A nonstochastic information theory for feedback

Abstract

This paper extends a recently proposed theory of nonstochastic maximin information to study feedback systems with erroneous channels. The concepts of conditional and directed maximin information are introduced, without assuming any statistical structure. It is proved that the zero-error feedback capacity of a stationary memoryless channel coincides with the largest rate of directed maximin information across it. This provides a nonstochastic counterpart to recent results in information theory. This characterization is then used to find a tight condition for exponential uniform stabilizability of a linear time-invariant plant over an erroneous channel, consistent with a result of Matveev and Savkin.

Development and Analysis of Deterministic Privacy-Preserving Policies Using Non- Stochastic Information Theory

“…Another symmetric measure of information is the maximin information. In order to define this measure of information, the notion of taxicab connectivity must be defined, borrowed from the pioneering works in [46], [48] on non-stochastic information theory.…”

“…There is a parallel less-studied (within tertiary colleges) theory of non-stochastic information theory [43]- [47], which has been recently used within engineering [48]- [50]. Nonstochastic information theory relies on uncertain variables and extension of analogues of probabilistic ideas, such as independence.…”

Quantized Control and Data Rate Constraints †

“…This suggests that there is no universal, operationally relevant information theory for feedback control over errorprone channels: such a theory must instead be tailored to match the underlying objectives and assumptions. For systems with nonstochastic disturbances, preliminary steps in this direction have been taken in Nair (2012Nair ( , 2013. The reader is also referred to You and Xie (2011) and Minero et al (2013) for information-theoretic analyses of stochastic linear systems controlled via Markov channels.…”

Invariance Feedback Entropy of Nondeterministic Control Systems

“…4.1] for noisy linear systems, defined as the average of the expected length of the transmitted symbols in the closed loop. We motivate the particular notion of data rate by two examples; one which illustrates that the time-varying definition [7] results in too large data rates and one which shows that the notion of data rate based on the framework of nonstochastic information theory, used in [27,28] for estimation [28] and control [27] of linear systems, leads to too small data rates. 5.1.…”

“…Zero-error capacity of uncertain channels. Alternatively to the definition of the data rate of a coder-controller in (18) we could follow [27,28] and define the data rate of a coder-controller as the zero-error capacity C 0 of an ideal stationary memoryless uncertain channel (SMUC) in the nonstochastic information theory framework presented in [28,Def. 4.1].…”