Geometric Theory for Infinite Dimensional Systems

Lecture Notes in Control and Information Sciences Edited by M.Thoma and A.Wyner

115 H. J. Zwart

Geometric Theory for Infinite Dimensional Systems

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo

Series Editors M. Thoma • A. Wyner

Advisory Board L. D. Davisson • A. G..I. MacFarlane • H. Kwakernaak J. L. Massey. Ya Z. Tsypkin • A. J. Viterbi

Author Hans J. Zwart Faculty of Applied Mathematics University of Twente P. O. B o x 217 7500 AE Enschede The Netherlands

ISBN 3-540-50512-1 $pdnger-Verlag Berlin Heidelberg New York ISBN 0-387-50512-1 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September g, 1965, in its version of June 24, 1965, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Bedin, Heidelberg 1989 Printed in Germany Offsetprinting: Mercedes-Druck, Berlin Binding: B. Helm, Berlin 216113020-543210

Geometry may sometimes appear to take the lead over analysis but in fact precedes it only as a servant goes before the master to clear the path and

light

him on his way.

Ja~ms Joseph Sylvester

PREFACE AND ACKNOWLEDGEMENT

In the spring of 1984 I started with my research on geometric theory for infinite

dimensional

systems.

The research

topic was suggested to

me by

Ruth Curtain~ who had done some preliminary investigations on this topic. Many questions were at that time still open and a more fundamental theory was still missing. We knew that the key-concept in geometric theory for finite

dimensional

systems,

that

is (A~B)-invariance,

has

lost

its

strength

for infinite dimensional systems. So I began to look for different concepts which would be more appropriate for infinite dimensional systems. It turned out that

these were the concepts of open-loop invariance and

frequency

invariance. Although the concept of frequency invariance had already been introduced for finite dimensional systems by Hautus, he did not give it any special

name. I have chosen this name,

since this expresses in a concise

way that this is an invariance concept in the frequency domain. Once the equivalence between open-loop, established, came

the

solvability

relatively

problems

are

easy.

of

In

studied:

frequency, and closed loop invariance was various

this

the

disturbance

monograph

Disturbance

three

decoupling

problems

disturbance

Decoupling

Problem

decoupling (DDP),

the

Disturbance Decoupling Problem with Measurement Feedback (DDPM) and the Disturbance

Decoupling

Problem

with

Measurement

Feedback

and

Stability

(DDPMS). The theory can easily be extended to other disturbance decoupling problems,

with

the

notable

exception

studied in the finite dimensional

of

the

'almost' version,

which are

case by Willems and Trentelman, see e.g.

[39]. The theory for the almost disturbance decoupling problems is one of the main still

open problems in geometric theory for

infinite

dimensional

systems. The monograph is addressed to researchers in the field of geometric theory of infinite dimensional the

infinite

dimensional

controllability,

initial

third

of

chapter

systems. In this book I shall use basic concepts of system

theory

as

observabUity, which are

Curtain

and

C0-semigroup~ covered

Pritchard [9]. This

in the

book

is

approximate second and self-contained

with respect to the notions of the geometric theory~ although sometimes we shall refer to the references for the finite dimensional case.

VI Although it may seem that writing a monograph and doing research is a solo occupation, in reality it is a team occupation and I owe my team members of the Groningen System Theory Group a great debt of gratitude. First of all I want to thank Ruth Curtain who found always the time and the patience to listen to my ideas. Her guidance made sure that my research would not wander off in queer directions. During the past four years it has been a great pleasure to share the office with Jan Bontsema. As a room-mate he always had a lending ear to listen to my (sometimes

obscure)

problems and his relativizing

way of

looking at

these problems really meant a lot to me. Furthermore I would like to thank the other members of the System Theory Group in Groningen; Harry Croon, Christiaan

Hey,

Hans

Nieuwenhuis,

Paula

Rocha,

Siep

Weiland

and

Willems, for the privilege of working with them. They all contributed

Jan in

their own way to this research and made our lunch breaks a very cosy hour. I also express my gratitude

to

Hans Schumacher

whose

insight

into

the

problem plus his remarks and ideas helped me to get my research started. Special thanks go to Erik Thomas, Malo Hautus and Luciano Pandolfi for the careful way they read this monograph. Their discussions and interest from different mathematical backgrounds all contributed to this research. I also want to university

of

thank

Groningen

the for

office of their

the mathematics

help

during

the

department last

years.

of the Special

thanks go to Janieta Schlukebir for typing part of this monograph. This research was sponsored by the Netherlands

Organization for Scientific

Research (N.W.O.), under grant no. 10-64-06 for which I am grateful. Hans Zwart

July, 1988

CONTENTS

Introduction Disturbance Decoupling Problem for Finite Dimensional Systems

1

4

Disturbance Decoupling Problem for Infinite Dimensional Systems

11

Outline of this Monograph

I,l

Chapter h Invarlance Concepts

15

1.1: A- and TA(t)-Invaziance

15

1.2: The Relation between TA(t)-Invariance and the Spectrum of A

18

Chapter Ih System Invarlance Concepts

20

II.l: System Invariance Concepts

21

II.2: Open Loop Invariance

23

II.3: Frequency Invariance

32

II.4: Equivalence

41

Chapter IIh Disturbance Deeoupling Problem

47

III.h DDP in Frequency Domain

48

III.2: DDP in Time-Domain

57

IiI.3: Properties of Controlled Invaxiant Subspaces

59

Chapter IV: Controlled Invarlance f o r Discrete Spectral Systems

62

IV.l: Discrete Spectral Operators

64

IV.2: Zeros and Invariance

69

IV.3: Characterization of all Invariant Subspaces for Spectra] Systems

72

IV.4: Examples

78

Chapter V: The Disturbance Decoupltng Problem with Measurement Feedback

87

V.I: Conditioned Invariance and (C,A,B)-Pairs

88

V.2: Disturbance Decoupling Problem with Measurement Feedback

97

Vlll Chapter Vh The Disturbance Decoupllng Problem with Measurement Feedback and Stability VLI: Stability, Stabilizability and Stabilizability Subspaces

107 107

VI.2: Disturbance Decoupling Problem with Measurement Feedback and Stability

116

Appendix E: Examples

127

E.I: Spectral Realisations of Delay Equations E.2: The Relation between

V*(K), Voz(K) and Vz(K)

E.3: On the Sum of two Controlled Invaxiant Subspaces

127 133 138

Conclusions

144

References

147

List of all Invarlance Concepts and their Relation

153

Notation

154

Index

156

INTRODUCTION

The aim of this monograph is to present a geometric approach to disturbance decoupling

problems

for

infinite dimensional

systems. Before

we

go

into

details we shall give an outline of the disturbance decoupling problems. By a disturbance type. Let E

decoupling

problem we mean

a problem

of the following

denote a s3~stem for which we can distinguish two

inputs, u(.) and

q(.), and

two

classes of outputs, y(.) and

classes of z(.), as

is

schematized in figure 1 by a signal flow graph. q

Z

.

E

U

,7

f~re In

this system

(disturbance)

on

outputs. Now

a

we

regard

the

system.

disturbance

the In

1

input general

decoupling

q(.) as

an

this input

problem

undesired

influence

will influence both

amounts

to constructing

a

second system El, which takes as input y(.) and gives as output a control input u(.), such that z(.) has become

independent of the disturbance

input

q(.). Pictorially we have

q

q~

Z

Z D

E 3

,Y

i

! i

I

[

E/

,

i..............................................

r.~z

figure 2 So

if we

system, Ed, disturbance

regard

the interconnection of the systems,

as schematized

in figure 2, then we

E

and

EI, as one

see the input q(.) as a

signa~ which should not influence the output z(.), and

the measurement

we

use

y(.) in order to design a control input u(.) which cancels

the effect of the q(.).

Therefore

disturbance

we shall

q(.) on z(.),

call u(.)

the

i.e. which decouples z(.)

control

input,

q(.)

the

from

disturbance

input, y(.) the measured output or measurement and z(.) the to be decoupled output.

The

next

example

shows that

this

problem

is solvable

for

some

systems ,~.

Example 1. In this example we consider the binary distilation column as studied by Takamatsu,

Hashimoto and Nakai [38]. The system is assumed to be in an

equilibrium and we want to make the composition of tile distilate composition of the

reboiler

independent of changes

in the

and the

composition of

the feed stream. As a model for this distilation column we take

f

(1)

x(t) =Ax(t)+Bu(t)+Eq(t)

E

y(t) =x(t) z(t) = Cx(t)

where x(.) = [xl(. ),x2(.),.. ,xll (.)~ t

xd.)

denotes

the

difference

between

the

liquid composition on the i-th tray and its equilibrium value,

u(.)=(ut(.),u2(.)]t; ul(.)

is the difference of the flow rate

of the liquid

stream and its equilibrium value, u2(. ) is the same, but now for the vapor stream,

z(. ) = (zl(.),z2(. )] t; zx(.) = xl(. ), z2(. ) =

x11(.).

Furthermore A=(ai,y ) is a

tri-diagonal

matrix,

of

which the

upper

diagonal,

the

diagonal and the lower diagonal are given by respectively, {aj,/+l} = (0.105, 0.469, O.529, 0.596, O.569, 0.718, 0. 799, 0. 901,1.021~ 1.142),

{aj,j} = - (0.174, 0. 943, 0. 991,1.051, i. 118,1.584,1.64,1.721,1.823,1.943, 0.171 ), {aj+l, j } = ( 0. 522, 0. 522~0. 522, 0. 522, 0. 522, O.922, O.922, O.922, 0.922, 0.115), B is given by

L0 -244

C=

-288 -304 -280 -232 -312 -382 -412 -396 -42

10000000000} OOOO00OO0O

and E - - ( 0

J

0 0 0 0 0.4 0 0 0 0 0 ) t .

1

It is shown in [35] that there exists a simple system

E!

which makes the

output z(.) independent of the disturbances. This system is given by

(2)

L'I: u(t)fFx(t),

where F is given by, (correct to five digits) 0 0 -330.06 0 0 0 0 0 0 470.17 0

F-~

0 0 -251.47 0 0 0 0 0 0 632.04 0 1

So Ul(.) = -330.06"x3(.) + 470.17"xm(.) and u=(.) = -251.47"x3(,) +632.04"xlo(.).

In figure 3 the output z(.) is drawn for the system ,U and for the system Ed. In both systems the same disturbance signal was used. 102

10-1

10-4

10-7

10-1o

lO-lS

10-18 /:" ° - ...........................................

10-19

- ....................................

~.!~[ .....................

e(

0

t

I

I

l

I

I

i

L

i

20

40

60

80

100

120

140

160

180

the output signal has become a

factor

1016

order

So

this

200

time - >

figure 3 Note that smaller

for

and

is

the system Ed of

the

same

as

the

machine accuracy.

in

example we can reject the effect of disturbances

A

naturaJ

question

decoupling possible

now and

axises:

how

can

under one

which

construct

C:]

conditions the

is

feedback

disturbance system

•17

Before we can solve this question we should specify which systems we want to

consider. A simple, but not

unimportant, class of

systems is the

class

4

of

linear

time-invariant,

finite

dimensional

systems.

For

this

class

we

shall give the solution of the di.qturbance decoupling problem as presented by Basile /o Marro [1] and Woaham [42].

Disturbance DecoupUng Problem f o r Finite Dimensional Systems The class of systems that we shall consider in this section is the class of systems that have the following representation

x(t) = Ax(t) +Bu(t) +Eq(t);

(3)

y(t)---Cx(t); z(t)=Dx(t); t > 0 ; x ( 0 ) = x 0

where x(t), R', ~ ,

u(t),

~,

~

q(t),

y(t) and z(t) are time trajectories

in respectively

and R~, and A, B, C, D and E are matrices of appropriate

size. Notice that the system of example 1 is of this class. In order to obtain some insight into the disturbance decoupling problem we assume that we measure the full state of the system, i.e. we assume that

y(t)=x{t), or equivalently C=I,. Furthermore we want the system ,U/ to be as simple

as

possible,

and

so

we

assume

it

to

be

time

invariant

and

memoryless. So we assume that (4)

u(t)=Fx(t)

/~I:

with F a matrix. components of

Or in other words, u(t) is a linear combination of the

the

state

at

time

feedback system (4) such that

t.' The problem

after

at

interconnecting

hand

is

(3) and

to

(4)

find

a

we have

that the disturbance input q(.) has no influence on the output z(.) for all disturbance

signals.

