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Now, it is required to design a control law (rotor voltages u
rd
and u
rq
) which
guarantees that lim
t
!1
X
¼
X
. Then, we will consider the reduced
order DFI-Motor model represented by the rotor
t
!1
u
rd
¼
u
rd
and lim
fl
flux and speed equations.
x
1
¼
_
a
5
ð
x
4
x
3
x
5
x
2
Þ
a
6
x
1
a
7
C
l
_
x
2
¼
a
3
x
2
þ
a
4
x
4
x
r
x
3
þ d
1
ð
x
1
;
x
2
Þþ
u
1
ð
18
Þ
x
3
¼
_
a
3
x
3
þ
a
4
x
5
þ x
r
x
2
þ d
2
ð
x
3
;
x
2
Þþ
u
2
with x
1
¼
X;
x
2
¼
u
rq
;
x
3
¼
u
rd
;
x
4
¼
u
sq
;
x
5
¼
u
sd
;
u
1
¼
u
rq
;
u
2
¼
u
rd
;
a
3
¼
R
r
=
L
r
r;
a
4
¼
=
L
r
L
s
r;
a
5
¼
=
JL
r
L
s
r;
a
6
¼
k
f
=
J and a
7
¼
=
d
i
R
r
M
pM
1
J.where
(i
2) are the unknown uncertainties and perturbations that can be naturally
generated from the parameter variations.
Backstepping design procedure (Krstic et al.
1995
) is used here for the con-
struction of the FABC which guarantees asymptotic tracking of rotor speed and
rotor
¼
1
;
flux reference signals. Then, the variables to be controlled in the model (
18
)
are the rotor speed (x
1
) and the rotor
fl
x
3
).
Step 1. For a continuous bounded reference signal x
1d
,wede
fl
ux (x
2
;
ne the tracking error
e
1
as follows
e
1
¼
x
1
x
1d
ð
19
Þ
Its derivative
e
1
is given by
_
e
1
¼
_
_
x
1
_
x
1d
ð
20
Þ
From the
first subsystem of (
18
), we can write
e
1
¼ a
5
x
4
x
3
a
5
x
5
x
2
a
6
x
1
a
7
C
l
x
1d
ð
21
Þ
t
1
as a desired
Choose a
5
x
5
x
2
as a virtual control to stabilize e
1
and select
reference signal for a
5
x
5
x
2
a
6
x
1d
_
t
1
¼
a
5
x
4
x
3d
þ
c
1
e
1
x
1d
a
7
C
l
ð
22
Þ
where
c
1
[
0
is a design constant.
However, the exact value of the external load torque
C
l
in (
22
) is generally
dif
cult to be known in advance for practical applications. Then, it cannot
be used in the virtual control signal. We can select the new virtual control
as follows
a
7
C
l
t
2
ð
z
0
Þ
¼
a
5
x
4
x
3d
þ
c
1
e
1
a
6
x
1d
_
x
1d
ð
23
Þ
where C
l
is the estimate of
x
4
; C
l
T
.
C
l
and
z
0
¼½
x
1
;
This leads to the following dynamics
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