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⎛
⎝
∂μ
⎞
n
l
l
n
w
i
(
pi
(
x
p
,
σ
pi
)
∂
)
x
l
l
⎠
.
=
,
k
=
p
μ
ki
(
x
k
,
σ
ki
)
(A.29)
∂
x
q
∂
x
q
p
=
1
k
=
1
Given that
∂
f
(
x
i
)
=
0
,
∀
i
=
q
,
(A.30)
∂
x
q
the equation (
A.29
) can be reduced to:
l
l
n
w
i
(
∂μ
qi
(
x
q
,
σ
qi
)
∂
x
)
l
l
=
q
μ
ki
(
x
k
,
σ
ki
).
(A.31)
∂
x
q
∂
x
q
k
=
1
,
k
=
On the other hand:
⎛
⎞
m
w
i
(
∂
u
)
∂
⎝
l
l
⎠
,
=
1
μ
ji
(
u
j
,
α
ji
)
(A.32)
∂
x
q
∂
x
q
j
=
that is:
m
m
∂
w
i
(
u
)
∂
x
q
l
1
i
(
u
1
,
α
l
1
i
)
j
=
1
,
j
=
1
μ
l
2
i
(
u
2
,
α
l
2
i
)
j
=
1
,
j
=
2
μ
=
∂μ
ji
)
+
∂μ
l
ji
(
u
j
,
α
l
l
l
ji
)
ji
(
u
j
,
α
∂
x
q
∂
x
q
m
l
mi
l
mi
+ ···+
∂μ
(
u
m
,
α
)
l
ji
l
ji
m
μ
(
u
j
,
α
),
(A.33)
∂
x
q
j
=
1
,
j
=
or more compactly:
⎛
⎝
∂μ
⎞
m
m
w
i
(
l
v
l
v
∂
u
)
i
(
u
v
,
α
i
)
l
l
⎠
.
=
μ
ji
(
u
j
,
α
ji
)
(A.34)
∂
x
q
∂
x
q
v
=
1
j
=
1
,
j
=
v
In a closed-loop fuzzy control system, control signals are generated by a fuzzy
controller, and can be calculated from (
A.35
).
n
u
j
=
c
kj
˜
x
k
,
(A.35)
k
=
0
l
ν
l
ν
Therefore, to calculate
∂μ
i
(
u
ν
,
α
i
)
in (
A.34
), is necessary to obtain
∂
u
ν
, since:
∂
x
q
∂
x
q
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