In the fiber winding process, strong disturbance, uncertainty, strong coupling, and fiber friction complicate the winding constant tension control. In order to effectively reduce the influence of these problems on the tension output, this paper proposed a tension fluctuation rejection strategy based on feedforward compensation. In addition to the bias harmonic curve of the unknown state, the tension fluctuation also contains the influence of bounded noise. A tension fluctuation observer (TFO) is designed to cancel the uncertain periodic signal, in which the frequency generator is used to estimate the critical parameter information. Then, the fluctuation signal is reconstructed by a third-order auxiliary filter. The estimated signal feedforward compensates for the actual tension fluctuation. Furthermore, a time-varying parameters fractional-order PID controller (TPFOPID) is realized to attenuate the bounded noise in the fluctuation. Finally, TPFOPID is enhanced by TFO and applied to control a tension control system considering multi-source disturbances. The stability of the method is analyzed by using the Lyapunov theorem. Finally, numerical simulations verify that the proposed scheme improves the tracking ability and robustness of the system in response to tension fluctuations.

In the fiber winding process, one of the most critical factors affecting the winding quality is the adjustment of fiber tension [

Researchers have conducted many investigations on constant tension control [

However, the framework of existing control methods cannot effectively suppress tension fluctuation by using the main characteristics of tension fluctuation, which is conservative. Because of multi-factor disturbance for speed in the winding process (the non-circularity, coulomb friction, and vibration of the guide roller) and strong coupling effect between fiber speed and tension, resulting in periodic tension fluctuation [

In order to reduce the tension fluctuation of fiber winding, the purpose of this technical description is to propose a tension fluctuation rejection strategy based on feedforward compensation. A composite control scheme of tension fluctuation observer and time-varying parameter fractional PID controller is developed using DOBC [

The structure of this paper is as follows:

The main contributions and innovations of this paper can be summarized as follows:

(1) The tension fluctuation generated by the actual tension system is analyzed, and the compensation signal of the uncertain disturbance is reconstructed by its characteristics. The equivalent form of the tension fluctuation of the uncertain bias harmonic compensation signal is given.

(2) A third-order auxiliary filter is constructed. The disturbance signal with unknown frequency is introduced, and the relationship between the disturbance frequency and the compensation coefficient is given. A tension fluctuation observer (TFO) is proposed to realize the accuracy and robustness of tension fluctuation estimation.

(3) By introducing the dynamic parameters into the fractional order PID controller, a time-varying parameters fractional-order PID controller (TPFOPID) is established, which can realize the efficient control of the tension system with complex disturbance.

(4) The proposed TFO observer and TPFOPID controller are independent of each other and remove the necessity of estimating the system state to complete the control target, making the system stability easier to analyze.

In this section, we consider the control model of fiber winding tension regulator. Additionally, based on actual tension data, the main types of disturbances causing tension fluctuations were identified. This serves to further inform the design of control strategies.

According to Xu et al. [

where

The transfer function for the tension regulating device can be expressed as:

Define the state vector

The state space control model describing the motion of the winding tension regulating device can be expressed as:

In the research problem shown in

We consider the tension experimental data from the actual winding system and fit the equivalent tension output curve that characterizes the tension fluctuation, as shown in

The composite disturbance causing tension fluctuation through fitting and comparison analysis has the following characteristics: random noise, unknown frequency simple harmonic wave disturbance, and bias disturbance. In order to facilitate the verification of the performance of the controller and observer proposed in this paper, in the following research, these disturbance factors are reasonably amplified and described. Therefore, the tension fluctuation

where,

The bias harmonic signal can be equivalent to a third-order external system,

where

Most of the current work needs to estimate the system state and signal parameters information in

This section describes the nominal model of the controlled system for cases where the frequency deviation harmonic signal is unknown. An inverse filter is then designed to stimulate signal features, and a frequency generator is used to estimate tension fluctuation frequency. Further, the fluctuation signal is reconstructed using a third-order auxiliary filter based on the estimated frequency information. Finally, feedforward compensation is implemented for practical.

