Image Processing Reference
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0
:
25P(3)
0
:
25
1
:
1862
K(2)
¼
5P(3)
¼
1862
¼
0
:
1862
1
þ
0
:
1
þ
0
:
5
1
:
0
:
25P(4)
0
:
25
1
:
1863
K(3)
¼
5P(4)
¼
1863
¼
0
:
1863
1
þ
0
:
1
þ
0
:
5
1
:
0
:
25P(5)
0
:
25
1
:
1875
K(4)
¼
5P(5)
¼
1875
¼
0
:
1875
1
þ
0
:
1
þ
0
:
5
1
:
0
:
25P(6)
0
:
25
1
:
2
K(5)
¼
5P(6)
¼
2
¼
0
:
2
1
þ
0
:
1
þ
0
:
5
1
:
0
:
25P(7)
0
:
25
1
:
3333
K(6)
¼
5P(7)
¼
3333
¼
0
:
3333
1
þ
0
:
1
þ
0
:
5
1
:
0
:
25P(8)
0
:
25
4
K(7)
¼
5P(8)
¼
4
¼
0
:
3333
1
þ
0
:
1
þ
0
:
5
5.3.3 S
TEADY
-S
TATE
A
LGEBRAIC
R
ICCATI
E
QUATION
When the time horizon N
!1
, the performance index is given by
2
1
k
¼
1
0
x
T
(
k
)
Qx
(
k
) þ
u
T
(
k
)
Ru
(
k
)
J
¼
(
5
:
73
)
Under this scenario, the gain matrix K approaches a constant and is obtained by
solving the steady-state Riccati equation. From Equation 5.66, we have
P
¼
A
T
PA
A
T
PBR
1
(
I
þ
B
T
PBR
1
)
B
T
PA
þ
Q
(
5
:
74
)
Once the positive de
nite matrix P is obtained, the feedback gain matrix K is
computed using Equation 5.71
K
¼
R
1
B
T
P
(
I
þ
BR
1
B
T
P
)
1
A
(
5
:
75
)
Example 5.7
Consider the control system given by
x(k þ
1)
¼
0
:
5x(k)
þ u(k)
with initial state x(0)
1.
Find the optimal control feedback law to minimize the performance index
¼
2
1
k¼
1
x
2
(k)
2u
2
(k)
J ¼þ
þ
0
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