A Time-Delay Approach to Extremum Seeking with Large Measurement Delays

In this paper, we present a time-delay approach to gradient-based extremum seeking (ES) with known large distinct measurement delays, for N-dimensional (ND) static quadratic maps. We assume that the Hessian has a nominal known part and norm-bounded uncertainty, the extremum point belongs to a known ball, and the extremum value to a known interval. By using the orthogonal transformation, we first transform the original static quadratic map into a new one with the Hessian containing a nominal diagonal part. By applying a time-delay approach to the resulting ES system, we arrive at the neutral type system with a nominal linear time-delay system. We further present this system as a retarded one and employ variation of constants formula for practical stability analysis. To obtain tight bounds we exploit positivity of the fundamental matrix that corresponds to the nominal system with delays. Explicit conditions in terms of simple scalar inequalities depending on tuning parameters and delay bounds are established to guarantee the practical stability of the ES control systems. An example from the literature illustrates the efficiency of the new approach.