This

problem

is

commonly known

as

th__~e disturbance

decoupling problem. Definition: Disturbance Decoupling Problem. The

Disturbance

Decoupling Problem

is

to

find,

if

possible,

for

the

system (3) a feedback system of the form (4) such that in the closed loop system

the

disturbance

disturbance signals.

input

q(.)

has

no

influence

on

z(.)

for

all

5 It is standard to refer to this problem by its initials, DDP, and we shall continue this tradition. For the class of systems defined in (3) we can give a precise formula for the

closed

loop

behaviour

of

the

system.

This

solution

is given

by

the

well-known variation of constant formula, t

z(t) =De(A+BF)txo+I De(A+BF)(t-S)Eq(s)ds

(5)

0 Since we assumed no prior knowledge about the disturbance must have

that

the

input q(.), we

DeiA+BF)tE" " is identically zero on [O,co), in

function

order to make z(.) independent of q(.). So DDP is solvable if and only if we can find a feedback law F such that This

last

problem

is

very

hard

De(A+BF)tE-O; t~O.

to

solve

directly.

Since

one

h~m

to

calculate e(.A+BF)t for all F. However there is a simple necessary condition which we can deduce from it, namely for t = O we have that De(A+BF)UE=I)E,--and tttis must be zero. So DDP is solvable, only if DE=O. This condition shows in particular

that

DDP is not

solvable for

every system in our

class. In

DE=O is not sufficient. De(A+BF}tE=o; t~O in the following system theoretic way, namely all trajectories of the system x(.)=(A+BF)x(.) that start in ImE will remain in the kernel of D. Let V denote the reachable subspace for the system (A+BF,E), where F is the feedback law that solves example 2 we shall see that the condition

We can interpret the condition

DDP. This subspace is defined as

(6)

V = span { e(A+BF)tEq ); where the span is taken over t>_O and q~Rq

By the solvability of DDP we must have

(7.a)

Ira E cV cKer D

ImE and KerD denote the image of E and the kernel of D respectively, and the semigroup property of e(A+BF}t implies that V also

where

satisfies

(7.b)

e(A+BF)tvcv; t>O

Ou the other

hand,

suppose that there exists a subspace VcR",

(7.a) and (7.b). Then DDP is solvable with e(A+BF)tlmEce(A+BF)tv cVcKerD, and thus De(A+BF)tE-O.

some F satisfies the conditions this F, since

whictt for

So in conclusion we

can

exists a subspace F and could argue

say that DDP

is solvable if and

only if there

a feedback F which satisfy (7.a) and (7.b). One

that this result makes

the problem

more

difficult; not

only

must one construct a feedback F, but also one must construct a subspace

VcR n with the properties (7.a) and (7.b). The next theorem which can be found in Basile & Marro [1] and W o r d m m

[42, p.88] shows that one only has

to construct a subspace V c R n with special properties which

are easy

to

check.

Theorem be a linear subspace of I~n. Then

Let V

the following properties are

equivalent:

i)

There exists an F such that

ii) iii)

There exists an F such that

e(A+BF)tvcv. (A+BF)VcV.

A VeV+Im B

Some remarks must be made about this theorem; first the feedbacks in i) and ii) can

be

feedback F

chosen

to be

the same,

and

second

the construction of the

iii) to ii) or i) is done by solving linear equations,

from

which is a relatively easy problem, see example 2. Before

we

concepts

the

continue

differential

the

DDP

There

we

shall briefly refer to

invariance

]c(t)=Ax(t)+Bu(t). Since the solution of

~c(t)= (A+BF)x(t);

equation

x(t)=e(A+BF)txo, i')

with

related with the system

x(0) = x0

is

given

by

the

differential

we can replace assertion i) by exists

equation

an

F

such

:~(t)=(A+BF)x(t)

that

all

solutions

of

which start in V will remain in V.

We can now pose the question whether one would gain more if one were to allow

for

'arbitrary'

inputs

instead

of

inputs

generated

by

feedback.

In

other words are there subspaces V which do not satisfy i'), but do satisfy: iv)

For every solution of

xoeV there exists x(t)=Ax(t)+Bu(t);

a continuous input u(t) such that the x(O)=x 0 remains in V.

The answer to this question is negative. Since if x(t) is in V for all t_>O, then x(t) is in F for all t>O. So for t=O we have that Ax0=Ax(O)=x(O)-Bu(O) e I/+ImS. Thus by the equivalence of i') and ii~) we have that there exists an F such that the solutions of

x(t)=(A+BF)x(t); x(O)=xo

remain in g.

7 From the theorem and the argument above we see that property iii) is of great importance. This property has been given a special name. Definition: (A,B)-lnvariance A subspace V is (A,B)-invariant if

A VcV+Im B

(S) So

with

respect to

the DDP

we

see that this problem

is equivalent

to

finding an (A,B)-invariant subspace V with property (7.a). Let ~(A,B;KerD) denote the class of all subspaces that axe (A,B)-invariant and contained in the kernel of D. Note that we are looking for an element in ~(A,B;KerD) which contains Im E. Trivially the zero subset is an element of ~(A,B;KerD).

Furthermore

it is

an easy exercise to prove that the sum of two elements in ~(A,B;KerD) again

an

element

of

~(A,B;KerD).

So

by

the

finite dimensionality

is of

KerDcK n there will exists a supremal element in ~(A,B;KerD), which we shall denote by P*(KerD). Thus for every element P' in ~(A,B;KerD) we have that VcP*(Ker.D). From this we have the following theorem as an easy corollary.

Theorem The DDP is solvable

if

and

only

if

/mEcP*(KcrD).

Furthermore the feedback that solves the DDP can be any feedback F that satisfies

(A+BF)I~*(KerD) cP~'(Ker D).

We have mentioned that calculating the feedback F is a linear problem. Now we shall see that

the calculation of P*(KerD)

is also fairly easy. Define

the sequence P~ according to (9)

p°=KerD; ll~=KerD n A-~(ImB+P~-I~; #--1,2,..

where A-t(X) is the set consisting of all elements y such that

AyeX.

By induction it is easy to show that I f c l ~ -t, and for some k 1}, see Kato [22, p.210]. From this lemma we have the following corollary. C o r o l l a r y 1.10. Suppose that the resolvent set of A is connected and that V is a closed A-invariant suhspace. Then V is TA(t)-invariant if and only if a(Av)ca(A). Proof: This

is

a

direct

consequence

of

the

above

lemma.

Since

connected, then p(A) = p®. So for example 1.6 this corollary implies that there a(Av)=C.

if p(A)

is D

CHAPTER Ih SYSTEM INVARIANCECONCEPTS

The theory of controlled invariance of a subspace has been investigated in detail

in the

[1],

[35]

[19],

dimensional [27]

and

case that

and

there

[32],

[42]. have

but

shall

investigate

case

of

finite

state the

been

many

this

the For

some

questions

invariance

rank

inputs.

is finite dimensional, see

that

the

preliminary remain

for

We

space

case

space

investigations

unanswered.

infinite

shall

state

In this

dimensional

consider

the

e.g.

is

infinite

in

[6]-[8],

chapter

systems

controlled

we

for

the

version

of

system (1.1):

(2.1)

x(t)=Ax+Bu; x(O)=x0, x~X, u~U,

where X and U are Banach spaces, A is a generator of a Co-semigroup, Ta(t), and furthermore

we shall impose the condition that B is a bounded linear

operator with ImB finite dimensional, so without loss of generality we may assume that ht=R m and B is injective. The reason for only considering finite rank of

inputs operators view

a

natural

Secondly there

is twofold. First of all it is from a practical point choice,

one

can

is the mathematical

only

implement

reason;

finitely many

we believe

that

inputs.

only

a

small

part of the theory, as will be presented here, will remain valid if ImB is not finite dimensional. For

the

invariance loop The

system

in section

invariance concept

finite infinite

of

and

dimensions.

invariance

concepts open

that

for

II.2.)

frequency

dimensions,

we

II.1, and

(section

subspaces, prove

(2.1)

In is

it

shall

discuss

we shall pay and

frequency

invariance turns

section

was

introduced concept

11.4.

the

relation

and

there

frequency invaxiance.

subspaces

closed

loop

we

of

attention

invariance

this

investigated,

kinds

particular

that

and closed loop invariance are these

various

open II.3.).

Hautus [19] a

key

between

the

show

to

(section

by

plays

system

that

for

role

is

in

various closed

equivalent. Furthermore invariance

for

equivalent

we to

21 Section ILl: System Invariance Concepts The theory of system invariance entails many definitions. Here we shall summarize some of them and give some important properties. We shall start with the strongest. By

TA+s~(t) we

A+BF.

shall denote the semigroup generated by

Since F is

a bounded operator we have from Curtain and Pritchard [9, p.38] that always generates a C0-semigroup and the domain of

A+BF

A+BF

is equal to the

domain of A.

Definition ILl: Closed Loop Invariance A subspace V of A' is called closed loop invariant

if there

exists a

bounded feedback law F such that

TA+BF(~)VcV

(2.2}

for all t in [0,oo).

Remark: So a

subspace

V is

invariant for the system

called

closed

x(t)= (A+BF)x(t)

loop

invariant

for some

if

it

is semigroup

Fe£(X,U).

Lemma II.2.

VeX is TA+BFl(t)-invariant, for a T~.BF2(t)-invariant for a bounded ImB(Fz-F=)]VnD(A)CV.

Assume that a closed linear subspace certain

bounded operator

operator F 2 if and only if

F 1. Then V is

Proof: See lemma 4 in Curtain [7].

D

C o r o l l a r y II.3. Let 8 be any subspace of If

a

closed

subspace

V

is

ImB

such that

closed

loop

B+(ImBnV)=lmB. invariant~

then

bounded feedback law F such that V is Ta.sF(t)-invariant and

there

ImBF

exists

a

VnD(A)CB"

Proof: Assume that V is Ta+B~.(t)-invariant , then from the fact that the range of

F

is

finite-dimensional

and

F

is

bounded,

BF

can

be

written

as,

22

(Kato [22, p.160]),

~bi+ i--I

~ bi,

where

i~q+l

span {bi}=/~ and i~l,

Defining F = t..,,['

0

j=l

Thus det(S(s0) ) =0, providing the contradiction. The following leman is the frequency domain version of leman II.10.

12

38 [,emma II.20. If a closed subspace V of X is frequency invaxiant, then every x in V

has a unique and

there

((s)

and

(~,w)-representation

exists an

w(s)

interval

with Bw(.}

[~,oo),

axe continuous

on

in B°_l(s) and ~(.) in V.l(s )

which is independent

this

interval,

of x,

this .~ is the

such same

that as

in

lemma II.19. Proof: Let ~ be the constant of lemma II.19 and suppose that g i e X '

= 6 i i , where {bj}m°1"= is a basis for B °. Furthermore let

gi[ v = 0 and be

an

satisfies

arbitrary

representation

element

with Bw(.}

of

V,

contained

t~en

by

lemma

in g°_l(s)

II.18

it

has

a

X

((,w)

. Thus oil an interval [rx,~ )

x can be written as

(2.24)

x=(s-A)~(s)-

mO

~, b#oi(s), ~(.)~V_I(s ) i=l

(2.24) implies: mo

(2.25)

(s-A)-tx=((s)-

~, (s-A)-tbioai(s) /=,

Calculating < g i , ( s - A ) - t x > ; i = 1,..,m o gives mO

(2.26)

= -~



wj(s)

mo

= - F, wj(s), since ((s) is in V. If we premultiply (2.26) by s we obtain; ml]



(2.27)

= F.

wj(s).

i=1

Or in matrix notation:

(2.2s)

=

-S(s)

< g,.0,s( s - A ) " x >

,

w.,o(s) j

where Sift s) = < g,, s(s - A )-'bj >. Note

that

S(s)

and

< g o s ( s - A ) - 1 x>

are

continuous

on

the

interval

39 [~,oo)¢p(A), and from lemma II.19 we have interval.

