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linear control deteriorates under such condition of control saturation. The time
optimal control or bang-bang control is a well known solution for point-to-point
maneuver with limited control authority. Practical realization of the bang-bang
control has several drawbacks such as control chatter and sensitivity to para-
meter variations. In practice, the head positioning servomechanism of HDD
uses variants of bang-bang control that achieve near-time-optimal performance
but avoid problems associated with time-optimal control. These solutions of-
ten combine time-optimal control for large errors with linear control for small
errors and ensure smooth transfer between the two.
2.6.1 Time Optimal Control
The nominal dynamics of the VCM actuator can be modeled as a double
integrator
y = au,
(2.33)
with an upper bound on the amplitude of the control input, i.e., |u| ≤ U m .
Since the magnitude of the input is upper bounded, the magnitude of the
acceleration is also limited by a maximum value (|y| ≤ aU m ). The objective of
the seek control is to move the states of the head positioning servomechanism
from its initial values to some final values in the shortest possible time. The
state vector (x) includes the position (y) of the read-write head and the velocity
(v). If we assume the actuator to be at rest before and after such minimum-
time maneuver then v i = v f = 0. So the seek control transfers the state
vector from [y i 0] T to [y r 0] T in minimum time. Defining the position error
e = y r −y, the double integrator model is re-written as
e = −au.
(2.34)
Then the time-optimal seek control brings the error from e(0) = y r − y i at
t =0toe(T f )=0sothatT f is minimized. Simple analysis of Newton's 2 nd
law of motion reveals that for a double integrator, time-optimal maneuver is
achieved by applying maximum acceleration to the actuator for exactly half of
its travel and then decelerating with maximum effort for the remaining half of
the travel. This is equivalent to driving the actuator with +U m (or −U m )for
half of the travel followed by −U m (or +U m ) for the remaining half. The sign of
the accelerating or decelerating input depends on the direction of displacement
the head is expected to go through. Many text books such as [23] and [11]
provide in-depth analysis of the time-optimal control law, not only for a double
integrator model but also for a general plant model.
If the actuator of equation 2.34 is subject to maximum acceleration for T
second followed by maximum deceleration for another T second then the total
displacement of the read/write head is aU m T 2 (according to Newton's 2 nd law

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