2018 3rd International Conference on Computational Modeling, Simulation and Applied Mathematics (CMSAM 2018) ISBN: 978-1-60595-035-8
Research on Adaptive Filtering Algorithm for Maneuvering
Target Tracking
Yi WANG and Qiang LI
Air Force Engineering University
Keywords: Target tracking, Adaptive filtering, Truncated normal probability density model.
Abstract: For the maneuvering target tracking problem, based on the truncation of the normal probability density model, the self-adaptive adjustment of the noise variance is achieved by the functional relationship between the target maneuvering condition and the change of the position estimator at the adjacent sampling time, and a new adaptive filtering algorithm which is based on truncated normal probability density model modified is proposed. The computer simulation results show that the algorithm has good tracking performance when tracking maneuvering targets, and greatly improves the ability to track non-maneuvering targets.
Introduction
Target tracking is a key and difficult issue in the field of data fusion research. In particular, the subject of maneuvering target tracking has been widely used in military and civilian fields. In order to track the maneuvering target, many scholars have proposed many methods to improve Kalman filtering. Chinese scholar Zhou H proposed an adaptive filtering method based on the current statistical model of maneuvering targets [1], which has a high reputation internationally. To this end, Cai Q and others improved the algorithm and proposed a truncated normal probability density model for target acceleration and its adaptive filtering algorithm [2], which also obtained very good results. But the maximum maneuvering acceleration amax of the model system parameters cannot be adjusted adaptively during the tracking process, and the tracking between the rapidity and accuracy of the tracking system is difficult to satisfy.
In order to solve the problem, this paper presents an adaptive filtering algorithm based on truncated normal probability density model (abbreviated as TGPMMKF). This algorithm combines the ideas of the literature [1]. When the maneuvering target is maneuvering with a certain acceleration, the acceleration value at the next moment is limited, and can only be in the “current” acceleration neighborhood. Base on this nature and accord to the maneuverability of acceleration components change between adjacent samples identifies targets online to achieve the online self-adaptive estimation of the variance of the process noise, which avoids the adverse effect of the presetting of the acceleration limit value in the truncated normal probability density model on the maneuvering target state estimation accuracy. Computer simulations show that the algorithm has good tracking ability for strong maneuvering targets and has the advantage of real-time performance.
Normal Truncation Probability Density Model Adaptive Tracking Algorithm
Non-zero Mean, Time-correlated Normal Truncation Probability Density Model
Target maneuver means that the target acceleration changes. Assume that the mean value of the target acceleration is not 0, ie it is not a stationary random process. Therefore, a non-zero mean correlation model can be used to describe the maneuver of the target:
( ) ( )
x t a a t (1)
( ) ( ) ( )
Where x t( ) is the non-zero mean associated random acceleration; ais the mean of the maneuvering acceleration; a(t) is zero mean colored acceleration noise; α is the reciprocal of the time constant of the maneuvering acceleration; ω(t) is white noise with a mean of 0 and a variance of2 2a2 .
Simulating the target's maneuvering acceleration as a normal distribution is easy to conceive. The question is how to deal with the relationship between the mean and variance of the normal distribution.
The maneuvering of the target is caused by unknown flight instructions, and the flight instructions are constrained by various factors such as atmospheric turbulence, ground fire change, and climate change. In addition, the operator and the aircraft itself in a piloted aircraft limited the acceleration tolerance. So when considering the state distribution of the target maneuvering acceleration and looking for the relationship between its mean and variance, make the following assumptions.
Assumption 1: The target maximum acceleration is bounded. At this stage, the target maneuver can reach 5~6g (g=11.8m/s2), assuming the target maximum accelerationamaxis 8g.
Assumption 2: When the target is maneuvering with acceleration a, the target maneuver range at the next moment is determined. If a is large, the range of change of target a at the next moment is small, and the reverse is also true.
According to Chebyshev's inequality, when random variables follow a normal distribution, the maximum probability that the deviation of random variables and its mathematical expectation falls outside the range of 3 times its mean square deviation is 0.003. So assume that:
max
|a |a|| 3 a (3) Then the relationship between the variance a2 and the mean a is
2
2 ( max | |)
9
a
a a
(4)
In summary, the truncated normal probability density of the target acceleration is expressed as follows:
2 2
( ) 2
max
1 1
( ,| | 3
( , ) 2 2 (3) )
0 others
a
a a
a a
e a a a a
p a a erf
(5)
where
3
2 2 2 /2
0
1
[ ] var[ ] ( 0.02666) (3) 0.49865
2 y
a a a
E a a a erf e dy
， ，
Adaptive Kalman Filtering Algorithm Based on Normal Truncation Probability Density Model
From the above analysis, the discretized state equation and observation equation of the target can be
X(k1)= ( ) ( ) k X k U k a W k( ) ( ) (6)
( ) ( ) ( )Z k H k X k V k
2 1
1 ( 1 )
1
( ) 0 1 (1 )
0 0
T
T
T
T T e
k e e , 2 1 1 ( ) 2 1 ( ) 1 T T T T e T e
U k T
e 2
( ) [ ( ) T( )] 2
a
Q k E W k W j q
Where q is a constant matrix anda2depends on the description of the target acceleration.
