System-theoretic and algebraic aspects of the rings of stable and proper stable rational functions

A new approach in transfer-function methods for solving a variety of control-theoretic problems is to work with fractional representations over the ring of stable or proper stable rational functions of various transfer matrices. The rings of stable and proper stable rational functions are well known to be Euclidean domains. These rings differ critically from the polynomial ring by the nonuniqueness of the remainders obtained. This major difficulty in extending the idea of polynomial models of Fuhrmann to the rings of stable and causal stable rational functions is circumvented by choosing a remainder in a special form. A natural realization theory is thus developed for matrix fraction representations of transfer-function matrices over these rings. As an application of the new theory developed, linear matrix equations (QX+RY=T and QX+YR=T) over the rings of stable and causal stable rational functions are reduced to finite sets of linear equations over the base field.