Reachable Set Estimation for T–S Fuzzy Markov Jump Systems With Time-Varying Delays via Membership Function Dependent H∞ Performance

Abstract

This article considers the reachable set estimation problem and membership function dependent <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> performance analysis for a class of fuzzy Markov jump systems (FMJSs) with mode-dependent time-varying delays and bounded external disturbances via sampled-data control. First, mode-dependent sampled-data control for the FMJS is designed using the Takagi–Sugeno (T–S) fuzzy method. Then, a novel stochastic Lyapunov–Krasovskii functional (LKF) is constructed in mode-dependent augmented form by taking full advantage of the variable characteristics related to the actual sampling pattern. At the same time, a membership function dependent <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> performance index is introduced for the first time to attenuate the impact of disturbances on the closed-loop FMJS. Based on the novel <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> performance index and LKF, new delay-dependent conditions are derived in the framework of linear matrix inequalities to ensure stochastic stability of the closed-loop system and its reachable set is bounded by an ellipsoid in the presence of bounded disturbances. Finally, two illustrated application problems validate theoretical results with less conservatism in the sense of enlarging the sampling period and minimizing the disturbance attenuation level.