The continuous-discrete time feedback particle filter

In this paper, the feedback particle filter (FPF) algorithm is introduced for the continuous-discrete time nonlinear filtering problem. As with the continuous-time FPF, the continuous-discrete time algorithm i) admits an innovation error-based feedback control structure, and ii) requires a solution of an Euler-Lagrange boundary value problem (E-L BVP). These solutions are described in closed-form for the linear Gaussian filtering problem. For the general nonlinear non-Gaussian case, an algorithm is described to obtain an approximate solution of the E-L BVP. Comparisons are also made to the particle flow filter algorithm introduced by Daum and Huang.