Fuzzy-Model-Based Non-Fragile Control for Nonlinear Singularly Perturbed Systems with Semi-Markov Jump Parameters

This paper researches the fuzzy-model-based non-fragile control problem for nonlinear discrete-time singularly perturbed systems with Markov jumping parameters. To remove the restriction of sojourn-time in widely used discrete-time Markov jump model, i.e. the sojourn-time is subject to a geometric distribution which leads that the transition probabilities is memoryless, the semi-Markov jump model in which the <formula><tex>${memory}$</tex></formula> property of the transition probabilities is considered, is adopted to describe the stochastic jump parameters in the nonlinear singularly perturbed systems. Based on the T-S fuzzy model approach and semi-Markov Kernel concept, several criteria ensuring <formula><tex>$\delta$</tex></formula>-error mean square stability of the underlying closed-loop system are established. With the help of those criteria, the design procedures of a resilient controller which can well deal with the fragility problem in the implementation of the proposed fuzzy-model-based controller is presented and a technique is developed to estimate the larger permissible maximum value of singularly perturbed parameter for discrete-time nonlinear semi-Markov jump singularly perturbed systems. Finally, the validity of the established theoretical results is illustrated by a numerical example and a modified tunnel diode circuit.