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

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.

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

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).

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 Figure 2. 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 𝐿_𝑖(𝑡)𝑑𝑇_𝑖(𝑡)𝑑𝑡=𝑉_𝑖(𝑡)𝑇_𝑖−1(𝑡)−𝑉_𝑖+1(𝑡)𝑇_𝑖(𝑡)+𝐴𝐸𝑉_𝑖+1(𝑡)−𝑉_𝑖(𝑡+)𝐴𝐸−𝑇_𝑖(𝑡)𝑑𝐿_𝑖(𝑡)𝑑𝑡𝑖=1,2. (1)

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 𝐿₁(𝑡)𝑇̇₁(𝑡)=𝑇₀−𝐴𝐸𝑅₁(𝑡)𝜔₁(𝑡)+𝐴𝐸−𝑇₁(𝑡)𝑅₂𝜔₂(𝑡)+𝐴𝐸−𝑇₁(𝑡)𝐿̇₁(𝑡),𝐿₂𝑇̇₂(𝑡)=𝑇₁(𝑡)−𝐴𝐸𝑅₂𝜔₂(𝑡)+𝐴𝐸−𝑇₂(𝑡)𝑅₃𝜔₃. (2)

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 𝑑𝐽𝑑𝑡^𝑖(𝑡)𝜔_𝑖(𝑡)=𝑇_𝑖(𝑡)−𝑇_𝑖−1(𝑡)𝑅_𝑖(𝑡)+𝐶_𝑖𝑢_𝑖(𝑡)−𝑓𝜔_𝑖(𝑡),𝑖=1,2. (3)

Because 𝑅₁(𝑡) and 𝐽₁(𝑡) are time-variant parameters for the unwinding roll and decrease with time, (4) can be obtained as follows: 𝑑𝐽𝑑𝑡¹(𝑡)𝜔₁(𝑡)=𝐽̇₁(𝑡)𝜔₁(𝑡)+𝐽₁(𝑡)𝜔̇₁(𝑡). (4)

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 𝑅₁(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: 𝐽₁(𝑡)𝜔̇₁(𝑡)=𝑇₁(𝑡)−𝑇₀𝑅₁(𝑡)+𝐶₁𝑢₁(𝑡)−𝑓𝜔₁(𝑡),𝐽₂𝜔̇₂(𝑡)=𝑇₂(𝑡)−𝑇₁(𝑡)𝑅₂+𝐶₂𝑢₂(𝑡)−𝑓𝜔₂(𝑡). (5)

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: 𝐿₁(𝑡)=𝐿₁−2𝐿_𝐷2𝜃(𝑡). (6)

Combining (6) with (2), (2) can be rewritten as follows: 𝐿₁−2𝐿_𝐷2𝜃(𝑡)𝑇̇₁(𝑡)=𝑇₀−𝐴𝐸𝑅₁(𝑡)𝜔₁(𝑡)+𝐴𝐸−𝑇₁(𝑡)𝑅₂𝜔₂(𝑡)−2𝐿_𝐷2𝐴𝐸−𝑇₁(𝑡)𝜃̇(𝑡),𝐿₂𝑇̇₂(𝑡)=𝑇₁(𝑡)−𝐴𝐸𝑅₂𝜔₂(𝑡)+𝐴𝐸−𝑇₂(𝑡)𝑅₃𝜔₃. (7)

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 𝑇₁(𝑡)=12𝐿_𝐷2𝐽_𝐷𝑎𝜃̈(𝑡)+𝑓_𝐷𝜃̇(𝑡)+𝐾𝐿²_𝐷1𝜃(𝑡). (8)

