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

Authors: Shanhui Liu; Xuesong Mei; Fanfeng Kong; Kui He
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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.Presently, the control synthesis methodology based on the proportional-integral-derivative (PID) algorithm is still the most common control strategy for the tension control system of the gravure printing machines. However, with the development of high-speed and high-precision printing technologies, a control strategy based solely on PID is insufficient for satisfying control requirements. Many researchers have focused their efforts on improving the traditional PID strategy [1–6]. With the development of modern control theories, a number of modern control methods have been applied to web tension control. Chung et al. [7] applied fuzzy logic control to tension control. Wang et al. [8] achieved decoupling control between tension and speed using neural networks. Abjadi et al. [9] designed a decoupling controller based on nonlinear sliding-mode control for the tension control system. Pagilla et al. [10, 11] proposed fully decentralized state feedback controller and verified the controller by experimental results. A decentralized adaptive controller with model based feedforward was developed and experimentally verified in [12]. As the web processing plants are known as highly resonant systems with strong coupling between the variables to control, several improvements of multivariable controllers based on robust control have been introduced to address the resonant effects and tension control while maintaining performance [13–20]. Although most of these methods resulted in acceptable control performance, they relied too heavily on the system model. However, in industrial practice, it is very difficult to establish a fully accurate mathematical model. Therefore, it was necessary to find a solution using a small amount of model information that could effectively address the nonlinear, time-varying, strong coupling, and uncertain nature of web tension control. One ideal candidate is the ADRC. The essence of ADRC is that the internal dynamics and external disturbances can be estimated and compensated in real time [21–24]. Due to its strong robustness and antidisturbance capability, ADRC has been successfully applied in many fields [25–28]. An initial evaluation of the application of ADRC for the regulation of web tension was performed with a linear transfer function model of the tension loop [29]. ADRC was then applied to design control methodologies for a truly nonlinear model of the tension dynamics in [30, 31]; acceptable performance was observed. Few researchers have addressed the design of tension decoupling control methodology based on ADRC, particularly, with regard to the unwinding system for gravure printing machines.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

Figure1 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).Figure 1
Experimental setup for the four-colour gravure printing machine.The schematic diagram of the unwinding system in the experimental setup is presented in Figure2. To achieve constant tension control, the unwinding system consists primarily of an unwinding component driven by M1 and a traction component driven by M2. A passive dancer roll is used in the unwinding component to reduce tension fluctuations and simultaneously measure the tension signal T1(t) by the position θ and tension observer [32]. 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 T2(t) is measured by the tension load cell. M3 is the first printing motor whose main task is to maintain a constant speed.Figure 2
Schematic diagram of the unwinding system.


### 2.2. System Model

The standard model of the relationship between tension and velocity has been presented in [4, 13, 32–34]. 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
(1)Li(t)dTi(t)dt=Vi(t)Ti-1(t)-Vi+1(t)Ti(t)+AE(Vi+1(t)-Vi(t))+(AE-Ti(t))dLi(t)dti=1,2.Equation (1) is a nonlinear tension model that considers variation in web length within a span. As shown in Figure 2, L2(t) is constant for L2. Substituting V(t)=R(t)ω(t) into (1), we can obtain
(2)L1(t)T˙1(t)=(T0-AE)R1(t)ω1(t)+(AE-T1(t))R2ω2(t)+(AE-T1(t))L˙1(t),L2T˙2(t)=(T1(t)-AE)R2ω2(t)+(AE-T2(t))R3ω3.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
(3)ddt(Ji(t)ωi(t))=(Ti(t)-Ti-1(t))Ri(t)+Ciui(t)-fωi(t),i=1,2.BecauseR1(t) and J1(t) are time-variant parameters for the unwinding roll and decrease with time, (4) can be obtained as follows:
(4)ddt(J1(t)ω1(t))=J˙1(t)ω1(t)+J1(t)ω˙1(t).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. BecauseR1(t) varies gradually over a short period of time, R1(t) and J1(t) vary gradually in comparison with the web dynamics. Hence, the differential of J1(t) can be neglected. As shown in Figure 2, (3) becomes (5) as the following:
(5)J1(t)ω˙1(t)=(T1(t)-T0)R1(t)+C1u1(t)-fω1(t),J2ω˙2(t)=(T2(t)-T1(t))R2+C2u2(t)-fω2(t).As shown in Figure2, the working process of the dancer roll can cause a change in web length. After analysis, (6) can be obtained as follows:
(6)L1(t)=L1-2LD2θ(t).Combining (6) with (2), (2) can be rewritten as follows:
(7)(L1-2LD2θ(t))T˙1(t)=(T0-AE)R1(t)ω1(t)+(AE-T1(t))R2ω2(t)-2LD2(AE-T1(t))θ˙(t),L2T˙2(t)=(T1(t)-AE)R2ω2(t)+(AE-T2(t))R3ω3.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(8)T1(t)=12LD2(JDaθ¨(t)+fDθ˙(t)+KLD12θ(t)).Combining (5), (7), and (8), the nonlinear mathematical model for the unwinding system is constructed as follows:
(9)J1(t)ω˙1(t)=(T1(t)-T0)R1(t)+C1u1(t)-fω1(t),J2ω˙2(t)=(T2(t)-T1(t))R2+C2u2(t)-fω2(t),2LD2T1(t)=JDaθ¨(t)+fDθ˙(t)+KLD12θ(t),(L1-2LD2θ(t))T˙1(t)=(T0-AE)R1(t)ω1(t)+(AE-T1(t))R2ω2(t)-2LD2(AE-T1(t))θ˙(t),L2T˙2(t)=(T1(t)-AE)R2ω2(t)+(AE-T2(t))R3ω3.The controlled variablesu1(t) and u2(t) of servo motors M1 and M2, respectively, are the input signals in the unwinding system, and tensions T1(t) and T2(t) are the outputs in the unwinding system. Because both u1(t) and u2(t) have an impact on T1(t) and T2(t), the unwinding system can be characterised as nonlinear, time-varying, multi-input multioutput (MIMO), and coupling.


