Stochastic passivity of discrete-time Markovian jump nonlinear systems

In this paper, we analyze stochastic passivity of discrete-time Markovian jump nonlinear systems (MJNS). We define the notions of general stochastic dissipativity, stochastic QSR dissipativity and then stochastic passivity for these systems. Based on these definitions, the discrete-time stochastic KYP lemma is derived, which gives a necessary and sufficient condition for such a Markovian jump nonlinear system to be stochastic QSR dissipative. Based on the stochastic KYP lemma, we prove that a Markovian jump nonlinear system is locally stochastic feedback passive if and only if its zero dynamics are locally stochastic passive. Furthermore, we extend these results to interconnected MJNS. Given such interconnected subsystems that are stochastic feedback passive, we design a feedback control law and obtain the conditions on the weighted Laplacian matrix between all the subsystems to stabilize the entire system in the stochastic sense.