Image Processing Reference
In-Depth Information
Equation 5.92 can be simpli
ed further as
e
(
k
þ
1
) ¼ (
A
21
LCA
11
GC
)
x
1
(
k
) þ (
A
22
LCA
12
)
x
2
(
k
)
þ (
B
2
LCB
1
H
)
u
(
k
)
Fx
2
(
k
) þ
FLCx
1
(
k
)
¼ (
A
21
LCA
11
GC
þ
FLC
)
x
1
(
k
) þ (
A
22
LCA
12
)
x
2
(
k
)
þ (
B
2
LCB
1
H
)
u
(
k
)
F
(
x
2
(
k
)
e
(
k
))
(
5
:
93
)
or
e
(
k
þ
1
) ¼
Fe
(
k
) þ (
A
21
LCA
11
GC
þ
FLC
)
x
1
(
k
)
þ (
A
22
LCA
12
F
)
x
2
(
k
) þ (
B
2
LCB
1
H
)
u
(
k
)
(
5
:
94
)
For the error to be independent of x
1
(
k
)
, x
2
(
k
)
, and u
(
k
)
, we must have
A
21
LCA
11
GC
þ
FLC
¼
0
A
22
LCA
12
F
¼
0
B
2
LCB
1
H
¼
0
(
5
:
95
)
Using the above three equations, we have
F
¼
A
22
LCA
12
(
5
:
96
)
H
¼
B
2
LCB
1
(
5
:
97
)
G
¼ [
A
21
LCA
11
]
C
1
þ
FL
(
5
:
98
)
with these choices for F, G, and H, Equation 5.94 is reduced to
e
(
k
þ
1
) ¼
Fe
(
k
) ¼ (
A
22
LCA
12
)
e
(
k
)
(
5
:
99
)
The error signal will approach zero as k
!1
if the eigenvalues of matrix
A
22
LCA
12
are inside unit circle in the complex plain. The gain L matrix is
designed by assigning eigenvalues to the matrix A
22
LCA
12
. This is similar to
the full-observer design with A
22
LCA
12
playing the role of A
LC. The following
simple example illustrates the design of a reduced-order observer:
Example 5.9
Consider the dynamic system given by
x(k)
u(k)
0
1
0
1
x(k þ
1)
¼
þ
0
:
72 1
:
7
y(k)
¼
½
10
x(k)
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