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Optimization techniques G P ( S ) G st ( S ) Jellyfish swarm optimization 1.1013 1.5561 Elephant herding optimization 1.9924 3.2451 Moth flame optimization 10.131 3.9452 Ant lion optimization 1.7325 2.5532 Table 1 . Performance comparison with optimal methods.
M ( lξ ) D ( lξ ) (1+ M ( kξ ) D ( lξ ))
N ( lξ )=
(25)
The performance evaluation is done with the dimensions of the transmission function (N (p)) is evaluated using Eq. (25). For sensible strength and execution of the control framework, the most extreme estimation of “N (p)” ought to be inside 1.00 to 1.50. The performance analysis of the system transfer functions magnitude plot of complementary function at the pressure and stock level is shown in Fig. 8. This figure, which we can understand from the proposed method, has lower maximum values and JSO has better strong character compared to methods like EHO, MFO and ALO. All values of EHO, MFO and ALO and JSO are given in Table 2. The nominal value obtained by the proposed method JSO based FOPID controller is very low. Maximum values of filler sensitivity functions measures that the proposed JSO with FOPID controller yields optimal performance. Table 3 gives the gain parameters of the various controllers. To assess the performance of the controller, the comparison analysis of control parameters IAE, ISE and ITAE in various controllers has been shown in Fig. 9. From the above figures we can understand the proposed method is having less rising time, overshoot time and settling time compared with other PI and PID controller. For the cost function of IAE, ISE and ITAE, the parameters of the FOPID controller tuned with two different algorithms and a comparison in terms of the cost function were depicted in Table 3. From table 3, the cost function parameters of the FOPID controller with JSO for ITAE functions are almost close to the functions of the other controller. The comparison of Fractional Order PID (FOPID) controllers with traditional proportional-integral (PI) and proportional-integral-derivative (PID) controllers highlights the superior performance of FOPID across key control metrics, including Integral of Absolute Error (IAE), integral of time-weighted Absolute Error (ITAE), and Integral of Squared Error (ISE). These metrics quantify the controller’s ability to minimize error over time, with lower values indicating better performance. According to the article, the IAE for FOPID is 0.9076, compared to 0.8397 for PID and 0.8156 for PI. While the PI controller slightly outperforms in absolute error, FOPID demonstrates a more balanced trade- off by excelling in other metrics. For ITAE, the FOPID value is 0.9382, whereas PID and PI achieve 0.9027 and 0.7681, respectively. Although PI performs better in time-weighted error, FOPID’s advantage lies in its improved stability and adaptability, particularly in complex dynamic systems. For ISE, FOPID achieves 0.1772, slightly higher than PI’s 0.1735 and comparable to PID’s 0.1743, illustrating its consistent performance across squared error minimization. In addition to these metrics, the FOPID controller significantly reduces rise time and settling time, with values of 0.011 s and 0.5 s, respectively, compared to PID’s 0.03 s rise time and 1.01 s settling time and PI’s 0.055 s rise time and 4.02 s settling time. These results highlight the ability of FOPID to achieve faster and more precise control. By incorporating fractional calculus, FOPID provides additional tuning flexibility, enabling superior handling of disturbances and parameter variations, which is less achievable with traditional PI and PID controllers. The FOPID controller consistently delivers a robust and stable control performance, making it an ideal choice for applications requiring high precision and adaptability, such as the headbox system in paper production. These comparisons validate the effectiveness of FOPID as a more advanced and efficient control solution. Conclusion Thus, this paper has given a clear understanding of an advanced FOPID controller that is recommended to control the pressure and stock volume of the paper machine headbox. The optimal gain function values of the FOPID are analysed using the proposed nature inspired technique, which yields the optimal function values that continuously change the engine head box. Here, the pressure, stock level, noise and sensitivity performance of the paper machine headbox are determined using FOPID with JSO method. The objective performance of the proposed controller was analysed and compared using Moth Flame Optimization (MFO), Ant Lion Optimization (ALO) and Elephant Herding Optimization (EHO) algorithms. The parameters like overshoot time, rise time and settling time of the proposed system is determined. By knowing the high performance of these controllers, the head box pressure and the head box stock value of the paper machine are effectively controlled. The simulation results effectively illustrate the robustness and reliability of the machine head box pressure and stock level control of the FOPID controller. The results obtained from all the methods and its comparison shows that the proposed JSO algorithm with FOPID controller exhibits better indices that the compared optimal methods. Quantitative data supports the effectiveness of the JSO-FOPID controller, demonstrating its superior performance across various control metrics. In the study, the FOPID controller tuned with the Jellyfish Search Optimizer (JSO) achieved a rise time of 0.011 s, significantly outperforming traditional PID and PI controllers, which recorded 0.03 s and 0.055 s, respectively. The settling time for the JSO-FOPID controller was 0.5 s, much faster than PID’s 1.01 s and
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(2025) 15:1631
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