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x Step 1: compute the degree of fulfilment of each rule for any given input
set (crisp) by
l
EP
P
"
P
,
x
x
x
l
l
l
n
G
1
G
2
G
n
1
2
where is the min or product operator
x Step 2: compute the each rule consequent set as given by
l
l
E
l
c
F
F
1
1
x Step 3: aggregate all consequent fuzzy sets as shown by
M
F
c
l
1
* "*
2
M
*
c
c
c
c
FFF
F
1
1
1
1
aggr
l
1
x Step 4: defuzzify the aggregated fuzzy set
F c using the COG method.
aggr
The defuzzified value of the aggregated fuzzy set is the crisp output value from the
Mamdani-type fuzzy model in response to the given input value. In Step 3 the
aggregation is the union (standard/Zadeh's union) of the consequent fuzzy sets.
4.4.2 Inferencing a Takagi-Sugeno type Fuzzy Model
The inference formula of the Takagi-Sugeno model is only a two-step procedure,
based on a weighted average defuzzifier. In the first step the degree of fulfilment,
or firing strength (also called the degree of activation), of each rule is computed
using the product operator. In the second step, the final output value of the system
is calculated using the weighted average defuzzifier. This can, for the inference
process of a Takagi-Sugeno type fuzzy logic system consisting of M rules, be
presented as
R 1 :
1
IF x 1 is
G and … and x n is
1
1
n
THEN
y
1
1
"
1
G
TT
x
T
x
0
1
1
nn
TS
R 2 :
2
IF x 1 is
2
1
and … and x n is
2
n
THEN
y
2
2
"
2
G
G
TT
x
T
x
0
1
1
nn
TS
:
:
:
:
:
R M :
IF
x 1
is
M
and
and
x n
is
M
n
THEN
G
G
1
M
MM
"
M
nn
y
TT
x
T
x
0
1
1
TS
The degree of fulfilment is now calculated using the product operator , as was
done when the set of Takagi-Sugeno rules with antecedent fuzzy sets and
parameters are known for a given set of inputs
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