Combining (5), (7), and (8), the nonlinear mathematical model for the unwinding system is constructed as follows: 𝐽₁(𝑡)𝜔̇₁(𝑡)=𝑇₁(𝑡)−𝑇₀𝑅₁(𝑡)+𝐶₁𝑢₁(𝑡)−𝑓𝜔₁(𝑡),𝐽₂𝜔̇₂(𝑡)=𝑇₂(𝑡)−𝑇₁(𝑡)𝑅₂+𝐶₂𝑢₂(𝑡)−𝑓𝜔₂(𝑡),2𝐿_𝐷2𝑇₁(𝑡)=𝐽_𝐷𝑎𝜃̈(𝑡)+𝑓_𝐷𝜃̇(𝑡)+𝐾𝐿²_𝐷1𝜃(𝑡),𝐿₁−2𝐿_𝐷2𝜃(𝑡)𝑇̇₁(𝑡)=𝑇₀−𝐴𝐸𝑅₁(𝑡)𝜔₁(𝑡)+𝐴𝐸−𝑇₁(𝑡)𝑅₂𝜔₂(𝑡)−2𝐿_𝐷2𝐴𝐸−𝑇₁(𝑡)𝜃̇(𝑡),𝐿₂𝑇̇₂(𝑡)=𝑇₁(𝑡)−𝐴𝐸𝑅₂𝜔₂(𝑡)+𝐴𝐸−𝑇₂(𝑡)𝑅₃𝜔₃. (9)

The controlled variables 𝑢₁(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: 𝑇̈₁(𝑡)=2𝐿_𝐷2𝑇₁(𝑡)−𝐴𝐸𝜃̈(𝑡)+2𝑇̇₁(𝑡)𝜃̇(𝑡)−𝑅₂𝜔₂(𝑡)𝑇̇₁(𝑡)+𝑇₀−𝐴𝐸𝜔₁(𝑡)𝑅̇₁(𝑡)×𝐿₁−2𝐿_𝐷2𝜃(𝑡)⁻¹+𝑅₁(𝑡)𝑇₀−𝐴𝐸𝑅₁(𝑡)𝑇₁(𝑡)−𝑇₀−𝑓𝜔₁(𝑡)𝐽₁(𝑡)𝐿₁−2𝐿_𝐷2𝜃(𝑡)+𝐶₁𝑅₁(𝑡)𝑇₀−𝐴𝐸𝐽₁(𝑡)𝐿₁−2𝐿_𝐷2𝜃(𝑡)𝑢₁(𝑡)+𝑅₂𝐴𝐸−𝑇₁(𝑡)𝑅₂𝑇₂(𝑡)−𝑇₁(𝑡)−𝑓𝜔₂(𝑡)𝐽₂𝐿₁−2𝐿_𝐷2𝜃(𝑡)+𝐶₂𝑅₂𝐴𝐸−𝑇₁(𝑡)𝐽₂𝐿₁−2𝐿_𝐷2𝜃(𝑡)𝑢₂(𝑡),𝑇̈₂(𝑡)=𝑅₂𝜔₂(𝑡)𝑇̇₁(𝑡)−𝑅₃𝜔₃𝑇̇₂(𝑡)𝐿₂+𝑅₂𝑇₁(𝑡)−𝐴𝐸𝑅₂𝑇₂(𝑡)−𝑇₁(𝑡)−𝑓𝜔₂(𝑡)𝐽₂𝐿₂+𝐶₂𝑅₂𝑇₁(𝑡)−𝐴𝐸𝐽₂𝐿₂𝑢₂(𝑡). (10)

