General derivation and analysis for input–output relations in interval type-2 fuzzy logic systems

In this paper, analytical closed-form expressions are derived for the input–output relation related to an interval type-2 fuzzy logic system. It has been assumed that the related fuzzy system possesses diamond-shaped type-2 fuzzy sets for each input and singletons for output. Moreover, the Nie–Tan inference engine that provides a closed-form is preferred. The footprint of uncertainty in diamond-shaped type-2 membership functions generates four times as many regions in analytical closed-form expression as generated by standard triangular type-1 membership functions. The derived mathematical relationships provide a chance to examine the internal structure of an interval type-2 fuzzy system. These extra regions may enhance the performance of an interval type-2 fuzzy logic system over the type-1 counterpart. An important advantage of the proposed technique is that the analytical input–output relations are applicable for any number of input fuzzy sets. Analytical structures of two special cases of interval type-2 fuzzy logic systems which use different number of membership functions for each input are derived in detail.