Method for Determining the Stability Regions of Stationary Oscillations of a Nonlinear MEMS Resonator under the Action of Phase-Locked-Loop and Automatic Gain Control Systems

The work is devoted to the study of the dynamics of a micromechanical resonator under the action of systems of phase-locked loop of the generator frequency and stabilization of the oscillation amplitude. Particular attention is directed to the study of the nonlinearity factor of the elastic restoring force of the resonator. It was found that the choice of control system parameters based on the analysis of the stability of the operating resonance mode for a mechanically linear model, in the general case, does not provide the required resonant phase adjustment and stabilization of the oscillation amplitude. Stable multifrequency modes of oscillations are found, an analytical study of the mechanisms of their appearance and evolution is carried out when the key parameters of the system change. The real regions of stable operation of the control system are determined (which do not coincide, as was found, with the regions of stability of the operating resonance mode, due to the presence of hidden attractors in the phase space of the system). A methodology for identifying such areas of stable operation has been developed. A significant complication of the structure of possible motions in the system with an increase in the Q-factor of the resonator is noted.