A Kind of Self-tuning Kalman Filter for the High Maneuvering Target Tracking System

For the high maneuvering target tracking system,in order to solve the filtering problem with unknown noise statistics, based on the jerk model, a kind of self-tuning filter is presented in this paper. System identification method is used to estimate the noise statistics information which is substituted into the optimal Kalman filter to get the self-tuning Kalman filter. It’s proved that this self-tuning Kalman filter converges to the optimal Kalman filter in this paper. A simulation example shows the effectiveness of this kind self-tuning filter.


Introduction
Kalman filter has been proved to be the best method to deal with the filtering problem in the target tracking field [1].In order to get the optimal convergence of the filter, noise statistics information should be precisely known.But in the practical applications, especially in the early stage of the filtering process, noise statistics information is always unknown [2].In order to solve the filtering problem with unknown noise statistics, self-tuning filter was presented in [3].It's basic principle is to estimate the unknown parameters and noise statistics information by system identification method.The estimations could be used to replace the real values in the Kalman filter [4].Based on the Dynamic Error System Analysis (DESA) method [5], it has been proved that the self-tuning filter has good convergence to the optimal Kalman filter.Deng presented self-tuning α-β filter [6] and α-β-γ filter [7], which could deal with the models with unknown parameters and noise ststistics information.In these filters, the target state vector includes position, velocity and acceleration.But when the target has high maneuverability, the tracking ability of these filters will be declined.Jerk model is known to be a accurate model with high order in these years.To get more accurate estimations, the acceleration rate was considered in the Jerk model filter [8], which could deal with the system with acceleration rate.This paper try to give a kind of self-tuning filter based on Jerk model to deal with the vector with acceleration rate.The convergence of this kind self-tuning filter is proved in this paper.An example shows the good convergence of this kind self-tuning Kalman filter.

Problem formulation
Consider the time-invariant system where t is the discrete time, ( ) τ is the interval of two discrete time, 2 j σ is the variance of ( ) j t ,α is the constant of the self-correlation function, when α is very small, 0 α → , ( ) j t can be treated as a constant value, F has the form ] The covariance of ( ) w t is σ is recorded as R.With the assumption 1-2, the objective is to get the self-tuning Kalman filter and its convergence analysis.

Noise Variance Estimation
Define backward shift operator 1 q − and 1 ( 1)   ( ) q X t X t − + = , From (1) we have Substituting ( 8) into (2), we can get From ( 3)-( 5), we can get From (10), we can get 2) ( ) q q q q q q q y t T q T q T q T q T q T q T w t q q q q q q q v t According to the Assumption 1, we define 2) ( ) r t q q q q q q q y t We define ( ) M q − and 1 ( ) N q − are both the polynomial matrices with the form x n is the highest order of 1 ( ) X q − .From ( 13)-( 16) we can get The correlation function of ( ) r t could be estimated by sampling method Which has the recursive formula From ( 18) we can get the correlation function Equation ( 21) can be treated as linear functions with unknown variables Q and R , this linear functions are compatible, so the estimations ˆ( ) Q t and ˆ( ) R t at time t could be obtained.

Self-tuning Kalman filter
Substituting ˆ( ) Q t and ˆ( ) R t ( Q and R ) into the optimal Kalman filter, we have the self-tuning Kalman filter In ( 22), we can get Riccati function When t is renewed, the values of the varibles are renewed in ( 22)-(23).

The Convergence analysis
Applying the ergodicity of stationary stochastic process, we can get . .1 w p is in short for "with probability 1", In(19), the estimations of Q and R are consistent, they converge to the real values, we can get So self-tuning Kalman filter could converge to the optimal Kalman filter.

Simulation Example
We take sampling period T = 1 s, input noise variance 2 0.8 . The self-tuning Kalman filter presented in this paper is used, Figure 1 is the comparison curve of self-tunning Kalman filter and optimal Kalman filter.In Figure 1, the curved line denotes the optimal Kalman filter, spots denote selftuning Kalman filter.It can be seen that self-tuning Kalman filter converges to the optimal filter.Figure 2 is the error curve,the curved line means the difference between the self-tuning Kalman filter and optimal filter.It has convergence to 0 as ( 27) is satisfied.

Conclusion
This paper presents a kind of self-tuning Kalman filter, which could deal with the target tracking model with accelerate rate.Jerk model is used to fit the state equations.The convergence of this algorithm is proved.The simulation example shows the good convergence of this algorithm.This kind of self-tuning Kalman filter could improve the tracking ability of the high maneuvering target, but the disadvantage is that the calculation is complicated.In the future, we can use the dimension reduction method to deal with the matrices to reduce the amount of computation.

Figure 1 .
Figure 1.The self-tuning Kalman filter and optimal Kalman filter

Figure 2 .
Figure 2. The difference between the self-tuning Kalman filter and optimal filter ) of the target.F , Γ and H are constant matrices.T is sampling period.
w t and ( ) i v t are uncorrelated white noises with zeros mean and variances 2 w σ and 2 vi σ respectively, the values of 2 w σ and 2 vi σ are both unknown.Assumption 2. The observational processes ( ) ,10] can be used to get the below Eq. 27.