Environmental Engineering Reference
In-Depth Information
Assume that the state-space realisation of the weighting function
W
v
is
A
v
B
v
W
v
=
.
C
v
D
v
Then
A
v
V
a
v
e
B
v
V
v
=
C
v
D
v
A
v
A B
1
B
2
C
a
D
1
a
D
2
a
u
B
v
=
C
v
D
v
⎡
⎤
u
A
0
B
1
B
2
⎣
⎦
=
B
v
C
a
A
v
B
v
D
1
a
B
v
D
2
a
.
(12.10)
D
v
C
a
C
v
D
v
D
1
a
D
v
D
2
a
Similarly, assume that the state-space realisation of the weighting function
W
u
is
A
u
B
u
C
u
D
u
W
u
=
,
then
A
u
B
u
C
u
D
u
u
N
V
u
=
⎡
⎤
u
A
0
B
1
B
2
B
u
C
1
b
A
u
B
u
D
11
b
B
u
D
12
b
D
u
C
1
b
C
u
D
u
D
11
b
D
u
D
12
b
⎣
⎦
=
.
(12.11)
Moreover, assume that the implementation of the low-pass filter
F
(
s
)is
A
F
B
F
C
F
D
F
F
(
s
)
=
,
then
A
F
B
F
C
F
D
F
i
c
+
ζ
V
i
=
I
n
A
F
B
F
C
F
D
F
A B
1
B
2
C
2
b
D
21
b
D
22
b
u
=
+
ζ
I
n
⎡
⎣
⎤
⎦
u
A
0
B
1
B
2
=
B
F
C
2
b
A
F
B
F
D
21
b
B
F
D
22
b
.
(12.12)
D
F
C
2
b
C
F
D
F
D
21
b
+
00
ζ
D
F
D
22
b
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