Asynchronous PID Control for T-S Fuzzy Systems Over Gilbert-Elliott Channels Utilizing Detected Channel Modes

This paper is concerned with the $H_{\infty }$ proportional-integral-derivative (PID) control problem for Takagi-Sugeno fuzzy systems over lossy networks that are characterized by the Gilbert-Eillott model. The communication quality is reflected by the presence of two channel modes (i.e., "bad" mode and "good" mode), which switch randomly according to a Markov process. In the "bad" mode, packet dropouts are governed by a stochastic variable sequence. Considering the inaccessibility of channel modes, a mode detector is utilized to estimate the communication situation. The relationship between the actual channel mode and the estimated mode is depicted in terms of certain conditional probabilities. Moreover, a comprehensive model is constructed to represent the probability uncertainties arising from statistical errors in channel mode switching, packet dropouts, and mode detection processes. Subsequently, a robust asynchronous PID controller, based on the detected channel mode, is proposed. Sufficient conditions are then derived to ensure the mean-square stability of the closed-loop system while maintaining the desired $H_{\infty }$ performance. Finally, the efficacy of the proposed design approach is demonstrated through a simulation example.