Hardware Reference

In-Depth Information

f
PTOS
reduces to the velocity profile of time-optimal control with magnitude

shifted by U
m
/k
2
.

The velocity profile of equation 2.40 and the control law of equation 2.41

divides the phase plane into 5 regions. These are labeled in Figure 2.28 as,

S
+

:

area above the upper dash-dot line,

S
−

:

area below the lower dash-dot line,

U
+

:

area bounded by two dash-dot lines and on the right of e
l
,

U
−

:

area bounded by two dash-dot lines and on the left of −e
l
,

L

:

area bounded by two dash-dot lines and two vertical dotted lines.

When the state trajectory is inside the linear region L, the velocity profile

is a linear function of the error (f
PTOS
=
k
k
2
e), and the control signal is u =

k
1
e+k
2
e which is same as the linear state feedback control of equation 2.39 u
l
=

k
1
x
1
+ k
2
x
2
. The areas labeled S
+
and S
−
are the areas of saturated control.

If the state trajectory is in one of these areas, then maximum control effort is

applied (U
m
in S
+
and −U
m
in S
−
). Such an input forces the state trajectory

move in a parabolic path to reach eventually the band around the velocity

profile f
PTOS
,markedU
+
or U
−
or L. The control signal is unsaturated

when the trajectory is inside the areas marked U
+
or U
−
, but it is computed

using nonlinear feedback rule. A trajectory, once inside U
+
or U
−
,remains

inside this band and finally enters the linear region L in finite time. Nature of

the trajectory inside the region L depends on the state feedback gains k
1
and

k
2
.

Simulated results for PTOS appliedtoaVCMactuatorareshowninFig-

ure 2.29. The double integrator model identified in section 2.2.2 is used for the

simulation,

³

´

s
2
=
3.6
×
10
7

G(s)=
k

µm

Volts

.

(2.44)

s
2

ItisverycommonintheHDDindustrytouseTracks as the unit of displace-

ment instead of metres or µm; the simulation results shown here also use this

unit.

Figure 2.29 shows displacement, control input, velocity, and phase-plane

for a 100-track seek. It is evident from the plot of the control input (u)that

saturated control is used only during the acceleration phase. Once the phase-

plane trajectory enters the band of unsaturated control, the control signal is

less than the maximum value and the control loop tries to make the trajectory

follow the deceleration profile of PTOS. Trajectory, deceleration profile and

the upper and the lower bounds are illustrated in Figure 2.30. Note that the

phase plane is plotted as velocity of the head-slider versus position error, not

de

dt
versus e showninFigure2.28.

If tracks per second is used as the unit of the velocity of head-slider then the

magnitude of velocity is extremely large. It is a common practice to express