Alternative formulation of the Kalman filter for correlated process and observation noise. Issue 5 (1st September 2014) - Chang, Guobin
Abstract : The Kalman filter for linear system with cross‐correlated process and observation noises at the same epoch is revisited. As the process noise is correlated with the observation noise, the probability of the former conditioned on the latter should have a higher concentration than its original probability, implying that the conditional one becomes more accurate. Inspired by this, an alternative formulation of the filter is derived that the conditional process noise rather than the original one is used in the prediction step. The conditional mean and covariance are derived in two different ways, that is, through state augmentation and conditionally Gaussian distribution. It is also proved that the proposed formulation is theoretically equivalent to the two available frameworks. However, this formulation has the merit of explaining why the estimate is improved taking the cross‐correlation into account, that is, the a posteriori covariance of the process noise is reduced compared to the original one implying that the process model becomes more accurate. A simple but illustrative example is simulated and the efficacy of the proposed framework is validated.