Tension control of a yarn winding system based on the nonlinear active disturbance-rejection control algorithm

Abstract

This research proposes a new system for controlling yarn tension in the winding operation based on active disturbance-rejection control (ADRC) technology, which controls and minimizes the fluctuation of tension during winding. The mechanical structure of the winding machine was analyzed to clarify the main factors affecting the yarn tension. Proportional–integral–derivative (PID) control is the most common control method of the winding machine. In order to compare the control effects of ADRC and PID control, the mathematical model of the yarn-tension control system was established on the MATLAB platform by means of system identification. Through the simulation on this model, the results showed that under the action of disturbance, the overshoot of the yarn tension controlled by the active disturbance-rejection controller was smaller, the adjustment time was shorter, and the steady-state error was smaller. An experimental platform was built to test the control effect of the controller under different parameters, thereby verifying the performance and stability of the active disturbance-rejection controller. The results showed that the controller had good performance and stability.

Keywords

Tension control active disturbance-rejection control yarn winding systematic identification

Winding is an important process in the weaving preparatory stage. Its function is to form a package of the desired shape and size with the required compactness.¹ Proper shape and stability of the yarn package allow the yarn to readily unwind in the next process and enable the package to withstand handling during transportation without the yarn sloughing off. The yarn tension in the winding process is a key factor affecting the quality of the yarn package.² The appropriate tension can make the yarn break at weak places, but does not strain it so much as to annihilate its elastic property.³ In this way, some defects in the yarn can be eliminated. In addition, the stability of the yarn tension is related to the winding density of the package. If the yarn tension fluctuates, the winding density of the package will be uneven, resulting in uneven dyeing in the next process and excessive tension fluctuation during unwinding.⁴ Therefore, improving the control accuracy and stability of yarn tension is of great significance to improving the quality of fabrics.

Researchers have carried out considerable investigation to acquire better control results. Zhao and Chang⁵ and Stanislav and Nace⁶ analyzed the principle, composition, and factors affecting the yarn tension of a winding machine to control the yarn-tension constancy of the yarn theoretical basis. Xu et al.⁷ used a charge-coupled device (CCD) image sensor, digital image processing technology, and the mathematical model of the yarn-tension and yarn balloon-shape parameters to calculate yarn tension. This non-contact detection method avoids the sensor's extra yarn tension. Duong et al.⁸ proposed an active winding system to replace the passive winding system of the brushless direct current (DC) coil winding machine, thus improving the performance of the winding machine.

Wang et al.⁹ designed a sensorless tension control method with proportional–integral (PI) parameters of adaptive speed controllers. A tension observer is used to replace the tension sensor with the tension observation value instead of the actual measurement value as feedback, which reduces the measurement delay caused by the tension sensor. Besides, the Landau discrete-time recursive algorithm is used to estimate the inertia of rewinding and unwinding motors, which can adjust the PI parameters of the controller. Thus, the control system can adapt to changes in inertia. Zhou et al.¹⁰ and Wang and Zhao,¹¹ based on the grey prediction model, studied the effect of chemical fiber spinning process parameters on winding tension. Appropriate process parameters were selected to conduct grey relational analysis on winding tension, and the feasibility of the grey prediction model in spinning tension prediction was verified through designed experiments. This application provides a suitable method for spinning tension prediction, which is of great significance to the tension control of chemical fiber products. Researchers have proposed many methods on the control algorithm, including fuzzy control,^12–17 sliding mode control,^18–20 neural networks,^21–26 iterative learning,^27–31 etc.

The above studies improve the performance and adaptability of the tension control system from different angles but ignore the influences of the mechanical structure and parameter changes on the yarn tension. The influence of different parameters on the yarn tension was investigated by analyzing the mechanical structure of the winding machine in the work, with a tension control method proposed.

The work is organized as follows. The second section introduces the mechanical structure of the system, and the influences of some parameters of the mechanical structure on the yarn tension. In the third section, the control strategy is proposed. Simulations are conducted for verifying the effect of the proposed controller by comparing it with that of the conventional proportional–integral–derivative (PID) controller in the fourth section. In the fifth section, the experimental study on the performance verification of the tension control system is performed by building an experimental platform.

Mechanical-structure analysis

Before designing the tension control system, the mechanical structure of the winding machine was analyzed. The winding process was divided into unwind, adjust, and rewind sections. Figure 1 shows the winding process system. The unwind section includes the fed yarn, lower porcelain ring, and lower idle roller. The yarn is unwound from the fed yarn under the traction of the overfeed roller, which causes a lot of shaking. The functions of the lower porcelain ring and the lower idle roller are to stabilize the yarn, reduce the sway of the yarn, and make the yarn enter the overfeed roller smoothly.

Figure 1.
Sketch of the winding process system.

The adjust section mainly includes the overfeed motor, overfeed roller, and tension sensor. The yarn is wound on the overfeed roller and overfeed idle roller two to three times, and the speed of the overfeed roller is controlled according to the feedback value of the tension sensor. Another function of this section is to draw the fed yarn into the machine, thus obtaining the required yarn tension and transport speed.

The rewind section includes the yarn-guide mechanism and the winding mechanism. The speeds of the spindle motor and yarn-guide motor are adjusted according to the set winding speed, the shape of the yarn package, and the real-time package radius, which can obtain the expected yarn package.

Unwind section

The above part briefly introduces the mechanical structure of the winding machine and the role of each section. Next, the relationship between the mechanical structure and tension of each section is analyzed in detail. In Figure 1, the unwind section is composed of the fed yarn, lower porcelain ring, and lower idle roller. The winding machine adopts the passive unwinding form, that is, the yarn is unwound under the traction of the overfeed roller. In this section, the yarn tension is mainly affected by the frictional resistance of the lower idle roller. Assuming that the contact between the yarn and the lower idle roller is uniform with equal friction coefficients of each contact point and no overlap in the yarn winding process, the analysis is as follows under the condition of ignoring the influence of stepping.

