Abstract:

In this paper, the Kalman filter robust to outliers and observation noise with unknown distribution is proposed. It is shown that a new criterion for deriving a Kalman filter can be constructed when the l₂ norm of the observation errors in its maximum posterior criterion is replaced by the Huber loss function. Since the Huber loss function can simultaneously describe both l₁ and l₂ norms of an observation error, it is illustrated that the Kalman filter derived from the new criterion is robust to outliers as l₁ norm and its performance is the same as the traditional one when the outliers are free. When the statistical distribution of the observation noise with outliers is unknown, a Gaussian mixture model with unknown parameters is used to describe such a distribution and the variational Bayesian is employed to infer these unknown parameters. In this way, a Kalman filter robust to outliers and unknown statistical distribution of observation noises is given. Both the correctness of the analytical results and the performance of the proposed algorithms superior to the robust Kalman filters reported in literatures are verified by simulations and experiments.

Key words: Kalman filter, Huber loss function, Gaussian mixture distribution, expectation-maximization algorithm, variational Bayesian