Multiple fading factors-based strong tracking variational Bayesian adaptive Kalman filter

doi:10.1016/j.measurement.2021.109139

Measurement

Volume 176, May 2021, 109139

https://doi.org/10.1016/j.measurement.2021.109139 Get rights and content

Highlights

•
The inverse Wishart distribution is chosen as the conjugate prior for MNCM.
•
Multiple suboptimal fading factors can make each filtering channel have different adjustment capability.
•
The computational complexity of filter will be increased because of the use of variational Bayesian approach.
•
The proposed filter has better estimation accuracy than the existing state-of-the-art algorithms.

Abstract

If the system model or the statistical characteristics of noise are inaccurate, the past measurements will directly affect the accuracy of current state estimation or even lead to filtering divergence. To overcome above difficulties, a multiple fading factors-based strong tracking variational Bayesian adaptive Kalman filter is proposed. Firstly, the inverse Wishart distribution is adopted to model the measurement noise covariance matrix. Secondly, the remodified measurement noise covariance matrix and the innovation covariance matrix estimated by exponential weighting method are employed to construct the scalar fading factor. Next, the multiple fading factors are calculated to correct the predicted error covariance matrix. Finally, the local optimal estimations of measurement noise covariance matrix and state are obtained by variational Bayesian approach. The target tracking simulations verify that the proposed algorithm has better tracking ability for the predicted error covariance matrix and the measurement noise covariance matrix compared with the existing filters.

Introduction

For the linear Gaussian state space model whose observation noise and system noise are stationary, the Kalman filter (KF) is a recursive optimal algorithm that employs the minimum mean square error (MMSE) as the estimation criterion and has been widely used in parameter estimation [1], [2]. The KF adopts the estimated state vector at previous moment and the measurements at current moment to obtain the current state, which is essentially a process of continuous prediction and correction [3], and it does not need to store a large amount of historical observation data. Due to its excellent parameter estimation performance, the KF has been widely used in various dynamic systems, such as target location and tracking, fault diagnosis and detection, navigation and guidance, risk index assessment and so on [4], [5], [6], [7], [8].

However, in practical applications, the noise statistical characteristics of the state space model are not always stationary [9], [10], [11], [12], [13]. The time-varying process noise covariance matrix (PNCM) and measurement noise covariance matrix (MNCM) usually affect the accuracy of the prior information obtained from noise statistics, and the wrong prior information will cause a large number of estimation errors or even filtering divergence [14], [15], accordingly diminish the filtering performance. Aiming at these problems, scholars put forward numbers of adaptive Kalman filters (AKF), e.g., innovation-based AKF (IAKF), fading AKF (FAKF), variational Bayesian-based AKF (VBAKF), strong tracking-based variational Bayesian AKF (ST-VBAKF) and so on [16], [17], [18], [19]. The IAKF solves the problem of imperfect prior information through a filtering learning process based on an innovative sequence, which has a significant improvement in performance over fixed filters [16]. However, the IAKF requires a fairly large data window to obtain a reliable estimation of the MNCM, which makes it unsuitable for rapidly changing MNCM. The FAKF decreases the weights of current observations by increasing the one-step prediction error covariances. However, the calculation process of the scalar fading factor is more cumbersome and it has the same adjustment ability for each filtering channel, which is not conducive to improve the stability and accuracy of the filter [20]. The VBAKF selects the inverse Wishart prior to model the measurement noise, and it uses the variational Bayesian (VB) approach to obtain the suboptimal estimations of the state vector and the slowly varying MNCM [18]. However, the PNCM is set as a fixed value which is not consistent with reality [21], thus, the filtering performance of VBAKF will decrease due to the inaccurate PNCM. On the basis of VBAKF, Huang et al. [15] proposed a novel variational Bayesian-based adaptive Kalman filter (N-VBAKF), which can estimate not only MNCM but also predicted error covariance matrix (PECM) in the process of variational iterative recursion, and good results have been achieved in the target tracking problem where both PECM and MNCM are slowly varying. However, the N-VBAKF also has the disadvantages of high computational complexity and time consumption. Tan [19] introduced the suboptimal fading factor into VBAKF and proposed a ST-VBAKF algorithm, which could adaptively track the MNCM in a linear Gaussian system with time-varying noise, and effectively overcome the influence of time-varying PNCM. The convergence speed and accuracy of the results are improved. However, as with FAKF, the scalar fading factor has regulatory limitations in ST-VBAKF.

In complex dynamic systems, the estimation accuracies of state variables represented by diagonal elements of PECM are different. Therefore, conventional FAKF can no longer meet the demand of accuracy since the scalar fading factor has the same ability to adjust the state variables. Zhou et al. [22] proposed an extended Kalman filter (EKF) with multiple suboptimal fading factors, which determined the weights corresponding to the fading factors through prior knowledge, and provided a new idea for the establishment of multiple fading factors. Geng et al. [23] scaled the PECM based on an assumption that the prediction residual vectors follow a chi-square distribution, and proposed a novel AKF with multiple fading factors for GPS/INS integrated navigation. The characteristics of this filter overcome the lack of traditional KF in robust estimation. Furthermore, scholars have designed a variety of multiple fading factors-based filters according to the characteristics of different dynamic systems to make the filters more applicable [24], [25], [26].

