Mathematics > Optimization and Control
[Submitted on 20 Jun 2014 (v1), last revised 1 Nov 2014 (this version, v2)]
Title:Faa di Bruno Hopf Algebra of the Output Feedback Group for Multivariable Fliess Operators
View PDFAbstract:Given two nonlinear input-output systems written in terms of Chen-Fliess functional expansions, it is known that the feedback interconnected system is always well defined and in the same class. An explicit formula for the generating series of a single-input, single-output closed-loop system was provided by the first two authors in earlier work via Hopf algebra methods. This paper is a sequel. It has four main innovations. First, the full multivariable extension of the theory is presented. Next, a major simplification of the basic set up is introduced using a new type of grading that has recently appeared in the literature. This grading also facilitates a fully recursive algorithm to compute the antipode of the Hopf algebra of the output feedback group, and thus, the corresponding feedback product can be computed much more efficiently. The final innovation is an improved convergence analysis of the antipode operation, namely, the radius of convergence of the antipode is computed.
Submission history
From: W. Steven Gray [view email][v1] Fri, 20 Jun 2014 13:23:39 UTC (769 KB)
[v2] Sat, 1 Nov 2014 09:54:17 UTC (771 KB)
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