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Discrete multidimensional systems over Zn
Systems & Control Letters 56 (2007) 702 – 708 www.elsevier.com/locate/sysconle
Discrete multidimensional systems over Zn Eva Zerz Lehrstuhl D für Mathematik, RWTH Aachen University, 52062 Aachen, Germany Received 10 August 2006; received in revised form 21 May 2007; accepted 1 June 2007 Available online 10 July 2007
Abstract We generalize the theory of multidimensional, discrete, linear, shift-invariant systems from the well-investigated case in which the signals take their values in a field, to the situation where these values belong to a finite ring of the form Z/nZ. Some basic systems theoretic properties are studied and characterized: autonomy, input–output decompositions, controllability, and observability. © 2007 Elsevier B.V. All rights reserved. Keywords: Multidimensional discrete linear shift-invariant systems; Systems over rings; Behavioral approach; Algebraic methods; Structural properties; Controllability
1. Introduction The study of multidimensional, discrete, linear, shiftinvariant systems in the behavioral framework originated in 1990 with Rocha’s thesis [15]. In the same year, a seminal paper by Oberst [12] introduced a rigorous algebraic approach to these (and other) systems, using the comprehensive language of category theory. The theory has subsequently been refined by various authors, for instance, see [16,17] and the references therein. A standing assumption in all these contributions is that the signals, which are multivariate sequences, take their values in a field. The present paper relaxes this restriction and studies signals with values in a finite ring of the form Zn := Z/nZ for some integer n > 1. Systems over finite rings have potential applications in coding theory, both in the 1D setting [3–5] and in the multidimensional case [6,8,10]. 2. Abstract linear systems Let D be a commutative ring (with 1), and let A be a D-module. For a given matrix R ∈ Dg×q , we consider the abstract linear system B := {w ∈ A | Rw = 0}. q
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The letter B has been chosen in allusion to the behavioral approach. One should think of A as a set of “signals”, and of D as a ring of “operators” acting on them. To say that A carries a D-module structure amounts more or less to the requirement that one can apply any operator d ∈ D to any signal a ∈ A to obtain a new signal da ∈ A. In the same way, the expression Rw from above becomes a well-defined element of Ag . The matrix R is called a representation matrix of B. For instance, let D = R[d/dt], which is the ring of linear differential operators with real coefficients, its elements being of the form d = rn
dn d + · · · + r1 + r 0 , dt n dt
where n ∈ N and ri ∈ R. A signal set with a D-module structure is given by A = C∞ (R, R), the set of smooth functions from R to R. Any operator d ∈ D can be applied to any signal a ∈ A, producing a function da which is again smooth, that is, da ∈ A. A set B as above is then the solution set of a system of linear ordinary differential equations with real coefficients. Another standard example is D = R[] and A = RN . Then a signal is a sequence a : N → R and d = rn n + · · · + r1 + r0 , where n ∈ N and ri ∈ R, acts on a ∈ A via (da)(t) := rn a(t+n) + · · · + r1 a(t+1)+r0 a(t) for all t∈N.
E. Zerz / Systems & Control Letters 56 (2007) 702 – 708
Clearly, da belongs again to A. A set B as above is then the solution set of a system of linear difference equations with real coefficients. Coming back to the abstract setting, an important observation by Malgrange [9] says that B is a D-module that is isomorphic to the module consisting of all D-homomorphisms from M := D1×q /D1×g R to A, that is, M → A , (1) B HomD (M, A), w → [x] → xw, where [x] denotes the residue class of x ∈ D1×q in M. The result itself is not hard to prove, but its importance lies in the fact that it draws attention to the algebraic object M, called the system module, and to the contravariant functor HomD (·, A) on the category of D-modules. One says that the D-module A is an injective cogenerator [7] if the functor F := HomD (·, A) is exact and faithful. This means that it preserves and reflects exactness, that is, a sequence of D-modules and D-homomorphisms f
g
M −→ N −→ P
B⊥ := {x ∈ D1×q | xw = 0 ∀w ∈ B} = D1×g R.
(3)
Given B = {w ∈ Aq | Rw = 0}, we can write B⊥ , which is a submodule of D1×q and thus finitely generated (since D is Noetherian), as B⊥ =D1×g1 R1 . Since every row of R belongs to B⊥ , we get D1×g R ⊆ D1×g1 R1 . Thus B ⊇ B1 := {w ∈ Aq | R1 w = 0}. On the other hand, any w ∈ B satisfies xw = 0 for all x ∈ B⊥ and thus R1 w = 0. We conclude that B = B1 . Thus for 0 → M → M1 (the map [x] → [x] is well-defined since D1×g R ⊆ D1×g1 R1 ), we have that 0 ← HomD (M, A) ← HomD (M1 , A) is exact, as it corresponds (via the Malgrange isomorphism) to id
Fg
F M ←− F N ←− F P is exact (that is, im(F g) = ker(Ff )), where (Ff )() = ◦ f for ∈ F N = HomD (N, A), and Fg is defined analogously. A well-known example is the case where D = F and A = F, where F is an arbitrary field (and thus also a commutative ring with 1, which can be considered as a module, or in this case, even vector space, over itself). Then HomF (·, F) is the functor that assigns to each F-vector space V its dual vector space V ∗ = HomF (V , F), and to every F-linear mapping f : V → W between two F-vector spaces its dual mapping f ∗ : W ∗ → V ∗ , → ◦ f . It is a standard fact of linear algebra that HomF (·, F) is an exact and faithful functor: formulated in the language of matrices, this simply amounts to im(A) = ker(B) ⇔ ker(·A) = im(·B) for A ∈ Fn×m , B ∈ Fp×n . Here, A is the map A : Fm → Fn , x → Ax and ·A denotes the map ·A : F1×n → F1×m , y → yA, and analogously for B. The injective cogenerator property is a very powerful tool for systems theory, because it enables us to translate any statement on abstract linear systems that can be formulated in terms of kernels and images, into an equivalent statement on D-modules. The most prominent consequences of the injective cogenerator property are: • If P ∈ Dg×p and Q ∈ Dh×g are such that im(·Q) := D1×h Q = ker(·P ) := {x ∈ D1×g | xP = 0}, then we have for all v ∈ Ag ∃y ∈ Ap : P y = v ⇔ Qv = 0.
• Assume that D is Noetherian. Then for an arbitrary representation matrix R of B, we have
0 ← B ←− B1 .
is exact (that is, im(f ) = ker(g)) if and only if the associated sequence Ff
703
(2)
This is called the “fundamental principle”, and it is nothing but the preservation of exactness, formulated in the language of matrices.
