Hybrid Minimum-Seeking in Synergistic Lyapunov Functions: Robust Global Stabilization under Unknown Control Directions

TL;DR

This paper introduces hybrid feedback laws that achieve robust global stabilization of control-affine systems with unknown control directions and topological obstructions, using Lyapunov functions and Lie-bracket averaging.

Contribution

It develops a novel class of hybrid feedback controllers based on synergistic Lyapunov functions, overcoming controllability issues and topological constraints in stabilization problems.

Findings

Successfully stabilizes systems on various manifolds with unknown control directions.

Demonstrates obstacle avoidance in vehicle kinematic models.

Shows robustness against dynamic uncertainties and topological obstructions.

Abstract

We study the problem of robust global stabilization in control-affine systems, focusing on dynamic uncertainties in the control directions \emph{and} the presence of topological obstructions that prevent the existence of smooth global control Lyapunov functions. Building on a recently developed Lie-bracket averaging result for hybrid dynamic inclusions presented in \cite{abdelgalil2023lie}, we propose a novel class of universal hybrid feedback laws that achieve robust global practical stability by identifying the minimum point of a set of appropriately chosen synergistic Lyapunov functions. As concrete applications of our results, we synthesize different hybrid high-frequency high-amplitude feedback laws for the solution of robust global stabilization problems on various types of manifolds under unknown control directions, as well as controllers for obstacle avoidance problems in vehicles characterized by kinematic models describing both holonomic and non-holonomic models. By leveraging Lie-bracket averaging for hybrid systems, we also show how the proposed hybrid minimum-seeking feedback laws can overcome lack of controllability during persistent (bounded) periods of time. Numerical simulation results are presented to illustrate the main results.