Image Processing Reference
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or
2
4
3
5
5)
k1
P
k
1
25)
k1
P
k
1
2
n
4
n
4(0
:
3(0
:
X
k
1
n¼
0
n¼
0
0
w
(k
1
n)Bu(n)
¼
5)
k1
P
25)
k1
P
k
1
k
1
n¼
2
n
4
n
8(0
:
9(0
:
n¼
0
n¼
0
2
4
3
5
2
k
4
k
5)
k1
1
25)
k1
1
4(0
:
2
3(0
:
1
1
4
¼
2
k
4
k
5)
k1
1
25)
k1
1
8(0
:
2
3(0
:
1
1
4
"
#
"
#
5)
k1
25)
k1
5)
k
25)
k
4(0
:
þ
4
þ
(0
:
4
8(0
:
þ
4(0
:
¼
¼
5)
k1
25)
k1
5)
k
25)
k
8(0
:
þ
12
þ
(0
:
12
16(0
:
þ
4(0
:
The total system response is the sum of zero-state and zero-input responses.
Therefore, we have
"
#
"
#
þ
X
k
1
5)
k
25)
k
5)
k
25)
k
4(0
:
3(0
:
4
8(0
:
þ
4(0
:
x(k)
¼w
(k)x(0)
0
w
(k
1
n)Bu(n)
¼
þ
5)
k
25)
k
5)
n
25)
k
8(0
:
9(0
:
12
16(0
:
þ
4(0
:
n¼
"
#
5)
k
25)
k
4
4(0
:
þ
(0
:
¼
5)
k
25)
k
12
8(0
:
5(0
:
The output signal is
5)
k
25)
k
4
5(0
:
þ
2(0
:
5)
k
25)
k
y(k)
¼
½
12
x(k)
¼
½
12
¼
28
25(0
:
2(0
:
5)
k
25)
k
12
10(0
:
2(0
:
4.9 CONTROLLABILITY OF LTI SYSTEMS
In general, a system is controllable if there exists an input that can transfer the
states of the system from an arbitrary initial state to a
finite interval of
time. In this section, we examine the controllability of LTI systems. Since the
results obtained are identical for continuous and discrete-time systems, we only
consider controllability of discrete-time LTI systems [6,7].
final state in a
4.9.1 D
EFINITION OF
C
ONTROLLABILITY
A discrete-time LTI system is said to be controllable at time k
0
if there exists an input
u
(
k
)
for k
k
0
that can transfer the system from any initial state x
(
k
0
)
to the origin in
finite number of steps.
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