Heide
Gluesing-Luerssen,
Linear
Delay- Differential
Systems with Commensurate
Delays: An Algebraic Approach
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
-
CIP-Einheitsaufnahme
Gltising-Ltierssen, Heide: delay differential systernswith commensurate'delays : an algebraic approach / Heide Gluesing-Lueerssen. Berlin; Heidelberg; New York; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Tokyo : Springer, 2002 (Lecture notes in mathematics ; 1770) Linear
-
ISBN 3-540-42821-6
Mathematics
Subject Classification (2000): 93CO5, 93B25, 93C23, 13B99, 39B72
ISSN 0075-8434 ISBN 3-540-42821-6
Springer-Verlag Berlin Heidelberg New York
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Preface
delay-differential equation was coined to comprise all types of differequations in which the unknown function and its derivatives occur with
The term ential
various values of the argument. In these notes we concentrate on (implicit) linear delay-differential equations with constant coefficients and commensurate
point delays. We present
an
investigation of dynamical delay-differential
sys-
tems with respect to their general system-theoretic properties. To this end, an algebraic setting for the equations under consideration is developed. A thorough
purely algebraic study shows that this setting is well-suited for an examination of delay-differential systems from the behavioral point of view in modern systems theory. The central object is a suitably defined operator algebra which turns out to be an elementary divisor domain and thus provides the main tool for handling matrix equations of delay-differential type. The presentation is introductory and mostly self-contained, no prior knowledge of delay-differential equations or (behavioral) systems theory will be assumed. people whom I am pleased to thank for making this grateful to Jan C. Willems for suggesting the topic "delaydifferential systems in the behavioral approach" to me. Agreeing with him, that algebraic methods and the behavioral approach sound like a promising combination for these systems, I started working on the project and had no idea of what I was heading for. Many interesting problems had to be settled (resulting in Chapter 3 of this book) before the behavioral approach could be started. Special thanks go to Wiland Schmale for the numerous fruitful discussions we had in particular at the beginning of the project. They finally brought me on the right track for finding the appropriate algebraic setting. But also later on, he kept discussing the subject with me in a very stimulating fashion. His interest in computer algebra made me think about symbolic computability of the Bezout identity and Section 3.6 owes a lot to his insight on symbolic computation. I wish to thank him for his helpful feedback and criticisms. These notes grew out of my Habilitationsschrift at the University of Oldenburg, Germany. The readers Uwe Helmke, Joachim Rosenthal, Wiland Schmale, and Jan C. Willems deserve special mention for their generous collaboration. I also want, to thank the Springer-Verlag for the pleasant cooperation. Finally, my greatest thanks go There
work
are
a
number of
possible.
I
am
VI
Preface
only for many hours carefully proofreading all making helpful suggestions, but also, and even more, for and so patient, supportive, being encouraging during the time I was occupied with writing the "Schrift". to my
partner, Uwe Nagel,
these pages and
Oldenburg, July
not
various
2001
Heide
Gluesing-Luerssen
Table of Contents
1
Introduction
2
The
Algebraic
3
The
3.1
Algebraic Structure Divisibility Properties
3.2
Matrices
3.3
Systems over Rings: A Brief Survey Nonfinitely Generated Ideals of Ho The Ring H as a Convolution Algebra Computing the Bezout Identity
3.4 3.5
3.6 4
5
................................................
Framework
over
Ho
..................................
of
Wo
,
............................
25 35
.........................
43
.....................
45
......................
51
Delay-Differential Systems
4.1
The Lattice of Behaviors
4.2
Input/Output Systems
4.3
Transfer Classes and Controllable
4.4
Subbehaviors and Interconnections
Assigning
4.6
Biduals of
..........
59
.....................
73
..................................
76
....................................
89
Systems
Nonfinitely Generated
...................
.........................
the Characteristic Function
First-Order
.......................
Ideals
5.1
Representations Multi-Operator Systems
5.2
The Realization Procedure of Fuhrmann
5.3
First-Order Realizations
5.4
Some
.....................
9&
104 115 129
................................
135
...................................
138
.....................
148
...................................
157'
...................................
162
......................................................
169
...........................................................
175
References Index
23
.........................................
..................
4.5
7
......................................
The
Behaviors of
I
Minimality Issues
Introduction
I
equations Delay-differential (DDEs, for short) arise when dynamical systems modeled. Such lags might for instance if some are being occur time-lags time is involved in the system or if the system needs transportation nonnegligible
with
a
amount of time
certain
feature
of
a
to
system
sense
information is that
time-lags
with
and react
on
dynamics
the
The characteristic
it.
at
a
certain
time
does
only depend on The dependence on the past can that of a constant for instance the a so-called retardation, point delay, describing reaction time of a system. More generally, the reaction time itself might depend on time Modeling such systems leads to differentialdifference (or other effects). also called equations, equations with a deviating differential argument, in which the instantaneous
not
of the system but also on some past take various shapes. The simplest type is
state
values.
the
unknown function
various
time
and its
instants
if the process under time interval. a certain
t--rk.
In this
theory
case
equations, the term
a
for
distributed
with
their
values
respective
at
on
form of past dependence arises the full history of the system over
ma*matical formulation instance integro-differential delay, as opposed to point
leads
general In equations.
delay,
to
has been
use the term type of past dependence. Wewill consistently delayfor differential differential'equation equations having any kind of delay involved.
coined
All
for
occur
different
depends
investigation
functional-differential control
derivatives
A completely
the
this
delay-differential
infinite-dimensional
twofold
equations systems.
way.
On the
as
abstract
described
The evolution
hand,
above
fall
of these systems the equations can, in certain
in
the
can
category
be described
of in
be circumstances, equations on an infinite-dimensional space. of all initial The space consists which in this case are segbasically conditions, interval of appropriate ments of functions over a time length. This description leads to an pperator-theoretic of the framework, well suited for the investigation of these systems. For, a treatment of DDEs based on funcqualitativeIbehavior tional methods we refer to the books Hale and Verduyn Lunel [49] and analytic Diekmann et al. [22] for functional-differential and to the introducequations linear tory book Curtain and Zwart [20] on general infinite-dimensional systems in control functions theory. On the other hand, DDEs deal with one-variable and can be treated with "analysis to a certain extent techon W' and transform of DDEs in this spirit to the books Bellman we refer niques. For an investigation, and Cooke [3], Driver and Norkin and [23], El'sgol'ts [28], and Kolmanovskii a
formulated
one
differential
H. Gluesing-Luerssen: LNM 1770, pp. 1 - 5, 2002 © Springer-Verlag Berlin Heidelberg 2002
Introduction
2
1
Nosov
[65]
at
and the references
analyzing with
time
All
t4erein.
the
the an
Our interest
behavior of their qualitative emphasis on stability theory.
monographs mentioned so far aim most of the respective equations,
DDEs is of
Our goal is an investigation nature. a different of by DDEs with respect to their general control-theoretic propTo this end, we will adopt an approach which goes back to Willems erties. (see for instance [118, 119]) and is nowadays called the behavioral approach to sysIn this framework, tems theory. the key notion for specifying is the a system of that system. This space, the behavior, can trajectories space -of all possible be regarded most intrinsic as the part of the dynamical system. In case the it is simply the corresponding dynamics can be described by a set of equations, solution all fundamental theory now introduces space. Behavioral system properties and constructions in terms of the behavior, that means at the level of the of the system and independent of a chosen representation. In order trajectories to develop a mathematical be able to deduce these properties must theory, one from the equations in governing the system, maybe even find characterizations terms of the equations. For systems governed by linear time-invariant ordinary differential this has been worked out in great detail and has led to equations the book Polderman a successful and Willems e. theory, see, g., [87]. Similarly for multidimensional differential or discrete-time systems, described by partial difference much progress has been made in this equations, see for direction,
systems
instance
troller, this
in
governed
most
framework.
Wood'et
[84],
Oberst the
important A controller
[123],
al.
tool
and
theory, system itself,
forms
a
and the
interconnection
of
leads
the intersection
of the two respective
to
a
Wood
of control
to-be-controlled
[122]. can
thus
The notion
of
a con-
also a
be incorporated family of trajectories,
system with
a
in
simply
controller
behaviors.
The aim of this
monograph is to develop, and then to apply, a theory which studied dynamical systems described by DDEs can be successfully from the behavioral of order In view. to this it is unavoidable point goal, pursue the relationship to understand between behaviors and their -describing equain full tions detail. For instance, we will relation need to know the (algebraic) between two sets of equations which share the 'same solution space. Restricting shows that
to
a
reasonable
gebraic systems coefficients
setting, we are
class
well
going
of systems, this for further
suited to
study
and commensurate
consists
can
indeed
be achieved
investigations. of
(implicit)
and leads
to
an
al-
To.be precise, the class of linear DDEs with constant
delays. The solutions being considered are in the space of C'-functions. all this in algebraic Formulating terms, one obtains where a polynomial a setting acts on a module of ring in two operators functions. However, it turns out that in order to answer the problem raised but rather has to be enlarged. above, this setting will not suffice, More specifcertain distributed ically, delay operators (in other words, integro-differential in our framework. These distributed equations) have to be incorporated delays have a very specific feature; just like point-delay-differential operators they are determined in fact they correspond to certain by finitely rational many data, point
1
Introduction
setIn order to get an idea of this larger in two variables. algebraic of scalar DDEs are needed. Yet, some properties only a few basic analytic indeed the careful to see that this provides are necessary algebraic investigations allows draw it In fact, framework. one to far-reaching subsequently appropriate the behavioral even for approach systems of DDEs, so that finally consequences,
functions
ting,
can be initiated.
of
As
which in
algebra
a
consequence,
the
is
fairly
our,
opinion
monographcontains interesting
by
a
considerable
part
itself.
remark that delay-differential systems have already been studpoint of view in the seventies, algebraic see, e. g., Kamen [61], These have initiated the theory of SysMorse [79], and Sontag [105]. papers of dynamical towards which developed tems over rings, an investigation systhe itself. in evolve the tems where Although this point of view trajectories ring whenever leads away from the actual system, it has been (and still is) fruitful the Furof are investigated. ring concerning solely operators system properties and difficult thermore it has led to interesting problems. purely ring-theoretic of sysit is not in the spirit Even though our approach is ring-theoretic as well, for simply the trajectories live in a function tems over rings, space., Yet, there between the theory of systems over rings. and our apexist a few connections proach; we will therefore present some more detailed aspects of systems over
We want ied
rings
to
from
an
later
in the
book.
of the book. Chapgive a brief overview of the organization the class of DDEs consideration under along with introducing A above. and relation bementioned the algebraic simple setting very specific differential and to tween linear a study equations DDEs'suggests ordinary ring of operators of point-delay-differential as certain as well operators consisting distributed delays; it will be denoted by H. In Chapter 3 we disregard the inthe ring 'H from a and investigate as delay-differential operators terpretation 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 transformain that that matrices with entries ring behave under unimodular Wenow ter
proceed
2 starts
Euclidean
domains.
The fact
that all operators in H question whether these data (that is to say, a desired operator) can be determined exactly. Wewill address this problem by discussing of the relevant constructions symbolic computability in that ring. of H as a convolution we will Furthermore, present a description of distributions with compact support. In Chapter 4 we fialgebra consisting nally turn to systems of DDEs. We'Start with deriving a Galois-correspondence between behaviors and the modules of annihilating on the one side operators on the other. of Among other things, this comprises an algebraic characterization systems of DDEs sharing the same solution space. The correspondence emerges from a combination of the algebraic of 'H with the basic analytic structure of scalar DDEs derived in Chapter properties 2; no further analytic study of tions are
like
to
with
matrices
determined
over
by finitely
many data
raises
the
1 Introduction
systems of DDEs is needed.*
machinery
for
addressing Therein,
sections.
quent
purely
Galois-correspondence system-theoretic problems
The
the
of the basic
some
constitutes
of systems
concepts
an
studied
in
the
theory,
efficient subse-
defined
of trajectories, will be characterized of by algebraic properties will We equations. mainly be concerned with the notions of conand the investigation partitions input/output (including causality)
in
terms
the associated
trollability,
of interconnection
theory,
control
well-known
of systems. touches upon the central The latter concept of control. The algebraic characterizations the generalize
feedback
results
for
described
by linear time-invariant ordinary difequations. finite-spectrum assignment problem,well-studied in the analytic framework of time-delay systems, will be given in the In the final algebraic setting. Chapter 5 we study a problem which is known as in case of systems of ordinary realization differential If we state-space equations. cast this for DDEs, the problem amounts to context concept in the behavioral form finding system descriptions, which, upon introducing auxiliary variables, DDEs of first -order and of retarded explicit (with respect to differentiation) we aim at transforming type. Hence, among other things, implicit system deinto explicit first order DDEs of retarded ones. scriptions Explicit type form the simplest kind of systems within our framework. -Of the various classes of DDEs in the literature, investigated they are the best studied and, with respect to the most important The construction of such a description ones. applications, in other (if it, exists) takes place in a completely polynomial setting, words, the methods of this chapter are different no distributed delays arise. Therefore, from what has been used previously. As a consequence and by-product, the construction class of systems including even works for a much broader for instance certain differential A complete characterization, partial equations. however, of first order description, will be derived only for systems allowing such an explicit systems
A
ferential
new
-
of the
version
DDEs.
A
more
detailed
We close of the
description
of the
the
first
introduction
applications
with
occurred
MacDonald of Volterra
early
ship
forties
the
glected. At
in
is
given
in
its
re-
basically
the
several
and automatic
point the similarity similarities occasions
[22]
unnoticed
DDEs got much at
existing
this
remarks
delays
the
in
great
reader
"tention steering.
feedback
interest
familiar
on
population
and Diekmann et al.
Because of the
structural out
[70],
remained
stabilization
tems
chapter
of DDEs. One applications dynamics, beginning with the the 1920s. Since population models are in this area and refer to the books Kuang [66],
some
models of Volterra in predator-prey not discuss we will general nonlinear,
the
of each
contents
introduction.
spective
with
in
and the references for
almost
therein. decades
The work
only in Minorsky [77] began to study He pointed for these sysout that mechanism can by no means be necontrol theory during that time and two
and
when
the
paper
[84]
of
Oberst
will
notice
the
of systems of DDEs to multidimensional systems. Wewill point and differences between these two types of systems classes on later
on.
1 Introduction
the
decades
rapid
work
of
Minorsky DDEs; for
of Kolmanovskii
239].
led
to
other
more
details
and Nosov
[65]
applications about
and
that list
and the
a
period of ap-
was Myschkis a [81] of a general equations and laid 'the foundations that appeared ever since theory of these systems. Monographs and textbooks include Bellman and Cooke [3], El'sgol'ts and Norkin [281, Hale [481, Driver [23], and Nosov [65], Hale and Verduyn Lunel [49], and Diekmann et Kolmanovskii A nice and brief overview of applications of DDEs in engineering al. [22]. can
plications
[23,
of
theory
preface
the
instance
the
of the
development
for
see
follow
to
Driver
in
pp.
introduced
of functional-differential
class
be found
the
in
book
Kolmanovskii
following examples of systems
the
who first
It
[65],
and Nosov
from
which
we
extract
and mixing processes natural time-lag arises
are engineering, because due a delay, needs to complete its job; see also Ray [89, Sec. 4.5] to the time the process function for an explicit form. Furthermore, example given in transfer any kind of system where substances, or energy information, (wave propagation in deep transmitted is certain to a being distances, experiences space communication) An additional time. time-lag might arise due to time-lag due to transportation
list.
In
chemical
standard
the time the
needed for
system
to
sense
certain
reactors
with
measurements
information
(ship stabilization) (biological models).
for
to be taken
or
it
A model
and react
on
delay equaengine, given by a linear system of five first-order in [65, variables can be found inputs and five to-be-controlled DDEsof neutral Sec. 1.5]. Moreover a system of fifth-order type arises as a linear model of a grinding Finally we would like to mention process in [65, Sec. 1.7]. model of the Mach number control in a wind tunnel a linearized presented in Manitius equations of first order with [75]. The system consists of three explicit but not in the input a time-delay only in one of the state variables occurring
of
a
turbojet
tions
with
channel.
three
In that
paper
Mach number is studied
the
problem of feedback and various
different
control feedback
for the
regulation
controllers
are
of the derived
by transfer function methods. This problem can be regarded as a special case of the finite-spectrum also be solved within assignment problem and can therefore 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 Delay-Differential 2 The
Framework
for
Equations
specific class of delay-differential equations we In this way are some basic, yet important, properties. make clear that, and how, the algebraic we hope to approach we are heading for depends only on a few elementary of the equations under analytic properties consideration. The fact that we can indeed proceed by mainly algebraic argufrom the structure ments results of the equations under consideration together with'the in. To be precise, restrict to we will type of problems we are interested linear with coefficients and constant commensurate delay-differential equations We are not aiming at solving these equaon the space C' (R, C). point-delays chapter
In this
we
interested
tions our
in
introduce
the
and derive
and expressing the solutions -in terms of (appropriate) initial data. For will suffice it know that the solution to DDE of a purposes (without space
conditions),
initial
"sufficiently polynomials
the
e.
kernel
of the
delay-differential
associated
operator,.
In essence, we need some knowledge about the exponential solution defined space; hence about the zeros of a suitably
function
order
in
the
in
characteristic
Yet,
L
rich".
is
in the
complex plane.
by algebraic
the
has to be appropriate setting handle also to driving goal systems of DDEs, in other words, matrix equations. In this chapter we will develop the of delay-differential a ring algebraic context for these considerations. Precisely, not operators acting on C1 (R, C) will be defined, only the pointcomprising induced by the above-mentioned delay differential operators equations but also distributed certain delays which arise from a simple comparison of ordinary differential and DDEs. It is by no means clear that the so-defined equations for studying operator ring will be suitable systems of DDEs. That this is indeed the case will turn out only after a thorough algebraic study in Chapter 3. In the with introducing that ring and providing present chapter we confine ourselves results about DDEs necessary for later some standard In particuexposition. under consideration lar, we will show that the delay-differential are operators on C1 (R, C). surjections found
first.
As the starting ear DDEwith
equation
to
pursue
force
The
point constant
of
our
in this
means,
direction
investigation,
coefficients
is
let
our
us
consider
and commensurate
of the type
H. Gluesing-Luerssen: LNM 1770, pp. 7 - 21, 2002 © Springer-Verlag Berlin Heidelberg 2002
a
point
homogeneous, delays, that
linis
an
2 The
Algebraic
Framework N
M
EEpijf( )(t-jh)=O, i=0
where
N,
delays
involved.
ME
length,
from
No, pij For
now on we
purposes
our
which
above reads
R, and h > 0 is the smallest of delays are integer multiples
c
Hence all
commensurate.
unit
tER,
j=0
it
suffices
to
easily be achieved only be concerned
will
assume
the
with
be important for R. Moreover,
focus
the solution
on
The choice
C
=
C'
:
=
is
ff
C
hence
a
of initial
(R, C),
is
satisfiedl.
1 (2. 1)
L
is considered
any kind
hence
that'equations
(N
=
differential
briefly
about
think
Let
us
requirement
the
minimum amount
for
E
that
solution
unique t
cover
particular
in
short)
for
for
as
'C is
invariant
corresponding ring of larger classes of functions be discussed occasionally
well
linear as
pure
time-invariant
delay equations
0).
the
a
(2.1) (ODEs,
of the type
equations
full but
the
over
In a certain delay-differential operators. way, however, be incorporated in the algebraic approach; this will the book. throughout
Observe
the
on
conditions
on
convenient,
very
module
can
ordinary
equation
(2.1)
equation-
C'
algebraically
shift,
and
E
the
imposing
space in
A C)
differentiation
not
we are
B
under
and the
I
tGR.
that
setting
our
axis
rather
=
j=0
i=0
will
h
delay to be of Therefore,
axis.
M
EEpjjf(')(t-j)=0, time
the time
case
point h, thus
as
N
It
of the
constant
the smallest
by rescaling
can
length the
[0, M],
and M is the
initial
solutions
'of
initial
(if any).
It
conditions be
data is
for
smooth,
it
should
natural
to
be
be in order
require
fo is some prespecified delay appearing in (2.1). amounts to solving the initial
where
largest
(2.1).
Equation
should
that
function
Disregarding
intuitively
(2.1)
for
f satisfy on
the
clear to
what
single
f (t)
=
interval
out
fo(t) [0, M]
on the finding a solution full time axis R value problem in both forward and backward direction. It also fails This, is, of course, not always possible. if with an arbitrary smooth initial one starts i. e. fo C- C' Q0, M], C), condition, 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 this will be shown in Proposition 2.14. The solvability of exists; this restricted initial value problem for the quite general equation (2.1) rests on the fact that we consider so that we have a sufficient amount of C'-functions, of the initial condition differentiability fo, necessary for solving the equation on
Then
the whole of R. Remark 2.1 It
is crucial
mensurate
for essentially delays. As it
all
parts
of
turns
out,
the
our
work to restrict
occurrence
to
DDEs with
of noncommensurate
com-
delays
Algebraic
The
2
Framework
delays of length 1 and V2_ or -7r) leads to serious obstacles preventing an approach similar to the one to be presented here; see [47, 109, 111, 26]. At this point in the general want to remark that we only case the according which will be derived properties operator ring lacks the advantageous algebraic These differences will be pointed for our case in the next chapter. out in some in later more detail chapters (see 3.1-8, 4.1.15, 4.3.13).
(like
e.
g.
algebraic
Remark 2.2 advanced
retarded PNO 0
4].
distinguishes
one
These notions
type.
(2.1),
say
of DDEs
theory
the
In
with
occurs
if PNO : 0 and PNj
a
describe
delayed
0 and PNj 0 0 for some
argument. 0 for
=
This classification
j
>
of
equations whether
j
=
1,
Precisely, M; .
.
.
,
0, and advanced
and retarded, neutral, in, highest derivative Equation (2.1) is called
the
not
or
said
it
is
in
all
other
if
be neutral
to
cases,
[28,
see
problems in forward how much differentiability direction. of the initial Roughly speaking, it reflects for (2.1) being solvable condition in forward on [0, M] is required see direction; the results for instance on p. [3, Thms. 6.1, 6.2, and the transformation 192]. with infinitely differentiable Since we are dealing functions and, additionally, these notions are not requite forward and backward solvability, really relevant p.
for
our
Let
us
the
(2.1)
Equation
rewrite
now
shift
the forward
af (t) and
when
:=
f (t
1),
-
in
f
where
differential ordinary 0, where
is a1unction D
operator
in
the
two
commuting B
For notational
reasons,
which
ker
will
corresponding
operators.
defined
on
d, Equation
=
dt
R,
(2.1)
reads
as
M
1: 1: i=0
a polynomial simply
of the
terms
N
is
value
length
of unit
a
p(D, a) is
initial
solving
purposes.
Introducing
p(D, u)f
is relevant
pij D'ai
(2.2)
j=0
D and
operators
p(D, a)
C
become clear
a.
The solution
(2.3)
L. in
space
a
moment, it will
be
conve-
polynomial ring R[s, z] algebraically independent elements s and z at our disposal. (The names chosen for the indeterminates should remind of the Laplace transform s of the differential operator D and the z-transform of the shift-operator in discrete7time Since the shift U is a systems.) be advantageous to introduce Laurent on L, it will even the (partially) bijection polynomial ring nient
to have
R[s,
an
z
abstract
Z-1
with
N i=O
pijSY j=m
Tn, MEZ N E=
No, pij
E R
10
Algebraic
2 The
Associating cluding
with
Framework
each Laurent
possibly
R[s, (of
polynomial
shifts)
backward
we
z-1]
z,
delay-differential embedding
the
EndC (,C),
)
(in-
operator
the ring
obtain
p
p(D, o,)
)
i
(2.4)
then the operator polynomial, p(D, 0') is not the the D and a are words, operator operators C). algebraically over R in the ring independent Endc(,C). Put yet another way, C is a faithful module over the commutative operator ring R[D, a, o-1]. if p is
course,
zero
a nonzero
Let
us
like
for
look
now
other
In
on
for
exponential
ODEs one has for
eA*
functions
in the
(NE E pjjDY) M
p(D, o,) (e A.)
i=O
solution
(2.3).
space
Just
A E C N
M
E.Y pjjA e-
(e\')
)
\j
,
j=M
i=O
e
A-
(2-5)
j=M
p(A, e--\)e"' Hence the
exponential
p(s, e-')
function
Before
providing
details
some more
only if
A is
is
it
function
we
of
function,
entire
an
of the
a zero
characteristic
polynomials,
exponential
on
by H(C) (resp. M(C)) on the full complex plane.
For
a
S C
subset
want to fix
H(C)
In
fl,
S
case
For q
=
0
.
q*
case
elements
f
.
.
01,
denote
,
:=
fj I
JA
denote
V(S)
variety
G
meromorphic) of all
the set
as
M(C)
entire,
we
call
the set zeros
f
all
E
func-
common
S}.
V (fl,
write
.
EN0 EM pijs'.zj j= j=
the
p(s, e_S) 0(s)
V(q*)
'
for
.
.
E
m
meromorphic
O(S)
is
the
=
0 for
simply
we
where p
by q*
=
M
H(C)
A e C.
