PI-D Æ : 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mi>I</mi></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>Æ</mi></mrow></msup></mrow></semantics></math></inline-formula>, designed to overcome key limitations of the derivative (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi></mrow></semantics></math></inline-formula>) term, such as noise amplification, derivative kick <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>D</mi></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula>, 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi></mrow></semantics></math></inline-formula> term based on the evolving relationship between system output and setpoint. This yields the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mi>I</mi></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>Æ</mi></mrow></msup></mrow></semantics></math></inline-formula> controller, which adapts in real time to changing conditions. The results show significant performance improvements. Simulation results on two systems demonstrate that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mi>I</mi></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>Æ</mi></mrow></msup></mrow></semantics></math></inline-formula> 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, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mi>I</mi></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>Æ</mi></mrow></msup></mrow></semantics></math></inline-formula> 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.

Abstract licence: CC BY

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