On Fliess models over a commutative ring

Fliess models come as natural generalizations of classical polynomial models (D,N). As known, they are defined in terms of finitely generated modules and provide an adequate description of arbitrary linear systems (not necessarily observable or controllable). In this note we extend Fliess’ approach to linear systems defined over an arbitrary commutative ring. We believe that the exposition will be of interest even for the field case. In this note we present a generalization of Fliess’ module-theoretic approach, as developed in [1], to the commutative ring case. Throughout, A is an arbitrary commutative ring, s an indeterminate, m an input number and p an output number. We shall denote by A(s) the ring of rational functions and by O the ring of proper rational functions (see [4]). A linear system is a quintuple (X;F,G,H, J), where X is a finitely generated A-module F : X → X, G : A → X, H : X → A are A-linear maps, and J is a polynomial p × m matrix. The rational matrix H(sI − F )−1 + J is called the transfer function. The reader is refered to [4] for the definitions of regularity, controllability and observability, and minimality. We define a Fliess model as a triple (M ; δ, ν), where M is a finitely generated A[s]-module, δ : A[s] → M a “generical” isomorphism and ν : A[s] → M an arbitrary homomorphism. The condition on δ means that it induces an isomorphism A(s) ' M ⊗ A(s). This implies that the module M is quite specific; in particular, all its torsion elements are anihilated by monic polynomials. A transformation of a Fliess model (M ; δ, ν) into a Fliess model (M1; δ1, ν1) is a homomorphism φ : M → M1 such that δ1 = φ ◦ δ and ν1 = φ ◦ ν. Obviously Fliess models together with transformations form a category. Let (M ; δ, ν) be a Fliess model. The rational matrix νδ−1 is called the transfer function. The model is called controllable if M = δA[s] + νA[s] and observable if M has no torsion. If the model is both controllable and observable it is said to be minimal. Example 1. Let T be a transfer function. Then (A[s] + TA[s]; id, T ) is a minimal Fliess model. (Notice that every minimal Fliess model is obtained this way. Indeed, if (M ; δ, ν) is such a model and T is its transfer function, then δ|M is an isomorphism of M onto A[s] + TA[s].)