  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.
