Information Technology Reference
In-Depth Information
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
Search WWH ::
Custom Search