Analysis and design of wire winding tension control system based on genetic algorithm and fuzzy PID

Abstract

Wire winding is a key process in transformer manufacturing, and the accuracy of tension control directly affects the quality of the products. To achieve the requirement of constant tension winding of wire, the influence of roll diameter, speed, and other factors on the dynamic performance of tension is analyzed by establishing the dynamic model of unwinding and winding as well as the pendulum part. A control strategy combining proportional-integral-derivative (PID) control with fuzzy control is adopted, and the parameters of the fuzzy PID controller are optimized using an improved genetic algorithm. Simulation results demonstrate that the optimized fuzzy PID controller significantly reduces tension overshoot and effectively suppresses the tension fluctuation when the speed and roll diameter change, and the tension fluctuation is controlled within 2%. Finally, the feasibility of the optimized fuzzy PID controller in the tension control system is experimentally validated.

Keywords

Wire winding tension control genetic algorithm fuzzy control PID

Introduction

In recent years, the electricity demand is increasing with the rapid development of science and technology and the economy. As power system infrastructure equipment, power distribution transformers have also grown in demand (Sun et al., 2023). The winding coil is an important part of the transformer, and winding quality directly determines the reliability of the transformer (Lu et al., 2018; Mahawan and Luo, 1999). The coil material for manufacturing coils is mainly metal wire and insulating paper, which has a certain degree of elasticity, and changes in winding speed and roll diameter will cause fluctuations in winding tension (Akkus and Genc, 2017). Therefore, stable tension control is the key technology in the wire winding process; if the tension is not controlled properly, it will directly affect the mechanical strength, pre-stress, and alignment accuracy of the winding products (Deng et al., 2021; Mertiny and Ellyin, 2002; Ren et al., 2008).

Factors such as speed and changes in coil diameter must be considered when designing tension control systems. The shape of the winding core mold and the speed can cause fluctuations in tension. As the wire is transferred from the unwinding roll to the rewinding roll, the tension changes with the variations in the radius of the unwinding roll and the core mold. If the tension is too low, the wire will slack during transmission, affecting the precision of the coil and leading to significant gaps between turns and even overlapping of the wire. If the tension is too high or experiences sudden changes, the wire may deform or even get damaged. Therefore, maintaining even tension is essential to ensure winding quality. This paper utilizes a connection between a cylinder and a pendulum arm to form a pendulum mechanism. This mechanism allows for the setting of initial tension and buffering of tension during the winding process. In addition, an angle sensor installed at the center of the mechanism is used to detect changes in tension.

The conventional proportional-integral-derivative (PID) algorithm is widely applied in industrial control systems due to its simplicity, strong adaptability, and ease of implementation (Imamura et al., 1999; Kuhm and Knittel, 2011; Roopa and Govinda, 2011). However, fixed-parameter PID controllers have limitations when dealing with time-varying, strongly coupled, and multi-disturbance control systems. In the face of changes in the unwinding roll radius, a method utilizing real-time detection of radius variations and employing feed-forward control was proposed to compensate for tension fluctuations caused by radius changes (Zhong and Pao (2008, 2009)). This method effectively suppresses tension fluctuations but increases the complexity of the control system and exhibits poor disturbance rejection capabilities. To suppress the interference from uncertain factors in the control system, the H-∞ control algorithm was introduced for tension control (Gassmann et al., 2011; Lai et al., 2020; Li et al., 2012). However, this algorithm has high computational complexity and is difficult to ensure real-time control. Adaptive PID controller designs are achieved using both model reference and relay feedback methods to achieve tension adaptive control which has been presented by Raul and Pagilla (2015). Wang et al. (2004) have applied neural network theory to the tension control system, overcoming the limitations of traditional PID control methods and effectively attenuating the coupling between velocity and tension in the tension control system.