So

on

[~,oo), {wi(.)},

i=l,..,m

that

S(s)

is invertible on this

is

the

tmique

o

solution

of

equation (2.28). By (2.25) we have that there is only one choice for ~(s), that is: mo

(s-A)-lx+(s-A)-t( ~ biwi(s) ) []

i--I

Let us remark that (2.28) implies that if ,1" is finite dimensional, and so A is a matrix,

wds)

then

is a

rational function, and with the

last line of

the proof of lemma I/.20 ~(s) is a rational function too. So if X is finite dimensional, then

definition H.15

is

the

same

as

if

we

were

to

restrict

ourselves to strictly proper rational functions, as in Hautus [19].

Before we can prove the equivalence between frequency and and closed loop invariance we need

some

properties

of

the

set

of

all

possible values

of

~(s) for x in V, which we shall denote by ~s. D e f i n i t i o n II.21: ~"

s

If

V

is

a

~t~VnD(A) such

frequency that

there

invariant exists a

subspace, x

then

in V with a

x=(s-A)~(s)-Bw(s), ~(. )~V_t(s), w(. )~ll_t(s),

Zatconsists (~,w)

of

all

representation;

such that ~(st)=~t.

Lemma II.22. If V is a closed subspace that then

we

have

that

there

exists

is frequency invariant and

a

real

.~ such

that

for

any

lmBnV={O}, st>~ , the

equalities (2.29)

I x=(sl-A)~t-Bwt

and

[ x=(sl-A)~2-Bw2,

when x, ~x and ~2 in V,

imply that ~t = ~2. Proof: Since V is a closed subspace of ,¥ and simple corollary lemma II.19

of

lemma II.19, and

ImB•V=(O},

this lemma is a

.~ in this lemma is the

same

as

in []

40 Lemma II.23. Let g be a closed frequency invaxia~t subspace with Im BoV= {0}, then:

S1 - - ~ s 2 )

for all s:,s2~[~,oo), where ~ is as in lemma II.19. Proof: Let ~l be an element of --h' then there exists a x in V with

x = ( s l - A ) ~ l - Bw(sl); ~t = ~(sl) Rewriting this equation gives x = (sL-s2+s2-A)~l-Bw(sx) , or

(2.30)

(s~-sl)~l+ x = (s2-A)~ 1-Bw(sl)

(s~-sl)~l+x is an element, of g thus it has a (~,w) representation. So there exists a pair (~,w) such that

(s2-sl)~,+x=(s-A)~(s)-O~(s)

(2.31)

From lemma II.20 we have that equation (2.31) holds on [.~,oo). Now relations (2.30}

and

(2.31)

with

lemma H.22

imply that

~(s2)=~r So _41c_sz.--~ By

symmetry we conclude that _ 41 _ - s2. - -

[]

_

Lemma II.24. Let V be a closed frequency invaxiant subspace with ImBnV={O}, : =~'-

4

is

closed

in

the

graph

norm

of

A,

for

definition

see

then Davies

[14, lemma 1.6.], where ~ is defined as in lemma II.19. Proof: Let ~n be a sequence in ~

such thai; ~n-~ y and A~n-~ z. Since A is a

closed operator) we have that yeD(A) and Ay=z. for ImB,

then since ImBnV={O}

Let {bl,..,bm0} be a basis

and V is a closed subspace there exist

gieX' such that 9i[ =0 and =6~i. Since ~,, is an element of ~, there V exist xn in V and wn in // such that m O

(2.32)

x. =

(~-A)~.-Bo2. = ( . ~ - A ) ~ , - ~ b ~ . # j=l

Since xn~V , we have that

41

O= < g , , x . > = < g i , ( ~ - A ) ~ n - B w . >

and thus

= - < g i , A ~ . > - wni

=-u,r= ~

So ~ni converges as n ÷ ¢0, i = l , . . , m 0 . Thus t¢n converges to say w e / / and since x n = ( ~ - A ) ~ n - B w n we have that x n converges to x. Since V is closed we have that x ~ V and so there exist ~{s) and w(s) such that x = (s-A)~(s)-B~(s).

By definition x is also equal to (.~-A)y-Bw. From lemma II.22 we have that y=~(.~), and thus y ~ S .

[]

S e c t i o n II.4: Equivalence In

the

closed

theory

loop

of

and

system

open

equivalence tells us that, stays in a subspace, trajectory this

stays

dimensional.

If

equivalence

was

bounded.

but

We

Furthermore frequency between

is

the

of

equivalence

great

between

importance.

This

the trajectory

then we can also find a feedback law such that subspace.

was

proved

the

state

proved

by

In Basfle & Maxro [1] and

in

the

case

space

is

Schmidt

that

the

infinite

and

state

space

dimensional,

Stern [32]

the

Wonham [42] is

finite

then

provided

that

the A

is

However, the interesting case in infinite dimensions is when A is

unbounded, section.

subspaces

invariance

if we can find an input such that

in this

equivalence

invariant

loop

generates

formulate

and

we

prove

shall

invariance open

a

and

C0-semigroup; prove that

for

closed

closed

loop

this

this

is

equivalence

closed

linear

and loop

subspaces

invariance

is lost

the

for

the

invariance and

focus

that

if the

of

system is

the

this (2.1).

equal

to

equivalence

subspace

is not

closed. We shall begin by showing that there is equivalence between (A,B)- and feedback (A,B)-

invariance.

and

However

closed loop

there

is

invariance even

in

general

no

equivalence

if we impose the

that VnD(A) is dense in V, as shown in Schmidt and Stern [32].

extra

between condition

42 Lemma II.25. If V~cD(A) is a linear subspace, closed with respect to the graph norm of A and V2c9:' is a closed linear subspace with (2.33)

AV, c

Vz+ImB ,

then there exists an A-bounded feedback law F such that (2.34)

(A + BF)V l c V~

Proof.. If X

is finite dimensional, then the

proof

can

be found in Basile &

Marro [1] and Wonham [42, p.88]. The general proof

that

will be presented

here is an adaptation of the proof given by Pandolfi [27]. Define X A to be the graph of A, with the graph

norm [[(x, Ax)[[A

=

[[x[[+[IAx[[ , where [[.[[ is the norm of X. A is a bounded operator from X a to X, and V1 is by definition closed in X A. If v l e V l , are

then there exists v 2 e V 2 and u e U such that A v l = v 2 + B u , and u and v2 J_ uniquely determined if we assume that u e [ K e r B ] (the annihilator of

Ker B) and B u e B ° ( V 2 ) .

Let F be defined by F v l = - u ;

V v l e V ~. The operator F is linear, since u is

uniquely determined. We shall show that F is a closed operator n

us assume that (vl,Fv~) converges to ( v L , - u ) , must prove that Fv I = - u . n

n

in ,~'A. Let n

thus v l - ~ v 1 and A v l - ~ Av 1. We

This is obvious since

rA

w : -- Av I - Bun = Av~ + BFv n converges to A v 1 - B u = : w,

and v l e V ,

since V1 is closed and B u e B ° ( V s ) , since B°(V2) is closed.

Thus F is a closed operator from the whole of V~, with induced norm of [[.[[4, to /L By the Closed Graph Theorem F is a bounded operator on VI, with norm [[-[[a, by the Hahn-Banach Theorem and the fact that, since /l is finiLe dimensional, F has finite dimensional range, F has a bounded extension on X A. From Kato [22, p.191 and 245] we have that F is A-bounded on X.

O

T h e o r e m II.26. If V is a closed subspace of X, then (A,B) and feedback invariance are equivalent.

43 Proof: This is a easy corollary of lemma ]:I.25 since if V is a closed subspace and A is a closed operator, then Vr~D(A) is closed with respect to Lhe graph norm of A.

[]

Now we have proved all the ingredients for our main result.

Theorem II.27. Let V be a closed linear subspace of X, then the following concepts of invariance are equivalent: a)

V is closed loop invaria~t

b)

V is open loop invariant

c)

V is frequency invariant

We remark that lemma 1.4 can be seen as a special case of this theorem i.e. no control action thus B =0. If X is finite dimensional, then the equivalence between these concepts are known, see for a) ,~ b) e.g. Basile & Maxro [1] and for a) ,~ c) Hautus [19]. Since we have equivalence between these invariance concepts we introduce a new concept that we shall use if a subspace satisfies II.27 a),b), or c).

D e f i n i t i o n II.28 Controlled Invariance A closed

linear subspace

V of

X

is called

controlled

invariant

if

it

satisfies II.27 a),b) or, equivalently c).

As we shall see in the next chapter the equivalence between open loop and closed loop invariance as well as the equivalence between frequency and closed loop invariance is lost in general if the subspace V is not closed. From the

previous chapter

between controlled and

we see

that

there

is no

hope

for

equivalence

(A,B) invariance in general. However there is one

case were this equivalence holds.

44 Lemma I1.29. If

V is a

closed

subspace

contained

in the

domain

of

A,

then

the

following assertions axe equivalent: a)

V is closed loop invariant

b)

V is open loop invariant

c)

V is frequency

d)

V is

invariant

e)

V is feedback invariant

(A,B) invariant

Proof: The equivalence between a) and e) is a consequence of lemma 1.7, the other equivalences follow from theorem 1/.26 and II.27. We shall prove theorem II.27 by showing a) =~ b) =~ c) =~ a). P r o o f o f T h e o r e m II.27:

a) => b)

TA+SF(t)V c V. x(t) as TA+BF(t)X0 and u(t) as FTA+BF(t)Xo, then Curtain and

Let F be the feedback law such that Defining

Pritchard [9, th.2.31] gives the desired result.

b)

=>

c)

Let x o be an element of V. Then from lemma II.11 we have that there exists an u(.)~C([0,oo);U) such that t

x(t}=T~(t)xo+ I TA(t-s)Bu(s)ds

(2.35)

0

is in V and I I ~ ( t ) l l _< 2~" ~eoa , f o r some H and ~ in R. The exponential boundedness of u(.) implies the same for x(.), and so we can take the Laplace transform of

equation

w(.)

the

(2.35).

Laplace

Define

transform

~(.)

of

to

be

the

u(t). Then

Laplace transform

equation

(2.35)

of

gives

x(t)

on

and

an

interval [r%,0o) the following relation (2.36)

~(s)=(s-A)-tXo + (s-A)-'Bw(s)

Since V is a closed subspace we have that Zemanian [46] we have that l i r a s-,.oo

(~,w) representation of x o.

f(s)eV, and with theorem 8.6-1 of

sw(s) = lira u(t) -- u(O). So ~(.), w(.) is a t¢ 0

45

c) => a) Since V is a closed subspace,

we see,

by lemma II.20, that

restrict our input operator B to B, such that B is injective, and ~ l ( V ) = I m B n V = { O } the

system

(A,B).

we may

B°(V)~.~°(V)

(see (2.8)). Then V is also frequency invariant for

So

we

may

assume

~tithout

loss

of

generality

that

Im Bn V = {0}. By assumption we have that all x in V admit the following decomposition.

(2.37) If s

x = (s-A)~(s)-Bw(s).

is larger

than

~ (see

lemma II.19 and

definition

II.21),

then

since

~{s)eEj,(2.37) implies that (2.38)

A ~ , c V + lm B.

From lemma II.24 we have that - = , = 3 . So (2.38) implies that (2.39)

A---cV+Im B.

