Prediction and optimization of chemical fiber spinning tension based on grey system theory

Abstract

Based on the grey prediction model, this paper studied the effect of the chemical fiber spinning process parameters on the winding tension. Suitable process parameters were selected to carry out grey incidence analysis with winding tension, and the feasibility of the grey prediction model in spinning tension prediction was validated by the designed experiments. The corresponding algorithm routines of various grey prediction models were written in MATLAB. The single-variable grey prediction model of GM(1,1) showed a higher prediction accuracy in the effect of the single process parameter changing on spinning tension; when multiple process parameters changed at the same time, the average modeling error of the MGM(1,n) multi-variable grey prediction model was 7.70%, and the maximum error was as high as 32.99%. The original MGM(1,n) model was optimized and the model background value was adjusted by using the auto-optimization and weighting method. The average modeling error of the improved model was reduced to 2.02%, which could meet the general accuracy requirement of tension prediction. Further combining fractional-order accumulation and adjusting the background value coefficient α and the cumulative order r jointly, the smallest prediction error was found among the 100,000 combinations, and the final error was further reduced to 1.30%. The results show that the grey prediction model is suitable and effective for predicting the spinning tension based on the process parameters. Appropriate model improvement mechanisms will increase the prediction accuracy significantly. This application provides a suitable method for spinning tension prediction, which has great significance for the tension control of chemical fiber products.

Keywords

chemical fiber spinning tension prediction grey prediction model grey incidence analysis background value optimization fractional-order accumulation

Chemical fiber is widely used in the textile, clothing, decoration, and industrial fields. The quality of chemical fiber products has increasingly come into the focus. Today the methods for controlling the quality of chemical fiber products include single spindle control, pre-alarming control, and online monitoring.^1,2 Xiang et al.³ studied the intelligent control model of spinning technology based on rough set theory (RST) and extracted logic rules from the decision table to select suitable fiber materials to ensure the quality of the product. Liu⁴ applied an orthogonal test and statistical analysis to the control of spinning quality, pointing out that an important means of managing and controlling spinning quality is to scientifically use statistical tools to collect valid test data and conduct mathematical analysis.

The product quality of chemical fiber spinning is inseparably related to the tension in the winding process.⁵ Consulting the first-line production experience of the chemical fiber industry and doing some research in the factory, it is known that the winding tension is an easy and effective way to detect and measure the quality of spinning products. Most of the tension measurement at this stage uses a contact tension meter, but direct contact with the fiber will affect the quality of the product, and the measurement requirements are more severe. For example, a slight tilt or jitter will cause changes in the results.⁶ Therefore, it is of great significance to explore a new approach to measure winding tension without contacting the fiber.

Yang and Hao⁷ used the theory of time series and established an Applied Regression (AR) mathematical model to predict and control the spinning tension, so as to achieve the purpose of controlling the spinning tension in real time. However, the prediction accuracy of this AR mathematical model is not high. Under the condition that the winding speed of chemical fiber spinning is constant, the spinning winding tension is greatly influenced by the following spinning process parameters: spinning hot roller temperature, spinning hot roller speed, side blowing speed, oil content, etc.^8,9 Based on these papers, we studied the effect of the spinning process parameters on the tension of chemical fiber spinning and established a model. The change of chemical fiber spinning tension was predicted through the change of process parameters in order to find the correlation between them. By adjusting the process parameters to control the winding tension within the ideal range, a new method that controls the quality of chemical fiber products was explored.

After studying many predictive models, we learned that the grey system theory is based on partial information, and takes uncertain issues, such as a small amount of data and grey information, as research objects. It generates some incomplete information to make the evolutionary features of the system distinct. Then it can scientifically evaluate, predict, and make decisions on the system. Grey theory can not only predict unknown data, but also can obtain continuous time series by processing discrete time series of the system to predict the grey systems with incomplete information.^10,11 So, it is very suitable for the spinning tension prediction in this paper.

The grey system theory uses the method of grey incidence analysis to evaluate the system. As a developmental system, the incidence analysis is a quantitative analysis of the dynamic development situation.¹² GM(1,1) is the most primitive single-variable grey prediction model with mature and extensive applications. When the variables increase from one-dimensional to multi-dimensional, the GM(1,n) multi-variable grey prediction model has a low prediction accuracy because of the forced linear assumptions between the parameters. The MGM(1,n) model is derived from the GM(1,1) model through increasing the dimension naturally. It is not a simple combination of n GM(1,n) models. The MGM(1,n) model establishes n n-variable differential equations and solves them, so that the parameters in the model can reflect the interactions and constraints between multiple variables.^13,14 In recent years, the MGM(1,n) model has been successfully applied to building analysis and prediction,^15–17 disaster prediction,^18,19 financial decisions,^20–22 and so on.

For different research objects, the grey prediction model needs to adopt corresponding improvement methods to achieve the desired prediction effect. The model accuracy is the key and the difficulty of modeling. Many scholars have proposed methods to improve accuracy, such as background value reconstruction,²³ background value coefficient optimization,²⁴ initial value optimization,²⁵ and the new information optimization model.²⁶ In order to facilitate the solution, the grey prediction model set the background value as the equal weight generation of a cumulative sequence, but there is no theory proving that the prediction accuracy of the model is the highest when α = 0.5. Finding the best background coefficient α using the auto-optimization and weighting method can effectively reduce the model error.

