Image Processing Reference
In-Depth Information
4.13
Consider the system de
ned by
2
3
2
3
2
3
2
3
"
#
x
1
(
k
þ
1
)
010
001
0
x
1
(
k
)
x
2
(
k
)
x
3
(
k
)
01
10
01
u
1
(
k
)
u
2
(
k
)
4
5
¼
4
5
4
5
þ
4
5
x
2
(
k
þ
1
)
x
3
(
k
þ
1
)
:
30
:
41
2
4
3
5
"
#
x
1
(
k
)
x
2
(
k
)
x
3
(
k
)
100
010
y
(
k
) ¼
(a) Is the system completely state controllable?
(b) Is the system completely state observable?
(c) Is the system completely output controllable?
4.14
Is the following system completely state controllable and observable?
2
3
2
3
2
3
2
3
x
1
(
k
þ
1
)
010
001
x
1
(
k
)
x
2
(
k
)
x
3
(
k
)
0
0
1
4
5 ¼
4
5
4
5 þ
4
5
u
(
k
)
x
2
(
k
þ
1
)
x
3
(
k
þ
1
)
3
1
2
2
4
3
5
x
1
(
k
)
x
2
(
k
)
x
3
(
k
)
y
(
k
) ¼
½
2
3
2
x
(
k
) þ
u
(
k
)
0
1
0
2
4.15
For
the
system x
(
k
þ
1
) ¼
and
0
:
125
0
:
75
y
(
k
) ¼
½
11
x
(
k
)
Y
(
z
)
U
(
z
)
.
(b) Is the system controllable? Is it observable? If not,
(a) Find the transfer function H
(
z
) ¼
find the uncontrollable
=
unobservable modes.
4.16
Consider the system given by
2
3
2
3
100
010
001
b
11
b
12
b
13
4
5
x
(
k
) þ
4
5
u
(
k
)
x
(
k
þ
1
) ¼
b
21
b
22
b
23
b
31
b
32
b
33
y
(
k
) ¼
x
1
(
k
)
(a) Find the transfer function
Y
(
z
)
U
(
z
)
.
(b) Find the necessary and suf
cient condition for controllability of this
system.
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