Active tension control of the conical winding system based on the neural network control algorithm of radial-basis functions

Abstract

A conical winding formation and tension control system was proposed in the doubling operation based on the yarn guide mode of a single spindle in this study. Conical winding formation realized the radial unwinding of wound package. An overfeed mechanism was introduced to achieve closed-loop control of yarn tension. The overfeed wheel was driven by a brushless Direct Current motor. The tension control system combined a Proportion Integration Differentiation controller with a radial-basis-function neural network, whose purpose was to meet the control requirements of the brushless DC motor. This system consisted of three main steps: Firstly, the radial-basis-function neural network was used to identify the system online. Secondly, the gradient descent method was used to adjust the node weight, center vector, and baseband width. Finally, incremental PID parameters online were adjusted according to the identified Jacobian information. A mathematical model of a control system was established in Matrix Laboratory. An experimental platform was designed for doubling winder to compare the control effects of Radial-basis-function-PID with traditional PID. The simulation results showed that the RBF-PID had a smaller overshoot of yarn tension, shorter adjustment time, and smaller steady-state error compared with the traditional PID controller in doubling operation by simulating the mathematical model. Experimental results showed the RBF-PID controller had good performance and stability and could be applied to yarns with different average linear velocity, yarn counts and strands . The yarn tension fluctuation will not exceed ±3% of the target value when the experimental materials and the cone angles are unchanged.

Keywords

Conical winding formation active tension control radial-basis-function neural network doubling winder

The doubling winder combines multiple yarns in the textile industry to improve yarn strength while reducing the yarn hairiness and breakage accident probability.¹ Therefore, the doubling winder technology determines the efficiency and quality of subsequent double-twisting yarns.² The package diameter has a positive and negative relationship with the linear density of the wound package and the yarn-crossing angle between the adjacent layers of the wound package, respectively.³ When the package diameter increases, the residual tension of the yarn winding layer will change accordingly,^4,5 which deforms the wound package or even the yarn near the bobbin surface unrolling.

At present, there are four mainstream different types of package: the cross-wound package,⁶ the precision wound package,⁷ the parallel wound package,⁸ and the near-parallel wound package.

The cross-wound package is widely used in drum winding machines, where the yarn is wound by friction of the grooved drum.⁹ The cross-wound package is universally used due to its stable package, constant density, and high withdrawal speed of 1200 mpm. The cross-wound package is widely used in double twisters, covering machines, and other equipment.¹⁰ However, friction exists between the groove drum and the wound package in process of the formation, which increases the hairiness of the wound package in the winding process¹¹ and affects the yarn quality and the production environment of the equipment. Besides, there are problems such as yarn folding and hard edge in the process of the cross-wound package.

The precision wound package means that each yarn layer is closely arranged in the same direction, which is used in the processes of re-winding, and doubling. The wound package with the precision winding formation has a uniform linear density and a stable forming process.¹² On modern precision winding machines, the settings are under microprocessor control which can be set easily and do not need any maintenance. However, with the increase of the diameter of the wound package, the winding cross angle decreases, and the density of the wound package increases, which can easily cause the outside of the wound package to be tight and the inside to be loose, and the structure to be unstable.

The parallel wound package means that the yarn is stacked in parallel along the radial direction of the package, which is mainly used for tape yarns. However, the stability of the yarn on the surface of the parallel package is poor, and the yarn at both ends can fall off easily, therefore, the parallel package requires the use of bobbins with edges to form, which is unsuitable for radial unwinding,¹³ double twisters, or other axial passive unwinding equipment.¹⁴

The near-parallel wound package comprises one or more threads which are laid very nearly parallel to the layers already existing on the package, such as “Milk bottle package,” which is designed to adapt to the production of ring doublers. The winding layers at the bottom of milk bottle package are nearly parallel, and the winding layers of the bottle body are parallel, which are characterized by good axial unwinding. In the process of forming the bottom, to increase the yarn capacity of the package, it is necessary to increase the yarn winding height and step rise each yarn winding layer to a certain extent, but they are smaller than those during the body forming, whose process is determined by the movement of the ring rail, traveler, and spindle.¹⁵ However, there is a major disadvantage in using ring doublers to make the milk package and that is the high level of power consumption.¹⁶

A conical winding formation is proposed to overcome the shortcomings of the above winding formations. The yarn guide moves along the axial direction of the bobbin, and the stroke of the yarn guide is not fixed. The yarn is wound and stacked in parallel along the radial direction of the bobbin. Unwound yarns do not rub with adjacent layers of yarns during the axial unwinding process of conical wound package, which solves the problem that the traditional parallel wound package cannot be used for axial unwinding. Compared with the wound package wound by groove drum, the wound package with conical formations has higher winding density and package capacity. The high package capacity is conducive to reducing the frequency of bobbin replacements during double-twisting, which improves productivity and reduces the number of workers. The winding density is inversely proportional to the package diameter for the wound package with the same weight and length. The power consumption of double twisters is directly proportional to the diameter of the twisting wound package, indicating that reducing the package diameter of the wound package can reduce the energy consumption of the twisting motor.

