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l
Z
j
min
E
,
,
j
"
1, 2,
,
K
.
max
R
lj
1
dd
lM
Z
ZZ Z
ª
12
,
,
"
,
K
º
¬
¼
¬¼
, a relational matrix of size
with
M
u
,
M
is the number of
given fuzzy rules and
k
is the number of output fuzzy sets/membership
functions that make the partitioning of the output domain or output
universe of discourse.
ªº
l
R
P
j
Mk
u
x Step 3: defuzzify the consequent fuzzy set by COG method to compute the
crisp output value
§
· § ·
k
k
j
y
j
y
¹ © ¹
,
j
¦
Z
¦
Z
¨
¸ ¨ ¸
0
©
j
1
j
1
where
j
j
th output fuzzy set for the
l
th rule, and
y
COG
l
,
l
FF
j
j
j
"
1, 2,
,
k
.
To illustrate the above inference mechanism of a relational fuzzy model, let us
again consider the
n
-input, single-output system described by the relational fuzzy
rule- based model
R1:
IF
x
1
is
G
,
THEN
y
is HIGH (0.9),
y
is MEDIUM (0.1),
y
is LOW (0.0)
G
and… and
x
n
is
1
1
R2:
IF
x
1
is
G
and… and
x
n
is
G
,
THEN
y
is HIGH (0.1),
y
is MEDIUM (0.8),
y
is LOW (0.0)
2
1
R3:
IF
x
1
is
G
,
THEN
y
is HIGH (0.0),
y
is MEDIUM (0.7),
y
is LOW (0.2)
3
1
and … and
x
n
is
3
G
If the antecedent's fuzzy sets,
i.e.
G
with
l
i
"
1, 2,
,
n
;
l
" and
M
= 3,
1, 2,
,
M
;
are given, then for given values of input variables
x
" we first
determine the output fuzzy set through the inferencing mechanism as stated in the
above three steps.
Now, the degree of fulfilment of the
l
th rule is computed using the product
operator
;
i
, ,
,
n
;
i
n
l
E
P
P
u
P
u u
"
P
x
x
x
x
l
l
l
l
G
i
G
1
G
2
G
n
i
1
i
1
2
n
Therefore,
n
1
E
P
P
u
P
u u
"
P
= 0.5 (say).
x
x
x
x
1
1
1
1
G
i
G
1
G
2
G
n
i
1
2
n
i
1
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