X, where is

F : H .... TM

the

map

given

in

natural

as

coordinates

F(x) - (x,f(x». Analogously, X is called a symmetry for x = f(x,u) if F.Xe with F : H X U

4

TH given as F(x,u}

a

X,

(x,f(x,u). and with Xn as in

the proof of Proposition 13.6. Let D be a distribution spanned by symmetries

Xl' ••.• Xk

distribution.

for x - f (x, u).

Conversely,

can

every

Show that D is an invariant invariant

distribution

be

written as the span of symmetries? (see [NvdS4]). 13.12 Prove Proposition 13.29, and show that N" as given by (13.120) can be also obtained by the general algorithm given above Proposition 13.29 (cf. (13.111) - (13.118».

14 Discrete-Time Nonlinear Control Systems

In the preceding chapters we have restricted ourselves to continuous-time nonlinear control systems, and their discrete-time counterparts have been ignored so far. Although most engineering applications are concerned with (physical) continuous-time systems, discrete-time systems naturally occur in

various

situations.

Most

commonly

discrete-time

nonlinear

systems

appear as the discretization of continuous-time nonlinear systems.

For a

continuous-time nonlinear system, locally described as

x

(14.1)

y

(14.2)

the discretization or sampled-data representation of (14.1,2) is formed in the following manner.

piecewise

Suppose the controls in (1l1.1,2)

constant fashion

[leT, (le+l)T) , le x(le) , u(lc)

=

and

so

that

u

is

constant

are applied in a

over

the

intervals

1,2, ... , where T> 0 is the sampling t:ime. Then, let:ting y(le)

denote

x(1cT) , u(lcT)

and

respectively,

y(kT)

one

obtains a discrete-time system x(lc+l)

y(k)

f(x(lc) ,u(1c»

(14.3)

h(x(k),u(k»)

=

The relation between (14.1,2) and (14.3,4) is then obviously given by the fact that x(1c+l)

= f(x(lc) ,u(k»

equals the solution at time (le+l)T of the

differential equation (14.1) starting at time leT in x(leT) - x(lc) and with a

constant

control

u = u

applied,

and

similarly,

y(k)

=

h(x(lc),u(k»

equals the output y evaluated at time leT. So we have

f(x,u) h(x,u)

where

fTu

(14.5)

h(x,u)

(14.6)

is the time T-integral of the vectorfield f (x) u

=

f(x,u). Clearly

the sampled-data representation (14.3,4) depends on the sampling time T. Discretizations of continuous-time

systems

are

especially

important

because in the control of continuous-time systems present-day technology often asks for digitally implemented controllers, and hence is operated in discrete

time.

This leads,

of course,

to several interesting questions

438

regarding

the

sampled-data

representation

(14.3,4)

of

(l4.l,2).

For

example,

what can be said about controllability and observability of

(14.3,4)

in relation

to

the

same properties of

(14.l,2}1 Or,

is

the

discrete-time system (14.3,4) input-output decouplable whenever (14.l,2) is? Here we will not pursue problems of this type, but instead refer to the relevant literature cited at the end of this chapter. The purpose of this chapter is to study the discrete-time nonlinear system

(VI.3,4)

per

and

se,

not

necessarily

as

the

sampled-data

representation of a continuous-time system. In this regard we emphasize that various models in biology, economics and econometrics are naturally formulated

in

discrete

time,

see

for

instance

Example

1.3

as

an

illustration. Several questions we l1ave studied so far for continuous-time nonlinear systems can also be stated for the discrete-time system (14.3,4). However, their solutions will not always results, since

typical

directly parallel

operations

associated

the continuous-time

with

system

a

in

continuous-time do not immediately pass over to a discrete-time system. We will

discuss

foregoing

here

only

chap ters .

some

In

of

the

Sec tion

continuous-time

14.1

we

deal

problems

wi th

the

of

the

feedback

linearization problem for the system (14. 3), which forms the discrete-time counterpart of Chapter 6.

Next,

in Section 14.2,

we study controlled

invariance and the Disturbance Decoupling Problem for the system (14.3,4), compare with Chapters 7 and 13. Finally we discuss in Section 14.3 the input-output decoupling problem for the system (14.3,4)

I

like we have done

in Chapters B, 9 and 13. 14.1 Feedback Linearization of Discrete-Time Nonlinear Systems Consider the smooth discrete-time nonlinear dynamics x(k+l) - f{x(k),u(k)) , where x - (xl •... ,Xn

)

(14.3)

and U - (u 1 , ...• um) are smooth local coordinates for

the state space f! and input space U respectively. Before defining feedback linearizability of static

state

(14.3)

feedback

for

we first (14.3).

introduce Similary

the notion of a as

for

a

regular

continuous-time

nonlinear system, see Chapter 5, we call a relation u

=

o(x,v)

(14.6)

a regular static state feedback, whenever :~(XIV) is nonsingu1ar at every

(x,v).

point

Notice

that

this

implies

locally

a

one-to-one

relation

between the old inputs u and the new controls v. Analogously to Definition 6.10, we formulate the feedback linearizability of (14.3).

Definition 14.1 Let

f(xo'u o )

(x o ' u o )

be and equilibrium point

The system (14.3)

= XO'

for

i. e.

(14.3),

is feedback linearizable around (xo,uo )

if there exist (i)

a

coordinate

transforma.tion

S: V c mn .... S(V) c mn

neighborhood V of Xo ldth S(x o ) - 0, (ii) a regular static state feedback u - c-(x,v) and

defined on

a

neighborhood V x 0

a.

on

satisfying a(xo ,0) ...

(xo ,0)

of

defined

with

~~(X,V)

Uo

non-

singular on V x 0, such that in the new coordinates z - Sex)

the closed loop dynamics are

linear z(lc+1) - Az(k) + Bv(k)

(14.7)

for some matrices A and B.

At, this point it is useful to note that a coordinate change z - Sex) transforms (14.3) in a different manner than for a continuous-time system. In the z-coordinates we obtain from (14.3) the discrete-time dynamics z(k+l) - S(f(S-l(z(k» ,u(Ie») , and so

the

(14.8)

feedback linearizability of Definition 14.1 amounts

to

the

equation S(f(S·'(Z(k», .(S·'(z(k»,v(k»») _ Az(k)

+

(14.9)

Bv(k),

(compare with equation (6.67». The

following sequence of distributions will be

solution 7r

:

of

the

feedback

linearization

problem

instrumental for

in the

(l4.3).

Let

H xU .... H be the canonical projection and K the distribution defined

by

K

ker f"

=

.

(14.10)

Algorithm 14.2 Assume fn has full rank around (xo ,uo)' Step 0 Define in a neighborhood of (xo'u n ) in H x U the distribution Do

·1

=>

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

<-1

(~) fe-I

chosen in such a way that rank again as

(x1"

.. ,x"') f

are the first p functions

can be written as

in

the

successively for t"'-2, .. " t

'"

f ~~ ,z ...

where

<-1

• Denoting

once

(Z1, •.. , Z"')

(f1, ... ,f"-1, X"-l)T,

xll:-

where

1

components of xr;-l and £1, ... ,£,,-1 are now the component

<

expressed

f(z,u) -

p

ax

[

z

coordinates.

Repeating

this

procedure

,2

<]

f

,u)

(14.24)

<-1

are

(Z1, .. , ,zl\.)

new

1 we arrive at the following expression for

the

local

coordinates

obtained in

the

last step.

Finally we perform a state feedback transformation as follows. Define v as v = (£1(z,u)

av

,u) where ii are m - Pl components of u selected in such a way

that rank au

In the (x,v) coordinates we obtain the system

III.

=

v (1e)

z(k+l)

v

with

~

=

1

denoting

(k)

(14.25)

the

first

Pl

components

of

v.