Since exactness is reflected, also 0 → M → M1 must be exact, and thus B⊥ = D1×g1 R1 = D1×g R. Thus, we have B1 = {w ∈ Aq | R1 w = 0} ⊆ B2 = {w ∈ Aq | R2 w = 0} if and only if there exists X ∈ Dg2 ×g1 such that R2 = XR 1 . Here, the “if” part is obvious, and the “only if” part follows from ⊥ 1×g1 B1 ⊆ B2 ⇒ B⊥ R1 ⊇ D1×g2 R2 . 1 ⊇ B2 ⇒ D
Note that this yields also a full characterization of the equivalence of representations: we have B1 = B2 if and only if there exist X, Y such that R2 = XR 1 and R1 = Y R 2 . In other words, two matrices represent the same behavior if and only if they have the same row module. In particular, there exists no 0 = d ∈ D with dA={0} (since dA={0} means {w ∈ A | dw=0}=A={w ∈ A | 0w=0}, and hence, we must have d = 0). Moreover, B = {0} holds if and only if any representation matrix R of B is left invertible. (Here, the “if” part is obvious, and the “only if” part follows from {w ∈ Aq | Rw = 0} = {0} = {w ∈ Aq | Iq w = 0}, which implies that Iq = XR for some X.) In many cases, the relevant rings D are variants of polynomial rings, which can be efficiently manipulated using modern computer algebra systems. Using the correspondence outlined above, one can then re-interpret the results of these computations using the language of systems theory. The rest of the paper is devoted to applying these general −1 ideas to the special case where D=C[1 , . . . , r , −1 1 , . . . , r ] r and A = C Z . Here, the signals are sequences, r-fold indexed by integers, taking their values in C. The ring D consists of all linear shift (or difference) operators with coefficients in C. A lot is known about these systems in the case where C is a field, for instance, see [12,15–17] and the references therein. However, in the present paper, we will study the case where C
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is a finite ring of the form Z/nZ for some integer n > 1. The main technical difficulty is that then D is no longer a domain (unless n is a prime number, but in this case, C is already a field, and we are back in the known situation). For instance, D has no quotient field, and thus we have no concept of transfer matrices, etc. 3. Multidimensional discrete systems over Z/nZ Let n > 1 be an integer. The ring C := Z/nZ is a quasiFrobenius ring [7, Section 15], i.e., it is Noetherian and selfinjective [7, Section 3B]. Moreover, it is a cogenerator ring [7, Section 19B]. This means that HomC (·, C) is an exact and faithful functor. −1 The Noetherian ring D := C[1 , . . . , r , −1 1 , . . . , r ] consists of all multivariate Laurent polynomials in the indetermir nates i , with coefficients in C. The set A := C Z consists of all functions Zr → C, that is, of all sequences, r-fold indexed by integers, taking their values in C. There is a natural D-module structure on A, which is given by (l11 · · · lrr a)(k1 , . . . , kr ) := a(k1 + l1 , . . . , kr + lr ) for all a ∈ A and ki , li ∈ Z. This is, the action of i on the sequence a=(a(k))k∈Zr is given by a “shift in the ith direction”. We will use the notation l := l11 · · · lrr for l ∈ Zr . There is an isomorphism of C-modules
B HomD (M, A) from (1), and the standard isomorphism A HomD (D, A) (that relates a D-linear map D → A to the image of the D-basis 1 of D), we find that j can be identified with HomD (j , A), where j : D → M,
1 → [ej ],
and ej denotes the jth natural basis vector of D1×q . Theorem 1. The following are equivalent: 1. B is weakly autonomous; 2. for all [ej ] ∈ M there exists 0 = dj ∈ D such that dj [ej ] = 0; 3. for any representation matrix R of B there exist d1 , . . . , dq ∈ D\{0} and X ∈ Dq×g such that diag(d1 , . . . , dq ) = XR. Proof. Weak autonomy is equivalent to the non-existence of a surjection j : B → A with j (w) = wj . In view of the injective cogenerator property, this is in turn equivalent to the nonexistence of an injection j : D → M with j (1)=[ej ]. Thus, weak autonomy amounts to the requirement that ker(j ) = 0 for all j, that is, ∀j ∃0 = dj : j (dj ) = dj [ej ] = 0. Finally, the third condition is merely the translation of Condition 2 into the language of matrices.
which relates a C-linear map : D → C to the image of the C-basis {l | l ∈ Zr } of D. The D-module structure of A can be transferred to the left-hand side by putting (d)(e) := (de) for d, e ∈ D. As pointed out above, HomC (·, C) is exact and faithful. Let M be a D-module. Because of the standard isomorphism [2, Chapter II, Section 4.1]
It is worth noting that the condition rank(R) = q, which is equivalent to weak autonomy in the case where C is a field, is only sufficient for weak autonomy in the present situation, ¯ 2 ∈ Z2×2 defines a and not necessary. For instance, R = 2I 4 ¯ and thus weakly autonomous system, but det(R) = 2¯ · 2¯ = 0, rank(R) = 1. Note that over a commutative ring (that is not necessarily a domain) there are at least two relevant notions of the rank of a matrix R ∈ Dg×q . For this, one considers the so-called determinantal ideals Js (R), which are the ideals generated by all s × s subdeterminants of R. We have
HomC (M, C) HomD (M, HomC (D, C)),
D= : J0 (R) ⊇ J1 (R) ⊇ J2 (R) ⊇ · · · ⊇ Jmin(g,q)+1 (R) := 0.
where we have used that M⊗D D M, we may conclude that also the functor HomD (·, HomC (D, C)) is exact and faithful. This functor can in turn be identified with HomD (·, A). Thus, we obtain that A is an injective cogenerator as a D-module. This result was already proven by Lu et al. [8]. The argumentation is similar to the one used in [12, Section 3]. Now consider B := {w ∈ Aq | Rw = 0}, where R ∈ Dg×q . Due to the injective cogenerator property, there is a duality between B and the D-module M := D1×q /D1×g R as pointed out in Section 2. We now proceed to investigate the most basic systems theoretic properties of such a “behavior” B.
Now the classical rank of R is given by
HomC (D, C) A,
→ ((l ))l∈Zr ,
4. Two notions of autonomy The first interpretation of “autonomy” to be considered is the concept of “absence of free variables”. More formally, we say that B is weakly autonomous if no component of w ∈ B is free, that is, if none of the projection maps j : B → A, w → wj is surjective. Using the Malgrange isomorphism
rank(R) := max{s ∈ N|Js (R) = 0}. On the other hand, the reduced rank of R [11, Section 3.2] is defined by red-rank(R) := max{s ∈ N|ann(Js (R)) = 0}, where ann(J ) := {d ∈ D | de = 0∀e ∈ J } is the annihilator of an ideal J. Both the rank and the reduced rank of R are welldefined properties of B (that is, they are independent of the choice of the representation matrix R). This is due to the fact that any two representation matrices R1 , R2 of B must satisfy D1×g1 R1 =D1×g2 R2 and thus, they have the same determinantal ideals [11, Section 1.4]. We clearly have red-rank(R) rank(R) for any matrix R. The inequality can be strict, for example, the ¯ has reduced rank 0, since J1 (R) = 2 matrix R = 2I2 ∈ Z2×2 4
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¯ The importance of the reduced rank lies in has annihilator 2. the fact that [11, Section 3.2] ∃0 = M ∈ Dq : RM = 0 ⇔ red-rank(R) < q. This will be crucial for our characterization of the second, stronger autonomy notion. In the 1D behavioral setting, autonomy is often identified with the concept of all trajectories being uniquely “determined by their past”. Since the notions of past and future do not make much sense in Zr for r > 1, we modify the concept as follows: we will call a system B strongly autonomous if it contains no non-zero trajectory with finite support, where the support of r w ∈ (C Z )q is defined by supp(w) := {k ∈ Zr | w(k) = 0}. It should be noted that, despite its appearance, this definition coincides with the classical one in the case where C is a field and r = 1. However, known results that are valid for other system classes (e.g., multidimensional systems given as the smooth solution spaces of linear PDE with constant coefficients) suggest that autonomy should be related to the nonexistence of non-zero trajectories with bounded support [13]. The following theorem is closely related to [4, Proposition 22], where an equivalent characterization is obtained for the case where r = 1 and n is a prime power. Theorem 2. B is strongly autonomous if and only if any representation matrix R of B has reduced rank q. Proof. If the reduced rank of R is less than q, then there exists 0 = M ∈ Dq such that RM = 0. Then we have M ∈ B for all ∈ A. Due to the cogenerator property of A, there exists ∈ A such that M = 0, say (M)(k) = 0 for some k ∈ Zr . Since (M)(k) depends only on a finite number of values of , ˜ defined by there exists a finite set U ⊂ Zr such that , (l) if l ∈ U, ˜ := (l) 0 otherwise
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maximal subset of free variables of B. Let m q be the largest integer such that there exists, possibly after a permutation of the components of w, a surjection : B → Am ,
w → (w1 , . . . , wm )T .
