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Gradient and Lie Bracket Estimation of Extremum Seeking Systems: A Novel Geometric-based Kalman Filter and Relaxed Time-dependent Stability Condition

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  • Control Theory and Applications
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Abstract

Extremum seeking control (ESC) is an adaptive control technique, introduced nearly a century ago, to drive a dynamic system to the extremum of an objective function that may not be known expression-wise. A rigorous stability analysis of the so-called classical ESC structure, using averaging and singular perturbation theory, has increased research topics and applications involving ESC. Another class of ESCs, control-affine in nature and analyzed using Lie bracket system-based approaches, has emerged, but with some limited theoretical advancements compared to classical ESC. Gradient estimation tools are not well-established for such control-affine ESC structures. Also, stability analysis can be challenging due to complex bounds and conditions. So, in this paper, we introduce a geometric-based extended Kalman filter (GEKF) for gradient and Lie bracket estimation in control-affine ESC systems. We also propose a time-dependent stability condition for control-affine ESC based on the Lie bracket system’s evolution with time. This enables real-time stability tracking. The potential and advantage of our results are demonstrated through numerical simulations of two ESC cases in the literature, including a multi-agent problem.

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Authors and Affiliations

  1. Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati, Cincinnati, Ohio, USA

    Sameer Pokhrel & Sameh A. Eisa

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  1. Sameer Pokhrel
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  2. Sameh A. Eisa
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Correspondence to Sameh A. Eisa.

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Sameer Pokhrel is a research assistant at the Aerospace Engineering and Engineering Mechanics (AEEM) Department, University of Cincinnati (UC), OH-USA. He is a member of the Modeling, Dynamics, and Control Lab (MDCL) at UC. He got his bachelor’s degree in mechanical engineering from Tribhuvan University, Nepal. His research interests lie in optimization, nonlinear dynamics, and control theory.

Sameh A. Eisa is both an applied mathematician and engineering scholar with B.Sc. degree in electrical engineering from Alexandria University, Egypt, and a Ph.D. degree in applied and industrial mathematics from New Mexico Tech, NM-USA. After that, he worked as a postdoctoral researcher and Lecturer at the University of California, Irvine until joining the University of Cincinnati (UC) in Fall 2021 as an assistant professor of aerospace engineering and engineering mechanics. Dr. Eisa’s research is interdisciplinary and broad, but centers around dynamical and control theory (geometric control, averaging, extremum seeking, and stability/sensitivity methods) with a fundamental approach to problems in bio-inspired behavior, biomimicry, unmanned systems, and renewable energies. Dr. Eisa is a 2023 University Research Council (URC) Scholar awardee, which is the most prestigious early-career research award given at UC. He also has other honors, highlights in major scientific news, co-organized events and served as a TC member in SIAM and IEEE conferences, and reviewed for leading journals such as IEEE Transactions on Automatic Control, Automatica and Nonlinear Dynamics, Springer.

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Pokhrel, S., Eisa, S.A. Gradient and Lie Bracket Estimation of Extremum Seeking Systems: A Novel Geometric-based Kalman Filter and Relaxed Time-dependent Stability Condition. Int. J. Control Autom. Syst. 21, 3839–3849 (2023). https://doi.org/10.1007/s12555-023-0224-y

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  • DOI: https://doi.org/10.1007/s12555-023-0224-y

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