Image Processing Reference
In-Depth Information
Assuming that the initial state is x
(
0
)
, then
x
(
t
) ¼
e
At
x
(
0
)
(
4
:
65
)
where e
At
is the matrix exponential function. To verify that this is the solution to the
homogeneous equation, we need to show that it satis
es the initial condition as well
as the differential equation. The initial condition is satis
ed since
) ¼
e
A0
x
(
x
(
0
0
) ¼
Ix
(
0
) ¼
x
(
0
)
(
4
:
66
)
Differentiating both sides of Equation 4.64, we have
x
(
t
) ¼
d
e
At
x
(
0
)
¼
Ae
At
x
(
0
) ¼
Ax
(
t
)
(
4
:
67
)
d
t
the homogeneous part of the solution is x
(
t
) ¼
e
At
x
(
Therefore,
0
)
. The matrix
exponential e
At
is called state-transition matrix and is denoted by
w(
t
)
. The state-
transition matrix
w(
t
)
is an N
N matrix and is the solution to the homogeneous
equation
w(
t
) ¼
A
w(
t
)
(
4
:
68
)
with initial condition
w(
0
) ¼
I. The state-transition matrix is given by
w(
t
) ¼
e
At
(
4
:
69
)
The state-transition matrix satis
es the following properties. These properties are
stated without proof:
(a) w(
0
) ¼
I
(
4
:
70
)
(b) w(
t
) ¼
e
At
¼ (
e
At
)
1
¼ w
1
(
t
)
(
4
:
71
)
(c) w(
t
1
þ
t
2
) ¼ w(
t
1
)w(
t
2
) ¼ w(
t
2
)w(
t
1
)
(
4
:
72
)
(d) w(
t
3
t
2
)w(
t
2
t
1
) ¼ w(
t
3
t
1
)
(
4
:
73
)
The proof is left as an exercise (see Problem 4.5).
4.5.2 C
OMPUTING
S
TATE
-T
RANSITION
M
ATRIX
There are several methods for computing the state-transition matrix. Here, we review
two methods, which are very useful.
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