In Figure 2, the yarn slides clockwise along the above cross-sectional view, and $F_{0}$ and $F_{1}$ are the yarn tensions before and after entering the lower idle roller, respectively. Assume $θ$ is the angle of the yarn passing the lower idle roller; micro-element $d θ$ is used for analysis; $R d θ$ is the length of this part of the yarn; $R$ is the radius of the lower idle roller. The forces received by the yarn are as follows: tension $F + d F$ on the positive side of the yarn sliding, tension $F$ on the opposite side, radial support force $d N$ , and sliding friction force $f$ perpendicular to the support force. In the steady state, the resultant force is zero, and the force balance in the vertical direction is as follows
$[(F + d F) + F] \sin \frac{d θ}{2} = d N$
(1)

Figure 2.
Cross-sectional view of the lower idle roller.

The force balance in the tangential direction is as follows
$(F + d F) \cos \frac{d θ}{2} = F \cos \frac{d θ}{2} + f$
(2)
where $μ$ is the friction coefficient between the yarn and the guide roller, and the sliding friction force $f$ is as follows
$f = μ d N$
(3)

Here, $d θ$ is tiny, so $\sin \frac{d θ}{2} \approx \frac{d θ}{2}$ , $\cos \frac{d θ}{2} \approx 1$ . Consequently, Equations (1)–( 3) can be expressed as
${\begin{cases} (2 F + d F) \frac{d θ}{2} = d N \\ d F = μ d N \end{cases}$
(4)

Ignoring the high-order small quantities in Equation (4), the equation can be denoted as
${\begin{cases} F d θ = d N \\ d F = μ d N \end{cases}$
(5)

Eliminate $d N$ in Equation (5) to get the following formula
$\frac{d F}{F} = μ d θ$
(6)

After integration processing, tension $F$ is
$\ln F = μ θ + C$
(7)
When $θ = 0$ , $F = F_{0}$ , so $C = \ln F_{0}$ . Equation (7) can be expressed as
$F = F_{0} e^{μ θ}$
(8)

According to the above mechanical analysis, the tension of the unwind section is mainly related to the friction coefficients among the yarn, lower idle roller, and winding angle of the yarns. The influence of the lower idle roller on the yarn tension is ignored in the subsequent control system design because the friction coefficient $μ$ is tiny.

Adjust section

The adjust section is mainly composed of an overfeed motor, overfeed roller, idle roller, and tension sensor. The overfeed roller is driven by the overfeed motor to track the rotation speed of the spindle and adjust the yarn tension using the speed difference between the two. In Figure 3, the feedback value of the tension sensor is compared with the reference tension, and the speed of the overfeed roller is changed according to the difference between the two, so the yarn tension tends to the reference tension.

Figure 3.
Sketch of the adjust section.

In the adjust section, the overfeed roller rotates under the combined action of the output torque of the overfeed motor, the yarn traction torque, and the frictional resistance torque. The external torque $M_{c}$ of the overfeed roller can be expressed as follows
$F_{q} R_{i} + M_{e} - M_{f} = M_{c} = J_{i} \frac{d ω_{i}}{d t}$
(9)
where $J_{i}$ is the inertia moment of the overfeed roller; $ω_{i}$ is the angular velocity of the overfeed roller; $F_{q}$ is the traction force of the yarn on the overfeed roller; $R_{i}$ is the radius of the overfeed roller; $M_{e}$ is the output torque of the overfeed motor; $M_{f}$ is the frictional resistance torque.

In Figure 4(a), assuming that there is no slippage between the yarn and overfeed roller, the traction force of the yarn on the overfeed roller can be expressed as the difference in yarn tensions before and after entering the overfeed roller: $F_{q} = F_{i - 1} - F_{1}$ . The frictional resistance torque can be denoted as $M_{f} = b_{i} ω_{i}$ , where $b_{i}$ is the friction coefficient.

Figure 4.
Cross-sectional view of the overfeed roller.

Therefore, Equation (9) can be expressed as
$J_{i} \frac{d ω_{i}}{d t} =_{1} (F_{i - 1} - F_{1}) R_{i} + M_{e} - b_{i} ω_{i}$
(10)

In the same way, the balance torque equation of the idle roller in Figure 4(a) is
$J_{d} \frac{d ω_{d}}{d t} = (F_{i} - F_{i - 1}) r_{i} - b_{d} ω_{d}$
(11)
where $J_{d}$ is the inertia moment of the idle roller; $ω_{d}$ is the angular velocity of the idle roller; $r_{i}$ is the radius of the idle roller; $b_{d}$ is the friction coefficient. Tension $F_{i}$ can be denoted as
$F_{i} = F_{1} + \frac{J_{d} {\dot{v}}_{d} + b_{d} v_{d}}{r_{i}^{2}} + \frac{J_{i} {\dot{v}}_{i} - M_{e} R_{i} + b_{i} v_{i}}{R_{i}^{2}}$
(12)

The balance torque equation of the overfeed roller in Figure 4(b) is
$J_{i} \frac{d ω_{i}}{d t} =_{1} (F_{i + 1} - F_{i}) R_{i} + M_{e} - b_{i} ω_{i}$
(13)

Tension $F_{i + 1}$ can be expressed as
$F_{i + 1} = F_{i} + \frac{J_{i} {\dot{v}}_{i} - M_{e} R_{i} + b_{i} v_{i}}{R_{i}^{2}}$
(14)

According to the above mechanical structure, in the adjust section, the main mechanical-structure factors affecting the yarn tension include the inertia moment $J_{i}$ , radius $R_{i}$ , friction coefficient $b_{i}$ of the overfeed roller, inertia moment $J_{d}$ , radius $R_{d}$ , and friction coefficient $b_{d}$ of the idle roller. The yarn tension is positively correlated with $J_{i}$ , $b_{i}$ , $J_{d}$ , and $b_{d}$ , and negatively correlated with $R_{i}$ and $R_{d}$ . Among the non-mechanical structural factors, it is mainly affected by the rotational speed of the overfeed roller, and the yarn tension is positively correlated with the acceleration ${\dot{v}}_{i}$ of the overfeed roller and the speed $v_{i}$ of the overfeed roller.

