Defuzzification is the process of converting a fuzzy output (a fuzzy set) into a crisp, single value. In fuzzy logic systems, the output is usually represented as a fuzzy set, where each element has a degree of membership between 0 and 1. However, real-world systems require a specific, concrete value (a crisp output) for actions such as controlling a device or making a decision.
Defuzzification takes this fuzzy output and transforms it into a single crisp value, which can be used in practical applications.
Centroid Method (Center of Gravity): The most widely used method of defuzzification is the centroid method, also called the center of area or center of gravity. In this method, the crisp value is the "center" of the area under the curve of the fuzzy set. It is computed as the weighted average of all the possible values, with the membership degrees as weights.
The formula for the centroid is:
Crisp value=∫x⋅μ(x) dx∫μ(x) dx\text{Crisp value} = \frac{\int x \cdot \mu(x) \, dx}{\int \mu(x) \, dx}
where μ(x)\mu(x) is the membership function and xx is the variable representing the fuzzy set.
This method results in a value that represents the "center" of the fuzzy set's membership distribution.
Bisector Method: The bisector method finds the point where the area under the curve is split into two equal parts. In this case, the crisp value is the point at which half of the total area of the fuzzy set lies to the left, and the other half lies to the right.
Mean of Maximum (MOM): The mean of maximum method defuzzifies by taking the average of the values corresponding to the maximum membership degrees. In other words, the crisp value is the average of all values xx for which μ(x)\mu(x) is at its maximum.
Largest of Maximum (LOM): The largest of maximum method selects the largest value xx for which the membership function reaches its maximum. This method is often used when the fuzzy set represents a "priority" or "preference" scale and the largest value is desired.
Smallest of Maximum (SOM): The smallest of maximum method chooses the smallest value xx for which the membership function reaches its maximum. This method is less commonly used but can be applied in certain situations where lower values are prioritized.
Weighted Average Method: In the weighted average method, the crisp output is calculated as the weighted average of the possible outputs, with each output weighted by its degree of membership. This method is particularly useful when dealing with several fuzzy sets with different membership values.
The formula is:
Crisp value=∑xi⋅μ(xi)∑μ(xi)\text{Crisp value} = \frac{\sum x_i \cdot \mu(x_i)}{\sum \mu(x_i)}
where xix_i are the possible output values, and μ(xi)\mu(x_i) are their corresponding membership values.
Defuzzification is commonly used in fuzzy control systems, where it helps in making decisions based on fuzzy rules. For example, in a fuzzy controller for a temperature system, the rules might produce a fuzzy output like "very hot" or "slightly cold," but the system needs to output a crisp value, such as a temperature setting or a heater power level. Defuzzification provides the method to convert these fuzzy values into actionable outputs.
In summary, defuzzification is the final step in a fuzzy logic system that transforms fuzzy results into clear and actionable outputs, making it essential for practical applications of fuzzy logic.