Ship Tracking with Static Electric Field Based on Adaptive Progressive Update Extended Kalman Filter

An adaptive progressive update extended Kalman filter is introduced for unknown noise in ship tracking using static electric field. The corresponding state space model is established; the algorithm is introduced, and the simulation is designed. The simulation results show that the adaptive algorithm can effectively improve the performance of the algorithm, when the noise covariance deviates from the real value; the finite number of noise covariance estimation is beneficial to the stability of the filter.


Introduction
In recent years, ship electric field is a hot topic of research. Ship tracking with static electric field has become a new research direction, which can be an effective supplement to acoustic tracking [1] . In the tracking algorithm, extended Kalman filter (EKF) and corresponding improvement algorithm are the best choice due to their simple form, low computational complexity, and the advantage of real-time tracking. Baoquan Sun [2] proposed a progressive update extended Kalman filter (PUEKF), which has better performance, and this article is based on the method. In Kalman filters, it is usually assumed that the statistical characteristics of environmental noise are accurately known and remain unchanged throughout the process, but in practice, environmental noise may be unknown and change. This may cause Kalman filters to lose its optimality and even cause filtering divergence. At present, in various tracking applications, scholars have put forward various methods to solve the problem of unknown statistical characteristics of environmental noise [3][4] . A noise covariance matrix estimation algorithm based on residuals is proposed in the reference [5]. The method is simple and feasible, and can make full use of the measurement information. This paper will study the adaptive Kalman filtering algorithm for ship electric field tracking based on it.
The reminder of the paper is organized as follows. In sectionⅡ ship motion state space model is established. In section Ⅲ, adaptive extended Kalman filter is described. In section Ⅳ, simulation to verify the performance of the method in ship tracking application is designed. a and h are some known functions.

Measurement model
The ship SE signal can be described by the point current array model with () 2 NN point currents at equal distance, in which the current density of each point current is pi I and the distance is d l . The ship SE is assumed to be the sum of electric fields of the N point currents [6] . Where, ( , ) pi K I P is the distance coefficient, reflecting the function between source and field point. In the condition shown in figure1 it is described as   is the seawater conductivity; m is the number of reflections, usually take 10~20 [6] ; 2 Then we can get E .  The target signal measured by electric field sensor ) can be modled as follows: is signal received by electric field sensor; is the background noise of the electric field sensor. The observation equation can be further obtained

State model
Ship target state vector at k time is defined as follows:

Adaptive PUEKF
The basic step of PUEKF is [2] : (1) Time update (2) Measurement update Steps1: Initialization, According to the principle of Kalman filter, the gain of the filter is: Multiplied both sides of the (12) by k H : According to the windowing method, the real-time estimation variance of residual error is [7] : Where, W is the length of sliding data window. The continuous change of k R is actually not conducive to the stability of filtering, so it will stop the estimation of Consider a point current array composed by 2 point currents. Consider the simulation scenario in Table 1  From figure3 we can see that it is not the bigger the i N is, the better. After an appropriate number of iterations, the estimation of k R is stopped, which is more conducive to filter stability.

Conclusions
In this paper, an adaptive PUEKF is introduced for the unknown noise. Simulation results show that the adaptive method can effectively improve the filtering performance. In addition, the continuous change of k R is actually not conducive to the stability of filtering, so it will stop the estimation of k R after a certain number of iterations when environmental changes are not drastic.

Funding Statement
This study is sponsored by Science and Technology on Near-Surface Detection Laboratory fund [TCGZ2017A007].