The tension fluctuation observer consists of three parts: inverse filter, third-order auxiliary filter and frequency generator. The control input is defined as

We can get the error equation transfer function from

(1) Compared with other disturbance rejection methods, it only needs to use the disturbance characteristics without estimating the disturbance state, which avoids the conservatism of the design.

(2) The tension fluctuation observer can be well combined with the designed time-varying parameter fractional order PID controller to control the controlled system, which can realize the independence of the observer and the controller.

(3) It can be applied to fractional order control systems to solve the problem of disturbance estimation and compensation in such nonlinear systems. At the same time, the rejection of a class of biased sinusoidal disturbances with unknown frequencies in the tension system is realized, and the large-scale asymptotic stability of the system is analyzed by using uniform ultimate boundedness.

The inverse filter is designed to excite the tension fluctuation characteristics. The equivalent input tension compensation form is derived by introducing the reconstructed fluctuation signal.

The inverse filter is constructed by using

where

According to

From

where

The noise signal

The decay term obeys

According to the above analysis, the error signal

Similarly, we can see that

Under the framework of the tension fluctuation observer, the output signal of the inverse filter

The structure of the third-order auxiliary filter is shown in

From

where

And

Proof. According to Theorem 1 in [

The bias harmonic signal

where

Thus, the bias harmonic reconstruction signal is expressed as follows:

where,

According to

Combining with

This section constructed a frequency generator to track the unknown constant scalar

where

From

We could see that if there is no bounded noise

According to

After obtaining the estimated reconstruction signal

In the existing methods of signal estimation, due to the uncertain tension fluctuation characteristics, the equivalent state of

As shown in

The next task of the tension fluctuation observer is to use the estimator

Furthermore, the input signal

We obtain the alternative form of

where

Further, when time

From

then the estimation error:

Define the following Lyapunov function:

where,

In consideration that

With the above design, the influence of the unknown frequency bias harmonic curve

For the uncertainty of the biased harmonic signal, we derive the derivative of

Given the above observer design, the new controller will be further introduced. In this section, an improved fraction-order PID controller is incorporated into the framework of tension fluctuation observer. A time-varying parameters fractional-order PID controller (TPFOPID) is realized to improve the system control performance and attenuate the bounded noise.

The control strategy is defined as:

TPFOPID reference the parameter self-coupling control strategy method [

where

According to [

The parameter

Proof. When the saturation of the controller integrator is limited,

For the tension control system

The controlled error system can be established by bringing

The Laplace change of

The characteristic polynomial of the transfer function of the closed-loop system under the error state can be defined as follows:

If to ensure that

It can be seen that the necessary and sufficient condition for the

When the integral control force is under the condition of

It can be seen from the previous derivation that the tension fluctuation observer effectively compensates and rejects the harmonic disturbance in the tension system, and the variance of the feedback

The output variance of the actual tension system:

We consider the scenario where random noise disturbance destroys the system. It is assumed that the bounded noise in the system obeys the Gaussian distribution and satisfies

The likelihood functions of y based on

Using the Bayes formula, the posterior distribution of y is expressed as:

Finally, we use the maximum likelihood method to obtain the estimated value of the system output:

From

The output variance under the controller:

According to

Ultimately, various simulations are carried out to prove the effectiveness of the proposed algorithm. The simulation is carried out on the MATLAB platform using SIMULINK and S function editor.

To verify the effectiveness of proposed method to deal with tension fluctuation, simulation considering different control target angles. The tension control system of

where

According to

The transfer function from

Therefore, according to

Next, the disturbance observer parameters are selected as:

Case 1 uses the traditional PID controller as the nominal controller, and its parameters are:

The random noise disturbance

Considering the high-frequency disturbance rejection, the parameters in the matrix

Case 2 uses the TPFOPID controller as the nominal controller, and its parameters are:

In

A comparative simulation of the disturbance estimation error is conducted in order to show the properties of the proposed tension fluctuation observer. In [

In the actual winding process, according to the process requirements, the winding tension required for different winding segments is different, and in the continuous winding process, the change of the target tension is instantaneous. In order to intuitively highlight the control effect of the proposed method, we tested the traditional PID controller [

The first red line represents a typical PID controller. This approach has a simple structure, but the system’s recovery capability is lagging, and the stability and robustness of the system are inadequate.

