Information Technology Reference
In-Depth Information
•
Rule
R
2
is represented by
J
L
(
(μ
P
21
(
x
1
), μ
P
22
(
x
2
)), μ
y
2
(
))
T
y
μ
P
21
(
x
1
)
·
μ
P
22
(
x
2
),
y
=
y
2
=
μ
P
21
(
x
1
)
·
μ
P
22
(
x
2
)
·
μ
y
2
(
y
)
=
0
,
y
=
y
2
Hence, the corresponding outputs under the CRI are:
μ
P
11
(
x
1
)
·
μ
P
12
(
x
2
),
y
=
y
1
μ
Q
1
(
)
=
y
0
,
y
=
y
1
μ
P
21
(
x
1
)
·
μ
P
22
(
x
2
),
y
=
y
2
μ
Q
2
(
y
)
=
0
,
y
=
y
2
Consequently:
⊧
⊨
x
1
)
·
μ
P
12
(
x
2
),
μ
P
11
(
y
=
y
1
x
1
)
·
μ
P
22
(
x
2
),
μ
Q
∗
=
Max
(μ
Q
1
(
y
), μ
Q
2
(
y
))
μ
P
21
(
y
=
y
2
⊩
0
,
otherwise
Eight Step: Defuzzification
Zadeh's CRI gives an output function
μ
Q
∗
, but what it is frequently needed, mainly
in control, is an output number as an “order” to be executed by the system. Hence,
this step consists in compacting in the best possible way, in a single real number,
the information on the system's behavior contained in
μ
Q
∗
. That is, the goal is to
defuzzify
μ
Q
∗
.
In the applications the most interesting cases are those in which, respectively,
either
μ
Q
∗
is a non-null continuous function, or it is a non-null function at only a
finite number of points in
Y
.
In the first case and among the diverse methods that have been suggested in the
literature, that known as “center of gravity” of the area below
μ
Q
∗
as well as the one
known as “center of area” are perhaps the most used ones. In the second case, if for
example,
⊧
⊨
ʱ
1
,
=
if
y
y
1
ʱ
2
,
if
y
=
y
2
μ
Q
∗
(
)
=
···
ʱ
n
,
y
⊩
if
y
=
y
n
0
,
otherwise
the most popular method of defuzzification is that consisting in taking the weighted
mean
μ
Q
∗
=
ʱ
1
·
y
1
+
ʱ
2
·
y
2
+···+
ʱ
n
·
y
n
ʱ
1
+
ʱ
1
+···+
ʱ
n
The case of
m
rules with numerical consequents and numerical inputs, with
Larsen's implication function and with defuzzification by the weighted mean, is the
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