Question

For a four-rule one-input one-output Tsukamoto fuzzy system, let X be the fuzzy set of the input and Y be the fuzzy set of the output of the fuzzy system system, and the four rules of the fuzzy system are defined by

Rule 1: If X is very dark then y1=min{10,max(2∗ w1,0)}
Rule 2: If X is dark then y2=min{10,max(2 ∗(1−w2)+4,0)}
Rule 3: If X is bright then y3=min{10,max(3∗ w3+7,0)}
Rule 4: If X is very bright then y4=min{10,max(3∗(1−w4)+6,0)}
Given the firing strengths of the four rules are w1=0.9,w2=0.8,w3=0.9,w4= 0.6 , and assuming weighted average is used for defuzzification, calculate the defuzzified output of the Tsukamoto fuzzy system using weighted average as Your Answer (rounded to 4 decimal places)

Answer 1

To calculate the defuzzified output of the Tsukamoto fuzzy system using weighted average, follow the steps: calculate the weighted sum of the output values, calculate the sum of the firing strengths, divide the weighted sum by the sum of the firing strengths. The defuzzified output for the given fuzzy system is 8.7465.

Step-by-step explanation:

To calculate the defuzzified output of the Tsukamoto fuzzy system using weighted average, we need to follow these steps:

1. Calculate the weighted sum of the output values of each rule by multiplying the firing strength of each rule with its corresponding output value.
2. Calculate the sum of the firing strengths.
3. Divide the weighted sum by the sum of the firing strengths to obtain the defuzzified output.

In this case, the weighted sum of the output values is:

y1 = min{10, max(2 * 0.9, 0)} = 10

y2 = min{10, max(2 * (1 - 0.8) + 4, 0)} = 8

y3 = min{10, max(3 * 0.9 + 7, 0)} = 10

y4 = min{10, max(3 * (1 - 0.6) + 6, 0)} = 8.4

The sum of the firing strengths is 0.9 + 0.8 + 0.9 + 0.6 = 3.2.

The defuzzified output is then:

output = (y1 * w1 + y2 * w2 + y3 * w3 + y4 * w4) / (w1 + w2 + w3 + w4) = (10 * 0.9 + 8 * 0.8 + 10 * 0.9 + 8.4 * 0.6) / 3.2 = 8.7465 (rounded to 4 decimal places).