Modules and Behaviours in nD Systems Theory

Multidimensional Systems and Signal Processing, 11, 11–48 (2000)c© 2000 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.

Modules and Behaviours in nD Systems Theory

JEFFREY WOOD* [email protected] Group, Dept. Electronics and Computer Science, University of Southampton, Southampton, SO17 1BJ, UK

Received January 15, 1999; Revised July 17, 1999

Abstract. This paper is intended both as an introduction to the behavioural theory of nD systems, in particular theduality of Oberst and its applications, and also as a bridge between the behavioural theory and the module-theoreticapproach of Fliess, Pommaret and others. Our presentation centres on the notion of a system observable, firstformally introduced by Pommaret, and uses this concept to provide new interpretations of known behaviouralresults. We discuss among other subjects autonomous systems, controllable systems, observability, transfermatrices, computation of trajectories, and system complexity.

Key Words: behavioural approach, system observables, module theory, duality, torsion

1. Introduction

In this paper we provide an introduction to the combined use of commutative algebra andbehavioural theory in the study of multidimensional systems. The main ideas and resultsin this paper are not new, and are due to many researchers; one of our aims here is to drawlinks between the approaches of different authors. The general approach and concepts ofbehavioural theory are due to Willems in the context of 1D systems, and were partiallyextended to the 2D case by Rocha. The systematic use of commutative algebra resultingfrom the duality theory to be discussed in section 2.4 is due to Oberst. This duality allowsthe extension of much of Willems’ and Rocha’s work to nD systems. Module theory isalso prevalent in the 1D/nD approaches of other authors who do not work with behaviours,including Fliess and Pommaret/Quadrat; we also discuss some of this work. Many othershave also contributed to the development of the behavioural/algebraic theory.

One of the principal advantages of behavioural theory as an approach to the study ofnDsystems is that it provides a framework in which it is possible to examine a huge rangeof system-theoretic concepts. Controllability, observability, poles and zeros, state-spacemodels, Rosenbrock system matrices, and many other concepts can all be analysed withinthe behavioural framework. The difficulty with the behavioural theory of nD systems isthat it is hard. Many fundamental results, which are not too difficult to prove in the 1Dcase, become remarkably tough even in 2 dimensions, and we require more sophisticatedtools to efficiently attack the nD case.

One such “sophisticated tool” is the theorem of Oberst which exhibits a duality betweenmodules and behaviours given by constant linear differential/difference equations. This en-ables the application of the huge and powerful machine of commutative algebra to problemsin nD linear systems theory.

* Jeffrey Wood is a Royal Society University Research Fellow.

12 J. WOOD

The use of module theory in control systems is not new, and goes back at least to thework of Kalman [23], in which the fundamentals of the classical 1D state space theory wasderived using algebraic techniques. The duality of Oberst provides a closer link betweenmodules and systems, and it is not surprising therefore that it has proved so effective inproducing basic new results. Modules also play a central role in the work of Fliess andco-workers (e.g. [3, 7, 9, 10, 11, 13, 27]), and in that of Pommaret and Quadrat (e.g. [38,39, 40, 41, 42]). Another fundamental algebraic theory is provided by Lomadze [26], whouses sheaves and modules to define 1D linear systems.

The main difficulty in using module theory is that it is highly abstract. However, theapproaches of Fliess and colleagues, and of Pommaret and Quadrat, illustrate that it ispossible to give a direct interpretation to much of the algebraic theory. Central to this is thenotion of a module element as a “system observable”, the explicit formula of which is dueto Pommaret [38, 39, 42].

The idea of a system observable has not previously been applied in conjunction withbehavioural theory, and this new perspective comprises the main novel contribution of thispaper. We also hope to provide a useful introduction to the behavioural approach to nDsystems, and to the use of algebra and duality within this approach. It is not possible to givea comprehensive overview of nD behavioural theory, as the area has already become quitelarge. Instead, we concentrate here on what we believe are the most fundamental conceptsand results of the theory.

In section 2 we introduce the behavioural approach to nD systems, the concept of a systemobservable, and the duality between the module of system observables and the behaviour.In section 3 we look at constrained observables (i.e. torsion elements) and autonomoussystems, and in section 4 at controllable systems. Section 5 looks at the structures whicharise when the system variables are partitioned, e.g. into inputs and outputs, or latent andmanifest variables. Section 6 examines the problems of constructing an initial condition setand of computing a system’s “complexity” using Gröbner basis theory. In section 7 we listreferences to some of the other works which have recently contributed to progress in this area.

Most of the theory described in this paper applies only to systems given by finite sets oflinear differential/difference equations with constant coefficients. The algebraic tools canbe extended to the non-constant case, as explained in the work of Fliess and of Pommaret andQuadrat (e.g. [10, 11, 13, 39, 41, 42]). Similar ideas can even be applied in the non-linearcase, using tools of differential algebra (e.g. [7, 9, 11, 38, 39, 50, 67]). To our knowledge,the only significant contribution to nD behavioural theory in the case of variable coefficientsis the paper [18] by Fröhler and Oberst, which extends the module-behaviour duality to onecase of interest. An alternative framework to the behavioural approach, which to date hasbeen developed further in the case of non-constant coefficients, is the formal theory of pdes.Beyond a few comments here and there, we will not discuss this alternative approach anyfurther; consult the work of Pommaret and Quadrat (e.g. [38, 39, 40, 41, 42]) for more details.

2. Observables and Duality

We begin by formally introducing differential/difference behaviours and their representa-tions and listing the signal spaces of interest. Then in section 2.2 we discuss the concept of

MODULES AND BEHAVIOURS 13

a system observable and the relationship (duality) between the module of observables andthe behaviour. In section 2.4 we review Oberst’s result giving the properties of the dualityfunctor.

2.1. Signals and Systems in the Behavioural Approach

The behavioural approach to 1D systems is due to Willems [37, 54, 55, 56], and centres onthe concept of the system behaviour, which is the set of associated trajectories. Formally,we define asystemas a triple(A,q,B), whereA is a set called thesignal space, q ∈ Z+is the number of components, andB ⊆ Aq is thebehaviour. The elements ofB are calledtrajectories. We will make little distinction between a system and its behaviour.

In all cases of interest (as listed below),A will have the structure of a module over asuitable ring of differential or difference operators; in particular,A is always a vector spaceover a fieldk, which is taken to beR or C. Specifically, throughout this paperA denotesone of the following signal spaces, with module structure as described:

1. The discrete signal spaceA = kNn, k = R,C, which is a module over the polynomialringR = k[z] = k[z1, . . . , zn], where the actionzi on a trajectoryw ∈ A is taken to bethe shift operatorσi , defined by:

(σiw)(t1, . . . , tn) := w(t1, . . . , ti−1, ti + 1, ti+1, . . . , tn) (1)

By extension, any element ofR has an action onA:

∀r ∈ k[z], w ∈ A r (z1, . . . , zn)w := r (σ1, . . . , σn)(w) (2)

2. The discrete signal spaceA = kZn, k = R,C, which a module over the Laurentpolynomial ringR = k[z, z−1] = k[z1, . . . , zn, z−1

z , . . . , z−1n ], wherezi acts as the

shift operatorσi , andz−1i as the inverse shiftσ−1

i .

3. The signal spaceA = C∞(Rn, k), k = R,C of all k-valuedC∞ functions onRn, whichis a module overR = k[z], wherezi acts as the partial derivative operator∂/∂ti . Thisaction is extended toR by:

∀r ∈ k[z], w ∈ A r (z1, . . . , zn)w := r (∂/∂t1, . . . , ∂/∂tn)(w) (3)

4. The signal spaceA = D′(Rn, k), k = R,C of all k-valued distributions onRn, whichis a module overR = k[z], wherezi again acts as the partial derivative operator∂/∂ti .

Throughout the paper, we consider the behaviourB ⊆ Aq to be the solution spacein Aq of a finite set of constant linear partial differential or difference equations inqdependent variables. Such behaviours will be calleddifferential behaviours anddifferencebehaviours(formerly,autoregressive behaviours) respectively.

The set of systems with differential/difference behaviours covers all linear time-invariantsystems dealt with by the classical 1D state-space framework and its nD analogues. Dif-ferential and difference behaviours are in particular linear and shift-invariant, and in the

14 J. WOOD

continuous case are also closed under partial differentiation. In each case any differentialor difference behaviourB is therefore a submodule ofAq; the ring action is the component-wise application of (2) or (3).

It is convenient to describe differential/difference behaviours using polynomial matrices(Laurent polynomial matrices whenA = kZn). The usage here is very similar to that inclassical systems theory; however, we do not consider Laplace or Z transforms; instead,the indeterminates are simply formalisms for denoting the derivative/shift operators. Thusin the continuous case, a polynomial matrix represents a differential operator in the senseof the formal theory of pdes (see e.g. [20, 39, 51]); in the discrete case, the situation isanalogous. Thus letE ∈ Rg,q; then the behaviour described byE is the submodule ofAq

consisting of allw satisfyingEw = 0, where the meaning ofEw is given by the action ofthe ringR onA. We say thatE is akernel representationof B, and we writeB = KerAE.Kernel representations are highly non-unique.

EXAMPLE:.

1. Consider the 3D differential behaviourB in 2 variables described by the equations

w1(t1, t2, t3)− ∂2w1

∂t2∂t3(t1, t2, t3)+ 2

∂w2

∂t1(t1+ 1, t2, t3) = 0 (4)

∂3w1

∂t21∂t3

(t1, t2, t3)− w2(t1, t2, t3)+ 3∂w2

∂t2(t1, t2, t3) = 0 (5)

We takeA to be the signal spaceC∞(R3,R). The behaviour is given by the polynomialmatrix

E =(

1− z2z3 2z1

z21z3 −1+ 3z2

),

i.e.

B = {(w1, w2) ∈ A2 | (4) and(5) hold} = KerAE

2. Fornasini–Marchesini model. Take the signal space to beRZ2(we consider the 2D

case); then the state-input behaviour described by the model is given by

Bx,u = KerA(I − z−11 A1+ z−1

2 A2 | z−11 B1+ z−1

2 B2), (6)

where A1, A2, B1 and B2 are matrices overR. We can of course also give a kernelrepresentation for the state-input-output behaviour.

3. Rosenbrock system matrix. With any (discrete or continuous) nD signal space, the

Rosenbrock system matrix

(T −UV W

)describes the equations

T(z)x(t) = U (z)u(t)

y(t) = V(z)x(t)+W(z)u(t)


The state-input-output behaviour is given by

Bx,u,y = KerA

(T −UV W

∣∣∣∣ 0−I

), (7)

and it is therefore not surprising that properties of the system are reflected in propertiesof the Rosenbrock system matrix.

For any matrixE ∈ Rg,q, we also define the following modules1:

ImRE := {v ∈ R1,q | v = x E for somex ∈ R1,g}, (8)

CokerRE := R1,q/ ImRE (9)

ImAE := {w ∈ Ag | w = El for somel ∈ Ag} (10)

Note the different subscripts used to denote different ring actions.

2.2. The Module of Observables

The most fundamental algebraic object associated with a system or behaviour is the moduleof system equations. Each system equation is an element ofR1,q, whereq is the numberof system variables. An element ofR1,q is a system equation precisely when it describes arelation which is satisfied by every trajectory. The set of system equations, also called theorthogonal module, is denotedB⊥ and formally defined [15, 55], [29, 2.18] by

B⊥ ={v ∈ R1,q

∣∣∣∣∣ q∑i=1

viwi = 0 for allw ∈ B}, (11)

whereviwi ∈ A is as usual interpreted as the signal obtained by applying the ring elementvi

to the signalwi , where the ring action is given by (2) or (3). Conversely, for any submoduleN ofR1,q, we define theorthogonal behaviour, denotedN⊥, by

N⊥ ={w ∈ Aq

∣∣∣∣∣ q∑i=1

viwi = 0 for all v ∈ N

}(12)

Any R-multiple of a system equation must be a system equation, and any finite sum ofsystem equations must be a system equation. In other words,B⊥ is a submodule ofR1,q.Furthermore, ifE is a kernel representation ofB then it is clear that each row ofE, ormore generally each element of ImRE, is a system equation. In fact, it follows from theexactness of duality to be discussed in section 2.4 that, for any kernel representationE,each system equation is anR-linear combination of the rows ofE, i.e.B⊥ = ImRE.

