Linear Delay-Differential Systems with Commensurate Delays: An Algebraic Approach 3540428216, 9783540428213

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Lecture Notes in Mathematics Edited by J.-M. Morel, F. Takens and B. Teissier

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Lecture Notes in Mathematics Editors: J.-M . Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1770

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

Heide

Gluesing-Luerssen,

Linear

Delay- Differential

Systems with

Delays: An Algebraic Approach Commensurate

4

11,11 4%

Springer

Author Heide

Gluesing-Luerssen

Department of Mathematics University of Oldenburg 26111 Oldenburg, Germany e-mail:

[email protected]

Cataloging-in-Publication Data available Die Deutsche Bibliothek

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CIP-Einheitsaufnahme

Gltising-Ltierssen, Heide: delay differential systernswith commensurate'delays : an algebraic approach / Heide Gluesing-Lueerssen. Berlin; Heidelberg; New York; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Tokyo : Springer, 2002 (Lecture notes in mathematics ; 1770) Linear

-

ISBN 3-540-42821-6

Mathematics

Subject Classification (2000): 93CO5, 93B25, 93C23, 13B99, 39B72

ISSN 0075-8434 ISBN 3-540-42821-6

Springer-Verlag Berlin Heidelberg New York

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Preface

delay-differential equation was coined to comprise all types of differequations in which the unknown function and its derivatives occur with

The term ential

various values of the argument. In these notes we concentrate on (implicit) linear delay-differential equations with constant coefficients and commensurate

point delays. We present

an

investigation of dynamical delay-differential

sys-

tems with respect to their general system-theoretic properties. To this end, an algebraic setting for the equations under consideration is developed. A thorough

purely algebraic study shows that this setting is well-suited for an examination of delay-differential systems from the behavioral point of view in modern systems theory. The central object is a suitably defined operator algebra which turns out to be an elementary divisor domain and thus provides the main tool for handling matrix equations of delay-differential type. The presentation is introductory and mostly self-contained, no prior knowledge of delay-differential equations or (behavioral) systems theory will be assumed. people whom I am pleased to thank for making this grateful to Jan C. Willems for suggesting the topic "delaydifferential systems in the behavioral approach" to me. Agreeing with him, that algebraic methods and the behavioral approach sound like a promising combination for these systems, I started working on the project and had no idea of what I was heading for. Many interesting problems had to be settled (resulting in Chapter 3 of this book) before the behavioral approach could be started. Special thanks go to Wiland Schmale for the numerous fruitful discussions we had in particular at the beginning of the project. They finally brought me on the right track for finding the appropriate algebraic setting. But also later on, he kept discussing the subject with me in a very stimulating fashion. His interest in computer algebra made me think about symbolic computability of the Bezout identity and Section 3.6 owes a lot to his insight on symbolic computation. I wish to thank him for his helpful feedback and criticisms. These notes grew out of my Habilitationsschrift at the University of Oldenburg, Germany. The readers Uwe Helmke, Joachim Rosenthal, Wiland Schmale, and Jan C. Willems deserve special mention for their generous collaboration. I also want, to thank the Springer-Verlag for the pleasant cooperation. Finally, my greatest thanks go There

work

are

a

number of

possible.

I

am

VI

Preface

only for many hours carefully proofreading all making helpful suggestions, but also, and even more, for and so patient, supportive, being encouraging during the time I was occupied with writing the "Schrift". to my

partner, Uwe Nagel,

these pages and

Oldenburg, July

not

various

2001

Heide

Gluesing-Luerssen

Table of Contents

1

Introduction

2

The

Algebraic

3

The

3.1

Algebraic Structure Divisibility Properties

3.2

Matrices

3.3

Systems over Rings: A Brief Survey Nonfinitely Generated Ideals of Ho The Ring H as a Convolution Algebra Computing the Bezout Identity

3.4 3.5

3.6 4

5

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

Framework

over

Ho

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

of

Wo

,

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

25 35

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

43

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

45

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

51

Delay-Differential Systems

4.1

The Lattice of Behaviors

4.2

Input/Output Systems

4.3

Transfer Classes and Controllable

4.4

Subbehaviors and Interconnections

Assigning

4.6

Biduals of

..........

59

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

73

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

76

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

89

Systems

Nonfinitely Generated

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

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

the Characteristic Function

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

Ideals

5.1

Representations Multi-Operator Systems

5.2

The Realization Procedure of Fuhrmann

5.3

First-Order Realizations

5.4

Some

First-Order

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

9&

104 115 129

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

135

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

138

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

148

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

157'

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

162

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

169

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

175

References Index

23

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

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

4.5

7

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

The

Behaviors of

I

Minimality Issues

I Introduction

Delay-differential equations (DDEs, for short) arise when dynamical systems time-lags are being modeled. Such lags might for instance occur if some nonnegligible transportation time is involved in the system or if the system needs

with

a

certain amount of time to

sense

information and react

on

it. The characteristic

feature of

a system with time-lags is that the dynamics at a certain time does not only depend on the instantaneous state of the system but also on some past values. The dependence on the past can take various shapes. The simplest type is

that of

retardation,

delay, describing for instance the generally, might depend on time (or other effects). Modeling such systems leads to differential- difference equations, also called differential equations with a deviating argument, in which the unknown function and its derivatives occur with their respective values at various time instants t--rk. A completely different form of past dependence arises if the process under investigation depends on the full history of the system over a certain time interval. In this case a ma*matical formulation leads to general functional-differential equations, for instance integro-differential equations. In control theory the term distributed delay, as opposed to point delay, has been coined for this type of past dependence. We will consistently use the term delaydifferential'equation for differential equations having any kind of delay involved. a

constant

reaction time of

All the

a

a

so-called point

system. More

the reaction time itself

delay-differential equations described above

fall in the category of

infinite-dimensional systems. The evolution of these systems can be described in a twofold way. On the one hand, the equations can, in certain circumstances, be formulated

abstract differential

equations on an infinite-dimensional space. conditions, which in this case are segover a time interval of appropriate length. This description leads to an pperator-theoretic framework, well suited for the investigation of the qualitativeIbehavior of these systems. For, a treatment of DDEs based on functional analytic methods we refer to the books Hale and Verduyn Lunel [49] and Diekmann et al. [22] for functional-differential equations and to the introductory book Curtain and Zwart [20] on general infinite-dimensional linear systems in control theory. On the other hand, DDEs deal with one-variable functions and can be treated to a certain extent with "analysis on W' and transform techniques. For an investigation, of DDEs in this spirit we refer to the books Bellman and Cooke [3], Driver [23], El'sgol'ts and Norkin [28], and Kolmanovskii and as

The space consists ments of functions

basically of

all initial

H. Gluesing-Luerssen: LNM 1770, pp. 1 - 5, 2002 © Springer-Verlag Berlin Heidelberg 2002

1 Introduction

2

[65]

t4erein. All the monographs mentioned so far aim analyzing qualitative behavior of their respective equations, most of the time with an emphasis on stability theory. Nosov

and the references the

at

Our interest in DDEs is of

systems

different nature. Our

a

goal

is

an

investigation of

governed by DDEs with respect to their general control-theoretic prop-

adopt an approach which goes back to Willems (see nowadays called the behavioral approach to systems theory. In this framework, the key notion for specifying a system is the space -of all possible trajectories of that system. This space, the behavior, can be regarded as the most intrinsic part of the dynamical system. In case the dynamics can be described by a set of equations, it is simply the corresponding solution space. Behavioral theory now introduces all fundamental system properties and constructions in terms of the behavior, that means at the level of the trajectories of the system and independent of a chosen representation. In order to develop a mathematical theory, one must be able to deduce these properties from the equations governing the system, maybe even find characterizations in terms of the equations. For systems governed by linear time-invariant ordinary erties. To this

for instance

end,

we

will

[118, 119])

and is

differential equations this has been worked out in great detail and has led to a successful theory, see, e. g., the book Polderman and Willems [87]. Similarly for multidimensional systems, described by partial differential or discrete-time difference equations, much progress has been made in this direction, see for instance Oberst

troller, the

[84], Wood'et

al.

[123],

and Wood

[122].

The notion of

a con-

important tool of control theory, can also be incorporated in this framework. A controller forms a system itself, thus a family of trajectories, most

and the interconnection of

a

to-be-controlled system with

a

controller

simply

leads to the intersection of the two respective behaviors. The aim of this

monograph is to develop, and then to apply, a theory which dynamical systems described by DDEs can be successfully studied from the behavioral point of view. In order to pursue this goal, it is unavoidable to understand the relationship between behaviors and their -describing equations in full detail. For instance, we will need to know the (algebraic) relation between two sets of equations which share the 'same solution space. Restricting shows that

to

a

reasonable class of systems, this

can

indeed be achieved and leads to

an

al-

gebraic setting, well suited for further investigations. To.be precise, the class of systems we are going to study consists of (implicit) linear DDEs with constant coefficients and commensurate point delays. The solutions being considered are in the space of C'-functions. Formulating all this in algebraic terms, one obtains a setting where a polynomial ring in two operators acts on a module of functions. However, it turns out that in order to answer the problem raised above, this setting will not suffice, but rather has to be enlarged. More specifically, certain distributed delay operators (in other words, integro-differential equations) have to be incorporated in our framework. These distributed delays have a very specific feature; just like point-delay-differential operators they are determined by finitely many data, in fact they correspond to certain rational

1 Introduction

functions in two variables. In order to get an idea of this larger algebraic setting, only a few basic analytic properties of scalar DDEs are needed. Yet, some

algebraic investigations are necessary to see that this provides indeed the appropriate framework. In fact, it subsequently allows one to draw far-reaching consequences, even for systems of DDEs, so that finally the behavioral approach careful

can be initiated.

of

As

algebra which in

a

consequence, the

our,

opinion

is

monographcontains

fairly interesting by

a

considerable part

itself.

delay-differential systems have already been studalgebraic point of view in the seventies, see, e. g., Kamen [61], Morse [79], and Sontag [105]. These papers have initiated the theory of Systems over rings, which developed towards an investigation of dynamical systems where the trajectories evolve in the ring itself. Although this point of view leads away from the actual system, it has been (and still is) fruitful whenever system properties concerning solely the ring of operators are investigated. Furthermore it has led to interesting and difficult purely ring-theoretic problems. Even though our approach is ring-theoretic as well, it is not in the spirit of systems over rings, for simply the trajectories live in a function space., Yet, there exist a few connections between the theory of systems over rings. and our approach; we will therefore present some more detailed aspects of systems over

We want to remark that ied from

an

rings later

in the book.

give a brief overview of the organization of the book. Chapintroducing the class of DDEs under consideration along with the algebraic setting mentioned above. A very specific and simple relation between linear ordinary differential equations and DDEs'suggests to study a ring of operators consisting of point-delay-differential operators as well as certain distributed delays; it will be denoted by H. In Chapter 3 we disregard the interpretation as delay-differential operators and investigate the ring 'H from a purely algebraic point of view. The main result of this chapter will be that the ring'H forms a so-called elementary divisor domain. Roughly speaking, this says that matrices with entries in that ring behave under unimodular transformaWe

now

proceed

to

ter 2 starts with

tions like matrices are

determined

over

Euclidean domains. The fact that all operators in H many data raises the question whether these data

by finitely

is to say, a desired operator) can be determined exactly. We will address problem by discussing symbolic computability of the relevant constructions in that ring. Furthermore, we will present a description of H as a convolution algebra consisting of distributions with compact support. In Chapter 4 we finally turn to systems of DDEs. We'Start with deriving a Galois-correspondence between behaviors on the one side and the modules of annihilating operators on the other. Among other things, this comprises an algebraic characterization of systems of DDEs sharing the same solution space. The correspondence emerges from a combination of the algebraic structure of 'H with the basic analytic properties of scalar DDEs derived in Chapter 2; no further analytic study of

(that

this

1 Introduction

systems of DDEs is needed.* The Galois-correspondence constitutes

an

efficient

machinery for addressing the system-theoretic problems studied in the subsequent sections. Therein, some of the basic concepts of systems theory, defined purely in terms of trajectories, will be characterized by algebraic properties of the associated equations. We will mainly be concerned with the notions of controllability, input/output partitions (including causality) and the investigation of interconnection of systems. The latter touches upon the central concept of theory, feedback control. The algebraic characterizations generalize the

control

well-known results for systems described by linear time-invariant ordinary differential equations. A new version of the finite-spectrum assignment problem,-

well-studied in the

analytic framework of time-delay systems, will be given in the Chapter 5 we study a problem which is known as state-space realization in case of systems of ordinary differential equations. If we cast this concept in the behavioral context for DDEs, the problem amounts to finding system descriptions, which, upon introducing auxiliary variables, form explicit DDEs of first -order (with respect to differentiation) and of retarded type. Hence, among other things, we aim at transforming implicit system descriptions into explicit ones. Explicit first order DDEs of retarded type form the simplest kind of systems within our framework. -Of the various classes of DDEs investigated in the literature, they are the best studied and, with respect to applications, the most important ones. The construction of such a description (if it, exists) takes place in a completely polynomial setting, in other words, no distributed delays arise. Therefore, the methods of this chapter are different from what has been used previously. As a consequence and by-product, the construction even works for a much broader class of systems including for instance certain partial differential equations. A complete characterization, however, of systems allowing such an explicit first order description, will be derived only for algebraic setting.

In the final

DDEs.

A

more

detailed

description of the

contents of each

chapter

is

given

in its

re-

spective introduction. We close the introduction with

some remarks on applications of DDEs. One applications occurred in population dynamics, beginning with the predator-prey models of Volterra in the 1920s. Since population models are in general nonlinear, we will not discuss this area and refer to the books Kuang [66],

of the first

MacDonald

[70],

and Diekmann et al.

of Volterra remained

basically

[22]

and the references

therein.

The work

unnoticed for almost two decades and

only in early forties DDEs got much at "tention when Minorsky [77] began to study ship stabilization and automatic steering. He pointed out that for these systems the existing delays in the feedback mechanism can by no means be neglected. Because of the great interest in control theory during that time and the

At this point the reader familiar with the paper [84] of Oberst will notice the similarity of systems of DDEs to multidimensional systems. We will point

structural

out the similarities and differences between these two

several occasions later

on.

types of systems classes

on

1 Introduction

the decades to follow the work of

rapid development

of the

of

Minorsky led to DDEs; for more

other

applications

and

a

theory period see for instance the preface of Kolmanovskii and Nosov [65] and the list of applications in Driver [23, pp. 239]. It was Myschkis [81] who first introduced a class of functional-differential equations and laid 'the foundations of a general theory of these systems. Monographs and textbooks that appeared ever since include Bellman and Cooke [3], El'sgol'ts and Norkin [281, Hale [481, Driver [23], Kolmanovskii and Nosov [65], Hale and Verduyn Lunel [49], and Diekmann et al. [22]. A nice and brief overview of applications of DDEs in engineering can be found in the book Kolmanovskii and Nosov

details about that

[65],

from which

we

extract

following mixing processes are engineering, with of because natural a delay, examples time-lag arises due systems to the time the process needs to complete its job; see also Ray [89, Sec. 4.5] for an explicit example given in transfer function form. Furthermore, any kind of system where substances, information, or energy (wave propagation in deep space communication) is being transmitted to certain distances, experiences a time-lag due to transportation time. An additional time-lag might arise due to the

list. In chemical

reactors and

standard

(ship stabilization) or for (biological models). A model

the time needed for certain measurements to be taken the system to of a turbojet

sense

information and react

on

it

engine, given by a linear system of five first-order delay equainputs and five to-be-controlled variables can be found in [65, Sec. 1.5]. Moreover a system of fifth-order DDEs of neutral type arises as a linear model of a grinding process in [65, Sec. 1.7]. Finally we would like to mention a linearized model of the Mach number control in a wind tunnel presented in Manitius [75]. The system consists of three explicit equations of first order with a time-delay occurring only in one of the state variables but not in the input

tions with three

channel. In that paper the problem of feedback control for the regulation of the Mach number is studied and various different feedback controllers are derived

by transfer function methods. This problem can be regarded as a special case of the finite-spectrum assignment problem and can therefore also be solved within our algebraic approach developed in Section 4.5. Our procedure leads to one of the feedback controllers (in fact, the simplest and most practical one) derived in

[75].

Algebraic Framework for Delay-Differential Equations 2 The

specific class of delay-differential equations we are some basic, yet important, properties. In this way we hope to make clear that, and how, the algebraic approach we are heading for depends only on a few elementary analytic properties of the equations under consideration. The fact that we can indeed proceed by mainly algebraic arguments results from the structure of the equations under consideration together with'the type of problems we are interested in. To be precise, we will restrict to linear delay-differential equations with constant coefficients and commensurate point-delays on the space C' (R, C). We are not aiming at solving these equaIn this

chapter

we

introduce the

interested in and derive

tions and

expressing the solutions

(appropriate)

-in terms of

initial data. For

purposes it will suffice to know that the solution space of a DDE (without initial conditions), L e. the kernel of the associated delay-differential operator,.

our

is "sufficiently rich". In essence, we need some knowledge about the exponential polynomials in the solution space; hence about the zeros of a suitably defined characteristic function in the complex plane.

Yet,

in order to pursue

by algebraic

means, the

appropriate setting has to be driving goal to handle also systems of DDEs, in other words, matrix equations. In this chapter we will develop the algebraic context for these considerations. Precisely, a ring of delay-differential operators acting on C1 (R, C) will be defined, comprising not only the pointdelay differential operators induced by the above-mentioned equations but also certain distributed delays which arise from a simple comparison of ordinary differential equations and DDEs. It is by no means clear that the so-defined operator ring will be suitable for studying systems of DDEs. That this is indeed the case will turn out only after a thorough algebraic study in Chapter 3. In the present chapter we confine ourselves with introducing that ring and providing some standard results about DDEs necessary for later exposition. In particular, we will show that the delay-differential operators under consideration are surjections on C1 (R, C). found first. The

force in this direction is

our

As the starting point of our investigation, let us consider a homogeneous, linear DDE with constant coefficients and commensurate point delays, that is an

equation of the type

H. Gluesing-Luerssen: LNM 1770, pp. 7 - 21, 2002 © Springer-Verlag Berlin Heidelberg 2002

2 The

Algebraic

Framework N

M

EEpijf( )(t-jh)=O, i=0

tER,

j=0

N,

where

M E No, pij c involved. Hence all

R, and h > 0 is the smallest length of the point delays are integer multiples of the constant h, thus commensurate. For our purposes it suffices to assume the smallest delay to be of unit length, which can easily be achieved by rescaling the time axis. Therefore, from now on we will only be concerned with the case h I and the equation

delays

=

above reads

as

M

N

EEpjjf(')(t-j)=0, i=0

(2.1)

tGR.

j=0

It will be

important for our setting that the equation- is considered on the full time axis R. Moreover, we are not imposing any kind of initial conditions but rather focus on the solution space in C C' (R, C), hence on B The choice C

C' A

:=

ff

E

L

1 (2. 1)

is

satisfiedl.