The yarn guide stroke is not fixed during the forming process of the wound package with conical formations, which causes complex fluctuations of the winding tension in a short time. The yarn tension should be kept constant and less than the yarn tension strength to avoid fiber breakage.¹⁷ Therefore, the control of the winding tension is the key to forming the wound package,^18,19 which directly affects the yarn quality and the production of the next process.²⁰

Researchers have tried to obtain better improvement effects as follows.

(1) Improvement of the doubling winder. Schärer Schweiter Mettler²¹ developed the CWX series high-speed doubling winder to realize the functions of length counting and yarn-breaking monitoring. Zhang et al.²² designed a general controller of the doubling winder based on the LPC1788 microcontroller, which uses RS-485 communication to complete the reliable transmission of data and instructions. The controller is stable and reliable to regulate yarn speeds in the doubling process. Cai et al.²³ proposed a doubling-winder supervision system based on EtherCAT real-time communication and embedded Linux online processing, which significantly improves the winding effect and merging quality of multi-strand yarns. However, the doubling winder and its corresponding control system developed by the above team are based on a groove drum to realize the formation. The formation has a smaller package with a large yarn cross angle compared with the electronic formation.

(2) Improvement of the winding tension control system. Feng et al.²⁴ designed a surface acoustic wave (SAW) yarn tension sensor. Multiple yarns can be detected quickly with the sensor, which overcomes the problems of the slow response speed and low resolution of traditional sensors. Xu et al.²⁵ proposed a novel tension control method to regulate the fiber tension and transport speed, which provides faster setting time and lower steady-state error than a conventional PID controller. Zhang et al.²⁶ designed a yarn tension control system based on the nonlinear active disturbance-rejection control (ADRC) algorithm for the winder, which can actively adjust the yarn tension by controlling the speed of the overfeed roller. The control system uses the ADRC algorithm to quickly track the speed of the overfeed motor, which realizes dynamic adjustment. The sensor and control methods developed by the above team have realized the closed-loop control of yarn tensions. However, the control effect of the above control methods in the acceleration phase is unsatisfactory. When the acceleration of the overfeed motor increases, the fluctuation range of yarn tension increases significantly. Therefore, the practicability of the system remains to be tested. Besides, the experiment was carried out when the yarn linear variety was relatively stable. It is inapplicable to the scene where the linear variety has a large periodic change. The above tension control system has high-complexity algorithms, which makes it difficult to realize online learning and adaptation in the embedded controller.²⁷

In this study, an overfeed mechanism is introduced to achieve closed-loop control of yarn tension. The overfeed wheel is driven by a brushless DC motor (BLDC), and the RBF-PID control algorithm is added to meet the control requirements of BLDC. Because the speed of the BLDC is high at the start-up stage, the yarn feeding amount is far greater than the winding speed, resulting in excessive winding of the remaining yarn on the overfeed wheel, which ultimately leads to yarn breakage. This work simplifies the “Milk bottle package” proposed by ring doublers. The yarn winding layers inside the cone wound package are parallel, whether at the cone bottom or cone body, to avoid the probability of yarn breakage in the start-up phase, which reduces the winding tension fluctuation in the winding process and improve the yarn quality.

The study was organized in the following steps to control the yarn tension in the process of conical winding formations in a real-time closed loop. Firstly, the winding motion was analyzed to reveal the relationship between the winding density and cross angle. Secondly, a method was designed for conical winding formations that adopted the electronic forming doubling winder based on the yarn guide mode of a single spindle, and the yarn model was based on the Kelvin model. Thirdly, a tension control system for conical winding formation was proposed to combine a PID controller with a radial-basis-function (RBF) neural network. It used the RBF neural network to complete the online identification of system parameters and adjusted PID parameters in real time according to the sensitivity information of the identification system. Finally, the mathematical model of the control system was established on MATLAB, and an experimental platform was designed for doubling winders. The results of the simulation and experiments showed that the control system had high feasibility.

Winding motion analysis

The winding process means that the yarn is wound into a spiral form, that is, the yarn is regularly stacked layer by layer along the radial direction of the bobbin.²⁸ M is the winding point during the winding process, indicating the contact point between the yarn and the wound package (see Figure 1), and V is the speed of the yarn at the winding point. According to the principle of vector velocity superposition, the winding process can be divided into the motion of two reference systems: winding movement and yarn-guide movement. The winding movement means the rotation of the yarn around the bobbin. The yarn-guide movement means the reciprocating motion of the yarn along the axial direction of the bobbin. $V_{1}$ and $V_{2}$ represent the speeds of the winding movement and yarn guide movement, respectively, and α is the yarn cross angle. The trigonometric function of α and the expressions of $V_{1}$ and $V_{2}$ are as follows:

\sin α = \frac{V_{2}}{V} = \frac{h_{n}}{π d}

(1)

\tan α = \frac{V_{2}}{V_{1}} = \frac{h}{π d}

(2)

\cos α = \frac{V_{1}}{V} = \frac{h_{n}}{h}

(3)

V_{1} = π d n_{k}

(4)

V_{2} = n_{k} h

(5)

where d is the diameter of the wound package; n_k the winding speed; h the axial pith; and h_n the tangential pitch.