The set of system equations is an algebraic object associated with the behaviour whichis not dependent upon the choice of kernel representation. Indeed, quite a lot of theoryresults from the correspondences betweenB andB⊥ alone, and some authors refer to the

16 J. WOOD

relationship betweenB⊥ andB as a “duality”. This is strongly related to but not the sameas the duality discussed in the next section.

The explicit notion of an observable is due to Pommaret (e.g. [38, 39, 42]), and is alsoimplicit in the work of Fliess and colleagues (e.g. [7, 9, 10, 11, 13]). Intuitively, a systemobservable is a measurable function of the system variables. It is mathematically convenientto restrict the class of such functions according to the class of systems under consideration;in the current case we have the following formal definition:

Definition 1. Consider a system(A,q,B) whereB is a differential/difference behaviourwith signal spaceA over the fieldk. A system observableis any mapping fromB toA which can be computed as ak-linear combination of the system variables and theirderivatives (continuous case) or their shifts (discrete case).

Note firstly that observables are only defined with respect to a behaviour. Secondly, sincean observable can be computed as ak-linear combination of system variables and theirshifts/derivatives, as an operator we can represent it as an element ofR1,q. Thirdly, anobservable is defined as a mapping, rather than as such a representation/computation. Thisis crucial, as an observable may have distinct representations. Furthermore, given two suchrepresentations of observablesv1, v2 ∈ R1,q, we see that they represent the same observableif and only if v1w = v2w for all w ∈ B, or equivalently if and only ifv1 − v2 ∈ B⊥. Forexample, on the discrete scalar behaviour of constant functions, we havez(w) = 1(w) forall system trajectories, i.e. “z” and “1” represent the same observable. Clearly(z− 1) is asystem equation.

Thus an observable is an equivalence class of elements ofR1,q given by equating to 0all elements ofB⊥. That is,the set of observables can be identified with the factor moduleR1,q/B⊥. We callR1,q/B⊥ themodule of observablesof the system/behaviour, and denoteit by M . Fliess callsM thesystem dynamics[10]. In the case of continuous, especially1D, systems, an alternative more intuitive notation is to denote each observable by thequantity it represents, e.g. the observablee1 + B⊥ is denoted byw1, and the observable(e1+ze2)+B⊥ byw1+w2, etc. We will however avoid this notation, as it may be confusedwith the notation for components of trajectories.

NowB⊥ andR1,q/B⊥ are both finitely generated modules. In fact, any finitely generatedmodule over a polynomial or Laurent polynomial ring can be written in the formR1,q/N forsomeq and some submoduleN ofR1,q, and also any suchN is equal toB⊥ for some (unique)behaviourB in Aq. Hence any finitely generatedR-module is the module of observablesof some system, and we have a one-to-one correspondence between finitely generatedmodules and differential/difference behaviours. The rest of section 2 will demonstrate thatthis correspondence has some very strong and useful properties.

2.3. Duality

The module-behaviour correspondence can be formalized as follows:

THEOREM 1 [29, 2.13] If B is a differential/difference behaviour with module of observ-ables M, thenB = HomR(M,A), whereA is the signal space ofB. If M is any finitely


generatedR-module thenHomR(M,A) has the structure of a differential/difference be-haviour (depending onA), and M is the module of observables of this behaviour.

It is convenient for any (finitely generated) moduleM to use the shorthand

B = D(M) := HomR(M,A), (13)

where the signal spaceA is assumed to be known from the context. The behaviourB =D(M) is called thedual of M .

The duality (13) is explained as follows. LetM be the module of observables of somebehaviourBwith signal spaceA. Now the elements of HomR(M,A) can be constructed byassigning elements ofA to each element of some generated set ofM . One such generatingset ofe1+B⊥,e2+B⊥, . . .eq+B⊥, wheree1, . . . ,eq ∈ R1,q are the natural basis vectors.An element of HomR(M,A) is therefore identified with an element ofw ∈ Aq. However,the values assigned to the generators must also satisfy the relations among the generators,i.e. the equations ofB⊥. We must therefore havew ∈ B. Conversely, an elementw ∈ B isinterpreted as an element of HomR(M,A) in the following way. For anyv+B⊥ ∈ M , say

v + B⊥ = v1(e1+ B⊥)+ · · · + vq(eq + B⊥), v1, . . . , vq ∈ R,the valuew(v + B⊥) ∈ A is given by

w(v + B⊥) = v1w1+ · · · + vqwq

In other words,w(v + B⊥) is the value of the observablev + B⊥ at the trajectoryw. Thisgives the correspondence between HomR(M,A) andB.

Also, the elements ofM are contained in HomR(B,A), by definition of an observable.In general, however, the set HomR(B,A) is much large thanM .

SinceR1,g for anyg is a free module, i.e. has a generating set with no relations, its dualis given by

D(R1,g) = Ag (14)

We remark that, in the discrete case, the dual can equivalently be given byD(M) =Homk(M, k), i.e. the classical dual from functional analysis [29, 3.11], [60].

2.4. Exactness of Duality

The mappingD(·) = HomR(·,A) which sends finitely generated modules into behavioursis useful because it has certain algebraic properties which create a strong relationshipbetweenM and D(M). In fact, the duality is functorial, and as such, there must also bea meaningful notion of the dual of a map. Therefore consider two finitely generatedR-modulesM1,M2 and their dual behavioursB1 = D(M1),B2 = D(M2), and suppose weare given a mapφ: M1 7→ M2 of R-modules. We construct a mapD(φ): B2 7→ B1 asfollows: for anyw2 ∈ B2,

D(φ)w2 := w2 ◦ φ, (15)

18 J. WOOD

wherew2 here is interpreted as a mapM2 7→ A. For example, the projection mapφ: R1,q 7→M dualizes to the inclusionB 7→ Aq. Also, the dual of the left natural action of ag× qR-matrix E: R1,g 7→ R1,q is the right action ofE on the trajectory spaceE: Aq 7→ Ag,i.e. the action in which we are interested.

It is easy to show that ifφ = τ ◦ γ for three mapsφ, τ, γ of finitely generated moduleswith appropriately compatible domains/codomains, thenD(φ) = D(γ )◦D(τ ). Therefore,given a commutative diagram of finitely generated modules, if we dualize each module anddualize each map (thus reversing the arrows) to obtain a commutative diagram of behaviours.ThereforeD(·) is acontrovariant functorin the language of category theory.

The next result, due to Oberst, is central to the development of further material, and ismuch harder.

THEOREM 2 [29, 2.54] Each moduleA listed in section 2.1 is an injective cogeneratorof the categoryR-modules. This signifies that duality is faithfully exact. In other words,given a complex of modules

· · · φi+2−−−−→ Fi+1

φi+1−−−−→ Fi

φi−−−−→ Fi−1

φi−1−−−−→ · · · (16)

and its dual complex

· · · D(φi−1)−−−−→ D(Fi−1)D(φi )−−−−→ D(Fi )

D(φi+1)−−−−→ D(Fi+1)D(φi+2)−−−−→ · · · , (17)

we have that (16) is exact (i.e.im φi+1 = kerφi for all i ) if and only if (17) is exact.

Theorem 2 is extraordinarily useful. The essential reason for this is that it is possibleto express a great variety of structural properties (both alegebraic and systems-theoretic)in terms of exact sequences; the duality then guarantees a correspondence between suchproperties. Since the theory of finitely generated modules over polynomial rings is verywell developed, we can make use of it to establish otherwise difficult results in nD systemstheory. Furthermore, the algebra brings along with it a large array of constructive techniquesbased on Gröbner bases, which can easily be used in systems problems.

Some of the more immediate and technical consequences of Theorem 2 are given inCorollaries 1–3, which will be useful later in the paper. The first of these results relatesB⊥andM to the behaviour’s kernel representations.

COROLLARY 1 [29, 2.23, 2.48–49, 2.61] IfB = KerAE is a differential/difference be-haviour thenB⊥ = ImRE and so the module of observables is given by M= CokerRE.If B1 = KerAE1 andB2 = KerAE2 are two behaviours in the same signal spaceAq thenthe following are equivalent:

1. B2 ⊆ B1.

2. B⊥1 ⊆ B⊥2 .

3. There exists a polynomial matrix L with E1 = L E2.

Furthermore,(B⊥)⊥ = B, and for any submodule N ofR1,q, (N⊥)⊥ = N.


The equivalence of conditions 1 and 3 in Corollary 1 is very important for further work,and remarkably difficult to prove without Theorem 2; see Rocha’s proof of the 2D result[44, Prop. II.9]. This result (and the result thatB⊥ = ImRE) does not hold for certain otherclasses of systems e.g. delay-differential systems; see [19, Ex. 2.3], [21]. The setting of theremainder of the paper is summarized by:

E ∈ Rg,q, M = CokerRE = R1,q/B⊥, B = KerAE = D(M) (18)

It follows from the functorial nature ofD(·) that, given any differential/difference be-havioursB′ ⊆ B, the factor spaceB/B′ also admits the structure of a differential/differencebehaviour (as more generally do images and kernels of maps between such behaviours)[29, Thm. 2.56(iii)]. This can be seen by choosing a kernel representationE′ ∈ Rg′,q forB′; then the coimage of the restriction ofE: Aq 7→ Ag′ to B is equal toB/B′, and so isisomorphic as a vector space to the imageEB′:

EB′ ∼= B/B′ (19)

Furthermore, this situation arises from dualizing a corresponding map of finitely generatedmodules, and therefore the modules of observables ofEB′ andB/B′ are isomorphic. Dueto strong structural links between a behaviour and its module of observables, we expectEB′ andB/B′ to share many system-theoretic properties. This leads to the following newdefinition.

Definition 2. Let B1 = M1 andB2 = M2 be two differential/difference behaviours withthe same signal space, but not necessarily the same number of components. IfM1 andM2 are isomorphic as finitely generatedR-modules, then we will say thatB1 andB2 areisomorphic differential/difference behaviours.

Throughout the paper, we will see that common system-theoretic concepts can generallybe characterized by algebraic properties, which are usually preserved by isomorphism. Inparticular, properties such as controllability and autonomy are preserved by isomorphism,since, as we will see later, they have algebraic characterizations which are preserved byisomorphism. We expect isomorphic behaviours to share important structural properties.A simple example of behavioural isomorphism is given by embedding a behaviourB ⊆ Aq

into the trajectory spaceAq+1 in the obvious way. Clearly such an embedding will notsignificantly affect many of the system’s properties. Another simple example is a change ofbasis in the space of the dependent variables, which again is an isomorphism of behaviours.

Next, we show that the dual of a submodule is a factor behaviour of the dual, and the dualof a factor module is a sub-behaviour of the dual. This is perhaps most clearly seen usingobservables:

COROLLARY 2 LetB′ ⊆ B be differential/difference behaviours. Then the set of observablesof B/B′ is B′⊥/B⊥, i.e. the submodule of the observables ofB which vanish onB′. IfB = D(M) and M′ is a submodule of M then D(M/M ′) is the sub-behaviour ofB on

20 J. WOOD

which the observables of M′ vanish. Hence we have a dual pair of exact sequences

0−→ M ′ −→ M −→ M/M ′ −→ 0 (20)

0−→ B′ = D(M/M ′) −→ B = D(M) −→ B/B′ = D(M ′) −→ 0 (21)

Proof: Let B′ ⊆ B, so thatB = D(R1,q/B⊥),B′ = D(R1,q/B′⊥). By Corollary 1, wehave an exact sequence

0−→ B′⊥/B⊥ −→ R1,q/B⊥ −→ R1,q/B′⊥ −→ 0

which, dualizing the sequence, implies thatD(B′⊥/B⊥) = B/B′. Thus the set of observ-ables ofB/B′ is equal toB′⊥/B⊥, i.e. to the subset of observables ofB which vanish onB′.