C)

is algebraically very convenient, for 'C is invariant shift, hence a module over the corresponding ring of delay-differential operators. In a certain way, however, larger classes of functions can be incorporated in the algebraic approach; this will be discussed occasionally throughout the book. =

under differentiation and

Observe that'equations of the type (2.1) cover in particular linear time-invariant ordinary differential equations (ODEs, for short) as well as pure delay equations

(N

=

0). briefly

Equation (2.1). Disregarding intuitively clear what the minimum amount 'of initial data should be in order for (2.1) to single out a unique solution (if any). It is natural to require that f satisfy f (t) fo(t) for t E [0, M], where fo is some prespecified function on the interval [0, M] and M is the largest delay appearing in (2.1). Then finding a solution on the full time axis R amounts to solving the initial value problem in both forward and backward direction. This, is, of course, not always possible. It also fails if one starts with an arbitrary smooth initial condition, i. e. fo C- C' Q0, M], C), and seeks solutions in L. But, if fo is chosen correctly (that is, with correct data at the endpoints of the interval [0, M]), a unique forward and backward &-solution exists; this will be shown in Proposition 2.14. The solvability of this restricted initial value problem for the quite general equation (2.1) rests on Let

us

think about initial conditions for

the requirement that solutions be

smooth,

it should be

=

the fact that

we

differentiability

consider

C'-functions, so that we have a sufficient amount of fo, necessary for solving the equation on

of the initial condition

the whole of R. Remark 2.1 It is crucial for

restrict to DDEs with

mensurate

of noncommensurate

essentially all parts of our work to delays. As it turns out, the occurrence

com-

delays

2 The

Algebraic

Framework

(like e. g. delays of length 1 and V2_ or -7r) leads to serious obstacles preventing an algebraic approach similar to the one to be presented here; see [47, 109, 111, 26]. At this point we only want to remark that in the general case the according operator ring lacks the advantageous algebraic properties which will be derived for our case in the next chapter. These differences will be pointed out in some more detail in later chapters (see 3.1-8, 4.1.15, 4.3.13). Remark 2.2 In the

theory of

DDEs

distinguishes equations of retarded, neutral,

one

and

advanced type. These notions describe whether or not the highest derivative in, say (2.1), occurs with a delayed argument. Precisely, Equation (2.1) is called

retarded if PNO : PNO 0 0 and PNj

4].

0 and PNj 0 for some

0 for

=

0

j

>

j M; it is said to be neutral if 1, 0, and advanced in all other cases, see [28, =

.

.

.

,

This classification is relevant when

solving initial value problems in forward Roughly speaking, it reflects how much differentiability of the initial condition on [0, M] is required for (2.1) being solvable in forward direction; see for instance the results [3, Thms. 6.1, 6.2, and the transformation on p. 192]. Since we are dealing with infinitely differentiable functions and, additionally, requite forward and backward solvability, these notions are not really relevant p.

direction.

for

Let

our

us

purposes.

now

Introducing

rewrite

af (t) and the

p(D, u)f

Equation (2.1)

the forward shift :=

f (t

-

a

1),

in terms of the

of unit

where

f

is a1unction defined

ordinary differential operator 0, where N

p(D, a)

is

a polynomial simply

in the two

D

=

R,

d, Equation (2.1)

dt

reads

as

M

(2.2)

j=0

commuting operators D and B

on

1: 1: pij D'ai i=0

is

corresponding operators.

length

ker p(D,

a)

C

a.

The solution space

(2.3)

L.

For notational reasons, which will become clear in a moment, it will be convenient to have an abstract polynomial ring R[s, z] with algebraically independent

disposal. (The names chosen for the indeterminates Laplace transform s of the differential operator D and the z-transform of the shift-operator in discrete7time systems.) Since the shift U is a bijection on L, it will be advantageous to introduce even the (partially) Laurent polynomial ring elements

s

and

z

at

our

should remind of the

R[s, z Z-1

N i=O

pijSY j=m

Tn, ME Z N E=

No, pij

E R

10

Algebraic

2 The

Associating

Framework

polynomial the delay-differential operator (including possibly backward shifts) we obtain the ring embedding with each Laurent

R[s, z, z-1] (of

)

EndC (,C),

p

p(D, o,)

)

i

(2.4)

polynomial, then the operator p(D, 0') is not the words, the operators D and a are algebraically operator C). R the in over independent ring Endc(,C). Put yet another way, C is a faithful module over the commutative operator ring R[D, a, o-1]. course, if p is

zero

Let

a nonzero

In other

on

exponential functions eA*

look for

us now

like for ODEs

in the solution space

(2.3).

Just

has for A E C

one

(NE

p(D, o,) (e A.)

i=O

M

N

M

E.Y pjjA e- \j )

E pjjDY) (e\')

e

A-

(2-5)

,

j=M

i=O

j=M

p(A, e--\)e"' Hence the

function the

exponential

p(s, e-')

function e,\* is

solution if and

a

only if

A is

a zero

of the

which therefore will be called the characteristic function of

0. Obviously, it delay-differential equation p(D, o,)f as exponential polynomial (or quasi polynomial).

is

=

an

entire

function,

known

Before some

providing

details

some more

on

exponential polynomials,

we

want to fix

notation.

Definition 2.3

(1)

tions

(2)

by H(C) (resp. M(C)) the ring on the full complex plane.

Denote

For

a

zeros

subset S C

H(C)

(3)

define the variety

fl,

S

case

.

.

.

,

JA

:=

fj I

is

C

finite,

=

P

I f (A)

In

case

q*

is

f

E

the set of all

as

we

call the set zeros

func-

common

E

S}.

m

meromorphic

p(s, e_S) 0(s)

V(q*)

'

.

.

E

.

,

fl)

for V (S).

R[s, z, z-']

function given

for

S

G

and

by

C\V(O).

the characteristic variety and its

of q.

and A E C let

ord.\ (f denote the

f

write V (fl,

j=

O(S) entire,

0 for all

EN 0 EM pijs'.zj j= the

M

H(C)

for A e C.

=

EN 0 EM Pijsie-i' j=

elements the characteristic For

V(S)

simply

we

=

0

q*(s)

(4)

E

R(s) [z, z-'], where p R[s]\f 01, denote by q* G M(C)

For q

(resp. meromorphic)

of S, thus

V(S) In

of entire

multiplicity of A '

,

minf k

E

as a zero

No I of

f.

f(k) (A) If

f

=-

76 0} 0,

we

put

ord.\(f)

=

oo

2 The

Algebraic

Framework

11

(1) of the next proposition is standard in the theory of DDEs. Just like ODEs, the multiplicities of the characteristic zeros correspond to exponen tial monomials in the solution space. As a simple consequence we include the fact that delay-differential operators are surjective on the space of exponential polynomials. Part

for

Proposition

(1)

2.4

R[s, z, z-'] \10}.

Let p e

No and A E C denote by ek,A tke,\'. Then

For k

ek,A (t)

,=o

In

particular, ek,X

C ker

H(C) is called operator p(D, o). (2) The operator p(D, o)

p(D, o)

is

ponential polynomials cisely, let a := ord,\(p*) !

monomial

only if ord,\ (p*)

k. The function

>

delay-differential

surjective endomorphism. on the space of exspan(C f ek,A I k E No, A E C}. More pre-

a

B

ao,

if and

the characteristic function of the

E

al+a E

exponential

(k)' (p*)()(A)ek-K,A.

p(D, u)ek,,%

p*

L the

E

:=

0.

Then, for

:

0 such that

C with al+a

all el,,\ E B there exist constants

( 1=0 +a

p(D, a)

E a,, e,,,,\

(2.6)

el,,\.

r.

PROOF:

(1)

Let p

verified in the

=

I:i,j pijs'zi

following

E

R[s, z, z-1].

The asserted

identity

is

easily

way:

di

(p(D, u)ek,.\) (t)

[(t

Pij Tti

_

E P'j Tti dAk (eA(t-j)

10

dk dAk

dk

di

j)k e)(t-j)]

1,3

(E pjjA'e'X(t-j) )

dk dAk

(p*(A)e\t)

1,3

k

E

=

K=o

The rest of

(1)

(k) (p*)( ')(A)ek-r.,A(t)K

is clear.

(2) It suffices to establish (2.6). We proceed by (p*) (a) (A). Then c : - 0 by assumption. For I =' 0 it follows from (1) that p(D, o) (c- 1 ea,,\)

induction

=

For 1 > 0 put al+a

1+a)-

1

a

c

-1

1+a

p(D, o,)(al+ael+a,,\)

=

al+a

E r.=a

.

Then, by

(1 a)

virtue of

eo,,\,

as

1. Put

c

desired.

(1),

+

K

on

el+a-r.,,X

=

el,,x +

1:'bjej,,\ j=o

2 The

12

Algebraic

Framework

some constants bj G C. By induction the functions bjej,,\ have preimages involving solely exponential monomials ei,.\ with i < 1 + a 1. Combining them El suitably with the equation above yields the desired result.

for

-

foregoing

play exactly the correspond to the exponentialmonomials in the solution space. The main difference to OI?Es is that the characteristic function has infinitely many zeros in the complex plane unless it degenerates to a polynomial. Since this property will be of central importance for the algebraic setting (in fact, this will be the only information about the solution spaces of DDEs we are going to need), we include a short proof showing The

same

role

how it in

as

ODEs,

in the

sense

that their

zeros

be deduced from Hadamard's Factorization Theorem. The estimate

can

(1)

part

considerations show that characteristic functions

for

below will be useful in

later section to embed

a

R[s, z, z-']

in

a

Paley-Wiener algebra. Proposition Let p E

(1)

2.5

R[s, z, z-1] .'Then

there exist constants

C,

jp*(S)1: (2)

0 and N G

C(I

+

No such that

ISI)N ealResi

for all

C

S

(C'

variety satisfies

the characteristic

#V(P*)

>

a

< 00 4==> P

=

Zko

for

some

k E Z and

0

E

R[s]\f01.

In the classical paper [88] much more details about the location of the zeros of p* can be found, see also [3, Ch. 13]. As we are not dealing with stability

issues, the above information (2) suffices for

(1) Letting p

PROOF:

=

:5

where C > 0 is

a

Theorem,

defined

Pij

estimate

we can


m. The function q* consequence, said to be the characteristic function of the operator 4.

As is

sum

2

:=

=

Proposition 2.4(l),

we

=

obtain

qf

=

and the desired result follows since

Pg

Em+kg,'Ev=0(v) (P _.,v=O

(p*) W(A)

=

K

0 for

n

K

< 1 +

k.

(A)e,-,.,,\ EJ

Remark 2.13

Notice that

we did not consider any expansions of solutions as infinite series exponential polynomials. Such expansions do exist, see [102] and [3, Ch. 6], the latter for solutions of retarded equations on R+. We will not utilize these facts since the only case, where the full information about the solution space is needed, is that of ODEs, see also (2.10). For the general case it will be sufficient for us to know which exponential monomials are contained in the solution space. Series expansions of the type above are important when dealing with stability of DDEs. We will briefly discuss the issue of stability in Section 4.5, where we will simply quote the relevant results from the literature.

of

We conclude

our

considerations

scalar DDEs with the

on

surjectivity of delay-

differential operators on L. This fact is well-known and can be found in [25, p. 697], where it is stated in a much more general context and proven with rather elaborate methods.

However,

we

shows what kind of initial conditions also

the opportunity to present the method for-solving initial value problems of DDEs.

gives

cedure

us

Earlier in this

be

would like to prove a version which also be imposed for the DDE (2.1). This

can

specified

chapter

we

in order for

briefly

(2.1)

of steps, the

standard pro-

addressed what kind of initial data should

single out a unique solution f. Apart from suggested that f has to be specified on an interval delay occurring in (2.1). For instance, a solution of the pure delay equation of 0 is determined completely by the restriction f fo := f 1 [0, 1). But in order that f be smooth, it is certainly necessary that the initial condition'fo can be extended to a smooth function on [0, 1] having equal (v) derivatives f0(v) (0) f0 (1) of all orders v E No at the endpoints of the interval. In other words, fo and all its derivatives have to satisfy the delay equation for 1. This idea generalizes to arbitrary DDEs and leads to the restriction given t smoothness requirements, of length M, the largest

we

-

=

=

to

=

2 The

18

Algebraic

(2.11) below,

in

Framework says that the initial condition has to be

simply

which

ble with the given

our

advanced equations of we could not find a reference for the result cedure is standard and

one

0 : 0, and

(1)

b]

:=

fo

E

Ej' o pj zi,

=

C' [0,

M]

there exists

f I [0,M]

PROOF:

--

(=- ker

unique function f

a

As

fo.

=

a

4 9 L satisfies f I [k,k+M]

g() (M)

for all

b.

[s],

0

po

:

pm,

=_

0 for

some

k c

(2.11)

No

p(D, o,)f on

g

and

L.

R, then f

-=

0.

b]

defined

M

0

-

g(') (t)

for all

v

E

No

(2.12)

j=o

for all t G

[a

+

M, b]

+

1]

which satisfies

be extended in

can

(2.12)

a

[a

on

-

unique way to a solution 1 + M, b + 1]. (Notice that the

initial condition given in the proposition is included To this

E

v

L such that

E

Epj (D)oif (v)) (t)

(t)

C' [a

a=O and

a or

-

0

1, b

G R

To prove the existence of f , we show: every fo C C' [a, length b a > M which satisfies the condition

(1)

(p(D, u)f(v)) -

0

pj,

consequence, the map 4 is surjective

interval of

on an

endpoints

satisfying

0

f

as

Furthermore,

(p(D, a)f ()) (M) (2)

b], (C)

Thms. 3.1 and 5.21. In f M for f C- C' [a, b]

[3,

well

as

0 1 where p let g E L

7io

G

M > 1.

For every

If

in the book

C' ([a,

2.14

po'

=

presented

course, to one-sided derivatives when taken at the

Proposition Let q

stated below. However, the prosimilarity of the proof given below

as

should notice the

for part (1) with, e. g., those the sequel the notation C' [a,

refers, of

compati-

approach comprises retarded, neutral, and also arbitrary order, and, additionally, requires smoothness,

DDE. As

as an

extreme

case

where

b=:M.) write po (s)

end,

=0

ai sz + Sr and

consider the

inhomogeneous ODE

M

po(D)f (t)

g(t)

=

1: pj (D) ci fo) (t)

-

(2.13)

j=1

for t G

[b, b + 1]

with initial condition

j(v) (b) (If a

r

=

0, then

po

unique solution

1 and c

=

no

(v)

f0

(b)

for

(b)

=

g (b)

=

0,..., r

-

(2.14)

1.

imposed). In any (2.13), (2.14) and j satisfies

initial condition is

C' [b, b +

1]

to

M

M

v

pj (D) ai

-

j=1

fo)

case, there is

r-1

(b)

-

E ai j(') (b) i=O

=

()

f0

(b).

Algebraic

2 The

Framework

19

(") Differentiating (2.13) and using (2.12) shows successively j(') (b) f0 (b) for all v E No. Therefore, the function f, defined by f, (t) for t E [a, b] and fo (t) f, (t) =j (t) for t E (b, b+1] is in C'[a, b+1] and, by construction, satisfies (2.12) on [a + M, b + 1]. In the same manner one can extend f, to a smooth solution =

=

on

[a

-

1, b

+

1];

one

takes the

unique solution of the ODE

M-1

pm(D)f (t)

=

g(t)

E pj(D)fj(t

-

-

j)

[a +

on

M

-

1,

a

+

M]

j=0

initial'data

with

(a

+

M)

fl(') (a)

=

for

v

=

0,

.

.

.

,

deg pm

-

1 and

puts

f2(t):=f(t+M)f6ra-1 0, that are linearly independent over Q. As shown in [47, Thms. 5.4 and 5.9], the algebraic approach leads to thepperator algebra

In

the

approach of

our

with noncommensurate

.

.

.

,

=

-

'H(j):=

f

G

f*(s)

above is due to Exa.

5.13]

ker

R[s,zj,...,zj],

p E

:=

[4].

q E

f (s, e-rls,...' e-"') Note that 'H

'H(j)

reveals that

is not

zout

identity by

kerfil

R[s]\fOl,v

and z'

a

algebraic approach

an

upon these issues in

common

divisor

equations with Chapter 4.

to

along with

a

Be-

matrices

H,

over

see

Exam-

3.2.3.

Example 3.1.9 For computational issues, which

will be addressed in Section

track of the coefficients of the indeterminates

starting with coefficients in Q. S2 Z + S2 1, q (1) Let p =

s

-

=

f (s) simple

=

s

requirement b* -

a* (s) (e-s +

choice

a

-1 G

=

8

over

ap

a

of the indeterminates Let p

=

Bezout

z, q

=

s

equation

+ 1 E I

the sole condition

=

-

a

Bezout

identity

Q(e).

as

_

are z

0

can

1) in

+

S

+

(Z

S2

+

_

82

0. The

identity

1) 82

Q (s) [z] n HO, that is, all the coefficients equation above are in Q.

in the

Q[s, z].

ap +

=

function

=

_

and

a*(-l)

p* (s) 'HO

over

a(z + 82

Ro suffices and leads to the Bezout

1

is

since

bq

H(C) forces a G Ro to be such that the 1) has a zero of multiplicity 2 at s

and b

s

s

+

S2

_(Z +'32

=

HO. Notice that

E

S2.

keep below,

will

we

as

q

Now the

3.6,

in the calculations

z

=

(p*)'(0) : 0. In this case, easily be found by rewriting it b

and

s

Q [s, z]. Then gcd,O (p, q) a Bezout identity s ap

G

=

-

and

(2)

H(C)

of the theorem above will be

addressed in the next section when

ple

E

q*

z1".....zl". The last identity

:=

simple examples. Part (b) considering

some

P*

=

greatest

a

N', 0

E

if -ri 1. A fairly simple example [47, Bezout domain whenever 1 > 1. As a

delays. We will touch

We illustrate the determination of

C

7i(j)

=

consequence, serious obstacles arise for

noncommensurate

4

H(C)

R(s, zl,..., zj) f

zvq wherenow

R[s, zj,..., zi],

p, q G

q

33

Divisibility Properties

3.1

Then p and q

bq one needs e-1. Hence

b*

are

coprime

1-a*e-'

8+1

E:

in

HO and for

H(C), leading

a

to

=

=

e-1Z

1

+

-

e-1z

S+1

desired. In this

case

(8+1) the coefficients of

8

and

z are

It is easy to see that no Bezout equation with coefficients in the field A of algebraic numbers exists. in the field

3 The

34

(3)

Let p

Algebraic Structure q

+1

e

e

identity

s

=

we

+

first let

0 have

=

a

Ro. The elements

G

z

e-\

and A +

of 'HO

=

are

coprime since the equations in C. To obtain

no common zeros

I and b

=

-

(s

bp

aq +

+

1)

s

+

=

a

Bezout

and get

(3.1.5)

e.

This is indeed the first step in the procedure given in the proof of Theo3.1.6(a) and corresponds to the elementary transformation

rem

I-(s The next step of the

R[s]

1)

+

I'll

q

.1

=

procedure would be

[0 where 6 E

01

1

-,

satisfies

+

8

-

e

transformation of the type

(s"+

e

-

8+1

(8+1)

e

1

a

(

' _e

S+1

J(-I)

e,

6(-e)

ee. Instead of

way, which would

require another step thereafter, Equation (3.1.5) implies

[q*(-e),p*(-e)] thus, by coprimeness of p and

with the

given

E

c

1-e

=

e-- e,

p, q, a, and b

As

E R.

a

a s

+ cp +

e

q+

s

+

e

-

-

going this as

follows.

