On feedback connections of analytic nonlinear systems and combinatorics on words

Given two analytic nonlinear input-output systems represented as Fliess operators, F_c and F_d, their feedback connection y=F_c[u + F_d[y]] can be described in terms of a feedback product of their corresponding generating series c and d, namely y=F_c@d[u]. A fundamental question is whether the operator F_c@d representing the closed-loop system is well defined, specifically, does its series representation converge in any sense given that the series representations of F_c and F_d are absolutely and uniformly convergent? In this paper, it is proven that _Fc@d is always well defined on an open ball in a suitable input signal space and over a nonzero interval of time. In the process of establishing this result, an interesting connection is derived between the radius of convergence and the asymptotic behavior of the sequence of Catalan numbers, C_n, or more specifically, the binomial transform of the sequence of Catalan numbers, S_n. This suggests a deeper connection between feedback structures of analytic systems and classical topics in algebraic combinatorics on words.