On $H_\infty$ Sliding Mode Control Under Stochastic Communication Protocol

This paper is concerned with the sliding mode control (SMC) problem for a class of uncertain discrete-time systems subject to unmatched external disturbances and communication constraints. In order to reduce the bandwidth usage between the controller and the actuators, the stochastic communication protocol (SCP) is utilized to determine which actuator should be given the access to the network at a certain instant. A key issue of the addressed problem is to design both the sliding surface and the sliding mode controller under the SCP scheduling. An updating rule on actuator input is first introduced and then a token-dependent SMC law is designed. Sufficient conditions are established for the resultant SMC systems such that not only the reachability with a sliding domain around the specified sliding surface is ensured, but also the stochastic stability with a prescribed <inline-formula><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> performance level is guaranteed. Based on these conditions, a set of coupled matrix inequalities is given to acquire the token-dependent parameter matrices in the proposed SMC law. Finally, a numerical example is presented to illustrate the effectiveness of the proposed <inline-formula><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> SMC scheme under the SCP scheduling.