Image Processing Reference
In-Depth Information
N
(a)
Modal Matrix Approach
: Let
i
¼
1
be the set of eigenvalues and
their corresponding eigenvectors of matrix A. Under the assumption of
independent eigenvectors, the state-transition matrix is given by
{
l
i
, x
i
}
w(
t
) ¼
M
L(
t
)
M
1
(
4
:
74
)
where
M
¼
x
1
½
x
2
x
N
is the N
N matrix of eigenvectors known as
modal matrix
L(
t
)
is an N
N diagonal matrix given by
2
4
3
5
e
l
1
t
0
0
e
l
2
t
0
0
L(
t
) ¼
(
4
:
75
)
.
.
.
.
.
.
e
l
n
t
00
The second approach which is more general is based on Laplace Transform.
(b)
Laplace Transform Technique
: In this method,
is computed using
the Laplace transform. Taking the Laplace transform from both sides of
Equation 4.62, we have
w(
t
)
s
F(
s
) w(
0
) ¼
A
F(
s
)
(
4
:
76
)
Since
w(
0
) ¼
I, then
F(
s
) ¼ (
sI
A
)
1
(
4
:
77
)
Therefore,
w(
t
) ¼
L
1
[(
sI
A
)
1
]
(
4
:
78
)
where L
1
stands for inverse the Laplace transform.
Example 4.8
Find the state-transition matrix of the following dynamic system using the two
methods described above:
x(t)
11
x(t)
¼
1
1
S
OLUTION
(a)
Modal Matrix Technique
: The eigenvalues of matrix A are
¼
jlI Aj¼
l þ
11
1)
2
(
l þ
þ
1
¼
0
! l
1
¼
1
þ j,
l
2
¼
1
j
1
l þ
1
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