A Nonstochastic Information Theory for Communication and State Estimation
It provides a novel framework for networked control, bridging communication and control theory by extending information-theoretic concepts to nonstochastic settings.
This paper develops a nonstochastic information theory for communication and state estimation, defining a maximin information functional without assuming probability distributions. It shows that the largest maximin information rate through a memoryless channel equals the zero-error capacity, and uses it to derive tight conditions for state estimation over such channels.
In communications, unknown variables are usually modelled as random variables, and concepts such as independence, entropy and information are defined 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, raising the question of whether it is possible to construct meaningful analogues of stochastic concepts such as independence, Markovness, entropy and information without assuming a probability space. This paper introduces a framework for doing so, leading to the construction of a maximin information functional for nonstochastic variables. It is shown that the largest maximin information rate through a memoryless, error-prone channel in this framework coincides with the block-coding zero-error capacity of the channel. Maximin information is then used to derive tight conditions for uniformly estimating the state of a linear time-invariant system over such a channel, paralleling recent results of Matveev and Savkin.