Similarly, let M ′ be a submodule ofM , so M ′ = N/B⊥ for some submoduleN ofR1,q which containsB⊥. By Corollary 1,B′ := N⊥ is a sub-behaviour ofB, and bydefinition it is the sub-behaviour on which the observables ofM ′ vanish. NowD(M/M ′) =D(R1,q/N) = B′ are required.

COROLLARY 3 Let M be a finitely generatedR-module with submodules N1, N2. Then:

D(M/(N1 ∩ N2)) = D(M/N1)+ D(M/N2) (22)

D(M/(N1+ N2)) = D(M/N1) ∩ D(M/N2) (23)

In particular, if N1 and N2 are submodules ofR1,q then:

(N1+ N2)⊥ = N⊥1 ∩ N⊥2 (24)

(N1 ∩ N2)⊥ = N⊥1 + N⊥2 (25)

Proof: Clearly the solution set ofN1+N2 is equal to the intersection of the solution sets ofN1 and ofN2, i.e. (24) holds. Similarly, for any sub-behavioursB1,B2 ofAq, (B1+B2)

⊥ =B⊥1 ∩ B⊥2 , which by the last claim of Corollary 1 yields (25).

We now show that the more general equations (22) and (23) follow from these. LetN1, N2 ⊆ M = R1,q/B⊥ for someB. Then we haveN1 = X1/B⊥ andN2 = X2/B⊥ forsome submodulesX1, X2 ofR1,q containingB⊥, and we now find

D(M/(N1 ∩ N2)) = D(R1,q/(X1 ∩ X2))

= (X1 ∩ X2)⊥

= X⊥1 + X⊥2 (from (25))

= D(M/N1)+ D(M/N2),

which establishes (22); the proof of (23) is analogous.


2.5. Observables and Observed Behaviours

Using the notion of an observable, we have given an interpretation of the elements of themoduleM to whichB is dual. From a behavioural perspective, an observable itself has anassociated behaviour, namely the set of all signals (scalar trajectories) which the observedvariable can exhibit:

Definition 3. LetB = D(M) ⊆ Aq be a differential/difference behaviour andv+B⊥ ∈ Ma system observable. Theobserved behaviouris the scalar behaviour

vB := {vw | w ∈ B},

wherevw is defined according to the usual action of the 1× q polynomial matrixv on thetrajectoryw, depending on the signal space.

The observable (a module element) and observed behaviour are formally related:

LEMMA 1 Let B = D(M) ⊆ Aq, and let v be any elementR1,q. Then the dual ofthe submodule of M generated by the observablev + B⊥ is isomorphic to the observedbehaviourvB:

D(R(v + B⊥)) ∼= vB, (26)

and furthermore the evaluation mapB 7→ A sendingw to vw, is dual to the mapR 7→ Msending r to r(v + B⊥).

More generally, if X∈ R1,q is a polynomial matrix with rowsv1, . . . , vl , then XB isisomorphic to the dual of the submodule of M generated by the observablesv1+B⊥, . . . , vl+B⊥:

D(R(v1+ B⊥)+ · · · +R(vl + B⊥)) ∼= XB, (27)

and again the evaluation mapB 7→ Al is dual to the mapR′ 7→ M, (r1, . . . , rl ) 7→∑li=1 ri (vi + B⊥).

Proof: We prove the general case (27) directly; letv1, . . . , vl and X be as stated. ByCorollary 2 we have

D(R(v1+ B⊥)+ · · · +R(vl + B⊥)) = D((ImRX + B⊥)/B⊥)= B/(ImRX + B⊥)⊥= B/(KerAX ∩ B (from (25)),

which is isomorphic toXB.We dualize the given mapφ: Rl 7→ M as follows. For anyw ∈ D(M) = B, D(φ)w ∈ Al

is given for each(r1, . . . , rl ) ∈ Rl by

(D(φ)w)(r1, . . . , rl ) = w(φ(r1, . . . , rl )) = yw (28)

22 J. WOOD

for y := φ(r1, . . . , rl ) ∈ M , since the meaning ofw(y) is the evaluation of the observabley at the trajectoryw. Now y is given by

y = φ(r1, . . . , rl ) =l∑

i=1

ri (vi + B⊥) = ((r1 · · · r .)X)+ B⊥ (29)

Combining (28) with (29) give us

(D(φ)w)(r1, . . . , rl ) = yw

= (r1 · · · rl )Xw

= (Xw)(r1, · · · , rl ),

treatingXw as an element ofAl with module of observablesR1,l . This completes the proof.

The duality HomR(·,A) between observable (or the module it generates) and observedbehaviour is not surprising; it states that the observed behaviour is the set of all assignmentsof signals to the observable which obey any algebraic restrictions on the observable. Wewill look at “algebraic restrictions on an observable” in the next section.

A first application of the concept of observables is the following new behavioural char-acterization of generating sets.

COROLLARY 4 LetB = D(M) ⊆ Aq be a behaviour with kernel representation E, and letX ∈ R1,q be a polynomial matrix with rowsv1, . . . , vl . Then the following are equivalent:

1. The observablesv1+ B⊥, . . . , vl + B⊥ generate M.

2. The observablesv1+B⊥, . . . , vl +B⊥ can detect whether a system trajectory is zero;,i.e. on each non-zero trajectory at least one of these observables does not vanish.

3. XB andB are isomorphic.

4.

(EX

)is zero right prime, i.e.

(EX

)(a) has full column rank for all a∈ Cn((C\0)n

forR = k[z, z−1]).

Proof: The given observables generateM if and only if ImRX + B⊥ = R1,q. ByCorollary 3, the dual result is KerAX ∩ B = 0. Condition 2 is a direct interpretation ofthis equation. Furthermore, KerAX ∩ B = KerA(ET XT )T , and from the exactness ofduality it is easy to show that(ET XT )T has kernel 0 if and only if it is zero right prime [29,Thm. 7.63]. This establishes equivalence of 1 and 4. Condition 3 follows from Lemma 1.

Note that the last condition in Corollary 4 is a result about interconnection of systems[56].


3. Torsion and Autonomy

In this section we consider the notion of a torsion observable or constrained observable,which is due to Pommaret. The relevance or torsion elements and torsion modules tosystems theory is also apparent in the work of Fliess and co-workers (e.g. [10, 13]). Inessence, torsion describes the structure of system variables or observed quantities whichare not free. In particular, we will see that it can be used to describe exponential trajectoriesand autonomous systems.

We begin this section by considering sets of module elements (observables) which satisfyno non-trivial algebraic relations. Such elements may be called independent observables.This concept is related to the freedom of variables in the system behaviour, which we nowdiscuss.

3.1. Independent Observables and Free Variables

Let us consider a set of observablesv1+ B⊥, . . . , vl + B⊥ ∈ M = R1,q/B⊥ which satisfyno non-trivial relations, i.e. for anyr1, . . . , rl ∈ R we have

l∑i=1

ri (vi + B⊥) = 0 ⇒ r1 = · · · = rl = 0 (30)

Such elements will be called(linearly) independent observables. We can interpret equa-tion (30) as meaning that the given observables provide in a sense uncorrelated information.This will be formalized in Corollary 5.

An important special case occurs when the observables in question are a subset ofe1 +B⊥, . . . ,eq + B⊥. In this case, linear independence is related to freeness of the systemvariables.

Definition 4. Let B ⊆ Aq be a differential/difference behaviour. The set of variables{wi | i ∈ 8} for some subset8 of {1, . . . ,q} is said to be aset of free variablesif themappingρ: B 7→ A|8| which projects onto the components8 is epic. The maximum sizeof a set of free variables ofB is called thenumber of free variablesof B, and denoted bym(B).

The following result is given in [60, proof of Lemma 5] for the discrete case, usingarguments which apply equally well in the continuous case.

LEMMA 2 Let B = D(M) ⊆ Aq be a differential/difference behaviour, and let8 ⊆{1, . . . ,q}. Then the set of variables{wi | i ∈ 8} is free if and only if the observables{ei + B⊥ | i ∈ 8} are independent. Hence the following are equivalent:

1. The number of free variables ofB.

2. The maximum number of independent observables.

3. q− rank E, where E is any kernel representation ofB.

24 J. WOOD

The equivalence of quantities 2 and 3 in Lemma 2 is standard by consideration of linearalgebra over the quotient fieldQ(R). The equivalence of quantities 1 and 3 is also givenin [29, Thm. 2.69]. It can also be shown that the number of free variables is additive, i.e.given an expert sequence of behaviours

0→ B1→ B2→ B3→ 0,

we havem(B2) = m(B1)+m(B3).The following result, which has not previously appeared in the literature, is a direct

generalization of Lemma 2.

COROLLARY 5 Let B ⊆ Aq be a differential/difference behaviour. Let X∈ Rl ,q be apolynomial matrix with rowsv1, . . . , vl for some l. Then the behaviour XB is equal toAl

if and only if the observablesv1+ B⊥, . . . , vl + B⊥ are linearly independent.

Proof: Let N be the submodule ofM generated byv1 + B⊥, . . . , vl + B⊥. By Lemma1, D(N) is isomorphic toXB. Applying Lemma 2 toXB gives the required result.

Thus independence of observables signifies that the observed trajectories can simultane-ously take on any values, and are therefore functionally independent.

3.2. Torsion Observables

We now consider the case of (linear) dependence of a single observable, i.e. the questionof whether a given observable satisfies a non-trivial algebraic constraint by itself. This isPommaret’s notion of a constrained or torsion observable (e.g. [38, 39, 42]), which comesfrom the standard notion of torsion in algebra (e.g. [5]):

Definition 5. LetB = D(M). Then an observablev+B⊥ ∈ M is called atorsion element,a torsion observableor aconstrained observableif there exists anr ∈ R, r 6= 0, suchthatr (v + B⊥) = 0 in M , i.e. r v ∈ B⊥. An observable which is not a torsion element issaid to beunconstrainedor free.

If every element ofM is a torsion element,M is called atorsion module. If no non-zeroelement ofM is a torsion element,M is calledtorsionfree. The set of torsion elements ofM forms a submodule, denotedt M , called thetorsion submodule.

For a given torsion element or module, we can obtain additional information by consid-ering the algebraic relations which it satisfies.

Definition 6. Let M be an arbitraryR-module. For anyv+B⊥ ∈ M , the ideal of elementsr ∈ R such thatr (v + B⊥) = 0 in M is called theannihilator of v + B⊥, and is denotedann(v + B⊥). Theannihilator of the moduleM itself if an ideal defined by

annM := {r ∈ R | r (v + B⊥) = 0 for all (v + B⊥) ∈ M} (31)


Note that the annihilator of a behaviourB is the set of all equations which are satisfiedby each system variable independently of the others.

An observable is constrained if and only if it has a non-zero annihilator. Similarly, amodule is a torsion module if and only if it has a non-zero annihilator, or equivalently ifand only if it has no linearly independent elements. The observed behaviour correspondingto a given torsion observable is the set of all signals satisfying corresponding constraints:

LEMMA 3 LetB = D(M), and letv + B⊥ ∈ M be a given observable. Then the observedbehaviourvB is given by

vB = {w ∈ A | rw = 0 for all r ∈ ann(v + B⊥)} (32)

In particular, the observed behaviour is determined by the annihilator of the observable.Also, the observed behaviour is equal to A if and only if the observable is unconstrained.

Proof: Be Lemma 1,vB = HomR(R(v + B⊥),A). An element ofφ ∈ HomR(R(v +B⊥),A) is identified with the scalar trajectorywφ = φ(v+B⊥) ∈ vB, and is determined byR-linearity from this trajectory. For the condition ofR-linearity to be met, it is necessaryand sufficient thatwφ satisfies anyR-linear relations onv + B⊥. This gives us (32), andhence the remaining results.

