Graphics Programs Reference
In-Depth Information
homogeneous solution (i.e.,
w
=
0
) to this linear system, assuming known
initial condition
x
()
at time
t
0
, has the form
x
()
Φ
t
=
(
)
x
t t
0
(
)
(9.27)
0
The matrix
Φ
is known as the state transition matrix, or fundamental matrix,
and is equal to
e
A
t t
0
(
)
Φ
t
(
)
=
(9.28)
0
Eq. (9.28) can be expressed in series format as
∞
∑
IA
t
A
2
t
2
2!
t
k
k
!
e
A
()
++ +
A
k
-----
…
----
Φ
t
(
)
=
=
=
(9.29)
0
t
0
=
0
k
=
0
Example:
Compute the state transition matrix for an LTI system when
01
0.5
A
=
1
Solution:
The state transition matrix can be computed using Eq. (9.29). For this pur-
pose, compute
A
2
A
3
…
and
. It follows
1
---
1
---
1
---
1
A
2
A
3
=
=
…
1
---
1
---
1
---
0
Therefore,
1
1
1
---
t
2
2!
---
t
3
3!
---
t
3
3!
t
2
2!
10
t
+
-------
++
------- …
0
+
t
-----
++
------- …
Φ
=
1
1
1
---
t
2
2!
---
t
3
3!
---
t
2
2!
0
t
3
3!
1
---
t
0
+
-------
------- …
+
1
t
+++
-------
------- …
The state transition matrix has the following properties (the proof is left as
an exercise):
Derivative property
1.
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