Realization Theory of Stochastic Jump-Markov Linear Systems

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.