Some scholars have studied the combination of fuzzy logic and PID controllers to realize the control of tension. Ponniah et al. (2014) have employed a hybrid approach based on fuzzy logic and PI control to predict disturbances from previous spans, reducing the impact of tension disturbances on the desired span moment; this method exhibits fast adjustment time and strong robustness. Eldessouky and Fahmy (2019) have proposed to dynamically adjust the affiliation functions in the fuzzy controller using a reference model, applied to fiber optic winding tension control. Simulation results demonstrate the effectiveness of this algorithm in suppressing tension disturbances caused by inertia changes during the fiber optic winding process. However, the performance of this method relies on the accuracy of the reference model. Huang et al. (2020) have presented a variable-domain fuzzy PI tension control algorithm, which effectively reduces tension disturbances caused by elliptical core belts and satisfies the tension control requirements at different winding speeds. However, determining the variable-domain functions poses challenges, and improper selection can lead to system uncontrollability. The fuzzy PID controller based on recursive extended Kalman filtering (REKF) proposed by Zhang et al. (2018) reduces control errors by online adjustment of the shape and position of membership functions, significantly enhancing the overall performance of the control system. The studies demonstrate the significant advantages of combining fuzzy control with PID in tension control systems.

Extensive simulation and experimental results indicate that the determination of the quantization factor and scaling factor greatly affects the static and dynamic performance of the fuzzy PID controller (Bi et al., 2000). However, in practical tuning, these factors are typically set by experts based on their experience, which is tedious and inefficient. Therefore, intelligent algorithms can be applied to achieve the search and optimization of controller parameters. Akihiro et al. (2008) have proposed a self-correcting PID controller with an estimator, constructed based on an adaptive particle swarm algorithm. Hashmia et al. (2023) have used the genetic algorithm (GA) for parameter self-tuning of conventional PID controller to obtain suitable parameters for k_p, k_i, and k_d and the simulation results proved the superiority of GA over the conventional methods. Xiao et al. (2021) have established a tension fuzzy PID model based on fuzzy theory and PID control method and optimized the initial parameters and fuzzy rules of the fuzzy PID using GA, which has good stability and can improve the precision of strip tension control, to make the tension during the operation of the rolling mill more stable. The above study shows that the optimization of PID parameters by the intelligent algorithm can improve the stability and fast response ability of the system to a certain extent, to improve the overall performance of the system.

In this paper, based on establishing the dynamic model of the wire winding system, the quantization factor and scale factor of fuzzy PID are optimized offline using the improved GA, and the optimization speed and the rightness of the optimization parameters of the improved GA are verified. The tension control system based on improved genetic algorithm optimized fuzzy PID (IGA-FUZZY-PID) is finally designed, and the performance of PID, FUZZY-PID, and IGA-FUZZY-PID in terms of static-dynamic performance, self-adaptation, and anti-interference is compared.

Structural analysis and dynamic modeling

The wire winding system studied in this paper is shown in Figure 1. The system consists of three mechanisms: the winding mechanism, the pendulum mechanism, and the unwinding mechanism. The wire is pulled out from the unwinding roll, passes through the pendulum mechanism, and reaches the winding roll. Both the unwinding roll and the winding roll are driven by servo motors, and the preset tension is set by adjusting the output force of the cylinder connected to the pendulum. In actual winding, the change of the coil diameter of the wire material, the non-circular winding mandrel, and the acceleration and deceleration process will cause the winding speed to rise and fall, which will cause fluctuation of the tension. The angle sensor installed at the center of the pendulum is used to detect changes in the tension value, thus adjusting the speed of the unwinding roll, keeping the pendulum mechanism in the preset position as much as possible, and realizing the continuous adjustment of the tension.

Unwinding roll modeling

The unwinding roll is driven by a servo motor; as the winding process is carried out, the roll diameter of raw materials will continue to decrease and the radius and inertia of the unwinding roll will change with time. See Figure 2 for the cross-section of the unwinding roll.

