Sum normal optimization of fuzzy membership functions

Given a. fuzzy logic system, how can we determine the membership functions that will result in the best performance? H we constrain the membership functions to a certain shape (e.g., t riangles or trapezoids) then each membership function can be parameteri'l:ed by a small number of variables and the membership optimization problem can be reduced to a parameter optimization problem. This is the approach that is typically taken, hut it results in membership functions that are not (in general) sum normal. That is, the resulting membership function values do not add up to one at each point in the domain. This optimization approach is modified in this paper so that the resultin