Shannon Zero Error Capacity and the Problem of Almost Sure Observability over Noisy Communication Channels

The paper addresses the state estimation problem involving communication errors and capacity constraints. Discrete-time partially observed unstable linear systems perturbed by uniformly bounded disturbances are studied. Unlike the classic theory, the sensor signals are transmitted to the estimator over a noisy digital communication link modelled as a stochastic stationary discrete memoryless channel. It is shown that the Shannon zero error capacity of the channel constitutes the border separating the cases where the plant is and respectively, is not almost surely observable.