Metaheuristic PID control of the forced Van der Pol oscillator across nonlinear and chaotic regimes

This paper investigates metaheuristic optimization for tuning classical PID controllers to suppress nonlinear and chaotic oscillations in the periodically forced Van der Pol oscillator. Particle Swarm Optimization (PSO), Grey Wolf Optimizer (GWO), and a hybrid PSO–GWO strategy are employed to automatically determine controller gains using a composite time-domain objective that balances transient response quality and steady-state vibration suppression. Six representative dynamical regimes, ranging from weak periodic excitation to fully developed chaos, are examined by varying the system nonlinearity and forcing parameters. Across all regimes, the optimized PID controllers achieve stable closed-loop behavior and effective suppression of self-excited oscillations, even under strong external forcing. The hybrid PSO–GWO strategy yields more consistent transient behavior in strongly nonlinear regimes, while standalone PSO and GWO remain competitive at lower computational cost. The results demonstrate that globally optimized classical PID controllers provide a practical, transparent, and robust solution for vibration reduction in nonlinear oscillatory systems, offering a viable alternative to more complex adaptive and chaos-control techniques. The proposed approach is directly applicable to vibration suppression in mechanically and electromechanically excited systems subject to self-excitation and periodic forcing.