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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
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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)