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Swarm The position of jelly fish swarm varies from each palce depending on its activity. The types 26 showcases its movement with respect to position along with each one. X L i ( t +1)= X L i ( t )+ γ ∗ rand (0 , 1) ∗ ( up − lp ) (16) The lower and upper boundaries are taken as lp and up respectively. Equations from 17 to 20 represents the timely action with respect to jelly fish movement in real time. → Step = X L i ( t +1) − X L i ( t ) (17) → Step = rand (0 , 1) ∗ → direction (18) → direction = X L j ( t ) − X L i ( t ) if f X L i f X L j X L i ( t ) − X L j ( t ) if f X L i <f X L j (19) X L i ( t +1)= X L i ( t )+ → Step (20) If more food is chosen in the jellyfish area than in the jellyfish area of interest, the jellyfish area of interest wins out. On the other hand, if there is less food available for jellyfish selection, the jellyfish will swiftly disappear from the zone of interest. The functionalities are given from Eq. 17 to 20. Jellyfish mechanism for switching movements When the wind or temperature shifts the speed of the sea, the jellyfish in the throng move to a different mass structure of jellyfish. A period control tool to simulate this movement category is shown in calculation (21). The jellyfish follow the water’s current when the time control capability value is greater than zero. They join the multitude when the exact value of the time control mechanism is unknown. The time control changes arbitrarily from zero to one. The Jellyfish Search Optimizer (JSO) offers distinct advantages over other metaheuristic algorithms, making it an ideal choice for optimizing the parameters of the FOPID controller. Inspired by the behavior of jellyfish in oceans, JSO excels in balancing exploration and exploitation during the optimization process, which is critical for avoiding local minima and ensuring global optimality. Unlike algorithms such as Particle Swarm Optimization (PSO) or Ant Lion Optimization (ALO), JSO dynamically adjusts its search strategy based on the distribution of solutions, enabling it to adapt to complex, high-dimensional search spaces effectively. This adaptability is particularly beneficial for nonlinear and multivariable systems like the paper machine headbox, where traditional algorithms may struggle to maintain performance under varying conditions. JSO also demonstrates faster convergence rates and improved accuracy in identifying optimal solutions compared to methods like Moth Flame Optimization (MFO) or Elephant Herding Optimization (EHO). Its ability to integrate environmental factors such as nutrient availability and jellyfish motion patterns enhances its robustness, making it highly effective for fine-tuning control parameters. The superior performance of JSO in both stable and transient states justifies its application in this research, ensuring a more reliable and efficient optimization process for FOPID controller tuning. c ( t )= 1 − t Max iter ∗ (2 ∗ rand (0 , 1) − 1) (21) taken, t as the t th iterations and Max iter is the possible higher number of iterations. Procedure of proposed algorithm Step 1: Initialisation . In this section, the FOPID regulator’s inputs randomly acquire the headbox’s limits. The underlying jellyfish population is created at random. The supplementary condition describes the data sources. γ =( γ 1 i , . . . ,γ d i , . . .γ n i ) (22) taken, n as the target space area, γ d i is the reality of the i th value in the d th time of measurement. Step 2: Performing fitness valuation . The fitness value is obtained by F i = min (Ψ) (23) With the help of Eq. 23 fitness value is calculated. Step 3: Parameters . The functionalities of jelly with with ocan currents are taken from (16) and (15). Step 4: Forms of motion . The inactive forms of jelly fish moves (16), obtained using Eqs. (17–20). Step 5: Update . The updated iterate values are used in order to determine c ( t ) with the values (21). Step 6: Updating place .
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