Intelligent Control of Magnetic Ball Suspension Systems via a Novel Hyperbolic Tangent PID Controller Tuned by the Artificial Lemming Algorithm

Magnetic ball suspension (MBS) systems are widely used as benchmark platforms in control engineering due to their nonlinear dynamics and inherent open-loop instability, which pose substantial challenges for conventional linear control strategies. The objective of this study is to investigate a hyperbolic tangent-based proportional-integral-derivative (tanh-PID) control structure for MBS systems and to assess the suitability of the artificial lemming algorithm (ALA) for tuning its parameters within a simulation-based benchmark framework. The proposed approach embeds smooth nonlinear signal shaping through the hyperbolic tangent function directly into the classical PID structure, while controller parameters are obtained via metaheuristic optimization using ALA. A performance index balancing overshoot suppression and tracking error minimization is adopted, and the controller is evaluated on a linearized MBS model to ensure comparability with existing studies. Simulation results demonstrate that the optimized tanh-PID controller achieves improved transient and steady-state performance, including a rise time of 0.0144 s, settling time of 0.0275 s, overshoot of 2.98%, and a steady-state error of 2.69 × 10^-5, when compared with classical PID, fractional-order PID (FOPID), and real PID with second-order derivative (RPIDD²) controllers under identical conditions. The results indicate that bounded nonlinear preprocessing combined with metaheuristic-based parameter tuning can provide an effective and practical control alternative for unstable nonlinear systems such as magnetic ball suspension systems.