On manifolds and in obstructed environments, topological obstructions rule out globally stabilizing continuous feedback. We design hybrid and synergistic feedback laws that overcome these obstructions, providing global, robust guarantees for stabilization and navigation. Current work addresses dynamic environments, in which the obstacles themselves move according to their own dynamics, and the interplay between these geometric methods and the transport-based ensemble control developed in our other thrusts.
References
2026
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Hybrid minimum-seeking in synergistic Lyapunov functions: Robust global stabilization under unknown control directions
M. Abdelgalil and J. I. Poveda
Automatica, 2026
We study the problem of robust global stabilization in control-affine systems, focusing on dynamic uncertainties in the control directions 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, 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.
2025
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On Lie-bracket averaging for hybrid dynamical systems with applications to model-free control and optimization
M. Abdelgalil and J. I. Poveda
IEEE Transactions on Automatic Control, 2025
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, ε)-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.
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On event-triggered extremum seeking via standard and Lie-bracket averaging: A hybrid dynamical systems approach
M. Abdelgalil and J. I. Poveda
In Proc. IEEE Conference on Decision and Control, 2025
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.
2023
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Multi-time scale control and optimization via averaging and singular perturbation theory: From ODEs to hybrid dynamical systems
M. Abdelgalil, D. E. Ochoa, and J. I. Poveda
Annual Reviews in Control, 2023
Multi-time scale techniques based on singular perturbations and averaging theory are among the most powerful tools developed for the synthesis and analysis of feedback control algorithms. This paper introduces some of the recent advances in singular perturbation theory and averaging theory for continuous-time dynamical systems modeled as ordinary differential equations (ODEs), as well as for hybrid dynamical systems that combine continuous-time dynamics and discrete-time dynamics. Novel multi-time scale analytical tools based on higher-order averaging and singular perturbation theory are also discussed and illustrated via different examples. In the context of hybrid dynamical systems, a class of sufficient Lyapunov-based conditions for global stability results are also presented. The analytical tools are illustrated through various new architectures and algorithms within the context of adaptive and extremum-seeking systems. These tools are suitable for the study of model-free optimization and stabilization problems that require the synergistic use of continuous-time and discrete-time feedback. The paper aims to acquaint the reader with a range of modern tools for studying multi-time scale phenomena in optimization and control systems, providing some guidelines for future research in this field.
2021
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On the motion planning and feedback stabilization of nonlinear systems with drift
M. Abdelgalil, A. Hassan, and H. Taha
In Proc. AIAA SciTech Forum, 2021
The motion planning problem (MPP) is the problem of finding a control input that steers a control system between two arbitrarily given points. Several algorithms have been developed for the motion planning of nonlinear driftless systems over the 1990’s using differential geometric control tools. However, little is known about MPP of nonlinear systems with drift. With the contemporary research flurry on motion planning in the robotics community, mostly using optimal control and machine learning techniques, we aim in this paper to demonstrate the capabilities of differential geometric control theory in solving the MPP for nonlinear systems with drift and also in designing feedback stabilization using Lie brackets for systems that fail Brockett’s necessary condition for stabilization using smooth feedback.