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⎛
⎝
∂ω
⎞
⎠
r
j
∂
x
q
ω
∂
w
i
N
j
M
i
∂
x
q
w
i
(
b
ji
)
b
l
ji
−
s
j
c
kj
)
c
kj
−
(
n
m
l
,
p
=
1
r
,
s
=
1
+
c
kj
+
b
ji
x
k
.
M
i
2
N
j
2
l
=
1
w
i
r
=
1
ω
k
=
0
j
=
1
r
j
(A.24)
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
, and the derivative of the degree of activation
of the rules of the plant with respect to each of the state variables,
∂
r
j
/∂
∂ω
x
q
, will be
determined below.
A.3 Derivative of the Degree of Activation of the Rules
of the Plant
n
m
w
i
(
l
l
l
l
x
,
u
)
=
1
μ
ki
(
x
k
,
σ
ki
)
1
μ
ji
(
u
j
,
α
ji
)
,
(A.25)
k
=
j
=
w
i
(
x
)
w
i
(
u
)
so that:
w
i
(
w
i
w
i
(
w
i
(
∂
)
∂
x
)
)
∂
u
)
∂
w
i
(
w
i
(
w
i
(
x
q
=
x
)
u
)
=
u
+
x
.
(A.26)
∂
∂
x
q
∂
x
q
∂
x
q
Given that:
n
w
i
(
∂
x
)
∂
l
l
=
1
μ
ki
(
x
k
,
σ
ki
)
,
(A.27)
∂
x
q
∂
x
q
k
=
and
n
n
w
i
(
l
l
l
l
∂
x
)
=
∂μ
1
i
(
x
1
,
σ
1
i
)
ki
)
+
∂μ
2
i
(
x
2
,
σ
2
i
)
l
l
l
l
1
μ
ki
(
x
k
,
σ
2
μ
ki
(
x
k
,
σ
ki
)
∂
x
q
∂
x
q
∂
x
q
k
=
1
,
k
=
k
=
1
,
k
=
n
l
l
+ ···+
∂μ
ni
(
x
n
,
σ
ni
)
l
l
n
μ
ki
(
x
k
,
σ
ki
),
(A.28)
∂
x
q
k
=
1
,
k
=
which can be represented more compactly by:
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