Hardware Reference
In-Depth Information
of motion). For a given initial error e(0), the time required to complete a
time-optimal maneuver is
s
e(0)
aU m .
T =
(2.35)
Open loop realization of the bang-bang control involves (1) pre calculation
of T of equation 2.35, (2) applying maximum acceleration for T/2 seconds
and (3) switching to deceleration and applying maximum deceleration for the
remaining T/2 seconds. Closed loop realization, on the other hand, controls the
velocity to follow a pre-defined deceleration velocity profile which is a function
of error
p
f toc = −sgn(e)
2aU m |e|.
(2.36)
When the actuator is subject to maximum deceleration from any initial state,
the state trajectory follows a path parallel to this profileshowninthephase
plane of Figure 2.24. The solid line in this figure represents the deceleration
velocity profile. The trajectories x 1 and x 2 (dashed lines) represent vectors
of error and error velocity for 2 different initial conditions but both with u =
+U m , while the trajectories x 3 and x 4 (dotted lines) are traced when u = −U m .
For u =+U m , the trajectories from any initial condition follow a path
parallel to that of x 1 or x 2 . The upper part of the deceleration error profile
is also a trajectory with u =+U m and it passes through the origin of the
phase plane. Similarly, u = −U m from any initial condition traces a trajectory
parallel to that of x 3 or x 4 , and the lower part of the deceleration trajectory is
one of them. From any initial condition not lying on the deceleration velocity
profile, it is possible with appropriate polarity of input u to make the trajectory
eventually intersect the deceleration profile. If u = −U m is applied when the
initial condition lies anywhere in the region below the deceleration profile,
the trajectory will intersect in finite time with the positive segment of the
deceleration profile. Similarly for any initial condition above the solid line,
u =+U m ensures intersecting the negative half of the velocity profile. If
switching from acceleration to deceleration is performed when the state vector
[e e] T is exactly on the deceleration profile, then the trajectory moves along
this curve and reaches the origin [0 0] T in finite time.
The actuator can be assumed to be at rest at the beginning of a track-to-
track seek maneuver, that is, the initial state lies on the horizontal axis. If
maximum acceleration is applied, the state trajectory follows a parabolic path
until it meets the deceleration profile when control is switched to the deceler-
ating phase so that the trajectory follows the deceleration velocity profile and
reaches [0 0] T in finite time. The control input is set to zero as soon as the
origin is reached. The control signal is generated using the rule
u = U m sgn(f toc − e),
(2.37)
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