On Symmetries in Analytic Input-Output Systems

TL;DR

This paper advances symmetry concepts for analytic input-output systems by defining coefficient reversible and palindromic symmetry for Chen--Fliess generating series and linking them to exchangeable series. It then develops a rigorous analysis of globally maximal and locally maximal palindromic SISO linear systems, proving that their generating series have infinite Hankel and Lie ranks and admitting infinite-dimensional state-space realizations, with explicit forms for impulse responses and zero dynamics. The results reveal fundamental limits of finite-dimensional realizations for these symmetric classes and provide constructive descriptions of zeroing inputs via Bessel- and hypergeometric-function-based expressions. The findings contribute to understanding time-reversal-like properties and stabilization in nonlinear analytic systems and illuminate the structure of complex, infinite-dimensional realizations relevant to advanced control and system-theoretic thermodynamics.

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.

Figures (6)

• Figure 1: Relationship between the classes of symmetric formal power series $CR(X)$, $P(X)$ and $E(X)$.
• Figure 2: Plot of $h(t,\tau)$ versus $\tau$ for $t=1,2,3$ and $K=M=1$ in the globally maximal case
• Figure 3: Input $u_k^\ast(t)=-kJ_k(2t)/t$, $k=1,2,3$
• Figure 4: Simulated output $y=F_{x_0+c_F}[u_1^\ast]$ when $K=M=1$
• Figure 5: Plot of $h(1,\tau)$ versus $\tau$ for $K=1$ and various $M$ in the locally maximal case

Theorems & Definitions (30)

• Definition 2.1
• Definition 2.2
• Definition 3.1
• Example 3.1
• Lemma 3.1
• Definition 3.2
• Lemma 3.2
• Theorem 3.1
• Definition 3.3
• Example 3.2