Stability Analysis and Performance Design for Fuzzy-Model-Based Control System Under Imperfect Premise Matching

Abstract

Abstract-This paper presents the stability analysis and performance design for nonlinear systems. The T-S fuzzy model is employed to represent the nonlinear plant to facilitate the stability analysis. A fuzzy controller, under imperfect premise matching such that the T-S fuzzy model and the fuzzy controller do not share the same membership functions, is proposed to perform the control task. Consequently, the design ffexibility can be enhanced and simple membership functions can be employed to lower the structural complexity of the fuzzy controller. However, the favourable characteristic given by perfect premise matching will vanish, which leads to conservative stability conditions. In this paper, under imperfect premise matching, the information of membership functions of the fuzzy model and controller is considered during the stability analysis. LMI-based stability conditions are derived to guarantee the system stability using the Lyapunov-based approach. Free matrices are introduced to alleviate the conservativeness of the stability conditions. LMI-based performance conditions are also derived to guarantee the system performance. Simulation examples are given to illustrate the effectiveness of the proposed approach.

Membership‐Function‐Dependent Stability Analysis of Interval Type‐2 Polynomial Fuzzy‐Model‐Base Control Systems

“…An IT2 polynomial fuzzy controller with c rules is employed to stabilize the nonlinear plant represented by an IT2 polynomial fuzzy model (5). The j-th rule is of the following format [29]:…”

“…In order to control the nonlinear system representing by T-S fuzzy model, a fuzzy controller, which is the average weighted summation of local linear controllers, is employed. In general, there are three types of fuzzy controllers [4][5][6][7], e.g. parallel distributed compensation (PDC) fuzzy controller, in which the fuzzy controller shares the same premise membership functions as the fuzzy model; partially matched fuzzy controller, in which it has the same number of rules as but different premise membership functions from the fuzzy model; and imperfectly matched fuzzy controller, in which neither the number of rules nor the premise membership functions are the same as the fuzzy model.…”

“…parallel distributed compensation (PDC) fuzzy controller, in which the fuzzy controller shares the same premise membership functions as the fuzzy model; partially matched fuzzy controller, in which it has the same number of rules as but different premise membership functions from the fuzzy model; and imperfectly matched fuzzy controller, in which neither the number of rules nor the premise membership functions are the same as the fuzzy model. In terms of imperfectly matched fuzzy controller, since it does not need to share the same premise membership functions with the fuzzy model and the number of rules can be freely chosen, design flexibility and robustness property can be enhanced [5]. Connecting a fuzzy controller with the nonlinear plant (represented by the T-S fuzzy model) in a closed loop forms the T-S fuzzy-model-based (FMB) control system.…”

Relaxed stability conditions for polynomial‐fuzzy‐model‐based control system with membership function information

“…With the study of the fuzzy-model-based control system with mismatched premise membership functions, the usage of numerical relationship between the model and the controller membership functions was taken into account to relax the conservativeness in the stability analysis and performance design [42]. For the purpose of extracting more membership function information, staircase membership functions [43] and piecewise linear membership functions [44] were used to approximate the original membership functions in the LMI approach.…”

Static output‐feedback control for interval type‐2 discrete‐time fuzzy systems

“…Recently, some results on type-2 fuzzy logic systems have been reported in [38][39][40][41][42][43][44][45][46]. More recently, the interval type-2 (IT2) fuzzy model-based (FMB) control systems have been developed [42,47,48]. An IT2 T-S fuzzy model was proposed to describe the T-S fuzzy systems with uncertain membership functions in [42], and it was proved that the IT2 fuzzy state feedback controller can obtain less conservative results than the usual type-1 PDC fuzzy state feedback controller.…”