Image Processing Reference
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placement. These eigenvalues should be located inside the unit circle closer to the
origin than the eigenvalues of A
BK
f
. With this eigenvalue assignment,
the
estimation error will approach zero faster than the dynamics of the system.
Example 5.10
Consider the dynamic system given by
x(k)
1
u(k)
0
1
0
x(k þ
1)
¼
þ
0
:
72 1
:
7
y(k)
¼
½
10
x(k)
Design a combined state feedback with a state observer for the system. Place the
eigenvalues of the closed-loop state feedback system at
l
1
¼
0
:
3 and
l
2
¼
0
:
2.
Place the observer poles at
l
1
¼
0
:
1 and
l
2
¼
0
:
1.
S
OLUTION
The characteristic polynomial of the open-loop system is
2
P(
l
)
¼jlI Aj¼l
1
:
7
l þ
0
:
72
Hence,
72.
The characteristic polynomial of the desired closed-loop system is
b
1
¼
1
:
7 and
b
2
¼
0
:
2
P
c
(
l
)
¼
(
l
0
:
3)(
l
0
:
2)
¼ l
0
:
5
l þ
0
:
06
Hence,
a
1
¼
0
:
5 and
a
2
¼
0
:
06. The transformation T is given by
b
1
1
10
01
11
1
71
10
:
10
01
¼
T ¼ QW ¼ BAB
½
¼
:
7
The feedback gain matrix Kf
f
is given as
1
10
01
T
1
K
f
¼ a
2
b
2
½
a
1
b
1
¼
½
0
:
12
1
¼
½
0
:
66 1
:
2
We now design a full-state observer for the system. The characteristic polynomial
of the observer is
2
P(
l
)
¼
(
l l
1
)(
l l
2
)
¼
(
l
0
:
1)(
l
0
:
1)
¼ l
0
:
2
l þ
0
:
01
The observability matrix is
¼
C
CA
10
01
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