Environmental Engineering Reference
In-Depth Information
Combining (12.8), (12.9), (12.10), (12.11) and (12.12), the state-space realisation of the
generalised plant
P
, with inputs
w
,
u
and outputs
z
,
y
, is obtained as
⎡
⎣
⎤
⎦
A
00 0
B
1
B
2
B
v
C
a
A
v
00
B
v
D
1
a
B
v
D
2
a
B
u
C
1
b
0
A
u
0
B
u
D
11
b
B
u
D
12
b
B
F
C
2
b
00
A
F
B
F
D
21
b
B
F
D
22
b
D
v
C
a
C
v
00
D
v
D
1
a
D
v
D
2
a
D
u
C
1
b
0
C
u
0
D
u
D
11
b
D
u
D
12
b
------------------------------------------------------------------------
C
a
P
=
.
00 0
D
1
a
D
2
a
00
C
F
D
F
D
21
b
+
00
ζ
D
F
D
22
b
D
F
C
2
b
12.2.2 State-space Realisation of the Closed-loop Transfer Function
Denote by
P
the transfer function from
u
to
z
y
. Then
⎡
⎤
A
0
B
1
B
2
B
F
C
2
b
A
F
B
F
D
21
b
B
F
D
22
b
C
a
0
D
1
a
D
2
a
C
1
b
0
D
11
b
D
12
b
-----------------------------------------------------------
C
a
⎣
⎦
⎡
⎤
A
B
1
B
2
=
P
⎣
⎦
.
C
1
D
11
D
12
=
C
2
D
21
D
22
0
D
1
a
D
2
a
D
F
C
2
b
C
F
D
F
D
21
b
+
00
ζ
D
F
D
22
b
Assume that the controller is realised as
A
K
B
K
C
K
D
K
K
=
,
0if
K
is obtained from the standard
H
∞
algorithm. However,
D
K
may
be non-zero after controller reduction. According to the star-product formula (Zhou and Doyle
1998), the transfer function from
where, usually,
D
K
=
w
to
z
is
⎡
⎤
A
B
2
RD
K
C
2
B
2
RC
K
B
1
+
B
2
RD
K
D
21
+
T
z
w
=
F
l
(
P
⎣
B
K
R C
2
B
K
R D
22
C
K
B
K
R D
21
⎦
,
,
=
A
K
+
K
)
C
1
+
D
12
D
K
R C
2
D
12
RC
K
D
11
+
D
12
D
K
R D
21
where
R
D
22
)
−
1
D
22
D
K
)
−
1
=
(
I
−
D
K
,
R
=
(
I
−
.
Search WWH ::
Custom Search