Digital Signal Processing Reference
In-Depth Information
Because we are solving for the vector
x
(
t
), the state transmission matrix is as-
sumed to be of the form
≥(t) =
K
0
+
K
1
t +
K
2
t
2
+
K
3
t
3
+
Á
so that
+
Á
)
x
(0)
x
(t) = (
K
0
+
K
1
t +
K
2
t
2
+
K
3
t
3
q
K
i
t
i
=
B
R
x
(0) = F(t)
x
(0),
(8.32)
a
i = 0
where the
n * n
matrices
K
i
are unknown and
t
is the scalar time. Differentiating
this equation yields
x
#
(t) = (
K
1
+ 2
K
2
t + 3
K
3
t
2
Á
)
x
(0).
+
(8.33)
Substituting (8.33) and (8.32) into (8.30) yields
x
#
(t) = (
K
1
+ 2
K
2
t + 3
K
3
t
2
Á
)
x
(0)
+
Á
)
x
(0).
=
A
(
K
0
+
K
1
t +
K
2
t
2
+
K
3
t
3
+
(8.34)
Evaluating (8.32) at
t = 0
yields
x
(0) =
K
0
x
(0);
hence,
K
0
=
I
.
We next
i = 0, 1, 2,
Á
,
t
i
equate the coefficients of
for
in (8.34). The resulting equations
are, with
K
0
=
I
,
K
1
=
AK
0
Q
K
1
=
A
;
A
2
2!
;
2
K
2
=
AK
1
=
A
2
Q
K
2
=
A
3
2!
Q
K
3
=
A
3
3!
;
3
K
3
=
AK
2
=
o o
.
(8.35)
Thus, from (8.32) and (8.35),
≥(t) =
I
+
A
(t) +
A
2
t
2
2!
+
A
3
t
3
3!
+
Á
.
(8.36)
It can be shown that this series is convergent [4].
In summary, we can express the complete solution for state equations as
t
0
≥(t - t)
Bu
(t) dt,
[eq(8.29)]
x
(t) =≥(t)
x
(0) +
L
