On Symmetries in Analytic Input-Output Systems

TL;DR

This paper introduces new symmetry concepts for analytic input-output systems, analyzes their properties, and explores the structure of systems with palindromic generating series, revealing their infinite-dimensional characteristics.

Contribution

It defines coefficient reversible and palindromic symmetries for Chen--Fliess systems and analyzes systems with linear time-varying, palindromic series with maximal coefficient growth.

Findings

Systems with palindromic series have infinite Hankel and Lie rank.

Such systems possess an infinite-dimensional state space realization.

The paper characterizes their relative degree and zero dynamics.

Abstract

There are many notions of symmetry for state space models. They play a role in understanding when systems are time reversible, provide a system theoretic interpretation of thermodynamics, and have applications in certain stabilization and optimal control problems. The earliest form of symmetry for analytic input-output systems is due to Fliess who introduced systems described by an exchangeable generating series. In this case, one is able to write the output as a memoryless analytic function of the integral of each input. The first goal of this paper is to describe two new types of symmetry for such Chen--Fliess input-output systems, namely, coefficient reversible symmetry and palindromic symmetry. Each concept is then related to the notion of an exchangeable series. The second goal of the paper is to provide an in-depth analysis of Chen--Fliess input-output systems whose generating series are linear time-varying, palindromic, and have generating series coefficients growing at a maximal rate while ensuring some type of convergence. It is shown that such series have an infinite Hankel rank and Lie rank, have a certain infinite dimensional state space realization, and a description of their relative degree and zero dynamics is given.