## 2.1. Experimental Setup

Figure1 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).Figure 1
Experimental setup for the four-colour gravure printing machine.The schematic diagram of the unwinding system in the experimental setup is presented in Figure2. To achieve constant tension control, the unwinding system consists primarily of an unwinding component driven by M1 and a traction component driven by M2. A passive dancer roll is used in the unwinding component to reduce tension fluctuations and simultaneously measure the tension signal T1(t) by the position θ and tension observer [32]. 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 T2(t) is measured by the tension load cell. M3 is the first printing motor whose main task is to maintain a constant speed.Figure 2
Schematic diagram of the unwinding system.


## 2.2. System Model

The standard model of the relationship between tension and velocity has been presented in [4, 13, 32–34]. 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
(1)Li(t)dTi(t)dt=Vi(t)Ti-1(t)-Vi+1(t)Ti(t)+AE(Vi+1(t)-Vi(t))+(AE-Ti(t))dLi(t)dti=1,2.Equation (1) is a nonlinear tension model that considers variation in web length within a span. As shown in Figure 2, L2(t) is constant for L2. Substituting V(t)=R(t)ω(t) into (1), we can obtain
(2)L1(t)T˙1(t)=(T0-AE)R1(t)ω1(t)+(AE-T1(t))R2ω2(t)+(AE-T1(t))L˙1(t),L2T˙2(t)=(T1(t)-AE)R2ω2(t)+(AE-T2(t))R3ω3.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
(3)ddt(Ji(t)ωi(t))=(Ti(t)-Ti-1(t))Ri(t)+Ciui(t)-fωi(t),i=1,2.BecauseR1(t) and J1(t) are time-variant parameters for the unwinding roll and decrease with time, (4) can be obtained as follows:
(4)ddt(J1(t)ω1(t))=J˙1(t)ω1(t)+J1(t)ω˙1(t).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. BecauseR1(t) varies gradually over a short period of time, R1(t) and J1(t) vary gradually in comparison with the web dynamics. Hence, the differential of J1(t) can be neglected. As shown in Figure 2, (3) becomes (5) as the following:
(5)J1(t)ω˙1(t)=(T1(t)-T0)R1(t)+C1u1(t)-fω1(t),J2ω˙2(t)=(T2(t)-T1(t))R2+C2u2(t)-fω2(t).As shown in Figure2, the working process of the dancer roll can cause a change in web length. After analysis, (6) can be obtained as follows:
(6)L1(t)=L1-2LD2θ(t).Combining (6) with (2), (2) can be rewritten as follows:
(7)(L1-2LD2θ(t))T˙1(t)=(T0-AE)R1(t)ω1(t)+(AE-T1(t))R2ω2(t)-2LD2(AE-T1(t))θ˙(t),L2T˙2(t)=(T1(t)-AE)R2ω2(t)+(AE-T2(t))R3ω3.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(8)T1(t)=12LD2(JDaθ¨(t)+fDθ˙(t)+KLD12θ(t)).Combining (5), (7), and (8), the nonlinear mathematical model for the unwinding system is constructed as follows:
(9)J1(t)ω˙1(t)=(T1(t)-T0)R1(t)+C1u1(t)-fω1(t),J2ω˙2(t)=(T2(t)-T1(t))R2+C2u2(t)-fω2(t),2LD2T1(t)=JDaθ¨(t)+fDθ˙(t)+KLD12θ(t),(L1-2LD2θ(t))T˙1(t)=(T0-AE)R1(t)ω1(t)+(AE-T1(t))R2ω2(t)-2LD2(AE-T1(t))θ˙(t),L2T˙2(t)=(T1(t)-AE)R2ω2(t)+(AE-T2(t))R3ω3.The controlled variablesu1(t) and u2(t) of servo motors M1 and M2, respectively, are the input signals in the unwinding system, and tensions T1(t) and T2(t) are the outputs in the unwinding system. Because both u1(t) and u2(t) have an impact on T1(t) and T2(t), 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 T1(t) and T2(t), the relationship between the inputs and outputs of the nonlinear model can be represented as follows:
(10)T¨1(t)=(2LD2[(T1(t)-AE)θ¨(t)+2T˙1(t)θ˙(t)]-R2ω2(t)T˙1(t)+(T0-AE)ω1(t)R˙1(t))×(L1-2LD2θ(t))-1+R1(t)(T0-AE)[R1(t)(T1(t)-T0)-fω1(t)]J1(t)(L1-2LD2θ(t))+C1R1(t)(T0-AE)J1(t)(L1-2LD2θ(t))u1(t)+R2(AE-T1(t))[R2(T2(t)-T1(t))-fω2(t)]J2(L1-2LD2θ(t))+C2R2(AE-T1(t))J2(L1-2LD2θ(t))u2(t),T¨2(t)=R2ω2(t)T˙1(t)-R3ω3T˙2(t)L2+R2(T1(t)-AE)[R2(T2(t)-T1(t))-fω2(t)]J2L2+C2R2(T1(t)-AE)J2L2u2(t).We define the dynamic coupling vectorf(t) and input matrix B(t) as follows:
(11)f(t)=[f1(t)f2(t)],f1(t)=(2LD2[(T1(t)-AE)θ¨(t)+2T˙1(t)θ˙(t)]-R2ω2(t)T˙1(t)+(T0-AE)ω1(t)R˙1(t))×(L1-2LD2θ(t))-1+R1(t)(T0-AE)[R1(t)(T1(t)-T0)-fω1(t)]J1(t)(L1-2LD2θ(t))+R2(AE-T1(t))[R2(T2(t)-T1(t))-fω2(t)]J2(L1-2LD2θ(t)),f2(t)=R2ω2(t)T˙1(t)-R3ω3T˙2(t)L2+R2(T1(t)-AE)[R2(T2(t)-T1(t))-fω2(t)]J2L2,B(t)=[C1R1(t)(T0-AE)J1(t)(L1-2LD2θ(t))C2R2(AE-T1(t))J2(L1-2LD2θ(t))C2R2(T1(t)-AE)J2L2].Substituting (11) into (10), (10) can be rewritten as follows:
(12)[T¨1(t)T¨2(t)]=f(t)+B(t)[u1(t)u2(t)].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 τ1(t) and τ2(t) as follows:
(13)[τ1(t)τ2(t)]=B(t)[u1(t)u2(t)].Combining (12) and (13), the nonlinear mathematical model for the unwinding system can also be rewritten as follows:
(14)[T¨1(t)T¨2(t)]=f(t)+[τ1(t)τ2(t)].Because the dynamic coupling vectorf(t) in the ADRC framework can be actively estimated and compensated in real time, the relationships between the virtual control quantities τ1(t) and τ2(t) and the controlled output variables T1(t) and T2(t) are single-input, single-output, and also decoupled, as shown in (14).In this work, becauseT0≪AE, T1(t)≪AE, and 2LD2θ(t)≪L1, so (15) can be got as follows:
(15)|B(t)|=C1C2R2R1(t)(T0-AE)(T1(t)-AE)J2L2J1(t)(L1-2LD2θ(t))≠0.SoB(t) is an invertible matrix.Therefore, the relationships between the virtual control quantitiesτ1(t) and τ2(t) and the actual control variables u1(t) and u2(t) can be expressed as
(16)[u1(t)u2(t)]=B-1(t)[τ1(t)τ2(t)],
where matrix B-1(t) is the decoupling plant of the unwinding system as follows:
(17)B-1(t)=[J1(t)(L1-2LD2θ(t))C1R1(t)(T0-AE)L2J1(t)C1R1(t)(T0-AE)J2L2C2R2(T1(t)-AE)].