We can easily display a kernel representation of the observed behaviourvB; if {r1, . . . , rl }is a generating set of ann(v + B⊥) then such a representation is given from equation (32)by

vB = KerA

r1...

rl

In Lemma 3, we see that the annihilators of the moduleR(v + B⊥) and of the dual

behaviourvB (also a module) are equal. This is a special case of the general law [57]:

LEMMA 4 Duality preserves annihilators. That is, given any finitely generated module M,we have

annM = annD(M) (33)

Proof: Let r ∈ R be arbitrary; then the conditionr ∈ annM is equivalent to exactness ofthe sequence

0−→ Mid−→ M

r−→ M,

where the last map is the ring action ofr . The conditionr ∈ ann D(M) is equivalent to

26 J. WOOD

exactness of the sequence

D(M)r−→ D(M)

id−→ D(M) −→ 0

By Theorem 2, the two conditions are equivalent.

Lemma 4 says simply that a constraint on each system observable is also a constraint oneach trajectory, and vice versa.

From Lemma 2, we know that a module CokerRE has a non-zero annihilator if and onlyif E has full column rank. We can identify certain elements of all annihilator from thematrix E.

LEMMA 5 [8, Prop. 20.6, 20.7] Let E∈ Rg,q,M = CokerRE. Then the ideal of qthorder minors of E is contained in the annihilator of M. Furthermore, these two idealshave the same radical; in particular, they vanish at the same points inCn((C\0)n f or R =k[z, z−1]).

The concept of an annihilator is already known from the theory of the 1D state-spacemodel. If we have a linear state-space modelx(t) = Ax(t) − Bu(t), then the behaviourwith inputu(t) equal to zero is given in the current formalism byBx,0 = KerA(z I−A). Theannihilator of this behaviour is equal to the ideal generated by the well-known annihilatingpolynomial or minimal polynomial [23] of the matrixA. In this context, Lemma 5 isexplained by the well-known fact that the minimal polynomial divides the characteristicpolynomial ofA, and furthermore these two polynomials have the same roots.

In [59], it is shown that the annihilator can be used to characterize the “degree of prime-ness” of a polynomial matrix, and in particular that the well-known concept of a Bézoutidentity is intimately connected with annihilators.

3.3. Exponential Trajectories

Of particular interest in systems theory is the case of a trajectory with annihilator equal toa maximal ideal. This is easiest to see when working over the fieldk = C, in which casethe maximal ideals are of the form

I (a) = R(z1− a1)+ · · · +R(zn − an)

for somea ∈ Cn((C\0)n forR = k[z, z−1]). In the discrete case(k = C), such trajectoriesare of the form

wj (t) = αat11 · · ·atn

n , α ∈ C\0 (34)

For each componentwj . In the continuous case(k = C), each trajectory component is ofthe form

wj (t) = αea1t1+···+antn, α ∈ C\0 (35)


In either case, a trajectory annihilated byI (a) is called anexponential trajectory withfrequencya. These definitions, and the following theory, can also be extended to the casek = R [31, 57]. It should however be noted that the frequencies of exponential trajectoriesare still inCn((C\0)n even in the casek = R.

The frequenciesa of exponential trajectories in a given behaviour can be characterized ina number of ways. The first three conditions of the following result are given in [57], andare derived from the theory of pdes, in which the term “characteristic variety” originates;see [2, p200, p340], and also [33, p138–139]. HereV(J) for an idealJ denotes the varietyof points inCn (or (C\0)n for R = k[z, z−1]) at which every element ofJ vanishes. Thefourth condition in the following result is new.

THEOREM3 LetB = D(M) be a behaviour with kernel representation E. Let abe a pointin Cn ((C\0)n). The following are equivalent:

1. a∈ V(annB) = V(annM).

2. E(a) has less than full column rank.

3. B contains a non-zero exponential trajectory with frequency a.

4. There is an observed behaviourvB, v ∈ R1,q, which contains a non-zero exponentialtrajectory with frequency a.

If a satisfies these conditions, then it is called acharacteristic point of B or of M. Thevariety V(annB) of all such points called thecharacteristic variety ofB or of M.

Proof: It only remains to prove the equivalence of condition 4 to the other conditions. IfB contains a non-zero exponential trajectoryw(t)with frequencya, then in particular somecomponent ofw(t), saywj (t)must be a non-zero exponential trajectory with frequencya.Then the observed behaviourejB containswj (t). Conversely, suppose that some observedbehaviourvB contains a non-zero exponential trajectory with frequencya. By the equiva-lence of conditions 1 and 3, and applying Lemma 4,a ∈ V(annvB) = V(ann(v + B⊥)).But V(ann (v + B⊥)) is contained inV(ann M), so a ∈ V(ann M) as required.

We see that the frequencies of the exponential trajectories contained in a behaviour canbe determined from the observed behaviours, or from the observables.

In the case of a Fornasini–Marchesini state-space model

Bx,u = KerA(I − z−11 A1+ z−1

2 A2 | z−11 B1+ z−1

2 B2),

the characteristic points of the zero-input behaviourBx,0 (the set of all system trajectorieswith inputu = 0) are given by the roots of the “characteristic polynomial” det(I −z−1

1 A1+z−1

2 A2).The concept of an exponential trajectory plays a significant role in many practical issues,

most particularly in the study of stability. In [57], the author, together with Oberst, Rogersand Owens, provides a theory of poles for nD linear systems with constant coefficients,which centres on this concept.

28 J. WOOD

3.4. Autonomous Systems

From Theorem 3, we see that a differential/difference behaviourB = 0 contains no non-zero exponential trajectories precisely whenV(annB) = V(ann M) = ∅, i.e. preciselywhen M = 0,B = 0. At the other end of the scale,B contains a non-zero exponentialtrajectory of every frequency precisely whenV(annB) = V(ann M) = Cn(or (C\0)n);equivalently, precisely when annB = annM = 0, i.e.M is not a torsion module, or has nolinearly independent elements. By Lemma 2, an equivalent condition is thatB has no freevariables.

The condition of having no free variables is intuitively connected with the concept ofautonomy. Formally, we can define an autonomous system in the discrete case as follows:

Definition 7. [15, 60] A difference behaviourB with signal domainT is autonomousifthere existsT1 ⊆ T such that any trajectory ofB is determined by its values onT1, and alsoT\T1 contains ann-dimensional cone.

Definition 7 says that a system is autonomous if there is a “small” initial condition set,the values of which determine any trajectory. The main characterization of such behaviours[60, Thm. 2] follows; similar results or special cases have also appeared in [15, 65].

THEOREM 4 [15, 60, 65] LetB = D(M) be a difference behaviour. The following areequivalent:

1. B is autonomous.

2. B has no free variables.

3. For any E withB = KerAE, E has full column rank.

4. annB = annM 6= 0; equivalently, M is a torsion module.

5. Every observable is constrained; equivalently, no observed behaviour is equal toA.

Proof: Equivalence of conditions 1–4 is established in [60, Thm. 2]; a sketch of theproof follows. IfB is autonomous (condition 1), then it clearly contains no free variables(condition 2). By Lemma 2, the rank of any givenE is q (condition 3) andM has nolinearly independent elements. Equivalently,M has a non-zero annihilator or is a torsionmodule (condition 4), and every observable is constrained (condition 5). The condition onobserved behaviours is immediate from Lemma 3. Finally, if condition 5 or equivalentlycondition 4 holds, we have annB 6= 0, so there is a recurrence relation on every componentof the trajectories ofB. From this it can be shown thatB is autonomous.

The importance of the torsion condition 4 is also shown in the work of Fliess [10, 13],where it is essentially used as a definition of autonomy. In the continuous case, Pillai andShankar have defined an autonomous system as one with a characteristic variety whichis not equal toCn [36, Defn. 4.2]; they show that this is equivalent to conditions 2–4 ofTheorem 4.


Combining Theorem 4 and Lemma 1 gives us the following:

COROLLARY 6 Let B = D(M) be a difference behaviour and X∈ R1,q a polynomialmatrix with rowsv1, . . . , vl ∈ R1,q. Then XB is autonomous if and only if the sub-moduleof M generated byv1 + B⊥, . . . , vl + B⊥ is a torsion module, or equivalently if and onlyif v1+ B⊥, . . . , vl + B⊥ are constrained observables.

Corollary 6 (and all following results in this paper) also applies to differential systems,using Pillai and Shankar’s definition of autonomy.

One special class of autonomous behaviours are thefinite-dimensional behaviours,which are those that are finite-dimensional as vector spaces over the fieldk [15, 32] (in [36]they are calledstrongly autonomous). These can be characterized as follows:

LEMMA 6 LetB be a differential or difference behaviour. Then the following are equivalent:

1. B is finite-dimensional.

2. The characteristic variety ofB is finite.

3. The set of observables ofB has a finite k-basis.

Proof: The equivalence of 1 and 2 is given in [32, 36]. The equivalence of 1 and 3 followsdirectly from [32].

In the 1D case, all autonomous behaviours are finite-dimensional, which accounts for thefinite-dimensionality of the state space of a 1D linear system. Forn > 2, this is not true,as a general autonomous system has an infinite set of initial conditions. In general, the“size” of an autonomous behaviour rises with the dimension of its characteristic variety.Intuitively, this occurs since the number of exponential trajectories grows with the dimensionof this variety, and the more general “polynomial exponential trajectories” [31] are densein the behaviour [31, 57]. The dimension ofV(annB) is equal to several other interestingquantities:

LEMMA 7 LetB ⊆ A1 be a behaviour with kernel representation E. The the following areequal:

1. The dimension of the characteristic variety ofB.

2. n minus the primeness degree of E, which is the height of the ideal generated by theqth order minors of E.

3. The degree of the affine Hilbert polynomial associated withB (see Definition 15).

Proof: The equality of quantities 1 and 2 is standard in algebra (given basic duality laws)and follows from Lemma 5. The remaining equality is proved in [58].

For the caseA = kNn, a further equal quantity isn minus the “autonomy degree” ofB[59], which is a natural measure of the extent to which the behaviour is restricted. Pommaret

30 J. WOOD

and Quadrat have used the dimension of the annihilator to grade the observables of a system[41].

4. Torsionfreeness and Controllability

In the previous section we have examined the concept of a constrained observable (a tor-sion element), and the condition of all observables being constrained (a torsion moduleor autonomous behaviour). In this section we will look at the complementary concept ofunconstrained observables and torsionfree modules. As several authors have discussed, thisis related to the concept of controllability.

4.1. Controllable Systems

We begin with an interpretation of torsionfree modules, extending by duality the ideas ofPommaret (e.g. [38, 29, 42]).

LEMMA 8 LetB ⊆ Aq be a differential/difference behaviour. Then the module of observ-ables is torsionfree if and only if every non-zero observed behaviour is equal toA.

Proof: By definition,M is torsionfree if and only if every non-zero observable is uncon-strained. The result now follows from Lemma 3.

Following an observation by Pommaret, Fliess has shown [11] that the torsionfree propertyis equivalent to Kalman’s definition of controllability in the classical 1D time-varying case.The torsionfree property is then a viable definition of controllability for more general classesof systems (see e.g. [11, 39]), particularly since it is clear that a system with a constrainedobservable cannot be fully controlled.

An alternative definition of controllability is given in the behavioural framework, and isdue firstly to Willems [55]. The behavioural definition was extended to the 2D discrete caseby Rocha [44], then to the nD discrete case by Wood, Rogers and Owens, and by Wood andZerz [60, 61], and to the continuous case by Pillai and Shankar [36].

Definition 8. [36, 44, 60, 61] A differential behaviourB is controllable if, for any twoopen setsT1, T2 ⊆ Rn with disjoint closures, and for any two trajectoriesw(1), w(2) ∈ B,there exists aw ∈ B such that

w(t) ={w(1)(t) if t ∈ T1

w(2)(t) if t ∈ T2(36)

A difference behaviourB with signal domainT = Zn or T = Nn is controllable if thereexists a real numberρ > 0 such that for any setsT1, T2 ⊆ T with d(T1, T2) > ρ, and foranyb1,b2 ∈ T , and any two trajectoriesw(1), w(2) ∈ B, there exists aw ∈ B such that

w(t) ={w(1)(t − b1) if t ∈ T1 andt − b1 ∈ Tw(2)(t − b2) if t ∈ T2 andt − b2 ∈ T

(37)


In the discrete case, we can takeb1 = 0 without loss of generality, and forT = Zn, we canalso takeb2 = 0. For both differential and difference behaviours, controllability expressesthe idea of being able to join with a system trajectory any two system trajectories definedon regions which are sufficiently far apart.

