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Given two nonlinear input-output systems written in terms of Chen-Fliessfunctional expansions, it is known that the feedback interconnected system isalways well defined and in the same class. An explicit formula for thegenerating series of a single-input, single-output closed-loop system wasprovided by the first two authors in earlier work via Hopf algebra methods.This paper is a sequel. It has four main innovations. First, the fullmultivariable extension of the theory is presented. Next, a majorsimplification of the basic set up is introduced using a new type of gradingthat has recently appeared in the literature. This grading also facilitates afully recursive algorithm to compute the antipode of the Hopf algebra of theoutput feedback group, and thus, the corresponding feedback product can becomputed much more efficiently. The final innovation is an improved convergenceanalysis of the antipode operation, namely, the radius of convergence of theantipode is computed.

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