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2 .Popular defuzzification methods • Under the defuzzification process a fuzzy output is converted to a crisp number. – Two of the more popular techniques are the MAXIMUM and the CENTROID. – With MAXIMUM one selects the maximum value of the fuzzy output as the crisp value. – There are several variations to the MAXIMUM theme. • One such variation is the AVERAGE-OF-MAXIMA which was used in the simple example and gave us a crisp value of 0.84. • With the CENTROID method, the center of gravity of the fuzzy output gives the crisp value. • Using PRODUCT inferencing and the SUM combination the CENTROID results in a crisp value of 0.56.
3 . Calculating the output of a rule
5 . Ways of Combining Fuzzy Logic Output Values There are four different techniques to combine the fuzzy logic rules output values. They are: • Maximizer • Average • Centroid • Singleton Observe that we no longer operate here in fuzzy interval [0, 1] Temperature not fuzzy value
6 . • With MAXIMU M one selects the maximum value of the fuzzy output as the crisp value. • With the CENTROID method, the center of gravity of the fuzzy output gives the crisp value.
7 .Another approach to Calculating the output of a rule • Rules have promises which are usually combined together by the connective and. • The output is calculated by taking the degree of membership of the lesser of all premises as the value of the combination and truncating the output fuzzy set at that level.
8 . Calculating the output of a rule (cont) Output Premise-1 0.45 Truncated at 0.45 0.2 Premise-2 0.2 • Rules have promises which are usually combined together by the connective and. • The output is calculated by taking the degree of membership of the lesser of all premises as the value of the combination and truncating the output fuzzy set at that level.
9 . Combining rule outputs • When all rules have been evaluated a single fuzzy set is calculated by combining all outputs. • The combination involves the connective or. • The single output is calculated by taking the maximum of their respective output fuzzy sets grades at each point along the horizontal axis.
10 . Combining rule outputs Output-1 Combined Output 0.2 0.8 Output-2 Think about the set of fuzzy rules as a network with fuzzy literals and operators MIN, MAX, etc. In primary inputs you have Shifted fuzzification, in primary outputs you have horizontally defuzzification. The network analogy is very useful in your thinking about fuzzy logic, mv logic, or any other kind of data structure to store knowledge. Do in class a complete example of designing a fuzzy logic network for a simple robot control with about 6 rules. Discuss various operators and fuzzifiers, defuzzifiers.
11 . Defuzzification The input for the defuzzification process is the aggregate output fuzzy set and the output is a single number. • The most popular one is the centroid technique. It finds the point where a vertical line would slice the aggregate set into two equal masses. Mathematically this centre of gravity (COG) can be expressed as:
12 .2/18 and Soft ng 9 Centroid defuzzification method finds a point representing the centre of gravity of the fuzzy set, A, on the interval, ab. A reasonable estimate can be obtained by calculating it over a sample of points.1.0 0.8 0.6 A 0.4 0.2 a b 0.0 X 150 160 170 180 190 200 210
13 .3/18 and Soft ng 9 Centre of gravity (COG): (0 10 20) 0.1 (30 40 50 60) 0.2 (70 80 90 100 ) 0.5 COG 67.4 0 .1 0.1 0.1 0.2 0 .2 0.2 0.2 0 .5 0 .5 0 .5 0.5 Degreeof Membership 1.0 0.8 0.6 0.4 0.2 0.0 0 10 20 30 40 50 60 70 80 90 100 67.4 Z
14 .Centroid Method
16 .Weighted Average
17 .Center of Sums
18 .Center of Largest Area
19 .First of Maxima, Last of Maxima
27 .What are the criteria for defuzzification?
28 .Criteria for Defuzzification
29 .Criteria for Defuzzification