  The goal of the paper is multi-fold. First, an explicit formula is derived to
compute the non-commutative generating series of a closed-loop system when a
(multi-input, multi-output) plant, given in Chen--Fliess series description is
in multiplicative output feedback interconnection with another system, also
given as Chen--Fliess series. Furthermore, it is shown that the multiplicative
dynamic output feedback connection has a natural interpretation as a
transformation group acting on the plant. A computational framework for
computing the generating series for multiplicative dynamic output feedback is
devised utilizing the Hopf algebras of the coordinate functions corresponding
to the shuffle group and the multiplicative feedback group. The pre--Lie
algebra in multiplicative feedback is shown to be an example of Foissy's
com-pre-Lie algebras indexed by matrices with certain structure.
