Derivative-free Huber–Kalman smoothing based on alternating minimization

We consider robust smoothing for nonlinear state space models in which both the process and measurement noises have heavy tails. To improve the robustness of Kalman smoothing, we formulate the robust smoothing problem by replacing the quadratic loss in the conventional Gaussian Kalman smoother by Huber's cost function. However, the Huber based smoother cannot be directly implemented within the Kalman smoothing framework, which has well recognized benefits in computational efficiency and stability. To address this issue, we introduce an auxiliary parameter to construct a surrogate function for the Huber cost function, leading to a reformulation of the robust Kalman smoothing problem. The reformulated robust smoothing is solved by an alternating minimization method, which iterates between a simple auxiliary parameter update step and a modified conventional Kalman smoothing step. Simulation results show that the proposed method achieves a performance improvement over several conventional and existing robust smoothers with a slight increase of computational time.