We define the dynamic coupling vector 𝐟(𝑡) and input matrix 𝐁(𝑡) as follows: 𝐟(𝑡)=𝑓₁(𝑡)𝑓₂(𝑡),𝑓₁(𝑡)=2𝐿_𝐷2𝑇₁(𝑡)−𝐴𝐸𝜃̈(𝑡)+2𝑇̇₁(𝑡)𝜃̇(𝑡)−𝑅₂𝜔₂(𝑡)𝑇̇₁(𝑡)+𝑇₀−𝐴𝐸𝜔₁(𝑡)𝑅̇₁(𝑡)×𝐿₁−2𝐿_𝐷2𝜃(𝑡)⁻¹+𝑅₁(𝑡)𝑇₀−𝐴𝐸𝑅₁(𝑡)𝑇₁(𝑡)−𝑇₀−𝑓𝜔₁(𝑡)𝐽₁(𝑡)𝐿₁−2𝐿_𝐷2𝜃(𝑡)+𝑅₂𝐴𝐸−𝑇₁(𝑡)𝑅₂𝑇₂(𝑡)−𝑇₁(𝑡)−𝑓𝜔₂(𝑡)𝐽₂𝐿₁−2𝐿_𝐷2𝜃(𝑡),𝑓₂(𝑡)=𝑅₂𝜔₂(𝑡)𝑇̇₁(𝑡)−𝑅₃𝜔₃𝑇̇₂(𝑡)𝐿₂+𝑅₂𝑇₁(𝑡)−𝐴𝐸𝑅₂𝑇₂(𝑡)−𝑇₁(𝑡)−𝑓𝜔₂(𝑡)𝐽₂𝐿₂,𝐁(𝑡)=𝐶₁𝑅₁(𝑡)𝑇₀−𝐴𝐸𝐽₁(𝑡)𝐿₁−2𝐿_𝐷2𝜃(𝑡)𝐶₂𝑅₂𝐴𝐸−𝑇₁(𝑡)𝐽₂𝐿₁−2𝐿_𝐷2𝜃(𝑡)𝐶₂𝑅₂𝑇₁(𝑡)−𝐴𝐸𝐽₂𝐿₂. (11)

Substituting (11) into (10), (10) can be rewritten as follows: 𝑇̈₁(𝑡)𝑇̈₂(𝑡)(𝑡)(𝑡)𝑢=𝐟+𝐁¹(𝑡)𝑢₂(𝑡). (12)

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: 𝜏₁(𝑡)𝜏₂(𝑡)(𝑡)𝑢=𝐁¹(𝑡)𝑢₂(𝑡). (13)

Combining (12) and (13), the nonlinear mathematical model for the unwinding system can also be rewritten as follows: 𝑇̈₁(𝑡)𝑇̈₂(𝑡)(𝑡)+𝜏=𝐟¹(𝑡)𝜏₂(𝑡). (14)

Because the dynamic coupling vector 𝐟(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, because 𝑇₀≪𝐴𝐸, 𝑇₁(𝑡)≪𝐴𝐸, and 2𝐿_𝐷2𝜃(𝑡)≪𝐿₁, so (15) can be got as follows: |𝐁(𝑡=𝐶)|¹𝐶₂𝑅₂𝑅₁(𝑡)𝑇₀−𝐴𝐸𝑇₁(𝑡)−𝐴𝐸𝐽₂𝐿₂𝐽₁(𝑡)𝐿₁−2𝐿_𝐷2𝜃(𝑡)≠0. (15)

So 𝐁(𝑡) is an invertible matrix.

Therefore, the relationships between the virtual control quantities 𝜏₁(𝑡) and 𝜏₂(𝑡) and the actual control variables 𝑢₁(𝑡) and 𝑢₂(𝑡) can be expressed as 𝑢₁(𝑡)𝑢₂(𝑡)=𝐁^−𝟏(𝑡)𝜏₁(𝑡)𝜏₂(𝑡), (16) where matrix 𝐁^−𝟏(𝑡) is the decoupling plant of the unwinding system as follows: 𝐁^−𝟏(𝑡)=𝐽₁(𝑡)𝐿₁−2𝐿_𝐷2𝜃(𝑡)𝐶₁𝑅₁(𝑡)𝑇₀𝐿−𝐴𝐸²𝐽₁(𝑡)𝐶₁𝑅₁(𝑡)𝑇₀𝐽−𝐴𝐸₂𝐿₂𝐶₂𝑅₂𝑇₁(𝑡)−𝐴𝐸. (17)