The number m will be called the input-dimension of B, and p := q − m will be said to be the output-dimension. In view of the injective cogenerator property, the input-dimension is the greatest integer m such that there exists, possibly after a permutation of the columns of R, an injection D1×m → M, ej → [ej ] for 1 j m. Here, ej denotes both the jth natural basis vector of D1×m and of D1×q , using extension by zeros. Therefore, the input-dimension can be algebraically characterized as being the largest number m such that we have, after a suitable permutation of the columns of R, D1×g R ∩ (D1×m × {0}) = {0}.
(4)
In other words, B=A is the only system with input-dimension q; and if the input-dimension m of B is less than q, then it is the smallest integer such that for any choice of m + 1 components of w, the system law Rw = 0 implies a restriction involving only these variables. In particular, B is weakly autonomous if and only if its input-dimension is zero. q
Theorem 3. Let m and p be the input- and output-dimension of B, respectively. Suppose that (4) holds (that is, suppose that the columns of R and the components of w have been suitably re-arranged). Define u := (w1 , . . . , wm )T and y := (wm+1 , . . . , wq )T . Let R=[−Q, P ] be partitioned correspondingly. Then we have ∀u ∈ Am ∃y ∈ Ap : P y = Qu and the system {y ∈ Ap | P y = 0} is weakly autonomous.
˜ will still satisfy (M )(k) = 0. On the other hand, 0 = M ˜ ∈ B will be zero on the complement of some finite set. Conversely, if B contains a non-zero trajectory w with finite support, then by setting 0 = M := w(l)−l ∈ Dq , we g will have RM = 0 in D . Thus the reduced rank of R is less than q.
Proof. Let u ∈ Am be given. In view of the fundamental principle (2), there exists y ∈ Ap with P y = Qu if and only if we have Qu = 0 for any ∈ D1×g with P = 0. However, P = 0 implies that R = [−Q, 0] ∈ D1×g R ∩ (D1×m × {0}), which implies, in view of (4), that Q=0. Hence the solvability condition for P y = Qu is satisfied by any u ∈ Am . The second statement follows directly from the choice of m as the largest integer with (4).
Summing up, we have the following implication chain (where f.c.r. means “full column rank”)
6. Controllability
strong autonomy ⇔ reduced f.c.r. ⇒ f.c.r. ⇒ weak autonomy.
The converse of the two implications does not hold in general. An in-depth study of autonomy properties of multidimensional discrete systems over Zn , containing many examples and further results, can be found in [18]. The converse problem of finding a model for given trajectories is treated in [19]. 5. Inputs and outputs A system that is not weakly autonomous has free variables. To speak about inputs and outputs, we have to identify a
We say that B = {w ∈ Aq | Rw = 0} := ker A (R) has an image representation if B = {w ∈ Aq | ∃ ∈ Am : w = M}=: imA (M) for some M ∈ Dq×m . Due to the injective cogenerator property, this is equivalent to R being a left syzygy matrix, i.e., the rows of R generate the left kernel of some matrix M, that is, im(·R)= D1×g R = ker(·M) = {x ∈ D1×q | xM = 0}. The following lemma shows that this is true if and only if the module M is torsionless [7, Section 4H], that is, if there exists, for any 0 = m ∈ M, a homomorphism f ∈ HomD (M, D) with f (m) = 0.
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Lemma 1. The following are equivalent: 1. B has an image representation; 2. M is torsionless; 3. any representation matrix R of B is a left syzygy matrix. Proof. The equivalence of Conditions 1 and 3 is an immediate consequence of the injective cogenerator property (we have im(·R) = ker(·M) if and only if B = ker A (R) = imA (M)). “3 ⇒ 2”: If im(·R) = ker(·M), then M = D1×q /im(·R) = 1×q D / ker(·M) im(·M) ⊆ D1×m , using the homomorphism theorem. Thus M can be embedded into the free module D1×m , from which it is easy to see that M must be torsionless. “2 ⇒ 3”: Suppose that R is not a left syzygy matrix. Let M be a right syzygy matrix of R, that is, ker(R) = im(M), and let R c be a left syzygy matrix of M, that is, ker(·M) = im(·R c ). Since RM = 0, the rows of R can be written as linear combinations of the rows of R c , that is, im(·R) ⊆ im(·R c ). If equality would hold, then R would be a left syzygy matrix, namely, of M. Thus there exists x0 ∈ im(·R c )\im(·R). Then any D-homomorphism M → D will map 0 = m0 := [x0 ] ∈ M to zero, since the homomorphisms from M to D are necessarily of the form M → D,
[x] → xy
for some y ∈ D which must satisfy xy = 0 for all x ∈ im(·R) for well-definedness. Therefore, we must have Ry = 0, or equivalently, y = Mz for some z ∈ Dm (since M is a right syzygy matrix of R). For any such y, we have x0 y = 0, because x0 ∈ im(·R c ) and R c M = 0. This shows that M is not torsionless. q
Corollary 1. The proof of the lemma also yields a test for M = D1×q /D1×g R being torsionless in terms of an arbitrary representation matrix R: let M be a right syzygy matrix of R, and let R c be a left syzygy matrix of M. Then M is torsionless if and only if R and R c have the same row module. Proof. By construction, we have ker(R) = im(M) and im(·R c ) = ker(·M). Then we have im(·R) ⊆ im(R c ). Now if im(·R) = im(·R c ), then R is a left syzygy matrix (namely, of M). Thus M = D1×q /im(·R) is torsionless. Conversely, if im(·R)im(·R c ), then (as we have seen in the proof above) M = D1×q /D1×g R cannot be torsionless. Reformulated in the language of homological algebra, the module M is torsionless if and only if Ext1D (N, D) = 0, where N := Dg /RDq . In the context of algebraic analysis, this important fact was first noted and extensively used by Pommaret and Quadrat, see e.g. [14]. If M is torsionless, then we have in particular: if dm = 0 for some non-zero-divisor d ∈ D and m ∈ M, then m must be zero. The latter property is called torsion-freeness. Note that when D is a domain, then “torsionless” and “torsion-free” are equivalent for finitely generated modules (considering Q as a Z-module, which is torsion-free but not torsionless, shows that the assumption of finite-generatedness is really needed). However, in the presence of zero-divisors in D, this does not
hold in general, and “torsionless” may be a strictly stronger property than “torsion-free”. It turns out, however, that for the rings under consideration here, the two properties are indeed equivalent. From the behavioral point of view, the existence of image representations is interesting because it is, for many relevant system classes, equivalent to the possibility of concatenating any two trajectories of a behavior by another system trajectory. This is the behavioral controllability paradigm. Here is our version: We call B controllable if for any w1 , w2 ∈ B, and any U1 , U2 ⊂ Zr that are sufficiently far apart there exists w ∈ B such that w coincides with wi on Ui . It is quite easy to see that if B has an image representation, then it is controllable. The converse is also true according to the next theorem. We need some preparation. Lemma 2. An element d ∈ D is a zero-divisor if and only if there exists 0 = c ∈ C = Zn such that cd = 0. The total ring of fractions of D, that is, n Q := n, d ∈ D, d non-zero-divisor d is a quasi-Frobenius ring, in particular, HomQ (·, Q) preserves exactness. Proof. If n is a prime number, i.e., C is a field, the statement is trivial: then d ∈ D is a zero-divisor if and only if d = 0, and Q is the field of rational functions of the i , with coefficients in C. Suppose that n is not prime. It is a standard fact in commutative algebra that a polynomial d = dt t ∈ C[1 , . . . , r ], where C is a commutative ring, is a zero-divisor if and only if there exists 0 = c ∈ C such that cd = 0, that is, cd t = 0 for all t [1]. It is easy to generalize this −1 result to C[1 , . . . , r , −1 1 , . . . , r ]. Given d ∈ D, let cont(d) be the greatest common divisor of all coefficients appearing in d, called the content of d. Then ˜ d = cont(d)d, where the coefficients of d˜ are coprime, and thus, d˜ is a nonzero-divisor. Thus every element q = n/d ∈ Q can be written as q = cont(n)u, where u ∈ Q is a unit. To show that Q (which is Noetherian since D is) is a quasi-Frobenius ring, it suffices to show [7, Section 15] that ann(ann(J )) = J for every ideal J ⊆ Q. Let J = n1 /d1 , . . . , nk /dk be given. Every generator ni /di can be replaced by cont(ni ), and thus J = Qc, where c ∈ C is the greatest common divisor of all cont(ni ). Then ann(J ) = Q ann(c), where ann(c) is the annihilator of c in C. Finally, ann(ann(J )) = Q ann(ann(c)), and since C is a quasi-Frobenius ring, we have ann(ann(c)) = c. Theorem 4. The following are equivalent: 1. B has an image representation; 2. M is torsionless; 3. any representation matrix R of B is a left syzygy matrix;
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4. B is controllable; 5. M is torsion-free.