Fliess has shown that, in the 1D case, the behavioural and module-theoretic (i.e. torsion-free) definitions are in fact equivalent [12]. This equivalence extends to nD systems andleads to other characterizations of controllability, some of which we omit here for brevity;see [57] for a list of all main characterizations to date.

THEOREM5 LetB = D(M) be a differential/difference behaviour. Then the following areequivalent:

1. B is controllable.

2. B is minimal in its transfer class (see section 5.2).

3. B has an image representationB = ImAF for some polynomial matrix F.

4. M is torsionfree.

5. Every non-zero observed behaviourvB is equal toA.

Proof: The equivalence of 1 and 3 is given in [36, Prop. 3.4, Thm. 3.9] for differentialbehaviours, in [60, Thm. 5] for difference behaviours onZn, and in [61] for general differ-ence behaviours. The equivalence of 3 and 4 has been given in [39, Prop. VIL.A.10], andthe equivalence of 2 and 4 in [29, Thm. 7.21]. Condition 5 is immediate from Lemma 8.

The equivalence of 3 and 4 can be established for variable coefficient systems using theformal theory of pdes [39, Prop. VIL.A.10]. A very similar equivalence holds even fornon-linear systems [67]!

In the special case where the behaviour has a full row rank kernel representationB =KerAE, it can be shown thatB is controllable if and only ifE is minor left prime [59, Cor.7.9]. So for example in the case of 2D Fornasini–Marchesini state-space model

Bx,u = KerA(I − z−11 A1+ z−1

2 A2 | z−11 B1+ z−1

2 B2),

the state-input behaviour is controllable if and only if

(I − z−11 A1+ z−1

2 A2 | z−11 B1+ z−1

2 B2)

is minor left prime.Combining Theorem 5 with Lemma 1 leads us to the following:

COROLLARY 7 Let B = D(M) be a differential/difference behaviour, and let X∈ R1,q

be a polynomial matrix with rowsv1, . . . , vl . Then XB is controllable if and only ifv1+ B⊥, . . . , vl + B⊥ generate a torsionfree submodule of M.

32 J. WOOD

We mention that an algorithmic test for torsionfreeness exists; see [42, Sec. 4.1] andalso [60], [65] for the equivalent task of testing whether a matrix is “generalized factor leftprime”.

4.2. The Controllable Part

Any differential/difference behaviour has a “controllable–autonomous decomposition”.The 2D result is given in [15]; the nD results for discrete behaviours are in [60, 65],and these generalize to the continuous case now that Theorem 5 is known. Alternatively,we can reason that the controllable–autonomous decomposition can be obtained from thewell-known algebraic result of primary decomposition by dualizing and summing terms;see section 7.1 of [57].

THEOREM6 Any differential/difference behaviourB can be written in the form

B = Bc + Ba (38)

for some differential/difference sub-behavioursBc,Ba, whereBc is controllable andBa isautonomous. Furthermore,Bc is uniquely determined by these conditions. The decom-position (38) is called acontrollable–autonomous decompositionof B;Bc is called thecontrollable part ofB, andBa is called anautonomous part.

The controllable–autonomous decomposition can be refined by breaking down the au-tonomous part into the “uncontrollable pole decomposition” of [57]. In the 2D case, Valcherhas a given further refinement of the controllable–autonomous decomposition [53].

Controllable systems and autonomous systems are respectively distinguished by the casesBc = B,Bc = 0. In the general case, we can also characterize the controllable part.Characterizations of the controllable part, or of the minimal element of the transfer class,have previously been given in [29, Thm. 7.21], [44, Lem. IV.14], [60, Cor. 6]. In the nextresult, condition 5 is new, and condition 3 is new for the continuous case.

THEOREM 7 Let B = D(M) be a differential/difference behaviour. The following areequal:

1. The controllable part ofB.

2. The minimal element of the transfer class ofB (see section 5.2).

3. The maximal controllable sub-behaviour ofB.

4. D(M/t M), where t M is the torsion submodule of M.

5. The unique controllable sub-behaviour ofB with the same number of free variables asB.

6. The sub-behaviour ofB consisting of all trajectories on which every constrained ob-servable vanishes.


Furthermore, a constrained observable ofB is precisely an observable ofB which vanishesonBc.

Proof: Equality of 1 and 4 is given in [60, Cor. 6], and equality of 2 and 4 in [29, Thm.7.21]. Equality of 1 and 5 proceeds as follows:Bc is controllable, and by condition 4B/Bc = D(t M), proving thatB/Bc has no free

variables by Lemma 2. By additivity of the number of free variables (see section 3.1),m(B) = m(Bc). Now letB′ be a controllable sub-behaviour ofB with m(B′) = m(B); wemust show thatB′ = Bc. Again by additivity,m(B/B′) = 0. Writing B/B′ = D(N) forN a submodule ofM , we have thatN has no linearly independent elements, i.e.N ⊆ t M .HenceM/t M is a factor module ofM/N, so by duality and the equivalence of 1 and 4,Bc ⊆ B′. But now we haveB = B′ +Ba for any autonomous partBa of B, proving thatB′is a controllable part ofB, and therefore equal toBc.

We now show equality of 1 and 3. LetB∗ be an arbitrary controllable sub-behaviour ofB;then it is easy to show thatB∗ + Bc is controllable. By the same argument as in the end ofthe previous paragraph,B∗+Bc must be a controllable part ofB, and therefore equal toBc ,from whichB∗ ⊆ Bc. This proves more strongly thatBc is the union of all controllable sub-behaviours ofB. This establishes equality of 1–5. Condition 6 follows from Corollary 2.

Note that the last claim of Theorem 7 leads back to the characterizations of controllableand autonomous systems by constrained observables.

The controllable part of a behaviour can be explicitly computed: in [29, Thm. 7.24] itis shown how to compute the minimal element of the transfer class of a behaviour, and in[60, Cor. 6] it is explained that computing the controllable part can be done by computinga “minimal right annihilator” of any given kernel representation matrix. An equivalentalgorithm in the related framework of the formal theory of pdes is given by Pommaret andQuadrat in [42, Sec. 4.1]; this version extends to the case of variable coefficients.

Note from Theorem 7 that we have two dual exact sequences:

0 −→ t M −→ M −→ M/t M −→ 0 (39)

0 −→ Bc −→ B −→ B/Bc −→ 0 (40)

and in particularB/Bc is autonomous as the dual of the torsion modulet M . By Corollary2, the set of observablet M is equal to the set of observables ofB which vanish onBc.Combining this observation with condition 4 of Theorem 3 gives us the following newresult:

COROLLARY 8 LetB be a differential/difference behaviour. Then ais a characteristic pointofB/Bc if and only ifB has an observablev +B⊥ which vanishes onBc and such thatvBcontains a non-zero exponential trajectory with frequency a.

As we will see in section 5.2,Bc has the same free input/output structures asB. Aconsequence of Corollary 8 is therefore that, ifa is a characteristic pointB/Bc, the observedbehaviourvB can take the value of a non-zero exponential trajectory with frequencyairrespective of the system input. Thus the characteristic points ofB/Bc are called the

34 J. WOOD

uncontrollable pole pointsof B, and are the behavioural equivalent of the classical inputdecoupling zeros. The paper [57] contains more information about these and other classesof zeros in the behavioural framework. See also [3], which provides a corresponding set ofdefinitions for 1D systems in the dual framework of module theory.

4.3. Strong Controllability

The concept of strong controllability first appears in [44] for 2D discrete systems, whereit is given a trajectory definition. Due to the intractability of this definition forn ≥ 3, wehave adopted instead the following definition, formerly termed “rectifiability” [48]. Recallthat a matrix over the ringR is said to beunimodular if it is square and has a determinantwhich is invertible withinR.

Definition 9. A differential/difference behaviourB is said to bestrongly controllable ifthere exists a unimodular matrixU such thatUB = (Ar ,0) for somer .

The characterizations of strong controllability are as follows:

THEOREM8 LetB = D(M) = KerAE. Then the following are equivalent:

1. B is strongly controllable.

2. M is a free module, i.e. has anR-linearly independent generating set.

3. E has constant rank, i.e.rank E(a) = rank E(z) for all a ∈ Cn((C\0)n for R =k[z, z−1]).

Proof: Equivalence of 1 and 3 is given in [48]; equivalence of 2 and 3 is well-known (e.g.[8, Prop. 20.8]); see also [24].

Condition 2 can roughly be interpreted as saying that there is a set of independent observ-ables, the values of which determine the system variables. This is the concept of flatness,which is of great practical significance in non-linear control (e.g. [14]). See also [29, Thm.7.53] for a discussion of free modules in the context of behavioural theory.

5. Inputs, Outputs, and Latent Variables

In this section we consider various issues which arise when the system variables are parti-tioned in some way. In particular, we consider input/output structures and latent variables.

5.1. Observability

Suppose we have a partitioning of system variables into variablesf and variablesh. Wewrite this asB = B f,h. It then becomes useful to consider the behaviour

B0,h :={(

0h

)∈ B f,h

}(41)


This is clearly a sub-behaviour ofB f,h, and we can consider the factor behaviourB f,h/B0,h,which is naturally isomorphic to

B f :={

f

∣∣∣∣ ∃ h such that

(fh

)∈ B f,h

}, (42)

i.e. to the behaviour of thef variables only. We can represent this relationship by the exactsequence

0−→ B0,h −→ B f,h −→ B f −→ 0 (43)

Now suppose that we have a kernel representationB f,h = KerA(−Q P), i.e. the behaviouris described byPh= Q f . Then a kernel representation ofB0,g is

B0,h = KerA

( −Q PI 0

)= KerA

(0 PI 0

)(44)

We do not usually distinguish betweenB0,h and KerAP. It is possible to construct a kernelrepresentation ofB f using the procedure for “elimination of latent variables” [25], [29, Cor.2.38].

In some cases, the variablesh may have the interpretation oflatent variables(e.g. states),whereas the variablesf may bemanifest variables. In this case,B f,h is said to be alatentvariables descriptionof B f , and converselyB f is called themanifest behaviourof B f,h.In this case, we will termB0,h theunobservable behaviourof B f,h for reasons that willshortly be clear.

The relationshipB f = B f,h/B0,h provides a useful context in which to re-examine Corol-lary 2. This result claims that the set of observables ofB f is equal to the set of observablesB f,h which vanish onB0,h. From equation (44),

(B0,g)⊥ = B⊥ +Re1+ · · · +Rec, (45)

wherec is the number of variablesf . Hence the set of observables ofB f,h which vanishonB0,h is equal to

(B⊥ +Re1+ · · · +Rec)/B⊥

These observables are generated bye1+ B⊥, . . . ,ec + B⊥, as expected.In the behavioural framework, observability is defined as follows [55], [44, Defn. IV.19],

[65]:

Definition 10. LetB f,h be a behaviour with two sets of variables,f, h. The variablesh aresaid to beobservablefrom the variablesf if ( f, h(1)), ( f, h(2)) ∈ B f,h impliesh(1) = h(2).

Thus the variablesh are observable from the variablesf if their values can be uniquelydetermined from the values of thef variables. The equivalence of the first three conditionsin the following result is given in the 2D, 1D, and nD cases respectively in [44, 55, 65].

36 J. WOOD

The fourth condition, which is new in the behavioural context, shows that “observability”and “observables” are indeed related.