The effective moment of inertia J₁(t) on the unwinding roll can be expressed as follows

J_{1} (t) = J_{m 1} + J_{ω 1} (t) + J_{c 0}

(1)

where J_m₁ represents the constant inertia on the motor side, J_c₀ represents the inertia of the winding shaft, J_m₁ and J_c₀ are constants. J_ω1(t) represents the inertia of the wire material, which varies with time. J_ω₁(t) is expressed as follows

J_{ω 1} (t) = \frac{1}{2} π H ρ (R_{1}^{4} (t) - R_{0}^{4})

(2)

where ρ is the density of the wire, H is the width of the wire roll, R₁(t) represents the real-time radius of the unwinding roll, and R₀ is the radius of the wire roll.

The dynamic model of the unwinding roll can be expressed as follows

{\overset{\cdot}{J}}_{1} (t) ω_{1} + J_{1} (t) {\overset{\cdot}{ω}}_{1} = T_{1} R_{1} (t) - M_{d 1} - b_{f 1} ω_{1}

(3)

where ω₁ represents the angular velocity of the unwinding roll, T₁ represents the wire tension, M_d₁ represents the input torque of the servo motor, b_f₁ represents the coefficient of friction in the unwinding roll shaft. From equation (1), the rate of change in J₁(t) is only related to J_ω1(t), and from equation (2), the rate of change of J₁(t) is given as follows

{\overset{\cdot}{J}}_{1} (t) = 2 π H ρ R_{1}^{3} (t) {\overset{\cdot}{R}}_{1} (t)

(4)

The relationship between the linear velocity and angular velocity of the unwinding roll is V₁ = R₁ω₁. Hence, ${\overset{\cdot}{ω}}_{1}$ can be expressed as follows

{\overset{\cdot}{ω}}_{1} = \frac{{\overset{\cdot}{V}}_{1}}{R_{1} (t)} - \frac{V_{1} {\overset{\cdot}{R}}_{1} (t)}{R_{1}^{2} (t)}

(5)

According to Formulas (2)–(5), equation (6) can be obtained

\begin{matrix} T_{1} = \frac{M_{d 1}}{R_{1} (t)} + b_{f 1} \frac{V_{1}}{{R_{1}}^{2} (t)} + J_{1} (t) (\frac{{\overset{\cdot}{V}}_{1}}{R_{1}^{2} (t)} - \frac{V_{1} {\overset{\cdot}{R}}_{1} (t)}{R_{1}^{3} (t)}) \\ + \frac{V_{1}}{R_{1}^{2} (t)} (2 π H ρ R_{1}^{3} (t) {\overset{\cdot}{R}}_{1} (t)) \end{matrix}

(6)

From equation (6), it can be observed that the tension on the unwinding side, T₁, is influenced by factors such as R₁(t), V₁, and J₁(t), which are time-varying. Therefore, this tension control system is a time-varying, complex, and nonlinear system.

The reduction of the wire of the unwinding roll by one layer requires the unwinding roll to rotate more than one turn, so its coil diameter in the same layer is approximated as a constant, which can be obtained as follows

T_{1} = \frac{M_{d 1}}{R_{1} (t)} + b_{f 1} \frac{V_{1}}{{R_{1}}^{2} (t)} + J_{1} (t) \frac{{\overset{\cdot}{V}}_{1}}{R_{1}^{2} (t)}

(7)

The unwinding roll acceleration $a_{1} = {\overset{\cdot}{V}}_{1}$ , equation (7) becomes

T_{1} = \frac{M_{d 1}}{R_{1} (t)} + \frac{V_{1}}{{R_{1}}^{2} (t)} b_{f 1} (t) + \frac{J_{1}}{{R_{1}}^{2} (t)} a_{1}

(8)

And we can get the tension fluctuation value ΔT₁

Δ T = \frac{b_{f 1}}{2 {R_{1}}^{2} (t)} a_{1}

(9)

According to equation (9), it can be observed that for the same radius, a higher acceleration value leads to more severe tension fluctuations. Conversely, when the acceleration is constant, a smaller radius has a greater impact on the tension.

