Quadratic control with partial information for discrete-time jump systems with the Markov chain in a general Borel space

This paper deals with the finite horizon quadratic optimal control problem of discrete-time Markov jump linear systems (MJLS) considering the case in which the Markov chain takes values in a general Borel space S. It is assumed that the controller has access to an output variable as well as the jump parameter. The goal is to design a dynamic Markov jump controller such that the closed loop system minimizes the quadratic functional cost of the system over a finite horizon period of time. It is shown that an optimal controller can be obtained from two S-coupled difference Riccati equations, one associated to a filtering problem and the other one associated to a control problem in which the state variable is fully available. By S-coupled it is meant that the difference Riccati equations are coupled via an integral over S. This result, which can be seen as a separation principle for discrete-time MJLS, generalizes previous ones that were restricted to the Markov chain taking values in a finite set.