Realization Theory of Stochastic Jump-Markov Linear Systems

TL;DR

This paper develops a comprehensive realization theory for stochastic jump-linear systems, providing conditions for realization existence, minimality, and an algorithmic approach based on formal power series.

Contribution

It introduces a complete realization framework for stochastic jump-linear systems, including minimality conditions and an algorithm, extending bilinear system realization theory.

Findings

Necessary and sufficient conditions for realization existence.

Algorithm for checking minimality.

Solution based on formal power series theory.

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.