Robustness of Stochastic Discrete-Time Switched Linear Systems With Application to Control With Shared Resources

Motivated by control applications relying on shared resources (such as computation time or bandwidth), we analyze the stability and robustness of discrete-time switched linear systems with stochastic commutations. We show that a wide class of shared resources control strategies can be modeled by a stochastic jump linear system involving two stochastic processes. The class of systems we study encompasses Markov chains and independent and identically distributed switching processes. For these systems, we recall existing definitions of stability and robustness, by relying on the input-to-state stability (ISS) property. We show that, for the class of systems under concern,
$\delta$
-moment stability is equivalent to
$\delta$
-moment ISS and that they both imply almost sure ISS. Several sufficient conditions are provided to guarantee these properties. Anytime control design for a translational oscillator/rotational actuator (TORA) system is used to illustrate all these concepts.