Optimal state estimation of linear discrete-time systems with correlated random parameter matrices

In this paper, dynamic systems with correlated random parameter matrices are considered. Firstly, we consider dynamic systems with deterministic transition matrices and temporally one-step correlated measurement matrices. The optimal recursive estimation of the state is derived by converting the problem to the optimal Kalman filter with one-step correlated measurement noises. Then, we consider a class of specific dynamic systems where both state transition matrices and measurement matrices are one-step moving average matrix sequences driven by a common independent zero-mean parameter sequence. The optimal recursive estimation of the state can be obtained by using the first order through the sixth order moments of the driving parameter sequence, which implies that when both transition matrices and measurement matrices are correlated, in general, it is impossible to obtain an optimal filter by only using the variance and covariance information of the transition matrices and the measurement matrices. Moreover, if only the state transition matrices in the dynamic system are temporally correlated, optimal filters can be given by using lower order moments of the driving parameters. Numerical examples support the theoretical analysis and show that the optimal estimation is better than the random Kalman filter with the correlation of parameter matrices ignored, especially for the case of both the transition matrices and the measurement matrices being correlated.