On the balanced truncation and coprime factors reduction of Markovian jump linear systems

Abstract

The paper deals with the balanced truncation and coprime factors reduction of Markovian jump linear (MJL) systems, which can have mode-varying state, input, and output dimensions. We develop machinery for balancing mean square stable MJL system realizations using generalized Gramians and strict Lyapunov inequalities, and provide an improved a priori upper bound on the error induced in the balanced truncation process. We also generalize the coprime factors reduction method and, in doing so, extend the applicability of the balanced truncation technique to the class of mean square stabilizable and detectable MJL systems. We provide tools to establish mean square stabilizability and detectability of the considered MJL systems. In addition, a notion of right-coprime factorization of MJL systems and methods to construct such factorizations are given. The error measure in the coprime factors reduction approach, while still norm-based, does not directly capture the mismatch between the nominal system and the reduced-order model, as is the case in the balanced truncation approach where mean square stable models are considered. Instead, the error measure is given in terms of the distance between the coprime factors realizations, and thus has an interpretation in terms of robust feedback stability. The paper concludes with an illustrative example which demonstrates how to apply the coprime factors model reduction approach.