Hardware Reference
In-Depth Information
where
1
1+G
p
G
c
,
S
0
=
(3.26)
1
1+T
0
C
p
,
S
F
=
(3.27)
G
p
G
c
1+G
p
G
c
.
T
0
=
(3.28)
The transfer function S
O
is automatically defined once the nominal controller
G
c
(s) is chosen. It is easily deduced from the above equations that by using the
controller structure of Figure 3.26, the narrow band disturbance compensator
C
p
can be designed to shape S
F
so that overall sensitivity transfer function
satisfy the desired performance [229].
Let us assume that the desired magnitudes of the peak and the baseline of
the frequency response of the peak filter are M dB and N dB, respectively, as
shown in Figure 3.27. Then a suitable peak filter can be constructed in the
s-domain according to the following [213],
C
p
(s)=
s
2
+2ζ
1
ω
p
s +ω
p
s
2
+2ζ
2
ω
p
s +ω
p
(3.29)
with
p
∆
2
+2∆
2(1+∆)
ζ
1
=
n
2
−1
ζ
2
=
ζ
1
m
(3.30)
θ = an
−1
m
−
1
2
√
m
where ω
p
is the center frequency of the peak filter in rad/sec, ζ
1
and ζ
2
are
the damping ratios with ζ
1
> ζ
2
, ∆ is the percentage of variation in center
frequency, m =10
M/20
,andn =10
N/20
. Possible phase loss due to inclusion
of the peak filter is estimated by θ. The inclusion of peak filter modifies both
the gain and phase of the open loop transfer function.
This design method is now illustrated with the help of the same plant
model used in previous sections. Let the desired magnitude at the peak and
the baseline of the bode plot of the peak filter be M=40 dB at 360 Hz and N=2
dB. To keep the 0-dB crossover frequency same as in the earlier designs, the
gain of G
c
is reduced by half to offset the gain introduced by C
p
(s). The bode
plots of L,andS are shown in Figures 3.28 and 3.29. The open loop frequency
response for this design shows an increase in gain by approximately 40 dB at
the disturbance frequency 360 Hz compared to the design that includes PID-
type controller and notch filter. This additional gain is provided by the peak
filter, which is effectively a bandpass filter. The increase in gain of the open
