Faa di Bruno Hopf Algebra of the Output Feedback Group for Multivariable Fliess Operators

TL;DR

This paper extends the Faa di Bruno Hopf algebra framework to multivariable systems, introduces a new grading for simplification, and provides an efficient recursive algorithm for feedback computations with improved convergence analysis.

Contribution

It presents the multivariable extension of the Hopf algebra approach, introduces a new grading for simplification, and develops an efficient recursive algorithm for feedback calculations.

Findings

Multivariable extension of the Hopf algebra framework.

Introduction of a new grading for simplification.

Recursive algorithm for antipode computation with convergence analysis.

Abstract

Given two nonlinear input-output systems written in terms of Chen-Fliess functional expansions, it is known that the feedback interconnected system is always well defined and in the same class. An explicit formula for the generating series of a single-input, single-output closed-loop system was provided by the first two authors in earlier work via Hopf algebra methods. This paper is a sequel. It has four main innovations. First, the full multivariable extension of the theory is presented. Next, a major simplification of the basic set up is introduced using a new type of grading that has recently appeared in the literature. This grading also facilitates a fully recursive algorithm to compute the antipode of the Hopf algebra of the output feedback group, and thus, the corresponding feedback product can be computed much more efficiently. The final innovation is an improved convergence analysis of the antipode operation, namely, the radius of convergence of the antipode is computed.