(14.25)

Clearly

is

14.6

Remark

Like

in

a

o

controllable linear system.

the

continuous-time

the

case,

coordinate

transformations used in the above proof change at each step the component 1

functions

f

However,

the

,r-K

, •••

f

of

(while

modifications

continuous-time,

are

in

leaving

f

on

general

in

them

of

the

discrete-time,

different,

see

form

(14.23».

respectively

(1l1.B),

in

respectively

(6.67) .

Example

14.7

Consider

as

in

Example

1.3

the

dynamics

of

a

controlled

closed economy, which are described by equations of the form (see (1.33»

{

Y(k+1)

=

fl

R(lc+l)

=

f2 O'(k) ,R(k) ,K(k) ,fv(1c) ,N(k»

K(lc+l) - f

J

(Y(k) ,R(k) ,K(k) ,fv(k) ,G(k»

(14.26)

(l'(k) ,R(k) ,K(k»

In (14.26) G(k) and N(/C) denote the controls on the system and fv(k) is an exogenous

variable.

Assume

we

are

working

around

an

equilibrium

(Y,R,K,G,H,W)

and suppose the exogenous variable fv(k) equals

In

see

order

to

if

the

model

(14.26)

is

feedback

W for

linearizable

point all

k.

around

444

(y,R,K,G,H,W)

we have to verify the conditions of Theorem 14.5. We make

the following assumptions

8tl _ _ _ _ _ ~(y,R,K,W,G)

0 ,

(14.27a)

8t'2. ~(Y.R,K,w,T!) ~ 0 ,

(14.27b)

Btl _ _ _

~

BtJ __ _

(ar-(Y,R,K) , ax-(Y.R,K») ~ (0,0) .

(14.27c)

Note that for the specific dynamics of Example 1.3 the condition (14.27a) is automatically satisfied. Define a preliminary feedback so that

(Y,R,K,W,G) +

Y

z - tz(Y,R.K,W,H) +

R

Ul U

-

t

1

(14.28)

Provided (14.27a,b) holds, the Inverse Function Theorem assures us that we can

achieve

by

(Ill. 28)

means

of

a

state

regular

feedback

The feedback modified system then

G - al(Y,R,K,W,Ul)' If - 0!2(Y,R,K,W,u z )'

has the form

Y(k+l) {

R(k+l) K(k+1) -

YRK-

u 1 (k) u 2 (k) , t

3

(14.29)

(Y(k),R(k),K(k»

-

K

The right-hand side of (14.29), seen as a mapping from the (l',R,K,u l ,u z )space into the (Y,R,X)-space, has full raclc, cf. (14.27c). We now apply Algorithm

14.2.

Denotin~

the

(Y-r.R-R,K-K)

stace variables

as

x,

we

compute the distribution K as in (14.10) as che 2-dimensiona1 distribution on the (x,u) space of the form (14.30)

with Xl and Xl satisfying iJt3

ax (x)X i (x)

~

(V,.31)

0 ,i = 1,2 .

Since, see (14.11), Do constant dimensional

spanl a/ilu I, we immediately have that Kl ... Do is a involutive distribution.

Using

again

(14.27c)

we

obtain from (14.12) that D1 is an involutive distribution of dimension 5 on the (x,u)-space. Therefore, the conditions (14.27a,b,c) guarantee that the

dynamics

point.

(1l1.26)

are

feedback

linearizable

about

the

equilibrium

o

445

14.2 Controlled Invariant Distributions

and

the

Disturbance

Decoupling

Problem in Discrete-Time

We

now

discuss

the

notion

of

local

controlled

invariance

for

the

discrete-time system (14.3). Afterwards we show how controlled invariant distributions

are

instrumental

in

the

solution

decoupling problem for discrete-time systems.

The

of

the

disturbance

theory we develop here

very much resembles the corresponding continuous-time theory of Chapters 7 and 13,

the

see in particular Sections 13.1 and 13.2,

proofs

and

results

will

only

briefly

be

and therefore some of

sketched.

The

following

definition is the discrete-time version of Definition 13.1. Definition 1'/,8 A dist:ribution D on N is invariant for the smooth dynamics (14.3) if

for every u E U

f.(·,u)DeD,

and locally controlled invariant

for

(14.32)

if there

(14.3)

regular state feedback, briefly feedback, u

=

locally

exists a

u(x,v) such that the closed

loop dynamics f(x(k), v(k»

x(k+l)

(14.33)

f(x(k),a(x(k) ,v(k»)

satisfies for every v.

[,,(·,v) DeD,

An involutive constant dimensional distribution D which under

(14.3),

induces

slightly differs

from

a

local

the

decomposition

continuous-time

for

case,

the

see

difference is due to the fact that if we apply f(·, u)

is

invariant

dynamics,

e.g.

which

(13.9).

This

to a point x in a

local coordinate chart, then f(x ,u) may leave the chart, no matter how the control was chosen. Let

We

therefore

introduce a pair of coordinnte charts.

(x o , u o ) E N x U and choose local coordinates on a neighborhood I' of

f(xo ,u o )'

local

Consider

coordinate

Denote

the

the

chart

coordinntes

referred to as pair permits us

open set [1(1')

V

in N x U

for

Ii

as

containing

(x, u)

the coordinate chart pair (V, V). to do

(x o ,u o )

local computations.

Note

the

local

coordinates for

coordinates

;reV)

on

V

a

Such a coordinate chart that

for

point (xo ,u o ) we may always select iT such that n(V) C V, case

and choose

(xo ,u o ) such that V C f~l(I'). and for V as x. This will be

about

may

be

chosen

in

an equilibrium so that in this that

the

and V coincide. The invariance condition (1/1.32)

such

way

for

446

an

involutive

constant

dimensional

distribution

D now

translates

as

follows. Using Frobenius' Theorem (Corollary 2.43) we select a coordinate chart

(V,V)

pair

Z x = (x1,x ). xl

such

and

(x1, .... x\<) 2

accordingly f ~ (f ,f l

D is

that

described x

2

D - span{~l, ax!

by

writing

Then,

(xk + 1 , ... ,xn ).

where

(14.32) implies the local decomposition

),

(14.35)

and

similarly for

the

closed

loop

dynamics

(14.33)

D is

if

locally

controlled invariant. The next theorem provides a coordinate-free test for checking the local controlled invariance of a given distribution D

Its

proof is very much inspired by the corresponding continuous-time result, cf. Theorem 13.7 and Corollary 13.8.

Theorem 14.9 Consider the smooth discrete-time system (14.3) and let D be an

involutive

f(x,·): U ... H £:1(D)

n

1f:

1

constant has

(O)

is

dimensional

constant constant

rank

distribution

for

every

dimensional.

Then

x, D

N.

-on and

is

Assume

suppose

locally

that

cont:rolled

invariant if and only if

Proof Let D

In

(xo ' uo )

E

N

X U

~ spanl~l. where x

and choose a coordinate chart pair so

2 1 (X ,X ),

axl these coordinates

the

Xl

~

(x1, . . . ,X k )

1r:

distribution

1

(D)

and x

Z

is

that

(Xk+1' •••• x n ).

then

given

as

spanl_8_!!.-) and the condition (14.36) is equivalent to

ax 1 'au

af z

8£Z

(14.37)

1m ax 1 (X,U) C 1m au-(x,u) for all (x,u) .

