Internal Process Controls - Control of Color Imaging Systems: Analysis and Design - page 490

The actuator updates are obtained from

U ( k ) ¼ U 0 þ u ( k )
u ( k þ

( : )

9 34

1 ) ¼ u ( k ) þ v ( k )

This closed-loop system equation is very similar to Equation 9.26.

9.6.1 J ACOBIAN M ATRIX FOR D EVELOPABILITY C ONTROL
Development of the Jacobian matrix for level 2 control loop is very challenging
because there are three actuators being coupled to the sensor signals as apposed
to two actuators in the case of electrostatic control. We can think of three methods
for extracting the Jacobian matrix. They are (1) an analytical method, (2) design of
experiments (DoE) methods [22,23], and (3) direct numerical methods. In the
analytical method, at first the Jacobian matrix of a development system is expressed

in terms of analytical expressions for the separation of interest using a charging,
exposure, and development model. The Jacobian with respect to the actuators of the
process models can be calculated using the symbolic toolbox in MATLAB. Apart
from the Jacobian matrix, this approach gives invaluable information about the
robustness of the model with respect to key process parameters of the print engine.
This information can be used to satisfy several other speci

cations and improve
the system performance. The direct experimental method involves running carefully
designed experiments to extract the nonlinear surfaces in the DMA outputs
with respect to actuator values. The DoE method allows systematic ways to select
actuator values in the vector, U 0 ¼ U ho U lo U b ½ 0 , between their operating limits,
so that they are orthogonal and yet minimize the number of experiments required
to determine the Jacobian matrix. We will describe the numerical method in
detail below.
The numerical method will be useful to understand the available operating space
for the DMA vector in a static machine when actuators are varied within their limits.
Figure 9.12 shows a three-dimensional (3-D) space covered by the 16 3 actuator
values for combinations sampled uniformly for a tandem printer example. Each point
on this plot represents the actuator vector, U. In DoE language, this type of design
matrix is called a

In Operations Research, this is known as an
exhaustive search. Three patches with low, mid, and high area coverages are devel-
oped under different actuator values for U spread over their limits. They are
measured with the optical DMA sensor, if actual printer is used (or calculated
when a virtual printer model is used). Figure 9.13 shows the volumetric space
covered by the DMA vector for the actuator values of Figure 9.12 shown for four
different orientations of the DMA axes. Figure 9.14 shows the full development TRC
for the same actuator values. Clearly, the available space for controlling the DMA
using the hierarchical level 2 controls is quite limited. The high DMA values show a
larger change, meaning higher sensitivity to actuator changes. However, careful
selection of actuators can further increase the available space, which will be covered
brie

'' full factorial design. ''

y in Example 9.7 and Section 9.14.