Realization Theory of Stochastic Jump-Markov Linear Systems

Abstract

In this paper, we present a complete stochastic realization theory for stochastic jump-linear systems. We present necessary and sufficient conditions for the existence of a realization, along with a characterization of minimality in terms of reachability and observability. We also formulate a realization algorithm and argue that minimality can be checked algorithmically. The main tool for solving the stochastic realization problem for jump-linear systems is the formulation and solution of a stochastic realization problem for a general class of bilinear systems with non-white-noise inputs. The solution to this generalized stochastic bilinear realization problem is based on the theory of formal power series. Stochastic jump-linear systems represent a special case of generalized stochastic bilinear systems.

This paper is partially based on the following conference paper: Mihaly Petreczky and Rene Vidal. Realization Theory of Stochastic Jump-Markov Linear Systems. 46th IEEE Conference on Decision and Control, 2007. In the new version some minor typos and mistakes have been corrected and Remark 9-10 were added