Environmental Engineering Reference
In-Depth Information
The corresponding plant transfer function is
⎡
⎤
⎦
=
P
y
w
A
B
1
B
2
P
yu
.
A
)
−
1
B
⎣
P
=
C
(
sI
−
+
D
=
C
1
D
11
D
12
(4.3)
C
2
D
21
D
22
4.3 Controller Design
4.3.1 Formulation of the H
∞
Control Problem
The
H
∞
control-based design procedure for repetitive controllers proposed in (Weiss and
Hafele 1999) can be applied to design the controller. It uses additional measurement informa-
tion from the plant. The block diagram of the control system is shown in Figure 4.2, where the
controller consists of an internal model
M
and a stabilising compensator
C
. The stabilising
compensator assures the exponential stability of the entire system, which implies that the
error
e
converges to a small steady-state error, according to (Weiss and Hafele 1999). The
three external signals (the components of
w
) are assumed to be periodic, with a fundamental
frequency of 50 Hz.
According to Chapter 2, the internal model
M
is obtained from a low-pass filter
W
(
s
)
=
ω
c
10000 rad/sec, cascaded with a delay element
e
−
τ
d
s
, where
with
ω
c
=
τ
d
is slightly
s
+
ω
c
less than the fundamental period
τ
=
20 ms given as
1
ω
c
=
τ
d
=
τ
−
19
.
9ms
.
A
w
0
−
ω
c
0
. The choice of
B
w
ω
c
A realisation of
W
is
W
=
=
ω
c
is based on a compromise:
C
w
1
if
ω
c
is too low, only a few poles of the internal model are close to the imaginary axis, leading
to poor tracking and disturbance rejection at higher frequencies; if
ω
c
is too high, the system is
−
s
W
(
s
)
e
d
P
i
d
+
e
u
g
w
plant
+
in
ternal
m
odel
u
ref
M
u
i
o
p
stabilising
compensator
C
Figure 4.2
Repetitive control for voltage tracking
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