Warp yarn tension coupling models and fuzzy PID control for enhancing warp beam unwinding tension stability in sizing machines

Abstract

The coupling model between unwinding tension and various factors is crucial for understanding warp beam unwinding characteristics and designing tension control systems. However, currently established models are simple and fail to reflect the complexity of the actual unwinding process. Additionally, the traditional unified PID control method for multiple warp beams is difficult to ensure stable unwinding tension for single warp beam, negatively impacting the quality and efficiency of the sizing process. To address these issues, coupling models between tension, radius, speed, turns and air pressure during the warp beam unwinding process were established and validated using constructed detection devices. Experimental results showed that the four types of measured warp beams fit the three coupling models with $R^2$ values exceeding 90%, confirming the accuracy of these models. Subsequently, a novel fuzzy adaptive PID control system for single warp beam unwinding tension was proposed based on these models. Unlike traditional methods that measure tension directly, the system utilizes unwinding turns detection device to monitor the real-time changes. Simulation results confirmed that the proposed system maintains constant tension under different reference tensions and unwinding radius. Compared to traditional PID control, it significantly reduces tension fluctuation, with reductions of 64.49% under different speed step disturbances and 23.66% under noise disturbances. The validated models and proposed tension control system in this paper provide a solid theoretical foundation and practical guidance for optimizing the new generation of sizing machines.

Introduction

Warp beam unwinding is a critical component of the sizing process. The stability of its tension plays a key role in ensuring the quality of sizing. During unwinding, tension can be easily affected by internal and external factors (such as unwinding radius, unwinding speed, and moment of inertia) to cause fluctuations, which have adverse effects on the final properties of the sized yarn. In practical production, current tension control mode still relies on the traditional unified PID control method for multiple warp beams. ¹ Only one tension sensor detects the combined tension and adjusts them collectively. This approach cannot independently control single warp beam. Moreover, PID control has limitations in managing time-varying and strongly coupled tension systems.^2,3 As a result, the tension of warp yarns remains uneven after control. Therefore, it is imperative to clarify the unwinding behavior of warp beam and propose new tension control mode to maintain stable tension throughout the entire unwinding process.

At present, Wang ⁴ and Xiao ⁵ theoretically and experimentally analyzed the factors affecting the unwinding tension. However, these studies only established simple model related to unwinding tension, leading to an incomplete reflection of the actual unwinding process. For new intelligent tension control method, researchers such as Alkhafaj and Uzun, ⁶ Li et al. ⁷ and Zhu et al. ⁸ combined particle swarm algorithms, neural networks, and fuzzy control with PID control to achieve adaptive tension control in sizing and related fields. In particular, fuzzy control has notable advantages. It applies fuzzy sets and rules for reasoning according to the control experience of production operators or the knowledge of relevant experts to realize precise tension control.^9–11 However, the above approaches still use the mode of measuring tension to control tension. Due to space limitations, the sizing machine cannot install independent tension sensors to control tension of single warp beam.

Therefore, this paper aims to address these issues and proposes the following research hypotheses: (1) A coupling model relationship exists between tension and factors such as speed, radius, and brake air pressure. (2) The unwinding turns sensor can be easily installed on each warp beam, enhancing tension control accuracy through real-time monitoring and feedback adjustment. Based on these hypotheses, this paper investigates the tension coupling models and introduces a new tension control mode for single warp beam, verifying the hypotheses through experiments.

The main contribution of our study is that we first established the coupling models between unwinding tension and various factors and verified them through the experimental data gathered from constructed detection devices. Subsequently, these models are applied to design a novel fuzzy adaptive PID control system to realize the constant tension of single warp beam. The system takes the unwinding length as the control objective and utilizes the unwinding turns detection device to monitor real-time changes in the unwinding process. Finally, the proposed control system is proven to be more effective than traditional PID control through simulations and analysis under varying conditions. Our research presents the following innovations:

• Tension coupling models of the unwinding process are developed and validated to provide important theoretical support for optimizing the performance of sizing machines.

• A novel single warp beam tension control system is designed to accurately control tension by measuring unwinding turns. The sensor is easy to install and cost-effective.

• Based on the coupling models, a fuzzy PID control method is developed to maintain constant unwinding tension for single warp beam under varying conditions.

Literature review

Warp tension, as a key factor in the sizing process, directly affects the quality of the sized yarns.^12,13 To achieve stable tension, it is necessary to establish a clear theoretical model that defines the relationship between tension and influencing factors, as well as to design an efficient tension control system. This section reviews previous research on tension theory models and control systems in sizing fields, aiming to identify research gaps and further highlight the innovations of this study.

Tension theory models

Tension theory models play an important role in the field of tension control. These models provide a theoretical foundation that helps to clarify the interactive influences of various factors on tension. ¹⁴

In the unwinding process of sizing, the related works for tension influencing factors are shown in Table 1. Zhang ¹⁵ established a theoretical relationship between unwinding tension, radius, and warp beam weight, exploring the root causes of the uneven tension. Next, Hu ¹⁶ applied Euler’s formula to determine the relationship between the encircling angle of sheet yarns and friction coefficients on unwinding tension, noting that proximity to the sizing tank correlates with higher tension in sheet yarns. Qu ¹⁷ established the relationship between unwinding tension, warp beam diameter, and friction force. The results showed that using materials with a lower coefficient of friction for the warp beam’s sliding bearings can result in more uniform unwinding tension. Similarly, Yu ¹⁸ derived a similar model using the moment balance equation and applied it to analyze tension variations during the unwinding of multiple warp beams.

Table 1.

The related works for tension theory models.

References	Factors considered in the models	Research gaps
Zhang 15	Unwinding tension, unwinding radius, and warp beam weight	Under static conditions Established simple model Neglected model validation
Hu 16	Unwinding tension, encircling angle, and friction coefficients
Qu 17	Unwinding tension, warp beam diameter, and friction force
Yu 18	Unwinding tension, warp beam radius, and friction force
Our study	Unwinding tension, unwinding radius, unwinding speed, unwinding turns, braking air pressure, and yarn characteristics	Under dynamic conditions Established coupled models Validated models through experiments

Tension control systems

Tension control systems are essential in industrial production processes. They keep tension within a preset stable range, thereby improving product quality and increasing production efficiency.^19–21

In the unwinding process on the sizing machines, related works for tension control systems are summarized in Table 2. Tong ²² used tension and elongation as detection objects, employing the PID control method to uniformly control multiple warp beams, thereby achieving stable tension in the unwinding area. Wang and Luo ²³ detected the real-time tension and speed of warp sheets from multiple warp beams to ensure constant tension. Li et al. ²⁴ proposed multi-neuron PID control method to improve warp tension stability, demonstrating a system transition time of approximately 0.5 s without overshoot. Afterward, Liu and Luo ²⁵ focused on warp yarn unwinding tension as the control target and utilized a fuzzy PI controller for adjustment. Their results showed that the method effectively controlled unwinding tension with robust dynamic response, anti-interference capability, and control accuracy.

