Adaptive Fuzzy Control for Stochastic Nonlinear Systems With Delayed Input and Delayed State via Improved Lyapunov–Krasovskii Function

The aim of this study is to address the challenges presented by the nonstrict feedback (NSF) stochastic nonlinear systems structure with delayed state and delayed input. The Pade approximation and an intermediate variable are used to handle the input delay, transforming the original system into one that is perturbed only by the state delay. Subsequently, a novel Lyapunov–Krasovskii function (LKF) related to all states is proposed to effectively compensate for the delay term. In addition, fuzzy logic systems (FLSs) and the variable separation concept are employed to manage the complexity of the system structure. Based on the fixed-time Lyapunov criterion, the closed-loop system bounded in probability is guaranteed within a fixed time, and all signals in the system are bounded. Finally, two stochastic nonlinear systems are utilized to demonstrate the effectiveness of the proposed control strategy.