Hankel Norm Approximation for Infinite-Dimensional Systems

Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari

277

Springer Berlin Heidelberg NewYork Barcelona Hong Kong London Milan Paris Tokyo

A. Sasane

Hankel Norm Approximation for Inˇnite-Dimensional Systems With 19 Figures

13

Series Advisory Board

A. Bensoussan · P. Fleming · M.J. Grimble · P. Kokotovic · A.B. Kurzhanski · H. Kwakernaak · J.N. Tsitsiklis

Author Amol Sasane University of Twente Signals, Systems and Control Dept. Faculty of Mathematical Sciences Postbus 217 7500 AE Enschede The Netherlands

Cataloging-in-Publication Data applied for Die Deutsche Bibliothek – CIP-Einheitsaufnahme Sasane, Amol: Hankel norm approximation for infinite-dimensional systems / Amol Sasane. Berlin ; Heidelberg ; NewYork ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Tokyo : Springer, 2002 (Lecture notes in control and information sciences ; 277) (Engineering online library) ISBN 3-540-43327-9

ISBN 3-540-43327-9

Springer-Verlag Berlin Heidelberg New York

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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution act under German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2002 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Digital data supplied by author. Data-conversion by PTP-Berlin, Stefan Sossna e.K. Cover-Design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN 10869676 62/3020Rw - 5 4 3 2 1 0

Preface This book is aimed primarily at system theorists working on the approximation of systems described in terms of linear partial diÿerential equations by systems described in terms of linear ordinary diÿerential equations. However, it will be of interest also to functional analysts with an interest in the classical problems of interpolation theory of analytic functions. Model reduction is an important engineering problem in which one aims to replace an elaborate model by a simpler model without undue loss of accuracy. The accuracy can be mathematically measured in several possible norms and the Hankel norm is one such. The Hankel norm gives a meaningful notion of distance between two linear systems: roughly speaking, it is the induced norm of the operator that maps past inputs to future outputs. It turns out that the engineering problem of model reduction in the Hankel norm is closely related to the mathematical problem of þnding solutions to the sub-optimal Nehari-Takagi problem, which we call \the sub-optimal Hankel norm approximation problem" in this book. Although the existence of a solution to the sub-optimal Hankel norm approximation problem has been known since the 1970's, in this book, we give explicit solutions and, in particular, we give new formulae for several large classes of inþnite-dimensional systems. The approach taken in this book is as follows. First we give a \frequency domain" solution to the sub-optimal Hankel norm approximation problem. We start with a complex matrix-valued function G deþned on the imaginary axis satisfying certain assumptions. In particular, we demand the existence of a solution to a certain Jÿspectral factorization problem. We then give a solution to the sub-optimal Hankel norm approximation problem in terms of the Jÿspectral factor. Furthermore, we give a parameterization of all solutions in terms of the Jÿspectral factor, the parameterizing set being the unit ball in a certain Hardy space. In this manner we give purely \frequency domain" solutions to the sub-optimal Hankel norm approximation problem in Chapter 4 of this book.

VI

Preface

Subsequently, we give \state-space" solutions to the sub-optimal Hankel norm approximation problem for important classes of inÿnite-dimensional linear systems. In Chapter 5 we consider the case where G is the transfer function of a well-posed linear system given by a triple of operators (A; B; C ). Under certain assumptions we solve the sub-optimal Hankel norm approximation problem for G = C (sI ÿ A)ÿ1 B by constructing a J ÿspectral factor in terms of the system parameters (A; B; C ) and verifying that this constructed J ÿspectral factor satisÿes the assumptions demanded in Chapter 4. In this manner, we obtain \state-space" solutions to the sub-optimal Hankel norm approximation problem for two important classes of well-posed linear systems: the smooth PritchardSalamon class of exponentially stable inÿnite-dimensional systems and the class of exponentially stable analytic systems. Finally, we also solve the sub-optimal Hankel norm approximation problem for certain classes of inÿnite-dimensional systems with a non-exponentially stable semigroup and for regular linear systems. This book is a slightly adapted and extended version of my Ph.D. thesis. It is the outcome of a period of four years at the Department of Mathematics of the University of Groningen, The Netherlands. My advisor during this period was Professor Ruth Curtain, and the results in this thesis are the product of cooperation with her. I would like to express my deepest gratitude for this. Finally, I wish to thank my family for everything.

Contents 1

Introduction 1.1

The sub-optimal Nehari problem

1.2

The sub-optimal Hankel norm approximation problem

. . . . . .

5

1.3

The model reduction problem . . . . . . . . . . . . . . . . . . . .

6

1.3.1

The Hankel norm of systems

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

6

1.3.2

The model reduction problem . . . . . . . . . . . . . . . .

8

1.3.3

Hankel norm approximation and model reduction . . . . .

8

1.4

2

3

1

Organization of the book

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

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

4

10

Classes of well-posed linear systems

13

2.1

Well-posed linear systems

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

14

2.2

Duality theory

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

16

2.3

The Pritchard-Salamon class

2.4

The analytic class

2.5

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

20

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

25

Transfer function algebras . . . . . . . . . . . . . . . . . . . . . .

52

Compactness and nuclearity of Hankel operators

63

3.1

Compactness of Hankel operators . . . . . . . . . . . . . . . . . .

66

3.2

Nuclearity of Hankel operators

74

3.3

L1 ÿerror of sub-optimal Hankel norm approximants .

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

80

VIII

4

Characterization of all solutions

85

4.1

The sub-optimal Hankel norm approximation problem . . . . . .

85

4.2

Main assumptions and a few useful consequences

89

4.3

All solutions to the sub-optimal Hankel norm approximation

4.4

5

6

7

8

Contents

. . . . . . . . .

problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

A comparison with existing results

99

State space solutions

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

101

5.1

J ÿspectral factorization for the

Pritchard-Salamon class . . . . . 101

5.2

J ÿspectral factorization for the

analytic class . . . . . . . . . . . 104

The non-exponentially stable case

109

6.1

A few useful results . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.2

The class of systems and the main result . . . . . . . . . . . . . . 111

6.3

State-space solutions . . . . . . . . . . . . . . . . . . . . . . . . . 114

The case of regular linear systems

119

7.1

Reciprocal systems . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.2

The sub-optimal Hankel norm approximation problem . . . . . . 121

7.3

The sub-optimal Nehari problem

Coda

. . . . . . . . . . . . . . . . . . 123

127

Bibliography

131

Index

139

Standard notation

141

Chapter 1 Introduction Systems and control theory is a subject in which beautiful and deep mathematical theorems hav e exactly matched the needs of a vital branch of technology. This conjunction of mathematics and engineering has been enric hing for both ÿelds.

In this book w e present a solution to the sub-optimal Hankel norm

approximation problem which has been studied b y both mathematicians and engineers. In this c hapter, among other things, w e explain the problem and the relev ance of its solution to the important engineering problem of model reduction, in which the aim is to replace a high order (possibly inÿnite-dimensional) system by a low order model, without undue loss of accuracy. Finally we will giv e a brief outline of the chapters in this book. But ÿrst we give a brief introduction to the history of the problem, based on Young [93]. In 1957, Zeev Nehari [56] studied the following natural problem. Suppose we are given a sequence does the inÿnite

fangnÿ0 of complex numbers. 2 3 a0 a1 a2 : : : 6 a1 a2 a3 : : : 7 H =6 64 a2 a3 a4 : : : 7 7 5 . . .

Hankel matrix 1

. .

. .

. .

..

Under what conditions

(1.1)

.

`

deÿne a bounded linear operator on 2 (N )? He found a striking answer: a necessary and suÆcient condition is that there

'þ on the unit circle T = fz 2 ÿ j jz j = 1g (parameterized as eiÿ j 0 ÿ ÿ ÿ 2þ ) suc h that th F ourier coeÆcient ' the n ^(n) = an ; for all n þ 0: (1.2) 1 That is, a matrix fckl gk lÿ0 , where ckl depends only on k + l and so ckl = ak+l for some sequence fan gnÿ0 .

should exist an essen tially bounded function

C

;

A. Sasane: Hankel Norm Approximation for Infinite-Dimensional Systems, LNCIS 277, pp. 1−12, 2002.  Springer-Verlag Berlin Heidelberg 2002

2

Chapter 1.

Introduction

Hermann Hankel (1839-1873).

(Courtesy of

http://www-groups.dcs.st-andrews.ac.uk/ÿhistory)

Furthermore, when (1.2) is satisÿed,

kH kL(`2 (N))

= inÿmum of the

2 6 H =6 6 4

Example 1.0.1 If

1 1 2 1 3

. . .

1 2 1 3 1 4

. . .

L1 þ norms of all such ': 1 3 1 4 1 5

. . .

::: ::: ::: ..

(1.3)

3 7 7 ; 7 5

.

ÿ þ ÿ. By taking ' eiÿ i(ÿ þ þ)eiÿ , we have a function with L1þnorm at most ÿ, and Z 2þ 1 '^(n) = i(ÿ þ þ)eiÿ eÿinÿ dþ 2ÿ 0 ý ü2þ Z 2þ þi(ÿ þ þ)eÿi(n+1)ÿ i = þ eÿi(n+1)ÿ dþ 2ÿ (n + 1) 2 ÿ ( n + 1) 0 0 the

Hilbert matrix, Schur proved that kH kL(`2 (N))

= for all

1

n+1

=

=

;

ÿ

n ý 0.

We can reformulate the above result as follows. Let

G(z ) =

1 X

ÿ1

n=

gn z n

and

gn = an

for all

n ý 0:

Introduction

3

In (1.3), the right hand side is the inÿmum of on the circle such that

k'k1

over all

L1 ÿfunctions '

'^(n) = an ; n þ 0: This class of

'

is exactly the coset

G + H1 (D )?

in

L1 (T),

where

H1 (D )?

L1 (T)) of functions whose nonnegative Fourier coeÆcients fz 2

is the subspace (of

vanish, which is the space of functions which are analytic and bounded in

C

j jz j > 1g.

Hence

kH kL(`2 (N))

inf kG + K k1 K 2H1 (D)? ÿ ?þ : distL1 (T) G; H1 (D )

=

=

H1 (D )?

So Nehari's theorem gives us an expression for the distance of a bounded function on the circle from

given function in

H1 (D )?

as the norm of a Hankel matrix, and so it gives a

lower bound on the error if we are ÿnding an approximation in

L1 (þ).

of a

1 (ÿ)

L

1(D )?

H

ÿÿ2 ÿÿ6 ÿÿ4 ÿ

G(z )

=

P

1

n=

1

a0

a1

:::

a1

a2

:::

gn z

. . .

n

+

. . .

..

P1

.

3ÿÿ 75ÿÿÿ ÿÿ

n=0 an z

n

Figure 1.1: Nehari's theorem.

L1 ÿapproximation on

More detailed information relating to this type of

the

basis of Hankel operator theory was subsequently obtained by V.M. Adamjan, D.Z. Arov and M.G. Krein [2], D. Sarason [74] and S.R. Treil [82]. An excellent introduction to the sub ject is Young [93], upon which this chapter is based, and a survey of results can be found in Nikol'skii [57]. The fact that results on such

L1ÿapproximation problems were relevant to the preoccupations of some

engineers was realized by a number of mathematicians in the seventies.

This

stimulated further research and the birth of an intensive period of research and

ÿ

interaction by both engineers and mathematicians (see Francis [32]).

the

H1 ÿera

of the 80's

4

Chapter 1.

1.1

The sub-optimal Nehari problem

ÿ

A reformulation of the question answered

of

2

by Nehari is the following: Given a

p m matrix-valued function G on the imaginary axis, ÿnd the distance

G from H1 (C pÿm )? in the L1 þnorm, where H1 (C pÿm )? denotes the space

bounded of

Introduction

p

ÿm

matrix-valued functions which are bounded and analytic in the open

left half-plane. That is, ÿnd

ÿ

L1 G; H1

dist

ÿm ý? ü

þ

Cp

=

=

where we use the notation

K (þ )2H1 (C pÿm ) inf

K (þ )2H1 (C pÿm ) inf

H1 (C pÿm )

kG iý (

K (iý)k1

) +

esssup

kG i! pÿm

!2R

for the space of

(

)+

K (i!)k;

matrix-valued

functions which are bounded and analytic in the open right half-plane. Another

G in the L1 and we seek the closest point or points in the closed subspace

way of phrasing the question is to say that we are given a point

H1 (C pÿm )? .

Banach space

It is perhaps not surprising that in any Banach space other than

a Hilbert space, best approximation problems are usually diÆcult and one does not expect to solve them exactly.

Fortunately for control theory, the Nehari

problem is one of the rare cases which does admit a precise solution. Let

H2 (C k ) denote the set of all analytic functions f û

kf k

2 := sup 0

ÿ>

1

ÿ

2

1 kf þ1

Z

:

þ + i!)k d! 2

(

ú 21

G 2 L1 (iR; C pÿm ) we deÿne the Hankel operator m p by HG , acting from H2 (C ) to H2 (C ), as follows: For

HG f = þ(MG fþ )

!C

k

such that

< 1:

with symbol

G,

denoted

f 2 H2 (C m ) ;

L2 (iR; C m ) induced by G, þ is the orp p thogonal pro jection operator from L2 (iR; C ) onto H2 (C ) and fþ (s) := f (þs). The operator HG is related to the inÿnite Hankel matrix H in (1.1): HG has

where

MG

for

C+ 0

is the multiplication map on

G 2 L1 (iR; C pÿm ),

a matrix of that form with respect to a suitable orthonormal basis. theorem says that for any

L1

dist

G; H1 (C pÿm )?

þ

ý

=

Nehari's

kHGk:

One is of course interested not only in the value of the distance, but also in characterizing the set of

K 's for which the distance is attained. In the case of K (see Glover [34]).

rational transfer functions there are explicit solutions for

However, for control applications, the focus is on the sub-optimal Nehari problem. The

2 The

sub-optimal Nehari problem

is the following:

matrix case was settled by Adamjan, Arov and Krein in [1].

1.2.

The sub-optimal Hankel norm approximation problem

If

ÿ >

kH k; G

5

Let G(iÿ) 2 L1 (R; C pÿm ): then ÿnd K (þÿ) 2 H1 (C pÿm ) such that kG(iÿ) + K (iÿ)k1 ý ÿ:

The solutions K to the sub-optimal Nehari problem are the key to the design of controllers which maximize robustness with respect to uncertainty or minimize sensitivity to disturbances of sensors (see McFarlane and Glover [54] and Georgiou and Smith [33]). For the purpose of \model reduction", a notion to be explained later in this chapter, one considers a natural generalization of the Nehari problem, and this problem turns out to be much harder.

1.2

The sub-optimal Hankel norm approximation problem

Let H1;l (C pÿm ) denote the space of all p ü m matrix-valued functions K of a complex variable deÿned in the right half-plane such that K = Gþ + F , where F is an element in H1 (C pÿm ) and Gþ is the transfer function of a ÿnite-dimensional system with MacMillan degree 3 at most l, with all its poles in the open right half-plane. H1;l (C pÿm ) is a subset of L1 (iR; C pÿm ). Now we recall the notion of singular values of a bounded linear operator from a Hilbert space H1 to a Hilbert space H2 . For k 2 f1; 2; : : :g the kth singular value of an operator H 2 L(H1 ; H2 ) (denoted by ÿk (H)) is deÿned to be the distance with respect to the norm in L(H1 ; H2 ) of H from the set of operators in L(H1 ; H2 ) of rank at most k þ 1. Thus ÿ1 (H) = kHk, and ÿ1 (H) û þ ÿ2 (H) û ÿ3 (H) û : : : û 0. If H is compact, then H H is compact and nonnegative, and so the spectrum of Hþ H consists of a pure point spectrum with countably many nonnegative eigenvalues. The square roots of these eigenvalues are then the singular values of H. If the Hankel operator with symbol G 2 L1 (iR; C pÿm ) is compact, we sometimes denote the singular values of HG , ÿk (HG ), simply by ÿk (G). The ÿk (G)'s are then referred to as the Hankel singular values of G. 3 The

MacMillan degree

of a proper rational function G is deÿned to be the minimal state

dimension of all possible (A; B; C; D)'s that realize G, that is

MacMillan degree of G =

min

2 Cn; mÿn ; B 2 C nÿp ; C 2 C pÿm D 2 C

A

fn j G(s) = C (sI ÿ A)þ1 B + Dg:

6

Chapter 1.

The following theorem

4

Introduction

of Adamjan, Arov and Krein [1] is a natural extension

of Nehari's theorem:

Theorem 1.2.1 For any G

2 L1 (iR; C pÿm ), ÿ k1 = ÿl+1 (G):

þ 2 1 Cÿ k ÿ

K(

)

inf

H

;l (

p

G(i ) + K (i )

m)

The Nehari theorem corresponds to the case

ÿ

üÿ

ü þÿ

ÿ

K ( ) = G ( ) + F ( ), where G (

l

=

0.

In Theorem 1.2.1, if

) is the rational transfer function of a ÿnite-

ÿm

dimensional system with MacMillan degree at most

þÿ) 2

in the open right half-plane, and F (

1

H

(

be an optimal Hankel norm approximant of G.

Cp

l

and with all its poles

), then

þGü (ÿ)

is said to

The sub-optimal Hankel norm

approximation problem is the following:

ÿ 2 L1 (R; C pÿm ): pÿm ) ÿl+1 < ÿ < ÿl ; then ÿnd K (þÿ) 2 H1 l (C such that kG(iÿ) + K (iÿ)k1 ý ÿ: Let

If

G(i )

;

K is then called a solution of the sub-optimal Hankel norm approximation problem.

1.3

The model reduction problem

An elaborate model of a physical system can sometimes be replaced by a simpler one, which is easier to analyze and use, without a signiÿcant loss of accuracy. In this section we shall see how a solution to the sub-optimal Hankel norm approximation problem gives rise to a ÿnite-dimensional reduced model to the original inÿnite-dimensional system.

1.3.1

The Hankel norm of systems

In practice, many physical systems and engineering devices are modelled by mathematical systems of the type

4 the

x(t)

=

Ax(t) + Bu(t);

y(t)

=

C x(t);

matrix case was proved in Kung and Lin [50].

ÿ t

ü 0;

1.3.

7

The model reduction problem

where T (t)

is the inÿnitesimal generator of a strongly continuous semigroup (C m ; X ) and C (X; C p ). 5 0 on the Hilbert space X , B

f gtÿ

2L

A

2L

If A is exponentially stable (that is, there exists an ÿ < 0 and a M > 0 ÿÿt for all t 0), then the impulse response h( ) = such that T (t) Me C T ( )B is an element belonging to L1 ([0; ); C pþm ), and h induces a compact time-domain Hankel operator þh (L2 ([0; ); C m ); L2 ([0; ); C p )), deÿned m as follows: given u L2 ([0; ); C ),

k kÿ

ý

2

1

(þh u)(t) =

Z

þ 1 2L 1

1

0

ý

1

( + þ )u(þ )dþ for all

h t

t

þ0

:

ü

The impulse response h has the Laplace transform G(s) = C (sI A)ÿ1 B with pþm ). This induces a compact Hankel operator G( ) : iR C pþm L1 (iR; C HG with symbol G, which is related to þh via Laplace transformation:

ý

!

2

ÿ

H

þ

G f^(s) = þd h f (s) for all

s

in the open right half-plane;

k k k kÿk k ÿk k üý 2 1

and þh = HG G 1 h L1 . If uý ( ) L2 ([0; ); C m ), and we have for t (þh u) (t) =

1

Z

0

ý2

þ0 u

( + þ )u(þ )dþ =

CT t

Z

L2

ü1 0] C

((

0

ÿ1

;

(

CT t

ü

;

)

ý

m ),

ý(

þ u

then u( ) :=

)

þ dþ :

Roughly speaking, if we run the system ( ) + Buý (t); ()

() = () =

x t

Ax t

y t

Cx t ;

ü1

ü1

from time to 0 with input function uý and \initial state x( ) = 0", and then cease to input anything and observe the output y ( ) from time 0 onwards, then (þh u) (t) = y (t) for all t 0 (see Figure 1.2).

