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Fig. 6 . Simulation analysis of the proposed system.
the optimization process. For JSO, the curves typically showcase faster convergence rates and a more consistent approach to the global optimum, reflecting its adaptability and robustness. By comparing these curves with other algorithms, it becomes evident how JSO effectively balances search strategies, maintaining superior performance in both stable and transient states. Benchmark functions such as Rastrigin, Rosenbrock, or Sphere function can be used to evaluate the performance across different optimization scenarios. Including these curves within the study not only strengthens the empirical validation of JSO but also underscores its advantages in terms of accuracy, reliability, and efficiency, making it a compelling choice for complex optimization problems like tuning FOPID controllers. Test 1: analysis of sensitivity function In this section, analysis of the sensitivity function connection between the reaction and load disturbance of a framework has been determined. It shows the minimized effects of load disturbance using sensitivity function. The Eq. (24) depicts that function: P ( lξ )= 1 1+ M ( kξ ) D ( lξ ) (24) Sensitivity function assumes a significant part in considering the strength of any control framework which is summed up underneath: • Find the transfer capability of the sensing function depicted by Eq. (24). • Outline the graph (size) of the system transfer function of P (p). • Compute the most peak value of magnitude plot of P (p). • The most extreme estimation of P (p) should put between 1.2 to 3.0. In Fig. 7, the pressure level and stock level in the paper machine headbox system are analyzed. These figures make it clear that the value of the proposed JSO-based sensitivity function is very low compared to the sensitivity function of other methods such as EHO, MFO and ALO, respectively. The lower values of the JSO recommended in Table 1 stipulate higher strength. Therefore, the JSO based FOPID controller is more optimal compared to other optimal methods such as EHO, MFO and ALO. The performance metrics of the proposed FOPID controller compared to traditional PI and PID controllers demonstrate its superior efficiency in dynamic control systems. The rise time, overshoot, and settling time are key parameters used to evaluate the responsiveness and stability of the control system. Based on the data from the study, the FOPID controller achieves a rise time of 0.011 s, significantly outperforming the PID controller’s 0.03 s and the PI controller’s 0.055 s. This rapid rise time indicates the FOPID controller’s ability to respond quickly to changes in the system. The overshoot for FOPID is effectively minimized, as indicated by its superior control precision, whereas PID exhibits 0.8 s and PI 0.46 s, reflecting the improved damping capability of the FOPID controller. For settling time, which measures how quickly the system stabilizes after a disturbance, the FOPID controller demonstrates a remarkable value of 0.5 s, compared to PID’s 1.01 s and PI’s 4.02 s. These metrics highlight the FOPID controller’s ability to maintain stability and minimize error efficiently. The numerical comparisons clearly show that the FOPID controller delivers faster response and greater stability under dynamic operating conditions. These improvements are attributed to the additional tuning parameters introduced by fractional calculus, which allow for more precise adjustments to system behavior. This comprehensive analysis reinforces the effectiveness of the FOPID controller, particularly in applications requiring
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(2025) 15:1631
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