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The above expression can be written as:
n
x
k
k
=
0
j
=
1
m
k
=
0
n
∂
b
ji
c
kj
˜
x
k
∂
a
ki
˜
J
iq
(
x
)
=
+
.
(A.5)
∂
x
q
∂
x
q
A.1 Derived from the First Term
x
k
k
=
0
n
∂
a
ki
˜
n
n
n
∂ (
a
ki
˜
x
k
)
∂
a
ki
a
ki
∂
˜
x
k
=
=
x
q
˜
x
k
+
x
q
.
(A.6)
∂
x
q
∂
x
q
∂
∂
k
=
0
k
=
0
k
=
0
In as much as
∂
˜
x
k
x
q
=
1if
k
=
q
and 0 in other case:
∂
n
a
ki
∂
˜
x
k
x
q
=
a
qi
.
(A.7)
∂
k
=
0
Considering the first term of (
A.6
):
M
i
M
i
l
=
1
l
=
1
w
i
a
l
ki
w
i
∂
∂
⎛
⎞
l
=
1
M
i
l
=
1
w
i
−
M
i
l
=
1
w
i
a
l
ki
M
i
w
i
a
l
ki
⎝
⎠
=
∂
a
ki
∂
∂
∂
x
q
∂
x
q
x
q
=
.
M
i
2
∂
x
q
l
=
1
M
i
l
=
1
w
i
w
i
(A.8)
The degree of activation of the rules of the plant,
w
i
, is dependent on the state
vector, while
a
l
ki
are constants, so:
M
i
l
=
1
w
i
a
l
ki
∂
M
i
w
i
∂
x
q
a
l
ki
=
,
(A.9)
∂
∂
x
q
l
=
1
and
M
i
l
=
1
w
i
∂
M
i
w
i
∂
=
.
(A.10)
∂
x
q
∂
x
q
l
=
1
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