Introduction to a Behavioral Approach of Continuous-Time Systems

Introduction to a behavioral approach of continuous-time systems S.J.L van Eijndhoven & J .M.Soethoudt Department of Mathematics and Computing Science Eindhoven University of Technology P.O. Box 513 5600 MB Eindhoven The Netherlands Abstract. In this paper we discuss an open problem in the behavioral approach to continuous-time systems: what conditions have to be imposed on the external signals such that the behavior can be described by a set of differential equations? To keep clear sight we will deal with a 5I50-system and make some smoothness-assumptions. Keywords. Behavior;continuous-time 5150 systems;

1

Introduction

taken in C(lR.), which means

From thye momen t of its introduction the behavioral approach to system theory has been developed in all kind of subareas. Nevertheless there are some basic open problems left. One of them is the transition from a behavior to an AR-relation for continuoustime systems. This problem has been solved completely in the discrete-time case (see [2)) . However we cannot extend the proof to continuous-time systems. So, the question is: what conditions have to be imposed on the external signals such that the behavior can be described by a finit.e set of differential equations? illostly times the signals are considered to be in lY(IR), the distribution-space ofSchwartz. We think that D'(lR) ha.s not the necessary structure to work with . Mostly, external signals are in C( lR) or in L1,loc(IR), which are both Frechet-spaces. It is in this context of Frechet-spaces, where we can use the closed-graph theorem, that we can find an explicit descript.ion for the behavior. In this paper we work with signals in C(IR) . However the results we give can be, mutatis mutandis, extended to behaviors wit.h Ll,loc(lR)-signals.

2

Behaviors with one input and one output

Consider a behavior B that contains two external continuous signals. \Ve assume that the system has one input and one output. Under this assumption the external signals can be transformed into the input and the output by a simple linear transformation. So that is where we start from. Note that such a simple transformation will not affect properties such as shift-invariance and c1osedness, which are the ones we are going to use . The behaviors we want to describe are linear and shift-i7lvariant. Suppose we choose WI to be the input and W2 the output. The input signals will be

,-

-.)

7rIB:= {WI E C(JR) 13w,ECCm)[(WI,W2) E Bl} Assume that CCO(JR) ~ 7rIB. The assumption on shift-invariance means

Since we consider behaviors with one input and one output one could get the idea to look at B as the graph of an operator . We want to describe B by a differential equation, like (WI,W2) E B <:? 51WI = 52W2 where .5\ and 52 arc linear differential operators . Definition 2.1 Let us define bac(IR) as Ihe subclass of righl-conli1l110us fU7lcliolls pet) such Ihat theTe exisls aTE JR, with

pet) { p(t)

=0 = peT)

t <-T t?. T

Consequently the support of p, supp(dp), [-T,T] .

IS

l1!

Lemma 2.2 A continuous linear functional on C(IR) corresponds uniquely to some p E bac(IR) In ih e fol/owing way V/EC(lR) [

£(J) =

fm f(t)dp(t) =: p(J) )

Definition 2.3 The convolution-operator CIl is defined for all f E C(JR) by (CIlf)(t) := p(atf). Suppose that N2 is finite-dimensional. Then it is a linear, shift-invariant and closed subspace in C(JR). From the theory of mean-periodic functions as described in [1) we have the following lemma: Lemma 2.4 If W is a linear, shift-invariant and closed subspace of C(JR) thell we only need two elements in bac(IR) to describe W. W equals the intersection of the kernels of the cont;oluiion-operators corresponding to these tIro elements.

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SYSTEM STR.UCTURE AND CONTHOL 3 . 5 September 1002. PRAGUE, CZECHOSLOVAKIA

Hence we have that N2 can be generated by some

what sense? Assume

jjl, jj2 E bac(IR) and because of the finite dimension

it is even true that we can characterize Nz as the kernel of some linear differential operator with constant coefficients, say Sz. S2 has order n if n:=dim N2 •

B is closed in C(JR) x G"(IR)

Lemma 2.6 If we define

iJ := { (wl,Szwz)1 then

Remark 2.5 With the theory of mean-periodic functions this assumption on the dimension of N2 can be translated into an assumption on the zeroes of the Fourier-transform of the jj'S (=cospectrum of jj) characterizing N2. 0 The interpretation of this assumption is that, if there is no input, B is finite-dimensional. To every Wl corresponds at most a finite-dimensional affine subspace of wz's. Note that we imposed smoothness-conditions on W2 here in order to apply S2. This could be done in some generalized sense, but we do not go into detail about that. Now suppose we have the following smoothnesscondition: :

{W2 E G(JR)

13w1E r. 1 B[(Wl, W2)

E Bl}

<; Cn(JR) (2)

iJ

(Wl,WZ) E B }

is closed in C(JR) x C(JR).

Now we have a very strong result to characterize V. Theorem 2.7 Let J( be a closed operator from C(JR) to C(JR), GOO(JR) <; V(I<) and J(atICOO(R) "t J(IcOO(R)' Then J( = Sl Cl' for some linear differential operator Sl and I-' E bac(JR).

