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## Mathematical Problems in Packaging Engineering

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Research Article | Open Access

Volume 2013 |Article ID 439797 | https://doi.org/10.1155/2013/439797

Shanhui Liu, Xuesong Mei, Fanfeng Kong, Kui He, "A Decoupling Control Algorithm for Unwinding Tension System Based on Active Disturbance Rejection Control", Mathematical Problems in Engineering, vol. 2013, Article ID 439797, 18 pages, 2013. https://doi.org/10.1155/2013/439797

# A Decoupling Control Algorithm for Unwinding Tension System Based on Active Disturbance Rejection Control

Revised18 Sep 2013
Accepted25 Sep 2013
Published17 Nov 2013

#### Abstract

This paper presents a new control methodology based on active disturbance rejection control (ADRC) for designing the tension decoupling controller of the unwinding system in a gravure printing machine. The dynamic coupling can be actively estimated and compensated in real time, which makes feedback control an ideal approach to designing the decoupling controller of the unwinding system. This feature is unique to ADRC. In this study, a nonlinear mathematical model is established according to the working principle of the unwinding system. A decoupling model is also constructed to determine the order and decoupling plant of the unwinding system. Based on the order and decoupling plant, an ADRC decoupling control methodology is designed to enhance the tension stability in the unwinding system. The effectiveness and capability of the proposed methodology are verified through simulation and experiments. The results show that the proposed strategy not only realises a decoupling control for the unwinding system but also has an effective antidisturbance capability and is robust.

#### 1. Introduction

Tension control stability is essential to the printing process of the gravure printing machines because the fluctuation and variation of tension have great influence on the precision of the printing register. The unwinding system is vital for generating web tension in the tension control system of the machine. In the process of unwinding, the diameter and rotational inertia of the unwinding roller are time-variant, which creates an indeterminate tension control system that is nonlinear and strong-coupling. Hence, accurate tension control is essential for ensuring adequate printing performance.

The objective of this research is to design a tension decoupling control methodology based on ADRC for the unwinding system in gravure printing machines. First, a nonlinear model is established based on the working principle of unwinding system. Next, a decoupling model is constructed in detail and an ADRC decoupling controller is designed to enhance the tension stability of the unwinding system. Last, to test the effectiveness of the ADRC controller, simulations and experiments with PID and ADRC controllers are carried out.

#### 2. Mathematical Modeling

##### 2.1. Experimental Setup

Figure 1 shows the experimental setup for a four-colour gravure printing machine that consists of an unwinding system, a four-colour printing system, and a rewinding system. All of the driving shafts are driven by Mitsubishi servo motors. The unwinding and rewinding rollers are driven by HF-SP102, and the other rollers are driven by HF-SE52JW1-S100. Tension signals are measured by passive dancer rolls with tension observer and Mitsubishi tension load cells (LX-030TD). The entire system is controlled by a multiaxis controller (Googoltech T8VME). The experimental web material is biaxially oriented polypropylene (BOPP).

The schematic diagram of the unwinding system in the experimental setup is presented in Figure 2. To achieve constant tension control, the unwinding system consists primarily of an unwinding component driven by and a traction component driven by . A passive dancer roll is used in the unwinding component to reduce tension fluctuations and simultaneously measure the tension signal by the position and tension observer . To simplify the mathematical model, the upstream and downstream tensions of the dancer roller and the idle rollers are considered equal. In the traction component, the value of is measured by the tension load cell. is the first printing motor whose main task is to maintain a constant speed.

##### 2.2. System Model

The standard model of the relationship between tension and velocity has been presented in [4, 13, 3234]. The model is constructed from equations that describe the web tension behaviour of two consecutive rolls and the velocity of each roll. The relationship between the tension and velocity of the unwinding system can be expressed as

Equation (1) is a nonlinear tension model that considers variation in web length within a span. As shown in Figure 2, is constant for . Substituting into (1), we can obtain

Assuming that the web does not completely slide on the roll and that the web velocity is equal to roll linear velocity and by applying Newton’s law to the rotational movement, (3) can be expressed as

Because and are time-variant parameters for the unwinding roll and decrease with time, (4) can be obtained as follows:

