Nonstochastic information concepts for estimation and control

Entropy and information are crucial notions in stochastic communication systems. However, they have arguably not been as central in control theory, which has a rich tradition of non-random models and techniques. This tutorial session aims to describe the key elements of certain non-probabilistic entropy and information concepts for state estimation and control. In this paper, which comprises the first half of the session, the focus is on a recently developed theory of nonstochastic information. Motivated by worst-case estimation and control, this framework allows non-statistical analogues of mutual independence, Markovness, information, and directed information to be rigorously defined. This yields powerful information-theoretic tools for finding fundamental bounds in zero-error communication and worst-case control systems. In the second half of this session, notions of entropy for deterministic nonlinear control systems are described, based on dynamical systems theory. These notions lead to characterisations of minimal feedback data rates for set-invariance. Taken together, the concepts discussed in this session give deterministic control theorists a way to use information and entropy ideas, without having to adopt a stochastic formulation.