Table 2.

The related works for tension control systems.

References	Detection objects	Control methods	Control objects	Research gaps
Tong 22	Tension and elongation	PID	Total tension of multiple warp beams	Unified control of multiple warp beams Tension control by measuring tension No independent sensor installation for single warp beams
Wang and Luo 23	Tension and speed	PID	Total tension of multiple warp beams
Li et al 24	Unwinding tension	Neural network PID	Total tension of multiple warp beams
Liu and Luo 25	Unwinding tension	Fuzzy-PI	Total tension of multiple warp beams
Our study	Unwinding turns	Fuzzy PID	Tension of single warp beam	Individual control of single warp beam Tension control by measuring unwinding turns The sensor is easy to install and cost-effective

Research gaps

The above studies have revealed significant research gaps in the tension theory models and control systems for the unwinding process on the sizing machines, which still require further investigation.

Firstly, previous studies primarily focused on simple tension theory models under static conditions and lacked model validation. The coupling models between the unwinding tension and influencing factors of the sizing machine need to be further improved. In our study, the coupling models between unwinding tension and factors such as unwinding radius, speed, turns, air pressure and yarn characteristics were established through dynamic analysis of the warp beam and yarn. Additionally, five types of detection devices were developed to analyze the behavior of each factor and verify the accuracy of the models. This provides a solid theoretical foundation for innovations in unwinding tension control technology for sizing machines.

Secondly, due to limited installation space, the traditional unified control method of multiple warp beams in sizing machines cannot install independent tension sensors to control each warp beam individually. However, the distinct variations in tension states across each warp beam necessitate further refinement. To address this, we propose a new control mode and method. Through the unwinding turns detection device, the real-time monitoring of the unwinding state of single warp beam is achieved to ensure constant tension for each warp beam. Figure 1 illustrates the traditional and proposed tension control modes. Furthermore, drawing on practical production experience, we designed a fuzzy adaptive PID controller to enhance the system’s intelligence and response speed. This offers strong technical support for optimizing and upgrading the sizing production process.

Figure 1.

Schematic diagram of the warp beam unwinding process. (a) Traditional control mode of measuring tension to control tension; (b) proposed control mode of measuring unwinding turns to control tension.

Theoretical models

The warp beam unwinding process of the sizing machine involves the simultaneous passive unwinding of warp sheets from the warp beam, encompassing multiple warp beams and yarn guide rollers. Specifically, schematic diagram of the single warp beam unwinding process is shown in Figure 2. The speed difference between the warp beam and the guide roller generates unwinding tension. As the unwinding radius of the warp beam decreases, its rotational speed and number of turns gradually increase. The air pressure of the brake cylinder is automatically adjusted by the warp beam unwinding tension control system to maintain tension stability. To comprehensively understand the interaction and influence between unwinding tension and various factors, it is necessary to establish and validate theoretical couple models for factors such as tension, speed, radius, turns, and air pressure.

Figure 2.

Schematic diagram of the single warp beam unwinding process. (a) Single warp beam unwinding; (b) empty warp beam; (c) single warp yarn before and after stretching.

Theoretical model 1: Unwinding tension, speed, radius and air pressure

According to Newton’s second law of rotational motion, the dynamic moment balance equation of the warp beam unwinding process at any time $t$ is expressed as follows ²⁶ :

$T\left(t\right)R\left(t\right)-M_f\left(t\right)-M_0\left(t\right)=\frac{d\left(J\omega \right)}{dt}$

(1)

where $T\left(t\right)$ is the unwinding tension of the warp yarn, $R$ ( $t$ ) is the radius of the warp beam, $M_f\left(t\right)$ is the frictional resistance torque, $M_0\left(t\right)$ is the braking torque, $J\left(t\right)$ is the moment of inertia of the warp beam, $\omega \left(t\right)$ is the angular speed of the warp beam.

Specifically, the braking torque $M_0\left(t\right)$ and the frictional resistance torque $M_f\left(t\right)$ can be given as:

$M_0\left(t\right)=K_{0}P\left(t\right)$

(2)

$M_f\left(t\right)=\beta \left(G_0+G_1\right)r=\beta \left[G_0+\pi b\rho \left(t\right)\left(R^2\left(t\right)-{R_0}^2\right)\right]r$

(3)

where $K_0$ is constant, $P\left(t\right)$ is the air pressure of brake cylinder, $\beta$ is the coefficient of friction, $r$ is the ruffle radius, $G_0$ is the weight of empty warp beam, $G_1$ is the weight of warp yarns on the warp beam, $b$ is the distance between flanges, $\rho \left(t\right)$ is the winding density of warp yarn, $R_0$ is the barrel radius.

The moment of inertia of the warp beam $J\left(t\right)$ consists of the moment of inertia of empty warp beam $J_0=\frac{1}{2}{G}_0{R}_0^2$ and the moment of inertia of warp yarn on the warp beam $J_1\left(t\right)=\frac{1}{2}\pi b\rho \left(t\right)\left(R^4\left(t\right)-{R_0}^4\right)$ . Thus:

$\frac{dJ\left(t\right)}{dt}=\frac{1}{2}\pi b\left(R^4\left(t\right)-{R_0}^4\right)\frac{d\rho \left(t\right)}{dt}+2\pi b\rho \left(t\right)R^3\left(t\right)\frac{dR\left(t\right)}{dt}$

(4)

Then, the radius of the warp beam $R\left(t\right)=R_1-\frac{\theta \left(t\right)}{2\pi }h$ and the angular speed of the warp beam $\left(t\right)=\frac{V\left(t\right)}{R\left(t\right)}$ , so the rate of change of $R\left(t\right)$ and $\omega \left(t\right)$ is:

$\frac{dR\left(t\right)}{dt}=-\frac{h}{2\pi }\frac{V\left(t\right)}{R\left(t\right)}$

(5)

$\frac{d\omega \left(t\right)}{dt}=\frac{1}{R\left(t\right)}\frac{dV\left(t\right)}{dt}-\frac{V\left(t\right)}{R^2\left(t\right)}\frac{dR\left(t\right)}{dt}$

(6)

where $R_1$ is the initial radius of warp beam, $\theta \left(t\right)$ is the angular displacement of warp beam, $h$ is the thickness of warp yarn, $V\left(t\right)$ is the unwinding speed of warp beam.

