Environmental Engineering Reference
In-Depth Information
Combining equations from (5.7) to (5.9), the generalised plant is then realised as
⎡
⎤
A
B
2
C
d
0
0
B
1
B
2
D
d
⎣
⎦
0
A
d
0
0
0
B
d
B
w
ξ
B
w
C
1
B
w
D
2
C
d
A
w
B
w
D
1
B
w
D
2
D
d
P
=
,
(5.10)
D
w
C
1
D
w
D
2
C
d
C
w
D
w
ξ
D
w
D
1
D
w
D
2
D
d
μ
---------------------------------------------------------------------
C
1
0
0
0
0
0
D
2
C
d
0
ξ
D
1
D
2
D
d
for which the stabilising controller
C
can be calculated using the well-known results on
H
∞
controller design (Zhou
et al.
1996).
5.2.4 Evaluation of System Stability
According to (Weiss and Hafele 1999; Weiss
et al.
2004), the closed-loop system in Figure 5.2
is exponentially stable if the closed-loop system from Figure 5.5 is stable and its transfer
function from
a
to
b
, denoted
T
ba
, satisfies
1.
Assume that the state-space realisation of the controller is
T
ba
∞
<
A
c
B
c
C
c
C
=
.
0
Note that the optimal controller obtained from the
H
∞
design is always strictly proper. The
realisation of the transfer function from
a
to
b
, assuming that
w
=
0, can be found as follows:
1
A B
2
C
1
D
2
W
d
C
−
1
T
ba
=
−
W
1
A B
2
C
1
D
2
A
d
B
d
C
d
D
d
A
c
B
c
C
c
−
1
A
w
B
w
=
−
0
C
w
0
⎡
⎤
−
1
A
2
C
d
B
2
D
d
C
c
0
A
w
⎣
⎦
0
A
d
B
d
C
c
0
B
w
=
0
0
A
c
−
B
c
C
w
0
C
1
D
2
C
d
D
2
D
d
C
c
1
⎡
⎤
A
2
C
d
B
2
D
d
C
c
0
A
w
⎣
⎦
0
A
d
B
d
C
c
0
B
w
=
B
c
C
1
B
c
D
2
C
d
A
c
+
B
c
D
2
D
d
C
c
B
c
C
w
0
C
1
D
2
C
d
D
2
D
d
C
c
1
⎡
⎣
⎤
⎦
A
2
C
d
B
2
D
d
C
c
0
0
0
A
d
B
d
C
c
0
0
A
c
+
=
B
c
C
1
B
c
D
2
C
d
B
c
D
2
D
d
C
c
B
c
C
w
0
.
0
0
0
A
w
B
w
C
1
D
2
C
d
D
2
D
d
C
c
C
w
0
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