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respond to the words 'low' and 'high' - how do we supply the output? That is where Defuzzification finishes the process.


From the previous step we have 'x' and 'y';
the input value belongs to low_light set to the degree of 'x' and to the high_light set to the degree of 'y'. These x and y values must be mapped on to the voltage fuzzy set - on the low and high voltage fuzzy linguistic sets. The output membership function values are:

m low_voltage(u) = y

m high_voltage(u) = x

Now we use the Weighted Average Method to
arrive at the crisp value. (A Weighted Average Method starts with a sequence of function values and a matching sequence of real numbers, called weights, where the sum total of all weights is one. The sum of all products of the weights times the function values is defined as the function values weighted average.) Please note that I choose this method for simplicity but there are other sophisticated methods that are mathematically rigorous (See Ross - 1995).

Z* = mC(`Z ) .`Z / mC(`Z )
(Ross, 1995)

Z* = x . l + y . k / (x + y)

So the Z* will be the crisp value given as the
output to the fuzzy energy saving bulb.

An Example

Assuming the fuzzy sets defined above, we try
the system with a crisp value of 11. First, convert the fuzzy value to a crisp value.
Since u (11) falls in between 'b' i.e. Since u(11)
falls in between 'b' i.e. 10 and 'c' i.e. 15 (see Fig 4) for the low_light fuzzy set we use:

(c-u) / (c-b) for b<= u <= c

(15-11) / (15-10) = 0.8

Since u (11) falls in between 'a' i.e. 10 and 'b'
i.e. 15 (see Fig 4) for the high_light fuzzy set we use:

(u-a) / (b-a) for a <= u <= b

11-10 / 15-10 = 0.2

Figure 4: Low Light Fuzzy Set
Figure 5: High Light Fuzzy Set
Figure 6: Defuzzification in the Example- mapping of the membership values ofthe light set onto the voltage set

Therefore, the input crisp value belongs to the low light and the high light sets with the membership 0.8 and 0.2 respectively. Consulting the Fuzzy Knowledge base, we realize that the voltages must be low and high. For the low light membership, the voltage must be high to the degree of 0.8. For the high light membership, the voltage must be low to the degree of 0.2. We project the membership values of the low and high light on to the low and high voltage sets (See Fig 6).

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16.4 2002

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