On the stability of the recursive Kalman filter for linear time-invariant systems

Stability of the Kalman filter for linear time-invariant systems is studied from the perspective of divergence of the estimation error under incorrect noise measurement. We provide testable, necessary and sufficient conditions for the filter to be stable, stable with respect to perturbations in the initial error covariance, or semi-stable, respectively meaning that the estimate error covariance is bounded, bounded for perturbations in the initial error covariance or that it does not diverge exponentially. Previous conditions present some degrees of conservativeness or address only stability.