### 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 Figure3. ADRC1 and ADRC2 are relevant to T1(t) and T2(t), 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).Figure 3
Structure of the ADRC decoupling control methodology.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 Figure3, v11 and v21 are tracking signals of T1ref and T2ref, respectively, and v12 andv22 are approximately differentiated signals of T1ref andT2ref, respectively. According to [22], the discrete forms of second-order TDi(i=1,2) are expressed as
(18)fhi=fhan(vi1(k)-Tiref(k),vi2(k),ri,h),vi1(k+1)=vi1(k)+hvi2(k),vi2(k+1)=vi2(k)+hfhi,
where ri is the velocity factor; h is the sampling step; andfhan(x1,x2,r,h) is the time-optimal control synthesis function defined as follows:
(19)d=rh;d0=hd;y=x1+hx2,a0=d2+8r|y|,a={x2+a0-d2sign(y),|y|>d0,x2+yh,|y|≤d0,fhan(x1,x2,r,h)=-{rsign(a),|a|>d,rad,|a|≤d.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 Figure3, z11 and z21 track the output variables T1(t) and T2(t), respectively; z12 and z22 estimate the differentiated signals of T1(t) and T2(t), respectively; and z13 and z23 actively estimate f1(t) and w1(t) and f2(t) and w2(t), respectively. The z13 and z23 variables are considered to be extended states of the system. According to [22], the discrete forms of third-order ESOi(i=1,2) are expressed as
(20)ei(k)=zi1(k)-Ti(k),zi1(k+1)=zi1(k)+h(zi2(k)-βi1ei(k)),zi2(k+1)=zi2(k)+h(zi3(k)-βi2fal(ei(k),0.5,δ)+τi(k)),zi3(k+1)=zi3(k)-hβi3fal(ei(k),0.25,δ),
where βi1, βi2, and βi3 are the ESO gains; δ is the interval length of the linear segment; and fal(e,α,δ) is a nonlinear function defined as follows:
(21)fal(e,α,δ)={eδ1-α,|e|≤δ,|e|αsign(e),|e|>δ.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 Figure3. The control law actively compensates for dynamic coupling and external disturbances that are estimated by the ESO in real time. According to [22], the discrete forms of NLSEFi(i=1,2) are defined as
(22)ei1(k)=vi1(k)-zi1(k),ei2(k)=vi2(k)-zi2(k),τ0i(k)=kipfal(ei1(k),0.5,δ)+kidfal(ei2(k),1.5,δ),τi(k)=τ0i-zi3(k)b0i,
where kip is the proportionality coefficient; kid is the differential coefficient; and b0i is the compensating factor.Combining (18), (20), and (22), the discrete algorithm of the ADRCi(i=1,2) is shown in
(23)fhi(k)=fhan(vi1(k)-Tiref(k),vi2(k),ri,h),vi1(k+1)=vi1(k)+hvi2(k),vi2(k+1)=vi2(k)+hfhi(k),ei(k)=zi1(k)-Ti(k),zi1(k+1)=zi1(k)+h(zi2(k)-βi1ei(k)),zi2(k+1)=zi2(k)+h(zi3(k)-βi2fal(ei(k),0.5,h)+τi(k)),zi3(k+1)=zi3(k)-hβi3fal(ei(k),0.25,h),ei1(k+1)=vi1(k+1)-zi1(k+1),ei2(k+1)=vi2(k+1)-zi2(k+1),τi(k+1)=kipfal(ei1(k+1),0.5,δ)+kidfal(ei2(k+1),1.5,δ)-zi3(k+1)b0i.After calculating the virtual control variablesτ1(t) and τ2(t) with the ADRC controller, the actual control variables u1(t) and u2(t) can be obtained by (16).Two diagrams are provided in Figures3 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 (w1(t) and w2(t)), dynamic coupling components of the system (f(t)), and unknown internal dynamics, is defined as an extended state (zi3) 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 vi2 and zi2 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.Figure 4
Structure of the PID control methodology.


