A characterization of the critical value for Kalman filtering with intermittent observations

In [1], Sinopoli et al. analyzed the problem of optimal estimation for linear Gaussian systems where packets containing observations are dropped according to an i.i.d. Bernoulli process, modeling a memoryless erasure channel. In this case the authors showed that the Kalman Filter is still the optimal estimator, although boundedness of the error depends directly upon the channel arrival probability, p. In particular they also proved the existence of a critical value, p_c, for such probability, below which the Kalman filter will diverge. The authors were not able to compute the actual value of this critical probability for general linear systems, but provided upper and lower bounds. They were able to show that for special cases, i.e. C invertible, such critical value coincides with the lower bound. This paper computes the value of the critical arrival probability, under minimally restrictive conditions on the matrices A and C. This paper also gives an example to illustrate that the lower bound is not always tight.