A Formal Power Series Approach to Multiplicative Dynamic and Static Output Feedback

TL;DR

This paper develops explicit formulas for the generating series of systems in Chen-Fliess form under multiplicative dynamic and static output feedback, revealing their interpretation as transformation groups acting on the plant.

Contribution

It introduces explicit formulas for closed-loop generating series with multiplicative feedback and interprets these feedback connections as transformation groups.

Findings

Derived explicit formulas for dynamic feedback generating series.

Established the group-theoretic interpretation of feedback connections.

Extended results to static feedback scenarios.

Abstract

The goal of the paper is two-fold. The first of which is to derive an explicit formula to compute the generating series of a closed-loop system when a plant, given in a Chen-Fliess series description is in multiplicative output feedback connection with another system given in Chen-Fliess series description. Further, the objective extends in showing that the multiplicative dynamic output feedback connection has a natural interpretation as a transformation group acting on the plant. The second of the two-part goal of this paper is same as the first part albeit when the Chen-Fliess series in the feedback is replaced by a memoryless map. The paper provides an explicit formula to compute the generating series of a closed-loop system in multiplicative static output feedback connection and shows that the static feedback has a natural interpretation as a transformation group acting on the plant.