Tuning Fuzzy Systems by Function Approximation

A prototype fuzzy system is quite easy to set up and modify with the techniques within fuzzy system theory when linguistic rules can be given for the desired behavior. Fine-tuning of such a system proves more difficult as well as building a system when no rules can be given. Because this tuning is a very specialistic job, during construction as well as in maintenance, ways were examined to do the tuning automatically. Function approximation is described as a general technique to tune parts of a fuzzy systems based on the desired input and output behavior. Two distinct function approximation techniques were taken into consideration: Techniques based on B-splines (analytically as well as numerically) and techniques based on sigmoids (only numerically, e.g. with the backpropagation procedure used in neural networks). Experiments were done to determine which technique gives the best results when inhomogeneously scattered data points with noise are used. The sigmoid technique seemed give the best approximation in areas were no data points were available (generalization). Some constraints could be given for the position and number of B-splines with respect to the distribution and the amount of the data respectively.