Ifxis a state variable of the Kalman filter, the output of the Kalman filter itself contains the statistical information x k k( / )of the target activation. According to the theory of optimal
estimation, the estimated value ˆ( / )x k k of the state xis the conditional mean of the given input.
ˆ( / ) [ ( ) / k]
x k k E x k Z
(8) Where, Zk { (1), (2), (3),Z Z Z , ( )}Z k
So acan be replaced by ( / )x k k . And then
2 2 ( max | ( / ) |)ˆ
9
a
a x k k
(9)
ˆ( 1/ ) ( ) ( / )ˆ
X k k k X k k
(10)
2
1 1
2
( ) 0 1
0 0 1
T T k T
Therefore, adopting the equations of state described in equation (6) and the equation described in equation (7),and using the standard Kalman filter recurrence relation, an adaptive Kalman filter algorithm (short for TGPPKF) based on the normal truncation probability density model is obtained. The basic process is as follows,
ˆ( 1/ ) ( ) ( / )ˆ
X k k k X k k
(11)
1 1
( 1/ ) ( ) ( / ) T( ) ( )
P k k k P k k k Q k
(12)
ˆ( 1/ ) [ (ˆ 1/ )]
Z k k F X k k (13)
( 1) ( 1) ( 1/ ) T( 1) ( 1)
S k H k P k k H k R k (14)
where,
ˆ ( 1/ )
( 1)
X k k
F H k X 1
( 1) ( 1/ ) T( 1) ( 1)
ˆ( 1/ 1) ˆ( 1/ ) ( 1)[ ( 1) ˆ( 1)/ ]
X k k X k k K k Z k Z k k (16)
( 1/ 1) ( 1/ ) ( 1) ( 1) ( 1/ )
P k k P k k K k H k P k k (17)
From the above analysis, it can be seen that the non-zero mean value of the target acceleration introduces the change of 2
a
to follow the change of ˆ( / )x k k , which affects the Q matrix and realizes the closed-loop calculation of the gain matrix. In addition, it also makes the Φ into Φ1,
which is equivalent to the effect that αT tends to 0, improves the tracking performance and achieves the purpose of adaptive filtering.
Adaptive Filtering Algorithm Based on Truncated Normal Probability Density Model
From formula a and formula b, it can be seen that the magnitude of the current acceleration variancea2is directly affected by the maximum value of the target maneuvering acceleration amax.
At the same time, the size of the variancea2 directly affects the variance matrix Q of the process noise. Therefore, it is not difficult to conclude that the ability to characterize external disturbances based on the normal truncation probability density model depends on the maximum value of the target maneuvering acceleration. At present, in the practical application of the normal truncation probability density model, the maximum value of the target maneuvering acceleration is generally artificially given. The problem with this approach is that if amaxis selected too large, the bandwidth of the tracking filter can be expanded, and the ability to track maneuvering targets can be improved. But at the same time, the steady-state error of the tracking filter will increase. If amax is chosen too small, it is bound to reduce the bandwidth of the tracking filter. Although the tracking filter's steady-state error can be reduced, it will reduce the ability to track the maneuvering target.
Based on the pre-set dependence of this acceleration limit value, based on the normal truncation probability density model, the self-adaptive adjustment of the noise variance is realized by the functional relationship between the target maneuvering condition and the change of the position estimator at the adjacent sampling time. Based on this, a new adaptive filtering algorithm based on truncated normal probability density model modified adaptive filtering algorithm (abbreviated as TGPMMKF) is proposed. The basic process of the algorithm is the same as that of the TGPPKF
algorithm. The key point is the improvement of the covariance
2
a
of the dynamic acceleration of the computer.
