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Fig. 1 . Control system’s architecture.
Fig. 2 . Block diagram of proposed FOPID controller for Paper machine headbox model.
proposed controller is associated as an FOPID controller. The controller has reached the objective function with the help of the efficient JSO algorithms. The controlling of the absolute headbox pressing factor and stock level of a paper-production headbox framework is clarified previously. At that point, the paper machine’s headbox work is clarified beneath. Fractional Order PID (FOPID) controllers provide significant advantages over traditional PID controllers, particularly in systems with complex, nonlinear, or multivariable dynamics. The inclusion of fractional calculus in FOPID introduces two additional parameters fractional-order integral and derivative parameters which allow for more precise tuning and greater flexibility in achieving stability under varying operational conditions. Traditional PID controllers, which often struggle with parameter variations, time delays, and external disturbances, FOPID controllers’ exhibit superior robustness and adaptability. The range of parameters for integrators and derivatives during optimization is crucial for ensuring the effective tuning of the FOPID controller. For the integrator, typically represented by the integral gain (K i ) and the fractional integral order (λ), the range is selected based on the system’s operational requirements, ensuring adequate integration over time to minimize steady-state errors. Common ranges for K i are between 0.1 and 10, while λ is often confined to values between 0 and 1 to maintain stability and avoid excessive overshooting. For the derivative parameters, including the derivative gain (K d ) and the fractional derivative order (μ), the range is generally set between 0.01 and 5 for K d and between 0 and 1 for μ. These ranges are chosen to provide a balance between sensitivity to error changes and noise suppression, ensuring robust control without introducing high-frequency noise into the system. The selection of these ranges during optimization is critical to achieve a well-tuned FOPID controller that meets the desired performance criteria across various operating conditions. This ensures better adaptability to varying operational conditions and disturbances, effectively handling time delays, and providing better noise rejection. Furthermore, FOPID controllers excel in nonlinear systems by dynamically adapting to system changes and reducing the need for frequent retuning. These attributes ensure enhanced stability and resilience, making FOPID controllers particularly effective in applications like the headbox control in paper production, where traditional PID controllers may falter. The model used in the proposed work, a low inlet pestering, controls the highest level of inventory adjustment and capacity utilization of the air pressure across the rear of the multivariable paper machine with two inlets and two outlets 41,42 . The functional block diagram representation of the paper machine headbox is shown in Fig. 2 and the transfer functions for the system is obtained as, Input Reference (R(t) Represents the desired setpoints for pressure and stock levels in the headbox. Error Signal e(t) Calculated as the difference between the
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(2025) 15:1631
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