Adaptive Learning Kalman Filter with Gaussian Process

This paper presents an adaptive Kalman filter for a linear dynamic system perturbed by an additive disturbance. The objective is to estimate both of the state and the unknown disturbance concurrently, while learning the disturbance as a stochastic process of the state vector. This is achieved by estimating the state according to the extended Kalman filtering applied to the marginal distribution of the state, and by estimating the disturbance from a backward smoothing technique. The corresponding pair of the estimated states and disturbances are fetched to a Gaussian process, which is constantly updated to resemble the disturbance process. The unique feature is that all of uncertainties in the estimated state and disturbance are accounted throughout the learning process. The efficacy of the proposed approach is illustrated by a numerical example.