Feedback transformation group for nonlinear input-output systems

The goal of this paper is to describe a feedback transformation group for the class of nonlinear input-output systems that can be represented in terms of Chen-Fliess functional expansions. There is no a priori requirement that these input-output systems have a state space realization, so the results presented here are independent of any particular state space coordinate system or state space embedding when a realization is available. Using an explicit formula for the generating series of the closed-loop system, an invariant series of this transformation group is described. The order of this invariant series corresponds to the relative degree of the system (defined purely from an input-output point of view) when it is well defined. But in general Brunovsky forms are not on all the orbits of this transformation group unless the system has an extended notion of relative degree.