ý

þ

u( )

y( )

ÿh

0

0

t

t

Figure 1.2: The time-domain Hankel operator. 5 More and

Y

generally, the input space

, respectively, and

B

and

C

Cm

and the output space

could be

unbounded.

Cp

could be Hilbert spaces

U

8

Chapter 1.

Introduction

In other words, ÿh is the mapping from past inputs to future outputs. The

Hankel norm of a system with transfer function kH k.

is deþned to be

G

G

(sometimes denoted by

kGk

H)

It is now evident that this is a physically meaningful

notion of closeness of systems. To say that

ÿ is close to

G

G

in the Hankel norm

means that the corresponding systems acting on a past input signal give future outputs which are close to each other.

1.3.2

The model reduction problem

In some areas of engineering, linear inþnite-dimensional state-space models of dynamic systems are derived and it is desirable to replace them by þnitedimensional systems without incurring too much error. This would make the subsequent control system design computationally less demanding and possibly numerically more reliable. A wide variety of methods for model reduction have been proposed over the years; here we list a few: 1. Balanced model reduction: See for example Moore [55] and Pernebo and Silverman [64] for the þnite-dimensional case and Curtain and Glover [36], Ober [58], Young [92] for the inþnite-dimensional case.

2

\

h L1 L2 with a nuclear Hankel operator: See Curtain, Glover and Partington [37].

2. Approximation of systems with

3. Other miscellaneous techniques: See Curtain [12], Glover, Lam and Partington [38], [39], Gu, Khargonekar and Lee [43]. However, in this book, we consider the

the Hankel norm:

Given a stable

G

þnd a stable rational such that

model reduction problem with respect to

and a nonnegative integer

ÿ

G

l;

of MacMillan degree at most

kG ÿ Gÿ k

H

l

is small.

We explain the relation of this question to the problem of sub-optimal Hankel norm approximation in the next section.

1.3.3

Hankel norm approximation and model reduction

Suppose that

+1

ÿl

G

2

< ÿ < ÿl , K

a compact Hankel operator and that if 1 ( R C pþm ) phas (ÿþ) 2 1 l (C þm ) is a solution to the sub-optimal Hankel L

i

;

H

;

norm approximation problem, that is,

kG(iþ) + K (iþ)k1 ý ÿ:

1.3.

ÿ

Let

9

The model reduction problem

K(

) =

ÿ (ÿ) +

G

ÿ

F(

), where

G

ÿ (þÿ) is the rational transfer function of a þÿ 2 1 þ þ ÿÿ

ÿnite-dimensional system with MacMillan degree at most l and with all its poles in the open right half-plane, and F ( ) H (C p m ). Then G ( ) (called a sub-optimal Hankel norm approximant) is a reduced order model of the original

system

k

G

G

in the Hankel norm, with error at most equal to

ÿ kH = k

+G

G

ÿ

+G +F

kH k =

G

+K

kH ý k ÿ G(i

ÿ.

Indeed, we have

ÿ k1 ý

) + K (i )

ÿ:

(1.4)

It is not clear that closeness in the Hankel norm is an appropriate criterion for a reduced order model. Most robust control designs are robust with respect

1 þnorm:

to the stronger

L

k kH ý k k1 G

G

. However, in practice the Hankel

norm approximants performed well. The explanation of this fortunate fact was provided by Glover in his seminal paper [34], which was generalized in [37].

P1

2

In Glover et al. [37] it was shown that if h L1 ([0; L2 ([0; ); C p m ), and if ÿk < , then for a given integer k=1

1

þ

1

ists an optimal Hankel-norm approximation

0

G

k þ 0 kH ý k þ 0 k1 ý G

G

G

G

of order

1 X

= +1

l,

1

þ

); C p m )

l,

\

there ex-

such that

ÿk (G):

k l

It is very commonly the case that the Hankel operator has a few sizeable singular values and the remaining tail away very quickly to zero. In such a case, the right hand side can be made very small, and one is assured that an optimal Hankel norm approximant is also good with respect to the

L

1 þnorm.

This norm is

extremely desirable because there exist many robustness theories with respect to this norm. In this book, we only consider the sub-optimal Hankel-norm approximation problem. In (1.4) above we showed how this yields a sub-optimal Hankel norm approximant

þ ÿÿ G

( ) which is close in the Hankel norm. However, for control

applications, we prefer closeness in the

1 þnorm.

L

Since we were unable to

ÿnd any results on this, even in the ÿnite-dimensional literature, we prove (in Section 3.3) that the error bound in Corollary 7.5 (see page 69, Partington [62])

P1

for an optimal Hankel norm approximant also holds for a sub-optimal Hankel norm approximant: If h L1 ([0; ); C p m ) and , then any subk=1 ÿk < optimal Hankel norm approximant G as in (1.4) of MacMillan degree l satisÿes

2

k ÿ G(i

1

ÿ

ÿ

ÿ k1 ý

) + G (i )

þ

4l

ÿl (G)

tends to zero as

l

!1

:= 4l

ÿl (G)

1 X

= +1

ÿ k ( G ):

k l

Moreover we will show that the error bound El

+2

1

+2

1 X = +1

ÿk (G)

k l

. So the sub-optimal Hankel norm approximant is also

a good approximation of

G

in the

L

1þnorm.

10

Chapter 1.

Introduction

While the natural formulation of the sub-optimal Hankel norm approximation problem is in terms of the transfer function, most systems in control applications are modelled in state-space form in terms of parameters

A, B , C .

The Hankel norm approximation problem has been solved for the rational case in Glover [34], Ball and Ran [6] and for the special class of Pritchard-Salamon inÿnite-dimensional systems in Curtain and Ran [17]. Our aim in this book has been two-fold:

1. To obtain suÆcient conditions for the existence of a solution to the suboptimal Hankel norm approximation problem in terms of a solution to a key

J ÿspectral factorization problem.

This allows a complete param-

eterization of all solutions in terms of the spectral factor (see Chapter 4). 2. For certain classes of well-posed linear inÿnite-dimensional systems, we use the frequency domain results established in Chapter 4 to obtain explicit solutions to the sub-optimal Hankel norm approximation problem in terms of the state-space parameters

A, B , C .

The classes considered

are exponentially stable, smooth Pritchard-Salamon systems (Section 5.1 of Chapter 5), exponentially stable analytic systems (Section 5.2 of Chapter 5), non-exponentially stable systems (Chapter 6) and regular linear systems (Chapter 7).

The above strategy is reminiscent of the strategy in Ball and Ran [6], Ran [67] and Curtain and Ran [17] with one important diþerence. In these three references, the starting point was to quote a result (Theorem 4.4.1 of Section 4.4 in Chapter 4) which was claimed to be a corollary of the results in Ball and Helton [5]. Since it is not directly clear to non experts and certainly not to the engineering community how this result follows from [5], we found it worthwhile to derive a simpler frequency domain result from ÿrst principles using elementary mathematics in Chapter 4.

We remark that an analogous

strategy has been followed earlier to solve the special case of the sub-optimal Nehari problem in Curtain and Zwart [27], [26], Curtain and Oostveen [16] and Curtain and Ichikawa [14].

1.4

Organization of the book

In this section, we give an overview of the results in this book.

Chapter 2:

We ÿrst deÿne well-posed linear systems, which forms the general

framework for studying inÿnite-dimensional systems with a state space representation. We recall a number of deÿnitions concerning this class of systems

1.4.

11

Organization of the book

which are relevant to the remainder of this book. Next we discuss two important classes of well-posed linear systems: 1. the Pritchard-Salamon class, and 2. the analytic class, which are central in this book. Some of their properties, which will be used in Chapter 5 are listed. Finally, in the last section of this chapter we will introduce several frequency domain spaces and prove a few elementary lemmas which will be used later.

Chapter 3:

In the theory of approximation of inÿnite-dimensional systems, the

nuclearity and compactness properties of the Hankel operator play an essential role. In this chapter we investigate the relationships between the exponential (or strong) stability of certain classes of regular linear systems and the compactness and nuclearity properties of the Hankel operator by means of considering illustrative examples. New suÆcient conditions for nuclearity are given for exponentially stable regular linear systems with an analytic semigroup. In the last section of this chapter we will derive a bound on the

L1 ÿerror of a sub-optimal

Hankel norm approximant of a system with a nuclear Hankel operator.

Chapter 4:

In this chapter, ÿrst we prove a few useful consequences of the main

(frequency domain) assumptions S1-5 in this chapter, and subsequently, under these assumptions, we give a characterization of all solutions to the sub-optimal Hankel norm approximation problem. These derivations are new. Later in Chapter 5, we will show that for the Pritchard-Salamon class and the analytic class, there do exist

J ÿspectral factors for each class satisfying

the assumptions in this chapter, hence solving the sub-optimal Hankel norm approximation problem for these classes of inÿnite-dimensional systems. We

J ÿspectral factor þ in terms of the stateA B , C ) of the original system.

will give an explicit formula for a space parameters ( ,

Chapter 5:

In Section 5.1, the sub-optimal Hankel norm approximation prob-

lem is solved for the smooth Pritchard-Salamon class of exponentially stable inÿnite-dimensional systems with ÿnite-dimensional input and output spaces by using the results from Section 2.3 of Chapter 2 and Chapter 4: we give a formula for a

J ÿspectral factor þ in terms of the state-space parameters, and

check that it has all the properties required in Chapter 4. These results were obtained earlier in Curtain and Ran [17] by appealing to the Ball and Helton result [5]. Next, in Section 5.1, the sub-optimal Hankel norm approximation problem is solved for the analytic class of exponentially stable inÿnite-dimensional systems with ÿnite-dimensional input and output spaces by using the results

12

Chapter 1.

Introduction

from Section 2.4 of Chapter 2 and Chapter 4. Again, we give a formula for a

J ÿspectral

factor ÿ in terms of the state-space parameters, and check that it

has all the properties required in Chapter 4. The results of this chapter are new and include the results on the Nehari problem in Curtain and Ichikawa [14] as

a special case.

Chapter 6:

In the previous chapters the sub-optimal Hankel norm approxi-

mation problem was solved for various classes of inþnite-dimensional systems under the assumption that

A

generates an

exponentially

stable strongly contin-

uous semigroup. In this chapter, we will solve the sub-optimal Hankel norm approximation problem for non-exponentially stable inþnite-dimensional systems in terms of a solution to the sub-optimal Hankel norm approximation problem for an exponentially stable system. The exponentially stable system is obtained by shifting the generator of the semigroup of the original system. The results in this chapter are new.

Chapter 7:

In this chapter, we will solve the sub-optimal Hankel norm approx-

imation problem for regular linear systems with generating operators (A; B; C ) satisfying some mild assumptions. Furthermore, in the special case of the suboptimal Nehari problem, we will also give an explicit parameterization of all solutions. The class of regular linear systems is a large class of inþnite-dimensional state-linear systems allowing for unbounded inputs and outputs; it includes the Pritchard-Salamon class and the analytic class. But the solution obtained in this chapter is diýerent from the form in the preceding chapters.

Chapter 2 Classes of well-posed linear systems

In this chapter we recall the notion of w ell-posed linear systems, which forms the general framework for studying inÿnite-dimensional systems in a state space con text. In particular, we introduce tw o importan t classes of well-posed linear systems which are central in this book and list some of their properties, which will be used in the sequel. Figure 2.1 shows the hierarchy of the various classes of systems that we will encounter in this chapter. Finally, in the last section of this chapter we will introduce several frequency domain spaces and prov e a few elementary lemmas which will be used in what follows.

well posed

regular well posed Pritc hard Salamon smooth Pritchard Salamon A A; B; C

bounded A

generates an

analytic semi group on

generates a

strongly continuous semigroup B

and

C

bounded

Z,

2 L(U; ZÿB ), C 2 L (Z ÿ C ; Y )

B

Figure 2.1: Various classes of systems.

A. Sasane: Hankel Norm Approximation for Infinite-Dimensional Systems, LNCIS 277, pp. 13−62, 2002.  Springer-Verlag Berlin Heidelberg 2002

14

Chapter 2.

2.1

Classes of well-posed linear systems

Well-posed linear systems

We assume that

is a Hilbert space and

:

A

0 on space X1 as D (A) with the norm x 1 = (ÿI ÿxed1 . The ÿ Hilbertÿspace ÿ Xÿ1 is the completion of 1 xÿ. We have x ÿ1 = ÿ(ÿI A)

kk

ÿ

T (t) t

þ þ

X1

! k

D (A)

f gÿ kk k ÿ

X

of a strongly continuous semigroup

X.

A)x

ÿ1 , see Weiss [86].

A,

We assume that

whose domain is

U

X,

so that

are bounded from

L2 ([0;

Z þt u :=

t

0

1 ÿ

); U ) to

T (t

x(t)

and

u

=

2

X -valued

1

is called

B

We assume that called an

ý

ÿ

2

Y

x(t)

=

T (t)x0

bounded

if

B

2L

t

0

We regard

ý

Kt

Z

The operator

þ

(

ÿ1 ) !

X

u

ý 2

f g

1

L2 ([0;

); U ):

(2.2)

k

ý ý

t

2

ÿ

0,

+ þt u:

(2.3)

(U; X ) (and

for

0 such that

C T (ý )x0

k ü k k 2

dý

2

x0

Kt

unbounded

2

f gþ T (t) t

for all

C

otherwise).

2L

(X1 ; Y ) is

0 , if for every

x0

2

t >

D (A):

0 there

(2.4)

2L

C is called bounded if it can be extended such that C (X; Y ). loc ([0; ); Y ) as a Frý echet space with the seminorms being the

L2

1

2N

L2 norms on the intervals [0; n], n that there is a continuous operator ü :

(üx0 )(t) =

.

X

C T (t)x0

Then the admissibility of C means loc L2 ([0; ); Y ) such that

!

for all

x0

2

1

D (A):

The operator ü is completely determined by (2.5), because 1 This

X

2L

is another Hilbert space. The operator

admissible observation operator

exists a number

ÿ

for all

function of t. We have that for all

The operator

:

X 1 , but the result is in X . If x is the solution 0, which is an equation in X 1 , with x(0) = x0 X ); U ), then x(t) X for all t 0. In this case, x is a continuous

Ax(t) + Bu(t), t

L2 ([0;

T (t) t

ý )Bu(ý )dý ;

The above integration is done in of

f gþ

for

X

A

B (U; Xÿ1 ) is said 0 , if the input maps þt tþ0 for all ÿnite t 0, where

is a Hilbert space. The operator

admissible control operator

is

(2.1)

f gþ

X

to be an

þ(A)

T (t) t 0 extends to a The generator of this extended

ÿ1 , denoted by the same symbol.

X

semigroup is an extension of

2

ÿ

X

densely and with continuous embeddings. The semigroup semigroup on

, where

with respect to the norm

X

ÿ1 ;

X

is the generator

X

We deÿne the Hilbert

norm is equivalent to the graph norm.

D (A)

(2.5) is dense in

X.

2.1.

B B

for

L L

is bounded.

f g ÿ0 T (t) t

If

f g ÿ0 1

is said to be an inÿnite-time admissible control operator for

is admissible and for any

X)

15

Well-posed linear systems

B

2L

2

u

1

); U ), the map

7!

t

T (t) t

ÿt u (from [0;

1

ÿ 0 k2 þ k 0k2

0

k

0 such that

K

C T (ÿ )x

dÿ

K

for all

x

x

2

0

D (A):

ÿ1 ) is an admissible control operator for f ( )g þ0 and 2 f ( )g þ0, then the transfer

(U; X

T t

(X1 ; Y ) is an admissible observation operator for

T t

C

t

t

functions of the system þ given by the triple (A; B; C ) are solutions

(U; Y ) of

ý

G(s)

for

and

s

ý

T ( t) t

G(ý )

=

ý ý (s

, (

I

ÿ1

A)

B

(U; X )

B

L

a

ý

(U; Y )

A)

I

,

and

Re(s)

s

ý

ýI

A)

ÿ1

:

þ(A)

!

(2.6)

B

is an admissible control operator for

B

analytic function. Both ( =

ÿ1 (

G

valued analytic function and since

T (t) t

right half-plane

A)

L ý f g þ0 ü ý ü ý ÿ1 ü ý f 2C j g is an

an admissible observation operator

C+ ÿ

ý

ý )C (sI

in þ(A). We remark that since

f g þ0 ü ý

if

) to

is said to be an inÿnite-time admissible observation operator

C

if there exists a

Z

2([0;

L

C( I

C( I

> ü

.

A)

A)

ÿ1 is a L(

ÿ1

X; Y

ý

)

C

is

valued

are analytic on some

Thus any transfer function is

valued function which is analytic in some

transfer functions diýer only by an additive constant, is that they need not necessarily be bounded on any

C+ ÿ . Moreover any two

D

2L

(U; Y ). The point

C+ ÿ . We impose this as

an extra assumption on the triple (A; B; C ) and call this well-posedness. The system þ given by the triple (A; B; C ) is said to be well-posed linear system if is an admissible control operator for operator for

f g þ0 T (t) t

f g þ0 T (t) t

,

C

B

is an admissible observation

and its transfer functions are bounded on some half-plane

C+ ÿ . For a well-posed linear system, the operator from the initial state and the

input function to the ünal state and the output function is bounded. The input and output functions

u

and

are locally

y

respectively. The state trajectory

x

2 functions with values in U

L

is an

X -valued

property mentioned earlier means that for every that

k k2 x(t)

(with

ct

Z +

independent of

t >

ÿ

k k 2 þ 2 k k2 0 t

y (ÿ )

x(0)

dÿ

and of

ct

u).

x(0)

and in

Y

,

function. The boundedness

Z +

0 there is a

k k2 0

þ

t

u(ÿ )

ct

dÿ

ÿ

0 such (2.7)

For the detailed deünition, background

and examples we refer to Salamon [72], [73], Staýans [75], [76], [79], Weiss [89], [88], Weiss and Rebarber [68] and Weiss2 [90]. ÿ (A; B; C ) u

y

Figure 2.2: Well-posed linear systems.

f g þ0 T (t) t

U

is called the semigroup of þ and

is called the input space ,

Y

A

is called its inÿnitesimal generator.

is called the output space and

X

is called the

16

Chapter 2.

state-space

called

Classes of well-posed linear systems

of the well-posed linear system. The well-posed linear system ÿ is if the limit lim G(ÿ)v = Dv (2.8)

regular

!+1

ÿ

2

exists for every v U , where ÿ is real (see [87], [89]). In this case, the operator (U; Y ) is called the feedthrough operator of ÿ, and there holds

D

2L

( ) = Cÿ (sI

G s

where

C

)ÿ1 B + D;

A

ÿ is an extension of C given by C

(ÿ

ÿ

2 R), for all

x

C ÿ(ÿI ÿ x0 = !lim +1 ÿ

ÿ

)ÿ1 x0

(2.9)

A

0 in the domain D

(Cÿ ) =

f 0 2 j the limit in (2.9) existsg x

X

(see Staþans [79]). The operators A, B , C , D are called the generating operators of ÿ, and we write ÿ = (A; B; C; D).

2.2

Duality theory

We will use the notion of \duality" in this book following the notation and results from Keulen [84]. For a short introduction about representations of dual spaces, duality pairings and pivot spaces we refer to Aubin [4]. The dual space of a Hilbert space Z is just the linear space of bounded linear functionals, that is, (Z; C ). We shall denote (Z; C ) by Z d . The vector space d 0 Z d, Z is a Banach space with the dual norm d deýned as follows: If z then 0 z (z ) 0 z d = sup :

L

kk

z

2

Z;z

L jj þ jj j j =6 0 k k

2

z

If Z is a Hilbert space, then there exists a surjective isometry J from Z to Z d , deýned by (J (z1 )) (z2 ) = z2 ; z1 Z for all z2 Z . The dual space Z d is a Hilbert space with the inner product deýned as follows: if z10 and z20 Z d , then

h

h 10 20 i z ;z

Zd

=

i

2

ÿ1 20 ÿ1 10

þ

ÿ

J

! !

z ;J

z

Z

= z10

ý

J

ÿ1 ( 20 ) z

2

ü

:

d We call the isometry J : Z Z the duality map 2 from Z to its dual Z d . ÿ 1 d d The inverse map J : Z Z is denoted by þZ and the inner product on Z satisýes 0 0 0 0 Z for all z10 and z20 in Z d: z1 ; z2 Z d = þZ z2 ; þZ z1

h

i

h

i

2 The duality map is conjugate linear, that is, it satisÿes ÿ2 J (z2 ) for all ÿ1 and ÿ2 in C , and all z1 and z2 in Z .

J (ÿ1 z1

+

ÿ2 z2 )

=

ÿ1 J (z1 ) +

2.2.

