A Robust $H_{\infty}$ Non-PDC Design Scheme for Singularly Perturbed T–S Fuzzy Systems With Immeasurable State Variables

This paper addresses an observer-based robust H
_∞
fuzzy controller design method for singularly perturbed nonlinear systems with external disturbances, which are represented by uncertain Takagi-Sugeno (T-S) fuzzy models. For a practical output feedback design, the premise variables of the fuzzy controller and the fuzzy observer are considered unknown in general. A fuzzy Lyapunov function is utilized for the synthesis of the nonparallel-distributed-compensation-based controller. The closed-loop singularly perturbed T-S fuzzy system is asymptotically robustly stable in the absence of disturbances and satisfies an H
_∞
-norm condition in the presence of disturbances for all positive values of the singular perturbation parameter within the given desired bound. Using the descriptor redundancy approach, some strict linear matrix inequality (LMI) conditions are derived for the H
_∞
robustness of the closed-loop system when uncertainties are simultaneously considered in all state-space matrices of each fuzzy subsystem. The main disadvantage of a fuzzy Lyapunov function-based design is that the bounds on the derivatives of the membership functions must be known a priori. Inspired by a recent work that provides the bounds of the membership derivatives for a given arbitrary compact set on the state space, a new method is proposed that guarantees that the closed-loop observer-based uncertain singularly perturbed T-S fuzzy system remains robustly stable in the compact set. Using Finsler's lemma, the resulting design conditions are converted to LMIs. Two numerical examples are provided to show the effectiveness of the proposed method.