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 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 Figure 3, 𝑣₁₁ and 𝑣₂₁ are tracking signals of 𝑇_1ref and 𝑇_2ref, respectively, and 𝑣₁₂ and𝑣₂₂ are approximately differentiated signals of 𝑇_1ref and𝑇_2ref, respectively. According to [22], the discrete forms of second-order TD𝑖(𝑖=1,2) are expressed as 𝑓ℎ_𝑖=fhan𝑣_𝑖1(𝑘)−𝑇_𝑖ref(𝑘),𝑣_𝑖2(𝑘),𝑟_𝑖,𝑣,ℎ_𝑖1(𝑘+1)=𝑣_𝑖1(𝑘)+ℎ𝑣_𝑖2(𝑘),𝑣_𝑖2(𝑘+1)=𝑣_𝑖2(𝑘)+ℎ𝑓ℎ_𝑖, (18) where 𝑟_𝑖 is the velocity factor; ℎ is the sampling step; andfhan(𝑥₁,𝑥₂,𝑟,ℎ) is the time-optimal control synthesis function defined as follows: 𝑑=𝑟ℎ;𝑑₀=ℎ𝑑;𝑦=𝑥₁+ℎ𝑥₂,𝑎₀=𝑑²𝑦,𝑥+8𝑟𝑎=²+𝑎₀−𝑑2sign𝑦,𝑦>𝑑₀,𝑥₂+𝑦ℎ,𝑦≤𝑑₀,fhan𝑥₁,𝑥₂(𝑎),|𝑎|𝑟,𝑟,ℎ=−𝑎𝑟sign>𝑑,𝑑,|𝑎|≤𝑑. (19)

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, 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 𝑒_𝑖(𝑘)=𝑧_𝑖1(𝑘)−𝑇_𝑖(𝑘),𝑧_𝑖1(𝑘+1)=𝑧_𝑖1(𝑘)+ℎ𝑧_𝑖2(𝑘)−𝛽_𝑖1𝑒_𝑖(𝑘),𝑧_𝑖2(𝑘+1)=𝑧_𝑖2(𝑘)+ℎ𝑧_𝑖3(𝑘)−𝛽_𝑖2fal𝑒_𝑖(𝑘),0.5,𝛿+𝜏_𝑖(𝑘),𝑧_𝑖3(𝑘+1)=𝑧_𝑖3(𝑘)−ℎ𝛽_𝑖3fal𝑒_𝑖(𝑘),0.25,𝛿, (20) where 𝛽_𝑖1, 𝛽_𝑖2, and 𝛽_𝑖3 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: ()=𝑒fal𝑒,𝛼,𝛿𝛿^1−𝛼,|𝑒||𝑒|≤𝛿,_𝛼sign(𝑒),|𝑒|>𝛿. (21)

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 [22], the discrete forms of NLSEF𝑖(𝑖=1,2) are defined as 𝑒_𝑖1(𝑘)=𝑣_𝑖1(𝑘)−𝑧_𝑖1(𝑘),𝑒_𝑖2(𝑘)=𝑣_𝑖2(𝑘)−𝑧_𝑖2(𝑘),𝜏_0𝑖(𝑘)=𝑘_𝑖𝑝fal𝑒_𝑖1(𝑘),0.5,𝛿+𝑘_𝑖𝑑fal𝑒_𝑖2(𝑘),𝜏,1.5,𝛿_𝑖(𝑘)=𝜏_0𝑖−𝑧_𝑖3(𝑘)𝑏_0𝑖, (22) where 𝑘_𝑖𝑝 is the proportionality coefficient; 𝑘_𝑖𝑑 is the differential coefficient; and 𝑏_0𝑖 is the compensating factor.

