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Therefore,
M
i
M
i
w
i
w
i
w
i
−
w
i
a
l
ki
l
=
1
M
i
∂
l
=
1
l
=
1
M
i
∂
l
=
1
a
l
ki
∂
x
q
∂
x
q
∂
a
ki
∂
x
q
=
M
i
2
l
=
1
w
i
(A.11)
w
i
w
i
l
=
1
M
i
p
=
1
M
i
∂
l
=
1
M
i
p
=
1
M
i
∂
a
l
ki
w
p
a
p
ki
w
p
−
i
i
∂
x
q
∂
x
q
=
,
M
i
2
l
=
1
w
i
Which can be simplified to:
w
i
M
i
∂
w
p
a
p
a
l
ki
−
i
(
ki
)
∂
x
q
∂
a
ki
∂
l
,
p
=
1
x
q
=
.
(A.12)
M
i
2
l
=
1
w
i
Thus, Eq. (
A.6
) can be written as:
⎛
⎝
⎞
⎠
x
k
w
i
M
i
∂
k
=
0
n
w
p
a
p
a
l
ki
−
i
(
ki
)
∂
a
ki
˜
n
∂
x
q
l
,
p
=
1
=
x
k
˜
+
a
qi
.
(A.13)
M
i
2
∂
x
q
l
=
1
k
=
0
w
i
The derivative of the degree of activation of the rules of the plant with respect to
each of the state variables,
w
i
/∂
∂
x
q
, will be determined later.
A.2 Derived from the Second Term
n
m
k
=
0
j
=
1
m
j
=
1
⎛
⎝
∂
˜
⎞
∂
b
ji
c
kj
˜
∂
x
k
b
ji
c
kj
n
n
m
x
k
⎠
,
=
x
k
+
˜
b
ji
c
kj
(A.14)
∂
x
q
∂
x
q
∂
x
q
k
=
0
k
=
0
j
=
1
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