Systems & Control Letters 37 (1999) 31–37

Notes on the de�nition of behavioural controllability

Je�rey Wooda ;∗;1, Eva Zerzb

aISIS Group, Department of Electronics and Computer Science, University of Southampton, Southampton, SO17 1BJ, UKbDepartment of Mathematics, University of Kaiserslautern, 67663 Kaiserslautern, Germany

Received 25 March 1998; received in revised form 1 November 1998

Abstract

We take another look at the behavioural de�nition of controllability for discrete one-dimensional (1D) systems, and itsextension to multidimensional (nD) systems de�ned on Zn or Nn. We suggest that the current de�nition for nD systems isinappropriate for systems de�ned on the domain Nn, and that care has to be taken even with the 1D de�nition. We propose anew de�nition which applies to all standard classes of discrete nD systems. c© 1999 Elsevier Science B.V. All rights reserved.

Keywords: Controllability; Behavioural approach; Multidimensional systems

1. Introduction

Controllability is a concept fundamental to anysystem-theoretic paradigm. The behavioural approachof Willems has introduced a new intuitive idea ofcontrollability in terms of the system trajectories [6].The de�nition of behavioural controllability given

by Rocha in [3] is an extension of the original 1Dde�nition to discrete systems de�ned on Z2, and gen-eralizes directly to systems de�ned on Zn. For suchsystems, it is a highly natural de�nition, and leads toa variety of useful characterizations of controllability[3, 7]. The de�nition given in [3, 7] also makes sensein the context of discrete systems de�ned on Nn,but in this case, the characterizations fail. Also, wehave undesirable phenomena such as the existenceof non-trivial behaviours which are both autonomousand controllable. This suggests that the de�nitionis not suitable for systems on Nn. Furthermore,

∗ Corresponding author. Tel.: +44-1703-595776; fax: +44-1703-594498; e-mail: jjw@ecs.soton.ac.uk.1 The �rst author would like to acknowledge the sponsorship

of the EPSRC, under grant no. GR=K 18504.

Rocha’s de�nition causes these problems even whenapplied to 1D systems on N, which indicates thatgreat care needs to be taken in the formulation of the1D de�nition. As discussed, for example, by Rosen-thal et al. [4], a change in the signal domain can a�ectmany important system-theoretic properties.In this paper we propose a new de�nition of be-

havioural controllability, which works equally well inthe cases of systems de�ned onZn and onNn. The uni-�ed de�nition can also be applied to systems de�nedon “mixed” signal domains Zn1×Nn2 . The new de�ni-tion is equivalent to Rocha’s de�nition in the Zn-case,but it admits the characterizations given in [7] for bothclasses of systems. In particular, this work establishesthat for any discrete nD behaviour, controllability isequivalent to minimality (in the transfer class).

2. Previous de�nitions of controllability

A system behaviour B is the set of its associatedtrajectories w [6]. A (1D, linear, time-invariant) dif-ference behaviour is a discrete behaviour that can bewritten as the kernel of a polynomial matrix R=R(z),

0167-6911/99/$ - see front matter c© 1999 Elsevier Science B.V. All rights reserved.PII: S0167 -6911(99)00004 -3

32 J. Wood, E. Zerz / Systems & Control Letters 37 (1999) 31–37

B = ker R(�). Here the e�ect of applying R is de-rived from the substitution of the left shift operator� for the indeterminate z [6]. Di�erence behavioursare called autoregressive behaviours by several au-thors; we avoid this nomenclature due to its undesiredstochastic connotation.A multidimensional (nD) system is one for which

the trajectories are functions of more than one inde-pendent parameter. Thus a discrete nD behaviour isa subset of (kq)T, where in the most general caseT = Zn1 ×Nn2 , q is the number of components (e.g.inputs plus outputs) and k is some �eld (normally Ror C). A trajectory w ∈ (kq)T is a multi-indexed se-quence taking its values in kq. We refer to the set Tof multi-indices as the signal domain, and we alwayswrite n= n1 + n2 (in much existing work either n1 orn2 equals zero).In the nD case, a (linear, shift-invariant) di�erence

behaviour is one which can be written as the kernelof a polynomial matrix R = R(z) in n indeterminatesz := (z1; : : : ; zn). The action of R on w ∈ (kq)T is givenby replacing each indeterminate zi by the backwardshift operator �i in the ith direction, i.e.

