Analysis of Footprint of Uncertainty of the IT2 Mamdani Controllers Using Different Type-Reducers

Appropriate Footprint of Uncertainties (FOUs) are beneficial to the performance of Interval Type-2 (IT2) fuzzy controller, revealing the effect of FOUs is a key problem. In our published work, as the FOUs increase, the IT2 Mamdani and TS fuzzy controllers, using KM or EKM type-reducer (TR), approach the constant and piecewise linear controllers, respectively, while they finally become constant and piecewise linear controllers. To verify the validation of the above results, when a different TR is used, in this study, the effects of other popular TRs (i.e., Nie-Tan, Wu-Mendel, Iterative Algorithm with Stop Condition) on output of IT2 Mamdani fuzzy controller, are explored. We proven that, (1) as the FOUs increase, irrespectively of the TRs used, the IT2 Mamdani fuzzy controllers approach constant controllers, (2) when all the FOUs are equal to 1 (i.e., at their maximum ), the fuzzy controllers using Nie-Tan and Iterative Algorithm with Stop Condition TR become constant controllers. The FOUs of the controllers using Wu-Mendel TR can be infinitely approaching 1 and cannot be equal to 1 (otherwise, the denominator of the TR output expression are equal to 0), hence when FOUs are infinitely approaching 1, the controller will approach the constant controller infinitely. These results imply regardless of which popular TR is used, the IT2 Mamdani fuzzy controller, when using larger FOUs, the fluctuation of the input variables have a limited impact on the output, the ability to deal with system uncertainties will deteriorate. Laboratory control experiments are provided to demonstrate these findings.