Nonlinear Bayesian Estimation： From Kalman Filtering to a Broader Horizon

This article presents an up-to-date tutorial review of nonlinear Bayesian estimation. State estimation for nonlinear systems has been a challenge encountered in a wide range of engineering fields, attracting decades of research effort. To date,one of the most promising and popular approaches is to view and address the problem from a Bayesian probabilistic perspective,which enables estimation of the unknown state variables by tracking their probabilistic distribution or statistics(e.g., mean and covariance) conditioned on a system's measurement data.This article offers a systematic introduction to the Bayesian state estimation framework and reviews various Kalman filtering(KF)techniques, progressively from the standard KF for linear systems to extended KF, unscented KF and ensemble KF for nonlinear systems. It also overviews other prominent or emerging Bayesian estimation methods including Gaussian filtering, Gaussian-sum filtering, particle filtering and moving horizon estimation and extends the discussion of state estimation to more complicated problems such as simultaneous state and parameter/input estimation.

Data-Driven Discovery of Stochastic Differential Equations

Stochastic differential equations(SDEs)are mathematical models that are widely used to describe complex processes or phenomena perturbed by random noise from different sources.The identification of SDEs governing a system is often a challenge because of the inherent strong stochasticity of data and the complexity of the system’s dynamics.The practical utility of existing parametric approaches for identifying SDEs is usually limited by insufficient data resources.This study presents a novel framework for identifying SDEs by leveraging the sparse Bayesian learning(SBL)technique to search for a parsimonious,yet physically necessary representation from the space of candidate basis functions.More importantly,we use the analytical tractability of SBL to develop an efficient way to formulate the linear regression problem for the discovery of SDEs that requires considerably less time-series data.The effectiveness of the proposed framework is demonstrated using real data on stock and oil prices,bearing variation,and wind speed,as well as simulated data on well-known stochastic dynamical systems,including the generalized Wiener process and Langevin equation.This framework aims to assist specialists in extracting stochastic mathematical models from random phenomena in the natural sciences,economics,and engineering fields for analysis,prediction,and decision making.