Membership-Function-Dependent Control Design and Stability Analysis of Interval Type-2 Sampled-Data Fuzzy-Model-Based Control System

This article investigates the design and stability analysis of the interval type-2 (IT2) sampled-data (SD) fuzzy-model-based (FMB) control system with the optimal guaranteed cost performance. An IT2 Takagi–Sugeno (T-S) fuzzy model is applied to describe the dynamics of the nonlinear systems where the parameter uncertainties are captured by the lower and upper membership functions. To conduct the stability analysis for the SD FMB control system, a looped-functional approach taking the advantage of the information about the sampling periods is employed. Because of the SD control strategy, the state will be sampled at each sampling instant and the control signal generated by the IT2SD fuzzy controller will be kept by the zero-order holder during the sampling period, which will result in mismatched membership grades between IT2 T-S fuzzy model and IT2SD fuzzy controller that leads to the complexity in carrying out stability analysis. Thanks to the imperfect premise matching concept, which allows the difference on the number of rules and the premise membership functions between model and controller, the design of the IT2SD fuzzy controller can be more flexible. To further relax the stability conditions and minimize the upper bound of the guaranteed cost index, the membership-function-dependent stability analysis approach which can make use of the features of the IT2 membership functions is adopted. The performance of the control system can also be adjusted through the choice of the weighting matrices in the cost function. The stability conditions building on the Lyapunov stability theory and the performance conditions building on the concept of the guaranteed cost control in the shape of linear matrix inequalities are established to assure the system stability and acquire the optimal guaranteed cost performance. The proposed IT2SDFMB control design is tested on the inverted pendulum system and the simulation results verify the effectiveness of the proposed approach.

I. Introduction

Nonlinearities are inevitable in various real-world systems, such as mass-spring-damping systems, teleoperation systems, and active magnetic bearing systems [1]–[5]. To handle the control problem of nonlinear systems which are ill-defined, the fuzzy control approaches have been successfully applied during the past years [6]. Among fuzzy control approaches, the fuzzy-model-based (FMB) control approach attracts much attention [7]. In the FMB control approach, Takagi–Sugeno (T-S) fuzzy model is an important tool which supports the stability analysis and control of nonlinear systems due to the favorable modeling property [8], [9]. The type-1 fuzzy model has been the mainstream model in the fuzzy control, but the lack of the ability to tackle uncertainties directly is a drawback [10], [11]. Under the circumstance, more attention is paid to type-2 fuzzy model which can capture uncertainties directly by the type-2 fuzzy sets [12]. However, the general type-2 fuzzy sets will result in the complex design process and high computational expense. Therefore, the interval type-2 (IT2) fuzzy sets which are the generalization of type-1 fuzzy sets and interval-valued fuzzy sets [13] are applied. The IT2 fuzzy model not only retains the ability of capturing uncertainties but also decreases the computational expense compared with the general type-2 fuzzy sets [14]. The work in [15] was the first paper proposing the IT2 fuzzy model, stability analysis, and control synthesis techniques in IT2 FMB control framework. Since then, it has drawn the attention from the fuzzy control community and led to many follow-up works regarding different techniques and control methodologies [7], [16].