Sparse Grid-Based Nonlinear Filtering

The problem of estimating the state of a nonlinear stochastic plant is considered. Unlike classical approaches such as the extended Kalman filter, which are based on the linearization of the plant and the measurement model, we concentrate on the nonlinear filter equations such as the Zakai equation. The numerical approximation of the conditional probability density function (pdf) using ordinary grids suffers from the "curse of dimension" and is therefore not applicable in higher dimensions. It is demonstrated that sparse grids are an appropriate tool to represent the pdf and to solve the filtering equations numerically. The basic algorithm is presented. Using some enhancements it is shown that problems in higher dimensions can be solved with an acceptable computational effort. As an example a six-dimensional, highly nonlinear problem, which is solved in real-time using a standard PC, is investigated.