Now suppose D is locally controlled invariant around (x o .lI a )· Then there is

locally

f(x,o(x. v»

feedback

a

u=o:(x,v)

Equivalently

ai z -(x,v)

=

such 0

that

i .. D c

D,

where

l(x,v)

-

or

axl

af z

af2 ax

a

+ au(X,U)1

-l(X,U)1

u=o:(x.v)

u=o(x,V)

. ~(x,v) 1

o

(14.38)

ax

for all (x,u) around (xo,u o )' Clearly (14.38) implies (14.37). Conversely let (lll.37) be satisfied. Then there exist smooth m-vectors bi(x,u), i

E~,

satisfying

447

af' -a (X,Ll) x, Let P

=

b i (x,u)

ann V,

then

and

f*P

11"

0, i E

=

(14.39)

~

"P are involutive co distributions on N x U

(see Exercise 2.14), and so also f"P + 7r"p is an involutive codistribution on H x U. In the given coordinates f*P +

Writing

;p - span(dx

2

-

b,

(b l i

, ...

af'

,

ax

1

, dx

af'

au

+

(14.40)

du}.

T

,bmd , we have in view of (14.39) that

span

b si

a ex, u)au;;-

, i E

+

te)

(14.41) Note

that

the

last

term

in

the

right hand

f:

constant dimensional distribution

1

side

(D) n 1I":-1(O},

of

(14.41)

equals

the

thus ker(i'P + 1r"P)

and

is a constant dimensional distribution. Now define ker(f"p + r/'P) ,

E

then it clearly follows that 1r"E

which

c

D

is

the

discrete-time

locally

the

follows

(compare

E n 1r~l(O) dim D D

=

Ie

=

feedback

and

a

with or

= 0,

u

we

=

analog

a(x,v)

the

of

makes

of

Theorem

proof

take

a

In

new

13.7).

coordinate

span! ax l )' and moreover by the fact that

order

D invariant

£;1(D) n 7t~1(0)

equivalently may

(13.56).

which

1r~:

=

to we

First O.

chart

(x,v)

such

that E

span{~).

=

The

assume

that

dim E

Then pair

such

=

that

E ..... D is an isomorphism,

there exists a state-dependent change of coordinates for U, coordinates

construct proceed as

input

resulting in

coordinate

change

axl

v

=

ii(x,u) is precisely the inverse of the desired feedback u

case E n 7t;1(0) of

Theorem

;o!

13.7

=

o(x,v). In

0, a slight modification as has been given in the proof again

yields

a

feedback

u - o(x,v)

which

renders

As

in

maximal

D

o

invariant.

the

continuous-time

(locally)

controlled

case

an

important

invariant

role

distribution

is

played

for

the

by

the

dynamics

(ILI.3) contained in some given involutive distribution J( on N. That such a

maximal locally controlled invariant distribution in

J(

exists, follows in

a similar way as in Chapter 7. see in particular Propositions 7.10 and

7.11 and Corollary 7.12. Without proof we give the following result. Proposition 14.10 Lec If Dl

(1)

and Dz

on

(14.36)

x

H

~

be an involutive dlstribucion on N. Then lI'e have

are discributicms on H conea.ined in '}{, U,

If D is a distribution on N contained in '}{,

(if)

f1 x U,

satisfying

tllen also DI + Dz satisfies (14.36) on H x U. satisfying (14.36) on

then the same holds true for the involutive closure

(iii) There

exists

a

largest

involutiva

distribution

D~,

existence

D-

D of

on

D. which

H

satisfies (14.36) on H x U.

Clearly.

the

distribution

whose

is

guaranteed

by

Proposition 14.10 (iii) is locally controlled invariant on the open and dense subsets of N x U where the constant dimension hypothesis of Theorem 14.9 are met.

The distribution D* may be computed analogously

to

the

continuous-time algorithm (7.53) in the following way. Define recursively "" T(/1 x U)

EO

EI1+l _

11'; 1 00

n

(X E TeN x U)

I

f"X E 'fr.E I1 + f" (11"" 1 (0»

on an open and dense subset of N x U) The following result is immediate. Corollary 14.11 Lee £" ~ lim E'L. ellen l1->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

-<

M now

also

depends

upon

the

disturbances q E Q, the i-dimensional disturbance manifold. Regarding the output invariance with respect to the disturbances q in eVI.lI6) we have the following result which parallels Proposition 13.20. Proposition 14.12 The out::pue y of the syscem (l4,46) is invariant under q if there exists em involutive constant: dimensional discribution D on f1 such

that

449

for all

f .. (·,u,q) Dc D

(ll,q) E

Ux Q ,

(14,47a) (14,47b)

Deiter d b(· ,u)

(Here TQ

denotes

(14.47c)

, for all u E U

the

i-dimensional

on N x U x Q given in

distribut:ion

a

ah

local coordinates (X,ll,q) by span(aq), and dxh(x,ll) :- ax(x,u)dx.) Proof Us lng Frobenius' that

such Xl

described

is

D

Theorem we selee t

and

(XIlo •• ,X):)

a

D-

by

coordinate chart pair

spanl __a__ ) ax'

where

Then

(xk + 1 , ... ,x n ).

xl =

x

(V, V)

(xl,x2),

=

accordingly

writing

f - (fl,fZ) we obtain from (14.47a) the local decomposition 1

1

x (lc+l) "" fl(X (Je) ,xz(k) ,uCk) ,q(k» 2

x {k+l}

=

Furthermore (14.47c)

2

r(x {1c) ,uCIe) ,q(k»

(14.4 7b)

it follows

implies

that hex,ll)

that y(k) - hexCk) ,uCk»

r

that

does

not

depend upon

does not depend on

, x ,

q

and

from

thereby implying

o

is invariant under q.

The Disturbance Decoupling Problem consists of finding a regular static state feedback u = o(x,v) for the system (1l1.lI6) such that in the feedback transformed system the output y

invariant under q,

is

cf.

Problem 7.7.

Analogously to Chapters 7 and 13 we search for a local solution of the Disturbance Decoupling Problem in the following manner. Let (XO,llO) be an arbitrary locally

point

in

solvable

f1 xU,

around

then

(xo,u o )

the if

Disturbance there

Decoupling

exists

a

Problem

feedback

II -

is

a(x,v)

defined around (xo ,u o ) such that in the feedback modified system x(k+1) ~ [(x(k) ,o(x(k), v(k) {

the

y(k)

~

output y(k)

,q(k)

h(x(k),o(x(k),v(k)))

is

-: ~(X(k), v(k), q(k)) -: h(x(k),v(k»

invariant under

q(k).

Let

11"

f1 x U x Q ... f1

:

be

the

canonical projection and define the distribution J{ as J{ =

n

ker d)l(. ,u)

(1l1.50)

UEU

Now, viewing the discrete-time dynamics of (14.46) as a system with inputs u

and

q,

we

know

that

according

to

Proposition

VI.lO

there

exists

a

maximal involutive distribution D~ in 11. on f1 which satisfies -1

f~(ii~ (D~»

-1

c if + f~(;~ (0»

(14,51)

450

With

the

aid

of

Proposition

14.12

and

using

the

results

about

local

controlled invariance we arrive at the next result.

Theorem 14.13 Consider the system (14.46) and assume f(x,','): U x Q -. H -1

has

constant rilnk,

distributions.

if

and

Then

and

locally

n f" (If,, (0»

about

regular static sCate feedback u system (14.49)

If

each

chere

(xo,uo)

exists

a

such thac the feedback modified

Q(x.v)

=

are constant dimensional

point

satisfies the conditions (14.47a,b,c)

around (xo,u o ).

if

and only if

c

fw (TQ)

if'

(14.52)

Proof The necessity of (14.52) follows directly from Proposition 14.12 nnd the

fact

that

disturbances. feedback

=

U

the

feedback

u

=

0: (x

for

Q(x,v)

(14.47a,b,c), is

11

which

the

for

not

the

modified

depend

upon

the

local existence

system

(14./,9)

of a

satisfies

consequence of the following facts. As in the proof of

the distribution D~

Theorem lLI,9

does

,v)

The sufficiency of (14.52)

induces a

E

distribution

on

x U x Q

}J

cf. (14.42,43),

such that,

(14.53)

i.E

C

If

This already implies ehe local existence of a feedback also depending upon the f"

disturbances,

which

(TQ) C jj~ i t follows

Proposition

13.21

makes

that 'l'Q C

it

follows

the

E.

distribution

If'

invariant.

