Image Processing Reference
In-Depth Information
For any nominal CMY input, we can approximately represent the dynamics of
the single-color reproduction system using a
rst-order
finite difference equation
given by
x
(
k
þ
1
) ¼
BV
(
k
) þ
x
0
(
8
:
17
)
where
x
0
is the L*a*b* vector corresponding to the nominal CMY
B is the Jacobian matrix given by
2
4
3
5
@
L
*
@
C
@
L
*
@
M
@
L
*
@
Y
@
a
*
@
C
@
a
*
@
M
@
a
*
@
Y
B
¼
(
8
:
18
)
@
b
*
@
C
@
b
*
@
M
@
b
*
@
Y
The Jacobian matrix B is computed at the nominal CMY using the coarse printer
model. The Jacobian matrix can also be measured directly on the printer using
techniques described in Ref. [7].
Since we are using an integrator
V
(
k
) ¼
V
(
k
1
) þ
u
(
k
)
(
8
:
19
)
substituting Equation 8.19 into Equation 8.17 results in
x
(
k
þ
1
) ¼
BV
(
k
) þ
x
0
¼
BV
(
k
1
) þ
Bu
(
k
) þ
x
0
(
8
:
20
)
Assuming that the Jacobian is not changing between two consecutive prints, Equa-
tion 8.17 can be written for print number k as
x
(
k
) ¼
BV
(
k
1
) þ
x
0
(
8
:
21
)
Subtracting Equation 8.21 from Equation 8.20 results in
x
(
k
þ
1
) ¼
x
(
k
) þ
Bu
(
k
) þ
Ax
(
k
) þ
Bu
(
k
)
(
8
:
22
)
where A is a 3
3 identity matrix. Using the state feedback, u(k)
¼
K[r
x(k)], the
closed-loop state equation becomes
x
(
k
þ
1
) ¼ (
A
BK
)
x
(
k
) þ
BKr
(
8
:
23
)
The gain matrix, K, can then be designed using pole-placement or optimal control
techniques, as described in Chapter 6.
Search WWH ::
Custom Search