The above models all belong to the integer-order grey prediction model, but actually there are many objects that satisfy the characteristics of fractional order. The use of fractional-order accumulation can reveal their essence and result in better behavior. Wu and Liu²⁷ proposed fractional-order accumulation grey model FGM(1,1), where the grey prediction error can be reduced by using fractional-order grey generation.²⁸

This paper analyzed the mechanism of the quality control of chemical fiber spinning, and studied the factors affecting the tension, providing a suitable and effective prediction model for the prediction of the spinning tension of the chemical fiber.^29,30 The single-variable spinning tension was predicted with the GM(1,1) model, and the incidence degrees between each parameter and the spinning tension were analyzed. In terms of multi-variable prediction, based on the modeling principle of the MGM(1,n) model, the original data was accumulated to increase the exponential law, and the randomness was reduced. The model parameters were obtained by the least squares method and the time response function was given. Taking the fractional order and background coefficient as design variables and the minimum average relative error as the objective function, an optimal modified FMGM(1,n) model was established, and each solving program was compiled based on MATLAB. The final improved model shows a strong adaptability and effectiveness. It has found a suitable model for the prediction of spinning tension and has promoted the tension control of chemical fibers. This is also a successful application of the grey prediction model in spinning tension prediction.

Except for the introductory part, the follow-up contents of this research are arranged as follows: the second section describes the establishment process of the general grey prediction model, the third section describes the experimental verification of the single-variable model, the fourth section describes the experimental verification and optimization of the multi-variable model, and the fifth section summarizes and discusses the article.

General grey prediction model

Grey modeling is the main task of the grey system theory, that is, processing and converting the original data sequence to obtain a new sequence with better laws, and then establishing a grey model to predict the system.

In the process of chemical fiber spinning, the spinning tension is affected by various spinning process parameters. Firstly, it is necessary to determine the process parameters that have a significant influence on the spinning tension. According to the characteristics of the fiber materials, chemical fiber spinning can be roughly divided into two categories: melt spinning and solution spinning. This paper predicted the winding tension of melt spinning based on the spinning parameters of polyester. By analyzing the polyester spinning process and combining first-line production experience, the six main process parameters, namely side blowing speed, oil content, first hot roller speed, second hot roller speed, first hot roller temperature, and second hot roller temperature, were determined. The general production process of polyester filament spinning is shown in Figure 1.

Figure 1.

Polyester spinning process.

The GM(1,1) single-variable grey prediction model was established to study the effect of single process parameters changing on the spinning winding tension. The single-variable model was used to study the influence of the above six process parameters on the spinning winding tension when the parameters were uniformly changed.

Taking the side blowing speed as an example, the corresponding winding tension value was recorded as the original one-dimensional data when the side blowing speed changed uniformly.

Supposing the original one-dimensional sequence is

X^{(0)} = (x^{(0)} (1), x^{(0)} (2), \dots, x^{(0)} (m))

After determining the original data sequence, the GM(1,1) model is implemented according to the following steps. Firstly, the original data is accumulated to eliminate the random disturbance and then the 1-AGO sequence $X^{(1)}$ is obtained

X^{(1)} = (x^{(1)} (1), x^{(1)} (2), \dots, x^{(1)} (m))

The accumulated generating operation is written as

x^{(1)} (k) = \sum_{i = 1}^{k} x^{(0)} (i), k = 1, 2, \dots, m

(1)

The system background value time series $Z^{(1)}$ is obtained from the background value generation formula (2), where α is the weight coefficient of the background value and α = 0.5 is generally taken in the model

z^{(1)} (k) = α x^{(1)} (k - 1) + (1 - α) x^{(1)} (k), k = 2, 3, \dots, m

(2)

Z^{(1)} = (z^{(1)} (2), z^{(1)} (3), \dots, z^{(1)} (m))

The grey system whitenization equation is a first-order differential equation expressed as Equation (3), which is processed discretely to obtain the single-variable grey prediction model (4)

\frac{d x^{(1)}}{dt} + a x^{(1)} = b

(3)

x^{(0)} (k) + a z^{(1)} (k) = b

(4)

where a is the system development coefficient, which shows the development trend of the model, and b is the grey effect coefficient of the model. The influence degree and polarity of each influence factor on winding tension can be determined according to the a value. The case with a negative a value illustrates that the influence of this parameter on tension is positively correlated. In other words, the increase of this variable value will increase the winding tension. Conversely, when the a value is positive, it is negatively correlated. Here, a and b are obtained by the least square method, set

a^{^} = (a, b)^{T}

The matrix solution of the parameter is

a^{^} = (B^{T} B)^{(- 1)} B^{T} Y

(5)

where

\begin{matrix} Y = [\begin{matrix} x^{(0)} (2) \\ x^{(0)} (3) \\ : \\ x^{(0)} (m) \end{matrix}] B = [\begin{matrix} - z^{(1)} (2) & 1 \\ - z^{(1)} (3) & 1 \\ : & : \\ - z^{(1)} (m) & 1 \end{matrix}] \end{matrix}

By substituting the parameter values a and b that are obtained from Equation (4) into Equation (3), the time response function (6) can be obtained, and the discretized form (7) can be written

x^{(1)} (t) = C e^{- at} + \frac{b}{a}

(6)

{x^{^}}^{(1)} (k + 1) = C e^{- ak} + \frac{b}{a}, k = 0, 1, 2, \dots, m - 1

(7)

$C = x^{(0)} (1) - b / a$ can be obtained by substituting $x^{(1)} (1) = x^{(0)} (1)$ as the initial value into (7), then substituting the corresponding k to obtain the fitted value ${x^{^}}^{(1)}$ of the system sequence. The reverse accumulated generating operation of ${x^{^}}^{(1)}$ is performed, as shown in Equation (8), to obtain the original sequence fitted value ${x^{^}}^{(0)}$

{x^{^}}^{(0)} (k + 1) = {x^{^}}^{(1)} (k + 1) - {x^{^}}^{(1)} (k)

(8)

Thus, the winding tension simulation values and prediction values of the original sequence are obtained. After the step is completed, the average relative error of the system, the mean absolute percentage error (MAPE), is calculated to verify the accuracy of the model, as shown in Equation (9)

MAPE (%) = \frac{1}{n} \sum_{k = 1}^{n} | \frac{x^{(0)} (k) - {x^{^}}^{(0)} (k)}{x^{(0)} (k)} | \times 100 %

(9)

The target of tension prediction error is within 2%. If the error exceeds 2%, further optimization is needed.