Since

Completely analogous to the proof of

that

there

locally

exists

a

feedback

u = a(x. v) which makes jjn invariant.

0

Theorem 14.13 is a local result, since the feedback u

= Q(x,v)

defined in a neighborhood of an initial point (x(O),u(O». may

apply

the

result

x(l) = f(x(D), u(O) ,q(O»

14.13

are

together

met), into

but a

again

about

the

point

is only

Of course one

(x(l) ,u(l»),

where

and so on (provided the assumptions of Theorem

in

generil1

globally

these

defined

local

feedback.

feedbac\{s Of

need

course,

not the

patch output

invariance with respect to q remains as long as the state and input evolve in such a way that we do not leave the neighborhood of (x(D), u(O» which the feedback u

a(x,v) was constructed.

on

This happens in particular

if the system evolves for all time on a sufficiently small neighborhood of an equilibrium point (xo ' u o ) .

451

14.3 Input-Output Decoupling in Discrete-Time We

next

discuss,

in analogy with Chapters

8 and

13,

the

input-output

decoupling problem for a square analytic discrete-time nonlinear system

f(x(lc),u(k»

x(Jc+l)

(14.3)

y(k) - h(x(k),u{k»

,

where u and yare m-dimensional control and output vectors respectively. (The smooth case can be treated more or less similarly, see Chapters Band 13.)

With

each

component

of

the

output,

Yi'

we

can

associate

characteristic number Pi in the following way. Let (x,u) E H X U,

a

then we

compute for i E m the derivative

(14.54)

From

the

analyticity of

the discrete-time

system

it

follows

that

this

vector is either nonzero for all (x,u) in an open and dense subspace 01

M x V,

case

or the vector (14.54) vanishes at all points (x,u). we

observing

define that

write hi (x, u)

=

-1,

Pi

whereas

in

function hi (x, u)

the

h~ (x).

the

latter

we

continue u;

so

Next we compute the vector :u h7 (f(x, u».

vector is nonzero at an open and dense subset 0i otherwise we continue with the function the number Pi -

case

does not depend upon

h;(x)

of

n

if it exists - determines the inherent delay

inputs and the i-th output; that is given a point

ex, u)

by

we may If this

x U we set Pi - 0,

h7(f(x,u».

=

of

In the first

In this way between

E 01 C

n

xU,

the then

we can "see" the input u(D) - u after Pi + 1 steps in the i-th output. case none of the iterated functions h~+l(x,u) - h~(f(X,u»

In

depends on u we

define Pi

Note that the above definition of a characteristic number

is

with

consistent

the

continuous-time

analogues,

Definitions

8.7

and

13.23. Next, assume the analytic discrete-time system (14.3,4) has finite characteristic numbers

PI""

'Pm'

Then define

the

decoupling matrix

of

(14.3,4) as (see also (8.25) and (13.87»

_

a au

P,

h, (f(x,u»

A(x,u)

(14.55)

[a

Pm:

au h m (f(x,u»

where, as before h~(x) = ht(x,u), h~(x) ~ h~(f(x,u»

etc., i Em.

452

In

analogy with

B.3

Definitions

and

13.23

we

say

that

the

system

(U.3,4) is locally strongly input-output decoupled about (xc ,u o ) if the decoupling matrix A(x,u) is a diagonal nonsingular matrix for all (x,u) on a

Since we only deal with strong input-output

neighborhood of (xo ,u o )'

decoupling

in

this

chapter

"strongly"

throughout.

The

we

will

regular

drop

for

static

simplicity

state

the

feedback

adj ective

input-output

decoupling problem now consists of finding a regular static state feedback u - a(x,v) such that the feedback modified system

(14.56)

x(k+1) '" f(x(k),cr.(x(k),v(k») -: f{x(k),v(k» - h(x(k) ,o(x(k) ,v(k») -' ~(x(k),v(k)

y(k)

is

input-output

solution

of

decoupled.

this

problem

As in

in

a

Section

14.1 we will

neighborhood

of

an

XO' This will be referred to

a

local

equilibrium

give

point

as the local regular

static state feedbaclc input-output decoupling problem around (xo ,u o )' We have the following result, which parallels Theorems

Theorem

01 , •••

Consider

14.14

characteristic ,Om

of

numbers

H

X

U.

the

analytic

Let

belonging to 0lf'1 •.... n Om'

system

defined

P1"",P m

be

(xo.u o )

an

on

B.13 and 13.24. Ivith

(14.3,4)

open

and

equilibrium

finite

dense

point

subsets

of

(14.3)

Then the local regular static state feedback

input-output decoupling problem is solvable around (xo ,u o ) if and only if ran1e A(xo ,u o )

=

HI

(14.58)

•

Proof First assume (14.58) holds and let hi(XD,u O) - YO!

' i Em. Consider

the set of equations PI

hl

(f(X,"»

YOl

v1

(14.59) Pm hm

By

the

YOm

Implicit Function Theorem

solution u Since

(f(x,u»

Vm

there

a(x,v) of (14.59), with

exists

Uo - a(xo'O)

Pi Yi (k+Pi +1) - hi (f(x(k) , u(k»), i

E~,

the

locally about and

~~(x,v)

above

(xo ' u e )

a

invertible.

defined

feedback

solves the input-output decoupling problem around (xo , U o ). Next, suppose u .. cr.(x. v) solves the input-output decoupling problem around (xo, u o ) . Consider the feedback modified system (14.56,57) for which the i-th input

only

Vi

influences

nonsingular

the

diagonal

i-th

output,

decoupling

decoupling matrix of (14.56,57).

implying

matrix.

Let

that

us

(lll.56,57)

write

li(x,v)

Then the decoupling matrix

has

a

as

the

(14.55)

and

A(x,v) are related via

-

A(x,v) ~ A(X,u)lu=o(x,V)'

Bo

av(x,v)

(llI-60)

in a neighborhood of (xo,u o )' To see that (14.60) is true, we first note that

the

charac teris tic

invariant under the

h, ... , Pm

numbers

application

Namely, let i E m, and suppose Pi

a

of

already

Pi ::.:: O.

Then

establishes the

(14.60)

functions

h~(x)

the

system

static

(14.3, 1,)

state

are

feedback.

-1. Then

=

Bh, B - av(hi(x,o(x,v») - au( x '

which

of

regular

in

U)I

(14.61)

u~o(x,v)'

case

Pl="

'=P m =

-1.

Now

suppose

P, hi (x,u), h:(x)

=

=

hi (f(x,u», .. ,hi (x)

Pi -1 =

hi

(f(x, u»

are

only

depending

applying a state feedback u

=

upon

x

and

not

on

u.

Obviously,

o(x, v) does not alter these functions.

Now

the i-th row of the matrix A(x, u) is given as P,

a

ah l

Pi

au hi

(f(x,U»

ax

-

Bf

(f(x,u»

(14.62)

au(x u)

whereas the i-th row of li(x, v) equals

axCombining

the

af ao (f(X,U»lu~o(xIV)' au(x,U)lu=o(x,v)' av(X,v).

expressions

Since in (lLI.60)

(14.62)

the matrix li(x,v)

and

(lll.63)

exactly

is nonsingular,

yields

(lll.60).

the result (lLI.5B)

is

o

immediate.

