Environmental Engineering Reference
In-Depth Information
u
dn
þ
T
2
u
dn
¼
v
ð
3
:
36
Þ
v
¼ð
k
2
þ
g
2
Þ
sgn
ð
s
2
Þ
ð
3
:
37
Þ
where g
2
¼
max
ð
T
2
u
dn
j
jÞ;
T
2
[ 0
;
k
2
[ 0.
3.5.3 q-Axis Current Controller Design
Define an error variable between the required and actual q-axis current:
e
3
¼
i
q
i
q
ð
3
:
38
Þ
Based on Eq. (
3.20
), the q-axis current error dynamics can be expressed by:
e
3
¼
R
L
i
q
þ
xi
d
þ
u
q
ð
3
:
39
Þ
L
A TSM manifold is designed as follows [
12
,
13
]:
s
3
¼
e
3
þ
c
3
e
q
3
=
p
3
ð
3
:
40
Þ
3
where c
3
[ 0, p
3
[ 0, q
3
[ 0, p
3
and q
3
are odd.
Theorem 3 If the TSM manifold is chosen as Eq. (
3.40
), and the control is
designed as follows, e
3
can converge to zero in finite time [
25
]:
u
q
¼
u
qeq
þ
u
qn
ð
3
:
41
Þ
u
qeq
¼
Ri
q
xLi
d
Lc
3
e
q
3
=
p
3
ð
3
:
42
Þ
3
u
qn
þ
T
3
u
qn
¼
v
ð
3
:
43
Þ
v
¼ð
k
3
þ
g
3
Þ
sgn
ð
s
3
Þ
ð
3
:
44
Þ
Þ;
T
3
[ 0
;
k
3
[ 0.
where g
3
¼
max
ð
T
3
u
qn
The proof of Theorems 2 and 3 is similar to that of Theorem 1 and hence is
omitted here.
In Theorem 2 and 3, the control signals, u
d
and u
q
, can be transformed to the
control signals s
d
and s
q
in Eq. (
3.12
) by using the following equations:
