Fuzzy Control for Nonlinear Multiple Time-Delay Systems by Dithers

This study presents a robustness design of H ∞ fuzzy control for nonlinear multiple time-delay (NMTD) systems. First, a neural-network (NN) model is constructed for the NMTD plant. The dynamics of the NN model is then converted into a linear differential inclusion (LDI) state-space representation. Next, in terms of Lyapunov’s direct method, a delay-dependent stability criterion is derived to guarantee the stability of NMTD systems. Subsequently, the stability condition of this criterion is reformulated into a linear matrix inequality (LMI). Based on the LMI, a fuzzy controller is synthesized to stabilize the NMTD system and achieve the H ∞ control performance at the same time. When the designed H ∞ fuzzy controller cannot stabilize the NMTD system, a high-frequency signal (commonly referred to as dither) is simultaneously introduced to stabilize it. If the dither’s frequency is high enough, the output of the dithered system and that of its corresponding mathematical model− the relaxed system can be made as close as desired. This makes it possible to obtain a rigorous prediction of the stability of the dithered system based on the one of the relaxed system. Finally, a numerical example with simulations is provided to illustrate the concepts discussed throughout this paper.