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suitable for tuning FOPID controllers 15 . This research addresses existing gaps by improving robustness, stability, and error minimization in headbox control while reducing rise time, overshoot, and settling time. A detailed block diagram enhances the understanding of the system, visually representing the FOPID controller’s structure and its integration with JSO. Additionally, the equivalence of Machine Direction (MD) and Cross Direction (CD) is rigorously justified mathematically and experimentally, ensuring uniformity in stock and pressure regulation. This equivalence is critical for maintaining the consistency and quality of paper production, aligning with industrial standards. The equivalence of Machine Direction (MD) and Cross Direction (CD) in paper production is fundamental to ensuring uniform material distribution and quality. This equivalence can be expressed mathematically by considering the mass flow rates in both directions. The mass flow rate in the Machine Direction (Q MD ) is defined as: Q MD = Total Stock Flow Effective Width of the Machine Similarly, the mass flow rate in the Cross Direction (Q CD) must maintain the same consistency to achieve uniform distribution: Q CD = Total Stock Flow Effective Length of the Machine For uniformity, the flow rates in both directions must be equal: Q MD = Q CD . This condition ensures that the stock flow is evenly distributed across the width and length of the headbox, maintaining consistent sheet formation. Ensuring the weight of the paper remains consistent across the width and length of the sheet. Monitoring pressure levels to verify even distribution throughout the headbox. By maintaining Q MD =Q CD , the headbox achieves the necessary balance for producing a high-quality and uniform paper sheet. Compression factor and weight control are important in the paper production because they determine the flow rate at which materials are delivered to the paper machine’s head box. The cut also ascertains the velocity of the water flow at the beginning and the cross-feed of the material 16 . The devices that are primarily used to control full compression factor and flow rate depend on the individual control of gas tension and material flow in the enclosure. Numerical models obtained from various real-world conditions of a particular framework do not contain aggregated data about the framework. It is necessary to demonstrate the inadequacy of a particular framework 17,18 . Although regulators target such frameworks, they cannot continuously manage computers as needed. Therefore, the proposed regulator should be scrutinized for both visible and dangerous patterns of the promised framework. Strength is the ratio of the regulator’s ability to withstand the weakness of a given framework 19,20 . Since the size of the solidity check cannot be foreseen due to its non-linearity, the PI/ PID controller may not have sufficient capacity, even as single input single output (SISO). Regular changes to the regulatory limits require co-ordination. General guidelines are used in a similar way for Internet testing or PC playback procedures 21–23 . As, this settled PID controller quite a while ago appeared to be observing and adjustment of the circles plan method, taking into account that the regular PID regulators aren’t proper for nonlinear projects and higher-requested paper machine head box programs 24,25 . In order to evaluate the achievement of the PID regulators the aggravation dismissal execution list IAE is of crucial consequence since the vast majority of the PID regulators work as controllers. Because, of the frameworks without defers Traditional techniques for PID boundary tuning are generally utilized 26 . Advancement of three boundaries is needed in manual tuning of PID regulators and this is a tedious assignment. In manual testing of PID regulators three extremities is needed but, this is unexciting. It is a great effort to develop effective tuning techniques to address this problem 27 . In latest solicitation, one can find the most sought after principles used to control the PID controller the exact techniques which are Cohen–Coon (CC) tuning, Tyreus-Luyben (TL), Likening Co-Efficient Strategy, Two Degree of Freedom Systems (2DOF) control plot, Security Examination Strategy, Streamlining Technique, Direct Synthesis Method, Internal Model Control (IMC) technique, Ziegler–Nichols (ZN) tuning, tuning rules dependent on the derogation of copious blunder rules and neural organizations based strategies 28,29 . In this manuscript, a JSO procedure is regularly dependent on the best auto change Fractional Order PID (FOPID) regulator for paper machine headbox. The development of this paper includes the modification of constraints, in paper machine headbox which is ideally perceived by the JSO strategy. The proposed system is regularly misusing the paper machine headbox which yield constraints to start the JSO capacities. The portion of the new examination works are explained in "A brief review of recent research works" section. The control procedure and proposed technique are clarified in Sects. 3 and 4. The outcomes and conversations are indicated in "Result and discussion" section. Many research works are formerly carried out and established on the sketch of the PI, PID controllers for the paper head. Few related works are assessed here. Saini et al. 30 . has presented a PI control plan for material strength in paper machine head box utilizing Particle Swarm Optimization (PSO). Here the PSO strategy tried its capacity to improve productivity. Corresponding coordinated regulator (PI regulator) for stock consistency, quite possibly the main boundaries for paper creation, and it is utilized as a SISO framework (single information single yield) PSO-based PI approach as a rule overshoot, recurrence reaction and relative edge (acquire Margin and stage Margin) Provides better
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