THEOREM 9 Let B f,h be a behaviour with kernel representation(−Q P). Suppose thatthe number of f variables is c, and the number of h variables is d. The following areequivalent:

1. The variables h are observables from the variables f .

2. B0,h = 0.

3. P is zero right prime, i.e. P(a) has full column rank for every point ain Cn((C\0)n forR = k[z, z−1]).

4. Each observable ej + B⊥f,h, j = c+ 1, . . . , c+ d, is an observable ofB f .

Proof: Equivalence of 1 and 2 holds by linearity of the system; equivalence of 2 and 3is elementary from the exactness of duality. We now prove the equivalence of 2 and 4.By Corollary 2 or the remarks preceding Definition 10, the observablesej + B⊥f,h, j =c+ 1, . . . , c+ d are observables ofB f if and only if they vanish onB0,h. This is clearlyequivalent to the conditionB0,h = 0.

In [40], Pommaret and Quadrat characterize observability by a condition correspondingto zero right primeness in the formal theory of pdes. Condition 4 of Theorem 9 is essentiallythe definition of observability used in [10, 40].

Note that this last condition can be interpreted by Lemma 1 as saying that each quantityhj , j = 1, . . . ,d, can be observed from the variablesf1, . . . , fc. In the case of a latent vari-able description, we see that the unobservable behaviourB0,h is the obstacle to observabilityof the latent variablesh from the manifest variablesf .

Consider the special case of a behaviour given by a Rosenbrock system matrix

Bx,u,y = KerA

(T −UV W

∣∣∣∣ 0−I

)We see that the variablesx are observable from the variables(u, y) if and only if (TT VT )T

is zero right prime. This accords with the classical notion of observability in the 1D context.Rocha also provides the concept of weak observability [44, Defn. IV.21]; see also [65].

Definition 11. The variablesh are said to beweakly observablefrom the variablesf if knowledge of f determinesh to within a finite-dimensional space, i.e. ifB0,h is afinite-dimensional behaviour.

This concept could be generalized to an “observability degree”, defined in analogy to the“autonomy degree” or “primeness degree” as the codimension of the characteristic varietyof B0,h. In this case, observability would correspond to observability degree∞, and weakobservability to observability degreen.


5.2. Input/Output Structures and Transfer Matrices

The preceding section on partitioning of system variables is a natural starting point fora discussion on input/output structures. In some applications, it may be appropriate toconsider an input/output structure as an arbitrary partitioning of system variables, i.e. itis not necessarily the case that a system’s inputs are free variables. For example, in anelectrical circuit it may be quite reasonable to assume that a given variableu(t), e.g. apotential difference, is of the form

u(t) = α cos(t + ψ)

for someα ∈ R; u is constrained by∂2u∂t2 (t) = −u(t), and is not a free variable. Nevertheless,

u can be chosen “freely” in the sense thatα andψ can be arbitrarily chosen, and can thereforebe treated as a system input using an oscilloscope. This example is due to Pommaret [39,p320–321].

Such situations can be examined by further development of the theory in section 5.1.However, the remainder of this section is devoted to the more traditional situation wherethe inputs are a maximal set of free variables of the system. To acknowledge the situationdescribed above, we will modify the current nomenclature to “free input/output structure”:

Definition 12. [29, Thm. 2.69], [44, Defn. IV.8], [60, Defn. 12] Afree input/outputstructure on the behaviourB is a partitioning of the system variablesw = (u, y), such thatthe set of variablesu is free and thezero-input behaviourB0,y is autonomous.

An equivalent definition is given by considering a corresponding partitioningE =(−Q P) of any kernel representationE of B. The columns ofQ correspond to the in-put variablesu, and the columns ofP to the output variablesy, and we have the condition

rank E = rank P = number of columns ofP, (46)

which is equivalent to Definition 12. It is easy to show that the number of inputs is equalto m, the number of free variables; the number of outputs is denoted byp and by Lemma 2is equal to the rank of any kernel representation of the behaviour. In particular, the numberof inputs and number of outputs of a behaviour is independent of the free input/outputstructure.

A different perspective arises from the consideration of observables. Suppose thereforethat we have a partitioning of the system variables intom variablesu, and p = q − mvariablesy. The condition that the variablesu are free is that the behaviourBu of the uvariables is equal toAm, which by Lemma 2 means that the observablese1+B⊥, . . . ,em+B⊥of M are linearly independent. The condition thatB0,y is autonomous means that all of itsobservables are constrained, by Theorem 4. However, sinceBu = B/B0,y, the module ofobservables ofB0.y is equal to

M

R(e1+ B⊥)+ · · · +R(em + B⊥) ,

38 J. WOOD

so the condition of autonomy means that each observableem+1 + B⊥, . . . ,em+p + B⊥is linearly dependent on the observablese1 + B⊥, . . . ,em + B⊥. This module-theoreticinterpretation of a free input/output structure is essentially the definition of independentinputs used in the 1D case by Fliess (e.g. [10, 11]).

Given a behaviour with a free input/output structureBu,y = KerA(−Q P), the condition(46) tells us that every column ofQ is linearly dependent on the columns ofP. Thus thereexists a (unique) rational function matrixG with PG = Q. This is thetransfer matrixof the system, as defined by Oberst [29, Thm. 2.69]. Note that, in the case where(−Q P)has full row rank, this reduces to the well-known case of a left matrix fraction descriptionG = P−1Q. In the domain of modules, the transfer matrix can be obtained by movingfrom M to Q(R)⊗R M whereQ(R) denotes the field of rational functions inn variables;the mapM 7→ Q(R)⊗R M is called theformal Laplace transform by Fliess [13]. Fliessshows that in the 1D case the transfer matrix can be defined in a purely module-theoreticway [13].

The following definitions and results are due to Oberst.

THEOREM 10 [29, Thms. 2.94, 7.17, 7.21] LetB = Bu,y be a differential/difference be-haviour with a given free input/output structure and the transfer matrix G with respect to thisinput/output structure. Then the set of all behaviours with the same given free input/outputstructure. Then the set of all behaviours with respect to this structure, is called thetransferclassof G, and denoted by [B]. The transfer class is independent of the free input/outputstructure chosen.

Two given behavioursB1 = D(M1),B2 = D(M2)with the same number of variables arein the same transfer class if and only if Q(R)⊗R M1 = Q(R)⊗R M2.

Each transfer class has a unique element which is minimal with respect to set inclusion.

Note that, as reported in Theorem 7, the minimal element in a given transfer class isequal to the controllable part of each behaviour in the class. This gives a one-to-onecorrespondence between controllable behaviours and transfer classes (transfer matrices,when an input/output structure is understood). Properties of a controllable behaviour areoften reflected in the properties of its transfer matrix; in particular, Oberst has shown [29,Cor. 7.29] that, for any behaviourB with transfer matrixG,

((Bc)0,y)⊥ = {v ∈ R1,p | vG ∈ R1,m}, (47)

where we treat (Bc)0,y here as a sub-behaviour ofAp rather than ofAq, in the obvious way.It follows [57] that ann (Bc)0,y is the ideal generated by the least common denominator ofthe entries ofG, and hence that the characteristic variety of(Bc)0,y is the set of all pointswhich are the poles of entries ofG. These points are called thecontrollable pole pointsof B [57], and are analogous to the transmission poles in the classical 1D theory.

6. Grobner Bases and the Computation of Solutions

Grobner bases play a huge role in computational commutative algebra. The literatureon Grobner bases provides algorithms for computation of torsion submodules, Hilbert


polynomials, finite free resolutions and many other objects, none of which can in general becomputed without the knowledge of a Gröbner basis or equivalent object. Due to the dualitybetween modules and systems, Gröbner bases likewise play a large role in the computationof various system-theoretic objects, e.g. the controllable part, the manifest behaviour of alatent variable description, or the complexity indices to be discussed in section 6.3.

In fact, the concept of a Gröbner basis, and the Buchberger algorithm for computingGrobner bases, were discovered in the context of partial differential equations before theyarose in algebra. As discusses by Pommaret in the introduction to [39], this earlier form ofthe Grobner basis is due to Janet [22]. However, the literature on Gröbner bases has grownenormously to date, whereas the equivalent work of Janet is not well-known. Both strandsof work are in fact superseded by the notion of “formal integrability” (e.g. [20, 51]), aconcept which comes from the formal theory of pdes and which has the advantage of beingindependent of the choice of basis. The concept of formal integrability and its relation tothe Janet (or Buchberger) algorithm is discussed by Pommaret in [39].

An exposition of algorithms for computing various objects in behavioural theory is beyondthe scope of this paper. Instead, we will give some examples of concepts which are verystrongly linked to Gröbner bases and which have a direct role to play in systems theory.

The discussion and results of this section do not apply to systems with signal spaceA = kZn.

6.1. Grobner Bases

We begin by defining the standard concepts of a monomial ordering and initial term module[1, 4, 6, 8]. ThroughoutR denotes the polynomial ringk[z].

Definition 13. Recall that amonomial of R1,q is any element of the formzaej for somea ∈ Nn, j ∈ 1, . . . ,q. A monomial ordering onR1,q is a total ordering of the monomialsof R1,q such that ifm1 andm2 are monomials ofR1,q andn 6∈ k is a monomial ofR thenm1 ≥ m2 impliesnm1 ≥ nm2 > m2.

Given an arbitrary elementv ofR1,q and a monomial ordering≥ onR1,q, we denote byin≥ v the initial term of v, i.e. the term ofv which is greatest under the given monomialordering. IfN ⊆ R1,q is a submodule, then in≥ N denotes the submodule ofR1,q generatedby elements of the form in≥ v, v ∈ N. We refer to in≥ N as theinitial term module of N.

A Grobner basisof N is a generating set forN whose initial terms are a generating setfor in≥ N. If the Grobner basis has the additional property that no term of any element is amultiple of an initial term of a distinct element, then it is called areduced Grobner basis.We will call a matrix E with q columns a(reduced) Grobner basis matrix for N if itsrows form a (reduced) Gröbner basis of theirR-span,N.

The importance of Gröbner bases comes from the strong link between the modulesNand in≥ N. Every submodule ofR1,q has a reduced Gröbner basis under any monomialordering. (Reduced) Gröbner bases can be computed by means of Buchberger’s algorithm[1, 4, 6, 8]. In the 1D case, a reduced Gröbner basis matrix is a polynomial matrix inHermite from [29, p102–104].

40 J. WOOD

6.2. Initial Conditions and Observables

The following classic theorem is due to Macaulay; see for example [8, Thm. 15.3].

THEOREM 11 Let B⊥ be a submodule ofR1,q with factor module M, and let≥ be amonomial ordering onR1,q. Then the monomials ofR1,q which are outsidein≥ B⊥ forma basis for M over the field k.

We denote byT(M,≥) the set of monomials ofR1,q which are outside in≥ B⊥. Theorem11 says that this (generally infinite) set of observables is independent overk, and anyobservable can be written as ak-linear combination of elements ofT(M,≥). Such a“normal form” for any given observable can be obtained via a reduced Gröbner basis byinductively subtracting multiples of elements of the basis, and it is easy to see that “normalforms” are unique.

This can be further interpreted as follows. In the continuous case, a monomial observablesimply picks out the value of a derivative of some system variable; in the discrete case,a shift. For ease of exposition we consider only the continuous case. Thus there is apartition of the set of derivatives of system variables into a setS1 of independent (overk)derivatives, and another setS2 of derivatives, each of which can be uniquely expressed asa k-linear combination of the values ofS1. In the formal theory of pdes, the first set arecalledparametric derivatives and the second setprincipal derivatives; the partition is a“cut” in the sense of Riquier [43]. See the book [39] by Pommaret for a further discussionof this topic in the context of the formal theory of pdes.

Theorem 11 therefore leads to the formal power series solution of any set of linear partialdifferential equations with constant coefficients; the parametric derivatives (correspondingto the monomials outside in≥ B⊥) can be chosen freely at the origin, and the values of theprincipal derivatives (corresponding to the monomials of in≥ (B⊥) at the origin are therebydetermined. The discrete equivalent of this is the solution of the canonical Cauchy problemfor systems with signal spacekNn , which is due to Oberst [29, para. 5]. The initial conditionsidentified by Oberst are the values of the trajectory at the pointsa ∈ Nn corresponding tothe monomialsza lieing outside in≥ B⊥. A trajectory is then determined from its values onthese initial conditions, e.g. using a reduced Gröbner basis.

