Enhanced Stabilization of a Nonlinear Magnetic Ball Suspension System Using BPBO-Tuned 2-DOF PID Control

Magnetic ball suspension systems represent a challenging nonlinear benchmark due to their open-loop instability and strong electromechanical coupling. This study proposes a birds of prey-based optimization (BPBO) framework for tuning a two-degree-of-freedom (2-DOF) PID controller applied to a nonlinear magnetic levitation system. The application of BPBO to this benchmark problem remains limited in the existing literature. The proposed approach combines the structural flexibility of the 2-DOF PID architecture with the search capability of BPBO to improve transient response characteristics and robustness.The tuning performance is evaluated in comparison with particle swarm optimization (PSO) and the recently proposed hiking optimization algorithm (HOA) under identical computational conditions. The results indicate that the BPBO-based tuning approach provides improved convergence behavior and consistent performance across multiple independent runs. Time-domain analysis shows that the proposed controller achieves favorable transient characteristics with reduced overshoot and settling time, along with low steady-state error and integral absolute error (IAE). Robustness evaluations under resistance and inductance variations further demonstrate the stability-preserving capability of the proposed method. In addition, comparisons with SCA-based PID and LsOBL-HGS-based FOPID controllers highlight the effectiveness of the proposed approach in enhancing control performance.