PI-DÆ: An Adaptive PID Controller Utilizing a New Adaptive Exponent (Æ) Algorithm to Solve Derivative Term Issues

This study proposes an enhanced derivative control strategy, named PI-DÆ, designed to overcome key limitations of the derivative (D) term, such as noise amplification, derivative kick (D-k), and tuning difficulties. These issues often arise in high-frequency or rapidly changing systems, in which traditional PID controllers struggle. The proposed solution introduces a novel adaptive exponent algorithm (Æ) that dynamically modulates the D term based on the evolving relationship between system output and setpoint. This yields the PI-DÆ controller, which adapts in real time to changing conditions. The results show significant performance improvements. Simulation results on two systems demonstrate that PI-DÆ achieves a 90% faster response time, a 35% reduction in peak time, and a 100% improvement in settling time compared with conventional PID controllers, all while maintaining a near-zero steady-state error even under external disturbances. Unlike more-complex alternatives such as fuzzy logic, neural networks, or sliding mode control, PI-DÆ retains the simplicity and robustness of PID, avoiding high computational costs or intricate setups. This adaptive exponent strategy offers a practical and scalable enhancement to classical PID, improving performance and robustness without added complexity, and thus provides a promising control solution for real-world applications in which simplicity, adaptability, and reliability are essential.