Environmental Engineering Reference
In-Depth Information
100
80
60
40
W
v
(s)
20
0
−20
W
u
(s)
−40
−60
−80
−100
10
−1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
Frequency (rad/sec)
Figure 12.5
Bode plots of the weighting functions
W
u
(
s
)and
W
v
(
s
)
12.4 Design Example
The parameters of the neutral leg are:
L
N
=
2
.
5 mH,
R
N
=
0
.
2
,
C
N
1
=
C
N
2
=
6600
μ
F and
1000
s
+
V
DC
=
850 V. The switching frequency is
f
s
=
10 kHz and
F
(
s
)
=
1000
. The tuning param-
eters are chosen to be:
k
=
0
.
01,
g
=
10,
ρ
=
0
.
01,
ζ
=
0
.
01,
ω
l
=
1 rad/s and
ω
h
=
10000
=
K
v
K
i
is
, the
H
∞
rad/s. Using the
μ
-analysis toolbox from MATLAB
R
controller
K
obtained as
10
11
)(
s
10
4
)(
s
10
4
)(
s
=
−
0
.
0023529(
s
−
2
.
407
×
+
8
.
87
×
+
+
80
.
03)(
s
+
74
.
28)
K
v
(
s
)
,
(
s
+
1
.
556
×
10
8
)(
s
+
1
.
096
×
10
5
)(
s
+
8327)(
s
+
76
.
57)(
s
+
1)
10
8
(
s
10
4
)(
s
8
.
63
×
+
+
1000)(
s
+
80)
K
i
(
s
)
=
57)
.
(
s
+
1
.
556
×
10
8
)(
s
+
1
.
096
×
10
5
)(
s
+
8327)(
s
+
76
.
This controller is somewhat unrealistic because the sampling frequency is usually limited. In
order to make the controller implementable, any zeros or poles which correspond to a corner
frequency much higher than
10
4
rad/s are substituted by a proportional gain, i.e. ignoring
the
s
. Then the controller is reduced to
ω
h
=
2
9457
(
s
+
10
4
)(
s
+
80
.
03)(
s
+
74
.
28)
(
s
+
10
4
)(
s
+
1000)(
s
+
80)
K
r
(
s
)
=
.
.
5
.
5475
(
s
+
8327)(
s
+
76
.
57)(
s
+
1)
(
s
+
1
.
096
×
10
5
)(
s
+
8327)(
s
+
76
.
57)
The Bode plots of these controllers are shown in Figure 12.6. It can be seen that the reduced-
order controllers are very close to the original controller, in particular, at low frequencies.
The Bode plots of the corresponding closed-loop transfer functions using the original
H
∞
controller
K
and the reduced controller
K
r
are shown in Figure 12.7. The curves are very close
to each other.
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