SISO Output Affine Feedback Transformation Group and Its Faa di Bruno Hopf Algebra

摘要

The general goal of this paper is to identify a transformation group that canbe used to describe a class of feedback interconnections involving subsystemswhich are modeled solely in terms of Chen-Fliess functional expansions orFliess operators and are independent of the existence of any state spacemodels. This interconnection, called an output affine feedback connection, isdistinguished from conventional output feedback by the presence of a multiplierin an outer loop. Once this transformation group is established, three basicquestions are addressed. How can this transformation group be used to providean explicit Fliess operator representation of such a closed-loop system? Is itpossible to use this feedback scheme to do system inversion purely in aninput-output setting? In particular, can feedback input-output linearization beposed and solved entirely in this framework, i.e., without the need for anystate space realization? Lastly, what can be said about feedback invariantsunder this transformation group? A final objective of the paper is to describethe Lie algebra of infinitesimal characters associated with the group in termsof a pre-Lie product.