The solution of the input-output cover problems

Abstract

The input-output cover problems involve the solution of an algebraic equation over the ring of strictly proper rational functions. This equation constitutes the input-output or polynomial formulation of the state-space cover problems introduced by Wonham. The cover problems provide a unifying formulation for a number of problems in linear, finite-dimensional, system theory, like the observer problem, the exact model matching problem, etc.

The approach adopted, consists of solving the equation over the field of rational functions first. The solutions over the ring of strictly proper rational functions are then obtained from the partial realizations of a sequence of constant matrices, by means of closed formulae.

In the context of linear systems, (strict) proper rationality of the transfer function is equivalent to (strict) causality of the underlying linear system. Our theory implies, that causality and partial realization are equivalent. This fact constitutes a new algebraic characterization of causality; it provides further insight into the fundamental relationships of linear system theory.