In order to verify the feasibility of the grey system theory in predicting spinning winding tension, a number of experiments were designed. Field experiments were conducted in a high-speed polyester filament spinning workshop and experimental data were collected to verify the theory.

The experiments are mainly divided into two categories. The first category is the effect of changing a single process parameter on the winding tension, which is described in the third section; the second category is the effect of various process parameters changing simultaneously on the winding tension, which is described in the fourth section.

Experimental verification of single-variable tension prediction

When only one process parameter is changed, a single-variable grey model is established to predict the winding tension. The effect of individual process parameters on the winding tension is shown in Table 1. The six sets of data indicate the corresponding changes in the winding tension of the chemical fiber spinning when the six process parameters (wind blow speed, oil content, first hot roller speed, second hot roller speed, first hot roller temperature, and second hot roller temperature) are changed, respectively.

Table 1.

Experimental data sheet of winding tension influenced by single process parameters

Number	F1	$x_{1}^{(0)}$	F2	$x_{2}^{(0)}$	F3	$x_{3}^{(0)}$	F4	$x_{4}^{(0)}$	F5	$x_{5}^{(0)}$	F6	$x_{6}^{(0)}$
1	0.3	15.9	10	15.8	1850	18.1	5050	18.8	86	15.6	127	16.5
2	0.35	16.1	13	15.9	1900	17.9	5060	17.9	88	16.4	129	16.6
3	0.4	16.1	16	16.1	1950	17.7	5070	15.9	90	16.3	131	16.1
4	0.45	16.7	19	16.1	2000	16.1	5080	14.7	92	17.6	133	16.3
5	0.5	17	22	16.3	2050	15.6	5090	13.8	94	17.5	135	16.4
6	0.55	17.3	25	16.4	2100	15.1	5100	12.5	96	18.1	137	16.2
7	0.38	16.1	15	16.1	1920	17.9	5066	17.2	89	16.3	130	16.5
8	0.42	16.5	17	16	1980	17.2	5072	15.8	91	16.6	132	16.4
9	0.48	16.9	21	16.3	2040	15.8	5088	14	93	17.5	134	16.4

In Table 1, F1 indicates the side blowing speed (m/s), F2 indicates the oil content (%), F3 indicates the first hot roller speed (r/min), F4 indicates the second hot roller speed (r/min), F5 indicates the first hot roller temperature (℃), F6 indicates the second hot roller temperature (℃), and $x_{i}^{(0)}$ indicates the corresponding spinning winding tension (N).

Since internal interpolation prediction of the grey prediction model has higher accuracy than the extrapolated prediction, we set the data range of the process parameters collected in the experiment as the maximum range for ensuring the normal process. Taking the side blowing speed as an example, when F1 is within 0.3–0.55 m/s, the spinning process can be performed normally. When the side blowing speed is lower than 0.3 m/s, the expected cooling effect cannot be achieved, and when the side blowing speed exceeds 0.55 m/s, the filament structure will be affected. So, the measurement range of the side blowing speed is set at 0.3–0.55 m/s. The values of parameters 1–6 in the table are successively increased, which are within the maximum range of the normal process, and are used to establish the model; data 7–9 are interpolation data in the range, which are used to verify the prediction accuracy of the internal interpolation.

The above six sets of experimental data all meet the service conditions of the grey prediction model, and the stepwise ratio test is eligible. In the process of system development, if the synchronization of change trends of the two factors is strong, it means that the two are highly related; on the contrary, it is lower.³¹ In order to verify the correlation between the above parameters and spinning tension, the grey incidence analysis was performed on the above six sets of data. Here is an example of the effect of the side blowing speed on the winding tension. The original data is

\begin{matrix} X^{(0)} = [\begin{matrix} x_{1}^{(0)} (1) & x_{1}^{(0)} (2) & \dots & x_{1}^{(0)} (M) \\ x_{2}^{(0)} (1) & x_{2}^{(0)} (2) & \dots & x_{2}^{(0)} (M) \\ \dots & \dots & \dots & \dots \\ x_{n}^{(0)} (1) & x_{n}^{(0)} (2) & \dots & x_{n}^{(0)} (M) \end{matrix}] \\ = [\begin{matrix} 0.30 & 0.35 & 0.40 & 0.45 & 0.50 & 0.55 \\ 15.9 & 16.1 & 16.1 & 16.7 & 17.0 & 17.3 \end{matrix}] \end{matrix}

Step 1: to make the original data comparable, preprocess it and eliminate the magnitude and dimension of the original data. The commonly preprocessing algorithms are initial conversion, average conversion, and standard conversion. The average conversion algorithm is used here, as shown in Equations (10) and (11)

x_{i}^{(0)} = \frac{\sum_{t = 1}^{M} x_{i}^{(0)} (t)}{M} (t = 1, 2, \dots, M)

(10)

x_{i}^{(1)} (t) = x_{i}^{(0)} (t) / x_{i}^{(0)} (i = 1, 2, \dots, n)

(11)

Solved:

X^{(1)} = [\begin{matrix} 0.7059 & 0.8235 & 0.9412 & 1.0588 & 1.1765 & 1.2941 \\ 0.9627 & 0.9748 & 0.9748 & 1.0111 & 1.0293 & 1.0474 \end{matrix}]