17

Duality theory

hÿ ÿi

The duality pairing

;

h i 0

=

z ; z hZ d ;Z i

Let

Z

0

z

0

2L

ÿZ j

tation of

h

(z ) =

Z

d

Z

0

onto

Z

d

z

0

2

i

0

z; ÿZ z

for all

Z;

0

þ! þ!Z j

Z

ÿ

d

Z.

=

Z

0

and

2

z

Z.

h

i

0

d

Z

Z

d

and

f g 0

The pair

hÿ ÿi

. We deÿne the duality pairing z ; z hZ 0 ;Z i

2

0

z

2

Z:

be a conjugate linear map which is an

j

(Z 0 ; Z ) is an isometry onto

h i

z

, that is, Z

0

for all

d ;Z i is deÿned by

be a Hilbert space and let

isometry from

and

hZ

;

Z ;j

h

0

= (j z )(z ) =

j z ; z hZ ;Z i

In this case, we also call

Z

is called a represen-

by

hZ 0 ;Z i

0

z; ÿZ j z

0

i

Z;

the dual space of

Z

(this

is standard abuse of terminology). We say that Z is identiÿed with its dual if the representation for Z d is chosen ÿ1 to be Z; ÿZ . It follows that in this case, the duality pairing corresponds to

f

g

the inner product on

Z:

h

i

=

z1 ; z2 hZ;Z i

for all

z1

and

z2

in

Z.

If

Z

h

z2 ; ÿZ ÿ

ÿ1

Z

i h =

z1 Z

i

z2 ; z1 Z

is identiÿed with its dual,

Z

is called a pivot space.

Usually, it is assumed that whatever choice is made for the representation

f g L L 0

Z ;j

of

(Z 0 )d =

given by

(Z; C ), the representation of the dual of this dual space (that is,

f

(Z 0 ; C )) is given by

cZ

=

ÿ1

ÿ

Z0

j

ÿ 1 ÿ1

ÿ

Z

Z; cZ

:

0 d

(Z ) (we could also say that

h

z

00

Z

00

;z

0

i

=

g

, where

Z þ! þ!Z þ! ÿ

Z ).

0

Z

0

z

00

z ;z

0

i

ÿ1

ÿ

Z0

ý

00

j

0

z ;j

ý

i

to

Z

d

Z

hZ;Z 0 i

;z

0

i

d

h(Z 0 ) ;Z 0 i

0

ÿ

ÿ1

ÿ1 ÿ1

ÿ

Z

ÿ1 ÿ1

ÿ

Z

z

00

; jz

=

h

=

(j z )(z )

Z

0

ÿZ j z ; z 0

h

= Z

0

;z

00

ÿ

=

h

ÿ

d

(cZ z )(z )

=

with

00

cZ z

=

Z

Z

h h

=

hZ 00 ;Z 0 i

=

given by

j

Z

This implies that

=

We identify the bidual of

is the isometry from

cZ

0

z ;z

00

0

i

z

z

00

00

ü

Z

þ

ü

0

(z )

Z0

d

Z

00

00

i

hZ 0 ;Z i :

itself and the duality pairing

hZ;Z 0 i

=

h

0

z ;z

00

i

hZ 0 ;Z i :

hÿ ÿi ;

hZ;Z 0 i

is

18

Chapter 2.

If we have two Hilbert spaces Here, we identify an element Similarly, if

w

2

0

W

d

2

j

z

satisÿes

and

W 0

w

0

Z

z

2

0

d

Z

and

2

w

h i 0

Z

=

Z

0

for

w1

h

0

and

in

w2

Z

=

Z

0

0

0

w1 ; w2

i

0

2

Z

d

,

f

ÿZ

0

W ;

k k ÿZ z

is given by

g

d

Z

#

Z

=

to

Z.

=

W0

Z

,

Z

1

ÿZ z

d

0

W

; w)

Z

Z

=

Z

0

Z

W

image of

V

d

i

0

we have

W

,

ü

= W

ÿZ

0

1

ÿ

z

0

û

7!

n

h 0

0

0

v1 ; v2

V ;

(ÿZ

i

Z

!

,

Let

W

W

W

0

j h i

1

ÿ

V

d)

D

o

(ÿZ

!

Z is continuous.

j

V

0

ýý

Zd

ü

ÿZ

(w ) =

d)

0

0

which is a repre-

, as follows. Recall

þ

0

w1

Zd

kk

=

0

z

1

ÿ

ü

w

1

ÿ

ÿ

Z

0

û

Wd:

to

ÿZ

V

1

ÿ

to mean W

ú

0

of

0

v2 ;

(ÿZ

ÿ

W

d

z

0

û

h i

(w ) =

V

with

V d

w; z

!

Z ,

such that to

ÿZ

hþ þi ;

j

(ÿZ

j

1

ÿ

V d)

1

ÿ

V

d)

v

0

hW 0 ;W i

(w )

is a representation of =

then

be the completion of

.

=

V0

we use the notation ,

x : W

W,

W :

d denote the restriction of

0

x

!

0

so that the duality pairing

=

w ; w hW 0 ;W i

v ; v hV 0 ;V i

3 Here

2

. Furthermore,

V

d

V

V 0

0

Z:

, we have

!

,

Z

=

and deÿne

Z

V

under this map. Furthermore, deÿne the inner product on

Then the pair is given by

j

ÿZ V

z

.

(W; C ).

. Furthermore, we can choose the pair

d

and we can deÿne a representation

as follows. Let

0

d

d

If in addition we have a third Hilbert space Z

w

W

0

,

ÿ

1

ÿ

W

!

0

ýý ÿ

0

w2 ; ÿ

!

d

W

0

W ,

Z

to

h

0

=

1

ÿ

for all

! L ,

0

!

ÿ

as a representation of

z ; w hW 0 ;W i

d

W

Z

d

=

has a unique conjugate linear extension

ÿZ

2 ý h i ! !

and for all (z

Finally,

0

is an isometry from

(ÿZ )ÿ1

ÿ

. We have

Z

Thus it follows that and

z

W

Z

d

z ; w hW d ;W i :

W ,

=

W0

ÿZ

z

z

then

h i

(w ) =

such that

W

Z

For all

0

Z,

0

is also denoted by

Z

with respect to the inner product

Z

0

(a constant)

to

denotes the canonical map from

ÿZ

=

z

! j 2 jj jj ,

is identiÿed with its dual, we can deÿne a space

sentation of the dual of that

jÿ

0

W

(Z; C ) with

=

we have

W

z ; w hZ d ;Z i

Now if

w

with 3

L

Z

(z )

the unique continuous extension of for all

d

Classes of well-posed linear systems

ù

V

0

v1

d

E

Zd

0

d

!

V 0

,

!

,

V

as the V

0

by

:

and the duality pairing

(v ):

Z , W is dense in Z , and the canonical injection

2.2.

19

Duality theory

2 ÿ

Hence for all (v 0 ; z ) 0

=

v ; z hV 0 ;V i

(v

0

V

; w)

0

! 2 ÿ ,

Z

V

0

=

0

we have

Z

ÿ

h i

and

0

V

!

Z ,

V

j

(ÿZ

V d)

1

ÿ

0

þ (z ) =

ý

ÿ

1

ÿ

Z

v

0

ü

h i

(z ) =

z; v

h i 0

h

=

v ; w hW 0 ;W i

w; v

Vd

V

0

0

=

Z

v ; w hV 0 ;V i :

Wd ÿZ

ÿZ Z

0

0

=Z

W

0

Figure 2.3: Interlacing of the dual spaces. and

X

representations of

hþ þi ;

hY 0 ;Y i

that A

ÿY k

from

2.

D (A

h

0

0

Y

ý

û )=

to

X

Y

2 j 9 2h h i 0

y

0

i

Y

A y ; x hX 0 ;X i

The adjoint of

A

x

0

=

þ

h

2

)

=

i

þ

x; A y X

þ

A

0

) from 0

y

=

h

=

0

D (A)

1

ÿ

(ÿY

ÿ

k)

x

0

2

0

D (A

1

Y

0

0

Y ;k

;

to

D (A)

)

ý

) and all

Y

to

);

1

ý 2

)

x ; x hX 0 ;X i

þ

ÿ

(ÿX j )

ù

=

2

0

D (A

h i

=

D (A

(ÿX j )A (ÿY k ) x

from

A

for all

k ) D (A

ù

f hþ þi

and

is an isometry from

ÿX j

i

0

) and

g

X

g

hX 0 ;X i 0

to

X

are and and

. The dual of a densely deÿned operator

Y

such that for all

0

(ÿY

=

D (A þ

0

0

A y

y

X

y ; Ax hY 0 ;Y i

þ

so that for all

to

y ; Ax hY 0 ;Y i

(denoted by

D (A

0

is an operator

Y

0

X ;j

, with the pairings denoted by

, respectively. Note that now

follows: 1.

and

d

is an isometry from

D (A)

0

X

f

are Hilbert spaces and that

Y

d

Z

i h i

Zd ÿZ jV d

Suppose that

0

. We depict this schematically in Figure 2.3. For all

we have

W

v

for all

y

y

X

2

0

, deÿned as

ú , and

2

x

X

D (A).

is related to

þ

D (A

A

0

by

);

we have 0

(ÿX j )A (ÿY k )

ø

y; Ax

i 2L

1

ÿ

ø

y; x

hX 0 ;X i

hY 0 ;Y i

Ax; y Y :

2L

For a bounded linear operator T (X; Y ) we have T 0 (Y 0 ; X 0 ), T þ = 0 ÿ1 (ÿX j )T (ÿY k ) (Y ; X ). Owing to the identiÿcation of the bidual of a Hilbert

2L

space with itself, we have that an operator

T

2L

(X; Y ) satisÿes (T 0 )0 =

The dual of a linear densely deÿned closed operator denoted by

0

A

:

0

D (A

) (

ý

Y

0

)

!

X

0

A

:

D (A)

(

T.

ý ! X)

Y

,

is closed and densely deÿned. Hence the

20

Chapter 2.

dual of

A

0

, denoted by (A0 )0 :

and moreover,

D ((A

0 0

))=

0 0

D ((A

)) (

ÿ

0 0

D (A)

and (A )

X

Classes of well-posed linear systems

00

=

x

=

X)

!

Y

for all

Ax

00

x

=

2

Y

is well-deÿned

D (A).

Since we identify the bidual of a Hilbert space with itself, it follows that the dual of an operator T

2L

T

0

2L

(X; X 0 ) satisÿes

(X; X ) self-dual if

T

is self-adjoint and satisÿes

2.3

=

h

T

0

T

2L

0

(X; X 0 ). We call an operator

. In this case the operator

i

h

=

y; S x X

i

:= (ÿX j )T

S

T x; y hX 0 ;X i .

2L

(X )

The Pritchard-Salamon class

In this section we introduce a special class of well-posed linear systems for which there exists a rich literature (see Salamon [72], [73], Pritchard and Salamon [66], Curtain et al. [15]). Most systems containing delays and some systems described by partial diþerential equations ÿt into this framework. This class generalizes the class of state linear systems with bounded input and output operators studied in [27]. Let

V

and

be separable Hilbert spaces with continuous, dense injections and

W

which satisfy

! !

W ,

Suppose ÿ þthat T

W

(t)

that

tÿ0

,

ÿ

A

Z ,

V:

is theþ inÿnitesimal generator of strongly continuous semigroups

T

Z

(t) T

tÿ0

V

(t)

j

and

Z

=

ÿ

T

T

Z

V

þ

(t )

on

tÿ0

(t) and

T

Z

W, Z

j

(t)

W

=

and

T

W

V

, respectively, such

(t):

Since these semigroups are consistent, we shall simply use the notation

f g

T (t) tÿ0 .

Assume further that

U

and

Y

are separable Hilbert spaces (the

input and output spaces), respectively.

1.

2L f g

(U; V ) is a Pritchard-Salamon admissible control operator for

B

T (t) tÿ0

for all 2.

u

2L f g C

if there exist a

ýZ ý ý ý

2

t

T (t

0

þ

t >

ý )Bu(ý )dý

ý ý ý ý

þ >

0 such that

ý kk þ

W

u L2 ([0;t);U )

(2.10)

L2 ([0; t); U ).

(W; Y ) is a Pritchard-Salamon admissible observation operator for

T (t) tÿ0

if there exist a

t >

z

2

0 and a constant

k ü k CT (

for all

0 and a constant

W.

)z

L2 ([0;t);Y

)

ü >

ý kk ü

z

V

0 such that (2.11)

2.3.

21

The Pritchard-Salamon class

Remarks: In the above deÿnition,

it is usual to take either

t

Z

=

V

or

Z Z

is not essential and in the applications

W.

=

Furthermore, we remark that if

(2.10) holds for some , then it can be shown that it holds for all

ÿ

t

t

t > 0, where

will depend on . Similarly, if (2.11) holds for some , then it can be shown

t > 0, where þ will depend on t in general.

that it holds for all

See for example,

Remark 2.10, page 11, Curtain et al. [15].

Under the above assumptions, the state linear system þ( Pritchard-Salamon system for any

D 2 L(U; Y ).

A; B; C; Dÿ) is þcalled a D AV ,! W ,

If, in addition,

A; B; C; D) is called a smooth Pritchard-Salamon system.

then þ(

When confusion may arise we use superscripts, for example denote the operators on on

X

by

!0X

to

V , Z or W . We denote the growth bound4 bounds !0W and !0V are diýerent in general). We

X

(the growth

AX , T X (t),

=

now enumerate a few properties of Pritchard-Salamon systems which have been proved in Curtain et al. [15] and in Curtain and Zwart [26] and will be used in the proofs.

1. For a smooth Pritchard-Salamon system,

ÿ þ ÿ þ ý AW = ý AV .

L 2 L Y; V fT t gtÿ0

2. Let

(

ÿ

ýp AW

þ

ÿ

ýp AV

=

þ

and

) be a Pritchard-Salamon admissible input operator for

A; B; C; D) is a Pritchard-Salamon sysA + LC; B; C; D) is also a Pritchard-Salamon system, where A + LC is the inÿnitesimal generator of fTLC (t)gtÿ0 which is the unique ( )

and assume that þ(

tem. Then þ( solution of

TLC (t)z = T (t)z + Furthermore,

fTLC t gtÿ0 ( )

We note that

TLC (t ÿ ü )LCT (ü )zdü:

0

is also the unique solution of

TLC (t)z = T (t)z + trol operator.

Z t

Z t

T (t ÿ ü )LCTLC (ü )zdü:

0

L 2 L(Y; W ) is always a Pritchard-Salamon admissible con-

A; B; C; D) has a well-deÿned G(s) given byý y^(s) = G ( u(s), for all s 2 C +ÿ := fs 2 ü s)^ V W j Re(s)> ûg, where û > max !0 ; !0 , and

3. A smooth Pritchard-Salamon system þ( transfer operator

C

G(s) = D + C sI ÿ AV ÿ

4 The growth bound of a semigroup fT (t)g

inf

t>0

log

kT (t)k

t

tÿ0

,

þÿ1

B:

denoted by

!0 ,

is deÿned to be

22

Chapter 2.

Furthermore,

ÿ

C

ÿþ

ÿ

L

+

I

A

2 !L

B

ý

þ

C ÿ ; (U; Y ) =

H1

þÿ1

V

G

+

Cÿ

:

(C + ; ÿ

H1

W

0

B

0

.

2L

(V 0 ; U ),

C

0

j

(U; Y )

4. Let us assume that the Hilbert spaces duals. Then

Classes of well-posed linear systems

2L

L

(U; Y )), where is analytic, and

G

sups2C + ÿ

Z, U , Y

(Y ; W 0 ),

D

0

k

k

G(s) L(U;Y )


0, and

CT (

þt

e

G

L

)B

.

and

exponen-

2L BB CC CB þ 2

(W 0 ; W ) and

LB

=

0

,

LC

=

G

C (sI

A)

0

ÿ1

. Its

observability

Hankel opera-

. For an exponen-

B

c

L

H1 (

(U; Y )),

deÿned in the closed right half-

(U; Y ), such that

G

is analytic in the open

is bounded and continuous in the closed right

0. Moreover,

ÿ 2

CT (

tain et al.

,

)z

); Y )) is deÿned by ý =

jslim j!1

(That

V

0

0

); Y )) is deÿned by

(U; Y )) denotes the space of maps

plane Re(s)

V

B

); U ); W ) is deÿned by

controllability Gramian LB 0

!

,

,

,

T (s)Bu(s)ds

0

tially stable, smooth Pritchard-Salamon system, c

1

0

0

1

=

(V ; V ) are deÿned by

Gramian LC

ý

B2L

controllability map

u

tor

D

0 W

0

then we say that the Pritchard-Salamon system þ(A; B; C; D ) is

tially stable.

for

û

generates an exponentially stable strongly continuous semigroup on

A

W,

:

are identiÿed with their

2L

If þ(A; B; C; D ) is a smooth Pritchard-Salamon system, then þ(A0 , 0

1

ü

k

G(s)

k

= 0:

(2.12)

Re(s)þ0

L2 ([0;

[15], where

1 L ÿ

CT (

);

(U; Y )) for some

ÿ

>

0 is shown in Cur-

)B has to be interpreted in the sense of Re-

mark 2.10.(iv) [15].

Using the Cauchy-Schwarz inequality, it follows that

CT (

(U; Y )). Finally, using Property A.6.2.g, page 636, Cur-

ÿ 2 )B

L1 ([0;

1 L );

tain and Zwart [27], we obtain the desired result.) We now quote the following lemma from Curtain and Zwart [26] on Lyapunov equations:

Lemma 2.3.1 Suppose that þ(A; B; C; D ) is an exponentially stable, smooth Pritchard-Salamon system. Then

2.3.

1.

23

The Pritchard-Salamon class

L

B

and

L

L

are the unique solutions in

C

of the following Lyapunov equations

h

z1 ; (ALB

1 2

for all z ; z

2

ÿ

(A0 )W

D

h1

z ;

1 2

for all z ; z

2

D

ý

A

0

(A

+ LB A + BB )z2 0

þ 0

+ LC A + C

LC

and ÿ

2L

(V ; W )

2L

ÿ

2L \L \L (V )

(LB LC ) =

ÿ

(V ; V 0 )

respectively

=0

hW 0 ;W i

(2.13)

0

C )z

2

i

=0

hV ;V 0 i

(2.14)

. Moreover,

ÿ

W

ý

D

ü

V

A

(W ),

(LB LC )

0 W

(A )

D

ÿ

V

i

L

and

;

LC

2. LB LC

0

ü V

LB

and

(W 0 ; W )

0

þ ;D

ÿ

(I + þLC LB )

ÿ

0 W

0

þþ :

2L \L ÿ 2 2 L \L 2 L \L

LC LB

1 1

ÿ

üþ

V

A

(A )

;D

and for

(I + þLB LC )

ý

(W 0 ; V 0 )

1

( LB LC ) ,

ÿ

ÿ

(V )

(V 0 )

\L

(W 0 ),

there holds

(W );

0

0

(W )

(V )

and

1

(I + þLB LC )

ÿ

(I + þLC LB )

ÿ

1

2 L ÿý 2 L

ý

D

üü

V

A

ÿ

D

;

0 W

(A )

0

þþ :

From Curtain and Zwart [26], we have

Lemma 2.3.2 Suppose that ÿ(A; B; C ) is an exponentially Pritchard-Salamon system. Then ý (þ r

k k2

(þþ þ) = þ

If

U

=

þ

, where r (

C m and

Y

=

)

þ

nf g

þ)

0 =

ý

(LB LC )

denotes the spectral radius.

nf g

stable,

0

smooth

and r (LC )

=

C p , then the Hankel operator of an exponentially stable

Pritchard-Salamon system is compact (Lemma 3.5 in Curtain et al. [15] and Glover et al. [37]).