I f (A)
finite,
is
the characteristic
E
C
E
fl)
,
R[s,
function
S
G
V (S).
for z,
z-']
given
and
by
C\V(O).
the characteristic
variety
and its
of q.
and A E C let
ord.\ (f
for
the
(resp.
of entire
EN0 EM Pijsie-i' j=
q*(s)
For
define
R(s) [z, z-'],
P
R[s]\f
In
the ring
S, thus
of
V(S)
(4)
the
2.3
zeros
(3)
if and
be called =
Denote tions
(2)
solution
a
will
notation.
Definition
(1)
is
therefore
0. Obviously, equation p(D, o,)f polynomial (or quasi polynomial).
delay-differential known as exponential the
some
e,\*
function
which
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
(1) of the next proposition of ODEs, the multiplicities
The
Algebraic
Framework
11
standard of DDEs. Just like in the theory zeros correspond to exponen characteristic monomials in the solution the tial space. As a simple consequence we include fact that delay-differential are surjective on the space of exponential operators polynomials. Part
for
Proposition
(1)
2.4
R[s,
Let p e For
k
ek,A
(t)
is
the
z-'] \10}.
z,
by ek,A
and A E C denote
No tke,\'.
p(D, u)ek,,% ,=o
In
(2)
particular,
ao,
is
polynomials a
:=
al+a
E
L the
exponential
monomial
(p*)()(A)ek-K,A. only if ord,\ (p*)
if and
surjective
a
B
ord,\(p*)
:
!
C with
(k)'
characteristic
the
E
ponential cisely, let
p(D, o)
C ker
ek,X
H(C) is called operator p(D, o). The operator p(D, o)
p*
E
Then
al+a
function
endomorphism. f ek,A I k E No,
=
span(C
0.
Then, for
:
0 such that
( 1=0
all
E B
el,,\
k.
>
of the
on
The function
delay-differential the
A E there
space
of
ex-
C}.
More pre-
exist
constants
+a
p(D, a)
E a,,
(2.6)
el,,\.
e,,,,\
r.
PROOF: verified
(1)
Let p
=
following
in the
(p(D, u)ek,.\)
pijs'zi
I:i,j
E
R[s,
z,
z-1].
The asserted
identity
is
easily
way:
di
(t)
Pij
[(t
Tt
i
_
j)k
e)(t-j)]
EP'j
10
di
dk
Tti dAk
(eA(t-j)
1,3
dk
(E pjjA'e'X(t-j)
dAk
)
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) (p*)
It suffices to establish on 1. (2.6). We proceed by induction (a) Then 0 c :by assumption. (A). For I =' 0 it follows from (1) that p(D, o) (c- 1 ea,,\) as desired. eo,,\,
Put
=
For 1 > 0 put
al+a
1+a)a
1 c
-1
1+a
p(D, o,)(al+ael+a,,\)
=
al+a
E r.=a
.
Then, by
(1 a)
virtue
of
(1),
+
K
el+a-r.,,X
=
el,,x
+
1:'bjej,,\ j=o
c
Algebraic
2 The
12
for
The
solely with
role
C.
G
exponential the equation
foregoing
same
bj
constants
some
involving suitably
Framework
By
ODEs, in the
that
sense
solution
bjej,,\
their
have preimages them
Combining
1.
-
El
play exactly
functions
characteristic
show that
in the
functions
the
ei,.\ with i < 1 + a the desired result.
yields
above
considerations for
as
induction
monomials
to the
correspond
zeros
the
exponen-
the to OI?Es is that complex plane unless it Since this property will be of central degenerates to a polynomial. importance for the algebraic about the setting (in fact, this will be the only information solution a short proof showing spaces of DDEs we are going to need), we include
tialmonomials
function
characteristic
how it in
(1)
(1)
z-1]
z,
exist
the characteristic
the
of
p* issues,
classical be
can
C,
C(I
< 00 4==> P
[88]
found,
see
(1) Letting
p
=
tion
suffices oo.
Theorem,
defined
a
much
[3, (2)
also
In
i=O
j =M
C(l
+
order
R[s,
z,
z-']
in
a
all
S
C
k E Z and
0
(C'
details
13].
about
As for
our
5y,
Pij
M
we are
the
R[s]\f
E
of the
location
dealing
not
01. zeros
stability
with
purposes.
we can
straightforwardly
estimate M
m. The function consequence, said to be the characteristic function of the operator 4.
As
G
k.
EJ
Remark 2.13
Notice of
that
the
latter
facts
did
we
consider
not
any
for
Such expansions of retarded equations
solutions
of solutions
expansions
polynomials.
exponential
do exist,
R+.
on
infinite
as
[102]
see
Wewill
and not
series
[3,
6],
Ch.
these
utilize
about the solution only case, where the full information space is 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. Wewill briefly discuss the issue of stability in Section 4.5, where we will simply quote the relevant from the literature. results since
needed,
the
is that
Weconclude
differential
697],
p.
our
where
it
elaborate
rather
shows what kind also
on
L.
stated
is
on
in
a
scalar
fact
This
much
methods.
However,
of initial
conditions
Earlier
in
we
chapter
we
briefly
more
general
would like
the
of
surjectivity
and
can
context
to prove
a
be found
delayin
[25,
and proven with version which also
imposed for the DDE (2.1). the method of steps, the standard
can
us
this
DDEs with well-known
is
the opportunity to present initial value problems for-solving
gives
cedure
considerations
operators
be
This pro-
of DDEs.
addressed
what
kind
of initial
data
should
single (2.1) unique f. Apart from that f has to be specified we suggested requirements, on an interval of length in (2.1). For instance, M, the largest delay occurring of a solution the pure delay equation 0 is determined the restriction of f completely by fo := f 1 [0, 1). But in order that f be smooth, it is certainly that the necessary specified
be
in
order
for
to
out
solution
a
smoothness
-
initial
condition'fo
In t
(v)
be extended to a smooth function on [0, (v) of all orders v E No at the endpoints (1) 0
can
f 0 (0) f other words, fo and all 1. This idea generalizes
derivatives
=
=
=
its to
derivatives
arbitrary
have to
satisfy
DDEs and leads
the
1] having equal of the interval.
delay equation
to the restriction
for
given
2 The
18
Algebraic
Framework
(2.11)
has to be compaticondition below, which simply says that the initial and also As DDE. ble with the given our approach neutral, comprises retarded, of advanced equations smoothness, requires order, and, additionally, arbitrary below. However, the profor the result as stated not find a reference we could of the proof given below cedure is standard and one should notice the similarity the book in those for part (1) with, e. g., [3, Thms. 3.1 and 5.21. In presented C' [a, b] := C' ([a, b], (C) as well as f M for f C- C' [a, b] the sequel the notation when taken at the endpoints a or b. refers, of course, to one-sided derivatives in
Proposition
2.14
po'
Let
q
0 :
0, and M>
(1)
For every
=
E
C'
[0, M]
(p(D, there
exists
f I [0,M]
(2)
f
If
an
(1)
interval
all
[a
f
g() (M)
for
all
0
po
:
pm,
L such
E
f I [k,k+M]
0 for
=_
some
(2.11)
No
E
v
p(D, o,)f
that
the map 4 is surjective
M
on
R, then f
k c
g
and
L. -=
0.
[a, b]
defined
(v))
(t)
0
g(') (t)
-
for
all
v
E
(2.12)
No
j=o
[a
+
M, b]
1]
which
given
be extended
can
(2.12)
satisfies in
the
in
unique way to a solution that 1 + M, b + 1]. (Notice
a
[a
on
-
is included
proposition
as
an
extreme
case
the
where
b=:M.)
end,
To this
--
Epj (D)oif
+
a=O and
(M)
consequence,
(t)
G
condition
initial
G R
-
1, b
t -
[s],
0
pj,
of f , we show: every fo C C' To prove the existence the condition satisfies of length b a > Mwhich
0
for
())
0
function
4 9 L satisfies
(p(D, u)f(v)) C'
a
zi,
pj
g E L
satisf ying
a)f
unique
a
As
fo.
(=- ker
PROOF: on
=
Ej' o
=
Furthermore,
1.
fo
0 1 where p let
7io
G
write
po
(s)
=0
consider
ai sz + Sr and M
po(D)f (t)
g(t)
=
1:
-
fo)
(D) ci
pj
inhomogeneous
the
ODE
(t)
(2.13)
1.
(2.14)
j=1
for
t G
[b,
b +
1]
with
condition
initial
j(v) (b) (If a
r
=
0, then
unique
po
solution
1 and c
C'
=
no
[b,
(v)
f0
(b)
1]
to
(b)
=
g (b)
pj
-
j=1
=
(2.13),
M
M
v
condition
initial
b +
for
(D) ai
fo)
0,...,
r
-
imposed). (2.14) and j
In any case,
is
satisfies
r-1
(b)
-
E ai j(') i=O
(b)
=
()
f0
(b).
there
is
Algebraic
2 The
Framework
19
(")
j(') (b) successively f 0 (b) for for t No. Therefore, E [a, b] and f, fo (t) by f, (t) satisfies f, (t) =j (t) for t E (b, b+1] is in C'[a, b+1] and, by construction, (2.12) In the same manner one can extend f, to a smooth solution on [a + M, b + 1]. of the ODE on [a 1, b + 1]; one takes the unique solution (2.13)
Differentiating all
and
(2.12)
using
shows
=
defined
the function
E
v
=
-
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
elementary
C'
xo is of class
the solution
to + 1. In order
to
ealculations
(2.15)
which satisfies
C[t]3
for
some
imposed
condition
For instance,
to] satisfying
xi
coefficient
(2.15)
for
[23]
in t >
is
discuss that
to + 1 is
on
b,
a,
(2.16)
C.
c
c
from
just
at
there
(to)
matrix
-
is
2X2 (to)
of
of view
x(t
single
one no
-
(to) : (2.15).
X3
-
1)
point
differentiable in
0-
where
only smooth In this particular being considered. example form for R by applying triangular elementary row our
point
are
we
have
83
0
0-
s
-2
0
--Z
8
1-
:=
2z 2s
1
0
0
0
1
0-
operator remains
from the =
fp
VR
and detV=
the
(D, or)
=
-2-
R(D, a)
explicitly
ker R,
derivatives,
for
S2+
too,
one can
of the form
-
Since
condition
some
singular
R,
V
(C3,
--4
Ct2
(-oo,
to achieve it is possible a indeed over R[s, z]; operations
-
1]
to +
to + 1 in all
(2.15)
solution.
of R
where
[to,
:
If the initial
=
C
on
Equation whole
t
initial
an
xo
in forward
conditions.
Ct2
2c)t
-
is due to the
study
now
No.)
+ Ct
X3
X2)
G
for
polynomial
backward
a
(Xl)
=
Of course,
(b
even
allow
not
solution
Let
b +
-
condition
[to, oo)
on
+ bt +
P (t)
might
(2.15)
version of by using the (corresponding is continuously differentiable on (to + 1, 00) Thm. 6.2]. the is of solution see [3, fact, (In
shown
is
function
a
a
1).
-
0_
2
direction
The solution
solvability,
differentiable
the interval
As
initial
in forward
that interval, (2.15) (to + k, oo) for each k 1] and satisfies (2.15) on [to, oo) and fulfills
on
[to,
every
_O
on
to + of class C' on
X(t
10 0
+
0.-
continuous
a
method of steps.
and satisfies class
X(t)
the purpose in [231 to present a system that is, nicely solvable but lacks backward solutions for most choices of initial
Indeed, solve
-000-
0-
0 0 -1
=
E
-2.
As a conon L3 V(D, u) is bijective unchanged under this transformation form. In fact, one obtains triangular
(C
form does not
it also exhibits space of (2.15), data can be imposed so that,
.
[t] 3I'p
is of the
form
(2.16)1.
(2.17)
only have the advantage of providing via its diagonal elements, where and of virtue by Proposition 2.14, forward
Framework
Algebraic
2 The
21
The We can go even one step further. solutions are guaranteed. is in fact a solubeing a finite-dimensional space of polynomials, This operator differential tion space of an ordinary can be determined operator. in (2.16). the linear equations governing the coefficients explicitly by exploiting show Elementary calculations and backward
(2.17),
kernel
R(D, a)
ker
R2 (D)
ker
=
53
0
0
s
-2
0
0
1
where R2
8-1
[44,
also
see
relating
tions
R2-R-1
in
227]
pp.
the
for
a
general
matrices
R(s, Z)3X3
method.
and obtain
UR
R2R-1
us
2z
easily
matrix
is
R2(D)
is obtained
Let
us
briefly
R(D, a)
from
via
draw
link
a
0
0
forward
the
to
transforma-
S.
2 and entries
unimodular
a
Ro. In Section 4.1 we will related like this. are always
the operator algebra ing the same kernel
-2
SZ
to have determinant
seen
simply
we
row
have to calculate
IS2 -2s+2-2z 2 S_1+Z
S3
This
the
2s
1
=
83 +2sz _S2 -2z+2Z2
way,
determine
end, R2 where
=
82+
-
U:=
Let
R and R2. To this
R[S]3x3;
E
see
that
solutions
Ro. Put another
in
row
transformation
discussed
over
matrices
operator
above.
shar-
Since
the
they pre0 in C1 ([to, 00), (C3) but only respect the serve the solution space of R(D, u)x of the solutions". A few time units have to elapse (the number depending "tails the in the matrices of and possibly amount of on the on z occurring degree the the for before solutions applying required differentiability transformations), of R(D, u)x This is 0. 0 turn into solutions of, say, R, (D, u)x exactly what transformation
matrices
the shift
V and U contain
operator
=
we
in
z,
,
=
observed
don't
=
(2.16)
and
(2.17).
about the example a first information det R .9 3. The by noticing'that D shows that the o solutions 3J3 equation (adjR) (D, a) immediately R(D, a) of degree at most 3, hence the solution of R(D, o,)f 0 on R are polynomials
Finally
we
should
C3 could
of R in
kernel
mention
that
this
in
have been obtained
=
=
=
space is finite-dimensional.
full
the in
solution
original
the
determinant
the
det R G
In this
space
equation of the
R(D, c7)f system
=
0.
in the
some more
information
that
about
sense
a
the
But not every polynomial matrix form. Fortunately, this can triangular
tion
a
matrices
having
entries
in
R,
as we
that
it
further
and determine
space of
in two variables
always
will
see
actually a sysadvantage for delay-differential
describes
form is of
triangular
solution
operator. to
even
of degree three general polynomial This idea, of course, applies whenever is in R[s] but fails in the general case
R[s, z].
tem of
getting
go
the
matrix
though the example is extreme ODEs, it should demonstrate
Even
could
one
case
by substituting
a
can
be achieved in the
next
be
with
chapter.
row
reduced
transforma-
Algebraic
3 The
Structure
of Wo
on the chapter. we will concentrate purely algebraic part of the theory of Ho. As the following will sections analyze the ring structure show, the but also operator ring Ho carries a rich algebraic structure, by itself, interesting nicely suited for an algebraic study of delay-differential and equations systems
In
this
and
later
thereof
The combination
of the two embeddings Ho 9 R(s)[z] and powerful tool for the upcoming investigations; the is a principal ideal domain, while H(C) is a Bezout domain (it is even known to be an elementary divisor domain, but for us the Bezout property will be the main tool). The inclusion Ho 9 H(C) has been exploited already once in Proposition 2.5(2), where we established that a proper delay-differential This fact in combination operator has infinitely zeros. with many characteristic of the finitely an easy handling denominator in many zeros of each possible will be a permanent ingredient 1 o C R(s')[z] for the arguments in Section 3.1. In that section the main results we derive about 'Ho. (The corresponding facts about the ring R, which is simply the localization (Ho), are readily derived.) On the one hand, it will be shown that Ho is a Bezout domain, that is, each Put another way, each two nonzero elements finitely generated ideal is principal. have a greatest common divisor can be expressed which, additionally, as a linear combination of the given elements with coefficients in Ro. On the other hand, also see that Ho is a so-called we will adequate ring, meaning that each element in a certain can be factored desired manner concerning its characteristic zeros. on.
H(C) ring R(s)[z] 7io g
In Section
chapter
and
domains. when
a
ring theory
domain.
triangular
Euclidean
be
to
3.2, general divisor
ementary admit
proves
This
diagonal
This
study
fact
is
says
applied essentially,
forms
in
a
to deduce that
that way
matrices
quite
be of fundamental
will
similar
lio
is
a
so-called
with
entries
to
matrices
in
el-
Ho over
in the next importance equations. Furthermore,
delay-differential to generalize the notions of greatest common divisors and least common multiples to matrices over Ro. As has just been indicated, the reason for focusing on matrix theory rather than general module theory over Ho is the fact that in the next chapter our main objects of study -will be systems of delay-differential hence operators which are equations, matrices over 'Ho or n. However, a translation and interpretation of the matrixtheoretic results into the language of general module theory will be given at the the
Bezout
we
property
end of Section
systems
of 'Ho will
of
be utilized
3.2.
H. Gluesing-Luerssen: LNM 1770, pp. 23 - 72, 2002 © Springer-Verlag Berlin Heidelberg 2002
Algebraic
3 The
24
In Section
3.3
order
will
it
study
generated
are
3.5
(R, C)
tributions
excursion
theory
into
is not
algebraic
the
ideals
structure
of HO follows
Section
in
HO has Krull-dimension
recall
we
of HO, 3.4. that
one,
meaning of H as
original
how it
a description Among other
all
is,
of the
things, prime
nonzero
the distributions
associated
can
with
also
as
an
be understood
of entire
algebra
Paley-Wiener
operator
an
be characterized
can
This
compact support.
suitable
a
the
and demonstrate
with
of H into
brief
maximal.
In Section
C'
a
of control
be shown that
ideals
on
take
area
further
to
nonfinitely
of Ho
the theory of systems over rings. related to our approach directly systems, it has been a helpful source for getting acquainted matrix-theoretic and their structures implications.
will
we
though this to delay-differential with various ring Even
In
Structure
of H will
the elements
as
functions.
algebra acting algebra of disan embedding
The structure
of
explicitly.
be given
required for question is posed whether the ingredients earlier be comcan actually objects introduced numerical on symbolic computability; puted. Wewill concentrate questions will calculations of the previous sections not be addressed. are Although thevarious work is necessary with respect in a certain sense constructive, some additional This is mainly due to the central role played by to their symbolic computability.
Finally,
in Section
the
calculation
the
zeros
suitable
3.6 the
of the
various
(transcendental)
of the
functions
Starting
ring elements.
R[s, e-']
in
functions
with
in,
say
of
for
the
Q[s, e-'],
of
construction
is
one
necessar-
Q. Interestingly
enough, it degree of of Besuch extensions, which, in case it is valid, enables symbolic computability of all relevant in HO and, consequently, constructions zout equations presented In this context it is also interesting in this chapter. to note that generically the of three or more polynomials difficulties for the Bezout identity computational ily
led
to
is
exactly
in
Q[s, z]
Wewill
transcendental
certain
Schanuel's
(still
do not arise make
Notation
use
field
due to the lack
of the
extensions
open) conjecture
about
of any
the transcendence
to be taken
common zeros
care
of.
following
3.1
Let
R be any commutative
(a)
By R'
we
denote
domain with
the
group
unity.
of units
of R.
We will
write
p
J."
q if
p di-
Any greatest common divisor of P1, pl c R (if it exists) will be denoted each expression conConsequently, by gcd,.(p1,...'p1). taining pl) has to be understood as up to units in the ring. gcd., (pi In the same way, a least common multiple (if it exists) will be denoted by pi). In case R R[s], we omit the index R. This will not cause lCMR. (P1 due to the specilic rings under consideration. any confusion vides
q in
R.
.
I...
I...
(b)
A matrix
,
=
is called
unimodular
if det U E R'
R\10}. The group of unimodular while E,,(R) is the subgroup
if det U E
nonsingular denoted by Gl,,(R),
.
)
I
U E R"'
.
and U is said matrices
over
to be
R is
Divisibility
3.1
E, (R)
G1, (R) I U is
-elementary
where
matrices
a
linite
understood
are
of
product
matricesl,
elementary
in the usual
25
sense,
e.g.,
see,
[55,
338].
p.
(C)
UE
Properties
The rank of matrix
a
independent
to be the rank
of R. Hence it columns
or
rows
is defined
q
field
the quotient
over
linearly
of
ME RPx
matrix
a
in
denotes
of Mregarded
the maximal
the matrix
as
number
M.
over R P, Q E RP left equivalent (resp. equivalent) Glp(R) (resp. U E Glp(R) and V (=- Glq(R)) such that UP Q (resp. UPV Q). Right equivalence is defined accordingly. For a polynomial x,,] let deg,,,, p denote the degree of p as p G R[xl,..., where p E R[s, z] and in xi. For q a polynomial E R(s) [z], R[s],
(d)
We call
two matrices
there
if
x
U C
exists
=
=
(e)
q
=
degs
define
(f)
For
degs
p
-
indeterminate
an
formal
q
=
Laurent
x
series
in
deg
R((x))
R the notation
over
x
coefficients
with
R, that
Z,
ri
0"
Erix'l
R((x))
L G
for
stands
in
the
of
ring
is,
Rj.
Cz
i=L
R xj
Likewise, series
:=
Divisibility
3.1
In this
section
tions.
To
in the
ring
f EjZo rix'
we
want to we
lay
comprise
division
with
remainder
the fact
that
two entire
establish
Rj'
denotes
down the foundations
present
Ho
can
E
ri
the
of formal
ring
power
Properties
begin with,
These rules
I
R.
over
the
some
q E
R(s)[z]
certain
types
close
to that
functions
same
basic,
for
our
I q*
E
algebraic rules
yet important,
for
considera-
calculating
H(C)J.
of factorizations in Euclidean
as
well
domains.
as
some
kind
of
Furthermore, using common divisor, we
always have a greatest Along with the division with remainder one Bezout domain, that is, the greatest common divisor do
for 'Ho.
result
that Ho is a of the given elements. On the other expressed as a linear combination ideal domain. A combination hand, it is a simple fact that Ho is not a principal of the embeddings 'Ho g R(s) [z] and Ho 9 H(C) will show that Ho finally in a sense to be made precise admits adequate factorizations below. obtains
even can
be
Remark 3.1.1
sequel
In the
functions
f,
g c
is
H(C)
we a
will
Bezout
make
frequent
domain
have a-greatest
[82,
use
p.
of the fact
136]. Precisely,
common divisor
d C
that
the ring
each two
H(C)
which
H(C)
nonzero
satifies
of entire
functions
Algebraic
3 The
26
Structure
ord,\ (d) and
expressed
be
can
d Let
us
Recall
with
start
(a)
linear
bg
for
A c C
all
combination
suitable
with
following
the
of
list
functions
H(C).
b E
a,
Comments will
properties.
from Definition
the notation
Proposition
+
minf ord.\ (f ), ord,\ (g) I
=
as
af
=
of HO
be given
below.
2.3.
3.1.2
For all p E Ho and A G C
we
have
(X)
p*
=
(A),
p*
where
complex
denotes
conjugation.
(b)
of Ho
Moreover,
fp
E
(C)
Let p, q G Ho and let
(d)
For all
ii) a
(e) (f)
either cz
R, iii)
The
For all
(h)
p
is not factorial
z'0 p,
q G
i)"
is
true
in
every
commutative
Algebraic
3 The
28
(e)
Consider
for
v
N
c-
irreducible
Z
I
-
Structure
E
w'-'
satisfy
Ro. The polynomials (s p, for all n c N. Hence G lio =
M-1 in ? to
factors
(f)
properly
infinite
an
is
simple
a
V (0)
coefficients
(g)
C
(0)
[21,
R, cf.
in
-
2,7riv)
+
G
infinitely
1 has
R[s]
many
'Ho.
(-k)ve-k,\
=
such
...
for
polynomial
a
A E
each root
5 exists,
with
even
371.
p.
Write
ELj=
P
R[s] (f),
pj, qj') consideration.
where
Pf:=P until
(i) Only
the
b G R[s,
p
7
and
0
Cz R
z]
follows.
as
[s]
Define
0
write
an
Weproceed
-
is
g
=
0102
=
in
of
ness
gcd H(C)
a,
bi
and
(a7L, 01 01 -K)
(=-
in
R[s]
bq*
appropriate
g
(h)
R[s,z]
the
to
G
(k) be use
is
a
#V(
implies
can
be shown
provided
for
a
of the Bezout
of
see
Let p
(c).
also
and q
=
-
where
steps.
=
a-
1,. (aig).
0
b
G
=
R[s]
in
by applying
possible
deg,
`6
a greatest common First, Thereafter extracted. only finitely many common producing a polynomial gcd in H(C). The details b. (a, b) c R[s, z] and let gal a, gbl gcd,,,,,,,,;,
a,
is
L
case,
such that
fP1
for
-
achieve
ii)
If
via
deg,
coefficients clidean a
M, for
f elementary =
E
k, we use Proposition 3.1.2(g) to accomplish Pk ::: : 7io and deg, Pk :5 deg, pl. Proceeding this way, we can that the degrees Of P2 operations p, are at most M1. 7 ....
deg,
=
...
(PlM,) P2M1 domain R[s]. Let
i
transformation
OT,
(J,,O,
...
I
5
matrix
[36,
see
31
some
some
pi
Properties
p,,
MI,
=
pnM1 T via
V G
En (R [s])
134].
