# A Faà di Bruno Hopf algebra for analytic nonlinear feedback control systems

### Luis A. Duffaut Espinosa

University of NSW at the Australian Defence Force Academy, Canberra, Australia### W. Steven Gray

Old Dominion University, Norfolk, USA

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## Abstract

In many applications, nonlinear input-output systems are interconnected in various ways to model complex systems. If a component system is analytic, meaning it can be described in terms of a Chen–Fliess functional series expansion, then it can be represented uniquely by a formal power series over a noncommutative alphabet. System interconnections are then characterized in terms of operations on formal power series. This paper provides an introduction to this methodology with an emphasis on feedback systems, which are ubiquitous in modern technology. In this case, a Faà di Bruno type Hopf algebra is dened for a group of integral operators, where operator composition is the group product. Using a series expansion for the antipode, an explicit formula for the generating series of the compositional inverse operator is derived. This result produces an explicit formula for the generating series of a feedback system, which had been an open problem until recently.