rule based fuzzy control system and function approximation

I am trying to implement a function approximator (aggregation) using a rule-based fuzzy control system. So as to simplify my implementation (and have better understanding) I am trying to approximate y=x^2 (the simplest non-linear function). As far as i understand i have to map my input (e.g. uniform samples over [-1,1]) to f开发者_Python百科uzzy sets (fuzzyfication) and then use a defuzzyfication method to take crisp values. Is there any simple explanation of this procedure because fuzzy control system literature is a bit mess.

This is sort of a broad question, but I'll give it a go since it has sat unanswered for so long.

First, I believe you need to refine your objective (at least as it stated here). I would hesitate to use the term "function approximation" in this context. If I follow your question correctly, the objective is map a non-linear function into another domain via fuzzy methods.

To do so, you first need to define your fuzzy set membership functions. (This link is a good example of the process.) Without additional information, the I recommend the triangular function due to its ease in implementation. The number of fuzzy sets, their placement and width (or support), and degree of overlap is application specific. You've indicated that your input domain is [-1,1], so you might find that three fuzzy sets does the trick, i.e Negative, Zero, and Positive.

From there, you need to craft a set of rules, i.e. if x is Negative then...

With rules in place, you can then define the defuzzification process. In short, this step weights the activation of each rule according to the needs of the application.

I don't believe I can contribute more fully until the output is better defined. You state "use a defuzzyfication method to take crisp values." - what does this set of crisp values mean? What is the range? Etc. Furthermore, you'll get more a response if you can identify the areas in which you are stuck (i.e. more specific questions).