Next we discuss as

in Chapters Band 13

the

dynamic sCaCe feedback

inpuc-output decoupling problem. For the analytic discrete-time nonlinear system

(14.3,4)

a

dynamic

s tate

given as

z(1<+I) - .(z(k),x(k),v(k» { uCk)

-

~(z(k),x(k),v(k»

feedback

or

dynamic

precompensator

is

454

wi th

Z

IRq and v E U, denoting the new inputs. The dynami,c state feedback

E

input-output decoupling

problem

consists

of

finding - if

possible - a

precompensator (14.63) such that the closed loop system x(k+l) ~ f(x(1c) ,I/J(z(k) ,x(k). v(le») ~

z (k+ 1) {

(14.65)

'P (z (k) ,x (k) • v (lc) )

y(k)

h(x(k),~(z(k),x(k),v(k))

is input-output decoupled. As before we will produce a local solution of this

problem

in

a

neighborhood

of

an

equilibrium

point

(xo ,u e ).

The

essential tool in obtaining a local solution is a discrete-time version of the dynamic extension algorithm as was discussed in Chapters Band 13.

Algorithm 14.15 (Discrete-time Dynamic Extension Algorithm) Consider the square analytic system (14.3,4). Step 1 Assume

the characteristic nwnbers of (lLI.3,4)

neighborhood containing (x o , u o ) decoupling matrix (cf. 14.55» the analyticity of (14.3,4), r dense

subset

81

H xU.

of

functions

h1

linearly

independent,

, ...

,hm

in

1

P:, .... p!.

and denote them as

is constant, say r

(x,u)

a

Ii

Write the

(x o ' u o ) E HI'

way

that

the

1 ,

on an open and

Reorder

first

r1

rows

the

output

of D1

are

h[) -

\Jrite

(h l , ... ,Il r 1) J }j0 - (lJ r 1+ l' ... ,11m) and Let h(xo ,u o ) = Yo and consider the equations

(/,y1).

correspondingly y =

defined in

as DI(x,u). Let rl(x,u) :- rank DI(x,u). By

Assume

such

are

PI

(f'X.U) )

hI

v1

YOl

(14.66) Pr 1

hr:

1

;

(f(x.u»

vr 1

YOrl

By the Implicit Function Theorem we can locally solve around (xo ,u o ) r 1 components of u as a function of x and v. So after a possible relabeling of the input components (u 1 u v

l 1

a1(x.v

=

1

(VI""

map

Leaving

.vrl ).

(14.34)

}jl(x, vI) := }j°(x,a (x,v

?(k) with

inputs

f1(x(k»

1

».

the

}jl (x(Jc) ,u (Ie» U ,

outputs

we obtain a "partial" state feedback where

remaining

u

inputs

ill

f\x,vl,il1):=

into

Renaming

,u1(lc) ,u1(lc»

1

-1

,um )

a1(xe ,a),

transforms l

X (1c+l )

, ...

u~ ~

with

)

VI

as u

1

1

= (u 1

""

,u r l )

and

unchanged

the

system

1 1 f(X,CJ (X,V ) ,ill)

and

we obtain the system

, (14.67)

,

?

and

where

U

1

is

regarded

as

a

set

of

455

By the above chosen "partial" feedback

parameters.

the first r 1 outputs

are already input-output decoupled, see Theorem III .14.

Step .2 + 1 Assume we have defined a

.£

I

ql :=

, -,

ril

and we

sequence of integers r 1

have a block partitioning of has

'11 dl"'I,'~r,,'.1 E.=., 'd Wit an correspon d'lng 1 y y

, •

(1/,11

2

.

,r.£ with

, ..•

(1 y, ... ,J £-1) ,Y .

=

1 -£ ,h ,h)

I''urth er-

more we have obtained a system x(k+l)

=

FE (xUc) ,

Vi - I ek) ,

-yl(k)

=

l/ (x(le) ,

V£-l(k) , u1!(Jc»

Here u

the

~

(new)

with

suitable

(14.67)

controls and

(u1, ... ,ui,lii)

l u (k), ;;l(k))

u

iJ£-l denotes i

s > 0, i

= 1, ... ,1-1.

into

together system

V1 - 1 (k)

1+1

and u (Jc). the

u (1c+s),

split

,/-1(k» The

and

inputs

correspondingly

(14.67) is regarded as a system with inputs U£-l, parametrized by 1

future

are

Ui-Iek) = (u 1(k), ...

Denote the characteristic numbers of (14.67) by Pq J+l"" decoupling

matrix

of

1 1 n +\x,V

by

(V•. 67)

,u

1

From

).

1+1 ,Pm

the

analyticity of the system (14.67) the rank of this matrix is constant, say r + on 1 1

an

belongs

to

func tions

open dense

~

of points

-1 -.I' (x,U ,u).

Assume

(xo,u o )

the projection of B n on to f1 xU. Reorder the output -1 ",+1 .I' h such tha t the firs t r i+l rows of n +1 are linearly

and

independent,

)/

subset B.I'+1

(hqi+l+ 1

. ,hm )

' .

write

and

qn + r

q 1+1

x

similarly

hi

1'1'

(Y

1+1 -.1'+1)

,y

=

,

(h q £+l""

,hql~l)'

-1

Y

=

=

q1 + 1, ... , q £+1 .

U

Consider the equations

i

By

the

Implicit Function Theorem we -1

components of u

can locally solve

r

-1

as a function of x, U

and (v::. + 1

j

(14.68)

1 +1

:= ql-

q1t1 ).

, ••• ,V

~'1

- q£

So after a

possible relabeling of the input components of u we obtain a parametrized "partial" feedback u 1 +1 = ol+1(x,V£,V£+l). Leaving the other inputs u.l'-n 1 1 unaltered and renaming again u + of the form x(Jc+l) -1+1

y

where

=

f.l'+1(x(k) ,VP.{k) ,u 1+1(k) ,u.l'+1(1e» -1+1

(k) = h

we

set

hJ!+I(x,l/ ,/-+1)

-J!

(x(lc) ,U (Ie) ,u

fJ!+1(x,V i

,V

J!+1

(14.69) (k»

i +1 ,uJ!+l)

,

:= f1(X,Vi-l,OJ!+1(X,Vl,vl+1) ,

:= h.l' {x ,V1 - 1 ,,/+1 (x ,V1 ,v£+l».

Notice

that

the

-1-+1 U

)

and

"partial"

456

feedback defined above achieves input-output decoupling between yf+l and

v

.11+1

Assuming

1 - 1.2 •...

that

the

project

open onto

Clnd a

dense

subsets

neighborhood

of

i

the

(x,02,u )-spaces,

(xo ,u o )

of

we

obtain

as

in

Algorithm B.IB a finite. list of integers qL.q2 •... 'qk such that

o<

q 1 < ..... <

qk

"'" qk + 1

(14.70)

•

and the integer q" '- q:.: will be called the rank of the system (14.3,4).

Remark 11•. 16 Although

the discrete-time

Dynamic Extension Algorithm as

givC!n here only works on a neighborhood of an equilibrium point (xo ' u o ) • it can also be used at an arbitrary point constant

rank

assumptions

are

met.

The

provided the

(x. u) E tf xU.

analyticity

of

the

system

guarantees that the rank q •• dete.mined on a neighborhood of an equilibrium point (xI), u o ), equals the rank of the system (14.3 , 1.) on an open and dense subset of tf xU. Therefore we may refer to q

as being the rank of the

system (14.3,4), compare with Chapter B, cf. (8.88).

Theorem 14.17 Consider

tile square ana1yt ie

system

(11•. 3,4)

around

the

equilibrium point (xo .u o )' Suppose all assumptions made in AlgOrithm 14.15 are saCisfied. Then the fol1oliing (1)

titfO

condit:ions are equiva1ellc:

The dyna.mic state feedback: input:-output decoupling problem is locally solvable around (xo

I

Uo

).

(Ii) The rank q. of r::he sys tern equals m.

