Environmental Engineering Reference
In-Depth Information
4.3.3 State-space Realisation of
T
e
w
Denote the central sub-optimal controller for a given
H
∞
-norm of
T
z
w
by
A
c
B
c
1
B
c
2
C
c
=
=
[
C
1
C
2
]
.
C
00
Note that the feed-through matrix is equal to 0. Assume in Figure 4.3 that
v
1
=
0 and
v
2
=
0
,
then
u
=
C
c
x
c
, where
x
c
satisfies
x
c
=
A
c
x
c
+
B
c
1
e
+
B
c
2
i
o
.
Substitute
u
=
C
c
x
c
into (4.1), then
x
=
+
B
2
C
c
x
c
+
B
1
w
Ax
and, from (4.2),
e
=
C
1
x
+
D
12
C
c
x
c
+
D
11
w,
i
o
=
C
2
x
+
D
22
C
c
x
c
+
D
21
w.
Furthermore,
x
c
=
(
A
c
+
B
c
1
D
12
C
c
+
B
c
2
D
22
C
c
)
x
c
+
(
B
c
1
C
1
+
B
c
2
C
2
)
x
+
(
B
c
1
D
11
+
B
c
2
D
21
)
w.
Hence, the transfer matrix from
w
to
e
is
⎡
⎤
A
B
2
C
c
B
1
⎣
⎦
.
T
e
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
D
11
+
B
c
2
D
21
(4.5)
C
1
D
12
C
c
D
11
4.3.4
State-space Realisation of
T
ba
Assume
w
=
0 and
v
2
=
0 in Figure 4.3, then
x
=
+
,
=
+
,
i
o
=
+
Ax
B
2
u
e
C
1
x
D
12
u
C
2
x
D
22
u
and
u
=
C
c
x
c
.
Hence
e
=
C
1
x
+
D
12
C
c
x
c
and
x
c
=
A
c
x
c
+
B
c
1
(
e
+
a
)
+
B
c
2
i
o
=
A
c
x
c
+
(
B
c
1
C
1
+
B
c
2
C
2
)
x
+
(
B
c
1
D
12
+
B
c
2
D
22
)
C
c
x
c
+
B
c
1
a
.
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