Identification of Phase-Locked Loop System from Its Experimental Time Series

Phase-locked loops (PLLs) are now widely used in communication systems and have been a classic system for more than 60 years. Well-known mathematical models of such systems are constructed in a number of approximations, so questions about how they describe the experimental dynamics qualitatively and quantitatively, and how the accuracy of the model description depends on the behavior mode, remain open. One of the most direct approaches to the verification of any model is its reconstruction from the time series obtained in experiment. If it is possible to fit the model to experimental data and the resulting parameter values are close to the expected values (calculated from the first principles), the quantitative correspondence between the model and the physical object is nearly proved. In this paper, for the first time, the equations of the PLL model with a bandpass filter are reconstructed from the experimental signals of the generator in various modes. The reconstruction showed that the model known in the literature generally describes the experimental dynamics in regular and chaotic regimes. The relative error of parameter estimation is between 2% and 50% for different regimes and parameters. The reconstructed nonlinear function of phase is not harmonic and highly asymmetric in contrary to the model one.