A quadrature-based method of moments for nonlinear filtering

According to the nonlinear filtering theory, optimal estimates of a general continuous-discrete nonlinear filtering problem can be obtained by solving the Fokker–Planck equation, coupled with a Bayesian update rule. This procedure does not rely on linearizations of the dynamical and/or measurement models. However, the lack of fast and efficient methods for solving the Fokker–Planck equation presents challenges in real time nonlinear filtering problems. In this paper, a direct quadrature method of moments is introduced to solve the Fokker–Planck equation efficiently and accurately. This approach involves representation of the state conditional probability density function in terms of a finite collection of Dirac delta functions. The weights and locations (abscissas) in this representation are determined by moment constraints and modified using the Bayes’ rule according to measurement updates. As demonstrated by numerical examples, this approach appears to be promising in the field of nonlinear filtering.