Winding roll modeling

During the winding process, the equation for the relationship between the winding roll speed V₂, and the tension T₂, is similar to that of the unwinding roll and can be expressed as follows

\begin{matrix} M_{d 2} = T_{2} R_{2} (t) + b_{f 2} \frac{V_{2}}{R_{2} (t)} + J_{2} (\frac{{\overset{\cdot}{V}}_{2}}{R_{2} (t)} - \frac{V_{2} {\overset{\cdot}{R}}_{2} (t)}{R_{2}^{2} (t)}) \\ + \frac{V_{2}}{R_{2}} (2 π H ρ R_{2}^{3} (t) {\overset{\cdot}{R}}_{2} (t)) \end{matrix}

(10)

Pendulum analysis

The pendulum mechanism employed in this design serves as an energy storage system, capable of buffering abrupt changes in wire tension and providing real-time feedback on tension variations through an angle sensor. The force analysis of the pendulum mechanism is illustrated in Figure 3, and the equilibrium equation for the pendulum moment is expressed as follows

\begin{matrix} J_{b} {\overset{\cdot}{ω}}_{b} (t) + F_{gas} [\cos (β - α)] + b_{f 0} = T_{1} \cos (θ_{1} - α) \\ + T_{2} \cos (θ_{2} - α) \end{matrix}

(11)

where J_b represents the moment of inertia of the pendulum, ω_b(t) represents the angular velocity at the center of the pendulum, F_gas represents the pressure output from the cylinder, β represents the angle between the pendulum and the cylinder, α represents the horizontal angle of the pendulum, b_f₀ represents the frictional force of the pendulum, T₁ and T₂ are the tensions at the ends of the pendulum wire, θ₁ and θ₂ are the angles between the wire direction and the pendulum. In the elastic range of the wire, the tension change caused by the tangential velocity difference is

Δ T = \frac{SE}{L} \int_{0}^{t} V_{2} - V_{1} dt

(12)

where V₁ and V₂ are, respectively, the linear velocities of the wires upstream and downstream of the pendulum; S represents the cross-sectional area of the wire; E represents Young’s modulus of the wire; L represents the length of the wire between points A and B; l₀ is half the length of the pendulum; and T represents the tension in the wire. From V(t) = α(t)l₀/t, we can get

Δ T = \frac{SE l_{0}}{L_{0} + 2 l_{0} \sin ϕ} \int_{t_{1}}^{t_{2}} \frac{α_{1} (t) - α_{2} (t)}{t} dt

(13)

T = T_{0} + Δ T

(14)

In the equation, L₀ represents the length of the wire between the two points when the pendulum is in the equilibrium position, l₀ is half the length of the pendulum, $2 l_{0} \sin ϕ$ is the length of wire released to cushion tension fluctuations for the pendulum mechanism, T₀ represents the preset tension, and T represents the tension in the wire.

According to equation (11), it can be observed that the tension in the wire outputted by the pendulum mechanism is influenced by factors such as the moment of inertia of the pendulum, angular acceleration, and the force output from the cylinder. Equation (12) indicates that the fluctuation in tension is the integral of the velocity difference. Equation (13) suggests that the fluctuation in tension can be reflected by the variation in the pendulum angle. Equation (14) states that the value of wire tension is obtained by adding the set value of the cylinder and the fluctuation in tension.

Controller design

In order to realize good static and dynamic performance of the tension control system even when it is disturbed by parameter changes and frequent acceleration and deceleration of the unwinding roll, a fuzzy algorithm is used to adjust the operating parameters of the PID controller. The initial parameters of the PID controller are determined using the Ziegler–Nichols method, while the selection of quantization and proportion factors in the fuzzy controller is based on theoretical calculations and engineering experience. This section presents the design of an adaptive fuzzy PID controller and the improvement of the GA for optimizing the fuzzy PID controller.

