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)