Through the derivation of Equations (1) to (6), the coupling model between unwinding tension and various factors is established:

$T\left(t\right)=\frac{K_{0}P\left(t\right)}{R\left(t\right)}+\frac{\beta r\left[G_0+\pi b\left(R^2\left(t\right)-{R_0}^2\right)\rho \left(t\right)\right]}{R\left(t\right)}+K_1\left[\frac{1}{R^2\left(t\right)}\frac{dV\left(t\right)}{dt}+\frac{h{V}^2\left(t\right)}{2\pi R^4\left(t\right)}\right]+K_2\frac{V\left(t\right)}{R^2\left(t\right)}\frac{d\rho \left(t\right)}{dt}-hb{V}^2\left(t\right)\rho \left(t\right)$

(7)

The expressions of $K_1$ and $K_2$ are:

$K_1=\frac{1}{2}{G}_0{R}_0^2+K_2\rho \left(t\right);\hspace{0.5em}{K}_2=\frac{1}{2}\pi b\left[R^4\left(t\right)-{R_0}^4\right]$

(8)

To sum up, equation (7) clarifies the coupling relationship between the unwinding tension, tension, speed, radius and air pressure. At the same time, it is shown that the unwinding tension control system is a nonlinear and time-varying system.

Theoretical model 2: Unwinding tension, speed and yarn characteristics

The distance between the warp beam and the guide roll remains constant during the unwinding process, so it is a typical mass flow system. Based on the mass conservation law of the yarn before and after stretching, the model between unwinding tension, speed and yarn characteristics can be written as ²⁷ :

$\frac{dT}{dt}=\frac{EA}{L}\left[V^{\prime }\left(t\right)-V\left(t\right)\right]-\frac{1}{L}T\left(t\right)V^{\prime }\left(t\right)$

(9)

where $E$ and $A$ are the elastic modulus and cross-sectional area of the warp yarn, $L$ is the distance between the warp beam and the guide roller, $V^{\prime }\left(t\right)$ is the speed of the guide roller.

Theoretical model 3: Unwinding speed, radius and number of turns

When the unwinding speed of the warp beam increases, the unwinding radius decreases more rapidly, and the number of unwinding turns accumulates faster. During the circular motion of the warp beam unwinding process, the relationship between these three factors is represented as:

$V\left(t\right)=2\pi R\left(t\right)\frac{N\left(t\right)}{t}$

(10)

where $N\left(t\right)$ is the unwinding turns of the warp beam.

Detection devices and experimental setup

Detection devices and principles

Based on the key factors involved in the aforementioned models, five types of detection devices were constructed, including air pressure, unwinding speed, radius, tension and turns detection devices. Figure 3 illustrates the installation schematic of detection devices on the warp beam. The measuring device and their actual installation are presented in Figure 4.

Figure 3.

Installation schematic of detection devices on the warp beam.

Figure 4.

Detection devices and their actual installation. (a) Warp beam measurement with application program; (b) air pressure measuring device; (c) unwinding speed measuring device; (d) unwinding radius measuring device; (e) unwinding tension measuring device; (f) unwinding turns measuring device.

The air pressure sensor is utilized to measure the air pressure of the brake cylinder, with a range of −0.1 to 1.0 Mpa and with an accuracy of 0.2% of full-scale value (FS), as depicted in Figure 4(b). From Figure 4(c), the unwinding speed is accurately determined using the roller speed sensor, which has an accuracy of 0.01% FS and is supported by its fixture. The operating principle is based on circumference measurement, and the diameter of the roller is 64 mm. During the measurement process, the speed sensor is consistently pressed firmly against the warp beam to ensure precision and reliability. In Figure 4(d), the ultrasonic distance sensor, along with the magnetic crab clamp gauge holder fixture, is employed for unwinding radius measurement. This sensor operates by measuring the time difference between the transmission and reception of ultrasonic waves. It has a detection range of 70 to 1000 mm and an accuracy of 0.1% FS. During installation, the sensor is positioned within its sensing range and aligned with the warp beam’s center of gravity.

The three-pulley yarn tension sensor, as shown in Figure 4(e), is based on the resistance-strain measuring principle. It offers a detection range of 0 to 20 N and has a high accuracy of 0.05% FS. This sensor measures the unwinding tension of 10 warp yarns at the leftmost end of the warp beam and then averages the obtained values. Furthermore, the warp sheet tension is derived from the number of warp yarns on the beam. Figure 4(f) illustrates the use of the laser reflection photoelectric sensor along with the tiger clamp fixture to measure the unwinding turns of the warp beam. The laser reflection photoelectric sensor is a widely used non-contact sensor that operates on the principle of detecting the presence or absence of objects through the reflection of a laser beam. Each time the laser beam is reflected, it is counted as one unwinding cycle. It can detect distances up to 0-10 m (with no blind spot), has a response time of 1 ms, and maintains an accuracy of 0.02% FS.

All fixtures utilize magnetic attraction and can be securely affixed to the warp beam frame. To enable simultaneous and real-time data collection from all five sensors, an application program is developed through the Python language with Modbus communication protocol. The serial port and baud rate of the corresponding sensors are set separately to read and save the data.

Experimental materials and methods

In the actual factory, the above devices were employed to monitor changes in various factors throughout the entire unwinding process of four warp beams. The sampling frequency of each device was set at 1 s. Real-time tension data needs to be mean filtered, while unwinding radius data requires correction. The detailed parameters of the measured warp beams are outlined in Table 3.

Table 3.

The detailed parameters of the measured warp beams.

Production varieties	Warp beam No.	Yarn linear density (tex)	Yarn type	Radius of the full warp beam (mm)	Radius of the empty warp beam (mm)	Number of warp yarns
1	WB-1	97.18	Ordinary cotton yarn	346	150	292
2	WB-2	97.18	Slub cotton yarn	302	150	293
3	WB-3	58.31	Ordinary cotton yarn	361	150	390
4	WB-4	58.31	Slub cotton yarn	331	150	385

Note: The warp yarns are all spun by ring spinning.

The raw yarns WB-1-Y, WB-2-Y, WB-3-Y, and WB-4-Y on the aforementioned warp beams underwent testing for changes in tension and elongation on the electronic universal testing machine. The stretching speed was set at 250 mm/min, with a distance of 250 mm between the clamps at both ends. During stretching, one end remained stationary while the other end moved until the yarn broke.