Fuzzy PID

The fuzzy PID model for the wire winding tension control system is illustrated in Figure 5. The voltage signal error e, from the angle sensor feedback, along with the derivative of the error ec, serves as the fuzzy input linguistic variables, while ΔK_p, ΔK_i, and ΔK_d are the fuzzy output linguistic variables. The input and output fuzzy universes are set to [−3, 3], and in practical applications, the values of the fuzzy universes are transformed into the actual universe using linear transformation formulas as shown in equations (15) and (16). The actual universe range is defined as [x_min, x_max], while the fuzzy universe range is also set as [X_min, X_max]. The parameter K represents the quantization factor or scaling factor

x = \frac{x_{max} - x_{min}}{2} + K (X - \frac{X_{max} - X_{min}}{2})

(15)

K = \frac{x_{max} - x_{min}}{X_{max} - X_{min}}

(16)

The fuzzy subset is set to {NB, NM, NS, ZO, PS, PM, PB} and the S-shaped membership function is used when the linguistic variables are NB and PB, and the triangular membership function is used for the other linguistic variables. The membership function for the controller inputs is shown in Figure 4.

Based on the experience of previous researchers in FUZZY-PID controller design, although interval type-2 (IT2) fuzzy set can theoretically provide higher uncertainty modeling capability (Gu et al., 2023; Sun et al., 2022), interval type-1 (IT1) fuzzy control is less computationally complex and easier to implement and debug than IT2 fuzzy control; therefore, the fuzzy set type is selected as IT1. The fuzzy rule base consists of if-then statements; according to the previous tension control as well as controller design experience (Huang et al., 2020), the fuzzy rules of ΔK_p, ΔK_i, and ΔK_d are established, as shown in Table 1.

Table 1.

Fuzzy rule table of ΔK_p, ΔK_i, and ΔK_d.

ΔK_p/ΔK_i/ΔK_d
e\ec	NB	NM	NS	ZO	PS	PM	PB
NB	PB/NB/PS	PB/NB/NS	PM/NM/NB	PM/NM/NB	PS/NS/NB	ZO/ZO/NM	ZO/ZO/PS
NM	PB/NB/PS	PBNB/NS	PM/NM/NB	PS/NS/NM	PS/NS/NM	ZO/ZO/NS	NS/ZO/ZO
NS	PM/NB/ZO	PM/NM/NS	PM/NS/NM	PS/NS/NM	ZO/ZO/NS	NS/PS/NS	NS/PS/ZO
ZO	PM/NM/ZO	PM/NM/NS	PS/NS/NS	ZO/ZO/NS	NS/PS/NS	NM/PM/NS	NM/PM/ZO
PS	PS/NM/ZO	PS/NS/ZO	ZO/ZO/ZO	NS/PS/ZO	NS/PS/ZO	NM/PM/ZO	NM/PB/ZO
PM	PS/ZO/PB	ZO/ZO/PS	NS/PS/PS	NM/PS/PS	NM/PM/PS	NB/PB/PS	NB/PB/PB
PB	ZO/ZO/PB	ZO/ZO/PM	NM/PS/PM	NM/PM/PM	NM/PM/PS	NB/PB/PS	NB/PB/PB

Defuzzification is performed using the weighted average method, as shown in equation (17)

\bar{x} = \frac{\sum_{i = 1}^{n} X_{i} U_{u} (x_{i})}{\sum_{i = 1}^{n} U_{u} (x_{i})}

(17)

In the equation, U_u(x_i) represents the membership degree of each variable.