Combining (18), (20), and (22), the discrete algorithm of the ADRC𝑖(𝑖=1,2) is shown in 𝑓ℎ_𝑖(𝑘)=fhan𝑣_𝑖1(𝑘)−𝑇_𝑖ref(𝑘),𝑣_𝑖2(𝑘),𝑟_𝑖,𝑣,ℎ_𝑖1(𝑘+1)=𝑣_𝑖1(𝑘)+ℎ𝑣_𝑖2(𝑘),𝑣_𝑖2(𝑘+1)=𝑣_𝑖2(𝑘)+ℎ𝑓ℎ_𝑖(𝑘),𝑒_𝑖(𝑘)=𝑧_𝑖1(𝑘)−𝑇_𝑖(𝑘),𝑧_𝑖1(𝑘+1)=𝑧_𝑖1(𝑘)+ℎ𝑧_𝑖2(𝑘)−𝛽_𝑖1𝑒_𝑖(𝑘,𝑧_𝑖2)(𝑘+1)=𝑧_𝑖2(𝑘)+ℎ𝑧_𝑖3(𝑘)−𝛽_𝑖2fal𝑒_𝑖(𝑘),0.5,ℎ+𝜏_𝑖(𝑘),𝑧_𝑖3(𝑘+1)=𝑧_𝑖3(𝑘)−ℎ𝛽_𝑖3fal𝑒_𝑖(𝑘),𝑒,0.25,ℎ_𝑖1(𝑘+1)=𝑣_𝑖1(𝑘+1)−𝑧_𝑖1(),𝑒𝑘+1_𝑖2(𝑘+1)=𝑣_𝑖2(𝑘+1)−𝑧_𝑖2(),𝜏𝑘+1_𝑖(𝑘+1)=𝑘_𝑖𝑝fal𝑒_𝑖1(𝑘+1),0.5,𝛿+𝑘_idfal𝑒_𝑖2()−𝑧𝑘+1,1.5,𝛿^𝑖3(𝑘+1)𝑏_0𝑖. (23)

After calculating the virtual control variables 𝜏₁(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 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 (𝑤₁(𝑡) and 𝑤₂(𝑡)), dynamic coupling components of the system (𝐟(𝑡)), and unknown internal dynamics, is defined as an extended state (𝑧_𝑖3) 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 𝑣_𝑖2 and 𝑧_𝑖2 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 Table 1. 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.

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.

4.1.1. Case A: Robustness against Dynamic Changes

To verify the robustness against dynamic changes of the developed ADRC control methodology, the 𝑇₁(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.

(a) 𝑇1(𝑡) with PID controller

(b) 𝑇1(𝑡) with ADRC controller

(c) 𝑇2(𝑡) with PID controller

(d) 𝑇2(𝑡) with ADRC controller

Figure 5 

Step response curves for 𝜔3=100 r/min.

(a) 𝑇1(𝑡) with PID controller

(b) 𝑇1(𝑡) with ADRC controller

(c) 𝑇2(𝑡) with PID controller

(d) 𝑇2(𝑡) with ADRC controller

Figure 6 

Step response curves for 𝑅1=0.2m.

As shown in Figure 5, 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, 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 𝑇₁(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.

(a) 𝑇1(𝑡) with PID controller

(b) 𝑇1(𝑡) with ADRC controller

(c) 𝑇2(𝑡) with PID controller

(d) 𝑇2(𝑡) with ADRC controller

Figure 7 

Decoupling response curves for 𝜔3=100 r/min.

(a) 𝑇1(𝑡) with PID controller

(b) 𝑇1(𝑡) with ADRC controller

(c) 𝑇2(𝑡) with PID controller

(d) 𝑇2(𝑡) with ADRC controller

Figure 8 

Decoupling response curves for 𝑅1=0.2m.

Figure 7 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 of 𝑇₀ 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.

(a) 𝑇1(𝑡) with PID controller

(b) 𝑇1(𝑡) with ADRC controller

(c) 𝑇2(𝑡) with PID controller

(d) 𝑇2(𝑡) with ADRC controller

Figure 9 

Antidisturbance curves for 𝜔3=100 r/min.

