Weak detectability and the linear-quadratic control problem of discrete-time Markov jump linear systems

The paper deals with the stochastic concepts of weak detectability and weak observability for Markov jump linear systems, which is an special class of composite linear systems. The concepts are explored here to strengthen the similarities with the corresponding concepts of deterministic detectability and observability. We introduce a collection of matrices, referred to as the observability matrices. We show that weak observability is equivalent to full rank of each matrix in the set of observability matrices. In addition, we present a stochastic counterpart of the well known result on the invariance of trajectories within non-observable subspaces. These characterizations allow us to clarify the relationship between weak detectability and mean square detectability and to provide a testable condition for weak detectability. Relying on the assumption of weak detectability, we develop a method for solving the linear quadratic problem that is based on iterations of uncoupled algebraic Riccati equations, which converges to the solution of the coupled algebraic Riccati equation if and only if the system is mean-square stabilizable. Numerical examples are included.