Image Processing Reference
In-Depth Information
The characteristic polynomial of the closed-loop system is
2
P(
l
)
¼ lI A c
j
j ¼ l
(
l
1
:
7
þ k 2 )
þ
0
:
72
þ k 1 ¼ l
þ
(k 2
1
:
7)
l þ
0
:
72
þ k 1
(5
:
10)
To improve the transient response of the closed-loop system, we select the closed-
loop poles to be at
l 1 ¼ l 2 ¼
0
:
2
(5
:
11)
This means that the characteristic polynomial of the closed-loop system should be
2
P(
l
)
¼
(
l
0
:
2)(
l
0
:
2)
¼ l
0
:
4
l þ
0
:
04
(5
:
12)
Comparing the two equations (Equations 5.10 and 5.12), we have
0
:
72
þ k 1 ¼
0
:
04
(5
:
13)
k 2
1
:
7
¼
0
:
40
Therefore
k 1
¼
0
:
68
(5
:
14)
k 2 ¼
1
:
3
The closed-loop system response is given by
k
0
1
(A BK) k
x(k)
¼
¼
x(0)
0
:
04 0
:
4
"
# 1
2) n
2) k
(1
k)(0
:
5k(0
:
¼
(5
:
15)
2) n
2) k
2
0
:
2k(0
:
(1
þ k)(0
:
Therefore the two states of the closed-loop system are
2) k
x 1 (k)
¼
(1
11k)(0
:
(5
:
16)
and
2) k
x 2 (k)
¼
(2
þ
2
:
2k)(0
:
(5
:
17)
The response of the
first state of the open-loop and closed-loop systems is shown
in Figure 5.4.
As it can be seen, the closed-loop response converges much faster than the open-
loop response. Clearly, the state feedback has led to a controllable response to the
state, which is one of the reasons why closed-loop systems are preferred over open-
loop design. Location of poles directly affects the state response.
In the next section, we discuss the pole-placement design for SISO systems using
state feedback [1
3].
-
Search WWH ::




Custom Search