Operating frequency and stable limit cycle regulation in active free-piston Stirling oscillators

• A high-fidelity dynamic model elucidates the complex thermomechanical coupling between displacer motion, power piston dynamics, and working fluid thermodynamics. • A rigorous integral backstepping control architecture is formulated for frequency regulation of active free-piston Stirling oscillators. • The proposed control methodology ensures precise frequency modulation of the displacer piston while inducing stable autonomous limit cycles in power piston oscillations. • Theoretical stability guarantees are established through Lyapunov analysis, demonstrating global asymptotic convergence of tracking errors. • Extensive numerical investigations validate the control efficacy, revealing robust frequency-dependent limit cycles and synchronized thermomechanical behavior. This paper introduces a novel strategy for regulating the operating frequency and stable limit cycle behavior in an active free-piston Stirling oscillator (AFPSO) utilizing an integral backstepping-based control scheme. The research develops a comprehensive dynamic model for active free-piston Stirling oscillators that effectively captures the complex interactions among the displacer piston, driven by a DC motor through a slider-crank mechanism, the power piston, and the working fluid pressure dynamics. The proposed control law is designed not only to precisely regulate the displacer piston’s reciprocating frequency, but also to ensure robustness against external disturbances or system parameter uncertainties, enabling reliable control of the AFPSO’s operating frequency and enforcing stable oscillatory behavior (limit cycles) in the power piston. The asymptotic stability of the tracking error for operating frequency is rigorously proven using the direct Lyapunov stability theorem. Simulations assuming system parameters akin to a B10-B free-piston Stirling oscillator confirm the effectiveness of this control strategy in regulating operating frequency and achieving stable limit cycle behavior in power piston motion. Frequency response analysis reveals distinct amplitude characteristics across different operating frequencies, with a resonant peak at 81.5 rad/s, which closely aligns with the analytically estimated natural frequency of the power piston. This finding provides valuable insight for optimizing system performance through frequency tuning.