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which leads to
C
l
¼
a
7
Þ bC
l
c
ð
x
1
þ
_
a
5
x
5
x
2
a
5
x
4
x
3
þ
a
6
x
1
a
7
e
1
ð
61
Þ
It is worth noticing that because of the integral structure of the adaptation law
(
61
), this updating law is implementable despite the presence of the time derivative
_
x
1
. To show that, let
'
s rewrite the adaptation law as
Z
t
ðÞ
a
7
C
l
¼
C
l
0
ð
x
1
t
ðÞ
x
1
0
ðÞ
Þ þ
h
ðÞ
d
s
ð
62
Þ
0
where
a
7
e
1
þ
a
7
¼
bC
l
þ c
h
ð
a
5
x
5
x
2
a
5
x
4
x
3
þ
a
6
x
1
Þ
ð
63
Þ
Consequently, the load torque adaptation law can be computed without the need
of using
x
1
.
_
C
l
¼
bC
l
þ c
Remark 3 From (
59
), we can rewrite
a
7
e
1
, this equation can be seen
as a standard disturbance observer. In fact, if e
1
converges to zero, then
C
l
also
converges to zero. Consequently,
C
l
converges to
C
l
.
To summary, Fig.
3
shows the block diagram of our FABC proposed. The
overall scheme of the controlled DFI-Motor is depicted in Fig.
4
in which the stator
is directly connected to the grid, and the DIF-Motor is controlled by acting on the
rotor winding.
In the following section, the effectiveness of the proposed FABC will be illus-
trated via some simulations results.
˅
u
z
Virtual control
˅
Eqs. (23) and (61)
2
1
u
Eqs.(37), (42) and (43)
0
2
Control
1
z
z
u
u
Eqs.(50), (55) and (56)
Control
2
2
2
Fig. 3 The proposed FABC
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