Model-Based Fuzzy $l_{2}-l_{\infty }$ Filtering for Discrete-Time Semi-Markov Jump Nonlinear Systems Using Semi-Markov Kernel

Abstract

This article concentrates on the model-based fuzzy <inline-formula><tex-math notation="LaTeX">$l_{2}-l_{\infty }$</tex-math></inline-formula> filtering problem of a discrete-time semi-Markov jump nonlinear system. The random jumps in the studied system are governed by the discrete-time semi-Markov process. Therefore, the storage characteristics of the transition probability between systems are fully considered. To analyze the <inline-formula><tex-math notation="LaTeX">$\sigma$</tex-math></inline-formula>-mean-square stability of the filtering error system, a mode-dependent filter, which is based on the discrete-time fuzzy semi-Markov jump model, is constructed to estimate the state of the system. Thereafter, based on the Lyapunov stability theory and the discrete-time semi-Markov kernel concept, a set of sufficient criteria to ensure the <inline-formula><tex-math notation="LaTeX">$\sigma$</tex-math></inline-formula>-mean-square stability and <inline-formula><tex-math notation="LaTeX">$l_{2}-l_{\infty }$</tex-math></inline-formula> performance are derived. In addition, by using the Takagi–Sugeno fuzzy model, the nonlinear problem is effectively solved. Two illustrative examples, including a numerical example and a tunnel diode circuit example, are demonstrated to reveal the practicability of the developed filtering strategy.