Asynchronous Sliding Mode Control for Nonlinear Markov Jumping Systems With PDT-Switched Transition Probabilities

This article studies the asynchronous sliding mode control for nonlinear Markov jump systems (NMJSs), whose transition probabilities is piecewise-constant and subjected to the persistent dwell-time switching rule. By resorting to the interval type-2 Takagi–Sugeno (IT2 T-S) fuzzy set theory, the nonlinearity and uncertainty features of the investigated systems are modeled. Taking the uncertain membership functions (MFs) of IT2 T-S fuzzy systems into consideration, the nonparallel distributed compensation strategy is employed for designing the MFs of the controller such that they do not need to be obtained in time. Furthermore, in view of the difficulty of accurately acquiring the mode of NMJSs with complex transition probabilities, the hidden Markov model is employed to provide a detected mode for constructing the sliding mode law. Afterward, a novel fuzzy sliding surface is proposed, and an asynchronous IT2-fuzzy-model-based sliding mode control law is designed for forcing the closed-loop systems to move on the predefined sliding surface. Thereafter, a novel two-mode-dependent Lyapunov-like function is conceived for the studied systems, where modes of NMJSs and transition probabilities are permitted to change simultaneously. By applying an improved technique to handle the parameteric matrix inequalities, some less-conservative sufficient conditions in the form of linear matrix inequality are obtained for ensuring the stochastic exponentially asymptotical stability of the sliding mode with
$H_\infty$
performance. Finally, a mass-spring mechanical system is presented to substantiate the validity of the proposed main results.