Solving Nonlinear Filtering Problems in Real Time by Legendre Galerkin Spectral Method

• 成果类型: Article
• 署名作者: Dong, Wenhui; Luo, Xue; Yau, Stephen S-T
• 署名单位: Tsinghua University; Shaanxi Normal University; Beihang University
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2020.3002979
• 发表日期: 2021
• 页码: 1559-1572
• 关键词: Real-time systems; Approximation algorithms; Mathematical model; Kalman filters; CONVERGENCE; Boundary conditions; Nonlinear systems; convergence analysis; forward Kolmogorov equation (FKE); Legendre spectral method (LSM); nonlinear filtering (NLF)

摘要

It is well known that the nonlinear filtering (NLF) problem has important applications in both military and civil industries. The central question is to solve the posterior conditional density function of the states, which satisfies the Kushner or the Duncan-Mortensen-Zakai (DMZ) equation after suitable change of probability measure. In this article, we shall follow the so-called Yau-Yau's algorithm to split the solution of the DMZ equation into on- and off-line part, where the off-line part is to solve the forward Kolmogorov equation (FKE) with the initial conditions to be the orthonormal bases in some suitable function space. Instead of the generalized Hermite function investigated by the second and the third author of this article, we shall explore the generalized Legendre polynomials. The Legendre spectral method (LSM) is used to numerically solve the FKE. Under certain conditions, the convergence rate of LSM is twice faster than that of the Hermite spectral method. Two two-dimensional numerical experiments of NLF problems (time-invariant and time-varying cases) have been numerically solved to illustrate the feasibility of our algorithm. Our algorithm outperforms the extended Kalman Filter and particle filter in both real-time manner and accuracy.