Coprime factorizations and stability of multivariable distributed feedback systems

Abstract

The stability of multivariable feedback systems presents different problems from the stability of single loop feedback systems, owing mainly to the complexities of "pole"-"zero" cancellation in the multivariate case. In this paper, the "coprime factorization" of a nonrational transfer function matrix is defined and is used in studying the stability of multivariable distributed feedback systems. However, the stability results based on coprime factorizations, though they are quite elegant, do not lead to readily applicable testing procedures. For this reason, we introduce the notion of "pseudo-coprime" factorizations. These also lead to many stability theorems. As a special case of these stability results, we obtain explicit necessary and sufficient conditions for the stability of a multivariable feedback system whose open loop transfer function contains a finite number of poles in the closed right half-plane, but is otherwise stable. These results significantly generalize those of Callier and Desoer [8].