Rewind section

In Figure 5, the rewind section is mainly composed of a yarn-guide mechanism and a winding mechanism. The yarn winding speed is adjusted by controlling the spindle speed, and the yarn tension is adjusted by controlling the speed difference between the overfeed roller and the spindle. According to Hooke's law, there is a linear relationship between the stress and strain inside the yarn during the rewind section
$\frac{F}{A} = E \frac{△ L}{L}$
(15)
where $E$ is the resilient modulus of the yarn; $L$ is the yarn length when the yarn is stretched without tensions; $△ L$ is the elongation of the yarn under the tension; $A$ is the force area of the yarn.

Figure 5.
Sketch of the rewind section.

As shown in Figure 5, $L$ is divided into two sections, one section is from the overfeed roller to the yarn guide, and the other section is from the yarn guide to the spindle, so $L$ can be expressed as
$L = L_{1} + \frac{L_{2}}{\sin θ}$
(16)
where $L_{1}$ the distance from the overfeed roller to the yarn guide; $L_{2}$ is the distance from the yarn guide to the spindle; $θ$ is the angle between the movement direction of the yarn and the vertical direction before winding;

The amount of yarn elongation $△ L$ depends on the speed $v_{1}$ of the overfeed roller feeding the yarn and the speed $v_{2}$ of the spindle winding the yarn. The length of the yarn delivered by the overfeed roller is $\int_{t}^{t + △ t} v_{1} d t$ , the length of the yarn to be wound is $\int_{t}^{t + △ t} v_{2} d t$ , and $△ L$ is the difference between the two
$△ L = \int_{t}^{t + △ t} (v_{2} - v_{1}) d t$
(17)

The yarn tension of the rewind section can be denoted as
$F = E A \frac{△ L}{L} = E A \frac{\int_{t}^{t + △ t} (v_{2} - v_{1}) d t}{L_{1} + \frac{L_{2}}{\sin θ}}$
(18)

It is assumed that $T$ is the period of longitudinal oscillation of the yarn guide; $2 L_{3} / T$ is the speed of yarn oscillation; $n_{1}$ is the speed of the overfeed roller; $n_{2}$ is the speed of the spindle; $R_{i}$ is the radius of the overfeed roller; $R_{w}$ is the spindle radius. The speed of the yarn can be denoted as
$v_{1} = 2 π R_{i} n_{1}, v_{2} = \sqrt{{(2 π R_{w} n_{2})}^{2} + {(\frac{2 L_{3}}{T})}^{2}}$
(19)

Therefore, Equation (18) can be expressed as follows
$F = E A \frac{Δ L}{L} = E A \frac{\int_{t}^{t + △ t} (\sqrt{{(2 π R_{w} n_{2})}^{2} + {(\frac{2 L_{3}}{T})}^{2}} - 2 π R_{i} n_{1}) d t}{L_{1} + \frac{L_{2}}{\sin θ}}$
(20)

According to the above mechanical-structure analysis, in the rewind section, the main mechanical structural factors affecting the yarn tension include the radius of the overfeed roller $R_{i}$ and distances $L_{1}$ and $L_{2}$ between the overfeed roller and the spindle, These parameters need to be adjusted according to the actual yarn characteristics in production. During the working process of the winding machine, the yarn tension is mainly adjusted by non-mechanical structural factors, such as the speed $n_{1}$ of the overfeed roller, the speed $n_{2}$ of the spindle, and the speed of the yarn guide. Here, $n_{1}$ and $n_{2}$ determine the yarn elongation $△ L$ , while $L_{1}$ and $L_{2}$ determine the yarn length $L$ . The yarn tension is positively correlated with $△ L$ and negatively correlated with $L$ . In addition, the yarn tension is positively related to the elastic modulus $E$ and the cross-sectional area $A$ of the yarn itself.

Design of the yarn-tension control method

The analysis of the mechanical structure in the second section shows that the main factors affecting the yarn tension include the type of yarns, the material and size of the overfeed roller, the radius of the spindle, the transportation route of yarns, and the running speed of each motor. The design of the control scheme should consider the following aspects, for example, coordinating the relationship among the speeds of the overfeed roller, yarn guide, and spindle and avoiding the influence of package radius changes on yarn tensions. The working process of the winding machine is divided into the start-up stage, stable stage, and stopping stage. The control scheme is also divided into three parts, as the main causes of yarn-tension fluctuations are different at the three stages.

In the start-up stage, the micro control unit (MCU) calculates the target speed of the spindle motor according to the set series of parameters and the empty tube radius of the yarn package, which controls the spindle motor to reach the target speed within the set acceleration time. The overfeed motor should start for yarn feeding before starting the spindle motor to reduce the sudden change of yarn tensions at the start-up stage. During the acceleration of the spindle motor, the feedback value of the tension sensor is used as the input to design the controller to control the speed of the overfeed motor, thereby reducing the fluctuation of yarn tensions. Besides, a tension range needs to be set according to the breaking strength of different yarns to prevent yarn breakage. When the yarn tension exceeds this range, the spindle motor will stop accelerating and keep at the current speed. The spindle motor continues to accelerate only when the feed motor adjusts the yarn tension back to this range.

At the stable stage, since the package radius is a time variable, the MCU needs to adjust the speed of the spindle motor according to the real-time radius of the yarn package to ensure the stability of the winding line speed. Similar to the start-up stage, the stable stage also takes the feedback value of the tension sensor as the input to design the controller, thus controlling the speed of the overfeed motor. Therefore, the yarn tension can be stabilized near the set target tension.