=

V satisfies the conditions in the theorem and therefore it is a convolution-operator times a differential operator. Thus (Wl' W2)

E B {:} S2WZ

= SlCI'Wl

(4)

for some jj E bac(JR) and Sl in LDO(IR). All together we have the following result: Theorem 2.8 The following statements are equivalent:

• B is a linear, shift-invariant closed subspace of C(JR) x cn(JR). Nz = {W2 E cn(IR) I (0, wz) E B} is finitedimensional.

then <::} S2W2 = SlCI'Wl (for some Jt E bac(IR) and some lin ca l' diffferenlial operators SI,S2'

• (Wl' tu2) E B

and to every tul E 7rl [) corresponds at most one S2W2' With this assumption we created the possibility of introducing an operator. At this moment we introduce the subs pace

\ Ve can consider iJ as the graph of an operator, say V, since for every tul E 7rl B there exists exactly one S2W2 E G(JR) such that (Wl' W2) E B. Hence, using this relationship, we can define V as an operator from C(JR) to G(JR) with V(V) = 7r l B, as follows:

Remark 2.9 Note that the operators SI and Cl' are not unique, because, if we think in terms of Fouriertransforms, it is possible to show that they can exchange zeroes (if there are). There is only one thing to be taken care of: since SI has real coefficients its characteristic roots i).. have to be symmetric with respect to the real a.xis. Hence exchanging is possible under the constraint that if not i).. E JR then its conjugate should be exchanged too. 0 This is not exactly the result we wanted to achieve. Nevertheless we can consider it as a generalization. Let us have a closer look at V = SI Cl" We can prove that the Fourier-transform of p, F Il , has either infinitly many zeroes or none at all.

So here we have, for the first time, a kind of representation for the behavior B . It remains to determine the operator V. From equation (3) we obtain that, since B is linear and shift-invariant, V is linear and should commute with the shift-operators at. To find an explicit expression for V we need a little more. It is some condition on the c10sedness of B. This is a common condition for continuous-time behaviors. Nevertheless the problem arises: what has to be closed in

Lemma 2.10 If FIl has no zeroes, then C Il is a conslant times a shift-operator. Our aim is to find a condition such that FI' has no zeroes, which means that C Il has to be injective and equals a constant times a shift-operator. This is also equivalent to IV:= {WI E 7rIBI (WI,O) E B} is finite-dimensional Hence we have the following corollary:

Corollary 2.11 The

follouling

statements

are

equivalent: • B is a linear, shift-invariant closed subspace of G(JR) x Gn(JR) (m arbitary). Ld NI = {WI E ?rIBI (WI,O) E B} and N2 = {W2 E C(JR) I (0, w:z) E B} be finitedimensional. u'here GOO(JR)

~

?rIB

~

C(JR)

• (WI' w:z) E B {::> S2W:z = SIUdWI (for some a E JR, J.l E bac{lR) and some linear differential operators SI, S2. Conjecture We think that the condition NI and N2 are finitedimensional can be replaced by: B is locally-specified (see [2]'p.184). This would be a consequence of Length(supp(J.l)) ::; L {::> B is L - complete (see [2]'p.184). Note that if J.l E bac{lR) has support in one point it is, upto a multiplicative constant, a shift-operator.

What we see from the result in theorem 2.8 is that :

topology. What we see from this example is that e.g. the behavior

cannot be described in this set up. This example explains the problems around the c1osedness-assumption. \Ve know that c10sedness (in some sense) is something unavoidable to ask for when dealing with continuous-time systems. Nevertheless, we do not find a systemtheoretical interpretation of it. We can show that in some generalized sense not only the smoothness-condition on W2 disappears but the problem on c10sedness has gone too. It involves introduction of new spaces that have the same structure as G(JR). The technicalities that go along with this introduction disturb the clear view on the idea we used. Hence we left out that part.

References [1] Schwartz,L. Theorie generale des fonctions moyenne-periodiques, Ann.of Math. 48 (1947), pp .857-929. [2] Willems,J .C. Models for dynamics, dynamics reported 2 (1988). pp.I71-269.

Remark 2.12 It goes without saying that the roles of Wl and W2 can be exchanged. So, it is not necessary to specify, in advance, what is the input and what is the output. \\'e only assume the existence of a relation between the signals. Once we have the description, we can define what is the input and what is the output; for instance, on basis of causality (if that is possible at all). 0

Now the work is done we will return to an assumption that is an eyesore to us. It is the assumption about the closedness of B in theorem 2.8. In spite of the fact that everything seems nice the closed nessassumption on B is not appropriate, since it excludes a lot of behaviors we do not want to exclude. Example: Consider the behavior B described by

Then B is closed in GI(JR) x Gl(JR). Clearly (Wl,W:?) (Itl,ltl+ 1) is not in B since It I is not a differentiable function. However, it is continuous and hence it is the limit of a sequence of polynomials, say (Pn(t» nE N' Thus (Pn(t),Pn(t) + 1) nE N is a sequence in B that converges to (Wl' W2). If we assume B to be closed in C(JR) x G(JR), then this behavior is excluded because it is not closed in this

=