In the actual production process of the gravure printing machine, the thickness of the web is very small compared with the radius of the rollers over which the web is wrapped. Because varies gradually over a short period of time, and vary gradually in comparison with the web dynamics. Hence, the differential of can be neglected. As shown in Figure 2, (3) becomes (5) as the following:

As shown in Figure 2, the working process of the dancer roll can cause a change in web length. After analysis, (6) can be obtained as follows:

Combining (6) with (2), (2) can be rewritten as follows:

According to Newton’s laws of motion and eliminating the influence of the weight of the dancer roll, the dynamic equation of the dancer roll is expressed as

Combining (5), (7), and (8), the nonlinear mathematical model for the unwinding system is constructed as follows:

The controlled variables and of servo motors and , respectively, are the input signals in the unwinding system, and tensions and are the outputs in the unwinding system. Because both and have an impact on and , the unwinding system can be characterised as nonlinear, time-varying, multi-input multioutput (MIMO), and coupling.

#### 3. Proposed ADRC Decoupling Controller

##### 3.1. Decoupling Model

According to (9) and after calculating the second derivative of and , the relationship between the inputs and outputs of the nonlinear model can be represented as follows:

We define the dynamic coupling vector and input matrix as follows:

Substituting (11) into (10), (10) can be rewritten as follows:

Equation (12) illustrates that the unwinding system of the gravure printing machines is a second-order coupling system with two inputs and two outputs. In this study, we define the virtual control quantities and as follows:

Combining (12) and (13), the nonlinear mathematical model for the unwinding system can also be rewritten as follows:

Because the dynamic coupling vector in the ADRC framework can be actively estimated and compensated in real time, the relationships between the virtual control quantities and and the controlled output variables and are single-input, single-output, and also decoupled, as shown in (14).

In this work, because , , and , so (15) can be got as follows:

So is an invertible matrix.

Therefore, the relationships between the virtual control quantities and and the actual control variables and can be expressed as where matrix is the decoupling plant of the unwinding system as follows:

##### 3.2. ADRC Decoupling Control Methodology

Based on the order and decoupling plant of the unwinding system, an ADRC decoupling control methodology is designed to enhance tension stability, as shown in Figure 3. ADRC1 and ADRC2 are relevant to and , respectively, for constant tension control. Because the unwinding system is a second-order system, two second-order ADRC controllers are needed. Each ADRC controller consists of a second-order tracking differentiator (TD), a third-order extended state observer (ESO), and a nonlinear state error feedback (NLSEF).

The second-order TD is a nonlinear component; given an input signal, we can acquire a corresponding tracking input signal and an approximately differentiated input signal, even for a nondifferentiable or noncontinuous input signal. As shown in Figure 3, and are tracking signals of and , respectively, and and are approximately differentiated signals of and, respectively. According to , the discrete forms of second-order are expressed as where is the velocity factor; is the sampling step; and is the time-optimal control synthesis function defined as follows:

The third-order ESO is the core of the ADRC, and it not only tracks system output variables and their differentiated signals but also actively estimates dynamic coupling and external disturbances in real time. As shown in Figure 3, and track the output variables and , respectively; and estimate the differentiated signals of and , respectively; and and actively estimate and and and , respectively. The and variables are considered to be extended states of the system. According to , the discrete forms of third-order are expressed as where , , and are the ESO gains; is the interval length of the linear segment; and is a nonlinear function defined as follows:

The NLSEF is a nonlinear combination of the resulting differences of the state variables and estimations generated by the TD and ESO, respectively, as shown in Figure 3. The control law actively compensates for dynamic coupling and external disturbances that are estimated by the ESO in real time. According to , the discrete forms of are defined as where is the proportionality coefficient; is the differential coefficient; and is the compensating factor.

Combining (18), (20), and (22), the discrete algorithm of the is shown in

After calculating the virtual control variables and with the ADRC controller, the actual control variables and can be obtained by (16).