Therefore the following statement is an immediate consequence of A being an injective cogenerator.
Proof. Since Conditions 1–3 are equivalent according to Lemma 1, and since the existence of an image representation implies controllability, it suffices to show that “4 ⇒ 5 ⇒ 2”. The crucial part of the proof is “5 ⇒ 2”: Suppose that M is torsion-free, that is, if dx ∈ im(·R) for some non-zero-divisor d, then x ∈ im(·R). Using Corollary 1, we let M be a right syzygy matrix of R, and R c a left syzygy matrix of M. Then im(·R) ⊆ im(·R c ) and for M to be torsionless, we need to show that this is actually an equality. We first note that
Theorem 5. The following are equivalent:
ker(R) = im(M) ⇒ ker Q (R) = imQ (M) and im(·R c ) = ker(·M) ⇒ imQ (·R c ) = ker Q (·M). Since Q is a quasi-Frobenius ring, we have ker Q (R) = imQ (M) ⇒ imQ (·R) = ker Q (·M). Thus we can conclude that imQ (·R) = imQ (·R c ). This means that there exists a Q-matrix Z with R c = ZR. Since Z = N/d for some D-matrix N and some non-zero-divisor d, we get dR c = N R. Due to torsion-freeness, we may conclude that R c = N1 R for some D-matrix N1 . Finally, we prove “4 ⇒ 5”: Suppose that M is not torsionfree, that is, there exists a non-zero-divisor d ∈ D and 0 = m ∈ M such that dm=0. This means that there exists x ∈ D1×q with dx ∈ D1×g R, but x ∈ / D1×g R. Then there exists, according to (3), a trajectory w ∈ B with xw = 0, say, (xw)(k) = 0 for some k ∈ Zr . Suppose that B is controllable. Let U1 ⊂ Zr be a finite set such that w = w¯ on U1 guarantees that (xw)(k) = (x w)(k). ¯ Let > 0 be large enough such that with U2 = {l ∈ Zr | dist(l, U1 ) > }, the trajectories w =: w1 and 0 =: w2 are concatenable.Here, dist(l, U1 ) := min{|l − k1 | | k1 ∈ U1 }, where |l| := ri=1 |li | for l ∈ Zr . Let w¯ ∈ B be a connecting trajectory, that is, w¯ coincides with w on U1 , and with 0 on U2 . Then 0 = x w¯ has finite support.On the other hand, we have dx w¯ =0, since w¯ ∈ B and dx ∈ D1×g R. Thus x w¯ is contained in {v ∈ A | dv = 0} which is strongly autonomous, since d is a non-zero-divisor, that is, red-rank(d) = 1. Thus we have a contradiction, in view of Theorem 2. 7. Observability Unlike autonomy and controllability, observability is not an intrinsic system property. It depends on a partition of the system variable w into two subvectors w1 , w2 . Then one says that w1 is observable from w2 if w1 w1 , ∈ B ⇒ w1 = w1 . w2 w2 Let R = [R1 , R2 ] be partitioned accordingly. Thus, w1 is observable from w2 if and only if {w1 ∈ Aq1 |R1 w1 = 0} = {0}.
1. w1 is observable from w2 ; 2. R1 is left invertible. 8. Conclusion We have extended the basic systems theoretic concepts from multidimensional discrete systems over a field to systems over the rings C = Z/nZ. The crucial tool is the fact (originally r established in [8]) that A=C Z is an injective cogenerator over −1 D = C[1 , . . . , r , −1 1 , . . . , r ]. To the author’s knowledge, the results are partially new even for the case r = 1. Several recent papers [3–6,8,10] suggest that there may be applications in (the academic foundations of) coding theory. From the constructive point of view, there are still several open problems, mostly concerning Gröbner basis theory with coefficients in Zn . Since several research teams are currently working on these problems, some progress in this direction can be expected soon. Acknowledgment The author would like to thank M. Kuijper for giving two inspiring talks [3,5], and J. Rosenthal for asking a good question. References [1] M.F. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, MA, 1969. [2] N. Bourbaki, Algebra I, Springer, Berlin, 1989. [3] M. Kuijper, R. Pinto, J.W. Polderman, Kernel representations for behaviors over finite rings, in: Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS), Kyoto, 2006. [4] M. Kuijper, R. Pinto, J.W. Polderman, P. Rocha, Autonomicity and the absence of free variables for behaviors over finite rings, in: Proceedings of the 7th Portuguese Conference on Automatic Control (Controlo), Lisbon, 2006. [5] M. Kuijper, X.-W. Wu, U. Parampalli, Behavioral models over rings—minimal representations and applications to coding and sequences, in: Proceedings of the 16th IFAC World Congress, Prague, 2005. [6] V.L. Kurakin, A.S. Kuzmin, V.T. Markov, A.V. Mikhalev, A.A. Nechaev, Linear codes and polylinear recurrences over finite rings and modules, in: M. Fossorier et al. (Ed.), Applied Algebra, Algebraic Algorithms and Error Correcting Codes, Lecture Notes in Computer Science, vol. 1719, Springer, Berlin, 1999. [7] T.Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, vol. 189, Springer, Berlin, 1999. [8] P. Lu, M. Liu, U. Oberst, Linear recurring arrays, linear systems and multidimensional cyclic codes over quasi-Frobenius rings, Acta Appl. Math. 80 (2004) 175–198. [9] B. Malgrange, Systèmes différentiels à coefficients constants, Semin. Bourbaki 15 (1962/63), 246 (1964) 11. [10] A.V. Mikhalev, A.A. Nechaev, Linear recurring sequences over modules, Acta Appl. Math. 42 (1996) 161–202.
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[11] D.G. Northcott, Finite Free Resolutions, Cambridge Tracts in Mathematics, vol. 71, Cambridge University Press, Cambridge, 1976. [12] U. Oberst, Multidimensional constant linear systems, Acta Appl. Math. 20 (1990) 1–175. [13] H.K. Pillai, S. Shankar, A behavioral approach to control of distributed systems, SIAM J. Control Optim. 37 (1999) 388–408. [14] J.-F. Pommaret, A. Quadrat, Algebraic analysis of linear multidimensional control systems, IMA J. Math. Control Inform. 16 (1999) 275–297. [15] P. Rocha, Structure and Representation of 2D Systems, Ph.D. Thesis, University of Groningen, 1990.
