A non-stochastic information theory for communication and state estimation over erroneous channels

In communications, unknown variables are usually modelled as random variables, with concepts such as independence, entropy and information given in terms of the underlying probability distributions. In contrast, control theory often treats uncertainties and disturbances as bounded unknowns having no statistical structure. The area of networked control combines both fields and raises the question of whether it is possible to construct meaningful analogues of important stochastic concepts such as independence, Markovianness, entropy, and information, without assuming a probability space. This paper introduces a framework for doing so, leading in particular to the construction of a maximin information functional for non-stochastic variables. It is shown that, in this framework, the largest maximin information rate through a memoryless, error-prone channel coincides exactly with its block-coding zero-error capacity. This leads to a tight condition for the achievability of uniform exponential convergence when estimating the state of an unperturbed linear system over such a channel, similar to recent results of Matveev and Savkin.