Zero-Error Feedback Capacity of Finite-State Additive Noise Channels for Stabilization of Linear Systems

It is known that for a discrete channel with correlated additive noise, the ordinary capacity with or without feedback is equal $ \log q-\mathcal{H}_{ch} $, where $ \mathcal{H}_{ch} $ is the entropy rate of the noise process and $ q $ is the alphabet size. In this paper, for a class of finite-state additive noise channels, it is shown that the zero-error feedback capacity is either zero or $C_{0f} =\log q -h_{ch} $, where $ h_{ch} $ is the {\em topological entropy} of the noise process. A topological condition is given to determine when the zero-error capacity with or without feedback is zero. We explicitly compute $ C_{0f}$ for several examples, including channels with isolated errors and a Gilbert-Elliot channel. Furthermore, the zero-error feedback capacity of a general channel model is revisited and {\em uniform zero-error feedback codes} are introduced. It is revealed that there is a close connection between zero-error communication and control of linear systems with bounded disturbances. A necessary and sufficient condition for stabilization of unstable linear systems over general channels with memory is obtained, assuming no state information at either end of the channel. It is shown that $ C_{0f} $ is the figure of merit for determining when bounded stabilization is possible. This leads to a "small-entropy theorem", stating that stabilization over finite-state additive noise channels can be achieved if and only if the sum of the topological entropies of the linear system and the channel is smaller than $\log q$.