Image Processing Reference
In-Depth Information
4.5
Show that the continuous-time state-transition matrix
w(
t
) ¼
e
At
satis
es the following properties:
(a) w(
) ¼
I
(b) w(
t
) ¼
e
At
0
¼ (
e
At
)
1
¼ w
1
(
t
)
(c) w(
t
1
þ
t
2
) ¼ w(
t
1
)w(
t
2
) ¼ w(
t
2
)w(
t
1
)
(d) w(
t
3
t
2
)w(
t
2
t
1
) ¼ w(
t
3
t
1
)
4.6
Show that the discrete-time state-transition matrix
w(
k
) ¼
A
k
satis
es the following properties:
(a) w(
) ¼
I
(b) w(
k
) ¼
A
k
0
¼ (
A
k
)
1
¼ w
1
(
k
)
(c) w(
k
1
þ
k
2
) ¼ w(
k
1
)w(
k
2
) ¼ w(
k
2
)w(
k
1
)
(d) w(
k
3
k
2
)w(
k
2
k
1
) ¼ w(
k
3
k
1
)
4.7
The state-transition matrix of a time-varying system is the solution to the
following partial differential equation:
@w(
t,
t)
¼
A
(
t
)w(
t,
t)
,
with boundary condition w(t
,
t) ¼
I
@t
Find the state-transition matrix of the following systems:
x
1
(
t
)
x
2
(
t
)
e
t
x
1
(
t
)
x
2
(
t
)
0
(a)
¼
0
2t
x
1
(
t
)
x
2
(
t
)
x
1
(
t
)
x
2
(
t
)
1
t
(b)
¼
0
1
4.8
An LTI system is described by the following differential equations:
x
1
(
t
)
x
2
(
t
)
u
(
t
)
x
1
(
t
)
x
2
(
t
)
64
0
1
¼
þ
20
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