Environmental Engineering Reference
In-Depth Information
s
1
s
1
¼
2
C
E
s
s
1
i
dn
¼
2
C
E
s
s
1
ð
i
dn
þ
T
1
i
dn
T
1
i
dn
Þ
¼
2
C
E
s
s
1
ðð
k
1
þ
g
1
Þ
sgn
ð
s
1
Þ
T
1
i
dn
Þ
¼
2
C
E
s
ðð
k
1
þ
g
1
Þ
s
jj
T
1
s
1
i
dn
Þ
2
C
E
s
k
1
s
jj
i.e.,
V
1
2
C
E
s
k
1
s
j
\0 for
j
s
1
j 6¼
0
The system (
3.22
) satisfies the sufficient condition of the existence of the sliding
mode. Therefore, the states of the system (
3.22
) can reach the TSM manifold
s
1
= 0 within finite time.
3.5.2 d-Axis Current Controller Design
Define an error variable between the required and actual d-axis current:
e
2
¼
i
d
i
d
ð
3
:
31
Þ
Based on Eq. (
3.20
), the d-axis current error dynamics can be expressed by:
e
2
¼
i
d
þ
R
L
i
d
xi
q
E
s
L
þ
u
d
ð
3
:
32
Þ
L
A TSM manifold is designed as follows [
12
,
13
]:
s
2
¼
e
2
þ
c
2
e
q
2
=
p
2
ð
3
:
33
Þ
2
where c
2
[ 0, p
2
[ 0, q
2
[ 0, p
22
and q are odd.
Theorem 2 If the TSM manifold is chosen as Eq. (
3.33
), and the control is
designed as follows, e
2
can converge to zero in finite time [
25
]:
u
d
¼
u
deq
þ
u
dn
ð
3
:
34
Þ
u
deq
¼
Li
d
Ri
d
þ
xLi
q
þ
E
s
Lc
2
e
q
2
=
p
2
ð
3
:
35
Þ
2
