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Chapter 4 / Target Tracking
_ _
µ +
:=
( 0.
70 0.
05 0.
20 0.
05 )
4-22
Equation 4.7-1
Switching between filters evolves according to a time-invariant, semi-
Markov chain defined by a constant switching probability matrix. This
process is characterised by the time between changes in the flight regime,
often referred to as the sojourn time. If the switching matrix is diagonally
dominant, and each element is close to 1, with small but equal off-diagonal
transitional probabilities, the sojourn time is random. This is unrealistic as
it implies that a target regime changes rapidly. The transition probabilities
chosen are biased towards the existing flight regime (the Singer filter is a
transitional mode).
Ξ
: =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0.
70
0
0.
25
0
,
,
,
,
0
0.
7
0.
25
0
,
,
,
,
0.
3
0.
3
0.
25
0.
3
,
,
,
,
0
0
0.
25
0.
7
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
Equation 4.7-2
The transitions between the filters are shown in Figure 4-6. Direct
transitions are prevented so that the Singer filter can expand the filter
uncertainties before another flight regime is established. The probability
that covariance expansion continues, or another flight regime is established,
being equally likely. The injection of system noise is expected to improve
the overall filter just after a
manoeuvre has started since
transition between specific
flight regimes is prohibited.
In formulations such as
renewal filters the sojourn
time depends on how long a
flight regime has been
dominant. The sojourn time
increases exponentially
implying that the longer a
flight regime has existed the
more likely it is that a mode
WEAVE
FILTER
VELOCITY
FILTER
SINGER
FILTER
ACCELERATION
FILTER
transition will occur. The transition probabilities are thus determined by a
sojourn time that depends on a Poisson or Gamma distribution rather than a
Normal distribution.
0.25
0.25
0.30
0.3 0.30
0.25
Figure 4-6
Mode Transition Probabilities