A Detector-Based Approach for the <inline-formula> <tex-math notation="TeX">$H_{2} $</tex-math></inline-formula> Control of Markov Jump Linear Systems With Partial Information

Abstract

In this paper, we study the $H_{2} $ -control for discrete-time Markov Jump Linear Systems (MJLS) with partial information . We consider the case in which we do not have access to the Markov jump parameter but, instead, there is a detector that emits signals which provides information on this parameter. A salient feature of our formulation is that it encompasses, for instance, the cases with perfect information, no information and cluster observations of the Markov parameter, which were previously analyzed in the Markov jump control literature. The goal is to derive a feedback linear control using the information provided by the detector in order to stochastically stabilize the closed loop system. We present two Lyapunov like equations for the stochastic stability of the system. In addition, we show that a Linear Matrix Inequalities (LMI) formulation can be obtained in order to design a stochastically stabilizing feedback control. In the sequel we deal with the $H_{2} $ control problem and we show that, again, an LMI optimization problem can be formulated in order to design a stochastically stabilizing feedback control with guaranteed $H_{2} $ -cost. We also present two special cases, one of them always satisfied for the limit case in which the detector provides perfect information on the Markov parameter, and the Bernoulli jump case, under which LMI conditions become necessary and sufficient for the stochastic stabilizability of the system and the LMI optimization problems provide the optimal $H_{2} $ cost. For the Bernoulli jump case we show that our formulation generalizes previous ones. The case with convex polytopic uncertainty on the parameters of the system and on the transition probability matrix is also considered. The paper is concluded with some numerical examples.