Image Processing Reference
In-Depth Information
The poles of the open-loop system or the eigenvalues of the A matrix are
l
1
¼
0
:
9
and
l
2
¼
0
:
8. The zero-input response of the system for the initial condition of
T
is given by
x(0)
¼
½
1
2
k
0
1
8)
k
9)
k
8)
k
9)
k
1
9(0
:
8(0
:
10(0
:
þ
10(0
:
x(k)
¼
x(0)
¼
8)
k
9)
k
8)
k
9)
k
0
:
72 1
:
7
2
7(0
:
7(0
:
8(0
:
þ
9(0
:
(5
:
6)
Therefore, the two states of the system are
8)
k
9)
k
x
1
(k)
¼
11(0
:
þ
12(0
:
(5
:
7)
and
8)
k
9)
k
x
2
(k)
¼
9(0
:
þ
11(0
:
(5
:
8)
Plot of x
1
(k) is shown in Figure 5.3. As is seen in this example, the response of the
system is slow and it takes more than 50 time steps to converge to the origin. In
order to improve the transient behavior of the system, we use state feedback.
The closed-loop system matrix A
c
is given by
k
1
0
1
0
1
0
1
A
c
¼ A BK ¼
½
k
2
¼
0
:
72 1
:
7
0
:
72
k
1
1
:
7
k
2
(5
:
9)
4
3.5
3
2.5
2
1.5
1
0.5
0
0
5
10
15
20
25
30
35
40
45
50
k
FIGURE 5.3
Transient behavior of the open-loop system.
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