## 3.1. Decoupling Model

According to (9) and after calculating the second derivative of T1(t) and T2(t), the relationship between the inputs and outputs of the nonlinear model can be represented as follows:
(10)T¨1(t)=(2LD2[(T1(t)-AE)θ¨(t)+2T˙1(t)θ˙(t)]-R2ω2(t)T˙1(t)+(T0-AE)ω1(t)R˙1(t))×(L1-2LD2θ(t))-1+R1(t)(T0-AE)[R1(t)(T1(t)-T0)-fω1(t)]J1(t)(L1-2LD2θ(t))+C1R1(t)(T0-AE)J1(t)(L1-2LD2θ(t))u1(t)+R2(AE-T1(t))[R2(T2(t)-T1(t))-fω2(t)]J2(L1-2LD2θ(t))+C2R2(AE-T1(t))J2(L1-2LD2θ(t))u2(t),T¨2(t)=R2ω2(t)T˙1(t)-R3ω3T˙2(t)L2+R2(T1(t)-AE)[R2(T2(t)-T1(t))-fω2(t)]J2L2+C2R2(T1(t)-AE)J2L2u2(t).We define the dynamic coupling vectorf(t) and input matrix B(t) as follows:
(11)f(t)=[f1(t)f2(t)],f1(t)=(2LD2[(T1(t)-AE)θ¨(t)+2T˙1(t)θ˙(t)]-R2ω2(t)T˙1(t)+(T0-AE)ω1(t)R˙1(t))×(L1-2LD2θ(t))-1+R1(t)(T0-AE)[R1(t)(T1(t)-T0)-fω1(t)]J1(t)(L1-2LD2θ(t))+R2(AE-T1(t))[R2(T2(t)-T1(t))-fω2(t)]J2(L1-2LD2θ(t)),f2(t)=R2ω2(t)T˙1(t)-R3ω3T˙2(t)L2+R2(T1(t)-AE)[R2(T2(t)-T1(t))-fω2(t)]J2L2,B(t)=[C1R1(t)(T0-AE)J1(t)(L1-2LD2θ(t))C2R2(AE-T1(t))J2(L1-2LD2θ(t))C2R2(T1(t)-AE)J2L2].Substituting (11) into (10), (10) can be rewritten as follows:
(12)[T¨1(t)T¨2(t)]=f(t)+B(t)[u1(t)u2(t)].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 τ1(t) and τ2(t) as follows:
(13)[τ1(t)τ2(t)]=B(t)[u1(t)u2(t)].Combining (12) and (13), the nonlinear mathematical model for the unwinding system can also be rewritten as follows:
(14)[T¨1(t)T¨2(t)]=f(t)+[τ1(t)τ2(t)].Because the dynamic coupling vectorf(t) in the ADRC framework can be actively estimated and compensated in real time, the relationships between the virtual control quantities τ1(t) and τ2(t) and the controlled output variables T1(t) and T2(t) are single-input, single-output, and also decoupled, as shown in (14).In this work, becauseT0≪AE, T1(t)≪AE, and 2LD2θ(t)≪L1, so (15) can be got as follows:
(15)|B(t)|=C1C2R2R1(t)(T0-AE)(T1(t)-AE)J2L2J1(t)(L1-2LD2θ(t))≠0.SoB(t) is an invertible matrix.Therefore, the relationships between the virtual control quantitiesτ1(t) and τ2(t) and the actual control variables u1(t) and u2(t) can be expressed as
(16)[u1(t)u2(t)]=B-1(t)[τ1(t)τ2(t)],
where matrix B-1(t) is the decoupling plant of the unwinding system as follows:
(17)B-1(t)=[J1(t)(L1-2LD2θ(t))C1R1(t)(T0-AE)L2J1(t)C1R1(t)(T0-AE)J2L2C2R2(T1(t)-AE)].