At the stopping stage, after the MCU receives the stop command, it controls the spindle motor to decelerate slowly, and the deceleration trend is determined by the set deceleration time. The overfeed motor needs to continue to feed the yarn in the deceleration process of the spindle motor to ensure the slow decrease of yarn tensions at the deceleration stage. When the spindle motor decelerates to the set range, the overfeed motor decelerates until it stops.

Figure 6 shows the tension controller described at each stage. It consists of two loops: the inner loop is the speed control loop, and the outer loop is the tension control loop. In the outer loop, the actual tension detected by the tension sensor is the feedback value, and a controller is designed for regulating the speed correction. The target speed of the overfeed roller is output in real-time by comparing and calculating the actual tension and reference tension. In the inner loop, an active disturbance-rejection controller is designed to adjust the input voltage of the overfeed motor by analyzing and calculating the reference speed and actual speed of the overfeed roller, thereby realizing the closed-loop speed of the overfeed motor. In summary, the closed-loop control of the adjust section of the yarn tension is completed by the tension control loop and the speed control loop.

Figure 6.
Tension control system. MCU: micro control unit; ADRC: active disturbance-rejection control.

The PID controller, having the advantages of strong robustness and simple structure, is one of the most widely used adjustment methods at present. However, the analysis in the second section shows that the relationship between the speed of the overfeed roller and the yarn tension is time-varying and nonlinear. Traditional PID control is a control method of linear weighted combination, which cannot obtain a good control effect. Therefore, a more suitable and efficient combination method can be found in the nonlinear range to obtain a better control effect. The active disturbance-rejection control (ADRC) algorithm provides a systematic method using human expertise and nonlinear combination. This method, overcoming the contradiction between the traditional PID rapidness and overshoot, can suppress and eliminate various forms of external disturbance. In Figure 7, the active disturbance-rejection controller is mainly composed of a tracking differentiator (TD), nonlinear state error feedback (NLSEF), and extended state observer (ESO).

Figure 7.
Tension control scheme based on the active disturbance-rejection controller. TD: tracking differentiator; NLSEF: nonlinear state error feedback; ESO: extended state observer.

Tracking differentiator

The TD is a signal processing link arranging the transition process according to the reference input and output simultaneously: reference speed $V_{0}$ is the reference input, tracking speed $V_{1}$ is the tracking signal of the reference input, and tracking acceleration $A_{1}$ is the differential signal of the reference input. The link aims to alleviate the overshoot phenomenon caused by the excessive initial error. The differential signal given by the traditional differentiator has a serious amplifying effect on noise and cannot be configured with a pure zero-point structure. A TD with a good noise-filtering ability can be used to approximate the zero-point system formed by the differential link. The discrete form of the second-order TD is as follows
${\begin{cases} V_{1} (k) = V_{1} (k - 1) + h A_{1} (k - 1) \\ A_{1} (k) = A_{1} (k - 1) + hfst (V_{1} (k - 1) - V_{0} (k), A_{1} (k - 1), δ, h) \end{cases}$
(21)
where $V_{1} (k - 1)$ and $A_{1} (k - 1)$ are the tracking speed and acceleration of the previous iteration, respectively; $k$ is the current iteration number and an integer, $k \in [1, + \infty]$ , $V_{1} (0) = 0$ , $A_{1} (0) = 0$ ; $h$ is the sampling period; $δ$ is the parameter to adjust the tracking speed; $fst (\cdot)$ is the fastest control synthesis function, described as follows
$fst (x_{1}, x_{2}, δ, h) = {\begin{cases} - δ sgn (a) | a | > d \\ - δ \frac{a}{d} | a | \leq d \end{cases}$
(22)

$a = {\begin{cases} x_{2} + \frac{a_{0} - d}{2} sgn (y) | y | > d_{0} \\ x_{2} + \frac{y}{h} | y | \leq d_{0} \end{cases}$
(23)
where $sgn (\cdot)$ is the symbolic function; $x_{1}$ and $x_{2}$ are the two independent variable parameters of the fastest control synthesis function; $a$ and $y$ are the two intermediate variables of the fastest control synthesis function
$d = δ h, d_{0} = h d, y = x_{1} + h x_{2}, a_{0} = \sqrt{d^{2} + 8 δ | y |}$
(24)

The TD involves adjustment parameters $h$ and $δ$ ; $h$ is the sampling period, that is, the integration step length; $b$ is the speed factor used to adjust the speed of the transition process. If the speed factor $δ$ is small, the output of the TD will track the reference input more smoothly; if $δ$ is large, the output of the TD will be closer to the original input. The choice of $δ$ depends on the bearing capacity of the controlled object and the control ability that can be provided. The bearing capacity of the object and the provided control ability are strong with large $δ$ , so the arrangement process is closer to the given reference input. Otherwise, $δ$ should be appropriately small so that the object can trace the original reference input as much as possible within the tolerated range.

Extended state observer

The negative feedback in the control system has a certain inhibitory effect on the disturbance, but it cannot eliminate its influence. From the perspective of completing the control, only the disturbance affecting the output of the controlled system needs to be eliminated, and the disturbance that cannot be controlled or that affects the controlled output is not considered. Since a certain kind of disturbance can affect the controlled output, its effect should be reflected in the information of the controlled output, Therefore, it is possible to process the controlled output appropriately to estimate its effect. Based on the estimated effect of the disturbance, it is possible to eliminate its effect utilizing compensation. Borrowing the idea of the state observer, the perturbation affecting the controlled output is expanded into a new state variable, and a special feedback mechanism is used to establish a state that can be expanded, that is, to observe the disturbance. The state observer of the expanded system is called the ESO.