5" is closed in the graph norm of A, so from lemma II.25 we have

the

existence of an A-bounded F such that (A+BF)EcV. Rearranging equation (2.37) gives x = (s-A)~(s)-Bw(s) = (s-A-BF)~(s)-B(w(s)-F~(s)) Thus B ( w ( s ) - F ~ ( s ) ) e V ,

so

by the assumption made in the beginning of this

proof w(s)=F~(s). So (2.40)

x = (s - A - BF)~(s)

It remains to prove that this feedback law is bounded. We shall begin by showing that the mapping x~-*sw(s) is a bounded operator from V to U for se[~,oo)np(A). Let for se[~,oo)np(A), F~ be defined as Fsx: =sw(s), where w(s) is the (unique)

input

from equation (2.37). It is easy to show that

these

46 operators

are

closed,

and

thus,

since

they

axe

defined

on

the

closed

subspace V, bounded. By the definition of //_l(s) we have that lira sw(s} exists, so for every s-~o x in V we can define the operator Fx by F x = l i m F,x= lira sw(s). Then by the $-t00

uniqueness

of w(.) ,~ is a linear

Boundedness theorem,

operator

3=)~0

defined

on V. By the Uniform

and since V is closed we have that F is a bounded

operator on V. The Hahn Banach Theorem gives that F can be extended ms a bounded operator to the whole of X. We shall show that on ~. F--F. Let ~0 be an element :7, then there exists a x0, ~(.) and w(.) such that

xo = (s-A)~(s)-Bw(s) and ~(~)=~o

(2.41)

Rearranging this equation gives that

[. o~(s)-~(,~} (2.42)

-e

~o= ( s - A )

.q-$

~(S ) -,% Define ~t(.)

and

wl(.)

to

be respectively

Then F~0=lim swl(s)=w(~)=F~(~)=F~o. operator

2-)¢o

F.

So

equation

(2.40)

and I .~-s ]" ~-s So on &" F is equal to the bounded

implies

that

for

(s-A-BF)-tx=~(s)eV, and lemma 1.4 concludes the proof.

sufficiently

large

s C:]

CHAPTER II1: DISTURBANCE DECOUPLING PROBLEM

tn

this

chapter

we

shall

consider

the

disturbance

decoupling problem

(DDP): given the system

(3.U

x(t)=Ax(t)+Bu(t)+Eq(t),

z(t)=Dx(t),

where A and B are the same as in (2.I) and E and D are bounded linear operators from respectively Q to ,1' and ,l" to Z, find an bounded feedback law such that, in the closed loop system, z(.) does not depend on q(.). So pictorially we have the following situation;

x = Ax U

+ B u + Eq

z = Dx

Thus the Disturbance Decoupling Problem is to design a feedback law F such that the transfer from q to z is zero i.e. D ( s - A - B F ) - I E - - O . Tile next theorem gives the link between controlled invariance and DI)P.

Lemma III.1. The

Disturbance Decoupling Problem

is solvable

if

and

only

if

there

exists a controlled invariant subspace V such that I m E c V c K e r D . Proof: See Curtain [6].

In

the

finite

dimensional

theory

it

(:outrolled invariant subspace contained in the calculated.

Here

we shall also

turns

out

kernel of

introduce this subspace,

that

the

D can but

largest

ahvays be

tile calculation

is a difficult problem in general. $

D e f i n i t i o n III.2: V (K). Let K be a closed subspace in X. Then we shall denote by I)*(K) the largest controlled invariant subspace contained in K.

48 If this subspace exists, then we have the following nice result. T h e o r e m III.3. If V*(KerD) exists, then DDP is solvable if and only if 12*(Ker D)~ Im E.

{3.2) Proof: See Curtain [6] In

Curtain [6]

Y•(K)

the

is posed.

ID question

of

We remark

sufficient

here

that

conditions

for

the

existence

~'*(K) need

not

always

of

exist, see

Pandolfi [27] for a counter example for delay equations, or see appendix E for

a

Y*(K)

counter exists,

example then

it

for

partial

must

be

differential

closed,

since

problem

will

equations.

Note

that

TA+sF(t)VcV implies

if that

rA+sr(t)VcV. The

Disturbance

Decoupling

be

frequency-domain as well as in the time-domain respectively.

In

section

HI.3

we

investigate

investigated

in

the

in section III.1 and III.2

some

properties

of

closed

loop invariant subspaces using the results from llI.1 and Ill.2.

Section III.l: DDP In Frequency Domain Keeping the equivalence

between closed loop-

and frequency invariance

in mind we define the natural candidate for 12*(K).

Definition I~.4: V~(K) We assume that K is a closed linear subspace. Let

PE(K)

be

the

subset

of

X

which

contains

all

x~X

with

a

(~,w)

representation with ~(.) in K(s), (see definitions II.12 and H.13).

The next lemma will show that P u(K) is the supremal frequency invariant subspace contained in the closed subspace K.

49

I.emma III.5 a)

Every

frequency

invariant

subspace

contained

in

K

is

contained

in

VE(K). b) Every closed loop invarianL subspace contained in K is contained in VE(K)c)

V£(K) c K.

d)

1)•(K) is the supremal frequency invariant subspace, contained in K.

Proof: a)

Obvious,

with the

definition of V u(K ) and the definition of

frequency

invaxiance. b)

Obvious,

with

theorem

II.27, since if V is closed loop

invaxiant,

then

]7 is it too. c)

Let (~(s),w(s)) and

x=lim

be

a (~,w) representation of x

with lemma TI.14 and the fact

in V F.(K),

then

E,(s)~K

that K is closed we conclude that

s~(s) is an element of K.

S-~O0

d)

Let

x

be

an

element

functions ~(s) and w(s)

of

I)~(K),

so

that

there

exist

strictly

proper

which are continuous on an interval [rx,oo ) and

such that 5(s) is in K and

(3.3)

for s>r~

x=(s-A)((s)-Bw(s);

(see

definition

II.15). Let

s o be

an

arbitrary,

but

fixed

point

in R

with s0>rx, then by using (3.3) we get

(3.4)

0 = ( s - A ) ~ ( s ) + ( s o - A ) ( - ~ ( s o ) ) - B w ( s ) +Bw(so)

,s A, i ,s, ,so,l +,s

=

,so, 1

So if s ~ So, then (3.4) implies that

(3.5)

~(so)

= (s-A)

I(~(s)-~(s°))lJ -s I (w(s)-w(s°)) 1 -

--

.[ -

B

S O

SO-S

Define for s > s o {l(s): =

and wl(s): = S o --S

It

is

not

hard

to

show

that

. S o -S

(~l,wl)

is

a

(~,w)-representation

of

5O ~=~(so)

limswl(s)=w(so).

with

Since ~(s) is in K, (l(s)

is in K. Thus by

8-~.¢O

the

definition

of

V u(K),

~(so)

is

in Vx:(K ). Since

so was

an

arbitrary

element of [r~,oo) we may conclude that x has a (~,w)-representation

with

~{.)eY]:(K). Now since x was an arbitrary element of 1)~(K) we have that V~(K) is frequency invariant. By c) we obtain

I;$(K)cK

and a} implies

that

Vx(K)

is the

largest

frequency invariant subspace with this property.

[]

With this lemma we can easily prove the next theorem. T h e o r e m III.6.

Let K be a closed subspace of k'. If

VF(K)

is closed, then

V"(K) exists

and is equal to

VF(K).

Proof: If conclude

~)U(K) is that

closed, VE(K ) is

then closed

by

lemma

loop

III.5.a) and c) it must be the largest and is equal to

III.5.d)

invariant.

and

theorem

Furthermore

with this property. So

VF(K).

II.27

from

we

lemma

V*(K) exists 0

Remark:

We remark that the converse of the theorem does not hold, i.e. if exists, then

VF_(K)

So the question which arises is under what conditions is

12E(K)

turns out that these conditions can he formulated in terms of

IZE(K),

V*(K)

need not be closed, see example E.10. closed. It

ImB

and

which we do in the next lemma. In this lenLma we shall use the

following notation. If V is a subspace of X, then

VJ={fe2d'[

such that

f(V)=O}.

V'cX'

this sequence is weak * convergent to f if



will denote the annihilator of V i.e.

Furthermore if {)¢,,} is a sequence in ,V', then

~

for all

xeX,

where

denotes the operation of the functional f on x. See for more details

Kato [22, p.136] and Yosida [44].

51 Lemma III.7. Let

13°(l;E(ll)) denote the largest subspace of IrnB that with llE(K), let {bD..,bm0} be a basis of 13o(Vz(K)) Irn BnV. Then the following assertions are equivalent. VE(K ) is closed.

has

intersection

BI(V) i)

be

ii)

(VE(K)) atnD(A')

iii)

There

exist

=6ij,

is weak * dense in ~xJE(K)).~ and m 0

(f~}i=t

functionals

zero let

BIO;~(K))= ~ I ( V - - ~ ) .1.

in

and

[V~:(K)) nD(A')

such

that

where D(A') is the domain of the dual operator of A, see

Yosida [44]. Remark: Before proving this theorem, we remark that the result of this lemma is independent

of

8°(V~(K)). condition

the

basis

Furthermore in

the

third

it

of

B°(]/E(K))

and

is

clear

simple

assertion

of

by

this

of

the

linear

lemma

is

actual

choice

of

algebra

that

the

to

the

equivalent

&

existence of functionals f i e (V~:(K)) nD(A') such that the matrix

S®=

is nonsingular. P r o o f o f lemma III.7:

0=>~0 .I.

So we must show that the annihilator of V~(K), lYZ(K) intersected with D(A') is sufficiently rich. If I)E(K ) is a closed subspace, then by theorem ILI.6 it is closed loop invariant. So by lemma 1.4.c) there exists a bounded feedback law such that

(A-A-BF)-IVE(K)cV~(K)

for all AeR sufficiently large. at

Let y be an arbitrary

(A_ A" _ F'B')-ty

element of

is an element of

[VE(K)) , then we shall show that

[VE(K)]~'nD(A ") and A(A_A'_F°B')-ly

is weak

J.

* convergent to y, A-,co. This will prove that is 0;~(K)] nD(A'} weak * dense .L

in ( v E ( g ) ) . Let x be an arbitrary element of VE(K), then since Vz:(K) is invariant we have that



=0

(A-A-BF) -1

52

(A-A'-F'B')'ty

Thus

is an element of (I,)$(K))±nD(A').

Now let x be an a r b i t r a r y element of X, then

lira A..t,~

= . ),.÷¢0

This proves the assertion.

ii)=>iil) 13*(PZ(K))=BI(VZ~O)

Since

we have from the well known Hahn-Banach mQ

theorem

the

matrix S ' =

existence

of

functionals

{gi}i= l



S=

such

that

the

is non singular. Since (VE(K)) nD(A') is weak * dense in m0

there

/.

3.

.L

0?~(K}~ ,

(VZ(K))

in

also

exisl;

..k

{fi}i=lc(Y,F,(K))nD(A')

functionals

is non singular. Taking linear combinations of

these

such

that

fi's

gives

the desired result. iii)=>i)

V u(K ) is frequency invariant and by lemma II.18, we obtain that every x ra 0

in Y ~ { K ) h a s

a (~,w) representation with

argument

in

as

representation. So

(3.61

the

proof

wi(s)

is given by (2.27)

"

of

= -S(s)

lemma

- 1

Bw(s)= ~ b,~J,(s). II.20

we

can

calculate

this

(~,w)

and

.

~w,,:(s)J

Using the same

ra 0

,(s)=($-A)'Ix-F (s-A)-I I ~ biw/($)]) i=1 where

Sifts)=



Let x be a element of V~(K), then there exists a sequence {xn} in Y~(R) which

converges

to

x. xn is

an

element

of

YECK) so

it

has

r=

representation with w " ( s ) =

wmo(s) rl

w.,(s)

, where m is the dimension of

a

ImB.

(~,w)

With

53

the above we may choose wm0+t(s)=,.. , =w~(s)~-0 and

"

as in (3.6) with

l W~o(S) n x

replaced

by

x n.

