Hardware Reference
In-Depth Information
The transfer function of the HDD servomechanism plant can be described
by the model [113]:
G
p
(s)=kP
d
[P
0
+ P
m
],
∙
e
−T
d
s
T
amp
s +1
r
0
(s
2
+2ζ
m0
ω
m0
s +ω
2
m0
)
= k
#
X
N
a
r
mi
(s
2
+2ζ
mi
ω
mi
s +ω
2
mi
)
+
,
(3.5)
i=1
where the loop gain k includes gains of various stages of the servo plant e.g.
the DAC (Digital-to-Analog Converter) gain
∗
,amplifier gain, torque gain, mass
and position gain. The transfer function P
d
(s)=
T
amp
s+1
e
−T
d
s
represents both
the dynamics of power amplifier with time constant T
amp
and the computa-
tional delay T
d
. The rigid body model of the actuator coupled with linearized
pivot friction is modeled as P
0
(s), where as P
m
(s)=
1
P
N
a
i=1
r
mi
s
2
+2ζ
mi
ω
mi
s+ω
2
mi
represents N
a
modes of mechanical resonances.
The computational delay in a typical HDD servomechanism is of the order
of T
d
=15µs. The phase lag of such a delay term is 2.7
◦
at 500 Hz and
5.4
◦
at 1 kHz. The phase lag of a typical amplifier with 40 kHz bandwidth
is 0.72
◦
at 500 Hz and 1.43
◦
at 1 kHz. Hence these dynamics are ignored
in the nominal model used for design of controller, but their effects must be
taken into consideration while evaluating the performances of the designed
closed loop system. For the sake of simplicity, only the first resonance mode is
considered in the design examples presented in this chapter, i.e., i =1inthe
model of equation 3.5, and the pivot friction is ignored so that P
0
=
s
2
.
The series of design examples begins with a simple PID type controller
followed by few compensators to be used with the basic PID type controller
for enhancement of performance.
3.2.1 Basic PID-type Controller
In order to design the nominal controller, which is the firststepintheprocess
of designing practical servo controller, the actuator resonances, power amplifier
time constant, and the delay caused by the digital control, shown in Figure 3.5,
are ignored. A rigid body, double integrator model k/s
2
with−40 dB/dec slope
is used to represent the nominal model of the servo plant. This simplification is
reasonable when the desired open loop servo bandwidth is very low compared
to the frequency of actuator resonance, power amplifier bandwidth, and the
sampling frequency so that the ignored dynamics contribute negligible gain
(approximately 0 dB) and phase (close to 0
◦
) for frequencies below the servo
bandwidth. For example, the resonant mode at ω
m1
=2π×5000 rad/s with a
damping ratio of 0.02 adds only
−0.23
◦
phase and 0.0101 dB gain at 500 Hz.
∗
the controller for modern HDD servomechanism is always implemented in discrete time
