Gaussian filters for nonlinear filtering problems

We develop and analyze real-time and accurate filters for nonlinear filtering problems based on the Gaussian distributions. We present the systematic formulation of Gaussian filters and develop efficient and accurate numerical integration of the optimal filter. We also discuss the mixed Gaussian filters in which the conditional probability density is approximated by the sum of Gaussian distributions. A new update rule of weights for Gaussian sum filters is proposed. Our numerical tests demonstrate that new filters significantly improve the extended Kalman filter with no additional cost, and the new Gaussian sum filter has a nearly optimal performance.

[1]  A. Jazwinski.  Stochastic Processes and Filtering Theory , 1970 .

[2]  H. Sorenson,et al.  Nonlinear Bayesian estimation using Gaussian sum approximations , 1972 .

[3]  V. Maz'ya,et al.  On approximate approximations using Gaussian kernels , 1996 .

[4]  Alain Bensoussan,et al.  Non linear filtering theory , 1990 .

[5]  Yaakov Bar-Shalom,et al.  Estimation and Tracking: Principles, Techniques, and Software , 1993 .

[6]  M. Piccioni,et al.  Convergence of implicit discretization schemes for linear differential equations with application to filtering , 1987 .

[7]  R. Wishner,et al.  Suboptimal state estimation for continuous-time nonlinear systems from discrete noisy measurements , 1968 .

[8]  R. Bucy.  Nonlinear filtering theory , 1965 .

[9]  Patrick Florchinger,et al.  Time-discretization of the Zakai equation for diffusion processes observed in correlated noise , 1990 .

[10]  K. Ito.  Approximation of the Zakai Equation for Nonlinear Filtering , 1996 .

[11]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[12]  R. Glowinski,et al.  Approximation of the Zakai¨ equation by the splitting up method , 1990 .

[13]  H. Kushner.  Approximations to optimal nonlinear filters , 1967, IEEE Transactions on Automatic Control.

[14]  S. Julier,et al.  A General Method for Approximating Nonlinear Transformations of Probability Distributions , 1996 .

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

[16]  ItoKazufumi.  Approximation of the Zakai Equation for Nonlinear Filtering , 1996 .

[17]  R. E. Kalman,et al.  A New Approach to Linear Filtering and Prediction Problems , 2002 .

[18]  H. Sorenson,et al.  Recursive bayesian estimation using gaussian sums , 1971 .

[19]  F. Gland,et al.  Time-discretization of the Zakai equation for diffusion processes observed in correlated noise , 1991 .

[20]  H.F. Durrant-Whyte,et al.  A new approach for filtering nonlinear systems , 1995, Proceedings of 1995 American Control Conference - ACC'95.

[21]  H. Banks.  Center for Research in Scientific Computationにおける研究活動 , 1999 .

[22]  H. Kunita,et al.  Stochastic differential equations for the non linear filtering problem , 1972 .