Image Processing Reference
In-Depth Information
x
(
1
) ¼
x
d
(
9
:
54
)
It can be seen from Equation 9.54 that the deadbeat control law can provide control
in one measurement-process-actuation cycle.
For a MIMO system, as in level 1 and 2 control, all elements of the closed-loop
system matrix B will not be equal to zero and is not diagonal. To determine the
number of measurement-process-actuation cycles required to achieve deadbeat con-
trol, the matrix B has to be expressed in diagonal or pseudo-diagonal (i.e., Jordan)
form (Chapter 3). Since there will be repeated eigenvalues during deadbeat control,
the B matrix can be written as follows:
BM
¼
J
M
1
(
9
:
55
)
Substituting Equation 9.55 in Equation 9.53
x
(
k
) ¼
MJ
k
M
1
x
0
þ
BKx
d
(
:
)
9
56
To make this problem simple, let us consider that the B matrix is of size 4
4. Now,
if
l
1
¼l
2
¼l
3
¼l
4
are eigenvalues of matrix B, then from linear algebra [28], J
k
can
be expressed as follows:
2
4
3
5
k
2
k
3
l
l
1
k
1
l
k
l
(
k
)
(
k
)
1
2
1
2
!
1
3
!
1
k
2
1
2
!
l
1
k
1
0
l
k
l
(
k
1
)
J
k
¼
(
9
:
57
)
1
1
k
1
0
0
l
k
l
1
1
0
0
0
l
when k
¼
1, from Equation 9.57
2
4
3
5
2
4
3
5
2
4
3
5
2
4
3
5
(
9
:
58
)
0100
0010
0001
0000
0010
0001
0000
0000
0001
0000
0000
0000
0000
0000
0000
0000
J
1
, J
2
, J
3
J
4
¼
¼
¼
,
and
¼
For k
¼
¼
x
d
. This implies that for a four-dimensional
(4-D) system, if B is diagonalizable to pseudo-diagonal form such as the Jordan
canonical form, a minimum of four measurement-process-actuation cycles are
required to reach the
4, we see that the states, x(4)
nal desired values. In most cases, in printers, the matrix B is
diagonalizable. Hence, theoretically speaking, it is possible to reach deadbeat in one
measurement-process-actuation cycle. However, in practice, noise and uncertainties
will make it dif
cult to achieve deadbeat. Loops must be designed away from the
deadbeat response.
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