Estimation with Information Loss: Asymptotic Analysis and Error Bounds

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. That work assumes 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 π_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 πg, as long as π_g> 0, the estimator is stable almost surely, i.e. for any given ε > 0, the error covariance matrix P_k is bounded by a finite matrix M, with probability 1 -ε. We also give explicit formula for the relationship between M and ε. We consider the case where the observation matrix is invertible