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5 Design of the Fuzzy Adaptive Backstepping Control
In this section, the stator
flux subsystem control is designed in order to achieve
asymptotic alignment of the stator
fl
flux vector with the d-axis of the line voltage
vector reference frame, consequently, the stator voltage and
fl
fl
ux vectors become
orthogonal.
Introduce
fl
flux stator tracking errors as
u
sd
¼
u
sd
u
s
;
u
sq
¼
u
sq
ð
13
Þ
u
s
where
is the d-axis
fl
flux reference trajectory.
Using (
8
), the stator
fl
flux dynamic equations in (
1
) can be written in error form as
a
1
u
s
þ
a
2
u
rd
þ x
s
u
sq
u
s
u
sd
¼
a
1
u
sd
ð
Þ
14
a
2
u
rq
x
s
u
sd
x
s
u
s
þ
u
sq
¼
a
1
u
sq
þ
u
s
where a
1
¼
R
s
=
L
s
r
, a
2
¼
R
s
M
=
L
r
L
s
r
.
To realize the required stator
fl
flux orientation, the d-axis component of rotor
fl
ux
u
rd
can be considered as control input in (
14
), and should be
1
a
2
a
1
u
s
þ u
s
u
rd
¼
ð
15
Þ
with the d-axis stator
fl
flux reference computed from the second equation of (
14
)
1
x
s
u
s
¼
u
s
þ
a
2
u
rq
ð
16
Þ
Using (
15
) and (
16
), (
14
) becomes
a
2
ðu
rd
u
rd
Þ
u
sd
¼
a
1
u
sd
þ x
s
u
sq
þ
ð
Þ
17
u
sq
¼
a
1
u
sq
x
s
u
sd
However, in a DFI-Motor, the rotor
fl
flux is not available as control input and
u
rd
u
rd
in (
15
) can only represent the d-axis rotor
u
rd
.
The rotor voltages u
rd
and u
rq
are the only physical available control inputs of
DFI-Motor. From (
17
), one concludes that the dynamic of the stator
fl
flux reference
for the real
fl
ux
fl
flux is expo-
t
!1
u
rd
¼
u
rd
.
Remark 1 From (
17
) and (
15
), it can be concluded that in the steady state (
t
!1
u
sd
¼
u
s
nentially stable (i.e. lim
and lim
t
!1
u
sq
¼
0) provided that lim
u
s
a
1
u
s
L
M
u
s
.
constant),
u
rd
¼
a
2
¼
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