A Novel Real-Time Filtering Method to General Nonlinear Filtering Problem Without Memory

Abstract

In this paper, the filtering problem for the general time-invariant nonlinear state-observation system is considered. Our work is based on the Yau-Yau filtering framework developed by S.-T. Yau and the third author in 2008. The key problem of Yau-Yau filtering framework is how to compute the solution to forward Kolmogorov equation (FKE) off-line effectively. Motivated by the supervised learning in machine learning, we develop an efficient method to numerically solve the FKE off-line from the point of view of optimization. Specifically, for the off-line computation part, the computation of the solution to a FKE is reduced to computing a linear system of equations by making the temporal inverse transformation and the loss function optimization, and we store the results for the preparation of on-line computation. For the on-line computation part, the unnormalized density function is approximated by a complete polynomial basis, and then the estimation of the state is computed using the stored off-line data. Our method has the merits of easily implementing, real-time and memoryless. More importantly, it can be applicable for moderate-high dimensional cases. Numerical experiments have been carried out to verify the feasibility of our method. Our algorithm outperforms extended Kalman filter, unscented Kalman filter and particle filter both in accuracy and costing time.

Keywords

• Duncan-Mortensen-Zakai equation
• nonlinear filtering
• forward Kolmogorov equation
• estimation theory
• numerical algorithms