A Note on Nussbaum-type Control and Lie-bracket Approximation

TL;DR

This paper introduces an adaptive control law for unknown scalar linear systems using Lie-bracket approximation, compares it with Nussbaum-type methods, and discusses stability challenges and future research directions.

Contribution

It proposes a novel adaptive control approach based on Lie-bracket methods and analyzes its stability properties compared to existing solutions.

Findings

Global stability of the Lie-bracket system is proven.

The stability of the proposed control law remains an open problem.

The paper highlights the connection to partial stability and Chen-Fliess expansion.

Abstract

In this paper, we propose an adaptive control law for completely unknown scalar linear systems based on Lie-bracket approximation methods. We investigate stability and convergence properties for the resulting Lie-bracket system, compare our proposal with existing Nussbaum-type solutions and demonstrate our results with an example. Even though we prove global stability properties of the Lie-bracket system, the stability properties of the proposed dynamics remain open, making the proposed control law an object of further studies. We elaborate the difficulties of establishing stability results by investigating connections to partial stability as well as studying the corresponding Chen-Fliess expansion.