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

TL;DR

The paper tackles the challenge of analyzing dynamical systems that combine fast oscillations, discrete events, and set‑valued dynamics by extending averaging theory to a second‑order framework for hybrid dynamical systems (HDS). It develops a second‑order average mapping $\bar{f}$ and proves a $(T,\varepsilon)$‑closeness: every solution of the original HDS is close to a solution of the second‑order averaged HDS $\mathcal{H}_2^{\text{ave}}$ on compact time intervals. Under suitable stability assumptions on $\mathcal{H}_2^{\text{ave}}$, the original HDS inherits semi/global practical asymptotic stability (SGAS/GPAS) as $\varepsilon\to0^+$, enabling robust and scalable design of hybrid Lie‑bracket based controllers. The framework is demonstrated through three model‑free control and optimization applications: switching synchronization of oscillators, Lie‑bracket extremum seeking under intermittence and spoofing, and switched global extremum seeking on smooth compact manifolds, illustrating practical resilience to faults and adversarial conditions. These results broaden the toolbox for high‑frequency, high‑dimensional hybrid control and optimization, with potential impact on adaptive robotics, networked control, and autonomous systems.

Abstract

The stability of dynamical systems with oscillatory behaviors and well-defined average vector fields has traditionally been studied using averaging theory. These tools have also been applied to hybrid dynamical systems, which combine continuous and discrete dynamics. However, most averaging results for hybrid systems are limited to first-order methods, hindering their use in systems and algorithms that require high-order averaging techniques, such as hybrid Lie-bracket-based extremum seeking algorithms and hybrid vibrational controllers. To address this limitation, we introduce a novel high-order averaging theorem for analyzing the stability of hybrid dynamical systems with high-frequency periodic flow maps. These systems incorporate set-valued flow maps and jump maps, effectively modeling well-posed differential and difference inclusions. By imposing appropriate regularity conditions, we establish results on $(T,\varepsilon)$-closeness of solutions and semi-global practical asymptotic stability for sets. These theoretical results are then applied to the study of three distinct applications in the context of hybrid model-free control and optimization via Lie-bracket averaging.

Figures (7)

• Figure 1: Automaton-like representation of switching seeking dynamics in nonholonomic vehicles with multiple operating modes: under spoofing ($z_1=1$), under no measurement ($z_1=2$), and under nominal operation ($z_1=3$).
• Figure 2: Trajectories of the vehicle starting from two initial conditions $(x_1(0),x_2(0))=(-4,4)$ (blue) and $(x_1(0),x_2(0))=(-4,-4)$ (red), for $\varepsilon = 1/\sqrt{10\pi}$, $J(x_p)=0.5 x_1^2+0.5x_2^2$, and for two different switching signals $z_1(t)$, shown in the left plot. The black dots indicate the initial conditions.
• Figure 3: Network topologies in the second case of Example 1.
• Figure 4: Simulation results of the first scenario in Example 1. The figure on the right is a flat embedding of the torus $\mathbb{T}^2\subset\mathbb{R}^4$ into the plane $\mathbb{R}^2$. Synchronization is achieved on the submanifold $\mathcal{S}$ corresponding to the diagonal line.
• Figure 5: Simulation results of the second scenario considered in Example 1.

Theorems & Definitions (26)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Remark 1
• Remark 2
• Definition 6
• Remark 3
• Lemma 1