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].
-
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