Derivation and characteristic investigation of a two-input, two-output interval type-2 fuzzy controller using product and operations

Interval type-2 (IT2) fuzzy logic, with the footprint of uncertainty (FOU) in membership functions, is very effective in dealing with disturbances and uncertainties of nonlinear systems. With this advantage, a two-input, two-output IT2 fuzzy controller with IT2 input and singleton type-1 fuzzy output sets is proposed for coupling systems. Firstly, the controller's input-output relationship is determined. Derivation results indicate that the controller is equivalent to the sum of a relay module and a local proportional-integral (PI) controller with variable gains. Next, analytical structure properties and gain variations are explored. It is demonstrated that the relay module is independent of the FOU, playing a dominant role in determining the control behavior of the IT2 controller. On the other hand, the control action of the PI controller diminishes when increasing the FOU and the number of fuzzy input sets. These properties contribute to the fast responsiveness of the controller and advantages in handling system uncertainty. Furthermore, the stability condition of this controller is established using the small gain theorem. Finally, both simulation and experiments are carried out to illustrate the applicability of the IT2 controller. The experimental results show that the controller's performance is embodied as the tracking errors of the two-linkage axes equal to +/- 4 mu m, while the repeated positioning accuracy values are within +/- 3 mu m.