Image Processing Reference
In-Depth Information
4
3
Open-loop response
2
1
0
Closed-loop response
-1
-2
0
5
10
15
20
25
30
35
40
45
50
k
FIGURE 5.4
Open-loop and closed-loop response.
5.2.2 P
OLE
-P
LACEMENT
D
ESIGN OF
SISO S
YSTEMS
Consider the SISO dynamic system described by the following state equation:
x
(
k
þ
1
) ¼
Ax
(
k
) þ
Bu
(
k
)
(
5
:
18
)
With the assumption that the system is completely state controllable using the state
feedback u
(
k
) ¼
Kx
(
k
)
, we wish to place the poles of the closed-loop system
at
l
N
. This means that the eigenvalues of the closed-loop matrix
A
c
¼
A
BK are at the desired locations given by
l
1
,
l
2
,
...
,
l
1
,
l
2
,
...
,
l
N
. Therefore the
characteristic polynomial of A
c
is
P
(l) ¼jl
I
A
c
j¼(l l
1
)(l l
2
) (l l
N
)
¼ l
N
N
1
N
2
þ a
1
l
þ a
2
l
þþa
N
(
5
:
19
)
By Cayley
Hamilton theorem (Section 3.11.3), we have
-
A
N
c
þ a
1
A
N
1
c
þ a
2
A
N
2
c
þþa
N
I
¼
P
(
A
c
) ¼
0
(
5
:
20
)
But
P
(
A
c
) ¼ a
N
I
þ a
N
1
A
c
þ a
N
2
A
2
c
þþ
A
N
c
¼ a
N
I
þ a
N
1
(
A
BK
) þ a
N
2
(
A
BK
)
2
N
þþ(
A
BK
)
(
5
:
21
)
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