Defuzzification

Center-of-Gravity (C-o-G) The C-o-G method (centroid defuzzification) is often referred to as the Center-of-Area method because it computes the centroid of the composite area representing the output fuzzy term. 

Center-of-Maximum (C-o-M) In the C-o-M method, only the peaks of the membership functions are used. The defuzzified crisp compromise value is determined by finding the place where the weights are balanced. Thus, the areas of the membership functions play no role and only the maxima (singleton memberships) are used. The crisp output is computed as a weighted mean of the term membership maxima, weighted by the inference results. 

Mean-of-Maximum (M-o-M) The M-o-M is used only in some cases where the C-o-M approach does not work. This occurs whenever the maxima of the membership functions are not unique and the question is as to which one of the equal choices one should take. 

For the above case of output membership function, the centroid defuzzification method is given by the expression 

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Defuzzification is the procedure for mapping from a set of inferred fuzzy control signals contained within a fuzzy output window to a non-fuzzy (crisp) control signal. The centre of area method is the most well known defuzzification technique, which in linguistic terms can be expressed as... 

Defuzzification of set memberships to give a predicted reaction rate. 

The most popular recipe for defuzzification, although not the fastest, is the centroid method. This finds the vertical line, which would divide the aggregated set determined in the previous step, into two equal portions. The center of gravity (cog) is defined by ... 

Essentially, the neurofuzzy architecture is a neural network with two additional layers for fuzzification and defuzzification. The fuzzification and input weighting are illustrated graphically in Fig. 9, adapted from the thesis of Bossley. It can be seen that there are similarities with the RBF network, except now the radial functions are replaced by the multivariate membership functions. 

Hellendoorn, H. and Thomas, C., Defuzzification in fuzzy controllers, J. Intel. Fuzzy Syst. 1 109-123,1993. 

Figure 57.17 shows the basic configuration of an FLC system, comprising four components a fuzzification interface, a knowledge base, a decision-making logic (control algorithm), and a defuzzification interface. 

The resulting set must be defuzzified or resolved to a single number (crisp value). Some defuzzification methods are center of area (CoA), bisector, middle of maximum (MOM), largest of maximum, and smallest of maximum (Teti and Kumara 1997). Perhaps the most popular defuzzification method is the center of area (CoA), which returns the center of area under the curve. 

In Fig. 2, the fuzzy inference flow from linguistic variable fuzzification to defuzzification of the aggregate output is shown it proceeds up from the inputs in the lower left, then across each row, or rule, and then down the rule outputs to end in the lower right. 

The last step in a fiizzy logic system is defuzzification. As the name suggests, defuzzification is the opposite of fuzzification, which produces crisp output f for a fuzzy logic system from the aggregated output of fuzzy set B. A number of defuzzifiers have been developed the most popular is the centroid defuzzifier, which finds a vertical line and divides an aggregated set into two equal portions. Mathematically the center of gravity (COG) can be defined by ... 

The rule viewer shown in Fig. 13 consists of the system s inputs and output column. The first and second column is the inputs which are maturity level and size while third column is the quality level output (defuzzification output). Based on the defuzzification results from the rule viewer in Fig. 13 where the maturity (= 0.66) is in matured set with membership grade 1 and size (= 0.5) is in big and medium set both with 0.5 membership grade. The defuzzification value of quality is calculated by using centroid method. Then the classification of fruit is determined based on crisp logic given in Table 3. 

Notice that the fuzzy probability distribution function presented in Fig. 4 represents the possibility function related to the probabilities of operator success and failure. Nevertheless, these possibilities functions are in the linguistic domain and must be translated to the real domain. To achieve this, the defuzzification process presented by Chen and KJien (1997) has to be used. 

Fuzzy control is reahzed within 3 processes fuzzification, inference and defuzzification. The fuzzification process describes variables by means of input values. Thus a fuzzy set is represented by a membership function. The membership function is constructed based on the variable s input values (laanineh Maijohann 1996). Accordingly every input value obtains a membership degree between one and zero. If the input value is clearly assigned to the description of the variable, it receives a membership degree one. It is a fiiU membership to a related fuzzy set. The membership degree zero means that an input value does not belong to the fuzzy set. Membership values between zero and one indicate a partial membership of an input value to the certain fuzzy set. 

Table 9. Final results by means of different defuzzification operators.

Type-2 fuzzy logic systems are rule-based systems that are similar to type-1 fuzzy logic systems in terms of the structure and components but a type-2 fuzzy logic system has an extra output process component which is called the type-reducer before defuzzification as shown in Fig. 5.3. The type-reducer reduces outputs type-2 fuzzy sets to type-1 fuzzy sets and then the defuzzifier reduces them to crisp outputs. The components of a type-2 Mamdani fuzzy logic system are [26] ... 

Defuzzifier Defuzzifier maps the reduced output type-1 fuzzy sets that have been reduced by type-reducer into crisp values exactly as the case of defuzzification in type-1 fuzzy logic systems. 

After the defuzzification, the weight of the point of the FM under test (ITpiviy) is known, and the current coordinates (Cp) can be calculated using (13.3) ... 

The rule set of the FLC contain 9 rules, which govern the input-output relationship of the FLC and this adopts the Mamdani-style inference engine. We use the center of gravity method to realize defuzzification procedure. In Table 5.3, we present the rule set whose format is established as follows ... 

M, and B are abbreviations for small, medium, and big, respectively. We used the phase clusterization method reported by Konstantin (68) and Kishimoto (66). First, the phase of culture growth was determined from the pattern of the state variable classified by each membe ip function. Second, the probability that the measured state fails into the phase was calculated. Third, the value of the operative variable was calculated for each situa tion. Sugeno s third inferraice method (69) was used for defuzzification. 

Taheri Shahraiyni (2007) developed new heuristic search, fuzzification and defuzzification methods for ALM algorithm. In the next sections of this chapter, ALM algorithm with these modifications is explained and the ALM abilities and applications are illustrated. 

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