Reconstruction of integrated equations of periodically driven phase-locked loop system from scalar time series

Abstract

Purpose of this work is to develop a reconstruction technique for the equations of a phase-locked loop system under periodic external driving from a scalar time series of one variable. Methods. Instead of the original model, we reconstructed a time-integrated model. So, since it is not necessary to evaluate the second derivative of the observable numerically, the method sensitivity to observation noise has significantly decreased. The external periodic driving is approximated with a trigonometric polynomial of time, the antiderivative of which is also a trigonometric polynomial. The assumption about continuity of an unknown nonlinear function is used to construct the target function for optimization. Results. It is shown that the proposed approach gives a significant advantage over the previously developed approach to the reconstruction of non-integrated equations, allowing to achieve acceptable parameter estimates with measurement noise being about 10% of the RMS deviation of the signal even in the presence of external driving. Conclusion. The described approach significantly extends the possibilities of reconstruction of phase-locked loop systems, allowing systems to be reconstructed under arbitrary periodic driving and at the same time significantly increasing noise resistance.