Lemma 2.3.3 Suppose that ÿ(A; B; C ) is an exponentially

ÿ1 ÿ1 ÿ [ 1

Pritchard-Salamon system with input space

+1

ýl

< ý < ýl .

The operator J

trum ý (J ) contained in consists of exactly

l

(

;

:=

Æ)

I

(Æ;

þ2

)

negative eigenvalues.

Cm

þþ þ

2L

1

stable,

and output space

(L2 ([0;

for some Æ >

0,

); C m )

and ý (J )

Cp

smooth and let

\ ÿ1 ÿ

has a spec-

(

;

Æ)

24

Chapter 2.

Classes of well-posed linear systems

ÿþ ÿ

ýÿ ü ú with 0 as the accu-

ÿÿ ÿ is compact and has a pure point spectrum with 0 as the accumuþ1 1 ÿÿ ÿ ÿI , it is easy to see lation point. Considering the resolvent I ÿ2 1 that J has a spectrum which is a shifted version of the spectrum of ÿ 2 ÿÿ ÿ. Proof

û

Finally, since ÿÿ ÿ has a pure point spectrum mulation point, and since

þl+1 < þ < þl , J

2

2

þ1 ; þ2 ; : : :

has exactly

l

negative eigenvalues.

Finally we give a simple example5 of a delay system which can be modelled as a smooth Pritchard-Salamon system.

ÿsÿ

e

Example 2.3.4 Let G(s) =

of

G.

Let

Z

=

1;2

þý ;

(

0. We will þrst give a realization

ý >

ÿ L2 (þý ; 0) (with the obvious inner product).

C

Sobolev space

W

, where

s+1

8 < 0) = :

2 L2 (þý ; 0) j

f

9 = distributions) is a regular distribution ; with 2 (þ 0) the derivative of

Tg ;

equipped with the inner product

g

Z h

f

L2

(in the sense of

W 1;2 (þþ;0) =

f (x)g (x)

;

ý;

0

hf; gi

Consider the

0

+ f (x)g 0 (x)

i

dx:

þþ

(With this inner product,

W

1;2

þý ; 0) is a Hilbert space: see for example, Propo-

(

sition 5, page 55, Yosida [91]. Furthermore, from Corollary 7.3 [80], it follows that

W

1;2

þý ; 0) ý C (þý ; 0).)

(

ù

Deþne

A

÷ù

with D (A)

=

ø

z1 z2

Then it can be shown that

A

z1

ø ùþ =

z2

z2

W.

5

ÿ

Let

W

z ~1

y (t)

= =

tý0

øô

z ~2

B

D (A)

z (t)

z (t

ÿ

+ u(t); ÿ ):

:

=

on

Z

(see for instance, Example

hz2 ; z~2 i

W 1;2 (þþ;0)

W

ù ø

ÿ

= z2 (0)

with the inner product

tý

Bu z (t)

fT (t)g

:=

fT (t)g 0 restricts to a 2 L(C ; Z ) be deþned as follows:

is a Hilbert space and on

ö z1

is the inþnitesimal generator of an exponentially

øù ;

;

0

z2

;

2.8, van Keulen [84]). The space z1

ø

2 Z j z2 2 W 1 2 (þý ; 0);

stable strongly continuous semigroup

õù

z2 (0)

=

u

0

;

strongly continuous semigroup

2.4.

25

The analytic class

Finally, if

:

C

D (A)

!C

is given by

ÿ

then

2L

and

B

C

ÿ

= z2 (

z2

(W; C ). We now choose

C

is explained that

þ

z1

C

=

V

Z.

ÿ );

In Pritchard and Salamon [66] it

are Pritchard-Salamon admissible control and obser-

vation operators, respectively. Finally we note that satisÿed since

=

W

D (A),

ý Vü A

!

,

W

is trivially

and so in fact we have a smooth Pritchard-Salamon

system, with a transfer function

2.4

D

eÿÿs . s+1

ÿ

The analytic class

In this section we introduce the class of exponentially stable analytic inÿnitedimensional systems and establish some basic system theoretic properties. First we recall the basic properties of exponentially stable analytic semigroups using the notation from Staþans [77] (see also Pazy [63]). 1. Let

ÿ

A

be the inÿnitesimal generator of an exponentially stable analytic

semigroup

f gtÿ 2R T (t)

in

0

Z.

k kZ

ý ü , we let6 Zÿ = Aÿÿ Z be D Aÿ with the norm û û þ ú ÿ ÿ ù ûAÿ z û and inner product z1 ; z2 Zþ = A z1 ; A z2 Z . Z

2. For each

3. The restrictions of for

þ


0 (which we still denote by

ÿ for all

ÿ 2L

ÿ

h

0 and the extensions of

A)

þ =

z

ÿ

A

to

ÿ

Z

generate analytic semigroups in

. The generator of the semigroup

(Zÿ +1 ; Zÿ ).

f gtþ T (t)

0

on

ÿ is then

Z

These semigroups are all similar to each other, and they commute with

þ for all ý T (t) tþ0 .

f g

A

2R

. We therefore denote all of them by the same symbol

2R

4. For each

t >

0 and

5. For each

þ

0, there exist

þ

þ

,

T (t)

ÿ into

maps

K1 >

û ÿ û ûA T (t)û

Z

0 and

ý

K1

ü >

ÿýt e ÿ

t

\þ2R

Z

þ.

0 such that

;

t >

0

where the norm represents the operator norm in any one of the spaces If

þ


û ÿ û ûA T (t)û

0 and

L(Z )

ý

ü >

0 such that

ÿýt ;

K2 e

ÿ.

Z

t >

0:

(This follows from Lemma 6.3, page 71, Pazy [63].) 6 For

the deÿnition of

Aÿ

are its related properties, we refer the reader to

x2.6

of Pazy [63].

26

Chapter 2.

6. The map

t

7!

T (t)

1 !L 2R

: (0;

)

Classes of well-posed linear systems

(Z ) is continuous in the norm topology.

(See for example, Proposition 2.1.1.(iv), page 35, Lunardi, [53].) This implies that for any

, the map

ÿ

7!

t

continuous in the norm topology.

T (t)

: (0;

7. The same conclusion as above can be repeated with 0

give another chain of Hilbert spaces 0

We identify that

=

Z0

where

0

=

Z0

Z.

For

h

z1 ; z2

hÿ ÿi ;

with the dual of

Zÿ

2

z1

i

hZ

Zÿ

and

D

=

ÿÿ ;Zÿ i 0

= (A )

by using

Zÿÿ

0 ÿÿ

Z

0 ÿÿ

z2

2

Z

(Zÿ ) is

replaced by

A

A

0

to

with similar properties.

we have

0 ÿÿ

ÿ

(A )

A z2 ;

)

as the pivot space. Note

Z

Zÿ ,

1 !L

E

z1

; Z

denotes the duality pairing.

0 ;Zÿ i ÿÿ

hZ

Next we analyze the system theoretic properties of the class of systems described by the triple (

A1.

þ

A

þ

A; B; C )

B

2L

C

f g

T (t) tþ0

on the Hilbert space

(U; Zþ ), where

satisfying A3.

2L

þ

1

< þ

ý

(Z; Y ), where

ý

under the assumptions

satisfying

þ

1

Z.

is a Hilbert space and

U

þ

is a ÿxed number

0. is a Hilbert space.

Y

2L ! ! !

We note that the assumption A2 implies that for every

Z,

is the inÿnitesimal generator of an exponentially stable analytic

semigroup

A2.

on the Hilbert space

< ý

ý

þ: Z ,

A

þ

Zþ ,

B

Zý ,

(U; Z ) and

B

Zÿ1 .

2L

(U; Zý )

Remark: We remark that what follows is similar to what was done in Curtain

and Ichikawa [14], but there are diþerences in assumptions (see Table 2.1). First we derive several results about

B

Lemma 2.4.1.1 below we prove that if

1

then the controllability map

B B

:

u

L2 ([0;

Z

=

ý

and its controllability Gramian. In 1 ý < 2 + þ, ); U ) Zý given by is a real number satisfying

!

1

0

T (t)Bu(t)dt

is a well-deÿned bounded linear map, that is,

f g

B

(2.15)

is an admissible input operator

(see Weiss [86]) for the semigroup T (t) tþ0 on the Hilbert space Zý , where 1 ý < 2 + þ. Moreover, in part 2. we show that if ý and ü are real numbers satisfying ý ü < 1 + 2þ, ý < 1 + þ, ü > (1 + þ), then the controllability

Gramian

þ

LB

:

0

Zü

!

Zý

given by

Z

LB z

=

þ

1 0

0

0

T (t)BB T (t) zdt

(2.16)

2.4.

1.

27

The analytic class

Curtain and Ichikawa [14]

Present approach

They considered

Here we simply assume that

A

to be a

ÿ

uniformly strongly elliptic

A

operator of order two with

is the inÿnitesimal

generator of an exponentially

smooth coeÆcients

stable, analytic semigroup on

deÿned on a bounded open domain þ of d .

Z.

Thus

ÿ

R

is the inÿnitesimal

A

generator of an exponentially stable, analytic, compact semigroup on 2.

B

2L

(U; Zÿ ),

=

2 (@ þ).

where 3.

U

Z.

L

ÿ

1

ÿ 34

< ÿ


þ 21 .

fT (t)g 0 tþ

on the Hilbert space

Zþ

for any

Theorem 2.4.3 Let þ

þ

2R

2R

satisfying

þ 21

satisfy

þ 12

< þ


in

ÿ(

þA),

þ ÿ(þ ) = þ(s þ þ )C (sI + A)ÿ1 (þ + A)ÿ1 B:

Finally we show that the system þ given by the triple (

4.

þ

g

is well-

C+ 0 :=

fs 2

0 . In order to do this, we use the exponential stability of the

generator of the semigroup and (2.29): We have for all

s

2 C +0

ÿ ÿ ÿ ÿ1 ÿ ÿ(sI + A)ÿ1 ÿ ÿ ÿ L(Zÿ ;Zÿ+1 ) ÿ A(sI + A) L(Zÿ ) ÿ ÿ1 ÿ = ÿI þ s(sI + A) ÿ L(Zÿ ) ÿ ÿ ÿ ÿ kI kL(Zÿ ) + jsj (sI + A)ÿ1 ÿL(Zÿ )

for some

M >

ÿ

1+

jsj

=

1 + M;

M

jsj

(using Theorem 5.2.(c), Pazy [63])

0. This completes the proof.

Remark: A more general result with unbounded C can be proven in a similar

manner. We quote the following from Staýans [77]:

þA generate an exponentially stable analytic strongly confT (t)g þ0 in Z , B 2 L (U; Z B ), C 2 L (Z C ; Y ) and D 2 ÿ ý < ý + 1. Fix any ü satisfying ý þ 12 < ü < ý + 12 .

Theorem 2.4.4 Let tinuous semigroup

L(U; Y ), where ý Then þA, B , C

B

X

=

t

C

ÿ

ÿ

B

B

C

generate a regular well-posed system on

Zþ . The transfer function is given by G(s)

=

C (sI

(U; Zþ ; Y ),

+ A)ÿ1 B + D ,

where and it

satisÿes

jslim j!1 G(s) = D: 2C +0

s

In the sequel, the operators

LB LC

and

LC LB

play an important role, so we col-

lect some properties here which readily follow from Lemmas 2.4.1.2 and 2.4.2.2.

2.4.

Corollary 2.4.5

1. 2.

43

The analytic class

LB and LC deÿned by (2.16) and (2.24), respectively, satisfy:

LB LC 2 L (Zÿ ) for any ÿ satisfying ÿ1 < ÿ < 1 + þ. L

C LB

2 Lÿ

Z

þ 0

þ

for any

satisfying

ÿ

ÿ

ý
0. Suppose that

wk

H

k k

f

wk

1

= 1. Deÿne vk =

g

k

ÿk S

ÿ ÿ

ify that the vk 's are orthonormal and S vk = ÿk wk . Let þ

2 onto span w1 ; w2 ; : : : ; wl+1 ; then

jection from

ÿ ÿ

Consider the following restriction of þ S :

f

ÿ ÿ

: span v1 ; v2 ; :::; vl+1

þ S

P

which has rank at most z =

l+1 k=1 ak vk ,

l,

P j j l+1 k=1

X

=

ak

ÿ

X

2

þ Sz

l+1

=

Thus

k þ ÿk ÿ S

S

k=1

ü 2 G

with symbol G, namely H

inf c

þü 2 k ü )

ü ÿü 1c C pþm ü k1 ÿ k (

G(i ) + K (i )

ÿ ÿ

ker (þ S ),

kk z

.

= 1. If

= 1, and

þ

2 2

ÿk ak

ÿ ÿ

þ S z

ÿ

k k

2 ÿl+1

1 R C pþm

L

(

;

X 2

=

l+1 k=1

ÿ

þ Sz

2

k

2

2

ak = ÿl+1 :

) be such that the Hankel operator

;l

ÿ

(C p

m)

k ü

ü k1 ÿ

ü

). Thus,

ÿ

G+G

l

G(i ) + K (i )

ÿ +1 :

ÿ þü

Let K ( ) = G ( ) + F ( ), where G ( H

2

g

S

, is compact. Then

K (ý )2H1

F(

span w1 ; w2 ; :::; wl+1

S

ÿl+1 .

Theorem 4.1.2 Let G(i )

Proof

2

S )

ak ÿk wk ;

k=1

k þ ÿk ÿ k S

f

þ ÿ k ý k þ ÿk

l+1

ÿ

þ Sz

S

g!

wk . It is easy to ver-

be the orthogonal pro-

þ (S

and hence there exists a z

then we have

ÿ

is an orthonormal sequence of eigenvectors of the

2

operator S S : S S wk = ÿk wk ,

+F

kH k =

) has MacMillan degree

ÿ kH

G +G

=

k

ý

l, and

G + HGþ k ÿ ÿl+1 (G):

H

The ÿrst equality follows easily from Lemma 8.1.2.c and Examples 8.1.3 (page 388, Curtain and Zwart [27]).

The last inequality is a consequence of Lemma

4.1.1 above and the classical result that the rank of the Hankel matrix of the transfer function of a ÿnite-dimensional system is equal to its MacMillan degree (see for instance Kalman et al. [48]).

4.1.

The sub-optimal Hankel norm approximation problem

ÿ 2C R C

Theorem 4.1.3 If G(i )

(

ÿ

p m ) with limit

;

þ 2 1 Cÿ

K(

Proof

1.

inf

H c ;l ( p

)

m)

k ÿ

0 at

ÿ k1 ý

G(i ) + K (i )

87

þ1

, then

i

ÿl+1 :

From Hartman's theorem (see for instance Corollary 4.10, page

46, Partington [62]), it follows that the Hankel operator corresponding to G is compact. Consequently, from Theorem 4.1.2 above, we have

þ 2 1 Cÿ

K(

2.

inf

H c ;l ( p

)

k ÿ

ÿ k1 ü

G(i ) + K (i )

m)

We now prove the reverse inequality.

f g 2N

ÿl+1 :

From Theorem A.6.11 (page 640,

Curtain and Zwart [27]), we have the existence of an approximating sequence Gn

n

2N k ÿû

üN

of rational transfer matrices with no poles on the imaginary axis for

G. Given any þ > 0, choose

N

large enough such that for all

G(i )

ÿ k1

Gn (i )


0 was arbitrary, the claim follows.

From the results shown in Theorems 4.1.2 and 4.1.3, we have the following:

ÿ 2C R C

Theorem 4.1.4 If G(i )

(

ÿ

p m ) with limit

;

þ 2 1 Cÿ

K(

)

inf

H

c

;l (

p

m)

k ÿ

û1

0 at

ÿ k1

, then

= þl+1 :

G(i ) + K (i )

In light of the above theorem, we are now ready to state the sub-optimal Hankel

norm approximation problem that we consider:

Let

ÿ 2C R C ÿ k ÿ

G(i )

(

p m)

;

þl+1 < þ < þl ;

If

þÿ 2 ÿ k1 ü

with limit

then ÿnd

such that

K(

0 at

)

1

H

G(i ) + K (i )

c

û 1ÿ Cp

;l (

:

m)

þ:

K is then called a solution of the sub-optimal Hankel norm approximation problem.

þÿ 2

In the following corollary, we show that any solution K0 (

problem in

þÿ 2 ÿ üÿ

ÿ

Corollary 4.1.5 If K (

)

1 l C pÿm ü þÿ

H

c

;

) is such that

(

þÿ 2

þl , then K ( ) = G ( ) + F ( ), where G (

l

ÿ

1

H

p m ). (C ;[l]

Proof degree

r

ÿ

ÿ

ü

k ÿ

r

þÿ 2 1 ýk ÿ and all

þl > þ

c

(

;

)

)

H

c

)

1 C pÿm

ÿ

H

c

(

ÿ

;l

ül ÿ k1 ý

G(i ) + K (i )

r

þ
þr+1 , which implies that

)

inf

H c ;r ( p

l < r + 1,

m)

G(i ) + K (i )

and so

. Hence

þr+1 :

=

.

4.2.

4.2

89

Main assumptions and a few useful consequences

Main assumptions and a few useful consequences

In this section, we state the assumptions S1-S5 under which we solve the suboptimal Hankel norm approximation problem. We will also prove a few elementary consequences of these assumptions which will be used in the sequel. We make the following assumptions:

S1. S2.

G(iÿ) 2 C (R; C pÿm ) with limit 0 at þi1. ÿl+1 < ÿ < ÿl , where ÿk 's denote the Hankel singular values of G.

S3. There exists a ÿ(

ý

=

S4. ÿ(

ýÿ

)

Ip 0 G(i!)þ Im ý þ Ip ÿ(i! )

V (ýÿ) s 2 C +0 .

0

2

ÿp

(p+m)

üý

exists a

0

!!ý1 ÿ(i! ) =

1

Ip

üý

!2R

Ip G(i!) 0 Im

m

ü

i!):

c H þ1

ÿ

(4.4)

C (p+m)

ÿp

( +m)

such that ÿ(

þ

ýs V ýs

, that is, there

)

(

) =

Ip+m

ü

0

ÿIm

0

2

ÿ(

m

such that for

ýÿ I

0

ýI

þ

0

ü

ÿ H c C (p+m)ÿ(p+m) ý

( +m)

Ip

) is invertible as an element of

for all

S5. lim

ýÿ 2 H1c ÿC

.

Remarks:

1. From Hartman's theorem (see for example Corollary 4.10, page 46, Partington [62]) and S1, we have that the Hankel operator with symbol

G is

compact.

2. From S5 and Lemma 2.5.9, it follows that

!1+

ýs

lim ÿ(

s

s2C 0

ý

) =

Ip 0

ü

ÿIm : 0

iÿ) at þ1. Indeed, if there exists a iÿ) has a limit, say ÿ1 at þ1, then

3. We remark that S5 is only a normalization condition. All we actually need is the existence of a unique limit of ÿ( ÿ0 satisfying S3 and S4 such that ÿ0 (

90

Chapter 4.

Characterization of all solutions

it can be easily seen that ÿ deþned by

ÿ

s

Ip

ÿ( ) = satisþes S3, S4

and

þ

0

ÿIm

0

ÿ1 ÿ0 (s)

1

for all

ÿ

s 2 Cÿ 0:

S5.

Lemma 4.2.1 If the assumptions S1-S5 hold, then

1.

lim s

!1 V (ÿs) =

ÿ

Ip

2C +0

2.

5.

6.

.

ÿ

V22 (i!) is invertible for all ! 2 R,

3. There exists a 4.

1 Im

0

s

þ

0

and

ýý ý V22 (iþ)ÿ1 V21 (iþ)ý1 < 1.