Hence
pp.
elementary
gcdR[.] (P'Ml
:=
I
-
the
-'fin
there
G
7
the
in
R[s]. Then V (PIMI P2MI
such that
highest
of
vector
transformations
PnMi)
-,,
-
7PJ
V(PI....
handle
we can
i
Eu-
exists
PnMi
....
7in0
C
and
deg, Pj Combining
these
M,
1.
whenever
common divisor
greatest
S2
Let p Z + and (p*)'(0) =
in
1,
-
:
q
=
Q. S2
by rewriting
=
it s
be
Exam-
f (s) simple
b* E
requirement s
-
a*
choice
(s) (e-s a
-1
=
8
over
HO. Notice
of the Let p
that
-
=
Bezout
z,
q
=
s
the sole
I
G
Ro suffices
and
ap +
a
Bezout
in the field in the
field
identity
Q(e).
a(z
as
over
=
0
can
82
+
are z
a
G
multiplicity
and leads S
1) in
+
+
(Z
S2
+
_
82
Q(s) [z] n HO, that equation above
in the
bq one needs e-1. Hence
b*
2 at
1-a*e-'
8+1
=
0. The
1) 82 is, all the coefficients in Q.
are
coprime
are
s
identity
the Bezout
to
Then p and q
the function
to be such that
Ro of
a zero
=
e-1Z
desired.
1
+ In
-
e-1z
S+1
E:
in
HO and for
H(C),
leading
a
to
this
case
(8+1) the
is easy to see that no Bezout numbers exists. A of algebraic
It
p* (s) 'HO
since
bq
=
1
is
_
Q[s, z].
a*(-l)
condition
has
_
and b s
=
forces
S2.
+ 1 E
equation
H(C) 1)
+
a
indeterminates
s
+
keep below,
S2
_(Z +'32
=
will
we
as
ap
-
3.6,
in the calculations
G
q
=
z
=
b
Now the
Section
in
and
s
Q[s, z]. Then gcd,O (p, q) s ap case, a Bezout identity
0. In this
be found
easily
be addressed
will
of the indeterminates
coefficients
with
starting
which
issues,
of the coefficients
track
(2)
simple
3.2.3.
Example 3.1.9 For computational
(1)
a
identity example [47,
The last
fairly
domain
upon these
touch
H(C)
E
q*
z1".....zl".
:=
algebraic
an
P*
N',0
E
1. A
=
examples. Part (b) when considering
section
next
for
of
simple
some
a
arise
Wewill
kerfil
and z' if -ri Bezout
7i(j)
not
the determination
Weillustrate zout
e-"') =
is
C
R[s]\fOl,v
q E
'H
'H(j)
that
4
H(C)
R[s,zj,...,zj],
Note that
reveals
ker
f
f (s, e-rls,...'
:=
[4].
above is due to
zj)
zl,...,
p E
zvq
zi],
zj,...,
33
Properties
coefficients
of
equation
with
8
and
z
coefficients
are
3 The
34
(3)
Let p
Algebraic q
+1
e
Structure
identity
we
first
+
let
G
z
e-\
and A +
e
s
=
Ro. The elements
0 have
=
a
of 'HO
=
I and b
=
This
indeed
is
3.1.6(a)
first
the
and
bp
I -(s The next
6 E
which
way,
the
1)
01
1 +
satisfies
require implies
(3.1.5)
Equation
by coprimeness
c
1-e
=
e--
altering
(3.1.5)
Equation a s
+ cp + -
e
q+
s
e)z + (e (e ee)(s -
with
The
coefficients
examples (2)
in
and
(3)
+ -
e
(3.1.5) given
(
8
+
a
of the type
8+1 e
Instead
ee.
thereafter,
a
we
e
of
proceed
going as
this
follows.
0,
follows
E
P*(-e) [ q*(-e)
imR
one
has
r-p*(-e)
C
q*(-e)
to
-
(s"+
e
'+P
consequence,
leads
of Theo-
e
transformation
(b*(-e))('
and b
q, a,
proof
' _e
(8+1)
step
the
in
transformation
6(-e)
e,
another
As
E R.
e,
a
Bezout
a
e.
S+1
of p and q, it
p,
-
e
a*(-e)) (b*(-e)) where
=
would be
J(-I)
the given
with
+
q
.1
( a*(-e) b*(-e)) Indeed,
s
I'll
[q*(-e),p*(-e)] thus,
in C. To obtain
and get
elementary
1
R[s]
1)
=
-,
would
+
procedure
to
step of the procedure
[0 where
the
in
step
corresponds
(s
-
aq +
rem
coprime since the equations
are
no common zeros
the Bezout
s+e
and
identity
--q ',_4+e
are
'HO and
in
P
ee)s + 1)(s
e -
+
e)
+
e
2
q +
(e
-
1)z + (,e 1)s + (e ee-) (s + e) -
e -
e
-
Q(e, ee). should
demonstrate how (successive) Bezout identities the field of coefficients, in this case from Q through step by step 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 force
to extend
Matrices
3.2
general
conjecture
present
in
of Schanuel
However,
3.6.5.
little
very
is
known about
35
theory, which we conjecture (just
number
in transcendental
WO
over
will
this
to
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 in Section 3.6. in 7to. Wewill identities turn to these questions stated
The results
respect
far
so
show
algebraic
their
to
being presented postponed until
another
next,
striking
a
Section
is the
one
of 'H
resemblance
But there
structure.
and
differences, of the rings
also
are
dimension
H(C)
2
with
of them
one
and has to be
3.4.
Remark 3.1.10 For
and a,, c R such that > 1
n
bn
345].
p.
It
.
is easy to
that
En+1 ciai i=2
a, +
138],
shows.
this
is
a
the
is not
Let a, z a Bezout =
-
in 'H and
unit
1 and a2
coefficlents In
[82,
the cl
p.
139]
roots
one
IC
this
to
(a,,
.
.
stable .
,
exist
in R. While
this
is true
for
an+
bnan+1),
the property
there
range
1)
e.g.,
for
that
all
Cn+1 c R such
C2,
ring H(C),
the
there
see
[82,
see
H and 'Ho, as the following example (s 1) (s 2) E X Then a, and a2 are coprime for the coefficients + C2a2 implies cial the
=
1
=
an +
rings
-
-
=
C2a2
-
clal
-
a,
of the
a2
denominators
it
can
be
and C2 can be a unit in X it has been proven that for every Bezout
of we
that
seen
domain with
elementary arrive
of the
neither
1 in the sta-
This
matrices.
at Theorem
3.1.6(b)
H(C).
Matrices
3.2
In
=
R
an+1)
matrices ble range unimodular are finite products result to the ring H(C) and applies in particular
for
1 is in the
says that
satisfying (a, + blan+l,..., is equivalent
(a,,...,
for
equation
=
this
=
-
Considering
,
unity
an+ 1 E R
R
R
case
Cl
.
.
that
see
an+1 E R satisfying
al,...,
p.
all
bl,...,
exist
[30,
domain R with
commutative
a
of R if for
section
over
we
turn
WO
our
attention
to
of the Bezout
matrices
that
over
'Ho.
First
of
all,
it
is
an
always achieve'left
property equivforms. RomTheorem 3.1.6(b) this can even be we know that triangular done by elementary But even more can be accomplished. row transformations. result that an adequate commutative It is a classical Bezout domain allows divia left and right for its matrices. In other words, equivalence agonal reductions admit a Smith-form, matrices just like matrices with entries in a Euclidean doeasy consequence
one can
alent
main.
This
will
be dealt
with
in the first
theorem
below and
some
consequences
Algebraic
3 'The
36
will
be
pointed
in that
matrix-theoretic
least
of the concepts of generalization for As our armatrices. multiples Bezout domains, the results will be a
common
over
with
start
us
present
we
and
commutative arbitrary The end of this section is devoted generality. in terms of general results module theory.
guments work
Let
of Ho
Thereafter
out.
common divisors
greatest
given
Structure
triangular
and
diagonal
to
summary of the
a
forms.
Theorem 3.2.1 Let IC be any of the
(a)
everymatrix that is, there
(b)
IC is
an
H
rings
Then
is left
UG
exists
'Ho.
or
P (=- 1C'11
Gl,,(IC)
equivalent
to
such that
UP is upper
elementary divisor domain, is equivalent to a diagonal the next one. Precisely, there
that
an
by definition,
is,
P Cz IC"I
matrix
divides
exist
triangular triangular,
upper
matrix, matrix
every
where each
diagonal
element
Gl,,(IC)
and Wc
Glm(IC)
V E
such that
VPW
=
(di,
diag,,
, m
.
.
_dl where
with
r
=
the
PROOF: Part
from
follows
di are
name
(3.2.1)
drj
(a) [51],
is
/C\f 01 satisfying
the
invariant
elementary type of diagonal
this
consequence
[64,
also
di 1,, di+1 for
factors
called
also
with
a
see
G
are
of rings
...
rrxr
elements
(They
0
d2
A
rk P and
of 1C.
units
0 01
d,)
-
L
diagonal
,
.
of Theorem
473],
p.
i
=
1,.
divisors
in
r
..,
of P and hence
-
unique
1.
The
up to
[51, 64], explaining
reduction.) 3.1.6(a).
The statement
where is has been proven domains; recall Theorem
in
(b)
adequate
that
domains are elementary divisor 3.1.6(c) for the the diagonal elements follows, adequateness of Ho. The uniqueness'of just like for Euclidean domains, from the invariance of the elementary divisors under left and which in turn is a consequence of the Cauchy-Binet theorem right equivalence, ideal (valid over every commutative domain), see e. g. [83, pp. 25] for principal Bezout
domains. It
C1
is worth
tative
and
mentioning
Bezout
that
domain is
is still
it
an
an
elementary
open
divisor
[68].
whether conjecture domain, see [17,
every p.
commu-
492,
7]
ex.
Remark 3.2.2 It
is worthwhile
over
an
arbitrary
noticing
that
commutative
left
equivalent
Bezout
triangular
domain
R. This
forms can
can
easily
be obtained be
seen
as
Matrices
3.2
follows. be
a
Let P
are
coprime
the
Bezout
Gln(R), to
a
(pij)
=
=
equation an), which, R, hence form a unimodular row (a,,..., unimodular be to a can completed always property, in
[12,
pp.
with
first
see
matrix
Theorem
3.1.6(a)).
Example
3.2.3
Consider
81].
This
Since
.
.
,
.
transform
can
one
way
(d, 0,
column
OT.
-
-,Pni) a,,
d
=: .
.
.
,
an
using again A E
matrix
equivalence
left
P via
37
Our by induction. use of made, implicitly, 3.1.2(g) (see the proof of
follows
The rest
simpler since slightly in Proposition as given
we
the matrix
[
P=
is
-
the first
proof of part (a) above is with remainder the division
(1)
and alpil+. gcd,,.(pjj, ..+a,,p,,l column of P. Then the coefficients
R"'
G
for
Bezout
'Ho
over
of P
the entries
are
82
8z
coprime
in
1
+ 1
8Z
Z-1
2
i
-
'Ho,
H2X2. 0
EE
an
form of P
divisor
elementary
given by
[0 P] [1 Z2(83._ 1
In order
=
a triangular riving equation
form.
8
be derived
+
=
z
gcd,. (s 2, 1
-
1)
_
82
S2 + 1
_
let
matrices,
82
Example 3.1.9(l).
in
as
Z('5
+
that
Notice
S
can
S)
0
the transformation
also
obtain
to
0
0
det
(Z
_
1)
*
us
begin with
=
s.
de-
The Bezout
1).
_
Hence
z
-
1
we
get the left
unimodular
transformation
82
1 S
S.
z
-
Sz
1
Sz
s+z-1
+ 1
2
S-
1-1z ,
0
(
ks Z +
(SZ
S
+
1)
+
(SZ2 S(SZ2
j
[0 b_ s a
To obtain there
and
a
exist x
=
(I
x, -
diagonal
[-by 1
(2)
form
that
notice
'Ho such that 1 2/3a)s-1 E Ho yields y G
[s b]
-
0
0
y
and
s xs
+ ya.
are
a
The
coprime in 'Ho, hence simple choice y 2/3 =
now
[I O's] [01
a
a
1_
=
-8
P]
0
=
o -b
det
-
The matrix
[
M is
in
ring
G12(R[s, z]) R0, however,
but
not
it
factors
M=
I Is
1-z
1 + -Z
2
1
E2(R[s, z]),
in
S2
8z
-
see
SZ] [16]
[97,
or
p.
into
0
11 I [ 0110, S] Is 1
-
0
1 -
1
-11
1
1
0
Z-1
i -
676].
Over the
Algebraic
3 The
38
Let
Structure
of 'Ho
return to the equivalence p* I ,(c ) q* 4* p 1, q for p, q E H, given (for ring Ho) in Proposition Using diagonal forms, this can easily be 3.1.2(c). to matrices. To this end, we extend the embedding H -+ H(C) to generalized matrices in the obvious entrywise way, thus us now
the
,Hpxq
(PQ)*
Clearly,
H(C)Pxq,
P*Q*
=
(P
and
P
Q)*
+
=
=
(Pij)
P* +
P*
(P -).V
:=
(3.2.2)
Q*, whenever defined.
3.2.4 Proposition Let Pi E Hpixq, i There exists F G H(C)P2XP1 such 1, 2, be two matrices. 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
xij X
X
E
(a) (b) (C) (d)
a
(xij)
G
of
3.1.2(c) x
r,
Qjj
yields
desired
the
[Q, Q']
Q'
xijdj
=
factor
left
0 and
for is
some
given
by
in the
Ho of A
case
of entries
divisible
not
are
'Ho guarantees
in
by
that
making Proposition
z,
the di3.1.2
(c)
consequence
of the
invertibility
for
right
diagonal
reduction
matrices
following
is the
char-
H.
over
3.2.5 P E 7jpXq
matrix
P has
right
a
P* has
a
rkP*(A)
right
right
(f)
The greatest
Furthermore,
inverse
be
all
completed
The
corresponding
that
one
is,
equivalent:
are
PM
=
Ip
for
matrix
some
ME 'Hqxp.
H(C).
over
to to
[Ip, 0]. unimodular
a
common divisor
each matrix
unique
H,
conditions
that
A E C.
Ae7jr,pisofrankpandP are
following
over
equivalent
P
can
the
inverse
=p for
(e)
and P
'HP2
E
=
Assume P2 V
10 01 ,thus
1001 'A 0
UPIV
,A* 0
F(U-')*
=
let
1:1
standard
P is
P2*V*
direction,
unimodular.
are
Q*. Proposition
rank condition
acterization
Corollary
the other
EHP2.XPl.
agonal elements dj applicable again.
For
U, V
Then
of
entries
The additional
Another
and
Defining
[X,O]U
=
(3.2. 1)
accordingly.
Q -,7,3 the
3
As for
is.obvious.
in
as
partitioned
is
d
A is
if-part
Q E 7jrxq C
matrix
of the full-size of rank p
Hpxq is right
,
can
[Qp]
E
Glq (H) of P is
minors
be factored
invertible
right resp. left equivalence. when H true are equivalences
over
as
a
Q
unit =
in 'H.
AP where
H. The matrices
up to
adds the condition
rk
R(s)
P(S) 0)
=
p in
is
replaced
the parts
(b)
by Ho provided and
(c).
A
Matrices
3.2
Ho
over
39
equivalences (a) 4= (c) '* (e), if formulated accordingly, a polynomial over are ring K [xi, XMI This is the celebrated in Theorem 5.1.12 5. also Chapter (K any field), too, see modules. Theorem of Quillen/Suslin on projective It
is worthwhile
the above
that
noticing
valid
"(a)
PROOF:
(b)
=>.
for
(c)"
=: ,
matrices
.
obvious,
is
and
"(c)
is
so
(f)",
#
,
.
.
recalling
the
[,A, 0] H(C), whence, by to [Ip, 0]. To establish A G Glp (H). Thus, P is right Remark 3.1.5, equivalent 1 with Then U let PU G Glq (H). Q [0, Iq-pj U- leads to [Ip, 0] (d) => (e) ", matrix. the asserted unimodular "(e) => (f)" follows from the Laplace expansion units
'H from Remark 3.1.5.
in
HPIP.
Then P* U*
0]
=
"
"(c)
As for
IQ]
P
along
the block
row
Q use
given by
unimodular
U and V. Then A
matrices
the desired
7-10,
condition
is not
of this
The second part
greatest
common
matrices
'H.
over
greatest
divisors
We will
formulate
domain,
common
nonsingular,
right
31-36].
literally given below
domains
Bezout
to
over
to
even
V with
[1p, 01V
=
which is not
z,
unit
a
of P. of the concepts of in H to
functions
matrices
over
ideal
principal by-product.
The result
as a
matrices
non-square
arbitrary proof The
an
being square and domains, see [71,
of them
one
over
and
dp) P
what is needed for the
theory multiple comes
in matrix
common left
A least
for
results
exactly
is
is trivial.
Xq
a
of two matrices,
divisor
is standard
to
the
this
as
minors
(a)"
=>
(di, dp) and
Udiag,
generalization from common multiples
is devoted
section
and least
Bezout
commutative
=
Udiagxp(dl,...,
of the full-size
common divisor
a
Q
form =
The uniqueness is straightforward. that for the ring Ho guarantees
result.
The additional
a
"(e)
implication
P. The
diagonal
of
yield
A G
in
unit
a
with
=
=
For the factorization
p.
is
=
of det
in
(d)",
=:>.
and hence detzA*
PU
let
carries
the
in
way
fairly standard, too, but seems since the precise to be less known. Wewould like to present a proof description between finitely will be needed later in Chapter 4, where a Galois correspondence generated submodules of Hq and solution spaces in fq of systems of DDEs will looks
This version
in Theorem 3.2.8.
be established. The
following
Definition,
(a)
Jn,q
ordered
(b)
For
:=
(c)
J(PI,
selection
a
-
-
-,
=
PnJ (pi, n
Accordingly, a
q
-
x
U -
-
Pn)
-,
-
of
selections
Let p of order of
helpful.
< q.
n
p
E
U1, Pn)
(pi,..
=
-
-,
-
-,
E
fiq-n} Jn,q.
A(P) denotes A.
-,
that
of A obtained
n-matrix
Nn
elements
n
selection,
complementary
fpl,
be
3.2.6
Let n, q E N and Let
will
notation
=
For
after
11
< p,
from
Pn)
E
is
11,
P -
..,
an n x
selecting
the minor
0.
Let
according
some
B e R7n X q be two
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
such is unique up to left equivalence. We write D gcrd(A, B). there exist MG Rrxl, N E Rrxn such that D MA+ NB and =
=
im
(b)
AT
+
Suppose If
r
-
M. As for
c
principal,
admissible
an
generated
=
lcm(ol,...'01)
where
erated,
admissible
3.4.8
Let =
associated
and the
E Mthere
finitely
is
Ho. Then is
some
-Ho, which implies
(po- 1),
=
(M)
q
h4>
=
E
MC
gen-
h-P-
for
V)
Ho such
Dp,
and
Mis finite.
Remark 3.4.9 It
is easy to
admissible
see
that
we can
Mof denominators
set
confine
sets of denominators. for p
one
without
ourselves
Indeed,
for
has. the
identity
p e
M=f0GDpj0monic,]0EM: is the
restriction
R[s, z] ((p)) (M)
all
=
((p)) (V)
where
of M.
saturation
completely describe the ideals in Ho. The presentation later generated given in part (3) below, will,be important where we study the solution of delay-differential spaces equations ideals
these
saturated
010}
Now we can
to
to
and each admissible
of
nonfinitely
in Section
4.6
corresponding
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:
(2) (3)
Mis I=
a
((p))
(1) (h, 0) gcd,. PROOF:
saturated
gcd,,(h,0)=,1andhP.E1}.
admissible
set
of denominators.
(M). The inclusion =
ah +
"C"
bo, where
is trivial. a, b G Ho
For "D" are
a Bezout multiply identity coefficients, by po-1
suitable
I E
Ho.
Algebraic
3 The
50
(2)
c Mand
If
0
Structure
R[s]
E
Hence Mis saturated
taking
least
Romthe
factorial,
above
multiples.
+
01 +
021 0102)
The inclusion
((p))(M)
gcd(ol
Since
(3) q
q G
hP- for
=
assume
gcd,. (h, 0)
02
1,
C
I, hence q some 0 Dp
let
converse
=
(01
=
7
we
+
P
02)-7--
I is immediate
E
((p))
Oi
0
and
and h (E
p is
by (1),
1
2)
E I-
(1)
that
from
the
0
=
0102
E M.
definition
of M. For the 1. Then
a
yields
0
((p))
Thus q c
G M.
one
can as
(M)
3.4.11
f 0} 7
I C
'Ho
is
[s]
c R
some a
PROOF: By and the
too,
1
=
EJ
Corollary .for
to
1CM(01 02)
gcd( ,,
desired.
If
G 1.
respect
=
of sandwich-polynomial Ho. Using Proposition 3.1.2(h),
since
1, which,
Mand
pR
=
0102
from
obtain
G
j I
where
,
P
then
0,
=
Mis closed.with
2. Hence
1,
i
Oi
Op
say
01, 02
let
01 2
may write know f G I for P
end,
To this
0,
to show that
remains
we
we
of
monic divisor
a
and there
common
Since R[s] is
is
of Ho
having
ideal
an
sandwich-polynomial
a
in
R[S],
then I
=
(a)
-
of Proposition 3.4.5(b), each other sandwich-polynomial of in Theorem 3.4.10 completes representation part (3)
R[s],
_T is in
the
proof. n
following
The
of Ho
(cf.
each
is,
ascending
3.1.6(c)).
an
alternative
reveals
It
of prime ideals Bezout domains
that
chain
one-dimensional
that
provides
theorem
Theorem
contrast, mention, that, ring Krull-dimension, yet still adequate [12, the
in
to
Ho
is
a
adequate
H(C) Thms.
length
1. It
[12,
95].
p.
of entire
3.17,
adequateness
the
one-dimensional
has maximal are
for
argument
ring,
that
is well-known
Wewould
functions
is
like
of infinite
3.18].
Theorem 3.4.12 Let
f 01 :
(a)
If I is
I C a
prime
irreducible
(b)
I is
the be
Let I be
the
a
(a)
for I n
that
is not
each
R[s]
Proposition
nonfinitely
C
sandwich-polynomial
since
ideal.
finitely
generated,
then
I
=
( p))
for
some
R[s, z]\R[s].
intersection
consequence a
with
of
p G
an
ideal
prime if and only if I is maximal.
PROOF: (a)
otherwise
be
lio
would
generated contain
prime ideal.
Then I n
irreducible
element
an
I, contradicting Proposition of 1, hence 19 ((p)). But
3.4.1. then
Let p I
even
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 in R[s, z]. polynomial 3.4.3(2), p can be chosen as an irreducible =
=
3.5
(b)
light
Ring 'H
The
a
as
Algebra
Convolution
51
3.4.1 we are reduced to show that each I Proposition ((p)), where 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 and 3.4.11 Remark 3.4.4. If q 0 R[s], then Proposition Corollary 3.4.3(2) applied with the irreducibility to ((p)) of p yields Hence I 9 &)) together ((P)) In
of
p E
R[s, zj R[s]
is
maximal
a
Weclose
=
ideal.
this
Proposition
b))
which
fying
((p))
This
let
is
sible
sets
ki
and
((-r2h))
More
that
place precisely,
this
to
The to
and
are
satisfying
a
(isomorphic Dirac-impulses
a
ring i acting
situation
we
support. out
the
operators
turn
form
E.R[s]
Ring Was
differential now
n
will
'the
associated
suitable
to)
in
=
E
MC
R[s]
be the
and
Dp
is
unique
ER[s]. contained
in
D,p
satis-
f 1}.
of the ideal
((p))
[s, z]
G R
and
monic,
are
ki
g
D.,
n
(M)
primitive
D,pi
polyn
as
are
Then rl
I
following
the
in
saturated r2,
=
sense:
'omials admis-
PIP2
1
E R
Algebra in
Chapter
2
as a
ring of delay-
(R, C) The main purpose of this section is the broader of convolution context operators. H as an algebra of distributions with compact C'
.
operators
q introduced
convolution
operators.
Paley-Wiener
Theorem,
the space of distributions
J(1) 0
0
-r=
D,
Mi
a
a
has been introduced
describe
delay-differential
be
and
Convolution
on
Let
of denominators
Pi
R[s]
some
D,, and put
k2-
=
recall
us
11
for
Furthermore,
z.
p.
set
where
(k2)
of denominators
3.5'The
Let
admissible and
=
in
for
JOEMI
gcd(0, a)
polynomialsri
[s, ;]
polynomial
D4,
Mn
(k) unique presentation
a
a
0 =
( -rf)))
=
(M)
as
such that
((-rip,)) the
is
E R
of denominators
set
saturated
a
provides
in z,
primitive
is
k 11
uniqueness of the lengthy but straight-
the
concerning Its proof
ideals.