With

this choice

[ ~l(s)]

converges to x, then ~ ( s )

•

is the

right

it can

easily be

converges, and wi(s):=

hand

side

of

(3.6).

o-~o

seen

that

if x n

n

lira wi(s)--O if i > m o and

From

(3.6)

it

is

easily

seen

~omo(s)J that

w(s) is continuous

on an

interval [r,oo) and since for

= +

all x c V £ ( K ) , we have that

=

, < A'/,~o, x >

SO(s) = 64i

S-~O0

Define ~(s)=(s-A)-Xx-(s-A)-XBw(s); x=(s-A)~(s)-B~(s) and ~(s) is the limit

then of

~(s)

is

continuous,

~n(s)=(s-A)-lx,~-(s-A)-lB~o'~(s),

as n ÷ oo, thus ~(s) is in K. So x has a (~,w) representation, with f(.) in K(s), thus x is in Y,u(K). The

importance

results,

not

in

of its

case of A bounded

this

theorem

direct

lies in its usefulness

application

to

specific

in deriving

systems.

Even

El further for

the

it is not clear if the conditions are fulfilled. However

in the next lemma we show that if A is a bounded operator, then

YE(K)

is

closed. Lemma 111.8. If A is a bounded linear operator, then 1)"(K) exists and it is equal to

vz( l. Proof: Suppose first that the following holds: (3.7) Then

A[ Y~--~] with

Schmidt

and

lemma III.5 ))•(K)=V*(K),

c

])E(K)+ImB.

Stern [32], but

V£(K) is controlled

V*(K)cV£(K),

invariant.

So

with

so VE(K)=VE(K). Thus it remains

to prove (3.7). Let

x

be

an

element

of

I/,u(K), then

there

exists

~(.)~K(s)

and

54 ~(.)e//_l(s) such that

(3.8)

x= (s-A)~(s)-Bw(s)

With lemma II.14 we have that x = l i m

s((s). Rearranging equation (3.8) gives

As~(s) =s2((s)-sx-Bsw(s) From lemma ffL5.d) it follows that ~(s)~VI:(K ).

So As~(s)eVz(K)+ImBcV,u(K)+ImB.

Since A is a

bounded

operator

and

s~(s)-*x as s ÷ 0% we have that Ax~VI~(K)+Im B=VE(K)+ImB , because lmB is finite dimensional. Thus AV~(K) ¢])z(K) +Ira B. Using once again the fact that A is a bounded operator and V~(K)+ImB is a closed subspace we have proved that ,4 lenm~a.

VI~IK) [--1

c VE(K)+ImB and thus this D

The next lemma will give sufficient conditions for theorem I]].6, which are easy

to

verify.

These

conditions

first

occurred

in

Curtain [6],

where

the

aim was to give sufficient conditions for the existence of V*(K). Lemma III.9. Let KerD be the kernel of a bounded linear operator D. Then Vz(KerD) is closed if either of the following conditions holds. a)

There exists a q in Nu{0} such that DA~ has a bounded extension defined on the whole of ?d (denoted by DA ~) for 0 x ~ 0. Then < d,A. > x does not have a bounded extension from Pg to C, but

furthermore that the

multiplicity

of

the

zeros

in

plus

infinity

of

the

transfer

function

is one. Using the characterization of III.6 we derive

the

following

l~E(tierD) given in lemma III.5 and theorem

equivalent

statements

for

the

solvability of

DDP. Theorem III.10.

Assume

that

YE(KerO) is closed,

then

the

following

statements

are

equivalent. a)

DDP is solvable.

b)

lm Ec l),u(Ker D).

c)

For every

q in Q

there

exists a w(.) in U_a(s ) (see definition II.12),

such that (3.13) d)

D(s- A)'tEq = - D(s- A)-IBw{s)

There exists a U ( s ) i n

(3.14)

I£,(Q,bl)]_~s)such that

D(s-A)-IBU(s) = D(s-A)-tE

57 e)

There exists a U I s ) i n I/:(Q,//)] (s} and X ( s ) i n ~ I I / : ( Q ' X ) ] - I ( s ) s u c h that

13.15/

-D

0

.

-U(s)J

0

Proof: This

follows

trivially

from

theorems

III.3

and

]]].6,

and

definitions

III.4 and I1.13.

E1

Remark:

Theorem ITI.10 is the same as theorem 3.4 and 3.6 in Hautus [19] for the finite dimensional case. We see from this theorem that under an extra condition on the system the

solvability of DDP is equivalent

matrix

equation.

One

can

easily

to

show

the

solvability of

that

the

solvability

a

meromorphic of

(3.14)

is

always a necessary condition for the solvability of DDP. However it can be seen from example E.10 in appendix E that

it is not sufficient; of course

in this case P~(Ker D) will not be closed.

Section III.2: DDP in T i m e - D o m a i n In

this

section

we

shall

state

similar

results

as

in

the

previous

section, however now we shall work in the time-domain. Analogous to r E ( K ) we can define the largest open loop invaxiant subspace.

D e f i n i t i o n III.11: Let that

Pol(K)

there

Vol(K )

be the subset of X which contains all

exists a continuous

input u(.)

such that

xoeX

with the property

the mild solution x(.}

t

of (2.1) i.e.

x(t)=TA(t)xo+fTA(t-s)Bu(s)ds

is in K for all t > 0 .

0

Remark:

If K is the kernel of a bounded operator D~ then this subspace contains

z(.)=O , ~(t)=Ax(t)+Bu(t); x(O)=zo; z(t)=Dx(t).

all initial values x 0 such that for some continuous u(.) the output where z is the output of the system

58

For this subspace we shall give similar results as for lZE(K ) in section ]:[I.1. Since

the

proofs

of

these

results

are

very

similar

we

shall

omit

them. Theorem III.12. Let Poz(K) be the subspace defined by ITI.11, then it has the following properties. a)

V~(K) is the largest open loop invariant subspace in K.

b)

If Vo~{K) is closed, then Y*(K) exists and is equal to Pot(K).

c}

Let

B°(12oz(K)) denote

intersection V~(K)

is

the

with Vo~(K) and closed

if

and

largest

let

only

subspace

of

[roB

{bl,..,b~,0} be a basis if

there

exist

that

has

zero

of /~°(Vol(K)). Then

functionais

{fi}

m0

in

D(A')

i =l

such that < f~, bi > = ~ii and fi[ d)

Let

BI(V)

denote

v~(K)

-- 0.

ImBnV.

Then

(Vo,(g)) ± nD(A') is weak * dense in e)

If A is

a

bounded

operator,

)J~(K)

is

dosed

if

and

only

if

(Vo,(K))x and B I (Yo,(K))=]3'(P--~ then

V*(K) exists and it is equal

to

Yo~(K). Remark: As in the frequency domain it is possible that ~2*{K) exists, but VodK) is not closed, see example E.12. Remark:

As an easy corollary of theorem III.12.d) one has the following result. If V is open loop invariant,

then 17 is controlled

invariant

provided

that

VXnD(A') is weak * dense in Vl and BI{V)=131(17). Lemma III.13. Let KerD be the kernel of a bounded linear operator D. Then V~(KerD) is closed if either of the following conditions holds. a)

There exists a q in Nu{0} such that DAi has a bounded extension from ,k' to Z (denoted by DA ~) for 0 < i < q + l , DAiB=O for O_l be a nest of closed, linear and controlled invariant

a)

subspaces.

Then

V: = 13 V n

is

controlled

invariant

if

there

exist

n_>l

functionals {fi}7°1 ha D(A') such that / j [ v = 0

and

=6#,

where bi is

a basis for B°( O V,). n>l

Consider the same nest Vn. Let B be one dimensional and suppose

b)

there exists a closed subspace K such that VncK and the controllability subspace,

, is not contained in K. Then V:= U v,, is controlled

invariant

if

n_>l

and

only

if there

exists

a

functional f

in D(A') such

that

f l y = 0 and = l . Proof: a)

Let V be U Vn, then V, is closed loop invariant and contained in V for n~'t

all n, nest,

thus by iemma 1TI.5.b) Vn is contained we

have

U vncyE(v). n>_l

By

definition

in YE(V). of

Since {Vn} is a

FE(V) we

have

that

61

V£(V)cV = U v. , U V,,CVF(V)cV. Thus V~:(V) = V. n>l

n>l

Let {bl,..bml } be

a basis for

B°(PE(V))

then

since

U V,, is contained n>l

in I)E(V ) we can

extend

this basis to

a

basis of

B°(UV,).

Denote this

n_>l

basis

by

{bw.bml,..,bmo }.

functionals {/~}¢D(A')

By

assumption

we

have

=6~j and lily=

such that

that

f,[

there v ~c( v)

=0.

exist From

lemma IlI.7.iii) we have that VE(V ) is a closed subspace, and thus ~)~(V)=V is controlled invariant.

[]

b) (if): see a) (only if): Suppose V is controlled invariant TA(t)-invaxiant.

Furthermore is

the

since

smallest

K

is

closed

Ta(t)-invaxiant

b~V, then V is also

and

we

subspace

have

that

containing

VcK. Im B

=

span{b}. So c g ¢ K. So 13°(V)=span{b}, and III.7 gives the desired result

O

In this chapter we have analyzed the problem of the existence of the largest controlled invariant

subspace. As we have

not

when

necessarily

largest

open

difficult to

exist

loop

or

and the

verify when

it

largest

does

exist

frequency

it exists. In the

seen, it

is

invariant

this subspace does not

necessarily

subspace

following chapter

and

the it

is

we consider

a

special class of systems for which we can give readily verifiable necessary and

sufficient

conditions

invariant subspace.

for

the

existence

of

the

largest

controlled

CHAPTER IV: CONTROLLED INVARIANCE FOR DISCRETE SPECTRAL SYSTEMS

In

this

chapter

we shall

consider

~gain

the

following

in a

general

linear

controlled

system described by the set of equations

Ax(t) +Bu(t) z(t)=Dx(t)

(4.1a)

x(t) =

(4.1b)

but

instead

assume the

of

that

input

considering

the state

space

chapter

A is a

operator

from

In

U is assumed

7/ to

chapter

chapter

2

ill

tile

of

this

the we

l]anach

one

space Z

introduced

shall

of

be

space

Hilbert space

dimensional.

and

As

and

since

the

we shall

furthermore

in

the

previous

bounded

linear

space

is one

input

Bu=bu with b ~ .

of

kernel

to

Banach

of a C0-semigrou p on H, D is a

the

we

we

characterization

zeros of

space ~/ is a separable

generator

dimensional we h a v e

this

this system

derive

all D.

derive

the a

controlled This

transfer

for

concept class

invariant

characterization

function

necessary

of

of

controlled

spectral subspaces

is

given

invariance.

systems of in

a

(4.1)

the

In

complete contained

terms

of

the

D(s-A)-1B of system (4.1). As a consequence and

sufficient

conditions

for

the

existence

of

/ ) ' ( K e r D), the largest controlled invariant subspace in the kernel of D That

there

subspaces

in

illustrated What

is

by the

exists

a

close

relationship

between

controlled

invariant

KerD and zeros is well known in finite dimensions and can be the form

following of

all

problem.

one

We

dimensional

can

ask

a

controlled

very

simple

invariant

question.

subspaces

in

the kernel o f D? l,et span{v}

be

such

a subspace,

A+BF invariant,

also lenmaa

1.3.a).

Thus

for v

is

some an

then

since

feedbzLck

eigenvector

it

law of

is controlled F,

see

A+BF.

invariant

del'iuitioa We

shall

it

is

11.2~ and distinguish

between two cases olle;

FV = 0

In this case v is an e i g e n v e c t o r of A and in the kernel of D. two:

Fv ~ 0

Fv=l. So that (A+BF)v=c~'u implies (c~l-A)v=b. If we assume that (~ is in p(A), then v=(c~l-A)-Ib. Since v is in the kernel of D we obtain D(cd-A)'tb=O. Thus c~ is a zero of the Without loss of generality we may assume that

63 transfer

function.

So

we

see

that

a

one-dimensional

controlled

invariant

subspace is either an eigenvector of A in the kernel of D or it is span{v}

Dv=O and (cd-A)v=b for some ¢~ in C. If this

where v satisfies the equations c~ is an

element

of

the

resolvent

set

of

A,

then

it

is a

zero

of

the

transfer function. For

finite

understood,

dimensional

see

e.g.

systems

Davison

&

the

concept

Wang [15].