0,

has

r-p*(-e) q*(-e)

'+P s+e

to the Bezout

C

and

identity

--q ',_4 +e

are

in

'HO and

P

e)z + (e ee)s (e ee)(s + 1)(s

-

one

consequence,

altering Equation (3.1.5) leads

proceed

P*(-e) [ q*(-e)

imR

a*(-e)) (b*(-e)) where

e

we

q, it follows

(a*(-e) b*(-e)) Indeed,

(' (b*(-e)) a

e

e

-

+

+

e)

e

2

q +

(e

-

1)z + (,e 1)s + (e ee-) (s + e) -

e -

e

-

Q(e, ee). (3) should demonstrate how (successive)

with coefficients in

examples (2) Bezout identities force to extend step by step the field of coefficients, in this case from Q through Q(e) to Q(e, ee). It seems unknown whether the transcendence degree of (Q(e, ee) is two, which is what one would expect. This is a very specific case of a more The

and

3.2 Matrices

general conjecture of Schanuel

WO

35

theory, which we will conjecture (just to

However, very little is known about this

in 3.6.5.

present

in transcendental number

over

2

give an example, it is only known that at least one of the numbers el or ee is transcendental, gee. [1, p. 119]). Handling of the successive field extensions forms an important (and troublesome) issue in symbolic computations of Bezout identities in 7to. We will turn to these questions in Section 3.6. The results stated

far show

so

algebraic

to their

respect

resemblance of 'H and

striking

a

being presented next, another postponed until Section 3.4.

is

one

H(C)

with

differences, one of them the dimension of the rings and has to be

structure. But there

also

are

Remark 3.1.10 For

commutative domain R with

a

of R if for all

bl,..., bn

exist

[30,

n

345].

p.

unity

says that 1 is in the stable range

one

satisfying R (a,, an+ 1) there blan+l,..., an + bnan+1), see e.g., equivalent to the property that for all

and a,, E R , an+ 1 c R such that R (a, + > 1

.

.

.

=

It is easy to

al,..., an+1 E R

see

satisfying

that this is

R

=

(a,,..., an+1)

=

En+1

.

there exist C2,

.

.

,

Cn+1 c R such

is a unit in R. While this is true for the ring H(C), see [82, that a, + i=2 ciai p. 138], this is not the case for the rings H and 'Ho, as the following example 1 and a2 shows. Let a, z in 'H and a Bezout equation 1 =

=

-

-

Cl

Considering coefficlents In

[82,

p.

(s

=

-

1) (s

-

2)

cial + C2a2

C2a2

E X Then a, and a2 are coprime implies for the coefficients -

a,

the roots of the denominators it

cl

139]

clal

-

a2 can

be

seen

that neither of the

and C2 can be a unit in X it has been proven that for every Bezout domain with 1 in the sta-

ble range unimodular matrices are finite products of elementary matrices. This result applies in particular to the ring H(C) and we arrive at Theorem 3.1.6(b) for IC

=

H(C).

3.2 Matrices

In this section

we

WO

over

turn

our

attention to matrices

over

'Ho. First of all, it is

an

easy consequence of the Bezout property that one can always achieve'left equivalent triangular forms. Rom Theorem 3.1.6(b) we know that this can even be

by elementary

transformations. But

even more can be accomplished. adequate commutative Bezout domain allows diagonal reductions via left and right equivalence for its matrices. In other words, matrices admit a Smith-form, just like matrices with entries in a Euclidean do-

done It is

a

row

classical result that

an

main. This will be dealt with in the first theorem below and

some

consequences

Algebraic Structure

3 'The

36

of Ho

pointed out. Thereafter we present a generalization of the concepts of greatest common divisors and least common multiples for matrices. As our arguments work over arbitrary commutative Bezout domains, the results will be given in that generality. The end of this section is devoted to a summary of the matrix-theoretic results in terms of general module theory. will be

Let

us

start with

triangular and diagonal forms.

Theorem 3.2.1 Let IC be any of the

(a) everymatrix that

(b)

IC is

is, there

rings H

'Ho. Then

or

P (=- 1C'11 is left

equivalent

Gl,,(IC)

exists U G

to

an

upper

such that UP is upper

triangular matrix, triangular,

elementary divisor domain, that is, by definition, every matrix equivalent to a diagonal matrix where each diagonal element divides the next one. Precisely, there exist V E Gl,,(IC) and W c Glm(IC) an

P Cz IC"I is

such that

VPW

diag,,

=

,m

(di,

.

.

_dl where

rk P and

di

elements

are

the

name

PROOF: Part

follows from

(They

(3.2.1)

drj /C\f01 satisfying di 1,, di+1

G

are

is

also

=

1,.

..,

r

-

unique

1. The up to

elementary divisors in [51, 64], explaining this type of diagonal reduction.)

also called

consequence of Theorem

a

see

for i

the invariant factors of P and hence

of rings with

(a) [51],

...

rrxr

=

units of 1C.

0

d2

A

diagonal

r

0 01

d,)

-

L with

,

.

[64,

3.1.6(a).

The statement in

(b)

473],

where is has been proven that adequate divisor domains; recall Theorem 3.1.6(c) for the

p.

are elementary adequateness of Ho. The uniqueness'of the diagonal elements follows, just like for Euclidean domains, from the invariance of the elementary divisors under left and right equivalence, which in turn is a consequence of the Cauchy-Binet theorem (valid over every commutative domain), see e. g. [83, pp. 25] for principal ideal

Bezout domains

domains. It is worth

C1

mentioning

that it is still

tative Bezout domain is

and

[68].

an

an

elementary

open

conjecture whether every commudomain, see [17, p. 492, ex. 7]

divisor

Remark 3.2.2 It is worthwhile over an

noticing that left equivalent triangular forms can be obtained arbitrary commutative Bezout domain R. This can easily be seen as

3.2 Matrices

follows. Let P

(pij)

=

R"' and alpil+. ..+a,,p,,l

G

over

gcd,,.(pjj,

=

-

-

'Ho

37

-,Pni)

=:

d

equation for the first column of P. Then the coefficients a,, , an unimodular form hence in row which, a using again are coprime (a,,..., an), R, the Bezout property, can always be completed to a unimodular matrix A E Gln(R), see [12, pp. 81]. This way one can transform P via left equivalence

be

a

Bezout

to

a

matrix with first column

.

(d, 0,

.

.

,

.

OT.

The rest follows

.

.

by induction. Our implicitly, use of (see the proof of

proof of part (a) above is slightly simpler since we made, the division with remainder as given in Proposition 3.1.2(g)

3.1.6(a)).

Theorem

Example

(1)

3.2.3

Consider the matrix

[

P=

Since the entries of P is

are

82

+ 1

8Z

Z-1

8z

coprime

2

i

-

1

'Ho,

in

H2X2. 0

EE

elementary divisor form of

an

P

given by

[0 P] [1 Z2(83._ S) 1

0

0

=

det

0

Z('5

+

8 S

can

be derived

as

gcd,. (s 2, z

form. Notice that

in

+

=

z

1

-

82

82

Example 3.1.9(l).

S2 + 1

_

matrices, let

In order to obtain also the transformation

riving a triangular equation

1)

_

(Z

_

*

us

begin with

=

s.

de-

The Bezout

1).

_

Hence

1)

-

1

get the left unimodular

we

transformation

82

1

S.

S

z

-

Sz

1

Sz

s+z-1 (

+ 1

2

S - ks

1-1z ,

0

S

(SZ

Z

+

+

1)

+

(SZ2 S(SZ2

j

[0 b_ s a

To obtain

a

diagonal

form notice that

there exist x, y G 'Ho such that 1 and x (I 2/3a)s-1 E Ho yields =

-

[-by 1_ [s b]

=

0

(2)

y

=

[I O's] [01

a

a

0

coprime in 'Ho, hence simple choice y 2/3

a are

+ ya. The

now

-

1

and

s

xs

o -b

-8

P]

0

=

-

det

The matrix

[

M is in

G12(R[s, z])

ring R0, however,

but not in

1 + -Z

S2

8z

2

1

E2(R[s, z]),

-

see

SZ] [16]

[97,

or

p.

it factors into

M=

I Is

1-z

0

11 I [ 0110, S] Is -

0

1 -

1

1

-11

1

1

0

Z-1

i -

676].

Over the

3 The

38

Let

Algebraic

Structure of 'Ho

equivalence p* I ,(c ) q* 4* p 1, q for p, q E H, given (for Proposition 3.1.2(c). Using diagonal forms, this can easily be Ho) generalized to matrices. To this end, we extend the embedding H -+ H(C) to matrices in the obvious entrywise way, thus return to the

us now

the ring

in

,Hpxq

Clearly, (PQ)*

H(C)Pxq,

P*Q*

=

and

(P

+

P

Q)*

=

=

(Pij)

P* +

P*

:=

(P V-).

(3.2.2)

Q*, whenever defined.

Proposition 3.2.4 Let Pi E Hpixq, i 1, 2, be two matrices. There exists F G H(C)P2XP1 such that FP,* P2* if and only if there exists X E 'HP2XP1 such that XPI P2. If P, and P2 have entries in Ho and P, satisfies rk rk then the P1 (S PI, 0) R(s) matrix X can be chosen with entries in Ho, too. =

=

=

=

1

PROOF: The where is

d

A is

if-part is.obvious. As for the other direction, let UPIV as

(3.2. 1)

in

and

U, V

are

partitioned accordingly. Then P2*V*

Q 7,3 -, the

3

xij X

X

E

Defining

[X,O]U

=

entries of

10 01 ,thus Q' ,A* 0

F(U-')*

=

EHP2.XPl.

agonal elements dj applicable again.

G

Ho of A

are

case

of entries in

not divisible

by

z,

'Ho guarantees that the di-

making Proposition

following

conditions

(a) P has a right inverse over H, that is, (b) P* has a right inverse over H(C). (C) rkP*(A) =p for all A E C. (d) P is right equivalent to [Ip, 0]. (e)

P

(f)

The greatest

can

be

Furthermore,

completed

to

common

a

The that

PM

=

are

Ip

unimodular matrix

char-

equivalent:

for

[Qp]

some

E

matrix M E 'Hqxp.

Glq (H)

divisor of the full-size minors of P is

each matrix Q E 7jrxq of rank p

Ae7jr,pisofrankpandP and P

following

3.2.5

matrix P E 7jpXq the

a

3.1.2 (c) 1:1

Another standard consequence of the diagonal reduction is the acterization of right invertibility for matrices over H.

Corollary

0 and

Q*. Proposition 3.1.2(c) yields Qjj xijdj for some x 'HP2 the desired E left factor is given by r, (xij)

The additional rank condition in the

For

'A 0

[Q, Q']

unimodular. Assume P2 V =

1001

=

C

can

be factored

Hpxq is right invertible

,

over

as

a

Q

unit in 'H. =

AP where

H. The matrices A

right resp. left equivalence. corresponding equivalences are true when H is replaced by Ho provided are

one

unique

up to

adds the condition rk

R(s) P(S)

0)

=

p in the

parts

(b)

and

(c).

3.2 Matrices

noticing that the above equivalences (a)

It is worthwhile

accordingly,

lated

valid for matrices

are

over a

any

Ho

(e),

'*

polynomial ring

field), too, see also Theorem 5.1.12 in Chapter Theorem of Quillen/Suslin on projective modules. (K

(c)

4=

over

39

if formu-

K [xi,

.

.

.

,

XMI

5. This is the celebrated

(f)", recalling the [,A, 0] with A G "(c) =:>. (d)", in hence is unit detzA* and a HPIP. Then P* U* H(C), whence, by 0] Remark 3.1.5, A G Glp (H). Thus, P is right equivalent to [Ip, 0]. To establish 1 (d) => (e) ", let PU [Ip, 0] with U G Glq (H). Then Q [0, Iq-pj U- leads to the asserted unimodular matrix. "(e) => (f)" follows from the Laplace expansion PROOF:

"(a)

(b)

=>.

(c)"

=: ,

obvious, and

is

"(c)

is

so

units in 'H from Remark 3.1.5. As for

#

let PU

=

=

"

=

=

of det

IQ] along P

the block

row

Q

For the factorization of

given by P.

use a

The

implication "(e)

diagonal form Q

unimodular matrices U and V. Then A

=

=>

Udiag, X q (di,

Udiagxp(dl,..., dp)

=

(a)"

and P

is trivial.

dp) V with [1p, 01V =

yield the desired result. The uniqueness is straightforward. The additional condition for the ring Ho guarantees that z, which is not in

7-10,

is not

unit

a

divisor of the full-size minors of P.

a common

The second part of this section is devoted to a generalization of the concepts of greatest common divisors and least common multiples from functions in H to matrices

over

'H. We will formulate the results for matrices

commutative Bezout

greatest p.

right

common

nonsingular,

31-36].

domain,

as

this is

exactly

divisor of two matrices,

theory multiple comes

is standard in matrix

A least

left

common

literally given below in Theorem

over

to Bezout domains and

over

one

of them ideal

principal by-product.

as a

arbitrary proof The

over an

what is needed for the

being square and domains, see [71, The result carries

to non-square matrices in the way

even

fairly standard, too, but seems present proof since the precise description will be needed later in Chapter 4, where a Galois correspondence between finitely generated submodules of Hq and solution spaces in fq of systems of DDEs will 3.2.8. This version looks

to be less known. We would like to

a

be established. The

following

notation will be

helpful.

Definition, 3.2.6 Let n, q E N and

(a)

Let

Jn,q

:=

n

< q.

J(PI,

-

-

-,

Pn)

E

Nn

11

< p,
0.

R1 X q B

U,

U4

B

0

for

some

R7n X q be two

Let

Gli+,n(R), partitioned according

1U3 U2] [A [D

e

to

U,

G

R'xl,

D G R'Xq.

(3.2-3)

Then

(a)

D is

a greatest common right divisor of A and B of full row rank r and as such is unique up to left equivalence. We write D gcrd(A, B). Moreover, there exist M G Rrxl, N E Rrxn such that D MA + NB and therefore, =

=

im AT +

(b) Suppose

im.6r rk A

=

=

im.

1,

Lir.

rk B

If r < 1 + m, then M

multiple of A and

:=

=

m.

U3A

B of full

=

row

-U4B rank.

E

R(1+1 -r)xq

F irthermore,

is

a

least

im AT n im

common

BT

=.

left

im MT.

3.2 Matrices

Every

least

multiple of A

left

common

and B in

over

R(1+1-1) xq

Ho

41

of full

row

equivalent to M. We write M lclm(A, B). If r 1 + m, the only common left multiple of A and B is the zero matrix; in particular, im AT n im, BT 0. It will be convenient to define lclm(A, B) Xq as the empty matrix in R' (The image, resp. kernel, of an empty matrix rank is left

=

=

=

.

is the

If rk A

zero

1,

=

subspace,

rk B

=

resp. the full

m, then rk A + rk B

space.) rk gcrd(A, B) =

+ rk lclm(A,

B).

all, recall that a matrix over an arbitrary commutative Bezout equivalent to an upper triangular form, see Remark 3.2.2. This

PROOF: First of

domain is left

guarantees the existence of the matrices U and D. It is not in triangular form; solely the full row rank is important.

(a) Using (3.2.3)

and

letting

[ U]

V2]

ui Q3 U4

we

V1 V3 V4

get the equations UjA + U2B

assertions of

As for

(b),

(a)

consider the

one

gets rk

of U

U3

=

case r




quotient

space M :=r

Rl+'/im [AT, _BTTtogether with the two maps from Rq

factoring through A and B, see [67, p. 59]. Indeed, with the proof above, it is easy to see that the map

into M in the

M,

V

)

I'lv+im['] -

V4

-B

notation

as

Algebraic

3 The

42

Structure of Ho

an embedding of Rl+'-' into M. Moreover, the finitely generated decomposes into its free part and its torsion submodule as follows:

is

M

(By

virtue of the

domains the in

.V)(R'+'-')

=

are

way

same

[67,

p.

free as

E)

v

+ im

[ ]I A

-B

I

a

c

R:

av

c im

module M

[A]j -B

-

'

finitely generated torsion-free modules over Bezout [97, p. 478], the decomposition above can basically be proven for principal ideal domains, for which a proof can be found fact,

that

533].)

Remark 3.2.10 We wish to end

our

matrix-theoretic considerations with

an

interpretation of

the results given above into module-theoretic terms. First of all, the Bezout property of H can simply be expressed as stating that every finitely generated a free 'H-module of rank one. Secondly, the left or right equivalent triangular forms for matrices over H (Theorem 3.2.1 (a)) imply that every finitely generated submodule of a free H-module is free. Indeed, if M is such a finitely generated module, we can assume without loss of generality that M C 'Hr for im Q for some matrix Q E Hr,,. Using a right some r C N and that M equivalent triangular form of Q, one can single out a full column rank matrix im , showing that M is free. Thirdly, the sum of two representation M finitely generated submodules N, and N2 of an H-module M is certainly finitely generated again, hence a free module if M is free. The construction of a greatest common right divisor in Theorem 3.2.8(a) presents a way of how to construct a basis for the sum N, + N2, given generating matrices AT and BT for N, and N2, respectively. More interesting from a module-theoretic point of view is the fact that also the intersection N, n N2 of two finitely generated submodules of a free H-module is finitely generated and free again. A basis for N, n N2 is given by the least common left multiple of generating matrices for N, and N2 (see Theorem 3.2.8(b)). Observe that all the above is true for arbitrary commutative Bezout domains (see Remark 3.2.2). In commutative ring theory the situation above is captured in a more general context by the notion of coherent rings

ideal of H is

=

=

and modules. A module M

over a

commutative

ring R

is called coherent if M

finitely generated and every finitely generated submodule N of M 'is finitely presented, hence there is by definition, an exact sequence F, --+ Fo --+ N - 0 with finitely generated free modules F0 and Fl. A commutative ring R is called coherent if it is a coherent R-module, hence if every finitely, generated ideal of R is finitely presented [38, Sec. 21. Since finitely generated free modules are trivially finitely presented, every commutative Bezout domain is coherent. It is known that if R is coherent, then every finitely generated submodule of a free R-module is finitely presented [38, Thm. 2.3.2]. This generalizes the situation is

for commutative Bezout domains where these modules turn out to be free

as we

have

submodules of

a

generalization, of the greatest left multiple. arrive at

a

even

above. Furthermore, sum and intersection of two coherent coherent module are coherent again [38, Cor. 2.2.4] and we

seen

common

divisor and the least

common

Systems

3.3

Systems

3.3

In this section

over

we

over

Rings: A Brief Survey

43

Rings: A Brief Survey

want to take

short excursion into the

a

area

of systems

over

rings. We present some of the main ideas and discuss the ring Ho with respect to some ring-theoretic properties arising in the context of systems over rings.

theory of systems over rings is a well-established part of systems theory, mainly by the papers [79, 105], in which it has been observed that in various types of systems, like for instance delay-differential systems, the main underlying structure is that of a ring. As a consequence, the properties of such systems can be studied, to a certain extent, in an algebraic setting. This in turn has led to several notions for rings, which, beyond their system-theoretic background, can be studied in purely algebraic terms. The book [12] provides not only an excellent overview of these various concepts, but also introduces a variety of rings to systems theory. Although our algebraic approach to delaydifferential systems is not in the spirit of systems over rings, the book [12] has been our main guide through the area of Bezout domains and elementary divisor The

initiated

domains. In the

sequel

we

want to survey

For the moment it

one

might simply

branch of the

serve as a

(weak)

systems theory. But there is also

a

where

topic.

