Entropy of Generating Series for Nonlinear Input-Output Systems and Their Interconnections

TL;DR

This paper introduces a new entropy measure for nonlinear input-output systems with Chen-Fliess series and explores how system interconnections affect this entropy, providing insights into system complexity and interconnection effects.

Contribution

It defines a novel entropy concept for Chen-Fliess series and analyzes how different interconnection types influence this entropy in control systems.

Findings

Interconnections can either preserve or increase entropy.

Entropy is well-defined for systems with Chen-Fliess series.

Introduction of an entropy ultrametric space.

Abstract

This paper has two main objectives. The first is to introduce a notion of entropy that is well suited for the analysis of nonlinear input-output systems that have a Chen-Fliess series representation. The latter is defined in terms of its generating series over a noncommutative alphabet. The idea is to assign an entropy to a generating series as an element of a graded vector space. The second objective is to describe the entropy of generating series originating from interconnected systems of Chen-Fliess series that arise in the context of control theory. It is shown that one set of interconnections can never increase entropy as defined here, while a second set has the potential to do so. The paper concludes with a brief introduction to an entropy ultrametric space and some open questions.