A New Insight on the Mappings of Type-2 Fuzzy Logic Systems

Abstract

In this paper, we provide a new insight on the mappings of Interval Type-2 (IT2) Fuzzy Logic Systems (FLSs) in comparison to its Type-1 (T1) counterparts. For the sake of simplicity, we focus on Single input IT2 (SIT2) FLSs that employ the Karnik Mendel (KM) or the Nie-Tan (NT) Centre of Sets Calculation Method (CSCM). Through theoretical investigations, it is shown that there does exist an equivalent SIT2-FLS representation with a Single input T1 (ST1) FLS that uses Rational Polynomial Functions (RBFs) in their rule consequents. It is concluded that SIT2-FLSs cannot be implemented with traditional ST1-FLSs. It is also revealed that the extra degree of freedom provided by IT2 fuzzy sets and CSCM results with RBFs that can generate much more complex mapping when compared to its T1 counterparts. It is also proven that SIT2-FLSs can be seen as adaptive ST1-FLSs that are tuned via a collection of ST1-FLSs. Thus, SIT2-FLSs have adaptive feature property that does not exists in ST1-FLSs. It is also shown that the SIT2-FLS employing the KM is more complex but also capable to accommodate a much wider range of mappings when compared its NT counterpart.