In

of

zeros

infinite

is

very

dimensions

well

however

only a few articles have been published, see Pohjolalnen [30].

P*(KerD) is based on the result on pole

The proof of the existence of placement

from

Sun [37].

To

get

an

between the existence of Y*(KerD)

idea that

there

exists

a

relationship

and the .problem of pole placement we I

refer and then

to

the

113])

finite

that

dimensional

if the

a(A+BF[ , In

)

this

It

is well-known

single input system is

V (KerD)

V-(KerD).

case.

fixed

paper

we

for

all

shall

{4.1) F

prove

(Wonham [42, p.112

is controllable

D#0,

(A+BF)Y*(KerD)c

satisfying a

and

similar

result

for

spectral

systems, see theorem IV.5. Otherwise we have from the results of Sun [37] a

a(A+BF), for some F. Hence especially on a(A+BFIv.(K~.o)) , and this will give a condition for the existence of l)*(KerD).

restriction on

all possible sets

the fixed part of

The organization of this chapter wiU be as follows. In

section

spectral

IV.1

we

operators

and

characterization of all Properties of

shah

recall

some

facts

this

class

of

for

and

properties

operators

we

of

shall

discrete derive

a

given

in

presented.

In

TA(t ) invariant subspaces.

the zeros of

the class of spectral systems wiU be

section IV.2. In section this

IV.3

section

subspaces

the

we

main

shall

in the

theorems

give

kernel

a

of

full

of

these theorems

chapter

description

/9. In particular

sufficient conditions for the existence of Application of

this

will be

of

will be all

controlled

we shall give

invariant

necessary

and

I}*(Ker D). given in section IV.4. The

examples

in this section were partly calculated by L. Nooitgedagt [26]. We restricted and

remark our

here

that

attention

initially observable,

in to

order systems

(see Curtain

to

improve

which

are

the

readability

approximately

& Pritchaxd [9, p.60

more general theory we refer the reader to Zwart [47].

and

we

have

controllable 69])

for

a

64 S e c t i o n IV.l: D i s c r e t e S p e c t r a l O p e r a t o r s

In this section we shall give the definition and some properties of discrete spectral operators.

For more detail about these operators we refer

the reader to Dunford & Schwartz [18]. D e f i n i t i o n IV.l: D i s c r e t e O p e r a t o r A linear o p e r a t o r A from H to H is discrete if there exists a number A in its resolvent set for which the resolvent R(2;AI:

=(AI-A) -1

is compact.

Lemma IV.2. If A is discrete, then a)

its spectrum,

a(A),

is a denumerable set of points with no finite limit

point; b)

The resolvent R(A,A) is compact for every A not in

c)

Every A0 in

e(A)

a(A).

is a pole of finite order 0(20) of the resolvent and

if, for some positive integer k, x satisfies the equation

(A-AoI)kx = 0 then x satisfies the equation

(A-AoI}O(Ao)x=O The set of all vectors x satisfying the equation

(A-AoI)O(A°)x=O is &

finite

space

dimensional

linear

space,

called

the

of

generalized

eigenvectors of A corresponding to the eigenvalue A0; d)

If

¢4.21

P¢201=

rI21- Al-ld2

where F is a small closed curve surrounding only the eigenvalue 20 and P is traversed

once in the positive sense, then P(20)

projects 7f onto

the space of generalized eigenvectors corresponding to 20. Proof: See Dunford & Schwartz [18, lemma XIX 2.2.].

t::l

65 Remark: The spectrum of A shall be denoted by {An} n > 1. Definition IV.3. Discrete Spectral Operator. A

discrete

operator

is

spectral

if

the

spectral

projections

P(),j)

defined by (4.2) satisfy a)

The

family

of

sums

of

finite

collections

of

b)

No non zero x in ~ satisfies all of the equations P(Ai)x=0, Aj in

P(Aj)

projections

is

uniformly bounded and

a(A).

Remark: The spectral projections P(Aj) are not necessarily selfadjoint. Lemma IV.4. If

A

is

{P(Aj), Aj in

a

discrete

a(A)

spectral

operator,

then

the

spectral

projections

} generate an uniformly bounded Boolean algebra with the

completeness property:

~ P(Aj)=I j=l

(4.3)

where the convergence is in the strong topology. Proof: See Dunford 8, Schwartz [18, XVIII.I.].

[]

If a subspace V of I~n is invariant with respect to a diagonal matrix, then V must be of the form span{ci,

ieJc{1..n)

vector

discrete

of

Rn.

For

the

class

of

} where ei is the i'th basis

spectral

operators

a

similar

theorem holds. First we shall recall a lemma of Dunford and Schwartz [18] that gives some invariance properties of the spectral projections, P(A~). Lemma IV.5.

a(A). AP(Aj)x=P(Ai)Ax

Let A be a discrete spectral operator and Aj an element of Then all x

D(A)~P(Aj)H, the subspace P(Aj)H in D(A) and a(AIP(Aj)H)={Aj}.

is A-invariant,

for

Proof: See Dunford & Schwartz [18, p.2294].

El

66 With these lemmas we can now give a complete characterization

of all

TA(t)-invariant subspaces for the class of discrete spectral generators. T h e o r e m IV.6. Let

tile

discrete

spectral

operator

A generate

the

C0-semigrou p

TA(t),

then a closed linear subspace V of 2-/ is TA(t)-invariant if and only if OO

V= ~ |Vi 4=1

(4.4) where

Wi

is

a

subspace

of

H

which

is

contained

in

P(Ai)H

and

is

A-invariant. 'File summation (4.4)

is in the strong topology,

i.e. for all x e l '

th(u'c exist

{wi;ieN, with wieWi} such that ~ wi convergcs to x for n -~ co. Oil th(' other i-----l

hand, if {wi;ieN, with wieWi} is such that ~ wi converges to x for n ÷ co, i=l

then the limit is in V. Furthermore

the spectrum of A restricted

to V is equal

to

the

set of

all A i e a ( A } such that the corresponding Wi is not the zero subspace. Proof; (if): Since the be

dimension of P(Ai)?/ is finite, the

finite. So IVi is a

contained

in

D(A),

closcd

and

with

linear subspace lemma

1.7

dimension of Wi must also

of

we

7/. Furthermore may

conclude

P(Ai)H is

that

Wi

is

TA(t)-invariant. Every x in V is the limit of a sequence xn, with xne ~ Wv St) with the i=l

above we have Ta(t)x n is in ~ Wi. TA(t ) is a bounded linear o p e r a t o r thus i=1 TA(t)x n converges to an element in R, but also to a n element of V since

TA(t)x n is in V. So V is Ta(t)-invariant. ( o n l y if): Let V be a TA(t)-invariant subspace. Since a(A)={Ai} , i e N we have that the

resolvent

set,

p(A),

is connected.

With lemma

1.4 this

iml)lies that

V

is also ( A I - A ) -t invariant for all A in the resolvent set of A. So

P(Ai)V=~----~[ (AI-A)-IV dAcV; see (4.2) 2 So

F

P(Ai)Vc(P(AI)~I)nV. Using the fact that

P(Ai)

is

a

projection

we

get

67

(P(Ai)H) n V = (P(Ai)H)n (P(Ai)V) c P(Ai)V. Thus P(Ai)V= (P(Ai)7/)nV. If we set Wi equal to P(Ai)V and xi:

xEV,

=P(Ai)x;

then A(Wi) = AP(Ai)V

= AP(Ai)P(Ai)V = AP(Ai)(P(Ai)~lnV} c AP(Ai){VnD(A)}=P(Ai)A(VnD(A)} AWicW i.

c P ( A i ) V = W fi see lemma IV.5. So By (4.3) ~ xi. ~=~

and definition IV.3 every

x in H can

be uniquely written

as

[ Wi" iF'~

Hence V=

We shall now prove the last assertion. Let d denote than

zero.

Ai~a(A )

the index set of all

WicP(Ai)l~

Since

is

finite

such that dim(W/) is larger

dimensional and

A-invariant

it

i~J } c a ( A ] v ). (M-A)-I]V is a bounded

must

contain an eigenvector corresponding to ~i, thus {Ai; From lemma 1.4 we have that for all ,~ep®,

linear

operator from V to V and since A is discrete it, is also a compact operator. Furthermore

it

is the

(M-AIr).

inverse of

A]v

So

is a

discrete

operator

Air is a pure point spectrum, l.'urthermore we have from corollary 1.10 that a(A]v)ca(A ). Let Ao he an element from a(A]v), then there exists a v in V such that A]v(v)=)~oV. From the above we have that Wo=P(Ao)V. We shall show that Wo=P{Ao)V is non zero, but this is obvious since P(Ao)v=v. So it is shown that { Ai; i~d }=a(AIv). rn and hence the spectrum of

With aheady

this

theorenl

known

but

we

our

can

prove

proof

some

is much

interesting

si,npler,

see

corollaries

Curtain

that

are

and

Pritchard

with

{(p,} an

[9, p. 6l]. OO I

Let

A be

of

the

following

A= ) i AzHI 2

exists,

then

A is

a

i=1

discrete

spectral

operator

with

spectral

projections

P(-'~i)= < ., g~i> ~i.

l'urthermore if s u p { A i l i ~ N }O.

C o r o l l a r y IV.7. If

A

and

D

satisfy

the

properties

as

stated

above,

then

V0

is

span(¢iID(~i)--0 }. So V0={0 } if and only if D ( ~ i ) ~ 0 for all i in t~. Proof: From Curtain [6] we obtain that V0 is the largest subspace of 7/ that is semigroup invariant and

in the

kernel of

special structure of A we have that

D. From theorem

IV.6 and

Vo=s~(d)i[ieJcN;D(¢i)=O }.

the

Combining

these results for V0 we conclude this corollary.

[]

If the nonobservable subspace equals the zero set, then the pair is

initially

observable (A,B)

observable, is

the

see

concept

is approximately

[9, p.69]. of

Dual

to

approximately

controllable if

the

the

concept

controllable,

set

of

all states

reached from zero with an arbitrary input is dense in H, or

of

i.e.

(D,A)

initially

the

system

that

can

by

{xEH[3t>_O,3u(.)

t

s.t x=lTA(t-s)Bu(s)ds

} is a

dense subset

of

~/. The

system

(A,B)

is

0

approximately

controllable

if

and

only

if

the

system

(B*,A*)

is

illitially

observable. For our special system we now have the following corollary.

C o r o l l a r y IV.8. If A satisfies element of only if

the

7~, then

#0

same

the

properties

system

(A,b)

as

in corollary

IV.7 and

b

is an

is approximately controllable if and

for all i in t~.

Proof: This is the dual of corollaxy IV.7.

As in theorem IV.6 we can pose ttle question what tile Ta+Br(t)-invariant subspaces look like. This question is in general not solvable even if A is a

discrete

operator.

spectral

operator

Furthermore

we

since are

A+BF not

need

interested

not

be

in

all

a

discrete

spectral

TA+BF(t)-invariant

subspaces but only in those which are in the kernel of D. Before we can give

a

complete

description

of

all

Tn+nf(t)-invariant

investigate the notion of the zeros of a transfer function.

subspace

we

must

69 Section IV.2: Z e r o s and I n v a r t a n c e

In this section we shall discuss the relation system (4.1) and its controlled invaxiant Although and

we

shall

approximately

only

consider

controllable

we

between the

subspaces in systems

shall

of the

the kernel of D.

that

define

zeros

axe

zeros

initially in

a

observable

more

general

setting.

Definition IV.9: Zero An element #

of C is called a zero for the system (4.1)

if there exist

nonzero xeT/ and a u e U such that

--0.0 Remark: This definition can also be found in Davison and Wang [15].

Remark: If

/.L~p(A)~ then

# is a zero if and only if

D(pI-A)-lB=O.