Kþ(ÿþ) 2 H1 l (C pým ) such that kG(iþ) + Kþ (iþ)k1 ý ÿ. c

V22 (ÿþ)ÿ1 2 H1 l (C pýp ). c

;

;

K0 (ÿþ) := V12 (ÿþ)V22 (ÿþ)ÿ1 2 H1 [l] (C pým ) K0 (iþ)k1 ý ÿ. c

;

kG(iþ) +

and it satisÿes

V12 (ÿþ); V22 (ÿþ)) is a right coprime factorization over MH1 of K0 (ÿþ) 2 H1 [l] (C pým ). c

(

c

;

7.

det(

C+ 0.

V22 (ÿþ))

Proof 1. From S5,

ÿ

V (ÿs) ÿ I0p 10I m

þ =

üÿ

V (ÿ s )

2.

Ip

0

þ

ÿIm

0

ÿ

and the fact that

l

has no zeros on the imaginary axis, and exactly

ûÿ

ÿ ÿ(ÿs)

Ip 0

0

zeros in

1 Im

þ

;

ÿ

V (ÿþ) 2 MH1 , the result follows. c

ÿ satisþes S3, and so taking inverses, we obtain

V (i!)

ÿ = for all

! 2 R.

ÿ

Ip 0

Ip G(i!) 0 ÿIm

þ

þ ÿIm V (i!) 0

þÿ1 ÿ

Ip 0

0

ÿ 12 Im

þÿ

ÿ

Ip 0 G(i!)þ ÿIm

þÿ1

:

(4.5)

! 2 R:

(4.6)

; ÿblock of the above yields

Considering the (2 2)

V21 (i!)V21 (i!)þ ÿ V22 (i!)V22 (i!)þ = ÿ 2 Im ; ÿ 1

where

4.2.

91

Main assumptions and a few useful consequences

Thus for

u 2 Cm

we have

kV22 (i!)ÿ uk2 = kV21 (i!)ÿ uk2 + 2 kuk2: ÿ 1

V22 (i!)ÿ u = 0, then u = 0. So it follows that V22 (i!)ÿ is invertible for all ! 2 R, or equivalently, V22 (i! ) is invertible for all ! 2 R. ÿ ÿ2 ÿ2 1 ÿ From (4.6), we have ÿV22 (i! )þ1 V21 (i! )uÿ ÿ kuk2 = ÿ ÿ 2 ÿV22 (i! )þ1 uÿ . Let M > 0 be such that kV22 (i! )k þ M for all ! 2 R, we obtain kuk2 þ ÿ ÿ ÿ ÿ kV22 (i!)k2 ÿV22 (i!)þ1 uÿ2 þ M 2 ÿV22 (i!)þ1 uÿ2 . Hence Thus if

ÿ ÿ ÿV22 (i! )þ1 V21 (i! )ÿ2 þ 1 ÿ and so we have 3.

From S3 it

ÿ ÿ ÿV22 (iý)þ1 V21 (iý)ÿ

G(iý) 2 C (R; C pým )

0. ÿ ÿ1 V21 (iÿ)ÿ1 = 1 þ r. It follows from Lemma 2.5.7 0 < r < 1 such that ÿV22 (iÿ) a Æ1 > 0 such that Æ1 < ÿ and for any þ satisfying 0 < þ < Æ1 , ÿthat there exists ÿ ÿV22 (þþ þ iÿ)ÿ1 V21 (þþ þ iÿ)ÿ ý 1 þ r . Similarly it follows from Lemma 2.5.7 2 1 that there exists a Æ2 > 0 such that Æ2 < ÿ and for any þ satisfying 0 < þ < Æ2 , r kU1(þþ þ iÿ)U2(þþ þ iÿ)k1 ý 1 + 1 þ4 r = 1 þ1 r : 4 4 Let Æ := min fÆ1 ; Æ2 g, and ÿx a þ satisfying 0 < þ < Æ . Deÿne ý(ü; s) = det (üV21 (þþ þ s)U1 (þþ þ s) + V22 (þþ þ s)U2 (þþ þ s)) ; where ü 2 [0; 1]. The zeros of det(

half-plane

C+ ÿ ,

where

a. We know that

ý(0; ÿ) ý(1; ÿ)

V22 (þþ þ ÿ)U2 (þþ þ ÿ)) and = det (V21 (þþ þ ÿ)U1 (þþ þ ÿ) + V22 (þþ þ ÿ)U2 (þþ þ ÿ)) + . are meromorphic (in fact analytic!) in C ÿþ=2 + b. ý(0; ÿ) has a nonzero limit at inÿnity in C 0 : det(V22 (þÿ)) has a nonzero limit + + at inÿnity in C 0 and det (U2 (þÿ)) has a nonzero limit at inÿnity in C 0 (see =

det (

(4.13)).

ý(1; ÿ) has a nonzero limit at inÿnity in C +0 , U1 (þÿ) is proper in C +0 , and the above.

since

V21 (þÿ)

is strictly proper,

ü; s) 7! ý(ü; s) : [0; 1] ü iR ! C is a continuous function, and ý(0; i!) = det (V22 (þþ þ i!)U2 (þþ þ i!)) = det(V22 (þþ þ i! )) det (U2 (þþ þ i! )) ; and ý(1; i!) = det (V21 (þþ þ i!)U1 (þþ þ i!) + V22 (þþ þ i!)U2 (þþ þ i!)) :

c. (

d. We have

ý(ü; i!) = det(V22 (þþ þ i!)) det (U2 (þþ þ i!)) ÿ ÿ1 V21 (þþ þ i!)U1 (þþ þ i!)U2(þþ þ i!)ÿ1 þ det I + üV22 (þþ þ i! ) 6= 0;

since

ý ý

þþ þ iÿ ÿ1V21 þþ þ iÿ Uý1 þýþ þ iÿ U2 þþ þ iÿ ÿ1ýý1 ý þþ þ iÿ ÿ1V21 þþ þ iÿ ý1 ýU1 þþ þ iÿ U2 þþ þ iÿ ÿ1 ý1 þ þr < ; 4

ý ýüV22 ( ý 1 ýV22 ( h ri 1

2

)

(

)

1

1

(

1

)

)

(

)

(

(

)

)

(

)

94

Chapter 4.

Characterization of all solutions

V22 (ÿÿ ÿ i!)) 6= 0 and det (U2 (ÿÿ ÿ i!)) 6= 0.

det(

þ(ý; 1) = 6 0, since V21 (ÿþ) is strictly proper, U1(ÿþ) is proper + det(V22 (ÿþ)) det (U2 (ÿþ)) has a nonzero limit at inÿnity in C 0 .

e.

in

C+ 0 , and

Thus the assumptions in Lemma A.1.18 (Curtain and Zwart [27], page 570) are satisÿed by

þ(1; þ)

þ,

and hence it follows that the Nyquist indices of

are the same.

þ(0; þ)

and

Consequently, the number of zeros are the same (the

þ(0; þ), þ(1; þ) are analytic in C +ÿ Æ ) and so the 2 + sum of the number of zeros of s 7! det(V22 (ÿÿ ÿ s)) in C 0 plus the number of zeros of s ! 7 det (U2(ÿÿ ÿ s)) in C +0 equals the number of zeros of s! 7 det (V21 (ÿ+ÿ ÿ s)U1(ÿÿ ÿ s) + V22 (ÿÿ ÿ s)U2(ÿÿ ÿ s)) (= det(M (ÿ + s), using 4.12) in C 0 . In particular, we obtain that the number of zeros of s 7! det (V22 (ÿÿ ÿ s)) in + C 0 is less than or equal to l. But since the choice of ÿ can be made arbitrarily + small, it follows that s 7! det (V22 (ÿs)) has at most l zeros in C 0 . Thus m ÿ m V22 (ÿþ) 2 H1 l (C ). number of poles is zero, as

c

;

K0(ÿþ) := V12 (ÿþ)V22 (ÿþ)þ1 . From Lemma 2.5.5 and part 4 above, it follows that K0 (ÿþ) is an element in H1 l (C pÿm ). We have

5.

Deÿne

c

ÿ

G(i!) + K0 (i!) Im

þ

= =

with

! 2 R,

;

ÿ

þÿ

þ

Ip G(i!) K0 (i!) 0 Im Im þ ÿ þ ÿ Ip G(i!) V (i!) 0 0 Im V22 (i!)þ1 ;

and so we have (for notational convenience, we restrict writing out

the argument

i!)

G + K0)ý (G + K0) ÿ ü2Im þý ÿ þÿ þ G + K0 I 0 G + K0 p = Im 0 ÿüþ2Imÿ Im þý ÿ ÿ þ þÿ þ ÿ ý 0 0 I I 0 I p G p p G ý = 0 Im 0 ÿ ü 2 Im 0 Im V V22þ1 V22þ1 V þ þý ÿ þÿ ÿ 0 0 Ip 0 = þ 1 þ 1 0 ÿIm V22 ; V22 ÿ

(

V . Thus it follows that K0 (i!))uk2 ÿ ü2 kuk2 = ÿ ýV22 (i!)þ1 uý2 for all u 2 C m and all ! 2 R. Hence it follows that kG(iþ) + K0 (iþ)k1 ý ü . where we have used S3 and the ýdeÿnition of ý

k G i! (

(

)+

Thus from Corollary 4.1.5, it follows that

K0 (ÿþ) 2 H1 [l](C pÿm ). c

;

4.3.

95

All solutions to the sub-optimal Hankel norm approximation problem

V ÿÿ ; V22 þÿ þÿ 2 H1 ÿ V12 þÿ V22 þÿ MH1c 22 þs V22 þs þ þ 21 þs V12 þs I; þ 21 þÿ MH1c 22 þÿ V12 þÿ ; V22 þÿ V12 þÿ V22 þÿ þ1 V22 þÿ + We now prove that ( 12 ( ) c (C p m ): of 0 ( ) ;[l]

(

ÿ

(

MH1c

6.

K

)) is a right coprime factorization over

(

) and

(

) belong to

. More-

over,

where ÿ (

7.

(

(

)

(

)

) and

(

ÿ

(

(

)

ÿ

(

) belong to

From Lemma 2.5.6, it follows that det(

All

(

)=

.

Hence it follows that

)) is a right coprime factorization of

axis, and exactly

4.3

))

l

zeros in

(

(

)

(

)

.

)) has no zeros on the imaginary

C0 .

solutions

to

the

sub-optimal

Hankel

norm approximation problem In this section we obtain a nice parameterization of all solutions to the suboptimal Hankel norm approximation problem under the assumptions S1-S5 listed in the previous section.

kQ iÿ k1 ý

K (þÿ) := R1 (þÿ)R2 (þÿ)þ1

Theorem 4.3.1 Suppose that S1-S5 hold.

( )

then

and

R1 (þÿ) R2 (þÿ)

ÿ

þ

þ1 := ÿ(þÿ)

Q(þÿ)

, where

ÿ

2

c (C pÿm ) H1

Q(þÿ) ; Im

satisÿes

þ

(4.14)

c (C pÿm ) and kG(iÿ) + K (iÿ)k ý ÿ . K (þÿ) 2 H1 1 ;[l]

Proof in

1,

If

C+ 0,

Step 1: We show that

det(

V21 (þÿ)Q(þÿ) + V22 (þÿ)) has exactly l zeros

and

they are contained in the open right half-plane.

By Lemma 4.2.1.7, we know that

s 7! det(V22 (þs)) has no zeros on the imagiþ > 0 such that all its

C+ 0 . So there exists an zeros are contained in the half-plane C + ÿ. nary axis, and exactly

l

zeros in

c (C pÿm ). Thus, we V22 (þÿ)þ1 V21 (þÿ) 2 H1 ;l V22 (þÿ)þ1 V21 (þÿ), that there exists a Æ , 0 < Æ < þ, such that whenever 0 ý ý ý Æ , ý ý þ1 sup ýV22 (þý þ i! ) V21 (þý þ i! )ý < 1: From Lemma 2.5.5, it follows that

have from Lemma 4.2.1.2 and Lemma 2.5.7 applied to

Fix such a

ý > 0.

!2R

Consider

ü(û; s) := det (ûV21 (þý þ s)Q(þý þ s) + V22 (þý þ s)) ;

96

Chapter 4.

ÿ + C ÿ) ÿ2

where in

2

;

Characterization of all solutions

þ(0; ÿ) and þ(1; ÿ) are meromorphic (actually analytic + + containing C 0 with nonzero limits at inÿnity in C 0 .

[0 1]. The maps

on an open set

We have: 1. (

ÿ; s) 7! þ(ÿ; s) : [0; 1] þ iR ! C

is a continuous function.

þ(0; i!) = det(V22 (ýý ý i!)), and þ(1; i!) = det(V21 (ýý ý i!)Q(ýý ý i!) + V22 (ýý ý i!)). Im + ÿV22 (ýý ý i!)ÿ1 V21 (ýý ý i!)Q(ýý ý i!) is invertible, since

2.

3.

ÿ ÿÿV22 ( ý ÿ ÿ ÿV22 ( ý

ü




1. c

c

det (U (

s)U

(

+

s))

ÿ

+V

det(V

s

s))

ÿ

s)

ÿ

, that is,

det (U (

s

s

!7 2 þ U

(

s)

Proceeding as in the proof of Lemma 4.2.1.4, we have

s

ÿ

using (4.17)) in of zeros of

H

!7 22 þ þ + 7! 2 þ þ C0 21 þ þ 1 þ þ 22 þ þ 2 þ þ l den l den C+ 0 + !7 þÿ C0 2þ þ

number of zeros of

7!

M

1 (C pÿm ), by showing that

that the sum of the number of zeros of s

[S ( )V

(

(Q) =

in

ÿ

s))

in

C+ 0 plus the

equals the number of zeros of

in

s)U

ÿ

(

(

s))

ÿ

, where

. Thus

(= det(M (ÿ + s),

(Q) denotes the number

Q(

C+ . But ÿ H (C p m ).

) has no poles in

0 can be chosen arbitrarily small, which implies that

Q(

þÿ 2 )

1 ÿ c

Consequently, our main result is the following:

Theorem 4.3.3 Suppose that S1-S5 hold.

k ÿ G(i

ÿ k1 ý

) + K (i )

for some Q(

þÿ 2 )

þÿ 1 þÿ 2 þÿ

þ iÿ K (

ÿ

R

(

)

R

(

)

1 (C pÿm )

H

c

)=

þ

þÿ ÿ þÿ þ1 þÿ k ÿ k1 ý )R

= þ(

)

2 (þÿ)þ1

1(

R

satisfying

Q(i

þÿ 2

Then K (

, where

Q(

)

Im

)

)

1.

þ

1 [l](C pÿm )

H

c

;

and

4.4.

99

A comparison with existing results

4.4

A comparison with existing results

It is appropriate to compare the results in Theorem 4.1.4 and Theorem 4.3.3 of this chapter with a corollary to the abstract results from Ball and Helton [5] as stated in Curtain and Ran [17]. Note that for an easier comparison we have chosen the version which is closer to our main result. Denote by ^(C p m ) ^ h the set h L1 ([0; ); C p m ) . ^l (C p m ) denotes the set of complex p m

n

j 2

1

matrix-valued functions

ÿ

þ

X(

o

A

A

ÿ

ÿ

ÿ

) of a complex variable deÿned in the closed right

half-plane with a decomposition

X

=

þ+

G

F,

þ is the matrix transfer

where

G

function of a system of MacMillan degree at most equal to l, with all its poles ^(C p m ). The following result follows in the open right half-plane, and F

2A

from Ball and Helton [5].

Theorem 4.4.1 Let G G. Then

2A

^

and let ÿk 's denote the Hankel singular values of

k þ þ k1 ýþ 2 A C ÿ ýþ 2 k þ þ k1 ü ýþ ýþ ýþ ý ÿ ýþ þ ýþ ý ÿ ýþ þ ýþ inf

K (ý )2A^ (C ÿ ) p

l

A

ÿ

m

G(i

If ÿl+1 < ÿ < ÿl , then there exists a

^ (C p m ) l

satisÿes

G(i

ýþ 2 Aÿ A

(4.4).

) + K (i ) R1 (

)

R2 (

)

) + K (i )

þ(

ÿ iþ K (

= þ(

=

)

1

)=

Q(

R1 (

such that K ( ) ) 1 , where

)R2 (

)

Im

k ýþ k1 ü ýþ 2 A C

ÿ

ÿl+1 :

^( (p+m) (p+m) )

)

^(C p m ) satisfying Q(i ) ) ^(C (p+m) (p+m) ) such that þ( ) 1

for some Q( element in

ÿ

ýþ

1. Moreover, þ( ) is any ^( (p+m) (p+m) ) and it satisÿes

ÿ

Notice that our result is for a more general class of transfer functions G in ( ; C p m ) with limit 0 at i , although we cannot exclude the possibility

CR

û1

ÿ

that our result could be deduced from the abstract results of Ball and Helton [5].

Further comparison reveals that our Theorem 4.3.3 only gives suÆcient

conditions, whereas Theorem 4.4.1 gives necessary and suÆcient conditions. ^ the necessity part is fairly easy to show using Theorem For the class G

2A

2.3 of Iftime and Zwart [47] (see also the elementary proof of part

)

b.

c.

of

Theorem 3.1 of [47]). This elementary proof relies basically on the existence of a spectral factorization, which is well-known for the Wiener class (see Clancey and Gohberg [10]). The major diýerence between Theorem 4.4.1 and our main result is that we have used elementary transparent proofs in contrast to the abstract methods in Ball and Helton [5] which subsequently need to be translated to this problem.

Chapter 5

State space solutions In this chapter, w e will sho w that for the Pritc hard-Salamon class and the analytic class, there do exist

J ÿspectral

factors for eac h class satisfying the

assumptions in Chapter 4, hence solving the sub-optimal Hankel norm approximation problem for these classes of inÿnite-dimensional systems. We will give an explicit formula for a

J ÿspectral

factor þ in terms of the state-space pa-

rameters (A; B; C ) of the original system and we will chec k that it has all the properties S1-5 required in Chapter 4.

J ÿspectral

5.1

factorization for the Pritchard-

Salamon class In this section, we show the existence of a solution to the key

J

ÿspectral fac-

torization problem (satisfying S1-S5 in Chapter 4) for the smooth Pritc hardSalamon class of exponentially stable inÿnite-dimensional systems with ÿnitedimensional input and output spaces, in terms of the system parameters C.

A, B ,

We list the assumptions below:

P1. ý(A; B; C ) is an exponentially stable, smooth Pritchard-Salamon system. P2.

U

P3.

ÿl

=

C m,

Y

=

C p.

+1 < ÿ < ÿl .

In Curtain and Ran [17] the sub-optimal Hankel norm problem was solv ed for the

A. Sasane: Hankel Norm Approximation for Infinite-Dimensional Systems, LNCIS 277, pp. 101−108, 2002.  Springer-Verlag Berlin Heidelberg 2002

102

Chapter 5.

Pritchard-Salamon class by ÿnding a solution to the problem considered in S3 with

G(s)

=

C (sI

ÿ

A)

State space solutions

J ÿspectral

ÿ1 B .

factorization

However, there the

starting point was to quote a result from Ball and Helton [5], which states that the sub-optimal Hankel norm approximation problem is equivalent to solving the

J

ÿspectral factorization problem in S3. Then a solution is constructed from

a given realization of

G.

However, if one looks for the result quoted from [5],

one realizes that this is not an obvious corollary of the very abstract and general theory in [5]. This motivated the self-contained proofs of the sub-optimal Hankel norm approximation problem in Chapter 4. Here we use just the results from Chapter 4 and verify that all the assumptions are satisÿed. The sub-optimal Nehari problem is a special case of the sub-optimal Hankel K( ) H (C pÿm ) satisfying G(i ) +

ÿÿ 2 k ÿ ÿ)k þ ÿ for a given ÿ > kþk. In the case that ÿ > kþk, a solution ý to the J ýspectral factorization problem S3 was constructed in Curtain and Zwart norm approximation problem; one seeks K (i

1

1

[26] for the Pritchard-Salamon class.

