Target Tracking - MATLAB Simulations for Radar Systems Design - page 426

αβ

9.8.1. The Filter

The tracker produces, on the observation, smoothed estimates for
position and velocity, and a predicted position for the observation.
Fig. 9.20 shows an implementation of this filter. Note that the subscripts Ð p Ñ
and Ð s Ñ are used to indicate, respectively, the predicated and smoothed values.
The tracker can follow an input ramp (constant velocity) with no steady
state errors. However, a steady state error will accumulate when constant
acceleration is present in the input. Smoothing is done to reduce errors in the
predicted position through adding a weighted difference between the measured
and predicted values to the predicted position, as follows:

αβ nth

( n + 1 ) th

αβ

x s () xnn = ( ) = x p () α x 0 + ( () x p – () )

(9.84)

β
---

ß s ß s () x ' nn = ( ) = ( n – 1 ) + ( x 0 () x p – () )

(9.85)

x 0 is the position input samples. The predicted position is given by

T ß s

x p () x s = ( nn – 1 ) = x s ( n – 1 ) + ( n – 1 )

(9.86)

The initialization process is defined by

x s () x p = () x 0 = ()

ß s

() 0 =

x 0 () x 0 () –

T

ß s () = --------------------------------

A general form for the covariance matrix was developed in the previous sec-
tion, and is given in Eq. (9.75). In general, a second order one-dimensional
covariance matrix (in the context of the

αβ filter) can be written as

C xx C x ß

C nn ( ) =

(9.87)

C ß x C ß ß

where, in general, C xy is

Exy {}

C xy =

(9.88)

By inspection, the αβ filter has

1 α – ( 1 α – ) T

A = (9.89)

– β ⁄ T ( 1 β – )