Piezoelectric Harvester Proportional–Derivative (PHPD) Control for Nonlinear Dynamics Reduction in Underactuated Hybrid Systems

This study investigates the nonlinear dynamics and control of an underactuated hybrid system consisting of a Duffing oscillator, a pendulum, and a piezoelectric energy harvester. A novel Piezoelectric Harvester Proportional–Derivative (PHPD) control scheme is introduced, which integrates the harvester’s electrical output directly into the feedback loop to achieve simultaneous vibration suppression and energy utilization. The nonlinear governing equations are derived and analyzed using the Multiple-Scale Perturbation Technique (MSPT) to obtain reduced-order dynamics. Bifurcation analysis is employed to identify stability boundaries and critical parameter transitions, while numerical simulations based on the fourth-order Runge–Kutta method validate the analytical predictions. Furthermore, frequency response curves (FRCs) and an ideal system are evaluated under multiple controller and system parameter configurations. Bifurcation classification is performed on the analyzed figure to detect various bifurcations within the system, along with the computation of the Largest Lyapunov Exponent (LLE). The results demonstrate that PHPD control significantly reduces vibration amplitude and accelerates convergence, offering a new pathway for energy-efficient, high-performance control in nonlinear electromechanical systems.