Image Processing Reference
In-Depth Information
U
g
D
l
Charge, exposure, development and
TC system at the operating
point
U
l
D
m
U
b
D
h
U
TC
FIGURE 9.52
Four-input three-output MIMO process control system.
0
is the
Referring to the block diagram of Figure 9.52, if, x
d
¼
D
l
D
m
D
E
h
[U
g
(k) U
l
(k) U
b
(k) U
TC
(k)]
0
is the hard
process actuator vector with charge, ROS intensity, developer bias, and the TC target
[49], respectively, and x(k)
desired target DMA vector, if U(k)
¼
[D
l
(k) D
m
(k) D
h
(k)]
0
is the DMA measurement
vector, then for the purpose of developing a four-input three-output stable process
control system, a linear state-space model can be written as
¼
x
(
k
þ
) ¼
Ax
(
k
) þ
Bu
(
k
)
y
(
k
) ¼
Cx
(
k
)
1
(
9
:
112
)
where
C is the identify matrix
y(k) is same as x(k)
The system matrix A, and sensitivity matrix B are expressed as
2
4
3
5
2
4
3
5
@
D
l
@
U
g
@
D
l
@
U
l
@
D
l
@
U
b
@
D
l
@
U
TC
100
010
001
@
D
m
@
U
g
@
D
m
@
U
l
@
D
m
@
U
b
@
D
m
@
U
TC
A
¼
, B
¼
(
9
:
113
)
@
D
h
@
U
g
@
D
h
@
U
l
@
D
h
@
U
b
@
D
h
@
U
TC
Note that the above linear representation can also be used for controlling a paper-
based density or L*or
D
E
a
*
measurements (extracted with respect to paper) as
pointed out in Section 9.9. We use the LQR for designing the feedback controller.
The linear quadratic controller minimizes a selected quadratic objective function. For
example, if our goal is to minimize the error vector between the target vector and the
measured vector, then the objective function should be formed with the sums of the
squares of the weighted error (or the state) vector. Another term can be included with
the sums of the squares of the actuator values in the objective function to appropri-
ately weigh the desired actuator.
Let the performance objective be de
ned in terms of the following function:
2
P
N
1
k
¼
0
a
J
¼
1
2k
x
T
(
k
)
Qx
(
k
) þ
u
T
(
k
)
Ru
(
k
)
½
(
9
:
114
)
Search WWH ::
Custom Search