Figure 1.

Yarn speed vector.

The winding density is the yarn weight per unit volume of the wound package, which can represent the package capacity of the wound package. The factors affecting the winding density of the wound package include the yarn variety and count, yarn tension, external pressure of the bobbin, and yarn cross angle. The yarn guide mode determines the yarn cross angle. The microelement volume of the yarn at the winding point is used for analysis. The variables a, b and c are the length, width, and thickness of the hexahedron formed by the crossed yarn; l is the length of yarn; 2α is the yarn cross angle, that is, the helical winding angle (see Figure 2).

Figure 2.

Yarn intersection.

The expression for volume $V$ of the yarn at the intersection is as follows:

V = abc = l^{2} c \cos α \sin α = \frac{c l^{2}}{2} \sin 2 α

(6)

where

g

and

r

are the weight and density of the yarn, respectively.

g = (l \cdot H) / 1000

(7)

r = \frac{g}{v} = \frac{H}{2000 \cdot l \cdot c \cdot \sin 2 α}

(8)

where

H

(tex) is the yarn count. According to equation (8), the winding density of the wound package is inversely proportional to the sine value of cross angle 2α of yarns. When 2α is 45°, the winding density is the minimum. When 2α is close to 0°, the winding mode is parallel winding, and the winding density is the maximum. The average distance between two adjacent yarns in parallel winding is the yarn diameter from the perspective of the winding mode. The gap formed by yarn cross leads to a large distance between adjacent yarns in cross windings. The winding density of parallel winding formations is greater than that of cross-winding formations, and the winding density of winding formations with large cross angles is less than that with small cross angles. The volume of the wound package with the parallel winding formation is 130–150% times the size of that with cross winding formations under the same tension.

Process of conical winding formations

As previously mentioned, although the wound package with parallel winding formations has significant advantages in winding density and package stability, radial or active unwinding should be used for the yarn unwinding process, which is inappropriate for textile machinery with passive axial unwinding. Combined with the above advantages and disadvantages, the study proposed a conical winding formation to retain the advantages of parallel winding formations with the large winding density and solve the problem that the wound package with the parallel winding formation was unable to unwind axially. The radius of the wound package changed greatly during the process of conical winding formations, so the difference between the minimum and maximum linear velocity was 2–4 times. The linear velocity changed periodically in the short term, which resulted in a large fluctuation of yarn tensions. Therefore, an overfeed mechanism was used to actively adjust the yarn tension in the process of conical winding formations.

Mechanical structure analysis

Figure 3 shows the winding process system in the doubling operation based on the yarn guide mode of a single spindle. The system is composed of the original barrel yarn, overfeed mechanism, tension sensor, yarn guide mechanism, and winding mechanism. There are three parts to the winding process: the unwinding section, adjusting section, and rewinding section.

Figure 3.

Sketch of the winding process system.

The yarn unwound from the barrel yarn is subject to air resistance, friction, and other effects during the unwinding process. The function of the lower guide roller in the unwinding section is to stabilize the yarn and make it enter the adjusting section smoothly.

The yarn tension of doubling winders mainly depends on adjusting and rewinding sections to adjust tensions, which uses the speed difference between the overfeed roller and the winding motor. The winding motor speed is set to a constant value. The overfeed roller changes the speed according to the difference between the tension value fed back by the tension sensor and the target value, which makes the yarn tension approach the target value.

The rewinding section mainly includes the yarn-guide mechanism and the winding mechanism. The function of the winding mechanism is to realize the yarn traction and control the yarn winding speed. The function of the yarn guide mechanism is to realize the yarn winding formation.

Yarn tension model of conical winding formation

Yarn is a viscoelastic material with the elastic and damping characteristics of ordinary solids. The Kelvin model,²⁹ widely used in viscoelastic theory, is composed of Hook spring and Newton damping in parallel (see Figure 4). It can better describe the relationship between yarn deformations and tensions during rapid stretching. The relationship between tension and deformation of the Kelvin model is as follows:

T = S \cdot (E ε + η \frac{d ε}{d t})

(9)

ε = \frac{Δ L}{L}

(10)

where

T

is the yarn tension;

E

the elastic coefficient of the yarn;

S

the cross-sectional area of the yarn;

Δ L

the amount of yarn elongation;

L

the length of the yarn; η the viscosity coefficient of the yarn; and ε the deformation ratio of the yarn.

Figure 4.

Kelvin model of the yarn.