The work used the actual speed $V_{2}$ of the overfeed roller as the input of the ESO and calculated the tracking signal $V_{3}$ , differential signal $A_{2}$ , and observation disturbance $Z_{1}$ output by the system. Here, $Z_{1}$ is the total disturbance inside and outside the system. Parameters $β_{01}$ , $β_{02}$ , and $β_{03}$ are appropriately chosen so that the ESO can estimate states $V_{3}$ , $A_{2}$ , and expanded state $Z_{1}$ of the controlled object in real-time. For the selection of parameters $β_{01}$ , $β_{02}$ , and $β_{03}$ , an efficient method using the bandwidth concept is adopted
$β_{01} = 3 ω, β_{02} = 3 ω^{2}, β_{03} = ω^{3}$
(25)
where $ω$ is determined according to the requirements of the system bandwidth or adjusted online. In general, it has a wide range of adaptation, so it is easy to adjust the appropriate parameters. The formula of the ESO is as follows
${\begin{cases} e = V_{3} - V_{2} \\ f e = fal (e, 0.5, γ) \\ f e_{1} = fal (e, 0.25, γ) \\ {\dot{V}}_{3} = V_{3} + h (A_{2} - β_{01} e) \\ {\dot{A}}_{2} = A_{2} + h (Z_{1} - β_{02} f e + B * U) \\ {\dot{Z}}_{1} = Z_{1} + h (- β_{03} f e_{1}) \end{cases}$
(26)
where $f e$ and $f e_{1}$ are the two independent variable parameters of the formula; $h$ is the sampling period. In order to avoid the phenomenon of high-frequency oscillation, the power function ${| e |}^{α} sgn (e)$ is transformed into a continuous power function with a linear segment near the origin, that is, the saturation function, which is expressed as
$fal (e, α, γ) = {\begin{cases} \frac{e}{γ^{α - 1}} | e | \leq γ \\ {| e |}^{α} sgn (e) | e | > γ \end{cases}$
(27)
where $e$ is the systematic error; $α$ is the exponent; $γ$ is the limit to distinguish the size of the error.

Nonlinear state error feedback

The NLSEF law adopts a nonlinear controller structure independent of the object model. NLSEF decides the control law of controlling the integrator sequence object according to state errors $E_{1}$ and $E_{2}$ of the system. For error-feedback control quantity $U_{0}$ , the compensation of observation disturbance $Z_{1}$ is used to determine the final control quantity $U$ . Here, $B$ , as an adjustable parameter, is the “compensation factor” determining the strength of compensation, while $λ_{1}$ and $λ_{2}$ are the gains of tracking the input signal and tracking the differential signal, respectively. If the speed of the ESO is fast enough, then this compensation component can accurately reflect the disturbance of the system. The NLSEF is denoted as
$E_{1} = V_{1} - V_{3}, E_{2} = A_{1} - A_{2}$
(28)

$U_{0} = λ_{1} f a l (E_{1}, α_{1}, γ) + λ_{2} f a l (E_{2}, α_{2}, γ)$
(29)

$U = U_{0} - \frac{Z_{1}}{B}$
(30)

Simulation

In the control system described in the third section, the active disturbance-rejection controller is used to adjust the yarn tension through the speed of the overfeed roller. The yarn-tension fluctuations of the above two controllers under different yarns, package diameters, and target tensions are simulated to compare the control effects of the PID controller and the active disturbance-rejection controller. Table 1 shows the parameters used in the simulation, and the parameters related to the yarn refer to 40D/12F nylon filament and 75D/24F polyester (partially oriented yarn).

Table 1.
Parameters used in the simulation

Parameter Description Value

Nylon filament Polyester

$ρ$ Yarn density $1 {.14 g/cm}^{3}$ $1 {.38 g/cm}^{3}$

$E$ Modulus of the yarn $21 .501 MPa$ $25.998 MPa$

$A$ Cross-sectional area of yarn $3.899 e^{- 3} {mm}^{2}$ $8.052 e^{- 3} {mm}^{2}$

$J_{i}$ Inertia of the overfeed roller $4 {.084 e}^{- 4} kg \cdot m^{2}$

$R_{i}$ Radius of the overfeed roller $0.082 m$

$L_{1}$ Distance between the overfeed roller and yarn guide $0.3 - 0.52 m$

$L_{2}$ Distance between yarn guide and spindle $0.08 m$

Table 2.
Transfer function corresponding to different conditions

Yarn type Package diameter (mm) Transfer function Identification coincidence Number of poles Number of zeros

Nylon filament 40 $\frac{4 {.67e}^{- 3} s^{2} + 2.265 e^{- 4} s + 6.02 e^{- 5}}{s^{3} + 0.9748 s^{2} + 0.1091 s + 0.01397}$ 90.07% 2 3

Nylon filament 55 $\frac{3.907 e^{- 3} s^{2} + 1.544 e^{- 4} s + 6.509 e^{- 5}}{s^{3} + 0.4742 s^{2} + 0.08574 s + 0.01036}$ 87.24% 2 3

Nylon filament 70 $\frac{4.649 e^{- 3} s^{2} + 1.387 e^{- 4} s + 7.448 e^{- 5}}{s^{3} + 0.4399 s^{2} + 0.06831 s + 8.504 e^{- 3}}$ 87.01% 2 3

Polyester 55 $\frac{3 .732 e^{- 3} s^{2} + 2.566 e^{4} s + 5.834 e^{- 5}}{s^{3} + 0.7763 s^{2} + 0.07801 s + 9.895 e^{- 3}}$ 88.72% 2 3

The first step of the simulation is to establish the transfer-function model of the speed of the overfeed roller and yarn tension with the method of system identification. The transfer-function model of the speed of the overfeed roller and the yarn tension was established with nylon package diameters of 40, 55, and 70 mm to explore the influence of the package diameter on the yarn tension. In addition, the transfer-function model of the polyester yarn package with a diameter of 55 mm was also established to test the control effect of the two controllers on different yarns. Firstly, the experiment was used to record the speed of the overfeed roller and the yarn tension during the process from the start to the stop of the winding machine. Then one obtains the transfer functions of different zero-poles through the method of system identification. Finally, the recorded speed of the overfeed roller was used as the inputs of these transfer functions. The similarities between the output results of different zero-pole transfer functions and the experimental results were compared to select the transfer-function model with the highest similarity.

