On the Monotonicity of Interval Type-2 Fuzzy Logic Systems

Qualitative knowledge is very useful for system modeling and control problems, especially when specific physical structure knowledge is unavailable and the number of training data points is small. This paper studies the incorporation of one common qualitative knowledge-monotonicity into interval type-2 (IT2) fuzzy logic systems (FLSs). Sufficient conditions on the antecedent and consequent parts of fuzzy rules are derived to guarantee the monotonicity between inputs and outputs. We take into account five type-reduction and defuzzification methods (the Karnik-Mendel method, the Du-Ying method, the Begian-Melek-Mendel method, the Wu-Tan method, and the Nie-Tan method). We show that IT2 FLSs are monotonic if the antecedent and consequents parts of their fuzzy rules are arranged according to the proposed monotonicity conditions. The derived monotonicity conditions are valid for the IT2 FLSs using any kind of IT2 fuzzy sets (FSs) (e.g., Trapezoidal IT2 FSs and Gaussian IT2 FSs) and stand for type-1 FLSs as well. Guidelines for applying the proposed conditions to modeling and control problems are also given. Our results will be useful in the design of monotonic IT2 FLSs for engineering applications when the monotonicity property is desired.