The Annals of Applied Probability

When multiplicative noise stymies control

Jian Ding, Yuval Peres, Gireeja Ranade, and Alex Zhai

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We consider the stabilization of an unstable discrete-time linear system that is observed over a channel corrupted by continuous multiplicative noise. Our main result shows that if the system growth is large enough, then the system cannot be stabilized. This is done by showing that the probability that the state magnitude remains bounded must go to zero with time. Our proof technique recursively bounds the conditional density of the system state to bound the progress the controller can make. This sidesteps the difficulty encountered in using the standard data-rate theorem style approach; that approach does not work because the mutual information per round between the system state and the observation is potentially unbounded.

It was known that a system with multiplicative observation noise can be stabilized using a simple memoryless linear strategy if the system growth is suitably bounded. The second main result in this paper shows that while memory cannot improve the performance of a linear scheme, a simple nonlinear scheme that uses one-step memory can do better than the best linear scheme.

Article information

Ann. Appl. Probab., Volume 29, Number 4 (2019), 1963-1992.

Received: December 2016
Revised: August 2017
First available in Project Euclid: 23 July 2019

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Mathematical Reviews number (MathSciNet)

Primary: 93E03: Stochastic systems, general

Control stability multiplicative noise


Ding, Jian; Peres, Yuval; Ranade, Gireeja; Zhai, Alex. When multiplicative noise stymies control. Ann. Appl. Probab. 29 (2019), no. 4, 1963--1992. doi:10.1214/18-AAP1415.

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