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to each of the parameters of the antecedents. Obviously, this calculation is depen-
dent on the type of membership function that is used for each antecedent, however,
is possible to express in a general form:
⎛
⎞
n
w
I
∂
σ
∂
∂
∂
σ
⎝
L
L
⎠
JI
=
1
μ
qI
(
x
q
(
k
),
σ
qI
)
(2.30)
L
L
JI
q
=
or a more developed form:
n
w
I
∂
σ
L
L
∂
=
∂μ
JI
(
x
J
(
k
),
σ
JI
)
L
L
J
μ
qI
(
x
q
(
k
),
σ
qI
).
(2.31)
L
JI
L
JI
∂
σ
q
=
1
,
q
=
L
L
Note that
∂μ
JI
(
x
J
(
k
),
σ
JI
)
represents the derivative of the membership function
L
JI
∂
σ
L
that is defined by the parameters set
JI
. Thus, the calculation of this derivative
depends on the type of membership function used and it can be performed from the
expression that defines it.
Note it is not necessary that the membership functions are differentiable, but
it is enough to be piecewise differentiable. Piecewise membership functions could
provide a jump discontinuity in its derivative, however, since the set of singular points
is a null set, in numerical implementations this is not a real problem. For example,
for a triangular membership function:
σ
⎧
⎨
x
−
a
if
a
<
x
≤
b
b
−
a
c
−
x
μ
Tri
[
a
,
b
,
c
]
(
x
)
=
(2.32)
if
b
≤
x
<
c
⎩
c
−
b
0
in other case
σ
represents the vector
[
a
,
b
,
c
]
, and the derivatives are:
⎧
⎨
x
−
b
∂μ
Tri
[
a
,
b
,
c
]
(
x
)
if
a
<
x
<
b
=
2
(2.33)
(
b
−
a
)
⎩
∂
a
0
in other case
⎧
⎨
a
−
x
if
a
<
x
<
b
(
b
−
a
)
2
∂μ
Tri
[
a
,
b
,
c
]
(
x
)
c
−
x
=
(2.34)
if
b
<
x
<
c
∂
⎩
b
2
(
c
−
b
)
0
in other case
and
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