Fuzzy systems are universal approximators for a smooth function and its derivatives

One of the reasons why fuzzy methodology is successful is that fuzzy system s are universal approximators, i.e., we can approximate an arbitrary contin uous function within any given accuracy by a fuzzy system. In some practica l applications (e.g., in control), it is desirable to approximate not only the original function, but also its derivatives (so that, e.g., a fuzzy con trol approximating a smooth control will also be smooth). In our paper, we show that for any given accuracy, we can approximate an arbitrary smooth fu nction by a fuzzy system so that not only the function is approximated with in this accuracy, but its derivatives are approximated as well. In other wo rds, we prove that fuzzy systems are universal approximators for smooth fun ctions and their derivatives. (C) 2000 John Wiley & Sons, Inc.