The

we

will

come

back to this

theory of systems

over

brief introduction into that

connection to Section 4.5 of

rings.

area

our

of

work

starting point for the theory of systems over rings is the description of dynamical system as an equation

a

linear first-order discrete-time

Xk+1

where A

E

R"' and B

C

=

R"'

AXk are

+

Buk, k

matrices

>

(3.3.1)

0,

over some

ring R and

Xk G R'

and Uk E R' are the sequences of the states and inputs, respectively (at this point there is no need to consider an -output equation). Rom a system-theoretic

point of

a lot of natural questions arise. The most basic one is whether possible for a given system (3.3.1) to steer it from one state to any other in finite time by suitable choice of the inputs Uk. This is the well-known notion of reachability and can be expressed solely in ring-theoretic terms. or

view

not it is

(1)

The pair (A,B) is called reachable, if im[B,AB,...'A'-'B] R', see [79, 529]. If R is a domain, the above is equivalent to [AI A, B] being right invertible over the polynomial ring R[A], see [46, Thm. 2.2.3]. =

p.

It is

a

-

purely algebraic

the internal modes Uk

=

can

-

result that for reachable systems (3.3-1) over a field arbitrarily by use of static state feedback

be altered

FXk. This problem of modifying the systems dynamics can equally well rings. In this case it falls apart into two subproblems.

be formulated for

(2) [105,

20]

A pair (A, B) E R1 I I x R1 x ' is called coefficient if for each monic polynomial a E R[A] of degree n there exists p.

assignable, a

feedback

Algebraic Structure

3 The

44

of Ho

loop system given by

matrix F E R"I such that the closed Xk+1

(3), [79,

=

p.

(A + BF)Xk has characteristic polynomial det(Al xn x Rnxm 530], [105, p. 20] A pair (A, B) G Rn

assignable, such that

if forallaj,...'an G R there exists

det(/\I

A

-

-

BF)

Flnj= 1 (,\

=

a

the

A

-

-

equation

BF)

=

is called

a.

pole

feedback matrix F e Rmxn

aj

_

It is easy to see that coefficient assignability is stronger than pole assignability which in turn implies reachability, see [105, p. 21]. Whether or not the converse is true,

depends strongly

on

the

underlying ring

R. This has led to the

following

notions.

67] A ring R is called a CA-ring (resp. PA-ring) if pair (A, B). is coefficient assignable (resp. pole assignable).

(4) [12,

p.

each reachable

a CA-ring. In the general case of systems over rings a particular 1. In this case, simple case arises if there is only one input channel, that is, M reachability of (A, b) simply says that b is a cyclic vector for the matrix A and one straightforwardly verifies that (A, b) is coefficient assignable.

Each field is

=

As

a

the

one can

consequence,

following

show that

ring

a

is

a

CA-ring

if it is

an

FC-ring

in

sense.

(5) [105, p. 21], [12, p. 74] A ring is said to be an FC-ring (feedback cyclization and ring), if for each reachable pair (A, B) there exists a matrix F G Rm a vector v G Rn such that (A + BF, Bv) is'reachable. xn

simple rings it is surprisingly difficult to see if they have one of the properties above. We confine ourselves with reporting the following results and open questions. Even for

(i)

The

polynomial ring R[z]

is

a

PA-ring [105,

p.

23],

but not

a

CA-ring.

For

instance, the pair

(A, B) is

reachable, but

det(/\I and [99].

-

ii)

The it is

(iii)

In

A +

( rzo- [-1 Z20 1] 0

0

does not allow

BF)

A2 + A

=

+

a

(3.3.2)

-

R[Z]2x2 such that R[z][A]; see [29, p. 111]

feedback matrix F (E

(Z2

+

ring C[z] is a CA-ring [11], but FC-ring [100].

z

+

it is

2)/4

an

E

open

question whether

or

not

an

[10,

Thm.

0.1]

polynomial ring K[z] is a CA-ring, taking square roots and even qth such that the qth roots of unity are contained in K.

it is shown

that, if

a

then the field K is closed with respect to roots for every

prime

(iv) Every elementary

q

divisor domain is

a

PA-ring [12,

Thm.

3.13].

ring Ho has any of the properties PA, CA, or'FC. From the general result quoted in (iv) we obtain immediately that HO is a 'PA-ring. Furthermore, it is not hard'to see that HO is not a CA-ring. Indeed, Ho is contained in. R(s) [z], which is not CA according to the result quoted in (iii). But applying the proof in [10] to the ring R(s)[z]

In this context it is interes'ting to

see

whether the

3.4 The

Nonfinitely

Generated Ideals of Ro

provides an example of a pair (A, B) c H2X2 0 over 7io but not coefficient assignable. Hence HO is not an FC-ring either. even

We will

back to

come

a

slightly different

H2X2 0

which is, reachable

not CA

and, consequently,

X

notion of coefficient

Section 4.5. The topic of realization theory for systems addressed in the introduction to Chapter 5.

Nonfinitely Generated

3.4 The

45

over

assignability in rings will be briefly

Ideals of WO

The Bezout property of Ho says that all finitely generated ideals are principal, they are completely described by one generator. In this section we focus

hence our

can

attention to the

be

set -of

fully

zeros

nonzero

nonfinitely generated ideals. As we will see, each such ideal by one "generating" polynomial along with a specified

described

(counting multiplicities).

prime ideals

are

As

a

consequence, it will turn out that all

maximal, in other words, the Krull-dimension of HO

is

one.

The results of this section

are

directly

not

related to

differential equations in the next chapter. However, for a further algebraic study of the ring RO. We restrict to Let tion

us

thering HO.

The results about 71

our

can

investigation of delaythey are interesting

think

we

readily

be deduced.

first rephrase the characterization of prime elements, given in Proposiin ideal-theoretic language. The following is an immediate conse-

3.1.2(d),

quence of the Bezout

property together with Proposition 3.1.2(d) and

Proposition 3.4.1 Let 10} 54 1 C Ho be

a

I is prime

begin

Then

I is maximal I

We

finitely generated ideal.

(i).

=

0)

for

some

irreducible

R[s]\101

or

I

=

(z).

investigation with an important class of nonfinitely generated They can be regarded as "generalized" principal ideals, for the information on such an ideal is contained completely in one (generating) polynomial. These ideals will serve as a sort of building block for all nonfinitely generated ideals. In the sequel a polynomial 0 E R[s] is called monic if 0 54 0 and its leading coefficient is 1. our

ideals in HO.

Definition 3.4.2 Let p E R[s, z). Deline Dp all admissible denominators of p.

R[s] 10

monic and

Furthermore,

let

0

1,0 p}

to be the set of

3 The

46

Algebraic

Structure of Ho

((p)) We call

((p))

:=

the full ideal

jhP

I

0

h E

((p)).

It is clear that

full ideal is indeed

a

((p)),

generate

Proposition

G

Dpj

7io.

C

generated by p and the polynomial

full generator of

to consider full ideals

7io, 0

an

generated by

p is said to be

ideal of Ho. Notice that there is

ho\R[s, z]

q E

q

as

=

P-

E

no

a

need

Ho would fully

too.

3.4.3

R[s, z]\101. b)) is at most countably generated 0. is, Vif I f E ((P))} (2) Let q E R[s, z]\10}. Then

Let p C

(1)

The ideal *

((q)) In

(3)

particular, ((q))

Let

and has empty variety, that

=

=

((p))

9

((p))

if and

b))

p

only if q

q.

V)p for

some

0

E

R(s).

p. Then

z

p E

R[s]\f01

=

Ho

-

R(s)

n

assume

some

E M

((p))(M)

q c

is

finitely

Ho. Then

there is

some

-Ho, which implies

(po- 1),

=

q

h4>

=

E

M C

gen-

h-P- for V)

Ho such

Dp,

and

Remark 3.4.9 It is easy to see that we can confine ourselves without restriction to saturated admissible sets of denominators. Indeed, for all p e R[s, z] and each admissible set M of denominators for p

one

has. the

identity ((p)) (M)

=

((p)) (V)

where

M=f0GDpj0monic,]0EM: 010} is the saturation of M.

Now we can completely describe the ideals in Ho. The presentation of nonfinitely generated ideals given in part (3) below, will,be important later in Section 4.6 where we study the solution spaces of delay-differential equations corresponding to these ideals.

Theorem 3.4.10 Let

101 :/ -

I C

'Ho be

an

ideal and p G

Put M:=

0

G

R[s, z]

Dp

be

a

sandwich-polynomial

of 1.

J:E Ij. E

Then

(1) M=f0EDpj3heHo: gcd,,(h,0)=,1andhP.E1}. (2) M is a saturated admissible set of denominators. (3) I= ((p)) (M). (1) (h, 0) gcd,. PROOF:

The inclusion "C" is trivial. For "D" =

ah +

bo, where

a, b G Ho

are

multiply a Bezout identity I coefficients, by po-1 E Ho.

suitable

Algebraic Structure

3 The

50

(2)

If

c M

and

0

R[s]

E

is

of Ho

monic divisor of

a

0,

say

Op

0,

=

P

then

=

pR

G 1.

Hence M is saturated and there remains to show that M is closed.with respect to taking least common multiples. To this end, let 01, 02 G M and 0 1CM(01 1 02) =

0 1 2

Since R [s] is factorial, we may write Rom the above we know f G I for i

1,

Oi

P

+

01 Since

(3)

gcd(ol

021 0102)

+

=

((p))(M)

The inclusion

7

02

1,

C

=

we

(01

+

where

,

j I

Oi

and

gcd( ,, 2)

=

1

2. Hence P

02)-7--

E I-

0102

obtain from

(1)

that

0

=

0102

E M.

I is immediate from the definition of M. For the

let q G I, hence q E ((p)) since p is a sandwich-polynomial of 1. Then hP- for some 0 Dp and h (E Ho. Using Proposition 3.1.2(h), one can

converse

q

=

assume

gcd,. (h, 0)

1, which, by (1), yields 0

G M.

Thus q c

desired.

f0} 7

.for

3.4.11

I C

some a

PROOF:

as

EJ

Corollary If

((p)) (M)

'Ho

c R

is

[s]

ideal

an

having

sandwich-polynomial

a

in

R[S],

then I

=

(a)

-

By Proposition 3.4.5(b), each other sandwich-polynomial of _T is in R[s], representation in part (3) of Theorem 3.4.10 completes the proof.

too, and the

n

The

following

of Ho

(cf.

theorem

Theorem

provides

3.1.6(c)).

an

alternative argument for the adequateness Ho is a one-dimensional ring, that

It reveals that

is, each ascending chain of prime ideals has maximal length that one-dimensional Bezout domains

are

adequate [12,

p.

1. It is well-known

95].

We would like

mention, that, H(C) Krull-dimension, yet still adequate [12, Thms. 3.17, 3.18].

of entire functions is of infinite

contrast, the ring

in

to

Theorem 3.4.12 Let

f01 :

(a)

If I is

(b)

I is

I C

lio be

an

ideal.

prime ideal that is not irreducible p G R[s, z]\R[s].

PROOF:

a

finitely generated,

then I

=

( p))

for

some

prime if and only if I is maximal.

(a)

Let I be

a

nonfinitely generated prime

otherwise the intersection would contain

an

ideal. Then I n

irreducible element

the consequence (a) C I, contradicting Proposition 3.4.1. Let p be a sandwich-polynomial of 1, hence 19 ((p)). But then even I

R[s] a

10} for R[s] with R[s, z]\R[s] ((p)) is true, =

G

OP -E I and the primeness of I together Dp we have p 10} implies R G I. Again by the primeness of I and by virtue of Proposition 3.4.3(2), p can be chosen as an irreducible polynomial in R[s, z]. since for each

with I n

R[s]

=

=

Ring 'H

3.5 The

(b)

as a

Convolution

Algebra

51

light of Proposition 3.4.1 we are reduced to show that each I ((p)), where p E R[s, zj R[s] is irreducible, is maximal. To this end, let I C J for some ideal J having sandwich-polynomial q E R[s, z]. The case q E R[s] can be handled with Corollary 3.4.11 and Remark 3.4.4. If q 0 R[s], then Proposition 3.4.3(2) applied to ((p)) 9 &)) together with the irreducibility of p yields ((P)) Hence I is

In

=

maximal ideal.

a

n

We close this section with the

following

result

concerning the uniqueness of the proof is lengthy but straight-

representation of nonfinitely generated ideals. Its forward and will be omitted.

Proposition

3.4.13

b)) (M)

Consider the ideal

R[s, z]

which is

9 'Ho where p

primitive

as a

E R

polynomial

in

[s, ;]

for

R [s] and

some

Furthermore,

z.

saturated admissible set of denominators for p. Let monic polynomial such that M n D4, D,, and put a

a

E

R[s]

M C

Dp

is

be the unique

=

0

k 11

Then

is

=

a

((-rip,))

in z, the

gcd(0, a)

JOEMI

and

ki

( -rf))) (k)

and

11

n

D,

=

us

D,p

satis-

unique presentation of the ideal ((p)) (M) in the following sense: ((-r2h)) (k2) where Pi G R [s, z] are primitive as polyn'omials

ki

g

D,pi

are

Then rl

I

saturated admis=

k2-

Ring W

3.5'The

Let

ER[s].

f 1}.

polynomialsri E.R[s] are monic, and satisfying Mi n D.,

=

0 a

sible sets of denominators and

-r=

saturated admissible set of denominators contained in

a

fying ((p)) (M) This provides let

=

recall that the

as a

ring i

r2,

PIP2

1

E R

Convolution Algebra has been introduced in

Chapter

2

as a

ring of delay-

differential operators acting on C' (R, C) The main purpose of this section is now to place this situation in the broader context of convolution operators. .

More

precisely,

will describe H

we

as an

algebra

of distributions with compact 2.9(2) will

support. The delay-differential operators q introduced in Definition turn out to be

form and

a

'the

associated convolution operators.

suitable

(isomorphic to)

can

with

the

are

Laplace see

trans-

that R is

rational expressions in the

J, and have compact support. The structure of these be exhibited in more detail by going through some additional and

explicit calculations. can

Using

it will be easy to

the space of distributions which

Dirac-impulses J(1) 0 distributions

Paley-Wiener Theorem,

In

particular, it will turn out that each such distribution of a piecewise smooth function and a distribution finite support, hence as a polynomial of Dirac-distributions. Algebraically,

be written

as

the

sum

3 The

52

Algebraic Structure of 'Ho

by the decomposition of the functions in H into their strictly polynomial part in a sense to be made precise below. For the algebraic approach to delay-differential equations this description is important

this is reflected

proper and their

because it allows

to abandon the restriction to C'-functions for the solu-

one

tions. Recall that from

view -the space Coo (R,

convenient to

is

turns out

an algebraic point of begin with, simply because it

(cf.

3.5-7)

Remark

that

over

a

module

over

C) is very R[s, z, .Z- 11. it

the proper part of H much

more

gen-

eral function spaces, for example Lj, are modules with respect to convolution, too. We will take this aspect into consideration when discussing input/output

systems in the

chapter.

next

For the main line df

of this section

are

of distributions sheds We

approach, where we restrict to Cl-functions, the results strictly necessary. Yet, we think the description in terms some new light on our investigations.

our

not

begin with fixing

notation. Let D' be the

some

vector-space of complex-

the space D := If E CI(R, C) I supp f is compactl, endowed with the usual inductive limit topology. Here supp, f denotes the sup-

valued distributions

port of

function

a

on

(or distribution) f. Furthermore,

D+'

T E D' -

supp T bounded

let

on

the

left}

and

E)c'

T

D' I supp T

compactl.

D,' with their extension to distributions on S, instead of L as in Chapter 2, is meant to indicate that the space 9 is endowed with the topology of uniform convergence in all S n D+' be the space of functions in E derivatives on all compact sets. Let S+ We will

identify

C' (R,

.6

C).

the distributions in

,

The notation

with support bounded

on

the Dirac-distribution at

the left.

a

Finally, denote by j(k) a

the k-th derivative of

Ei R. Recall that the convolution S*T of distributions

is well-defined and commutative if either both factors

are

in

V+

or

if at least

one

factor is in D. Moreover, convolution is associative if either all three factors are C is an R-algebra in V+ or if at least two of them are in D,. Finally, (D+, without p.

zero

divisors and with J0

as

identity [104,

p.

14,

p.

28/29]

or

[128,

124-129].

setting, differentiation (resp..forward-shift) corresponds to convolution Sy E R[s,z,z-1] and with J(1) j:L 1 ENO (resp. Ji). Precisely, for p 0 i= Pij In this

f

G

.6

we

j=

have

ff

=

XW E Y, Apo j=1

Notice that

R[JO(l), Ji, 6-1]

is

approach

Jj)

*

f

=

WO(l), Ji)

*

f

-

(3.5.1)

subring of D+' and isomorphic to R[s, z, z-1]. already in [61], where it was utilized for a delay-differential systems. a

This observation has been made

transfer function

*

i=O

to

3.5 The

In the

subsequent discussions

we

3 tk G

f

PC'

:

R

Convolution

as a

Algebra

53

will also consider the function space

R, k

E

C liMk-cx) tk

--+

Ring R

f I (tk,tk+ I ]

such that tk


q(J(1), Ji) 0

part.

r

refers to the left-derivative of g. Since p and 0 have real consequently jb(g) are actually real-valued and

the function g and

(3.5.6)

is

a

the coefficient of

decomposes

into

a

in R [JO(l), Ji, 5- ], which q(k), J,)..It vanishes if and only

polynomial

the distribution

J(N-r)

we

1

0

is

nonzero).

As

a

call the

if N
." notice that rk

the existence of m

+p

-

r

a

r Q Q, resulting that the maximality of m yields

nonsingular

free variables

so

r x

=

r-submatrix of

of all, by (2) the formal transfer function

-Q-1P

E

R(s, z)Pxm

C

R(s)((z))Pxm

nonanticipation, dealing with inputs having their support bounded to operator given by the distribution (- Q P) (JO(l), Jj) G (D+)P x acting on ET, see Theorem 3.5. 1. Precisely, for all u FT satisfying u- 0, there exists a unique output y E S+P given exists. For the

left,

it is most convenient to utilize the convolution '

=

by

y

=

(-Q-'P) (J(1), Jj) 0

* u.

If

-Q-1P

has support in [0, oo) and thus yconverse follows from Lemma 3.5.4.

=

0,

C

R(s) jz Pxm,

then

too. Hence B is

(-Q-'P) (k), Jj) 0

nonanticipating.

The 0

Rom the above it is immediate that every behavior can be turned into an i/o-system by suitably reordering the external variables. It turns out that the same

is true

to comment

even on

for

nonanticipation. Before proving that assertion, nonanticipation given above.

the characterization of

we

want

4 Behaviors of

92

Delay-Differential Systems

Remark 4.2.4 For

an

i/07systern

B

=

kerC [P, Q] the formal transfer function

and induces the distribution It therefore T

Since

D+'

(D+)'

S+

*

C

utilized this fact

(D+)P +

E

Q

P exists

see

Theorem 3.5. 1.

* U.

