# Defuzzification Methods (PID and Fuzzy Logic Toolkit)

LabVIEW 2010 PID and Fuzzy Logic Toolkit Help

Edition Date: June 2010

Part Number: 370401G-01

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Defuzzification is the process of converting the degrees of membership of output linguistic variables into numerical values. When you use the FL New Fuzzy System VI or the Fuzzy System Designer, you can select from the following defuzzification methods.

 Note  A fuzzy logic controller uses an implication method to scale the output membership functions based on the corresponding rule weights before performing defuzzification. The following examples use the Minimum implication method, in which the fuzzy logic controller truncates the output membership functions at the value of the corresponding rule weights.

## Center of Area

In the Center of Area (CoA) defuzzification method, the fuzzy logic controller first calculates the area under the scaled membership functions and within the range of the output variable. The fuzzy logic controller then uses the following equation to calculate the geometric center of this area.

where CoA is the center of area, x is the value of the linguistic variable, and xmin and xmax represent the range of the linguistic variable.

The following figure illustrates the CoA defuzzification method:

In the previous figure, μ is the degree of membership, and the shaded portion of the graph represents the area under the scaled membership functions.

## Modified Center of Area

The modified Center of Area defuzzification method is similar to the Center of Area defuzzification method. However, the fuzzy logic controller considers the full area under the scaled membership functions, even if this area extends beyond the range of the output variable. The fuzzy logic controller uses the following equation to calculate the geometric center of the full area under the scaled membership functions.

where mCoA is the modified center of area.

The interval of integration is between the minimum membership function value and the maximum membership function value. Note that this interval might extend beyond the range of the output variable.

The following figure illustrates the modified CoA defuzzification method:

## Center of Sums

In the Center of Sums (CoS) defuzzification method, the fuzzy logic controller first calculates the geometric center of area for each membership function, as the following figure illustrates:

The fuzzy logic controller then uses the following equation to calculate a weighted average of the geometric center of area for all membership functions.

where CoAn is the geometric center of area of the scaled membership function n and arean is the area of the scaled membership function n.

## Center of Maximum

In the Center of Maximum (CoM) defuzzification method, the fuzzy logic controller first determines the typical numerical value for each scaled membership function, as the following figure illustrates. The typical numerical value is the mean of the numerical values corresponding to the degree of membership at which the membership function was scaled.

The fuzzy logic controller then uses the following equation to calculate a weighted average of the typical values.

where xn is the typical numerical value for the scaled membership function n and μn is the degree of membership at which membership function n was scaled.

## Mean of Maximum

In the Mean of Maximum (MoM) defuzzification method, the fuzzy logic controller first identifies the scaled membership function with the greatest degree of membership. The fuzzy logic controller then determines the typical numerical value for that membership function. The typical numerical value is the mean of the numerical values corresponding to the degree of membership at which the membership function was scaled.

The following figure illustrates the MoM defuzzification method:

Refer to the LabVIEW PID and Fuzzy Logic Toolkit User Manual for more information about defuzzification methods.

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