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.

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