A novel delay-affected two degree of freedom PID controller using physics-inspired optimization for robust control applications

This paper proposes a delay-affected two-degree-of-freedom proportional-integral-derivative (DE-2DOFPID) controller for accurate speed regulation of brushless direct-current (BLDC) motors operating under time-delay conditions. In the proposed architecture, the delay is explicitly embedded into the control law through a weighted delayed-error feedback term. At the same time, two reference-weighting coefficients reshape the proportional and derivative paths. This structure extends the classical 2DOFPID controller by enriching the instantaneous error response with memory-based information and tunable reference coupling, which strengthens both transient behavior and robustness in delay-dominated regimes. The controller gains are tuned using the physics-inspired Rime-inspired metaheuristic (RIME) algorithm, which is well-suited to the highly multimodal, nonconvex search landscape induced by the composite performance index. The objective function combines multiple criteria, including integral absolute error, integral time-weighted absolute error, integral squared error, steady-state error, rise time, settling time and overshoot. For a BLDC speed-control benchmark, comparative optimization studies are carried out against recent metaheuristics, including the educational competition optimizer, Fata morgana algorithm, Slime mould algorithm, Moss growth optimizer, and Ruppell's Fox optimizer, as well as RIME-based tuning of PI-FF, PID, PID with anti-windup, and conventional 2DOFPID controllers. The proposed RIME-tuned DE-2DOFPID reduces the settling time by 8.03% and the lower cost by 6.99% relative to PID-based methods, while maintaining overshoot below 3%. Real-time experiments on a laboratory BLDC drive confirm that the delay-affected 2DOFPID architecture preserves its advantages under measurement noise, parameter uncertainty and external disturbances, demonstrating its practical relevance for robust motion-control applications.