Filtering S-coupled algebraic Riccati equations for discrete-time Markov jump systems

This paper deals with a set of S-coupled algebraic Riccati equations that arises in the study of filtering of discrete-time linear jump systems with the Markov chain in a general Borel space S. By S-coupled it is meant that the algebraic Riccati equations are coupled via an integral over S. Conditions for the existence and uniqueness of a positive semi-definite solution to the filtering S-coupled algebraic Riccati equations are obtained in terms of the concepts of stochastic detectability and stochastic stabilizability. This result is then applied to solve the infinite horizon minimum mean square linear Markov jump filtering problem. The obtained results generalize previous ones in the literature, which considered only the case of the Markov chain taking values in a finite state space.