Polynomial Chaos Expansion Based Nonlinear Filtering of Stochastic Processes

David Bordenkircher, Ruixin Niu

In Dynamic Data Driven Applications Systems (DDDAS), non-linear continuous-discrete (CD) tracking algorithms have been proposed to recursively estimate stochastic processes which follow continuous-time stochastic differential equations (SDE) using non-linear discrete-time measurement sequences. In this paper, a new filter in this class which is based on the polynomial chaos expansion (PCE) is proposed as an alternative solution to these tracking problems. Using the orthogonality properties of the PCE basis, PCE coefficient-wise prediction and update steps are derived via the Galerkin projection. This differs from previous PCE based filters where collocation points are independently propagated through the motion model and the PCE is recomputed at each time step. For this reason, we call the proposed filter the CD-PCE coefficient filter (CD-PCE-CF). A case study is provided where the CD-PCE-CF and the CD-Extended Kalman Filter (CD-EKF) are used to track a ballistic object undergoing process noise using radar measurements. It is shown that the proposed method significantly outperforms the CD-EKF in terms of estimation accuracy and stability.