9 'Ho where p
(M)
admissible
polynomial
Then
result
3.4.13
saturated
monic
following generated
be omitted.
the ideal
R[s, z] a
the
nonfinitely
of
and will
Consider
with
section
representation forward
n
it
will
which
are
in
the
be easy to
rational
2.9(2)
Definition
Using
Laplace see
that
expressions
will trans-
R is in the
The structure of these J, and have compact support. in more detail by going through some additional calculations. In particular, it will turn out that each such distribution explicit can be written as the sum of a piecewise smooth function and a distribution with finite hence of as a Dirac-distributions. support, polynomial Algebraically, distributions
can
and
be exhibited
Algebraic
3 The
52
Structure
of 'Ho
of the
by the decomposition polynomial part in approach to delay-differential
is reflected
this
and their
proper
algebraic because
allows
it
convenient
(cf.
function
spaces,
too.
We will
take
systems in the
df
of distributions
fixing
distributions
of
port
the
D
space
when
D+'
D'
If
:=
be the E
respect
is
z, .Zmore
very
11.
it
gen-
convolution,
to
input/output
discussing
the results
limit
T E D'
on
f
compactl,
is
f denotes the
Here supp, let
supp T bounded
-
supp
in terms
of complex-
vector-space
CI(R, C) I
topology. f. Furthermore,
inductive
(or distribution)
function
a
modules with
Let
notation.
some on
the usual
endowed with
of H much
part
proper
R[s,
over
the solu-
C)
to Cl-functions, approach, where we restrict Yet, we think the description strictly necessary. some new light on our investigations.
sheds
with
We begin valued
module
a
Coo (R,
space
our
not
are
are
is
consideration
into
aspect
the
over
view -the
for
chapter.
next
section
that
example Lj,
for
this
For the main line
of this
3.5-7)
Remark
eral
algebraic point of simply because it
an
with,
begin
to
out
turns
from
that
Recall
tions.
important
is
C'-functions
to
For the
below.
description
this
equations
strictly
their
H into
in
be made precise
abandon the restriction
to
one
functions to
a sense
sup-
left}
the
and
E)c' identify
.6
C'
(R, C).
that
the
derivatives
the distributions
bounded
support
if
or
without
124-129].
In
this
f
G
.6
and with
Ji).
This
transfer
in
are
J0
for
Precisely,
=
E Y, j=1
Notice
both factors
as
that
R[JO(l),
observation
function
Ji, 6-1]
XW*
Jj)
*
f
=
WO(l),
if at least
one
factors
are
three
all
is
to
Sy
E
Ji)
f
*
an
R-algebra
28/29]
p.
corresponds i=
of
[128,
or
convolution
R[s,z,z-1]
-
and
(3.5.1)
i=O
is
to R[s, subring of D+' and isomorphic already in [61], where it was utilized delay-differential systems. a
has been made
approach
V+ or
in
if either
j:L j= 1 ENOPij
derivative
S*T of distributions
D,. Finally, (D+, identity [104, p. 14,
p
Apo
are
k-th
the
a
is associative
have
ff
by j(k)
the convolution
(resp..forward-shift)
differentiation
(resp.
we
if either
divisors
denote
that
convolution
two of them
setting,
J(1) 0
with
a
Ei R. Recall
least
at
zero
p.
Finally,
the left.
on
at
Moreover,
is in D. C
V+
compactl.
,
and commutative
is well-defined
in
supp T
The notation
the Dirac-distribution factor
I
in D,' with their extension to distributions on S, instead of L as in Chapter 2, is meant to indicate of uniform in all convergence space 9 is endowed with the topology S n D+' be the space of functions in E on all compact sets. Let S+
Wewill
with
D'
T
to
z,
z-1]. for
a
3.5
subsequent
In the
discussions
3 tk G
f
PC'
also
R, k
liMk-cx)
R --+ C
:
will
we
I
ff
PC+' By
of left-derivatives
use
for P E R[s, z, z-1] does not hold true for
a
JW f i
I
supp
anymore,
the
Observe,
known from
f (tk +)
5
also
serve
(3.5-3)
that
'H is the Heaviside The next
left}
the
on
J1, 5-1]
0
larger into
D+.
D+'
delay-differential operators p(D, a) that for f E PC' Equation (3.5.1) s and f the Heaviside-function. p with
data
as
(3.5.2)
in
(tk))
(tk+)
37/38].
p.
C
=
f
[103,
see
has the
(3-5.3)
tk+3'
J(1) 0
identity
one
aj f W E PC'
Note that
the well-known
embedding
an
into
D+. Actually,
)((Ji))
of formal
(1)
Recall
0,
generalizes
gives
R(Jo
field
=
and'bounded
V. Let
C
*
H
=
J0,
-
Ob-
where
function.
theorem
R[JO(l),
of
if i
sum vanishes
'E
everywhere
Ll loc
C
jL=0 kEZ
where the
and
tk+j
0
can
C
C
(T, eo,-.,)
s
from
'D,:
f
H(C)
C:
onto
the
13C, IAS)l
The constant
:
a
Paley-
Wiener
algebra
>
0, N c No Vs
:5
C(l
be chosen such that
+
E
C:
j,j)NajResj
supp T C
we
distributions
to
[-a, a].
-
Fourier
because.
identity on.6.
it
Now
2.81
where
differential
by
Theorem 3.5.6
(i)
q(50(1),
Each distribution The transform
(ii)
jq* I
(iii)
The
H}
q c
=
JI)
Pw((c)
monomorphism
R(J('))
c
(see
delay-
that
simply
convolution
admits
a
[39,
also
states
the
acting
compact support
[JI, J-1]
0
are
H
57
op-
S.
on
transform.
Laplace
by q*.
given
is
algebra
(iv)
2.9(2)
with
distributions
Algebra
Convolution
a
of the
in Definition
introduced certain
as
Part
result
the
operators induced
erators
description following appeared first).
the
present
we can
Thm.'
Ring 71
The
3.5
n
q
f q* I
q E
R(s) [z,
q(JO(l),
Jj)
from
1-4
z-
R(s, z)
D+'
into
induces
the iden-
tities
H
Ho
I q(J(l)0 Jj) G D'j, I q G H I supp q(J(')0, JI) C [0, oo) 1, PC0+0 J1 q G Ho I q (Jo jq
=
=
Ho,sp
q(J('),
(iv)
Jj)
0
(i)
PROOF:
tempered the
Let
as
where h is
(3.5.6)
The second
the
Laplace
(ii)
"C"
For q G R[s, For q
Using the
M.
[57,
p.
z, =
follows
assertion
z-1] po'
with
G
H there
10(s)l
one
Jj)
0
is
a
be deduced from
can
compact
a
for
all
is
follows
a
consequence
from
(ii)
of
E
s
p*,
the numerator
and
we
K the
of
Jj('),.
has been given in Proposet K C C having V(O)
C\K obtain
and for
in
(ii)
the
third
q*
is
is obvious.
one
following (3.5.6). po-1 E H and f E S. Choose g E & such that 2.9(2). Use of (3.5.1) and P(g) by Definition which o(J,( guarantees associativity Jj)
estimate
function
Paley-Wiener
and the
some con-
desired
a
continuous
inclusion
while
Lemma 3.5.4,
of
multiplicativity
is the transform
estimate
exists
> M> 0
for
s'e-j'
that
the fact
characterizing
the
estimate
identity
The first
This
linearity
from
q* valid on C\K. Since on the compact set bounded, we obtain q* E PW(C). The converse
(iii)
231].
==
for
The second
eo,_,_q(J(1),
be shown that
has to
R, cf.
S.
The impulsive part has compact support Ph(t) for t > L p-g(t) regular part satisfies this term can Since Ph is an exponential polynomial,
and hence
interior
its
stant
E
The
along
transform
2.5(l).
sition
f
too.
of the
part
C
follows.
as
(3-5.4).
in
tempered,
be made
It G
some c
tempered.
1
q G H and
all
(3.5.5).
in
as
for
representation
and is therefore
in
for
q be
R(s, _ ,)
,
f
distribution
c
Theorem:
is
immediate
from the discussion
(iv)
Let
q
=
4(f)
implies of'supp perform,
=
(p(J('),
0
for
This
compactness each step
imply
q(J('),
0
61)
f
(p(J(1),
0
Ag) which
(g) the
is what
we
wanted.
=
JI) 4'(f
*
O(J(1) 0
)- ) 1
*
(0 Wl)) 0
*
g)
=
P(J(,),
0
61)
*
g
to
Algebraic
3 The
58
Next,
would
we
R(W),
like
JI) using
0
Structure
of Ho
draw
to
specific
some
the calculations
and
conclusions
representations
for
distributions
in
above.
given
Remark 3.5.7
As
special 4 operator
of the
case
a
is
(3.5.6)
decomposition
convolution
a
we
remark
for
that
q E
Hosp
the
of the form L
g (-r)
df
r)
f
d-r
0
with
g E PC'
kernel
4
quence,
Lj,
L1,
-
functions
Ho,,p by 'Ho,p, If
restrict
be the
to
notes
(see
As
a
((61))
,,
Hence these
same
for
is true
section
Proposition
3.5.8
Let
g E
PC+'
g
q(J(1),
JI)
0
restricted
be
with
(1)
), ((Ji))
underlying
R(s)
left
to the
are
function
deg
+
and
modules
as
following
the
description
b
de-
coefficients
R((Ji))
D+,
9
consisting PCT' + over R(s), ((z)).
for
having no differentiation analogues. Wewill come back
input/output
deg
0. Weseek to factor
over
(n-1)
(n-1)
x
in
No,
Proposition Corollary 3.2.5(f)
of
=
At
det A
with
for
greatest If
m=
Ho is
and
0,
T is
done.
we are
Assume
some
t
sm and
=
in
the matrix
common divisor
3.6.4.
see
full-size
the
assume
may
we
Q[s, z].
virtue
T
detail),
more
Then their
some m G
invertible
A
in
69
(Q(s) [z]
E
n
7jo)(n-1)xn.
(3.6.5)
can over TQ 0. The factorization 'HO and satisfies right invertible Assume for the general procedure as follows. accomplished by an iterative (n- 1) x (n- 1) T A, T, where Al E step that we have already a factorization s' for some r E No and T, is written as with det A,
Then T is
=
be
=
=
Tj
Rs-1
=
(recall
deficiencies We"have
Tj* (0)
another
have rk
Tj* (0)
only Elementary row equal where
if rk
we
if and
T
ft,
=
=
1)
E
491 R, entrywise, and aq
n
=
=
Q[S, z](n-l)xn
rk
case
(n-1)
and in this
q E
Tj* (0)
way
AjV-1diag(n-1)x(n-1)(Si
1i
of the full
a is
the
derivation
t'
Q). Writing
1)
=
Q[s, z]
on
Tjjs'zj,
Ej'j
invertible
over
HO
proceed as follows. Gln- 1 (Q) produce -a first T a factorization A2T2
< n
1
-
V G
....
-LR* dst
is such that
xn
obtain
we
size
merely with possible rank 3.2.5 (c). see Corollary
Hen.ce T, is right
suitable
some
and 1 c No
common divisor
where
(Ej Toj).
rk
1. In
-
(Q[S, Z] 0 for
=
with
t(O, 1)
in
t
-z,
operations
zero
A2
where R E
the greatest
where
t(O, 1)
rk
=
row
to
=
.1, Oz
=
Since
t(O, 1)/l!,
=
way,
by (9s
defined
(n-l)xn
power of s, we have to be concerned (0) in order to achieve right invertibility,
Tj*
of
HO)
n
3.6.4(a)).
Proposition of T, is a
minors
(put
(Q(s) [z]
E
we
=
(Q[S](n-1)x(n-1)
E
and
T2
diag(n-1)x(n-1)(S-11
:=
with
which
T
At satisfying
=
invertible
and
(b)
Let
for
some
QI
=
T E
U
=
[Ul,
-
Denoting obtain
(3.6.5),
(qj,
.
-
.
3.6.4
,
(z-
,
thus =
qn
[Ur, TrT
exists n
full-size
a
field
'Ho')lxn
Q.
factorization
right
is
=
ith
additional
d:=
'Ho. a
(F1 (s) [z]
n
Ho)
uiqid-1 and
row
column
is
omitted,
sign consideration G
R\f 01.
0
vector
gcd,. (ql,...,
n
-lujT(j)
TQ Moreover,
have
we
over
F, of Q and
UQ,
of T where the an
Rom(a)
invertible
extension
such that
along with
T]
Gln
a
nHo)(n-l)xn
(Q(s)[z]
being right
minor
U
G
e
column of
be the first
there
T^
Ho)(n7l)xn
n
ends with
steps the process
the matrix
nHo)(n-l)xn
(F1 (s) [z]
the
m
(Q(s)lz]
1)VT,,c
....
0.
from Lemma3.,2.7(l)
det
Thus
.
(Q(s)[z]
UnI by T(j) -
TQ
yields
by Proposition
After
proceed.
we can
11
qn)we
70
3 The
Algebraic
[ U] U
Q
T
Structure
"O
=
Q'
where
Qf
0
of Ro
Wecan
proceed by induction.
At the
end of this
identity
in which the
section
want
we
(Q(s) [z]
E
consider
to
7jO)(n-1)x(m-1)
n
special
a
of the
case
Bezout
difficulties do not occur. In fact, a particular computational nice situation arises if the given polynomials p,.... Pn G F[s, z] (where F C Cis n a coefficient field) are coprime in HO and satisfy a Bezout identity 1 i=1 aipi where even the coefficients in the are This means that polynomial ring F[s, z]. ai the greatest of pl, common divisor without pn. E HO can be represented denominators introducing (and possible field extensions for the coefficients). By Hilbert's Nullstellensatz [67, Ch. X, Thm. 2.4] this is the case if and only if the associated variety 1
.
V(Pi, is
empty.
that pi,
As
only .
.
.
,
-,Pn)
a
(A' tt)
we
will
show
set
of
measure
pn leads
the
opposite is the forms a common variety
just
To make these space of all
ideas
Definition
below,
for
of
set
of C. For
of all
set
for
n
be the set of all
lists
common divisor
in
should
of
For
exists.
with
n
=
2
nonempty
(finite-dimensional)
the
some
parameter
prescribed
number.
E
be quite generically
the intersection
F[s, z] I tdegp
p with
FL,
ml
degree tdegp
total
the coefficients space
3
n
in
zero
precise,
polynomials
C2 I pi(A,
C
where
to cases,
,
-
.
of
-
.,Pn)
E
TX I V(Pi,
polynomials
F[s, z] intuitive
-
is
a
unit
that
of total
-
two affine i
.,Pn)
degree
and satisfies
C2 (thus (P1 P2) is generically empty. This in
-
a
plane
Z2), can
=
C
FnL
at most m, whose
Bezout
identity defined
curves
while
01
for
more
be made precise
greatest
within
by
p,
than as
F[s, z].
and P2 do two
follows.
curves
Computing the
3.6
Identity
Bezout
71
Theorem 3.6.14
(a)
Let
n
(b)
If
>- 3, the set Zn contains
n
Then Z2 is contained
2,
=
PROOF: For p E
(a) p.
Wewill
112].
make
If
two
somewhere w
E F
[s
,
F[s, z]
on
w].
z,
define
line
Then'
I
...
Since
i
4
Z[X11
w
=
V(P)
=
Tm2 I V(P1
algebraic polynomials
the
of p. curves
[35,
intersect
Let
coefficients
C3}
in
A.
=:
R Cby the resultant 4, see [19, Ch. 3, Thm, 2.3]. be regarded P E as a polynomial
i
can
of p,
P2) 4) 76 f 01
7
defined
variety P1 P2,
resultant
f (coeff(pi),
:=
G
i
an
in the
homogenization
for
constant}
f (PI P2)
of the
fixed,
is
X2L]
...
A
X2L+31
to be the
.
P1 7 P2 not
A describes
set
Z[Xli
w]
z,
.
obtain
C
The
F[s,
of FnL.
subset
,
at
we
Z2
G
i
c
Zariski-open
of F 2L
subset
plane projective do not intersect in C2 they polynomials This can be exploited as follows. infinity
nonconstant
f (P1 P2)
P
Zariski-closed
proper
a
of the Theorem of Bezout
use
the
a
in
and
and P2, and thus
coeff(P2))
GF
2L
I
P
(coeff(pi),
coeff(P2))
=
0}
2L
CF
The
because the
is proper,
complement of A is certainly least one polynomial pi assertion (a) is proved.
neglected part of Z2, algebraic variety itself,
the
forms
(b)
V(P)
variety
Since
an
where at
empty. constant,
not
is
Z, (pl,...,pn) This time we may V(P14243) : k 101 resultant use the and obtain E Z[X17 'X3L] of these three polynomials 3L (coeff(pi),coeff(P2)icoeff(P3))EV(Q)CF 'seeagain[19,Ch.3,Thm.2.3]. 11 of F U C FnL and the assertion follows. Again, V(Q) is a proper subvariety In
In
case
3
>
n
V Q
may argue
we
therefore
the
has
one
variety
(s
-
3)z
gcd-Ho (P1 P2)
Changing
(PI P2) Q. [ ,, i
=
i
P2 into
Z].
.
that
three
admit
a
or
in F[s, z] polynomials with coefficients identity
more
Bezout
computed using Gr6bner bases (provided that above only says that polynomials pl, P2, the result (C2 is not Of C this even in V(PliP2) empty. course,
the
pi
(C3
be
can
(rather unlikely) gcd-Ho (Pl) P2) with coefficients case
If
in
For two
F is
generically
follows.
...
above theorem says context,the coprime in Ho and even generically
F[s, z], which computable).
as
Z3 and
our
are
in
the
(Pl,P2,N)
then
p2
=
Note that
(s
+
s
-,-:::
(z
-
might
situation
F [s,
in
1)z
+ E
(PI
1) (z
in both
V(P1 P2)
=
i
-
2
+
z] 2,
P2
P2)Q[5,zp
that
occur
exists.
For
=
as
(z, is
-
1)(z
easily
-
the
V(pi, fi2)
algebraic =
2)
E
verified
variety
f (0, 1), (0, 2)}.
for
equation
for
instance,
2) 2, however, one'obtains
cases
Bezout
a
Q[s, z] MAPLE.
using
gcdH0 (PI P2) 7
is of the
form
=
S
V
72
The first
3 The
Algebraic
Structure
of 7JO
points are exactly the zeros of the associated expocondition for the a necessary p*,1 p*,2 p*.2 This is certainly polynomials in Q[s, z], but, as just illustrated, of a Bezout equation existence not sufficient. We will but close with the remark, not dwell that upon these considerations is in L G This a 0. T,,2, generically pair (PI,P2) HO, coprime e., V(pl*,p2*) be clear, and can formally should intuitively be established by parametrizing the set of noncoprime pairs. appropriately Together with part (a) of the theothe Bezout equation of two polynomials rem above this implies that generically cannot be solved in the polynomial ring F[s, z]. nential
coordinates
of its
Delay-Differential
of
4 Behaviors
Systems
equations in the framework possible to turn directly to systems of DDEs. As being indicated by, the title of this chapter, we now start the system-theoretic approach. Let us study in terms of the so-called behavioral briefly introduce the main ideas of this part of systems theory. In the behavioral it declares possible, framework, a system is specified by the set of all trajectories is If the laws governing the system are known, the behavior called the behavior. compatible with these laws. This point of view simply the set of all trajectories in systems and control has been introduced theory by Willems in the eighties, of a system as described above is completely The basic ,idea see e. g. [118]. The latter notion. from the "classical" different system as regards a control results this a device (in most input signals into output signals; transferring has also Such a system description cases) in the concept of a transfer function. of the system, hence conditions about the initial the information to comprise into under which a certain the circumstances, a certain input is transferred of all the collection is "simply" In the behavioral a system theory, output. of the specific circumstances feasible leading to pairs, regardless input/outputs the behavioral Furthermore, viewpoint goes even beyond the any of these pairs. As it was pointed out by Willems by some itself. of inputs and outputs notion in which it standard examples of control theory, there are certain situations and This between to distinguish a priori inputs outputs. might be misleading external when the same' are variables, systems, sharing applies in particular of the interconnection interconnecte ,d. In general it depends on the structure
Wenow resume the
of
Chapter
of the
which
will
act
as
of
investigation
delay-differential
2. Thanks to the Bezout
variables
will
act
as
property
inputs
for
of H it is
one
of the
components
and which
of
behavioral
outputs.
With the set of all
trajectories
being the central
concept
a
system,
defined in are System properties the tasks. to Firstly, following immediately in terms and hopefully these properties to understand, one wants characterize, of the chosen representatio equations say. This goal, ap,n, the set of describing of of certain the notion for to to the feasibility or instance, controllability, plies, feedback interconnections as to as well which, if structures, any cause/effect second the in lead of to notion a step they exist, systems. Coninput/output if it transfer from arises certain a function, of, and exists, properties sequently,
theory terms
begins,
of course,
of the trajectories.
at
This
this
very
stage.
leads
H. Gluesing-Luerssen: LNM 1770, pp. 73 - 134, 2002 © Springer-Verlag Berlin Heidelberg 2002
74
4 Behaviors
relations
might tionship
want
a
switch
to
from
point
system
"simply"
we
would
like
the
set
as
another.
to
one
Hence
one
clarify
has to
the
rela-
descriptions.
between the various
At this
Systems
the components of the (vector-valued) -in the betrajectories of system descriptions variety might be possible and one
between, Secondly,
havior.
Delay-Differential
of
theory before Willems' work. sidered, called the input/output are properties distinguishing
that
mention
to
of all
its
the
[7,
In the book
of
idea
describing
51]
p.
a
in
of this
variant
control
a
has been around
trajectories
systems is
set
of the system, even though no with the various components
relation associated
of the trajectories. inputs and outputs) However, we think is moreconvincing because of its consequence in pursuing
con-
specific
(named
Willems'
approach the idea to explain in terms of trajecto(say, the properness of a transfer function) every notion ries. Moreover, the behavioral approach has the advantage that by avoiding any the fundamental structure notions of systems theory prespecified input/output of systems) often come out in much simpler, or composition (like controllability therefore
much
more
form.*
transparent,
chapter we will develop a theory for studying systems described by delayequations from the behavioral point of view. Hence we assume that the system have already been determined the laws governing and were found to In this
differential
(at
be DDEs havior for
will
least
turn
the operators
Definition Fix
q C-
solution
7ipxq
the
in
out
F
modeled
on
L
situation).
sufficiently
to be
CI(R, C)
=
space of such that
system of
a
E
of
a
Definition
be2.9
X
R is of the
Wl,...,Wq ables of the system.
sequel
we
said
trajectories
will
the behaviors
use
just
a
system),
if it R
matrix
is
the
(rij)
=
E
=
0,
i
=
1,...,p
j=1
be
to
a
exists
q
E Fijwj
Wq The matrix
(or simply
behavior
a
DDEs, that is, if there
Lq
that
r
definition Recall
purposes.
our
4.1
N. A set B C Lq is-called
W1
In the
where
following
The
for
rich
the
kernel-representation
a
in
B
names
defined
are
called
are
behavior in
the
of B. external
and system
general
described
The coordinates
(or manifest)
interchangably. by an implicit
vari-
Notice
system
of DDEs. For sake of
also
machines
object, see
completeness
we
would
been used in the seventies
[27,
(dynamical systems over finite is,'the set of all trajectories 121.
that p.
like
to
by'Eilenberg
also
remark in the
that
context
It structures). (called'successful
the term
of finite
describes
paths)
'behavior'
has
automata
and
exactly of
an
the
same
automaton,
4 Behaviors
At first the
that
the definition
sight,
Delay-Differential
above appears to be rather of a delay-differential
Systems
75
for it requires It seemingly
restrictive
kernel
be the
behavior
of
operator.
variables help of some auxiliary that sitIn operators. of) delay-differential like, in the variables are certain uation only regarded describing equations appearing make up the behavior. -and only their variables manifest as the trajectories The the model wants to describe. These are the variables,whose trajectories resulted have have been introduced other variables or from, modeling. All for, also latent variables variables such auxiliary are called [87, Def.1.3.41 for (see full For in with variables latent of definition a dynamical a generality). system of behaviors mind in that have to it or our purposes suffices preimages images variable of latent under delay-differential are examples descriptions. operators in the sense of DefiPition Wewill see in Section 4.4, that they are behaviors 4.1,
systems, which are specified for instance, images of (matrices
excludes
therefore
which Notice
that
consist
of the
the
the smoothness also
briefly
DDEs. Wewill
chapter
is
as
appears.
behavior
this
Definition
B in
relations
between
w
C
function
other
the
3.5
qualify
spaces
4.1
variables that
we saw
solution
as
of
input/output
of
our
only
does not
external
'Cq. In Section
idea in the context
in Section
organized
it
condition
resume
functions
and transfer
the
as
the
DDEs
circumstances
certain
restrictive
as
description.of causing
the
also includes
The
is not
with
but under
for
spaces
structures
4.2.
The foundations
follows.
approach
laid
are
in Lq
in
family of all finitely to the lattice generated anti-isomorphic to is given by passing from behaviors submodules of Hq. The anti-isomorphism other characterize modules. their we algebraically Among things, annihilating This is of fundamenshare the same behavior. when two kernel-representations in terms of (the for our goal of describing tal importance system properties fact that the R is an eleto kernel-representations. "Due highly non-unique) of those for of results reminiscent this the section are domain, mentary.divisor of Euclidean the is of ODEs a operators ring domain). How(where systems the'close the lattice decided to structure we and, consequently, emphasize ever, the first stitutes
Therein,
section.
a
which
lattice
between
connection
for
properties exposition
sion
different that of
allows
ODEs)
one
when
of DDEs. This ties
of the
it
is shown that
of all
the
behaviors
is
for
constructions
certain
representing as in, say,
matrices
[87].
on
systems
the
on
the other.
