The rings of stable rational functions: Algebraic properties

A recent approach for treating a variety of control problems is to work with matrix fraction representations over the rings of stable or causal stable rational functions of the transfer matrices. This approach is very similar in nature to the polynomial fractional methods. This paper extends the idea of the polynomial models of FUHRMANN to the rings of stable and causal stable rational functions. A natural realization theory is thus developed for matrix fraction representations of transfer 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 number of linear equations over the base field.