Sliding mode control for singularly perturbed Markov jump descriptor systems with nonlinear perturbation

This paper develops a stochastic integral sliding mode control strategy for singularly perturbed Markov jump descriptor systems subject to nonlinear perturbation. The transition probabilities (TPs) for the system modes are considered to switch randomly within a finite set. We first present a novel mode and switch-dependent integral switching surface, based upon which the resulting sliding mode dynamics (SMD) only suffers from the unmatched perturbation that is not amplified in the Euclidean norm sense. To overcome the difficulty of synthesizing the nominal controller, we rewrite the SMD into the equivalent descriptor form. By virtue of the fixed-point principle and stochastic system theory, we give a rigorous proof for the existence and uniqueness of the solution and the mean-square exponential admissibility for the transformed SMD. A generalized framework that covers arbitrary switching and Markov switching of the TPs as special cases is further achieved. Then, by analyzing the stochastic reachability of the sliding motion, we synthesize a mode and switch-dependent SMC law. The adaptive technique is further integrated to estimate the unavailable boundaries of the matched perturbation. Finally, simulation results on an electronic circuit system confirm the validity and benefits of the developed control strategy.