Globally optimal Kalman filtering with finite-time correlated noises

In this paper, an extension of the standard Kalman filtering for the dynamical systems with white noises to finite-time correlated noises is addressed. Although one can augment the state vector with white noises in a time-variant moving average process which models the finite-time correlated noise, and then use standard Kalman filtering to obtain the optimal state estimate in the mean square error sense, a direct recursion for the optimal estimate of original state in general cases was pursued owing to the lower computational complexity. By decomposing the original Kalman gain to two recursively represented factors and increasing some recursive terms (for more than one-step correlated noises), we directly provide recursive algorithms for the globally optimal estimate of original state for stochastic linear dynamic systems with (i) multi-step correlated process noises; (ii) multi-step correlated observation noises; and (iii) multi-step correlated process and observation noises. There is no any limitation on all involved matrices in the model and algorithms. The new development for Kalman filtering is expected to further promote practical applications of dynamic system theory and methods.