Discrete-Time Coupled Riccati Equations for Systems with Markov Switching Parameters

This paper deals with a set of coupled Riccati equations which arises in the study of filtering and quadratic optimal control of discrete-time linear systems with Markov switching parameters. In this case the duality between the filtering and optimal control equations is not straightforward and a generalization of the concepts of detectability and stabilizability, as well as a discussion of how the duality between these concepts should be understood, is presented. Conditions for existence and uniqueness of a positive semi-definite solution to a set of coupled Riccati equations in a generic form are provided in terms of these concepts of detectability and stabilizability. Applications to the filtering and optimal control problems are presented, giving a unified and rather complete picture of coupled Riccati equations for discrete-time Markovian jump linear systems. Interpretations for the sub-optimality of the filtering problem are also provided.