Hardware Reference
In-Depth Information
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
Time (ms)
Figure 3.19: Closed-loop step response corresponding to Figure 3.15. Note
that after using the notch filter, the step response is more oscillatory.
nomenon is also evident in the smooth plot of the complementary sensitivity
transfer function shown in Figure 3.10. However, oscillation caused by a step
change in input disturbance is not affected by inclusion of this filter. This
can be easily concluded by observing the shock transfer function, which still
shows peak at the frequency of actuator resonance (Figure 3.11). Therefore,
any oscillation caused by a disturbance entering the loop at the input, such
as windage induced vibrations, is not sufficiently suppressed at the frequencies
of actuator's structural resonant modes. Furthermore, use of notch filter to
suppress resonances leads to higher order controllers, requiring more compu-
tational power for realization and thus causing higher cost of implementation.
An aternative method of suppressing resonance with a fairly low order
controller was proposed in [114] which was named as the phase stable design.
We can explain the underlying concept using an example and let us consider
again the plant model of equation 3.5. In the phase stable design, a low-
order compensator C(s) such as a PI-lead compensator which can take the
form of equation 3.9 is used to provide phase lead in the region near the 0
dB crossover frequency and to lift the gain in low frequency range. While a
notch filter reduces the open loop gain in the frequencies around the resonant
frequency, in the phase stable design the filter
1
T
ps
s +1
,
F(s)=F
ps
=
(3.24)
added to the controller C(s) makes the phase and gain of the open loop trans-
fer function about 360
◦
and higher than 0 dB at the resonant frequency. This
causes the sensitivity transfer function to have a notch which provides extra
vibration suppression at the actuator resonant frequency. This design also pro-
vides attenuation of the peak in the shock transfer function near the resonant
