# On a class of generating vector fields for the extremum seeking problem: Lie bracket approximation and stability properties

## TL;DR

This paper introduces a broad class of control functions for extremum seeking that unify existing methods, enable new designs, and provide a constructive way to ensure stability through Lie bracket approximations.

## Contribution

It generalizes and unifies extremum seeking strategies based on Lie brackets and offers a new approach for analyzing their stability and asymptotic behavior.

## Key findings

- Unified framework for extremum seeking controls
- Constructive procedure for frequency selection ensuring stability
- Proven Lyapunov stability for the proposed class

## Abstract

In this paper, we describe a broad class of control functions for extremum seeking problems. We show that it unifies and generalizes existing extremum seeking strategies which are based on Lie bracket approximations, and allows to design new controls with favorable properties in extremum seeking and vibrational stabilization tasks. The second result of this paper is a novel approach for studying the asymptotic behavior of extremum seeking systems. It provides a constructive procedure for defining frequencies of control functions to ensure the practical asymptotic and exponential stability. In contrast to many known results, we also prove asymptotic and exponential stability in the sense of Lyapunov for the proposed class of extremum seeking systems under appropriate assumptions on the vector fields.