Amplitude Resonance Tracking Using PLL and ESC for Nonlinear Systems

{This work presents a hybrid control strategy for resonance tracking in nonlinear structural systems, combining classical phase-locked loop (PLL) control with an outer-loop extremum seeking control (ESC) scheme. Traditional PLL-based methods rely on enforcing phase quadrature between excitation and response—a condition that is not always satisfied for closely-spaced mode and high damping situations — leading to inaccurate backbone identification. To address this limitation, the proposed approach adaptively adjusts the PLL’s reference phase in real time to directly maximize the response amplitude. The ESC-enhanced PLL is embedded in a continuation framework that incrementally increases the excitation level while tracking amplitude resonance at each step. The method is validated on a two-degree-of-freedom system and compared against reference solutions obtained via harmonic balance. Results show excellent agreement in both amplitude and phase, while highlighting the discrepancy between phase quadrature and true amplitude resonance. The approach enables accurate backbone extraction and shows strong potential for experimental implementation in nonlinear system identification and fatigue testing applications.