Environmental Engineering Reference
In-Depth Information
Combining equations (3.8), (3.9) and (3.10), the realisation of the generalised plant is then
obtained as
⎡
⎤
A
0
0
B
1
B
2
⎣
⎦
B
w
ξ
B
w
C
1
A
w
B
w
D
1
B
w
D
2
P
=
0
C
w
0
0
0
.
(3.11)
μ
-----------------------------------------------
C
1
0
0
0
0
0
ξ
D
1
D
2
The stabilising controller
C
can be calculated using the results on
H
∞
controller design (Zhou
et al.
1996) for the generalised plant
P
.
3.2.3 Evaluation of the System Stability
According to (Weiss and Hafele 1999; Weiss
et al.
2004), the closed-loop system in Figure 3.2 is
exponentially stable if the closed-loop system from Figure 3.4 is stable and its transfer function
from
a
to
b
, denoted
T
ba
, satisfies
T
ba
∞
<
1. Assume that the state-space realisation of the
controller is
A
c
B
c
C
c
D
c
C
=
.
(3.12)
Note that the controller obtained from the
H
∞
design is always strictly proper. However, after
controller reduction, the reduced controller may not be strictly proper. The realisation of the
transfer function from
a
to
b
, assuming that
w
=
0 and noting that
D
2
=
0, can be found as
follows:
1
A B
2
C
1
D
2
C
−
1
T
ba
=
−
W
1
A
A
c
B
c
C
c
D
c
−
1
A
w
B
2
B
w
=
−
C
1
0
C
w
0
⎡
⎣
⎤
⎦
.
A
+
B
2
D
c
C
1
B
2
C
c
B
2
D
c
C
w
0
B
c
C
1
A
c
B
c
C
w
0
=
0
0
A
w
B
w
C
1
0
C
w
0
Once the controller
C
is obtained, the stability of the system can be verified by checking
T
ba
∞
. The realisation of the transfer function from
w
to
e
, assuming that
v
=
0, can be
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