In this paper, we present a constructive approach to extremum seeking (ES) by using a time-delay approach to averaging. We consider gradient-based ES of static maps in the case of one and two variables, and we study two ES methods: the classical one and a more recent bounded ES method. By transforming the ES dynamics into a time-delay system where the delay is the period of averaging, we derive the practical stability conditions for the resulting time-delay system. The time-delay system stability guarantees the stability of the original ES plant. Under assumption of some known bounds on the extremum point, the extremum value and the Hessian, the time-delay approach provides a quantitative calculation on the lower bound of the frequency and on the upper bound of the resulting ultimate bound. We also give a bound on the neighborhood of the extremum point starting from which the solution is ultimately bounded. When the extremum value is unknown, we provide, for the first time, the asymptotic ultimate bound in terms of the frequency in the case of bounded ES. Moreover, our explicit bound on the seeking error of ES control systems allows to select appropriate tuning parameters (such as dither frequency, magnitude, and control gain). Two numerical examples illustrate the efficiency of our method. Particularly, our quantitative bounds are more efficient for the classical ES than for the bounded one. However, the latter bounds correspond to a more general case with unknown extremum value.
- Extremum seeking