(�iw)(t1; : : : ; tn) = w(t1; : : : ; ti−1; ti + 1; ti+1; : : : ; tn):

For any a ∈ T , we write �a for the compound shift�a11 · · · �ann . Thus a di�erence behaviour takes the formB= ker R(�) = ker R(�1; : : : ; �n)

= {w ∈ (kq)T: R(�)w = 0}:In the following, we deal exclusively with di�erencebehaviours. The matrix R is called a kernel represen-tation matrix ofB. The module generated by its rowsdetermines and is uniquely determined by B, i.e. itdoes not depend on the particular choice of the repre-sentation matrix [1, Corollary 2:63, p. 36]. Similarly,if

B= imM (�) = {w ∈ (kq)T: ∃l ∈ (km)T such thatw =M (�)l}

for some polynomial q× m matrix M , we say that Bhas an image representation with image representa-tion matrix M .The property of having an image representation is

important for many classes of systems [2, 3, 6–8] andit is equivalent to several other interesting properties.In particular, a behaviour has an image representationif and only if it is minimal in its transfer class (combine[1, Theorem 7:21, p. 142] with [7, Theorem 5] or [8,Theorem 1]). Two behaviours are said to be transferequivalent if the rows of their kernel representation

matrices have the same span over the �eld of rationalfunctions. The class [B] of all behaviours that aretransfer equivalent to B is called the transfer class ofB. There exists a unique minimal element Bmin ⊆Bin [B], and Bmin has an image representation. We saythatB itself isminimal in its transfer class ifB=Bmin.For systems de�ned on Z [6] or Z2 [3], the existence

of an image representation has also been characterizedin terms of concatenability of system trajectories thatare speci�ed on subsets of the signal domain that are“su�ciently far apart”. This is a natural concept ofcontrollability in the behavioural setting.As we have indicated, there are some subtleties in

the original 1D de�nition of controllability [6], whichreveal themselves upon applying the de�nition to thesignal domain N. Due to these �ne points, we willpresent several versions of Willems’ original de�ni-tion. Throughout the paper, di�erent de�nitions ofcontrollability will be distinguished by indices.

De�nition 1. A 1D di�erence behaviour B (de�nedon T =Z) is said to be controllable(1) if there exists� ∈ Z+ such that for all w(1); w(2) ∈ B, there existsw ∈ B such that

w(t) =

{w(1)(t) if t ¡ 0;

w(2)(t − �) if t¿�:(1)

The original de�nition of Willems [6] allows thatlength � of the transition period depends on the tra-jectories w(1); w(2) to be concatenated. For di�erencebehaviours however, � can be chosen uniformly forall trajectories.Controllability(1) is obviously unsuitable for sys-

tems de�ned on N, since it allows speci�cation ofthe �rst trajectory only for negative times. We canget around this problem by moving the region oftransition, i.e. the time interval [0; �], to an arbitrarylocation.

De�nition 2. A 1D di�erence behaviour B (de�nedon T = Z or T = N) is said to be controllable(2) ifthere exists � ∈ Z+ such that, for any w(1); w(2) ∈ B,and any t0 ∈ T , there exists w ∈ B such that

w(t) =

{w(1)(t) if t ¡ t0;

w(2)(t − (t0 + �)) if t¿t0 + �:(2)

Clearly, controllability(2) is equivalent to controlla-bility(1) for di�erence systems with T = Z. Ingeneralizing controllability(2) to T = Z2 and hence

J. Wood, E. Zerz / Systems & Control Letters 37 (1999) 31–37 33

to T = Zn, Rocha [3] observed that for these systemsthe shift of w(2) in Eq. (2) is not important. The ex-isting nD de�nition [3, 7] requires an arbitrary metric,but we will �nd it convenient to introduce the follow-ing speci�c one:

d(T1; T2) = min{|t1 − t2|: t1 ∈ T1; t2 ∈ T2};where |a|=∑n

i=1 |ai| for a ∈ Zn.

