Coprime Factorizations for Linear Systems over Rings and Realization of Precompensators by Feedback

This paper develops a general factorization theory of transfer matrices for linear systems defined over rings. Restricting the rings to the class of unique factorization domains (UFDs) and introducing a notion of denominator sets and a generalized notion of coprimeness, a coprime factorization 1 ( ) ( ) ( ) H z W z V z − = for the transfer matrix ( ) H z of a system defined over a UFD is developed where ( ( ), ( )) W z V z is a pair of coprime transfer matrices such that all of their denominators belong to a given denominator set. Various properties of coprime factorizations are presented. In particular, necessary and sufficient conditions for the coprimeness are characterized in terms of algebraic properties of the matrices ( ) W z and ( ) V z . Further, the problem of realizing precompensators by state or output feedback is examined, and necessary and/or sufficient conditions for the realizability are presented.