  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.
