Line Integral Approach to Extended Dissipative Filtering for Interval Type-2 Fuzzy Systems

This article is concerned with the problems of extended dissipativity analysis and filter design for interval type-2 (IT2) fuzzy systems. Based on the line integral Lyapunov function, a sufficient condition of asymptotic stability and extended dissipativity of the systems under consideration is established. A LMI-based equivalent condition to the obtained one in a nonlinear form is provided by combining congruence transformation with change of variables. This LMI condition obtained is more general than the one which is based on the common quadratic Lyapunov function. Meanwhile, in terms of parameterization, the extended dissipative filter is developed which guarantees the asymptotic stability and extended dissipativity for the filtering error system. Furthermore, our filter obtained by the parameterization method includes the one obtained by the equivalent transformation method as a special case. Two simulation examples are provided to show the merits and effectiveness of the proposed approach.

I. Introduction

In the past few decades, type-1 Tagaki–Sugeno (T–S) fuzzy systems [1] have been investigated which can represent complex nonlinear systems by a set of local linear systems [2]–[5]. However, type-1 fuzzy systems are not able to directly handle uncertainties of the membership functions. So, type-2 fuzzy models were introduced to deal with the uncertainties of the fuzzy systems [6]–[8]. The interval type-2 (IT2) fuzzy models not only can handle the uncertainties of the membership functions involved in the fuzzy sets but also can reduce the burden of computation for type-2 fuzzy models. So some research has been done on the stability analysis and controller design of IT2 fuzzy systems in [9]–[16]. Stability conditions and control synthesis were provided for IT2 fuzzy systems by using a quadratic Lyapunov function in [10]. Unlike the one in [10], a new IT2 controller was proposed in [9], of which the membership functions and the number of rules could be chosen freely. In [15], stochastic stability analysis and control design for uncertain stochastic IT2 fuzzy systems with unmatched premises were investigated by using the Lyapunov function method and space decomposition approach.