A novel DIO optimized sigmoid-activated 2DOF-PID controller for robust control of magnetic levitation systems.

Among the classical testbeds in control engineering, magnetic levitation systems occupy a prominent place owing to their strongly nonlinear character and intrinsic open-loop instability. Stabilizing a freely suspended ferromagnetic object requires careful regulation of the electromagnetic force, which makes these systems particularly demanding for controller design. To address this challenge, the present work introduces a sigmoid-activated two-degree-of-freedom PID (SA-2DOF-PID) controller, wherein a sigmoid-based nonlinear term is embedded within a standard 2DOF-PID architecture. This integration yields smoother actuation signals and increases overall closed-loop robustness. Controller parameter selection is carried out via the dholes-inspired optimization (DIO) algorithm, a recently proposed nature-inspired method. A multi-objective fitness function incorporating peak overshoot, settling time, and cumulative tracking error guides the tuning procedure. Simulation experiments performed under both nominal plant conditions and deliberate parametric perturbations serve to assess the proposed scheme. Quantitative comparisons against secant optimization algorithm (SOA)-, honey badger algorithm (HBA)-, and particle swarm optimization (PSO)-based controllers reveal that the DIO-tuned SA-2DOF-PID achieves markedly lower overshoot, quicker settling, and higher steady-state precision. Robustness evaluations involving intentional variations in system parameters further confirm that the closed-loop behavior remains well-damped with only marginal deterioration. Additional benchmarking against whale optimization algorithm combined with simulated annealing (WOA-SA)-based proportional-integral-derivative (PID), opposition-based artificial electric field algorithm (ObAEF)-based fractional-order PID (FOPID), and manta ray foraging optimization (MRFO)-based real PID with second-order derivative (RPIDD²) controllers corroborates the practical advantages of the proposed approach. Taken together, the study demonstrates that the SA-2DOF-PID framework tuned by DIO constitutes an effective, structurally lean, and computationally tractable strategy for managing nonlinear and open-loop unstable plants.