Lie bracket approximations of RMSprop for Extremum Seeking Control

Abstract

This paper presents novel extremum seeking algorithms inspired by RMSProp dynamics and based on Lie bracket approximation techniques. To inherit the behavior of the classical continuous-time RMSProp algorithm, we introduce an extremum seeking scheme that exploits second-order Lie brackets, allowing the excitation of squared gradient terms in the resulting Lie bracket system. Moreover, we propose two modified RMSProp-type algorithms that can be implemented using only first-order Lie bracket approximations, thereby reducing the complexity of the extremum seeking control design. The three resulting extremum seeking algorithms are compared through numerical simulations.

arXiv commentExtended version of the paper accepted for presentation at the 65th IEEE Conference on Decision and Control (CDC 2026), with the complete proof of Theorem 2 provided in the Appendix