多元Fliess算子输出反馈群的Faa di Bruno霍普夫代数
Faa di Bruno Hopf Algebra of the Output Feedback Group for Multivariable Fliess Operators
摘要：
给定两个用Chen-Fliess函数展开式表示的非线性输入输出系统，已知反馈关联系统总是定义良好且属于同一类.前两位作者在早期的工作中通过霍普夫代数方法给出了单输入单输出闭环系统的生成级数的显式公式。本文是续篇。它有四个主要创新。首先，充分的多变量扩展的理论。接下来，一个主要的简化的基本设置介绍了使用一种新的类型的分级，最近出现在文献中。这种分级也便于全递归算法来计算输出反馈组的霍普夫代数的对极，因此，可以更有效地计算相应的反馈乘积。最后的创新点是对对极运算的收敛性分析进行了改进，即计算了对极的收敛半径。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
