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,
area above the upper dash-dot line,
area below the lower dash-dot line,
area bounded by two dash-dot lines and on the right of e l ,
area bounded by two dash-dot lines and on the left of −e 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
s 2 = 3.6 × 10 7
ItisverycommonintheHDDindustrytouseTracks as the unit of displace-
ment instead of metres or µm; the simulation results shown here also use this
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
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