Image Processing Reference
In-Depth Information
z
z
w
(
z
) ¼
1
e
s
(
z
)
(
9
:
67
)
The gains of the PI controller are designed by
first expressing Equation 9.66 in state-
space form as follows:
Substituting the error signal e
s
(z)
t
c
(z) in Equation 9.67 and taking the
inverse z-transform, we get the following difference equation:
¼
t
d
(z)
w
(
k
þ
1
) ¼
w
(
k
) þ
t
d
(
k
þ
1
)
t
c
(
k
þ
1
)
(
9
:
68
)
For a TC systemwith no dispenser lag (i.e.,
m¼
0) and no consumption (i.e., v(k)
¼
0),
Equation 9.63 can be written as
t
c
(
k
þ
) ¼
t
c
(
k
) þ
gu
(
k
)
(
:
)
1
9
69
Now, Equations 9.68 and 9.69 are arranged in matrix form as follows:
t
c
(
k
)
w
(
k
)
u
(
k
) þ
t
d
(
k
þ
t
c
(
k
þ
1
)
10
g
g
0
1
¼
þ
1
)
(
9
:
70
)
w
(
k
þ
1
)
11
Equation 9.70 can be written in more general matrix difference (state space) form for
a single-input system as
x
(
k
þ
1
) ¼
Ax
(
k
) þ
Bu
(
k
) þ
Er
(
k
þ
1
)
(
9
:
71
)
, A
¼
, B
¼
, E
¼
,andr(k)
t
c
(
k
)
w
(
k
)
10
g
g
0
1
with x
(
k
) ¼
¼
t
d
(k).
11
The desired TC value is considered constant (i.e., t
d
(k
þ
¼
t
d
(k)
¼
constant).
In some systems, the TC target is adjusted at a lower rate as a function of toner age in
order to achieve better control for low area coverage images. Due to the repeated
mechanical stresses during the mixing process, the toner will start to lose its
charging property. This is called toner aging [31], which has a negative impact on
development and transfer performance. Fresh toner has higher additive coverage
whereas the aged toner has almost no additives on the toner surface. As a result of
this, there is a signi
1)
cant difference in toner transfer to the paper.
For simplicity, let us assume a constant TC target.
For a constant TC target, the amount of dispensed toner can be expressed as a
function of the controller gains:
u
(
k
) ¼ t
K
p
e
s
(
k
) þ
K
i
w
(
k
)
(
9
:
72
)
Equation 9.72 can then be written in matrix form as
e
s
(
k
)
w
(
k
)
u
(
k
) ¼t
K
p
K
i
(
:
)
9
73
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