Hardware Reference
In-Depth Information
Command Shaping
Use of notch filter improves the performance of the servomechanism during
track-following, but it does not solve the problem when the resonances are
excited by large jumps in the control signal from one polarity to another during
the execution of any seek algorithm, e.g., the PTOS. Such excitation of the
resonant modes is inevitable, but as long as they diminish before the end of
seek, the tracking performance is not affected much. With continuing demand
for lighter actuators, the resonant modes continue to be less and less damped
and, therefore, it takes longer for the modes to decay. The problem of switching
induced vibration is more severe for short seek as the seek algorithm ends before
the vibration is reduced to negligible amplitude. These vibrations also extend
the access time as it takes longer for the heads to settle on the target track.
The switching-induced vibration can be removed using a scheme called
Command Shaping or Input Shaping [176]. In this method, the reference track
is not changed abruptly at the beginning of the seek operation, but a filtered
reference command is injected into the servo loop. This command generation
scheme convolves the reference command with a sequence of impulses called the
input shaper to create a command signal that cancels the vibration produced
by the system to which it is applied. The concept of the command generation
is explained next using a simple example.
If an impulse input is applied to an underdamped second order system at
t = 0 then its response is a decaying sinusoid. After the application of the
first impulse, if a second impulse, with suitable relative amplitude and phase
with respect to the first, is applied, the oscillation due to the first impulse can
be completely eliminated by the response to the second impulse. The relative
amplitude and the phase of the second impulse can be determined considering
the resultant output as the superposition of the responses to two impulses such
that the condition for no residual vibration is satisfied. The relative amplitude
and phase depend on the natural frequency and the damping coefficient of
the resonant mode to be suppressed. Let us consider the simplest case with
only one resonant mode. If the undamped natural frequency and the damping
coefficient are ω
0
and ζ, respectively, then two impulses of magnitude
1
1+K
K
1+K
separated by ∆T satisfy the condition of zero-vibration for ∀t>∆T
where,
and
−
ζπ
√
K = e
1
−ζ
2
π
∆T =
p
1−ζ
2
.
(2.61)
ω
0
This sequence of impulses can be convolved to an arbitrary reference command
to obtain vibration reduced output. For example, if the actual reference com-
mand is a step function, the convolved reference becomes a staircase signal.
The responses of an under damped second order system to a step input and
the corresponding staircase input are shown in Figure 2.35. The dotted lines
