H∞ Stabilization of Discrete-Time Nonlinear Semi-Markov Jump Singularly Perturbed Systems With Partially Known Semi-Markov Kernel Information

Abstract

In this paper, the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> stabilization problem is studied for discrete-time semi-Markov jump singularly perturbed systems (SMJSPSs) with repeated scalar nonlinearities. As the exact statistical information of the sojourn time or the mode transition is difficult to obtain, the case with only partial semi-Markov kernel information available is considered. Furthermore, introducing an external disturbance or nonlinearity into the analysis of discrete-time semi-Markov jump systems (DTSMJSs) meets critical obstacles, since the relation between the system state vectors at two nonadjacent instants is difficult to determine. To address this issue, the variation trend of the Lyapunov function for a semi-Markov jump sequence is analyzed in detail. Subsequently, criteria of mean-square exponential stability (MSES) for DTSMJSs are established for the first time based on the Lyapunov stability theory. By virtue of the criteria obtained and the cone complementary linearization algorithm, a controller ensuring MSES and H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance for discrete-time nonlinear SMJSPSs is constructed. Finally, the effectiveness and applicability of the proposed method are validated by simulation examples including an inverted pendulum model.