Real-Time Solution of Time-Varying Yau Filtering Problems via Direct Method and Gaussian Approximation

Direct method for Yau filtering system has been studied since 1990s and all these work are limited in time-invariant systems. In this work, we extend the direct method so that it is applicable to time-varying cases. We need less assumptions compared with our previous work. The novelty of this work is that we propose several transformations on the forward Kolmogorov equation so that it can be solved by means of solving some ordinary differential equations if the initial distribution is Gaussian. The corresponding results for any non-Gaussian initial distributions can be obtained via Gaussian approximation. It can be seen that our new scheme direct method can treat nearly most general Yau filtering problems under natural assumptions. Our algorithm has been compared with the extended Kalman filter, multilevel particle filter, and ensemble Kalman filter by numerical examples and the simulation results show the efficiency of our method.

[1]  Stephen S.-T. Yau,et al.  Finite-dimensional filters with nonlinear drift X: explicit solution of DMZ equation , 2001, IEEE Trans. Autom. Control..

[2]  N. Gordon,et al.  Novel approach to nonlinear/non-Gaussian Bayesian state estimation , 1993 .

[3]  S. S.-T. Yau *,et al.  Classification of finite-dimensional estimation algebras of maximal rank with arbitrary state–space dimension and mitter conjecture , 2005 .

[4]  Stephen S.-T. Yau,et al.  Finite-dimensional filters with nonlinear drift. XV. New direct method for construction of universal finite-dimensional filter , 2002 .

[5]  Shing-Tung Yau,et al.  Solution of Filtering Problem with Nonlinear Observations , 2005, SIAM J. Control. Optim..

[6]  Stephen S.-T. Yau,et al.  Explicit solution of DMZ equation in nonlinear filtering via solution of ODEs , 2003, IEEE Trans. Autom. Control..

[7]  R. E. Kalman,et al.  New Results in Linear Filtering and Prediction Theory , 1961 .

[8]  Nando de Freitas,et al.  An Introduction to Sequential Monte Carlo Methods , 2001, Sequential Monte Carlo Methods in Practice.

[9]  Shing-Tung Yau,et al.  Real Time Solution of the Nonlinear Filtering Problem without Memory II , 2008, SIAM J. Control. Optim..

[10]  M. Zakai.  On the optimal filtering of diffusion processes , 1969 .

[11]  Yan Zhou,et al.  Multilevel Particle Filters , 2015, SIAM J. Numer. Anal..

[12]  Geir Evensen,et al.  The Ensemble Kalman Filter: theoretical formulation and practical implementation , 2003 .

[13]  D. Crisan,et al.  Fundamentals of Stochastic Filtering , 2008 .

[14]  T. Faniran.  Numerical Solution of Stochastic Differential Equations , 2015 .

[15]  G.-G. Hu.  Finite-dimensional filters with nonlinear drift. , 1997 .

[16]  P. Fearnhead,et al.  Particle filters for partially observed diffusions , 2007, 0710.4245.

[17]  Stephen S.-T. Yau,et al.  Direct method for Yau filtering system with nonlinear observations , 2018, Int. J. Control.

[18]  Jeffrey L. Anderson,et al.  A Monte Carlo Implementation of the Nonlinear Filtering Problem to Produce Ensemble Assimilations and Forecasts , 1999 .

[19]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[20]  Stephen S.-T. Yau,et al.  New direct method for Kalman-Bucy filtering system with arbitrary initial condition , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[21]  Xue Luo,et al.  Complete Real Time Solution of the General Nonlinear Filtering Problem Without Memory , 2012, IEEE Transactions on Automatic Control.

[22]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[23]  Xue Luo,et al.  Direct Method for Time-Varying Nonlinear Filtering Problems , 2017, IEEE Transactions on Aerospace and Electronic Systems.

[24]  Stephen S.-T. Yau.  Recent results on nonlinear filtering: new class of finite dimensional filters , 1990, 29th IEEE Conference on Decision and Control.

[25]  T. Duncan.  PROBABILITY DENSITIES FOR DIFFUSION PROCESSES WITH APPLICATIONS TO NONLINEAR FILTERING THEORY AND DETECTION THEORY , 1967 .

[26]  Wen-Lin Chiou,et al.  Finite-Dimensional Filters with Nonlinear Drift II: Brockett's Problem on Classification of Finite-Dimensional Estimation Algebras , 1994 .

[27]  Jeffrey L. Anderson.  An Ensemble Adjustment Kalman Filter for Data Assimilation , 2001 .

[28]  S.S.-T. Yau,et al.  Nonlinear filtering and time varying Schrodinger equation , 2004, IEEE Transactions on Aerospace and Electronic Systems.

[29]  James Ting-Ho Lo,et al.  Synthetic approach to optimal filtering , 1994, IEEE Trans. Neural Networks.

[30]  Neil J. Gordon,et al.  A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking , 2002, IEEE Trans. Signal Process..

[31]  Stephen S.-T. Yau,et al.  Real time solution of nonlinear filtering problem without memory I , 2000 .