A direct quadrature approach for nonlinear filtering

The nonlinear filtering problem consists of estimating states of nonlinear systems from noisy measurements and the corresponding techniques can be applied to a wide variety of civil or military applications. Optimal estimates of a general continuous-discrete nonlinear filtering problem can be obtained by solving the Fokker-Planck equation, coupled with a Bayesian update. This procedure does not rely on linearizations of the dynamical and/or measurement models. However, the lack of fast and efficient algorithms for solving the Fokker-Planck equation presents challenges in real time applications. In this paper, a direct quadrature method of moments is introduced which involves approximating the state conditional probability density function as a finite collection of Dirac delta functions. The weights and locations, i.e., abscissas, in this representation are determined by moment constraints and modified using the Baye´s rule according to measurement updates. As compared with finite difference methods, the computational cost is lower without a compromising in accuracy. As demonstrated in two classical numerical examples, this approach appears to be promising in the field of nonlinear filtering.