An algebraic analysis approach to linear time-varying behaviors

This paper introduces an algebraic analysis approach to time-varying systemsgiven by linear ordinary differential equations with meromorphic coefficients.The analysis is carried out in a generic sense, i.e., the signals are considered outside a discrete set of possible singularities, where they are smooth.The algebra is based on a normal form for matrices over the resulting ring ofdifferential operators, which is a non-commutative analogue of the Smith form.It is used to establish a duality between linear time-varying behaviors on theone hand, and modules over the ring of differential operators on the other.This correspondence leads to algebraic characterizations of the basic systems theoretic properties such as autonomy, controllability, and observability.