ÿl+1 < ÿ < ÿl , but The proof of the following

For the sub-optimal Hankel norm problem we have nonetheless, the ý factor is precisely the same.

lemma is analogous to that of Lemma 2.11 in Curtain and Zwart [26]:

Lemma 5.1.1 Suppose that ü(A; B; C ) satisÿes P1-P3 and let G(s) = C (sI þ

A)

1B.

ý

Then

ÿ 0

ü

A ;

þ

C

ý

CB

0

L

ü

1

ý 2 ÿ

;

ýC LB ÿB

û

ü

ÿ 0

N ;

0

Ip

0

0

ÿIm

ûú

is an exponentially stable, smooth Pritchard-Salamon system and its transfer operator is

ý(

ü

ÿ=

ù

I

û

0

Ip

ý(s) =

N

ýÿ), where

0

+

ÿIm

ø ý ÿ12 LB LC 1 þ

ÿ

0

ü

A ;N

ÿ 0

þ

1

ü

2 ÿ

ÿB

and s

C

ýC LB þ

1þ

ÿ (sI + A )

N

0

2 C0 .

1 ÿ LC B

0

û 0

0 þ

C

0

CB

L

ý

(5.1)

;

Moreover,

ý

;

1 ÿ

2

ü

ýC LB ÿB

û ü ;

0

Ip

0

0

ÿIm

ûú

is an exponentially stable, smooth Pritchard-Salamon system and its transfer operator is V

ü V

(s) =

Ip

0

ýÿ), where

(

0

1 ÿ Im

û

1

ý 2 ÿ

ü

ýC LB B

0

û

1N þ ÿ

0 þ

(sI + A )

0

C

0

1 ÿ LC B

ý

;

s

2 C0 :

Furthermore, 1.

ýÿ) 2 MH c

ý(

1

is invertible in

MH c , and ý(ýÿ) 1 þ

1

is equal to V

ýÿ).

(

þ

J ÿspectral factorization for

5.1.

2. ÿ given by (5.1) is a solution of the in S3.

!1+ ÿ(ÿs) =

ÿ

3. lim s

2C 0

s

Ip

0

0

ÿIm

103

the Pritchard-Salamon class

J ÿspectral

factorization problem (4.4)

þ

.

Remarks:

1. Note that while it is algebraically straightforward to verify that ÿ(s) solves the

J

ÿspectral factorization problem, it is important to justify the well-

posedness of each term at all steps. Use is made of the Lyapunov equations from Lemma 2.3.1, but to make sense of all terms, it was necessary to make use of the smoothing properties of

and

LB

LC

from Lemma 2.3.1.

Of course, these smoothing properties are only known to hold for smooth Pritchard-Salamon systems. 2. If

jjþjj, it was shown in Curtain and Zwart [26] that ÿ11 (ÿþ) belongs to MH1 , is invertible in MH1 and ÿ11 (ÿþ)ÿ1 is the transfer operator of an exponentially stable smooth

ÿ >

(a) (b) (c)

c

c

Pritchard-Salamon system.

In the case that

ÿ11 (ÿþ)ÿ1

ÿl

+1 < ÿ < ÿl , ÿ11 (ÿþ)ÿ1 still exists, but it is not stable:

is the sum of a stable part in

rational part with at most

l

1 (C pþp )

H

unstable poles.

c

and an antistable

Finally we give the main result of this section.

ÿ1

Suppose that ý(A; B; C ) satisÿes P1-P3 and let Let ÿ be given by (5.1). Then we have:

Theorem 5.1.2 A)

B.

ÿþ) 2 H1 l (C pþm ) ÿ1 , where R1 (ÿþ)R2 (ÿþ) c

K(

;

ÿ

for some Proof

satisÿes

1 (ÿþ) R2 (ÿþ) R

þ

kG(iþ)

+

ÿ1 = ÿ(ÿþ)

þ)k1 ý

K (i

ÿ

Q(

ÿþ)

ÿ

G(s)

iþ

=

C (sI

ÿþ)

K(

ÿ =

þ

Im

ÿþ) 2 H1 (C pþm ) satisfying kQ(iþ)k1 ý 1.

Q(

c

The assumptions S1-S5 in Chapter 4 hold. S1 is satisüed since

G

is the

transfer operator of an exponentially stable smooth Pritchard-Salamon system (see 2.12) and S2 holds. Finally, Lemma 5.1.1 shows that S3-S5. Now the result follows from Theorem 4.3.3. Example 2.3.4 (continued) All the results in this section apply to the Example

2.3.4 considered earlier in Chapter 2.

ÿ

104

Chapter 5.

J ÿspectral

5.2

factorization

for

State space solutions

the

analytic

class In this section, we show the existence of a solution to the key

J ÿspectral

fac-

torization problem satisfying S1-S5 in Chapter 4 for the exponentially stable analytic class of inÿnite-dimensional systems discussed in Section 2.4 of Chapter 2 with ÿnite-dimensional input and output spaces: throughout this chapter,

ÿ

we consider the class of systems described by the triple ( A; B; C ) satisfying m p (see page 26). We list all the assumptions U = C , Y = C

A1, A2, A3 with below:

A1.

ÿ

A

is the inÿnitesimal generator of an exponentially stable analytic

semigroup

fT (t)gtÿ0 on the Hilbert space Z .

A2.

B

2 L(C m ; Zÿ ), where ÿ is a ÿxed number in (ÿ1; 0].

A3.

C

2 L(Z; C p ).

A4.

þl

+1 < þ < þl .

We construct a solution þ to the

J

ÿspectral factorization problem (??) which

has the properties S3-S5. The spectral factor þ is the same one as in Curtain and Ichikawa [14] (in which they consider the Nehari problem for a similar class with some diýerences; see also Table 2.1), but for the sub-optimal Hankel norm

approximation problem we have þl+1 < þ < þl . Extensive use of the results of Section 2.4 is made to show that all the conditions in Chapter 4 are satisÿed. The candidate solution is described by the quadruple

ÿ

ÿA ; 0

þ

C

0

L

ý

CB

1

;

þ

ü

2

CL

B

ÿþB

û

ü

þ 0

N ;

0

ûú

Ip

0

0

þIm

:

First we show that this deÿnes a regular linear system (see Section 2.1), and that its transfer function is invertible.

ÿA; B; C ) satisfy A1-A4 and let G(s) = C (sI +

Lemma 5.2.1 Let the triple (

1 B . Let ý be a ÿxed number satisfying ÿ1 < ý < ÿ. ÿý ÿ 21 < ü < 21 . Then A)

ÿ

ü1 = and

ü2 =

ÿ

ÿA ; 0

þ

C

ÿ

ÿA ; Nþ 0

0

0

þ

C

L

C

0

B

ý

;

1 þ

2

ü

CL

B

ÿþB

0

Choose any

û

ü

satisfying

ú þ 0

N ;

0

;

ü û ú 1 LC B ý ; 1 C LB ; 0 þ 2 ÿB þ 0

(5.2)

(5.3)

J ÿspectral factorization for

5.2.

105

the analytic class

C p+m ,

ÿ 0

are regular linear systems with state space Z , input space p+m . Denote the transfer function of ÿ by space C 1

ÿ) =

G1 (

ÿ

1 ÿ

þ

B

CL

þÿB

2

N

0

þ (ÿI + A )

0 ÿ1

0

ý

C

ü

CB

0

L

2 Hc

and output

û

C (p+m)þ(p+m)

1

ú :

(5.4) Then we have

!1 G1 (s) = 0:

lim

s

(5.5)

s C+ 2

Furthermore, if

þs) :=

ÿ

þ(

then

c

H1

ù

0

0

ÿIm

þÿ

þ( ) is an element in ø C (p+m)þ(p+m) and its inverse

V

and G2

C (p+m)þ(p+m) ÿ

1

G2 (s)

for all s

Proof

þÿ) =

=

ÿ

(

ù

2 Hc

1

ø

+ G1 (s)

c

H1

for all s

ù (p+m)þ(p+m) ø C ,

is given by V

þ

0

Ip

it

þÿ), where

B

is

þB

invertible

over

(

þ

0

(5.6)

þ G2 (ÿ);

1

þ Im

0

2 C +0 ;

ÿ2 :

denotes the transfer function of

CL

2 ÿ

þ

Ip

0 ÿ1

(sI + A )

N

þ 0

ý

C

0

þ LC B 1

ü

(5.7)

2 C +0 .

BûNþ , úwe ýrst note by appealing to Lemma 2.4.8 that Nþ 2 L (Zý ) and Nþ 2 L Z ý are well-deýned for any ù ø þ satisfying þ1 < þ < ý. From Lemma 2.4.1.2, it follows that LB 2 L Z ÿ ; Z ú û for ü < 1 + 2ý, and so LB Nþ 2 L Z ý ; Z for þ < 1 + 2ý. Here we remark that if þ satisýes þ1 < þ < ý, then in particular þ < 1+2ý. Since B 2 L (C m ; Zý ) for ú û þ1 < þ < ý, we have B Nþ 2 L Z ý ; C m . Furthermore, if þ1 < þ < ý, then using Lemma 2.4.2.2, we obtain that LC 2 L (Zý ; Z ), and so LC B 2 L (C m ; Z ). To make sense of the terms

0

þ

B N

0

and

0

L

0 ÿ

0

0 ÿ

0

0

Consequently, if

ý

C

0

We now ýx a possible, since

L

ü

0 ÿ

0

0 ÿ

þ1 < þ < ý, then we have:

CB

ü

2

ù ø L C p+m ; Z ;

satisfying

þ1 < þ .

þþ þ 21

ÿ

CL

B

þÿB

< ü
0, there exists a Æ > 0 such that sup kG(i! ) ÿ G(þ + i! )k < ÿ for all þ satisfying 0 þ þ þ Æ:

Lemma 6.1.1 If

continuous, then given any

!2R Proof

It follows from Theorem 5.18 (page 96, M. Rosenblum and J. Rovnyak

[69]) that

G(þ + i!) = Since for

þ > 0,

Choose a

ü > 0 such that

þ ý

1 G(it) dt; þ1 (t ÿ !)2 + þ 2

Z

þ > 0:

1 1 dt = 1; þ1 (t ÿ !)2 + þ 2 and we have ÿ Z 1 ÿ ÿþ G(it) ÿ kG(i!) ÿ G(þ + i!)k = ÿ ý þ1 (t ÿ !)2 + þ 2 dt ÿ G(i!)ÿÿÿ ÿ ÿ Z 1 ÿþ G(it) ÿ G(i!) ÿÿ ÿ dt : = ÿ ý þ1 (t ÿ !)2 + þ 2 ÿ þ ý

Z

kG it ÿ G i! k < ÿ ( )

Now choose a

Thus

(

)

2

for all

t

Æ > 0 such that for any þ ÿ Z ÿþ ÿ ÿ ÿ ý Rn[!þÿ;!+ÿ] (t

ÿ

and

!

jt ÿ !j < ü:

satisfying

satisfying 0

ÿ ÿ 1 ÿ< dt ÿ !)2 + þ 2 ÿ

þþþÆ

, we have

ÿ

4

kG iý k1 : ( )

kG i! ÿ G þ i! k ÿ ÿ Z þ ýþ !þÿ kGt itÿ !ÿ2G i!þ 2 k dt ýþ Rn[!þÿ;!+ÿ] kGt itÿ !ÿ2G i!þ 2 k dt Z Z þ þ þ ! +ÿ 2 þ ý !þÿ t ÿ ! 2 þ 2 dt ý Rn[!þÿ;!+ÿ] t ÿkG! i2ý k1þ 2 dt þ ÿ kG iý k1 kG ÿiý k1 ÿ: (

=

)

( +

)

ÿ Z 1 ÿ ÿþ G(it) G(i!) ÿÿ ÿ dt ÿý (t ! )2 + þ 2 ÿ þ1 Z !+ÿ ( )

(

(

2

1+2

(

) +

( )

)

) +

4

( )

+

(

2

+

( )

(

=

(

) +

( )

) +

)

6.2.

111

The class of systems and the main result

Since the choice of

! is arbitrary, this completes the proof.

G(iÿ) : R ! C pÿm has limits Gþ1 at þ1, then given any ÿ > 0, there exists sup! 2R kG(i! ) ý G(þ + i! )k < ÿ whenever 0 ü þ ü Æ . G 2 H1 (C pÿm )

Corollary 6.1.2 If

Proof

Given any

and

sup

!2(ý1;M1 ]

[

Æ>

0

such that

ÿ > 0, there exist M1 > 0 and M2 > 0 such that sup kG(i!) ý G1 k < ÿ ; and !2[M2 ;1)

Moreover, since

is continuous and a

G(iÿ)

:

R

kG i! ý Gý1 k (

)

(6.1)

2

ÿ


0, there exists a Æ such that 1 > Æ > 0 !1 and !2 lie in [M1 ý 1; M2 + 1] and j!1 ý !2 j < Æ,

and whenever

kG i! ý G i! k < ÿ: (

1)

(

2)

(6.3)

Thus it follows from (6.1), (6.2) and (6.3) that whenever

R and

j! ý ! j < Æ 1

2

, then

kG i! ý G i! k < ÿ (

1)

(

2)

!1 and !2 belong to G(iÿ) : R ! C pÿm is

. Hence

uniformly continuous, and so the result follows from Lemma 6.1.1.

6.2

The class of systems and the main result

The speciÿc class of systems we consider in this chapter is deÿned below:

B1.

1. 2.

h 2 L2 ([0; 1); C pÿm ). G(iÿ) 2 C (R; C pÿm ) and has a unique limit G1 at þ1, where G := F (h), where F denotes the Fourier(-Plancherel) transform.

B2. There exists a 1.

G ( þ + ÿ)

Æ > 0, such that for every þ

2 H1 Cÿ ÿ ÿ ý û ý û ::: ÿ Gþ ÿ ý 0 such that 0

ÿ

ÿ ! 0, and S2[ ] = S2[ ]

ku t k dt < ý : þÿ

"

where

[ÿ ]

[ÿ ]

and

ÿ

ÿ

( )=

0

0

ÿIm

is such that K

[ÿ ]

)

R2

(

)

+ÿ

þÿ 2

(

)

)=

#

þÿ þÿ

(

þ

Ip

þÿ

(

R1

[ÿ ]

0

such that for every þ satisfying

K

[ÿ ]

0

[ÿ ]

L

C

i B

ÿ.

This is a consequence of Theorem 5.1.2 and Theorem 6.2.4.

Proof

ÿ

C

ÿ2

[ÿ ]

R1

h

[ÿ ]

= ÿ

ÿ

þ

CL

l] (C

c

1;[

)

þÿ

(

)

þh

[ÿ ]

B

ÿB

H

þÿ

(

h

i

ÿ1

pþm ) and

þÿ

(

ÿ

)

Q

and G(s)

k ÿk ý Q(i

)

1

< þ < Æ,

ÿ1

;

þÿþ

(

þ)

þ

[ÿ ]

i

h

ÿþ þ ÿ ÿ ýý ý 0

(

0

I

þI)

A

ý ýG(i ) + K [ÿ ] (i )

ÿ1

1

C

0

L

[ÿ ] C

i B

ÿ.

This is a consequence of Theorem 5.2.3 and Theorem 6.2.4.

Proof

Example 3.1.2 (continued) Example 3.1.2 in Chapter 3 þts in both classes

and

B

1.

= 1,

Im

Nþ

0

[ÿ ]

R2

i

B

A

considered above: A

þ

BB

ý

,

and

B

:=

C

B

ý

are all bounded, and so ý(A

smooth Pritchard-Salamon system with 2. Since

ü

A

ÿ

AÿBB t

e

3. Since

A

þû

+

BB

tü0 ý

A

ý

is bounded,

. Moreover,

B

W

=

V

þ

ý

BB ; B; C )

is a

= `2 (N ).

it generates the analytic semigroup ( m ; Zý ) with m = 1 and ý = 0.

2L C

= 0, it follows from Corollary 2.2.3 (page 33, Curtain and

Zwart [27]) that

A

generates a contraction semigroup. Thus from Lemma

2.2.6 (page 19, Oostveen [61]), it follows that the system given by the triple ý ý ý (AÿBB ÿ ) (A BB ; B; B ) is output stable, that is, the map z B e z

L

þ

(`2 (Z); L2(0;

ÿ

h(

) =

ý

B e

1

)). (AÿBB ÿ )

in Chapter 3 that

So in particular, with

2 ÿ

B

G(i

L2 (0;

1

z

=

B

7! 2

2

Z), we obtain

`2 (

). Furthermore, we have already veriþed

) is continuous with the unique limit 1 at

ü1

.

So the results in this section apply, solving the sub-optimal Hankel norm approximation problem for this non-exponentially stable example.

ÿ

Chapter 7

The case of regular linear systems In this c hapter, w e will solve the sub-optimal Hankel norm approximation problem for regular linear systems with generating operators (A; B; C ) satisfying some mild assumptions. In the special case of the sub-optimal Nehari problem, an explicit parameterization of all solutions is obtained in terms of the system parameters

A, B

and

C.

The limiting factor to extending the results from Chapter 5 to the general case of well-posed linear systems is the diÆculty in manipulating with the unbounded operators

B

and

C.

In general, it is not clear that the candidate

spectral factor given b y (5.1) is ev en well-posed (see Staÿans [78]). In this c hapter w e translate the problem to the analogous one for reciprocal systems which we deþne in Section 7.1. In Section 7.2, we will

solve the sub-

optimal Hankel norm approximation problem for regular linear systems. The sub-optimal Nehari problem is a special case in which ÿ > ÿ1 . In this special case, using the results from Curtain and Oostveen [16], we can give a parameterization of

7.1

all

solutions, and we do this in the last section of this chapter.

Reciprocal systems

recipr oal system

The

of the regular linear system with generating opera-

tors (A; B; C ) such that 0 of

ÿ

A),

ÿ

A

2

þ(A)

(here

þ(A)

denotes the resolvent set

is the regular linear system with the bounded generating operators þ 1 ; A 1 B; C A 1 . The key to our approach is the relationship betw een

ÿ

ÿ

ÿ

A. Sasane: Hankel Norm Approximation for Infinite-Dimensional Systems, LNCIS 277, pp. 119−125, 2002.  Springer-Verlag Berlin Heidelberg 2002

120

Chapter 7.

The case of regular linear systems

ÿ

the transfer function of the original regular linear system G(s) = Cÿ (sI 1 B and that of its reciprocal system with the transfer function G (s) = r ÿ þ 1 sI 1 1 A 1 B . Note that A 1 , A 1 B , C A 1 are all bounded CA A ÿ 1þ operators and for all s ÿ(A) ÿ A there holds

ÿ

A)

ÿ

ÿ

ÿ ÿ

ÿ

Cÿ (sI

ÿ

2

ÿ

ÿ

\

ÿ

ÿ A)ÿ1 B

=

ÿCÿ Aÿ1 B ÿ C Aÿ1

G(s)

=

Gr

=

Gr

ý ü 1 s

ý

ÿ

ÿ Aÿ1

1 s

üÿ1 A

ÿ1 B;

(7.1)

ÿ Cÿ Aÿ1 B

ý ü 1 s

+ G(0):

(7.2)

1L

2

1L

2

From (7.2), it is clear that G H ( (U; Y )) iÿ Gr H ( (U; Y )). In addition, the reciprocal system has the same controllability and observability Gramians, which we prove in the following lemma.

Let (A; B; C ) be generating operators of a regular linear system with transfer function G. Suppose that 0 ÿ(A) and Gr is the transfer function of its reciprocal system with generating operators (A 1 ; A 1 B; C A 1 ). Then the following hold: Lemma 7.1.1

1.

2

ÿ

ÿ

ÿ

ÿ

ÿ

ÿ

is an inÿnite-time admissible observation operator for A iþ C A 1 is an inÿnite-time admissible observation operator for A 1 . If either C or 1 is inÿnite-time admissible (for the semigroups generated by A or CA 1 A , respectively), then the observability Gramians are identical.

