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.
Highlights
Ever since Kalman and Bucy proposed the famous Kalman filter which has been widely used in various fields of industry in the 1960s [1], [2], numerous researchers have devoted many efforts to the study of nonlinear filtering (NLF) theory and practical NLF algorithms
particle filter (PF) is well known for its ability to be used in general NLF problem while it cannot be implemented in real-time for high-dimensional systems
In the late 1960s, Duncan, Mortensen and Zakai independently derived the famous Duncan-Mortensen-Zakai (DMZ) equation for nonlinear filtering problem, which is satisfied by the unnormalized conditional density function of the VOLUME 9, 2021
Summary
Ever since Kalman and Bucy proposed the famous Kalman filter which has been widely used in various fields of industry in the 1960s [1], [2], numerous researchers have devoted many efforts to the study of nonlinear filtering (NLF) theory and practical NLF algorithms. PF is well known for its ability to be used in general NLF problem while it cannot be implemented in real-time for high-dimensional systems. J. Shi et al.: Novel Real-Time Filtering Method to General NLF Problem Without Memory states [7]–[9]. In 1990s, Lototsky, Mikulevicius and Rozovskii gave a recursive in time Wiener chaos representation of the optimal nonlinear filter [20] All these methods require that the drift term and observation term, i.e. f (x) and h(x) in system (1) are bounded. The proposed filtering method is easy-to-implement, fast, efficient and can be applied for moderate-high dimensional case. It inherits the merits of real-time and memoryless.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
