A note on smooth approximation capabilities of fuzzy systems

Modeling and prediction in some systems requires the simultaneous approximation of mappings and their derivatives to a certain finite order. In this paper, universal approximation capabilities of fuzzy systems are extended to this situation, by showing the denseness of some general classes of fuzzy models in appropriate function spaces where distance between functions is defined in terms of their derivatives. Requirements are generally very mild, and are often limited to imposing sufficient differentiability conditions on the classes of fuzzy systems to be used. The cases covered by this paper include additive fuzzy systems with Gaussian membership functions and general additive fuzzy models with realistic membership functions. Some potential application fields are considered, including examples from economics and time series prediction.