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l
ν
l
ν
l
ν
l
ν
∂μ
i
(
u
ν
,
α
i
)
=
∂μ
i
(
u
ν
,
α
i
)
∂
u
ν
x
q
.
(A.36)
∂
x
q
∂
u
∂
ν
l
ν
l
ν
∂μ
i
(
u
ν
,
α
i
)
can be obtained directly from the definition of the membership
∂
u
ν
function. Thus,
x
k
k
=
0
n
∂
c
k
ν
˜
n
n
∂
u
∂
c
k
ν
∂
c
k
ν
∂
˜
x
k
ν
x
q
=
=
x
q
˜
x
k
+
x
q
,
(A.37)
∂
∂
x
q
∂
k
=
0
k
=
0
ergo,
n
∂
u
∂
c
k
ν
∂
ν
x
q
=
x
q
˜
x
k
+
c
q
ν
.
(A.38)
∂
k
=
0
Substituting (
A.22
)into(
A.38
):
⎛
⎝
⎞
⎠
∂ω
N
ν
r
ν
s
c
k
ν
−
c
k
ν
)
x
q
ω
ν
(
n
∂
∂
u
ν
r
,
s
=
1
x
q
=
c
q
ν
+
x
k
.
˜
(A.39)
N
ν
2
∂
r
=
1
ω
k
=
0
r
ν
Finally, substituting (
A.31
) and (
A.34
)into(
A.26
), the expression that calculates
the derivative of the degree of activation of the rules of the plant in a closed-loop
fuzzy control system is obtained:
m
l
l
n
w
i
(
j
=
1
μ
∂μ
qi
(
x
q
,
σ
qi
)
k
=
1
,
k
=
q
μ
∂
x
,
u
)
l
l
l
l
=
ji
(
u
j
,
α
ji
)
ki
(
x
k
,
σ
ki
)
∂
x
q
∂
x
q
⎛
⎝
∂μ
⎞
n
m
m
l
v
l
v
i
(
u
v
,
α
i
)
⎠
,
l
l
l
l
+
1
μ
ki
(
x
k
,
σ
ki
)
μ
ji
(
u
j
,
α
ji
)
∂
x
q
k
=
v
=
1
j
=
1
,
j
=
v
(A.40)
l
ν
l
ν
where
∂μ
i
(
u
ν
,
α
i
)
is calculated using (
A.39
) over (
A.36
) and the derivatives of
∂
x
q
the membership functions can be obtained directly from the definition of the function
itself.
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