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The goal of this paper is to compute the generating series of a closed-loopsystem when the plant is described in terms of a Chen-Fliess series and anadditive static output feedback is applied. The first step is to consider theso called Wiener-Fliess connection consisting of a Chen-Fliess series followedby a memoryless function. Of particular importance will be the contractivenature of this map, which is needed to show that the closed-loop system has aChen-Fliess series representation. To explicitly compute the generating series,two Hopf algebras are needed, the existing output feedback Hopf algebra used todescribe dynamic output feedback, and the Hopf algebra of the shuffle group.These two combinatorial structures are combined to compute what will be calledthe Wiener-Fliess feedback product. It will be shown that this product has anatural interpretation as a transformation group acting on the plant andpreserves the relative degree of the plant. The convergence of theWiener-Fliess composition product and the additive static feedback product arecompletely characterized.

Formal Power Series Approach to Nonlinear Systems with Additive Static Feedback