Towards Finding the Critical Value for Kalman Filtering with Intermittent Observations

TL;DR

This paper investigates the critical packet arrival probability for Kalman filtering with intermittent observations, providing a method to compute this threshold under broad conditions, which determines filter stability.

Contribution

It introduces a way to compute the critical arrival probability for Kalman filters with intermittent observations under minimal assumptions on system matrices.

Findings

Derives a formula for the critical probability p_c.

Shows p_c can be explicitly computed for general linear systems.

Establishes conditions under which the Kalman filter remains stable.

Abstract

In [1], Sinopoli et al. 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, pc, for such probability, below which the Kalman filter will diverge. The authors are not able to compute the actual value of this critical probability for general linear systems, but provide upper and lower bounds. They are 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.