On feedback connections of analytic nonlinear systems and combinatorics on words
William Steven Gray, Yaqin Li · 2005
Given two analytic nonlinear input-output systems represented as Fliess operators, F/sub c/ and F/sub d/, their feedback connection y=F/sub c/[u + F/sub d/[y]] can be described in terms of a feedback product of their corresponding generating series c and d, namely y=F/sub c@d/[u]. A fundamental question is whether the operator F/sub 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/sub c/ and F/sub d/ are absolutely and uniformly convergent? In this paper, it is proven that /sub 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/sub n/, or more specifically, the binomial transform of the sequence of Catalan numbers, S/sub n/. This suggests a deeper connection between feedback structures of analytic systems and classical topics in algebraic combinatorics on words.