Environmental Engineering Reference
In-Depth Information
between the current reference and the current injected into the grid. The plant
P
can then be
described by the state equation
x
=
Ax
+
B
1
w
+
B
2
u
(3.2)
and the output equation
y
=
e
=
C
1
x
+
D
1
w
+
D
2
u
(3.3)
with
⎡
⎤
R
f
+
R
d
R
d
L
f
1
L
f
−
−
⎣
⎦
L
f
R
d
L
g
R
g
+
R
d
1
L
g
−
A
=
,
L
g
1
C
f
1
C
f
−
0
⎡
⎣
⎤
⎦
1
L
f
⎡
⎣
⎤
⎦
,
0
1
L
f
0
0
1
L
g
B
1
=
,
B
2
=
−
0
00
C
1
=
0
10
,
D
1
=
01
,
−
D
2
=
0
.
The corresponding plant transfer function is then
=
D
1
D
2
+
A
)
−
1
B
1
B
2
.
P
C
1
(
sI
−
(3.4)
In the sequel, the following notation is used for transfer functions:
A B
1
B
2
C
1
D
1
D
2
P
=
.
(3.5)
Detailed manipulation of state-space realisations of systems can be found in (Zhong 2006).
3.2.2 Formulation of the Standard H
∞
Problem
In order to guarantee the stability of the system, an
H
∞
control problem, as shown in Figure 3.4,
is formulated to minimise the
H
∞
l
(
P
norm of the transfer function
T
z
w
=
F
,
w
=
C
) from ˜
[
[
z
1
z
2
]
T
, after opening the local positive feedback loop of the internal model
and introducing weighting parameters
vw
]
T
to
z
=
ξ
and
μ
. The closed-loop system can be represented as
z
y
P
˜
u
=
,
(3.6)
u
=
C y
,
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