Fuzzy PID tension control system and method

The fuzzy PID control combines fuzzy control with PID control. By utilizing the experience gained from production practice and relevant expert expertise, it establishes the fuzzy control rules and then applies fuzzy reasoning to realize the optimal adjustment of PID control parameters. This approach is particularly effective for uncertain, nonlinear, and time-varying tension control systems. The anti-interference ability of the system is significantly enhanced by the time-varying control parameters $K_P$ , $K_I$ and $K_D$ . Furthermore, the aforementioned theoretical Model 2 and Model 3 illustrate the relationship between the unwinding turns, unwinding speed, unwinding tension and unwinding length. Therefore, a novel fuzzy adaptive PID control system for single warp beam unwinding tension of the sizing machine is further designed. It takes unwinding length as the control objective and employs the unwinding turns detection device to monitor real-time changes in the unwinding process.

Fuzzy PID tension control system

The control schematic of warp beam unwinding tension based on fuzzy adaptive PID controller is shown in Figure 5. It uses the unwinding length as the control target and employs unwinding turns detection device to monitor the real-time unwinding state of the warp beam. This approach differs from traditional tension control systems, which rely on directly measuring tension for regulation.

Figure 5.

The warp beam unwinding tension control based on fuzzy adaptive PID controller.

When there is an error between the reference value and the actual value $L_a\left(s\right)$ of the unwinding length, the fuzzy control outputs adjustments $∆K_p\left(s\right)$ , $∆K_i\left(s\right)$ and $∆K_d\left(s\right)$ of the proportional, integral and differential coefficients of the PID control parameters to momentarily adjust the PID control parameters. Next, the PID controller calculates the control signal $S\left(s\right)$ and sends it to the brake. The brake generates the corresponding braking torque $M_0\left(s\right)$ to change the warp beam’s unwinding speed $V\left(s\right)$ . Thus, when the speed of the guide rollers $V^{\prime }\left(s\right)$ is constant, the speed difference between the guide rollers and the warp beam can be changed to regulate the tension $T\left(s\right)$ . During the unwinding process, the self-developed unwinding turns detection device monitors real-time changes in the unwinding status, forming a closed-loop control of the system to ensure the stability of the tension.

Therefore, the main components of the control system include computer, fuzzy adaptive PID controller, brake, warp beam and measuring device, as illustrated in Figure 6. Based on their respective characteristics and mathematical models, the transfer function of each control element is derived. Transfer functions are critical tools for analyzing, designing, and simulating control systems.

Figure 6.

The fuzzy PID control system for warp beam unwinding tension.

The brake is the actuating element. The output braking torque $M_0\left(s\right)$ is driven through an amplifier based on the input regulation $S\left(s\right)$ . This process is simplified to a proportional link. The transfer function is:

$G_1\left(s\right)=\frac{M_0\left(s\right)}{S\left(s\right)}=K_3$

(11)

The warp beam serves as the controlled object. In Model 1, both unwinding radius and moment of inertia are time-varying parameters, which are related to the thickness of the yarn $h$ . However, the thickness is at the micrometer scale. Therefore, the variation of radius and moment of inertia per unit time is negligible. Equation (1) is further simplified and Laplace transformed to obtain the transfer function:

$G_2\left(s\right)=\frac{V\left(s\right)}{TR-M_0\left(s\right)}=\frac{R}{Js+\beta }$

(12)

The measuring device utilized is a laser photoelectric sensor. The actual unwinding speed is obtained by detecting the unwinding turns of the warp beam (Model 3). The relationship between the unwinding speed and unwinding length is integral. Thus, the transfer function of the detection device can be described as:

$G_3\left(s\right)=\frac{L_a\left(s\right)}{V\left(s\right)}=\frac{K_4}{s}$

(13)

The relationship between unwinding tension and speed is expressed in Model 2. The transfer function is derived through the application of Laplace transformation to this model, which can be expressed as:

$G_4\left(s\right)=\frac{T\left(s\right)}{V^{\prime }\left(s\right)-V\left(s\right)}=\frac{K_5}{s+C}$

(14)

where $K_5=n\frac{EA}{L}$ , $C=\frac{V^{\prime }}{L}$ , $n$ is the total number of yarns on the warp beam.

Fuzzy PID control method design

The core of the fuzzy adaptive PID control lies in utilizing the fuzzy controller to adjust the PID control parameters online. ²⁸ It can be stated as follows ²⁹ :

$\{\begin{array}{l}{K}_p\left(t\right)=K_{p0}+∆K_p\left(t\right)\\ K_i\left(t\right)=K_{i0}+∆K_i\left(t\right)\\ K_d\left(t\right)=K_{d0}+∆K_d\left(t\right)\end{array}$

(15)

where $K_{p0}$ , $K_{i0}$ and $K_{d0}$ are initial values of $K_p\left(t\right)$ , $K_i\left(t\right)$ and $K_d\left(t\right)$ , respectively. $∆K_p\left(t\right)$ , $∆K_i\left(t\right)$ and $∆K_d\left(t\right)$ are defuzzified outputs of $K_p\left(t\right)$ , $K_i\left(t\right)$ and $K_d\left(t\right)$ , respectively.

The fuzzy adaptive controller can be described as ³⁰ :

$\{\begin{array}{l}e\left(t\right)=L_p\left(t\right)-L_a\left(t\right),∆e\left(t\right)=e\left(t\right)-e\left(t-1\right)\\ x\left(t\right)={\left[e\left(t\right)/K_e,∆e\left(t\right)/K_{ec}\right]}^T\\ u\left(t\right)={\left[∆K_p\left(t\right)/K_{up},∆K_i\left(t\right)/K_{ui},∆K_d\left(t\right)/K_{ud}\hspace{0.17em}\right]}^T\end{array}$

(16)

where $e\left(t\right)$ is the error between the preset and the actual unwinding length, $∆e\left(t\right)$ is the error change. $x\left(t\right)$ is the input variable, it includes $e\left(t\right)$ and $∆e\left(t\right)$ . $K_e$ and $K_{ec}$ are the quantization factors of $e\left(t\right)$ and $∆e\left(t\right)$ , respectively. $u\left(t\right)$ represents the output variable, encompassing $∆K_p\left(t\right)$ , $∆K_i\left(t\right)$ and $∆K_d\left(t\right)$ . $K_{up}$ , $K_{ui}$ and $K_{ud}$ are the scaling factors of $∆K_p\left(t\right)$ , $∆K_i\left(t\right)$ and $∆K_d\left(t\right)$ , respectively.

