# Probabilistic Performance of State Estimation Across a Lossy Network

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

Automatica, 44(12):3046-3053, 2008 (to appear)

We consider a discrete time state estimation problem over a packet-based network. In each discrete time step, a measurement packet is sent across a lossy network to an estimator unit consisting of a modified Kalman filter. Using the designed estimator algorithm, the importance of placing a measurement buffer at the sensor that allows transmission of the current and several previous measurements is shown. Previous pioneering work on Kalman filtering with intermittent observation losses is concerned with the asymptotic behavior of the expected value of the error covariance, i.e. E[Pk ] < ∞ as k → ∞. We consider a different performance metric, namely a probabilistic statement of the error covariance Pr[Pk ≤ M ] ≥ 1 − ϵ, meaning that with high probability the error covariance is bounded above at any instant in time. Provided the estimator error covariance has an upper bound whenever a measurement packet arrives, we show that for any finite M this statement will hold so long as the probability of receiving a measurement packet is nonzero. We also give an explicit relationship between M and ϵ and provide examples to illustrate the theory.

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Richard Murray (murray@cds.caltech.edu)