Aerospace and Electronic Systems Magazine September 2017 - 15

Layh and Gebre-Egziabher

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FlightControlSystem
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C
C

xˆ : A priori state vector estimate

y −2, k = H 2 xˆ 2, k + v 2, k

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Mapping matrices M1 and M2, which map x1 and x2 into x, respectively, are used to shape x1,k and x2,k into xk. We call the shaped
vectors s1,k and s2,k, respectively. This can be stated mathematically
as follows:
x k = M1x1, k  s1,k

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x k = M1x 2, k  s 2,k

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Our goal is to form an optimal estimate of x at t = tk, denoted as
xˆ +k , by a linear fusion of all xˆ i+, k or their shaped equivalents sˆ1,+ k and
sˆ +2, k. Mathematically, what we are looking for is

)

(

)

= B1 sˆ1,+ k + B 2 sˆ 2,+ k

The following variable definitions are used in the derivation of the
blender equations:

xˆ +k : A posteriori state vector estimate

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(

DERIVATION OF BLENDER EQUATIONS

xk: True (unknown) state vector

y1,− k = H1xˆ 1, k + v1, k

xˆ +k = B1 M1xˆ 1,+ k + B 2 M 2 xˆ 2,+ k

APPENDIX A

C

These are linear models. Even though, for clarity, an explicit dependence on k was not indicated, the matrices in the preceding
equation can be time varying. Thus, these equations are also valid
for an EKF in which the matrices would represent the Jacobians of
the dynamic model [23]. Similarly, the measurement models associated with the dynamics models given earlier can be written as
follows:

(25)
(26)

where the blending weights B1 and B2 are selected to satisfy the
following requirements of optimality:
29. The estimate xˆ +k is unbiased.
30. The estimate xˆ +k has minimum variance.
We examine the implication of these two requirements next.

−
k

xˆ i+, k: The ith filter a posteriori state vector estimate of the
ith filter

UNBIASED ESTIMATE

C

xˆ i−, k: The ith filter a priori state vector estimate of the ith filter

Mathematically, the requirement for an unbiased estimate can be
written as follows:

C

yi,k: Measurement vector for the ith parallel filter

C

vi,k: Measurement error on the ith measurement vector

C

ui,k: The ith filter input

C

{

{ }

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In terms of B1 and B2, this requirement can be written as follows:

wi,k: Uncertainty (unmodeled error) on the ith filter dynamic
model
It is assumed that each blender receives only two inputs. Given
these definitions and assuming only two parallel filters per blender,
we can write the dynamic models for them as follows:
C

xˆ 1,− k +1 = F1xˆ 1,− k + G1u1, k + w1, k

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xˆ −2, k +1 = F2 xˆ −2, k + G 2u 2, k + w 2, k

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SEPTEMBER 2017

}

E xˆ +k − x k = 0  E xˆ +k = x k

{ }

{

}

E xˆ +k = E B1sˆ1,+ k + B 2sˆ 2,+ k = x k

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We assume that both parallel estimates are unbiased; thus,
E sˆ1,+ k = x k and E sˆ +2, k = x k. This allows us to write the following:

{ }
{

{ }

}

E B1sˆ1,+ k + B 2sˆ 2,+ k = ( B1 + B 2 ) x k = x k

(29)

This implies the following:
B1 + B 2 = I or B1 = I − B 2

IEEE A&E SYSTEMS MAGAZINE

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Table of Contents for the Digital Edition of Aerospace and Electronic Systems Magazine September 2017
