Derivation and Analysis of the Analytical Structures of the Interval Type-2 Fuzzy-PI and PD Controllers

Research results on type-2 (T2) fuzzy control have started to emerge in the literature over the past several years. None of these results, however, are concerned with the explicit input-output mathematical structure of a T2 fuzzy controller. As the literature on type-1 (T1) fuzzy control has demonstrated, revealing such structure information is important as it will deepen our precise understanding of how and why T2 fuzzy controllers function in the context of control theory and lay a foundation for more rigorous system analysis and design. In this paper, we derive the mathematical structure of two Mamdani interval T2 fuzzy-proportional-integral (PI) controllers that use the following identical elements: two interval T2 triangular input fuzzy sets for each of the two input variables, four singleton T1 output fuzzy sets, a Zadeh and operator, and the center-of-sets type reducer. One controller employs the popular centroid defuzzifier, while the other employs a new defuzzifier that we propose, which is called the average defuzzifier. The advantages of using the latter defuzzifier are given, which include the fact that the derivation method originally developed by us in previous papers for the T1 fuzzy controllers can be directly adopted for the T2 controller, and the results are general with respect to the design parameters. This is not the case for the other T2 controller, for which we have developed a novel derivation approach partially depending on numerical computations. Our derivation results prove explicitly both controllers to be nonlinear PI controllers with variable gains (i.e., the expressions are different). We analyze the gain-variation characteristics and extend these findings to the corresponding T2 fuzzy-proportional-derivative (PD) controllers. Our new results are consistent with the relevant structure results on the T1 fuzzy-PI and PD controllers in the literature and contain them as special cases. We discuss how the new structure information can be utilized to design and tune the T2 controllers, even when the mathematical model of the system to be controlled is unknown. Neither derivation method is restrictive only to the T2 controllers in this paper-they are directly applicable to other T2 controllers with more complex configurations.