Polynomial approximations and convergence rate of Mamdani fuzzzy controllers

The analytical expression of a fuzzy controller has been studied when the output fuzzy membership functions are equally spaced and triangular or trapezoidal. The cases, when a fuzzy set of the output is extreme - with respect to the universe of discourse - or not extreme have been considered. When considering extreme sets, the controller can be approximated with a good precision by a second-degree polynomial and, in the contrary, by a first-degree polynomial. The modelling of the output by a second-degree multivariable polynomial has been shown to affect strongly the convergence rate, which is enlightened by an example for which the closed-loop stability is not exponential.