Pre-Lie algebra characterization of SISO feedback invariants

Transformation groups have been used extensively in system theory since its inception. Recently, a feedback transformation group for a nonlinear input-output system characterized by a Chen-Fliess functional expansion was described by the authors. In particular, an algorithm was given to identify a class of feedback invariant series. Their relationship to series having well-defined relative degree was also developed. This paper is a continuation of that work, but with three innovations. First, newly developed algebraic tools from the field of pre-Lie algebras are applied to give more insight into the invariance theory. The role of relative degree in this paper is diminished in favor of systems with arbitrary generating series. Finally, dynamic output feedback, as described by the feedback product, is considered explicitly.