ÿ

C

ÿ

ÿ

ÿ

ÿ

is an inÿnite-time admissible control operator for A iþ A 1 B is an inÿnite-time admissible observation operator for A 1 . If either C or A 1 B is inÿnite-time admissible (for the semigroups generated by A or A 1 , respectively), then the observability Gramians are identical.

2.

B

3.

G

ÿ

ÿ ÿ

2 H1 (L(U; Y )) iþ Gr 2 H1 (L(U; Y )).

Proof 1. From Hansen and Weiss [45] (see also Grabowski [41]), we know that C

is an inþnite-time admissible observation operator iÿ the Lyapunov equation

hAz1 ; LC z2 i + hLC z1 ; Az2 i = ÿ hC z1 ; C z2 i for all

z1

and

z2

in

D (A),

(7.3)

has a nonnegative deþnite solution

C

L

=

The equation (7.3) is clearly equivalent to the Lyapunov equation

û

C Aÿ1 x2

x1 ; L

ú

+

û

C Aÿ1 x1 ; x2

L

ú

=

ÿ

û

ÿ1

CA

ÿ1

x1 ; C A

x2

ú

L

þC þ 0. (7.4)

for all x1 and x2 in X , which establishes the equivalence. Moreover, the observability Gramians are the smallest positive solution and so the Gramians are identical.

7.2.

121

The sub-optimal Hankel norm approximation problem

2. This is dual to part 1 above. 3. This follows from (7.2).

The idea is then to translate the sub-optimal Hankel norm approximation problem for the regular linear system with transfer function with transfer function

Gr ,

the latter having

now elaborate on this. Clearly,

ÿÿ) 2 1 l (C pÿm ) iÿ

K(

are related by

H

;

Consequently we have

=

1

K

G(i

)

ÿ k1 k ÿ

) + K (i )

ÿk (G)

=

=

ÿk (Gr )

G

into one for the system

generating operators. We

1 l (C pÿm ), where

H

+ G(0); for all

s

k ÿ

orem 1.2.1, it follows that

7.2

þÿ 2

Kr (

ÿ þ

Kr (s)

bounded

;

s

2 C +0 ÿ k1

and

Kr

:

Gr (i ) + Kr (i ) for all k N.

2

K

. Thus from The-

The sub-optimal Hankel norm approximation problem

In this section we will solve the sub-optimal Hankel norm approximation problem for regular linear systems, by þrst translating the problem to its reciprocal system with bounded generating operators, albeit a nonexponentially stable semigroup. Subsequently we use Theorem 6.3.1 from Chapter 6 in order to obtain explicit formulae for solutions to the sub-optimal Hankel norm approximation problem for such systems. We will solve the sub-optimal Hankel norm approximation problem for the regular linear system on a Hilbert space

X

with generating operators (A; B; C )

under the following assumptions:

H1. H2. H3. H4. H5.

U

0

=

2

C m and

þ(A)

Y

and

=

C p.

\ C +0

ÿ (A)

is empty.

f g þ0 f g þ0

B

is an inþnite-time admissible control operator for

C

is an inþnite-time admissible observation operator for

ÿ

G(

) =

C

ÿþ

ÿ( I

unique limit at

ý1

A)

ý1

.

B

2

H

1 (C pÿm ),

G(i

ÿ 2C R C )

(

;

T (t) t

ÿ

.

T (t) t

.

p m ) and it has a

122

Chapter 7.

The case of regular linear systems

H1 ÿ H5 are satisÿed ÿ 3 of Class A on page ÿ ÿ1 ). Furthermore,

First of all, we will show that under if the assumptions by the original system (A; B; C ), then the assumptions 1 115 are satisÿed by the reciprocal system (A 1 ; A 1 B;

ÿ

ÿ

CA

from (7.2) and Theorem 3.0.11, the Hankel operator of the original system and that of the reciprocal system

H

Gr

H

Section 7.1, it follows from Theorem 1.2.1 that the Hankel singular values of and

G

are both compact. Also, as discussed in G

are equal. Applying Theorem 6.3.1, we give solutions to the sub-optimal

Gr

Hankel norm approximation problem for the reciprocal system and hence also to the original system. But ÿrst we will prove the following elementary result. Lemma 7.2.1 If

ÿ

ÿ

(A 1 ; A 1 B;

ÿ

(A; B; C )

H1 ÿ H5,

satisfy

ÿ1 ) satisÿes assumptions1

CA

1

ÿ3

then

the

of Class

reciprocal

A.

system

1 is obvious.

Proof

ÿ

ÿ

2 : Owing to the inÿnite-time input admissibility of C A 1 for the reciprocal 1 T ( )x system, it is clear that for all x X, CA L2 ([0; ); C p ). Conse1 1 p m quently, CA T ( )A B L ([0; ); C ). Furthermore, the continuity of

þ

ÿ

3 : It follows from

H2

Gr

ÿ

ÿ

2

2

2

1

ÿ

ÿ

þ 2

ÿ

on the imaginary axis follows from equation (7.2), and

H5.

and the spectral mapping theorem ÿ þfor the resolvent

þ1 \ C +0 is empty. þ1 is bounded, it satisÿes the spectrum determined growth

(see 1.13.(i), page 243, Engel and Nagel [31]) that Furthermore, since

1

ÿ

A

A

assumption (see for instance Corollary 2.4, page 252, Engel and Nagel [31]). Thus it is clear that the growth bound of the semigroup generated by nonpositive, and so

A

þ1 ÿ

þI

A

þ1 is

is the inÿnitesimal generator of an exponentially

stable, strongly continuous semigroup. We end this section by giving our main result. Theorem 7.2.2 Suppose that (A; B; C ) satisÿes

ÿþ 2

and Q(

1 (C pÿm )

)

H

that for every þ satisfying

0

ý ý

þ(s)

=

"

Ip

0

0

ÿIm

þ1

2 CA

1 see page 115

ü

)

ÿ

s)

s)

s)

:=

ÿ

1(

R

ü

ÿ

= þ(

1.

ÿ þ1 ÿÿ

2(

s)R

þ1

s)

ý

H1 ÿ H5.

Let ÿl

+1

< ÿ < ÿl

Then there exists a Æ >

Q

s)

(

0

such

;

s

ü

þ)

;

Im

+

# [þ ] h iý h ÿ þ B þý Nÿ[þ ] sI + Aþ1 ý þ 1 A B

ÿ1ÿ1ÿ ÿ

ÿ ÿ

1( R2 ( R

Q(i

< þ < Æ,

Kr (

where

k þ k1 ý

, and

L

ÿ

þI

iþ1h ÿ

þ1 þý

CA

[þ ] þ1 CA B

L

i ;

7.3.

123

The sub-optimal Nehari problem

ÿÿ) 2 1 l (C pÿm ) þ1 þ þ1 þ þ1 þ

is such that Kr (

H

[] ÿ

. Here LB

;

[] ÿ

and LC

denote the controllability

iþ1 þ 12 L[ ]L[ ] .

Gramian and the observability Gramian, respectively, of the exponentially stable system

ÿ

A

ÿI; A

B;

[] ÿ

, and Nþ

CA

more, deÿning

ý ü K (s)

=

Kr

1 s

h

:=

I

þ

ÿ

ÿ

B

C

Further-

þ G(0);

(7.5)

þÿ) 2 H1 l (C pÿm ) and kG(iÿ) + K (iÿ)k1 ý þ.

we have K (

;

Proof From Lemma 7.2.1 it follows that the reciprocal system þ1 þ þ1 ) satisÿes the assumptions 1 þ 3 of Class A. Let r denote ( þ1 þ1 þ þ1 ). Moreover, we have the transfer function of ( þ1 A

;A

B;

CA

G

A

(Gr ) =

þl

;A

B;

CA

+1 (G) = þl+1 (Gr ) :

þl (G) > þ > þl

Consequently, using Theorem 6.3.1, we have that

þÿ) 2 H1 l (C pÿm ) 2

Kr (

;

and

kGr (iÿ) + Kr(iÿ)k1 ý þ. Finally it is clear that K deÿned by (7.5) is such that pÿm ) and kG(iÿ) + K (iÿ)k = kG (iÿ) + K (iÿ)k K (þÿ) 2 H1 l (C r r 1 1 ý þ. ;

7.3

The sub-optimal Nehari problem

Finally, in this section we solve the sub-optimal Nehari problem for the regular linear system on a Hilbert space

X

with generating operators (A; B; C ) under

the following assumptions:

N1. = C m and = C p . N2. 0 2 ( ). N3. is an inÿnite-time admissible control operator for f ( )g ý0. N4. is an inÿnite-time admissible observation operator for f ( )g ý0. N5. (ÿ) = ÿ (ÿ þ )þ1 2 1(C pÿm ). U

Y

ý A

B

T t

C

G

t

T t

C

I

A

B

t

H

The sub-optimal Nehari problem can be thought of as a special case of the sub-optimal Hankel norm approximation problem with

l

= 0. So in principle,

the results of the previous section apply to this case. However, we do not do 2 Note that from Theorem 6.3.1, it follows that that

Kr (þ ) 2 H1 l (C pÿm ) ;

Kr (ÿ ) 2 H1 l (C pÿm ), but this implies c

;

124

Chapter 7.

that in this section.

The case of regular linear systems

This is because the Theorem 6.3.1, which was used in

the previous section, can be now replaced (in the

special

case of the Nehari

problem) by a more powerful result from Curtain and Oostveen [16], which will enable us to even obtain a

of

parameterization

all

solutions to the sub-optimal

Nehari problem. Also, we ÿnd that with this alternate approach we can solve the problem under weaker assumptions than those demanded in the previous

N2 versus H2, and N5 versus H5).

section (notice the diþerences in

However,

the broad approach in both sections remains the same: instead of looking at the original system, we translate the problem to the reciprocal system, solve it, and ÿnally retrieve solutions to the original problem. Suppose that

ÿ > ÿ1 (G)

=

ÿ r (þÿ) 2 1

ÿ)k1 < ÿ, then K

K (i

H

Kr (s)

Conversely, if K

) H (C p m ) satisÿes

ÿ (þÿ) 2 1 H

:=

ÿ þ 1

K

H

G

G i

i

K

i

< ÿ

+ G(0):

s

1 (C pÿm )

þÿ 2

Kr (

ÿÿ 2 1 (C pÿm ) satisÿes k ( ÿ) + k r ( ÿ) + r ( ÿ)k1 , where

(Gr ). If K ( ) (C p m ) satisÿes

ÿ1

satisÿes kGr (iÿ) + Kr (iÿ)k1 kG(iÿ) + K (iÿ)k1 < ÿ, where

ÿ þ

K (s)

:=

Kr

1 s


ÿ1 and let

ý

ý(s) = where L

B

ý

be given by

Ip

0

0

ÿIm

and L

C

fT (t)gtý0.

ü

ý +

þ ÿ12 C LB 1B

ÿ

ü

ü

ü

ü ]þ1 û ü

ÿ [sI + A

N

C

CB

L

ú

:

þ

denote the controllability Gramian and the observability

Gramian, respectively, of the system

(A; B; C ),

ÿ

and N

:=

û

I

þ ÿ12 LB LC

ú 1

.

7.3.

ÿÿ) 2 1 (C pÿm ) þ1 2 (ýÿ)

Then K ( R1 (

125

The sub-optimal Nehari problem

, where

ÿ

þ

ýs) R2 (ýs) R1 (

ýs) 2 H1 (C pÿm )

Q(

for some

kG(iÿ) + K (iÿ)k1 þ

satisÿes

H

ýs)R

= ÿ(

satisfying

N

ÿ

ýs)þ1

ÿ

ýs)

K(

iþ

=

þ

ýs)

Q(

Im

kQ(iÿ)k1 þ 1.

ýN

Indeed, under the assumptions 1 5, the reciprocal system has its transfer 1 are inþnite-time function in H (C p m ) and the operators A 1 B and CA

1 ÿ

admissible with the same Gramians

þ

B

and

L

þ

ý

C

as the original system. So in

L

light of what has been said above, upon applying Theorem 7.3.1 to the reciprocal system (A 1 ; A 1 B; C A 1 ), we obtain the following result:

þ

þ

þ

ý

Theorem 7.3.2 Suppose that (A; B; C ) satisÿes

ýÿ) 2 H1 (C pÿm ), and kQ(iÿ)k1 þ 1.

Q(

ÿ

ÿ(s) =

þ

ÿ

Ip

0

0

ÿIm

B

and L

+

C

where L

ý ÿ1ý2 C Aþ1üLB 1

ÿ

þ1 ý

A

B

þ

ý

N1 ý N5

Let ÿ > ÿ1 and

.

be given by

þ1 üý iþ1û ý þ1 üý

sI + A

CA

C Aþ1 B

L

ú

þ

;

denote the controllability Gramian and the observability

ýÿ) 2 H1 (C pÿm )

Gramian, respectively, of the system Then K (

ýh

ÿ

N

ÿ

Let

satisÿes

(A; B; C ),

and N

ÿ

:=

kG(iÿ) + K (iÿ)k1 þ ÿ

û

iþ

I

ý ÿ12 LB LC

ú 1

.

ù ø

K (s)

=

Kr

1 s

+ G(0);

ýs) = R1 (ýs)R2 (ýs)þ1 , and

where Kr (

ÿ

for some

ýs) R2 (ýs) R1 (

ýs) 2 H1 (C pÿm )

Q(

þ

þ1 = ÿ(ý ) s

satisfying

While it is tempting to try to write ÿ and

ÿ

Q

ýs)

(

þ

Im

;

kQ(iÿ)k1 þ 1. K

in terms of its reciprocal, we know

that this will not (in general) be well-deþned (see Staýans [78]). So we leave the explicit solution as it stands.

Chapter 8

Coda In this last chapter, w e discuss the approach follo w edin this book to solve the sub-optimal Hankel norm approximation problem for inÿnite-dimensional systems. In Chapter 4, we ga ve suÆcient frequency domain condi to ions

solve the

sub-optimal Hankel norm approximation problem. Subsequently in Chapter 5, w e solv ed the sub-optimal Hankel norm appro

ximation problem for the expo-

nentially stable smooth Pritchard-Salamon class and the class of exponentially stable analytic systems, by explicitly verifying the frequency domain conditions given in Chapter 4.

In Chapter 6, under certain conditions, w e ga v ea solu-

tion to the sub-optimal Hankel norm approximation problem in the case when the inÿnitesimal generator is not necessarily exponentially stable and ÿnally, in Chapter 7 w e solv ed the sub-optimal Hankel norm approximation problem in the case of regular linear systems. The main advan tage of our approac h in this book is that it is a self-contained, elementary solution to the sub-optimal Hankel norm approximation problem. Moreov er, our results in Chapter 4 hold for a class larger than the Wiener class. Although our results in Chapter 4 hold for a class larger than the Wiener class, unfortunately, except for the results in Chapters 6 and 7, most of our examples

h 2 L1 !

ha v e

One might conjecture that an approach similar to the one in this book w ould yield solutions to the sub-optimal Hank el norm approximation problem for more general classes of well-posed linear systems than the ones considered in this book. How ev er, w e suspect this task to be a formidable one, since ewneed the smoothness properties of the controllabilit y and observ abilit y Gramians to v erify the candidate spectral factor formulas given in this book.

A. Sasane: Hankel Norm Approximation for Infinite-Dimensional Systems, LNCIS 277, pp. 127−129, 2002.  Springer-Verlag Berlin Heidelberg 2002

128

Chapter 8.

Coda

This book also gives alternate state-space formulae for the sub-optimal Hankel norm approximation problem for ÿnite-dimensional systems. We conclude this book by explicitly working out an elementary ÿnite-dimensional example, which elucidates the procedure involved and the theorems in Sections 5.1 and 5.2 of Chapter 5.

(An elementary ÿnite-dimensional example.) Consider the ra-

Example 8.0.3

tional function G(s)

=

1

ÿ

s

þ

; 1 2 + 2

which has the minimal (and hence controllable) state-space realization of MacMillan degree 2 given by

ý

ÿ 12

=

A

ü

1 1 2

ÿ

0

The impulse response is

h(t)

ý ;

=

=

B

Ce

0

ü and

1

A B = teÿ 12 t

t

for

and observability Gramians are given by

ý

B

=

L

respectively. We have L

B LC

ý

=

3

4

2

3

2

1

1

1

ü

and

ü ;

ÿ

C

L

(LB LC ) =

ý =

n

=

C

t

1

0

ú

:

ÿ 0 and its controllability

1

1

1

2

3+2

û

ü ;

p

2; 3

þ2

po 2

;

and so the Hankel singular values of the system are ÿ1

Let

l=1

and

ÿ

2

= 1

norm approximant of

ý

V

(s)

=

G

2

p

2+1

þ 1;

p

> ÿ2

þ

=

p

þ 1:

2

2 + 1 . Thus we seek a sub-optimal Hankel

of MacMillan degree at most 1. We have

ý ü 1 þC LB (sI + A ) 1 N û C 1 L B ú þ ÿ ÿ C 2 0 B ÿ ÿ Im ý ü ý üý ü 1ý 1 1 1 üý 1 0 þ 2 þ1 sþ 0 þ 1 2 2 2 þ 0 1 1 1 0 1 1 s þ þ 1 0 2 2 3 2 1 s 1 2 64 1 þ (s 12 )2 þ (s 121)2 7 5: s 2 2 1þ 2 1 1 s s ( ( ) ) Ip

0 1

ü

ÿp

=

0 ÿ

0

ÿ

=

ÿ

=

ÿ

ÿ

With

0

0

ÿ

ÿ

2

2

þs) = 0, we obtain

Q(

þs)

K0 (

=

þs)V22 (þs)

V12 (

1

ÿ

1 2

ü

Coda

129

=

s

ÿ

s

=

þ 1 2 + 2

ÿ

1þ

1 + 2

p12

ÿ

þ 1 2 + 2

+1

ih

p

G0 (s)

kG þ G0 kH ý 1.

i

1 1 + 2 + 2 1 + 1 1 þ 1 2 2 2 2 2 2 + 1 1 1 + 1 : s+ s+ 2 þ 2 2 2

þ

s

p p

p p

Thus one sub-optimal Hankel norm approximant of

and

#ÿ1

1 2

s

s

h

s

=

"

+1

=

p p

1 1 2 + 2 2 1 1 s þ 2 + 2

G

is

;

ÿ

Bibliography [1] V.M. Adamjan, D.Z. Arov, and M.G. Krein. Analytic properties of the Schmidt pairs of a Hankel operator and the generalized Schur-Takagi problem. Math. Sb. (N.S.), 86:34{75, 1971. [2] V.M. Adamjan, D.Z. Arov, and M.G. Krein. Inÿnite Hankel block matrices and related extension problems. American Mathematical Society Transla-

tions, 111:133{156, 1978. [3] W. Arendt and C.J.K. Batty. Tauberian theorems and stability of oneparameter semigroups. Transactions of the American Mathematical Soci-

ety, 306:837{841, 1988. [4] J-P. Aubin. Applied Functional Analysis. John Wiley and Sons, 1979. [5] J.A. Ball and J.W. Helton. U (m; n)

A Beurling-Lax theorem for the Lie group

which contains most classical interpolation theory.

Journal of

Operator Theory, 9:107{142, 1983. [6] J.A. Ball and A.C.M. Ran. Optimal Hankel norm model reductions and Wiener-Hopf factorization I: The canonical case. SIAM Journal on Control

and Optimization, 25(2):362{382, 1987. [7] J.S. Baras and R.W. Brockett.

H

2

ÿfunctions and inÿnite-dimensional real-

ization theory. SIAM Journal on Control and Optimization, 13(1):221{241, 1975. [8] A. Bensoussan, G. Da Prato, M.C. Delfour, and S.K. Mitter. Representa-

tion and Control of Inÿnite-Dimensional Systems, volume 1, 2. Birkh auser Verlag, Boston, 1993. [9] C. Bonnet and J.R. Partington. Stabilization and nuclearity of fractional diþerential systems. In Proceedings of the 14th International Conference

on Mathematical Theory of Networks and Systems MTNS 2000, Perpignan, France, June 2000.

Bibliography

132

[10] K.F. Clancey and I. Gohberg. gular Integral Operators, plications.

