Image Processing Reference
In-Depth Information
implements the Ackermann algorithm for SISO systems. For MIMO
systems, the
place
''
''
algorithm uses extra degrees of freedom to obtain a robust
solution for matrix K. It minimizes the sensitivity of the closed-loop poles to the
variations in system
place
''
''
s parameters A and B. The command to implement pole place-
ment in MATLAB is K
¼ place(
A,B,P
)
'
, where A and B are system matrices and P is
a vector containing desired poles of the closed-loop system.
Example 5.5
Consider the MIMO dynamic system given by
x(k)
u
1
(k)
u
2
(k)
0
1
1
1
01
x(k þ
1)
¼
þ
0
:
12
1
Design a state feedback to place the closed-loop poles at
l
1
¼
0
:
3
l
2
¼
0
:
2
S
OLUTION
The system matrix of the closed-loop system is
K
11
K
12
K
21
K
22
0
1
1
1
01
A
c
¼ A BK ¼
0
:
12
1
K
11
þ K
21
1
K
12
þ K
22
¼
0
:
12
K
21
1
K
22
The characteristic polynomial of the closed-loop system is
P
c
(
l
)
¼ lIA
c
j
j
2
¼l
þ
(K
11
þK
22
K
21
þ
1)
lþK
11
(1
þK
22
)
þ
0
:
12(1
þK
22
)
K
12
(0
:
12
þK
21
)
Since the desired closed-loop poles are at
l
1
¼
0
:
3, and
l
2
¼
0
:
2, then
2
P
c
(
l
)
¼
(
l
0
:
3)(
l
0
:
2)
¼ l
0
:
5
l þ
0
:
06
Comparing the two equations of the closed-loop characteristic polynomials, we have
K
11
þ K
22
K
21
þ
1
¼
0
:
5
K
11
(1
þ K
22
)
þ
0
:
12(1
þ K
22
)
K
12
(0
:
12
þ K
21
)
¼
0
:
06
As can be seen, we have two equations and four unknowns. Therefore the solution
is not unique. For example if we choose K
11
¼ K
22
¼
0, we have
K
21
þ
1
¼
0
:
5
0
:
12
K
12
(0
:
12
þ K
21
)
¼
0
:
06
The solutions to the above equations are
K
21
¼
1
:
5
K
12
¼
0
:
037
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