Improved GA to optimize fuzzy PID

To ensure stability, controllability, and adaptive modification of PID parameters in the control system, the initial values (K_p₀, K_i₀, K_d₀) of the PID parameters are used as the center point, and the center point is adjusted through a fuzzy algorithm. The final output of the controller is represented by equations (18)–(20). The value and relative relationship of the quantization factor and scaling factor in the fuzzy controller has a significant impact on its control performance, even altering its output characteristics. Unsuitable parameter tuning can result in suboptimal system performance and poor adaptability. Traditional fuzzy PID controllers rely on personal experience for parameter tuning. To overcome the uncertainty caused by subjective factors and meet the requirements of reliability, real-time capability, and fast response in practical engineering applications, an improved GA is employed offline to optimize the parameters (K_e, K_ec, K_ΔKp, K_ΔKi, K_ΔKd) in the fuzzy PID controller, as shown in Figure 5. The optimized parameters are then loaded into the controller to achieve optimal control performance

K_{p} = K_{p 0} + K_{Δ kp} Δ K_{p}

(18)

K_{i} = K_{i 0} + K_{Δ ki} Δ K_{i}

(19)

K_{d} = K_{d 0} + K_{Δ kd} Δ K_{d}

(20)

GAs exhibit excellent parallel optimization characteristics. To address the issues of slow convergence and susceptibility to local optima in early GAs, this study dynamically adjusts selection probability, crossover probability P_c, and mutation probability P_m to enhance the optimization speed of the GA. To suppress overshoot and reduce the settling time, an improved fitness evaluation function is selected in this paper:

Real-valued encoding is selected due to its simplicity, fast optimization speed, and ability to achieve high optimization accuracy.

Improvement of the fitness function. The magnitude of the fitness function value is the sole criterion for evaluating the quality of individuals. The rationality of the chosen fitness function determines the performance of the GA. To improve convergence speed and achieve the goal of obtaining optimal solutions, the following improved fitness function is chosen in this study

J = \int_{0}^{\infty} (w_{1} | e (t) | + w_{2} u^{2} (t) + w_{4} | e (t) |) dt + w_{3} t_{u}

(21)

In the equation, e(t) represents the error, t_u is the rise time, u(t) denotes the output of the controller, and w₁, w₂, w₃, and w₄ are the weighting factors. This function comprehensively reflects the response speed, overshoot, steady-state error, and other performance indicators of the control system.

3. Improvement of fitness value, crossover, and mutation probabilities. In order to increase the selection probability of individuals and promote population diversity in the early stages of optimization, while accelerating convergence speed in the later stages, the fitness value is improved as follows

f' (x_{i}) = af (x_{i}) + \frac{e - e^{k / k_{max}}}{e + e^{k / k_{max}}} (f_{max} - f_{min})

(22)

In the equation, f_max represents the maximum fitness value, f_min represents the minimum fitness value, k represents the current iteration number, k_max represents the maximum number of iterations, and a > 0 is a constant.

To enhance the speed of optimization and the precision of the solutions, dynamic crossover probability P_c and mutation probability P_m are adopted, and their expressions are as follows

P_{c} = {\begin{matrix} k_{1} \frac{\arcsin (\frac{f_{avg}}{f_{max}})}{π / 2} \arcsin (\frac{f_{avg}}{f_{max}}) < \frac{π}{6} \\ k_{1} (1 - \frac{\arcsin (\frac{f_{avg}}{f_{max}})}{π / 2}) \arcsin (\frac{f_{avg}}{f_{max}}) \geq \frac{π}{6} \end{matrix}

(23)

P_{m} = {\begin{matrix} k_{2} (1 - \frac{\arcsin (\frac{f_{avg}}{f_{max}})}{π / 2}) \arcsin (\frac{f_{avg}}{f_{max}}) < \frac{π}{6} \\ k_{2} \frac{\arcsin (\frac{f_{avg}}{f_{max}})}{π / 2} \arcsin (\frac{f_{avg}}{f_{max}}) \geq \frac{π}{6} \end{matrix}

(24)

In the equation, f_avg represents the average fitness value, and k₁ and k₂ are both positive constants.

The optimization process of the improved GA is illustrated in the following steps:

Step 1. Coding to generate the initial population.

Step 2. Adjusting selection probability, crossover probability, and mutation.

