On Rosenbrock models over a commutative ring

Rosenbrock’s notion of system equivalence is general in nature; it is a kind of equivalence which in algebra is often termed as stable. We have shown recently that Fuhrmann’s notion of system equivalence can be viewed as a homotopy equivalence, and as such is also general in nature. This note deals with a generalization of the theory of system equivalences from the field case to the commutative ring case. In this note we generalize the classical theory of Rosenbrock models and their equivalences, given in [2,6,7], to the commutative ring case. We shall follow very closely our recent paper [3]. For other generalization the reader is refered to [5]. Throughout, A is an arbitrary commutative ring, s an indeterminate, m an input number and p an output number. We define A(s) to be the localization of A[s] with respect to polynomials with invertible leading coefficient. Elements of this ring will be called rational functions. A rational function f/g will be said to be proper if deg(f) ≤ deg(g). We let O denote the ring of proper rational functions. One has the notion of a finite A[s]-module, the notions of left and right coprimeness. (For details, see [4].) A Rosenbrock model is a quintuple (Z;T, U, V,W ), where Z is a finitely generated Amodule, T : Z[s] → Z[s] is a “generical” isomorphism and U : A[s] → Z[s], V : Z[s] → A[s], W : A[s] → A[s] are arbitrary homomorphisms. (The condition on T means that it induces an isomorphism Z(s) ' Z(s).) The transfer function is defined as the rational matrix V T−1U +W . A model is called regular if its transfer function is proper. A transformation of a model (Z;T, U, V,W ) into a model (Z ′;T ′, U ′, V ′,W ′) is a quadruple (K,L,M,N) consisting of homomorphisms K : Z[s]→ Z ′[s], L : A[s] → Z ′[s], M : Z[s]→ Z ′[s], and N : Z[s]→ A[s] such that [ M 0 N I ] [ T U −V W ] = [ T ′ U ′ −V ′ W ′ ] [ K −L 0 I ] ; that is, MT = T ′K, MU = −T ′L+ U ′, NT − V = −V ′K, NU +W = V ′L+W ′. If Φ1 = (K1, L1,M1, N1) and Φ2 = (K2, L2,M2, N2) are two transformations such that the range of the first one is equal to the domain of the second, then their composition is defined to be Φ2 ◦ Φ1 = (K2K1, K2L1 + L2,M2M1, N2M1 +N1).