Fusion of Neural Networks, Fuzzy Systems and Genetic Algorithms: Industrial Applications Fusion of Neural Networks, Fuzzy Systems and Genetic Algorithms: Industrial Applications
by Lakhmi C. Jain; N.M. Martin
CRC Press, CRC Press LLC
ISBN: 0849398045   Pub Date: 11/01/98
  

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Definition 4 (Fuzzy Switching Function)   A fuzzy switching function is a mapping of [0, 1] → [0, 1] described by a fuzzy formula.

Definition 5 (Clause)   A clause is a disjunctive combination of at least two variables. (Combination with +)

Definition 6 (Phrase)   A phrase is a conjunctive combination of at least two variables. (Combination with *)

Definition 7 (Disjunctive Normal Form DNF)   A fuzzy formula is in DNF if S = P1 + P2 + . . . + Pm and every Pi is a phrase.

Definition 8 (Conjunctive Normal Form CNF)   A fuzzy formula is in CNF if S = C1 + CP2 + . . . + Cm and every Ci is a clause.

Using these definitions above it is now possible to describe every fuzzy formula in DNF or CNF.

To define whether a rule and therefore a fuzzy formula (fuzzy switching function) is valid, the terms fuzzy-valid and fuzzy-inconsistent have to be defined.

Definition 9 (Fuzzy-Consistent)   A fuzzy formula F(x) is fuzzy-consistent if the output of the formula is ≥ 0.5 ∀x.

Definition 10 (Fuzzy-Inconsistent)   A fuzzy formula F(x) is fuzzy-inconsistent if it is not consistent.

These definitions ensure that every rule of the knowledge base can uniquely be transformed into fuzzy switching functions. With these fuzzy switching functions, the algorithm can be described.

Design of the Algorithm

It is assumed that certain initial rules have been generated. That means that for each value of the residual a number of fuzzy sets are defined. From this definition of the fuzzy sets, the number of rules are determined. The key question is now [16]: is it possible to distinguish between all defined faults using the given rules ? To answer this question it is necessary to prove whether or not a distinction between the faults can be made. If all premises of two fault descriptions have the same fuzzy values, a distinction is not possible. Two faults are distinguishable if they have at least one different definition in the premise of the rule. To illustrate this fact, the following example is given:

If Res1 is positive and Res2 is positive then f1
If Res1 is positive and Res2 is normal then f2

For these two rules, a unique distinction between f1 and f2 is possible. In addition, it can be mentioned that f1 and f2 cannot occur at the same time because Res2 can just be positive or normal.

To illustrate two distinguishable faults, which can occur at the same time, the following example is given:

If Res1 is positive and Res2 is positive then f1
If Res1 is positive and Res3 is negative then f2

These rules allow the unique decision whether f1 or f2 has occurred, but both faults can occur at the same time, because f1 and f2 use two different residuals for the second premise, in this case Res2 and Res3 while the first premise uses Res1 identically for both faults. As a consequence, the rules have to be modified so that the fuzzy set positive in residual 1 is divided into two different fuzzy sets, f1 and f2. Now both faults are uniquely distinguishable and cannot occur at the same time.

From these examples it can be seen that the rule base has to be checked for such inconsistencies. If there are certain rules which lead to inconsistent fault behavior, these fuzzy sets have to be subdivided into (at least) two fuzzy sets. The task of the algorithm is now to check all faults against all others. For the first fault, this can be described as follows:

f1 occurs, but none of the other faults

This has to be defined for all faults and leads to the following description:

fk occurs, but none of the other faults

To handle this with the algorithm, each rule has to be transformed into a fuzzy switching function. If the result of the fuzzy switching function is ≥ 0.5 then the rule is fuzzy-consistent. If the result is < 0.5 the rule is fuzzy-inconsistent. That means that for fuzzy-inconsistent rules the compatibility degree is < 0.5 for all x. This implies that there exists at least one phrase Pi in the fuzzy switching function with the following structure:

The task can now be specified as follows:

Find phrases with the form shown in Equation (5)


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