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Appendix H / Steady State Tracking Filters
_ _
21.1 Introduction
Kalata [K.7] produced an α−β filter in 1984 which minimised the sum of the
expected PV errors for a filter driven by piecewise constant acceleration.
He also introduced the tracking index (TI) concept that has become a key
element in selecting static filter gains. Sudano [S.17] was responsible for the
α−β−γ filter driven by piecewise continuous jerk minimising expected
PVA error in 1993. The constant acceleration dynamics was replaced by
manoeuvre driven filters first by Sudano [S.16] (1 st order Gauss-Markov in
1995), and by Vorley [V.3] (2 nd order Gauss-Markov in 1991). In these cases
the filter gains are complex functions of the TI requiring iterative
numerical solution, the TI representing the ratio of system to measurement
noise, balancing measurement smoothing with accurate tracking.
21.2 Constant Velocity Filter
An α−β filter with states {X,d(X)/dt}, minimising the Performance Index
used by Kalata [K.7] ,
∑
2 2
2
( E ( ∆X
) + ∆t
⋅ E ( ∆X
) )
J : = min
&
Resulting in the recursive state equations,
( 1 − α ) ⋅ ( X + X&
⋅ ∆t
) + α ⋅ X ( k )
Xk 1 : =
k k
I
+
X&
β
k + 1 : = ⋅
⋅
∆t
( XI
( k ) − Xk
) + ( 1 − β ) X&
k
21-3
Equation 21.2-1
Equation 21.2-2
Equation 21.2-3
Kalata also introduced the tracking index (TI) on which the filter gains in
many subsequent papers on this subject rely. The TI relates the
uncertainty in the X-state (σS) to the uncertainty in the measurement of
that state (σM),
TI
: =
σS
⋅ ∆t
σ
M
2
Equation 21.2-4
The noise characteristics, and the update rate of the filter (∆t), define the
tracking index from which the gains are computed. The filter gains are a
function of the number of measurements processed (n), Quigley [Q.1] ,