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In-Depth Information
Finally
⊨
0
.
4
,
if
y
=
2
3
.
,
=
0
if
y
6
μ
Q
∗
(
y
)
=
max
(μ
Q
1
(
y
), μ
Q
2
(
y
), μ
Q
3
(
y
))
=
⊩
0
.
6
,
if
y
=
8
0
,
otherwise
the output is given by the graphics,
Q*
In many applications, the output should be converted into a single numerical
value: it should be 'defuzzified'. In this cases with numerical input and numerical
rule's consequents (the most used in
fuzzy control
), such number is easyly obtained
by averaging the values of
μ
Q
∗
,intheform
3
2
×
0
.
4
+
6
×
0
.
+
8
×
0
.
6
7
.
5998
=
3333
=
5
.
6999
5
.
7
1
.
3
0
.
4
+
0
.
+
0
.
6
Hence, the numerical output that corresponds to the input
x
0
=
0
.
4, is
y
0
=
5
.
7.
Remark 3.4.7
Notice that once a system of rules linguistically describing the
behavior of a system is given, and where the consequents of the rules are numerical,
at each numerical input
x
0
in
X
does correspond a numerical output
y
0
in
Y
. In that
way, a function CRI:
X
Y
is defined. As it will be later on commented, were
the system's behaviour previously known by a continuous function
f
ₒ
:
X
ₒ
Y
,the
function CRI approaches, under some additional conditions, the function
f
.
Remark 3.4.8
Look how important is to properly select the T-conditionals repre-
senting the rules.
Given the rule 'If x is small, then y is big', with
X
=
Y
=[
0
,
1
]
, and
μ
S
(
x
)
=
1
−
x
,μ
B
(
y
)
=
y
,
J
(
a
,
b
)
=
max
(
1
−
a
,
b
)
,
it follows
J
(μ
S
(
x
), μ
B
(
y
))
=
max
(
x
,
y
)
, that could be interpreted as '
x
is big
or y
is big'.
With
J
(
a
,
b
)
=
min
(
1
,
1
−
a
+
b
)
,itfollows
J
(μ
S
(
x
), μ
B
(
y
))
=
min
(
1
,
x
+
y
)
=
W
∗
(
x
,
y
)
, also interpretable as '
x
is big
or y
is big'. But with
J
(
a
,
b
)
=
min
(
a
,
b
)
,
is
J
(μ
S
(
x
), μ
B
(
y
))
=
min
(
1
−
x
,
y
)
, interpreted as
'x is small and y is big'
.
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