Design of the footprints of uncertainty for a class of typical interval type-2 fuzzy PI and PD controllers

The Footprint of Uncertainty (FOU) for an Interval type-2 (IT2) input fuzzy set is a key parameter of IT2 fuzzy PI and PD (FPI and FPD) controllers because it provides additional options to handle uncertainties. Because the effects of FOUs on IT2 FPI and FPD controller are still unclear, it is necessary to provide a guideline for designing the FOUs. In this study, from the input–output relationship of IT2 FPI and FPD controllers using typical and popular configuration, the variation trend of variable gains, in relation to the increase of FOUs, was analyzed. It is mathematically proven that variable gains of typical controllers decrease or not increase as the FOUs increase. These theoretical results can be extended to provide a guideline for designing FOUs of typical IT2 FPI and FPD controllers. Real-time response of linear voice coil motor is used to show the effectiveness of our results. •The effects of FOUs are analyzed for typical IT2 FPI and FPD controllers.•The analytical structure, around equilibrium point, is same as PI and PD controller.•The variable gains are decreasing or non-increasing as the FOUs increase.•A guideline for designing FOUs of the class of controllers is provide.