Step 2: calculate the absolute error $Δ_{ij} (t)$ between the reference sequence $x_{i}^{(1)} (t)$ and the compared sequence $x_{i}^{(1)} (t)$ , find the maximum $Δ max$ values and minimum values $Δ min$ of $Δ_{ij} (t)$ , and calculate the incidence coefficient $L_{ij} (t)$

Δ_{ij} (t) = | x_{i}^{(1)} (t) - x_{j}^{(1)} (t) | (j = 1, 2, \dots, n; j \neq i)

(12)

L_{ij} (t) = (Δ min + k Δ max) / (Δ_{ij} (t) + k Δ max)

(13)

where the coefficient k is 0.5, calculated as

Δ max

= 0.2568

Δ min

= 0.0336,

L_{12} = [\begin{matrix} 0.4206 & 0.5793 & 1.0000 & 0.9198 & 0.5878 & 0.4319 \end{matrix}] .

Step 3: calculate the incidence degree $R_{ij}$

R_{ij} = \frac{\sum_{t = 1}^{M} L_{ij} (t)}{M} (t = 1, 2, \dots, M)

(14)

Calculate the incidence degree between the winding tension and the first spinning process parameter from formula (14): R₁ = 0.6565.

Similarly, calculate the grey incidence degree between the other five groups of data and the winding tension. The results are

\begin{matrix} R_{2} = 0.7064; R_{3} = 0.7378; R_{4} = 0.6032; \\ R_{5} = 0.6160; R_{6} = 0.6718 \end{matrix}

The analysis and comparison of the incidence degree between the six process parameters and the spinning winding tension show that all the incidence degree values are greater than 0.6. This indicates that these six process parameters have a greater impact on the spinning winding tension and these six parameters are reasonable to be selected. In addition, the first hot roller speed has the greatest effect on the spinning tension in these parameters.

Write the corresponding MATLAB program of the single-variable grey prediction model, substituting these six groups of data respectively, and run the program to obtain tension fitting values and interpolation prediction values. The results are shown in Tables 2 –4 (each table incorporates two parameters).

Table 2.

Side blowing speed (F1) and oil content (F2)

Number	F1 (m/s)	k	Real value	Simulation values	Simulation error	Number	F2 (%)	k	Real value	Simulation values	Simulation error
1	0.3	1	15.9	15.90	0.00%	1	10	1	15.8	15.80	0.00%
2	0.35	2	16.1	15.99	0.71%	2	13	2	15.9	15.92	0.13%
3	0.4	3	16.1	16.31	1.28%	3	16	3	16.1	16.04	0.38%
4	0.45	4	16.7	16.63	0.40%	4	19	4	16.1	16.16	0.37%
5	0.5	5	17.0	16.97	0.20%	5	22	5	16.3	16.28	0.13%
6	0.55	6	17.3	17.31	0.04%	6	25	6	16.4	16.40	0.00%
MAPE1					0.44%	MAPE1					0.17%

				Forecast values	Forecast error					Forecast values	Forecast error
7	0.38	2.6	16.1	16.18	0.48%	7	15	2.67	16.1	16.00	0.62%
8	0.42	3.4	16.5	16.44	0.39%	8	17	3.33	16.0	16.08	0.49%
9	0.48	4.6	16.9	16.83	0.40%	9	21	4.67	16.3	16.24	0.37%
MAPE2					0.42%	MAPE2					0.49%

MAPE: mean absolute percentage error.

Table 3.

First hot roller speed (F3) and second hot roller speed (F4)

Number	F3 (r/min)	k	Real value	Simulation values	Simulation error	Number	F4 (r/min)	k	Real value	Simulation values	Simulation error
1	1850	1	18.1	18.10	0.00%	1	5050	1	18.8	18.80	0.00%
2	1900	2	17.9	18.06	0.87%	2	5060	2	17.9	17.66	1.34%
3	1950	3	17.7	17.23	2.66%	3	5070	3	15.9	16.19	1.81%
4	2000	4	16.1	16.44	2.12%	4	5080	4	14.7	14.84	0.94%
5	2050	5	15.6	15.69	0.57%	5	5090	5	13.8	13.60	1.43%
6	2100	6	15.1	14.97	0.85%	6	5100	6	12.5	12.47	0.25%
MAPE1					1.18%	MAPE1					0.96%

				Forecast values	Forecast error					Forecast values	Forecast error
7	1920	2.4	17.9	17.72	1.00%	7	5066	2.6	17.2	16.76	2.56%
8	1980	3.6	17.2	16.75	2.62%	8	5072	3.2	15.8	15.91	0.70%
9	2040	4.8	15.8	15.84	0.23%	9	5088	4.8	14.0	13.84	1.14%
MAPE2					1.28%	MAPE2					1.47%

MAPE: mean absolute percentage error.

Table 4.

First hot roller temperature (F5) and second hot roller temperature (F6)

Number	F5 (℃)	k	Real value	Simulation values	Simulation error	Number	F6 (℃)	k	Real value	Simulation values	Simulation error
1	86	1	15.6	15.60	0.00%	1	127	1	16.5	16.50	0.00%
2	88	2	16.4	16.27	0.78%	2	129	2	16.6	16.42	1.08%
3	90	3	16.3	16.71	2.54%	3	131	3	16.1	16.37	1.68%
4	92	4	17.6	17.17	2.46%	4	133	4	16.3	16.32	0.12%
5	94	5	17.5	17.63	0.76%	5	135	5	16.4	16.27	0.79%
6	96	6	18.1	18.11	0.06%	6	137	6	16.2	16.22	0.12%
MAPE1					1.10%	MAPE1					0.63%

				Forecast values	Forecast error					Forecast values	Forecast error
7	89	2.5	16.3	16.49	1.17%	7	130	2.5	16.5	16.40	0.64%
8	91	3.5	16.6	16.94	2.04%	8	132	3.5	16.4	16.34	0.36%
9	93	4.5	17.5	17.40	0.58%	9	134	4.5	16.4	16.29	0.64%
MAPE2					1.26%	MAPE2					0.55%

MAPE: mean absolute percentage error.