De�nition 3. An nD di�erence behaviour B (de�nedon T = Zn1 × Nn2 ) is said to be controllable(3) ifthere exists �¿ 0 such that, for any w(1); w(2) ∈ B,and any regions T1; T2⊂T with d(T1; T2)¿�, thereexists w ∈ B such that

w(t) ={w(1)(t) if t ∈ T1;w(2)(t) if t ∈ T2: (3)

Controllability(3) is equivalent to the earlier de�ni-tions for T =Z. In the case T =Zn, we can prove [7]:

Theorem 4. A behaviour B with signal domain Zn iscontrollable(3) if and only if it has an image repre-sentation.

To see that controllability(3) is inappropriate forsystems de�ned on T = Nn, we need the notion ofautonomy. A set of free variables (inputs) [7, 8] ofa behaviour B⊆(kq)T is a set of components of wwhich are collectively unrestricted by the system lawsR(�)w= 0. The maximum size of such a set is calledthe number of free variables of B, and it has beenshown [1, Theorem 2.69, p. 38] to equal q− rank(R),where R is an arbitrary kernel representation matrixof B. An autonomous behaviour is one that is devoidof free variables, or equivalently, one whose kernelrepresentation matrices have full column rank.Consider the following 1D examples:

B1 = {w ∈ kN: w(t + 1) = 0 for all t ∈ N}= ker R1(�); R1 = (z):

B2 = {w ∈ (k2)N: w1(t + 1) + w1(t) = w2(t);w1(t) = w2(t)− w2(t + 1) for all t ∈ N}

= ker R2(�); R2 =(z + 1 −11 z − 1

):

B3 = {w ∈ (k2)N : w1(t + 3)− w1(t + 2)+ w2(t + 4) + w2(t + 3) = 0 for all t ∈ N}

= ker R3(�); R3 = (z2(z − 1) z3(z + 1)):

B4 = {w ∈ (k3)N : w1(t) = w2(t); w2(t + 1)=w3(t + 2) for all t ∈ N}

= ker R4(�); R4 =(1 −1 00 −z −z2

):

B1 is zero everywhere except at time t = 0, when itcan take any value. B2 is also zero except at t = 0; 1.These two behaviours are obviously autonomous, butare also controllable(3), since in this example the con-catenation conditions of controllability(3) are trivial.Note also that both representations can be regarded asbeing in classical state-space form w(t + 1) = Aw(t),where A= 0 in the �rst example and

A=(−1 1−1 1

)

in the second. As systems without inputs, they arecertainly not state-point controllable in the classicalsense. We see, furthermore, that neither B1 nor B2 isminimal in its transfer class (the trivial behaviour {0}is the minimal element in each case), so the charac-terization of Theorem 4 must fail for T =Nn.The behavioursB3 andB4 are also controllable(3),

which can be seen taking separation distances of �=52 ;32 , respectively. However, they do not satisfy the

conditions of controllability(4) to be de�ned below,and therefore as we will see they admit no image rep-resentations (alternatively, we can argue that R3 andR4 are not left prime).The existence, in particular, of autonomous control-

lable non-trivial behaviours is counterintuitive, andsuggests a problem with the de�nition of controlla-bility. Trajectories in a system with signal domainNn can behave in a di�erent way close to the originthan arbitrarily far from it, and controllability(3) def-inition does not require that the concatenating trajec-tory w exhibits this close-to-the-origin behaviour ofthe trajectory w(2). Therefore, if the control problemrequires the reproduction of an entire signal w(2), thecontrollability(3) condition will be insu�cient. Essen-tially, this complication comes from the fact that �i isnot an invertible operator on (kq)T for T =Nn, unlikein the Zn case. The de�nition of controllability needsto be adapted to take account of this.

3. A new de�nition of controllability

We will shortly present our new de�nition of be-havioural controllability. This requires some prelimi-nary notation. We will �nd it convenient to de�ne the

34 J. Wood, E. Zerz / Systems & Control Letters 37 (1999) 31–37

following obvious action of a shift on any subset T1of T :

�aT1 := (−a+ T1) ∩ T = {t ∈ T : t + a ∈ T1}:The diameter of a bounded set T1⊂T is�(T1) = max{|t − t′|: t; t′ ∈ T1}:Finally, given a polynomial matrix M , its support isde�ned as follows: For M =

∑a∈Nn Maz

a, with coef-�cient matrices Ma over k,

supp(M) = {a ∈ Nn: Ma 6= 0}:A monomial ordering is a total order on the set ofmonomials {za: a ∈ Nn} such that 16za, and za ¡ zb

implies za+c ¡ zb+c, for all a; b; c ∈ Nn.