This
one
side
and divi-
results
in
slightly machinery
provides standard to proceed in a fairly is, like (that way the basic concepts of behavioral theory discussing
will
behavior
con-
be initiated
which
lead
Yet,
the first
in
Section
to
_ distinction
4.2.
section
Here
we
of the
discuss
external
a
for for those
a
systems systems proper-
variables
into
structures. possible nonanticipating including cause/effect The characterizations, generalize those* given in terms of kernel-representations, for systems of ODEs in a straightforward systems the way. For input/output with (formal) transfer function is introduced in the usual way and investigated
inputs
respect
and outputs,
to
nonanticipation. any inputs,
tems without
Autonomous systems arise hence without
any
possibility
as an
extreme
to control.
case
of sys4.3
In Section
76
of
4 Behaviors
Delay-Differential
Systems
to their More precisely, structure. classify systems according input/output the relation induced the transfer function. It turns equivalence investigate by classes constitute out that the equivalence sublattices of the lattice of all systems element. and contain This particular element is shown to be the unique a least in its controllable class. The notion of controllability equivalence system refers, of course, to behavioral that is the to drive controllability, ability any system into any other in finite time. Various characterizations of trajectory (algebraic) derived. Section 4.4 devoted is the to are of interconnection controllability systems. Adding some regularity this can be regarded as the behavioral condition, we we
of the
version
to-be-controlled
a
of systems which one
system with
a
controller.
usually in order to derive might want to eliminate Webegin wi ,th this step by presenting an elimination
overall
system repTesentation. Thereafter
of
connection
interconnection
to the
turn
we
of two systems forms ask which subsystems
leads
variables
The
in the model for a
the
kernel-
theorem.
of systems. Since the interconnection of either of its components, it is natural
interconnection
subsystem
a
to latent
to
intercongiven system can be achieved as a (regular) in other words by connecting controller. Wepresent various a suitable nection, is purely in terms of the trajectories; in fact', one of which it characterizations, of controllability. At the end. of the section can be seen as a generalization we which can be regarded as the dual of achievability turn to a question of subsysof behaviors. This problem might not sum decompositions tems, namely direct be of system-theoretic but from a mathematical directly significance, point of view it arises quite naturally in this context. As we will show, direct sum decomrelated to the skew-primeness of certain are closely matrices involved. positions In Section 4.5 we briefly address the issue of stability for autonomous systems, before we turn to the question of constructing autonomous interconnections with prescribed As a particular (say, stable) characteristic polynomial. case, the finite-spectrum assignment problem via feedback control for first-order sys-
algebraic
a
Weshow how the
tems is studied. our
of
framework.
problem final
In'the
can
Sectign
be formulated 4.6
we
slightly
and solved
the nonfinitely generated ideals in under taking biduals with respect to the they are invariant of these ideals in' Chapter 3, obtained L.. Using the description
of view
and reconsider
whether on
for
invariance
in terms
In most parts
forward
Only
when concerned
ring
section
ciating
with
cause/effect
Ho c R(s)[z]
The Lattice
In this
with
we
zeros
in
order
it is
structures
avoid
to
action a
of H
criterion
is derived.
the operator ring H c R(s) [z, is the natural choice for the
chapter shift,
and backward
the smaller
4.1
of the
of the characteristic
within
change our point R. It is investigated
z`], containing both algebraic description.
more
backward
to utilize
convenient
shifts.
of Behaviors
analyze
each behavior
the structure the
of the -set
space of all
of all
annihilating
behaviors
equations,
in
Lq
we
.
Asso-
obtain
a
correspondence
one-one
with
of Hq
submodules
generated
q columns
determine
rowspace
qj1Xq
same
in
behaviors
between
.
on
the But
the
same
the
on
Precisely,
other.
behavior
in
one
Cq if and
77
hand and
only
if
finitely
R, and R2
two matrices
they
The results
be achieved.
more can
even
Cq
in
of Behaviors
The Lattice
4.1
share the
derived
in
see actually correspondence Chapter 3 provide an easy of behaviors In particular, of lattices. sum and intersection anti-isomorphism are are kernel-represpntations given by a least common left again behaviors, of the given representations, divisor reand common right a greatest multiple in the terms and This description particularly Galois-correspondence, spectively. will be of fundamental of representing importance for this chapter and matrices, A lot of situations the one to follow. arising later on can be subsumed in this correspondence. It is worthwhile remarking that these results about systems of DDEs can (and further without deduced be Indeed, thanks analysis of delay equations. will) about scalar DDEs, results of H, the basic analytical to the Bezout property for the matrix case as well. derived in Chapter 2, are sufficient whether or not a given behavior also discuss the question We will permits a the useful context be in will information This kernel-representation. polynomial of first-order systems to be dealt with in the next chapter. for systems results of related The section will be closed with a short presentation with noncommensurate delays.
that
way to
this
is
an
.
Let
us
start
with
the
R
(rij)
Each matrix
rise
of maps,
two kinds
to
HP,
,H q
and submodules
between behaviors
correspondence E HpXq gives
h
Of Hq
namely
Rh
and W1
W1
'CP'
,Cq
ij Wj
the
both
note
operators
Fj_j
are
simply
by
R and
maps
ker,c R, imCR) for It would certainly
C is
more
and the consistent
all
a
we
an
believe
R-module,
matrices
unimodular
the
2.9(2).
Definition
with
We will
kerH R, imij
notation
(resp.
image of the first
that
Definition
and
2.9,
we
meaning of R is always clear
the
R
second) probably,
de-
(resp. map.
less
R and S
matrix
The R-module
Uc
structure
over
Qlq(H) on
from the context.
have
RS=RoSas maps for
in
as
use
the second operator by k The disadvantage of that choice when dealing with block matrices. somewhat cumbersome notation
Furthermore, Since
be
defined
to denote
confusing, would be
kernel
the
j=1
Wq
Wq and where
q
R
H of acts
L induces
on
,Hq and
compatible bijectively
on
As
sizes. on
theH-bilinear
rq
both
a
consequence,
Hq and Lq.
map
each
4 Behaviors
78
of
Delay-Differential jjq
Systems
Cq
X
which in turn
gives
rise
hTw,
(h, w)
L,
.
the spaces
to
MI =fWELq I Jw=Ofor all hEMj forM CRq, B' fh E] Hq I 17w 0 for all w E 131 for L3 C fq. =
M-L is the solution by M C Hq, while B'
Notice
that
induced of the
functions
It
is clear
one
(-Ml
1
Bj-'
and
+
easily
Renwk
the
these
to be the
spaces the duals of M and B. Furbiduals of Mand B, respectively.
H-submodules
and
M2-L,
n
(,61
in Definition
=
+
Lq,
4.1
i
=
appear
(im?j
B2)jas
the
verify
the
generator
C"o in
paper some
(Rk' C).
the
[84] In
category
similar
brief
The
1
1,
n
(4.1.3)
this
With
2.
L321
the
notation,
the duals
we
give
a
RT
X
(4.1.4)
q.
F-#
ring,
to
"C)
hTw,
where
presentable This
T
systems.
with
of
coset
respect
has been utilized In that
Ok] acting
say
the
is
modules
observation
multidimensional
[84, (54), of C[(91,..
Calois-correspondence mensional.
on
polynomial
in this
book
Hom?j(-,'L).
functor
contravariant
like
B C BJ-
isomorphism
with w G kerL R the mapping Thus, behaviors are duals of finitely
by Oberst in his erator algebra is space
1311
=
where R E ?jp
limH
to
J-,
the identities
Hq kerL R --- HomH
h EE Hq.
MC MJ-
that
4.1.1
is easy to
associating
infinitely many) equations annihilating equations
of all
space
said
Mi g Hq and Bi
introduced
kerL R
It
defines
are
derives
(possibly
space of the
We call
are
M2)-L= MI-L
'H-submodules
behaviors
B C Lq.
MI and B'
that
Moreover,
for
in
Mj-
thermore,
(4.1.2)
=
case on
a
the
op-
function
33] it is shown that C' (Rk' C) is a coThis deep result allows a ak]-modules. Theorem 4.1.5 below. (In Example 5.1.3 later p.
-,
of the these structural
overview
results
for
multidi-
systems.)
isomorphism
above reveals
and the module-theoretic
a
connection
approach
between the behavioral
framework
the latter systems theory, being pursued by Fliess and coworkers. In their context, differential a linear system with dea finitely lays is, by definition, generated module over R[s, z], hence the cokernel
R[s, Z]q /. 1
distributerT
of
would lead to quotients
proach
matrix
some FF delays were taken
is to consider
to
R with into
of the form the coordinates
entries
R[s, z],
in
consideration,
Hq
/imli
gr.
this The
of the vectors
in
see
[32,
quite.
underlying
R[s, z]q
p.
162].
abstract
If
also
concept
idea of this as
representations
ap-
for
system variables,
the
that
governing are
not
Let
us
which
incorporated
by of
79
equations (the matrix R) evolving in time, the'system,
the
model.
generated
finitely
to
return
in this
restricted
trajectories
are
The actual
system.
of Behaviors
The Lattice
4.1
submodules
of ?jq
and their
duals
intro-
as
duced above. Definition
4.1.2
Fix
Denote
q G N.
ordered
Observe that the Bezout is
free,
is
simply
B is
of Hq As
a
.
the
the
consequence,
that
of the
each
Thus,
generated
the set of all
of duals
set
of H implies
Remark 3.2.10.
also
allfinitely
of
set
Moreover, denoted by B.
inclusion.
property
see
M the
by
by by inclusion, ordered
partially
R in
fact
in
(4.1.4)
in
Furthermore,
M.
in
generated
Mconsists
matrix
modules
finitely
Of Hq, Lq, partially
submodules
behaviors
of all
free
submodules
be chosen with
can
of Hq
submodule
full
row
rank.
Proposition Mis
4.1.3
(non-complete)
a
modular
PRooF: Mis modular
that
the
again,
while
the
of the
Bezout
obvious
This
is
follows
gcd,
to
respect
finitely
intersection
is
=
of 'Hq.
submodules is
if q
1.
It
a
consequence
spaces,
see
the for
same
Proposition (a, b), lcmH (a,
3.1.2
c)),
p.
along which
the
as
way
[56,
instance
463].
with is
true
the in
of the
non-distributivity For
q
'e law
lcm-H(a,gcd,(b,c))
identity
=
Bezout
commutative
every
lattice
distributiv
the
1,
=
main.
It
doEl
is worth
that even the lattice of all ideals mentioning commutative Bezout domains arbitrary
is shown for
this
is
generated
The of H, see Theorem 3.2.8(b) and Remark 3.2.10. immediate from of M is of the existence nonfinitely gener3.4. For q > 1, the lattice is not distributive. see Section
from
(IcmH
of all
submodules
generated
of Mwith
closedness
exactly
in
seen
of the lattice
finitely
only
if and
is distributive
property
non-completeness ated submodules, of vector
It
sublattice
as a
of two
sum
lattice.
in in
'H is
[58,
distributive;
Thm.
1].
to be derived ordered set B will anti-isomorphism, next, the partially result chartoo. Weneed the following a modular lattice, preparatory of matrices of delay-differential and injectivity acterizing surjectivity operators.
Via the
into
turn
Proposition
4.1.4
Let R EE HpXq Then .
(a)
imCR =,CP Hand only ifrkR
(b)
kerc R .
=
R(A, e-A)
10} from
if
and
(3.2.2).
only
=
p.
if rk R*
(A)
=
q for
all
A E
C; recall
R*
(A)
80
PROOF: Since
Pelay-Differential
of
4 Behaviors
Systems
act bijectively on LP resp. Lq, we may asdiagonal form. Then (a) follows from the scalar case given 2.14. The only-if in Proposition part of (b) is a consequence of Lemma2.12(a), follows while the if-part from the left invertibility in Corolover 'H, as derived lary 3.2.5.
that
sume
theorem
The next
viewed
unimodular
matrices
R is in
the
the
contains
of this
results
main
theory
of the
Part
section.
(a)
can
be
going to develop. The. charof the inclusion of behaviors acterization via right division of the according matrices the main reason for passing from polynomial to was, to some extent, in Chapter 2. Recall that the ring H more general delay-differential operators in such a way that the inclusion was constructed kerc 0 9 kerc p for 0 G R[s] and p E R[s, z] is true if and only if po' Thanks to the algeG H, see (2.10). of H this generalizes braic structure of delayto matrices immediately arbitrary This is differential without much about the possible even operators. knowledge of such operators, solutions like for instance series expansions into exponential Observe that, polynomials. by virtue of Proposition 3.2.4, part (a) below could well be for as as In 13, some F E H((C)P2XPIL. 9 L32 expressed just FR,* R2* this formulation, the implication "=>" is a special case of [72, Thm. 3], where the result is stated in much more generality for distributions on R' having compact back will to We this the end of at come the when discussing section support. the situation for systems with noncommensurate delays. as
cornerstone
we are
=
Theorem 4.1.5 For i
(a)
Ri
2 let
1,
=
131 9 132 If rk Ri
XR1 for
pi
=
B1
(b) (c) (d)
A
n
132
a
(a)
R1. Let URIV
such
ker,c
L32 '#=>PI
pi for
modified
PROOF:
see
".,W' r
=
[P,
P2 and
=
Bi
Put
E
kerL Ri
Then
B.
c
JJP2 XP1.
R1, R2
left
are
equivalent.
R2)
10 01
1,
=
Then L31 + L32
2.
sublattice
a
from
kerc lclm(Ri,
=
of the lattice
of this
version
follows rk
ZI 0
=
i
B is
appeared first
result
(4.1.1).
of all
For
"=>"
we
R2)
submodules
[42, Prop. 4.4].
in
make
of L q
of
use
diagonal
a
form
R, and where
Theorem 3.2. 1 (b). a
=
kerc gcrd(Ri,
=
consequence,
slightly
for
=
Ri
Let rk
As
be two matrices.
R2 for some X 1, 2, then
=
imH RJ.
(Bi) B,
i
=
Xq
Hp j
C
P way that Thus Q Q].
=
U, V
Put
(Pij) =
R2 V has
0 and
unimodular
are
r
=:
[P, Q]
columns.
kerc dj
and
A
where the Then
g kerL
=
matrix
is
kerc (UR,V)
Pij
for
(dl,.
diag,,,
all
i
.
.
d,))
in partitioned kerc [A, 0] 9 1) P2 and
=
==
,
...
I
4.1
j tion
Using R2*2 for
Lemma 2.12,
r.
FU*Rl*
=
The consequence
3.2.4.
(c)
follows
(a) along with 3.2.8(a).
from
Pi*j
and the
if
all
This
i, j.
follows
result
with
81
implies Proposi-
only if kerL R, g kerL
and
R2)
gcrd(RI,
representation
a
for
of Behaviors
is standard.
Rj)-L
(b) For every a E Hq one has a E (kerL Hence the result is a consequence of (a). derived
dj*
obtain
we
H(C)P2xP' stated in (a)
F G
some
The Lattice
=
T
a
MR, + NR2 as
Theorem
in
(d)
In order
left
equivalence
obtain
to
into
lclm(RI,
an a
full
R2),
[ U2] U1 Q3 U4
transform
we
rank part.
row
Precisely,
[Rj, RjF via [Rj, RjT and let
the matrix let
1
=
rk
Glp,+P2 (H)
G
be such that
IU3 U2] I R11 R2] V1
U4
FDJ L 0"]
for
some
gcrd(RI, R2) and U4R2 lclm(Rl, the operator D is surjective 4.1.4(a),
Then D sition
=
=
D G 7j1Xq
R2) by
=
of rank
1.
Theorem 3.2.8.
and therefore
one
By Propo-
gets for
w
E
Lq
the. equivalences w
kerL R1
e
kerc R2
+
0 0
4= '
E
R1
0
0
R2
Iq
Iq
imC
W
-V1 U2 0 U3 U4 0 -
0
0
-
w
The assertion
B
being
a
G
1q_
lattice
imc
0
0
0
0
follows
D U2R2 G
0
im'C
_O
W
kerC lclm(Rl,
=
Iq
0
\W
kerC U4R2
0 -R, R2 R2
now
from
R2)
(c)
and
U4R2
Iq _
-
(d).
Remark 4.1.6
(i)
The sole least
reason
for'the left
rank condition
multiple The proof Theorem 3.2.8(b). ker,C R2 kerC U4R2 is true. common
is
defined
in
only
shows that
(d)
part for
in any
of the theorem
full case
row
the
rank
identity
is that
the
see matrices, ker'C R, +
=
(ii)
The theorem
above is true without if one replaces R by any modifications This is, of course, a wellrepresenting ordinary diff6rential operators. known result, see, e. g., [7, pp. 91] for part (a). But one can also recover this special case from Theorem 4.1.5, since it is easy to see that for Ri C R[s]P` 11q the matrix X in (a), if it exists, with entries can be chosen in R[s], too. The same is true for the gcrd and lclm.
R[s],
4 Behaviors
82
Wewould like
Example
(a)
illustrate
(a) by
part
some
examples.
4.1.7
A first
in Example 2.16 example was derived by elementary considerations Chapter 2. Therein two matrices in R[s, Z]3X3 having the same kernel V, were presented. The left equivalence over 'HO was directly verified.
of
,
in
(b)
to
Systems
Delay-Differential
of
R
Let ril
(rij)
=
G
R[s, Z]2X2
where
1)S2, (Z3 Z2 + 1)85 + (Z3 2Z2),94 + (Z 1)S2, (Z2- 3z + 2) 82 + (Z2 2z + 2)s + 1, (Z4- 3Z3 + 2z 2)83 + (Z4 2Z3 + 2Z2 + Z 1)S2 +(2z
(z
=
r12=
-
2)s
3
(Z
+
_
_
r2l
=
r22
:--::
_
-
-
_
-
1)s
-
-
1.
=_,S4 I and kerc R C kerc (841) Thus and consists of polynomials of degree the general form into at most 3. Wecould calculate a basis by substituting less work is necessary by using the following 0. A little Rw argument. column of R are easily seen to be coprime The entries r1l and r2l of the first in Q[s, z] and therefore, they are also coprime 3.6.4(b), using Proposition in HO. Thus the matrix to some matrix R is left equivalent Then det R
-s4,
=
(adjR)R
hence
.
kerc R is finite-dimensional
the kernel
=
A
Using Pioposition 3.1.2(g), degree less than 4, Say P ker,c R kerc A is given by
H2X2. 0
E
S4
we can
P2S2+ P03.
(-PO ), (
-pot
pot3
3plt2
P1
-
1
t
_
), ( _Pot2
6P2t
-
-
6P3
t3
-1
successively
+ 2s
-
polynomial leads
to
p the
the
UR=: B c
applies).
can
R[8]2x2 (B
functions
that
the
with
be verified
also
RA-'
matrix
consequence
R[s]
is no
in
In both
examples
det R G
R[s]
has
. but
Ue
R,
operator that not
in
would
=
of
ciated
differential
guided by
finite-dimensional
ordinary
kernel
gets
one
With R[s,
Z]
the 2
differential
operator
the argument that a matrix kernel which, consequently,
operator
is calculated
(see explicitly
also
X
p
given
2. This
G12 (R[s, z]) exists such that kerc A and Remark 4.1.6(ii)
satisfy kerc B to some pure differential equivalent operator, in 'HO, but not in R[S, z]. has entries matrix
we were a
and has
behavior
2P2
2PIt t2
directly,
G12 ( io),
matrix
)
-
Hence R is left
the transformation
an
these
3s2 + 3S3. It
p G
Hence the
the space
=
Checking
that
arrange
even
PO + PlS +
=
span,c
[0' P1
:=
[44,
p.
227]
where
R E R` with 0 has to be the where
from the prescribed
an
asso-
solution
4.1
space).
Together
H to
over
direct
a
The results
entries
this
in Lemma4.1-10
generated
which
PROoF:
By
show that
with
they
are
in
and likewise,
as
even
)M1,
B
B''
we
Galois-correspondence
a
modular
lattices;
the
maps
have B-L G M for of each
inverses onto
follows.
For i
In
sums.
This
2 let
1,
=
(imH lclm(RI,
imH R2T)
n
the
using
Now there
modular
M1
anti-homomorphisms.
kerc R,
=
a
4.1.5(b),
map intersections
and 4.1.5
T f (im-H R,
lattice
with
below.
Ri
all B E B, so f We other, see also (4.1.4). it suffices light of (4.1.3), can
G
be derived
Hpj
Xq
from
the
be two matrices
Then
rank pi.
imw Rj.
equivalent
be established
submodules.
M,
maps and
they
Theorems 3.2.'8
B
of Theorem
virtue
and g are well-defined have to show that to
g:
)B,
83
of each other.
inverses
are
f:
R is left
that
4.1.8 Corollary The partially ordered sets B and M are anti-isomorphic is given by taking anti-isomorphism duals, that is by the
M
of Behaviors
(and will)
can
be summarized
can
finitely
and
implies
This
in
of Theorem 4.1.5
behaviors
4.1.5(a) R[s].
Theorem
with
calculations
matrix
between
with
matrix
The Lattice
lattice
gcrd,
only
is
a
one
remains
J_
=
kerc lclm(Rl,
n
kerc R2)
R2)
kerc R2
obtains to
lattice
modular
+
R2 fl
g(kerc R,
observe itself.
the
that
But this
=
anti-isomorphic is
a
im7-t Rj + image of
standard
exercise
in
theory.
El
Remark 4.1.9 The
lir)j-
(imij
identity
the roles of H kerc R is also valid if we interchange but be Incan seen directly. part preceding corollary, deed, using a diagonal form for R E Hp X q, we see that the module ker-H R C jjq is finitely C Cq is a behavior. generated and that imcff Moreover, both are Jrelated above with 'C and H interby (imc ff ) kerH R, which is the identity in the following changed. As a consequence, L satisfies the fundamental principle for matrices R C HpXq and S C Hqxl one has the equivalence sense: and L.
This
=
of the
is not
=:
-
S
7tl
Hq
R )
HP is exact
.4==>
LP
'j -
)
Lq
S )
L'
is exact.
This result
if combined with the fact that delay-differential might look surprising .6 (the F--> on continuously q(J('),0 Jj) * f, see Theomap f is continuous Thm. 27.3]). on .6 by [107, It tells in particular 3.5.6(iv),
operators rem
that
act
operators
surjectivity
in
in the
7 pXq
have
scalar
case
a
closed
range.
(Proposition
But
this
2.14) along
follows with
indeed a
triangular
from
the
form.
Delay-Differential
of
4 Behaviors
84
Systems
under which conditions investigate with the speciai We start kernel-representation. in determinant trices having R[s]. The following Next
will
we
result
that
was
guiding
examples
the
a
behavior
allows
a
polynomial
of square nonsingular malemma provides the general
case
in 4.1.7.
Lemma4.1.10 Let
A
-Hn
G
be
x I
Then R[s]\fO}. B c R[S]nx t.
det A
that
to the
equivalent
Zk 0 for
=
(over 'H)
equivalent
left
A be left
PROOF: Let
such
matrix
a
A is
to
an
triangular
upper
some
k G Z and
triangular
upper
matrix
matrix
a,
A=
E
an-
-
(see det
Theorem 3.2.1
A
is
a
Note that
in
unit
(a)). Then R(s) [z, z- 1].
elements
the
A
det
G
rIn i=
=
We may
H be such
ai
I
an
uo for
=
assume
diagonal
above the
p,z') 0-1
Let p'=
Hnxn
without
some u
element
in
virtue
aj
3.1.2(f), R[s]. This
restriction
way
generalize
The lemma does not
be demonstrated
Example Consider
=
entries
in
z
-
1 is
R[s, z].
10,
z'-b,p,,aj aj,0
E
choose
contained
HnR(s)
R[8]nxn. determinant
R[s, z]
in
will
as
polynomial see this,
a
Z-1
-
H2X2.
I
but R is
To
suppose
[a db]
=
c-
c
such that
UR
=
[a
c
bs-1,
with
L}\10} is not
4.1.11
U
b
RIS].
the matrix
Thus det R
Then a,
B c
matrix
matrices
to
E
ai
next.
