Graphics Programs Reference
In-Depth Information
y
()
Gx
()
v
()
=
+
(9.61)
The homogeneous solution to this system is given in Eq. (9.27) for continuous
time, and in Eq. (9.40) for discrete time.
x
t
()
v
∫
G
Σ
Σ
u
x
y
A
Figure 9.18. An LTI system.
The state transition matrix corresponding to this system can be obtained
using Taylor series expansion of the vector
x
. More precisely,
T
2
2!
ß
-----
ßß
…
xx
=
+++
(9.62)
ß
ß
T
ßß
…
=
ßß
++
ßß
…
=
+
It follows that the elements of the state transition matrix are defined by
T
ji
÷
(
ji
)!
1
≤
ij n
,
≤
Φ
i
[]
=
(9.63)
0
ji
<
Using matrix notation, the state transition matrix is then given by
T
2
2!
1
T
----- …
Φ
=
(9.64)
01
T
…
00 1…
…………
The matrix given in Eq. (9.64) is often called the Newtonian matrix.
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