Image Processing Reference
In-Depth Information
The gain matrix K is given as
1
21
T
1
K ¼ a
2
b
2
½
a
1
b
1
¼
½
0
:
13
1
:
6
0
:
12 1
¼
½
0
:
0292
1
:
5708
Example 5.4
Consider the dynamic system given by
x(k)
u(k)
0
1
0
1
x(k þ
1)
¼
þ
0
:
12
1
Design a state feedback to place the closed-loop poles at
l
1
¼ l
2
¼
0
S
OLUTION
The characteristic polynomial of the open-loop system is
2
P(
l
)
¼ l
þ l þ
0
:
12
Hence
12.
The characteristic polynomial of the desired closed-loop system is
b
1
¼
1, and
b
2
¼
0
:
2
P
c
(
l
)
¼ l
Hence,
a
1
¼ a
2
¼
0 and the transformation T is given by
11
10
b
1
1
10
01
1
10
01
T ¼ QW ¼ BAB
½
¼
¼
1
The gain matrix K is given as
1
10
01
T
1
K ¼ a
2
b
2
½
a
1
b
1
¼
½
0
:
12
1
¼
½
0
:
12
1
The closed-loop state equations are
x(k)
Kx(k)
0
1
0
1
x(k þ
1)
¼
0
:
12
1
x(k)
x(k)
0
1
0
1
01
00
¼
½
0
:
12
1
x(k)
¼
0
:
12
1
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