Avoiding exponential parameter growth in fuzzy systems

Abstract

For standard fuzzy systems where the input membership functions are defined on a grid on the input space, and all possible combinations of rules are used, there is an exponential growth in the number of parameters of the fuzzy system as the number of input dimensions increases. This "curse of dimensionality" effect leads to problems with design of fuzzy controllers (e.g., how to tune all these parameters), training of fuzzy estimators (e.g., complexity of a gradient algorithm for training, and problems with "over parameterization" that lead to poor convergence properties), and with computational complexity in the implementation for practical problems. We introduce a fuzzy system whose number of parameters grows linearly depending upon the number of inputs, even though it is constructed by using all possible combinations of the membership functions in defining the rules. We prove that this fuzzy system is equivalent to the standard fuzzy system as long as its parameters are specified in a certain way. Then, we show that it still holds the universal approximator property by using the Stone-Welerstrass theorem. Finally, we illustrate the performance of the fuzzy system via an application.