Defuzzifier

Again, a similar inferenee proeess oeeurs with x and x. Following defuzzifieation, a erisp eontrol foree F(t) is obtained. 

A defuzzifier is the opposite of a fuzzifier it maps the output sets into crisp numbers, which is essential if a fuzzy logic system is to be used to control an... 

FAMs map fuzzy sets (inputs) to fuzzy sets (outputs). The output fuzzy sets should be defuzzified into numerical values to control the plant. Due to the powerful ability of fuzzy sets in describing system linguistic and qualitative behavior and imprecise and/or uncertain information, many industrial process behavior and control laws can be modeled by fuzzy logic-based approaches. Fuzzy logic has been applied in a wide range of automated systems, including ... 

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. 

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 ... 

In addition to centroid defuzzifiers, maximum defiizzifiers and means of maxima defuzzifiers are also commonly used. 

Maximum Defuzzifier This defuzzifier chooses f as the point at which associated membership functions achieve their maximum values. 

Mean of Maxima Defuzzifier This defuzzifier examines fuzzy set B, determines values for which associated membership functions achieve their maximum values and computes the mean of these values as its output j . 

In the present case, c = 0 and d = 1, which delimitates the domain of probabilities. Using equation (15), one cans defuzzify the fuzzy probability distributions of success and error presented in Fig. 4. The results obtained are 17.5% for error probability and 8.,5% for success probability. Based on this result, which was considered by the Petrobras managers as very high unsuccessful chances, one decided to not proceed with optical monitoring. The reservoir control will be done by traditional electrical cable and sensors. 

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] ... 

Inference Engine Inference Engine maps input type-2 fuzzy sets into output type-2 fuzzy sets by applying the consequent part where this process of mapping from the antecedent part into the consequent part is interpreted as a type-2 fuzzy implication which needs computations of union and intersection of type-2 fuzzy sets. The inference engine in Mamdani system maps the input fuzzy sets into the output fuzzy sets then the defuzzifier converts them to crisp outputs. The rules in Mamdani model have fuzzy sets in both the antecedent part and the consequent part. For example, the /th rule in a Mamdani rule base can be described as follows ... 

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. 

The defuzzified output value has been created by using the MOM (Mean of Maximum) defuzzification method. 

Center of gravity. The defuzzified value of a fuzzy set C is its fuzzy centroid. 

Finally, Figure 4 shows the TE s FuBI, which defuzzified lower and upper bounds are 2.59E-3 and 1.65E-2 respectively. 

The last layer defuzzifies the computed values of y, using an average weighting procedure. A backpropagation training method can be employed to find the optimal values for the parameters of the membership functions and a least squares procedure for the linear parameters of the fuzzy mles, in such a way as to minimize the error between the calculated output and the measured output. 

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. 

Figure 6.18 shows the membership function of the riskiness of an event on an arbitrary scale, which would later be used to defuzzify the fuzzy conclusion and rank the risk according to a priority number. The membership function used is a triangular function. Unlike the trapezoidal function, the membership value of 1 in the triangular function is limited to (mly one value of the variable on the x-axis. 

The defuzzification process creates a single assessment from the fiizzy cmiclusion set expressing the risk associated with the event, so that corrective actions can be prioritised. Several defuzzification techniques have been developed (Runkler and Glesner (1993)). One conunon technique is the weighted mean of maximum method, which is illustrated here. This technique averages the points of maximum possibility of each fuzzy conclusion, wei ted by their degrees of truth. Hence, if the conclusion fixrm the risk evaluation phase is, for example, 0.5 Low, 0.1 Low and 0.5 Mod, the maximum value for each linguistic term is taken. This reduces the conclusion to 0.5 Low and 0.5 Mod to be defuzzified. 

From this result the priority for attention of this particular event can be numerically expressed as being 0.078. This method of defuzzification has been discussed in Chapter 6. Similarly all the potential failure modes identified in the FMEA can be analysed in this manner to produce a ranking such that the highest value of the defuzzified conclusion reflects the highest priority for attention. 

For the application of this matrix in an FMEA study, the value of x,(fc) represents the defuzzified crisp number describing each linguistic variable considered for the identilled failure modes. For example, consider three failure events. A, B and C, where the linguistic terms have been assigned for the three variables considered as seen in Table 7.5 and assume that the values in brackets represent the defuzzified value for the associated linguistic term. The information in Table 7.5 can be represented in a matrix form to reflect the comparative series. 

Table 7.11, give the results of the modified FMEA. These results are then defuzzified using the WMoM method to obtain a ranking as shown in Table 7.12. 

Using the fuzzy rule base generated in Table 7.11, Rule 7 will apply to the first event. This rule is interpreted to read as, if the probability of occurrence is Remote, severity is High and detectability is High, then priority for attention is 0.58 Low, 0.68 Fairly low . The conclusion 0.58 Low, 0.68 Fairly low can be defuzzified using the WMoM method to produce a crisp number as shown here ... 

The priority for attention for the first event can be represented numerically by 0.274. Similarly, all other events are analysed and the corresponding priorities for attention are obtained such that the higher the value of the defuzzified results, the higher the priority in the ranking series. From the analysis and the results presented in Table 7.12, the failure event with the highest priority is failure component - hydraulic, failure mode - system loss, with a defuzzified result of 5.353. The lowest in the series is identified to be failure component - shaft propeller, failure mode - propeller blade failure, with a defuzzified result of 0.055. 

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