De�nition 5. Let B be an nD di�erence behaviourwith signal domain T=Zn1×Nn2 . ThenB is said to becontrollable(4) if there exists �¿ 0 such that for allT1; T2⊂T with d(T1; T2)¿�, and for all w(1); w(2) ∈B, and all b1; b2 ∈ T , there exists w ∈ B such that

�b1w = w(1) on �b1T1 and �b2w = w(2) on �b2T2;

(4)

i.e.

w(t) ={w(1)(t − b1) if t ∈ T1 and t − b1 ∈ T;w(2)(t − b2) if t ∈ T2 and t − b2 ∈ T:

(5)

In that case, we also say that B is controllable(4)with separation distance �.

The de�nitions of controllability which we have al-ready presented are easily seen to be special casesof controllability(4). In particular, controllability(3),which works perfectly well for T = Zn, is seen to becontrollability(4) for b1 = b2 = 0 (and it is easy toshow that these de�nitions are equivalent for such T ).For T =N, controllability(4) is in fact equivalent tocontrollability(2).As commented above, to derive previous character-

izations of controllability in the case T = Zn for thegeneral case, it is su�cient to prove the following:

Theorem 6. Let B be an nD di�erence behaviourwith signal domain T =Zn1 ×Nn2 . The following areequivalent:

1: B is controllable(4);2: B has an image representation.

Moreover; ifM is an image representation matrix forB; thenB is controllable(4) with separation distance�(supp(M)).

Proof. “1 ⇒ 2”: Let Bmin denote the minimal ele-ment of the transfer class of B. Then Bmin has animage representation. Let R and Rmin be kernel rep-resentation matrices of B and Bmin, respectively;then there is a rational function matrix X such thatRmin = XR. Write X = N=d with a polynomial matrixN and a polynomial d.Suppose that B is controllable(4) with separation

distance �. We will prove that B =Bmin, and hencethat B has an image representation, as required. IfBmin = (kq)T (that is, Rmin = 0), or if d is a constantpolynomial, this is trivial, so assume otherwise. Letw(1) denote the zero trajectory and let w(2) be an arbi-trary trajectory of B. We will prove that w(2) ∈ Bmin.Next, let deg(d) denote the exponent in Nn corre-

sponding to the initial term of d (with respect to anarbitrary monomial ordering), and de�ne

T1 = {a+ s: a ∈ supp(Rmin);s ∈ Nn\(deg(d) +Nn)}; (6)

T2 = b2 +Nn; (7)

where b2 is chosen such that the distance betweenT1 and T2 is greater than �. Although T1 and T2 arecontained in Nn, we consider them as subsets of T .Finally, let b1 = 0. Apply the new de�nition of con-trollability; let w ∈ B be a connecting trajectory. Notethat R(�)w=0, and so (dRmin)(�)w=(NR)(�)w=0.Let Rmin(z)=

∑a R

mina za, where the summation runs

over all a ∈ supp(Rmin). As w = w(1) on T1 we havethat (Rmin(�)w)(s) =

∑Rmina w(a+ s) = 0 for all s ∈

Nn\(deg(d) + Nn). But d(�)(Rmin(�)w) = 0, henceRmin(�)w = 0 on the whole of Nn⊆T . This is due tothe fact that, for any monomial ordering, a solution vof (d(�)v)(t)=0 for all t ∈ Nn, is uniquely determinedby the values of v on the set Nn\(deg(d) +Nn).Since �b2T2 =Nn, w(2) is equal to a shift �b2 of w

on all of Nn, and so (Rminw(2))(t) = 0 for all t ∈ Nn.This argument can be re-applied in each hyperquadrantof Zn1 × Nn2 , for appropriate choices of T1; T2 andhence w, and so Rminw(2) vanishes on the whole of T .Therefore w(2) ∈ Bmin, and so B=Bmin.“2⇒ 1”: Suppose thatM is an image representation

matrix of B. We will prove that B is controllable(4)with separation distance � := �(supp(M)), thus estab-lishing the �nal claim also. This part of the proof