(a) 𝑇1(𝑡) with PID controller

(b) 𝑇1(𝑡) with ADRC controller

(c) 𝑇2(𝑡) with PID controller

(d) 𝑇2(𝑡) with ADRC controller

Figure 10 

Antidisturbance curves for 𝑅1=0.2m.

Figures 9 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 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.

Table 3 

Controller parameters in experiments.

4.2.1. Case A: Robustness against Dynamic Changes

To determine the practical robustness against dynamic changes of the ADRC controller, the 𝑇₁(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.

(a) 𝑇1(𝑡) with PID controller

(b) 𝑇1(𝑡) with ADRC controller

(c) 𝑇2(𝑡) with PID controller

(d) 𝑇2(𝑡) with ADRC controller

Figure 11 

Step response curves for 𝜔3=100 r/min.

(a) 𝑇1(𝑡) with PID controller

(b) 𝑇1(𝑡) with ADRC controller

(c) 𝑇2(𝑡) with PID controller

(d) 𝑇2(𝑡) with ADRC controller

Figure 12 

Step response curves for 𝑅1=0.2m.

Figures 11 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, the 𝑇₁(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.

(a) Tension with PID controller

(b) Tension with ADRC controller

(c) Tension with PID controller

(d) Tension with ADRC controller

Figure 13 

Decoupling response curves for 𝜔3=100 r/min.

(a) Tension with PID controller

(b) Tension with ADRC controller

(c) Tension with PID controller

(d) Tension with ADRC controller

Figure 14 

Decoupling response curves for 𝑅1=0.2m.

Figures 13 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 of 𝜔₃=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. 𝑆_𝑇1 and 𝑆_𝑇2 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_𝑇1 and 𝑆_𝑇2 in various test conditions.

(a) 𝑅1=0.2 m

(b) 𝑅1=0.05 m

(c) 𝑅1=0.2 m

(d) 𝑅1=0.05 m

Figure 15 

Tension experimental curves for 𝜔3=100 r/min.

(a) 𝑅1=0.2 m

(b) 𝑅1=0.05 m

(c) 𝑅1=0.2 m

(d) 𝑅1=0.05 m

Figure 16 

Tension experimental curves for 𝜔3=500 r/min.

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 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.

Nomenclature

𝑇0:	Constant tension of the unwinding roll (N)
𝑇𝑖:	Web tension in the 𝑖th span (N)
𝑤𝑖:	External disturbances that influence 𝑇𝑖
𝑉𝑖:	Tangential velocity of the 𝑖th roll (m/s)
𝜔𝑖:	Rotational speed of the 𝑖th roll (r/min)
𝑢𝑖:	Torque control signal of the 𝑖th servo motor (Nm)
𝐿𝑖:	Nominal span length of the web in the 𝑖th span (m)
𝐿𝐷𝑖:	The 𝑖th arm length of the dancer roll (m)
𝑅𝑖:	Radius of the 𝑖th roll (m)
𝐽𝑖:	Inertia of the 𝑖th roll (kgm2)
𝐽𝐷𝑎:	Inertia of the dancer roll (kgm2)
𝐶𝑖:	Constant of the 𝑖th servo motor
𝑇𝑖ref:	Reference inputs of 𝑇𝑖(𝑡) (N)
𝑉ref:	Reference inputs of 𝑉3 (m/s)
𝐾:	Young’s modulus of the spring (N/m)
𝐴:	Cross-sectional area of the web (m2)
𝐸:	Modulus of elasticity of the web material (Pa)
𝑓:	Viscous friction coefficient (Nm/(rad/s))
𝑓𝐷:	Viscous friction coefficient of the dancer roll (Nm/(rad/s))
𝜃:	Angle of the dancer roll arm from a neutral position (rad)
𝑀𝑖:	Servo motors driving the 𝑖th 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.