CHAPTER 8

ESTIMATION OF STATIC NONLINEAR FUNCTIONS FROM DATA

In Chapters 3 and 6, we saw that some fuzzy systems have nonlinear memoryless input–output characteristic functions. The nonlinear characteristic function of these fuzzy systems is a result of the rules, T-norm, membership functions, and defuzzification method chosen by the designer. In these systems, the input and output membership functions are fixed a priori.

It is also possible, given a particular nonlinear function, to find a fuzzy system whose input–output characteristic matches it. In the fuzzy system, the output and in some cases input membership functions are not fixed a priori, but are adjusted so that the input–output characteristic most closely matches the nonlinear function in some sense. The determination of a fuzzy system to approximate a given nonlinear function is done utilizing well-known results from estimation theory. The estimation methods introduced in this chapter are the least-squares and gradient approaches. These are introduced because they have direct applications to control.

It will be seen in Chapter 9 that the ability to model static nonlinear functions as fuzzy systems makes it possible to model dynamic nonlinear systems as fuzzy systems. The motivation for modeling a dynamic nonlinear system as a fuzzy system is that with such a model, the parallel distributed control methods of the previous chapter can be employed to control the system. This enables a very powerful type of control known ...