Optimal necessary conditions for general SISO Mamdani fuzzy systems as function approximators within a given accuracy

In this paper, necessary conditions are investigated for a single input/single output (SISO) Mamdani fuzzy systems as function approximators of continuous functions within a given accuracy. Since general SISO Mamdani fuzzy systems are monotonic on subintervals, the optimal configuration of fuzzy systems is that the number of division points is at least the times of its monotonicity changes. Thus with the extreme of the desired continuous function, necessary conditions are obtained through generating intervals that contain division points and pruning redundant intervals. Furthermore, a dynamically constructive method is proposed to show the conditions are optimal. It has been shown that existing results concerning necessary conditions are only special cases of our results. Finally, simulation examples are given to illustrate the conclusions, the strength of the fuzzy systems as function approximators are analyzed.