The diameter of the wound package changes correspondingly during the winding process. As a result, there is a speed difference between yarn speed $v_{c}$ on the wound package and yarn speed $v_{f}$ on the overfeed roller; $\int_{0}^{t} v_{c}$ is the wound yarn length; $\int_{0}^{t} v_{f}$ the length of the yarn delivered by the overfeed roller; and $Δ L$ is the difference between the two above, which expression is as follows:

Δ L = \int_{0}^{T} (v_{f} - v_{c}) d t

(11)

Therefore, the expression of yarn tensions during the winding process is as follows:

T = E S \frac{\int_{0}^{t} (v_{f} - v_{c}) d t}{L} + S η \frac{v_{f} - v_{c}}{L}

(12)

According to equation (12), yarn tension control can be regarded as motor speed control. The system controls the yarn tension by actively adjusting the speed of the overfeed motor when the speed of the winding motor is set to be constant (see Figure 5).

Figure 5.

Model of the overfeed active tension-adjusting system.

As the tension in viscosity interval $a$ is constant in the area where the yarn contacts the overfeed roller, and there is relative movement between the yarn and the overfeed roller in interval $g$ , the yarn tension is affected by the force of the overfeed roller.³⁰

The deformation ratio of the moving yarn is as follows:

ε (x, t) = {\begin{array}{l} ε_{1} (t) if x < a \\ ε_{2} (t) e^{u (x - a)} if a \leq x \leq a + g \\ ε_{2} (t) if a + g \leq x \leq L_{m} \end{array}

(13)

where

u

is the friction coefficient between the yarn and the overfeed roller;

L_{m}

is the yarn length after deformations, and

L_{m} = a + g + L

It is assumed that the sectional area and weight of the yarn remain unchanged during the winding process:

ρ_{1} S L_{1} = ρ_{0} S L_{0}

(14)

L_{1} = L_{0} (1 + ε)

(15)

where

L_{0}

and

ρ_{0}

are the length and density of the unstressed yarn, respectively;

L_{1}

and

ρ_{1}

are the length and density of the stressed yarn, respectively; and

S

is the sectional area of the yarn.

As the yarn is a medium with continuous density, the continuity equation is as follows:

\frac{\partial ρ_{1}}{\partial t} + \frac{\partial (ρ_{1} v)}{\partial x} = 0

(16)

where

v

is the yarn speed difference between the spindle and overfeed roller, and

v = v_{f} - v_{c}

The volume integral of the yarn between the initial contact point of the overfeed roller and the bobbin is as follows:

\int_{V} \frac{\partial}{\partial t} (\frac{1}{1 + ε}) d V = - \int_{V} \frac{\partial}{\partial x} (\frac{v}{1 + ε}) d V

(17)

Assume that the sectional area of the yarn does not change before and after the force is applied:

d V = Sdx

(18)

Substitute equation (18) into equation (17) and integrate $x$ , where the integral interval is $[0, I_{t}]$ . Then equation (17) can be expressed as follows:

\frac{\partial}{\partial t} (\int_{0}^{I_{t}} \frac{1}{1 + ε (x, t)} d x) = - \int_{0}^{I_{t}} \frac{\partial}{\partial x} (\frac{v (x, t)}{1 + ε (x, t)}) d V

(19)

According to equation (13) and assuming $a + g ≪ L_{1}$ , then equation (19) can be changed into the following expression:

\int_{0}^{I_{t}} \frac{1}{1 + ε (x, t)} d x = \frac{L_{1}}{1 + ε (x, t)}

(20)

Set $ε (0, t) = ε_{1}$ , $ε (I_{t}, t) = ε_{2}$ , $v (0, t) = v_{1}$ , $v (I_{t}, t) = v_{1}$ ; then equation (20) can be expressed as follows:

\frac{d}{d t} (\frac{L_{1}}{1 + ε_{2}}) = \frac{v_{1}}{1 + ε_{1}} - \frac{v_{2}}{1 + ε_{2}}

(21)

Assuming $ε (x, t) ≪ 1$ , the following expression can be obtained:

\frac{1}{1 + ε (x, t)}; 1 - ε (x, t)

(22)

Substituting equation (22) into equation (21) and combining with equation (12), then the simplified expression of yarn tension can be obtained as follows:

L \frac{d T_{2}}{d t} = E S (v_{c} - v_{f}) + T_{1} v_{c} - T_{2} v_{f}

(23)

where

T_{1}

and

T_{2}

are the yarn tensions before and after entering the overfeed roller, respectively.

Control method of conical winding formations

Figure 6 shows that the process of conical winding formations is divided into the conical preliminary forming stage and the columnar forming stage, and the red dotted line represents the boundary between the two stages. Besides, the alternating grey and green blocks under the red dotted line represent the conical preliminary forming stage, while the alternating blue and purple color blocks above the red dotted line represent the columnar forming stage.

Figure 6.

Conical forming process.

It is necessary to set the linear velocity $v_{c}$ of the yarn, diameter $d_{\min}$ of the empty bobbin, diameter $d_{\max}$ of the full wound package, winding ratio $i$ of the yarn, transmission coefficient $m$ of the doubling winder, and the soft edge distance and coefficient before starting the system.

