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The above expression can be written as:
n
x k
k = 0
j = 1
m
k = 0
n
b ji c kj ˜
x k
a ki ˜
J iq (
x
) =
+
.
(A.5)
x q
x q
A.1 Derived from the First Term
x k
k = 0
n
a ki ˜
n
n
n
∂ (
a ki ˜
x k )
a ki
a ki ˜
x k
=
=
x q ˜
x k +
x q .
(A.6)
x q
x q
k
=
0
k
=
0
k
=
0
In as much as ˜
x k
x q =
1if k
=
q and 0 in other case:
n
a ki ˜
x k
x q =
a qi .
(A.7)
k
=
0
Considering the first term of ( A.6 ):
M i
M i
l = 1
l = 1
w i a l ki
w i
l = 1
M i
l = 1 w i
M i
l = 1 w i a l ki
M i
w i a l ki
=
a ki
x q
x q
x q =
.
M i
2
x q
l = 1
M i
l = 1
w i
w i
(A.8)
The degree of activation of the rules of the plant, w i , is dependent on the state
vector, while a l ki are constants, so:
M i
l = 1
w i a l ki
M i
w i
x q a l ki
=
,
(A.9)
x q
l
=
1
and
M i
l = 1
w i
M i
w i
=
.
(A.10)
x q
x q
l
=
1
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