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where,
w
(
i
1
...
i
n
)
(
)
=
μ
1
i
1
(
x
1
)μ
2
i
2
(
x
2
)...μ
ni
n
(
x
n
)
x
(1.6)
the membership function that corresponds to the fuzzy set
M
i
j
j
being
μ
ji
j
(
x
j
)
.
. The para-
meters of the fuzzy system can be calculated as a result of minimizing a quadratic
performance index:
Let m be a set of input/output system samples
{
x
1
k
,
x
2
k
,...,
x
nk
,
y
k
}
m
2
2
J
=
1
(
y
k
−ˆ
y
k
)
=
Y
−
XP
(1.7)
k
=
where
=
y
1
y
2
...
y
m
T
Y
(1.8)
p
(
1
...
1
)
0
T
p
(
1
...
1
)
1
p
(
1
...
1
)
2
p
(
1
...
1
)
p
(
r
1
...
r
n
)
0
p
(
r
1
...
r
n
)
P
=
(1.9)
...
...
...
n
n
β
(
1
...
1
)
1
β
(
1
...
1
)
1
x
11
...β
(
1
...
1
)
x
n
1
... β
(
r
1
...
r
n
)
...β
(
r
1
...
r
n
)
1
x
n
1
X
=
1
1
(1.10)
β
(
1
...
1
)
x
1
m
...β
(
1
...
1
)
... β
(
r
1
...
r
n
)
...β
(
r
1
...
r
n
)
1
...
1
β
x
nm
x
nm
m
m
m
m
m
and
w
(
i
1
...
i
n
)
(
x
k
)
β
(
i
1
...
i
n
)
k
=
r
1
i
1
=
1
...
r
n
(1.11)
1
w
(
i
1
...
i
n
)
(
x
k
)
i
n
=
If X is a matrix of full rank, the solution is obtained as follows:
2
T
J
=
Y
−
XP
=
(
Y
−
XP
)
(
Y
−
XP
)
(1.12)
X
T
X
T
Y
X
T
XP
∇
J
=
(
Y
−
XP
)
=
−
=
0
(1.13)
X
T
X
)
−
1
X
T
Y
P
=
(
(1.14)
1.3 Restrictions of TS Identification Method
The method proposed in Takagi and Sugeno (
1985
) arises serious problems as it can
not be applied in the most common case where the membership functions are those
shown in Fig.
1.1
.
The membership functions
b
i
−
x
i
b
i
x
i
−
a
i
b
i
μ
i
1
(
x
i
)
=
and
μ
i
2
(
x
i
)
=
are defined in
−
a
i
−
a
i
an interval
[
a
i
,
b
i
]
which should verify:
μ
i
1
(
a
i
)
=
1
μ
i
1
(
b
i
)
=
0
(1.15)
μ
i
2
(
a
i
)
=
0
μ
i
2
(
b
i
)
=
1
(1.16)
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