Internal Process Controls - Control of Color Imaging Systems: Analysis and Design - page 549

cases? Close the loop with the gain matrix of Problem 9.1. Determine the
eigenvalues. Comment on the stability.

9.3 Consider the electrostatic control system of Figure 9.4. The charging system is
characterized by the Jacobian matrix B, at the nominal operating point.
a. Write the open- and closed-loop transfer functions of the system. Use any
convenient signals as inputs and outputs.
b. Determine the steady-state error between the desired exposed and unexposed
voltages to the measured voltages.
c. Replace the integrator with the unity gain. Determine the open and closed-
loop transfer functions for the new structure.
d. Assume the gain matrix as K ¼ B 1 when the integrator is replaced with the
unit gain. Determine the steady-state error.
e. Compare the results of b with d. Which controller gives the lowest steady-
state error?
f. If all the elements of matrix B are increased by 20% and the gain matrix, K,is
equal to that used in d, determine the percentage change in steady-state errors
with and without integrators in the loop.
g. If B matrix has not changed, but the voltages at the nominal operating point,
V ho and V lo , have changed by 20%, determine the percentage change in
steady-state errors with and without the integrators.

9.4 Consider the development control system of Example 9.5. The development
system is characterized by the Jacobian matrix B, at the nominal operating point.
Find the expression for the steady-state actuator U. Is the steady-state actuator
vector dependent on the gain matrix? Explain the reasons for your answer.

9.5 The cost of an electrostatic sensing system can be reduced by removing
the charge sensor (ESV) and associated level 1 control loop. It is now required
to design a three-input three-output developability control loop using three
process actuators, the grid voltage, U g (k) ¼ V g (k), the exposure intensity,

U l (k) ¼ V bias (k), and measurements

from the DMA sensor. Let D l (k), D m (k), and D h (k) represent the three
different DMA measurements shown in the state vector, x(k), measured
every pitch, indicated by the parameter, k. The linear state variable descrip-
tion of the control system is characterized by the Jacobian matrix at the
nominal operating point {U go ¼

¼ X(k), and the development bias, U b (k)

cm 2 , U bo ¼ 600 V, U lo ¼ 4ergs = 200 V}

as shown below:

2 3 2 3

100
010
001

0 : 06813 0 : 06047 5 : 69587

4 5 x ( k ) þ 4 5 v ( k )

10 3 x ( k þ 1 ) ¼ 0 : 27346 0 : 23242 21 : 52804

1 : 55592 1 : 43732 94 : 84767

2 3

100
010
001

4 5 x ( k )

y ( k ) ¼