On Lie-Bracket Averaging for a Class of Hybrid Dynamical Systems with Applications to Model-Free Control and Optimization

Dynamical systems characterized by oscillatory behaviors and well-defined average vector fields have traditionally been subjects of stability analysis through methodologies rooted in averaging theory. Such tools have also found application in the stability analysis of systems that combine continuous-time dynamics and discrete-time dynamics, referred to as hybrid dynamical systems. However, in contrast to the existing results available in the literature for continuous-time systems, averaging results for hybrid systems have mostly been limited to first-order averaging methods. This limitation prevents their direct application for the analysis and design of systems and algorithms requiring high-order averaging techniques. Among other applications, such techniques are necessary to analyze hybrid Lie-bracket-based extremum seeking algorithms and hybrid vibrational controllers. To address this gap, this paper introduces a novel high-order averaging theorem for the stability analysis of hybrid dynamical systems with high-frequency periodic flow maps. The considered systems allow for the incorporation of set-valued flow maps and jump maps, effectively modeling well-posed differential and difference inclusions. By imposing appropriate regularity conditions on the hybrid system's data, results on $(T,\varepsilon)$-closeness of solutions and semi-global practical asymptotic stability for sets are established. In this way, our findings yield hybrid Lie-bracket averaging tools that extend those found in the literature on ordinary differential equations. These theoretical results are then applied to the study of three distinct applications in the context of model-free control and optimization.