Generally, the implementation of a fuzzy controller consists of three main steps: input fuzzification, establishment of fuzzy control rules and defuzzification of the output.

Fuzzification

The process of fuzzification is to convert precise inputs into fuzzy sets through the membership functions of linguistic variables and their corresponding linguistic values. Specifically, the linguistic variable of $e\left(t\right)$ is $E$ , the linguistic variable of $∆e\left(t\right)$ is $EC$ , while the fuzzy domains of them are [-11, 11] and [-15, 15] respectively. To describe the state of tension error throughout the entire unwinding process, the linguistic values assigned to these variables include NB (Negative Big), NM (Negative Medium), NS (Negative Small), ZO (Zero), PS (Positive Small), PM (Positive Medium), and PB (Positive Big).

The linguistic variables of $∆K_p\left(t\right)$ , $∆K_i\left(t\right)$ and $∆K_d\left(t\right)$ are denoted as $KP$ , $KI$ , and $KD$ , respectively. The fuzzy theory domains for all these variables are [0, 3], and the linguistic values are ZO, PS, PM, and PB. In this case, Gaussian membership functions are employed for the input linguistic values NB and PB as well as the output linguistic values ZO and PB. On the other hand, triangular membership functions are used for the remaining linguistic values. Figure 7 shows the membership functions for input and output linguistic variables. The X-axis represents the fuzzy domain of the input and output variables, while the Y-axis indicates their membership value, ranging from 0 to 1. This membership value describes the degree to which the input and output variables belong to a specific fuzzy set. The system uses these membership values to match and activate the appropriate rules from the fuzzy rule base.

Figure 7.

The membership functions for input and output linguistic variables. (a) $E$ ; (b) $EC$ ; (c) $KP$ ; (d) $KI$ ; (e) $KD$ .

Fuzzy rules

The fundamental of fuzzy control is fuzzy inference based on fuzzy rules. These rules consist of a series of if-then rules. Based on expert knowledge and operator experience, we have gained a deep understanding of the roles of the three PID control parameters and mastered how to adjust them in different situations to achieve optimal control.

Specifically, the role of the proportionality coefficient KP is to speed up the system’s response and improve its regulation accuracy. The integral coefficient KI serves to eliminate the system’s steady-state error. The function of the differential coefficient KD is to enhance the system’s dynamic characteristics and stability by predicting changes in deviation in advance. Therefore, when the error $E$ is large, indicating a significant deviation from the desired setpoint. The fuzzy rules recommend taking a larger value of $KP$ to rapidly reduce the error. At the same time, a smaller value of $KI$ is chosen to avoid over-correction and potential oscillations, while $KD$ is maintained at a moderate level to provide damping and stability. Conversely, when the error $E$ is small, suggesting that the system is close to the desired setpoint. The fuzzy rules advise opting for a smaller value of $KP$ to prevent overshooting and fine-tune the system. Larger values of $KI$ and $KD$ may be chosen to enhance the system’s ability to eliminate residual errors and improve steady-state performance. The change of the error $EC$ follows the value of the error $E$ . $EC$ provides insights into the trend of $E$ , enabling the fuzzy controller to predict future changes and adjust control actions accordingly. In this study, the Mamdani inference method is employed for inference.

The fuzzy control rules for the output variables of KP, KI, and KD are presented in Table 4, totaling 49 rules. For example, the first rule in Table 4: If (E is NB) and (EC is NB) then (KP is PB) (KI is ZO) (KD is PS). It indicates when the current input variable E and EC are NB of the system, the output variable KP, KI and KD will be set to PB, ZO and PS, respectively. All other rules follow similar principles. The specific settings of KP, KI and KD are decided according to the fuzzy states of input variables E and EC. In practice, the system’s controller applies the Mamdani inference method to determine the fuzzy values of the output variables. This process ensures that the PID controller can flexibly adjust its control parameters according to the system state to achieve better control results.

Table 4.

Fuzzy rules for $KP$ , $KI$ and $KD$ .

Defuzzification

The center of gravity method is used to convert a fuzzy conclusion into a precise value. It refers to the center of gravity of the area enclosed by the membership function curve and the abscissa. The formula of the center of gravity method is shown as ³¹ :

$u\left(t\right)=\frac{{\displaystyle \sum }_{k=1}^m{\mu \left(u_k\right)u}_k}{{\displaystyle \sum }_{k=1}^m\mu \left(u_k\right)}$

(17)

In contrast to the commonly employed maximum membership method, the center of gravity method has smoother output inference control. Even for minor changes in the input signal, the output undergoes proportional adjustments. Notably, this method offers high calculation accuracy and realistic results. Figure 8 illustrates the control surface of $KP$ , $KI$ , and $KD$ .

Figure 8.

The control surface of $KP$ , $KI$ , and $KD$ . (a) $KP$ ; (b) $KI$ ; (c) $KD$ .

Results and discussions

Experimental validation of theoretical model

According to the test results of the monitored changes in various factors during the unwinding process of the four warp beams and the corresponding raw yarn tensile test results, we have comprehensively validated the established models. Specifically, the measured relevant data of tension, radius, speed, and air pressure measured during the warp beam unwinding process were fitted to Model 1 and Model 3. Additionally, the tension and elongation data of the warp yarn before breakage verified Model 2.

Testing data analysis

The test results of WB-1, WB-2, WB-3, and WB-4 under the PID tension control method during the actual unwinding process are illustrated in Figure 9. From Figure 9 (a1-2) (b1-2) (c1-2) and (d1-2), when the warp beams maintain at relatively constant speed during the unwinding process, the unwinding tensions of the four warp beams fluctuate around a constant value. At present, the warp beam unwinding tension control system utilizes PID closed-loop control. When the unwinding tension deviates from the set value, the braking torque of the warp beam is adjusted by modifying the air pressure to maintain stable tension. The sizing machine detects the yarn tension of multiple warp beams in the same sizing tank, and then collectively regulates all warp beams. However, differences in warping production, braking mechanisms, among other factors, contribute to variations among individual warp beams. Therefore, the uniform air pressure control for each warp beam leads to inaccurate control and periodic fluctuations in unwinding tension.

Figure 9.

Changes in warp beams unwinding tension, speed, air pressure and turns with radius under the PID tension control method during the actual unwinding process. (a) WB-1; (b) WB-2; (c) WB-3; (d) WB-4.

