Fast Error-Free Algorithms for Polynomial Matrix Computations

Polynomial matrices over fields and rings provide a unifying framework for many control system design problems. These include dynamic compensator design, infinite dimensional systems, controllers for nonlinear systems, and even controllers for discrete event systems. An important obstacle for utilizing these powerful mathematical tools in practical applications has been the non-availability of efficient and fast algorithms to carry through the precise error-free computations required by these algebraic methods. Recently, with the advent of computer algebra this has become possible. In this paper we develop highly efficient, error-free algorithms, for most of the important computations needed in linear systems over fields or rings. We show that the structure of the underlying rings and modules is critical in designing such algorithms. We also discuss the importance of such algorithms for controller synthesis.