A Robust Time-Delay Approach to Extremum Seeking via ISS Analysis of the Averaged System

For N-dimensional (ND) static quadratic map, we present a time-delay approach to gradient-based extremum seeking (ES) both, in the continuous and, for the first time, the discrete domains. As in the recently introduced (for 2D maps in the continuous domain), we transform the system to the time-delay one (neutral type system in the form of Hale in the continuous case). This system is O($\varepsilon$)-perturbation of the averaged linear ODE system, where $\varepsilon$ is a period of averaging. We further explicitly present the neutral system as the linear ODE, where O($\varepsilon$)-terms are considered as disturbances with distributed delays of the length of the small parameter $\varepsilon$. Regional input-to-state stability (ISS) analysis is provided by employing a variation of constants formula that greatly simplifies the previously used analysis via Lyapunov-Krasovskii (L-K) method, 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.