In order to further improve the performance of ST-VBAKF, a multiple fading factors-based strong tracking variational Bayesian adaptive filter (MST-VBAKF) is proposed. Compared with ST-VBAKF, the improvements of proposed method are reflected in two aspects. Firstly, an exponential weighting method based on fading memory is introduced to improve the utilization weight of current observations, thus the innovation covariance matrix is estimated more accurately. Secondly, the multiple fading factors are employed to promote the tracking ability for PECM and the filter robustness is enhanced. Simulation results show that the estimation accuracy of the new algorithm is better than ST-VBAKF and the other filters.

The rest of this paper is organized as follows. Section 2 formulates the problem and gives the KF solution. Section 3 introduces the formation process of the MST-VBAKF in detail, including the selection of prior distributions for process and measurement noise, the construction of multiple fading factors and the derivation of variational measurement update. Section 4 performs simulations to verify the proposed filtering algorithm, in which the influences of weakening factor and forgetting factor on MST-VBAKF and the robustness of different nominal noise covariance settings are analyzed, respectively, and then the proposed filtering algorithm is compared with the existing filters. Section 5 summarizes the performance of the new algorithm.

Section snippets

Problem formulation and KF solution

The state space model of discrete linear stochastic system includes state equation and measurement equation, which is defined as [27], [28] $X_{k} {= F}_{k - 1} X_{k - 1} {+ w}_{k - 1}$ $L_{k} {= H}_{k} X_{k} {+ v}_{k}$ where $X_{k}$ and $L_{k}$ are the state and measurement vectors, respectively; $F_{k - 1}$ is the system transition matrix; $H_{k}$ is the design matrix; $w_{k - 1}$ and $v_{k}$ are the process and measurement noise vectors, respectively. It should be noted that the initial state vector $X_{0}$ is assumed to be the Gaussian distribution of mean vector ${\hat{X}}_{0 | 0}$ and

Strong tracking variational Bayesian adaptive Kalman filter based on multiple fading factors

In this section, the multiple fading factors are introduced into VBAKF and a novel filtering method named MST-VBAKF is proposed, which can adjust the PECM $P_{k | k - 1}$ and inaccurate MNCM ${\tilde{R}}_{k}$ simultaneously. The selection of prior distribution, the construction of multiple fading factors and the variational measurement update process of MST-VBAKF are introduced in detail, respectively.

The experimental simulation

Like many existing researches [15], [18], [19], this paper adopts the continuous white noise acceleration model in two-dimensional (2D) Cartesian coordinates to verify the performance of the proposed MST-VBAKF algorithm, where the PNCM and MNCM of the target change slowly with time. It is assumed that the motion process is monitored in real time by sensors such as the global navigation satellite systems (GNSS). The position and velocity of target are respectively denoted as $(x_{k}, y_{k})$ and $({\dot{x}}_{k}, {\dot{y}}_{k})$

Conclusions

When the PNCM and MNCM of the dynamic system are time-varying, the KF is not ideal for parameter estimation accuracy, while existing filter algorithms are not able to weaken the influence of inaccurate PNCM on filtering performance or have limited ability to weaken. On the basis of existing researches, this paper proposes a strong tracking variational Bayesian adaptive filtering algorithm with multiple fading factors, namely MST-VBAKF, which can adjust the PECM and MNCM simultaneously.

CRediT authorship contribution statement

Cheng Pan: Conceptualization, Methodology, Formal analysis, Writing - original draft. Jingxiang Gao: Supervision, Project administration, Funding acquisition. Zengke Li: Software, Visualization. Nijia Qian: Investigation, Writing - review & editing. Fangchao Li: Investigation, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (grant numbers 41974026, 41874006), the China University of Mining and Technology (grant number 2020WLJCRCZL052), and the Jiangsu Education Department (grant number KYCX20_2058). The authors are grateful for the three anonymous reviewers for their constructive comments on the manuscript.

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View full text

Multiple fading factors-based strong tracking variational Bayesian adaptive Kalman filter

Highlights

Abstract

Introduction

Section snippets

Problem formulation and KF solution

Strong tracking variational Bayesian adaptive Kalman filter based on multiple fading factors

The experimental simulation

Conclusions

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgements

Automatica

IFAC Proc.

Measurement

Measurement

Chinese J. Aeronaut.

The Kalman filter as the optimal linear minimum mean-squared error multiuser CDMA detector

IEEE Trans. Inf. Theory.

Kalman filtering implementation with Matlab

Target Tracking Using Machine Learning and Kalman Filter in Wireless Sensor Networks

IEEE Sens. J.

A New Adaptive Extended Kalman Filter for Cooperative Localization

IEEE Trans. Aerosp. Electron. Syst.

Experimental Test of a Two-Stage Kalman Filter for Actuator Fault Detection and Diagnosis of an Unmanned Quadrotor Helicopter

J. Intell. Robot. Syst.

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A Variational Bayesian-Based Robust Adaptive Filtering for Precise Point Positioning Using Undifferenced and Uncombined Observations

Adv. Sp. Res.

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IEEE Trans. Aerosp. Electron. Syst.

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2015 IEEE 18th Int Conf. Intell. Transp. Syst.

Target motion analysis with unknown measurement noise variance

2017 20th Int. Conf. Inf. Fusion

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J. Geod.