Proof The proof of t:his result completely parallels the continuous-time result, cf. proof of Theorem 8.19, once we have observed that at each step in Algorithm 14.15 the vector

Vr (1e)

can be replaced by a suitable s-fold

integrator defined for each of the inputs u1 •... ,u qr in

vr.

Namely let for

i l , ... ,qr •

Zij(k+l)

(14.71a) (14.71b)

where sr stands for the largest value s for which some ui(Jc+s) appears in iY(Je).

Using (lLl-7la,b)

the vector fY(k)

(zll(k)"",Zls/Jc)"",z'Ir1(lc).....

can be replaced by

/Jc).

result follows similarly as in Theorem 8.19. We terminate with an illustrative example.

w1 (lc) •...

,t.''1

r

(k)

the vector and

the

0

457

Example 14.18 (see Example 14.7) The dynamic equations for a controlled closed economy have the form Y(k+l) R(1<+l)

{

As

f, (Y(k) ,R(k) ,K(k) ,rICk) ,G(k)) ~

(14.26)

f, (Y(k) ,R(k) ,K(k) ,N(k))

K(k+l) - f, (Y(k) ,R(k) ,K(k)).

in Example 14.7 we assume we are working around an equilibrium point

(r,R,K,G,H,W)

and

furthermore

equals W for all k.

the

exogenous

variable

Similarly as in Example 1. 3,

rv'(k)

in

(14.26)

see (1. 31,) we consider

outputs of the form Q, (k) - Y(k)

,

Qz(lc) = h(W,Y(lc),K(lc»

Note that both target variables Q1 and Q'l

are not directly influenced by

the

Pi

control

variables

G and

H,

and

so

~

0, i

=

1,2.

In

order

to

determine Pi and Pz we make the following assumptions

(14.73a)

(14.73b)

Then PI

~

Pz - 0 and the decoupling matrix of the system (14.26,72)

is

given as af,

""BG(y,R,K,W,G) A{l',R,K,G,H,W) =

[

~~(W,r,K),

(14.74)

:;l(l',R,K,fj,G)

which clearly has rank 1 about the equilibrium point. According to Theorem 14.14 the system (14.26,72) is therefore not input-output decoupab1e via static state feedback.

So we have to use the Dynamic Extension Algorithm

to see whether or not the system is

input-output decoupab1e by dynamic

state feedback. As in step 1 of Algorithm 14.15 we introduce a "partial" state feedbac1c as the inverse relation of (14.75)

G.

As

in the proof of Theorem 14.17 we introduce an s-fold delay on the input

G,

yielding the input G as a state-dependent function of the new input

where

s

depends

on

the

characteristic

numbers

in

each

step

of

the

algorithm. In this specific example we will see that

1 suffices. The

5

precompensated system (14.26) then reads as

Y(k+l)

I

Z(k)

Z(k+l) ~ G(lc) (14.76)

R(1e+l)

- f2(Y(k),R(k),K(k),~(k)H(k»

K(k+l) = fl(Y(k),R(k),K(k»

where G and H are the new inputs of the system. Note that G(k+l) - G(k). Clearly,

in

the

precompensated

system

(14.76,72)

we

already

input-output decoup1ing between the control G and the output

Q1'

have

Next we

determine the characteristic number of the second output with respect

to

the input H. We compute Qz (1c+1)

which implies

(14.77)

h(W, Y(k+l) ,K(k+l) = h(W,Z(k) ,f'J ('fUe) ,R(k) ,KUc»)

pi

~ 0, and

Q2 (lc+1) - h(~,Z(Jc+l) ,iJ (1'(/(+1) ,R(k+l) ,K(k+1»)

(14.7B)

From (14.78) and (14.76) it follows that Q2 (k+2) explicitly depends upon

H(k) provided the following assumption holds

at l

_

_

at l

_

____ _

(~,Y,K)' aR(Y,R,K). aJ1(Y,R,K,rv,l't)

¢

a ,

(14.79)

or, equivalently,

afJ

_

_

af z

_

____ _

M(Y,R,K) '" 0, aJ1(Y,R,K,rv,H) '"

a .

(14.80)

In case (14.73a,b), and (14.79) or (14.80) hold, the system (14.26,72) is locally

dynamic

(Y,R,K,G,B,W). output

input-output

decouplable

about

the

equilibrium

point

Notice that the forementioned conditions for dynamic input-

decoupling

imply

that

feedback linearizab1e about

the

discrete-time

the equilibrium point,

dynamics cf.

the

(14.26)

conditions

o

(14.27a,b,c) of Example 14.7. Finally we

note

that,

completely

is

analogous

to

the

continuous-time

theory of Chapters 8 and 13, a discrete-time nonlinear system which can be locally input-output decoupled via either static state feedback or dynamic state feedback possesses,

after applying such a decoupling feedbac1c,

a

local "normal form". We leave the details for the reader (see Exercise 14.6).

459

Notes and References

This

chapter

discrete-time

has

been

nonlinear

devoted

to

a

discussion

systems which

of

already have

previous chapters in the continuous-time case.

some been

problems

treated

for

in

the

We have limited ourselves

to some results which more or less parallel the continuous-time theory. A different approach for

based

upon

discrete-time

[HN3], [MNlj, [KoJ. [Sol], [SR). by

studying discrete-time nonlinear

generating

series,

Polynomial discrete-time

see

e.g.

systems

systems

is

[No], [HNIJ, [HN2] ,

have

been

treated

in

Invertible discrete-time nonlinear systems have been studied with

associating

IFNI, IJNI, IJSI·

In

[FlJ

the

family

a

system

discrete-time

systems

of

vectorfields, been

have

5

see

tudied

using

tools from difference algebra. As already noticed, discrete-time nonlinear systems may appear as the discretization of a system in continuous-time. Various

questions

continuous-time

regarding

the

sampled-data

system have been studied,

results on sampling of linear systems,

see

representation

e.g.

of

[ABS) , [Ch], [GS]

and [AJU1S],[HN3],[S02]

a for

for their

nonlinear versions. The feedback linearization problem of Section VI. 1 has been studied in [Gr4].

Other

results

[LAM], [illl] , [Ut2].

about

Gontrolled

this

problem

invariant

may

be

distributions

found for

in

{JA],

disc·rete-time

nonlinear systems have been introduced in [Crl), [Cr2], see also [HN2] and [NijlJ. A solution of the disturbance decoupling problem using controlled invariant

distributions has been given

in

[Crl],

see

[Ko]

for

another

approach. A geometric approach to the input-output decoupling problem has been given in [Gr3},[GNj. The solution of Section 14.3 has been taken from [Nij2], see [HNI)

for a similar treatment. The Examples il.7 and 4.18 are

based upon [WKJ and have appeared in [Nij3], [Nij4].

[AHS]

K.J. Astrom, P. Hagander, J. Sternby, "Zeros of sampled systems", Automatica 20, pp, 31-38, 1984, [AJUIS] A. Arapostathis, B. Jakubczyk, H.G. Lee, 5.1. Harcus, E.D. Sontag, "The effect of sampling on linear equivalence and feedback linearization", preprint, 1989. [Gh) G.T. Ghen, Linear System Theory and Design, Holt, Rinehart and Winston, New York, 1984. [FN) 11. F1iess, D. Normand-Gyrot, "A group theoretic approach to discrete-time nonlinear controllability", Proc. 20th. IEEE Gonf. Decision Gontrol, San Diego, pp. 551-557, 1981. [F1J 11. F1iess, "Esquisses pour une theorie des systemes nonlin;§.aires en temps discret", Rend. Sem. Mat. Univers. Po1itechnico Torino, pp. 55-67, 1986.

