Image Processing Reference
In-Depth Information
Therefore, the transfer matrix is
H
(
s
) ¼
C
(
sI
A
)
1
B
þ
D
(
4
:
58
)
Example 4.7
Find the transfer function corresponding to the state equations given by
¼
x
1
(t)
x
2
(t)
þ
u(t)
(4
x
1
(t)
x
2
(t)
01
2
3
x
1
(t)
x
2
(t)
and y(t)
¼
½
45
:
59)
5
6
S
OLUTION
The transfer function is
1
s
1
2
3
¼ C(sI A)
1
B þ D ¼
H(s)
½
41 5
(4
:
60)
5 s þ
6
Hence,
s þ
61
2
3
1
H(s)
¼
5
45
½
s
2
þ
6s þ
5
s
2s þ
15
1
23s þ
10
¼
5
45
½
¼
(4
:
61)
s
2
þ
6s þ
s
2
þ
6s þ
5
3s
10
4.5 SOLUTION OF LTI CONTINUOUS-TIME STATE EQUATIONS
Consider an LTI system described by the state-space equations:
x
(
t
) ¼
Ax
(
t
) þ
Bu
(
t
)
(
:
)
4
62
with the output equation,
y
(
t
) ¼
Cx
(
t
) þ
Du
(
t
)
(
4
:
63
)
To obtain the output of this system for a given input and an initial state, we
rst
solve the state equation (Equation 4.62) and then the solution is substituted into
algebraic equation (Equation 4.63) in order to
. The solution to the
state equation has two parts, homogeneous solution and particular solution [4].
We
nd the output y
(
t
)
first consider the homogeneous solution.
4.5.1 S
OLUTION OF
H
OMOGENEOUS
S
TATE
E
QUATION
The homogeneous solution is the solution of state equations to an arbitrary initial
condition with zero input. The homogeneous state equation is given by
x
(
t
) ¼
Ax
(
t
)
(
4
:
64
)
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