Set Δa as the change in target maneuvering acceleration, that is the absolute value of the change
in acceleration from time k to k+1.Then, the change relationship (
2
2
T
d a
)between the
displacement and the acceleration change is used to obtain the position change of the maneuvering target in two adjacent sampling periods due to the target maneuver. The value of dcan be obtained from the difference in the position estimate between adjacent sampling moments, ie
ˆ( 1/ 1) ˆ( / )
d X k k X k k
(18)
Where ˆ (X k1/k1) represents the estimated value of the position component at time k+1, and ˆ ( / )
X k k represents the estimated value of the position component at time k. Therefore, from the above analysis, it can be seen that the expression between a and d is,
2
ˆ ˆ
2( ( / )X k k X k( 1/k 1))
a
T
the current maneuver acceleration follows the normal truncation distribution.
a a a (20)
2
ˆ ˆ
2( ( / ) ( 1/ 1)) 3 a
X k k X k k
a
T
(21)
2 2
4
ˆ ˆ
4( ( / ) ( 1/ 1)) 9
a
X k k X k k
T
(22) The functional relationship between the covariance of the maneuvering acceleration and the change of the position estimator is established by equations (18)-(22).From equation (22), it can be seen that when the target does not maneuver, Δd is relatively small, so the covariance 2
a
value of the perturbation of the maneuvering acceleration at the next moment is small, and when the target maneuvers, Δd will increase sharply, and with the greater the deviation of the maneuver, the greater the covariance 2
a
of the perturbation of the maneuvering acceleration, and the faster the algorithm converges. In addition, from the formula 2
( ) 2 a
Q k q, it can be seen that the essence of the noise variance determination is to use the change of the acceleration of the first two moments to predict the maneuver condition of the target at the next moment to complete the noise variance
Q(k)adaptive estimation. Combining equations (6) to (17), an improved maneuvering target tracking algorithm with process noise variance self-adaptation is obtained. Through the derivation process of the above algorithm, it is known that the improved algorithm effectively avoids the preset problem of limit acceleration amax in truncating the normal probability density model and increases the range
of the target target maneuver acceleration .
Simulation Results and Analysis
In order to test the effectiveness of the algorithm, the trajectory tracking of the same target with the TGPMMKF algorithm and TGPPKF algorithm under the same simulation conditions is given. The simulation program uses 100 Monte Carlo simulations to compare the TGPMMKF algorithm and the TGPPKF algorithm presented in this paper. Since the purpose of the simulation is to examine the performance of the algorithm for position tracking, the program chooses to perform specific simulation tests in rectangular coordinates.
Target start state: x0=[77000m,-426m/s,2000m,0m/s],the target movement process lasts 120s.
[image:5.595.51.543.585.682.2]The target maneuver moment and acceleration are shown in the table.
Table 1. Target maneuver moment and acceleration.
Moment that target
maneuvering occurs t=31s t=38s t=61s t=71s t=91s
x direction acceleration 10 m/s2 0 m/s2 -5 m/s2 -10 m/s2 50 m/s2
y direction acceleration -10 m/s2 -10 m/s2 30 m/s2 0 m/s2 -2 m/s2
In the simulation process, assume that the radar sampling interval is T = 1s, ranging error
100m
, angle error 0.03rad, autocorrelation time constant 1 / 20, and the maximum
acceleration 2
max 100 /
a m s , 2
max 100 /
a m s .
maneuvering state of the target and the change of the position estimation vector between the adjacent sampling moments to establish a self-adaptive estimation method of the process noise variance, effectively avoids the dependence of the state-estimation algorithm on the acceleration limit value of the adaptive filtering algorithm based on the truncated normal probability density model,and greatly enhances the dynamic range of the tracking maneuvering target.
Conclusions
Theoretical analysis and simulation results show that the modified maneuvering acceleration variance can better reflect the target's actual maneuverability, and its adaptive filtering algorithm has stronger ability to track the maneuvering target and the effect is better. Compared with the original adaptive filtering algorithm, the improved adaptive filtering algorithm shortens the dynamic delay of maneuvering target, effectively overcomes the problem of presetting the acceleration limit
amax, improves the tracking accuracy, and expands the tracking. The maneuver range of
maneuvering targets has excellent tracking performance for maneuvering targets.
References
[1] Zhou Hongren, Jing Zhongliang, Wang Peide. Maneuvering Target Tracking [M]. Beijing: National Defense Industry Press, 199l: 135-153.
[2] Cai Qingyu, Xue Yi, Zhang Boyan. Phased array radar data processing and simulation technology [M]. Beijing: National Defense Industry Press, 1997: 123-127.
[3] Li Tao, Wang Baoshu, Qiao Xiangdong. Improved target tracking algorithm based on truncated normal probability model [J]. Systems Engineering and Electronics Technology, 2003, 25(10): 1289-1291.