[,emma IV.10. Let V be a closed subspace the kernel of D. Assume further

A+BFIv ,

then X is a zero for the

TA+BF(t)-invaziant and that Aeap(A+BF]v), the point system (D,A,B). that

is

contained

in

spectrum of

Proof: By assumption t h e r e exists an x in V such that

u=-Fx,

then

=

D system

0

(A+BF)x=Ax.

=

Now define

. So ~ is a zero for the

Dx

(D,A,B).

[]

Remark: As can be seen from the proof we have not used the special structure of system

(4.1).

So

the

result

remains

true

if

our

state

space

is

a

general

Banach space.

In the sequel of additional

assumptions

this section we shall consider system and

discuss

the

concept

of

zeros

(4.1) for

with some this

special

7O kind of system. The additional ~sumptions we make on the system (4.1) are: (&l)

The

generator

A

is

a

discrete

spectral

operator

with

spectrM

decomposition A = ~ ;~P(A~) iml

where Ai#A j (for all i # j ) generality

we

may

and dimP(Ai)=l (i>__1). Without loss of

assume

that

the

P(Ai)

(i_>Á) are

selfadjoint

operators in ~/, see Wermer [40]. The normalized eigenvector of A corresponding to P(Ai) will be denoted by ¢i ( i > l ) ,

(zx2)

b ~ : = < b , O i > H ~ 0 , for all i>_l.

(A3)

For all i eN, D4~ii~ O. Let us remark

that

(zX2) is the controllability assumption and

(A3) is

the observabflity assumption, see corollary 13/.7 and IV.8. Lemma IV.11. Assume that

the system (4.1) satisfies conditions (A1),

If p is a zero of the system (4.1), then

(ZX2) and

(A3).

pep(A) and D(p-A)-IB=O.

Proof: Let # be a zero, then there exists a

xeb(A), x#O and uell such that

'1[:] 00 = If u were to be zero, then x would be an eigenvector of ,4 in the kernel of D. However this would imply by assumption contradiction. So uia0. Assume that

(ZX3) that

x=O, providing the

pea(A), then O=P(#)O=P(I.~)((p-A)x+bu}

P(#)(~-A)x+P(p)bu = (#-A)Pl#)x+P(#)bu = O+P(p)bu. Since u # 0 this implies that P ( p ) b = 0 , but this is not possible by assumption (A2). So pep(A) and the remark below definition IV.9 gives the desired result.

C]

With this lemma we can define controlled invariant subspaces associated with a zero of the system (4.1).

71 k Definition IV.12: Z~ Let the system (4.1) satisfy assumptions (A1), (A2) and (A3). For

a

zero

#

of

this

system we shall

define

the

following nest

of

subspaces, /1:

-n

Z~=span{(p-A) b; l _

b

we

]

.. b~

achieved for respectively A.

index such that

b-

where .~j is the subsequence of

single

terms

for

in

_>

sum.

j=30,45,60,75,90

n=89,133,177,222 and .

this

By

j • {1,.., 100}/{30, 45, 60, 75, 90}

Vn~gl and for

- #j

is determined by

bnj

A~ as in theorem IV.16. We shaft first investigate calculations

#i>11i+1. ))*(NarD)

A m i . i - #j

266. Let

Vn~N.

simple that

this minimum is

mini

We then

denote the have

that

btainj _~0 A n j=l b,j

is larger

than

for

j=l

every

subsequence

bml"i

{hi}. In the next table we have listed the partial sums, Sl(m): = ~

[ A ~ , j - #j

2

In figure 4.1 the numbers of table 4.1 are plotted and we see that in this example we have (numerical) evidence that

Y*(KcrDl)

does not exist.

81

Table 4.1

S~(m)

m

.91(m)

m

1

3. 8211e÷003

20

2.8479e÷009

2

7. 5736e+004

30

1.8650e+010

3

4. 5039e+005

40

8.0862e+010

4

1. 6463e+006

50

2.3042e+011

5

4. 5795e+006

60

5.8681e+011

6

1. 0677e+007

70

1.2725e+012

7

2.1990e+007

80

2.3695e+012

8

4. 1308e÷007

90

4.1805e+012

9

7. 2273e+007

i00

7.2350e+012

10

1. 1949e+008

xlOZ2

Partial sums for GI(S)

A I

v

I

I

I

10

20

30

J 40

50

60

70

80

90

,00

m ->

Figure 4.1 [] For our second example we shall take two measurement functions d 2 and d3 which are relatively close to each other with ~ 0 and ~ 0 , but the

82 the first is an element of D(A') and the second is not. Example IV.22. In this second example we shall investigate the existence of ~)'(KerD) for two measurement

functions that

we shall see

sequel,

in

the

are close in the L2(0,1)-norm,

P*(KerD)

only exists for

one

of

but,

them.

as The

measurements functions we shall consider are D2 and D3, where D,= < . , d , > with (4.21)

d2(t/) = l[0,t/~l(r/)

and

=f

-1007/2 + 2Or/ 1

(4.22)

da(r])

;

0, and thus

X E ~oorth"

[]

Remark: With a similar proof one can show that g is open loop invariant for the system

(A,B) and Sorth is open loop invariant for the system (A',C'}.

Remark: The

projection

of

a

dosed

linear

subspace

is

not

necessarily

closed

subspace. This is easy to see from the next example. Let A:H~-,7/ be a

bo~mded linear operator

range is unequal to 7/. Then the graph of A; ~e~,

with dense range,

and

this

Ve:={(x, Ax)[xe~}, is closed in

but the projection of this graph on the second coordinate gives the

range of A, which was by assumption not closed.

Remark: If Soreh and V axe lemma V.3.

The finite

both closed subspaces,

then

by

theorem

II.27 and

(S,V) is a (C~A,H}-pair.

next

lemma will show

dimensional

i.e.

14/ is

that a

if the

finite

feedback

processor

dimensional space~

then

in (5.7} S

defined by (5.10) and (5.11) are closed subspaces and so in this case is a

and

is V

(S,V)

(C~A,B)-pair.

Lemma V.7. Suppose

that

Vt

is a

closed Ta~(t)-invariant subspace and

Iq is finite

(S,V) of subspaces defined by (5.10) and (5.11} (C,A,B)-pair and dim(VnS')

=

<x,(s-A)-XEq + (s-A)-'BX(s)C(s-A)-aEq> =0, since x e V ~. Since q was an arbitrary element of Q, we have that E*~(s)=O. So summarizing we have the existence of a pair of subspace, S~-~h and V, such that V is frequency invariaat for the system (A,B) and VcKerD, So~h is frequency invariant for the system (A*,C*) and S~thcKerE* , S'~orthc V Combining these results give that

S* (lm £) = S~orthc V c l)* (Ker D)

Now we shall investigate under which conditions we obtain the existence of a finite dimensional compensator. T h e o r e m V.12. The following assertions are equivalent i)

DDPM is solvable with a finite dimensional compensator

ii)

There exists a (C,A,B)-pair (S,V) with dim(VnS~)-6} TA(t)-invariant

system

(A,B)

Ha and

~/u such

for

subspace

C_6,_:={seC[Res0.

have

So ~ = { 0 }

is Since

from and

that

dim(?'/u)_I and 6>0. ~ in ~Q. The result of The

next

ii)

result

is

finite

dunensional.

such that H=~u$~a and

So

there

[[TA(t)[~_ls][<Me-6t

is H~ plus all stable (generalized) eigenvectors

follows now easily from theorem VI.6. will show that

a

similar

decomposition

as

in

6)

a

closed

of

theorem VI.3 holds for all semigroup invariant subspaces. Lemma YI.5. Suppose

that

(A,B)

is

stabilizable.

Then

if

V

is

Ta(t)-invariant subspace, then V can be decomposed in

V=Vu~V, where V~ and V~ are both TA(~)-invariant, dim(Vu)0 such that

IIT,t÷nr(t)lvll_0

Remark: In stable.

Bazile,

Marro

and

Piazzi [2]

this

notion

Combining the equivalence between open closed loop invariance with theorem VI.3. gives

loop,

was

called

frequency

internally

domain

and

following assertions

are

Theorem VI.IO. Let V be a closed subspace equivalent. i)

V

is

a

ii)

For every

stabilizability

of 7/. Then

the

subspace.

x 0 in V there

exists

a

u(t)~C([O, co);Rm]nL~([O,oo);Rm}

112

such that t

x(t) =TA(t)xo+f

(6.6)

Ta(t-s)Bu(s)ds

o

is contained in V and x(.)eL2([O,~);V). For every x o e V there exists a ((.)~tI2(V) and w(.)e//z(N m) such that

iiil

lira swls) exists, ((s)~VnD(A) and

(6.7)

seE+: = {se(:l/~e(s ) >0}.

x o = (s-A)~(s)-Bw(s);

Proof:

i)~ii)From definition VI.9 we have the existence of a bounded feedback law F such

that. the

closed

loop

system

satisfies

Ta+Br(t ) V c V

and

A+/3FIv

is

stable. From Curtain & Pritchard theorem 2.31 we have that t

(6.8)

TA +BF(t )Xo = Tx(t )Xo + ] TA(t - s)BFT A +13~'(s)xods 0

Defining x(t)=Ta+tjt.(t)x o and u ( t ) = F l ' a + ~ . ( t ) x o gives the desired result.

Taking the Laplace transform of (6.6) gives (6.9)

~ ( S ) ----(8 - A ) - l x o

+ (.5 - A)-IB~(s)

where ~(s) and w(s) are respectively the Laplace transfornl of x(.) and of u(.).

Since

x(.)

and

u(.)

are

square

integrable,

~(.)

and

w(.)

are

//2

functions. By the definition of the Laplace transform we have that ~ ( s ) e V and equation (6.9) gives that ~(s)eD(A). Rewriting equation (6.9) gives

xo= (s-A)~(s)-Bo4s)

(6.10)

and since ~(.) and w(.) are H2 functions we have that (6.10) holds on C+. The

strictly

properness

of

continuous, Doetsch [16, p.226].

w(.)

follows

from

the

fact

that

u(.)

is

113

iii)~i): From theorem 1I.27 we have the existence of a bounded feedback law Fl such that

TA÷BFI(t)VcV.

So we have the invariance, but no further stability

properties yet. From equation (6.7) we have that

Xo=(s-A-BF,)~(s)-B{w(s)-Fl~(s)}.

(6.11)

TA÷BFI(t) we

By the invariance of V under

B(ta(s)-FI~(s)}~V.

Consider now the system there

VI.3.iv)

exists

with

a

(s-A-BF1)(VtaD(A))cV.

Let 121 denote the subset of Rm such that

{(A+BFI))B }

(6.11)

bounded

So

B(Ui)=Pv ImB.

{w(s)-Fl~(s)}eU 1.

Then by definition of 121 we have that

Theorem

have that

with state space V and input space 121.

implies that

feedback

law

this

system

F~:V~/21

such

is stabflizable. Thus that

(A+BFI+BF2)

generates a stable semigroup on V. Now we have to construct a feedback law on the

whole state

space such that

V is a stabilizability subspace.

Define

Fas

F[v=FI+F 2 and FIv, =Ft (BF-BF1)x=BFzPvxeV, since F2(V)cUt. So V is TA+BF(t) invariant, furthermore BF(V)= (BFI+BF2~(V). So with lemma V.1 we have that TA+Bf(t)Iv=TA+BFx+BF2(t)Iv and so V is stabilizabflity subspace. []

Then

The

next

stabilizability under

rather

technical

subspaces

in

lemma

relation

will with

show

the

maintaining

finite dimensional extensions. We shall need this for

usefulness

of

stabilizability the

disturbance

decoupling problem. [,emma VI.11. Let

(A,B)

be

stabllizable, and let V be a stabilizability subspace. For

a space )¢ with dim(lq)0 t

(6.18)

0

and Tae(t ) is stable, i.e. there exists M, 6>0, such that

the

output

117 (6.19)

IITA~Ct)II-<Me -6'

where Dt=(D,O), Et= [0 E] and TAC(t) is the semigroup generated by the closed loop system operator A * (6.5). Pictorially our

configuration is given in figure 6.1.