[16] J. Wood, Modules and behaviours in nD systems theory, Multidimensional Systems Signal Process. 11 (2000) 11–48. [17] E. Zerz, Topics in Multidimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences, vol. 256, Springer, Berlin, 2000. [18] E. Zerz, Autonomy properties of multidimensional linear systems over rings, in: Proceedings of the 5th International Workshop on Multidimensional Systems (NDS), Aveiro, 2007. [19] E. Zerz, The discrete multidimensional MPUM, Multidimensional Systems Signal Process. (2007), submitted for publication.
Discrete multidimensional systems over Zn Eva Zerz Lehrstuhl D für Mathematik, RWTH Aachen University, 52062 Aachen, Germany Received 10 August 2006; received in revised form 21 May 2007; accepted 1 June 2007 Available online 10 July 2007
Abstract We generalize the theory of multidimensional, discrete, linear, shift-invariant systems from the well-investigated case in which the signals take their values in a field, to the situation where these values belong to a finite ring of the form Z/nZ. Some basic systems theoretic properties are studied and characterized: autonomy, input–output decompositions, controllability, and observability. © 2007 Elsevier B.V. All rights reserved. Keywords: Multidimensional discrete linear shift-invariant systems; Systems over rings; Behavioral approach; Algebraic methods; Structural properties; Controllability
1. Introduction The study of multidimensional, discrete, linear, shiftinvariant systems in the behavioral framework originated in 1990 with Rocha’s thesis [15]. In the same year, a seminal paper by Oberst [12] introduced a rigorous algebraic approach to these (and other) systems, using the comprehensive language of category theory. The theory has subsequently been refined by various authors, for instance, see [16,17] and the references therein. A standing assumption in all these contributions is that the signals, which are multivariate sequences, take their values in a field. The present paper relaxes this restriction and studies signals with values in a finite ring of the form Zn := Z/nZ for some integer n > 1. Systems over finite rings have potential applications in coding theory, both in the 1D setting [3–5] and in the multidimensional case [6,8,10]. 2. Abstract linear systems Let D be a commutative ring (with 1), and let A be a D-module. For a given matrix R ∈ Dg×q , we consider the abstract linear system B := {w ∈ A | Rw = 0}. q
E-mail address: [email protected]. 0167-6911/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2007.06.002
The letter B has been chosen in allusion to the behavioral approach. One should think of A as a set of “signals”, and of D as a ring of “operators” acting on them. To say that A carries a D-module structure amounts more or less to the requirement that one can apply any operator d ∈ D to any signal a ∈ A to obtain a new signal da ∈ A. In the same way, the expression Rw from above becomes a well-defined element of Ag . The matrix R is called a representation matrix of B. For instance, let D = R[d/dt], which is the ring of linear differential operators with real coefficients, its elements being of the form d = rn
dn d + · · · + r1 + r 0 , dt n dt
where n ∈ N and ri ∈ R. A signal set with a D-module structure is given by A = C∞ (R, R), the set of smooth functions from R to R. Any operator d ∈ D can be applied to any signal a ∈ A, producing a function da which is again smooth, that is, da ∈ A. A set B as above is then the solution set of a system of linear ordinary differential equations with real coefficients. Another standard example is D = R[] and A = RN . Then a signal is a sequence a : N → R and d = rn n + · · · + r1 + r0 , where n ∈ N and ri ∈ R, acts on a ∈ A via (da)(t) := rn a(t+n) + · · · + r1 a(t+1)+r0 a(t) for all t∈N.
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Clearly, da belongs again to A. A set B as above is then the solution set of a system of linear difference equations with real coefficients. Coming back to the abstract setting, an important observation by Malgrange [9] says that B is a D-module that is isomorphic to the module consisting of all D-homomorphisms from M := D1×q /D1×g R to A, that is, M → A , (1) B HomD (M, A), w → [x] → xw, where [x] denotes the residue class of x ∈ D1×q in M. The result itself is not hard to prove, but its importance lies in the fact that it draws attention to the algebraic object M, called the system module, and to the contravariant functor HomD (·, A) on the category of D-modules. One says that the D-module A is an injective cogenerator [7] if the functor F := HomD (·, A) is exact and faithful. This means that it preserves and reflects exactness, that is, a sequence of D-modules and D-homomorphisms f
g
M −→ N −→ P
B⊥ := {x ∈ D1×q | xw = 0 ∀w ∈ B} = D1×g R.
(3)
Given B = {w ∈ Aq | Rw = 0}, we can write B⊥ , which is a submodule of D1×q and thus finitely generated (since D is Noetherian), as B⊥ =D1×g1 R1 . Since every row of R belongs to B⊥ , we get D1×g R ⊆ D1×g1 R1 . Thus B ⊇ B1 := {w ∈ Aq | R1 w = 0}. On the other hand, any w ∈ B satisfies xw = 0 for all x ∈ B⊥ and thus R1 w = 0. We conclude that B = B1 . Thus for 0 → M → M1 (the map [x] → [x] is well-defined since D1×g R ⊆ D1×g1 R1 ), we have that 0 ← HomD (M, A) ← HomD (M1 , A) is exact, as it corresponds (via the Malgrange isomorphism) to id
Fg
F M ←− F N ←− F P is exact (that is, im(F g) = ker(Ff )), where (Ff )() = ◦ f for ∈ F N = HomD (N, A), and Fg is defined analogously. A well-known example is the case where D = F and A = F, where F is an arbitrary field (and thus also a commutative ring with 1, which can be considered as a module, or in this case, even vector space, over itself). Then HomF (·, F) is the functor that assigns to each F-vector space V its dual vector space V ∗ = HomF (V , F), and to every F-linear mapping f : V → W between two F-vector spaces its dual mapping f ∗ : W ∗ → V ∗ , → ◦ f . It is a standard fact of linear algebra that HomF (·, F) is an exact and faithful functor: formulated in the language of matrices, this simply amounts to im(A) = ker(B) ⇔ ker(·A) = im(·B) for A ∈ Fn×m , B ∈ Fp×n . Here, A is the map A : Fm → Fn , x → Ax and ·A denotes the map ·A : F1×n → F1×m , y → yA, and analogously for B. The injective cogenerator property is a very powerful tool for systems theory, because it enables us to translate any statement on abstract linear systems that can be formulated in terms of kernels and images, into an equivalent statement on D-modules. The most prominent consequences of the injective cogenerator property are: • If P ∈ Dg×p and Q ∈ Dh×g are such that im(·Q) := D1×h Q = ker(·P ) := {x ∈ D1×g | xP = 0}, then we have for all v ∈ Ag ∃y ∈ Ap : P y = v ⇔ Qv = 0.
• Assume that D is Noetherian. Then for an arbitrary representation matrix R of B, we have
0 ← B ←− B1 .
is exact (that is, im(f ) = ker(g)) if and only if the associated sequence Ff
703
(2)
This is called the “fundamental principle”, and it is nothing but the preservation of exactness, formulated in the language of matrices.