R=
with
L
r-v=l,v:Ao
Then P
the desired
obtain
we
well.)
hence
negative powers of z. the, say, jth column of A. of aj multiple appropriate
ip
equally
but works
in
70,
may contain
of Proposition 3.1.2(f) we can subtract an in R[s]. from p, to obtain a polynomial Indeed, for v G J,, e R[s] such that "-61 E H. (The case where v is negative
By
c
c
E
d
R[s, z] ds-1
=
and it
for
is
some
a'
+
0-'
b(z
+
d(z
3
8
easy
b, d
c
to
see
-
-
not to
left
equivalent
the contrary
that
to
some
there
matrix
exists
G12(H)
1)] 1)
that
c
R[s, Z]2x2.
b and d have to
R[s, z] satisfying
(4.1.5) be of the
form
(0)
b*
(4.1.5)
Now, Equation
has to be
As it
(ad
s-1
det U
unit
a
will
yields
the
out,
turn
ment in
Theorem 4.1-13
izations
of
polynomial
+
for
we
matrices,
+
the
=
make
(4.1.6).
which
we
Y]pxq
be
polynomial corresponding result concerning
kernel-
of the
of
use
d+cb
d-
of
existence
proof
a
But
+
a+b
c))
because of
For
will
(c
s
b(d
-
possible
condition
(4.1.6)
and
b)d
+
is not
below
6)
85
(0).
d*
---
strengthened.
has to be
representations
s-'((a
R, which
in
0
(a
s
bc)
-
=
of'Behaviors
The Lattice
4.1
a
first.
present
want to
state-
factor-
Theorem 4.1.12 and R G F[x,
Let F be any lield with
Put N
rank p.
exists
D E
matrix
a
R
(a)
R is minor prime,
(b)
R is
=
left-factor
PROOF: A
following
is
Rom this
AN
by &
a
divisible
formula
right
the
as
left
F [x,
Cor.
1,
non-square
ideal
follows:
=
MN. Then there
f?
Df?,
and the fact or
up to
sired. where
implies
F[x, y]pXq
where D is
for
the
-
such
square
a
case
where R is
be deduced
easily
can
the
Q
=
Q
=
d if
in
the
the
from
introduced
AjBj)
each full-size
that
sign
-
det
Hence, applying D is
a
square
immediately
Ai
[31,
for
that
the
in
the
(P)
-
[Aj,...,ANI(p)
minor i
some
Cor.
matrix
Binet-Cauchy
Bi G F[x, y]PxP, be a matrix in Definition 3.2.6, one obtains
Ajl3j,
A.-
of every
ring F[x, y]PxP)
the
B,
det
any chosen
the matrices
determinant
the
be deduced
can
of R in
117],
p.
AN (within
by A,,-,
EN j=1
[91,
to
determinant
latter
the notation
p-submatrices
x
According
PGJ'p,Np
-
the full-size
I-
=:
127]
p.
case
the p i.
generated
N
det
all
with
But
Let
y]PxP
for
divisor
the right by d, too.
Vsing
ideal.
E
(det Aj)
common
Q in
matrix
R
y]
equivalent:
are
MN)
whenever
in [31,
be found
can
Then d
have
some
conditions
(mj,...,
is',
of m,....
d and
=
two variables
in
MN G F [x,
way.
Denote
order.
proof
matrix.
square
det D
following
that
prime,
I
...
matrix
D is unimodular.
then
matrix,
with
gcd,,.,Y,
is
mi,
any common divisor
the
that
polynomial
a
by
and denote
Dk Consequently,
that
a
:=
F[x, y] be F[x, y]PxP
of R. Let d G
minors
(P) q
1,
p.
having matrices
BNJ
of the
1,
127],
.
[A,,
matrix .
.
,
we
N
.
implies
obtain
determinant
Ai form the
a
d.
-
.
d
AN] Y,
x
is
either
(det Q)
zero as
de-
DAi, Ai of D nonsingularity of some p-submatrices
factorization
The p
.
=
RE
Delay-Differential
4 Behaviors
of
F[x, y]pXq (in
the
86
same
Systems
chosen order
for
as
Aj),
the
finally
that
so
R
=
DR. o
An alternative
(45,
in
is
be found
for
the
is
property of
case
an
given
algebraically
that
than
more
proven
Now
in
in
we are
polynomial
of
can
field.
mentioning
in
matrices
[78]
the result
the preceding result for polynomial is not true for an example see [126]. In [117, 3.2.7] variables; that a polynomial for its ring S[y] has the factorization property the above sense if and only if S is a principal ideal domain.
worth
is
matrices it
2].
proof of the factorization
constructive
more
In
coefficient
closed It
and
Thm.
representations
two
for the to present condition a position a sufficient We also show, that kernel-representations. polynomial be reduced to full row rank ones. can always
existence
kernel-
Theorem 4.1.13
(1)
HpXq be
R c
Let
a
right
invertible
If
matrix.
all
full-size
equivalent (overH) R[s, z, z-11, R[s, Z]pXq. As a consequence, kerc R kerc R'. Let R E R[s, Z]pxq be.a matrix where rk R r < full row rank matrix R (E R[s, Z]rXq such that R is to [Ar, OT. As a consequence, kerc R kerc k R is left
then
in
are
to
of R
minors
R'
matrix
some
C:
'
=
(2)
Then there
p.
=
left
exists
equivalent
a
H
over
=
(1)
PROOF: a
We use
maximal
restriction
we
is written
as
factorization The
=
"numerator
part
R c-
'Hpxq 0
0
where
k(p)
of the
remaining that
may assume
R
satisfy
minors
a
factor.
left
OPR(p)
for
E
R[s]
all
will
(that
is,
no
R'.
desired
z-'
is
Without
involved)
and
,
and R E
p E
of R to extract
matrix"
be the
Jp,q,
R[s, Z]pXq.
and the
Then the
assumption
on
full-size the
full
that OP is a common divisor of the full-size minors f?(p) f? Theorem obtains factorization AR, one a Using 4.1.12, R[s, z]. A and R' over R[s, z] and det A matrices with suitable OP. Hence R(P) R(p) for all p C- Jp,q and, consequently, the matrix R' is right invertible over H, too The identity R 0-'AR' yields O-1A E Glp('H) and so R 3.2.5(f)). (Corollary and R' are left equivalent. size
minors
in the
of R implies
ring
=
=
=
=
(2) Again,
we are
restriction
that
R
[R3 R4] R, R2
=
Denoting minors
satisfies
going
E
G R[s, z] [RI, R2], we
det D
=
a
various
use
R[s, Z]pXq
by d of
to
where Ri E R[s,
1
-R3Q,-'
0-
I_
z]
We may
r x r
is such that
assume
rk
R,
without
=
r.
(within R[s, z]) of the full-size [RI, R2] D[Ql, Q21 where D G R[s, Z]rxr [Q1, Q2] R[s, Z]rxq. The rank r of the matrix
greatest
common divisor
may factor
d and where
factorizations.
=
PI Q2] [Q1 R R3 R4
0
Q2 R4
-
R3Qj
IQ2]
The Lattice
4.1
yields
that
[R3 R41
Consider the equation R3Q Q2 R4, thus R3Ql 1Q2 is polynomial. and notice that is minor R3Ql [Ql Q2] [Ql Q21 prime by construc7
Cramer's
R3QJ_l
rule
R
Again,
by
i
applied
polynomial
is
full-size
87
=
=
7
tion.
of Behaviors
itself
Theorem
submatrix
square
establishes
that
Hence
A[Ql, Q21
=
each full-size
to
4.1.12,
one
of A to the
[R3QlD_ I
where A:=
extract
can
R[s, z]P".
c
I
-
greatest
a
All
where All
0
By virtue G1, (R),
right.
of the
divisor
common
A we may write Precisely, AlB for some B cz R[s, zJ111 such that the matrix A, E R[s, z]P" is minor prime. This yields minors of A, have a greatest that, if considered over 71, the full-size common divisor has only finitely and a E R, which thus is in a even many zeros, R[S] by Proposition 2.5(2). As a consequence, Al is left equivalent over H to a matrix minors
of Lemma4.1.10 F E R[S]
r I
r
G
=
7-t"'
we can
and det F
=
a.
All
and det
factor finally Putting f?
a
=
All
c
R[s]\10}. All
as
=
CF where C G
[FBQ,,'FBQ2]
=
the assertion
,
follows. Remark 4.1.14 It
should
tion
be noted
space
L
=
As
mark 2.15.
and the next
a
which
about
of this
results
replaced
is
the
consequence,
section
we
systems
with
Recall for
from
(4.1.7)
would like
some
reference
equivalent
matrix
to
the
of this
whole
X with
formulate
Ri
right
with
systems
the results
if a
X It
in in
from the
results
some
delays.
We restrict
we
7im
presented
ab I f
a,
a
is
are
with
matrices
entries
divisor
of
R2, that
natural
to
ask whether
noncommensurate
G
R(s,
R[s,
zl,...,
zi,
delays.
ring
zi],
....
zi)
f
*
kerL b E
H(C)J,
C
kerL
a
in
is, XR1
remark.
the operator b G:
results
(4.1.7)
Remark 4.1.15 In Remark 3.1.8
existing
to
kerL R2
C
that
entries to
quote
to
the inclusion
R, being
to
generalizes we
applies
comment
noncornmensurate
4.1.5(a)
Theorem is
acterization
valid, when the funcanalogue, see also Re-
remain
real-valued
its
same
kerL R,
then
section
by
characterizing
with
concerned
are
all
(R, C)
chapter.
At the end of this literature
that
C'
I
this For
=
'h
R2
char-
future
Q-1inearly forward I
Let rem
[47].
from
taken
respect
In order
of uniform
(1)
Pi
ready
i
(4.1-7)
related
results
to the
in all
If
R,
rk
Since p,
q c
H(j) Hy)
is
not
which in this
principal, and R2
1. Then the
=
is
for
be solved
all
solutions
als
contained
case
=
(1)
(3)
Just
a
like
special the
in
Laplace
the
fined
case
transform for
at least
H(j)
q E
taken
in
can
the
F-+
see
=
cas
p*
(p,
to be the
full
(4)
More
generally,
arbitrary
it
kerg
q.
and
we
get
=
Since
ibility
Hy) R,
Put
is
=
not
[p, qT
[102,
5],
Thm.
polynomi-
0,
V(p*, q*)
such
R(j)
In the
a
principal
same
solution
has been established
it
paper
......
-rl
principal
not
in the
PW(C)
algebra rl
>
by de-
is shown
0, functions
(p*, q*)pw(c) [111, Prop. 2.6]. As a PW(C) exists satisfyOf course, the (4.1.8).
0 and the ideal
V(p*, q*) is
be embedded
can
Paley-Wieher
the
X c
H(C)1X2
of entire
ring in
[72,
p.
are
with
because
the
ideal
functions.
282,
p.
318]
that
for
operators
inclusion
induced
9
kerg R2,
of the retardations
Ri the
q), j(,)
functions
exponential
of
p n
f 01
algebra
does have
and q is
follows.
as
exist
synthesis
limits.
are
=
R2*
seen
(4.1-8)
'e, the algebra
in
be found such that
=
be
there
ring.
X with entries in not even a matrix consequence, version ing the Laplace transformed XRI* R2* of
XRI* generated by p
x"
R2
[111, Prop. 2.3].
cases
Paley-Wiener
equation
need
(4.1.7).
of
p
3.5.5, particular
can
by spectral
9
kerg
commensurate
in Theorem
that p,
is
also
the equation =
do not exist
kerg R, which
3.5
convolution.
X E H" (1)
5.13],
Exa.
not
But,
.
0 in
=
some
This
true.
[47,
means
7 1X2
qw
Section
Wewill
sets.
S via
on
0 and the ideal
=
intersection
polynomials
exponential
compact
R2 for
=
above says that
X E
of pw in the
from
C) equipped with the topology acts
not
XR, cannot
is
recall
results.
domain
V(p*, q*)
such that
however, first
us
all
on
XR,
Bezout
a
straight-
then
pl,
=
equivalence
the
< pi,
This,
Each such distribution
If rk R,
be characterized
let
C'(R,
space
derivatives
are
light of Theoby right diin general not
In
matrices.
two
can
hy).
ring
operator
kers R, 9 kers R2
(2)
Tj which are
case.
1, 2, be
=
that
quote the following
to
[47, Prop. 4.7]
xq,
some
6'.
dual
commensurate
expect the
to
convergence
are
Hy)
which refers
topological
Now we
the
quote
to
9,
the notation its
E
of
might
one
with
true.
Ri
1 and
41.5(a)
vision
shifts of positive lengths zj represent * retardations. The notation kerL a and f
The variables
independent generalizations
>
Systems
Delay-Differential
of
4 Behaviors
88
(4.1.7)
(recall
by convolution) condition
X.FRj
that
implies =
E
the
for
FR2 for
(Slyi
(4.1.9)
Xq
kernels
the Fourier some
X c
respect
transforms
H(C)P2xPll.
to
the
maps
_FRj the divisThe
converse
Input/Output
4.2
is
rkR1
if
true
direction
identity has
X*
R,
closed
a
the
in
Remark 4.1.9
commensurate
triangular
forms
mensurate
case,
analogous
the lack
is
the closed
for
range
R, with is
a
entries
H,
in
surjectivity
the scalar
operators
we
observed
of the existence
consequence
of scalar
and the
implication equivalent
of left
that
case,
that
Input/Output
4.2
in
EPI.
in
range
In
the
Ri as R2 for
=
89
in the [26, Thin. 4.1] this has been generalized the inclusion is to an equivalent (4.1.9) (4.1-7) 'PI X if if and the c (E')P2 some only operator R,
In
pi.
=
for
that
Systems
In
the
operators. as well surjective [24, Thin. the range of a matrix-operator fails
onto
are
-
of
noncom-
5],
but
due to
forms.
triangular
Systems
around the system-theoretic of inputs and outnotions ,centers Capturing these concepts in the behavioral language amounts to the task in terms of the trajectories. their essential of defining Once this is properties wants to understand, settled,,one equations, probably in terms of describing
This
section
puts.
structure. a given system is endowed with an input/output and understand causal to describe one wishes fashion, (that is, between inputs and outputs. The incorporation relationships nonanticipating) in the behavioral of all these notions approach has been elaborated by Willems dynamical sys[118, 119], see also [87]. The concepts are defined for arbitrary Of all system classes, tems in terms of the trajectories. however, linear systems described by ODEs are those, for which these notions are best understood and are known, see [87]. algebraic characterizations
whether In
the
not
or
same
4.2.1 for our situation the concepts in Definition of delay-differential in terms of kernel-representations, given in Thesystems. The characterizations orem 4.2.3, are fairly simple and standard, which is due to the fact that we are
Werecall
dealing
with
ODEs, in for
a
C'-trajectories straightforward
input/output
in this
more
Note first
systems
general
that
to E R, where for
arbitrary
the
definition
only. way.
(a
w) (t)
functions
and present
a
also
the
sufficient
the criteria
case
known for
(L' )+-trajectories
of
loc
for
condition
nonanticipation
situation.
B C Cq
behaviors to
generalize
The results We discuss
=
(t
w
w on
below is just
a
-
time-invariant, to) 'is the forward
that
are
R. Therefore
the
time
of choice
matter
shift
a
by to
instant
and has
is
no
to
(8)
time
to
=
specific
-
units
8 for
all
defined
0 occurring
itself. For the
causality
considerations
we
will
W_ :=
make
WJ(
.
....
use
0]
in
meaning by
of the notation
(4.2.1)
for
of the function
the restriction
Occasionally
will
it
R(s, z)
tions
in
from
C to the
Definition
topological
autonomous
Let
q
m+
=
parti'tioned
p and
into
w
an
0 and
=
where
if
,
variables
G
u
pass
(simply,
free
to'be
an
implies
0
--
(w,....
=
free,
5
w
0.
=
wm+p
are
free)
be
to
for
if
all
c
(ilo-)
that
variables
said
is
u
input/output
maximally
of external
wi,,,,
w
(uT, rT
Y E LP such that
is
w-
Lm and y G LP.
if
is,
with
system free
is
u
input
and
is free
which
exists
As
all
with
input
u
Lm satisfying
u
(uT, FT
u
selec-
no
and satisfies
and output y. 0 there u-
the
the free
it
system-theoretic of a trajectory
no
relationship
intime
exists
y G
nonan-
LP such that
meaning of these
can
can
is
be set
be considered
arbitrarily,
while
the
In
notions.
an
determined completely by its On the other hand, in freely. variables as controlling (the
of the
consists
output
bound
the setting chosen for the input. reflects Nonanticipation (causal with respect to time) between input and output:
"The past of the output In terms of input/output occur
variable
variables
be set
can
processes
Then B is called
=:
B.
consequence,
a
i/o-system which input), variables;
ilo-system
system the future
an
cannot
B the condition
external
called
are
u
describe
briefly
autonomous
causal
the
u
G
w
assume
B is said y,
if for
ticipating
past.
func-
M.
Let B be
us
if for all
(uT, r
exists
The behavior
7h >
a
=
in
Lm there
E
and output tion (wi, ....
Let
(cf.
01
oo,
behavior.
The variables
y-
(-
line
4.2.1
B is called
(d)
half
the left
to
the interpretation of rational Section 3.5). In that context we will
space E.
(a)
(c)
R,
on
utilize
to
distributions
as
a
u
w, defined
be convenient
Let B C Lq be
(b)
Systems
Delay-Differential
of
4 Behaviors
90
by the future
is not restricted maps
prior
to
(cf.
the
4.2.4),
it
assume
that
Remark
[87,
input."
of the
p.
89].
the effect
says that
simply
cause.
Remark 4.2.2 It a
is not
priori
the last
in the
quite in
an
p that
ordering
behavioral
spirit
of outputs. consideration.
to
only Instead, it would be the first
such that
mcan
the external
play
more
natural
Since that would add merely orderings into this additional freedom and to the setting, we disregard if possible, has already been carried out. reordering,
Clearly, turn
the
out
that
number of ation
as
maximum number of free
this
number
equals
variables
a
is
uniquely external
this we
is
will
permutation
assume
the number of all
Observe that independent equations. algebra over fields. Moreover,
in linear
variables
are
of inputs and to take arbitrary
the role
matrix a
suitable
determined. variables
simply see
that
that
It
the classical every
will
minus the situ-
collection
Input/Output
4.2
of free
variables
be extended
can
to
free
maximally
a
given below. As to be sequence of the rank criteria closely related to the size of the retardations acting
This
one.
expected,
is
a
91
trivial
con-
nonanticipation inputs and outputs.
the
on
Systems
is
Theorem 4.2.3 B
Let
=
m, p > 0. Definition
(1)
(2)
E are
if and
only if
Q
rk
an ilo-system -Q-'P E R(s, z)Pxm the ilo-system B. Let B be an ilo-system.
B defines
(3)
variables
Notice
that the
the
in
scalar
if and
only if
Then B is
rk
case
triangular,
upper
r
Q acting
Q
(2)",
Obviously, the system kerc I
systems in Cq form
autonomous =
10}
proof
the
completing
its
as
a
transfer
having
class
the trivial
element.
least
It is significance. system-theoretic a controllable system in the sense that it is capable of steering every trajectory finite within time and without into every other trajectory leaving the behavior. combine to is the possibility Put another any past of the way, controllability
The least
first
a
need
Definition
a
notion
transfer
(far)
any desired
system with we
Bc of
element
for
class
is of
of the system. combining functions.
In order
future
W(t) W,(t)
(wAt.w')(t) Using concatenations, Definition
(see [87,
4.3.4
subspace
there
time
exists
some
wAocAtoutOw' the
concatenation
E
5.2.2]
Def.
B ofCq
instant
of
to
>
a
to
t >
to
and w' at time
expressed
as
function
c :
to
as
follows.
given therein)
interpretation if
controllable
called
0 and
t
0
controllability
time
f?(B)
to
exists 0 and
>
Lemma 4.3.6
function
a
have the entries
concatenating in trajectory to
and
there
some
of
instants.)
can
of
f?
and
(At
c.
in
'Ho in Since
be steered
R(B).
to
Delay-Differential
of
4 Behaviors
110
Systems
Remark 4.4.5
Note that
the map
f?(ker,c is
with
controllability
image of
the
lary 4.3.7),
the
controllable
a
following
behaviors
above could
be
can
be iden-
variables)
number of external
expressed
in
and
terms
of
,behavior
additional
is
controllable
characterization
(see
again
Corol-
from the the-
is immediate
that
Notice
understood
in
(Definition regular
via
(b)
part
kerc R
+
can now be by part (b) below the term controllability it describes the steer to trajectories ability way. Firstly, of all subsystems 4.3.4), and secondly, it expresses the achievability In other words, it guarantees of the very existence interconnections.
above.
orem
in
"quotient
different
a
w
behavior.
the quotient
Since
condition
Rw
k,
Therefore,
(with
behaviors"
real
R/kerc
kerc
of H-modules.
isomorphism
an
tified the
R)
a
twofold
controllers. 4.4.6 Corollary The following conditions
(a) (b) (c)
1
each subbehavior
f 0}
C B
can
Remark 4.4.7
Consider
once
regular the
C
are
equivalent.
outputs
suitable
choice
both
regular
regular
interconnection
from B,
from B.
interconnection
of Theorem 4.4.4.
the situation
the output number of 8 is, numbers of the components
In
by
B n B, is
that
case
definition
of
B and B'.
This,
that
regularity, however,
the
,
of the
two
i/o-systems,
nonanticipating
then
the
B'
controller
can
be chosen in
form, (and, of course, such that the outputs match). This can the same way as described be shown in exactly for systems of ODEs in for Thm. 9]; see also Proposition the condition of nonanticipation. 4.2.5(b) this
is
too
worth
strongly fails'even
that
general mentioning possible nonanticipating i/o-systems (see Remark 4.2.4) at the for systems of ODEs as can be seen by the example kerc
In this that one
case
strong
is the
output
in
11
it
2s 3 +1 ,
s
2
external
of B. But it
of
8
variable
is not
system B' having
is
1]
82 s+
nonanticipation
the second and third
interconnecting
a
outputs of the given subsystem 8 are made up But this can always be achieved by a components. of the component B'. Even more can be accomplished. If 8 C B
gua rantee
the
via
via
-
more
interconnection, of the output
does not
are
Cq
C
B can be achieved
be achieved
sum
by
system 13
on a
controllable,
B is
C
not
B
to
kerc [2, '93 + 1,
and B requires are the output
possible
the third
have all
to find
variable
a as
8,
strongly output.
[120, It
components
same
time.
This
S21..
by Proposition of
easily
while
4.2.5(c) the second
nonanticipating
Subbehaviors
4.4
After
these be
considerations
regarded,
interconnections
on
and Interconnections turn
we now
to
ill
problem,
a
that
via below, as the dual of achievability Given a behavior regular interconnections. 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
161
simply
made
L32 for
0)
B,
call
precise
some
direct
L32
behavior
C
(4.4-3)
130-
of Bo. In terms
of the duals Mi can Bi -Hq, question posed as follows: given finitely generated mpdules.A4o C A41 g -Hq, find a finitely generated submodule M2 C -Hq such that M1 + -A42 Hq and M, n -M2 of Mo. This is exactly the condition I
case
we
a
above
the
C
term
=
achievability
(see
modules above but
also
direct
on
=
regular
via
(4.4.2)
see
we
where
interconnections
believe
Example 4.4.8 (a) For Bo =,Cq, the clas's
the
for
might
terms
nevertheless
=
be
not
be of
of all
direct
behaviors
replaced by problem stated significance by itself,
condition).
regularity
is natural
it
now
system-theoretic to be
are
The
investigated.
of Bo is
immediately seen to be the C-q is equivkerc R, E) kerc R2 alent to gcrd(RI, R2) Iq and lclm(Rl, R2) being the empty matrix. But this simply means that [Rj, RjT is unimodular so that 3.2.5 by Corollary and Theorem 4.3.8 the behaviors kerc R, and kerc R2 are controllable. class
of all
controllable
terms
Indeed,
systems.
=
=
(b)
In
the previous section system is always a ,direct see
Theorem 4.3.14.
lable
(c)
subsystem
Consider
is
has been shown that
it
term,
the
The theorem a
direct
complementary below will
the
controllable
term
part
of
a
being autonomous,
show that
even
each control-
term.
system Bo 9 Ll q
kerc A, hence given by Bo Choose a frequency A E C with nonsingular. k > 0. It is intuitively clear that there exists an exponential ord,x (det A*) solution woe,\' in Bo. We will show even more. By some matrix w(t) calculations it is possible to derive of kerC.A that a direct decomposition extracts exactly the solutions having frequency A. To this end, let U, V E A is diagonal. Glq('H) such that UAV diagqxq(al ...... aq) Extracting from each a the (possible) root A with maximal multiplicity, we obtain a the
autonomous
an
matrix
=
,
A E Hq Xq is
=
=
factorization A
==
diagq
Xq
('al
i
....
aq)
-
diagq
x
q
((S
_
A)kj
)(S
_
A)k,)
3
-
A
0. In particular, k. The eti E H and &,i* (A) we have Ejq_ ki of and induces the direct eti coprimeness sum decompositions A)ki (S ker,c 6,j E) kerc (s kerc ai for the components, see Theorem 4.1.5(c) A) ki and (d). This in turn implies kerc A kerc 3 E) kerc A and we finally get
where
=
-
=
=
the direct
sum
decomposition ker,c A
=
kerc
(3V-1)
(D
kerc
(AV-').
(4.4.4)
4 Behaviors
112
det(AV-1)
Since
eratorAE
R[s]9
other
tain
any
from
.
solely hand,
the first
This
system.
solutions
and I
this
case
is also
p(t)eA' (ZAV-1) in
this
way
vector
space
a
many various
the
A; this
For
successively
finitely
the
to
by op-
frequency
of
(det(,AV-1)1q).
g kerL
know
where p c(C[t]q. (4.4.4) does not
=
polynomial
(,AV-')
we
differential
k-dimensional
a
w(t)
component kerL
H),
in
somepurely
On con-
follows
autonomous
complete direct of frequencies
well-known
polynomials.
expansion
Remark that
A implies the identities lclm(.AV-1,AV-1) by virtue of Theorem 4.1.5. In this particular from the fact that A and A are commuting.
