On the control of jump linear Markov systems with Markov state estimation

We analyze a jump linear Markov system being stabilized using a linear controller. We consider the case when the Markov state is associated with the probability distribution of a measured variable. We assume that the Markov state is not known, but rather is being estimated based on the observations of the variable. We present conditions for the stability of such a system and also solve the optimal LQR control problem for the case when the state estimate update uses only the last observation value. In particular we consider a suboptimal version of the casual Viterbi estimation algorithm and show that a separation property does not hold between the optimal control and the Markov state estimate. Some simple examples are also presented.