Two Extensions of Topological Feedback Entropy

TL;DR

This paper extends the concept of topological feedback entropy to partially observed and discontinuous systems, establishing that the generalized TFE quantifies the minimal feedback rate needed for controlled invariance.

Contribution

It introduces two new generalizations of TFE for broader classes of systems and proves their equivalence to minimal feedback bit rates for invariance.

Findings

Generalized TFE matches minimal feedback rate for partially observed systems.

Generalized TFE matches minimal feedback rate for discontinuous systems.

Provides theoretical foundation for feedback control in complex systems.

Abstract

Topological feedback entropy (TFE) was introduced in 2004 to measure the intrinsic rate at which a continuous, fully observed, deterministic control system generates information for controlled set-invariance. In this paper, we generalise this notion in two directions, first to continuous, partially observed systems and then to discontinuous, fully observed systems. In each case, we show that the corresponding generalised TFE coincides with the the smallest feedback bit rate that allows a form of controlled invariance to be achieved.