The second orange line represents the active disturbance rejection technique, which is highly adaptive to complex perturbations and can ensure a certain robustness and anti-disturbance ability. However, when the target value of the system changes abruptly, the controller has a long adjustment time, which is easy to cause process errors.

The third cyan line represents a fuzzy controller, which can deal with the nonlinear and time-varying problems in the tension system and can significantly suppress tension fluctuations. However, the complexity and calculation of fuzzy reasoning are enormous, and the control accuracy still needs to be improved.

The TFO + TPFOPID control method proposed in this paper is shown in the blue line. Compared with other methods, this method can effectively reject and attenuate the bias harmonic signals and noise in the multi-source tension fluctuation, ensuring lower tension fluctuation rate and better robustness.

However, it should be noted that the control force of the TPFOPID needs to be improved by adjustable parameters. The frequency estimation speed of the TFO is related to the constant value

This paper studies the constant tension output control in the fiber winding process, and a tension fluctuation rejection strategy based on feedforward compensation is proposed. A compound control method combining TFO and TPFOPID is designed. The TFO is devised as a frequency generator and a third-order auxiliary filter to estimate and feedforward compensate the unknown state periodic signal in the tension fluctuation. Based on the time-varying parameter coupling theory, a TPFOPID controller is designed to attenuate noise disturbance and improve system robustness. Finally, its stability is proved by the Lyapunov stability theory. Numerical simulation verifies the effectiveness of the proposed method in constant tension control. In the following work, we will implant the proposed method into the platform to further study its effectiveness.

The content of this paper provides a theoretical basis for future research and exploration in practice, but the following discussion still needs to be made. The tension fluctuation generated by the fiber winding process is a complex signal formed by uncertain perturbations of the external environment and internal model. The control scheme proposed in this paper is based on analyzing the actual winding tension data, fitting the tension fluctuation curve, and extracting the components of the perturbation signal. The constant tension control objective is realized by designing TFO observer and TPFOPID controller. However, the complexity of constant tension control in natural systems mainly stems from the diversity and difficulty in characterizing the disturbance components. During the realization of the proposed control scheme, it may face the problems of inaccurate fitting of different signals, the rejection of the actual disturbance signals still with errors, and the challenges of compatibility and reliability of the proposed control strategy in practical control system applications.

In the actual implementation stage, some problems still need to be considered: for example, the mechanical structure defects of the existing equipment and the low sensitivity of the sensors make it challenging to realize high-precision control; the existing technology and energy power make it impossible for the mechanical devices to respond quickly in real-time. Finally, our work provides a control scheme and theoretical basis for solving the tension fluctuation suppression problem in natural winding systems. The next step can be combined with DSPACE and other tools to build a semi-physical simulation platform, which can help us to realize the proposed control scheme in the actual winding equipment.

The authors wish to express sincere appreciation to the reviewers for their valuable comments, which significantly improved this paper.

This research is funded by the National Natural Science Foundation of China (Grant Number 52075361), Shanxi Province Science and Technology Major Project (Grant Number 20201102003), Lvliang Science and Technology Guidance Special Key R&D Project (Grant Number 2022XDHZ08), National Natural Science Foundation of China (Grant Number 51905367), Shanxi Natural Science Foundation General Project (Grant Numbers 202103021224271; 202203021211201), Shanxi Province Key Research and Development Plan (Grant Number 202102020101013).

The authors confirm contribution to the paper as follows: study conception and design: Yujie Duan, Jianguo Liang, Xinyu Wen; data collection: Yujie Duan, Jianglin Liu; analysis and interpretation of results: Yujie Duan, Xinyu Wen; draft manuscript preparation: Yujie Duan, Haifeng Gao, Yinhui Li, Jinzhu Zhang. All authors reviewed the results and approved the final version of the manuscript.

The data that support the findings of this study are available from the first and corresponding authors upon reasonable request.

The authors declare that they have no conflicts of interest to report regarding the present study.

_{∞}performance analysis: Design and experimental results