A canonical form for discrete-time systems defined over \(\mathbb{Z}_+\) (Q1969536)

It is shown that for each member \(G\) of a large class of causal time-invariant nonlinear input-output maps, with inputs and outputs defined on the nonnegative integers, there is a functional \(A\) on the input set such that \((Gs)(k)\) has the representation \(A(F_ks)\) for all \(k\) and each input \(s\), in which \(F_k\) is a simple linear map that does not depend on \(G\). More specifically, this holds -- with an \(A\) that is unique in a certain important sense -- for any \(G\) that has approximately finite memory and meets a certain often-satisfied additional condition. Similar results are given for a corresponding continuous-time case in which inputs and outputs are defined on \(\mathbb{R}_+\). An example shows that the members of a large family of feedback systems have these ``\(A\)-map'' representations.