460

IGrl]

J.W. Grizzle, "Controlled invariance for discrete-time nonlinear systems with an application to the disturbance deeoupling problem". IEEE Trans. Aut. Contr. AC-30, pp. 868-874. 1985. {Gr2 J J. W. Grizzle, "Distributions invariantes commandees pour les systemes nonlineaires en temps discret", C. R. Acad. Sci. Paris, t.300, Serie I, pp. 447-450, 1985. [Gr3] J.W. Grizzle, "Local input-output decoupling of discrete-time nonlinear systems", Int. J. Contr. 43, pp. 1517-1530, 1986. [Gr4] J.W. Grizzle. "Feedback linearization of discrete-time systems", in Analysis and Optimization of Systems (eds A. Bensoussan, J.L Lions) Lect. Notes Contr. Inf. Sci. 83, Springer, Berlin, pp. 273-281, 1986. (GN) J.W. Grizzle, H. Nijmeijer, "Zeros at infinity for nonlinear discrete-time systems", Math. Syst. Th. 19, pp. 79-93, 1986. [GS I J . W. Grizzle, M. H. Shor, "Sampling, infinite zeros and decoupling of linear systems". Automatica 24, pp. 387-396, 1988. [JNJ B. Jakubczyk. D. Normand-Gyrot, "Orbites de pseudo-groupes de diffeomorphismes et commandabi1ite des systemes nonlineaires en discret", C.R. Acad. Sci. Paris, t.298, Serie I, pp. J 1984. [Ja} B. Jakubczyk, "Feedback linearization of discrete-time systems", Systems Control Lett. 9, pp. 411-416, 1987. [J8J B. Jakubczyk, E.D. Sontag, "Controllability of nonlinear discretetime systems: a Lie-algebraic approach". Report SYCON-88-09, Rutgers Center for Systems and Control, 1988. [Ko 1 U. Kotta, "The disturbance decoupling problem in nonlinear discrete time systems", Preprints IFAC-Symposium Nonlinear Control Systems Design, Capri, Italy. pp. 59-63, 1989. [LAM] H.G. Lee. A. Arapostathis. 5.1. Marcus, "On the linearization of discrete-time systems". Int. J. Contr., 45. pp. 1803-1822, 1987. {Un] H.G. Lee, 5.1. Hareus, "Approximate and local linearizability of nonlinear discrete-time systems", Int. J. Contr., 44, pp. 1103-1124, 1986. (l.M2J H.G. Lee, 5.1. Harcus, "On input-output linearization of discretetime nonlinear systems", 5yst. Contr. Lett. 8, pp. 249-260, 1987. [HN1] S. Honaco, D. Normand-Gyrot, "Sur la commande non interactive des systemes nonlineaires en temps discret" , in Analysis and Optimization of Systems (eds. A. Bensoussan, J.L. Lions) Lect. Notes Contr. Inf. Sci. 63, Springer, Berlin. pp. 364-377, 1984. 1!1N2] S. Honaco. D. Normand-Cyrot, "Invariant distributions for discrete time nonlinear systems", Systems Control Lett. 5, pp . . 191-196, 1984. [HN3] S. Honnco. D. Normand-Cyrot. "Zero dynamics of sampled nonlinear systems", Systems Control Lett. 11, pp. 229-234, 1988. [1'IN1] S. Monaco, D. Normand-eyrot. I. Isola, "Nonlinear decoupling in discrete time", Preprints IFAC-Symposium Nonlinear Control Systems Design. Capri, Italy, pp. 48-55, 1989. (Nijl] H. Nijmeijer, "Observability of autonomous discrete-time nonlinear systems, a geometric approach", Int. J. Contr. 36, pp. 867-874.

1982. (Nij2J

[Nij3]

[Nij4)

Nijmeijer, "Local (dynamic) input-output deeoupling of discrete-time nonlinear systems", IHA J, 11ath. Contr. Inf. 4, pp. 237-250. 1987. H. Nijmeijer, "On dynamic decoupling and dynamic path controllability in economic systems", J. Econ. Dyn. Contr. 13, pp. 21-39, 1989. H. Nijmeijer, "Remarks on the control of discrete-time nonlinear systems", in Perspectivas in control theory (eds. B. Jakubczyk. K. Halanowski, W. Respondek), Birkhauser, Boston. 1989. H.

D. Normand-eyrat, "Theorie et pratique des systemes nonlineaires en temps discret", These de Docteur d'Etat, Universite de Paris-

[No[

Sud, Centre d'Orsay, 1983. E.n. Sontag, Y. Rouchaleau, "On discrete-time polynomial systems",

[SR)

J. Nonl. Anal. 1, pp. 55-64, 1976. E.n, Sontag, Polynomial response maps, Springer, Berlin, 1979.

[Sol] [ 502]

E.n.

Sontag, controllability 313-316, 1986. H.W. Wohltrnann, controllability 315-330, 1984.

[WK]

"An eigenvalue condition for sampled weak of bilinear systems", Systems Gontrol Lett. 7, pp. W. Kromer, "Sufficient conditions for dynamic path of eco'nomic systems", J. Econ. Oyo. Contr. 7, pp.

Exercises 14.1

Prove that Algorithm 14.2 is feedback invariant.

14.2

Consider wi th

the discrete-time nonlinear system x(k+l)

(x 1 u) E IR

respect

,.I ,iIt (X)tl

to

n

given

Let z

i .

Assume

x iRm.

the

~

that

the

coordinates

S(x)

dynamics

f(x(k),

are

i.e.

(x,u),

~

u(k»

affine

wi th

f(x,u) = lex) +

be a coordinate transformation.

Show that

in general the system in the z-coordinates is not affine.

1',,3

Consider

the

(14.1)

system

with

together

sampled-data

the

representation (14.3) wi th sampling time T.

Prove tha t

feedback

linearization

under

feedback

linearizability

is

not

preserved

of

does

not

in general

sampling, imply

i.e.

feedback

linearizability of (14.3). 14.4

Consider an analytic

E: X"" f(x) + g(x)u, y{k)

=

h(x(k»

be

single

input

y = hex).

single

output nonlinear

system

:Ed: x(k+l) = f{x(k),u(k»,

Let

the sampled-data representation of this

system.

Let P and Pd denote the characteristic numbers of :E respectively "d' m implies (aJ Prove P Pd ~

~

O.

(b) Prove that whenever P is finite then Pd =

14.5

Consider

the

discrete-time nonlinear

system

characteristic numbers Pi' Prove that Pi 14.6

Consider

a

requirements

smooth of

discrete-time

Theorem

continuous-time result

(cf.

14.14.

~

(1l1,3,4)

nonlinear Derive,

Chapter 13,

with

finite

n - 1.

system

in

meeting

analogy

(13.91»,

a

with

the the

local normal

form for the input-output decoupled system. 14,7

Consider an analytic discrete-time nonlinear system (14.3,4) an

equilibrium

point

Assume

Du

111 =

p,

ahout

Let

AXk + Buk , Yk = CXk + k , be the linearization of (14.3,4) around (xo ,u a )· Prove that under generic conditions the local regular

=

462

static

feedback

inpuc-output

decoupling

problem

for

the

system

(14.3,4) is solvable around (xo,u o ), if and only if the staCie state feedback input-output decoup1ing problem for 14.8

Consider

the

system

of

(1 11.26,72)

~i

Example

is solvable.

l4.1B.

Interpret

the

exogenous variable fJ(k) as a. not necessarily constant, disturbance for

the system.

ah arJ

Assume that

~

0 in equation (14.72).

Show that

the local Disturbance Decoupling Problem is not solvable for

Bf] Bfl

system if 14.9

this

Bf] Bfz

av-'aw- + ~'aw- ~

O.