J4.4.4)
gcrd(.AV-1,
=
is
unit
a
of course, nothing else but finite sums of exponential
is, into
decomposition
the
to
for
behavior
of the type
according
decomposition
of the
==
(up
kerLA
exponential
kerc
inclusion
R[s]
c
of ODEs one can derive
systems sum
A)k (AV-')
Hence this
xq
(vector-valued)
the
Systems
-
kerL
of functions
consisting the
(s
==
that
Lemma4.1.10
the
Delay-Differential
of
=
AV-')
clear
the question posed above let us first rewrite (4.4-3). Choosing Bi kerL Ri, we see that, as in the previkernel-representations the decomposition ous example, (4.4.3) is equivalent to gcrd(RI, R2) 1. and Ro. Let furthermore, Ro XR, be the factorization lclm(Rl, R2) implied the B, C B0. In the scalar case the existence of R2 satisfying by the inclusion above requirements is identical of X and R1. In the matrix to the coprimeness case this generalizes to some skew primeness between these two matrices, which for a direct then provides in terms of the given data R, a criterion sum (4.4.3) 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
full
rank
row
=
=
=
The
=
(straightforward)
(a)
equivalence
two-dimensional
for
result
#
discrete-time
(b)
is the
systems in
analogue of
[108,
Thm.
a
corresponding
18.3.4].
Theorem 4.4.9
",
Let Ri G RP' ated
behaviors
thus
B,
(a) (b)
is
0, 1, be two matrices with full kerL Ri g cq and assume XR,
B0. Then the following
C
B,
i
Bi
a
direct
the
matrices
7P,
xP0
term
Ri
and G E 'HqxP'
there
exists
a
matrix
are
H G -Hgxq.
and in
case
Define some
the associ-
X E 'HPO
xP1,
equivalent:
are
skew-prime,
=
G G I-PxPi
Bo Furthermore,
rank.
Ro for
that
is,
there
exist
matrices
F G
such that
lp:
(c)
=
B0,
of
X and
conditions
row
=
FX + R,
(4.4.5)
G,
such that
Bi
ED GRI
(Bo).
term B, C 13o is of the form B1 every direct H(BO) for some subbehavior B, is a direct term of BO, Moreover, every controllable term of Bo is controllable, too. B0 is controllable, every direct =
Subbehaviors
4.4
Remark that
does not depend skew-primeness condition to being of full row rank, are left equivalent
the
and
R0, which, representation. PROOF:
full
row
"(a)
(b)
=: ,
rank.
B0
Let
"
L31
=:
E)
Then Theorem 4.1.5
XR1. RomTheorem 3.2.8
we
form
132 where 132 yields gcrd(RI,
get that
leftmost
,HqXpi and
Again hence by
matrix
Theorem 3.2.8
.
the
of
loss
that
(which
matrix
column
least
3.2.5) a
"(b)
=-
(c)"
can
be
completed
priately,
again
arrive
at
and Z. For the verification for
and calculate
(i) (ii)
RI(I
IP2
equation
of the
=:::
sum
in
(c)
we can
GE
R2)
assume
a
unimodular
some
elementary
to
of the form
(4.4.6)
Choosing with
to
lclm(Ri,
an
(b).
we use
and
[(SF, Xrf
completions
appro-
[RI, F]
the
suitable
matrices
the
identity
R2, N, Y, Ro XR1 =
the directness
=
=
=
-
RoGRiwo Ro(GR1 GRI (Bo) 9 Bo. =
(a)"
=: ,
In order
=
to
-
I)wo
guarantees
=
that
X(RIG
hence
0,
GRI (Bo)
I)Rlwo
-
of the sum,
Bo
is contained
is
a
=
behavior,
0
by (ii),
the
thus
implication
is clear.
establish
the representation
B1
=
H(BO)
and define
consider
H
:=
for
given direct term B, ZR2 GR1. The inclusion the converse follows from B, E_
B0, again (4.4.6) Bi D H(Bo) is immediate by (ii) above, while ker,c GRI g kerc (I ZR2). The remaining assertions of
after
both matrices
FX) R1 wo R, wo, implying FXRlwo (I RIG)Rlwo
-
Since Theorem 4.4. 1 (a)
"(c)
R2)
]
0
0
is
multiple
get
shows
(4.4.6)
of the direct
has
lclm(RI,
the sum,
in
(iii)
-
=
left
identity
L
matrices.
Equation
Xq
B0
wo c
(I GRI)wo
R1 GRI wo
shows that
unimodular
to
=
This
sizes.
(4.4.5)
The equation
we
fitting
F and N of
matrices
'HP2
according
R1
[RJ, RiF
[Ip,
Ri
with
an
partitioned
we
matrix
[R2 NJF] [G[X YJZ]
q and
-
matrix
common
by Corollary if necessary,
the
Completing
X.
=
and
that
of the
and
G
chosen
0
Glp,+P2(H)
implies
uniqueness
generality is possible transformations,
without
in
is
1.
of R,
other
every
[Iq]
R2
C Y
R2)
=
113
the choice
on
kerc R2 and R2
=
P1 + P2
po
[G Z] [Ri]
where the
and Interconnections
=
a
I
-
above in combination
with
Theorem 4.3.8
and Cor 4.3.7.
-
are
consequences
of the El
Remark 4.4.10 not able to provide we are characterization for B, an intrinsic Unfortunately in of the terms being a direct term of B0, that is to say a criterion purely trajectories. However, the skew-primeness of the matrices X and R, can be given Note that the existence a behavioral of a direct interpretation. decomposition does not only require the splitting of the exact sequence
of
4 Behaviors
114
Delay-Differential 0
but also the
be
can
(4.4.5)
RI(BO)
that
For
equally
well
for
sketching
For the details
situation.
behavior
is
a
direct
B0,
in
from
B0,
of
term
if it
solution,
a
Since
exists.
(apart
from computational where K is a field, a nice
issues,
the will
be summarized
over
also for R. The result the main idea
problem
a
the
Thanks to
7i-isomorphism
an
B,
01.
criterion K[x], [94]. Studying the proof in [94], one remarks and, as a consering H(C) of entire functions
matrices
has been derived
works
it
quence, to
3.6).
Section
a
as
B0 that,
in
know that
One has to check the solv-
term.
and to find
equation is not
we
in
space
not
or
complementary
a
contained
contained
embed this
to
how to check whether
this
is linear
solvability
for
Rj(Bo)
the operator Ginduces GRI(Bo) C L3 0 C fq.
above tells
equation
see
possible
if so, how to determine of the skew-primeness
this
behavior
a
B1. Precisely,
the behavior
onto
The theorem
and, ability
to
to
0,
)
B1. From Remark 4.4.5
behavior
the
B01B,
)
isomorphic
with
is indeed
it
complementary
L30
)
to be
trivially regarded as
intersects
quotient Equation
B,
)
B01131
quotient
additionally,
Systems
in
of the
proof
Wewill
next.
[94] along
in
with
confine
ourselves
adaptation
to
our
[94].
is asked to consult
the reader
its
Theorem 4.4.11
H1 x n,
Let A E
B EE Hn x
m, and C
2
x n
7in
be given
Then the matrix
matrices.
equation C=FA+BG is solvable
over
Ii
if and
only
(4.4.7)
if the matrices
[B qA] [13 AO] 0
,
E
0
-H(n+1)x(,rn+n)
(4.4-8)
equivalent.
are
up to
that divisor form (Theoby the uniqueness of the elementary of matrices H can easily be checked (easily over equivalence again the invariant factors of the practical computational issues) by calculating
given
matrices.
We remark
3.2.1),
rem
SKETCHOF PROOF: Wefollow
1) Necessity
follows
easily
(over
the steps
[I -.r] [B AC] 0
2)
For
sufficiency
the
matrices
and
bl,
.
ing (4.4.7)
.
.
,
one
A and B
may are
0 A
1
assume
in
-1
0
.
rkA
diagonal
Hence a, ba, respectively. reduces to finding fij and
taken
=
form
since
] r =
(4.4.7) 0"
>
0, rk B
with
invariant
a
implies
0 A
a,
gij
[94].
in
domain)
every
such that
and bi
=
3
>
factors
0 and that a,....
b,3. Now,
,
a,
solvL
Assigning
4.5
fijaj
(cij)
bigij
+
the
Characteristic
cij
=
Function
115
(4.4.9)
,
of (4.4.9) a and i >,3. The, solvability of the by showing that the equivalence in (4.4.8) matrices polynomial -Y E K[x] which. implies that for each irreducible with maximal power r in aj and bi, the element -yr is also a divisor of cij. occurs Thus, cij is in the ideal generated by aj and bi. As for the ring h, one can use
where C
=
and aj in [94] for
is established
the
same
0
=
of arguments
line
bi for j ring K[x]
to show that
minf ord,\ (aj*),
ci*j
Hence and
is
ord,\ (V)}
=
the
Bezout
A E C.
by aj*
and of 'H
property
bi*
in
yields
H(C) E
cij
11
is not suitable
procedure
as a
for
(4.4.7)
solving
of A and B, which would comprise
for
it
requires a diagonal For certain the computations. matrices over the polynomial square nonsingular for alternative the procedures skew-prime equation are given solving ring K[x] in [121]. These procedures that',the skewwere motivated by the observation arisen in in has several the over see prime equation places K[x] systems theory; reduction
introduction
[121]
in
4.5
Assigning
This
section
to
design
mial.
and the references
devoted
is
autonomous
The first
therein.
Characteristic
the
to
a
Function
special-case
of with
interconnections
requirement,
autonomy,
of
the main bulk
regular interconnection. characteristic prescribed
We want
a
simply
says that
all
inputs
of the
polynooriginal
L e. no free variables in the inare left by the controller, is a system of the form implies that the interconnection In this case, the characteristic matrix. kerL A, where A Ej Hqxq is a nonsingular detA* G H(C) provides function structural information about the some first whether it is finite-dimensional, hence a system of ODEs, system; for instance, and if so, whether it is stable, which can be-seen from see Proposition 4.2.7(b), the location of the zeros of det A* in the complex plane. It is natural to ask whether a stability criterion in terms of the characteristic is also true for zeros autonomous delay-differential systems. This will be dealt with in the first part
system
restricted
are
This
terconnection.
-
of this
section.
Thereafter
we
turn
ask ourselves
suitable additional
as
choice
to
More
terconnections.
the problem of assigning characteristic precisely, given a system kerL R, where
to which
functions
of the controller
properties
of the
a
CE
controller,
G
H
achievable
are
-H(q-p)Xq. like
One a
functions R G ?jp
as a
might
(nonanticipating)
=
also
X
via
q,
we
in-
will
det[RT, CTT by ask for
certain
i/o-structure.
of
The existence
Delay-Differential
of
4 Behaviors
116
controller
a
Systems
such that
the
interconnection
stable
is
turns
out
stabilizability. Following [87] we will define stabilizof a behavior to steer its trajectories as the possibility ability asymptotically it is not clear whether this In contrast to systems of ODEs, however, zero. of stabilizing controllers. to the existence results will Only partial equivalent given below. be related
to
the
In
last
:t
type
so-called
to
of this
part
A((7)x
=:=
section
B(o,)u
+
specific polynomial of
controllers
prescribed problem of finite
of infinite-dimensional
approach
type
a
sought
are
[s, z]
('+').
such that
In the
-
case
solution
a
Definition
of
stability
with
that
delay-differential
(b)
w'(t) We should
the
that to
Since
stay bounded,
stabilizability and,
[s],
this
a
well-kno
is the
algebraic delay-differential
the
as a
w(t)
limt--->c,.
if
all
methods
'wn
with
a
systems.
0 for
if
lim
w'(t)
t < 0 and
>00
wB.
all
for
all
=
w
C
B there
0.
above is usually called asymptotic in the sense dealing with stability skip the adjective asymptotic.
defined
as
not
we are we
=
stabilizable
t
says that
zero
will
every
consequence,
in 13
trajectory
asymptotically
can
be steered
asymp-
other
trajectory
condition
for
to every
the behavior.
stability Clearly, does not bility
implies
:=
Wecall
f
A c C
a
I
matrix
Re (A)
exponential
is
such that
kerc A
following
necessary
the notation
given
sta-
in Definition
2.3.
1
rn
case:
With
Delay-Differential
of
4 Behaviors
124
is obtained.
Remark 4.5.12
proof
The
(in
that
of
Q)
is
true.
be
Let
single-input
solving
a
Bezout
are
rational
coefficients
all
symbolic
solution
multi-input According to [113]
the
case
this
can
the
case
numbers
be achieved
in
the
revisit
.
symbolically
accomplished us
-
.
if the initial
proof above for
two
in
feedback
where qj E R[s, , ql*), of a greatest amounts to the determination V (ql*,
varieties
or
algorithmically
additional
data
special
of
computation In Section
equation.
be found
can
the
In
certain this
the
in
to
essence
case
a
found.
shows that in
amounts
z],
3.6
we
have shown
field
extensions
if Schanuel's
conjecture
certain
matrix
finitely
many
have to
K needs to be
steps
in which
be determined.
divisor, computable
common
have
controller
a
this
again
As can
coefficients.
cases.
Remark 4.5.13
(1)
Firstly, single
we can
input
In order
to
Since
this
in
the coefficients 1x
do so, case
ri
reachability
proof
above the well-known
fact
for
that
to coefficient equivalent assignability. let (A, B) be a reachable single-input pair, hence m 1. the matrix over A, -B] is right invertible [sl R[s, z], in (4.5.8) are even in R[s, d, G z] and, consequently, q
is
=
-
=
and q, Zj is monic and has R[s,
from the
recover
systems
n
=
0. Thus
degree a
=
det
c
=
n,
and
[
d
rn+ we
1
obtain
sI -A -B 1
+ hB c R[s,
1
z] has to be one, since the familiar static feedback
a
where d G R[z]'x'
while the converse is true implies coefficient reachability assignability of Theorem 4.5.11 with reachability arbitrary systems. Due to the failure in place of controllability, the above does not generalize to multi-input systems. However, at the end of this section we will show that for reachable multi-input systems one can always achieve coefficient assignment with
Hence for
F CHmxn and G O'P
=
0.
Assigning
4.5
(2)
A
particular
(A, B)
simple
is in R"'
procedure
of the
case
R[Z]n,
x
obtain
a
Bezout
R[s]
with
E, ai l-"P'-----'-Pn
be formulated
then
be solved
the
first
particular, is
Thus
we
i
the
some
We illustrate
Example
(a)
det(sI
=
A)
-
is in
R[s],
can
one
of the
strictly
choice
of a,,.
cen.
=
This
interpolation problems for ai, which can by the desired Multiplying Equation (4.5.12) a
that
R[s]
G
proof
shows that
the
of Theorem 4.5.10
part
proper
we see
(4-5.12)'
+ anPn + an+lPn+l
many
R[s].
within
is
q,
Using
zero.
remainder
the
vector
is
once
(4.5.9)
d in
q
actually is
=
more a
(rl,...'rn)
R[S]lxn.
in
that
constant
SI
In -
A
vector.
get finally
d G Rlxn
[
sI-A
-B
d
I
-
g]
=
R[s]
c
a
and g G Ho,sp.
by
the situation
following
the
4.5.14
Consider
when
This
where F is constant.
of via appropriate
care
finitely
det
for
above arises
the requirement Indeed, 1,...,n. an+1 only finitely many zeros Of Pn+1 (including
=
needs
'Ho as
case
R[S]nxn,
in
for
+...
alpi
polynomial
characteristic in
=
to be taken
can
125
equation I
G
P",+,
multiplicities)
proof
the
in
Function
is, if there is just one input channel and the In this situation, a prescribed one can achieve
that
delays occur only in the input. (4.5.4) finite spectrum even with a controller Since the polynomial be seen as follows. p,,+, can
Characteristic
the
the matrix
[sI-A,
-B]
examples.
1-18,
1 0Z]
0
1
=
-
s-
.
A is unstable
The matrix
wish to assign the stable characteristic polynomial a (s + 1) (S + 2). minorsp, Z(S-1)5 P? -Z5 P3 s(s-1) of the matrix [sI-A, -B] in H0, showing that the system is controllable. are coprime Using the idea of the preceding remark, one easily finds the Bezout equation and
we
The
=
=
=
1
=
-PI
-
=
eSP2 +
I +
(Z ez)s S(S-1) -
-
Z
Hence 0
-8
-Is-1
a=det
-a
0
(1+(z-ez)s-z)a
esa
S(S-1)
8
0
-1
S-1
6e-2
6e
-
det
-
-Z
-Z
1
0 (6ez-2z-4)s+2z-2 _
P3.
4 Behaviors
126
where
the
which
produce
volution
Delay-Differential
of
last
Systems
follows
expression
the
in
constants
associated
operator
after
first
with
elementary
transformations
row
of the last row. The con(6ez-2z-4)s+2z-2 2(1-z) 6(ez-1) + S 8-1 S(S-1) This leads finally to the (stabilizing) two entries
g
=
.
.
be obtained
can
Example
from
2.7.
controller
u(t) (b)
In
the
6e)xl(t)
-
special
very
R[s],
s+ ao c
(2
=
the
case
n
u
=
m
=
procedure
g
-b(e -A)-l
b(z) Example 2.7) E.
g.
for
which for L
Finally, it
=
AL
-e,
=
(A
ao)x
+
I has been obtained
consider
want to
we
has been derived
as
a
1
-
C
as
(see again
different
methods
reads
L
eA-r u(- -,r)dr, completely
with
a
HO'Sp.
simply
fo
ao)
earlier
the
[sI where mark
+a
A, -B]
-
,
r,,
a,
and
4.5.13(2)
c
length
R[s]
one.
of
Put b
==
S2 + b,
A Bezout -
We want
A, -B]
('82
=
s
8
0
W2
+
3. It
s
+
there
since
[75],
from
taken
where in
a
takes
be useful
[s].
bo
c R
It
det
L
+
+ is
a) +,3,
easily
S
0
W2 ra
"
Notice
delay
-'
S+Ct
s
a
0 0
2 w _W2 0
so
0
that
bi, bo,'O
(A, B)
2 w
Re-
matrix
A.
the
delay polynomial
form
common divisor
0
+
in the
that
in the
prescribed
where
-1
-raz
0
I
a
checked that
for
a
R[s, Z]3x4
parameters. occurs
to express
the greatest the simple form s
G
2 w _W2
been normalized
bis +'bo)(s
equation
+
already assign an arbitrarily
to
will
0 0
nonzero
are
apply
0 -1
-naz
the model has
degree a
pair.
example,
model of the Mach number control
0
R
w
does not
We assume that has
following
linearized
s
[sI
+
R[z],
controller
Let
wind tunnel.
a
(A
E
in the
+ g4
A
-
b(z)
=
results
ao)x
+
equation
-
R, B
-,r)d-r.
[76, (2.13),(2.16)].
in
(c),
ZL the controller
=
u
(A
s
6e')u(t
-
4.5.13(2)
b(e -A) -'b(z)
(A+ ao)
=
(2
1 and A G
=
of Remark
where
1
fo
6eX2(t)'+
-
rzae',
E R.
is
a
controllable
of the minors
of
Assigning
4.5
2nael
since
is
w
proof subtracting
an
Multiplying multiple
appropriate
+
s
2rae aa
W
Since
b is
finally
of
order
in
Ka)3`"-'S+a
degree two, obtain
to
0
s
+ nae'b
proper
of
row
-w
0
perform
j
steps
two
last
rational
and
02
2w 0
have to
we
a
+
the
a
0
-1
W
in
derives
one
row,
82
0
polynomial
a
by
row
127
proceed as polynomial
the
-Kaz
-
Function'
we can
of the first
a
,3
transformations
the last
0
det
--
From this
constant.
nonzero
a
of Theorem 4.5.10.
the Characteristic
This
row.
leads
to S
a
det
=
-Kaz
0
0
8
-1
0
2 W
+
a
K1
K21 + K22
where the constants
are
)3
K,
-
Hence the
Kaeclw2
=
b
)3
K22
1
-
W.2
is of the
controller
I
j
given by
K21
.,
2
-w
K3
S+a
-
0
2 u)
+
s
0
W2
K3
bi
2w
-
=
W2
form 1
u
=
-Klxl
e"rX2(*
K21X2 + K22
-
7')dr
K3X3-
-
0
is the
This
same
controller
controllers
the various
problem since required in order to ment
in this
by different
obtained
as
derived
[75],
in
X2 is the
case
determine
this
the
input
[75, (24)].
methods in
Of
for the assign-
simplest one only variable whose integration is the
is
u.
Remark 4.5.15 In the
next
admits
a
chapter
so-called
it
will
be shown that
first-order
(A, f3, 0,b)
the controller
representation,
E
R[z]'Xr
x
L
R[z]
rxn
x
e.
given
one can
R[z],Xr
x
find
R[z]
in
(4.5.4)
always
matrices mXn
such that
[-F,
ker,c
Using
such
I a
G]
-
?b
the
[A
for
in,
[46]
g.,
WE
Lr:
Bb BC_ b Aconnection
=
Aw + f3x;
(x)
[b
w
to
rings, state-space systems with respect to stabilizability, over
b
u
=
6w +
of the interconnection
equations
+
system shows the close
feedback e.
I (xT, uTT 111
representation,
(.:t) This
=
the
classical
0,
[46,
p.
W
framework
39].
given by
(x)
which has been studied see
are
bxj.
dynamic extensively
of
128
4 Behaviors
Notice
that
will
in
deg,
R[Z]nxn
c-
of B
entries
form
the x,
u
Fx,
=
(4.5.6).
see
As
constant.
4.5.16
(A, B)
with
Systems
0. It
=
Corollary Let
Delay-Differential
of we derived a controller Example 4.5.14(c) feeds back a segment of the trajectory simply show next, this is always possible if the matrix B is
hence G we
of
a
there
n
=
R[Z]nx' in R[z]
x
coprime
are
exists
be
controllable
a
Then for
-
feedback
a
[
det
pair monic
every
F E
matrix
and suppose
polynomial
Ho,pxn
that
such that
A -B
sl-F
(4-5-13)
a.
I
the
[s, z]
E R
a
-
particular,
In
the above conditions
PROOF: Let
that
131 get
a
(4.5.11). hence
first
=
(3, 0,...,
I
det
Adding of
[-F,
Smith-form
s'
-
det
=
G]
I
this
leads
sI
for
[sl
-
some
det U
=
det V
F E H ?nxn and G (=- 'H 0'P
=
1 such
"
as
0'sp
(s p)g e Ro,p for all p G R[z] UAU-1, -Bi], multiplied by 3-1g, to
yields
-
assertion
and
-
the
I
V1
in
Ho,p. Consequently,
-UB
F, entries
[-VFIU sI-
det
=
A -B I
]
VF1U
G
7
on
reachable
the
following
pairs
is
easily
seen
by resorting
x
0'P
n'
to
a
B.
the section
case
in
to
UAU-'
-
with
it
as
(4.5.13).
system
a
easy
to
obtain
for
R[s, z],
a,,
ao (=-
R[z],
is
+ ao G
(A, B)
above to the pair
assignable
F=
satisfying
G]
of g
F, which has
for
coefficient + ais
-
of
row
Example 4.5.17 Let us apply the result In
I
properness
first I
The additional
not
with
(4-5-13).
establishing
Weclose
c
-F
matrix
some
(R[z])
By the assumption on B, the first row of 131 is Rlxm where,3 =7 - 0. As in the proof of Theorem 4.5.10
UAU-1 -B.1
-
the
a
for
0)
sl
The strict
row
and V G G1 ..
(A, 13).
pairs
UBVis in Smith-form.
:=
of the form we
Gln(R[z])
U G
by reachable
met
are
1
al
-
which is reachable
but 3.3.
every
the controller
ao'-' 0
Hence the feedback
01
ao
.