Two diagrams are provided in Figures 3 and 4 that show the differences between PID and ADRC control synthesis methodology for tension decoupling control of the unwinding system. First, the total disturbance in the ADRC methodology, which includes the external disturbances ( and ), dynamic coupling components of the system , and unknown internal dynamics, is defined as an extended state () of the system. The total disturbance can be actively estimated by ESO in real time and compensated in the corresponding control signal in each sampling period. However, in PID methodology, the total disturbance is managed passively by parameter tuning. Second, because the differentiated signals and are obtained by the TD and ESO, respectively, in ADRC methodology, the errors of the differentiation are not produced by direct differentiation of the inputs and the outputs. This attribute makes the ADRC algorithm less sensitive to discontinuities in the inputs and noise in the outputs. However, due to noise sensitivity, PID methodology is often implemented without the derivative component.

#### 4. Simulation and Experiments

To investigate the performance of the proposed ADRC decoupling control methodology, a comparison of the unwinding systems of PID and ADRC control strategies was performed through simulation and experiments. The parameters of the unwinding system used in the simulation and experiments are summarised in Table 1. All parameters of ADRC and PID were determined at  r/min and m. Although the tests were conducted under different operating conditions, particularly for different unwinding diameters, the parameters remain constant.

 Parameters Value Units 1 m 0.35 m 5 × 10−6 m2 0.064 m 0.128 m 0.03 m 0.03 m 1.6 × 108 Pa 1 × 104 N/m
##### 4.1. Simulation Results

MATLAB simulation models were constructed to test the performance of the ADRC and PID control strategies. The simulation adopted a fixed-step size mode; the fixed-step size is 10 ms. According to [22, 30, 32], the adjusted parameters of the ADRC are listed in Table 2. The adjusted parameters of the PID are also listed in Table 2.

 Controller Controller parameters ADRC1 = 100, = 5560, = 81200, = 920, = 760, = 60 ADRC2 = 100, = 2870, = 51600, = 3460, = 750, = 60 PID1 = 30, = 2.5, = 12 PID2 = 65, = 32, = 5
###### 4.1.1. Case A: Robustness against Dynamic Changes

To verify the robustness against dynamic changes of the developed ADRC control methodology, the and step change is from 50 N to 80 N when is equal to 100 r/min and is equal to 0.2 m. The simulation behaviours of the PID and ADRC controllers are shown in Figures 5 and 6.

As shown in Figure 5, when is equal to 0.2 m, 0.1 m, or 0.05 m, both and quickly reach 80 N without overshoot or oscillation in the ADRC controller. However, in the PID controller, a decrease in leads to deterioration in the control abilities of and . For example, when is equal to 0.05 m, demonstrates an overshoot of approximately 6.38% and arrives at 80 N after 4 s. Figure 6 illustrates that when is equal to 100 r/min, 300 r/min, or 500 r/min, both and quickly reach 80 N without overshoot or oscillation in the ADRC controller. In contrast, an increase in contributes to deterioration in the control abilities of and in the PID controller.

The simulation results show that once the ADRC controller is set up properly, it can handle a wide range of dynamic changes. Compared with the PID controller, Figures 5 and 6 show that the ADRC controller can obtain better robustness against dynamic changes in the unwinding system. The primary reason is that uncertainties and variations in the internal model parameters can be actively estimated by the ESO and compensated by the NLSEF in the ADRC controller; the PID controller does not have these capabilities.

###### 4.1.2. Case B: Decoupling Performance

To determine the decoupling performance of the proposed controller, the step change is from 50 N to 80 N at 5 s and from 80 N to 50 N at 10 s. Figures 7 and 8 illustrate the simulation performance of the PID and ADRC controllers when is equal to 100 r/min and is equal to 0.2 m, respectively.

Figure 7 indicates that when is equal to 0.2 m, 0.1 m, or 0.05 m, variations in have no influence on in the ADRC controller. In contrast, in the PID controller, is continuously influenced by variations in , and variations in increase with a decrease in . As shown in Figure 8, when is equal to 100 r/min, 300 r/min, or 500 r/min, variations in have no influence on in the ADRC controller. However, in the PID controller, is continuously influenced by variations in , and variations in increase with an increase in . For example, when is equal to 500 r/min, variations in at 5 s and 10 s cause to vary by 6.5 N.

The simulation results illustrate that the designed ADRC controller has better decoupling capability than the PID controller. Because the dynamic coupling components and the unknown coupled dynamics can be combined as the total disturbance estimated and compensated in real time, the decoupling control of the unwinding system in the ADRC controller can be realised, depending on the system order which can be easily obtained and the decoupling plant which may be inaccurate. However, the decoupling control cannot be realised, only depending on the imprecise of decoupling plant in the PID controller.

