Elsevier

Signal Processing

Volume 171, June 2020, 107496
Signal Processing

Short communication
A novel robust Student’s t-based Gaussian approximate filter with one-step randomly delayed measurements

https://doi.org/10.1016/j.sigpro.2020.107496Get rights and content

Highlights

  • By introducing a Bernoulli random variable and using state augmentation approach, the conditional likelihood function is transformed into an exponential multiplication form.

  • The augmentation state vector together with the auxiliary random variables and the Bernoulli random variable are inferred based on variational Bayesian approach.

  • The superiority of the proposed filter as compared with the existing filters is demonstrated in a target tracking simulation.

Abstract

In this paper, a novel robust Student’s t-based Gaussian approximate filter (RSTGAF) is proposed to solve the filtering problem of the nonlinear system with one-step randomly delayed measurements (ORDM) and heavy-tailed measurement noise. The conditional likelihood function is transformed into an exponential multiplication form after using state augmentation approach and introducing a Bernoulli random variable, and then the augmentation state vector, the auxiliary random variables and the Bernoulli random variable are jointly estimated based on the variational Bayesian (VB) approach. The simulation results demonstrate the superiority of the proposed filter, as compared with the existing filters, to address the ORDM and heavy-tailed measurement noise.

Introduction

The Gaussian approximate filter (GAF) has been successfully applied in engineering applications based on the fact that the measurements should be available for a stochastic system in real time. However, the performance of the GAF may degrade when the measurements information are not updated in real time [1], [2], [3], [4], [5]. Lots of filtering algorithms have been proposed to address the scenarios with randomly delayed measurements, such as the GAF with ORDM [6]. The GAF with ORDM is an good solution to address the scenarios with ORDM in which the measurement noise vector is inferred based on the Bayesian rule [6]. However, in many actual applications such as the agile targets tracking problem, the heavy-tailed non-Gaussian measurement noise may be induced due to the measurement outliers provided by unreliable sensors [7]. The performance of the existing GAF with ORDM in these scenarios may dramatically degrade.

To improve the estimation accuracy, the Student’s t distribution is used to model the heavy-tailed measurement noise [8], [9], [10], [11], [12], [13], [14]. However, the Student’s t distribution will degrade to a Gaussian distribution when the degree of freedom (dof) parameter increases, and the weighted sum of two Student’s t distributions can’t guarantee that the posterior distribution is of the same functional form as the prior distribution. Therefore, the existing likelihood function is a weighted sum of two Student’s t distributions under the case of ORDM and the heavy-tailed measurement noise, which is a unclosed and unconjugated distribution and difficult to use the Bayesian inference directly. The main idea of this paper is to transform the weighted sum of two Student’s t distributions into an exponential multiplication form after using the augmentation approach and introducing a Bernoulli random variable, and then the augmentation state vector together with the auxiliary random variables and the Bernoulli random variable are inferred based on the VB method.

In this paper, a novel RSTGAF is proposed to address the scenarios with ORDM and the heavy-tailed measurement noise. Firstly, the conditional likelihood function is transformed into an exponential multiplication form after using state augmentation approach and introducing a Bernoulli random variable. Secondly, a novel RSTGAF is designed for nonlinear systems with ORDM and heavy-tailed measurement noise by choosing the prior information, in which the probability mass function (PMF) of the Bernoulli random variable and the probability density functions (PDFs) of the auxiliary random variables are respectively modeled as a Bernoulli distribution and Gamma distributions. Thirdly, the augmentation state vector together with the auxiliary random variables and the Bernoulli random variable are estimated using the VB method. Finally, the superiority of the proposed RSTGAF is shown by a target tracking simulation scenario with ORDM and heavy-tailed measurement noise.

Section snippets

Problem formulation

Consider a nonlinear state-space model as follows{xk+1d=fk(xkd)+wkzk+1=hk+1(xk+1d)+vk+1dyk+1=(1ξk+1)zk+1+ξk+1zk,k>0;y1=z1where k denotes the discrete time, and fk(·)Rn×n represents the state transition function, and hk+1(·)Rm×n is known as observation function, and zk+1Rm and yk+1Rm, respectively, represent the ideal and actual measurement vectors, and wkRn represents the process noise vector assumed to has a Gaussian distribution with zero mean vector and covariance matrix Qk, and vk+1dR

An exponential form of the likelihood function

Exploiting (1), we obtainyk+1=(1ξk+1)[hk+1(xk+1d)+vk+1]+ξk+1[hk(xkd)+vk]

Using (3), the likelihood probability density function (PDF) p(yk+1|xk+1d,xkd) can be formulated asp(yk+1|xk+1d,xkd)=ξk+1=01p(yk+1,ξk+1|xk+1d,xkd))=Pr(ξk+1=0)p(yk+1|xk+1d,xkd,ξk+1=0)+Pr(ξk+1=1)p(yk+1|xk+1d,xkd,ξk+1=1)=(1pk+1)pvk+1d(yk+1hk+1(xk+1d))+pk+1pvkd(yk+1h(xkd))where pvk(·) is the PDF of the measurement noise vector.

Since the likelihood PDF p(yk+1|xk+1d,xkd) in (4) is a unclosed and unconjugated distribution, it

Simulations

The proposed filter is compared with the existing filters in a target tracking simulation with ORDM and the heavy-tailed measurement noise. The nonlinear stochastic system is formulated as follows [14]{xk+1d=[1sinΩΔtΩ0sinΩΔt1Ω00cosΩΔt0sinΩΔt001cosΩΔtΩ1sinΩΔtΩ00sinΩΔt0cosΩΔt000001]xkd+wkzk+1=[rk+1θk+1]=[ιk+12+κk+12tan1(ιk+1κk+1)]+vk+1where the state vector xk+1d=[ιk+1ι˙k+1κk+1κ˙k+1Ω]T, ιk+1 and κk+1 represent the positions in the X axial and Y axial of the cartesian coordinates, and ι˙k+1

Conclusion

In this paper, a novel RSTGAF was proposed for nonlinear systems with ORDM and heavily-tailed measurement noise, where the augmentation state vector together with the auxiliary random variables and the Bernoulli random variable were estimated by the VB approach. Simulation results demonstrated the superiority of the proposed RSTGAF algorithm as compared with the existing filtering algorithms for the case of ORDM and heavily-tailed measurement noise.

Declaration of Competing Interest

The authors declare that there is no conflict of interests regarding the publication of this article.

References (18)

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This work was supported in part by the National Natural Science Foundation of China under Grants 61903097, 61773133 and 61633008, in part by the Fundamental Research Funds for the Central Universities under Grants GK204026025901.

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