(4.2.2)

rise to the convolution operator

gives :

(- Q -'P) (JO(l), Ji) 0

-

)

(D+')P,

u

i

)

(-Q-'P)(60(1)'6J)

6+, the operator can be restricted to a map .6+1 already in the proof of part (3) above.) In this way,

9+'. (We

T may be associated with as an the regarded input/output (ilo-) operator system B. The of the restriction the to is B of all one-sided graph subspace n .6+m exactly

.67+P

trajectories

in B. The distribution

(-Q_1P)(JO(1), 61) is

E

(D+)Px"'

usually called the impulse response since its columns are the responses to the 6o ei E (D+')', where ei,...' en denote the standard basis

Dirac inputs ui vectors in R'.

According

=

to Theorem

4.2.3(3),

the operator T

(or

rather its

graph

in

9

rn+p

is nonanticipating if and only if -Q-1P C R(s)JzJPx'. As a consequence, each purely differential behavior kerc [P, Q] (that is, [P, Q] C R [s]P x (m+P)) is a nonanticipating i/o-system provided that Q is nonsingular. In this context no requirement like -Q-1P being a proper rational matrix arises. This is simply due to the fact that we allow C'-functions only, so that differentiation .(the polynomial part of a rational matrix) causes no particular difficulties. The situation isidifferent when taking other functions into consideration. In Remark 3.5.7 we discussed the possibility of more general functions spaces. Let us consider the case of (LI,,)+ -functions being fed into the system. Then, in order to avoid impulsive parts in the output, -Q-1P has to be proper in the sense

that map

-Q-1P C R(s),((z))Pxm (see (4.2.2) specializes to

Remark 3.5.7 for the

notation).

Then the

P

+

nonanticipating iff -Q-1P is a power series (rather than merely series) over the ring R(s),. For systems of ODEs this has been described in [120, p. 333]. We will call a system kerL [P, Q] satisfying the condition -Q-1P E R(s)jz 'xm a strongly nonanticipating ilo-s'ystem. At this point a main difference between behaviors defined by DDEs and those given by ODEs arises. The latter ones can always be turned into strongly nonanticipating i/o-systems by suitably reordering the external variables, see also [87, Thm. 3.3.22]. This is not true for delay-differential systems. For instance, for the behavior B given by [p, q] [5 92Z, I 'S3 z] neither q-1p nor p-lq is in

which, again, a

is

Laurent

=

R(s), ((z)). Thus, systems. But defines

a

on

B

_

_

neither way be regarded a strongly nonanticipating i/othe other hand, both quotients are in R(s) Jz , so the behavior B can

nonanticipating i/o-system (over C') either

way.

Input/Output Systems

4.2

following proposition provides

The

some

information how to read off

93

directly

expanding -Q-1P into a series, whether or not [P, Q], the system is (strongly) nonanticipating. The criteria take their best formulation by choosing a normalized form for [P, Q] in the sense that the matrix has no negative powers of z and a constant coefficient (with respect to z) of full row rank. Part (a) below shows that'each behavior admits such a normalized kernel-representation. The criterion for Q-'P being a matrix over R(S) zj is from the matrix

without

then very natural: the constant coefficient of Q has to be nonsingular. The normalization is a,Iso implicitly contained in the assumption of part (c) leading to a

strongly nonanticipating system. Although

we

will not dwell

on

the

case

of

(L,I,) +-trajectories

later on, we would like to include this particular criterion. It will be utilized later to demonstrate that the systems arising in Chapter 5 as well

the controller used for spectrum

as

assignment

in Section 4.5

are

actually

strongly nonanticipating systems.

deg's q for rational functions q G R(s) [z] given in 3. 1. For R(S)[Z]pXq we denote by M(s, 0) the matrix in R(s)PXq obtained 0 into M. We call M normalized if rkR(s)M(SI 0) substituting z R

Recall the definition a

matrix M E

after

=

Proposition

(a)

U e

(b)

4.2.5

For each matrix

L

e.

Let

=

71" (m+P) with rank p there exists a matrix (m+p) and such that U[P, Q] is in Hpx normalized, 0

[P, Q]

Glp(R[s, z, z-1]) rkR(s)(U[PQD(S,0) =P[P, Q] E Hop (m+p)'be a normalized matrix and Q x

Q-1P (c)

E

Let

[P, Q]

4==>

det

Q(s, 0):

0.

=

EL

=

Q-1P

Then

nonsingular. Then

zi with and det Q -A 0. Write det Q j= 0 qj (s) and Moreover, R(s) suppose deg, (det Q) deg, q0. supQ) is maximal among all degrees of the full-size minors of c

deg, (det

[P, Q].

R(s)[zJPxm

(m+p) Hpx 0

G

coefficients qj pose

E

be

G

R(s), jz px'.

by (a) and (b) every system can be i/o-system by reordering the external variables. Notice that

turned into

a

nonanticipating

PROOF: (a) It is enough to establish a denominator free version, i. e., [P, Q] G R[s, z]PI (m+P). Assume rk R(s) [P) Q1 (Si 0) < p. Then there exists a row transformation U E Glp (R [s]) such that the last row of U [P, Q] (s, 0) is identically zero. Hence the matrix

1IP-1 il U[P, Q1 0

0

has entries in R [s, ceed in the

on

as

the other

=:

[p, Qi]

[Pi, Q 1 ] (s, 0) p we are done. Otherwise we can pro[PI, Q1]. This way we can build a procedure which the current matrix [P1, Q1] satisfies rk [Pi, Q1] (s, 0) < p.

If rk

same manner

keeps running But

z].

Z_

=

with

long as hand, the procedure

must

stop after finitely many steps since

94

4 Behaviors of

the full

rank of

row

Delay-Differential Systems

[P, Q] guarantees

that the maximaldegree in z of the fullstrictly decreasing sequence of nonnegative numbers. the desired matrix after finitely many steps, which proves the

size minors constitutes

Thus

obtain

we

a

assertion.

(b)

Notice that both P and

Q are matrices over the ring R(s)IzT and Q is in only if det Q is a unit in R(s) zT, hence iff det Q(s, 0) = k 0. This proves For "=>." observe that P Q Aj (s) zi with coefficients Aj c R(s)Px"n implies P(s, 0) Q(s, O)Ao(s), which together with the normalization rk [P(s, 0), Q(s, 0)] p yields rk Q(s, 0) p. vertible

such if and

as

=

=

=

(c)

Let

us

=

start with the scalar

Q_iP

=

since ao

zi Too _j=0 aj (s)

=

q-lpo 0

The matrix

for

R(s),

c

1. Write P p EjM=0 pj (s) z3 where degs (det Q) reads as deg, qO ! deg, pj Using (b), we have Q-'P c R(s) zj, say G R (s). Now the result follows by induction

case m

pj cz R,(s). Then the assumption and deg, qo ; degs qj for each j. some a

i

and aj

=

=

=

=

on

q-lpj 0

j=

q lqjaj_j

G

R(s),.

consequence of the scalar case along with Cramer's rule. where Indeed, the entry (Q-'P)ij is of the form (det Q)-' det is the matrix obtained by replacing the ith column of Q with the jth column of P. case is a

jj,

Hence

Qjj

is

a

full-size minor of

tions combined with the scalar

[P, Q]

Oij

and the result follows from the assumpEl

case.

Remark 4.2.6 For normalized matrices i

=

1, 2,

reads

the

E

Hpox (m+p),

hence rk

[Pi, Qj] (s, 0)

kernel-representations

p for in Theorem 4.1.5 (a) =

as

ker,c [Pi, Qi] This

[Pi, Qj]

uniqueness result about

can

=

kerc [P2) Q2]

be verified

: XEGlp(Ho): [P2iQ2]=X[P17Q1]-

straightforwardly.

We close this section with

an algebraic characterization of autonomy. It is immediate from the definition that autonomous systems have no free variables. The converse is true as well and follows from the identity kerC R C kerL (det R 1q), -

where R is we

nonsingular, together Proposition 2.14(2). completeness, special case of finite-dimensional systems, which can easily by use of a diagonal form together with the scalar case in Corolwith

also include the

be derived

lary 2.6(a) and Lemma Proposition

4.1.10.

4.2.7

Let R EE -Hpxq be

(a) (b)

For

a

matrix with associated behavior B

B is autonomous if and

B is finite-dimensional

only

(as

if rk R

R-vector

=

kerc R

C Lq

-

Then

q.

space)

if and

nonsingular purely differential operator, nonsingular T E R[S]qxq. some

=

only

i. e., B

if B is the kernel of

kerc

T for

some

Systems

4.3 Transfer Classes and Controllable

4.3 rJ[ransfer Classes and Controllable

In Section 4.1

95

Systems

equality of behaviors

equivalence of we a weaker equivkernel-representations over alence relation on the lattice B, which will be called transfer equivalence. This notion refers to the fact that for i/o-systems each equivalence class is going to consist of the systems with the same formal transfer function. However, the equivalence itself can easily be handled.without use of any input/output partition, which is merely a reordering of the external variables, anyway. In particular, there is no need for giving an interpretation of -Q-1P as an operator. It will be shown that each equivalence class is a sublattice of B with a (unique) least element. This particular element can be characterized algebraically, but also purely in terms of its trajectories. It turns out to be a controllable system meaning that every trajectory of the behavior can be steered into every other within finite time without violating the laws governing the system. Finally, a direct decomposition of behaviors into their controllable part and an autonomous subsystem will be derived. characterized the

we

associated

H. Now

via left

will tum to

Definition 4.3.1

(a)

For B

o(B) (b)

kerc R, where R

=

:=

For systems define

B1

-

HpXq

E

define the output number of 8

by

rkR.

Bi

kerc Ri, where Ri

O(BI)

132

R2

=

=

O(L32)

MR1 for

Z

HpiXq have full

row

rank,

i

=

1, 2,

and a

nonsingular

matrix

M'C-

R (s,

z)PI xP,-.

provides an equivalence relation on the lattice B. We call two B, and B2 transfer equivalent if B, B2. The equivalence class behavior B will be denoted by [B] and is called its transfer class. This

tems

-

sys-

of

a

The output number is well-defined by Theorem 4.1.5(a). It does indeed count the number of output variables of the system, see Theorem 4.2.3(2). Observe that transfer equivalence simply means that the kernel-representations share the same rowspace as R(s, z)-vector spaces. Since R(s, z) is the quotient field of

the operator ring

B1 It is

-

L32

easily

seen

H, transfer equivalence

O(Bl) AR2 that for

=

=

O(B2)

can

just

as

well be

expressed

as

and

BRI for nonsingular

i/o-systems

transfer

matrices

equivalence

A,

B E RPI

is the

same as

IPI

equality

of the formal transfer functions. In the next theorem Nye describe the structure of the transfer classes. Among other things, we obtain thatbehaviors with right' invertible erators.

kernel-representations

are

exactly

the

images of delay-differential

op-

Delay-Differential Systems

4 Behaviors of

96

Theorem 4.3.2 Let B G B have output number o(S) p. Then the transfer class [B] of B is a sublattice of B. It contains a least element Bc and can therefore be written as =

[B] For

(1) (2) (3)

system B'

a

13' B'

G

=

[B]

=

13c,

=

kerc R' for

S' has

f B'

I o( 3')

B

Cz

following

the

full

row

an

right invertible R'

some

L31 132

E

1

[B]

B,

be given

-

rem

3.2.8, rk lclm(Ri, R2)

and

(d),

[B]

obtain

with respect to

(B1

[B] satisfying (2).

rank.

+

L32

Q

(4.3.1)

=

L32)

a

least

To this

Using Corollary

3.2.5

Bi

as

B,

-

for

some

Q

C-

Hqx(q-p)

[R2] Ri

some

rk R1

=

=

Ri

E

rk R2

-HpX q having

and, by

Theo-

R2) p, too. Using Theorem 4.1.5 (c) n B2), which implies the closedness of =

(BI

-

imCQ

be chosen left invertible.

kerc Ri for

=

rk gcrd(Ri,

-HpXq.

E

is B=

can

it follows rk

taking finite

As for the existence of in

Bc 9 13'1.

equivalent:

are

image-representation, that

rank. From

we

and

the least element.

of full column rank. The matrix

PROOF: Let

o(B)

=

and intersections.

sums

element, we first show that there exists a behavior kerc R where R G Hp X q has full row end, let B =

we

may factor R

R

=

as

(4.3.2)

BRc

where B G HPxP is

nonsingular

and

Rc

7jpxq is right invertible.

.

(4-3-3)

Now

Bc is

a

system in

To show the

:=

kerc Rc

G

[B]

[B] satisfying (2). implication "(2) = . (3)", let B' Completing R' to U:=

(see Corollary 3.2.5)

and

U-1 obtains kerc R'

=

V

=

Hence kerc R' C

R'Q

=

0.

kerc R'

=

invertible matrix R' EE 7jpXq.

one

(4.3.4)

=

[R']

partitioning the

U-1UV

imcQ

inverse

[Q', Q] according

imcQ. Indeed, for =

and the

[Q" Q]

G

[B]

for

some

v

to G

(0) W

converse

Q

(4.3.5) as

G

7jqX(q-p)

kerc R' and =

right

unimodular matrix

Glq(H)

E

U/

a

QW

c

w

(4-3-6) Uv

one

has

iM'CQ.

inclusion follows from the

identity

4.3 T ansfer Classes and Controllable

Systems

97

Hqx(q-p) imCQ for some matrix A is nonand invertible left is QA Q Q The matrix observe imc we Q can imCQ. singular. Using Proposition 4.1.4, be completed to a'unimodular matrix, say U-1 as in (4.3.6) and U as in (4.3.5), and the argument above leads again to B' kerc R', where R' is a imcQ matrix. invertible right For the

implication "(3)

=: ,

(2)"

,

of full column rank and factor

let B'

=

=

where

=

=

=

In order to prove "(2) (1)", we first remark that the system Bc defined in (4.3.4) is the unique system in the transfer class [B] with a right invertible kernel-representation. To see this, let MRc NRc, where R'c G Hpxq =

is

verses,

one

Al,

and

right invertible, too,

N G 'HPxP

N-1M, M-1N Rc showing that kerc Rc'

obtains that

are

'HPxP,

E

nonsingular. Using right thus

Rc'

=

(N-'M)Rc

in-

is left

kerc Rc by Theorem 4.1.5 (a). Now kerL Rc in [B]. We know kerC R' be any imc Q for some matrix Q . Let B' already that kerc lic LR for some nonsingular matrices K, L c 'HPxP behavior in [B]. Then KR'

equivalent

to

there remains to establish the

=

minimality of Be

=

=

=

=

0 and LBRc by (4.3.2) and (4.3.3). This yields R'Q latin the element least B'. Hence Bc is the (unique) thus imCQ C kerL R' the well as tice [B]. Together with Theorem 4.1.5 (a) we get (4.3, 1) as implication

and hence KR'

=

=

=

"(1)

=;>

(2)", completing

Obviously, the system kerc I

autonomous =

10}

as

proof

the

systems in Cq form

a

transfer class

having

the trivial

its least element.

system-theoretic significance. It is a controllable system in the sense that it is capable of steering every trajectory into every other trajectory within finite time and without leaving the behavior. Put another way, controllability is the possibility to combine any past of the The least element Bc of

system with we

a

transfer class is of

(far)

future of the system. In order to make this precise notion for combining functions.

any desired

first need

a

Definition 4.3.3 For w, w' E Cq and to G R deline the concatenation of the function wAt,,w' : R --+ (Cq given by

(wAt.w')(t)

W(t) W,(t)

Using concatenations, trajectory steering

(see [87,

Definition 4.3.4 A time-invariant there exists

some

wAocAtoutOw' Note that the

E

Def.

5.2.2]

can

w

and w' at time to

as

for t < to for t > to

be

and the

expressed

as

follows.

interpretation given therein)

subspace B ofCq is called controllable if for all w, w' c B time instant to > 0 and a function c : [0, to) ---+ (Cq such that

B.

requirement

wAocAt,,ot0w' utowf(to)

concatenation is smooth. Since

E =

B implies in particular, that the

w'(0),

the concatenation switches

98

4 Behaviors of

Delay-Differential Systems

exactly from w(O) to w'(0) but allows for some finite time to ! switching smooth and compatible with the laws of the system.

0 to make the

Remark 4.3.5

The definition of

controllability given above appears to be the most intrinsic merely refers to the collection of all trajectories of the system and does not make use of any kind of representation, for instance, a kernelrepresentation or a state space representation. A slightly different version of controllability, yet also based solely on the set of possible trajectories, has been introduced in the algebraic approach to systems theory in [125, p. 153]. In this case, the notion resorts to input/output partitions, which makes the concept of controllability more technical than the definition above. one

possible.

It

Of course, the space Lq is controllable. It is even controllable in arbitrarily short time, that is, for all w, w' EE rq and all to > 0 there exists a function c such that wAocAto, to Wt C Lq.

verify (straightforwardly) that the image U(wAtow') wAtow' under a delay-differential operator U is aconcatenation of U(w) and U(w') and some intermediary piece. Its length is determined by the size of the maximal retardation appearing in the operator U. In the next lemma

of

we

smooth concatenation

a

Lemma 4.3.6 Let w, w' EE Lq and to C- R be such that matrix U E

HpXq 0

Then there exists

be written a

function

PROOF: First of we

a)

proceed

all,

in two

Assume first U

Ej=o Ujzj

as c

[to, to

U(wAtow')

wAtow'

L

U

=

L)

+

E

Lq

Furthermore,

with coefficients

Uj

G

let the

R(s)PXq.

CP such that

--+

U(w)AtcAto+LU(W1)

it is clear that

.

U(wAtow')

C

E

Lp-

LP. As for the concatenation;

steps. G

R[s, Z]pXq

,

hence

Uj

R[S]pXq

G

L

Then

L

E Uj(wAt.w)(t

U(wAtow')(t)

.

-

j)

1: (Uj(w)AtoUj(w'))(t

=

j=o

-

j)

j=o

-_jL

=0

Uj (w') (t

[ELi= OUj (W) (t

-

-

j)

j)

U(w') (t)

=

=

U (W) (t)

if t > to + L if t

to

and the desired result follows.

b)

For the

put V

U( FD) Using

=

=

general 3

V(V),

the

VjO-' Uj R[s, Z]pxq Then

case

ELj=o V-z3

let

=

where V E Lq satisfies

appropriate

Vj G R[S]pXq and Vo-' and for all 70

where

C:

U

Lq

=

R[s]\10}. we

have

O(V) entrywise. Let fowAtow'. initial, conditions at to, one observes that one may find =

fv-

=

Systems

4.3 'h-ansfer Classes and Controllable

;V-

G

vAt,,v' proof yields U(Cv)

Lqsuch that V

part a)

of the

U(W)At0CAto+LU(WI) [to, to + 4 One obtains

=

for

some

w. But then w and 0(v') 0(v) V(D) V(vAt,,v') V(v)At,,CAt,,+LV(VI)

where =

99

=

=

=

=

=

suitable function

c

defined

the interval

on

13

immediately

Corollary

4.3.7

Let B be

a

time-invariant controllable

the space

U(B)

C

LP is

controllable,

subspace of Lq Then for all U .

E

HP'q

too.

