Continuous-Time Kalman Filtering with Implicit Discrete Measurement Times

A new approximation to a continuous-time Kalman filter has been developed, one that is useful for systems with implicitly defined measurement update times. This Kalman filter is applicable to radio navigation problems in which a state-dependent range delay must be determined by iterative solution of an implicit equation in order to compute the relevant Kalman filter measurement update time. The new filter consists of three main elements. The first element is a dense-output numerical integration method that outputs a continuous description of the state over each integration interval. The second element is a new process noise model that approximates the underlying continuous-time white noise as a finite order, piecewise polynomial. The third element is a new dynamic propagation/measurement update calculation that sensibly combines a dense-output numerical integration scheme with the new process noise model and that estimates the process noise model's polynomial coefficients as discrete-time random variables such that the model optimally approximates continuous-time white noise. After developing the necessary theory, the method is demonstrated in simulation for an example tracking problem.