Interval Type-2 Fuzzy PID Controller: Analytical Structures and Stability Analysis

In this paper, we derive the analytical structure of a class of the interval type-2 fuzzy proportional-integral-derivative (IT2F-PID) controller. In the proposed IT2F-PID controller structure, a rule-based Mamdani-type-2 fuzzy system is used to select the gains of the classical PID controller. The mathematical input-output relationships for the IT2F-PID controller are derived by dividing the input space and identifying the input-output relationship for each region. The IT2F-PID controller uses the following identical elements: two interval T2 triangular input fuzzy sets for each of the two input variables, two interval triangular output fuzzy sets for each output, the Mamdani interval type-2 fuzzy rule based, a Zadeh AND T-norm, a Lukasiewicz OR T-conorm and a new method for type-reduction that we propose, which called simplified type-reduction method. This new method of type-reduction reduces the computational complexity of the output processing for interval type-2 fuzzy logic system. We prove that the proposed IT2F-PID controller is a nonlinear PID with variable gains changing as the input variables values vary. Moreover, the sufficient conditions for the bounded-input bounded-output stability of the IT2F-PID control system is established using the well-known small gain theorem. To show the robustness of the proposed type-reduction method, a comparison is carried with the other types of type-reduction methods.