Prove Proposition 14,10,

14 .10 Consider nonlinear

an

analytic

characteristic

the

single-input

single-output

system x(1c+1) "" f(x(1c) , u(k», y(k)

D" = ker dho n ker dh

number l

of

this

=

discrete-time

h(x(k».

Let

p

Prove

system.

n .,. ker dh P ,

14.11 ([Ja]) Consider the discrete-time system x(k+l) - f(x(k),u(k» (x. u)

E

1Rt!

X

1R!l1

A(x,U) (X,U)

about

an

u) and B(x,u)

distributions tl i + 1

-

tl i (x,u),

i

equilibrium ~

point

at

au(x,u), Define on W.

E!2.,

via

1

tll (x,u)

A- (x,u)(tl 1 (f(x,u),u) + 1m B(x,u»),

system is

n

independent of u and dim

~

A1""'~

=

n.

La t

the u-dependent

£l(x,u)Im B(x,u), where

locally feedback linearizah1e about

only if the distributions

with

(xo • U o ) .

the pre-image of A, i. e, Afv' = V. Assume tha t rank f" the

be

that

fV' - A-IV

is

n, Prove that (xo • u o )

if and

are of constant dimension and

Subject Index accessibility

algebra 78 distribution

79 rank condition 81 Ackermann's formula 205 admissible controls 13, 74 affine 16 analytic

29

bilinear systems 91, 123 block-input block-output decoupling 286 block input-output decoupling 27L!, 279, 434 Brunovsky normal (canonical) form canonical coordinates

186

359

canonical coordinate transformation 361 canonical mapping 361 center manifold 311, 33l, Center Hanifold Theorem 311 characteristic number 246, 376, 416, 451 clamped dynamics 332 closed economy 8, 443, 457

codistributioo

65

collocated 353 compensator 169, 265 complete 15, {IS complete lift 401, 402, 403 concatenation 13 conditioned invariant 236

conservation law 387 conservative 354 conserved quantity 386 constrained dynamics 332, 376, 424 constrained dynamics algorithm 325, 326 constraints 332 continuous-time system 12 controllable 7Lj controllability 74 controllability indices lsl" 186 controllability subspace 279 controlled invariant distribution 212 controller canonical form 205 coordinate chart 30 neighborhood 30 function 30 transformation 30, 31, 149, 178 coordinate transformation into a linear system cotangent bundle 62, 369 space 61 vector 61

o*-algorithm

223 decoupling matrix

254, 417, l,51

150, 155, 159

332, 425 structure 31 differential 61 differential one-form 62 exact 64 closed 64, 361 diffeomorphism 34 Dirac bracket 381 discrete-time system 19, 437 discretization 437 dissipation 355 forces 349 55

disturbnnce decoupling problem 220 divergence 397 drift vectorfield 73 extension algorithm 25B, 420, 454 state feedback 169 strong input-output decoup1ing problem (by dynamic state feedback) 257, 419 453 353 19

6, BB, IBB, 307, 335, 357 equations 2, 349 system 190, 406, 415, 423 external differential 132, 135 external representation 126 feedback lienarizab1e 17B, 190, 439, 462 fiber bundle 16, 426 fiber respecting coordinates 429 first integral 386 first method of Lyapunov 299 flat distribution 57 Fliess functional expansion 124 flow 1,5 Flow-box Theorem 48 foliation 58, 102 formal zeros at infinity 289, 347 Frobenius' Theorem 57, 59 function group 365 function space 365 generalized configuration coordinates 349, 355 momenta 351, 355 velocities 3LI9 general nonlinear system 401 global feedback 11 28 Hamiltonian Hamiltonian Hamiltonian Hamiltonian Hamiltonian Hausdorff Hirschorn's hyperbolic

control system 351 equations 351 function 351, 356 (input-output) vectorfield 30 algorithm 344 300

353, 362

465

immersed submanifold 37, 58 immersion 35 Implicit Function Theorem 32 infinitesimal Poisson automorphism 357 input-linear 16 input-output decoupled 243 input vectorfield 73 instantaneously 138, 244 integrability conditions 218, 408, 411 integrable distribution 56 integral manifold 56 353 interaction (coupling) Hamiltonians interconnection 337, 425 invariant codistribution 105 distribution 103, 211, 400, 445 folation 102 function group 396 set 300, 305 submanifold 323 subspace 102 Inverse Function Theorem 32 inverse system 342, l,35 inversion 339 involutive 56 involutive closure 60 Jacobi-identity

49, 355

Kalman rank condition 76, 97 Kalman decomposition 107, 365 kinematics direct 4 inverse 4 349, 370 kinetic energy Lagrangian control system 351 Lagrangian function 349 left inversion 339 Legendre transform 351 Leibniz' rule 355 Lie algebra 49 Lie bracket 48 Lie derivative 46 Lie-Poisson bracket 394 linearizable eror dynamics 161 linearization 75 local coordinates 30 222, 415, 449 local disturbance decoupling problem local feedback stabilization problem 302 local generator 55 local Lipschitz condition 14 locally accessible 81 locally asymptotically stable 299 214, 401, 407, 412, 430, 445 locally controlled invariant distribution locally controlled invariant submanifold 324, 422 locally strongly accessible 85 locally strongly input-output decoupled 244, 417, 452 locally observable 94

locally stable 299 local representative 30, 44, 63 Lyapunov function 300 manifold topological 30 smooth 31 analytic 31 maximal controlled invariant distribution 222, 331 maximal locally controlled invariant output-nulling submanifold 325, 424, 425 mixed-culture bioreactor 10, 201 model matching 240, 347, 435 modified disturbance decoupling problem 236 nntural outputs 353 391 Noether's theorem non-degenerate Poisson bracket 359 noninteracting condition 275 normal form (for an input-output decoupled system)

252, 272, 333, 418, 461

observability 93 canonical form 160 codistribution 95, 106 rank condition 96 observability indices 157 observable 93 observation space 94, 363 observer 161 orders of zeros at infinity 287, 288, 289 output controllabili ty 276, 29B output feedback 168. 171 output invariance 136, 142. 219, 414, 448 output nulling 325 output tracking 346 PO control 375 periodic orbits 19, 20 Poisson automorphism 357 Poisson bracket 355, 369 Poisson manifold 355 Poisson structure 355 Poisson system 363 polar group 365 positive definite function 300, 370 potential energy 349, 370 prolongation 401, 402, 403, 404 qualitative behavior

20

ranll.: 34 Rank Theorem 37 rank of a square analytic system 263, 343, 421, 456 Rayleigh's dissipation function 355, 372 375 reachable set 80 real analytic 29 regular static state feedback 166, 171, 401, 438 relative order 246 J

467

reproducibility 266 robot-arm control 1 robot manipulator 1, 100, 352, 36ll, 375, 384 sampled-data representation 437, L.61 second (or direct) method of Lyapunov simple Hamiltonian system 370 Singh's algorithm 344 smooth 29 spacecraft attitude control 4, 242 standard Poisson bracket 356

state space transformation strict static state feedback

148 165

strong accessibility algebra 85, 364 distribution 85, 106 rank condition 87, 275

strong input-output decoupling problem (by static state feedback)

246, fill

strongly accessible dynamics 108, 368 strongly input-output decoupled 244 strong model matching 347 Structure algorithm 344 structure matrix 359a submanifold 36 submersion 35 symmetry 391, 436 symplectic form 362 362 symplectic manifold s~nplectic submanifold 377

tangent bundle 42 mapping 42 space 38 vector 38 topological space 29 triangular decoupling problem uncontrollable modes unobservable dynamics

336 108, 135, 368

vector bundle 428 vectorfield 43 vertical lift 402, 403 voltage fed induction motor Volterra kernel 121 weight function 289 Wiener-Volterra series

297

121

zero dynamics 332, 382, 435 zeros at infinity 287, 289

300