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as the “theoretical” (linear) behavior of the system
(
x
,
y
)
. For each
x
0
∈
X
,thevalue
CRI
(
x
0
)
∈
Y
is the defuzzified value that corresponds to
x
0
.
Remark 3.4.14
Systems of fuzzy rules behave as
universal approximators
.This
means the following. Suppose a system
(
x
,
y
)
, with
x
∈[
a
,
b
]
,
y
∈[
c
,
d
]
, that
behave by following the continuous function
f
0,there is
always a system of fuzzy rules and a defuzzification method for the output, giving a
function CRI:
X
(
x
)
=
y
. For each
ʵ>
ₒ
Y
such that
|
f
(
x
)
−
CRI
(
x
)
|
<ʵ
, for all
x
∈[
a
,
b
]
This theorem (whose proof is here avoided) is simply an existential one, since
there is no general method for obtaing neither a fuzzy representation of the system
of rules, not the defuzzifiction method. It simply shows that it is possible to find a
CRI approaching enough well
f
for all points in
[
a
,
b
]
.
3.5 Deffuzification
How to defuzzify non discrete outputs
μ
Q
∗
? Let us proceed with two examples
without computational difficulties.
1st Example.
Rules,
•
r1: If
x
is big, then y is small
•
r2: If
x
is small, then y is big
with
X
=[
0
,
1
]
, and
Y
=[
0
,
10
]
. Take,
y
10
,μ
S
(
y
10
,
μ
B
(
x
)
=
x
,μ
S
(
y
)
=
1
−
x
)
=
1
−
x
,μ
B
(
y
)
=
and
J
(
a
,
b
)
=
min
(
a
,
b
)
-Mamdani-. Notice that, with the observation
x
0
=
0
.
5,
y
10
), μ
Q
2
(
y
10
).
μ
Q
1
(
y
)
=
min
(
0
.
5
,
1
−
y
)
=
min
(
1
−
0
.
5
,
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