Exponentially Stable Linear Time-Varying Discrete Behaviors

Abstract

We study implicit systems of linear time-varying (LTV) difference equations with rational coefficients of arbitrary order and their solution spaces, called discrete LTV-behaviors. The signals are sequences, i.e., functions from the discrete time set of natural numbers into the complex numbers. The difference field of rational functions with complex coefficients gives rise to a noncommutative skew-polynomial algebra of difference operators that act on sequences via left shift. For this paper it is decisive that the ring of operators is a principal ideal domain and that nonzero rational functions have only finitely many poles and zeros and grow at most polynomially. Due to the poles a new definition of behaviors is required. For the latter we derive the important categorical duality between finitely generated left modules over the ring of operators and behaviors. The duality theorem implies the usual consequences for Willems' elimination, the fundamental principle, input/output decompositions, and controlla...