A Formal Power Series Approach to Multiplicative Dynamic Feedback Interconnection

TL;DR

This paper develops a formal power series framework for analyzing multiplicative dynamic feedback interconnections in control systems, providing explicit formulas, group interpretations, and computational tools based on Hopf algebras.

Contribution

It introduces a novel formal power series approach for multiplicative feedback, including explicit formulas and a Hopf algebra-based computational framework.

Findings

Derived explicit formulas for the generating series of closed-loop systems.

Interpreted multiplicative feedback as a transformation group.

Established a Hopf algebra framework for computation.

Abstract

The goal of the paper is multi-fold. First, an explicit formula is derived to compute the non-commutative generating series of a closed-loop system when a (multi-input, multi-output) plant, given in Chen--Fliess series description is in multiplicative output feedback interconnection with another system, also given as Chen--Fliess series. Furthermore, it is shown that the multiplicative dynamic output feedback connection has a natural interpretation as a transformation group acting on the plant. A computational framework for computing the generating series for multiplicative dynamic output feedback is devised utilizing the Hopf algebras of the coordinate functions corresponding to the shuffle group and the multiplicative feedback group. The pre--Lie algebra in multiplicative feedback is shown to be an example of Foissy's com-pre-Lie algebras indexed by matrices with certain structure.