~ ( t ) = Ax( t) + Bu( t) + Eqlt)

q(t}

, z(t)

y(t) = Cx(t)

-u(t). ,

z(t)

y(t)

= Dx( t )

6(t ) = N ~ ( t ) + My(t} u(t)

= L ~ ( t ) + Ky(t)

figure 6.1 Now we can state finite

dimensional

our main result which is completely similar to

case.

For

finite

dimensional

systems

the

the

condition

dim(V/S)<eo is trivially satisfied, and thus is omited there. T h e o r e m VI.15. DDPMS is solvable with a finite dimensional compensator if and only if i)

(A,B) is stabilizable

ii) iii)

There exists a stable (C,A,B)-pair, (S,V) such that dim(V/S) Iz 0

independent of

i. So A5 is also

satisfied

Remark: The properties A1, A 4 and /x5 are properties of the system matrix. So if b and d is another pair such that f(s)ffi and b satisfies /x2 and d satisfies A3, then this realisation satisfies also A1 up to AS.

We shall illustrate this theorem by the following easy example.

Example

E.8.

Consider the transfer function f ( s ) =

1

. Then by theorem E.7

4 s ( - 4 s + e -4~' )

we have that the spectral realisation of f is given by

AX=~ Zn<X,en>en; n=l

O(A)={xe~21~ {z.[2t <x,e.> I~i are the zeros of ( - 4 s + e - 4 s ) ; Z l = 0 and

b = a = {1/2,1/(t6~=(

Furthermore this realisation satisfies A1 up to theorem

IV.16

a

~,.. ).

-: - 4z=)) ~,.., 1/(:6z#( - :- 4zj))

complete

characterization

A5, of

and so we have from all

controlled

invariant

subspaces in the kernel of D. Since the transfer function has no zeros we have

from

theorem IV.16 that

the

unique controlled invariant subspace

in

the kernel of D is the zero subspace. So we have for this realisatioa that

P*(KerD)

exists and it is equal to the zero subspace. Note that we do not

state anything about the more usual realisation in the space h r . In

the

next

section

equations in order

the

state

space

to

we

shall

use

spectral

obtain counter examples for

is finite dimensional

infinite dimensional case.

these

but

do

not

f~

realisations properties

alway

of

that

hold

delay hold if

for

the

133

Section E.2: The Relation between g (K), Vol(K ) and V2~(K ) The next example will show that V'(K) need not exist. Example E.9. Let ~/ be L2(0,1) and A is the "heat operator",

A = d2 with domain, dz 2

D(A) = {x~L2(O,1)lx"~L2(O,1); x(O)=x(1)=O},

8:=b(z)= [

o

; ze[0,½]

sin(27r z); ze[½,1] K = {xeL2(O,1}l x(z)=O; a.e. or~ [0,½]}. Notice that K ~ I m B . So B°(K)={0}, and from lemma II.19 we may conclude that, if K is controlled (or equivalently frequency} invariant, then every x in K has

an

unique

(~,w)

representation

with

Bw(s)--0,

which

means

that

(s-A)-IK c K. Thus K would be TA(t)-invaxiant , but this is in contradiction with example 1.6. So K cannot be controlled invariant. Define for n > 1 0 en( z ) = .

1

{sin(21rn z) + n( - 1)nsin(2zr z)}; z ~ [~, 1 ]

4ff2n(-l)n(n2- 1)

and /~. in C by /zn =-47r2n 2. 1)

Then

span{e~} is controlled invariant for all n a n i=2..n

2) span {ci} is K.

i.~/{l}

P r o o f o f D" By a simple calculation one can 8how that ( A - p ~ ) e , = b . So

(c.6) Thus

(s-A)e.f-b+(~-~.)~. span{e,}

is

controlled

o,.

=.=(~-A)se~.

invariant.

By

-

lemma

span{ei} is controlled invariant for all h e N . i~2. ,n

P r o o f o f 2): 0

on [0, I

If x a K , then x =

, with £~L~(,1). So on [~, 1]

b

i

pn-S

HI.14

we

have

that

134

~ 2 t 2sin(27rnz) and .=x L (~,t)

~(z)=

II~ll 2

~

.

[<x(z),2sln(21rnz)>

=

Let x . l . e , V n > l ~

<x,e,>

2t

L (;,1)

2 I

2

L (~,1)

[ < oo

=0

< ~(z),sin(21rn z) +n( - 1)'~sin(27r z) > L2t! 1~ = 0. So ~2,*/

(e.7)

L2({,1)=-n(-l)nL2(~,l)

(e.8)

~ 2> oo> Ilxll

,1)]2= ~ n2lL2(~ ,1)[2 IL2(~

~ n=2

(e.8)

implies

that

V n>l

n=2

2 1

L (~,1)

=0,

and

(e.7)

with

we

have

that

L2(~,l)=O , for all u in N/{0}, so ~ = 0 . Thus x -

0 is the only vector

in K perpendicular oil all en, so

span {e~}

is K. If I)*(K) were

to exist,

then it would necessarily

in K and by 1) it must contain

be closed,

span {e~}, for all h e N . i=2..

contained

This together with

n

2) would imply that 1)*(K) equals K. This contradicts the fact that K is not controlled invariant. Thus Y*(K) c a n n o t exist in this example.

"l'he next example will show that unequal

to

IIu(K ). Note

that

it is possible that

theorem

H

I;*(K) exists

III.12 implies that

but

it is

VE(li ) cam lot

he

closed then.

Example E.10. In

this

example

we

shall

study

the

delay

transfer

function

as

introduced in example E.8. So

(e.9)

f(s) =

1 4s( - 4s+e -4" )

From example

E.8 we have

that

this system

has a

spectral

realisation

(D,A,B) which satisfies the conditions A1 up to A5 and we have that, D is

135

d={i/2,1/[16z2(-1-4z2))g~,.,I/[16zj(-l-4zj))~,.};

given by D = < . , d >

where

zi, j > 2

th zero of

is the j - I

- 4 s + e -as. Furthermore we have proved that

]2*(KerD) is the zero subspace and thus by theorem ITI.3 the DDP is only solvable if E = 0 , i.e. no disturbances. Now we shall show that there exists a bounded operator E and a strictly proper

D(s-A)'IBU(s) = D(s-A)'IE.

(e. 10) The

existence of

Firstly;

since

equations solvability

such

the

disturbance

D.D.P.

(3.13), of

a

(3.14), these

is

non

and

input

solvable,

(3.15)

equatiorm

(3.14)

or

(3.15)

is

m.lo.

but

axe

is

in

solvability of DDP. So the condition that omitted i n - t h e o r e m

operator

a

necessary

the

solvable,

general

meromorphic we

not

two

have

facts. matrix

that

the

to

the

equivalent

l)£(KerD) is closed can not be

It easy to prove

always

will prove

that

condition

the solvability of for

the

{3.13),

solvability

of

unequal

to

DDP.

Secondly,

we

have

that

I)"(KerD)

can

exists,

and

it

is

P~(KerD). Namely from equation (e.10) we have that Eq is contained in VE(Kcr D), since

(e.11)

D(s-A)-IEq=D(s-A)-IBU(s)q, so (s -A)-lEq = ~(s) + (s-A)-lBU(s)q, with ~(s) e Kcr D.

Thus

(¢.12)

Eq = (s-A)~(s)+BU(s)q

and by definition this shows that

Eqe))r(KerD), and thus {O}~)~(KerD).

Now we shall define this operator E. Let E:C~-~ 2 be defined by

Eq=eq~ where

(e.13) 3

From (e.3) we have that the j th component of e is of the order j-~, so e ~ $ 2, and thus E is a bounded operator. Furthermore

it

is

from

lemma

E.5

easy

to

see

that

(D,A,E)

is

a

136 realisation of the transfer function -$

D(s_A)-tE=

(e.14)

e 4s( - 4s + e -4~ )

If

we

define

U(s)

as

e"s,

then

the

operator

E

and

the

function

U(s)

satisfies equation (e.lO) and so we have constructed the counter example.

Similar

P*(KerD)

to

this

exists,

example

but

it

subspace in the kernel of

we

is

shall

unequal

to

construct the

an

largest

example open

13

such

loop

that

invariant

D, Yoz(KerD).

Example E.1L Again we shall consider the function

f(s)=

1 . 4s( - 4 s + e "4s) realisation of f is given by

By

spectral realisation of example

E.8

we

have

the that

delay transfer the

spectral

m x = ~ Zn <X~en>en] D(A)---{x~d21~

Iz.121 <x,e.> 12 l are the zeros of ( - 4 s + e - 4 ° ) ;

n

z 1 = 0 and b = d = {1/2,1/(16z~(

- i - 4z~

I) ~,..,

1/C16zj( - 1 - 4z~ I~ v~,..

Now we shall construct an initial value continuous input function

u(t)

such that

xoeKerD

}.

such that

the solution of

there

exists a

~c(t)=Ax(t)+bu(t);

x(0) =x0 remains in the kernel of D. As xQ we take

(e.14)

Xo={(1+2a)/2, (-(4z2)~-2a(z2)V~-a))/((z~-1)(16z2(-1-4z~))V'~,. ., ( - { 4 z S ' -2a{zj) ~ - a~) / (lz~-1) (l~zjc -1 - 4zj)) ~ ,... }

1#here

(e. 15)

a=

sinh( - 1) sinh( 2 )

and with u(t) defined as

137

f (e.16)

u(t)=

asiuh(t) sinh(t-1)+a sinh(t)

;0e n;

D(A)={xe~Z({-1,O,..})I~ Iz.l~l <x,e.> t22

z _ l = - 1 , z 0 = l , Zl=O and

are the zeros of ( - 4 s + e - 4 ~ ) ;

de = {X, 1,1/2,1/[16zz( - 1- 4zz)) ~ , . . , 1 / 0 6 z j { xn=

e+ane:+an) (e-l+ane-2+an) 8(4+e4)

'

8(4-e-')

- 1 - 4z j)) ~ , . . }.

(l+2an) ' 2

-(4z2)¼-2an(Z2 i -a,,

'4(z~-l)

(z2(-1-4z2))

v~''"

- ( 4 z i)W-2a,~( z i ) ~ - a ,

"" ( z~_l) Ct 6zi(_l_4z i))V,'"} If n converges to infinity, then gan(s ) converges for fixed s e C

to

-#a(S).

Furthermore it is easy to see that x" converges to x~: ={O,O,x0} in the norm of

~2({-1,0,1,..}).

So

De(s-Ae)-1x~=D(s-A)-lxo.

for

fixed

seC

D~(s-A~)-tx" converges g~(s)=D(s-A)-lxo.

So for fixed s e e we have that

to

O

S e c t i o n E.3: On t h e Sum o f two C o n t r o l l e d I n v a r l a n t Subspaces In this section we shall show by means of a counter example that the closure of the sum of two controlled invariant subspaces is not necessarily controlled

invariant.

In this

example we shall use the

spectral

realisation

of a delay equation an derived in section E.1. The delay equation that will be considered is given by

F(s)=(se"

(e.23)

~+s2+l )

(se-S+s2

-1)

(s~-I)(e-'+s)(e -2"+ s) We shall

prove

that

the

spectral

realisation of this equation

satisfies

the

139 conditions

of

associated

with

the

with

zeros

of

the

Ilowcver

the

chapter

sum

4.

zeros

Purtherniore

se-S+sZ+l

of

se-S+s2-1 of

these

is

as

controlled

subspaces

will

pole of F(s) associated

since to e v e r y

we

shall

l)rove

well as

that

the

subspace

invariant, riot

(see

remain

with (e-a+s)

the

associated

theorem

coutrolled

there

subspace

are

two

IV.16).

invariant, zeros of

the system> see theorem IV.16. Before we can p r o v e our exanlple we need some results on the distribution of the poles and zeros of F(s).

Lemma E.13. Let h(.) and r/(.) satisfies on £ the following relation

(e.24)

h( s ) = h( so) + ( s - so)h'( so) + ( s - So)Tl(s ).

where h'(so) # O.

Ih(so)l Assume

further

inequality holds

that

on

tim circle C : = { z :

17/(z)l