Graphics Programs Reference
In-Depth Information
∂
Φ
t
(
)
=
A
Φ
t
(
)
(9.30)
0
0
∂
t
Identity property
2.
Φ
t
0
(
t
0
)
Φ
()
I
=
=
(9.31)
Initial value property
3.
∂
Φ
t
(
)
=
A
(9.32)
0
∂
t
t
=
t
0
Transition property
4.
Φ
t
2
(
t
0
)
Φ
t
2
=
(
t
1
)
Φ
t
1
(
t
0
)
;
t
0
≤≤
t
1
t
2
(9.33)
Inverse property
5.
)
Φ
1
Φ
t
0
(
t
1
=
(
t
1
t
0
)
(9.34)
Separation property
6.
)
Φ
t
()
Φ
1
Φ
t
1
(
t
0
=
t
()
(9.35)
The general solution to the system defined in Eq. (9.25) can be written as
t
∫
x
()
Φ
t
=
(
)
x
t
()
Φ
t
+
(
τ
)
Bw
()τ
d
(9.36)
0
t
0
The first term of the right-hand side of Eq. (9.36) represents the contribution
from the system response to the initial condition. The second term is the contri-
bution due to the driving force . By combining Eqs. (9.26) and (9.36) an
expression for the output is computed as
w
t
∫
y
()
C
e
A
t t
0
(
)
x
t
()
C
e
A
t
τ
(
)
BD
δ
t
=
+
[
(
τ
)
]
w
()τ
d
(9.37)
t
0
C
e
A
t
BD
δ ()
Note that the system impulse response is equal to
.
The difference equations describing a discrete time system, equivalent to
Eqs. (9.25) and (9.26), are
x
n
(
+
1
)
=
Ax
()
Bw
()
+
(9.38)
y
()
Cx
()
Dw
()
=
+
(9.39)
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