It is helpful to introduce theinitial condition behaviour B, defined for any given mono-mial ordering≥ by

in≥ B := D(R1,q/in≥ B⊥) (48)

In the discrete caseA = kNn, in≥ B is the set of all trajectories ofAq which are free on theinitial condition points and zero elsewhere. In the continuous case, in≥ B is the set of alltrajectories ofAq for which the system variable derivatives corresponding to the monomialsof in≥ B⊥ vanish. Note that the notation in≥ B⊥ is unambiguous, as in≥ (B)⊥ = in≥ (B⊥).

It is also interesting to note that the initial conditions can be regarded as the values ofa given set of observables (those in the basisT(M,≥)) at the origin. The values of theelement ofT(M,≥) therefore play the role of states, a point of view which effectively isfurther explored by Kleon and Oberst in [24].


EXAMPLE. Consider the 2D Fornasini–Marchesini model with signal spaceRN2. Thestate-input behaviour is given by

Bx,u = KerAE, E = (z2A1+ z1A2− z1z2I z2B1+ z1B2),

whereA1, A2, B1 and B2 are matrices overR. Let N be the number of states andm thenumber of inputs. For any monomial ordering onR = R[z1, z2] which is a refinement ofthe partial ordering by total degree, it is easy to see thatE is a Grobner basis matrix. Hencethe observables ofBx,u have ak-basisT(M,≥) = T1(M,≥) ∪ T2(M,≥), where

T1(M,≥) := {ei + B⊥, za11 ei + B⊥, za2

2 ei + B⊥ | i = 1, . . . , N,a1,a2 ∈ Z+}T2(M,≥) := {za1

1 za22 ej + B⊥ | j = N + 1, . . . , N +m,a1,a2 ∈ N}

Thus T1(M,≥) are observables corresponding to the classical initial conditions, andT2(M,≥) are observables corresponding to the free variables.

The solution of the Cauchy problem for more general discrete systems, and especially forsystems with signal spacekZn, has been derived by Pauer, Unterkircher and Zampieri [34,35, 62], and by Zerz and Oberst [66]. These papers also contain algorithms for generatinga trajectory from the initial conditions. Further work by Kleon and Oberst [24] looks inmore detail at trajectory computation.

6.3. Filtrations and Complexity

LetR1,q0 ⊆ R1,q

1 ⊆ R1,q2 ⊆ · · ·denote the standard filtration ofR1,q, i.e.

R1,qs := {v ∈ R1,q | v has degree at mosts} (49)

Each setR1,qw has the structure of a vector space overk, but is not a module. Now a

submoduleB⊥ of R1,q has a standard filtration given byB⊥s := R1,qs ∩ B⊥. This leads

naturally to a filtration

M0 ⊆ M1 ⊆ M2 ⊆ · · · (50)

of the factorM = R1,q/B⊥, given by

M2 := R1,qs + B⊥B⊥

∼= R1,qs

B⊥s(51)

We now introduce thedegree of an observablev + B⊥, denoted deg (v + B⊥), as theminimums such thatv + B⊥ ∈ Ms. We have

deg(v + B⊥) := min{degs | s ∈ R1,q andv − s ∈ B⊥}, (52)

i.e. the degree is the minimum degree in terms of which the observable can be expressed.In the continuous case, we can think of this as the minimum number of sequential differ-entiations needed to compute it (derivatives of distinct system variables being computablein parallel).

42 J. WOOD

Now in the discrete case, for anys ∈ N let Ts denote the set of all lattice points inNn

corresponding to monomials ofR with degree at mosts, i.e.

Ts :={

a ∈ Nn

∣∣∣∣∣ n∑i=1

ai ≤ s

}(53)

Now letB(s) denote the restriction ofB to this “hypertriangle”Ts, i.e.

B(s) := {v ∈ (kq)Ts | ∃ w ∈ B with w(a) = v(a) for all a ∈ Ts} (54)

The continuous counterpart, which we will also denote byB(s), is the space of formal powerseries solutions of the system, truncated to degrees. This space, represented geometricallyas a kernel of jet bundle maps, plays a central role in the formal theory of pdes, as discussedin [39, Chs. III,IV].

As observed by Pommaret and Quadrat [41] in the context of the formal theory,B(s) andMs are dual in the following sense.

LEMMA 9 For any s∈ N, we have the duality

B(s) = Homk(Ms, k) (55)

In particular, the space of solutions on the hypertriangle Ts has the same dimension as thespace of observables of degree at most s.

Proof: There is an orthogonality of vector spaces betweenB⊥s andB(s). The relationship(55) now follows from (51).

In the discrete case, we have the notion of the complexity indices of the behaviour, duein the 1D case to Willems [54], extended to nD systems with signal spacekNn in [58]. Herewe extend the definition to continuous systems in the obvious way.

Definition 14. Given a discrete behaviourB with signal spacekNn, or a continuousbehaviourB with signal spaceC∞(Rn, k) or D′(Rn, k), thecomplexity indicesof B aredefined as

cs(B) := dimk B(s) (56)

Thus the complexity measures the growth of the behaviour on increasingly large regionsof Nn (discrete case), or the growth of the space of truncated formal power series solutions(continuous case). By Lemma 9,cs = dimk Ms.

The following definition is well-known in algebra, but is usually given in textbooks forgraded modules rather than for filtrations, e.g. [8, p42]. The filtration form can be foundfor the case of an ideal in [6, Defns. 9.3.2, 9.3.5].

Definition 15. Let M0 ⊆ M1 ⊆ M2 ⊆ · · · be the standard filtration (50) of a finitelygenerated moduleM over the ringR = k[z]. Then theaffine Hilbert function of M is the


mapHM(s): N→ N given by

HM(s) := dimk Ms (57)

For sufficiently large,s, HM(s) agrees with a polynomial function ofs of degree at mostn,called theaffine Hilbert polynomial of M and denoted byPM(s).

The existence of the Hilbert polynomial is well-known, and proved for example in [8,Theorem 1.11] for the graded case. It is now clear that the complexity indices of a behaviourare given by the values of the Hilbert function of the module of the observables. The factthat the Hilbert function agrees with a polynomial of degree at mostn guarantees that the“structure indices”, derived from the complexity indices, are finite in number. This andrelated results are discussed in [58]. The Hilbert polynomial also provides the module with“characters”, which are of great importance in the formal theory of pdes (e.g. [39, p89–91]).

In [7], Delaleau and Fliess use filtrations to establish results on invertibility and decouplingof 1D time-varying systems.

6.4. Computing the Complexity

Our final problem is to compute the complexity indices of a behaviour, or equivalently tocompute the affine Hilbert function of a finitely generatedR-module.

A first attempt is to generate the spacesB⊥s from the equations of an arbitrary kernelrepresentation. Suppose thatE is a kernel representation ofB for which the maximum ofthe degrees of the rows isd. We defineB⊥d to be thek-span of the rows ofE. Then weobtain another filtration ofB⊥; now write

B⊥s :={

0 s< d

Rs−dB⊥d s ≥ d(58)

This is most easily interpreted in the continuous case when the rows ofE are treated asdifferential equations:B⊥s is the set of all equations of degree at mosts which can beobtained from the rows ofE by differentiating at mosts− d times.

Unfortunately, in the general case we have

B⊥s ( B⊥s ,

i.e. some equations of degrees can only be obtained by further differentiation and can-cellation of the highest terms. This whole issue is the basis for the concept offormalintegrability , which is central to the formal theory of pdes (e.g. [20, 39, 51]). Formalintegrability can however be defined in an intrinsic, i.e. basis-free manner, and we will sayno more about it here.

THEOREM12 [58] Let B ⊆ Aq be a differential behaviour or a difference behaviour withsignal space kNn, and let≥ be alag-priority monomial ordering onR1,q, i.e. a monomial

44 J. WOOD

ordering which is a refinement of the partial ordering by degree. Then

cs(B) = cs(in≥ B) (59)

Proof: A proof using homogenization arguments is given in [58]. The following demon-stration is considerably simpler.

By lemma 9,cs(B) is equal to the dimension ofMs, the space of observables of degree atmosts. On the other hand,cs(in≥ B) is equal to the number of elements of degree at mosts in T(M,≥), the basis forM obtained from the monomial ordering≥. Clearly, thek-spanof those elements ofT(M,≥) is contained inMs, and so

cs(in≥ B ≤ cs(B)

Now let v1 + B⊥ ∈ Ms be arbitrary. By definition ofT(M,≥), we can writev1 + B⊥ =v2 + B⊥, where each monomial occurrring inv2 is outside in≥ B⊥, andv2 is uniquelydetermined. It remains to show that degv2 ≤ s. However if degv2 > s ≥ deg v1,then since≥ is a lag-priority ordering we would have thatv2 − v1 ∈ B⊥ has initial termequal to that ofv2. This contradicts the definition ofv2, which proves the theorem.

Theorem 12 is not generally true for non-lag-priority monomial orderings [58].In the discrete case, equation (59) says that thesth complexity index ofB is equal to the

number of initial conditions in the hypertriangleTs; in the continuous case,cs(B) is equalto the number of parametric derivatives of degree at mosts. In either case, we have thatcs(B) is equal to the number of monomials of degree at mosts in T(M,≥). The complexityindices can now easily be computed from a Gröbner basis with respect to≥. All of theseresults depend on the fact that≥ is a lag-priority ordering.

7. Conclusions

Due to the vase size of the area, we have not been able to provide a comprehensive overviewof the use of modules and behaviours in nD systems theory. We have concentrated on themore fundamental theoretical results, and also on some of the connections between theapproaches of different authors. In particular, we have shown that the concept of a systemobservable is (in some cases, implicitly) common to all approaches, and proves the closestpoint of connection between behaviours and modules.

The behavioural theory we have discussed is largely applicable only to constant linearsystems. However, much of the algebraic theory can be generalized to the 1D/nD variablecoefficients case (e.g. [10, 13, 39, 41, 42]), and even to 1D/nD non-linear systems (e.g. [7,9, 11, 38, 39, 67]).

Further work on nD behavioural theory includes: state-space models for 2D behaviours(e.g. [44, 46]), system identification [30, 63], 2D model reduction [45], stability and stabi-lizability [52], pole placement [36], causality [28, 64], and the beginnings of a formal theoryof nD control in the behavioural framework [47]. Another huge area for further research


is in the connection between behaviours and convolutional codes (e.g. [16, 17, 49]). Weanticipate that the tools of behaviours and module theory will continue to be highly effectivein nD systems research in the coming years.

Acknowledgments

I would like to thank Ulrich Oberst, Jean-Fran¸cois Pommaret, Paula Rocha, Eric Rogers,Eva Zerz and other colleagues for many interesting and valuable discussions.

Notes

1. Note that the modules KerRE, ImRE,CokerRE are defined using row vectors, and do not therefore requiretransposition ofE, contrary to the author’s previous practice.

References

1. T. Becker and V. Weispfenning,Grobner Bases: A Computational Approach to Commutative Algebra,volume 141 ofGraduate Texts in Mathematics, Springer-Verlag, 1993.

2. J.-E. Bjork, Rings of Differential Operators, North-Holland Publishing Company, 1979.

3. H. Bourles and M. Fliess, “Finite Poles and Zeros of Linear Systems: An Intrinsic Approach,”Int. J. ofControl, vol. 68, no. 4, 1997, pp. 897–922.

4. B. Buchberger, “Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory,” In N. K. Bose,editor,Multidimensional systems theory, D. Reidel, 1985, pp. 184–232.