In the table, Real value represents the measured tension value, Simulation values represent the fitted tension value of the model, Simulation error represents the absolute value of the relative percent error of the fitting tension, Forecast values represent the predicted tension value of the model, Forecast error represents the absolute value of the relative percent error of the predicted tension, MAPE1 represents the average of the simulation error, and MAPE2 represents the average of the forecast error. The k value of numbers 7–9 is a decimal number, which is converted according to the ratio of its parameter values, and is used to realize the prediction of the internal interpolation.

The system development coefficients of the six single-variable grey prediction models in the model are as follows

\begin{matrix} a 1 = - 0.0199; a 2 = - 0.0074; a 3 = 0.0468; \\ a 4 = 0.0870; a 5 = - 0.0268; a 6 = 0.0031 \end{matrix}

In the grey prediction model, the model error increases rapidly with the increase of the absolute value of the development coefficient a. When |a| is less than 0.3, the model accuracy can reach more than 98%; if |a| is less than 0.5, the model accuracy can reach more than 95%; if |a| is greater than 1.5, the model accuracy is generally lower than 50%.³² The development coefficient of the six groups of data is less than 0.1, the average relative fitting error is within 1.2%, the average relative prediction error is within 1.5%, and the maximum error of all data in the model is only 2.66%. The prediction accuracy of the model has reached a relatively high level and meets the actual forecasting requirements.

It can be seen that the GM(1,1) model is successful in predicting the single-variable problem of tension. Using any parameter values within the allowable range of the process, the corresponding spinning tension can be accurately predicted when other influencing factors remain constant.

Experimental verification and optimization of multi-variable tension prediction

The above is the prediction of tension when a single process parameter changes. When multiple process parameters change at the same time, a multi-variable grey prediction model is needed to solve the problem. The following is a discussion of the changes in multiple process parameters.

Due to the limitations of experimental conditions in the factory, we chose three process parameters from multiple process parameters to collect data, which targets three parameter changes simultaneously. The other variables remained constant. The three selected related factors are the first hot roller speed, the second hot roller speed, and the first hot roller temperature. In the experiment, nine sets of four-dimensional data were measured. The four dimensions of each set of data represent the spinning winding tension

x_{1}^{(0)} (N)

, the first hot roller speed

x_{2}^{(0)} (r / min)

, the second hot roller speed

x_{2}^{(0)} (r / min)

and the first hot roll temperature

x_{4}^{(0)} (^{\circ} C)

. The model was built using the first six groups of data, and the last three groups were used to test the interpolation prediction accuracy. Table 5 records the experimental data of the effect of these three different process parameters changing simultaneously on the spinning tension.

Table 5.

Effect of simultaneous changes of multiple process parameters on winding tension

Number	k	$x_{1}^{(0)}$	$x_{2}^{(0)}$	$x_{3}^{(0)}$	$x_{4}^{(0)}$
1	1	19.4	1900	5050	80
2	2	17.2	1950	5060	88
3	3	15.3	1980	5070	90
4	4	15	2000	5080	92
5	5	13.2	2050	5090	98
6	6	12.1	2100	5100	108
7	2.5	16.2	1965	5065	89
8	3.5	15.1	1990	5075	91
9	5.5	12.3	2075	5095	103

Firstly, the data is analyzed using the MGM(1,n) model. The first-order n-variable ordinary differential equations of the MGM(1,n) grey prediction model are expressed as

\begin{matrix} {\begin{matrix} \frac{d {x_{1}}^{(1)}}{dt} = a_{11} {x_{1}}^{(1)} + a_{12} {x_{2}}^{(1)} + \dots + a_{1 n} {x_{n}}^{(1)} + b_{1} \\ \frac{d {x_{2}}^{(1)}}{dt} = a_{21} {x_{1}}^{(1)} + a_{22} {x_{2}}^{(1)} + \dots + a_{2 n} {x_{n}}^{(1)} + b_{2} \\ : \\ \frac{d {x_{n}}^{(1)}}{dt} = a_{n 1} {x_{1}}^{(1)} + a_{n 2} {x_{2}}^{(1)} + \dots + a_{nn} {x_{n}}^{(1)} + b_{n} \end{matrix} \end{matrix}

(15)

where n is the number of variables and m is the number of sequences for each variable. The solution and transformation methods are similar to those of the GM(1,1) model, and finally the model prediction solution is

{X^{^}}^{(1)} (k) = (X^{(1)} (1) + {A^{^}}^{- 1} B^{^}) e^{A^{^} (k - 1)} - {A^{^}}^{- 1} B^{^}

(16)

After the model is restored, the predicted value of the MGM(1,n) model can be obtained.

The MGM(1,4) grey prediction model was established, which was denoted as Model 1, and the corresponding MATLAB program was written. The data of Table 5 was substituted into the program for operation. The tension fitting and prediction of the model are shown in Figure 2.

Figure 2.

Tension prediction of Model 1.

The circles marked B in Figures 2, 3, and 5 represent the actual tension values, the curve marked C represents the fitting and predicted tension value curve of the model, and the triangles marked D represent the actual tension prediction value.

Figure 3.

Tension prediction of Model 2.

Figure 4.

Improved algorithm flow chart of Model 3. MAPE: mean absolute percentage error.

Figure 5.

Optimal solution tension prediction of Model 3.

