Stabilizing Sampled-data Linear Systems with Markovian Packet Losses and Random Sampling ⋆

We consider the stability properties and stabilizing problem of sampled-data networked controlled linear systems with Markovian packet losses. A binary Markov chain is used to characterize the packet loss phenomenon of the network. Then with the Markovian packet loss assumption, we obtain a discrete-time augmented Markov jump linear system which describes the continuous-time linear system evolving in deterministic discrete time. Furthermore, we show that the sampled-data system under consideration can also be considered as a randomly sampled system with an i.i.d. random sampling period. A number of necessary and sufficient conditions for the stochastic stability properties are established by using the known results of Markov jump linear systems and randomly sampled systems. Those conditions are based on the relationships of stability properties between the systems evolving in deterministic continuous time, deterministic discrete time, and random discrete time. In addition, the asymptotic stability of the system is also studied by using Lyapunov exponent method. Numerical examples are used to illustrate the main results of the paper.