Inferring Global Exponential Stability Properties using Lie-bracket Approximations

TL;DR

This paper presents a new method to determine global exponential stability of input-affine systems using Lie-bracket approximations, with applications in adaptive control and numerical validation.

Contribution

It introduces a novel approach to infer global exponential stability from Lie-bracket systems, enhancing stability analysis in control theory.

Findings

Established a theoretical link between Lie-bracket systems and input-affine system stability.

Applied the method to adapt dither frequencies in adaptive control.

Provided numerical simulations supporting the theoretical results.

Abstract

In the present paper, a novel result for inferring uniform global, not semi-global, exponential stability in the sense of Lyapunov with respect to input-affine systems from global uniform exponential stability properties with respect to their associated Lie-bracket systems is shown. The result is applied to adapt dither frequencies to find a sufficiently high gain in adaptive control of linear unknown systems, and a simple numerical example is simulated to support the theoretical findings.