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b. Find the characteristic polynomial of the open-loop system
N
N
1
N
2
P(
l
)
¼jlI Aj¼l
þ b
1
l
þ b
2
l
þþb
N
(5
:
29)
c. Find the transformation T given by
T ¼ QW
(5
:
30)
where Q is the controllability matrix and W is given by
2
4
3
5
b
N1
b
N2
b
1
1
b
N1
b
N1
10
.
.
.
.
W ¼
(5
:
31)
b
1
1
00
1
0
00
d. Find the gain matrix K using the following equation:
T
1
K ¼ a
N
b
N
½
a
N1
b
N1
a
1
b
1
(5
:
32)
Example 5.3
Consider the dynamic system given by
x(k)
u(k)
0
1
1
1
x(k þ
1)
¼
þ
0
:
12
1
Design a state feedback controller to place the closed-loop poles at
l
1
¼
0
:
3
j0
:
4 and
l
2
¼
0
:
3
þ j0
:
4
S
OLUTION
The characteristic polynomial of the open-loop system is
¼ l
j¼
l
1
2
P(
l
)
¼ lI A
j
þ l þ
0
:
12
(5
:
33)
0
:
12
l þ
1
Comparing Equation 5.33 with Equation 5.29, we have
b
1
¼
1, and
b
2
¼
0
:
12.
The characteristic polynomial of the desired closed-loop system is
2
P
c
(
l
)
¼
(
l
0
:
3
þ j0
:
4)(
l
0
:
3
þ j0
:
4)
¼ l
0
:
6
l þ
0
:
25
Hence
a
1
¼
0
:
6, and
a
2
¼
0
:
25. The transformation T is given by
11
10
b
1
1
10
11
1
21
T ¼ QW ¼ BAB
½
¼
¼
1
:
12
0
:
12 1
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