Nonlinear 2-DOF PID controller optimized by artificial lemming algorithm for robust engine speed regulation in spark-ignition systems.

Achieving precise and stable engine speed regulation in spark-ignition (SI) systems remains a challenging task because of the inherent nonlinearities, time-varying characteristics, and external disturbances of internal combustion engines (ICEs). Conventional proportional-integral-derivative (PID) controllers often fail to simultaneously ensure fast tracking and robust disturbance rejection under dynamic operating conditions. To address this limitation, a nonlinear two-degree-of-freedom (2-DOF) PID controller has been developed and optimized using the artificial lemming algorithm (ALA) which is a recent bio-inspired metaheuristic that mimics lemming population behaviors to balance exploration and exploitation adaptively through an energy-driven mechanism. The proposed controller was implemented on a detailed mathematical model of the SI engine, encompassing throttle dynamics, manifold pressure variation, combustion torque generation, and crankshaft motion. A multi-term cost function combining normalized overshoot, steady-state error, and stability coefficients was minimized to determine optimal controller gains. Extensive experiments were conducted, including statistical robustness evaluation, transient and steady-state analyses, trajectory tracking, and disturbance-rejection tests. ALA exhibited the lowest mean and standard deviation of the cost function (4.7170 and 0.1429, respectively), confirming its strong convergence stability compared to the starfish optimization algorithm, parrot optimizer, coati optimization algorithm, and dwarf mongoose optimizer. The ALA-optimized controller achieved a rise time of 0.3114 s, a settling time of 2.4313 s, an overshoot of only 0.0027%, and an extremely small steady-state error of 2.62 × 10⁻¹¹%. Furthermore, the controller demonstrated superior trajectory-tracking accuracy and exceptional disturbance-rejection capability, maintaining speed deviations below 0.5% under abrupt load torque perturbations. The results confirm that the ALA-based nonlinear 2-DOF PID controller provides a robust and energy-efficient solution for nonlinear engine speed regulation, outperforming recent metaheuristic-based approaches in both accuracy and reliability. Owing to its adaptive and scalable design, the proposed control framework is well-suited for integration into real-time embedded engine control units, hybrid powertrains, and other nonlinear dynamic systems requiring high-precision regulation under uncertainty.