Image Processing Reference
In-Depth Information
where matrix J is the Jacobian computed at u
¼
u
0
J
¼
Ju
ðÞ
(
9
:
14
)
and
D
u
(
k
) ¼
u
(
k
þ
1
)
u
0
(
9
:
15
)
However, this equation is still not in the typical state-space form of Equation 9.3b,
which is the most desired form. To get an equation of the form of Equation 9.3b, we
use a discrete integrator for
D
u(k) and de
ne
D
u
(
k
) ¼ D
u
(
k
1
) þ
v
(
k
)
(
9
:
16
)
D
u(k) in Equation 9.15)
or equivalently (by the de
nition of
u
(
k
þ
1
) ¼
u
(
k
) þ
v
(
k
)
(
9
:
17
)
Substituting Equation 9.16 into Equation 9.13, we get
x
(
k
þ
1
) ¼
x
0
þ
J
(D
u
(
k
1
) þ
v
(
k
))
(
9
:
18
)
Replacing k
þ
1withk in Equation 9.13, we obtain
x
(
k
) ¼
x
0
þ
J
D
u
(
k
1
)
(
9
:
19
)
Solving Equation 9.19 for x
0
and then substituting it into Equation 9.18 we get
x
(
k
þ
1
) ¼
x
(
k
) þ
Jv
(
k
)
(
9
:
20
)
which is of the desired form. Putting everything together, we have
x
(
k
þ
) ¼
Ax
(
k
) þ
Bv
(
k
)
y
(
k
) ¼
Cx
(
k
) þ
Dv
(
k
)
1
(
9
:
21
)
where A
¼
I, B
¼
J, C
¼
I, and D
¼
3 identity matrix.
From the analysis above, we observe that the use of a discrete integrator results
in a very neat and elegant linear state-space representation. This is schematically
presented in Figure 9.1 below for an example system from Chapter 8. Note that as
shown v(k)isde
0. Matrix I is a 3
¼
ke(k).
Although Equation 9.21 is fairly simple from a theoretical perspective, it is
ned as v(k)
dif
cult to apply in practice. This is so because the models describing the printing
system are very complicated and have some very challenging nonlinearities. Most of
the dif
culty stems from the fact that the process models for each separation are not
independent and additionally because the color model employed presents a very
complex dependency on the toner masses. These are the same reasons why it would
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