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8.2.2 Takagi-Sugeno of Order 1
The method of Takagi-Sugeno of order
n
is an immediate generalization of the last
example with numerical input, numerical consequents,
ML
-implication and defuzzi-
fication by a weighted mean. It is the case in which in place of consequents
y
=
y
i
,
x
n
and with the
given coefficients, appearing in the rules' antecedents. For the sake of brevity, and
without any loss of generality, let us consider the case of
m
y
is taken as a polynomial of degree
n
in the
n
variables
x
1
,
x
2
,...,
=
2 rules, with
n
=
2
variables:
Let us shorten
y
=
ʱ
1
x
1
+
ʱ
2
x
2
+
ʱ
3
by
q
1
, and
y
=
ʲ
1
x
1
+
ʲ
2
x
2
+
ʲ
3
by
q
2
.
with
q
1
=
ʱ
1
x
1
+
ʱ
2
x
2
+
ʱ
3
and
q
2
=
ʲ
1
x
1
+
ʲ
2
x
2
+
ʲ
3
.Itfollows:
1
x
1
and
x
2
=
x
2
,
if
x
1
=
μ
x
1
x
2
(
x
1
,
x
2
)
=
0
,
otherwise
1
,
if
q
1
μ
q
1
(
y
)
=
0
,
otherwise
1
,
if
q
2
μ
q
2
(
y
)
=
0
,
otherwise
As rule
R
1
is represented by
J
1
=
μ
P
11
(
x
1
)
·
μ
P
12
(
x
2
)
·
μ
q
1
(
y
)
and rule
R
2
by
J
2
=
μ
P
21
(
x
1
)
·
μ
P
22
(
x
2
)
·
μ
q
2
(
)
y
, it follows:
μ
Q
1
(
y
)
=
Sup
x
∈
X
,
y
∈
Y
Min
(μ
x
1
x
2
(
x
1
,
x
2
), μ
P
11
(
x
1
)
·
μ
P
12
(
x
2
)
·
μ
q
1
(
y
))
μ
P
11
(
x
1
)μ
P
12
(
x
2
),
if
q
1
x
1
)μ
P
12
(
x
2
)μ
q
1
(
=
μ
P
11
(
y
)
=
0
,
otherwise
μ
P
21
(
x
1
)μ
P
22
(
x
2
),
if
q
2
x
1
)μ
P
22
(
x
2
)μ
q
2
(
μ
Q
2
(
y
)
=
μ
P
21
(
y
)
=
0
,
otherwise
Hence,
⊧
⊨
⊩
x
1
)μ
P
12
(
x
2
),
μ
P
11
(
if
q
1
x
1
)μ
P
22
(
x
2
),
μ
Q
∗
(
y
)
=
Max
(μ
Q
1
(
y
), μ
Q
2
(
y
))
=
μ
P
21
(
if
q
2
0
,
otherwise
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