Steepest Descent Adaptation of Min-Max Fuzzy If-Then Rules

A new kchniqae for adaptation of fuzzy membership functions in a fuzzy inference system is proposed. T h e technique relics upon the isolation of the specific membership iunction t ha t con-tribu red to t h e final decision, followed by the updating of this function's parameters using steepest dexen t. The e m r measure ubed is thus back propagated from output to input, through the min and mur opera tom used during the inference stage. This is feasible because the operations of min and max arc continuous differentiable functions and therefore can be placed in a chain of partial derivatives for stcepest descent backpropagation adaptation. More interestingly, the partials of min and max (or any other order statistic, for that matter) act as 'pointers ' with the result that only the function t ha t gave rise t o the min or max is adapted; the others are not. T o illustrate, let a = m a x p l, A,-.., @#I. Then acr/aa, = 1 when 8, is the maximum and is otherwise zero. We apply this property to t h e fine tuning of membership functions of fuzzy min-max decision processes rad i l lustrate with ra estimation example.