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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
1
x 2 (k)
2u 2 (k)
J ¼þ
þ
0
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