Sampled-data Bounded Extremum Seeking via Time-delay Approach to Averaging

In this paper, we introduce a novel time-delay approach for sampled-data bounded extremum seeking (BES) targeting single-variable static quadratic maps. This approach provides a streamlined and less restrictive analysis of practical stability. Assuming that the Hessian <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$H$</tex> is within a known range, we employ a time-delay approach to averaging to convert the original sampled-data BES system into a time-delay system. This transformed system represents a perturbation of a neutral type system with fixed coefficients. We further describe this system using a linear ordinary differential equation (ODE) framework and utilize the variation of constants formula for analyzing its practical stability. We establish conditions, presented as simple scalar inequalities that rely on tuning parameters, to guarantee the practical stability of the sampled-data BES control system. This new method simplifies the conditions for stability and improves results compared to the traditional Lyapunov- Krasovskii (L-K) technique. The efficiency of the new approach is demonstrated by an example.