The expression of winding ratio $i$ is as follows:

i = \frac{n_{1}}{n_{2}}

(24)

where

n_{1}

is the number of winding cycles per minute; and

n_{2}

the reciprocating times of yarn guide hooks per minute.

The yarn at the conical preliminary forming stage is wound from inside to outside along the package radial direction and the package axial direction simultaneously (see Figure 6(a)–(b)). Increased single-layer yarn $T$ is fixed because the speed of the yarn guide motor is set to be constant (see Figure 6(c)):

T = h_{a} \sin θ

(25)

where

θ

is the cone angle; and

h_{a}

is the axial increase of the single-layer yarn, which is obtained by dividing the average axial height of the full wound package and the average layer value of the full wound package.

H_{\max}

and

L_{\max}

are obtained through multiple tests under the same parameters:

h_{a} = \sum_{i = 1}^{n} H_{\max} / \sum_{i = 1}^{n} L_{\max}

(26)

where H_max is the axial height of the full wound package; and L_max the layer value of the full wound package.

$X_{Start}$ is the initial point of yarn guide hook travel; $L_{t}$ the number of winding layers; $X_{End}$ the end point of yarn guide hook travel. The conical preliminary forming stage ends, and the columnar forming stage starts (see Figure 6(b)). The angular velocity of the winding motor is constant at the columnar forming stage (see Figure 6(d)). $X_{Start}$ and $X_{End}$ change with $L_{t}$ , indicating that $X_{Start}$ and $X_{End}$ move up synchronously to ensure that the wound package can reach the yarn layers of the full wound package. $L_{H}$ is yarn guide travel and increases according to the change of $L_{t}$ :

L_{H} = X_{End} - X_{Start} = (\begin{matrix} L_{t} \cdot h_{a} \begin{matrix} (L_{t} < L_{c}) \end{matrix} \\ L_{c} \cdot h_{a} \begin{matrix} (L_{t} \geq L_{c}) \end{matrix} \end{matrix}

(27)

where

L_{c}

is the number of boundary layers between two winding stages.

Problems such as hard edge, yarn unseating, and even yarn breakage occur due to the response delay and the vibration influence during the reversing process of the yarn guide motor, so soft edge parameters should be set to avoid the above situations. $X_{Start}$ , $X_{End}$ , and soft edge parameters determine the travel of each reciprocating yarn-guide motor. Soft edge parameters consist of the soft edge coefficient and soft edge distance. The soft edge coefficient and soft edge distance represent the proportion and difference between the long and short edges, respectively.

A large diameter difference exists between different positions of the winding bobbin in a short time. $v_{c}$ is the average yarn linear velocity on the bobbin within the set time interval; $ω_{c}$ the angular velocity of the winding motor determined by $v_{c}$ , $d_{\max}$ , and $d_{\min}$ . The expression of $ω_{c}$ is as follows:

ω_{c} = \frac{2 v_{c}}{(d_{\max} - d_{\min}) π}

(28)

The rotary speed of the yarn guide motor $v_{2}$ is calculated by $i$ and $m$ . The transmission coefficient of doubling winder $m$ denotes the distance of yarn guide hook movement when the yarn guide motor rotates for one circle. It is determined by the size of the timing belt and timing wheel. The rotary speed $v_{2}$ remains constant during conical winding formations to ensure the uniform thickness of each yarn layer:

v_{2} = \frac{i ω_{c}}{m}

(29)

Design of the yarn-tension control method

The yarn tension mainly depends on the linear velocity and overfeed speed. Besides, the yarn tension is related to the yarn density, yarn elasticity, wound package diameters, friction coefficients, and other factors.³¹ The linear velocity will change in a short time during the operation of the doubling system due to the changing position of the yarn guide hook according to the conical forming process, where the maximum linear velocity is 2–4 times the minimum linear velocity. It is difficult for the traditional PID controller to meet the control accuracy requirement of the system for the winding tension because the tension control system for conical winding formations is a nonlinear dynamic system. The RBF neural network has nonlinear mapping, parallel processing, and self-learning, which is suitable for such nonlinear dynamic systems. The adaptive PID controller based on the RBF neural network uses the RBF neural network to locally approximate the nonlinear dynamic system and then adjusts incremental PID parameters online according to the identified Jacobin information.

Process of forwarding outputs

The control system for conical winding formation adopts incremental PID to facilitate the adjustment of PID parameters. The RBF neural network system has three nodes in the input layer. System inputs are as follows:

X = [Δ u (k), y (k), y (k - 1)]

(30)

where

Δ u (k)

is the increment of the system input;

y (k)

the system output; and

y (k - 1)

the identification error.

The hidden layer is designed as six nodes, and the activation function of output $h_{j}$ is the Gosky function expressed as follows:

h_{j} (k) = \exp (- \frac{{‖ u (k) - c_{j} ‖}^{2}}{b_{j}^{2}})

(31)

where

u (k)

is the system input;

c_{j}

the center vector of the

j_{t h}

node; and

b_{j}^{}

the bandwidth parameter of the

j_{t h}

node.

