Short communicationA novel robust Student’s t-based Gaussian approximate filter with one-step randomly delayed measurements☆
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 followswhere k denotes the discrete time, and represents the state transition function, and is known as observation function, and and respectively, represent the ideal and actual measurement vectors, and represents the process noise vector assumed to has a Gaussian distribution with zero mean vector and covariance matrix Qk, and
An exponential form of the likelihood function
Exploiting (1), we obtain
Using (3), the likelihood probability density function (PDF) can be formulated aswhere is the PDF of the measurement noise vector.
Since the likelihood PDF 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]where the state vector and represent the positions in the X axial and Y axial of the cartesian coordinates, and
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)
- et al.
Nonlinear state estimation under bounded noises
Automatica
(2018) - et al.
Extended and unscented filtering algorithms using one-step randomly delayed measurements
Appl. Math. Comput.
(2007) - et al.
Unscented filtering algorithm using two-step randomly delayed measurements in nonlinear systems
Appl. Math. Model.
(2009) - et al.
Gaussian filter for nonlinear systems with one-step randomly delayed measurements
Automatica
(2013) - et al.
A variational Bayesian approach to robust sensor fusion based on Student-t distribution
Inf. Sci.
(2013) - et al.
Kalman filtering for multiple time-delay systems
Automatica
(2008) - et al.
Optimal control of continuous-time linear systems with a time-varying, random delay
Syst. Control Lett.
(2005) INS/GPS Integration system using adaptive filter for estimating measurement noise variance
IEEE Trans. Aerosp. Electron. Syst.
(2012)- M. Roth, E. Özkan, F. Gustafsson, A Student’s filter for heavy-tailed process and measurement noise, Proceedings of the...
Cited by (17)
An improved two-phase robust distributed Kalman filter
2024, Signal ProcessingAdaptive multinoulli-based Kalman filter with randomly unknown delayed and lost measurements
2022, Digital Signal Processing: A Review JournalCitation Excerpt :Wang et al. [17] proposed an improved Kalman filter (IKF), and an exponential form is obtained by combining two Gaussian distributions with a discrete Bernoulli random variable, which has better estimation accuracy on the basis of unknown time-varying delay probability. In order to tackle the problem of filtering for systems that have one-step randomly delayed measurements and heavy-tailed measurement noise, Jia et al. [20] used the state augmentation approach, and jointly estimated the augmentation state vector, auxiliary random variable and Bernoulli random variable exploiting VB, which has better performance in different delay probabilities. Some filters are proposed to solve the delay measurement problem for nonlinear systems [21–27].
A novel robust Kalman filter with adaptive estimation of the unknown time-varying latency probability
2021, Signal ProcessingCitation Excerpt :However, the accuracy of the above algorithms is reduced significantly when used in systems with ORDM. To address systems with both ORDM and HMN, [23] developed a Student’s t-based Gaussian approximate filter using the VB technique [23]; unfortunately, it assumes that the latency probability is known. The motivation of this short communication:
Morphology-preserving reconstruction of times series with missing data for enhancing deep learning-based classification
2021, Biomedical Signal Processing and ControlCitation Excerpt :One solution to this issue and achieving better generalization is augmenting data with synthetically created sequences. As an effective remedy for data paucity, this could expand the amount of available data without any requirement for collecting new ones [11,12]. Not all the synthetizing schemes are always feasible, especially if a sequence of samples ordered in time.
Nano satellite attitude determination with randomly delayed measurements
2021, Acta AstronauticaCitation Excerpt :In practice, the sensor noise may not be Gaussian. Ref. [10–15] separately studied the one-step delay filtering algorithm for measuring colored noise and heavy-tailed noise. Zhang proposed a new Gaussian approximate (GA) filter for colored noise systems.
- ☆
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.