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⎛
⎝
∂ω
⎞
⎠
r
j
∂
x
q
ω
w
i
∂
x
q
N
j
M
i
∂
w
i
(
b
ji
)
b
l
ji
−
s
c
kj
−
c
kj
)
j
(
n
m
l
,
p
=
1
r
,
s
=
1
+
c
kj
+
b
ji
x
k
,
˜
M
i
2
N
j
2
l
=
1
r
=
1
ω
=
=
k
0
j
1
w
i
r
j
which can be rewritten as:
m
J
iq
=
a
qi
+
b
ji
c
qj
j
=
1
⎛
⎝
⎞
⎠
w
i
M
i
∂
∂
x
q
w
i
(
a
ki
)
a
l
ki
−
l
,
p
=
1
M
i
2
l
=
1
w
i
n
⎛
⎝
⎞
⎠
∂ω
+
x
k
˜
r
j
w
i
N
j
M
i
∂
x
q
w
i
(
b
ji
)
k
=
0
b
l
ji
−
s
j
c
kj
−
c
kj
)
x
q
ω
(
∂
∂
j
=
1
m
l
,
p
=
1
r
,
s
=
1
+
c
kj
+
b
ji
M
i
2
N
j
2
l
=
1
w
i
r
=
1
ω
r
j
(A.45)
Finally, the elements of the Jacobian matrix of the closed-loop fuzzy control
system (
A.2
) can be calculated by:
⎛
⎞
c
kj
B
ji
m
n
m
b
ji
C
kj
⎝
A
ki
q
⎠
˜
J
iq
=
a
qi
+
b
ji
c
qj
+
+
+
x
k
,
(A.46)
q
q
j
=
1
k
=
0
j
=
1
being
w
i
M
i
∂
w
i
(
a
ki
)
a
l
ki
−
∂
x
q
l
,
p
=
1
A
ki
q
=
,
(A.47)
M
i
2
l
=
1
w
i
w
i
M
i
∂
w
i
(
b
ji
)
b
l
ji
−
∂
x
q
l
,
p
=
1
B
ji
q
=
(A.48)
M
i
2
l
=
1
w
i
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