The reference output of the network is as follows:

y_{m} (k) = w_{1} h_{1} + w_{2} h_{2} + \dots + w_{6} h_{6}

(32)

where

w_{1}, w_{2}, \dots, w_{6}

are node-network connection weights.

Process of reverse regulation

The gradient descent method is used to adjust the output weight, node center, and bandwidth parameters in reverse regulation. The gradient descent method is used for reverse regulation due to its forms, efficiency, and local optimization effects.³²

The node weight adjustment is expressed as follows:

w_{j} (k) = w_{j} (k - 1) + η (y (k) - y_{m} (k)) h_{j} + α (w_{j} (k - 1) - w_{j} (k - 2))

(33)

where

η

is the learning rate; and

α

is the momentum factor.

The node-bandwidth parameter adjustment is expressed as follows:

Δ b_{j} = (y (k) - y_{m} (k)) w_{j} h_{j} \frac{{‖ X - C_{j} ‖}^{2}}{b_{j}^{3}}

(34)

b_{j} (k) = b_{j} (k - 1) + η Δ b_{j} + α (b_{j} (k - 1) - b_{j} (k - 2))

(35)

The expressions of the center vector adjustment are as follows:

Δ c_{j i} = (y (k) - y_{m} (k)) w_{j} \frac{X_{j} - c_{j i}}{b_{j}^{2}}

(36)

c_{j i} (k) = c_{j i} (k - 1) + η Δ c_{j i} + α (c_{j i} (k - 1) - c_{j i} (k - 2))

(37)

PID controller based on the RBF neural network

Figure 7 shows the structure of the PID control system based on the RBF neural network. $K_{p}$ , $K_{i}$ , and $K_{d}$ adjust online to overcome model changes by the nonlinear characteristics of the tension control system, yarn variety differences, and operating conditions.

Figure 7.

Radial-basis-function (RBF)-Proportion Integration Differentiation (PID) structure.

The index function of RBF neural network performance is as follows:

E (k) = \frac{1}{2} e {(k)}^{2} = \frac{1}{2} {(y (k) - y_{m} (k))}^{2}

(38)

(38) where

e (k)

is the system identification error;

y (k)

the system output; and

y_{m}

the reference output of the network (the predictive value).

The sensitivity of system identification is expressed as follows:

\frac{\partial y (k)}{\partial u (k)} \approx \frac{\partial y_{m} (k)}{\partial Δ u (k)} = \sum_{i = 1}^{m} w_{i} h_{i} \frac{c_{i j} - Δ u_{k}}{b_{i}^{2}}

(39)

where

y (k)

is the system output;

u (k)

the system input;

Δ u (k)

the increment of the system input;

y_{m}

the predictive value;

m

the number of network nodes;

w_{i}

the node network connection weight;

h_{i}

the output of the activation function;

c_{i j}

the center vector; and

b_{i}

the node bandwidth parameter.

PID parameters are updated by identified system sensitivity, which is expressed as follows:

Δ K_{p} (k) = - η \frac{\partial E (k)}{\partial K_{p} (k - 1)} = η e (k) \frac{\partial y (k)}{\partial Δ u (k)} x_{c 1} (k)

(40)

Δ K_{i} (k) = - η \frac{\partial E (k)}{\partial K_{i} (k - 1)} = η e (k) \frac{\partial y (k)}{\partial Δ u (k)} x_{c 2} (k)

(41)

Δ K_{d} (k) = - η \frac{\partial E (k)}{\partial K_{d} (k - 1)} = η e (k) \frac{\partial y (k)}{\partial Δ u (k)} x_{c 3} (k)

(42)

The expressions of $x_{c 1}$ , $x_{c 2}$ , and $x_{c 3}$ are as follows:

x_{c 1} (k) = e (k) - e (k - 1)

(43)

x_{c 2} (k) = e (k)

(44)

x_{c 3} (k) = e (k) - 2 e (k - 1) + e (k - 2)

(45)

The output of the PID controller is $u (k)$ and is expressed as follows:

u (k) = u (k - 1) + K_{p} (k - 1) x_{c 1} (k) + K_{i} (k - 1) x_{c 2} (k) + K_{d} (k - 1) x_{c 3} (k)

(46)

Simulations and experiments

MATLAB was used to establish a mathematical model of the control system (see Figure 8), whose input and output are the rotation speeds of the overfeed motor and yarn tension, respectively. The input target value of the tension is a step function. The tension control mode adopts traditional PID and RBF-PID, respectively.

Figure 8.

Simulation blocks of the yarn tension control system.

The learning rate $η$ of the RBF neural network is 0.5; momentum factor $α$ is 0.5; initial PID parameters are in [10,0.2,0.5]; the tension target value is 50 cN. Figure 9 shows the step response of the system. Compared with the traditional PID controller, the maximum overshoot of the PID controller based on the RBF neural network is reduced by 60% under the same initial parameters. Besides, the stabilization time is earlier, and the steady-state error amplitude is significantly reduced after 2.5 s.

