# Packet-based Control with Norm Bounded Uncertainties

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In this lecture, we study the effect of data loss on the performance of the Kalman filter for discrete-time linear systems. Observations are lost according to a bernoulli independent process, modeling this way the presence of a lossy networks between the sensors and the estimator. We first prove that the Kalman filter is still optimal in this new scenario. We then provide asymptotic results on the performance of the filter. In particular, we show that a transition from boundedness to instability arises if the arrival probability is lower that a critical value, that depends on the unstable eigenvalues of the system.

## Lecture Materials

## Reading

Kalman Filtering with Intermittent Observations, B. Sinopoli, L. Schenato, M. Franceschetti, K. Poolla, M. Jordan and S. Sastry. This is the paper where all the proofs reside. Below I posted Chapter 3 of my thesis, which is essentially the same, but the notation is more consistent with the next two lectures.

Optimal Estimation in Lossy Networks This is chapter 3 of my thesis. Content is almost the same as the paper above, but notation is slightly modified to be consistent with the control part.

## Additional Resources

The Kalman Filter, G. Welch and G. Bishop. A webpage with many links on Kalman filter.

Optimal Filtering, B.D.O Anderson and J.B. Moore. Dover Books on Engineering, 2005. A reissue of a book from 1979. It contains a detailed mathematical presentation of filtering problems and the Kalman filter. A very good book.