In Figure 8, the highest similarity with the experimental results is the transfer function with three zeros and two poles and the transfer function with three zeros and three poles. Under different package diameters, the similarity between the two transfer functions and the experimental results are kept in the forefront, and the similarity between the two is close. Therefore, the transfer function with three zeros and two poles was selected as the transfer-function model used in subsequent simulations in the work. The transfer function models under different conditions are shown in Table 2.

Figure 8.
Measured and simulated model output. (a) Yarn: nylon filament, Reference tension: 15 cN, Package diameter: 40 mm; (b) yarn: nylon filament, Reference tension: 15 cN, Package diameter: 55 mm; (c) yarn: nylon filament, Reference tension: 15 cN, Package diameter: 70 mm and (d) yarn: polyester, Reference tension: 15 cN, Package diameter: 55 mm.

Simulations were run for studying the tension response at the start-up stage. The package diameter was 55 mm, and the reference tensions were 5, 10, and 15 cN. Figure 9 shows the results of the conventional method and the proposed method. The overshoot of the active disturbance-rejection controller is much smaller than that of the PID controller in the start-up stage, with a faster settling time and lower steady-state error. Comparing the control effects of the controller under different target tensions in Figure 9, the active disturbance-rejection controller is hardly affected by the change of targeted tensions. The settling time of the PID controller becomes longer with increased targeted tensions. The active disturbance-rejection controller has a better control effect in the above cases by comparison.

Figure 9.
Tension responses for reference tensions at 5, 15, and 30 cN. ADRC: active disturbance-rejection control; PID: proportional–integral–derivative. (a) Yarn: nylon filament, Reference tension: 5 cN, Package diameter: 55 mm; (b) yarn: nylon filament, Reference tension: 10 cN, Package diameter: 55 mm and (c) yarn: nylon filament, Reference tension: 15 cN, Package diameter: 55 mm.

A step signal of the overfeed roller speed is introduced to simulate external disturbance at T = 2 s to study the anti-disturbance capability of the controller. In Table 3, the simulation is divided into six cases based on yarn type, target tension, and package diameter. Figure 10 shows the results.

Table 3.
Tension, radius, and speed parameters

Serial number Yarn type Reference tension (cN) Package diameter (mm)

1 Nylon filament 5 55

2 Nylon filament 10 40

3 Nylon filament 10 55

4 Nylon filament 10 70

5 Nylon filament 15 55

6 Polyester 10 55

Figure 10.
Tension responses with the effect of a step disturbance. ADRC: active disturbance-rejection control; PID: proportional–integral–derivative. (a) Yarn: nylon filament, Reference tension: 5 cN, Package diameter: 55 mm; (b) yarn: nylon filament, Reference tension: 10 cN, Package diameter: 40 mm; (c) yarn: nylon filament, Reference tension: 10 cN, Package diameter: 55 mm; (d) yarn: nylon filament, Reference tension: 10 cN, Package diameter: 70 mm (e) yarn: nylon filament, Reference tension: 15 cN, Package diameter: 55 mm and (f) yarn: polyster, Reference tension: 10 cN, Package diameter: 55 mm.

In Figure 10, a step disturbance of the speed causes fluctuations in the yarn tension, and Figure 11 shows the peaks of overshoot and undershoot. In the six cases, compared with the PID controller, the active disturbance-rejection controller can reduce the tension overshoot and undershoot caused by the speed disturbance. Meanwhile, it can be faster and more stable with smaller steady-state errors. Comparing Figures 10(a), (c), and (e), the increased target tension slightly increases the peak tension overshoot of the active disturbance-rejection controller as well as the peak tension overshoot and undershoot of the PID controller. The change of the target tension has few effects on the anti-disturbance ability of the above two controllers. The main effect is that the increased target tension prolongs the time for the PID controller to reach a steady state in the start-up stage. Comparing Figures 10(b)–(d), in Figure 10(d), in the face of speed disturbance, the yarn tension controlled by the active disturbance-rejection controller has a higher overshoot compared to Figures 10(b) and (c), while the yarn tension overshoots under the control of the PID controller in the three figures are basically the same. In the start-up stage, the change of the package diameter has almost no effect on the active disturbance-rejection controller. For the PID controller, the transition time for the yarn tension to the target value is longer as the package diameter increases. Comparing Figures 10(c) and (f), different yarns are simulated under the same parameters, and the control effects of the active disturbance-rejection controller on the two yarns are basically the same, while the transition time of the PID controller in the start-up stage and the undershoot caused by the speed disturbance are different.

Figure 11.
Tension errors in six cases. ADRC: active disturbance-rejection control; PID: proportional–integral–derivative.

The above simulation results show that compared with the PID controller, the active disturbance-rejection controller has a smaller overshoot and undershoot at the start-up stage, a shorter time to reach stability, and lower steady-state errors. In the face of the tension fluctuation caused by the speed disturbance, the active disturbance-rejection controller has better adjustment ability. Different yarn types and different parameters also have less influence on the control effect.

Experimental details

A winding machine was built for the experiment to further verify the stability and accuracy of the tension control system. Figure 12 shows the winding machine. In the unwind section, the fed yarn enters the adjust section under the traction of the overfeed roller. The function of the porcelain ring and the guide roller is to reduce the vibration of the yarn and ensure that the yarn can enter the overfeed roller smoothly.