###### 4.1.3. Case C: External Antidisturbance Ability

To demonstrate the external antidisturbance ability of the proposed controller, a step disturbance of is set at 5 s with a value of 10 N, and a step disturbance of is set at 10 s with a value of 2 N when is equal to 100 r/min and is equal to 0.2 m. The simulation behaviours of the PID and ADRC controllers are shown in Figures 9 and 10.

Figures 9 and 10 illustrate that with a decrease in and an increase in , the web tension in the ADRC controller exhibits a trend similar to that of the PID controller, with a gradual increase in tension fluctuation. The range and duration of the tension fluctuation in the ADRC controller are much smaller than the range and duration of the tension fluctuation in the PID controller under similar simulation conditions. For example, when is equal to 0.05 m, the range and duration of the fluctuation in at 5 s are 0.5 N and 1.5 s, respectively, in the ADRC controller. However, in the PID controller, the range and duration of the fluctuation in at 5 s are 0.9 N and 4 s, respectively. In the ADRC controller, when is equal to 500 r/min, the range and duration of the fluctuation in at 5 s are 0.8 N and 0.8 s, respectively. However, the range and duration of the fluctuation in under similar simulation conditions in the PID controller are 1.2 N and 2 s, respectively.

The simulation results indicate that the proposed ADRC control methodology has a better external antidisturbance ability than the PID controller because the ADRC algorithm is less sensitive to disturbance.

##### 4.2. Experimental Results

To establish practical application results of the ADRC control methodology, comparative experiments between the ADRC and PID control methodologies are performed in the experimental setup, as shown in Figure 1, with the servo cycle and sampling period set to 10 ms. According to the parameters used in the simulation, the adjusted parameters of the PID and ADRC in experiments are presented in Table 3.

 Controller Controller parameters ADRC1 = 100, = 6560, = 84700, = 830, = 520, = 68 ADRC2 = 100, = 2950, = 54600, = 2760, = 670, = 57 PID1 = 43, = 3.2, = 7 PID2 = 81, = 46, = 8
###### 4.2.1. Case A: Robustness against Dynamic Changes

To determine the practical robustness against dynamic changes of the ADRC controller, the and step change is from 50 N to 80 N at 5 s when is equal to 100 r/min and is equal to 0.2 m. The performance of the PID and ADRC controllers in the experiments is shown in Figures 11 and 12.

Figures 11 and 12 show that with a decrease in and an increase in , the control abilities of and deteriorate in the PID controller. However, when is equal to 100 r/min or when is equal to 0.2 m, and quickly reach 80 N without overshoot in the ADRC controller. The experimental results are consistent with the simulation results and indicate that the proposed ADRC controller is superior to the PID controller in practical robustness against dynamic changes in the unwinding system. This finding is mainly due to the fact that uncertainties and variations in the internal model parameters can be actively estimated and compensated in the ADRC controller but are passively managed by parameters tuning in the PID controller.

###### 4.2.2. Case B: Decoupling Performance

To establish the practical decoupling performance of the proposed controller, the step change is from 50 N to 80 N at 5 s and from 80 N to 50 N at 10 s when is equal to 100 r/min and is equal to 0.2 m. Figures 13 and 14 show the performance of the PID and ADRC controllers.

Figures 13 and 14 illustrate that is always influenced by variation in and the variation in increases with a decrease in and an increase in in the PID controller. Unfortunately, when is equal to 100 r/min or is equal to 0.2 m, the variation in has no influence on in the ADRC controller. The experimental results agree with the simulation results and illustrate that the designed ADRC controller has better decoupling capability than the PID controller. Because the dynamic coupling components and the unknown coupled dynamics can be combined as the total disturbance, which is estimated and compensated in real time, the decoupling control of the unwinding system can be realised, depending on the system order and the decoupling plant in ADRC controller. However, only depending on the decoupling plant, the decoupling control cannot be realised in the PID controller.

