Aerospace and Electronic Systems Magazine September 2017 - 17

Layh and Gebre-Egziabher
P −
= 1  12, k
 0

APPENDIX B

0 T
+
 2 = P21, k .
0

(61)

ACCOUNTING FOR INTERFILTER CORRELATION
Now all that remains is to show how to compute the matrices that
are required to form B2 and the role of interfilter correlation on this.
The covariances P1,+k and P2,+ k shown in (52) are readily available
from the measurement update of the parallel filters. That is,

(

xˆ 1,+ k = xˆ 1,− k + K1, k y k ,1 − H1xˆ 1,− k

(

)

xˆ +2, k = xˆ −2, k + K 2, k y k ,2 − H 2 xˆ 2,− k

(53)

)

(54)

where K1,k and K2,k are the Kalman gains associated with the first
and second parallel filters, respectively. To determine expressions
+
+
for P12,
k and P21, k , it is easy to show that, combining (21) with (53)
and (22) with (54):
 Δxˆ − 
Δxˆ 1,+ k = 1  1, k  1T
 v1, k 

Δxˆ

+
2, k

(55)

 Δxˆ − 
= 2  2, k  2T
 v 2, k 

(56)

where
1 = ( I − K1, k H1 ) K1, k 

(57)

2 = ( I − K 2, k H 2 ) K 2, k  .

(58)

The only thing left is to develop a recursive expression that can
−
be used to determine P12,
k . We can do this easily if we write (19)
and (20) as follows:

 xˆ 1,− k +1  F1 0   x1,− k   G1 
 − =
  −  +   uk
 xˆ 2, k +1   0 F2   xˆ 2, k  G 2 
 w1, k 
+

 w1, k 

(62)

= χ k −1 + u k −1 + ξ k −1

(63)

where the definitions of  , , χ, and ξ are obvious. Given these
definitions, it is easy to show the following:

k1 k  T  k

(64)

−
 P−

P12,
k +1
=  1,− k +1

−
 P21, k +1 P2, k +1 

(65)

In the preceding equation, we assume that we have one time
update followed by a measurement update. If there are several time
updates between each measurement update, then (64) is valid only
for the first time update following a measurement update. For the
subsequent time updates, the equation is modified as follows:


k1 k  T  k

Thus,

{(

+
P12,
Δxˆ 1,+ k
k = E

)( Δx ) }
+
2, k

T
 −
−
  Δx1, k Δx 2, k
= 1E  
T
  v1, k x −2, k



( )
( )

SEPTEMBER 2017

T


x1,− k vT2, k  
 T
 2
v1, k vT2, k  


(59)

(66)

Finally, in practice, the measurement updates on the parallel filters may not be synchronized. If this is the case, during the
blending, a filter that has not had a measurement update will simply have its gain set to zero.

(60)

IEEE A&E SYSTEMS MAGAZINE

17



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