In this paper, we derive the analytical structure of the interval type-2 fuzzy proportional – integral – derivative (IT2F-PID) controller, which consists of a parallel combination of the IT2F-PD controller and the IT2F-PI controller. The IT2F-PID controller uses the following identical elements: two interval T2 triangular input fuzzy sets for each of the two input variables, three interval triangular output fuzzy sets, 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 cost of the output processing for the interval type-2 fuzzy logic controller (IT2-FLC). We relate the resulting structure to conventional PID control theory and 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 (BIBO) stability of the IT2F-PID control system have established using the well-known small gain theorem. The simulation results show that the IT2F-PID controller based on the proposed type-reduction method is able to improve the system performance compared with other type-reduction methods. 
Derivation and stability analysis of the analytical structures of the interval type-2 fuzzy PID controller