###### 4.2.3. Case C: Tension Control Ability of Steady Speeds

Because tension control at steady speeds is essential for the performance of the gravure machine, further attention should be paid to tension fluctuation at steady speeds. To investigate the steady performance of the ADRC controller, tension control experiments were performed at motor steady speeds of  r/min and  r/min. In these experiments, the steady tension was 50 N and the test time was 20 s. The performance of the PID and ADRC controllers in the experiments is shown in Figures 15 and 16. and are standard deviations of and , respectively, and were obtained under different test conditions, just as shown in Table 4.

 Conditions Controller type PID controller ADRC controller (N) (N) (N) (N) = 100 r/min, = 0.2 m 0.3683 0.2949 0.2400 0.2163 = 100 r/min, = 0.05 m 0.5540 0.4271 0.2686 0.2350 = 500 r/min, = 0.2 m 0.9326 0.6572 0.4880 0.4064 = 500 r/min, = 0.05 m 1.3349 0.8514 0.5831 0.4658

Figures 15 and 16 and Table 4 demonstrate that as the disturbance due to web speed and system dynamics increases, the web tension in the ADRC controller exhibits a trend similar to that of the PID controller, with a gradual increase in tension fluctuation. However, compared with the PID controller under similar experimental conditions, the resulting tension fluctuation in the ADRC controller is considerably smaller. For example, when is equal to 0.05 m and is equal to 500 r/min, the standard deviations of and in the PID controller are 1.3349 N and 0.8514 N, respectively. Under the ADRC controller, when is equal to 0.05 m and is equal to 500 r/min, the standard deviations of and are 0.5831 N and 0.4658 N, respectively. The experimental results indicate that the proposed ADRC controller has better tension control at steady speeds than the PID controller. The main reason is that the ADRC algorithm is much less sensitive to internal and external disturbances than the PID controller.

#### 5. Conclusions

Based on ADRC, an innovative control synthesis methodology is proposed for the design of a tension decoupling controller for the unwinding system of the gravure printing machine. First, according to the working principle of the unwinding system, a nonlinear mathematical model is established. Second, a decoupling model is constructed to obtain the order and decoupling plant of the unwinding system, which establishes a basis for the ADRC decoupling control methodology. Last, the results of the simulation and experiments demonstrate that the proposed control methodology can achieve not only good decoupling control performance but also better stability and robustness than the traditional PID controllers. From this study, it can be found that in the ADRC framework, all of the unknown coupling terms that may contribute to the total disturbance can be actively estimated and cancelled out in the corresponding control signal in real time; this greatly simplifies the design of the decoupling control methodology.

The main contributions of this paper are as follows. (1) We have addressed the design of a tension decoupling controller based on ADRC for the unwinding system of gravure printing machines. (2) Simulation and experiments of the proposed decoupling controller for the unwinding system were completed. (3) The study demonstrates that the proposed method is a promising solution for the problem of tension control in the unwinding systems of gravure printing machines because it is inherently robust against internal and external disturbances and decoupling control over the unwinding system of the gravure printing machine can be achieved in a simple manner.

#### Nomenclature

 : Constant tension of the unwinding roll (N) : Web tension in the th span (N) : External disturbances that influence : Tangential velocity of the th roll (m/s) : Rotational speed of the th roll (r/min) : Torque control signal of the th servo motor (Nm) : Nominal span length of the web in the th span (m) : The th arm length of the dancer roll (m) : Radius of the th roll (m) : Inertia of the th roll () : Inertia of the dancer roll () : Constant of the th servo motor : Reference inputs of (N) : Reference inputs of (m/s) : Young’s modulus of the spring (N/m) : Cross-sectional area of the web () : Modulus of elasticity of the web material (Pa) : Viscous friction coefficient (Nm/(rad/s)) : Viscous friction coefficient of the dancer roll (Nm/(rad/s)) : Angle of the dancer roll arm from a neutral position (rad) : Servo motors driving the th roll.

#### Conflict of Interests

The authors declare that they have no financial or personal relationships with other people or organizations that can inappropriately influence their work; there is no professional or other personal interest of any nature or kind in any product or company that could be construed as influencing the position presented in, or the review of, the paper.

#### Acknowledgments

This research is supported by the Program for Changjiang Scholars (IRT1172) and the National Science and Technology Pillar Program of China, under Grant no. 2012BAF13B06.

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