Spectral Decomposition in Kalman Filter Algorithm for Homogeneous Atomic Clock Ensembles
Existing studies have pointed out numerical instability in the Kalman filter of atomic clocks, but the reasons for such instability have not been clarified mathematically. In this paper, we mathematically clarify the reason for the numerical instability by a new approach of spectral decomposition of the error covariance matrix in the Kalman filter. In particular, we reveal the fact that the error covariance matrix for homogeneous undetectable atomic clock ensembles can be decomposed into a diverging part and a converging part. Furthermore, the Kalman gain is solely influenced by the converging part, but not the diverging part, meaning that the Kalman gain converges to a steady-state value if ideal computation is possible without computation error. We present an alternative method to the conventional Kalman filter to avoid numerical instability and reduce computation cost where the covariance of Kalman filter can be computed rigorously only using three n-dimensional Riccati iterations instead of an nN- dimensional Riccati iterations for an n-order clock model with N clocks. A numerical example is provided to illustrate the efficacy of our approach.
