Formal power series approach to nonlinear systems with additive static feedback

Abstract

The goal of this paper is to compute the generating series of a closed-loop system when the plant is described in terms of a Chen–Fliess series and an additive static output feedback is applied. The first step is to consider the so-called Wiener-Fliess connection consisting of a Chen–Fliess series followed by a memoryless function. To explicitly compute the generating series, two Hopf algebras are needed: the existing output feedback Hopf algebra used to describe dynamic output feedback and the Hopf algebra of the shuffle group. These two combinatorial structures are combined to compute what will be called the Wiener–Fliess feedback product. It will be shown that this product has a natural interpretation as a transformation group acting on the plant and preserves the relative degree of the plant. The convergence of the Wiener–Fliess composition product and the additive static feedback product are characterised.

Keywords:

• Chen–Fliess series
• additive static output feedback
• Fréchet spaces
• Hopf algebra
• Wiener–Fliess connection
• convergence

Acknowledgments

The authors want to acknowledge Dr Alexander Schmeding of Nord Universitet for providing his insights and discussions on the topic of Fréchet spaces.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

This work was supported by the National Science Foundation (NSF) [grant number CMMI-1839378].