Dynamics of a Nonlinear Energy Harvesting System in Time-Delayed Feedback Control under Stochastic Excitations

A time-delayed feedback control is applied to a nonlinear piezoelectric energy harvesting system excited by additive and multiplicative Gaussian white noises to improve its energy harvesting performance. An equivalent decoupling system can be obtained by using a variable transformation. Based on the standard stochastic averaging method, the Fokker–Plank–Kolmogorov equation and the stationary probability density functions of the amplitude, displacement, and velocity of the harvester are obtained, respectively. In addition, the approximate expressions of mean square electric voltage and the mean extracted output power are derived. Finally, the paper explores the influences of parameters on the mean square electric voltage. The results show that noise intensity, time delay, feedback strength, time constant ratio, and coupling coefficients have great influences on the mean square electric voltage. The accuracy of the theoretical method is verified by the Monte Carlo simulation.