Observability and detectability of discrete-time stochastic systems with Markovian jump

This paper studies detectability and observability of Markov jump discrete-time linear stochastic systems with multiplicative noise (MJDLS for short). The relations between some concepts of detectability and observability are established. These relations, together with equivalent expressions of observability Gramian allow us to obtain some other sufficient and necessary conditions of detectability and observability. As applications, discrete-time stochastic Lyapunov equations and stochastic discrete algebraic Riccati equations (SDARE) are discussed.