Since exactness is reflected, also 0 → M → M1 must be exact, and thus B⊥ = D1×g1 R1 = D1×g R. Thus, we have B1 = {w ∈ Aq | R1 w = 0} ⊆ B2 = {w ∈ Aq | R2 w = 0} if and only if there exists X ∈ Dg2 ×g1 such that R2 = XR 1 . Here, the “if” part is obvious, and the “only if” part follows from ⊥ 1×g1 B1 ⊆ B2 ⇒ B⊥ R1 ⊇ D1×g2 R2 . 1 ⊇ B2 ⇒ D
Note that this yields also a full characterization of the equivalence of representations: we have B1 = B2 if and only if there exist X, Y such that R2 = XR 1 and R1 = Y R 2 . In other words, two matrices represent the same behavior if and only if they have the same row module. In particular, there exists no 0 = d ∈ D with dA={0} (since dA={0} means {w ∈ A | dw=0}=A={w ∈ A | 0w=0}, and hence, we must have d = 0). Moreover, B = {0} holds if and only if any representation matrix R of B is left invertible. (Here, the “if” part is obvious, and the “only if” part follows from {w ∈ Aq | Rw = 0} = {0} = {w ∈ Aq | Iq w = 0}, which implies that Iq = XR for some X.) In many cases, the relevant rings D are variants of polynomial rings, which can be efficiently manipulated using modern computer algebra systems. Using the correspondence outlined above, one can then re-interpret the results of these computations using the language of systems theory. The rest of the paper is devoted to applying these general −1 ideas to the special case where D=C[1 , . . . , r , −1 1 , . . . , r ] r and A = C Z . Here, the signals are sequences, r-fold indexed by integers, taking their values in C. The ring D consists of all linear shift (or difference) operators with coefficients in C. A lot is known about these systems in the case where C is a field, for instance, see [12,15–17] and the references therein. However, in the present paper, we will study the case where C
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is a finite ring of the form Z/nZ for some integer n > 1. The main technical difficulty is that then D is no longer a domain (unless n is a prime number, but in this case, C is already a field, and we are back in the known situation). For instance, D has no quotient field, and thus we have no concept of transfer matrices, etc. 3. Multidimensional discrete systems over Z/nZ Let n > 1 be an integer. The ring C := Z/nZ is a quasiFrobenius ring [7, Section 15], i.e., it is Noetherian and selfinjective [7, Section 3B]. Moreover, it is a cogenerator ring [7, Section 19B]. This means that HomC (·, C) is an exact and faithful functor. −1 The Noetherian ring D := C[1 , . . . , r , −1 1 , . . . , r ] consists of all multivariate Laurent polynomials in the indetermir nates i , with coefficients in C. The set A := C Z consists of all functions Zr → C, that is, of all sequences, r-fold indexed by integers, taking their values in C. There is a natural D-module structure on A, which is given by (l11 · · · lrr a)(k1 , . . . , kr ) := a(k1 + l1 , . . . , kr + lr ) for all a ∈ A and ki , li ∈ Z. This is, the action of i on the sequence a=(a(k))k∈Zr is given by a “shift in the ith direction”. We will use the notation l := l11 · · · lrr for l ∈ Zr . There is an isomorphism of C-modules
B HomD (M, A) from (1), and the standard isomorphism A HomD (D, A) (that relates a D-linear map D → A to the image of the D-basis 1 of D), we find that j can be identified with HomD (j , A), where j : D → M,
1 → [ej ],
and ej denotes the jth natural basis vector of D1×q . Theorem 1. The following are equivalent: 1. B is weakly autonomous; 2. for all [ej ] ∈ M there exists 0 = dj ∈ D such that dj [ej ] = 0; 3. for any representation matrix R of B there exist d1 , . . . , dq ∈ D\{0} and X ∈ Dq×g such that diag(d1 , . . . , dq ) = XR. Proof. Weak autonomy is equivalent to the non-existence of a surjection j : B → A with j (w) = wj . In view of the injective cogenerator property, this is in turn equivalent to the nonexistence of an injection j : D → M with j (1)=[ej ]. Thus, weak autonomy amounts to the requirement that ker(j ) = 0 for all j, that is, ∀j ∃0 = dj : j (dj ) = dj [ej ] = 0. Finally, the third condition is merely the translation of Condition 2 into the language of matrices.
which relates a C-linear map : D → C to the image of the C-basis {l | l ∈ Zr } of D. The D-module structure of A can be transferred to the left-hand side by putting (d)(e) := (de) for d, e ∈ D. As pointed out above, HomC (·, C) is exact and faithful. Let M be a D-module. Because of the standard isomorphism [2, Chapter II, Section 4.1]
It is worth noting that the condition rank(R) = q, which is equivalent to weak autonomy in the case where C is a field, is only sufficient for weak autonomy in the present situation, ¯ 2 ∈ Z2×2 defines a and not necessary. For instance, R = 2I 4 ¯ and thus weakly autonomous system, but det(R) = 2¯ · 2¯ = 0, rank(R) = 1. Note that over a commutative ring (that is not necessarily a domain) there are at least two relevant notions of the rank of a matrix R ∈ Dg×q . For this, one considers the so-called determinantal ideals Js (R), which are the ideals generated by all s × s subdeterminants of R. We have
HomC (M, C) HomD (M, HomC (D, C)),
D= : J0 (R) ⊇ J1 (R) ⊇ J2 (R) ⊇ · · · ⊇ Jmin(g,q)+1 (R) := 0.
where we have used that M⊗D D M, we may conclude that also the functor HomD (·, HomC (D, C)) is exact and faithful. This functor can in turn be identified with HomD (·, A). Thus, we obtain that A is an injective cogenerator as a D-module. This result was already proven by Lu et al. [8]. The argumentation is similar to the one used in [12, Section 3]. Now consider B := {w ∈ Aq | Rw = 0}, where R ∈ Dg×q . Due to the injective cogenerator property, there is a duality between B and the D-module M := D1×q /D1×g R as pointed out in Section 2. We now proceed to investigate the most basic systems theoretic properties of such a “behavior” B.
Now the classical rank of R is given by
HomC (D, C) A,
→ ((l ))l∈Zr ,
4. Two notions of autonomy The first interpretation of “autonomy” to be considered is the concept of “absence of free variables”. More formally, we say that B is weakly autonomous if no component of w ∈ B is free, that is, if none of the projection maps j : B → A, w → wj is surjective. Using the Malgrange isomorphism
rank(R) := max{s ∈ N|Js (R) = 0}. On the other hand, the reduced rank of R [11, Section 3.2] is defined by red-rank(R) := max{s ∈ N|ann(Js (R)) = 0}, where ann(J ) := {d ∈ D | de = 0∀e ∈ J } is the annihilator of an ideal J. Both the rank and the reduced rank of R are welldefined properties of B (that is, they are independent of the choice of the representation matrix R). This is due to the fact that any two representation matrices R1 , R2 of B must satisfy D1×g1 R1 =D1×g2 R2 and thus, they have the same determinantal ideals [11, Section 1.4]. We clearly have red-rank(R) rank(R) for any matrix R. The inequality can be strict, for example, the ¯ has reduced rank 0, since J1 (R) = 2 matrix R = 2I2 ∈ Z2×2 4
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¯ The importance of the reduced rank lies in has annihilator 2. the fact that [11, Section 3.2] ∃0 = M ∈ Dq : RM = 0 ⇔ red-rank(R) < q. This will be crucial for our characterization of the second, stronger autonomy notion. In the 1D behavioral setting, autonomy is often identified with the concept of all trajectories being uniquely “determined by their past”. Since the notions of past and future do not make much sense in Zr for r > 1, we modify the concept as follows: we will call a system B strongly autonomous if it contains no non-zero trajectory with finite support, where the support of r w ∈ (C Z )q is defined by supp(w) := {k ∈ Zr | w(k) = 0}. It should be noted that, despite its appearance, this definition coincides with the classical one in the case where C is a field and r = 1. However, known results that are valid for other system classes (e.g., multidimensional systems given as the smooth solution spaces of linear PDE with constant coefficients) suggest that autonomy should be related to the nonexistence of non-zero trajectories with bounded support [13]. The following theorem is closely related to [4, Proposition 22], where an equivalent characterization is obtained for the case where r = 1 and n is a prime power. Theorem 2. B is strongly autonomous if and only if any representation matrix R of B has reduced rank q. Proof. If the reduced rank of R is less than q, then there exists 0 = M ∈ Dq such that RM = 0. Then we have M ∈ B for all ∈ A. Due to the cogenerator property of A, there exists ∈ A such that M = 0, say (M)(k) = 0 for some k ∈ Zr . Since (M)(k) depends only on a finite number of values of , ˜ defined by there exists a finite set U ⊂ Zr such that , (l) if l ∈ U, ˜ := (l) 0 otherwise
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maximal subset of free variables of B. Let m q be the largest integer such that there exists, possibly after a permutation of the components of w, a surjection : B → Am ,
w → (w1 , . . . , wm )T .