Let U PROOF: Since B is time-invariant, it is enough to consider U E 'Hp"'. 0 as in Lemma 4.3.6. We have to show that for all w, w' c B the images

be

U(w')

U(w)

and

have

0,LWI

i7v

can

be concatenated within

U(B). By assumption

on

B

we

B and there exists to > 0 together with a function c such that wAocAt00,to+Lw' E B. Now Lemma 4.3.6 provides some intermediary

:=

G

function cl such that

U(fv-)

=

=

completing Now

we are

proof

the in

U(wAocAt0ato+LW/) U(W)A0C1Ato+L U(Oto+LWf) U(W)A0C1Ato+LU to+LU(WI), =

since

position

a

U(fv-)

E

U(B).

to establish the

following

characterization of control-

lable behaviors. Theorem 4.3.8

kerc R where R G HpXq is a matrix of rank r. Then B is controllable r for all s c C. only if rk R* (s) Bc, where Bc is the least consequence, B is controllable if and only if B

Let B

=

if and

As

a

=

=

element in its transfer class

[B].

Notice that the rank condition does not

depend

on

the choice of the kernel-

representation R.

Sufficiency follows from Corollary 4.3.7 together with the existence of image representations as derived in Theorem 4.3.2. For necessity we first prove the assertion for the case B C L, hence R E X Let w G B be any trajectory. By controllability there exist to > 0 and a function c such that 0 and v wAocAt,,O G kerc R. Using twice Proposition 2.14(2), we obtain v desired shows R and Lemma 2.12 0 'H R 0. c w 1, as Therefore, kerc (cf. 1 PROOF:

=

Remark

3-1.5).

For the

general

and V

are

case use a

diagonal form URV

unimodular matrices and dl,..., d, E

=

diagp

H\101.

xq

(di,

.

.

.

,

d,)

where U

Since kerc R is control-

kerc diagpX q (di, lable, the same is true for the system V-'(kerc R) dr) see Corollary 4.3.7. This implies the controllability of kerc di C f- for each =

.

.

.

,

7

100

i

=

4 Behaviors of

1,

r

and

now

Delay-Differential Systems the rank condition

on

R follows from the first part of the

proof. The second part of the assertion can be deduced from Theorem 4.3.2(2) by using a full row rank kernel-representation and resorting to the rank criterion in

Corollary 3.2.5(c)

for

right invertibility.

El

Remark 4.3.9

Reconsidering

the arguments above

of B is equivalent to the zero.

we see

with

capability of steering

hindsight

each

that

trajectory

controllability

in finite time to

Precisely, B is controllable

V

-

w

G

B 3 to

such that

In the next remark

we

want to relate the

! 0,

[O'.to)

C :

wAocAtOO

E

controllability

__4

(Cq

B.

criterion above to

some

other results in the literature. Remark 4.3.10

(i)

The criterion for Thm.

5.51.

In the

controllability special

representations it has been

case

in Theorem 4.3.8

appeared first

of behaviors

proven

having a polynomial by completely different methods

The result

in

[42,

kernelin

[91].

generalizes the well-known Hautus-criterion for systems of ODEs to delay-differential systems; see [50] for state-space systems and [118, Prop. 4.3] for behavioral controllability of ODEs. For certain time-delay Ax+Bu with matrices A, B over R[z] or even Ho,p, systems of the form b it is also known to characterize spectral controllability [6, 74, 73], a notion referring to the controllability of certain finite-dimensional systems associ=

ated with the

zeros

of

det(sl

-

A* (s)). In

[85,

Thm.

1]

it has been shown

that spectral controllability is identical to null controllability. The latter that for every piecewise continuous initial condition there exists a continuous control u of bounded support in [0, oo) such that the corresponding solution x is of bounded support. means

piecewise

(ii)

easily seen that the constant rank assumption equivalent to the quotient module M :=

It is is

on

Hq/iM'J

R* for

controllability torsion-free. being gr

The connection between the system kerc R and the module M has been explained in Remark 4.1.1. Recall in particular that for R being polynomial, the quotient T is taken as the definition of a R[s, Z]q

/iMR[s,z]ff

delay-differential system in [32, 80]. In [80], controllability, depending on an R[s, z]-algebra A, is defined algebraically as the torsion-freeness of the module A (&R[,,,,l T. Since M H OR[s,z] T, behavioral controllability coincides with the algebraic notion of H-torsion-free controllability in [80]. (iii) For systems of PDEs, or generally for multidimensional systems, the notion of controllability or concatenability does not come as straightforward =

as. for onedimensional

systems

(like

ODEs and

DDEs).

Various notions of

4.3 Transfer Classesl and Controllable

Systems

101

controllability have been suggested in [124] (see also [129, Sec. 1.4]) and characterized algebraically and in structural terms similar to our Theorems 4.3.2 and 4.3.8. Some of the structural characterizations appeared first in [84, PP. 139]; controllability of smooth systems of PDEs has been investigated in detail also in [86].

(iv)

It is

For systems of DDEs with noncomMensurate be summarized in Remark 4.3.13 below.

an

in Cq

delays the existing results will

immediate consequence of Theorem 4.3.8 that two controllable systems transfer equivalent if and only if they are identical. Put another way,

are

the formal transfer function, taken after a suitable input/output partition, determines the (unique) controllable behavior Be in the transfer class [B]. The

ptoof of the not

shows, see (4.3.2), (4.3.3) and (4.3.4), how this controlobtained from a given system kerc R, namely by cancelling

Theorem 4.3.2

lable behavior

can

be

nonsingular left factors (if any) of R (which for change the formal transfer function -Q-'P).

R

=

[P, Q],

of course, does

minimality of Be in the transfer class can be rephrased as follows: a system B if and only if it has no proper subsystem with the same number of free variables.. As we will show next, there is another way to characterize Be. It says that Be is simply the controllable part of B in the sense that it is the The

is controllable

maximal controllable subbehavior contained in B. Recall from Remark 4.1.9

that ker,7-j R is

Proposition

finitely generated for

every matrix R.

4.3.11

Let R E -Hpxq be

a

matrix and put B

=

kerL R. Let Be be the (unique)

con-

trollable system in the transfer class [B]. Moreover, let kerjj R im-H T C ? q for some T (E HqXt. Then Bc imLT. Furthermore, one has B' C Be for every =

=

controllable behavior B' contained in B. We call Be the controllable part of B.

By Theorem 4.3.2(3), each controllable behavior B' has an image0 imLT' for some T' (E -HqXr.- Hence B' C 8 implies RT' representation B' TX for some X E -Htxr and B' so that T' imLT' C imcT. As a special BRc is factored as case, we obtain Be C imLT. On the other hand, if R 0 and in (4.3.2) and (4-3.3), then kerH R kerh Re imH T, whence RcT D Be. This concludes the proof. iMLT C kerL Rc PROOF:

=

=

=

=

=

=

=

=

=

Remark 4.3.12

Another characterization of controllable behaviors

can

be found

in

[111, C'

" only C and the of D Cl-functions 9 is having compact support where, again, space --c denotes the closure with respect to the topology on S. The only-if part follows in essence from the existence of image-representations and the denseThm.

3.5].

A behavior B C Sq is controllable if and

if B

=

B n Dq

102

4 Behaviors of

ness*of

D in S. The

Delay-Differential Systems

proof of

the other direction

form to the scalar case, where then kers p n D (Proposition 2.14(2)) is the key argument.

can

be reduced via

0 1 for each

a

diagonal

nonzero

p E 'H

Remark 4.3.13 In the

same

paper

[111],

controllable behaviors have been

tems with noncommensurate

type discussed in Remark

delays and

4.1.15(4).

investigated for

sys-

for convolution systems of the In this generality, it is not known whether even

the properties (a) controllability, (b) having a kernel-representation with constant rank on C, (c) having an image-representation, and (d) being the closure of its compact support part, Thms. 3.5, 3.6] that for R G next

equivalent. However, it has been shown in [111, each of the following conditions implies the

one:

(i) kerE R (ii) ker,, R (iii) kere (iv) (V)

are

(SI)pXq

R

kers R

=

im

,

Q for

some

Q

E

is controllable in the =

kers R

n J)q

-6 =

im,,Q

for

(S') q 1, x

sense

of Definition 4.3.4,

-0 I

some

Q

E

(Ef)qX1

LR(s) is constant on C, where LR denotes the Laplace (in this case, kerg R is called spectrally controllable).

rk

If R has full

one

a

R E=_ HpXq

row rank, then delay-differential operator

transform of R

"(v) =>. (iv)". In the special case of (see the Remarks 3.1.8 and 4.1.15) it is proven in [41, Thm 3.12] that "(iii) - * (iv) 4#. (v)", regardless of any rank constraint. The implication "(v) =* (h)", however, does not hold for general x operators R c Hp q, see the example in [41, Ch. 4]. Controllable systems are, in

also has

just the extreme opposite of ausystems. Controllability capability to switch from any trajectory to any other, in other words, the past of a trajectory has no lasting implications on the far future. On the other side, autonomy prohibits any switching at all, because, by definition, the past of a trajectory determines completely its future. These two extreme points on a scale of flexibility for behaviors can also be expressed in module-theoretic terms. It is easy to see that a system A kerc A is autonomous if and only if its annihilator in 71 is not trivial (indeed, if A is nonsingular, then det A e ann(A)\f0}; the other direction follows from Theorem 4.1.5(a)). On the other hand, it is not hard to show that a behavior B is controllable if and only if it is a divisible li-module, that is, if each a G H\f01 is a surjection on B. Next we show that each behavior can be decomposed into a direct sum of its controllable part and an autonomous subsystem. tonomous

=

a

certain sense,

describes the

4.3 T ansfer Classes and Controllable

Systems

103

Theorem 4.3.14 Let 8 C

behavior with controllable part Bc. Then there exists system A C Lq such that

Cq be

autonomous

a

B

Furthermore, let B

(4.3.3).

=

=

Bc

kerL R where R

(4.3.7)

(D A

BRc

=

an

(2

-Hpxq is factored

as

in

(4.3.2),

decomposition B Bc,nt, ED Baut into a controllable and an autonomous subsystem, the controllable system is given by Be ntr Bc, while the autonomous part is of the form Baut kerLA for some A Ej HqXq det B, up to units in H. satisfying det A Then in every direct

=

=

=

=

PROOF: Consider the factorization R

given by Be

controllable part of B is unimodular matrix

Re

U and partition the inverse

the.

matrix A

nonsingular

IU/Re] [QI7 Q]

the, identities

immediately verifies RcA

Iq

=

R

=

[Q', Q]

U-1

:=

GIq (H)

UI

U-1

as

BRc in (4.3.2), (4.3.3). Hence the kerL Re by (4.3.4). Complete Re to a

=

=

:---

as

[RI U/

IU/I Re

B

Thus

by

(4.3.7)

Theorem

3.2.8, 1

=

follows from Theorem

Consider

now a

with Theorem

gcrd(Rc, A) and BRc 4.1.5(c) and (d).

given decomposition 13

4.1.5(d)

one

and A

verifies

kerL A. Using

:=

Q'BRc

=

Define

QU',

+

one

=

on

13contr

the

one

once more

0(6contr)

(4-3.8)

Ip+q-

Q A

of the controllable term, observe that tion 4.3.11. On the other hand, using

Hqx(q-p)

E

as

[ Rc'] [ 0, Re] Q' QU

Q

HqXq and put A

G

[QIj Q]

well

such that

=

p

(D

=

R

lclm(Rc, A)

--

and

Baut. As for the uniqueness Bcontr 9 Be by Proposi-

hand

Theorem 3.2.8 in combination so

that

Bcontr

G

[B]

and there-

fore Be C Bcont, by Theorem 4.3.2. Hence Bcontr Be is the controllable part of B. As for the autonomous part, write Baut =: kerc A where A G Hq x q. We have =

to show that det

such that RcW

A

=

=

det B up to units in R. To this

[1p, 0]

and VAW

firstly, kerL RcW n kerL VAW tion 4.1.4. Secondly, one has kerL BRcW

=

=

Hence

[A,, 0]

f 01 gives

end, let V, W

where A4 E det A4 c

=

E

Glq (71)

H(q-p) X (q-p) Then, .

H',

see

kerL lclm(RcW,

also

Proposi-

VAW)

kerc [A,, 0].

and BRcW

R, which

=

kerL RcW + kerc VAW

are

left

divisor of the full-size minors of to units in

=

[A3A40] A,

is what

we

equivalent. Since det B is BRcW, this yields det A wanted.

the greatest =

det

A,

common

det B up 1:1

4 Behaviors of

104

Delay-Differential Systems

We close the section with Remark 4.3.15

(a)

The

decomposition (4.3.7)

is

quite standard

in behavioral

systems theory,

[87, 5.2.14] for systems described by ODEs. The sum can also be derived for multidimensional systems given by PDEs, but in this case the Thm.

see

decomposition

(b)

To

always direct,

is not

see

[123, Thm. 5.21.

regard the direct decomposition (4.3.7) as the "classical" decomposition of a system into its forced and free motions, see, e. g., [52, Prop. 3.1] in a slightly different context. Indeed, denoting by L+ the H-submodule of L consisting of all functions having support bounded on the left, it is easy to derive from (4.3.7) and (4.3.8) the relation kerL R n Lq kerC Rc n Lq extent,

some

one

can

"behavioral version" of the

=

This space

be viewed

the set of all forced motions of the system (including the forcing input, starting at some finite time to G R), while A kerL A contains the free motions (including input which has been actcan

as

=

ing

on

the system

forever).

In case,

kerc R

kerc [P, Q]

=

is

an

i/o-system

HP' (+P) and det Q =7 0, we know from Remark 4.2.4 that kerL R n Lq+ is the graph of the convolution op-

with

kernel-representation

[P, Q]

E=-

Q `P) (JO(l), Ji) restricted to LT. This way, we observe given by again that the formal transfer function is related merely to the controllable part of the system. Consequently, nonanticipation, as well, is a property related to the controllable part only. erator

4.4 Subbehaviors and Interconnections

So far

we

have

only been concerned with the analysis of

this and the next section two

systems,

will direct

our

a

single system.

In

attention to the interconnection of

of which

one

to-be-designed

we

controller.

being regarded the given plant, the other one the Indeed, a controller does constitute a system itself.

It processes (part of) the output of the to-be-controlled system and computes (part of) the inputs for that system with the purpose to achieve certain desired

properties of the overall system, like for instance stability. Thus, the system and the controller are interconnected to form a new system. In the behavioral framework the interconnection

ably defined behaviors.

can

be written

as

the intersection of two suit-

underlying idea is simply, that the trajectories of the interconnection have to satisfy both sets of equations, those governing the system and those imposed by the controller. Depending on the type of interconnection or on the description of the components, the resulting system might be described with the help of some auxiliary (latent) variables, which hopefully can be eliminated in a second step so that one ends up with a kernel-representation The

for the external variables of the interconnection. This elimination

procedure will be dealt with

at the

beginning

of the section.

4.4 Subbehaviors and Interconnections

Thereafter

we

turn to the interconnection of

105

systems and investigate the achiev-

ability of a given subsystem via regular interconnections from the overall system. The notion of regularity can be understood as requiring, in a certain sense, most efficient controllers. At the end of the section the dual of regular interconnections will be treated, are direct sum decompositions of behaviors. It will be shown that the

these

existence of direct

sum

decompositions

closely

is

related to the notion of skew-

primeness for matrices. The

following

theorem shows that

(and how)

considered

latent variables

can

be eliminated

exactly showing up in typparticularly important in the next latent variable where we chapter study systems of a specific type. In that conwill role be a text, special played by polynomial kernel-representations; therefore we also include the polynomial case in the theorem below. For the term "latent variable" we would like to recall the discussion following Definition 4.1 in the introduction to this chapter. in certain situations. The

cases

those

are

ical interconnections. The theorem will be

Theorem 4.4.1

(a)

The

image of a behavior under a delay-differential operator is again a bePrecisely, if Ri G Hp;,Xq for i 1, 2 are matrices of full row rank,

havior.

=

then

R1 (kerc R2)

=

kerc X,

where the matrix X C H"P-1 is such that XR, is

multiple of R,

(b)

least

a

common

left

and R2.

Moreover if Ri E

R[s, Z]pi,

the matrix X

be chosen in

can

Ri E RPxPi, i Fbrthermore, assume Let

V

V2

R2

qand

[M0-

Ri

rkC

R2

two matrices and

[VjT, V2TF

=

rkR(s,z)

Rj* (s)

for all

R2

s

E

C,

R[s, z]txP'.

1, 2, be

=

[ 1]

X

for

assume

rk

[Ri, R21

=

P-

is such that

c

some

M (,- HrXP2 with rank

r.

Then

B:= If s

we

E

have

C,

fw

c

LP'

additionally Ri V2

then the matrix

G

I Rjw

E

imLR21

R[s, z]PxPi

can

=

kerL (V2Rj)-

and rk R(s,,) R2 be chosen with entries in

=

rk CR* for all 2 (S)

R[s, z],

too.

As the proof will show, the condition on the gene?7ic rank of RI and R2 in (a), and hence also in (b), is not inherently necessary. It simply allows to

part

use of the least common left multiple, which has been defined for this only. Note that also the (extreme) case where rk [Rj, Rjf P1 + P2 is encompassed in the statement above, as in this situation the least common left multiple is the empty matrix while R, (kerC R2) is indeed all of LPI.

make case

=

4 Behaviors of

106

Delay-Differential Systems

I in (b) shows again that imCR2 is a behavior, a fact special case R, in Remark 4.1.9. In light of Theorem 4.3.2 we see that indicated already being the systems of this form (that is, having an image-representation), are just the

The

=

controllable systems.

possible

It is not

drop

to

pointwise rank condition imposed for the poly-

the

Z

nomial kernel-representations. For instance, imC

Proposition 4.3.11) representation can be found

in this

PROOF

(a)

and because of Theorem

as

THEOREm 4.4. 1:

OF

-

1

kerc [1,

8

4.1.5(a)

no

(by

polynomial

kernel-

case.

The first part is fairly standard and can be seen we know that there exist matrices Ui

By the Bezout property of H,

follows.

such that

U2] [R2Ri] [D]

U, U3 U4

=

for

0

and the leftmost matrix is in

by

Theorem

for

w

Ez-

3.2.8(b). Using

some

D CHrXq with rkD

Glp, +p2 (H).

provides lclm(Rl, R2) D, see Proposition 4.1.4,

This

surjectivity of

the

(4.4.1)

r

U3R,

=

we

get

Lq

w

E

(w)

R2)

R, (kerc

C

o

c

w

11

m'C

===,,

R2

(Ulw) U3W

E

m'C

[D0

kerL U3,

which proves the first assertion of (a). Let us now turn to R, and R2 being polynomial matrices. The existence of a polynomial kernel-representation for

R, (kerc R2) will be can

any

be chosen

proven

polynomial.

have established that

once we

This

accomplished

be

can

equation of the type (4.4. 1). Notice that

Theorem

4.1.13(2)

we

matrix

[R2 ki

[f?J, f? 2TT

[R

know that

T 1 ,

RTT 2

r

is

=

rk

as

[U3, U4]

(4.4. 1)

in

follows. We start with

[RIT, RiT. By

right equivalent

virtue of

over

H to

a

-

0

R[s, Z] (pl +p2)

G

0

x

q

-

The rank assumption on [R 1T , RTF and 2 the invariance of'the invariant factors under equivalence imply the coprimeness and

has full column rank

of the full-size minors of

[f?,T, f?jf

r.

in X

equation

[U3, U4]

4.1.13(l)

[03, 1 4].