In Figure 9 (a3-4) (b3-4) (c3-4) and (d3-4), it is evident that as the unwinding radius decreases, the air pressure gradually decreases while the number of unwinding turns increases. According to equation (7), when the unwinding speed and braking torque are basically constant, the warp beam unwinding tension will gradually increase with the decrease of the unwinding radius until the yarn breaks. Therefore, the warp beam braking torque is reduced by decreasing the air pressure to ensure that the tension is always stabilized. In addition, from equation (10), when the unwinding speed is maintained at a certain level, the number of unwinding turns increases as the unwinding radius decreases.

In summary, during the uniform unwinding process, the unwinding radius of the warp beam gradually decreases and the number of unwinding turns gradually increases. The tension control system reduces the braking torque of the warp beam by constantly decreasing the air pressure to keep the constant tension.

Theoretical model 1 validation

In the actual production process, the warp beam related parameters $\beta ,r,b,\rho ,h,R_0$ and $G_0$ can be treated as constants. Therefore, the change rate of winding density $\frac{d\rho \left(t\right)}{dt}=0$ . When the unwinding speed is essentially constant, equation (7) of Model 1 can be simplified as:

$T\left(t\right)R\left(t\right)=AP\left(t\right)+B{R}^2\left(t\right)+C\frac{V^2\left(t\right)}{R^3\left(t\right)}+D{V}^2\left(t\right)R\left(t\right)+E$

(18)

where $A,B,C,D$ and $E$ are all constant. The expressions of them are:

$\{\begin{array}{l}{A=K}_0\\ B=\beta r\pi b\rho \\ C=\frac{G_0{R}_0^{2}h-\pi b\rho R_0^{4}h}{4\pi }\\ D=-\frac{3}{4}b\rho h\\ E=\beta r{G}_0-\beta r\pi b\rho R_0^2\end{array}$

(19)

Model 1 employed the nonlinear least squares fitting method through the lsqnonlin function in Matlab. This method is better suited for handling complex nonlinear relationships. The fitting quality was assessed using $R^2$ (coefficient of determination), which ranges from 0 to 1. The $R^2$ value closer to 1 indicates a higher degree of model fit, while a lower value suggests poorer fitting performance. Figure 10 and Table 5 show the fitting of equation (18) to the experimental results. The $R^2$ values of WB-1, WB-2, WB-3 and WB-4 were 90.53%, 90.39%, 95.09% and 94.06%, respectively. Consequently, it is evident that the relationship between the unwinding tension, speed, radius, and air pressure adheres to the coupling Model 1. Based on this model, precise air pressure adjustments can be obtained to ensure constant tension throughout the unwinding process as the actual unwinding radius and speed change. The consistency of tension significantly contributes to improving sizing quality and productivity.

Figure 10.

Validation of model 1. (a) WB-1; (b) WB-2; (c) WB-3; (d) WB-4.

Table 5.

Fitting results of Model 1.

Warp beam no.	Fitted equation	Fit metric $R^2$ (%)
WB-1	$T*R=65P+24.6R^2+0.000001\frac{V^2}{R^3}-0.0077V^{2}R+0.30$	90.53
WB-2	$T*R=55P+29.3R^2+0.000003\frac{V^2}{R^3}-0.0043V^{2}R+0.52$	90.39
WB-3	$T*R=40P+8.0R^2+0.000002\frac{V^2}{R^3}-0.0028V^{2}R+0.20$	95.09
WB-4	$T*R=80P+11.4R^2+0.000001\frac{V^2}{R^3}-0.0062V^{2}R+1.01$	94.06

Note: In the fitted equation, $R$ represents unwinding radius. The fit metric of the fitted equations is $R^2$ .

Theoretical model 2 validation

During the warp stretching process, $V^{\prime }\left(t\right)$ operates at a constant value and $V\left(t\right)=0$ . Therefore, the equation (9) of Model 2 is further simplified to:

$T=\frac{EA}{L+V^{\prime }}∆L$

(20)

where $\frac{EA}{L+V^{\prime }}$ is constant, $∆L$ is the difference in elongation produced by the difference in speed.

The linear fitting method in Origin software was used to fit Model 2, as it is simple and applicable to this model. The results of the tensile tests for WB-1-Y, WB-2-Y, WB-3-Y and WB-4-Y and the corresponding fitting results are presented in Figure 11 and Table 6. From Figure 11, the tension of the warp yarn gradually increases with increasing elongation during the stretching process. Table 6 shows that the $R^2$ values for each warp beam after fitting using Model 2 are 99.92%, 99.85%, 99.96% and 99.95%, respectively. The value of $\frac{EA}{L+V^{\prime }}$ varies among yarns due to differences in spinning method, modulus of elasticity, and cross-sectional area.

Figure 11.

Validation of model 2. (a) WB-1-Y; (b) WB-2-Y; (c) WB-3-Y; (d) WB-4-Y.

Table 6.

Fitting results of Model 2.

Warp yarn no.	Fitted equation	Fit metric $R^2$ (%)
WB-1-Y	$T=140.328∆L$	99.92
WB-2-Y	$T=79.894∆L$	99.85
WB-3-Y	$T=52.744∆L$	99.96
WB-4-Y	$T=54.785∆L$	99.95

Theoretical model 3 validation

The equation (10) of Model 3 can be written as:

$N\left(t\right)=\frac{V\left(t\right)t}{2\pi R\left(t\right)}$

(21)

The theoretical number of turns( $N_{the}\left(t\right)$ ) was calculated using equation (21) based on the measurements of the unwinding radius and speed at any consecutive moments during the unwinding process. Subsequently, we utilized linear fitting methods in Origin software to fit the Model 3. The results of fitting the measured unwinding turns ( $N_{act}\left(t\right)$ ) to the theoretical values are illustrated in Figure 12 and Table 7. It demonstrated that the $R^2$ values of the WB-1, WB-2, WB-3 and WB-4 are 99.58%, 99.68%, 99.52% and 99.47%, respectively. Although the number of unwinding turns is an integer, there is a slight difference between theory and practice. Therefore, a Model 3 relationship exists among the three variables.

Figure 12.

Validation of model 3. (a) WB-1; (b) WB-2; (c) WB-3; (d) WB-4.

Table 7.

Fitting results of Model 3.

Warp beam no.	Fitted equation	Fit metric $R^2$ (%)
WB-1	$N_{act}=N_{the}-0.698$	99.58
WB-2	$N_{act}=N_{the}-0.589$	99.68
WB-3	$N_{act}=N_{the}+0.112$	99.52
WB-4	$N_{act}=N_{the}+0.108$	99.47

In summary, the experimental results demonstrated that the $R^2$ values for the three models all exceeded 90%, confirming the strong relationships between unwinding tension, speed, radius, turns and air pressure. Additionally, the detection device for unwinding turns offers the lowest cost and easiest installation compared to other detection devices.

