Environmental Engineering Reference
In-Depth Information
The transfer matrix from
a
to
b
can then be written as
⎡
⎤
A
B
2
C
c
0
⎣
⎦
T
ba
=
W
B
c
1
C
1
+
B
c
2
C
2
A
c
+
(
B
c
1
D
12
+
B
c
2
D
22
)
C
c
B
c
1
C
1
D
12
C
c
1
⎡
⎣
⎤
⎦
.
A
B
2
C
c
0
0
B
c
1
C
1
+
B
c
2
C
2
A
c
+
(
B
c
1
D
12
+
B
c
2
D
22
)
C
c
0
B
c
1
=
(4.6)
B
w
C
1
B
w
D
12
C
c
A
w
B
w
0
0
C
w
0
It is worth noting that the formulae (4.4), (4.5) and (4.6) are valid for the general case, regard-
less of the dimension of the measurement vector, which is the scalar
i
o
here, and for any
W
and
W
u
.
4.4 Design Example
The parameters of a system are shown in Table 4.1. The switching frequency for the IGBT
bridge is 10 kHz. A nearly optimal controller, for which the Bode plots are shown in Figure
4.4, was obtained for
W
u
=
−
10000
0
.
05
,
W
−
10000
0
,
1
10000
5. The
Bode plots show that this controller is not realistic, because it has a very large bandwidth.
Normally, the high-frequency poles can be reduced using various controller/model reduction
techniques, which is a topic of wide interest. Here, a different approach from that used in
(Naim
et al.
1997; Zhong
et al.
2004, 2006) is adopted to decrease the controller bandwidth:
instead of minimising the
H
∞
-norm of
=
ξ
=
37 and
μ
=
0
.
−
500
1
F
l
(
P
,
C
) but finding a suboptimal controller
C
such
F
l
(
P
that
,
C
)
∞
≤
25 (which is larger than the minimal value 17
.
37). Such a controller that
was obtained using the
hinfsyn
routine in MATLAB
R
is:
⎡
⎣
⎤
⎦
T
405(
s
2
10
6
)(
s
2
10
7
)
156751
.
+
2172
s
+
1
.
185
×
+
5274
s
+
3
.
767
×
(
s
+
9533)(
s
+
2331)(
s
+
989
.
2)(
s
2
+
2
.
76
×
10
4
s
+
3
.
07
×
10
8
)
C
(
s
)
=
.
(4.7)
10
4
)(
s
266474
.
3105(
s
+
+
2958)(
s
+
989
.
3)(
s
+
20
.
43)
(
s
+
9533)(
s
+
2331)(
s
+
989
.
2)(
s
2
+
2
.
76
×
10
4
s
+
3
.
07
×
10
8
)
Table 4.1
Parameters of the inverter for simulation
Parameter
Value
Parameter
Value
R
f
0
.
053
R
g
0
.
1
L
f
1
.
3mH
L
g
0
.
3mH
r
f
30
.
5
r
g
7
R
5
C
50
μ
F
L
5mH
u
g
230 V
,
50 Hz
r
500
V
DC
850 V
Search WWH ::
Custom Search