Now

R,

=

now

Lemma

3.2.7(l)

to the

0

are polynomial so that by Theoequivalent to a polynomial matrix replace the unimodular matrix in (4.4.1) by

the matrix we can

I A21

[U3, U4]

shows that the full-size minors of rem

Applying

[U3, U41

is left

[ l 2] U3 U4

E

Glp, +P2 M)

4.4 Subbehaviors and Interconnections

and obtain from the first part of the

hence

(b)

a

proof the identity R, (kerC R2) polynomial kernel-representation.

(a) by observing

follows from

[R j j 0

that the matrix Let

us now

-R2

start with the

Definition 4.4.2

that B

[Ip, 0] (kerc [Ri, -R21).

=

=

107

kerC 6r31

Note also

has constant rank whenever R*2 has.

investigation of interconnecting systems.

(see [120,

p.

332])

The interconnection of two systems B1, B2 C- B is defined to be the system B : 13, n B2. The interconnection is called regular if o (B) o (131) + 0(132) =

=

-

The concept of a regular interconnection is rather natural in the behavioral setting as can be seen by Theorem 4.2.3. Indeed, the number q of external variables minus the rank of a kernel-representation represents the number of input variables of

system. If one thinks of one of the interconnecting', components as it is natural to require that each linearly independent equation of the controller should put a restriction onto one additional input channel, for the

a

controller,

otherwise the controller would be inefficient. Put another way, restrictions are imposed on what is not yet restricted. As a consequence, the resulting inter-

B, and B2 is left with regularity condition.

connection of

exactly

the

Using

once more

0(131)

+

0(132)

q

-

o(BI)

Theorem 4.1.5 and 3.2.8,

one

-

o(B2) input variables,

obtains

which is

o(Bj n132) +O(B1 +B2)

and'theiefore

o(Bj

n

B2)

=

Hence the interconnection is

O(BI)

+

regular

0(132) if and

B,

+

132

=

(4.4.2)

Lq

only if the components add

up to

the full space rq.

As

an

example

we

want to discuss the classical

feedback-configuration

of two

systems. It also exhibits how "interconnected" variables may,turn into latent variables of the interconnection in the sense that they are not describing the external behavior of the

Example

new

system.

4.4.3

Given the two systems B,

=

(U)

C

Lq

Yi

I

Qiyi

piui +

x (m+P) where q and p + m and [Pl, Q 1 j c RP sical feedback-interconnection given by u := ul =

by

ker,C

0

-1 1

OQ1 Pi

_O

P2

0

01,

[P2 Q21

the system 1

=

0

Q2_

7

Y27 Y1

E

HM X (P+m). The clas-

=

U2

=:

y is described

Delay-Differential Systems

4 Behaviors of

108

(U) Y

for the variables

and y

only,

i

U 11

Y2)

If

-

one

is interested in the

1 0

00100-0 -

B:

one can

B

:=

[U3 U2] U1

G

U4

kerC

0

Using Theorem 4.4.1(a),

where U

new

eliminates the latent variables ul and Y2

one

_O P2 find the

G1,,+p(H)

u

projection

0

Q2_

0

kernel-representation

kerc [U4P1, Q3P2

=

the

-1 1

Q1 P1

0

external variables

by taking

+

U4Q1],

is such that U

[Q2] [D01

for

=

P,

some

full

row

governing the external variables (u, y) of easily be seen that the external behavior B is an i/osystem with output y if and only if det(I Q, 1P1Q2 1P2) 6 0. This is the usual well-posedness condition for this type of feedback-configurations in the classical transfer function approach. In the same way one can handle series- and parallel-interconnections. As this is completely analogous to the case of systems described by ODEs in [87, rank matrix D. It describes the laws

the

system. It

new

can

-

Exa. 6.2.9, Ex. 6.3, Ex.

6.4],

the details will be omitted.

Obviously, an interconnection is a subsystem of either of its components. It is fairly simple to characterize algebraically those subsystems of a given system, which can be achieved as regular interconnection from that system. But it is also not hard to give a dynamical characterization purely in terms of the trajectories involved.

Theorem 4.4.4 Let

8

B C ,Cq be two behaviors and

C

Then the

(a)

following

There exists

a

statements

system B'

are

C

B, C V of B

assume

8

=

kerC A where A

G

H,25 x q.

equivalent: 8 n BI is

f q such that

a

regular

intercon-

nection of B and

(b) (C) (d)

the

8

image

=

B is a

Bc

+

A(8) 8, where Bc c :

[0, to)

_._

w

G

point of view, part (d)

us

we

is the most

with an of trajectories

provides

B there exist to

! 0, lb

E

B,

and

Cq such that wAo cAto ?b C B.

equivalent conditions is satisfied, regular interconnection from B.

behavioral

a

since it

controllable,

8-controllable, that is, for each

function

If any of these achievable via a

From

is

denotes the controllable part of B,

intrinsic criterion for

say the

subsystem 8

is

important characterization

regular interconnections;

it is

and does not resort to any kind of representation purely of the behaviors. Observe that 8-controllability can be understood as the capain terms

bility

to steer every

trajectory of B

into the

subspace 8

in finite time. In

light

4.4 Subbehaviors and Interconnections

of Remark 4.3.9 is the

same as

controllability

that

we see

101-controllability.

in the

sense

109

-

of the previous section

The characterization above is close to what

has been obtained for multidimensional systems in [92, Thm. 4.2] more the structural analogy between these classes of systems.

showing

once

equivalence of (a) and (b) can be derived by taking the duals of.the behavconsidering the corresponding problem in terms of finitely generated submodules of Hq. However, we think it is reasonable to stay on the systems side in order to use one and the same language throughout the proof The

iors and

PROOF is

THEOREm 4.4.4: Let 13

OF

rank. We may also

row

co'ntained

assume

Hop^xq ,thus f?

in

kerL R for

=

does not contain any

simplify the application of k onto a inclusion L C B implies a relation Xf?'=

R E Hpxq

some

f?

without restriction that

has full

having full

powers of

negative

latter will

concatenation

The

R where X E HPxf' is

rank matrix. Note that R

lclm(f?, R)

=

R(B)

and therefore

rank and

row

later

=

The

z.

in the

proof.

full

a

kerL X

row

CP

C

by

Theorem 4.4.1 (a).

"(a)

(b)"

=)>

ker,c f?

Let B'

8

=

=

kerc

=

kerc R' where R'

[R]

Hence Theorem 4.1.5 (a) Thus X is

"(b)

=,,.

block

a

controllable

by

and

R'

row

P

yields that the

of

a

"(b)

= ,

to R

is

matrices

f?

a

the form L

=

BRc be factored

lclm(Rc, k) (up Af?

E

Pick

a

=

"(c) = (d)"

i'v'E B sdch that

trajectory

v

:=

which proves

"(d) =: ,'(b)"

left

equivalent. =

f?(B)

is

3.2.5.

unimodular matrix

[XT, YTT (see

as

in

(4.3.2)

and

to unimodular left

factors),

w

right invertibility of X, IpXq and a right divisor of R

=

(4.3.3), Bc+B

see

Xf?

=

equivalent

4.1.5(d). lelm(Rc, f?) is of

Theorem

since every =

thus Bc

is

BRc.

G B.

By assumption there exist wc G 13c and trajectory w wc +,Cb. Controllability of 13c implies the existence of a =:

wcA'OcAt,,O

c-

Bc. As

a

consequence,

v

+ &

=,wA0'c'At0tb

G

B,

(d). Let

trajectory Cv

f?w

v

E

suitable function

c

c

f?(B)

for

such that w, defined on

kwi

point

Then

=

Let R

an

are

R'

unimodular matrix and therefore kerc X

But the latter follows from

this

[R]

and

Corollary

p'.

of the interconnection.

is the controllable part of B. Then the condition B

being

obtain

matrix of rank

a

p' by regularity

(a)" 'follows by completing X to Yk and defining R' (c)"

-HP'xq

virtue of Theorem 4.3.8 and

Corollary 3.2.5) kerL Rc

p +

=

E

kb

R(B)

B.

wAocAt.7-b

tO,to).

c

c

some w

:=

Now

for

By assumption there E

we can

some

t,

B for

some

to

apply Lemma

> 0

and

a

4.3.6 and

function

it is convenient, but not necessary, to have the entries of

exists 0 and

>

f?

c.

in

(At

'Ho in

order to avoid any backward shifts of the concatenating time instants.) Since f?t^v 0, the last part shows that every trajectory in can be steered to

f?(B)

=

zero, which

by Remark

4.3.9 is

equivalent

to

controllability of R(B).

4 Behaviors of

110

Delay-Differential Systems

Remark 4.4.5

Note that the map

f?(ker,c R) is

an

kerc

Rw

w

+

kerc R

isomorphism of H-modules. Therefore, "quotient behaviors

tified with real behaviors"

controllability

the

R/kerc k,

(with

a

different number of external

condition in part

(b)

above could be

can

be iden-

variables)

expressed

and

in terms of

the quotient behavior.

Since the image of a controllable,behavior is controllable again (see Corollary 4.3.7), the following additional characterization is immediate from the theorem

above. Notice that

(Definition via

regular

by part (b) below the

term

controllability

can now

be

twofold way. Firstly, it describes the ability to steer trajectories 4.3.4), and secondly, it expresses the achievability of all subsystems

understood in

a

interconnections. In other

words,

it

guarantees the very existence of

controllers.

Corollary 4.4.6 The following conditions

on a

system 13

C

Cq

are

equivalent.

controllable, (a) (b) each subbehavior 1 C B can be achieved via regular interconnection from B, (c) f 0} C B can be achieved via regular interconnection from B. B is

Remark 4.4.7

Consider

-

once more

the situation of Theorem 4.4.4. In

case

that

B n B, is

a

regular interconnection, the output number of 8 is, by definition of regularity, the sum of the output numbers of the components B and B'. This, however,

does not guarantee that the outputs of the given subsystem 8 are made up by the outputs of the two components. But this can always be achieved by a ,

suitable choice of the component B'. Even more can be accomplished. If 8 C B are both nonanticipating i/o-systems, then the controller B' can be chosen in this

form,

too

Thm.

9];

is worth

(and,

of course, such that the outputs match). This can easily the same way as described for systems of ODEs in [120,

exactly also Proposition 4.2.5(b) for the condition of nonanticipation. It mentioning that in general it is not possible to have all components

be shown in

see

strongly nonanticipating i/o-systems (see Remark 4.2.4) at the as can be seen by the example

same

time. This

fails'even for systems of ODEs

kerc In this

case

11

2s 3 +1 ,

s

2

82 s+

1]

C

kerc [2, '93 + 1,

B

strong nonanticipation of 8 and B requires by Proposition

that the second and third external variable

4.2.5(c)

the output of 8, while the second to find a strongly nonanticipating

are

is the output of B. But it is not possible interconnecting system B' having the third variable

one

S21..

as

output.

4.4 Subbehaviors and Interconnections

After these considerations be

regarded,

interconnections

on

we now

turn to

ill

problem,

a

that

precise below, as the dual of achievability via regular interconnections. Given a behavior 80 with subbehavior B, C Bo, we ask for conditions which guarantee that B, is a direct summand of Bo in the "behavioral sense", that is can

in

a sense

L30 In this I

case we

161

made

L32 for

0)

simply call B,

Bi -Hq, mpdules.A4o that M1 + -A42

a

some

(4.4-3)

behavior L32 C 130-

direct term of Bo. In terms of the duals Mi

=

question above can be posed as follows: given finitely generated C A41 g -Hq, find a finitely generated submodule M2 C -Hq such

the

C

achievability

Hq and M, n -M2

=

regular

=

Mo. This

is

interconnections where

exactly

the condition of

behaviors

now are replaced by regularity condition). The problem stated above on direct terms might not be of system-theoretic significance by itself, but nevertheless we believe it is natural to be investigated.

via

(see

modules

also

see

(4.4.2)

for the

Example 4.4.8 (a) For Bo =,Cq, the clas's of all direct terms of Bo is immediately seen to be the class of all controllable systems. Indeed, kerc R, E) kerc R2 C-q is equivalent to gcrd(RI, R2) Iq and lclm(Rl, R2) being the empty matrix. But this simply means that [Rj, RjT is unimodular so that by Corollary 3.2.5 and Theorem 4.3.8 the behaviors kerc R, and kerc R2 are controllable. =

=

(b)

In the

previous section it has been shown that the controllable part of a system is always a,direct term, the complementary term being autonomous, see

Theorem 4.3.14. The theorem below will show that

lable

(c)

subsystem

Consider

is

a

even

each control-

direct term.

system Bo 9 Llq given by Bo

kerc A, hence nonsingular. Choose a frequency A E C with k > 0. It is intuitively clear that there exists an exponential ord,x (det A*) solution w(t) woe,\' in Bo. We will show even more. By some matrix calculations it is possible to derive a direct decomposition of kerC.A that extracts exactly the solutions having frequency A. To this end, let U, V E A is diagonal. Extracting Glq('H) such that UAV diagqxq(al ......aq) from each a the (possible) root A with maximal multiplicity, we obtain a an

autonomous

the matrix A E

Hq Xq

,

=

is

=

=

factorization A

==

diagq

Xq

('al

where eti E H and

aq) diagq -

i ....

&,i* (A) (S

coprimeness of eti and

ker,c 6,j and

E)

(d).

kerc (s

-

A) ki

This in turn

the direct

sum

0. In

A)ki

x

q

((S

_

A)kj

particular,

)(S we

_

have

induces the direct

A)k,)

Ejq_

sum

3

-

A

k. The ki decompositions =

kerc ai for the components, see Theorem 4.1.5(c) implies kerc A kerc 3 E) kerc A and we finally get =

=

decomposition ker,c A

=

kerc

(3V-1)

(D

kerc

(AV-').

(4.4.4)

112

4 Behaviors of

Since

Delay-Differential Systems

det(AV-1)

==

Lemma 4.1.10 that

eratorAE R[s]9 xq

(s

kerL .

A)k c R[s] (up to a unit in H), we know by (AV-') = = kerLA for somepurely differential op-

-

Hence this behavior is

k-dimensional vector space

a

consisting solely of functions of the type w(t) p(t)eA' where p c(C[t]q. On the other hand, the first component kerL (ZAV-1) in (4.4.4) does not contain any (vector-valued) exponential polynomial of frequency A; this follows =

from the inclusion kerc

systems of ODEs

(,AV-')

g kerL

(det(,AV-1)1q).

can derive this way

For autonomous

successively complete direct finitely many various frequencies of the system. This is, of course, nothing else but the well-known expansion of the solutions into finite sums of exponential polynomials. Remark that the decomposition J4.4.4) implies the identities A lclm(.AV-1,AV-1) and I gcrd(.AV-1, AV-') by virtue of Theorem 4.1.5. In this particular case this is also clear from the fact that A and A are commuting. sum

one

decomposition according

a

to the

=

=

question posed above let us first rewrite (4.4-3). Choosing kerL Ri, we see that, as in the previkernel-representations Bi ous example, the decomposition (4.4.3) is equivalent to gcrd(RI, R2) 1. and Ro. Let furthermore, Ro XR, be the factorization implied lclm(Rl, R2) by the inclusion B, C B0. In the scalar case the existence of R2 satisfying the above requirements is identical to the coprimeness of X and R1. In the matrix case this generalizes to some skew primeness between these two matrices, which then provides a criterion for a direct sum (4.4.3) in terms of the given data R, and Ro. This is the content of Theorem 4.4.9 below. The role played by the quotient 80113, will be discussed in Remark 4.4.10 right after the proof In order to attack the

full

row

rank

=

=

=

The

=

(straightforward) equivalence (a)

#

(b)

is the

analogue of

[108,

result for two-dimensional discrete-time systems in

Thm.

a

corresponding

18.3.4].

Theorem 4.4.9

Let Ri G RP' ", i ated behaviors

Bi

0, 1, be two matrices with full row rank. Define the associkerL Ri g cq and assume XR, Ro for some X E 'HPO xP1, =

thus B, C B0. Then the

following

conditions

(a) B, is a direct term of B0, (b) the matrices X and Ri are skew-prime, 7P,

xP0

equivalent:

that is, there exist matrices F G

and G E 'HqxP' such that

lp:

(c)

are

there exists

a

=

(4.4.5)

FX + R, G,

matrix G G I-PxPi such that

Bo

=

Bi

ED

GRI (Bo).

Furthermore, every direct term B, C 13o is of the form B1 H(BO) for some H G -Hgxq. Moreover, every controllable subbehavior B, is a direct term of BO, and in case B0 is controllable, every direct term of Bo is controllable, too. =

4.4 Subbehaviors and Interconnections

113

skew-primeness condition does not depend on the choice of R, R0, which, being of full row rank, are left equivalent to every other chosen representation. Remark that the

and

"(a)

PROOF:

full

row

=: ,

(b)

"

Let

B0

=:

L31

E) 132 where

rank. Then Theorem 4.1.5

XR1. Rom Theorem 3.2.8

we

form

132

kerc R2 and R2

=

yields gcrd(RI, R2)

get that po

P1 + P2

-

=

1.

and

q and

an

G

'HP2 X q has

lclm(RI, R2)

=:::

equation of the

[G Z] [Ri] [Iq] C Y

where the leftmost matrix is in

R2

0

Glp,+P2(H)

partitioned according to G E Again Theorem 3.2.8 implies that the matrix R1 is an lclm(Ri, R2) and hence by the uniqueness of the least common left multiple we can assume without loss of generality that X. Completing [RJ, RiF to a unimodular matrix (which is possible by Corollary 3.2.5) we get after some elementary column transformations, if necessary, a matrix identity of the form ,HqXpi

and

.

=

[R2 NJF] [G[X YJZ] Ri

with matrices F and N of

"(b)

= -

(c)"

The equation

fitting

(4.4.5)

=

[Ip, L

0

0

IP2

sizes. This shows

]

(4.4.6)

(b).

shows that both matrices

[RI, F]

and

[(SF, Xrf

completed to unimodular matrices. Choosing the completions appropriately, we arrive again at Equation (4.4.6) with suitable matrices R2, N, Y, and Z. For the verification of the direct sum in (c) we use the identity Ro XR1 and calculate for wo c B0 can

be

=

(i) R1 GRI wo (I (ii) RI(I GRI)wo =

-

-

FX) R1 wo R, wo, implying FXRlwo (I RIG)Rlwo

=

=

=

-

the directness of the sum, =

0, hence Bo

is contained

in the sum,

0 by (ii), thus (iii) RoGRiwo Ro(GR1 I)wo X(RIG I)Rlwo GRI (Bo) 9 Bo. Since Theorem 4.4. 1 (a) guarantees that GRI (Bo) is a behavior, the implication "(c) =: , (a)" is clear. In order to establish the representation B1 H(BO) for a given direct term B, of B0, consider again (4.4.6) and define H := ZR2 I GR1. The inclusion Bi D H(Bo) is immediate by (ii) above, while the converse follows from B, E_ ker,c GRI g kerc (I ZR2). The remaining assertions are consequences of the =

-

=

=

-

=

=

-

-

above in combination with Theorem 4.3.8 and Cor 4.3.7.

