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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
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 P 12 (
x 2 ),
if q 1
x 1 P 12 (
x 2 q 1 (
= μ P 11 (
y
) =
0
,
otherwise
μ P 21 (
x 1 P 22 (
x 2 ),
if q 2
x 1 P 22 (
x 2 q 2 (
μ Q 2 (
y
) = μ P 21 (
y
) =
0
,
otherwise
Hence,
x 1 P 12 (
x 2 ),
μ P 11 (
if q 1
x 1 P 22 (
x 2 ),
μ Q (
y
) =
Max
Q 1 (
y
), μ Q 2 (
y
)) =
μ P 21 (
if q 2
0
,
otherwise
 
 
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