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Fig. 8 . Complementary Sensitivity function analysis of system ( a ) Level of Pressure and ( b ) Level of stock.
PI’s 4.02 s, showcasing its ability to stabilize rapidly after disturbances. Overshoot was effectively minimized with JSO-FOPID, ensuring smoother system responses compared to PID and PI controllers, which exhibited overshoots of 0.8 s and 0.46 s, respectively. These quantitative metrics highlight the JSO-FOPID controller’s robustness and adaptability, especially in nonlinear and multivariable systems like the paper machine headbox. The JSO algorithm’s dynamic optimization capability contributed to these outcomes by fine-tuning the FOPID parameters to achieve optimal performance. For future research, exploring other optimization techniques such as hybrid algorithms or advanced metaheuristic methods like Grey Wolf Optimization or Whale Optimization
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