Environmental Engineering Reference
In-Depth Information
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
w
i
N
V
0
n
100
0
w
=
ρ
to
z
, which is
Hence, the closed-loop transfer function from
0
00
ζ
the relevant closed-loop transfer function in terms of the original variables, is
⎡
⎣
⎤
⎦
.
10 0
0
1
ρ
0
00
1
ζ
T
z
w
=
T
z
w
12.3 Selection of Weighting Functions
It is not easy to choose suitable weighting functions for a specific
H
∞
control problem. These
functions have to reflect the relative weight of different signals, their frequency characteristics,
and at the same time they must be chosen so that the problem is solvable. It is difficult to
find specific guidelines in the literature although some general guidelines can be found in, e.g.
(Green and Limebeer 1995; Zhou and Doyle 1998).
The weighting function
W
v
has to be large for the range of significant disturbance frequencies
(50 Hz and a few multiples of it) and it has to decay for high frequencies, which cannot be
controlled. Anyway, no significant high-frequency variations of
V
a
v
e
is expected because of
the two capacitors. The weighting function
W
v
(
s
) can then be chosen as
−
ω
l
g
ω
h
−
ω
l
g
g
s
+
ω
h
W
v
(
s
)
=
+
ω
l
=
,
s
where
g
is a tuning parameter to move the Bode plot of
W
v
up or down.
The signal
u
N
is closely related (almost proportional) to
u
p
. In order to prevent
u
from
becoming too large, especially at high frequencies,
W
u
is chosen small at low frequencies
and large at high frequencies. Moreover, the normal
H
∞
algorithm requires that the matrix
=
D
v
D
2
a
D
u
D
12
b
has full column rank. Since
D
2
a
=
0,
D
u
cannot be zero. Hence,
W
u
has to be
non-strictly proper. It can be chosen as
−
ω
h
k
s
+
ω
l
k
W
u
(
s
)
=
+
ω
h
=
,
ω
l
−
ω
h
k
s
where
k
is a tuning parameter to move the Bode plot of
W
u
up or down. In this application, the
fundamental frequency is 50 Hz and the high frequency components under consideration can
be up to the 31st harmonics. Therefore,
000 rad/s. The Bode plots of
the weighting functions
W
u
(
s
) and
W
v
(
s
) are shown in Figure 12.5 for
k
ω
h
is chosen to be 10
,
=
0
.
01 and
g
=
10.
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