Figure 9.

System step responses.

Firstly, to compare the closed-loop control effect of PID and RBF-PID, the experiment was carried out without using the overfeed mechanism to adjust yarn tension. The yarn was guided by the spring tension clamp and tension sensor, and the average linear speed of the yarn was changed by changing the rotation speed of the winding motor and testing the change of the yarn tension of the system. This experiment tested the fluctuation of yarn tension under the average winding speed of 300 m/min, 500 m/min, 700 m/min and 900 m/min respectively. The yarn material was cotton; the number of yarn counts was 41.67 tex; the number of yarn strands was 2; the cone angle is 35°; the diameter of the empty wound package was 50 mm; the diameter of the full wound package was 136 mm.

The experimental results showed that when the yarn was tested without using the overfeed mechanism to adjust the tension (see Figure 10), the yarn tension fluctuation amplitude at the average linear speed of 300 m/min was ± 7.81 cN, the yarn tension fluctuation amplitude at the average linear speed of 500 m/min was ±12.08 cN, and the yarn tension fluctuation amplitude at the average linear speed of 300 m/min was ±14.26 cN, the fluctuation range of yarn tension at the average linear speed of 900 m/min was ±17.67 cN.

Figure 10.

Experiment results of adjusting yarn tension without over feeding mechanism. (a) 300 m/min; (b) 500 m/min; (c) 700 m/min and (d) 900 m/min.

The yarn tension changes obviously in the acceleration and deceleration phases, and the larger the linear speed is, the larger the range of yarn tension fluctuation is. In the critical area of the acceleration and deceleration phases, the tension changes slowly, and then increases and decreases dramatically. The yarn tension in the system is mainly affected by the unwinding tension of the balloon and the additional tension of the yarn guide movement. In addition, because the linear velocity of the yarn changes periodically during the winding process, and the sign changes of linear acceleration in deceleration and acceleration phases, the yarn inertia force and unwinding tension wave received are large. When the yarn guide hook reverses, the speed change is relatively gentle at the beginning, resulting in a relatively slow tension change in a short period of time, but in the subsequent half process, the tension rapidly increases with the increase of yarn linear speed and unwinding balloon.

According to the requirements of the Chinese textile industry standard FZ/T 93059-2015 for the quality of yarn doubling: the tension between the doubling yarn and the yarn is uniform, the tension difference between two yarns on the same spindle is not more than ±3 cN, and the tension difference between two spindles on the same yarn is not more than ±4 cN. The experimental results show that the yarn after doubling does not meet the national standard. Therefore, this system introduces a closed loop control of yarn tension with overfeed mechanism.

An experimental platform for doubling winders was designed to verify the control effect of RBF neural network PID. The experimental platform included an overfeed mechanism, a winding mechanism, a yarn guide mechanism, and a tension sensor (see Figure 11). The sampling frequency of the tension sensor was 100 Hz. The tension was fed back to the PID controller after first-order filtering, with average value taken every three times and transmitted to the PC terminal.

Figure 11.

Experimental platform.

To compare the effect of traditional PID and RBF-PID in controlling yarn tension. The test experiment was conducted for the following different varieties of yarns with different linear velocities (see Table 1). The diameter of the empty bobbin was 50 mm at the columnar forming stage; the diameter of the full wound package was 136 mm; the minimum linear velocity was 270 m/min; the maximum linear velocity was 730 m/min; the winding period was 8 s; the tension value was 80 cN. Figures 12 –17 show the test results, where the cone angle of the wound package in experiments 1 to 5 was 30°, and that of experiment 6 was 35°.

Table 1.

Yarn test parameters

Number	Material type	Yarn count (tex)	Yarn strands	Elasticity coefficient (MPa)	Viscosity coefficient (GPa · s)	Average linear velocity (m/min)	The cone angles (°)
1	Cotton	41.67	2	18	4.4	500	30
2	Cotton	41.67	2	18	4.4	700	30
3	Cotton	41.67	3	27	6.6	500	30
4	Cotton	31.25	2	13.5	2.9	500	30
5	Polyester	41.67	2	22	5.1	500	30
6	Cotton	41.67	2	18	4.4	500	35
Distance between the overfeed roller and yarn guide						0.3–0.52 m
Distance between yarn guide and spindle						0.08 m

Figure 12.

Result of experiment 1.

Figure 13.

Result of experiment 2.

Figure 14.

Result of experiment 3.

Figure 15.

Result of experiment 4.

Figure 16.

Result of experiment 5.

Figure 17.

Result of experiment 6.

When the yarn winding positing changes from the starting winding diameter to the target winding diameter and the linear velocity increases, the amplitude of yarn tensions increases significantly in a short time (see Figures 12 –17). In addition, the tension falls back to the target value after the first second of the winding diameter change cycle. When the winding position of the yarn changes from the previous target winding diameter to the next starting winding diameter and the linear velocity decreases, the yarn tension amplitude decreases significantly. The tension returns to the target value after the first second of the winding-diameter changing period. The fluctuation amplitude of RBF-PID is smaller than that of traditional PID under each test parameter.

