Aerospace and Electronic Systems Magazine September 2017 - 16
Graceful Degradation and Recovery from GNSS Interruptions
We can now use this constraint on the weights to write the following:
Pk T
xˆ +k = ( I − B 2 ) sˆ1,+ k + B 2sˆ 2,+ k
where
(31)
sˆ +
B 2 1,+ k
sˆ 2, k
= I − B 2
(32)
This can be more conveniently expressed in terms of the parallel
filter states xˆ 1,+ k and xˆ +2, k as follows:
xˆ +k = ( I − B 2 ) M1xˆ 1,+ k + B 2M 2 xˆ 2,+ k
(33)
xˆ +
= ( I − B 2 ) M1 B 2M 2 1,+ k
xˆ 2, k
(34)
Alternatively, we can write this in a form similar to the KF measurement update equation as follows:
(
xˆ +k = M1xˆ 1,+ k + B 2 M 2 xˆ 2,+ k − M1xˆ 1,+ k
)
(35)
To determine B2, the second requirement of minimum variance is
imposed.
{(
)(
)}
T
(36)
+
k
Δx = ( I − B 2 ) M xˆ
+
k
+
1 1, k
= ( I − B 2 ) xˆ
+
1, k
+ B 2M xˆ
+
2 2, k
{(
)( Δx ) }
(43)
{(
)( Δx ) }
(44)
{(
)( Δx ) }
(45)
{(
)( Δx ) }
(46)
P1,+k = E Δ x1,+ k
+
P12,
Δx1,+ k
k = E
T
+
2, k
+
P21,
Δx 2,+ k
k = E
P2,+ k = E Δx 2,+ k
T
+
1, k
T
+
1, k
T
+
2, k
(
)
T
+
+
. We show how these covariances can be
Note that P21,
k = P12, k
computed later. The trace of (40) is minimized when
∂B 2
) = 0.
(47)
If we expand this equation and group like terms, we get:
0 = 2B 2Γ1 − 2B 2Γ 2 − Γ3
(48)
− xk
(37)
(
Γ1 = M1P1,+k M1T + M 2 P2,+ k MT2
)
(
+
+
T
T
Γ 2 = M 2 P21,
k M1 + M1P12, k M 2
(
(49)
)
)
+
T
Γ3 = M1P1,+k M1T − M1P12,
kM2 .
+ B 2 xˆ
+
2, k
− x k + B 2M1x1, k
−B 2M 2 x 2, k
= ( I − B 2 ) M1Δx1,+ k + B 2M 2 Δx 2,+ k
(38)
(39)
where we have used the fact that x = M1x1 = M 2 x 2 , Δx1,+ k = xˆ 1,+ k − x k,
and Δx +2, k = xˆ 1,+ k − x k. We can use this result in (36) to rewrite Pk+ as
16
(42)
where
and Δx = xˆ − x k is the a posteriori estimation error. Using (33),
the a posteriori estimation error can be written in terms of xˆ 1,+ k and
xˆ +2, k as follows:
+
k
(41)
+
+
+
and the covariances P1,+k, P12,
k , P21, k , and P2, k are defined as follows:
(
The requirement that the estimate have a minimum variance means
that it must minimize the trace of the a posteriori covariance or
Tr Pk+, where
P12,
k
P2,k
= ( I − B 2 ) M1 B 2M 2
∂ Tr Pk+
MINIMUM VARIANCE ESTIMATE
Pk+ = E Δx +k Δx +k
P
1,k
P21,k
(40)
(50)
(51)
When this is solved for B2, it gives the following:
B 2 = Γ3 ( Γ1 − Γ 2 )
−1
(52)
Even though the preceding notation does not indicate that B1 and
B2 vary with k, their dependence on the covariance matrices implies that they are time varying.
IEEE A&E SYSTEMS MAGAZINE
SEPTEMBER 2017
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