The number m will be called the input-dimension of B, and p := q − m will be said to be the output-dimension. In view of the injective cogenerator property, the input-dimension is the greatest integer m such that there exists, possibly after a permutation of the columns of R, an injection D1×m → M, ej → [ej ] for 1 j m. Here, ej denotes both the jth natural basis vector of D1×m and of D1×q , using extension by zeros. Therefore, the input-dimension can be algebraically characterized as being the largest number m such that we have, after a suitable permutation of the columns of R, D1×g R ∩ (D1×m × {0}) = {0}.
(4)
In other words, B=A is the only system with input-dimension q; and if the input-dimension m of B is less than q, then it is the smallest integer such that for any choice of m + 1 components of w, the system law Rw = 0 implies a restriction involving only these variables. In particular, B is weakly autonomous if and only if its input-dimension is zero. q
Theorem 3. Let m and p be the input- and output-dimension of B, respectively. Suppose that (4) holds (that is, suppose that the columns of R and the components of w have been suitably re-arranged). Define u := (w1 , . . . , wm )T and y := (wm+1 , . . . , wq )T . Let R=[−Q, P ] be partitioned correspondingly. Then we have ∀u ∈ Am ∃y ∈ Ap : P y = Qu and the system {y ∈ Ap | P y = 0} is weakly autonomous.
˜ will still satisfy (M )(k) = 0. On the other hand, 0 = M ˜ ∈ B will be zero on the complement of some finite set. Conversely, if B contains a non-zero trajectory w with finite support, then by setting 0 = M := w(l)−l ∈ Dq , we g will have RM = 0 in D . Thus the reduced rank of R is less than q.
Proof. Let u ∈ Am be given. In view of the fundamental principle (2), there exists y ∈ Ap with P y = Qu if and only if we have Qu = 0 for any ∈ D1×g with P = 0. However, P = 0 implies that R = [−Q, 0] ∈ D1×g R ∩ (D1×m × {0}), which implies, in view of (4), that Q=0. Hence the solvability condition for P y = Qu is satisfied by any u ∈ Am . The second statement follows directly from the choice of m as the largest integer with (4).
Summing up, we have the following implication chain (where f.c.r. means “full column rank”)
6. Controllability
strong autonomy ⇔ reduced f.c.r. ⇒ f.c.r. ⇒ weak autonomy.
The converse of the two implications does not hold in general. An in-depth study of autonomy properties of multidimensional discrete systems over Zn , containing many examples and further results, can be found in [18]. The converse problem of finding a model for given trajectories is treated in [19]. 5. Inputs and outputs A system that is not weakly autonomous has free variables. To speak about inputs and outputs, we have to identify a
We say that B = {w ∈ Aq | Rw = 0} := ker A (R) has an image representation if B = {w ∈ Aq | ∃ ∈ Am : w = M}=: imA (M) for some M ∈ Dq×m . Due to the injective cogenerator property, this is equivalent to R being a left syzygy matrix, i.e., the rows of R generate the left kernel of some matrix M, that is, im(·R)= D1×g R = ker(·M) = {x ∈ D1×q | xM = 0}. The following lemma shows that this is true if and only if the module M is torsionless [7, Section 4H], that is, if there exists, for any 0 = m ∈ M, a homomorphism f ∈ HomD (M, D) with f (m) = 0.
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Lemma 1. The following are equivalent: 1. B has an image representation; 2. M is torsionless; 3. any representation matrix R of B is a left syzygy matrix. Proof. The equivalence of Conditions 1 and 3 is an immediate consequence of the injective cogenerator property (we have im(·R) = ker(·M) if and only if B = ker A (R) = imA (M)). “3 ⇒ 2”: If im(·R) = ker(·M), then M = D1×q /im(·R) = 1×q D / ker(·M) im(·M) ⊆ D1×m , using the homomorphism theorem. Thus M can be embedded into the free module D1×m , from which it is easy to see that M must be torsionless. “2 ⇒ 3”: Suppose that R is not a left syzygy matrix. Let M be a right syzygy matrix of R, that is, ker(R) = im(M), and let R c be a left syzygy matrix of M, that is, ker(·M) = im(·R c ). Since RM = 0, the rows of R can be written as linear combinations of the rows of R c , that is, im(·R) ⊆ im(·R c ). If equality would hold, then R would be a left syzygy matrix, namely, of M. Thus there exists x0 ∈ im(·R c )\im(·R). Then any D-homomorphism M → D will map 0 = m0 := [x0 ] ∈ M to zero, since the homomorphisms from M to D are necessarily of the form M → D,
[x] → xy
for some y ∈ D which must satisfy xy = 0 for all x ∈ im(·R) for well-definedness. Therefore, we must have Ry = 0, or equivalently, y = Mz for some z ∈ Dm (since M is a right syzygy matrix of R). For any such y, we have x0 y = 0, because x0 ∈ im(·R c ) and R c M = 0. This shows that M is not torsionless. q
Corollary 1. The proof of the lemma also yields a test for M = D1×q /D1×g R being torsionless in terms of an arbitrary representation matrix R: let M be a right syzygy matrix of R, and let R c be a left syzygy matrix of M. Then M is torsionless if and only if R and R c have the same row module. Proof. By construction, we have ker(R) = im(M) and im(·R c ) = ker(·M). Then we have im(·R) ⊆ im(R c ). Now if im(·R) = im(·R c ), then R is a left syzygy matrix (namely, of M). Thus M = D1×q /im(·R) is torsionless. Conversely, if im(·R)im(·R c ), then (as we have seen in the proof above) M = D1×q /D1×g R cannot be torsionless. Reformulated in the language of homological algebra, the module M is torsionless if and only if Ext1D (N, D) = 0, where N := Dg /RDq . In the context of algebraic analysis, this important fact was first noted and extensively used by Pommaret and Quadrat, see e.g. [14]. If M is torsionless, then we have in particular: if dm = 0 for some non-zero-divisor d ∈ D and m ∈ M, then m must be zero. The latter property is called torsion-freeness. Note that when D is a domain, then “torsionless” and “torsion-free” are equivalent for finitely generated modules (considering Q as a Z-module, which is torsion-free but not torsionless, shows that the assumption of finite-generatedness is really needed). However, in the presence of zero-divisors in D, this does not
hold in general, and “torsionless” may be a strictly stronger property than “torsion-free”. It turns out, however, that for the rings under consideration here, the two properties are indeed equivalent. From the behavioral point of view, the existence of image representations is interesting because it is, for many relevant system classes, equivalent to the possibility of concatenating any two trajectories of a behavior by another system trajectory. This is the behavioral controllability paradigm. Here is our version: We call B controllable if for any w1 , w2 ∈ B, and any U1 , U2 ⊂ Zr that are sufficiently far apart there exists w ∈ B such that w coincides with wi on Ui . It is quite easy to see that if B has an image representation, then it is controllable. The converse is also true according to the next theorem. We need some preparation. Lemma 2. An element d ∈ D is a zero-divisor if and only if there exists 0 = c ∈ C = Zn such that cd = 0. The total ring of fractions of D, that is, n Q := n, d ∈ D, d non-zero-divisor d is a quasi-Frobenius ring, in particular, HomQ (·, Q) preserves exactness. Proof. If n is a prime number, i.e., C is a field, the statement is trivial: then d ∈ D is a zero-divisor if and only if d = 0, and Q is the field of rational functions of the i , with coefficients in C. Suppose that n is not prime. It is a standard fact in commutative algebra that a polynomial d = dt t ∈ C[1 , . . . , r ], where C is a commutative ring, is a zero-divisor if and only if there exists 0 = c ∈ C such that cd = 0, that is, cd t = 0 for all t [1]. It is easy to generalize this −1 result to C[1 , . . . , r , −1 1 , . . . , r ]. Given d ∈ D, let cont(d) be the greatest common divisor of all coefficients appearing in d, called the content of d. Then ˜ d = cont(d)d, where the coefficients of d˜ are coprime, and thus, d˜ is a nonzero-divisor. Thus every element q = n/d ∈ Q can be written as q = cont(n)u, where u ∈ Q is a unit. To show that Q (which is Noetherian since D is) is a quasi-Frobenius ring, it suffices to show [7, Section 15] that ann(ann(J )) = J for every ideal J ⊆ Q. Let J = n1 /d1 , . . . , nk /dk be given. Every generator ni /di can be replaced by cont(ni ), and thus J = Qc, where c ∈ C is the greatest common divisor of all cont(ni ). Then ann(J ) = Q ann(c), where ann(c) is the annihilator of c in C. Finally, ann(ann(J )) = Q ann(ann(c)), and since C is a quasi-Frobenius ring, we have ann(ann(c)) = c. Theorem 4. The following are equivalent: 1. B has an image representation; 2. M is torsionless; 3. any representation matrix R of B is a left syzygy matrix;
E. Zerz / Systems & Control Letters 56 (2007) 702 – 708
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4. B is controllable; 5. M is torsion-free.