## 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 Figure3. ADRC1 and ADRC2 are relevant to T1(t) and T2(t), 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).Figure 3
Structure of the ADRC decoupling control methodology.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 Figure3, v11 and v21 are tracking signals of T1ref and T2ref, respectively, and v12 andv22 are approximately differentiated signals of T1ref andT2ref, respectively. According to [22], the discrete forms of second-order TDi(i=1,2) are expressed as
(18)fhi=fhan(vi1(k)-Tiref(k),vi2(k),ri,h),vi1(k+1)=vi1(k)+hvi2(k),vi2(k+1)=vi2(k)+hfhi,
where ri is the velocity factor; h is the sampling step; andfhan(x1,x2,r,h) is the time-optimal control synthesis function defined as follows:
(19)d=rh;d0=hd;y=x1+hx2,a0=d2+8r|y|,a={x2+a0-d2sign(y),|y|>d0,x2+yh,|y|≤d0,fhan(x1,x2,r,h)=-{rsign(a),|a|>d,rad,|a|≤d.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 Figure3, z11 and z21 track the output variables T1(t) and T2(t), respectively; z12 and z22 estimate the differentiated signals of T1(t) and T2(t), respectively; and z13 and z23 actively estimate f1(t) and w1(t) and f2(t) and w2(t), respectively. The z13 and z23 variables are considered to be extended states of the system. According to [22], the discrete forms of third-order ESOi(i=1,2) are expressed as
(20)ei(k)=zi1(k)-Ti(k),zi1(k+1)=zi1(k)+h(zi2(k)-βi1ei(k)),zi2(k+1)=zi2(k)+h(zi3(k)-βi2fal(ei(k),0.5,δ)+τi(k)),zi3(k+1)=zi3(k)-hβi3fal(ei(k),0.25,δ),
where βi1, βi2, and βi3 are the ESO gains; δ is the interval length of the linear segment; and fal(e,α,δ) is a nonlinear function defined as follows:
(21)fal(e,α,δ)={eδ1-α,|e|≤δ,|e|αsign(e),|e|>δ.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 Figure3. The control law actively compensates for dynamic coupling and external disturbances that are estimated by the ESO in real time. According to [22], the discrete forms of NLSEFi(i=1,2) are defined as
(22)ei1(k)=vi1(k)-zi1(k),ei2(k)=vi2(k)-zi2(k),τ0i(k)=kipfal(ei1(k),0.5,δ)+kidfal(ei2(k),1.5,δ),τi(k)=τ0i-zi3(k)b0i,
where kip is the proportionality coefficient; kid is the differential coefficient; and b0i is the compensating factor.Combining (18), (20), and (22), the discrete algorithm of the ADRCi(i=1,2) is shown in
(23)fhi(k)=fhan(vi1(k)-Tiref(k),vi2(k),ri,h),vi1(k+1)=vi1(k)+hvi2(k),vi2(k+1)=vi2(k)+hfhi(k),ei(k)=zi1(k)-Ti(k),zi1(k+1)=zi1(k)+h(zi2(k)-βi1ei(k)),zi2(k+1)=zi2(k)+h(zi3(k)-βi2fal(ei(k),0.5,h)+τi(k)),zi3(k+1)=zi3(k)-hβi3fal(ei(k),0.25,h),ei1(k+1)=vi1(k+1)-zi1(k+1),ei2(k+1)=vi2(k+1)-zi2(k+1),τi(k+1)=kipfal(ei1(k+1),0.5,δ)+kidfal(ei2(k+1),1.5,δ)-zi3(k+1)b0i.After calculating the virtual control variablesτ1(t) and τ2(t) with the ADRC controller, the actual control variables u1(t) and u2(t) can be obtained by (16).Two diagrams are provided in Figures3 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 (w1(t) and w2(t)), dynamic coupling components of the system (f(t)), and unknown internal dynamics, is defined as an extended state (zi3) 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 vi2 and zi2 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.Figure 4
Structure of the PID control methodology.


## 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 Table1. All parameters of ADRC and PID were determined at ω3=100 r/min and R1=0.2m. Although the tests were conducted under different operating conditions, particularly for different unwinding diameters, the parameters remain constant.Table 1
Mechanical parameters of the unwinding system.
Parameters
Value
Units
L
1
1
m
L
2
0.35
m
A
5 × 10−6
m2
L
D
1
0.064
m
L
D
2
0.128
m
R
2
0.03
m
R
3
0.03
m
E
1.6 × 108
Pa
K
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.Table 2
Controller parameters in simulation.
Controller
Controller parameters
ADRC1
β
11 = 100, β12 = 5560, β13 = 81200, k1p = 920, k1d = 760, r1 = 60
ADRC2
β
21 = 100, β22 = 2870, β23 = 51600, k2p = 3460, k2d = 750, r2 = 60
PID1
K
1
p = 30, K1i = 2.5, K1d = 12
PID2
K
2
p = 65, K2i = 32, K2d = 5


#### 4.1.1. Case A: Robustness against Dynamic Changes

To verify the robustness against dynamic changes of the developed ADRC control methodology, theT1(t) and T2(t) step change is from 50 N to 80 N when ω3 is equal to 100 r/min and R1 is equal to 0.2 m. The simulation behaviours of the PID and ADRC controllers are shown in Figures 5 and 6.Step response curves forω3=100 r/min.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerStep response curves forR1=0.2m.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerAs shown in Figure5, when R1 is equal to 0.2 m, 0.1 m, or 0.05 m, both T1(t) and T2(t) quickly reach 80 N without overshoot or oscillation in the ADRC controller. However, in the PID controller, a decrease in R1 leads to deterioration in the control abilities of T1(t) and T2(t). For example, when R1 is equal to 0.05 m, T1(t) demonstrates an overshoot of approximately 6.38% and arrives at 80 N after 4 s. Figure 6 illustrates that when ω3 is equal to 100 r/min, 300 r/min, or 500 r/min, both T1(t) and T2(t) quickly reach 80 N without overshoot or oscillation in the ADRC controller. In contrast, an increase in ω3 contributes to deterioration in the control abilities of T1(t) and T2(t) 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, Figures5 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, theT1(t) 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 ω3 is equal to 100 r/min and R1 is equal to 0.2 m, respectively.Decoupling response curves forω3=100 r/min.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerDecoupling response curves forR1=0.2m.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerFigure7 indicates that when R1 is equal to 0.2 m, 0.1 m, or 0.05 m, variations in T1(t) have no influence on T2(t) in the ADRC controller. In contrast, in the PID controller, T2(t) is continuously influenced by variations in T1(t), and variations in T2(t) increase with a decrease in R1. As shown in Figure 8, when ω3 is equal to 100 r/min, 300 r/min, or 500 r/min, variations in T1(t) have no influence on T2(t) in the ADRC controller. However, in the PID controller, T2(t) is continuously influenced by variations in T1(t), and variations in T2(t) increase with an increase in ω3. For example, when ω3 is equal to 500 r/min, variations in T1(t) at 5 s and 10 s cause T2(t) 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 ofT0 is set at 5 s with a value of 10 N, and a step disturbance of T1(t) is set at 10 s with a value of 2 N when ω3 is equal to 100 r/min and R1 is equal to 0.2 m. The simulation behaviours of the PID and ADRC controllers are shown in Figures 9 and 10.Antidisturbance curves forω3=100 r/min.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerAntidisturbance curves forR1=0.2m.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerFigures9 and 10 illustrate that with a decrease in R1 and an increase in ω3, 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 R1 is equal to 0.05 m, the range and duration of the fluctuation in  T1(t) 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 T1(t) at 5 s are 0.9 N and 4 s, respectively. In the ADRC controller, when ω3 is equal to 500 r/min, the range and duration of the fluctuation in T1(t) at 5 s are 0.8 N and 0.8 s, respectively. However, the range and duration of the fluctuation in T1(t) 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 Figure1, 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.Table 3
Controller parameters in experiments.
Controller
Controller parameters
ADRC1
β
11 = 100, β12 = 6560, β13 = 84700, k1p = 830, k1d = 520, r1 = 68
ADRC2
β
21 = 100, β22 = 2950, β23 = 54600, k2p = 2760, k2d = 670, r2 = 57
PID1
K
1
p = 43, K1i = 3.2, K1d = 7
PID2
K
2
p = 81, K2i = 46, K2d = 8


