On Kalman Filtering for Detectable Systems With Intermittent Observations

We consider the problem of Kalman filtering when observations are available according to a Bernoulli process. It is known that there exists a critical probability pc such that, if measurements are available with probability greater than pc, then the expected prediction covariance is bounded for all initial conditions; otherwise, it is unbounded for some initial conditions. We show that, when the system observation matrix restricted to the observable subspace is invertible, the known lower bound on pc is, in fact, the exact critical probability. This result is based on a novel decomposition of positive semidefinite matrices.