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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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