Internal Process Controls - Control of Color Imaging Systems: Analysis and Design - page 510

2 3 2 3

U 1
U 2
:
:
U N

r 1
r 2
:
:
r N

4 5 4 5

U 0 ¼ and x d ¼ ( 9 : 50 )

Also, in this case, the Jacobian matrix, J, contains the first derivatives of the tone

values at each of the patch area coverages and the structure of the matrix is diagonal
because during level 3 control each patch tone value can be varied without interact-
ing with the other patches. That is, development of each patch can be regarded
independent of the other patch. This assumption holds good, provided the level 2
actuators do not change (i.e., have fully settled) while level 3 control loop is running.
The measurement-process-actuation interval for level 3 control is assumed quite high
as compared with level 1 and 2 controls, preserving the time hierarchy described
earlier in the chapter.
The derivatives are obtained at the patch area coverages for nominal operating
conditions. For N number of level 3 control patches with area coverages

U 1 , U 2 , , U N , the Jacobian matrix is written as

...

2
4

3
5

@ x 1
@ U 1

0 0

@ x 2
@ U 2

0 0

J ¼ ( 9 : 51 )

. .

.

.

. .

00 .

@ x N
@ U N

Since the Jacobian is diagonal for level 3 control, the gain matrix (Equation 9.49) will
be diagonal, meaning the dynamic control loop will have a SISO form, that is, each
reference tone value is controlled independent of the others. This independence is
exactly the reason why a nonmonotonic TRC can be inverted using control-based
approach. Once the level 3 loop converges to a stable state with a near zero error
vector, e(k)

¼ x d x(k), the measured tone values converge to the reference tone
values. Since the measured tone values are present in the control vector, U(k), the
inverse tone values can also be obtained from the same vector. A smooth curve is
constructed to pass through the tone values in the control vector U(k) and the end
points (0 and 255). This smooth curve represents the

final inverse TRC used to process
the pixels in the images. The approach is described below with a numerical example.

Example 9.9

Let the vector, U 0 , contain the input patch gray levels (or area coverages) for
controlling DMA on the photoconductor for cyan separation shown in Table 9.1.
The vector x contains the normalized values of the measurements, the vector x d
contains the tone values of the desired reference TRC. Both vectors are shown as a
function of the patch gray level.