Image Processing Reference
In-Depth Information
The controllability matrix Q is
1
1
:
25
Q ¼ BAB
½
¼
1
1
:
75
Since det (Q)
¼
3
6¼
0, Q is full rank and the system is completely state control-
lable.
Example 4.15
Consider the SISO system
2
4
3
5
x(k)
2
4
3
5
u(k)
0
510
00
:
1
x(k þ
1)
¼
75 1
000
:
þ
1
2
:
8
The controllability matrix Q is
2
4
3
5
1
0
:
51
¼
Q ¼ BABA
2
B
11
:
25
2
:
54
21
:
61
:
28
Since det (Q)
¼
9
:
74
6¼
0, Q is full rank and the system is completely state
controllable.
Example 4.16
Consider the MIMO system
2
4
3
5
x(k)
2
4
3
5
100
010
001
10
01
u
1
(k)
u
2
(k)
x(k þ
1)
¼
þ
12
The controllability matrix Q is
2
4
3
5
101010
010101
¼
Q ¼ BABA
2
B
12
12
12
Since rank(Q)
¼
2, Q is not full rank and the system is not completely state
controllable.
4.10 OBSERVABILITY OF LTI SYSTEMS
In general, a system is completely state observable if the states of the system can be
found from the knowledge of inputs and outputs of the system. Since we know the
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