Information Technology Reference
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In the next step, the crisp input and output values are fuzzified. For any input
value X ki , or output value Y k , the f ( X ki ), or f ( Y k ), is calculated such that
^
`
2
2
P
(
)
{
f
(
)
p(
)
2
V
,
X
X
X
c
j
ki
ki
ki
j
Gj
j
(4.3)
^
`
2
2
P
()
{
f
() p (
)
2
V
Y
Y
Y
c
j
k
k
k
j
Gj
j
where i = 1, 2, 3, 4 (corresponding to the first, second, third and fourth inputs in
our case), j = 1, 2, 3, ..., n (corresponding to G 1 to G n ), k = 1, 2, 3, ..., m/2 ( i.e.
corresponding to the k th row of the XIO matrix and m being the total number of
rows in XIO matrix in Equation (4.1)), and
X P is the degree of membership
P of X ki in the j th Gaussian fuzzy set G j . Hence, for any particular X ki (or Y k ), i.e.
for input X 11 (or output Y 1 ), when i = k = 1, then
j Gki
G X P will have n values
(because j = 1, 2, ..., n ) within the range [0, 1] and the same should be arranged in a
column vector form of size
11
j
1
n u .
Similarly, the same procedure should be adopted for other inputs and output
X 12 , X 13 , X 14 and Y 1 , i.e.
X P should be computed for all i = 1, 2, 3, 4, and,
thereafter, should also be arranged in
j Gki
1
n u column vector form. When such
column vectors, each of size
1
n u for all the inputs ( X 11 , X 12 , X 13 , X 14 ) and output
( Y 1 ), are arranged side by side sequentially, this results in a Mu-matrix of size
ni u , i.e. of
n u size for our four-inputs one-output system. Now, the
maximum value of degree of membership from each column of the Mu-matrix is
selected and the corresponding row number is recorded.
For example,
r Gki
1
5
max
P
X
is to be found out such that
0
P
X
d and
1
r Gki
, for all j = 1, 2, 3, ..., n , the integer value of r (1 r n )
should then be recorded, which is the key point of the automated rules generation
algorithm. Once the r values are computed for all X ki and Y k , for i = 1, 2, 3, 4 and k
= 1, 2, 3, ..., m /2, they should be recorded. For instance, it may be the case that
P
X
max
P
X
Gki
Gki
r
j
P G 3 ( X 11 ) = max(P G j ( X 11 )) = 0.95, i.e. r = 3 implies G 3
P G 5 ( X 12 ) = max(P G j ( X 12 )) = 0.80, i.e. r = 5 implies G 5
P G 2 ( X 13 ) = max(P G j ( X 13 )) = 0.98, i.e. r = 2 implies G 2
P G 6 ( X 14 ) = max(P G j ( X 14 )) = 0.90, i.e. , r = 6 implies G 6
P G 2 ( Y 1 ) = max(P G j ( Y 1 )) = 0.97, i.e. r = 2 implies G 2
In the next step, the fuzzy rules are built based on the values of r and degrees of
membership. For instance, in the above example the degree of membership (P) of
X 11 assumes a maximum value of 0.95 in G 3 (because r = 3). Similarly, the degrees
of membership (P) of X 12 , X 13 , X 14 and Y 1 assume maximum values of 0.8 in G 5
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