Figure 12.
Winding process system.

Figure 13.
Experimental winding machine.

In the adjust section, the yarn is wound on the overfeed roller and guide roller and then enters the rewind section through the tension sensor, which is installed between the overfeed roller and the yarn guide to detect the tension of this section of the yarn in real-time. Then the speed of the overfeed roller is changed according to the detected yarn tension to adjust the yarn tension and form a tension control closed loop.

In the rewind section, the yarn is evenly wound on the package under the traction of the yarn guide. The spindle motor changes the speed in real-time according to the radius of the yarn package to ensure the constant linear speed of the yarn winding. The yarn-guide motor determines the movement stroke of the yarn guide according to the set type of the yarn package and the radius of the yarn package. In terms of motor selection, the yarn-guide motor adopts a DC torque motor generating constant torque, while the overfeed motor and spindle motor are driven by alternating current (AC) servo motors. The actual picture of the winding machine is shown in Figure 13.

The active disturbance-rejection controller and PID controller are used for experiments under different parameters. The parameters used in the experiment are shown in Table 3. As shown in Figure 14, the yarn tension of the winding machine from start to stop was collected in the experiment to analyze and compare the control effects of the two controllers under different parameters. Since the fluctuation of yarn tension in the stable stage in Figure 14 is not intuitive enough, in this paper, the experimental data from 20 to 40 s in each of the above experiments are processed, and the fluctuation of the yarn tension in the stable stage is reflected from four aspects: mean value, variance, maximum value, and minimum value. Table 4 and Figures 15 and 16 present the analysis results.

Figure 14.
Tension responses under different experimental parameters. ADRC: active disturbance-rejection control; PID: proportional–integral–derivative. (a) Yarn: nylon filament, Reference tension: 5 cN, Package diameter: 55 mm; (b) yarn: nylon filament, Reference tension: 10 cN, Package diameter: 40 mm; (c) yarn: nylon filament, Reference tension: 10 cN, Package diameter: 55 mm; (d) yarn: nylon filament, Reference tension: 10 cN, Package diameter: 70 mm; (e) yarn: nylon filament, Reference tension: 15 cN, Package diameter: 55 mm and (f) yarn: polyster, Reference tension: 10 cN, Package diameter: 55 mm.

Table 4.
Analysis of experimental results

Serial number Yarn type Parameters
Controller Maximum value Minimum value Mean value Variance

Reference tension Package diameter

1 Nylon filament 5 55 ADRC 11.9682 0 4.7416 2.6266

PID 17.1025 0 5.7085 3.8391

2 Nylon filament 10 40 ADRC 15.6819 4.0353 9.7170 2.3151

PID 19.9293 3.5336 9.7261 3.3368

3 Nylon filament 10 55 ADRC 16.6996 5.2792 10.1368 2.6723

PID 22.6855 2.1908 9.8834 3.6684

4 Nylon filament 10 70 ADRC 17.9915 4.3095 9.9160 2.7858

PID 21.1590 0.9894 9.8823 3.8145

5 Nylon filament 15 55 ADRC 21.3661 9.2898 15.0925 2.2643

PID 22.8975 8.6219 14.7230 2.9327

6 Polyester 10 55 ADRC 16.3251 4.6643 9.9588 2.4120

PID 23.7456 1.6961 10.1859 4.3011

ADRC: active disturbance-rejection control; PID: proportional–integral–derivative.

Figure 15.
Variance of tensions in six cases. ADRC: active disturbance-rejection control; PID: proportional–integral–derivative.

Figure 16.
Tension-fluctuation range in six cases. ADRC: active disturbance-rejection control; PID: proportional–integral–derivative.

As shown in Figure 14, the largest tension fluctuations during the entire winding process occur during the start-up stage, which is why yarn breakage usually occurs during the start-up stage. It is obvious that in the experiments with different parameters, the fluctuation amplitude of the yarn tension controlled by the active disturbance-rejection controller is smaller in the start-up stage, which can effectively reduce the damage to the yarn caused by the huge tension fluctuation in the start-up stage. Comparing Figures 14(a), (c), and (e), with the increase of the target tension, the amplitude of the tension fluctuation in the start-up stage is also increased. Comparing Figures 14(b)–(d), an increase in the diameter of the package also leads to a small increase in the amplitude of the tension fluctuation during the start-up stage. Comparing Figures 14(c) and (f), under the same parameters, the tension fluctuations of different yarns at the start-up stage are significantly different.

As shown in Table 4, in each experiment with different parameters, the variance of the tension value under the control of the active disturbance-rejection controller is smaller, which indicates that the active disturbance-rejection controller can effectively reduce the fluctuation of yarn tension. Then comparing the mean value, maximum value, and minimum value of the tension value under the control of the two controllers, the difference between the mean value and the target tension value is similar, but the maximum value corresponding to the PID controller is larger, and the minimum value is smaller, which means that the active disturbance-rejection controller can effectively reduce the amplitude of the tension fluctuation in the stable stage. Comparing the data of No. 1, 3, and 5 groups, as the target tension increases, the difference between the average value of the tension value and the target tension will decrease slightly and the variance of the tension value is also reduced, especially the variance between the two groups of No. 3 and 5, which is greatly reduced. Comparing the data of No. 2–4 groups, as the diameter of the package increases, the variance of the tension value and the range of the tension fluctuation (the range between the maximum value and the minimum value) increase significantly, which means that an increase in the diameter of the package will increase the fluctuation of the yarn tension. Comparing the data of No. 3 and 6 groups, under the same parameters, the tension fluctuations of different yarns are significantly different.

In the stopping stage, the yarn-tension fluctuations did not change significantly, and different parameters had little effect on the tension fluctuations in this stage.

