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 in a second-moment sense. 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 (instead of focusing on the second moment) 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. In this paper, we show that while memory cannot improve the performance of a linear scheme, a simple non-linear scheme that uses one-step memory can do better than the best linear scheme.
 Control strategies for systems with information bottlenecks often follow an estimate-then-control paradigm. This paper presents a “non-coherent” system where this strategy cannot work and provides an alternative. The paper considers the estimation and control of a discrete-time linear system with continuous random observation gain, i.e. through a non-coherent channel. It is shown that such an unstable system is not mean-squared observable regardless of the density of the random observation gain: the mean-squared estimation error for any estimator must go to infinity. This is surprising in the context of threshold results for rate-limited estimation. In contrast to other results with rate-limited feedback, the paper shows that the system can be closed-loop mean-square stabilized in a certain parameter regime even though its open-loop counterpart is not mean-square observable. Finally, carry-free models (generalized deterministic models) provide an intuitive interpretation for the results. This paper considers optimum (MMSE) linear recursive estimation of stochastic signals in the presence of multiplicative noise in addition to measurement noise. Often problems associated with phenomena such as fading or reflection of the transmitted signal at an ionospheric layer, and also situations involving sampling, gating, or amplitude modulation, can be cast into such formulation. The different kinds of estimation problems treated include one-stage prediction, filtering, and smoothing. Algorithms are presented for discrete time as well as for continuous time estimation.