J. Wood, E. Zerz / Systems & Control Letters 37 (1999) 31–37 35

follows the lines of Rocha’s original proof [3]. Letw(1) = M (�)l(1); w(2) = M (�)l(2) be given, and letb1; b2 ∈ T be arbitrary. Let T1; T2⊆T be such thatd(T1; T2)¿�. Now for any t1 ∈ T1; t2 ∈ T2 and anys; s′ ∈ supp(M), we must have|t1 − t2|¿d(T1; T2)¿�= �(supp(M))¿|s− s′|:This yields that

(T1 + supp(M)) ∩ (T2 + supp(M)) = ∅and now we see that the following is well de�ned:

l(a) =

l(1)(a− b1)ifa ∈ T1 + supp(M) and a− b1 ∈ T;

l(2)(a− b2)if a ∈ T2 + supp(M) and a− b2 ∈ T;

0 otherwise:

Then w :=M (�)l ∈ B and

w(s) = (M (�)l)(s) =∑

a∈supp(M)Mal(a+ s):

For t ∈ �b1T1, say t = s− b1, s ∈ T1, we have(�b1w)(t) =w(s) =

∑a∈supp(M)

Mal(a+ s)

=∑

a∈supp(M)Mal(1)(a+ s− b1)

=w(1)(s− b1) = w(1)(t);so �b1w=w(1) on �b1T1, and similarly �b2w=w(2) on�b2T2. Thus B is controllable(4).

Combining the new result of Theorem 6 with exist-ing characterizations of behaviours with image repre-sentations [1, 7, 8], we obtain a further corollary; thisrequires a preliminary de�nition. A polynomial ma-trix R is generalized factor left prime (GFLP) withrespect to a ring D (k[z ]⊆D⊆ k[z; z−1]) if the exis-tence of a factorization R= LR1 with polynomial ma-trices L, R1 and rank(R)=rank(R1), implies that thereexists a D-matrix E such that R1 = ER.

Corollary 7. The following are equivalent:

1: B is controllable(4);2: B has an image representation;3: B is minimal in its transfer class; i.e. B=Bmin;4: Any kernel representation matrix R of B isGFLP with respect to the ring D that corre-sponds to the signal domain of B; that is

D= k[z1; : : : ; zn1 ; zn1+1; : : : ; zn; z−11 ; : : : ; z

−1n1 ]

for T = Zn1 ×Nn2 ;

5: There is no proper sub-behaviour of B with thesame number of free variables;

6: B is divisible; i.e. for any r ∈ D\{0} it must holdthat B= r(�)B.

Moreover; if B is controllable(4) and autonomous;then B= {0}.

This result applies to all standard classes of discretesystems. For completeness we mention that Pillai andShankar [2] have established the equivalence of 1 and2 for continuous systems, and in fact, this is su�cientto establish Corollary 7 for all classes of systems con-sidered in [1].An obvious question to consider now is the relation-

ship between controllability(3) and controllability(4).In fact, we can characterize this, which requires thefollowing de�nition:

De�nition 8. An nD di�erence behaviourB is said tobe permanent if �iB=B for i = 1; : : : ; n.

Permanence coincides with shift-invariance in thecase of behaviours de�ned on Zn, and is therefore triv-ial for di�erence behaviours with that signal domain.For behaviours over Nn however, shift-invarianceimplies only �iB⊆B, and permanence is strictlystronger than that. For a 1D behaviourB overN, withkernel representation matrix R, we mention withoutproof that permanence is equivalent to the conditionthat R does not lose rank at the origin.

Lemma 9. An nD di�erence behaviour B is contro-llable(4) if and only if it is controllable(3) and per-manent.