The calculation results show that the mean fitting error of the MGM(1,4) model on the tension value is 7.70%, the average prediction error is 5.72%, and the maximum error is as high as 32.99%. Because this model has a large prediction error for this problem and fails to meet the forecasting accuracy requirements, the model needs to be optimized accordingly.

Here, the model parameter α is improved. The model accuracy is increased by modifying the model background value, and the most suitable α value is determined by using the auto-optimization and weighting method to make the prediction accuracy of the model the highest. This method belongs in the category of modifying the internal modeling mechanism.

Set the initial background value of the model to 0, find the average relative error of the model, then add a tiny amount Δα greater than zero at a time, and repeat the above process until α = 1. Using the above method, we can obtain tension prediction values under different weights, and take the weight when the average relative error is the smallest as the optimal weight of background value, and use the grey prediction model with the optimal weights to predict the tension.

The MGM(1,4) model with the optimized background value is named Model 2, and the data in Table 5 is substituted into Model 2. The results obtained are as follows: when α = 0.44, the average fitting error MAPE1 is the smallest, at this time, MAPE1min = 2.02%. The average prediction error MAPE2 = 1.43%. Figure 3 shows the tension prediction results.

As can be seen from Figure 3, the model curve after improving the background value is closer to the actual tension value. When the precision of the spinning system prediction is not high, the model can meet the tension prediction requirements.

The above models all belong to the integer-order cumulative model, but in reality the data sequences satisfying the fractional features are more common. Using fractional accumulation to describe the objects with fractional attributes can reveal their essence better.³³ In order to meet the demand for higher accuracy predictions, further optimizations are combined with the fractional order.

The sequence of background values in the fractional multi-variable grey prediction model is generated by fractional-order accumulation, and the original data matrix is expressed as

\begin{matrix} X^{(0)} = {X_{1}^{(0)} X_{2}^{(0)} \dots X_{n}^{(0)}}^{T} \\ = [\begin{matrix} x_{1}^{(0)} (1) & x_{1}^{(0)} (2) & \dots & x_{1}^{(0)} (M) \\ x_{2}^{(0)} (1) & x_{2}^{(0)} (2) & \dots & x_{2}^{(0)} (M) \\ \dots & \dots & \dots & \dots \\ x_{n}^{(0)} (1) & x_{n}^{(0)} (2) & \dots & x_{n}^{(0)} (M) \end{matrix}] \end{matrix}

$X^{(r)}$ is the r-order cumulative generation of $X^{(0)}$ , and its matrix form is

\begin{matrix} X^{(r)} = [X^{(0)} (1) X^{(0)} (2) \dots X^{(0)} (M)] \\ \times [\begin{matrix} 1 & 1 & \dots & 1 \\ 0 & 1 & \dots & 1 \\ \dots & \dots & \dots & \dots \\ 0 & 0 & \dots & 1 \end{matrix}]^{r} \end{matrix}

(17)

Combined with the number of combinations, r is extended from an integer to a fraction to obtain a fractional cumulative generator matrix. When r is a fraction, the following formula can be obtained³⁴

{x_{i}}^{(r)} (k) = \sum_{j = 1}^{k} \frac{Γ (r + k - j)}{Γ (k + 1 - j) Γ (r)} {x_{i}}^{(0)} (j)

(18)

where

k = 1, 2, \dots, M; i = 1, 2, \dots, n; Γ

is a gamma function.

The r-order n-variable white differential equations of the list system whitening equations are expressed as

\begin{matrix} {\begin{matrix} \frac{d {x_{1}}^{(r)}}{dt} = a_{11} {x_{1}}^{(r)} + a_{12} {x_{2}}^{(r)} + \dots + a_{1 n} {x_{n}}^{(r)} + b_{1} \\ \frac{d {x_{2}}^{(r)}}{dt} = a_{21} {x_{1}}^{(r)} + a_{22} {x_{2}}^{(r)} + \dots + a_{2 n} {x_{n}}^{(r)} + b_{2} \\ : \\ \frac{d {x_{n}}^{(r)}}{dt} = a_{n 1} {x_{1}}^{(r)} + a_{n 2} {x_{2}}^{(r)} + \dots + a_{nn} {x_{n}}^{(r)} + b_{n} \end{matrix} \end{matrix}

(19)

The solution process is not repeated here, and the model calculation value of FMGM(1,n) is

{X^{^}}^{(r)} (k) = (X^{(r)} (1) + {A^{^}}^{- 1} B^{^}) e^{A^{^} (k - 1)} - {A^{^}}^{- 1} B^{^}

(20)

Then the results of the above formula are restored by the corresponding r-order reduction to obtain the model prediction value.

We combined the auto-optimization and weighting method with the fractional-order accumulation method to find the model with the smallest relative error among the 100,000-type coefficient combinations, which is denoted as Model 3. Figure 4 is a flowchart of the improved algorithm, and the corresponding MATLAB program is written. The data of Table 5 is substituted into the operation. Several representative sets of coefficients are selected from the running results and are listed in Table 6.

Table 6.

Model predictions when α and r change

Coefficient α	Order r	1	2	3	4	5	6	MAPE
0.44	1.000	19.4	16.9	15.9	15.0	13.2	11.3	2.02%
0.44	0.500	19.4	24.8	24.2	12.6	23.3	72.5	180.99%
0.44	0.900	19.4	18.5	17.4	14.8	8.9	1.9	23.32%
0.44	1.200	19.4	13.8	13.3	16.0	22.7	34.5	49.33%
0.44	1.008	19.4	16.8	15.7	14.8	13.2	11.3	2.15%
0.42	1.008	19.4	16.8	15.7	14.8	13.3	12.0	1.30%

MAPE: mean absolute percentage error.

