Extremum seeking control for systems with time-varying extremum

Most extremum seeking control methods consider time-independent input-to-output mappings. Correspondingly, these methods are suitable for systems where the variations of the mapping are much slower than the adaptation dynamics. The few existing results for the case where the mapping is time-dependent require the knowledge of the structure of the time-dependency. In this paper, an extremum seeking algorithm is proposed for mappings where the optimal point is time-varying. The time variations are considered to be of a general form. The proposed extremum seeking control method basically consists of two separate components, i.e., (1) real-time gradient estimation and (2) gradient search for tracking a time-varying optimal point. The gradient is estimated by adding a sinusoidal perturbation to the searching signal and then demodulating the measured output. The estimated gradient signal is then fed into a novel gradient search method, which is proven to move the searching signal toward the optimal value.