Proof. “if”: Let B be controllable(3) with separationdistance �. Let T1, T2⊂T be such that d(T1; T2)¿�and let w(1); w(2) ∈ B and b1; b2 ∈ T be given. By per-manence, there exists v(i) ∈ B such that w(i) = �bi v(i)

for i = 1; 2. Let w ∈ B be the connecting trajec-tory of v(1), v(2) with respect to T1; T2 according tocontrollability(3). It is now easy to see that w is alsothe desired connecting trajectory of w(1); w(2) with re-spect to T1; T2 and b1; b2.“only if”: Controllability(4) implies controllabi-

lity(3) (b1 = b2 = 0) and, by condition 6 of Corollary7, we have that controllability(4) implies B = �iBfor all i.

The result of Lemma 9 should be compared toStaiger’s [5] characterization of “remergeability”, as

36 J. Wood, E. Zerz / Systems & Control Letters 37 (1999) 31–37

discussed in [4]. Given this relationship between thetwo types of nD controllability, it is now natural toask whether controllability(3) admits an algebraic orpolynomial matrix characterization. Controllability(4)is equivalent to the condition that R is GFLP over k[z ].Given Lemma 9, it is tempting to conjecture that theweaker notion of controllability(3) is equivalent to Rbeing GFLP over k[z; z−1]. This is supported by theexamples of the previous section, where the represent-ing matrices are GFLP over k[z1; z−11 ], but not overk[z1]. But although controllability(3) does indeed im-ply generalized left factor primeness over the Laurentpolynomial ring (see Lemma 10 below), the converseis not true in general: Consider the 2D behaviour

B= {w ∈ k(N2) : w(t1 + 1; t2) = 0; w(t1; t2 + 1)=w(t1; t2) ∀t1; t2 ∈ N}

= ker R(�1; �2); R=(

z1z2 − 1

):

Its trajectories are constant along the t2-axis, and zeroeverywhere else. Thus, B is not controllable(3) al-though R exhibits the property of being GFLP overk[z1; z2; z−11 ; z

−12 ], but not over k[z1; z2].

Thus, controllability(3) of a behaviour with signaldomain T =Nn corresponds to a property of its ker-nel representation matrices that is weaker than “GFLPover k[z ]” and stronger than “GFLP over k[z; z−1]”.This supports our suggestion that controllability(3) isnot appropriate for systems de�ned on Nn at all.

Lemma 10. Let B be an nD di�erence behaviourwith signal domain T = Zn1 × Nn2 . If B = ker(�)is controllable(3); then R is GFLP with respect tok[z; z−1].

Proof. Let Rmin be a kernel representation matrix ofBmin. Then Rmin is GFLP over k[z ] (and thus overk[z; z−1]). Let g and g′ denote the number of rowsof R and Rmin, respectively. A variant of the proof “1⇒ 2” in Theorem 6 shows that ifB is controllable(3),then there exists b ∈ Nn such that �bB⊆Bmin,that is, w ∈ B ⇒ �bw ∈ Bmin ⊆B. It follows thatker Rmin(�)⊆ ker R(�)⊆ ker �bRmin(�), or

k[z ]1×g′zbRmin ⊆ k[z ]1×gR⊆ k[z ]1×g′Rmin :

This implies that the rows of R and Rmin generate thesame module over k[z; z−1], and so the behaviours onZn represented by R and Rmin coincide. Since Rmin isGFLP over k[z; z−1], R must also be.

In the proof of Theorem 6, we used controllability(4)only with b1=0 in order to prove thatB had an imagerepresentation. Thus controllability(4) is equivalent tothe same property but with the restriction b1 = 0. Forthe signal domain T =Zn, we can take b2 =0 as well,returning us to controllability(3). This is, of course,not possible for T =Nn, though in this case a di�erentchoice of b2 or of b1 characterizing controllability(4)is possible, as we see following the next de�nition.

De�nition 11. For ∅ 6= T1⊆Nn; let the cw-in�mumof T1; denoted by �(T1); be de�ned as the minimalelement of the smallest ( possibly in�nite) interval ofNn containing T1.