Factorization of Matrix Functions and Sin-

volume 3 of

Operator Theory: advances and ap-

Birkh auser Verlag, 1981.

[11] R.R. Coifman and R. Rochberg. Representation theorems for holomorphic

Lp .

and harmonic functions in

Asterisque,

77, 1980.

[12] R.F. Curtain. A new approach to model reduction in ÿnite-dimensional control design for distributed parameter systems. Submitted. [13] R.F. Curtain. SuÆcient conditions for inÿnite-rank Hankel operators to be nuclear.

IMA Journal of Mathematical Control and Information,

2:171{182,

1985. [14] R.F. Curtain and A. Ichikawa. The Nehari problem for inÿnite-dimensional systems of parabolic type.

Integral Equations and Operator Theory,

26:29{

45, 1996. [15] R.F. Curtain, H. Logemann, S. Townley, and H.J. Zwart. Well-posedness, stabilizability and admissibility for Pritchard-Salamon systems. Mathematical Systems, Estimation and Control,

Journal of

7(439-476), 1997.

[16] R.F. Curtain and J.C. Oostveen. The Nehari problem for nonexponentially stable systems.

Integral Equations and Operator Theory,

[17] R.F. Curtain and A. Ran.

Explicit formulas for Hankel norm approxi-

mations of inÿnite-dimensional systems. Theory,

31:307{320, 1998.

Integral Equations and Operator

13:455{469, 1989.

[18] R.F. Curtain and A.J. Sasane. Compactness and nuclearity of the Hankel operator and internal stability of inÿnite-dimensional state linear systems. To appear in the International Journal of Control. [19] R.F. Curtain and A.J. Sasane. Sub-optimal Hankel norm approximation for the analytic class of inÿnite-dimensional systems. To appear in Integral Equations and Operator Theory. [20] R.F. Curtain and A.J. Sasane. Inertia theorems for operator Lyapunov equations. In

Proceedings of the 38th IEEE Conference on Decision and

Control, Phoenix, Arizona,

December 1999.

[21] R.F. Curtain and A.J. Sasane. Sub-optimal Hankel norm approximation for inÿnite-dimensional systems. In France,

Proceedings of the MTNS 2000, Perpignan,

2000.

[22] R.F. Curtain and A.J. Sasane. Sub-optimal Hankel norm approximation for the analytic class of inÿnite-dimensional systems (extended version). Technical report, IWI 2000-4-01, University of Groningen, 2000. Available at

http://www/math.rug.nl/ÿamols/index.html.

Bibliography

133

[23] R.F. Curtain and A.J. Sasane.

Sub-optimal Hankel norm approxima-

tion for the Pritchard-Salamon class of non-exponentially stable inÿnitedimensional systems. In

Proceedings of the 39th IEEE Conference on De-

cision and Control, Sydney, Australia,

[24] R.F. Curtain and A.J. Sasane. inequalities.

December 2000.

Inertia theorems for operator Lyapunov

To appear in the Systems and Control Letters,

2001.

[25] R.F. Curtain and A.J. Sasane. Sub-optimal Hankel norm approximation for the Pritchard-Salamon class of inÿnite-dimensional systems. Equations and Operator Theory,

[26] R.F. Curtain and H.J. Zwart.

Integral

2001. The Nehari problem for the Pritchard-

Salamon class of inÿnite-dimensional linear systems: a direct approach. Integral Equations and Operator Theory,

[27] R.F. Curtain and H.J. Zwart. Linear Systems Theory.

18:130{153, 1994.

An Introduction to Inÿnite-Dimensional

Springer-Verlag, New York, 1995.

[28] R.F. Curtain and H.J. Zwart. Riccati equations and normalized coprime factorizations for strongly stabilizable inÿnite-dimensional systems. tems and Control Letters,

[29] L. Dumortier.

Sys-

28:11{22, 1996.

Partially Stabilizing Linear-Quadratic Optimal Control for

Stabilizable Semigroup Systems.

PhD thesis, Facultes Universitaires Notre-

Dame De La Paix, 1998. [30] N. Dunford and J.T. Schwartz.

Linear Operators,

volume 2. John Wiley

and Sons, 1964. [31] K-J. Engel and R. Nagel. Equations.

[32] B. Francis.

One-Parameter Semigroups for Linear Evolution

Springer, 2000. A Course in

H1

Control Theory,

in Control and Information Science.

volume 88 of

Lecture Notes

Springer-Verlag, Berlin, 1987.

[33] T.T. Georgiou and M.C. Smith. Robust stabilization in the gap metric: Controller design for distributed plants. Control,

IEEE Transactions on Automatic

37(8):1133{1143, 1992.

[34] K. Glover. All optimal Hankel-norm approximations of linear multivariable systems and their

L1

error bounds.

International Journal of Control,

39:1115{1193, 1984.

[35] K. Glover. A tutorial on Hankel norm approximation. In J.C. Willems, editor,

From Data to Model.

Springer-Verlag, 1989.

[36] K. Glover and R.F. Curtain. Balanced realizations for inÿnite-dimensional systems. In 1985.

Workshop on operator theory and its applications,

Amsterdam,

Bibliography

134

[37] K. Glover, R.F. Curtain, and J.R. Partington. Realization and approximation of linear inÿnite-dimensional systems with error bounds. Journal on Control and Optimization,

SIAM

26:863{898, 1988.

[38] K. Glover, J. Lam, and J.R. Partington.

Rational approximation of a

class of inÿnite-dimensional systems, I: Singular values of Hankel operators. Mathematics of Control, Signals and Systems,

[39] K. Glover, J. Lam, and J.R. Partington.

3:325{344, 1990.

Rational approximation of a

class of inÿnite-dimensional systems, II: Optimal convergence rates of approximants.

Mathematics of Control, Signals and Systems,

L1

4:233{246,

1991. [40] I. Gohberg and S. Goldberg. [41] P. Grabowski.

Basic Operator Theory.

Birkh auser, 1981.

On the spectral-Lyapunov approach to parametric opti-

mization of distributed parameter systems. Control and Information,

IMA Journal of Mathematical

7:317{338, 1990.

[42] M. Green, K. Glover, D. Limebeer, and J. Doyle. torization approach to

H1 control.

A

J ÿspectral

fac-

SIAM J. Control and Optimization,

28(6):1350{1371, November 1990.

[43] G. Gu, P. Khargonekar, and E.B. Lee. dimensional systems.

Approximation of inÿnite-

IEEE Transactions on Automatic Control,

34(6),

June 1989. [44] P.R. Halmos.

A Hilbert Space Problem Book.

Springer-Verlag, 2nd edition,

1982. [45] S. Hansen and G. Weiss. New results on the operator Carleson measure criterion.

IMA Journal of Mathematical Control and Information,

14:3{32,

1997. [46] E. Hille and R.S. Phillips.

Functional Analysis and Semigroups,

vol-

ume 31. American Mathematical Society Colloquium Publications, Providence, R.I., 1957. [47] O.V. Iftime and H.J. Zwart. Nehari problems and equalizing vectors for inÿnite-dimensional systems.

Submitted to the

Systems and Control Letters,

2001. [48] R.E. Kalman, P.L. Falb, and M.A. Arbib. Theory.

Topics in Mathematical System

McGraw-Hill, New York, 1969.

[49] T. Kato.

Perturbation Theory of Linear Operators.

Springer Verlag, New

York, 2nd edition, 1966. [50] S.Y. Kung and D.W. Lin. Optimal Hankel-norm model reductions: multivariable systems.

IEEE Trans. Autom. Control,

26:832{852, 1981.

Bibliography [51] S. Lang.

135

Complex Analysis.

Number 103 in Graduate Texts in Mathematics.

Springer-Verlag, New York, fourth edition edition, 1999. [52] I. Lasiecka and R. Triggiani.

Control theory for partial diÿerential equa-

tions: continuous and approximation theories,

volume I.

Cambridge Uni-

versity Press, 2000. [53] A. Lunardi.

lems,

Analytic Semigroups and Optimal Regularity in Parabolic Prob-

volume 16 of

Applications.

Progress in Nonlinear Diÿerential Equations and Their

Birkh auser, 1995.

[54] D.C. McFarlane and K. Glover.

Robust Controller Design using Normalized

Coprime Factor Plant Descriptions, and Information Sciences.

Lecture Notes in Control

volume 138 of

Springer-Verlag, 1989.

[55] B.C. Moore. Principal component analysis in linear systems: Controllability, observability, and model reduction.

Control,

IEEE Transactions on Automatic

AC-26:17{32, February 1981.

[56] Z. Nehari. On bounded bilinear forms.

Ann. Math.,

65:153{162, 1957.

[57] N.K. Nikol'skii. Ha-plitz operators: a survey of some recent results. In S.C. Power, editor, [58] R.J. Ober.

Operators and Function Theory.

A note on a system theoretic approach to a conjecture by

Peller-Khrushchev. [59] R.J. Ober.

Reidel, Boston, 1985.

Systems and Control Letters,

8:303{306, 1987.

A parametrization approach to inÿnite-dimensional balanced

systems and their approximation.

and Information,

IMA Journal of Mathematical Control

4:263{279, 1987.

Strongly Stabilizable Inþnite-Dimensional Systems.

[60] J.C. Oostveen.

PhD

thesis, Rijksuniversiteit Groningen, January 1999. [61] J.C. Oostveen.

Strongly Stabilizable Distributed Parameter Systems.

SIAM,

Philadelphia, 2000. [62] J.R. Partington.

An Introduction to Hankel Operators.

London Mathe-

matical Society Student Texts. Cambridge University Press, Cambridge, 1988. [63] A. Pazy.

Semigroups of Linear Operators and Applications to Partial Dif-

ferential Equations.

Springer-Verlag, New York, 1983.

[64] L. Pernebo and L.M. Silverman. Model reduction via balanced state space representations.

IEEE Transactions on Automatic Control,

AC-27(2):382{

387, 1982. [65] J.W. Polderman and J.C. Willems.

Theory.

Introduction to Mathematical Systems

Texts in Applied Mathematics. Springer-Verlag, 1998.

Bibliography

136

[66] A.J. Pritchard and D. Salamon. The linear quadratic control problem for inÿnite-dimensional systems.

SIAM Journal on Control and Optimization,

25:121{144, 1987.

[67] A. Ran. Hankel norm approximation for inÿnite-dimensional systems and Wiener-Hopf factorization. In R.F. Curtain, editor,

and Sensitivity Reduction in Control Systems,

Modelling Robustness

NATO ASI Series, pages

57{70. Springer-Verlag, 1986.

[68] R. Rebarber and G. Weiss. Optimizability and estimatability for inÿnitedimensional linear systems.

SIAM J. Control and Optim.,

[69] M. Rosenblum and J. Rovnyak.

Functions. [70] W. Rudin.

to appear.

Topics in Hardy Classes and Univalent

Birkh auser Advanced Texts. Birkh auser Verlag, Basel, 1994.

Real and Complex Analysis.

[71] W. J. Rugh.

Linear System Theory.

[72] D. Salamon. observation:

McGraw-Hill, 3

rd

edition, 1987.

Prentice Hall, 1993.

Inÿnite dimensional systems with unbounded control and

Trans. Amer. Math. Soc.,

A functional analytic approach.

300:383{431, 1987.

[73] D. Salamon.

Theory,

Realization theory in Hilbert space.

Mathematical Systems

21:147{164, 1989.

[74] D. Sarason.

Generalized interpolation in

127:179{203, 1967.

H1

.

Trans. Amer. Math. Soc.,

[75] O.J. Staþans. Quadratic optimal control of stable well-posed linear systems.

Transactions of the American Mathematical Society,

349:3679{3716, 1997.

[76] O.J. Staþans. Coprime factorizations and well-posed linear systems.

Journal on Control and Optimization,

SIAM

36:1268{1292, 1998.

[77] O.J. Staþans. Quadratic optimal control of a parabolic equation. In G. Picci A. Berghi, L. Finesso, editor,

Proceedings of the Mathematical Theory of

Networks and Systems symposium, [78] O.J. Staþans.

Admissible factorizations of Hankel operators induced by

well-posed linear systems.

[79] O.J. be

Staþans.

published,

pages 535{538, Padova, July 1998.

Systems and Control Letters,

37:301{307, 1999.

Well-Posed Linear Systems I: General Theory. 2002.

Draft

available

on

the

world

wide

web

To at

http://www.abo.fi/ staffans/publ.htm. [80] E.G.F. Thomas. Distributietheorie I en II. Vakgroep Wiskunde, Rijksuniversiteit Groningen, The Netherlands.

Bibliography

137

[81] E.G.F. Thomas. operator-valued Information,

Vector-valued integration with applications to the

H1

space.

IMA Journal of Mathematical Control and

14:109{136, 1997.

[82] S.R. Treil. A vector version of the Adamyan Arov Krein theorem (Russian). Funktsional. Anal. i Prilozhen.,

20(1):85{86, 1986.

[83] B. van Keulen. Hankel operators for non-exponentially stabilizable inÿnitedimensional systems. [84] B. van Keulen. space Approach.

[85] J. Weidmann.

H1

Systems and Control Letters,

15:221{226, 1990.

-Control for Distributed Parameter Systems: A State-

Birkh auser, Boston, 1993.

Linear Operators in Hilbert Spaces.

Springer-Verlag, 1980.

[86] G. Weiss. Admissibility of unbounded control operators. Control and Optimization,

SIAM Journal on

27:527{545, 1989.

[87] G. Weiss. The representation of regular linear systems on Hilbert spaces. In W. Schappacher, F. Kappel, and K. Kunisch, editors, tion of Distributed Parameter Systems,

Control and Estima-

pages 401{416. Birkh auser Verlag,

Basel, 1989. [88] G. Weiss. Regular linear systems with feedback. Signals and Systems,

Mathematics of Control,

7:23{57, 1994.

[89] G. Weiss. Transfer functions of regular linear systems, part 1: Characterizations of regularity.

Transactions of the American Mathematical Society,

342:827{854, 1994. [90] M. Weiss and G. Weiss. Optimal control of stable weakly regular linear systems.

Mathematics of Control, Signals and Systems,

[91] K. Yosida.

Functional Analysis.

10:287{330, 1997.

Springer Verlag, 6th edition, 1980.

[92] N.J. Young. Balanced, normal, and intermediate realizations of nonrational transfer functions.

IMA Journal of Mathematical Control and Information,

3:43{58, 1986. [93] N.J. Young. The Nehari problem and optimal Hankel norm approximation. In

Analysis and optimization of systems: state and frequency domain ap-

proaches for inÿnite-dimensional systems,

number 185 in Lecture Notes in

Control and Information Sciences. Springer, Berlin, 1993. [94] H.J. Zwart, R.F. Curtain, J.R. Partington, and K. Glover. Partial fraction expansions of delay systems. 1988.

Systems and Control Letters,

10:235{243,

Index ÿ

3

lines theorem, 57

G, 15 H2 (C k ), 4 HG , 4 H1 [l](C pÿm ), 52 H1 (C pÿm ), 52 H1 l (C pÿm ), 5 H1 (C pÿm ), 4 H1 (C pÿm ), 52 Xþ1 , 14 X1 , 14 Z , 25 c

;

c

;l

;

c

ÿ

ÿh , 7

bounded observation operator, 14

conjugate linear, 16 contraction semigroup, 115 controllability Gramian, 22, 26 controllability map, 22, 26

dual of an operator, 19 duality map, 16 duality pairing, 17

exponentially stable, 7 exponentially

þt , 14

C00 ; 1 C0 iR; C ÿ MH1 R R1 S1 ÿ+ (0

), 47

p m ),

(

stable

Pritchard-

Salamon system, 22

ý, 14

c

bounded control operator, 14

feedthrough operator, 16 63

, 53

, 53

, 53

generating operators of a well-posed linear system, 16 growth bound, 21

, 53

^, 7

Hankel, 2

Cr Cr

, 52

Hankel matrix, 1

, 52

Hankel norm, 8

þ

!0 , 21 C + , 52

þ r

Cr

, 52

ÿk (G), 5 ÿk (H ), 5 r(ÿ), 23

Hankel operator, 4, 22, 43 Hankel singular values, 5, 64 Hartman's theorem, 63 Hilbert matrix, 2

G

AAK theorem, 6 admissible control operator, 14 admissible observation operator, 14

impulse response, 7, 43 inünite-time admissible control operator, 15 inünite-time admissible observation operator, 15 inünitesimal

Bezout identity, 53

generator

of

a

well-

posed linear system, 15

140

Index

Schur, 2

input space, 15

semigroup of a well-posed linear sysLaurent matrix, 72

tem, 15

Lyapunov equation, 79

singular value, 5 skew-symmetric, 65

MacMillan degree, 5

smooth Pritchard-Salamon system,

model reduction, 8

21 Sobolev space, 24

Nehari, 1

solution to the sub-optimal Hankel

Nehari's theorem, 3

norm approximation prob-

nuclear, 65

lem, 6, 88 spectrum

observability Gramian, 22, 34 observability map, 22, 34 optimal Hankel norm approximant, 6

growth

as-

state-space, 16 strongly stable, 64 sub-optimal Nehari problem, 4

output space, 15

sub-optimal Hankel norm approxi-

output stable, 117

mant, 9 sub-optimal Hankel norm approxi-

pivot space, 17 Pritchard-Salamon admissible control operator, 20 Pritchard-Salamon

determined

sumption, 115

admissible

mation problem, 6, 88

time-domain Hankel operator, 7 ob-

transfer functions, 15

servation operator, 20 Pritchard-Salamon system, 21

unbounded control operator, 14

reciprocal system, 119

well-posed linear system, 15

regular well-posed linear system, 16 representation of the dual space, 17 right coprime, 53 right coprime factorization, 53

Standard notation Here we list some standard notation that we use in the thesis. Non standard symbols which are deÿned in the thesis can be looked up in the Index.

Set theory 1.

N:

2.

{ S:

f

the set of natural numbers complement of the set

g

1; 2; 3; : : : .

S.

Linear algebra Throughout this thesis, unless otherwise speciÿed, when considering a vector

C . In a matrix, blank entries are always to be

space, the underlying ÿeld is understood to be zeros.

V

1. dim( ), where

V

.

V

is a ÿnite-dimensional vector space: The dimension of

V !V

2. ker(T ) and ran(T ), where the vector space of

T,

T : 1 2 is a linear transformation from to the vector space 2 : the null space and the range

V

1

respectively.

V

Linear analysis 1.

LH H (

2 ),

1;

where

H

and

1

H

2

are Hilbert spaces: the Banach space of

all bounded linear operators with the norm 2.

ÿ(A),

þ (A),

p (A),

þ

linear operator, and

H

c (A), where

þ

A

:

kk T

D (A)

(

= sup(06=)x2H1

ÿH ! H )

H

, then

ÿ

A

is a closed

is a Hilbert space: the resolvent set, spectrum,

point spectrum and continuous spectrum, respectively, of dense in

kT xkH2 kxkH1 .

denotes the adjoint of

A.

A.

If

D (A)

is

142 3.

Standard notation

C

X ;X

X

( 1 2 ), where 1 continuous maps from

C

instead of 4.

`2 (N )

X1 ; C ).

(

`1 :

are topological spaces: the space of If

X2

=

X1 a b

n=1

ÿ

n n

respectively

C

(

X1 )

! 1 X ha; bi2 = anbn : n=ÿ1

the Banach space of summable complex sequences indexed by

kak1

=

1 X

n=1

the Banach space of complex sequences indexed by

kak1

N with

janj:

in absolute value with the norm

7.

C , we simply write

is

N (respectively Z), with the inner product

the norm

6.

X2 X2 .

`2 (Z)): the Hilbert space of square summable complex sequences

ha; bi2 `1 :

to

(

indexed by

5.

and

X1

= sup

n2N

N and bounded

jan j:

Lloc 2 (ÿ; X ), where ÿ(ÿ R) is measurable and X is a Banach space: ý þ measurable and : f : ÿ ! X j Rf iskfBochner-Lebesgue 2 K (! )k d! < 1 for any compact set K (ÿ ÿ)

þp