Conclusions

(1) According to the mechanical-structure analysis of the winding machine in the second section, the mechanical factors affecting the yarn tension mainly include the positional relationship between the overfeed roller, the yarn guide, and the spindle, the inertia, and the size and material of the overfeed roller itself. The non-mechanical factors mainly include the rotation speed of the overfeed roller, the speed of the yarn guide, and the winding speed of the spindle. In addition, the properties of the yarn, such as elastic modulus, linear density, smoothness, etc., are also important factors affecting the yarn tension.

(2) According to the simulation and experimental results, the change of the target tension will have a significant impact on the yarn tension. On the one hand, the increase of the target tension will lead to an increase in the amplitude of the tension fluctuation in the start-up stage, but on the other hand, a larger target tension will reduce the tension fluctuation in the stable stage to a certain extent. In the experiment, the increase of the diameter of the package will aggravate the fluctuation of the yarn tension in the stable stage.

(3) This paper proposes a controller technology based on ADRC, which can effectively reduce the amplitude of the tension fluctuation during the start-up stage of the winding machine, reduce its damage to the yarn, and reduce the occurrence of yarn breakage. The controller also reduces the tension fluctuation during the working process of the winding machine, and improves the stability of the yarn-tension control, which has a certain significance for improving the quality of the fabric. In addition, after simulation and experiments with different parameters and different yarns, the controller can show a better control effect under different conditions, and the influence of these changes is smaller.

Parameter	Description	Value
$ρ$	Yarn density	$1 {.14 g/cm}^{3}$	$1 {.38 g/cm}^{3}$
$E$	Modulus of the yarn	$21 .501 MPa$	$25.998 MPa$
$A$	Cross-sectional area of yarn	$3.899 e^{- 3} {mm}^{2}$	$8.052 e^{- 3} {mm}^{2}$
$J_{i}$	Inertia of the overfeed roller	$4 {.084 e}^{- 4} kg \cdot m^{2}$
$R_{i}$	Radius of the overfeed roller	$0.082 m$
$L_{1}$	Distance between the overfeed roller and yarn guide	$0.3 - 0.52 m$
$L_{2}$	Distance between yarn guide and spindle	$0.08 m$

Yarn type	Package diameter (mm)	Transfer function	Identification coincidence	Number of poles	Number of zeros
Nylon filament	40	$\frac{4 {.67e}^{- 3} s^{2} + 2.265 e^{- 4} s + 6.02 e^{- 5}}{s^{3} + 0.9748 s^{2} + 0.1091 s + 0.01397}$	90.07%	2	3
Nylon filament	55	$\frac{3.907 e^{- 3} s^{2} + 1.544 e^{- 4} s + 6.509 e^{- 5}}{s^{3} + 0.4742 s^{2} + 0.08574 s + 0.01036}$	87.24%	2	3
Nylon filament	70	$\frac{4.649 e^{- 3} s^{2} + 1.387 e^{- 4} s + 7.448 e^{- 5}}{s^{3} + 0.4399 s^{2} + 0.06831 s + 8.504 e^{- 3}}$	87.01%	2	3
Polyester	55	$\frac{3 .732 e^{- 3} s^{2} + 2.566 e^{4} s + 5.834 e^{- 5}}{s^{3} + 0.7763 s^{2} + 0.07801 s + 9.895 e^{- 3}}$	88.72%	2	3

Serial number	Yarn type	Reference tension (cN)	Package diameter (mm)
1	Nylon filament	5	55
2	Nylon filament	10	40
3	Nylon filament	10	55
4	Nylon filament	10	70
5	Nylon filament	15	55
6	Polyester	10	55

Serial number	Yarn type	Parameters	Controller	Maximum value	Minimum value	Mean value	Variance
1	Nylon filament	5	55	ADRC	11.9682	0	4.7416	2.6266
PID	17.1025	0	5.7085	3.8391
2	Nylon filament	10	40	ADRC	15.6819	4.0353	9.7170	2.3151
PID	19.9293	3.5336	9.7261	3.3368
3	Nylon filament	10	55	ADRC	16.6996	5.2792	10.1368	2.6723
PID	22.6855	2.1908	9.8834	3.6684
4	Nylon filament	10	70	ADRC	17.9915	4.3095	9.9160	2.7858
PID	21.1590	0.9894	9.8823	3.8145
5	Nylon filament	15	55	ADRC	21.3661	9.2898	15.0925	2.2643
PID	22.8975	8.6219	14.7230	2.9327
6	Polyester	10	55	ADRC	16.3251	4.6643	9.9588	2.4120
PID	23.7456	1.6961	10.1859	4.3011

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Hua Zhang

Yikun Wei

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Serial number	Yarn type	Parameters		Controller	Maximum value	Minimum value	Mean value	Variance
Serial number	Yarn type	Reference tension	Package diameter	Controller	Maximum value	Minimum value	Mean value	Variance
1	Nylon filament	5	55	ADRC	11.9682	0	4.7416	2.6266
1	Nylon filament	5	55	PID	17.1025	0	5.7085	3.8391
2	Nylon filament	10	40	ADRC	15.6819	4.0353	9.7170	2.3151
2	Nylon filament	10	40	PID	19.9293	3.5336	9.7261	3.3368
3	Nylon filament	10	55	ADRC	16.6996	5.2792	10.1368	2.6723
3	Nylon filament	10	55	PID	22.6855	2.1908	9.8834	3.6684
4	Nylon filament	10	70	ADRC	17.9915	4.3095	9.9160	2.7858
4	Nylon filament	10	70	PID	21.1590	0.9894	9.8823	3.8145
5	Nylon filament	15	55	ADRC	21.3661	9.2898	15.0925	2.2643
5	Nylon filament	15	55	PID	22.8975	8.6219	14.7230	2.9327
6	Polyester	10	55	ADRC	16.3251	4.6643	9.9588	2.4120
6	Polyester	10	55	PID	23.7456	1.6961	10.1859	4.3011