Image Processing Reference
In-Depth Information
Equation 4.127 can be written in matrix form as
2
4
3
5
2
4
3
5
2
4
3
5
2
3
0
AB
AB
þ
B
.
y
(
0
)
u
(
0
)
C
CA
CA
2
.
CA
N
1
y
(
1
)
u
(
1
)
4
5
y
(
2
)
u
(
2
)
¼
x
(
0
) þ
C
þ
D
(
4
:
128
)
.
y
(
N
.
u
(
N
P
N
2
0
A
N
2
n
B
1
)
1
)
n
¼
or
2
4
3
5
2
3
2
3
2
3
0
AB
AB
þ
B
.
y
(
0
)
u
(
0
)
C
CA
CA
2
.
CA
N
1
4
5
4
5
y
(
1
)
u
(
1
)
4
5
y
(
2
)
u
(
2
)
x
(
0
) ¼
C
D
(
4
:
129
)
.
y
(
N
.
u
(
N
P
N
2
0
A
N
2
n
B
1
)
1
)
n
¼
Equation 4.129 can be used to solve for x
(
if and only if matrix P is full rank. Once
the initial state is obtained, the states of the system at any other time can be obtained
by solving the state equations.
0
)
Example 4.17
Consider the following system
x(k)
1
u(k)
1
0
:
25
1
x(k þ
1)
¼
þ
1
:
5
0
:
25
y(k)
¼
½
10
x(k)
The observability matrix P is
¼
C
CA
10
1
P ¼
0
:
25
Since det (P)
¼
0
:
25
6¼
0, P is full rank and the system is completely state
observable.
Example 4.18
Consider the SISO system
2
4
3
5
x(k)
2
4
3
5
u(k)
0
510
10
:
1
x(k þ
1)
¼
:
75
1
þ
1
2
2
10
:
8
y(k)
¼
½
10
2
x(k)
þ
3u(k)
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