On Approximation Properties of Smooth Fuzzy Models

This paper addresses the approximation properties of the smooth fuzzy models. It is widely recognized that the fuzzy models can approximate a nonlinear function to any degree of accuracy in a convex compact region. However, in many applications, it is desirable to go beyond that and acquire a model to approximate the nonlinear function on a smooth surface to gain better performance and stability properties. Especially in the region around the steady states, when both error and change in error are approaching zero, it is much desired to avoid abrupt changes and discontinuity in the approximation of the input–output mapping. This problem has been remedied in our approach by application of the smooth compositions in the fuzzy modeling scheme. In the fuzzy decomposition stage of fuzzy modeling, we have discretized the parameters and then calculated the result through partitioning them into a dense grid. This could enable us to present the formulations by convolution and Fourier transformation of the parameters and then obtain the approximation properties by studying the structural properties of the Fourier transformation and convolution of the parameters. We could show that, irrespective to the shape of the membership function, one can approximate the dynamics and derivative of the continuous systems together, using the smooth fuzzy structure. The results of the paper have been tested and evaluated on a discrete event system in the hybrid and switched systems framework.