R
law is given
ul=al(o,)xl+fl(ao(o,)xl)(---r)d-r+ao((7)X2) 0
(4.5.7),
ring R[z], see (i) in Section monic polynomial prescribed
in
the
over
2X2
0'P
by U2=0-
a
Biduals
4.6
Nonfinitely
of
Biduals
4.6
of
Generated
Nonfinitely
Ideals
129
Ideals
Generated
chapter we want to return to the Galois-correspondence 4.1. We saw in Corolin -Section derived and behaviors, J- JA4 C Hq In submodule for that M M 4.1.8 finitely generated lary every I I is true also I Jwhether or not the identity this section we will investigate for ideals of H that are not finitely generated. This question is not quite in the of this chapter about behaviors, since, spirit
At
en& of this
the
submodules
between
=
-
=
IJis not
in the
behavior
a
ing equations specific context
I
cL
pw
=
all
p G
If many defin-
4.1, where only finitely definition
that
But
time-invariant)
(linear
0 for
of Definition
sense
allowed.
were
of
Jw
=
was
tailored
anyway
to
our
types of delays. In from a general yet convenient,
DDEs with
certain
artificial, of a Using the more general and natural definition the space IJbehavior as simply being a set of trajectories [87, Sec. 1.3/1.4], time-invariant in the class of linear, of course, (autonomous) behaviors. falls, to these quite But even without general ideas, we believe an investiresorting IJ-' in our work, because a description I fits naturally gation of the identity of the nonfinitely generated ideals is already available from Section 3.4. In fact,
this
sense,
Definition
behavioral
4.1
is somewhat
of view.
point
=
that
in Theorem 3.4.10
we saw
I
each ideal
hP_
(M)
0
19 H is of the form
Ih
G
H,
E
MI,
and Mis an admissible set of denominators where p E R[s, z] is some polynomial it depends decisively show by some simple examples, for p. As we will on the characteristic
identity
the
(in
ideal-theoretic
give on
I
indication
an
the
of the
zeros
not
characteristic
=
I-L
-L
p and the
polynomial holds
In
terms, say) appears of how to translate The
zeros.
denominator
set
Mwhether
or
characterization
particular, algebraic the examples be impossible. Instead, IJ-J- into a condition I the identity general case can then be carried out almost
true.
an
to
=
straightforwardly. Due to the infinite
ing IJ-
arises.
sections C
L
we
character In order
have to make
to the of the situation, one main difference about solution the further information get
to
use
of
topological argument. determined by its exponential variety of I. This is what some
precedspace
we precisely, monomials, or, in one would certainly
More
IJ- is completely words, by the characteristic theorem on expect, but for a formal proof one has to make use of Schwartz's translation-invariant generated ideals (or modules) it was subspaces. For finitely in X possible to circumvent these arguments due to the division properties
will
need that
other
Let
us
begin with
4 Behaviors
130
Definition
of
Delay-Differential
Systems
4.6.1
Let I C H be any subset.
Define
the characteristic
n v(P-)
v(r)
of I to be
variety c
c.
PEI
The elements
V(I*)
of
ord,\ (1*)
define
are
minpc,
1
called
the characteristic
ordx (p*)
of the set I.
zeros
For A G C
No
E
Remark 4.6.2 Let
I
H be
C
admissible
ideal
an
given
ord), (I*)
I
as
of denominators
set
ord,\ (p*)
=
((p))
=
for p. It
where p E R[s, easyto see that
max
-
ord,\ (0)
from Proposition 3.4.8 that in the special ated, the set M is finite, say M 01}. M E V) ICM(01, 01) (see the proof of 3.4.8) =
for
.
all
.
.
,
A G C. This
Now
nential
(4.6.1)
with
It
follows
in
I
=
ord,\ (1*)
and
an
(4.6.1)
where I is
case
and M is
A E C.
all
finitely
gener-
(pV)-')
where
=
ordx (p* 0
above.
the dual Ijprepared to describe precisely This in leads turn variety V(I*,). directly I-L 1 Recall the notation the bidual ek,A (t)
we are
characteristic elements
coincides
for
OEM
Recall
=
z]
(M),
is
9 C in terms to =
.
of the
of the description tkeAt for the expo
a
monomials.
Theorem 4.6.3
9
Let
(R, C), equipped
C'
=
compacta in all
ii-
=
derivatives.
nker_,
p
=
the
with
Then for
spanCjek,A
I
of uniform
topology
every
A G
subset
V(I*),
I CH
0 < k
0 0 and E Go. In case R is a field, the relationship between rational functions and their realizations is fully understood, including and uniqueness In particular, issues. each proper rational minimality matrix is realizable. For the general case, realizability is always guaranteed, but the too, results and uniqueness depend on the ring. Since we will concerning minimality take a slightly different approach, we will not go into the details but refer the operators
and the references
taken
are
therein.
into
On the other
=
=
-
=
-
=
=
reader
[12,
to
Ch.
=
4].
For systems over fields, an alternative the transfer funcapproach for realizing has been proven very fruitful, It is known as the polynomial too. model of Fuhrmann or simply the Fuhrmann-realization. Unlike the above-mentioned tion
approach, it does not realize factorization a polynomial
the sequence Q-1P of
on
construction
Let
us
ipating function
in
now
return
i/o-systern is
detail to
in
Section
DDEs. It
with
given by C(sI
input -
of coefficients
G,
5.2 where it is u
A)-'B
easily
seen
see
will
that
[33,
Gi but is rather
34].
We will
be utilized
(5.1)
is
for a
present our
strongly
based this
purposes.
nonantic-
and output the formal transfer y. Moreover, which looks formally + E G R(s,,z)P` just
5 First-Order
the transfer
like
we are
if
function
possible,
for
discrete-time
realizing given system kerC [P, Q]
not interested a
ker,C [P, Q]
8"' (A, B, C, E)
where
In Section
be found.
general
a
the ring
but rather
R[z].
137
However, realize,
want to
behavior
B"t (A, B, C, E),
(5.2)
and
that
the formal
A, B, C,
the
about
(5.3)
and E
transfer
are
the
matrices
-Q-'P
function
system because it 'neglects
realizing
behaviors
the transfer
realizing
over
function
external
as
=
consequence,
than
stronger
in
as
we saw
information
As
part.
is
4.3
full
the
contain
not
tonomous
systems
the transfer
in
Representations
in the
sense
of
to
does
the
(5.3)
au-
is in
function.
realization. to behavioral our approach sequel we -wish to explain briefly the operator representation (5.1) is completely polynomial, ring R with its nice algebraic turns out to be of little we properties help. Instead, first will the problem for systems kerC [P, Q] with a polynomial treat kernelAs menrepresentation [P, Q]. This brings us back to the Fuhrmann-realization. tioned utilizes above, that procedure, developed for systems over fields, polyIn the
Since
the
factors,
nomial
As
will
Q say, for realizing
P and
Section
transfer
the
function
G
=
-Q-1P.
procedure of Fuhrmann also works the more general of DDEs, and, even'more, context provides a behavioral alization. The latter is somewhat surprising since the procedure takes place of the delay -differential a completely polynomial only the surjectivity setting; we
want
order
In
one.
to
the
5.2,
very
be needed to establish
will
operators havioral
in
see
to
the realization
present
the transfer
strength
the
prove
in
an
even
function
realization
as
a
of Fuhrmann's
general
more
construction, In fact, setting.
in re--
in
bewe
as we
show, the procedure works for arbitrary systems where a polynomial ring of the on a module A, representing mutually commuting operators acts surjectively function that the operators underlying are algebraically space. It will be crucial for this will allow us to apply the theorem of Quillen/Suslin independent, on modules over projective polynomial rings so that we get a free module as an will
abstract tion
state
along
differential
is
In
reason
twofold.
provides
for
passing
On the
also
framework
abstract
classes
Section
general
in this
out
this
introduce
concrete
of systems, delays as well
noncommensurate
equations.
be carried The
various
(possibly)
with
Wewill
space.
with
eventually,
5.2
such as
the
as
in the next
differential
certain
sec-
systems
of partial procedure will
systems
realization
framework.
to this
quite
hand
general
think
that
setting
instead
of
sticking
this
to DDEs
situation, generality is needed exactly what kind of structure to work. On the other the context does more hand, general advanced methods. It is literally the same construction as it one
more
for
clarity
as
we
in
more
exhibits
it
the procedure more not'require would be for systems of DDEs.
Having mial
delays
finished
systems, in
our
we
Section
considerations
will
return
5.3.
Only little
to
in the general setting delay-differential systems extra
of abstract with
work is needed to derive
polyno-
commensurate a
criterion
for
Representations
5 First-Order
138
of kerc [P, Q], along with realizability with entries is an arbitrary operator
for
Fuhrmann-realization quence of the
the
R. For
in
"numerator
procedure
elimination
procedure, sufficiency matrix"; necessity
where
realization
a
of -Section
we
will
utilize be
will
[P, Q]
now
a
the conse-
4.4.
will of minimality be addressed. Unin this direction, answers one of which partial in a certain the Fuhrmann-realization is that for yields, sense, the best result kernel-representation. systems with a polynomial
Finally, fortunately,
the
in
last
In this
section
we
Systems
be
with
model of systems for which a realization the classes of systems reasons
the abstract
be
procedure being described systems
introduce
presented by this model illustrated throughout
will
As will
question
only provide
Multi-Operator
5.1
the
section
we can
later
For obvious
on.
be called
simply
will this
point-delays (The investigation
equations. partial Chapter 4 will be resumed in Section result concerning the formal transfer
5.3.)
also
close
of the
systems
of
framework
of
certain
of DDEs in the
Wewill
function
cover
but
noncommensurate
even
differential
multi-operator systems. not only differential
they
section,
this
section
systems
under
with
a
first
considera-
tion.
Let is
us
a
now
fix
field
arbitrary by definition, left multiplication. an
model for
abstract
the
commutative
polynomial K and
ring
a nonzero
multi-operator
the
K [zl,
.
.
sl K[zl,...,
zj,
,
.
divisible
polynomial
p induces
systems.
All
we
1 + 1 indeterminates
in
Z1,
s]-mo
over
,dule A Hence, map on A by
surjective The indeterminate s is distinguished merely because, which are explicit realizations and of first construct we will every
nonzero
a
need
in the
order section, for DDEs. For the time being there is no to (5.1) with respect to s, analogous particular meaning to s. Wewill also use the notation K[z] := K[zl,..., zi] for and K [z, s] for K [zi, the polynomial ring in the first 1 indeterminates Z1, S]. next
.
A matrix
RE
K[z, S]Pxq
the two
induces
K[z, S]q
K [z,
K[z, s]-linear s] P,
.
.
,
maps
Rp
P
and
AP,
Aq Just
like
for
delay-differential
and the notation in the
The carries
obvious
surjectivity over
kerK[z,,l
systems, R and iM
Ra.
a
both
K[z,,IR,
maps will resp.
kerA
simply
be denoted
R and im ARwill
by
way.
of the to matrices.
map
a
-4
pa for
each
nonzero
p G
R
be used
K[z, s] immediately
Systems
Multi-Operator
5.1
139
Lemma5.1.1
R E K[z,
Let
,Ipxq
K[z, s]-module.
be
For the verification
and utilizes
one
AR
=
simply
full
row
and A be any divisible
rank
AP.
selects
Q(adjQ)
identity
the
with
matrix
a
Then im.
nonsingular
a
(det Q)Ip.
=
full-size
Q of R
submatrix
.
model consists abstract of a polynomial our Summarizing, ring of 1 + I algedivisible module A. The following on a independent operators acting braically this model indeed show that concrete examples delaysystems, including covers differential equations with even noncommensurate delays as well as certain partial differential Webegin partial difference equations. equations or discrete-time with
Example A
Let
(ai f ) (t)
(Delay-Differential (R, C) and denote by f (t -ri). Then R[aj,
Systems)
5.1.2
C'
=
=
-
delay-diff rential
..
.
,
ai
the
shift
(71,
D]
is the
of length -ri > 0, i. ring of all linear, time-invariant
operator
e.
of the form
operators
N
P"jo'j'
P
1
0
o
...
o-,"
o
D',
E
p,,i
(5.1.1)
R,
i=O
E'
where
means
this
sum
R[aj,...'
of'an
ture
al,
being
The space A naturally carries for p as in (5.1.1) and
finite.
DI-module.
Precisely,
the struc-
f
E
A
one
has N
Epv,i
pf (t)
(')
f
(t
-
(v, -r)),
t E
R,
VEN' i=O
E,1=1
(v,r)
where 0'1'
and D E
al,
independent elements in the ring linearly the
the standard
denotes
Endc (A) mutually over Q, then al,
.
.
.
,
al,
product..It
scalar
commute.
D
Endc(A). To see this, let p Endc (A) implies in particular
in
operator
zero
vj-rj
is obvious
Moreover, algebraically
if -ri ......
rl
that
E R are
independent be as in (5.1.1). Then p being for the exponential functions are
the identity
eo,.\
0
pv,jA'e-A(vI'r)
peo,.\(t)
=
IEN'
(v, -r
Since
Thus,
zero.
elements
:
p.
6971
The
following
whenever
al, D] delay-di 'fferential
are
[25,
(p,,r)
R[a,, it
.
.
.
,
is known that
class
have been studied
is
v
a
for
alltERand
: p in N', all coefficients polynomial ring in 1 + with
operators the operators
of systems a unified
in
e\t
all
A EC.
i=O
arises
are
1 noncommensurate
surjective
in multidimensional
manner
in
[84].
p,,i
G R must
1 indeterminates.
on
be Its
delays.
From
theory.
They
A.
systems
140
Representations
5 First-Order
Example
(a)
following
the
Let
K be
(b)
R
(possibly
finite)
field
1 a(n)
1+1
of formal
K-algebra
the tackward
shifts
with
be the on
or
ring A
R1+1;
on
Kj,
E
where
n
(nl,...,
=
in 1 + I indeterminates
power series
nj+j),
over
K. Via
truncation
a(nl,.
zi
]
aXj+j
and let
nEN'+1
be the
'9
49xl
=
or
a(n)tnj.....tn,+, 1
A:=
'9
K[
C and let
or
acting on A C'(R'+',K) distributions complex-valued
operators
the space of real-
K be any
Let
fields
differential
D'(R'+'),
Systems)
situations.
of the
one
partial
of
(Multidimensional
5.1.3
Consider
nl+,)tni
..'
)
tnt+1
1
1+1
nEN1+1L
a(nl,...,
nl+,)tni
+
ni
tnt+,, 1+1
1
nEN1+1
the space A can be endowed with the structure This is usually the framework for discrete-time
It
the
back
in
Ehrenpreis
of
cogenerator for.
consequences
kernels
itself
[84]
to
the
details.
the
in
RIT
In
if R G
particular,
[84, (61),
36]
p.
=:
mark 4.1.9
analogous for
Remark 2.11,
acting
on
paper
[84].
be needed.
the
case
L, preventing
(1)
of multidimensional
(1)
and
Ri
kerA R2
=
XR, for
does not
hold
on
=
1, 2,
for
correspondence to
that
to extract
one
for the
has
AR1. AP.
one
has
S]P2 XPI. and Re4.1.5(a), (R, C) As we saw in operator ring R[U, -!dt2L]
=
C' the
-
to
be covered
only the suriectivity
section
merely for
systems in Examples 5.1.10
=
the
X E K [z,
systems
in the next
im
refer
Theorem
L
true
are
K[z, s]'xP
AR
im
some
4.1.4,
point-delay-differential be used
i
=
we
us
similar
quite
and R2 c-
K[z, S]piXq,
in
(3). will,
is
and
for
Wewould like
4.1.
Rj
E
of PDEs. The purposes
our
result,goes large
of this
Part
case
has rank p, then
R2
For the construction.
Parts
SjPXq
K[z, S]pxq
Proposition where H is acting
results
property
E
K[z,s]
K[z, S]pxq
kerA R, 9 kerA R2 the
im
For matrices
the
More important A. In essence, the
operators acting Aq and -operators in K[z, discussed in Section systems on
kerK[z, ,]
33].
P.
in
some
large
a
is not needed for
for
delay-differential from [84] for future reference. following (1) [84, (46), p. 30] For matrices R,
Recall
systems,
strong algebraic injective cogenerator
have
[84, (54)
see
and Palamodov
property
reader
interested
between
(2) (3)
z1+1]-module.
-,
in 1 + I indeterminates.
ring
situations
A constitutes
module
the
K[z, s]-modules,
of
work
injective the
common:
category to
polynomial
main result
structure in
above the operator ring is a of [841 that these
cases
the
is
K[zl,..
a
multidimensional
[123, 122].
cf. In all
of
a more
and 5.2.6.
detailed
by the
(2)
will
discussion
Multi-Operator
5.1
In
Theorem
R(s, z)P11 the formal
setting
crucial
role.
Example A
=
an
we
the
for
following
the
situation
1 + 1 indeterminates.
in
ring
K[z, s]-module
natural
a
-Q-1P
function
be introduced
context
141
same
general will
E
way
poly7 play a
Functions)
polynomial
for
The same is true
In this
transfer
C'+P of DDEs. In the
C
(and will)
can
section.
(Transfer
be any carries
formal
the
kerL [P, Q]
function
of this
5.1.4
K[z, s] K(z, s)
introduced
i/o-system
transfer
nomial
Let
4.2.3
of
Systems
Then the
space
given by multiplication.
structure
the space N
I
fis
NE
Z, fi
c
K(z)J
i=-00
of formal spaces
this
Laurent
are
will
in
s-1 with
coefficients
K[z, s]-modules,
behavioral
setting,
as we
series
divisible
make precise
thus
in the field
theory coincides with Example 5.1.8.
the
K(z).
both Clearly, For approach applies.
abstract
our
transfer
function
framework
in
Remark 5.1.5 it Throughout this section, Even distinguished. more, if the is true for over K, same
xi,
x1+1
...'
yi,
.
.
G11+1(K)
yl+
,
.
y1+1T
(yi,...,
play
does not
=
1,
having one of the variables algebraically independent elements
any role
are
where
x1+1T
A(xi,...' K1+1.
+ b
particular, K[yl,...,Yl+,] the polynomial ring can also be presented where the shift as R[D,ol we 1,...,ol replaced 1], operators difference and changed the ordering of by the corresponding operators the indeterminates. In this the list reads as of,operators case, (zl,...,zi,s) 1 is the distinguished The 1, (D, a, a, 1), so that s al operator. realization with procedure of the next section would then result in a first-order respect to the last difference 1, provided that certain operator ol necessary for
some
A
E
K[xl,...,xl+l].
For
and
instance,
in
b
E
-
-
-
.
.
.
,
In
Example 5.1.2, -
=
-
-
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 of all trajectories in Aq that space, it consists are governed e. g., by a system of (higher order) equations, delay-differential differential difference or partial equations, partial equations, 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 these systems formally we introduce along with the desired first-order representations. Let
us
return
K[z, S]pXq
the
to
the
kernel
5 First-Order
142
Definition
5.1.6
s]
Let R E K[z,
(a)
Representations
('+P)
"
be any matrix.
The module
kerAR is called
(b)
there
(or
behavior
a
exists
a
kerA R,
The behavior a
number
(A, B, C, E)
system) simply
or
E
A+P I Ra
in
A+P.
01
R,
the matrix
said
is
to
realizable,
be
if
E N and matrices
n
E
fa
=
K[Z]nxn
x
K[Z]nxm
K[Z]pxn
x
K[z]Px'
x
such that
kerA R
=
8 '(A,
B, C, E)
(5.1.2)
where
BA'(A, In
1(yU)
B, C, E)
such matrices
case
exist,
A-+P
E
we
3
G
EUI
sx=Ax + Bu
An
:
Y=Cx +
(A, B, C, E)
quadruple
the
call
x
(5.1-3)
a
realization
of kerA R. The system
said
is
to
8 '(A,
=
Ax + Bu,
y
=
Cx + Eu
first-order
a
B, C, E)
the internal
vector
x
is called
C(sI-A)-'B+E (5.1.4)
The term
first-order
that
induced
by
itdoes
the first
or
not
make
where the matrices
cases,
A few remarks
are
issaid
behavior
length n of (A, B, C, E).
of the realization
or first-order representation is linear equation in (5.1.4)
sense
the
The
to be theformal
transfer
(5.1.3).
of
As has been discussed
s.
K(z, s)Pxl
R and
(5.1.4).
of
behavior
the dimension E
kerA
of
representation
The matrix of
(5.1.4)
the external
is called
function
the fact
ter,
be
sx
to call are
system refers, respect
with
DDEs in the introduction
for
(5.1.4)
a
this
constant,
to
of course, to the operator
Chap-
to this
system. Only for certain be might appropriate.
state-space
in order.
Remark 5.1.7
(i)
It
is not
admit
a
clear
whether
each external
kernel-representation,
always examples above except possibly delays, where this is unknown. be eliminated.
Wewill
behavior
in other see
words,
below that
for
of
a
first-order
whether this
delay systems
is indeed
with
system does
latent
variables
the
case
for
can
the
noncommensurate
Multi-Operator
5.1
(ii)
Remember the differential
rally
the
that y E
of operators acting Rom the surjectivity
variables
the
AP such that
with
possible
the
variables
are
u
the
tute
u
system;
ces
That
Chapter means
4,
that
Again,
equations.
s]PI
('-4-P)
to
with.
start
accordance
4.2.1),
tion
of the
external
point
restrictive
Let
us
always place see also variables;
behavior
the external
ext
B
for
-Q-1P
=
C(sI'-
the list
exists
R'has
that
section
applies restricted
way,
we
only
to
with
S]
eliminate
for
a
is
com-
not
(see
m
a
lineaxly
kerA R,
behavior
systems
Remark 4.2.2
in
assume
K [z,
the first
into
full
independent
and systems since
to
matri-
to
will
linearly
p
functions
variables
We know from
see
Defini-
components
comment
on
this
of
examples above.
Functions) of
where A is either
(5.1.4)
(Y) U
(A, B, C, E) =
where
there
of view.
(nansfer
5.1.8
the
-
implies
input/output
of
the free
again Example 5.1.4
Consider case,
definition
our
the definition
discuss
Example
with
we
A'
5.1.2
we are
is indeed crucial: delays, this restriction ideal domain, it is in general not possible principal the associated dependent rows of R without.changing Example 5.1.10 below.
In
natu-
to
the number of outputs equals this will be true in more gen-
mensurate
(iii)
E
delay-'
immediate
is
it
below.
by exactly
the system is governed Except for the case of transfer
equations.
A'
this
that
Put another
that
5.2
on
discussion
procedure in the next kernel-representation, meaning that
R E K [z,
A
-
the realization
rank
Section
the
see
of
systems
rank p, see Theorem 4.2.3. the number of independent
row
of sI
meaning that for each u ker.A R. For the examples
,
of the
outputs
erality. However,
a
5.1.4, again (u' exception of systems with noncommensurate delays, the constieven maximally free, so that the last p variables
delay-differential
the
free,
are
of
variables
143
These concepts generalize on A and can be applied
4.2.1.
context
(5.1.3).
behavior
free
maximally
and
Definition
from
system
to
of free
notions
Systems
c-
is
K(z, s)
or
K(z)((s-1)).
In this
simply
Am+P
y
=
(C(sl
-
A)-'B
+
E)u
kerA [P, Q],
A)-'B
+ E is
any factorization
of the
formal
trans-
matrices polynomial Thus, the ex(which, of course, exists). ternal behavior B admits full rank a row B, C, (A, E) kernel-representation a be[P, Q] E K [z, s]Px 4+P). Obviously, for this special choice of A, realizing havior kerA [P, Q] is the same as realizing the rational function -Q-1P, that is, matrices as finding -Q-1P (A, B, C, E) satisfying 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
5.1.9
Systems)
of
D and Example 5.1.2, where s of noncommensurate lengths -rl ...... the first-order rl, In the situation
E A,c'x Y
we use
If 1
1,
=
the
we
fact
in
a
behavior
[P, Q]
al"'1
the
in
o
a,"
o
...
4.4.1(a)
that
0 1
B, C, E)
and AZ/
1
,
B, C, and E,,
the external
(kerA [sI
CE
are
A,
-
behavior
-BI)
of Definition
sense
always admits Q is nonsingular.
5.3.1 that it and C- 'HP' ('+P)
as
VEN'
:=
know from Theorem
Proposition
reads
in R.
entries
137 (A, is
a'
notation
with
matrices
operators
1]'E,,o,'u,
+
VEN'
constant
shift
(5.1.4)
are
VENI
E'C,,o,'x
=
al
system in
E'B,o,'u,
+
vEN1
where
a,,
=
4. 1. Moreover, see in we will kernel-representation kerA [P, Q] where In particular, u is maximally free, see
a
Theorem 4.2.3. It
remains
an
question
open
cf.
[127',
delays,
Example
(Multidimensional
5.1.10
similar
whether
noncommensurate
234]
p.
results
Then each external
structure.
mits
a
kernel-representation This
system.
be
can
seen
as
are
for
true
with
systems
3.1].
Sec.
Systems)
A be any of the spaces in Example 5.1.3
Let
[41,
and
"
BA
behavior
the corresponding moduleof ada B, C, (A, E) system (5.1.4) with
of rank p, the number of output Define the matrix
variables
y in
the
follows.
sl -A -B M:=
IM
0 -
Since each submodule
[Y, P, Q]
E K [z,
of
K[z, s]'+'+P
s]
for
some
T
kerK[z,s]M It
follows
rk
0
=
p.
rk
=
Furthermore,
p
In
(sl
-
easily
A)x; verifies
see
Lemma5.1.1.
Put
y
=
Cx + Eu.
-
A)P
Mu x
G
=
A'
Then Pu +
=
0 for
(5.1.9)
0.
all
u
such that
Qy
=
E
A'.
Bu
0 and
=
one
Multi-Opeiator
5.1
Mu
(5.1.9)
hence
follows.
det(sl
=
This
-
considered
(b)
as
an
equation
Write
R
again
completed
to
a
C(sI Q :7
det
[P, Q]. By
=
unimodular
0 and
K(z, s). Since both (5.1.8) is established. the
Gl,+m+p(K[z
8])
c
R
I
be rewritten
as
:=
have full
matrices
[-X, R]
matrix
be
can
91 -A -B
U1 U2 R
-X
0
IM
C
E
rIP1
P
I
0
_
+U2
01] [CE
(A, B, C, E)
