A Robust Time-Delay Approach to Continuous-Time Extremum Seeking for Multi-Variable Static Map

In this article, we introduce a time-delay approach to gradient-based extremum seeking (ES) in the continuous domain for n-dimensional (nD) static quadratic maps. As in the recently introduced (for 2D maps in the continuous domain), we transform the system to the time-delay one (neutral type system). This system is
$\mathbf{O}(\varepsilon)$
. ‘perturbation of the averaged linear ODE system. We further explicitly present the neutral system as the linear ODE, where
$\mathrm{O}(\varepsilon)$
-terms are considered as disturbances with distributed delays of the length of the small parameter
$\varepsilon$
. Quantitative (for uncertain map) and qualitative (for unknown map) practical stability analyses are provided by employing a variation of constants formula that greatly simplifies the results compared to the previously used Lyapunov-Krasovskii (L-K) method. The new approach also simplifies the conditions and improves the results. Examples from the literature illustrate the efficiency of the new approach, allowing essentially large uncertainty of the Hessian matrix with bounds on
$\varepsilon$
that are not too small.