Extremum Seeking for the First Derivative of Nonlinear Maps with Constant Delays via a Time-Delay Approach

• Dr. Jalen T. Rovimar

  Department of Control Systems and Automation, Technical University of Munich, Germany

  Author

This paper introduces a novel extremum seeking (ES) scheme specifically designed for identifying the first derivative of an unknown nonlinear map, particularly in systems subject to constant transmission delays. Traditional approaches often rely on predictor-based methods to compensate for delays, which can introduce significant complexity. In contrast, our research focuses on enhancing the delay-robustness of the ES system by employing a recently developed time-delay approach. This methodology transforms the original ES system into a nonlinear retarded-type plant, effectively incorporating disturbances into the model. A critical aspect of our work involves the derivation of stability conditions, which are presented in the form of linear matrix inequalities (LMIs). This provides a rigorous analytical framework for ensuring system stability. Furthermore, the paper addresses scenarios where the precise bounds of the nonlinear map are not explicitly known, offering a robust practical stability proof for such "black box" systems. More significantly, when prior knowledge regarding the nonlinear map is accessible (i.e., a "grey box" scenario), our time-delay approach facilitates quantitative calculations for crucial design parameters. These parameters include the maximum allowable delay, a quantifiable upper bound for the dither period, and a precise estimation of the ultimate seeking error. The practical utility and effectiveness of the proposed method are comprehensively validated through several numerical examples, demonstrating its applicability in real-world control systems. This approach offers a significant advancement in ES control by providing both qualitative and quantitative stability analyses, particularly for systems with inherent time delays.

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