Image Processing Reference
In-Depth Information
The observability matrix P is
2
4
3
5
1
1
1
1
1
0
2
4
3
5
¼
C
CA
CA
2
21
:
51
:
6
P ¼
1
0
:
51
3
:
62
:
75
2
:
46
0
1
:
25
0
:
1
Since rank(P)
¼
3, P is full rank and the system is completely state observable.
PROBLEMS
4.1
Find the state-space representation of the following dynamical systems in
controllable and observable canonical forms:
3
2
(a)
d
y
(
t
)
d
t
3
2
d
y
(
t
)
d
t
2
3
d
y
(
t
)
þ
þ
d
t
þ
y
(
t
) ¼
4u
(
t
)
(b)
y
(
k
þ
2
) þ
y
(
k
þ
1
) þ
0
:
8y
(
k
) ¼
8u
(
k
þ
1
) þ
6u
(
k
)
4.2
Find the state-space representation of the following dynamical systems:
3
2
(a)
d
y
(
t
)
d
t
3
3
d
y
(
t
)
d
t
2
6
d
y
(
t
)
2y
(
t
) ¼
d
u
(
t
)
þ
þ
d
t
þ
d
t
þ
8u
(
t
)
(b)
y
(
k
þ
3
) þ
0
:
5y
(
k
þ
2
) þ
y
(
k
þ
1
) þ
0
:
89y
(
k
) ¼
2
:
4u
(
k
)
4.3
Obtain the response y
(
t
)
of the following system, where u
(
t
)
is the unit step
function:
x
1
(
t
)
x
2
(
t
)
u
(
t
)
x
1
(
t
)
x
2
(
t
)
1
0
:
75
0
1
¼
þ
1
0
1
0
x
(
0
) ¼
x
1
(
t
)
x
2
(
t
)
y
(
t
) ¼
½
10
k
4.4
Obtain the response y
(
k
)
of the following system, when u
(
k
) ¼ (
)
for k
1
0:
x
1
(
k
)
x
2
(
k
)
u
(
k
)
x
1
(
k
þ
1
)
x
2
(
k
þ
1
0
:
25
1
0
¼
þ
1
)
1
0
1
0
x
(
0
) ¼
x
1
(
k
)
x
2
(
k
)
y
(
k
) ¼
½
1
1
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