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Fuzzy Composition

 

Once the membership functions have been described and the rules for a FIS generated then some method for combining the rules has to be found. Zadeh [36] extends the modus ponens of traditional logic and introduces the compositional rule of inference or the generalized modus ponens:

.5 true in

Ant1: If x is A then y is B

Ant2: x is



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Cons: y is

where x and y are objects and are fuzzy concepts represented by fuzzy sets. Note that this reduces to modus ponens when and . Mizumoto and Zimmerman [24] also describe generalized modus tollens:

.5 true in

Ant1: If x is A then y is B

Ant2: y is



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Cons: x is

which, when and , reduces to modus tollens. There are various fuzzy relations that allow for the handling of ``If x is A then y is B''. In [24] the many proposed approaches are all considered with a view to their suitability for generalised modus ponens and generalised modus tollens. This very detailed piece of theoretical comparison and the similar work by Lee [20], although interesting in themselves, do not indicate how a FIS developer might choose an implication function.

In practice the method commonly adopted is known as the min-max method [5]. Given a set of fuzzy rules of the form described in equation 13 the process is as follows:

  1. For each of the antecedents find the minimum of the membership function for the input data. Apply this to the consequent.

  2. For all rules construct a fuzzy set that is a truncated set using the maximum of the membership values obtained.
Suppose there are two fuzzy rules IF and then IF and then where x and y are inputs and A1, A2, B1, B2, C1 and C2 are fuzzy sets. Fig. 4 shows how this method works.

  
Figure 4: The min-max method

This method is easy to compute and whilst other approaches are available and their theoretical properties are well known there is little reported in the literature on how a FIS developer may choose a composition method.



next up previous contents
Next: Defuzzification Up: No Title Previous: Fuzzy Rule Generation



Bob John
Fri Oct 25 14:41:29 BST 1996