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Appendix A
Jacobian Matrix of a Fuzzy Control System
Let be a completly general autonomous control system given by:
x
1
(
˙
t
)
=
χ
1
(
x
1
(
t
),
x
2
(
t
),...,
x
n
(
t
))
x
2
(
˙
t
)
=
χ
2
(
x
1
(
t
),
x
2
(
t
),...,
x
n
(
t
))
(A.1)
.
x
n
(
˙
t
)
=
χ
n
(
x
1
(
t
),
x
2
(
t
),...,
x
n
(
t
)),
its jacobian matrix is defined as:
⎛
⎝
⎞
⎠
∂χ
1
(
x
)
∂χ
1
(
x
)
...
∂χ
1
(
x
)
∂
x
1
∂
x
2
∂
x
n
∂χ
2
(
)
∂χ
2
(
)
...
∂χ
2
(
)
x
x
x
∂
∂
∂
x
1
x
2
x
n
J
(
x
)
=
.
(A.2)
.
.
.
.
.
.
∂χ
n
(
)
∂χ
n
(
)
...
∂χ
n
(
)
x
x
x
∂
x
1
∂
x
2
∂
x
n
⎛
⎞
n
m
⎝
a
ki
+
⎠
˜
x
i
˙
=
b
ji
c
kj
x
k
,
(A.3)
k
=
0
j
=
1
so, the jacobian matrix of a closed-loop fuzzy control system is:
⎛
⎛
⎞
⎞
n
m
)
=
∂χ
i
(
x
)
∂
⎝
⎝
a
ki
+
⎠
˜
⎠
.
J
iq
(
x
=
b
ji
c
kj
x
k
(A.4)
∂
x
q
∂
x
q
k
=
0
j
=
1
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