On the design of pole modules for inverse systems

Abstract

When a linear dynamical system admits more than one inverse, it is known that the pole module of any inverse must contain, either as a submodule or as a factor module, a module of fixed poles isomorphic to the zero module of the original system. Design of the pole module for such an inverse system is resolved by introducing a variable pole module for the inverse, by determining necessary and sufficient conditions for a desired module to be a variable pole module, and by studying the manner in which the fixed and variable modules assemble into the pole module of the inverse. If the fixed and variable pole spectra are disjoint, the pole module of the inverse system is a direct sum of the fixedand variable-pole modules; if not, procedures for addressing the Jordan structure are presented. Subject only to mild necessary conditions, design of inverse systems with specified characteristic polynomial (and, therefore, specified state-space dimension) is possible even in the latter case. For pedagogical reasons, most of the paper is cast over polynomial rings. However, one section discusses generalizations to a wider class of useful rings. Examples are provided for three types of rings.