Shannon zero error capacity in the problems of state estimation and stabilization via noisy communication channels

The paper addresses state estimation and stabilization problems involving communication errors and capacity constraints. Discrete-time partially observed unstable linear systems perturbed by stochastic exogenous disturbances are studied. Unlike the classic theory, the sensor signals are transmitted to the estimator or controller over a noisy digital communication link modelled as a stochastic stationary discrete memoryless channel. It is shown that the capability of the noisy channel to ensure almost sure stabilizability/observability of the plant is identical to exactly its capability to transmit information with zero probability of error. Specifically, it is demonstrated that the standard numerical characteristic of the latter capability, i.e., the Shannon zero error capacity of the channel, constitutes the border separating the cases where the plant is and respectively, is not stabilizable/observable with probability 1.