Image Processing Reference
In-Depth Information
Now, designing K to place the eigenvalues of the closed-loop matrix (ABK)at
locations
0.25 will lead to a stable solution provided the control-
lability matrix (Equation 4.121)
l
1
¼
0.2,
l
2
¼
1
8
11
Q ¼ BAB
½
¼
(9
:
82)
1
2
is of full rank, which we can see by inspection (the two columns of Q are inde-
pendent) is the case which is full. Hence both states can be controlled using a single
actuator, u(k). To
find the gains let us use the procedure outlined in Example 5.4.
The characteristic polynomial of the open-loop system is
¼
¼jlI Aj¼
l
0
2
P(
l
)
(
l
1)(
l
1)
¼ l
2
l þ
1
¼
0
(9
:
83)
1
l
1
Comparing the open-loop characteristic polynomial, Equation 9.83, with Equation
5.21,
b
1
¼
2,
b
2
¼
1. The characteristic polynomial of the desired closed-loop
system is
2
P
c
(
l
)
¼
(
l
0
:
2)(
l
0
:
25)
¼ l
0
:
45
l þ
0
:
05
(9
:
84)
Comparing the closed-loop characteristic polynomial, Equation 9.84, with Equa-
tion 5.19,
a
1
¼
0.45, and
a
2
¼
0.05. The transformation T (Equation 5.30) is
given by
1
8
1
8
b
1
1
10
11
21
10
11
0
T ¼ QW ¼ BAB
½
¼
¼
1
2
1
(9
:
85)
The gain vector K (Equation 5.32) becomes
"
#
1
1
8
1
8
T
1
K ¼ a
2
b
2
½
a
1
b
1
¼
½
0
:
95 1
:
55
¼
[7
:
6
4
:
8]
1
8
0
(9
:
86)
Check: To know whether our computation is correct,
find the eigenvalues of
(ABK) using above gain vector. It should give
l
1
¼
0.2,
l
2
¼
0.25.
Case (b): Applying similar pole-placement techniques for SISO design, gain matrix
is calculated to place the closed loop eigenvalues at
0.65.
The characteristic polynomial of the desired closed-loop system is
l
1
¼
0.6,
l
2
¼
2
P
c
(
l
)
¼
(
l
0
:
6)(
l
0
:
65)
¼ l
1
:
25
l þ
0
:
39
(9
:
87)
Hence
0.39. The transformation T is the same as in Equation
9.85. Hence the gain matrix K is given as
a
1
¼
1.25, and
a
2
¼
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