Table 6 lists the prediction results of several groups of different α and r combinations. The analysis results show that when α takes the optimal cumulative order of 0.44, the order r has a great influence on the prediction error of the model, indicating that choosing an appropriate order plays a crucial role in model accuracy. When the background value coefficient α = 0.42 and the order r = 1.008, model error is minimal. Then α is not an integer-order cumulative best value of 0.44, indicating that there is a certain interaction between the coefficient α and order r in the search for the optimal solution.

Take the background value coefficient α = 0.42 and the order value r = 1.008 as the optimal model parameters, then the average fitting error is the smallest, 1.30%. The prediction results are shown in Figure 5.

Table 7 compares the multiple prediction results before and after the improvement of the multi-variable grey forecasting model. Among them, Model 1 represents the original MGM (1,4) model, Model 2 represents the optimized MGM (1,4) model after the auto-optimization and weighting method, and Model 3 represents the optimal model by combining the auto-optimization and weighting method and the fractional accumulation method.

Table 7.

Comparison of prediction results of various models

Number	Real value	Model 1		Model 2		Model 3
Number	Real value	Simulation values	Simulation error	Simulation values	Simulation error	Simulation values	Simulation error
1	19.4	19.4	0.00%	19.4	0.00%	19.4	0.00%
2	17.2	16.8	2.16%	16.9	1.47%	16.8	2.38%
3	15.3	15.7	2.55%	15.9	3.83%	15.7	2.69%
4	15.0	14.7	2.14%	15.0	0.27%	14.8	1.09%
5	13.2	12.4	6.37%	13.2	0.10%	13.3	0.66%
6	12.1	8.1	32.99%	11.3	6.46%	12.0	0.99%
MAPE1			7.70%		2.02%		1.30%

		Forecast values	Forecast error	Forecast values	Forecast error	Forecast values	Forecast error
7	16.2	16.0	1.23%	16.2	0.00%	16.2	0.00%
8	15.1	15.3	1.32%	15.5	2.65%	15.3	1.32%
9	12.3	10.5	14.6%	12.1	1.63%	12.2	0.81%
MAPE2			5.72%		1.43%		0.71%

MAPE: mean absolute percentage error.

Supposing the variances of the original sequence and the residual error sequence are $S_{1}^{2}$ and $S_{2}^{2}$ , there are

S_{1}^{2} = \frac{1}{n} \sum_{k = 1}^{n} (x^{(0)} (k) - {\bar{x}}^{(0)})^{2}, S_{2}^{2} = \frac{1}{n} \sum_{k = 1}^{n} (e (k) - \bar{e})^{2}

The posterior error ratio C = S₂/S₁.^35,36 Set the posterior error ratio of Models 1–3 as C₁, C₂, C₃

\begin{matrix} C_{1} = 1.456 / 2.420 = 0.602; C_{2} = 0.418 / 2.420 = 0.173; \\ C_{3} = 0.249 / 2.420 = 0.103 \end{matrix}

The accuracy of the grey model is higher when the value of C is smaller, and the model is rated as first-order accuracy when C ≤ 0.35. The posterior error ratio shows consistency with MAPE in the result.

Through the comparison of the above models, it can be concluded that the auto-optimization and weighting method greatly improves the accuracy of the model. The fractional-order accumulation compensates for the lack of integer-order accumulation, and further makes the model closer to reality. The joint adjustment method of the background value and order value shows a strong adaptability and effectiveness, and has a good significance of popularization.

Conclusions

This paper proposes a new idea for predicting and controlling the tension of chemical fiber spinning, which is verified through experiments. The single-variable grey prediction model shows high precision on the prediction of spinning tension that is influenced by a single process parameter. The tension prediction error of each parameter is within 1.5%. For the multi-variable problem, the mean tension prediction error of the MGM(1,n) model is 5.72%. After using the auto-optimization and weighting method to optimize the background value of the model, the error is reduced to 1.43%. The effect of accuracy improvement is obvious. Then, the fractional-order accumulation method is used to find the optimal combination of the background value and order value, and the final average tension prediction error is only 0.71%.

The improved MGM(1,n) model in this paper can achieve higher accuracy tension prediction of the multivariate with less sample data. It can solve tension prediction and control problems based on process parameters. The model predicts the spinning tension with adjustable and monitorable process parameters, which avoids the direct contact with the chemical fiber filaments. It can control the winding tension within an ideal range by adjusting the process parameter values, ensuring the stable quality of the chemical fiber filaments. It has a positive guiding effect on production, which is also a successful application of the grey prediction model in spinning tension prediction.

This article uses the internal prediction of the grey prediction model, and the extrapolative prediction also has guiding significance. For example, the temperature of the first hot roll does not exceed 98℃ in a normal process. It can be predicted from this model that if the temperature of the first hot roll reaches 100℃, the winding tension is approximately 19.11 N.

However, the process parameters that have influence on the tension in this paper may not be comprehensive. Further exploration and experiments are worthy to supplement and perfect the case, so as to establish a grey prediction model with more dimensions.

The method of adjusting the background value and order value jointly proposed in this paper shows a strong adaptability and can be extended to other fields. The grey forecasting model has developed rapidly in recent years and various improved models have been proposed. Appropriate improvement methods can adapt the grey forecasting model to more application fields.

Footnotes

Acknowledgements

Thanks to everyone who participated in the experiment, and thanks to Jiangsu Hengli Chemical Fiber Company Limited for technical and equipment support.

Declaration of conflicting interests

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

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Shanghai Natural Science Foundation (16ZR1401800) and the National Key R&D Program of China (2017YFB1304000).

References

. Trial analyses of quality control technology of modern spinning production. Cotton Text Technol 2006; 34: 225–228.

Krupincovãi

. Quality of new kind of yarns produced by original spinning system. J Text Inst Proc Abstr 2015; 106: 295–302.

