Regular Papers

International Journal of Control, Automation, and Systems 2023; 21(12): 3839-3849

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

© The International Journal of Control, Automation, and Systems

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

Sameer Pokhrel and Sameh A. Eisa*

University of Cincinnati

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.

Keywords Chen-fliess, control-affine, extremum seeking, gradient estimation, Kalman filter, Lie brackets.

Article

Regular Papers

International Journal of Control, Automation, and Systems 2023; 21(12): 3839-3849

Published online December 1, 2023 https://doi.org/10.1007/s12555-023-0224-y

Copyright © The International Journal of Control, Automation, and Systems.

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

Sameer Pokhrel and Sameh A. Eisa*

University of Cincinnati

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.

Keywords: Chen-fliess, control-affine, extremum seeking, gradient estimation, Kalman filter, Lie brackets.

IJCAS
February 2024

Vol. 22, No. 2, pp. 347~729

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