The yarn tension using the traditional PID controller fluctuates within ±8% of the target value in multiple cycles under the average linear velocity of 500 m/min for two-strand cotton-yarns with 41.67 tex (see Figure 12). The yarn tension using the RBF-PID controller fluctuates within ±3% of the target value. According to the comparison between Figures 12 and 13, the yarn count is 41.67 tex with two yarn strands. Compared with the cotton yarn under a linear velocity of 500 m/min, the tension fluctuation range of the cotton yarn is large within ±12% of the target value under a linear velocity of 700 m/min controlled by the traditional PID controller. The yarn-tension fluctuation range of the RBF-PID controller is the same as the linear velocity of 500 m/min.

The cotton-yarn count is 41.67 tex and the average linear velocity is 500 m/min through comparison between Figures 12 and 14. Compared with the cotton yarn with two strands, the tension fluctuation range of the cotton yarn with three strands controlled by the traditional PID controller is larger within ± 10% of the target value. The tension fluctuation range of the three-strand yarn by the RBF-PID controller is the same as that of the two-strand yarn.

According to the comparison between Figures 12 and 15, the linear velocity is 500 m/min with two-yarn strands. The tension fluctuation range of the 31.25 tex cotton yarn controlled by the traditional PID controller is larger compared with the 41.67 tex cotton yarn. The yarn-tension fluctuation range of the RBF-PID controller is smaller, especially in the deceleration phase. The RBF-PID control effect is better, and the dynamic error of yarn tensions will not exceed ± 3% of the target value.

When the material is changed to polyester, and the yarn count, number of strands, linear velocity, and the cone angle of the wound package remain unchanged (see Figure 16), Compared with the cotton-yarn tension under the traditional PID control (see Figure 12), the fluctuation is obviously larger, where the fluctuation range is ±16% of the set value, while the RBF-PID tension fluctuation amplitude is basically unchanged, with yarn tension fluctuation within ±3% of the target value, and only the tension fluctuates significantly when the guide hook reverses.

When the taper angle of the wound package becomes 35° (see Figure 17), and the yarn material, yarn count, yarn strands and linear speed remain unchanged, the yarn tension fluctuation range of traditional PID control is about ±3% of the target value in multiple cycles, while the yarn tension fluctuation range controlled by RBF-PID will not exceed ±2% of the set value. Compared with the packaging cone angle of 30° (see Figure 12), the fluctuation of yarn tension under the traditional PID and RBF-PID control is significantly smaller, where the fluctuation of yarn tension under the traditional PID control is reduced by ±5%, and the fluctuation of yarn tension under the RBF-PID control is reduced by ±1%. Besides, the RBF-PID controller can also maintain the yarn tension fluctuation within 2% for different packaging cone angles.

Conclusions

The study proposed a method of conical winding formations for the wound package, which were stacked layer by layer along the radial direction of the package. It overcame the axial unwinding problem of the wound package with traditional parallel winding formations. A system to control the yarn tension through the active overfeed regulation was designed to implement the conical winding formations. The viscoelastic model was used to analyze the yarn tension in the winding process, the relationship between the tension changes of yarns in the winding process, and the difference between overfeed and winding speeds.

As there were many factors affecting the yarn tension, the tension control system was time-varying, nonlinear, and accidental, and the linear velocity varied 2–4 times in the yarn guide cycle during conical winding formations. The tension control system using a traditional PID controller could not stabilize the yarn tension within ±3% of the target value, which did not meet the accuracy requirements of the system for the tension control in the yarn doubling process. Therefore, a PID controller was introduced based on a RBF neural network to approximate the system locally and adjust incremental PID parameters according to identified Jacobin information.

According to the yarn tension model of conical winding formations, the RBF-PID controller was used to control the speed of the overfeed motor. Compared with the tension control system without over feeding mechanism, the system has better yarn tension control capability. Compared with the traditional PID controller, the RBF-PID controller had a faster response speed and smaller steady-state error, and the yarn tension using the RBF-PID controller was less affected by the linear velocity under different yarn strands and yarn counts. When the experimental materials are unchanged, the RBF-PID controller can be applied to the yarn with different average linear velocity, yarn counts, and strands, and the yarn tension fluctuation will not exceed ±3% of the target value. When the experimental material is changed to polyester and other experimental conditions remain unchanged, the RBF-PID controller can still control the yarn tension fluctuation within ±3% of the target value. In addition, when the packaging cone angle is 35° and other conditions remain unchanged, the yarn tension fluctuation under RBF-PID control is kept within ±2%.

Future work

Fully realize the microprocessor control, realize the data communication between the microcontroller and the general computer through wireless technology, and easily set parameters, to reduce space to meet and expand commercial applications.

Develop a design anti-stacking algorithm for the cone wound package to avoid ribbon formation during winding process.

Design appropriate tension control strategies to reduce the phenomenon of internal tightness and external looseness in the wound package without affecting the twist.

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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