At present, the conventional tension control method typically focuses on controlling tension by measuring tension. However, due to constraints in actual installation space, the individual detection of single warp beam unwinding tension has not been realized. It is worth noting that differences exist among warp beams. Consequently, to achieve uniform unwinding tension across the entire sheet yarns, it is fundamental to realize the stability in tension of single warp beam.

Based on actual production testing and the theoretical models outlined, we propose a new tension control approach. Specifically, Model 3 illustrates the relationship between the unwinding turns and unwinding speed. Moreover, there is also a relationship between the unwinding speed and the unwinding tension (Model 2), as well as the unwinding length ( $\int Vdt$ ). Therefore, it is proposed to take the unwinding length as the control target and adopt the unwinding turns detection device to realize the constant tension control of single warp beam.

Simulation verification of fuzzy PID tension control system

Based on the designed control system, along with the established transfer functions of each system component and the formulated fuzzy rules, we conducted a simulation analysis in Matlab Simulink. This analysis aimed to verify the feasibility and effectiveness of the single warp beam unwinding tension control system under various conditions, including different unwinding radius, reference tensions, speed step disturbances, and noise interferences. The values of simulation parameters are mainly taken from relevant references. ³² and actual data, as illustrated in Table 8.

Table 8.

Parameters used in the simulation.

Parameter	Description	Value
$\beta$	Friction coefficient	0.07
$G_0$	Weight of empty warp beam	180 kg
$b$	Distance between the flanges	1800 mm
$R_1$	Initial radius of warp beam	340 mm
$J$	Initial moment of inertia of warp beam	17.66 kg·m2
$R_0$	Barrel radius	150 mm
$r_0$	Ruffle radius	122.5 mm
$\rho$	Winding density of warp beam	0.43 g/cm3
$h$	Thickness of warp yarn (10S)	0.0301 mm
$E$	Modulus of warp yarn	2.06 × 109 N/m2
$A$	Cross-sectional area of warp yarn	6.27 × 10−8 m2
$L$	Distance between warp beam and guide roller	1 m
$n$	Number of warp yarns	400
$K_3$	Proportional coefficient of brake	10
$K_4$	Integral coefficient of measuring device	1
$K_5$	Relationship coefficient between tension and speed	51665
$K_{p0}$	Initial proportional coefficient of PID controller	800
$K_{i0}$	Initial integral coefficient of PID controller	3836
$K_{d0}$	Initial differential coefficient of PID controller	24

According to the relevant simulation parameters, the total unwinding length of the warp beam can be calculated to be 9708.4 m. When the speed is 1.5 m/s, the total unwinding time of the warp beam is 6472 s. Considering the total radius of the warp beam unwinding is 0.34 m, representative values of 0.31 m, 0.25 m and 0.18 m are selected for the large, medium and small warp beams, respectively, to simulate and analyze the entire unwinding process. Based on equation (7), the unwinding radius and moment of inertia at different moments can be obtained, as shown in Table 9.

Table 9.

The unwinding radius and moment of inertia of the warp beam at different times.

Warp beam	Time (s)	$R$ (m)	$J$ (kg·m2)
Large	1360	0.31	12.66
Medium	3703	0.25	6.19
Small	5802	0.18	2.69

Control effect under different reference tensions

To achieve constant unwinding tension with the unwinding length as the control target, simulations were conducted to investigate the tension response during the steady-state stage. When the reference unwinding tension $T_1$ = 120 N and the speed of the guide roller $V^{\prime }$ = 1.5 m/s, the unwinding speed of warp beam $V$ = 1.496515 m/s can be calculated by Model 2. Similarly, under a reference unwinding tension of $T_2$ = 200 N and the same guide roller speed, the unwinding speed of the warp beam is 1.494195 m/s. That is, the set unwinding length is $\underset{0}{\overset{t}{{\displaystyle \int }}}Vdt.$

The simulation results of single warp beam unwinding tension control under the reference unwinding tension of 120 N and 200 N are depicted in Figures 13 and 14, respectively. From Figure 13(a) and 14(a), it can be seen that the input and output unwinding length are basically error-free after the proposed fuzzy adaptive PID method and traditional PID control method. Additionally, both methods achieve constant tension at different unwinding radius and reference tensions, as shown in Figures 13(b) and (c) and 14(b) and (c). However, it is observed that during constant-speed unwinding of large, medium, and small warp beams, the tension fluctuations were notably more pronounced when using the PID control method than the fuzzy adaptive PID method. The reason for this difference lies in the design of two control strategies. The fuzzy adaptive PID control continuously adjusts its control parameters in real-time based on the system’s changing conditions, whereas the traditional PID control operates with fixed control parameters during the entire unwinding process. Consequently, the fuzzy adaptive PID control demonstrates superior control effectiveness.

Figure 13.

The simulation results of single warp beam unwinding tension control under 120 N reference unwinding tension. (a) Unwinding length; (b) unwinding tension of large warp beam; (c) unwinding tension of medium warp beam; (d) unwinding tension of small warp beam.

Figure 14.

The simulation results of single warp beam unwinding tension control under 200 N reference unwinding tension. (a) Unwinding length; (b) unwinding tension of large warp beam; (c) unwinding tension of medium warp beam; (d) unwinding tension of small warp beam.

Control effect under different speed step interferences

To further evaluate the anti-interference capabilities of fuzzy adaptive PID control, the speed step interference was introduced to the guide roller while the reference unwinding tension of 200 N. The detailed parameters are outlined in Table 10. The unwinding speed of the warp beam will follow the speed change of the yarn guide roller to achieve constant tension control. The preset unwinding length is ${\int }_0^{t}Vdt$ .

Table 10.

Parameters of speed step interferences under 200 N reference unwinding tension.

Warp beam	Case no.	Step time (s)	The speed of guide roller (m/s)		The speed of warp beam (m/s)
			Initial	Final	Initial	Final
Large	1	1360	1.5	1.55	1.494195	1.544195
Medium	2	3703	1.5	1.55	1.494195	1.544195
Small	3	5802	1.5	1.55	1.494195	1.544195
Large	4	1360	1.5	1.45	1.494195	1.444195
Medium	5	3703	1.5	1.45	1.494195	1.444195
Small	6	5802	1.5	1.45	1.494195	1.444195

The simulation results are presented in Figures 15 and 16. From Figure 15(a)–(c) and 15(d)–(f), when the speed of the guide roller suddenly increases from 1.5 to 1.55 m/s or decreases to 1.45 m/s, these changes in speed lead to unwinding tension fluctuations. However, as the unwinding radius decreases, the effect of the step disturbance becomes smaller. Both control methods can adjust to restore the tension to the desired state, but the PID control results in greater tension fluctuations.

