Global non-quadratic D-stabilization of Takagi–Sugeno systems with piecewise continuous membership functions

Transient performance improvement of Takagi‐Sugeno fuzzy systems by modified non‐parallel distributed compensation controller

“…In this example, the proposed conservativeness of Theorem 1 with ρ ( x ( t )) = 0 is compared to other approaches that provide the global stabilization for the TS fuzzy system of Example 1. Besides Theorem 1, the QLF‐based approach [1], the line integral Lyapunov function‐based approach [30], polynomial line integral Lyapunov method [13], the NQLF with auxiliary membership function [15], and global D‐stability [24] are considered.…”

“…Since the line integral Lyapunov function is constructed by multiple Lyapunov matrices, its use reduces the conservativeness in comparison with the QLF and, thereby, the line integral Lyapunov function considered in [13,30] provides a feasible solution for a broader region than the QLF, although, the structural constraints of the line integral Lyapunov function still increase the conservativeness of the controller design based on it. Since such constraints are avoided in the modified NQLF approaches [15,24], these approaches lead to less conservative results and a wider feasibility region. However, the highest feasible points are computed via the proposed Theorem 1.…”

“…Feasibility region of the system of Example 1 (quadratic [1] by “+”, line integral [30] by “+ and o”, polynomial line integral [13] by “+, o, and ●”, non‐quadratic [15] by “+, o, ●, and ×”, global D‐stability [24] by “+, o, ×, ●, and □”, and proposed approach “+, o, ×, ●, □, and ■”) [Color figure can be viewed at wileyonlinelibrary.com]…”

“…On the other hand, achieving the desired transient performance of the closed‐loop nonlinear system is an important issue [16–19]. Unfortunately, the transient performance issue of the closed‐loop TS fuzzy systems is not a focus of prior research, although a few methods to improve the transient of the TS fuzzy systems have been presented, including the guaranteed cost function control [20–22], the D‐stability concept [23,24], the reset control [25], and finite‐time stability [26–28]. In [23,24], the fuzzy controller is designed based on the D‐stability concept, in which the poles of the each of the closed‐loop linear subsystems are located in a pre‐chosen region.…”

“…Unfortunately, the transient performance issue of the closed‐loop TS fuzzy systems is not a focus of prior research, although a few methods to improve the transient of the TS fuzzy systems have been presented, including the guaranteed cost function control [20–22], the D‐stability concept [23,24], the reset control [25], and finite‐time stability [26–28]. In [23,24], the fuzzy controller is designed based on the D‐stability concept, in which the poles of the each of the closed‐loop linear subsystems are located in a pre‐chosen region. The region can be selected in such a way that the linear systems experience a low oscillation and a fast convergence, but the overall closed‐loop TS fuzzy does not have the same properties as its linear subsystems and such pole placement cannot guarantee the same behavior of the overall nonlinear system.…”