arXiv subjects

Mahmoud Abdelgalil

Publications and source records attributed to Mahmoud Abdelgalil.

25 records · Page 2Linked to original sources

arXiv2024

Collective steering in finite time: controllability on $\text{GL}^+(n,\mathbb{R})$

We consider the problem of steering a collection of n particles that obey identical n-dimensional linear dynamics via a common state feedback law towards a rearrangement of their positions, cast as a controllability problem for a dynamical system evolving on the space of matrices with positive determinant. We show that such a task is always feasible and, moreover, that it can be achieved arbitrarily fast. We also show that an optimal feedback control policy to achieve a similar feat, may not exist. Furthermore, we show that there is no universal formula for a linear feedback control law to achieve a rearrangement, optimal or not, that is everywhere continuous with respect to the specifications. We conclude with partial results on the broader question of controllability of dynamics on orientation-preserving diffeomorphisms.

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arXiv2024

Initialization-Free Lie-Bracket Extremum Seeking in $\mathbb{R}^n$

Stability results for extremum seeking control in $\mathbb{R}^n$ 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 \emph{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.

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arXiv2022

Sea urchin sperm exploit extremum seeking control to find the egg

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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arXiv2022

Singularly Perturbed Averaging with Application to Bio-Inspired 3D Source Seeking

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.

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arXiv2022

Recursive Averaging with Application to Bio-Inspired 3D Source Seeking

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.

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arXiv2021

Lie Bracket Approximation-Based Extremum Seeking with Vanishing Input Oscillations

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.

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arXiv2020

A Geometric Approach to Modeling, Simulation and Control

In this work, we utilize discrete geometric mechanics to derive a 2nd-order variational integrator so as to simulate rigid body dynamics. The developed integrator is to simulate the motion of a free rigid body and a quad-rotor. We demonstrate the effectiveness of the simulator and its accuracy in long term integration of mechanical systems without energy damping. Furthermore, this work deals with the geometric nonlinear control problem for rigid bodies where backstepping controller is designed for full tracking of position and orientation. The attitude dynamics and control are defined on \textbf{SO(3)} to avoid singularities associated with Euler angles or ambiguities accompanying quaternion representation. The controller is shown to track large rotation attitude signals close to $180^\circ$ achieving almost globally asymptotic stability for rotations. A Quad-rotor is presented as an example of an under-actuated system with nonlinear model on which we apply the backstepping control law. In addition, an aerodynamic model aiming at deriving the aerodynamic forces and torques acting on rotors is added for realistic simulation purposes and to testify the effectiveness of the derived control method.

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