#### 4.2.1. Case A: Robustness against Dynamic Changes

To determine the practical robustness against dynamic changes of the ADRC controller, theT1(t) and T2(t) step change is from 50 N to 80 N at 5 s when ω3 is equal to 100 r/min and R1 is equal to 0.2 m. The performance of the PID and ADRC controllers in the experiments is shown in Figures 11 and 12.Step response curves forω3=100 r/min.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerStep response curves forR1=0.2m.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerFigures11 and 12 show that with a decrease in R1 and an increase in ω3, the control abilities of T1(t) and T2(t) deteriorate in the PID controller. However, when ω3 is equal to 100 r/min or when R1 is equal to 0.2 m, T1(t) and T2(t) 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, theT1(t) step change is from 50 N to 80 N at 5 s and from 80 N to 50 N at 10 s when ω3 is equal to 100 r/min and R1 is equal to 0.2 m. Figures 13 and 14 show the performance of the PID and ADRC controllers.Decoupling response curves forω3=100 r/min.
(a)
Tension with PID controller
(b)
Tension with ADRC controller
(c)
Tension with PID controller
(d)
Tension with ADRC controllerDecoupling response curves forR1=0.2m.
(a)
Tension with PID controller
(b)
Tension with ADRC controller
(c)
Tension with PID controller
(d)
Tension with ADRC controllerFigures13 and 14 illustrate that T2(t) is always influenced by variation in T1(t) and the variation in T2(t) increases with a decrease in R1 and an increase in ω3 in the PID controller. Unfortunately, when ω3 is equal to 100 r/min or R1 is equal to 0.2 m, the variation in T1(t) has no influence on T2(t) 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 ofω3=100 r/min and ω3=500 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. ST1 and ST2 are standard deviations of T1(t) and T2(t), respectively, and were obtained under different test conditions, just as shown in Table 4.Table 4
S
T
1 and ST2 in various test conditions.
Conditions
Controller type
PID controller
ADRC controller
S
T
1 (N)
S
T
2 (N)
S
T
1 (N)
S
T
2 (N)
ω
3 = 100 r/min, R1 = 0.2 m
0.3683
0.2949
0.2400
0.2163
ω
3 = 100 r/min, R1 = 0.05 m
0.5540
0.4271
0.2686
0.2350
ω
3 = 500 r/min, R1 = 0.2 m
0.9326
0.6572
0.4880
0.4064
ω
3 = 500 r/min, R1 = 0.05 m
1.3349
0.8514
0.5831
0.4658Tension experimental curves forω3=100 r/min.
(a)
R
1
=
0.2 m
(b)
R
1
=
0.05 m
(c)
R
1
=
0.2 m
(d)
R
1
=
0.05 mTension experimental curves forω3=500 r/min.
(a)
R
1
=
0.2 m
(b)
R
1
=
0.05 m
(c)
R
1
=
0.2 m
(d)
R
1
=
0.05 mFigures15 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 R1 is equal to 0.05 m and ω3 is equal to 500 r/min, the standard deviations of  T1(t) and T2(t) in the PID controller are 1.3349 N and 0.8514 N, respectively. Under the ADRC controller, when R1 is equal to 0.05 m and ω3 is equal to 500 r/min, the standard deviations of T1(t) and T2(t) 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.


## 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.Table 2
Controller parameters in simulation.
Controller
Controller parameters
ADRC1
β
11 = 100, β12 = 5560, β13 = 81200, k1p = 920, k1d = 760, r1 = 60
ADRC2
β
21 = 100, β22 = 2870, β23 = 51600, k2p = 3460, k2d = 750, r2 = 60
PID1
K
1
p = 30, K1i = 2.5, K1d = 12
PID2
K
2
p = 65, K2i = 32, K2d = 5


