Kalman filtering with intermittent observations: Critical value for second order system

Sinopoli et al. (2004) analyze 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 show 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 prove the existence of a critical value, p_c, for such probability, below which the Kalman filter will diverge.While it has been shown that the critical value for diagonalizable systems with eigenvalues of different magnitude coincides with the lower bound determined by Mo and Sinopoli (2008), the problem is still open in the case where some eigenvalues have equal magnitude. This paper provides a complete characterization of the critical arrival probability for diagonalizable second order systems with eigenvalues of equal magnitude. In general the critical value for these systems is higher than the lower bound, unless the transmission from the sensor includes both the current and the previous measurement. In this case it is possible to construct a filter that whose critical value achieves the lower bound. Although clearly restrictive, the analysis of second order systems presented herein can be used to bound the critical value of higher dimensional systems of this kind.