Corollary 12. Let T=Nn. The behaviour B iscontrollable(4) if and only if the following twoequivalent conditions are satis�ed:(a)There exists �¿ 0 such that for all T1; T2⊂Nn

with d(T1; T2)¿� and for all w(1); w(2) ∈ B; thereexists w ∈ B such that

��(T1)w = w(1) on ��(T1)T1

and

��(T2)w = w(2) on ��(T2)T2;

i.e.

w(t) ={w(1)(t − �(T1)) if t ∈ T1;w(2)(t − �(T2)) if t ∈ T2:

(b)There exists �¿ 0 such that for all T1; T2⊂Nnwith d(T1; T2)¿� and for all w(1); w(2) ∈ B; thereexists w ∈ B such that

w = w(1) on T1

and

��(T2)w = w(2) on ��(T2)T2;

i.e.

w(t) ={w(1)(t) if t ∈ T1;w(2)(t − �(T2)) if t ∈ T2:

Proof. Both conditions are special cases of the con-dition controllability(4). Conversely, if a behaviourBis controllable(4), then by Theorem 6 it has an imagerepresentation. By applying a variant of the proof “2⇒ 1” of Theorem 6, we can easily establish thatB sat-is�es (a) or (b). Note that for T1; T2 de�ned accordingto (6), (7), we have �(T1)=b1=0 and �(T2)=b2.

Note how condition (b) coincides with controllabi-lity(2) in the 1D case (T=N) when we consider “past”

J. Wood, E. Zerz / Systems & Control Letters 37 (1999) 31–37 37

T1 = {0; : : : ; t0 − 1} and “future” T2 = {t0 + �; : : :}.Clearly, d(T1; T2)¿� and �(T2) = t0 + �.

4. Behavioural and classical de�nitions

Following Willems [6], we now describe the re-lationship between behavioural controllability and“state-point controllability”, i.e. classical controlla-bility (or reachability) in terms of the state space.Let Bs; i denote the state-input behaviour of a 1D

classical state-space representation, i.e.

Bs; i = ker R(�) with R= (zI − A ...− B);where A and B are matrices over k. In other words,Bs; i contains all pairs (x; u) that satisfy x(t + 1) =Ax(t) + Bu(t) for all t ∈ T . The behaviour Bs; i issaid to be state trim if any state x0 occurs as an initialstate in the sense that there exists (x; u) ∈ Bs; i suchthat x(0) = x0. For T = N, Bs; i is always state trim,since for any state x0 we can simply set x(0) := x0 andallow the trajectory w = (x; u) to evolve according tothe system equations.

Lemma 13. (1) For T=Z; (A; B) is state-point con-trollable if and only if Bs; i is controllable(4) andstate trim.(2) For T =N; (A; B) is state-point controllable if

and only if Bs; i is controllable(4).

Proof. The proof is essentially due to Willems [6].For ease of exposition we consider only the cases k=R;C. Other �elds can be dealt with in a similar way,or by using an alternative trajectory proof.In view of Theorem 6, we only need that – in the

1D case – Bs; i has an image representation i� R is leftprime. In the case T =N, this signi�es that

R(�) = (�I − A ...− B)has full row rank for all � ∈ C, which is the classicalHautus condition for controllability of the matrix pair(A; B). For T =Z, left primeness of R is equivalent toR(�) having full row rank for all 0 6= � ∈ C. So weneed the additional requirement “(A

... B) has full rowrank”, corresponding to �=0. But this is precisely thecondition for Bs; i to be state trim [6].

For the case T =N, it is easy to see that Lemma 13does not apply to controllability(3): We can take

the simple earlier example B = {w :w(t + 1) =0; t¿0}, treated as a state-space representation withno inputs and A= 0; this behaviour is controllable(3)but certainly not state-point controllable.

5. Conclusions

The concept of controllability is central to the be-havioural approach. Great care has to be taken withthe behavioural de�nition of controllability, as equiv-alent de�nitions for the signal domain T =Zn becomeinequivalent for T=Nn. In particular, the existing def-inition for multidimensional systems is inappropriatein the latter case, as it leads to undesirable phenom-ena such as the existence of non-trivial controllableautonomous systems.We have proposed a new de�nition which ap-

plies to both cases. The new property is equivalentto many characterizations already given for the caseT = Zn, and is also equivalent to state-point control-lability in the setting of the classical 1D state-spacemodel.

Acknowledgements

We would like to thank the anonymous reviewersfor their helpful comments, and in particular one re-viewer for proposing Lemma 9.

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