Accurate Smoothing Methods for State Estimation of Continuous-Discrete Nonlinear Dynamic Systems

In this note, we extend the accurate continuous-discrete extended-cubature Kalman filter and continuous-discrete cubature Kalman filter to deal with the problem of Bayesian optimal smoothing in nonlinear dynamic systems. The dynamics can be modeled with nonlinear stochastic differential equations (SDEs) and the noise corrupted measurements are obtained at discrete time instants. To be consistent with the literature, the resulting nonlinear smoothers are referred to as the accurate continuous-discrete extended-cubature Kalman smoother and the continuous-discrete cubature Kalman smoother, respectively. We first present two approximation methodologies to solve the SDE encountered in the prediction step of continuous-discrete filter. Then, two types of novel Gaussian approximation smoothing methods are derived based on the fixed-interval Rauch-Tung-Striebel smoother, which computes the smoothing solution according to the stored filtering results. The performances of the proposed smoothing methods are demonstrated in a simulated application and the numerical results show that the newly presented approaches are more flexible and robust than other smoothing algorithms with lower computational cost.