Image Processing Reference
In-Depth Information
where
D
u
(
k
) ¼
u
(
k
þ
1
)
u
(
k
)
(
9
:
8
c)
In the rest of this chapter, we will focus on the discrete-time domain, where k refers
to index of the print. The index k can also refer to the sample measurements within a
single page, if the entire sensing-processing-actuation cycle is implemented within
the printed page.
This model is trivial to put into the state-space form of Equation 9.3b. First
de
ne
v
(
k
): ¼ D
u
(
k
) ¼
u
(
k
þ
1
)
u
(
k
)
(
9
:
9
)
which is mathematically equivalent to the use of a discrete-time integrator for
the calculation of u(k) from v(k) (since we have access to the input u(k) for control),
that is
u
(
k
þ
1
) ¼
u
(
k
) þ
v
(
k
)
(
9
:
10
)
Substituting into Equation 9.8b, we get
x
(
k
þ
1
) ¼
x
(
k
) þ
J
(
u
(
k
))
v
(
k
)
(
9
:
11
)
which is a linear time-varying state-space model of the form
x
(
k
þ
) ¼
A
(
k
)
x
(
k
) þ
B
(
k
)
v
(
k
)
y
(
k
) ¼
C
(
k
)
x
(
k
) þ
D
(
k
)
v
(
k
)
A
(
k
) ¼
I,
1
0
(
9
:
12
)
B
(
k
) ¼
J
(
u
(
k
))
,
C
(
k
) ¼
I,
D
(
k
) ¼
where, the only time-varying component is the system
is Jacobian matrix, which has
to be reevaluated at each step. Thus, Equation 9.12 is a generalized vector linear
time-varying representation of a printing system with actuators (inputs) and outputs.
It is vital to recognize that for this approximation to hold true, with respect to the
reference point of linearization, x(k) and v(k) must have relatively small deviations
between consecutive time instants. The size of the deviation depends on the local
nonlinearity of the model.
The Equation 9.8b represents a typical linearization scheme for a nonlinear
discrete-time dynamic system around a time-varying point. For the special case
where the system is linearized about a
'
fixed, time-invariant nominal value of the
actuators, namely u
0
(which gives in turn a nominal value of the output, namely x
0
)
Equation 9.8b becomes
x
(
k
þ
1
) ¼
x
0
þ
J
D
u
(
k
)
(
9
:
13
)
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