An exact inversion method for multi-input-single-output decomposable TS fuzzy systems

Abstract

Almost all exact inversion methods provide inverse solutions for only one input variable of fuzzy systems. These methods have certain limitations on the fuzzy system structure such as monotonic rule bases, singleton rule consequents, and invertibility check. These requirements limit the modeling capabilities of the fuzzy systems and also may result in poor application performances. In this study, an exact analytical inversion method is presented for multi-input-single-output decomposable TS fuzzy systems with either singleton or linear consequents. In the proposed method, fuzzy system structures do not need to have monotonic rule bases, singleton rule consequents, or any invertibility conditions. Thus, more flexible fuzzy systems can be used in inverse model based applications. The proposed method provides a simple and systematic way to obtain unique inverse solutions of all input variables simultaneously with respect to any desired system output value. For this purpose, an inversion trajectory approach that guarantees the existence and uniqueness of the inverse solutions is introduced. The inversion trajectory consists of a set of paths defined on the specific edges of universe of discourses of the decomposed fuzzy subsystems. Using this approach, the inverse definition of the overall fuzzy system can easily be derived only by inverting the related decomposed fuzzy subsystems on this inversion trajectory and then combining their inverse definitions. In this way, the inverse definition of the overall fuzzy system is obtained as consisting of analytical solutions of linear and quadratic equations for singleton and linear consequent cases, respectively. Simulation studies are given for the inversion of two and three-input-single-output fuzzy systems, and the exactness and effectiveness of the proposed method are demonstrated.