On Event-Triggered Extremum Seeking via Standard and Lie-Bracket Averaging: A Hybrid Dynamical Systems Approach

We introduce and analyze the stability of a class of event-triggered extremum-seeking algorithms designed to solve resource-aware, model-free, optimization problems. Leveraging recent advances in Lie-Bracket Averaging for hybrid systems, we demonstrate that the proposed controllers can be formulated as well-posed multi-time-scale hybrid systems that satisfy key regularity, stability, and robustness properties. In extremum-seeking systems, exploration and exploitation are inherently coupled. This coupling necessitates careful consideration in the design of the event-triggered controller. To address this challenge, we incorporate a low-pass filter into the algorithm and carefully design the flow and jump sets of the resulting hybrid system. The resulting controller renders the optimal point semi-globally practically asymptotically stable with solutions exhibiting a uniform semi-global dwell time. We also demonstrate how the proposed event-triggered scheme can be modified to allow analysis using traditional averaging tools for hybrid systems by introducing two independent tunable parameters in the controller. Numerical simulations are presented to validate and illustrate the theoretical results.