Fine Tuning of Membership Functions for Fuzzy Neural Systems

This paper presents a new method for fine-tuning the Gaussian membership functions of a fuzzy neural network (FNN) to improve approximation accuracy. This method results in special shape membership functions without the convex property. We first recall that any continuous function can be represented by a linear combination of Gaussian functions with any standard deviation. Therefore, the Gaussian membership function in the second layer of the FNN can be replaced by several small Gaussian functions; the weighting vectors of this new network (called FNN5) can then be updated using the backpropagation algorithm. The proposed method can adapt proper membership functions for any nonlinear input/output mapping to achieve highly accurate approximation. Convergence analysis shows that the weighting vectors of the FNN5 eventually converge to the optimal values. Simulation results indicate that (a) this approach improves approximation accuracy, and (b) that the number of rules can be reduced for any given level of accuracy. For the purpose of illustrating the proposed method, the FNN5 is also applied to tune PI controllers such that gain and phase margins of the closed-loop system achieve the desired specifications.