Formal Power Series Approach to Nonlinear Systems with Additive Static Feedback

TL;DR

This paper develops a formal power series framework for analyzing nonlinear systems with additive static feedback, utilizing Hopf algebras to explicitly compute the system's generating series and establish convergence properties.

Contribution

It introduces the Wiener-Fliess feedback product combining two Hopf algebras to analyze static feedback in nonlinear systems, extending existing algebraic methods.

Findings

Explicit computation of the generating series for closed-loop systems

Introduction of the Wiener-Fliess feedback product as a transformation group

Complete characterization of convergence for the composition and feedback products

Abstract

The goal of this paper is to compute the generating series of a closed-loop system when the plant is described in terms of a Chen-Fliess series and an additive static output feedback is applied. The first step is to consider the so called Wiener-Fliess connection consisting of a Chen-Fliess series followed by a memoryless function. Of particular importance will be the contractive nature of this map, which is needed to show that the closed-loop system has a Chen-Fliess series representation. To explicitly compute the generating series, two Hopf algebras are needed, the existing output feedback Hopf algebra used to describe dynamic output feedback, and the Hopf algebra of the shuffle group. These two combinatorial structures are combined to compute what will be called the Wiener-Fliess feedback product. It will be shown that this product has a natural interpretation as a transformation group acting on the plant and preserves the relative degree of the plant. The convergence of the Wiener-Fliess composition product and the additive static feedback product are completely characterized.