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A.4 Derivative of the Degree of Activation of the Rules of the
Controller
r
j
n
∂ω
∂
∂
r
r
x
q
=
1
μ
kj
(
x
k
,
kj
)
,
(A.41)
∂
x
q
k
=
so that:
r
j
1
j
(
1
j
)
n
2
j
(
2
j
)
n
∂ω
∂μ
x
1
,
∂μ
x
2
,
r
r
r
r
x
q
=
1
μ
kj
(
x
k
,
kj
)
+
2
μ
kj
(
x
k
,
kj
)
∂
∂
x
q
∂
x
q
k
=
1
,
k
=
k
=
1
,
k
=
r
r
n
∂μ
nj
(
x
n
,
nj
)
k
=
1
,
k
=
n
μ
r
r
+ ···+
kj
(
x
k
,
kj
).
(A.42)
∂
x
q
given that
∂
(
x
i
)
f
=
,
∀
=
,
0
i
q
(A.30)
∂
x
q
the expression (
A.42
) can be simplified to:
r
j
r
r
n
∂ω
x
q
=
∂μ
qj
(
x
q
,
qj
)
r
r
q
μ
kj
(
x
k
,
kj
),
(A.43)
∂
∂
x
q
k
=
1
,
k
=
where the derivatives of the membership functions can be obtained directly from the
definition of the function itself.
A.5 Final Expression of the Jacobian Matrix of a Closed-Loop
Fuzzy Control System
The jacobian matrix can be obtained substituting (
A.13
) and (
A.24
)in(
A.5
):
⎛
⎝
⎞
⎠
w
i
M
i
∂
w
p
a
p
a
l
ki
−
i
(
ki
)
n
∂
x
q
m
)
=
∂χ
i
(
x
)
∂
l
,
p
=
1
(A.44)
J
iq
(
x
=
a
qi
+
x
k
˜
+
b
ji
c
qj
M
i
2
x
q
l
=
1
k
=
0
j
=
1
w
i
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