# Estimation with Information Loss: Asymptotic Analysis and Error Bounds

### Ling Shi, Michael Epstein, Abhishek Tiwari and Richard M. Murray

2005 Conference on Decision and Control (CDC)

In this paper, we consider a discrete time state estimation problem over a packet-based network. In each discrete time step, the measurement is sent to a Kalman filter with some probability that it is received or dropped. Previous pioneering work on Kalman filtering with intermittent observation losses shows that there exists a certain threshold of the packet dropping rate below which the estimator is stable in the expected sense. In their analysis, they assume that packets are dropped independently between all time steps. However we give a completely different point of view. On the one hand, it is not required that the packets are dropped independently but just that the information gain pi_g, defined to be the limit of the ratio of the number of received packets n during N time steps as N goes to infinity, exists. On the other hand, we show that for any given pi_g, as long as pi_g > 0, the estimator is stable almost surely, i.e. for any given epsilon > 0 the error covariance matrix P{k is bounded by a finite matrix M, with probability 1 â epsilon. Given an error tolerance M, pi_g can in turn be found. We also give explicit formula for the relationship between M and epsilon.