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Top1. Introduction
PID and type-1 fuzzy controller were widely used in industry processes, but they were unable to handle uncertainties in these practical processes (Kumar, et al.2017). Many researches tried to solve these problems by a novel fuzzy controller, called type-2 fuzzy controller. The type-2 fuzzy controller was based on type-2 fuzzy sets introduced by Zadeh (Zadeh 1975). The main difference between type-1 and type-2 fuzzy sets was that type-2 fuzzy sets contained a type reduction procedure, this was a key step in type-2 fuzzy logic systems. The type reduction first reduced a type-2 fuzzy sets to a type-1 and then defuzzification was operated for type-1 fuzzy sets to get the crisp output. For many years, as the computation complexity of type reduction was very high, there were little applications about type-2 fuzzy logic systems.
Interval type-2 fuzzy sets provided a convenient condition for the developments of type-2 fuzzy logic systems. The secondary membership degree of interval type-2 fuzzy sets was set to 1, which simplified the type reduction procedure for interval type-2 fuzzy sets and the commonly applied type reduction algorithm was Karnik-Mendel (KM) algorithm (Karnik and Mendel.2001). The simulation results and some practical control problems showed that interval type-2 fuzzy controller had better control effects than type-1 fuzzy controller or PID controller. Many researchers tried to derive the analytical structure of interval type-2 fuzzy controller. The frequently used analysis method was input combination (IC) method, which derived from type-1 fuzzy controller structure analysis (Siler and Ying.1989; Ying et al.1990; Ying 2006). This method applied zadeh AND operator and divided the input space into several regions, the number of regions was decided by parameters of primary membership function and each region shared the same mathematical expression.
Du and Ying. (2010) analyzed a class of Mamdani interval type-2 fuzzy-PI/ PD controllers structure based on zadeh AND operator and average defuzzifier method. Ni and Tan. (2012) presented the analytical structure of a class of Mamdani interval type-2 fuzzy-PI/PD controllers that had symmetrical rule base and symmetrical consequent sets based on Zadeh AND operator and KM type reduction. El-Nagar and El-Bardini in 2014 analyzed a class of Mamdani interval type-2 fuzzy-PID controllers structure based on zadeh AND operator and a new type reduction method. Aliasghary et al. (2015) obtained the input–output relations of interval type-2 fuzzy logic systems based on Zadeh AND operator and NT type reduction for diamond-shaped primary type-2 membership functions. Raj and Mohan. (2020)described a T-S interval type-2 fuzzy-PI/PD controller structure analysis method using Zadeh AND operator and KM type reduction. Both primary membership functions in related articles were symmetrical. Zhou et al. in(2013,2017,2019) extended the primary membership function of interval type-2 fuzzy controller and obtained a more general Mamdani and T-S fuzzy controller structure analysis methods, also used Zadeh AND operator and KM type reduction.
In most situations, the primary membership functions were all linear, Lei (2016) applied a nonlinear primary membership function and analyzed the structure of interval type-2 fuzzy PI/PD controller. Long (2016) analyzed structure of a class of interval type-2 fuzzy controller using product operator, and proved that such interval type-2 fuzzy controller was equivalent to the sum of two nonlinear PI (or PD) controllers. In these literatures, KM type reduction and Zadeh AND operator were commonly applied. KM type reduction algorithm was an iterative search process, which was time consuming and lacked close-form solution. Zadeh AND operator needed to divide the input space, the number of input space was different according to controller parameters, and the space dividing process was complicated.