ǤǤ Ȁ Ȁ
input reference (R(t)) and the actual system output (Y(t)). FOPID Controller Processes the error signal using fractional-order integral and derivative operations. The controller generates the control signal u(t), which drives the headbox system. Jellyfish Search Optimizer (JSO) Optimizes the parameters of the FOPID controller (Kp, Ki, Kd, λ, μ) to ensure optimal performance under varying operating conditions. Objective Function Evaluates the performance of the FOPID controller based on error indices like Integral of Squared Error (ISE), Integral of Time-weighted Absolute Error (ITAE), and Integral of Absolute Error (IAE). This function guides the JSO in finding the best parameter set. Headbox Dynamics (GHB(s)) represents the transfer function of the headbox system, which models the relationship between control inputs and system outputs (pressure and stock level). Feedback Loop Continuously measures the system output (Y(t) and feeds it back to compute the error signal, enabling closed-loop control. G HB ( S )= 0 . 528 e − 0 . 6 s 2 . 2 s +1 0 . 081 1 . 89 s +1 1 . 49 ∗ 10 − 4 e − 1 . 5 s s − 7 . 0 ∗ 10 − 4 e − 2 s s (1) The decoupling control technology is utilized in this work for the control plan, so the decouple design using the standard method of decoupling is considered as d 1 = − 0.1534 and d 2 = 0.2129. Therefore, the models with decoupled method are listed underneath, G P ( S )= − 0 . 288 s 2 +0 . 8825 s +0 . 5452 1 . 247 s 3 +5 . 365 s 2 +4 . 39 s +1 (2) G St ( S )= 5 . 421 s 2 +1 . 693 s − 7 . 229 7500 s 3 +17500 s 2 +10000 s (3) Where, G P ( S ) and G St ( S ) are represents the relay assignments of pressures and stock level loop of the headbox accordingly. The multi-target work is characterized depending on the headbox pressure and headbox stock control procedure of paper machine headbox. The multi target work is characterized to limit the blunder pressing factor and supply of paper machine headbox. The required multi-target work is depicted in the accompanying condition (4). Ψ = min Ω 1 Ω 2 (4)
Where,
Ω 1 =
2 =
w i Ψ 1 and Ω
w i Ψ 2
Ψ 1 = P ∗ HB − P Ψ 2 = St ∗ HB − St
(5) (6)
act HB
act HB
w i =1
i = P,S
(7)
From the above conditions, Ψ 1 and Ψ 2 are pressure and stock errors respectively; w P and w St are the weight values of objective components of pressure and stock errors taken separately. The imaginative coefficient methodology is taken to put together supported objective limit. Constraints The accompanying limitations are used to accomplish the target work and the regulator requires are used to improve their practices which dependent on their benefit boundaries. The increase boundaries of the regulators are Proportional addition, Integral increase, Derivative addition, Integral request and Derivative request are characterized to set the base, greatest restriction by utilizing the accompanying equation,
min P
max P
(8)
K P K
K
min I
max I
(9)
K I K
K
min D
max D
(10)
K D K
K
min min
max
(11)
λ λ μ μ
λ
max
(12)
μ
By using the above limitations the objective is attained and the process is repeated with number of iterations by utilizing the JSO algorithm, which is illustrated in the following section.
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(2025) 15:1631
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