### 4.1.1. Case A: Robustness against Dynamic Changes

To verify the robustness against dynamic changes of the developed ADRC control methodology, theT1(t) and T2(t) step change is from 50 N to 80 N when ω3 is equal to 100 r/min and R1 is equal to 0.2 m. The simulation behaviours of the PID and ADRC controllers are shown in Figures 5 and 6.Step response curves forω3=100 r/min.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerStep response curves forR1=0.2m.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerAs shown in Figure5, when R1 is equal to 0.2 m, 0.1 m, or 0.05 m, both T1(t) and T2(t) quickly reach 80 N without overshoot or oscillation in the ADRC controller. However, in the PID controller, a decrease in R1 leads to deterioration in the control abilities of T1(t) and T2(t). For example, when R1 is equal to 0.05 m, T1(t) demonstrates an overshoot of approximately 6.38% and arrives at 80 N after 4 s. Figure 6 illustrates that when ω3 is equal to 100 r/min, 300 r/min, or 500 r/min, both T1(t) and T2(t) quickly reach 80 N without overshoot or oscillation in the ADRC controller. In contrast, an increase in ω3 contributes to deterioration in the control abilities of T1(t) and T2(t) 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, Figures5 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, theT1(t) 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 ω3 is equal to 100 r/min and R1 is equal to 0.2 m, respectively.Decoupling response curves forω3=100 r/min.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerDecoupling response curves forR1=0.2m.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerFigure7 indicates that when R1 is equal to 0.2 m, 0.1 m, or 0.05 m, variations in T1(t) have no influence on T2(t) in the ADRC controller. In contrast, in the PID controller, T2(t) is continuously influenced by variations in T1(t), and variations in T2(t) increase with a decrease in R1. As shown in Figure 8, when ω3 is equal to 100 r/min, 300 r/min, or 500 r/min, variations in T1(t) have no influence on T2(t) in the ADRC controller. However, in the PID controller, T2(t) is continuously influenced by variations in T1(t), and variations in T2(t) increase with an increase in ω3. For example, when ω3 is equal to 500 r/min, variations in T1(t) at 5 s and 10 s cause T2(t) 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 ofT0 is set at 5 s with a value of 10 N, and a step disturbance of T1(t) is set at 10 s with a value of 2 N when ω3 is equal to 100 r/min and R1 is equal to 0.2 m. The simulation behaviours of the PID and ADRC controllers are shown in Figures 9 and 10.Antidisturbance curves forω3=100 r/min.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerAntidisturbance curves forR1=0.2m.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerFigures9 and 10 illustrate that with a decrease in R1 and an increase in ω3, 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 R1 is equal to 0.05 m, the range and duration of the fluctuation in  T1(t) 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 T1(t) at 5 s are 0.9 N and 4 s, respectively. In the ADRC controller, when ω3 is equal to 500 r/min, the range and duration of the fluctuation in T1(t) at 5 s are 0.8 N and 0.8 s, respectively. However, the range and duration of the fluctuation in T1(t) 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.


## 4.1.1. Case A: Robustness against Dynamic Changes

To verify the robustness against dynamic changes of the developed ADRC control methodology, theT1(t) and T2(t) step change is from 50 N to 80 N when ω3 is equal to 100 r/min and R1 is equal to 0.2 m. The simulation behaviours of the PID and ADRC controllers are shown in Figures 5 and 6.Step response curves forω3=100 r/min.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerStep response curves forR1=0.2m.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerAs shown in Figure5, when R1 is equal to 0.2 m, 0.1 m, or 0.05 m, both T1(t) and T2(t) quickly reach 80 N without overshoot or oscillation in the ADRC controller. However, in the PID controller, a decrease in R1 leads to deterioration in the control abilities of T1(t) and T2(t). For example, when R1 is equal to 0.05 m, T1(t) demonstrates an overshoot of approximately 6.38% and arrives at 80 N after 4 s. Figure 6 illustrates that when ω3 is equal to 100 r/min, 300 r/min, or 500 r/min, both T1(t) and T2(t) quickly reach 80 N without overshoot or oscillation in the ADRC controller. In contrast, an increase in ω3 contributes to deterioration in the control abilities of T1(t) and T2(t) 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, Figures5 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, theT1(t) 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 ω3 is equal to 100 r/min and R1 is equal to 0.2 m, respectively.Decoupling response curves forω3=100 r/min.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerDecoupling response curves forR1=0.2m.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerFigure7 indicates that when R1 is equal to 0.2 m, 0.1 m, or 0.05 m, variations in T1(t) have no influence on T2(t) in the ADRC controller. In contrast, in the PID controller, T2(t) is continuously influenced by variations in T1(t), and variations in T2(t) increase with a decrease in R1. As shown in Figure 8, when ω3 is equal to 100 r/min, 300 r/min, or 500 r/min, variations in T1(t) have no influence on T2(t) in the ADRC controller. However, in the PID controller, T2(t) is continuously influenced by variations in T1(t), and variations in T2(t) increase with an increase in ω3. For example, when ω3 is equal to 500 r/min, variations in T1(t) at 5 s and 10 s cause T2(t) 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 ofT0 is set at 5 s with a value of 10 N, and a step disturbance of T1(t) is set at 10 s with a value of 2 N when ω3 is equal to 100 r/min and R1 is equal to 0.2 m. The simulation behaviours of the PID and ADRC controllers are shown in Figures 9 and 10.Antidisturbance curves forω3=100 r/min.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerAntidisturbance curves forR1=0.2m.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerFigures9 and 10 illustrate that with a decrease in R1 and an increase in ω3, 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 R1 is equal to 0.05 m, the range and duration of the fluctuation in  T1(t) 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 T1(t) at 5 s are 0.9 N and 4 s, respectively. In the ADRC controller, when ω3 is equal to 500 r/min, the range and duration of the fluctuation in T1(t) at 5 s are 0.8 N and 0.8 s, respectively. However, the range and duration of the fluctuation in T1(t) 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 Figure1, 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.Table 3
Controller parameters in experiments.
Controller
Controller parameters
ADRC1
β
11 = 100, β12 = 6560, β13 = 84700, k1p = 830, k1d = 520, r1 = 68
ADRC2
β
21 = 100, β22 = 2950, β23 = 54600, k2p = 2760, k2d = 670, r2 = 57
PID1
K
1
p = 43, K1i = 3.2, K1d = 7
PID2
K
2
p = 81, K2i = 46, K2d = 8


