Many control problems must be solved without an accurate model: the objective is available only through measurements, and the system can only be probed, not predicted. We develop feedback laws that optimize or stabilize dynamical systems in this setting. A central tool is extremum seeking, in which small periodic or stochastic perturbations reveal gradient information online; we analyze such schemes through Lie-bracket averaging and singular perturbation theory, and extend them to hybrid, event-triggered, and resetting architectures that provide robust global guarantees, including under unknown control directions.
This thrust also covers feedback optimization of stochastic plants, zeroth-order optimization on manifolds, and the stability of learning dynamics driven by non-conservative vector fields, such as those arising in games. Applications range from bio-inspired source seeking, motivated by how sperm cells locate an egg, to the real-time optimization of engineered systems.
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.
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Persistently resetting integrators for gradient learning: A framework for model-free feedback optimization
M. Abdelgalil and J. I. Poveda
Automatica, 2026
We study a novel class of algorithms for solving model-free feedback optimization problems based on persistent resetting integrators for gradient learning (PRINGL), which are integrators that are reset at the same frequency at which the plant is dithered. It is shown that PRINGLs serve as core mechanisms for real-time gradient estimation when only cost function measurements are available. Unlike existing approaches such as averaging, PRINGLs can produce global real-time gradient estimates with uniformly bounded perturbations of arbitrarily small magnitude. In this sense, PRINGLs serve as robust hybrid “oracles” suitable for interconnection with discrete-time optimization algorithms in closed-loop operation. It is proven that PRINGLs yield global stability properties for a broad class of cost functions, surpassing the local or semi-global guarantees offered by classical approaches. The proposed framework naturally bridges physical systems, modeled as continuous-time plants where continuous exploration is essential, with digital algorithms, represented as discrete-time optimization methods. This integration of physical and digital components allows for the application of a broad range of well-established optimization techniques, each offering unique performance benefits suited to different applications—such as accelerated convergence through momentum, safety assurances via projections, etc. Closed-loop stability guarantees are established using unifying tools from hybrid systems theory. The main results are illustrated using different numerical examples.
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A separation principle for stochastic feedback optimization
M. Tang, M. Abdelgalil, and J. I. Poveda
IEEE Control Systems Letters, 2026
We study the problem of online feedback optimization for stochastic linear plants and quadratic performance costs in the presence of an unknown disturbance and persistent white noise. For this problem, we introduce a novel two-timescale control architecture consisting of a static feedback term and a dynamically tuned feedforward term. Through singular perturbation analysis, we show that the steady-state mean of the system converges to the feasible optimizer of the problem, independently of the feedback term, and the steady-state covariance converges to the standard LQR baseline up to a small bias of order 𝒪(ε), where ε is the learning rate of the optimization dynamics. In this sense, our results highlight a novel separation principle that justifies a sequential design approach for stochastic feedback optimization: the feedback term may be designed offline via LQR techniques to minimize the covariance penalty, e.g., using a Riccati equation, while the dynamic feedforward leverages a gradient-based architecture that rejects the unknown disturbance from the steady-state mean, all while incurring only an 𝒪(ε)-optimality gap. Numerical simulations are provided to validate the theory.
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 the instability of Nesterov’s ODE under non-conservative vector fields
D. E. Ochoa, M. Abdelgalil, and J. I. Poveda
IEEE Control Systems Letters, 2025
We study the instability properties of Nesterov’s ODE in non-conservative settings, where the driving term is not necessarily the gradient of a potential function. While convergence properties under Nesterov’s ODE are well-characterized for optimization settings with gradient-based driving terms, we show that the presence of arbitrarily small non-conservative terms can lead to instability, a phenomenon previously observed empirically via numerical studies in optimization and game-theoretic problems. Our instability analysis combines multi-time scale techniques, such as averaging via variations-of-constants formula, and Floquet Theory, focusing on systems where the vector field is linear and its Helmholtz decomposition reveals a non-vanishing non-conservative component. To resolve the instability issue, the dynamics under non-vanishing non-conservative components, we study a regularization mechanism based on restarting. The resulting system is a hybrid dynamical system that mirrors Nesterov’s ODE during intervals of flow, and implements resets of the momentum state through discrete periodic jumps. For this hybrid system, we establish novel explicit bounds on the resetting period that ensure the decrease of a suitable Lyapunov function, guaranteeing not only stability but also "accelerated" convergence rates under suitable smoothness and strong monotonicity properties on the driving term. Numerical simulations support our theoretical results.
