Aerospace and Electronic Systems Magazine December 2017 - 75
Zhang et al.
where β is the softening factor, which can be selected according to the
empirical value. The error covariance of the innovation is obtained by
v (1)v T (1)
V0 (k + 1) = ρV0 (k + 1) + v (k + 1)v T (k + 1)
1+ ρ
k =0
k ≥1
where ρ is the preselected forgetting factor.
ADAPTIVE FILTER
For the 2-HFSWR multisensor surveillance system, the statistical
characteristics of the process noise can be acquired through a series
of experiments. However, due to the complex and volatile external
environment, it is difficult to obtain the statistical characteristics
including the mean and the covariance of the measurement noise,
which can be defined as
E{V (k + 1)} = 0
Cov{V (k + 1)V (k + 1)} = R(k + 1)δ j (k + 1)
Figure 2.
In order to solve this problem, the Saga-Husa adaptive filter is
applied to identify the properties in real time, and the simplified
equation is described by
R(k 1) (1 d (k )) R(k )
d (k )[v (k 1)v T (k 1) H (k 1) P (k 1| k ) H T (k 1)]
(1)
with
complicated situations but increases the calculation complexity. To
improve the computation efficiency of the algorithm, the judging
condition to determine whether the conventional IMM approach
is invalid is added to decide when it is necessary to start the IMMASTF algorithm. The criterion is defined by
Tr{v (k + 1)v T (k + 1)} > s ⋅ Tr{E(v (k + 1)v T (k + 1))}
d (k ) = (1 − b) / (1 − b k +1 ), 0 < b < 1
where b is the forgetting factor, which can be obtained according
to the empirical value.
Nevertheless, when the dimension of the system is high, the
Saga-Husa adaptive filter is prone to divergence because the covariance R cannot guarantee positive definiteness in the process of
iteration. To improve the stability of the filter, the biased estimation of the modified adaptive filter is given by
R(k + 1) = (1 − d (k )) R(k ) + d (k )v (k + 1)v T (k + 1)
Flow graph of IMMASTF algorithm.
(2)
The biased estimation is used if R is not positive definite, otherwise the unbiased estimation is started. The steps are outlined
below.
1. Judge whether R is positive definite.
2. If 1) is true, (1) is adopted to identify the statistical characteristics.
3. If 1) is false, (2) is adopted to identify the statistical characteristics.
where s ( s 1) is the adjustable parameter; v is the innovation.
IMMASTF ALGORITHM
In summary, assume there are n (n 2) models in the established
model set. The structure and the process of the presented IMMASTF algorithm can be illustrated in Figure 2.
As shown in Figure 2, the fusion data are sent to the model set
for interactive initialization. Then, the judging criterion is started
to determine whether to apply the adaptive algorithm. When the
system satisfies the judging criterion, the IMMASTF approach
with the idea of STF and Saga-Husa adaptive filter algorithm
starts work; otherwise, the conventional IMM approach algorithm is adopted. Next, the probability of the model is updated.
Finally, when the state estimation reaches convergence state,
the estimated value of the whole model set can be obtained by
weighted summation of the estimation result from each model.
CRAMER-RAO LOWER BOUND
The nonlinear filtering problem has the form
JUDGING CRITERION
X (k + 1) = AX (k ) + BW (k )
The IMMASTF approach with the idea of STF and Saga-Husa
adaptive filter ensures the stability and accuracy of tracking in
DECEMBER 2017
Z (k + 1) = H ( X (k + 1)) + V (k + 1)
IEEE A&E SYSTEMS MAGAZINE
73
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