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In this article, a particular approach to deriving recursive state estimators for linear state space models is generalised, namely the weighted least-squares approach introduced by Duncan and Horn in 1972, for the case of the two noise processes arising in such models being cross-correlated; in this context, the fact that in the available literature two different non-equivalent recursive algorithms are presented for the task of state estimation in the aforementioned case is discussed. Although the origin of the difference between these two algorithms can easily be identified, the issue has only rarely been discussed so far. Then the situations in which each of the two algorithms apply are explored, and a generalised Kalman filter which represents a merger of the two original algorithms is proposed. While, strictly speaking, optimal state estimates can be obtained only through the non-recursive weighted least-squares approach, in examples of modelling simulated and real-world data, the recursive generalised Kalman filter shows almost as good performance as the optimal non-recursive filter.
Andreas Galka (Sat,) studied this question.