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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.
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Stochastic Lie bracket approximations for zeroth-order optimization on manifolds
M. Abdelgalil and J. I. Poveda
In Dynamics at the Frontiers of Optimization, Sampling, and Games Workshop, Conference on Neural Information Processing Systems (NeurIPS), 2025
2024
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Initialization-free Lie-bracket extremum seeking
M. Abdelgalil and J. I. Poveda
Systems & Control Letters, 2024
Stability results for extremum seeking control in ℝⁿ have predominantly been restricted to local or, at best, semi-global practical stability. Extending semi-global stability results of extremum-seeking systems to unbounded sets of initial conditions often demands a stringent global Lipschitz condition on the cost function, which is rarely satisfied by practical applications. In this paper, we address this challenge by leveraging tools from higher-order averaging theory. In particular, we establish a novel second-order averaging result with global (practical) stability implications. By leveraging this result, we characterize sufficient conditions on cost functions under which uniform global practical asymptotic stability can be established for a class of extremum-seeking systems acting on static maps. Our sufficient conditions include the case when the gradient of the cost function, rather than the cost function itself, satisfies a global Lipschitz condition, which covers quadratic cost functions. Our results are also applicable to vector fields that are not necessarily Lipschitz continuous at the origin, opening the door to non-smooth Lie-bracket ES dynamics. We illustrate all our results via different analytical and/or numerical examples.
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.
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Singularly perturbed averaging with application to bio-inspired 3D source seeking
M. Abdelgalil, A. Eldesoukey, and H. Taha
In Proc. American Control Conference, 2023
We analyze a class of singularly perturbed high-amplitude, high-frequency oscillatory systems that arises in extremum seeking applications. We provide explicit formulas for averaging and establish the convergence of the trajectories of this class of systems to the trajectories of a suitably averaged reduced order system by combining the higher order averaging theorem with singular perturbation techniques. Finally, we propose a novel bio-inspired 3D source seeking algorithm and establish its singular practical stability.
2022
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Sea urchin sperm exploit extremum seeking control to find the egg
M. Abdelgalil, Y. Aboelkassem, and H. Taha
Physical Review E, 2022
Sperm cells perform extremely demanding tasks with minimal capabilities. The cells must quickly navigate in a noisy environment to find an egg within a short time window for successful fertilization without any global positioning information. Many research efforts have been dedicated to deriving mathematical principles that explain their superb navigation strategy. Here we show that the navigation strategy of sea urchin sperm, also known as helical klinotaxis, is a natural implementation of a well-established adaptive control paradigm known as extremum seeking. This bridge between control theory and the biology of taxis in microorganisms is expected to deepen our understanding of the process. For example, the formulation leads to a coarse-grained model of the signaling pathway that offers new insights on the peculiar switching-like behavior between high and low gain steering modes observed in sea urchin sperm. Moreover, it may guide engineers in developing bio-inspired miniaturized robots with minimal sensors.
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Recursive averaging with application to bio-inspired 3D source seeking
M. Abdelgalil and H. Taha
IEEE Control Systems Letters, 2022
Also presented at the 2022 IEEE Conference on Decision and Control.
We analyze a class of high-amplitude, high-frequency oscillatory systems in which periodicity occurs on two distinct time scales and establish the convergence of its trajectories to a suitably averaged system by recursively applying the averaging theorem. Moreover, we introduce a novel bio-inspired 3D source seeking algorithm for rigid bodies with a collocated sensor and prove its practical stability under typical assumptions on the source signal strength field by combining our averaging results with singular perturbation.
2021
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Lie bracket approximation-based extremum seeking with vanishing input oscillations
M. Abdelgalil and H. Taha
Automatica, 2021
In recent years, an approach to extremum seeking control made it possible to design control vector fields that lead to asymptotic stability of the minimum point provided that the minimum value of the function is known a priori. In this work, we aim to relax that assumption. We propose an extremum-seeking control law that converges to the minimum point with vanishing control oscillations, without access to the minimum value of the cost function. We provide a numerical example to support our results.