Therefore the following statement is an immediate consequence of A being an injective cogenerator.
Proof. Since Conditions 1–3 are equivalent according to Lemma 1, and since the existence of an image representation implies controllability, it suffices to show that “4 ⇒ 5 ⇒ 2”. The crucial part of the proof is “5 ⇒ 2”: Suppose that M is torsion-free, that is, if dx ∈ im(·R) for some non-zero-divisor d, then x ∈ im(·R). Using Corollary 1, we let M be a right syzygy matrix of R, and R c a left syzygy matrix of M. Then im(·R) ⊆ im(·R c ) and for M to be torsionless, we need to show that this is actually an equality. We first note that
Theorem 5. The following are equivalent:
ker(R) = im(M) ⇒ ker Q (R) = imQ (M) and im(·R c ) = ker(·M) ⇒ imQ (·R c ) = ker Q (·M). Since Q is a quasi-Frobenius ring, we have ker Q (R) = imQ (M) ⇒ imQ (·R) = ker Q (·M). Thus we can conclude that imQ (·R) = imQ (·R c ). This means that there exists a Q-matrix Z with R c = ZR. Since Z = N/d for some D-matrix N and some non-zero-divisor d, we get dR c = N R. Due to torsion-freeness, we may conclude that R c = N1 R for some D-matrix N1 . Finally, we prove “4 ⇒ 5”: Suppose that M is not torsionfree, that is, there exists a non-zero-divisor d ∈ D and 0 = m ∈ M such that dm=0. This means that there exists x ∈ D1×q with dx ∈ D1×g R, but x ∈ / D1×g R. Then there exists, according to (3), a trajectory w ∈ B with xw = 0, say, (xw)(k) = 0 for some k ∈ Zr . Suppose that B is controllable. Let U1 ⊂ Zr be a finite set such that w = w¯ on U1 guarantees that (xw)(k) = (x w)(k). ¯ Let > 0 be large enough such that with U2 = {l ∈ Zr | dist(l, U1 ) > }, the trajectories w =: w1 and 0 =: w2 are concatenable.Here, dist(l, U1 ) := min{|l − k1 | | k1 ∈ U1 }, where |l| := ri=1 |li | for l ∈ Zr . Let w¯ ∈ B be a connecting trajectory, that is, w¯ coincides with w on U1 , and with 0 on U2 . Then 0 = x w¯ has finite support.On the other hand, we have dx w¯ =0, since w¯ ∈ B and dx ∈ D1×g R. Thus x w¯ is contained in {v ∈ A | dv = 0} which is strongly autonomous, since d is a non-zero-divisor, that is, red-rank(d) = 1. Thus we have a contradiction, in view of Theorem 2. 7. Observability Unlike autonomy and controllability, observability is not an intrinsic system property. It depends on a partition of the system variable w into two subvectors w1 , w2 . Then one says that w1 is observable from w2 if w1 w1 , ∈ B ⇒ w1 = w1 . w2 w2 Let R = [R1 , R2 ] be partitioned accordingly. Thus, w1 is observable from w2 if and only if {w1 ∈ Aq1 |R1 w1 = 0} = {0}.
1. w1 is observable from w2 ; 2. R1 is left invertible. 8. Conclusion We have extended the basic systems theoretic concepts from multidimensional discrete systems over a field to systems over the rings C = Z/nZ. The crucial tool is the fact (originally r established in [8]) that A=C Z is an injective cogenerator over −1 D = C[1 , . . . , r , −1 1 , . . . , r ]. To the author’s knowledge, the results are partially new even for the case r = 1. Several recent papers [3–6,8,10] suggest that there may be applications in (the academic foundations of) coding theory. From the constructive point of view, there are still several open problems, mostly concerning Gröbner basis theory with coefficients in Zn . Since several research teams are currently working on these problems, some progress in this direction can be expected soon. Acknowledgment The author would like to thank M. Kuijper for giving two inspiring talks [3,5], and J. Rosenthal for asking a good question. References [1] M.F. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, MA, 1969. [2] N. Bourbaki, Algebra I, Springer, Berlin, 1989. [3] M. Kuijper, R. Pinto, J.W. Polderman, Kernel representations for behaviors over finite rings, in: Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS), Kyoto, 2006. [4] M. Kuijper, R. Pinto, J.W. Polderman, P. Rocha, Autonomicity and the absence of free variables for behaviors over finite rings, in: Proceedings of the 7th Portuguese Conference on Automatic Control (Controlo), Lisbon, 2006. [5] M. Kuijper, X.-W. Wu, U. Parampalli, Behavioral models over rings—minimal representations and applications to coding and sequences, in: Proceedings of the 16th IFAC World Congress, Prague, 2005. [6] V.L. Kurakin, A.S. Kuzmin, V.T. Markov, A.V. Mikhalev, A.A. Nechaev, Linear codes and polylinear recurrences over finite rings and modules, in: M. Fossorier et al. (Ed.), Applied Algebra, Algebraic Algorithms and Error Correcting Codes, Lecture Notes in Computer Science, vol. 1719, Springer, Berlin, 1999. [7] T.Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, vol. 189, Springer, Berlin, 1999. [8] P. Lu, M. Liu, U. Oberst, Linear recurring arrays, linear systems and multidimensional cyclic codes over quasi-Frobenius rings, Acta Appl. Math. 80 (2004) 175–198. [9] B. Malgrange, Systèmes différentiels à coefficients constants, Semin. Bourbaki 15 (1962/63), 246 (1964) 11. [10] A.V. Mikhalev, A.A. Nechaev, Linear recurring sequences over modules, Acta Appl. Math. 42 (1996) 161–202.
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