Aerospace and Electronic Systems Magazine December 2017 - 76
Target Tracking With 2-HFSWR Multisensor Surveillance System
where the noises W(k) and V(k + 1) obey Gaussian distribution
with zero mean and invertible covariance matrices Q and R , respectively. For the above system model, the CRLB can be derived.
From [22], we get
22
21
11
−1
(4)
In addition, for the IMM algorithm, there are two transition
matrices in the system, hence different CRLB from each system
should be fused to get the final CRLB.
D ( k ) = E{−Δ log p( xk +1 | xk )}
xk
xk
D12 (k ) = E{− Δ xxkk +1 log p( xk +1 | xk )} = [ D 21 ( k )] T
SIMULATION
D 22 (k ) = E{− Δ xxkk ++11 log p ( xk +1 | xk )} + E{− Δ xxkk ++11 log p ( zk +1 | xk +1 )}
− log p ( xk +1 | xk ) = c1 + 12 [ xk +1 − Axk ] T BQ −1B[ xk +1 − Axk ]
− log p ( zk +1 | xk ) = c2 + 12 [ zk +1 − h( xk )] T R −1[ zk +1 − h( xk )]
where c1 and c2 are constants. We can get
D11 (k ) = AT BQ −1BT A
D12 (k ) = − AT BQ −1BT = [ D 21 ( k )] T
D 22 (k ) = ( BQBT ) −1 + E{[−∇ xk h( xk )] T R −1[−∇ xk h( xk )]}
(3)
The initial information submatrix J0 can be calculated from the
a priori probability function
J 0 = E{− Δ log p ( x0 )}
x0
x0
In our algorithm, since the adaptive algorithm is combined, the
measurement noise covariance R is time-varying which is different from the traditional CRLB algorithm. Equation (3) should be
written as
D 22 (k ) = ( BQBT ) −1 + E{[−∇ xk h( xk )] T R(k ) −1[−∇ xk h( xk )]}
where
( x − a)2 + y 2
1
[(
x
a
)
v
y
v
]
−
⋅
+
⋅
x
y
2
2
( x − a) + y
h( x) =
2
2
( x + a) + y
1
[( x + a ) ⋅ vx + y ⋅ v y ]
2
2
( x + a) + y
xa
x
a)2 y 2
(
v v ( x a) 2 ( x a ) yv
x
y
x
( x a)2 y 2
xk h( xk )
H (k )
xa
x
a)2 y 2
(
2
vx vx ( x a) ( x a) yv y
( x a)2 y 2
74
D 22 (k ) = ( BQBT ) −1 + H T R (k ) −1 H
12
J (k + 1) = D (k ) − D (k )[ D (k ) + J (k )] D (k )
11
We can get Jacobi matrix H(k) by partial derivative, so (3) can be
replaced with
Combining with the 2-HFSWR multisensor surveillance system,
the tracking problem of a two-dimensional maneuvering target is
taken into consideration. Simulations compare the tracking performance of the proposed IMMASTF and that of the conventional
IMMEKF. The true trajectory of the target is composed based on
CV and CT models with the process noise covariance Q and the
measurement noise covariance R.
The simulation scenarios are as follows.The model transition
probability matrix is
0.9 0.1
P
0.1 0.9
Set the initial position of the target as (25, 23) km and the initial
velocity (15, 15) m/s. The target bears a constant velocity at [0,
300] s, [600, 900] s, [1,200, 1,500] s, and a constant turning with
the angular velocity 0.003 rad/s at [300, 600] s, [900, 1,200] s. The
statistics characteristics of the measurement noise are unknown
under practical condition. The initial covariance of the tracking filter is set different from the true covariance; set the initial position
of the filter as (25, 20) km.
The root mean square error (RMSE) is used to show the tracking
accuracy of these two approaches by using 100 Monto Carlo runs.
The RMSE of the position estimation of the two tracking filters
and the corresponding CRLB are shown in Figure 3, from which
the RMSE of the IMMASTF approach converges faster and stabilized at a minimized value. Hence, despite there is great deviation
in the initial parameters, the proposed IMMASTF filter approaches
to CRLB more proximately, which has a better tracking accuracy
and a faster convergence rate than that of the IMMEKF method.
The RMSE of the velocity estimation is accumulated to reflect
the tracking accuracy of the two approaches. As shown in Figure 4,
y
0
( x a)2 y 2
xa
v y v y y 2 ( x a) yvx
( x a)2 y 2
( x a)2 y 2
y
0
( x a)2 y 2
xa
v y v y y 2 ( x a) yvx
( x a)2 y 2
( x a)2 y 2
IEEE A&E SYSTEMS MAGAZINE
y
( x a) 2 y 2
0
y
( x a) 2 y 2
0
DECEMBER 2017
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