Xiang

Yang

et al.

Mining rule of quality control for spinning process with rough set theory. Appl Mech Mater 2011; 80: 1021–1026.

Liu

. Application of orthogonal test and statistical analysis in spinning quality control. Text Accessor 2016; 43: 64–66.

Xie ZQ. Non-contact dynamic measurement of the yarn tension in winding process. PhD Thesis, Donghua University, CN, 2009.

Mehrabad

Johari

Aghdam

. Finite-element and multivariate analyses of tension distribution and spinning parameter effects on a ring-spinning balloon. Arch Proc Inst Mech Eng C J Mech Eng Sci 2010; 224: 253–258.

Yang

Hao

. Apply time sequence model to predict and control spinning tension. Basic Sci J Text Univ 2002; 15: 232–235.

Huang

. Study on the influence factors on inherent quality evenness of polyester tow. Synth Fiber Chin 2007; 43: 43–45.

Sun

Chen

. Multivariate analysis of curve parameters to predict fabric stiffness handle from a pulling-out test. Text Res J 2018; 88: 863–872.

10.

Aimoqbel A and Wang Y. Carbonate reservoir characterization based on grey prediction theory. In: Aimoqbel A, et al. (eds) 72nd EAGE conference and exhibition incorporating SPE EUROPEC 2010. Barcelona, Spain: Society of Petroleum Engineers, 14 June 2010, pp. 4190–4195.

11.

Liu

Xie

. Grey system theory and its application, 4th ed. Beijing: China Science Press, 2008.

12.

Zhou Q. Monitoring and optimizing for carbon fiber spinning process based on intelligent algorithms. PhD Thesis, Donghua University, CN, 2012.

13.

Cui LZ. Grey forecast technology and its application research. PhD Thesis, Nanjing University of Aeronautics and Astronautics, CN, 2010.

14.

Ren

. GM(1,N) method for the prediction of anaerobic digestion system and sensitivity analysis of influential factors. Bioresour Technol 2017; 247: 1258–1261.

15.

Zhao

Tong

. Application of MGM(l, N) on concrete temperature prediction of RCC dam. Adv Mater Res 2012; 368: 2406–2410.

16.

Liu

Xiang

Nguyen

. A multivariable grey model based on optimized background value and its application to subgrade settlement prediction. Appl Mech Mater 2013; 256: 1721–1725.

17.

Guo

Liu

. A multi-variable grey model with a self-memory component and its application on engineering prediction. Eng Appl Artif Intell 2015; 42: 82–93.

18.

Guo

. Application of multivariable non-equidistance gray forecast model in open-pit slope disaster warning. Sci Survey Mapp 2014; 39: 83–86.

19.

Xiao

Song

. Forecasting of gas emissions at fully mechanized caving face in gassy mine based on multi-variable gray model. Safe Coal Mine 2013; 44: 11–13.

20.

Liu

. Modelling and forecasting CO₂ emissions in the BRICS (Brazil, Russia, India, China, and South Africa) countries using a novel multi-variable grey model. Energy 2015; 79: 489–495.

21.

Wang

Hao

. An improved grey multivariable model for predicting industrial energy consumption in China. Appl Math Model 2016; 40: 5745–5758.

22.

Wang

Liu

. R&D investment forecasting analysis based on MGM(1, n) model. Stud Sci Sci 2005; 23: 93–96.

23.

Xiong

Dang

. Optimization of background value in MGM(1,m) model. Contr Decis 2011; 26: 806–810.

24.

Han

Dong

. The method and grey MGM(1,n) optimizing model and its application to metal cutting research. Nat Sci J Xiangtan Univ 2008; 30: 117–120.

25.

Luo YX and Li JY. Application of multi-variable optimizing grey model MGM(1,n,q,r) to the load-strain relation. In: Luo YX, et al. (eds) The 2009 IEEE international conference on mechatronics and automation. Changchun, China: IEEE, 18 September 2009, pp. 4023–4027.

26.

Luo YX and Xiao WY. New information grey multi-variable optimizating model NMGM(1,n,q,r) for the relationship of cost and variability. In: Luo YX, et al. (eds) 2009 international conference on intelligent computation technology and automation. Changsha, Hunan, China: IEEE, 16 October 2009, pp. 120–123.

27.

Liu

. Grey system model with the fractional order accumulation. Commun Nonlin Sci Numer Simulat 2013; 18: 1775–1785.

28.

Wen

. The feasibility of fractional power in grey generating for error analysis in GM(1,1) model. J Grey Syst 2017; 29: 1–13.

29.

Gordon

Yang

. Accurate prediction of cotton ring-spun yarn quality from high-volume instrument and mill processing data. Text Res J 2017; 87: 1025–1039.

30.

Kim

. Effect of fabric structural parameters and weaving conditions to warp tension of aramid fabrics for protective garments. Text Res J 2018; 88: 987–1001.

31.

. Grey incidence analysis model of classification variables and its application on innovation and entrepreneurship education in Jiangsu. J Grey Syst 2018; 30: 123–128.

32.

Liu

Deng

. The range suitable for GM (1,1). Syst Eng Theor Pract 2000; 11: 131–138.

33.

Luo

Liu

. The non-homogenous multi-variable grey model NFMGM(1,n) with fractional order accumulation and its application. J Grey Syst 2017; 29: 39–52.

34.

Meng

Zeng

. Fractional order operator and grey prediction model, Beijing: China Science Press, 2015.

35.

Cong

Shang

. Scheme evaluation method for complex product based on the ideal solution of grey information. J Grey Syst 2018; 30: 75–94.

36.

Zhu

Guan

. Modified grey variable weight clustering method based on standard deviation and its application. J Grey Syst 2018; 30: 1–13.