Improving the speed of GA’s optimization search, new population sample is obtained.

Step 3. Feed the iterative population results into the fuzzy controller and calculate the fitness value.

Step 4. Determine whether the fitness value is less than the set value or whether the maximum number of iterations is exceeded. If the target is not reached, return to Step 2 to continue the evolution, otherwise output the optimization parameters to complete the PID optimal parameter output.

Simulation analysis

GA iterative analysis

To verify the effectiveness of the improved GA for optimizing the fuzzy PID controller in the tension control system, a control system model based on the proposed system in Figure 5 was constructed using MATLAB. The population size in the GA is set to 80, the crossover probability (P_c) is set to 0.99, the mutation probability (P_m) is set to 0.01, the maximum number of iterations (W₃) is set to 50, k₁ is set to 0.9, and k₂ is set to 0.9. The iteration curves of the two optimization algorithms are shown in Figure 6. It can be observed from the figure that the improved GA converges in approximately 18 iterations, while the traditional GA requires approximately 36 iterations for convergence. Therefore, the optimization speed of the improved GA is twice that of the traditional GA.

Controller dynamic performance

To evaluate the static and dynamic performance of the improved controller and validate the accuracy of the improved GA optimization, the parameters obtained from both the traditional GA and the improved GA were loaded into the controller. In conjunction with the actual tension control system, the unwinding roll radius was set to 0.11 m, and the tension was set to 40 N. When the pendulum mechanism was in the equilibrium position, a step reference voltage signal with an amplitude of 5 V was applied to the angle sensor input. The tension step response curves are shown in Figure 7. All three control algorithms meet the requirements for stability and steady-state error control. The IGA-FUZZY-PID algorithm exhibits the smallest overshoot, within 3%, and the fastest settling time, demonstrating superior dynamic performance. The FUZZY-PID control performs slightly worse, while the traditional PID control exhibits the poorest performance, with an overshoot of 15.0%.

Due to the traditional GA easily falling into the local optimal solution, there is a slight difference between the results of the two algorithms in this optimization process, but this slight difference will lead to a decrease in the performance of the system. As observed from the response curves, the improved GA effectively enhances the accuracy of parameter optimization.

Controller anti-interference performance analysis

Combined with the actual tension control system, the maximum unwinding radius of the wire is set to 0.11 m, and the reference tensions are set at 30 N, 40 N, and 50 N. To verify the anti-interference capability of the control algorithm, a disturbance in the form of changes in the winding roller speed, as shown in Figure 8(a), is introduced. The tension response curves are shown in Figure 8(b)–(d). When the system is subjected to disturbance, all three control algorithms can restore stability within 0.55 s.

From 4.5 to 5 s, the winding roll speed increases from 3 to 6 m/s. When the reference tension is 30 N, the traditional PID overshoot is 1.8%, the FUZZY-PID overshoot is 1.1%, and the IGA-FUZZY-PID overshoot is 0.4%. When the reference tension is 40 N, the traditional PID overshoot is 1.5%, the FUZZY-PID overshoot is 0.9%, and the IGA-FUZZY-PID overshoot is 0.3%. When the reference tension is 50 N, the traditional PID overshoot is 1.1%, the FUZZY-PID overshoot is 0.7%, and the IGA-FUZZY-PID overshoot is 0.2%.

From 9.8 to 10.4 s, the winding speed decreases from 6 to 3 m/s. When the reference tension is 30 N, the traditional PID overshoot is 1.7%, the FUZZY-PID overshoot is 1.2%, and the IGA-FUZZY-PID overshoot is 0.5%. When the reference tension is 40 N, the traditional PID overshoot is 1.5%, the FUZZY-PID overshoot is 0.8%, and the IGA-FUZZY-PID overshoot is 0.3%. When the reference tension is 50 N, the traditional PID overshoot is 1.1%, the FUZZY-PID overshoot is 0.7%, and the IGA-FUZZY-PID overshoot is 0.3%.

