Environmental Engineering Reference
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respectively. Another two weighting factors
I
0
and
I
n
, which are proportional to
V
0
and
n
via the factors
, are also introduced to adjust the relative importance of the three
disturbances
i
N
,
V
0
and
n
in the
H
∞
-norm minimisation process. The
H
∞
control problem is
then formulated to minimise the
H
∞
-norm of the transfer function from
ρ
and
ζ
w
=
i
N
I
0
I
n
T
to
=
V
v
V
u
T
,
denoted
T
z
w
=
F
l
(
P
,
z
K
), as shown in Figure 12.4. The closed-loop system
can be represented as
z
y
P
u
=
,
u
=
K
y
,
where
P
is the generalised plant and
K
is the controller to be designed. A nearly optimal
K
can
then be obtained with the standard
H
∞
control algorithm; see (Zhou and Doyle 1998; Green
and Limebeer 1995) for details.
12.2.1 State-space Realisation of
P
If the state variables of the original plant are chosen to be the inductor current
i
L
and the
voltage
V
c
=
, and if the control input
u
is
p
, then the following state
equations can be obtained from Figure 12.4:
i
L
V
c
V
a
v
e
−
V
0
, i.e.
x
=
x
=
Ax
+
B
1
w
+
B
2
u
(12.8)
with
⎡
⎤
⎡
⎤
L
N
⎡
⎤
R
N
L
N
1
L
N
V
DC
2
L
N
0
0
−
⎣
⎦
,
⎣
⎦
,
⎣
⎦
.
A
=
B
1
=
B
2
=
1
C
N
1
+
1
C
N
1
+
00
−
0
0
C
N
2
C
N
2
The output equations are
V
a
v
e
=
C
a
x
+
D
1
a
w
+
D
2
a
u
,
(12.9)
u
N
=
C
1
b
x
+
D
11
b
w
+
D
12
b
u
,
i
c
=
C
2
b
x
+
D
21
b
w
+
D
22
b
u
,
with
C
a
=
01
,
D
1
a
=
0
0
,
D
2
a
=
0
,
ρ
V
DC
2
C
1
b
=
01
,
D
11
b
=
0
0
,
ρ
D
12
b
=
,
C
2
b
=
−
10
,
D
21
b
=
100
,
D
22
b
=
0
.
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