Observer-Based Fuzzy Integral Sliding Mode Control For Nonlinear Descriptor Systems

This paper investigates observer-based stabilization for nonlinear descriptor systems using a fuzzy integral sliding mode control approach. Observer-based integral sliding mode control strategies for the Takagi-Sugeno (T-S) fuzzy descriptor systems are developed. A two-step design approach is first developed to obtain the observer gains and coefficients in the switching function using linear matrix inequalities (LMIs), and the results are used to facilitate the development of a single-step design approach, which is seen to be convenient but introduces some conservatism in the design. The potential application to a class of mechanical systems is also considered. Since the descriptor system representation of mechanical systems is adopted, it is shown that in contrast to the existing fuzzy sliding mode control methods based on the normal system representation, the resulting T-S fuzzy system does not contain different input matrices for each local subsystem and the required number of fuzzy rules is consequently markedly reduced. Finally, the balancing problem of a pendulum on a car is numerically simulated to demonstrate the effectiveness of the proposed method.

I. Introduction

The Takagi–Sugeno (T-S) fuzzy model [1] has been extensively utilized as a popular and convenient tool to deal with complex nonlinear systems. With the T-S fuzzy model, nonlinear systems with smooth nonlinearities can be exactly represented in a compact set of the state space by a set of linear subsystems connected by corresponding normalized weighted coefficients. Then, approaches to systematic analysis and synthesis for the resulting T-S fuzzy systems can be developed within the frame of conventional control technology and fuzzy logic control. As a result, this T-S fuzzy approach has attracted significant attention from the control community [2]– [12]. Despite the superiority of the T-S fuzzy model, the fuzzy rules will exponentially increase with the nonlinearities arising in the system representation. This increase in the number of fuzzy rules can produce a linear matrix inequality (LMI) condition that is infeasible or unnecessarily complicated. Recently, Taniguchi et al. [13] presented a T-S fuzzy descriptor model that can be regarded as a generalization of the T-S fuzzy normal model (). The advantages of the T-S fuzzy descriptor model are a reduction in the number of fuzzy rules and a tighter nonlinear system representation [14]. Over recent years, a great deal of effort has been devoted to the study of T-S fuzzy descriptor systems and many significant results have been achieved in diverse areas, for example, stability and stabilization [15]– [23], observer-based control [24] –[25], guaranteed cost control [26] –[27], and dissipative control [28]– [31], and fault-tolerant control [32] .