Abstract
This paper considers the stabilization of an unstable discrete-time scalar 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 gain is large enough the system cannot be stabilized in a mean-squared sense.
It was known that a system with multiplicative observation noise can be stabilized using a simple linear strategy if the system growth is bounded (this bound depends on the noise parameters). However, it was not clear whether non-linear controllers could overcome arbitrarily large growth factors. We use a non- standard approach to provide a proof-of-concept converse. One difficulty with multiplicative noise is that the mutual information per round between the system state and the observation is potentially unbounded. We handle this by providing the controller with side information about the magnitude of the state.
This is joint work with Jian Ding and Yuval Peres. The authors met through the Simons Institute.