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Fig. 1.1
Membership
functions of the fuzzy
system
μ
i
1
(
x
i
)
+
μ
i
2
(
x
i
)
=
1
(1.17)
For this case which is widely used, it can be easily demonstrated (Jiménez et al.
2012
) that the matrix X is not of full rank and therefore
X
T
X
is not invertible,
which makes the mentioned method of TS invalid. This result can be easily proven
as follows:
Supposing that it exists:
n
f
:
−→
(1.18)
y
=
f
(
x
)
(1.19)
in which each row of the matrix X is of the form:
X
k
=
μ
1
(
x
k
x
k
)μ
1
(
x
k
)
x
k
μ
2
(
x
k
)μ
2
(
x
k
)
b
−
x
k
b
x
k
b
−
x
k
b
x
k
−
a
b
x
k
−
a
b
=
x
k
(1.20)
−
a
−
a
−
a
−
a
verifying that:
⎡
⎣
⎤
⎦
=
−
a
b
−
x
k
b
x
k
1
−
b
−
x
k
x
k
−
a
x
k
−
a
x
k
0
(1.21)
−
a
b
−
a
b
−
a
b
−
a
b
1
The rank of X in this case is 3 in other words, the columns of X are linearly depen-
dent which in turn makes impossible the use of the above mentioned identification
method proposed in Takagi and Sugeno (
1985
). Analyzing another example of two
variables:
2
f
:
−→
(1.22)
y
=
f
(
x
1
,
x
2
)
(1.23)
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