Robust stability analysis of PD type single input interval type-2 fuzzy control systems

In this paper, the robust stability of a PD type Single input Interval Type-2 Fuzzy Logic Controller (SIT2-FLC) structure will be examined via the well-known Popov criterion and Lyapunov´s direct method approach. Since a closed form formulation of the SIT2-FLC output is possible, the type-2 fuzzy functional mapping is analyzed in a two dimensional domain. Thus, mathematical derivations are presented to show that type-2 fuzzy functional mapping is a symmetrical function and always sector bounded. Consequently, the type-2 fuzzy system can be transformed into a perturbed Lur´e system to examine its robust stability. It has been proven that the stability of the PD type SIT2-FLC system is guaranteed with the aids of the Popov-Lyapunov method. A robustness measure of the type-2 fuzzy control system is also presented to give the bound of allowable uncertainties/ nonlinearities of the control system. Moreover, if this bound is known, the exact region of stability of the type-2 fuzzy system can be found since SIT2-FLC output can be presented in a closed form. An illustrate example is presented to demonstrate the robust stability analysis of the PD type SIT2-FLC system.