It is evident that the proposed IGA-FUZZY-PID control method effectively reduces the tension overshoot compared to the traditional PID control. The tension fluctuation remains within ±2 N, with the overshoot consistently controlled within 2%. The IGA-FUZZY-PID controller meets the requirements of tension control during the wire winding process.

Experimental verification

To validate the real effectiveness of the IGA-FUZZY-PID controller in tension control, an experimental platform for the tension control system based on Mitsubishi PLC was constructed, as shown in Figure 9(a). The parameters optimized by the improved GA offline were loaded into the controller, and the tension values were set to 30 N, 40 N, and 50 N. In the initial stage of the experiment, the winding roller speed abruptly increased from 0 to 3 m/s, and after running for 4 s to stabilize, the speed increased again from 3 to 6 m/s and continued to run for 5 s before the speed decreased in two cycles and finally stopped. Figure 9(b) shows the actual speed curve of the two-cycle changes. Figure 9(c)–(e) records the variation of wire tension during the experiment. As the tension fluctuated significantly in the initial stage of the experiment, it is necessary to analyze the performance indicators of the interference response curve.

Based on the experimental data, it can be observed that the IGA-FUZZY-PID controller exhibits significant improvements in dynamic performance and disturbance rejection compared to the traditional PID controller. In the experiment, when the tension was set at 30 N, the traditional PID controller exhibited a maximum overshoot of 19.6% with the maximum tension fluctuating within 13 N. On the other hand, the IGA-FUZZY-PID controller demonstrated a maximum overshoot of 9.9% with the maximum tension fluctuating within 6 N. When the tension was increased to 40 N, the traditional PID controller showed a maximum overshoot of 12.4%, and the maximum tension fluctuated within 9 N. Conversely, the IGA-FUZZY-PID controller displayed a maximum overshoot of 7.1%, with the maximum tension fluctuating within 5.8 N. Finally, at the tension of 50 N, the traditional PID controller had a maximum overshoot of 9.7%, and the maximum tension fluctuated within 9.1 N. On the other hand, the IGA-FUZZY-PID controller demonstrated a maximum overshoot of 5.8%, with the maximum tension fluctuating within 5.5 N. These indicators show a decreasing trend as the set tension value increases. In addition, there was a slight discrepancy between the tension fluctuation range in the experimental data and the simulation. This can be attributed to the unstable operation of the winding speed during the experiment and the delayed adjustment of the cylinder during the buffering of tension changes, resulting in fluctuating wire tension. The difference in overshoot compared to the simulation results is due to the machine experiencing jitter caused by sudden changes in the winding speed, affecting the instantaneous overshoot variation. However, once the speed stabilizes, the tension fluctuations are observed to be limited to within 3 N, meeting the tension control requirements for wire winding. Therefore, overall, the IGA-FUZZY-PID controller exhibits significant advantages over the traditional PID controller and meets the requirements of tension control in practical operations.

Conclusion

During the wire winding process, frequent acceleration and deceleration, as well as changes in parameters such as roll diameter, can cause significant tension fluctuations. To address the issues of multiple disturbances, time-varying characteristics, and nonlinearity, this paper proposes a pendulum mechanism to buffer tension changes and feedback tension variations. An IGA-FUZZY-PID control algorithm is designed for the tension control system. Through a comparison with PID and FUZZY-PID, the results demonstrate that the IGA-FUZZY-PID control method achieves the shortest settling time of 0.26 s and the minimum overshoot. It effectively suppresses tension fluctuations during the winding process, indicating good adaptability and strong disturbance rejection capabilities of the system. In addition, the improved GA exhibits fast optimization speed and high accuracy compared to traditional GAs. Compared to traditional PID control methods, the control method used in this study can maintain actual tension fluctuations within 2%, significantly improving the coil winding quality.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by Heilongjiang Province Applied Technology Research and Development, grant number GA20A401.

ORCID iD

Shiyi Zheng

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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