Figure 15.

Control effect of speed step interferences in fuzzy PID method and the traditional PID method under 200 N reference unwinding tension. (a) Case 1; (b) Case 2; (c) Case 3; (d) Case 4; (e) Case 5; (f) Case 6.

Figure 16.

Tension error under different speed step interferences in case 1-6.

Figure 16 shows the specific tension errors. It is evident that the unwinding tension overshoot and downshoot of the fuzzy PID control for the large, middle and small warp beam are smaller than those of the PID control. After applying the fuzzy PID control method, the tension fluctuation range of Case 1-6 is significantly reduced by 64.49%, 71.40%, 72.25%, 65.92%, 70.14% and 72.59%, respectively. Thus, the proposed fuzzy PID control method is superior to traditional PID control, demonstrating better anti-interference ability.

Control effect under different noise disturbances

Typically, systems are susceptible to noise disturbance originating from various sources such as the external environment, internal components, signal transmission, and other factors. In order to verify the anti-noise interference capability of the system, noise disturbance signals are introduced into the intermediate links of the system. When the reference tension is 200 N, the relevant parameters of different noise signals are shown in Table 11.

Table 11.

Parameters of different noise signals under 200 N reference unwinding tension.

Warp beam	Case no.	Noise power	Sample time(s)
Large	1	1	0.1
Medium	2	1	0.1
Small	3	1	0.1
Large	4	5	0.1
Medium	5	5	0.1
Small	6	5	0.1
Large	7	10	0.1
Medium	8	10	0.1
Small	9	10	0.1

The control effects of the fuzzy PID control method and the traditional PID control method under various noise signal disturbances are illustrated in Figure 17. Throughout the entire unwinding process, tension exhibits varying magnitudes of fluctuation above and below a constant reference value after control by both methods in response to different noise disturbances. Obviously, the fuzzy PID control method further reduces the fluctuation range compared to the PID control method.

Figure 17.

Control effect of different noise signals in the fuzzy PID method and the traditional PID method under 200 N reference unwinding tension. (a) Case 1; (b) Case 2; (c) Case 3; (d) Case 4; (e) Case 5; (f) Case 6; (g) Case 7; (h) Case 8; (i) Case 9.

Moreover, it is observed that the magnitude of tension fluctuation increases with the intensity of noise. In contrast to Figure 17(a)–(c), (d)–(f) or (g)–(i), as the unwinding radius of the warp beam decreases, the fluctuations in tension under the same noise conditions also diminish. The specific tension error values for Cases 1-9 are detailed in Figure 18. Compared to traditional PID control, the fuzzy PID control method reduces the tension fluctuation range by 29.41%, 29.33%, 24.89%, 30.13%, 28.07%, 24.29%, 31.02%, 27.81%, and 23.66% for Cases 1-9, respectively. Hence, the proposed method has better anti-noise interference capability.

Figure 18.

Tension error under different noise disturbances in case 1-9.

In summary, the fuzzy PID control can achieve constant tension control in the entire unwinding process of single warp beam by controlling the unwinding length. When there are speed step interferences and noise disturbances, the fuzzy PID control has better control effects than traditional PID. Therefore, the proposed single warp beam unwinding tension control method exhibits stability, robustness, and anti-interference ability.

Cost-benefit analysis of the system

In this paper, constant unwinding tension control for single warp beam is successfully realized by measuring the unwinding turns and applying the fuzzy PID control method. The proposed system does not require any modification of the warp beam frame of the sizing machine. Only unwinding turns sensor, controller, proportional valve and other auxiliary components need to be purchased. The initial investment for single warp beam is estimated to be a maximum of US$198. In contrast, the traditional tension control mode of measuring tension requires modification of the warp beam frame, resulting in significantly higher costs. More importantly, the cost of a tension sensor (about US$150) is much higher than that of an unwinding turns sensor (about US$12). Additionally, components such as the controller and proportional valve also need to be purchased. Therefore, the proposed system is low-cost and easy to implement.

Moreover, this system effectively reduces tension fluctuation during the entire unwinding process by precisely controlling the unwinding tension of single warp beam. The quality and efficiency of the sizing process is significantly improved. If this system is applied to the entire warp beam frame, it will ensure consistent unwinding tension across multiple warp beams, reducing residual yarn at the end of the unwinding process and conserving resources. ²⁷ Furthermore, the improvement of sizing quality contributes to enhanced weaving efficiency and ensures the uniformity of the fabric surface during the weaving process. ³³ It enhances product quality and increases company revenue. The specific costs and revenues will depend on the type of yarn used in production. In the future, we will expedite the practical application of this system and continue to monitor the specific cost-benefit data.

Conclusions and future research

In this study, experimental data on tension, speed, radius, turns, and air pressure during the warp beam unwinding process are successfully obtained through the construction of detection devices to verify the established coupling models between these factors. Subsequently, a novel fuzzy adaptive PID control system for single warp beam unwinding tension is designed based on these models, and its performance is evaluated through simulations. The main conclusions are summarized as follows:

(1) The tension coupling models are established and verified. Experimental data of the four measured warp beams are fitted to verify the established coupled model relationships between unwinding tension, speed, radius, turns and air pressure. Specifically, the fit metrics $R^2$ values for these models all exceed 90%, demonstrating their accuracy and reliability.

(2) Based on the above models, a novel fuzzy adaptive PID control system for single warp beam unwinding tension is proposed with the unwinding length as the control target. This system uses the unwinding turns detection device to monitor real-time changes in the warp beam, thereby maintaining constant tension. It offers a new control mode of measuring turns to control tension and differs from the traditional method of measuring tension to control tension.

(3) The proposed system demonstrates stability and robustness in controlling the unwinding tension of single warp beam through simulation analysis. It can maintain constant tension under different reference tensions and unwinding radius. Compared to traditional PID control, the fuzzy adaptive PID control method significantly reduces tension fluctuation, with reductions of 64.49% under different speed step disturbances and 23.66% under noise disturbances.

Therefore, the proposed tension coupling models and novel tension control method have been proven effective. Based on these findings, future research should prioritize further experimental validation in real-world conditions and explore new control methods (such as predictive control) to optimize the system. Additionally, achieving consistent tension for multiple warp beams in sizing machines is essential. It is also advisable to broaden the application of this technology across the textile industry to improve production efficiency and product quality.