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.
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