The Level Set Kalman Filter for State Estimation of Continuous-Discrete Systems

We propose a new extension of Kalman filtering for continuous-discrete systems with nonlinear state-space models that we name as the level set Kalman filter (LSKF). The LSKF assumes the probability distribution can be approximated as a Gaussian and updates the Gaussian distribution through a time-update step and a measurement-update step. The LSKF improves the time-update step compared to existing methods, such as the continuous-discrete cubature Kalman filter (CD-CKF), by reformulating the underlying Fokker-Planck equation as an ordinary differential equation for the Gaussian, thereby avoiding the need for the explicit expression of the higher derivatives. Together with a carefully picked measurement-update method, numerical experiments show that the LSKF has a consistent performance improvement over the CD-CKF for a range of parameters. Meanwhile, the LSKF simplifies implementation, as no user-defined timestep subdivisions between measurements are required, and the spatial derivatives of the drift function are not explicitly needed.