5. P. M. Cohn,Algebra, volume 2. Chinchester, UK, John Wiley and Sons, 2nd edition, 1989.

6. D. Cox, J. Little, and D. O’Shea,Ideals, Varieties and Algorithms,Springer-Verlag, 2nd edition, 1996.

7. E. Delaleau and M. Fliess, “Algorithme de Structure, Filtrations et Découplage,”Computes Rendus Hebdo-madaires des Seances de l’Academie des Sciences, Serie I, vol. 315, 1992, pp. 101–106.

8. D. Eisenbud,Commutative Algebra with a View Toward Algebraic Geometry, volume 150 ofGraduate Textsin Mathematics, Springer-Verlag, 1995.

9. M. Fliess, “Generalized Controller Canonical Forms for Linear and Nonlinear Dynamics,”IEEE Trans. onAuto. Contr., vol. 35, 1990, pp. 994–1001.

10. M. Fliess, “Some Basic Structural Properties of Generalized Linear Systems,”Systems and Control Letters,vol. 15, 1990, pp. 391–396.

11. M. Fliess, “Controllability Revisited,” In A. C. Antoulas, editor,Mathematical System Theory(the influenceof R. E. Kalman), pp. 463–474, Springer-Verlag, 1991.

12. M. Fliess, “A Remark on Willems’ Trajectory Characterization of Linear Controllability,”Syst. Contr. Lett.,vol. 19, 1992, pp. 43–45.

13. M. Fliess, “Une Interprétation Algebrique de la Transformation de Laplace et des Matrices de Transfert,”Linear Algebra and Its Applications, vol. 203–204, 1994, pp. 429–442.

14. M. Fliess, J. Lévine, P. Martin, and P. Rouchon, “Flatness and Defect of Non-Linear Systems: IntroductoryTheory and Examples,”International Journal of Control, vol. 61, no. 6, 1995, pp. 1327–1361.

15. E. Fornasini, P. Rocha, and S. Zampieri, “State Space Realization of 2-D Finte-Dimensional Behaviours,”SIAM J. on Contr. and Opt., vol. 31, no. 6, 1993, pp. 1502–1517.

46 J. WOOD

16. E. Fornasini and M. E. Valcher, “Algebraic Aspects of 2D Convolutional Codes,”IEEE Trans. in Inf. Tech.,vol. 33, 1993, pp. 1210–1225.

17. E. Fornasini and M. E. Valcher. “nD Polynomial Matrices with Applications to Multidimensional SignalAnalysis,”Multidimensional Systems and Signal Processing, vol. 8, no. 4, 1997, pp. 387–408.

18. S. Frohler and U. Oberst, “Continuous Time-Varying Linear Systems,Syst. Contr. Lett., vol. 35, no. 2, 1998,pp. 97–110.

19. H. Glusing-Luerßen, “A Behavioural Approach to Delay-Differential Systems,”SIAM J. Control Optim.,vol. 35, no. 2, 1997, pp. 480–499.

20. H. Goldschmidt, “Integrability Criterion for Systems of Nonlinear Partial Differential Equations,”J. Differ-ential Geom., vol. 1, 1969, pp. 269–307.

21. L. C. G. J. M. Habets. “System Equivalence for Delay-Differential Systems with Incommensurable Delays,”Presented at the 13th International Symposium on theMathematical Theory of Networks and Systems, Padua,Italy, 1998.

22. M. Janet, “Les Systèms d’Equations aux Dérivees Partielles,”J. Math. Pure Appl., vol. 3, 1920, p. 65.

23. R. E. Kalman, P. L. Falb, and M. A. Arbib,Topics in Mathematical Systems Theory,International Series inPure and Applied Mathematics, McGraw-Hill, 1969.

24. S. Kleon and U. Oberst, “Transfer Operators and State Spaces for Discrete Multidimensional Linear Sys-tems,”Acta Applicandae Mathematicae, vol. 57, 1999, pp. 1–82.

25. J. Komornik, P. Rocha, and J. C. Willems. “Closed Subspaces, Polynomial Operators in the Shift, andARMA Representations,”Applied Mathematics Letters,vol. 4, no. 3, 1991, pp. 15–19.

26. V. Lomadze, “Finite-Dimensional Time-Invariant Linear Dynamical Systems: Algebraic Theory,”ActaAppliacandae Mathematicae, vol. 19, 1990, pp. 149–201.

27. H. Mounier,Proprietes Structurelles des Systemes Lineaires A Retards: Aspects Theoriques et Pratiques,PhD thesis, Université Paris-Sud, 1995.

28. M. Napoli and S. Zampieri, “2D Proper Rational Matrices and Causal Input/Output Representations of 2DBehavioural Systems,”SIAM J. on Control and Optimization, vol. 37, 1999, pp. 1538–1552.

29. U. Oberst, “Multidimensional Constant Linear Systems,”Acta Applicandae Mathematicae, vol. 20, 1990,pp. 1–175.

30. U. Oberst, “On the Minimal Number of Trajectories Determining a Multidimensional System,”Mathematicsof Control, Signals and Systems, vol. 6, 1993, pp. 264–288.

31. U. Oberst, “Variations on the Fundamental Principle for Linear Systems of Partial Differential and DifferenceEquations with Constant Coefficients,”Applicable Algebra in Engineering, Communication and Computing,vol. 6, no. 4/5, 1995.

32. U. Oberst, “Finite Dimensional Systems of Partial Differential or Difference Equations,”Advances in AppliedMathematics,vol. 17, 1996, pp. 337–356.

33. V. P. Palamodov.Linear Differntial Operators with Constant Coefficients, Springer-Verlag, 1970.

34. F. Pauer and A. Unterkircher, “Gröbner Bases for Ideas in Laurent Polynomial Rings and Their Applicationto Systems of Difference Equations,”Applicable Algebra in Engineering, Communication and Computing,vol. 9, 1999, pp. 271–291.

35. F. Pauer and S. Zampieri, “Gröbner Bases with respect to Generalized Term Orders and Their Applicationto the Modelling Problem,”J. Symbolic Computation, vol. 21, 1996, pp. 155–168.

36. H. Pillai and S. Shankar, “A Behavioural Approach to Control of Distributed Systems,”SIAM Journal onControl and Optimization, vol. 37, 1999, pp. 388–408.

37. J. W. Polderman and J. C. Willems,Introduction to Mathematical Systems Theory: A Behavioural Approach,volume 26 ofTexts in Applied Mathematics, Springer-Verlag, 1997.

38. J.-F. Pommaret, “New Perspectives in Control Theory for Partial Differential Equations,”IMA Journal onMathematical Control and Information, vol. 9, 1992, pp. 305–330.


39. J.-F. Pommaret,Partial Differential Equations and Group Theory: New Perspectives for Applications,volume 293 ofMathematics and Its Applications, Dordrecht, 1994.

40. J.-F. Pommaret and A. Quadrat, “Formal Elimination for Multidimensional Systems and Applications toControl Theory,” Technical Report 97-108, CERMICS/INRIA ENPC, France, 1997. Submitted for publi-cation.

41. J.-F. Pommaret and A. Quadrat, “Algebraic Analysis of Linear Multidimensional Control Systems,”IMA J.on Math. Control and Information, vol. 16, 1999, pp. 275–297.

42. J.-F. Pommaret and A. Quadrat, “Generalized Bezout Identity,”Applicable Algebra in Engineering, Com-munication and Computing, vol. 9, no. 2, 1998, pp. 91–116.

43. Ch. Riquier,La Methode des Fonctions Majorantes et les Systemes d’Equations aux Derivees Partielles,volume XXXII of Memorial Sci. Math., Gauthier-Villars, Paris, 1910.

44. P. Rocha,Structure and Representation of 2D Systems, PhD thesis, University of Groningen, The Nether-lands, 1990.

45. P. Rocha, “State/Driving-Variable Representations and Model Reduction forl2 2D Systems,” InProc. of theEuropean Control Conference 1997, 1997.

46. P. Rocha and J. C. Willems, “State for 2-D Systems,”J. Linear Algebra Appl., vol. 121–123, 1989, pp. 1003–1038.

47. P. Rocha and J. Wood, “A Foundation for the Control Theory of nD Behaviours,” Presented at the 13thInternational Symposium on theMathematical Theory of Networks and Systems, Padua, Italy, 1997.

48. P. Rocha and J. Wood, “A New Perspective on Controllability Properties for Dynamical Systems,”AppliedMathematics and Computer Science, vol. 7, no. 4, 1997, pp. 869–879.

49. J. Rosenthal, “Some Interesting Problems in Systems Theory which are of Fundamental Importance inCoding Theory,” InProc. of the 36th IEEE Conference on Decision and Control, IEEE, 1997, pp. 4574–4579.

50. J. Rudolph, “Duality in Time-Varying Linear Systems—A Module Theoretic Approach,”Linear Algebraand Its Applications, vol. 245, 1996, pp. 83–106.

51. D. C. Spencer, “Overdetermined Systems of Partial Differential Equations,”Bull. Am. Math. Soc., vol. 75,1965, pp. 1–114.

52. M. E. Valcher, “Characteristic Cones and Stability Properties of Two-Dimensional Autonomous Behaviours,”To appear inIEEE Trans. Circ. and Sys. Part I, 2000.

53. M. E. Valcher, “On the Decomposition of Two-Dimensional Behaviours,” Submitted for publication, 1998.

54. J. C. Willems, “From Time Series to Linear System—Part I: Finte-Dimensional Linear Time InvariantSystems,”Automatica, vol. 22, no. 5, 1986, pp. 561–580.

55. J. C. Willems, “Paradigms and Puzzles in the Theory of Dynamical Systems,”IEEE Trans. on Auto. Contr.,vol. 36, no. 3, 1991, pp. 259–294.

56. J. C. Willems, “On Interconnections, Control, and Feedback,”IEEE Trans. on Auto. Contr., vol. 42, no. 4,1997, pp. 458–472.

57. J. Wood, U. Oberst, E. Rogers, and D. H. Owens, “A Behavioural Approach to the Pole Structure of 1D andnD Linear Systems,” To appear inSIAM J. on Contr. and Opt., 1998.

58. J. Wood, P. Rocha, E. Rogers, and D. H. Owens, “Structure Indices for Multidimensional Systems,” Toappear in theIMA J. of Mathematical Control and Information, vol. 10, 1999, pp. 33–69.

59. J. Wood, E. Rogers, and D. H. Owens, “A Formal Theory of Matrix Primeness,”Mathematics of Control,Signals and Systems, vol. 11, no. 1, 1998, pp. 40–78.

60. J. Wood, E. Rogers, and D. H. Owens, “Controllable and Autonomous nD Systems,”Mult. Systems andSignal Processing, vol. 10, 1999, pp. 33–69.

61. J. Wood and E. Zerz, “Notes on the Definition of Behavioural Controllability,”Sys. and Contr. Lett., vol. 37,1999, pp. 31–37.

48 J. WOOD

62. S. Zampieri, “A Solution of the Cauchy Problem for Multidimensional Discrete Linear Shift-InvariantSystems,”Linear Algebra and Its Applications, vol. 202, 1994, pp. 143–162.

63. S. Zampieri, “A Behavioural Approach to Identifiability to 2D Scalar Systems,”Automatica, vol. 33, no. 1,1997, pp. 49–61.

64. S. Zampieri, “Causal Input/Output Representation of 2D Systems in the Behavioural Approach,” theSIAMJournal on Mathematical Control and Optimization, vol. 36, 1998, pp. 1133–1146.

65. E. Zerz, “Primeness of Multivariate Polynomial Matrices,”Systems and Control Letters,vol. 29, no. 3, 1996,pp. 139–146.

66. E. Zerz and U. Oberst, “The Canonical Cauchy Problem for Linear Systems of Partial Differnece Equa-tions with Constant Coefficients Over the Completer -Dimensional Integral LatticeZr , Acta ApplicandaeMathematicae, vol. 31, no. 3, 1993, pp. 249–273.

67. Y. F. Zheng, J. C. Willems, and C. S. Zhang, “Common Factors and Controllability of Nonlinear Systems,”In Proceedings of the 36th IEEE Conference on Decision and Control,1997, pp. 2584–2589.

Documents

Modules and Behaviours in nD Systems Theory