Internal Process Controls - Control of Color Imaging Systems: Analysis and Design - page 529

by multiplying the controller gain matrix with elements arranged along the diagonal
with the desired TC value. That is,

x target ¼ K c x desired ( 9 : 107 )

where x desired 2 R mþ 1 , x target 2 R mþ 1 ,andK c 2 R ( mþ 1) ( mþ 1)

is the controller gain

matrix with elements along the diagonal.

2
4

3
5

t d
t d
:
t d

x desired ¼ , K c ¼ diag K 1 K 2 K mþ 1 ( 9 : 108 )

9.12 PROCESS CONTROLS UNDER LIMITED ACTUATION
In modern electrophotographic printers, exposure, development, and TC control
systems use some form of integral control to reduce steady-state errors. This is
because, over time, an integral controller integrates even small variations of the
error signal between the sensed value and the target value, to provide an increasing
control signal that brings the process output back to the desired target.
There are, however, several aspects of an integral controller that can potentially
have ill effects on the performance of the system when the dynamic range of the
actuators is bounded by some practical limits [42 48]. For example, in the TC

control system, dispense rate is limited to low and high values. At the low end, the
rate is set to a minimum value to avoid the small rates that cannot be provided
reliably by the dispenser motor. At the high end, the maximum rate is limited by the
admix of the material package. Similarly, practical limits are set for charge voltage,
exposure raster output scanner (ROS) intensity and the development bias voltage due
to cost and other considerations. When an integrator is used in the control loop, a
phenomenon known as integrator windup occurs whenever the actuator signal is
driven to its maximum limit by the controller in an effort to bring the output to its
desired target. If the output does not return to the desired target, there will be a nonzero
error that will continue to be integrated by the integrator and this phenomenon is known
as

-

As a result, the actuator will remain at its limit until the system output
crosses the target value and suf

'' windup. ''

cient error of opposite sign is integrated to remove the
windup. Windup can decrease the overall performance of the control system and cause
large overshoot, longer settling time, and large steady-state errors. It can also occur in
other situations like when the loop dynamics are changed abruptly, are switched off for
some valid reason, and when a sensor fails or when there is inaccurate feedforward
compensation due to measurement errors, etc. These short comings of the integral
controllers can be minimized by including an antiwindup compensation. This is neces-
sary to sustain reliable operation of production printing systems.
Although an optimization-based antiwindup design for MIMO control system is
still an active area of research and many questions remain unanswered, many