El

Remark 4.4.10

Unfortunately we are not able to provide an intrinsic characterization for B, being a direct term of B0, that is to say a criterion purely in terms of the trajectories. However, the skew-primeness of the matrices X and R, can be given a behavioral interpretation. Note that the existence of a direct decomposition does not only require the splitting of the exact sequence

4 Behaviors of

114

Delay-Differential Systems 0

)

B,

)

L30

B01B,

)

)

0,

B01131

to be isomorphic to a behavior contained in B0 that, trivially with B1. From Remark 4.4.5 we know that the quotient can be regarded as the behavior Rj(Bo) contained in 01. Thanks to Equation (4.4.5) it is indeed possible to embed this space as a behavior in B0, complementary to B1. Precisely, the operator G induces an 7i-isomorphism from

but also the

additionally,

RI(BO)

quotient

intersects

onto the behavior

GRI(Bo)

C

L30

C

fq.

The theorem above tells how to check whether

or not B, is a direct term of B0, and, if so, how to determine a complementary term. One has to check the solvability of the skew-primeness equation and to find a solution, if it exists. Since this equation is linear this is not a problem (apart from computational issues, see Section 3.6). For matrices over K[x], where K is a field, a nice criterion for solvability has been derived in [94]. Studying the proof in [94], one remarks that it works equally well for the ring H(C) of entire functions and, as a conse-

quence, also for R. The result will be summarized next. We will confine ourselves to

sketching

the main idea of the

proof

in

[94] along

with its

situation. For the details the reader is asked to consult

adaptation

to

our

[94].

Theorem 4.4.11

H1 x n,

Let A E

B EE Hn x m, and C 2 7in x n be

given matrices. Then the

matrix

equation C=FA+BG is solvable

over

Ii if and

only

if the matrices

[B qA] [13 AO] 0

(4.4.7)

,

-H(n+1)x(,rn+n)

E

0

equivalent.

are

We remark that rem

by

the

3.2.1), equivalence

uniqueness of the elementary divisor form (Theoover H can easily be checked (easily again

of matrices

up to

practical computational issues) by calculating the

given

matrices.

SKETCH

OF

1) Necessity

0

For

sufficiency

one

the matrices A and B

and

bl,

.

.

.

,

invariant factors of the

PROOF: We follow the steps taken in [94]. follows easily (over every domain) since (4.4.7)

[I -.r] [B AC] 2)

(4.4-8)

0 A

1

may are

assume

in

-1 .

rkA

diagonal

ba, respectively. Hence a, to finding fij and

ing (4.4.7) reduces

0

=

] r =

a

implies

0"

0 A

>

0, rk B

=

3

>

0 and that

form with invariant factors a,.... , a, a, and bi b,3. Now, solv-

gij such that

L

4.5

fijaj

(cij)

Assigning +

bigij

=

the Characteristic Function

cij

115

(4.4.9)

,

bi for j > a and i >,3. The, solvability of (4.4.9) ring K[x] by showing that the equivalence of the matrices in (4.4.8) implies that for each irreducible polynomial -Y E K[x] which. occurs with maximal power r in aj and bi, the element -yr is also a divisor of cij. Thus, cij is in the ideal generated by aj and bi. As for the ring h, one can use

where C

and aj 0 is established in [94] for the

the

=

same

=

=

line of arguments to show that

minf ord,\ (aj*), ord,\ (V)} Hence and

ci*j

is in the ideal


. (d)" is in Proposition 4.5.5 since kerc A C kerc (b1q) is stable. R

=

=

=

=

b.

=

=

Notice that condition or

(a)

above is

if the set of rank deficiencies,

satisfied, if, for instance, kerL R

fA c C

I

rk R* (A)
0, i. e. -ri). Then R [aj, (71, D] is the ring of all linear, time-invariant

5.1.2

C' (R,

=

f (t

-

.

,

..

delay-diff rential operators of the form N

P"jo'j'

P

1

0

...

o-,"

o

o

D',

p,,i E R,

(5.1.1)

i=O

where

E'

ture of'an

means

this

sum

being

finite. The space A naturally carries the strucfor p as in (5.1.1) and f E A one

R[aj,...' al, DI-module. Precisely,

has N

E pv,i f (') (t

pf (t)

-

(v, -r)),

t E

R,

VEN' i=O

E,1=1 vj-rj denotes the standard scalar product..It is obvious that

(v,r)

where

Endc (A) mutually commute. Moreover, if -ri ......rl E R are linearly independent over Q, then al, al, D are algebraically independent elements in the ring Endc(A). To see this, let p be as in (5.1.1). Then p being the zero operator in Endc (A) implies in particular for the exponential functions eo,.\ the identity al, and D E

0'1'

.

0

=

.

.

,

pv,jA'e-A(vI'r)

peo,.\(t)

e\t

for alltERand all A EC.

IEN' i=O

Since zero.

(v, -r : (p,,r) R [a,,

Thus,

elements

are

6971

whenever

v

:

p in

N',

all coefficients p,,i G R must be

al, D] is a polynomial ring in 1 + 1 indeterminates. Its delay-di'fferential operators with 1 noncommensurate delays. From .

[25,

p.

The

following

.

.

,

it is known that the

operators

are

surjective

on

A.

class of systems arises in multidimensional systems a unified manner in [84].

have been studied in

theory. They

140

5 First-Order

Example

5.1.3

Let K be of

(Multidimensional Systems)

following

Consider the

(a)

Representations

situations.

of the fields R

one

or

'9

'9

K[ 49xl

C and let

aXj+j

]

be the

ring

partial differential operators acting on A C'(R'+',K) or on A the space of real- or complex-valued distributions on R1+1; =

D'(R'+'), (b)

Let K be any

(possibly finite)

field and let

1

a(n)tnj.....tn,+, a(n) 1 1+1

A:= nEN'+1

be the

K-algebra of formal

E

Kj,

where

n

=

(nl,..., nj+j),

power series in 1 + I indeterminates

over

K. Via

the tackward shifts with truncation

a(nl,.

zi

..'

)

tnt+1

nl+,)tni 1

1+1

nEN1+1L

a(nl,..., ni

tnt+,, 1+1

nl+,)tni 1

+

nEN1+1

the space A can be endowed with the structure of a K[zl,.. -, z1+1]-module. This is usually the framework for discrete-time multidimensional systems,

cf. In all

[123, 122].

cases

above the operator ring is a polynomial ring in 1 + I indeterminates. [841 that these situations have some strong algebraic

It is the main result of

structure in

common:

the module A constitutes

a

large injective cogenerator

category of K[z, s]-modules, see [84, (54) P. 33]. Part of this result,goes back to work of Ehrenpreis and Palamodov in the case of PDEs. The large in the

injective cogenerator property itself is

not needed for

our

purposes and

we

refer

[84] for the details. More important for us are the for. the operators acting on A. In essence, the correspondence consequences between kernels in Aq and -operators in K[z, SjPXq is quite similar to that for the interested reader to

delay-differential systems discussed in Section following from [84] for future reference.

(1) [84, (46),

p.

30]

For matrices

kerK[z, ,] RIT

=:

im

R,

E

4.1. We would like to extract the

K[z, S]pxq

K[z,s] Rj

and R2 c-

kerA R2

(2) In particular, if R G K[z, S]pxq has rank p, then (3) [84, (61), p. 36] For matrices Ri E K[z, S]piXq, i kerA R, 9 kerA R2 Recall the

analogous

R2

=

XR, for

some

one

has

im AR1.

=

im AR =

K[z, s]'xP

1, 2,

=

AP.

one

has

X E K [z,

S]P2 XPI.

Proposition 4.1.4, Theorem 4.1.5(a), and Remark 4.1.9 for the case where H is acting on L C' (R, C) As we saw in Remark 2.11, property (1) does not hold true for the operator ring R[U, -!2L] dt acting on L, preventing point-delay-differential systems to be covered by the paper [84]. For the construction. in the next section only the suriectivity (2) will be needed. Parts (1) and (3). will, be used merely for a more detailed discussion of multidimensional systems in Examples 5.1.10 and 5.2.6. results in

=

-

Multi-Operator Systems

5.1

141

we introduced the formal transfer function -Q-1P E i/o-system kerL [P, Q] C C'+P of DDEs. In the same way transfer function can (and will) be introduced for the general poly7

In Theorem 4.2.3

R(s, z)P11 the formal

nomial

of

an

setting of this

section. In this context the

following

situation will

play

a

crucial role. 5.1.4 (Transfer Functions) K[z, s] be any polynomial ring in 1 + 1 indeterminates. Then the space K(z, s) carries a natural K[z, s]-module structure given by multiplication.

Example Let

A

=

The same is true for the space N

I

fis

N E

Z, fi

c

K(z)J

i=-00

of formal Laurent series in s-1 with coefficients in the field spaces

are

divisible

K[z, s]-modules,

thus

our

abstract

K(z). Clearly,

both

approach applies. For

setting, behavioral theory coincides with the transfer function framework we will make precise in Example 5.1.8.

this as

Remark 5.1.5

Throughout this section, it does distinguished. Even more, if xi, over K, the same is true for yi, .

play

not

x1+1

...'

.

yl+ 1, where

,

.

(yi,..., y1+1T

having one of the variables algebraically independent elements

any role

are

=

A(xi,...' x1+1T

+ b

and b E K1+1. In particular, K[yl,...,Yl+,] instance, in Example 5.1.2, the polynomial ring can also be presented as R[D,ol 1,...,ol 1], where we replaced the shift operators by the corresponding difference operators and changed the ordering of the indeterminates. In this case, the list of,operators (zl,...,zi,s) reads as 1 is the distinguished operator. The 1, (D, a, a, 1), so that s al procedure of the next section would then result in a first-order realization with respect to the last difference operator ol 1, provided that certain necessary for

some

A

E

K[xl,...,xl+l].

G11+1(K)

For

-

-

-

.

.

.

,

-

=

-

-

conditions are satisfied.

general case of a divisible K[z, s]-module A. For R E kerA R is a submodule of Aq and can be regarded as an abstract version of a behavior of a dynamical system, generalizing those of Definition 4.1. If A is a function space, it consists of all trajectories in Aq that are governed by a system of (higher order) equations, e. g., delay-differential equations, partial differential equations, or partial difference equations in case of the examples above. In the general case, for instance in Example 5.1.4, there is no interpretation of kerA R in terms of trajectories.. In the following definition we introduce these systems formally along with the desired first-order representations. Let

us

return to the

K[z, S]pXq

the kernel

5 First-Order

142

Representations

Definition 5.1.6 Let R E K [z,

(a)

s]

"

('+P) be any matrix.

The module

fa

E

A+P I Ra

system)

in

A+P.

kerAR is called

(b)

a

behavior

(or

The behavior

kerA R,

there exists

number

a

(A, B, C, E)

a

or n

=

simply

R,

is said to be

realizable,

if

E N and matrices

K[Z]nxn

E

the matrix

01

K[Z]nxm

x

x

K[Z]pxn

x

K[z]Px'

such that

kerA R

8 '(A, B, C, E)

=

(5.1.2)

where

1(yU)

BA'(A, B, C, E) In

case

such matrices exist,

A-+P 3 x

E

we

call the

G

An

EUI

sx=Ax + Bu :

Y=Cx

+

quadruple (A, B, C, E)

a

(5.1-3) realization

of kerA R. The system

is said to

be

a

8 '(A, B, C, E)

sx

=

Ax + Bu,

y

=

Cx + Eu

first-order representation of kerA R and the behavior (5.1.4). The length n of

is called the external behavior of

the internal vector

x

is called the dimension of the realization

The matrix C(sI-A)-'B+E E

function of

The term

(5.1.4)

(5.1.4)

or

of

K(z, s)Pxl

(A, B, C, E).

issaid to be theformal transfer

(5.1.3).

first-order representation

or

first-order system refers, of

course,

to

the fact that the first equation in (5.1.4) is linear with respect to the operator induced by s. As has been discussed for DDEs in the introduction to this Chap-

ter, itdoes not make

sense

cases, where the matrices

A few remarks

are

to call

are

(5.1.4)

a

state-space system. Only for certain

constant, this

might be appropriate.

in order.

Remark 5.1.7

(i)

It is not clear whether each external behavior of

a

first-order system does

kernel-representation, always be eliminated. We will see examples above except possibly for delay systems with delays, where this is unknown. admit

a

words, whether latent variables can below that this is indeed the case for the

in other

noncommensurate

5.1

(ii)

Remember the notions of free and

Multi-Operator Systems

maximally free variables of

143

a

delay-'

differential system from Definition 4.2.1. These concepts generalize naturally to the context of operators acting on A and can be applied to the

(5.1.3).

A on A' it is immediate surjectivity of sI that each for u E A' there exists free, meaning the such that R. For AP E examples 5.1.2 5.1.4, again ker.A (u' y with the possible exception of systems with noncommensurate delays, the variables u are even maximally free, so that the last p variables constibehavior

Rom the

that the variables

-

u are

-

,

tute the

outputs of the system;

see

the discussion below. We know from

delay-differential systems of Chapter 4, that this implies that R'has rank p, see Theorem 4.2.3. That means that the number of outputs equals the

the number of

erality. However, rank

row ces

independent equations. Again,

this will be true in

more

procedure in the next section applies only kernel-representation, meaning that we are restricted to

the realization

R E K [z,

s]PI ('-4-P)

to start with. Put another way,

we

will

gen-

to

full

matri-

assume

in

Section 5.2 that the system is governed by exactly p linearly independent equations. Except for the case of transfer functions and systems with commensurate delays, this restriction is indeed crucial: since K [z, S] is not a principal ideal domain, it is in general not possible to eliminate lineaxly dependent rows of R without.changing the associated behavior kerA R, see Example 5.1.10 below.

(iii)

In accordance with tion

4.2.1),

we

our

definition of

always place

of the external variables; restrictive point of view.

Let

us

input/output systems (see

the free variables into the first

see

also Remark 4.2.2 for

discuss the definition for the list of

ext

(Y) U

(A, B, C, E) =

where

-Q-1P

=

c-

Am+P

m

components

comment

on

this

examples above.

Example 5.1.8 (nansfer Functions) Consider again Example 5.1.4 where A is either K(z, s) case, the external behavior of (5.1.4) is simply

B

a

Defini-

y

=

(C(sl

-

or

K(z)((s-1)).

In this

A)-'B + E)u

kerA [P, Q],

C(sI'- A)-'B

+ E is any factorization of the formal trans-

polynomial matrices (which, of course, exists). Thus, the external behavior B (A, B, C, E) admits a full row rank kernel-representation [P, Q] E K [z, s]Px 4+P). Obviously, for this special choice of A, realizing a behavior kerA [P, Q] is the same as realizing the rational function -Q-1P, that is, as finding matrices (A, B, C, E) satisfying -Q-1P C(sI A)-'B + E. Note also that in this case u is maximally free. fer function into

ext

=

-

5 First-Order

144

Example

Representations

(Delay-Differential Systems)

5.1.9

In the situation of

D and a,, Example 5.1.2, where s al are shift operators the first-order lengths -rl ......rl, system in (5.1.4) reads as =

of noncommensurate

E A,c'x + E'B,o,'u, vEN1

Y

VENI

E'C,,o,'x + 1]'E,,o,'u,

=

VEN'

VEN'

where

the notation a'

we use

:=

al"' 1

o

o

...

a,"

and A Z/ ,

1

B, C, and E,,

are

constant matrices with entries in R.

If 1

=

1,

we

is in fact

a

know from Theorem

4.4.1(a)

137 (A, B, C, E)

C E

behavior in the

that the external behavior

0 1

sense

(kerA [sI

-

of Definition 4. 1.

A,

-BI)

Moreover,

we

will

see

in

5.3.1 that it always admits a kernel-representation kerA [P, Q] where C- 'HP' ('+P) and Q is nonsingular. In particular, u is maximally free, see

Proposition

[P, Q]

Theorem 4.2.3. It remains

an

open

question whether similar results

noncommensurate

delays, cf. [127',

Example

(Multidimensional Systems)

5.1.10

Let A be any of the spaces in

p.

234]

Example

a

[41,

are

true for

systems with

3.1].

Sec.

5.1.3 with the

structure. Then each external behavior B

mits

and

"

A

corresponding moduleB, C, (A, E) of a system (5.1.4) ad-

kernel-representation of

system. This

can

be

seen as

rank p, the number of output variables y in the follows. Define the matrix

sl -A -B M:=

IM

0 -

Since each submodule of K[z,

[Y, P, Q]

E K [z,

s]

s]'+'+P is finitely generated, for

some

T

kerK[z,s]M It follows rk rk

0

=

p.

=

p

(5-1.5)

.

E_

C


(i) is trivial and (ii) == ' (iii) as (i) =: , (iii) are simple consequences of the Binet-Cauchy formula for as the minors of matrix products. SKETCH

PROOF: The

OF THE

well

The assertion

lp,q,

(iii)

= ,

(i)

M(p), of M

by Mp, thus det Mp

see

such that

EPE-Tp,q CAP)

K[z, S]qxp

where

Ep

with indices p

rows

are given by corresponding p x p-submatrices M(p). By assumption there exist numbers cp E K

be

can

seen as

follows: The minors of M

Definition 3.2.6. Denote the

P E

Then C constitutes

cz =

a

=

E pEl cpEp adiMp E identity 1p sitting on the elsewhere, hence MEp Mp.

1. Define the matrix C

=

:=

P,,

Kqxp is the matrix with the

(pi,

right

.

.

.

,

pp)

and

zeros

=

inverse of M.

The remaining implication (i) = - (ii) as well as the alternative formulation is the celebrated result of Quillen/Suslin, see [67, pp. 491]; we also want to mention [69]

for

an

algorithm computing. a unimodular completion.

PROOF

OF

PROPOSITION 5.1.11:

(a)

Rom

(5.1.2)

we

El

will first derive the iden-

tity M:=

QCadj(sI

-

A)B

+

det(sl

-

A)QE

+

det(sl

-

A)P

=

0.

(5.1.9)

0 for all u E A'. fact, by divisibility of A it is enough to show that Mu Thus, let u c A' be an arbitrary element and pick x G A' such that Bu Cx + Eu. Then Pu + Qy 0 and one (sl A)x; see Lemma 5.1.1. Put y easily verifies In

=

=

-

=

=

5.1

Mu

hence

(5.1.9)

det(sl

=

A) (QCx

QEu

+

Pu)

+

=

as an

rank, Lemma

equation

3.2.7

yields

(b)

0,

C(sI

over

det

the field

Q :7

0'

A)-'B + E

-

0 and

K(z, s). Since both matrices (5.1.8) is established.

Write again R [P, Q]. By Theorem 5.1.12, the matrix matrix unimodular to a completed

[-X U2] UI

c

R

and the assumptions

I

as

the matrix

91 -A -B

U1 U2 -X R

0

IM

C

E

Uj[sI-A,-B] +U2

be

identity

rIP1

P 0

-

:=

can

Gl,+m+p(K[z 8])

be rewritten

can

have full

[-X, R]

=

where T

147

follows. This in turn implies

[P' Q1 considered

-

Multi-Opeiator Systems

I

_

01] [CE

E

K[z, .] (n+m) x (n+m)

is

nonsingular.

Hence

(U),

sI -A -B

0 G

Y

B t (A, B, C, E)