Hardware Reference
In-Depth Information
frequencies, there is a need to define the second phase margin and gain margin.
1.5
1
0.5
0
−0.5
−1
−2
−1.5
−1
−0.5
0
0.5
1
Figure 3.25: Phase stable design Nyquist plot. Solid line: notch based design.
Dashed-line: phase stable based design.
De
fi
nition: The second phase margin of a Nyquist plot is an angle from the
negative real axis to a vector from the origin to the point where the Nyquist
plot crosses the unit circle from the inside to the outside [114].
For the example system we have above, the second phase margin is about
40
◦
. In [114], it is recommended that the phase of an open loop characteristic
could be secured within −360
◦
±90
◦
at the main resonance frequency, a phase
margin could be designed at 30
◦
or more, and a second phase margin could be
secured at 40
◦
or more.
Using this phase stable design approach, because the resonance mode is
not compensated, there is a high suppression of the vibration at this frequency
as shown in Figure 3.22. Notch filter based design is relatively flat for this
frequency as can be seen in Figure 3.10. The phase-stable design, however,
does not suppress the oscillation induced by step change in reference com-
mand. This issue can be resolved by using two degree-of-freedom (2-DOF)
controller, discussed in section 3.5.1, where proper shaping of the reference
input command is used to reduce the residual vibration of the actuator.
Like in all other design, the robustness of the compensator obtained us-
ing phase-stable design approach should be examined carefully. As have been
discussed in most literature [90], the actuator resonant frequency may have
a ±5% variation from part to part or due to changes in environmental con-
ditions. The damping of the actuator resonant mode is also variable. These
factors could make a nominally stable phase-stable design unstable. Actua-
