The Singularity Problem of LMV Fusion in Multisensor Kalman Filtering

The linear minimum variance (LMV) fusion rule, while widely employed in multi-sensor Kalman filtering fusion, relies typically on the implicit assumption of a nonsingular collective error covariance matrix (ECM). In practice, however, redundant information among local Kalman filters can result in a singular collective ECM. This paper aims to elucidate the fundamental causes behind the potential singularity of the collective ECM. We establish a necessary and sufficient condition for collective ECM non-singularity through reachability analysis, and demonstrate that redundancy arises when state estimates depend exclusively on model prediction. Furthermore, we derive intuitive sufficient or necessary conditions expressed directly in terms of system parameters. These results provide theoretical guarantees for LMV fusion and practical guidelines for designing multi-sensor systems.