Image Processing Reference
In-Depth Information
S
OLUTION
The homogeneous solution or the zero-state response is given by
1
0
¼
e
t
cos t e
t
sin t
e
t
sinte
e
t
cos t
e
t
sin t
x
h
(t)
¼ w
(t)x(0)
¼
t
cos t
The zero-state response or the particular solution is
ð
ð
d
t
0
w
t
e
tþt
cos(tt
) e
tþt
sin(tt
)
1
0
x
p
(t)
¼
(tt
)Bu(
t
)d
t¼
t
e
tþt
sin(tt
) e
tþt
cos(tt
)
0
2
4
3
5
e
t
cos t
Ð
t
tþe
t
sin t
Ð
t
e
t
cos
e
t
sin
t
d
t
d
t
ð
d
t
e
tþt
cos(tt
)
e
tþt
sin(tt
0
0
¼
t
e
t
sin t
Ð
tþe
t
cos t
Ð
)
t
t
e
t
cos
t
d
e
t
sin
t
d
t
0
0
0
"
#
t
0
t
0
e
t
cos t[0
5e
t
cos
5e
t
sin
þe
t
sin t[
5e
t
cos
5e
t
sin
:
tþ
0
:
t
]
j
0
:
tþ
0
:
t
]
j
¼
t
0
t
0
e
t
sin t[0
5e
t
cos
5e
t
sin
þe
t
cos t[
5e
t
cos
5e
t
sin
:
tþ
0
:
t
]
j
0
:
tþ
0
:
t
]
j
e
t
cos t[0
5e
t
cos tþ
5e
t
sin t
þe
t
sin t[
5e
t
cos tþ
5e
t
sin tþ
:
0
:
0
:
5]
0
:
0
:
0
:
5]
¼
e
t
sin t[0
5e
t
cos tþ
5e
t
sin t
þe
t
cos t[
5e
t
cos tþ
5e
t
sin tþ
:
0
:
0
:
5]
0
:
0
:
0
:
5]
5e
t
cos tþ
5e
t
sin tþ
¼
0
:
0
:
0
:
5
5e
t
cos t
5e
t
sin t
0
:
0
:
0
:
5
Therefore,
e
t
cos t
e
t
sin t
5e
t
cos t þ
5e
t
sin t þ
0
:
0
:
0
:
5
x(t)
¼ x
h
(t)
þ x
p
(t)
¼
þ
5e
t
cos t
5e
t
sin t
0
:
0
:
0
:
5
5e
t
cos t þ
5e
t
sin t þ
0
:
0
:
0
:
5
¼
5e
t
cos t
5e
t
sin t
0
:
1
:
0
:
5
The output y(t)is
5e
t
cos t þ
5e
t
sin t þ
0
:
0
:
0
:
5
y(t)
¼
½
1
1
x(t)
¼
½
1
1
5e
t
cos t
5e
t
sin t
0
:
1
:
0
:
5
2e
t
sin t þ
¼
1
4.6 STATE-SPACE REPRESENTATION OF DISCRETE-TIME SYSTEMS
4.6.1 D
EFINITION OF
S
TATE
The state of a discrete-time dynamic system is the minimum number of variables
called state variables such that the knowledge of these variables at time k
¼
k
0
together with input for k
k
0
uniquely determines the behavior of the system for
k
k
0
.IfN variables are needed, then these N variables are considered as compon-
ents of an N-dimensional vector x called state vector. The N-dimensional space
whose coordinates are the states of the system is called state space. The state of a
system at time n is a point in the state space [5].
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