October 2011
Intermediate to advanced
749 pages
23h 36m
English
Content preview from Optimal Estimation of Dynamic Systems, 2nd Edition
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380 Optimal Estimation of Dynamic Systems
for 0 ≤ t ≤ T. We now concentrate our attention on the expectation in Equa-
tion (5.273). Substituting Equation (5.270) into the expectation in Equation (5.273)
gives
E
x(t)e
T
f
(
τ
)
= −E
x(t)
˜
x
T
f
(
τ
)
H
T
(
τ
)+E
x(t)v
T
(
τ
)
(5.274)
Substituting x(t)=
ˆ
x
f
(t) −
˜
x
f
(t) into Equation (5.274) gives
E
x(t)e
T
f
(
τ
)
= E
˜
x
f
(t)
˜
x
T
f
(
τ
)
H
T
(
τ
) −E
ˆ
x
f
(t)
˜
x
T
f
(
τ
)
H
T
(
τ
)
+ E
x(t)v
T
(
τ
)
(5.275)
Since the state estimate is orthogonal to its error and since the measurement noise is
uncorrelated with the true state, then Equation (5.275) reduces down to
E
x(t)e
T
f
(
τ
)
= E
˜
x
f
(t)
˜
x
T
f
(
τ
)
H
T
(
τ
) ≡ P
f
(t,
τ
)H
T
(
τ
) (5.276)
where P
f
(t,
τ
) ≡ E
˜
x
f
(t)
˜
x
T
f
(
τ
)
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