October 2011
Intermediate to advanced
749 pages
23h 36m
English
Content preview from Optimal Estimation of Dynamic Systems, 2nd Edition
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580 Optimal Estimation of Dynamic Systems
to obtain
Φ(t,t
0
)=I +
t
t
0
F(
τ
1
) d
τ
1
+
t
t
0
F(
τ
1
)
τ
1
t
0
F(
τ
2
)Φ(
τ
2
,t
0
) d
τ
2
d
τ
1
(A.22)
One can now re-use Equation (A.20) to write
Φ(
τ
2
,t
0
)=I +
τ
2
t
0
F(
τ
3
)Φ(
τ
3
,t
0
) d
τ
3
(A.23)
which, when substituted into the final integrand of Equation (A.22), yields
Φ(t,t
0
)=I +
t
t
0
F(
τ
1
) d
τ
1
+
t
t
0
F(
τ
1
)
τ
1
t
0
F(
τ
2
) d
τ
2
d
τ
1
+
t
t
0
F(
τ
1
)
τ
1
t
0
F(
τ
2
)
τ
2
t
0
F(
τ
3
) d
τ
3
d
τ
2
d
τ
1
+ ···
(A.24)
This procedure is known as the Peano-Baker Method; as is shown by Ince (1926),
11
uniform and absolute convergence is guaranteed. Whether or not this process is prac-
tical depends, of course, upon how difficult the elements of the F(t) are to integrate,
and how quickly convergence ...
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