### 4.2.1. Case A: Robustness against Dynamic Changes

To determine the practical robustness against dynamic changes of the ADRC controller, theT1(t) and T2(t) step change is from 50 N to 80 N at 5 s when ω3 is equal to 100 r/min and R1 is equal to 0.2 m. The performance of the PID and ADRC controllers in the experiments is shown in Figures 11 and 12.Step response curves forω3=100 r/min.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerStep response curves forR1=0.2m.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerFigures11 and 12 show that with a decrease in R1 and an increase in ω3, the control abilities of T1(t) and T2(t) deteriorate in the PID controller. However, when ω3 is equal to 100 r/min or when R1 is equal to 0.2 m, T1(t) and T2(t) 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, theT1(t) step change is from 50 N to 80 N at 5 s and from 80 N to 50 N at 10 s when ω3 is equal to 100 r/min and R1 is equal to 0.2 m. Figures 13 and 14 show the performance of the PID and ADRC controllers.Decoupling response curves forω3=100 r/min.
(a)
Tension with PID controller
(b)
Tension with ADRC controller
(c)
Tension with PID controller
(d)
Tension with ADRC controllerDecoupling response curves forR1=0.2m.
(a)
Tension with PID controller
(b)
Tension with ADRC controller
(c)
Tension with PID controller
(d)
Tension with ADRC controllerFigures13 and 14 illustrate that T2(t) is always influenced by variation in T1(t) and the variation in T2(t) increases with a decrease in R1 and an increase in ω3 in the PID controller. Unfortunately, when ω3 is equal to 100 r/min or R1 is equal to 0.2 m, the variation in T1(t) has no influence on T2(t) 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 ofω3=100 r/min and ω3=500 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. ST1 and ST2 are standard deviations of T1(t) and T2(t), respectively, and were obtained under different test conditions, just as shown in Table 4.Table 4
S
T
1 and ST2 in various test conditions.
Conditions
Controller type
PID controller
ADRC controller
S
T
1 (N)
S
T
2 (N)
S
T
1 (N)
S
T
2 (N)
ω
3 = 100 r/min, R1 = 0.2 m
0.3683
0.2949
0.2400
0.2163
ω
3 = 100 r/min, R1 = 0.05 m
0.5540
0.4271
0.2686
0.2350
ω
3 = 500 r/min, R1 = 0.2 m
0.9326
0.6572
0.4880
0.4064
ω
3 = 500 r/min, R1 = 0.05 m
1.3349
0.8514
0.5831
0.4658Tension experimental curves forω3=100 r/min.
(a)
R
1
=
0.2 m
(b)
R
1
=
0.05 m
(c)
R
1
=
0.2 m
(d)
R
1
=
0.05 mTension experimental curves forω3=500 r/min.
(a)
R
1
=
0.2 m
(b)
R
1
=
0.05 m
(c)
R
1
=
0.2 m
(d)
R
1
=
0.05 mFigures15 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 R1 is equal to 0.05 m and ω3 is equal to 500 r/min, the standard deviations of  T1(t) and T2(t) in the PID controller are 1.3349 N and 0.8514 N, respectively. Under the ADRC controller, when R1 is equal to 0.05 m and ω3 is equal to 500 r/min, the standard deviations of T1(t) and T2(t) 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.


## 4.2.1. Case A: Robustness against Dynamic Changes

To determine the practical robustness against dynamic changes of the ADRC controller, theT1(t) and T2(t) step change is from 50 N to 80 N at 5 s when ω3 is equal to 100 r/min and R1 is equal to 0.2 m. The performance of the PID and ADRC controllers in the experiments is shown in Figures 11 and 12.Step response curves forω3=100 r/min.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerStep response curves forR1=0.2m.
(a)
T
1
(
t
) with PID controller
(b)
T
1
(
t
) with ADRC controller
(c)
T
2
(
t
) with PID controller
(d)
T
2
(
t
) with ADRC controllerFigures11 and 12 show that with a decrease in R1 and an increase in ω3, the control abilities of T1(t) and T2(t) deteriorate in the PID controller. However, when ω3 is equal to 100 r/min or when R1 is equal to 0.2 m, T1(t) and T2(t) 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, theT1(t) step change is from 50 N to 80 N at 5 s and from 80 N to 50 N at 10 s when ω3 is equal to 100 r/min and R1 is equal to 0.2 m. Figures 13 and 14 show the performance of the PID and ADRC controllers.Decoupling response curves forω3=100 r/min.
(a)
Tension with PID controller
(b)
Tension with ADRC controller
(c)
Tension with PID controller
(d)
Tension with ADRC controllerDecoupling response curves forR1=0.2m.
(a)
Tension with PID controller
(b)
Tension with ADRC controller
(c)
Tension with PID controller
(d)
Tension with ADRC controllerFigures13 and 14 illustrate that T2(t) is always influenced by variation in T1(t) and the variation in T2(t) increases with a decrease in R1 and an increase in ω3 in the PID controller. Unfortunately, when ω3 is equal to 100 r/min or R1 is equal to 0.2 m, the variation in T1(t) has no influence on T2(t) 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 ofω3=100 r/min and ω3=500 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. ST1 and ST2 are standard deviations of T1(t) and T2(t), respectively, and were obtained under different test conditions, just as shown in Table 4.Table 4
S
T
1 and ST2 in various test conditions.
Conditions
Controller type
PID controller
ADRC controller
S
T
1 (N)
S
T
2 (N)
S
T
1 (N)
S
T
2 (N)
ω
3 = 100 r/min, R1 = 0.2 m
0.3683
0.2949
0.2400
0.2163
ω
3 = 100 r/min, R1 = 0.05 m
0.5540
0.4271
0.2686
0.2350
ω
3 = 500 r/min, R1 = 0.2 m
0.9326
0.6572
0.4880
0.4064
ω
3 = 500 r/min, R1 = 0.05 m
1.3349
0.8514
0.5831
0.4658Tension experimental curves forω3=100 r/min.
(a)
R
1
=
0.2 m
(b)
R
1
=
0.05 m
(c)
R
1
=
0.2 m
(d)
R
1
=
0.05 mTension experimental curves forω3=500 r/min.
(a)
R
1
=
0.2 m
(b)
R
1
=
0.05 m
(c)
R
1
=
0.2 m
(d)
R
1
=
0.05 mFigures15 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 R1 is equal to 0.05 m and ω3 is equal to 500 r/min, the standard deviations of  T1(t) and T2(t) in the PID controller are 1.3349 N and 0.8514 N, respectively. Under the ADRC controller, when R1 is equal to 0.05 m and ω3 is equal to 500 r/min, the standard deviations of T1(t) and T2(t) 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.



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*Source: 439797-2013-11-17.xml*