Continuous-time Extremum Seeking for Scalar Systems via a Time-Delay Approach to Averaging

In this paper, we present a constructive approach to continuous-time extremum seeking (ES) by using a time-delay approach to averaging. We consider gradient-based ES of static maps in the case of single-input, 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 dither, 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 the assumption of some known bounds on the extremum value and the Hessian, the time-delay approach provides a quantitative calculation on the lower bound of the frequency and on the resulting ultimate bound. We also give a bound on the neighbourhood of the extremum point starting from which the solution is ultimately bounded. When the Hessian and the extremum value bounds are 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).