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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