Enhanced approximation capabilities of the fuzzy systems using variable universes of discourse

Abstract

The fuzzy systems with variable universe of discourse (VUD) have received increasing attention due to their fine control performance. VUD fuzzy systems are ones in which universes of discourse can be online tuned by means of a set of nonlinear contraction–expansion factors. One advantage of the VUD fuzzy systems lies in that higher control accuracy can be obtained without the aid of integral elements. Another advantage of the VUD fuzzy systems is that the number of fuzzy rules can be reduced significantly for a given application, compared with the fuzzy system based on fixed universes of discourse. Utilizing the multivariable Taylor’s expansion formula, it is proven that the VUD fuzzy system possesses higher approximating power as the input variables approach the equilibrium point. The approximation error converges to zero when the input variables reach the equilibrium point. It is also proven that the same approximating power can be obtained even when two fuzzy sets are applied on any universe of discourse. In addition, a sufficient condition is given for the VUD fuzzy systems as approximators for any given accuracy. Two examples show that the approximating capability of the VUD fuzzy systems can significantly outperform the fuzzy systems with fixed universes of discourse and that the shape of membership functions can influence the approximation effect. This paper provides researchers with some explanations why the VUD fuzzy systems hold the two advantages mentioned above from the standpoint of the functions approximation theories.