Image Processing Reference
In-Depth Information
D
v
u
min
u
B-LD
C
Integrator
u
max
e
(
z
)
A
-
LC
-
L
FIGURE 9.51
Block diagram representation of a MIMO antiwindup compensator with PI
controller.
When the integrator is not saturated, v(k)
¼
u(k), that is, from Equation 9.109
u
(
k
)
Cx
(
k
)
De
(
k
) ¼
0
(
9
:
110
)
Multiplying Equation 9.110 by the gain matrix, L, and adding the resulting equation
with Equation 9.109 we get a new controller state equation
x
(
k
þ
) ¼ (
A
LC
)
x
(
k
) þ (
B
LD
)
e
(
k
) þ
Lu
(
:
)
1
9
111
Proper choice of matrix L can make the controller matrix (A
LC) stable. This
expression represents a combined system of an antiwindup compensator and a PI
controller, which is very similar in structure to a state observer (Section 5.4) for
which the closed-loop dynamics are dependent on the eigenvalues of the matrix
(A
LC). The eigenvalues can be placed at arbitrary locations by selecting the anti-
windup gain matrix, L, if the observability matrix P
¼
(A
0
)
n
1
C
0
]
(with n number of states) is of full rank. Figure 9.51 shows the block diagram repre-
sentation of the MIMO PI controller with antiwindup compensator. The compensated
controller has new parameter matrices, A-LC and B-LD.
[C
0
A
0
C
0
9.13 OPTIMAL CONTROLS FOR SELECTIVE STATES
As the photoconductor and developer materials age, or the machine environment
changes, the TC target needs to be adjusted. These adjustments can be provided at a
reasonably lower update cycle compared with the basic TC loop described in Section
9.11. In some xerographic printers, toner age correction is done on both the TC target
and the TC sensor. This is because the desired TC value is adjusted as a function of
toner age in order to stabilize the development system, particularly when images
with low area coverage are printed. Another common approach is to augment the
level 2 process actuators with a TC target and then develop a four-input three-output
control system with low-, mid-, and high-density DMA measurements. A practical
developability controller operating in Xerox black and white printers through the use
of LQR and adaptive parameter estimator is described in Ref. [16].
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