Image Processing Reference
In-Depth Information
In terms of state-transition matrix, the total solution is given by
X
k
1
x
(
k
) ¼ w(
k
)
x
(
0
) þ
0
w(
k
1
n
)
Bu
(
n
)
(
4
:
119
)
n
¼
The total solution consists of two terms, the
rst term
w(
k
)
x
(
0
)
is referred to as the
zero-input response and the second term
P
k
1
is called zero-state
response. The zero-input response is the response of the system due to the initial
conditions only and the zero-state response is the response due to the input with zero
initial conditions. In the following example, we compute the zero-state and zero-
input responses of a second-order system.
n
¼
0
w(
k
1
n
)
Bu
(
n
)
Example 4.13
1 k
0
Find the output of
the following dynamic system if
u(k)
¼
and
0 k <
0
:
1
x(0)
¼
1
"
#
x(k)
"
u(k)
1
0
:
25
1
1
x(k þ
1)
¼
þ
1
:
5
0
:
25
y(k)
¼
½
12
x(k)
S
OLUTION
First we
find the zero-input response:
1
¼
5)
k
25)
k
5)
k
25)
k
5)
k
25)
k
3(0
:
2(0
:
(0
:
þ
(0
:
4(0
:
3(0
:
w
(k)x(0)
¼
5)
k
25)
k
5)
k
25)
k
1
5)
k
25)
k
6(0
:
6(0
:
2(0
:
þ
3(0
:
8(0
:
9(0
:
Next, we
nd the zero-state response:
"
#
"#
X
¼
X
w
11
(k
1
n)
w
12
(k
1
n)
1
k
1
k
1
0
w
(k
1
n)Bu(n)
w
21
(k
1
n)
w
22
(k
1
n)
1
n¼
n¼
0
"
#
¼
X
w
11
(k
1
n)
w
12
(k
1
n)
k
1
w
21
(k
1
n)
w
22
(k
1
n)
n¼
0
"
#
5)
k1n
25)
k1n
X
k
1
4(0
:
3(0
:
¼
5)
k1n
25)
k1n
8(0
:
9(0
:
n¼
0
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