On the Stabilization of Singular, Strictly Proper Transfer Function Matrices

Abstract

This work focuses on the computation of stabilizing controllers for singular transfer function matrices. Stabilization of systems is a thoroughly studied subject, and numerous methodologies that enable the determination of stabilizing controllers have been developed over the years. In this work, we present a framework that unifies these methodologies. More explicitly, we utilize equivalence relations of rational matrices, allowing us to reduce the problem of stabilizing a singular transfer function to the problem of stabilizing a nonsingular transfer function matrix of smaller dimensions, thus connecting the stabilization of singular and nonsingular MIMO transfer functions. Furthermore, it is shown that the aforementioned nonsingular matrix can always be diagonal, in which case the problem of stabilizing it is reduced to stabilizing each of its diagonal elements, which are SISO transfer functions. Finally, it is proved that using this equivalence relation, similar results hold for the case of strong stabilizing controllers, that is, stabilizing controllers that are stable themselves.