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In-Depth Information
Each row of the matrix X is of the form:
X
k
=[
μ
11
μ
21
μ
11
μ
21
x
1
k
μ
11
μ
21
x
2
k
μ
11
μ
22
μ
11
μ
22
x
1
k
μ
11
μ
22
x
2
k
μ
12
μ
21
μ
12
μ
21
x
1
k
μ
12
μ
21
x
2
k
μ
12
μ
22
μ
12
μ
22
x
1
k
μ
12
μ
22
x
2
k
]
(1.24)
It can be noticed that the columns 1, 3, 4 and 6 have the same formas in the previous
example multiplied by a constant
μ
11
and therefore they are linearly dependent as
well. The same thing happens with the columns 6, 9, 10 and 12, etc. In fact, the rank
of the matrix in this case is 8.
The solution proposed in Takagi and Sugeno (
1985
) avoids the occurrence of this
situation. In order to identify a function in the interval [
a
i
,
b
i
] using TS method,
certain intermediate points are chosen of the form:
a
i
b
i
] and
b
i
∈
[
a
i
,
∈
[
a
i
,
b
i
]
(1.25)
and they use membership functions which verify:
x
i
−
b
i
a
i
−
b
i
a
i
≤
x
≤
b
i
μ
i
1
(
x
)
=
(1.26)
b
i
0
≤
x
≤
b
i
0
a
i
a
i
≤
x
≤
μ
i
2
(
x
)
=
(1.27)
x
i
−
a
i
b
i
−
a
i
≤
x
≤
b
i
a
i
and thus:
b
i
)
=
μ
i
1
(
a
i
)
=
1
μ
i
1
(
0
(1.28)
a
i
)
=
μ
i
2
(
0
μ
i
2
(
b
i
)
=
1
(1.29)
which impedes that the domains of these functions being overlapped and therefore
it can be observed that, except for some isolated points,
μ
i
1
(
x
i
)
+
μ
i
2
(
x
i
)
=
1
(1.30)
and thus, in general, the matrix X will be of full rank and the method is applicable.
This solution can be clearly seen in Takagi and Sugeno (
1985
) where the authors find
the optimummembership functions minimizing the performance index and reducing
the problem to a nonlinear programming one. For this reason, they use the well-
known complex method for the minimization. This can obviously be observed in the
illustrative examples selected by the authors in Takagi and Sugeno (
1985
) where all
the identified memberships are non overlapping ones.
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