Article
An Adaptive Filter for Nonlinear Multi-Sensor
Systems with Heavy-Tailed Noise
Xiangxiang Dong
1,2,3
, Luigi Chisci
4
and Yunze Cai
1,2,3,
*
1
Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China; js.danesir@sjtu.edu.cn
2
Key Laboratory of System Control and Information Processing, Ministry of Education of China,
Shanghai 200240, China
3
Key Laboratory of Marine Intelligent Equipment and System of Ministry of Education,
Shanghai Jiao Tong University, Shanghai 200240, China
4
Department of Information Engineering, Università di Firenze, 50139 Firenze, Italy; luigi.chisci@unifi.it
* Correspondence: yzcai@sjtu.edu.cn
Received: 25 August 2020; Accepted: 23 November 2020; Published: 26 November 2020
Abstract:
Aiming towards state estimation and information fusion for nonlinear systems with
heavy-tailed measurement noise, a variational Bayesian Student’s t-based cubature information filter
(VBST-CIF) is designed. Furthermore, a multi-sensor variational Bayesian Student’s t-based cubature
information feedback fusion (VBST-CIFF) algorithm is also derived. In the proposed VBST-CIF,
the spherical-radial cubature (SRC) rule is embedded into the variational Bayes (VB) method for a
joint estimation of states and scale matrix, degree-of-freedom (DOF) parameter, as well as an auxiliary
parameter in the nonlinear system with heavy-tailed noise. The designed VBST-CIF facilitates
multi-sensor fusion, allowing one to derive a VBST-CIFF algorithm based on multi-sensor information
feedback fusion.
The performance
of the proposed algorithms is assessed in target tracking scenarios.
Simulation results demonstrate that the proposed VBST-CIF/VBST-CIFF outperform the conventional
cubature information filter (CIF) and cubature information feedback fusion (CIFF) algorithms.
Keywords:
nonlinear multi-sensor system; heavy-tailed noise; student’s t distribution; spherical-radial
cubature rule; information fusion
1. Introduction
The Kalman filter (KF) is an optimal state estimator for linear state-space systems [
1
–
3
]. It is widely
used, owing to its optimality, in many applications like, e.g., localization, control, target tracking,
and signal processing [
4
–
17
]. In reality, however, systems are usually characterized by strong
non-linearities which make the conventional KF inappropriate. To this end, nonlinear filtering methods
have been developed like, e.g., function approximation, deterministic sampling, and Monte Carlo
estimation methods [
18
–
20
]. The function approximation method adopted by the extended Kalman
filter (EKF) approximates the nonlinear system equations through truncated Taylor expansions [
21
].
However, the Jacobian matrix is not computable for systems with non-smooth non-linearities [
21
].
As for the deterministic sampling method, its main representatives are the unscented Kalman filter
(UKF), cubature Kalman filter (CKF), etc. [
22
,
23
]. Unfortunately, the state error covariance matrix
(SECM) in UKF may result a non-positive definite for high-dimensional systems, possibly leading
to filter divergence [
23
]. To overcome the drawbacks of UKF, Arasaratnam and Haykin proposed
CKF in 2009 [
23
]. CKF approximates the posterior probability density function (PDF) by means
of the spherical-radial cubature (SRC) rule, thus resulting in the improved filtering accuracy for
high-dimensional systems [
24
]. Moreover, compared to the Monte Carlo estimation approach exploited
by particle filters (PFs), CKF is not affected by particle depletion issues.
Sensors 2020, 20, 6757; doi:10.3390/s20236757 www.mdpi.com/journal/sensors
