A tiger by the tail: When multiplicative noise stymies control

This paper considers the stabilization of an unstable discrete-time linear system that is observed over a channel corrupted by continuous multiplicative noise. The main result is a converse bound that shows that if the system growth is large enough the system cannot be stabilized in a mean-squared sense. This is done by showing that the probability of the state magnitude remains bounded must go to zero with time. It was known that a system with multiplicative observation noise can be stabilized using a simple linear strategy if the system growth is suitably bounded. However, it was not clear whether non-linear controllers could overcome arbitrarily large growth factors. One difficulty with using the standard approach for a data-rate theorem style converse is that the mutual information per round between the system state and the observation is potentially unbounded with a multiplicative noise observation channel. Our proof technique recursively bounds the conditional density of the system state (instead of focusing on the second moment) to bound the progress the controller can make.

[1]  Bruno Sinopoli,et al.  Foundations of Control and Estimation Over Lossy Networks , 2007, Proceedings of the IEEE.

[2]  Bruno Sinopoli,et al.  Kalman filtering with intermittent observations , 2004, IEEE Transactions on Automatic Control.

[3]  Amos Lapidoth,et al.  Capacity bounds via duality with applications to multiple-antenna systems on flat-fading channels , 2003, IEEE Trans. Inf. Theory.

[4]  Sekhar Tatikonda,et al.  Control under communication constraints , 2004, IEEE Transactions on Automatic Control.

[5]  Robin J. Evans,et al.  Feedback Control Under Data Rate Constraints: An Overview , 2007, Proceedings of the IEEE.

[6]  Heinrich Meyr,et al.  Digital communication receivers - synchronization, channel estimation, and signal processing , 1997, Wiley series in telecommunications and signal processing.

[7]  B. E. Eckbo,et al.  Appendix , 1826, Epilepsy Research.

[8]  M. Athans,et al.  The uncertainty threshold principle: Fundamental limitations of optimal decision making under dynamic uncertainty , 1976 .

[9]  Anant Sahai,et al.  Control capacity , 2015, 2015 IEEE International Symposium on Information Theory (ISIT).

[10]  M. Srinath,et al.  Optimum Linear Estimation of Stochastic Signals in the Presence of Multiplicative Noise , 1971, IEEE Transactions on Aerospace and Electronic Systems.

[11]  Anant Sahai,et al.  Carry-free models and beyond , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[12]  Massimo Franceschetti,et al.  Data Rate Theorem for Stabilization Over Time-Varying Feedback Channels , 2009, IEEE Transactions on Automatic Control.

[13]  A. Sahai,et al.  Intermittent Kalman filtering: Eigenvalue cycles and nonuniform sampling , 2011, Proceedings of the 2011 American Control Conference.

[14]  Anant Sahai,et al.  Non-coherence in estimation and control , 2013, 2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton).