An ADRC Parameters Self-Tuning Controller Based on RBF Neural Network for Multi-Color Register System

Abstract

To improve the control precision of the nonlinear register system for flexographic printing, a feedforward active disturbance rejection control (ADRC) parameter self-tuning decoupling control strategy based on radial basis function (RBF) is proposed to address the existence of coupling interference and multiple working conditions. Firstly, according to the structure of the flexographic printing equipment system and registration principle, a nonlinear mathematical model of the global registration system is established and linearized using the small deviation method. Secondly, the decoupled controller of the register system is designed by integrating feedforward control, ADRC, and RBF, in which the feedforward control is used to eliminate the registration errors caused by modeled disturbances, the ADRC performs the estimation and compensation of unmodeled disturbances, and the RBF realizes the self-tuning of the ADRC controller parameters. Finally, different operating conditions are simulated to compare and verify the control performance of the proposed controller. Simulation results show that the designed controller has a better performance compared to traditional PID and ADRC control, and its register error peak is reduced by about 32% compared to ADRC, achieving the high accuracy control of a multi-color register system.

1. Introduction

Flexographic printing equipment has the advantages of high printing accuracy, a wide range of substrates, green and environmental protection, and is widely used in the fields of pharmaceuticals, food, and the packaging of daily necessities. At present, the register error of flexographic printing equipment is mostly controlled at ±0.02 mm, and the high-fidelity printing of products cannot be guaranteed if the control accuracy is poor during printing. Nevertheless, the multi-color register system shows characteristics such as non-linearity and strong interference, and the detection system has time lag, which makes it difficult to control. Therefore, it is urgent to propose a control method to decouple the multi-color register system of flexographic printing equipment and to cope with the time-varying control performance of the controller with working conditions, so that it can meet the industry demand for high-precision control.

Register system research mainly includes both register image detection and processing and register error analysis and control. In terms of register error analysis and control, many scholars have modeled multi-span register systems considering the effects of speed, strain, temperature, material, and mechanics. Kang analyzed the error generation principle and established a nonlinear model of three-layer register error with direct compensation with a servo motor using the law of mass conservation [1]. Chen analyzed the relationship of register error between neighboring rollers and web tension fluctuations, established a mechanical model of the acceleration phase of the R2R printing system, and quantified the dynamic relationship between tension, speed, and register error during the acceleration stage [2]. Lee analyzed the effect of thermal and elastic deformation of PET materials due to drying temperature on register error and established a register error model using system identification (SI) techniques [3,4]. Kang considered the lateral motion of the material and the incidence angle and established an oblique-machine-direction (OMD) error model to analyze the correlation between lateral and longitudinal errors using the plate roll translation and time delay characteristics [5]. Based on the mass conservation law, Kim modeled the register error from the perspective that the material (web) strain and the phase difference of the plate roll jointly generate the register error, and analyzed the accumulated phase difference between the printed layers and the linear variation term of the register error [6]. Liu developed a nonlinear registration model based on the mass conservation law of the material to reveal the relationship between velocity, strain, and registration error, and performed a serial extension on a coupled model of a multilayer registration system based on the previous color [7].

In terms of the decoupling control of register systems, PID control, ADRC, and other novel control methods have been applied to register systems for their nonlinear and strongly coupled characteristics. Chen proposed a feedforward PD control scheme based on the Brandenburg double-layer register model to reduce the register error to ±0.1 mm at a printing speed of 3 m/s [8,9] and proposed a model-based fully decoupled proportional-derivative (FDPD) control algorithm in [10], which controls the register error within ±0.05 mm. Lee considered the inherent characteristics in the roll-to-roll system when designing the controller for the register model obtained through the SI technique, and used PI controller control to reduce the register error to ±0.03 mm at a printing speed of 0.033 m/s [4,11]. Later, to improve the controller response speed, a register controller based on response acceleration input (RAI) was proposed to continuously control the register error within ±10 μm for the PET material [12]. Jung used an active motion-based roller (AMBR) combined with a PID control to compensate for tension disturbances and material stretching and converged the register error to less than 15 μm at a printing speed of 0.033 m/s [13]. It is clear that PID and the improved controller based on it cannot meet the requirements of register precision at high speed. Liu used feedforward control combined with ADRC technology to design a register system controller to suppress and compensate for tension fluctuations and speed fluctuations to continuously control the register error within ±20 μm at a printing speed of 6.67 m/s [7], but a solution for ADRC multiparameter adaption was not given.

With the advantages of rapid convergence and high approximation accuracy, RBF neural networks have been applied by many scholars to variable-parameter nonlinear systems with exogenous disturbances similar to register systems for accurate high-quality control. Asgharnia improved the control performance of a variable-pitch wind turbine based on RBF-rectified fractional-order PID (FOPID) controller parameters [14]. Li applied an RBF neural network to the PID parameter adjustment of a motor motion controller to improve the adaptability, stability, and dynamic-static performance of the servo system [15]. Liu designed an RBF-based ADRC for online tracking and the parameters’ real-time adjustment of a three-motor synchronous control system to achieve overshoot-free speed regulation [16]. Kumar verified that an RBF neural network has a higher accuracy when controlling servo motors compared to other methods (BP neural network) [17]. Therefore, for the characteristics of the register system with strong disturbances, large time delays, and the features of printing variable working conditions, RBF can deliver the real-time adjustment of the controller and linearized output of a nonlinear system for the register system.

In summary, this paper designs a feedforward ADRC parameter self-tuning control strategy based on RBF for the coupled disturbances and variable working conditions of the register system of unit-type flexographic printing equipment. The structure is as follows: Section 2 analyzes the structure of the flexographic equipment register system, builds a mathematical model, and linearizes it; Section 3 eliminates the modeled disturbances via feedforward, applies ADRC to estimate and compensate the unmodeled disturbances, and uses RBF for the real-time adjustment of ADRC parameters to optimize the control performance; Section 4 compares the performance of the designed controller with PID and ADRC for verification; Section 5 concludes and gives directions for future work.

2. Modeling and Linearizing of the Global Register System

The specific structure of the unit-type flexographic printing equipment is shown in Figure 1, including the composition of the unwinding unit, infeeding unit, four-color printing units, multi-level oven system, outfeeding unit, and rewinding unit. Among them, the register system mainly consists of printing units, oven systems, and inspection components, which are coupled in series through substrates to form a multi-input system with non-linear, strong coupling, and multiple interference characteristics.

The four-color register model shown in Figure 2 is divided into three spans. The first color speed is the reference, the color of the later unit follows the previous color, and the velocity of this unit is adjusted according to the error feedback signal detected by the photoelectric eye, to ensure that the longitudinal error is basically 0 where the plate roller is driven by a servo motor operating in the speed mode. According to the literature [7] and the law of the conservation of mass, the coupling model of the register system of the flexographic printing equipment can be written as:

$$\{\begin{array}{l}\frac{\mathrm{d}{e}_{aout1}(t)}{\mathrm{d}t}=R_2{\omega }_2(t)-R_1{\omega }_1(t-t_{\tau 1})-\frac{T_1(t)}{AE}{R}_1{\omega }_1(t)+\frac{T_0(t-t_{\tau 1})}{AE}{R}_1{\omega }_1(t-t_{\tau 1})\\ \frac{\mathrm{d}{e}_{aout2}(t)}{\mathrm{d}t}=R_3{\omega }_3(t)-R_2{\omega }_2(t-t_{\tau 2})-\frac{T_2(t)}{AE}{R}_2{\omega }_2(t)+\frac{T_1(t-t_{\tau 2})}{AE}{R}_2{\omega }_2(t-t_{\tau 2})\\ \frac{\mathrm{d}{e}_{aout3}(t)}{\mathrm{d}t}=R_4{\omega }_4(t)-R_3{\omega }_3(t-t_{\tau 3})-\frac{T_3(t)}{AE}{R}_3{\omega }_3(t)+\frac{T_2(t-t_{\tau 3})}{AE}{R}_3{\omega }_3(t-t_{\tau 3})\end{array}$$

(1)

where the following notations are used: e_aouti(t) is the register error of print unit i + 1 relative to the previous print unit i, R_i is the plate roller radius of unit i, ω_i(t) is the real-time angular velocity of print unit i (ω* is the reference velocity, Δω_i(t) is the compensation amount relative to the reference velocity), A is the cross-sectional area of the substrate, E is the modulus of elasticity of the substrate at room temperature, T_i(t) is the tension of the ith span, t_τi is the delay time of the color mark passing from unit i to unit i + 1, $t_{\tau i}=L_i/R_i{\omega }^{*}$ (L_i is the span material length), and i is the number of active rollers from left to right (i = 1, 2, 3, …).

According to the actual workshop equipment processes, by giving each color unit the plate roller radius with a 0.02–0.03 mm grade difference, to approximately ignore the material length changes due to installation errors or other hardware factors, it can be considered that span L_i is approximately the same. According to the actual control program design, the plate roller radius grade difference relative to the span material length L_i is extremely small, so it can be considered that the plate roller radius R_i is approximately the same. Then, there is auxiliary formula A as follows:

$$\{\begin{array}{l}{L}_1\approx L_2\approx L_3=L\\ R_1\approx R_2\approx R_3\approx R_4=R\\ t_{\tau 1}\approx t_{\tau 2}\approx t_{\tau 3}=\frac{L}{R{\omega }^{*}}=C\end{array}$$

(2)

Equation (1) is a first-order multi-input time-lag system. According to the multi-color register process, assuming that the length of the substrate does not change abruptly with temperature, each tension T_i(t), plate roller speed ω_i(t), and register error e_aouti(t) fluctuates slightly around the steady-state value during the printing process. Then, auxiliary equation B can be obtained according to the small deviation method:

$$\{\begin{array}{l}{T}_i(t)=T^{*}+\Delta T_i(t)\\ {\omega }_i(t)={\omega }^{*}+\Delta {\omega }_i(t)\\ e_{aouti}(t)=e^{*}+\Delta e_{aouti}(t)\end{array}$$

(3)

where the following notations are used: T*, ω*, and e* are the steady state values of tension, plate roller speed, and register error, respectively; ΔT_i(t), Δω_i(t), and Δe_aouti(t) are the small fluctuations near the respective steady-state values. By combining auxiliary equations A, B, and Formula (1), neglecting the higher order minima and omitting Δ, (1) can be simplified to the linear model as follows:

$$\{\begin{array}{l}\frac{\mathrm{d}{e}_{aout1}(t)}{\mathrm{d}t}=R{\omega }_2(t)+\frac{R{\omega }^{*}}{AE}(T_0(t-C)-T_1(t))-\frac{R{T}^{*}}{AE}{\omega }_1(t)+(\frac{T^{*}}{AE}-1)R{\omega }_1(t-C)\\ \frac{\mathrm{d}{e}_{aout2}(t)}{\mathrm{d}t}=R{\omega }_3(t)+\frac{R{\omega }^{*}}{AE}(T_1(t-C)-T_2(t))-\frac{R{T}^{*}}{AE}{\omega }_2(t)+(\frac{T^{*}}{AE}-1)R{\omega }_2(t-C)\\ \frac{\mathrm{d}{e}_{aout3}(t)}{\mathrm{d}t}=R{\omega }_4(t)+\frac{R{\omega }^{*}}{AE}(T_2(t-C)-T_3(t))-\frac{R{T}^{*}}{AE}{\omega }_3(t)+(\frac{T^{*}}{AE}-1)R{\omega }_3(t-C)\end{array}$$

(4)

In printing, the substrate strain is extremely small (T* << AE, $\frac{T^{*}}{AE}~0$). The Laplace transform of Formula (4) obtains the expression of the transfer function of the four-color register system as:

$$\{\begin{array}{l}{E}_{aout}{}_1(s)=G_{\mathrm{A}}(s){\omega }_2(s)+G_{\mathrm{B}}(s){\omega }_1(s)+G_{\mathrm{C}}(s)T_1(s)+G_{\mathrm{D}}(s)T_0(s)\\ E_{aout2}(s)=G_{\mathrm{A}}(s){\omega }_3(s)+G_{\mathrm{B}}(s){\omega }_2(s)+G_{\mathrm{C}}(s)T_2(s)+G_{\mathrm{D}}(s)T_1(s)\\ E_{aout3}(s)=G_{\mathrm{A}}(s){\omega }_4(s)+G_{\mathrm{B}}(s){\omega }_3(s)+G_{\mathrm{C}}(s)T_3(s)+G_{\mathrm{D}}(s)T_2(s)\end{array}$$

(5)

$$\{\begin{array}{l}{G}_{\mathrm{A}}(s)=\frac{R}{s}{G}_{\mathrm{B}}(s)=-(\frac{R{T}^{*}}{AEs}+\frac{R}{s}{e}^{-Cs})\\ G_{\mathrm{C}}(s)=-\frac{R{\omega }^{*}}{AEs}{G}_{\mathrm{D}}(s)=\frac{R{\omega }^{*}}{AEs}{e}^{-Cs}\end{array}$$

(6)

Formulas (5) and (6) reveal the linear model of register error between neighboring print units, with the previous plate roller as the reference. In the actual printing process, the register error is balanced by adjusting the angular velocity ω_i+1 of the latter unit. Therefore, the angular velocity ω_i of the former unit is the velocity coupling interference, and the tension T_i and T_i−1 of the adjacent span is the tension coupling interference.

3. Design of Decoupled Controller for Register System

3.1. Design of Feedforward Controller

The angular velocity ω_i(t) can be measured in real-time by the motor encoder, and tension T_i(t) and T_i−1(t) can be detected by the tension sensor. Since velocity-coupled and tension-coupled disturbances are measurable and uncontrollable, the feedforward controller can be designed by invariance to eliminate the modeled disturbances.

Combining the mathematical model with the literature [7], the feedforward controller structure is shown in Figure 3. G_A(s) is the transfer function of the main loop, G_B(s), G_C(s), and G_D(s) are the transfer functions for the velocity-coupled disturbance ω_i(t), and the tension-coupled disturbances T_i(t) and T_i−1 (t), respectively; G_ADRCi(s) is the ADRC controller expression, while G$*$_ωi(s), G$*$_Ti(s), and G$*$$*$_Ti−1(s) are the feedforward controller expressions for velocity-coupled and tension-coupled disturbances, respectively. According to the linear system superposition principle, the expression for E_i(s) can be written as:

$$\{\begin{array}{c}{E}_{aout1}(s)=\frac{G_{\mathrm{A}}(s)G_{{\mathrm{ADRC}}_1}(s)}{1+G_{\mathrm{A}}(s)G_{{\mathrm{ADRC}}_1}(s)}{E}_{rin1}(s)+\frac{G_{\mathrm{B}}(s)+G_{\mathrm{A}}(s)G_{{\omega }_1}^{*}(s)}{1+G_{\mathrm{A}}(s)G_{{\mathrm{ADRC}}_1}(s)}{\omega }_1(s)+\frac{G_{\mathrm{C}}(s)+G_{\mathrm{A}}(s)G_{T_1}^{*}(s)}{1+G_{\mathrm{A}}(s)G_{{\mathrm{ADRC}}_1}(s)}{T}_1(s)+\frac{G_{\mathrm{D}}(s)+G_{\mathrm{A}}(s)G_{T_0}^{**}(s)}{1+G_{\mathrm{A}}(s)G_{{\mathrm{ADRC}}_1}(s)}{T}_0(s)\\ E_{aout2}(s)=\frac{G_{\mathrm{A}}(s)G_{{\mathrm{ADRC}}_2}(s)}{1+G_{\mathrm{A}}(s)G_{{\mathrm{ADRC}}_2}(s)}{E}_{rin2}(s)+\frac{G_{\mathrm{B}}(s)+G_{\mathrm{A}}(s)G_{{\omega }_2}^{*}(s)}{1+G_{\mathrm{A}}(s)G_{{\mathrm{ADRC}}_2}(s)}{\omega }_2(s)+\frac{G_{\mathrm{C}}(s)+G_{\mathrm{A}}(s)G_{T_2}^{*}(s)}{1+G_{\mathrm{A}}(s)G_{{\mathrm{ADRC}}_2}(s)}{T}_2(s)+\frac{G_{\mathrm{D}}(s)+G_{\mathrm{A}}(s)G_{T_1}^{**}(s)}{1+G_{\mathrm{A}}(s)G_{{\mathrm{ADRC}}_2}(s)}{T}_1(s)\\ E_{aout3}(s)=\frac{G_{\mathrm{A}}(s)G_{{\mathrm{ADRC}}_3}(s)}{1+G_{\mathrm{A}}(s)G_{{\mathrm{ADRC}}_3}(s)}{E}_{rin3}(s)+\frac{G_{\mathrm{B}}(s)+G_{\mathrm{A}}(s)G_{{\omega }_3}^{*}(s)}{1+G_{\mathrm{A}}(s)G_{{\mathrm{ADRC}}_3}(s)}{\omega }_3(s)+\frac{G_{\mathrm{C}}(s)+G_{\mathrm{A}}(s)G_{T_3}^{*}(s)}{1+G_{\mathrm{A}}(s)G_{{\mathrm{ADRC}}_3}(s)}{T}_3(s)+\frac{G_{\mathrm{D}}(s)+G_{\mathrm{A}}(s)G_{T_2}^{**}(s)}{1+G_{\mathrm{A}}(s)G_{{\mathrm{ADRC}}_3}(s)}{T}_2(s)\end{array}$$

(7)

According to the invariance principle, the speed-coupled and tension-coupled disturbances need to be eliminated, which means each disturbance is set to 0. Then, the expression for the feedforward controller of the flexographic machine register system designed for disturbances is:

$$\{\begin{array}{c}{G}_{{\omega }_1}^{*}(s)=G_{{\omega }_2}^{*}(s)=G_{{\omega }_3}^{*}(s)=\frac{T^{*}}{AE}+e^{-Cs}\\ G_{T_0}^{*}(s)=G_{T_1}^{*}(s)=G_{T_2}^{*}(s)=-\frac{{\omega }^{*}}{AE}{e}^{-Cs}\\ G_{T_1}^{**}(s)=G_{T_2}^{**}(s)=G_{T_3}^{**}(s)=\frac{{\omega }^{*}}{AE}\end{array}$$

(8)

3.2. Design of ADRC Controller

For the register system model with the latter angular velocity as the control quantity and the register error as the output, the first-order ADRC controller is selected and its system structure is shown in Figure 4, consisting of TD, ESO, and NLSEF, with each part acting as follows:

TD enables the fast tracking of the input signal, extracting a continuous differentiable signal for the system non-differentiable reference input e_rini and transitioning the input smoothly without overshooting. For the register system, the register error e_aout is expected to be 0, which means that the input error signal e_rin is required to be stable to 0. Therefore, TD should be ignored.

ESO observes the system state with the total disturbance (uncertain nonlinear dynamics inside the system and the external random disturbances to which it is subjected) mainly based on the input and output. A second-order ESO discretization algorithm is used as follows:

$$\{\begin{array}{l}{e}_{ESO}(k)=z_1(k)-e_{aout}(k)\\ f{e}_{ESO1}=fal\left[e_{ESO}(k),a,\delta \right]\\ fal(e,a,\delta )=\{\begin{array}{ll}(|e{|}^{a}sign(e)& ,\left|e\right|>\delta \\ e/{\delta }^{-a}& ,\left|e\right|\le \delta \end{array}\\ z_1(k+1)=z_1(k)+h\left[z_2(k)-{\beta }_1{e}_{ESO}(k)+b_{0}u(k)\right]\\ z_2(k+1)=z_2(k)+h\left(-{\beta }_{2}f{e}_{ESO}{}_1\right)\end{array}$$

(9)

where the following notations are used: e_aout(k) is the register error feedback output, z₁ is the state variable of the system object, z₂ is the real-time action of the unknown disturbance and uncertainty model, known as system expansion state, β₁ and β₂ are the gain parameters of ESO, δ is the interval length of the linear segment, b₀ is the compensation factor, fal(e, a, δ) is the function that guarantees the fast and smooth convergence of ESO, and a is a constant (0 < a < 1) that determines the tracking speed and filtering effect, with a = 0.5 being taken.

The NLSEF is the nonlinear state error control rate to achieve a nonlinear control combination of errors. The combined discrete equation of the NLSEF control law for a first-order system is as follows:

$$\{\begin{array}{l}{e}_1(k+1)=e_{rin}(k+1)-z_1(k+1)\\ u_0(k+1)=k_{\mathrm{NL}}fal(e_1(k+1),\alpha ,\delta )\\ u(k+1)=u_0(k+1)-\frac{z_2(k+1)}{b_0}\end{array}$$

(10)

where the following notations are used: e_rin is the reference register error and k_NL is the NLSEF gain coefficient.

In summary, the control law of the ADRC controller organically combines ESO and NLSEF to complete the effective compensation of unknown system dynamics and external disturbances, and finally obtain a good control performance.

3.3. ADRC Parameter Tuning Based on RBF

The RBF neural network is a three-layer feedforward neural network that can approximate any nonlinear function with arbitrary accuracy, and its fast convergence speed meets the real-time requirements of the register system. As shown in Figure 5, the RBFNN neural network is constructed for ADRC with the real-time rectification of β₁, β_2, and k_NL, where the inputs into the identification system include the input u(k) and output e_aout(k) of the register system, as well as the network sample output error e_RBF(k).

For the mathematical model of the flexographic printing press register system, the RBF neural network system structure is selected as 3-7-1. As shown in Figure 6, the network input vector is x= [x₁, x₂, x₃]^T = [u(k), e_aout(k), e_RBF(k)]^T, and the network output is y_vo.

$$y_{vo}=WH={\displaystyle \sum _{i=1}^m{w}_i{h}_i},\left(0<i\le m=7\right)$$

(11)

where W is the node weight matrix of the hidden layer, W= [ω₁, ω₂, …, ω_i, …, ω_m]^T (m = 7), and the radial basis vector of the hidden layer is H = [h₁, h₂, …, h_i, …, h_m]^T (m = 7). h_i is chosen as a Gaussian function to realize the nonlinear mapping from the input layer to the hidden layer, which is calculated as follows:

$$h_i=\exp \left(-\frac{{\Vert \mathit{x}-{\mathit{C}}_i\Vert }^2}{2b_i{}^2}\right),\left(i=1,2,\cdots m\right)$$

(12)

where the center vector of the ith node of the network is C_i, C_i = [c_i1, c_i2, …, c_ii, …, c_im]^T, b_i is the base width parameter of the ith node, and the base width vector of the network is b = [b₁, b₂, …, b_i, …, b_m]^T. The network training algorithm uses the negative gradient descent method with a good local search effect. For the register system model, the output weight ω_i, the base width parameter b_i, and the center vector c_i are tracked iteratively with the specific discrete adjustment rules as follows:

$$\{\begin{array}{l}\Delta w_i(k)=-\eta \frac{\partial E_{\mathrm{RBF}}(k)}{\partial w_i(k)}=\eta e_{\mathrm{RBF}}(k)\frac{\partial {\displaystyle \sum _{i=1}^7{w}_i(k)h_i(k)}}{\partial w_i(k)}=\eta \left[e_{aout}(k)-y_{vo}(k)\right]h_i(k)\\ w_i(k)=w_i(k-1)+\Delta w_i(k)+\alpha [w_i(k-1)-w_i(k-2)]\\ \Delta b_i(k)=-\eta \frac{\partial E_{\mathrm{RBF}}(k)}{\partial b_i(k)}=\eta e_{\mathrm{RBF}}(k)\frac{\partial {\displaystyle \sum _{i=1}^7{w}_i(k)h_i(k)}}{\partial b_i(k)}=\eta \left[e_{aout}(k)-y_{vo}(k)\right]w_i(k)h_i(k)\frac{{\Vert \mathit{x}-{\mathit{C}}_i\Vert }^2}{b_i{}^3(k)}\\ b_i(k)=b_i(k-1)+\Delta b_i(k)+\alpha [b_i(k-1)-b_i(k-2)]\\ \Delta c_{ii}(k)=-\eta \frac{\partial E_{\mathrm{RBF}}(k)}{\partial c_{ii}(k)}=\eta e_{\mathrm{RBF}}(k)\frac{\partial {\displaystyle \sum _{i=1}^7{w}_i(k)h_i(k)}}{\partial c_{ii}(k)}=\eta \left[e_{aout}(k)-y_{vo}(k)\right]w_i(k)h_i(k)\frac{x_i(k)-c_{ii}(k)}{b_i{}^2(k)}\\ c_{ii}(t)=c_{ii}(k-1)+\Delta c_{ii}(k)+\alpha [c_{ii}(k-1)-c_{ii}(k-2)]\end{array}$$

(13)

where the following notations are used: E_RBF is the network recognition performance index (E_RBF = 0.5 e_RBF²), e_RBF is the sample output error, e_aout is the desired sample output or the actual output of the register system, y_ov is the network output, which tracks the actual system output e_aout (e_RBF = e_aout − y_ov), η is the learning rate, 0 < η < 1, and α is the momentum factor, 0 < α < 1.

With the help of the RBF neural network to determine the amount of parameter correction, which is consistent with the network parameters training. Set the indicator function E_control to adjust β₁, β₂, and k_NL so that the actual register error e_aout approximates the reference register error input e_rin (E_control = 0.5 e_control² = 0.5 (e_rin(k) − e_aout(k))²). The increments of parameters β₁, β₂, and k_NL can be calculated according to the negative gradient descent algorithm as:

$$\{\begin{array}{c}\Delta {\beta }_1(k)=-{\eta }_1\frac{\partial E_{control}(k)}{\partial {\beta }_1(k)}={\eta }_1{e}_{control}(k)\frac{\partial e_{aout}(k)}{\partial {\beta }_1(k)}={\eta }_1{e}_{control}(k)\frac{\partial e_{aout}(k)}{\partial z_1(k+1)}\frac{\partial z_1(k+1)}{\partial {\beta }_1(k)}\\ \Delta {\beta }_2(k)=-{\eta }_1\frac{\partial E_{control}(k)}{\partial {\beta }_2(k)}={\eta }_1{e}_{control}(k)\frac{\partial e_{aout}(k)}{\partial {\beta }_2(k)}={\eta }_1{e}_{control}(k)\frac{\partial e_{aout}(k)}{\partial z_2(k+1)}\frac{\partial z_2(k+1)}{\partial {\beta }_2(k)}\end{array}$$

(14)

$$\Delta k_{\mathrm{NL}}(k)=-{\eta }_2\frac{\partial E_{control}(k)}{\partial k_{\mathrm{NL}}(k)}={\eta }_2{e}_{control}(k)\frac{\partial e_{aout}(k)}{\partial k_{\mathrm{NL}}(k)}={\eta }_2{e}_{control}(k)\frac{\partial e_{aout}(k)}{\partial u(k)}\frac{\partial u(k)}{\partial {\beta }_1(k)}$$

(15)

Combining Formulas (16) and (17) with Formulas (11)–(14), the adjustments ∆β₁, ∆β₂, and ∆k_NL for parameters β₁, β_2, and k_NL can be obtained based on Jacobian information:

$$J=\frac{\partial e_{aout}(k)}{x_{in}}\approx \frac{\partial y_{vo}(k)}{x_{in}}={\displaystyle \sum _{i=1}^m{w}_i(k)h_i(k)\frac{c_{ii}(k)-x_{in}(k)}{b_i{}^2(k)}}$$

(16)

$$\{\begin{array}{ll}\Delta {\beta }_1(k)& ={\eta }_1\left[e_{rin}(k)-e_{aout}(k)\right]\frac{\partial e_{aout}(k)}{\partial z_1(k+1)}\frac{\partial z_1(k+1)}{\partial {\beta }_1(k)}\\ & ={\eta }_{1}h\left[e_{rin}(k)-e_{aout}(k)\right]e_{ESO}(k){\displaystyle \sum _{i=1}^m{w}_i(k)h_i(k)\frac{c_{ii}(k)-z_1(k+1)}{b_i{}^2(k)}}\\ \Delta {\beta }_2(k)& ={\eta }_1\left[e_{rin}(k)-e_{aout}(k)\right]\frac{\partial e_{aout}(k)}{\partial z_2(k+1)}\frac{\partial z_2(k+1)}{\partial {\beta }_2(k)}\\ & ={\eta }_{1}h\left[e_{rin}(k)-e_{aout}(k)\right]f{e}_{ESO}{}_1{\displaystyle \sum _{i=1}^m{w}_i(k)h_i(k)\frac{c_{ii}(k)-z_2(k+1)}{b_i{}^2(k)}}\end{array}$$

(17)

$$\{\begin{array}{ll}\Delta k_{\mathrm{NL}}(k)& ={\eta }_2\left[e_{rin}(k)-e_{aout}(k)\right]\frac{\partial e_{aout}(k)}{\partial u(k)}\frac{\partial u(k)}{\partial k_{\mathrm{NL}}(k)}\\ & ={\eta }_2\left[e_{rin}(k)-e_{aout}(k)\right]fal(e_1(k+1),\alpha ,\delta ){\displaystyle \sum _{i=1}^m{w}_i(k)h_i(k)\frac{c_{ii}(k)-u(k)}{b_i{}^2(k)}}\end{array}$$

(18)

$$\{\begin{array}{l}{\beta }_1(k)={\beta }_1(k-1)+\Delta {\beta }_1(k)\\ {\beta }_2(k)={\beta }_2(k-1)+\Delta {\beta }_2(k)\\ k_{\mathrm{NL}}(k)=k_{\mathrm{NL}}(k-1)+\Delta k_{\mathrm{NL}}(k)\end{array}$$

(19)

Formula (21) is the parameter update expression for parameters β₁, β_2, and k_NL. The flow of the RBF neural network for the ADRC parameters tuning during system operation is shown in Figure 7, and the procedure is as follows:

• The initial values of network parameters (center vector c_i, base width vector b_i, network weights ω_i, learning rate η, momentum factor α) and the initial values of ADRC controller parameters (NLSEF gain coefficient k_NL, ESO gain parameters β₁, β₂) are set.
• The ADRC controller output u(k) is received. Then, the network output y_vo is calculated and the actual output of the register system sampling e_aout is obtained.
• The discrimination model error indicator E_RBF is calculated. The RBF network parameters of center vector, base width vector, and network weights are updated. Next, after calculating the system error indicator E_control and combining it with Jacobian information, the ADRC controller parameter corrections β₁, β₂, and k_NL are updated.
• After judging, the actual system error e_control to be within an acceptable error range (±C), it is decided to carry out a k + 1 cycle or end.

After the above study, the controller structure designed for the four-color register control system is shown in Figure 8. The designed controller structure consists of three parts: (1) the ADRC controller, which performs the control of the three-span register system, with its output u, combined with the feedforward compensation quantity, being input to this span register system as the main loop control quantity; (2) the feedforward controller, which performs the compensation of speed and tension disturbances in this and the previous span; (3) the RBF identification system, which performs the parameter search and adjustment of the key parameters of the ADRC controller in real-time. The control performance of the designed controller will be verified in Section 4.

4. Simulation and Analysis

For the performance of the designed controller, the control model simulation of the register system was verified using MATLAB2019b. The steady-state tension of the system was set to 50 N, and the synchronous speed was set to 150 m/min or 300 m/min to represent different working conditions, and the register performance was compared with that under PID and ADRC control.

The simulation steps were 1 × 10⁻⁴, the simulation time was 20 s, the mechanical simulation parameters were consistent with the actual measured parameters of the flexo machine (see

Table 1. Parameters of the model.

Parameters	Value	Unit
A	2.7 × 10−5	m2
E(PET)	4.89 × 109	Pa
R1, R2, R3, R4	0.5	m
L1, L2, L3	3.05	m
tτ1, tτ2, tτ3	0.295	s
T*	50	N
ω*(ω1)	300,600	rad/min
e*	0	mm

* Note: Motor motion planning is PT mode, the plate roller is driven directly by the motor (reduction ratio is 1:1).

), and the simulation substrate was PET film.

Table 2. Parameters of each RBF-ADRC controller.

Parameters	RBF-ADRC1	RBF-ADRC2	RBF-ADRC3
β1	1000	990	960
β2	160,000	180,000	182,000
kNL	150	120	50
δ	0.8	0.8	0.8
b0	0.8	1	1
η	0.25	0.42	0.42
η1	0.1	0.3	0.35
η2	0.02	0.035	0.035
α	0.4	0.4	0.05
ci	[30] 3×7	[30] 3×7	[30] 3×7
bi	[40] 7×1	[40] 7×1	[40] 7×1
wi	[1] 7×1	[1] 7×1	[1] 7×1

shows the designed RBF-ADRCi controller parameters, and the network parameters and controller parameters were adjusted in real-time for optimal control.

The comparison was verified with PID and ADRC controllers under different working conditions (line speed 150 m/min and 300 m/min), where the PID and ADRC controller parameters are shown in

Table 3. Parameters of each PID controller.

Parameters	PID1	PID2	PID3
kp	160	150	180
ki	960	850	680
kd	0.0001	0	0.0001

and

Table 4. Parameters of each ADRC controller.

Parameters	ADRC1	ADRC 2	ADRC 3
β1	1000	990	1020
β2	90,000	98,000	121,000
kn	150	120	80
δ	0.8	0.8	0.8
b0	1	1	1

.

4.1. Analysis of Anti-Speed Interference Performance

In actual high-speed printing, the linear velocity of the plate rolls is not synchronized, as the out-of-roundness or mechanical deviation of the plate rolls may produce register errors. To verify the velocity decoupling performance of the controller, the velocity fluctuation working condition in the actual printing process was simulated with pulse disturbance.

The simulation conditions were as follows: assuming the tension was stabilized at 50 N (no change) after the system was running steadily, ω₃ was allowed to produce an overshoot speed fluctuation of 0.25 m/min at the 2nd s. After lasting for 8 s, the speed disturbance was reduced by 0.125 m/min, and at the 15ths, the steady-state speed of 300 m/min was restored. The three-span register error curves of PID, ADRC, and the designed controller at high speed (300 m/min) are shown in Figure 9.

In comparison with Figure 9, the velocity fluctuations of ω₃ caused register deviations in the 2nd and 3rd spans, and the error ranges controlled by different controllers are shown in

Table 5. Control range of each controller under speed disturbance.

Performance Index	eaout1 (μm)	eaout2 (μm)	eaout3 (μm)
PID	0	(−71.5,47.5)	(−18.1,23.5)
ADRC	0	(−4.3,2.9)	(−3.1,2.35)
Designed Controller	0	(−2.9,1.9)	(−0.3,0.2)

. When the conventional PID controller was used, the maximum three-span register error was −71.5 μm; when the ADRC controller was adopted, the maximum value was −4.3 μm; when the designed controller was applied, the maximum error was −2.9 μm, and the peak register error in the case of speed disturbance was reduced by about 32% compared to the ADRC.

As shown in Table 5, the performance of the designed controller is better than that of the ADRC controller and the PID controller when the speed disturbance is generated or disappears, and it has a good control ability for the error caused by the speed variation and realizes the decoupling control of speed.

4.2. Analysis of Anti-Tension Interference Performance

Tension fluctuations during equipment operation will spread backward along the direction of substrate movement, which causes multiple span register instability. To effectively verify the anti-tension disturbance capability of the designed controller, a short-time random disturbance simulation was conducted by emulating the tension fluctuation conditions in the actual printing process (during material change or flip frame rotation).

The simulation conditions were as follows: after the system entered the stable printing phase (steady-state tension of 50 N), a 10 N tension pulse disturbance was generated in the infeeding tension T₀ for 10 s. The three-span register error curves of PID, ADRC, and the designed controller are shown in Figure 10 and Figure 11 under different working conditions (printing speeds of 150 m/min and 300 m/min).

Comparing Figure 10 and Figure 11, the register error from tension disturbance increases with the increase in speed. The increased ratio of the error is different under the control of different controllers, and the specific control range is shown in

Table 6. Control range of each controller under tension disturbance.

Performance Index		eaout1 (μm)	eaout2 (μm)	eaout3 (μm)
PID	150 m/min	(−6.7,6.8)	(−7.2,7.2)	(−6.5,6.5)
	300 m/min	(−13.3,13.4)	(−14.5,14.4)	(−13.1,13.0)
ADRC	150 m/min	(−0.46,0.4)	(−0.53,0.5)	(−0.65,0.65)
	300 m/min	(−0.89,0.89)	(−1,1)	(−1.3,1.3)
Designed Controller	150 m/min	(−0.31,0.3)	(−0.13,0.1)	0
	300 m/min	(−0.6,0.6)	(−0.32,0.3.2)	0

.

For example, at 150 m/min printing speed (Figure 10), the register error peaks in the 3 spans were −13.4 μm, −14.5 μm, and −13.1 μm when the tension disturbance was generated and 13.4 μm, 14.4 μm, and 13.0 μm when the disturbance was over. When the conventional ADRC was controlled, the register errors were −0.46 μm, −0.53 μm, and −0.65 μm when the tension disturbance was generated, and 0.4 μm, 0.53 μm, and 0.65 μm when the disturbance disappeared. When the designed controller was applied, the errors were −0.31 μm,−0.13 μm, and 0 μm when the disturbance occurred, and 0.3 μm, 0.1 μm, and 0 μm when the disturbance ended.

The performance of the designed controller is better than the ADRC controller and PID controller when the tension disturbance is generated or disappeared. At a 150 m/min printing speed, the control peak of register error is reduced by about 32% relative to ADRC, and the decoupling control of tension disturbance is achieved.

4.3. Analysis of RBF Parameter Adjustment

For the above speed disturbance conditions, Figure 12 shows the real-time adjustment curves of the parameters of ESO and NLSEF under 300 m/min printing conditions, respectively. The RBF can adjust β₁, β_2, and k_NL in real-time when the speed ω₃ generates a disturbance, where the 2nd span controller k_NL is adjusted to 128 at the 10ths and 129.8 at the 15ths.

For the above tension disturbance conditions, Figure 13 and Figure 14 show the online adjustment curves of β₁, β_2, and k_NL at 150 m/min and 300 m/min, respectively. When T₀ changes, each controller β₁ and β₂ is slightly adjusted, and the k_NL adjustment of the 1st span controller is 4 and 32.6, which are adjusted to 154 and 182.6, respectively.

The RBF performs the online correction of controller parameters when disturbances are generated, and improves parameter values to reduce regulation time and computation. The problem that the control performance of the ADRC controller varies with the working conditions is solved, and achieves a high-precision register control.

5. Conclusions

High accuracy control of the register system is the guarantee of high-quality printing. To improve the control accuracy of the nonlinear register system for multi-color flexographic printing, this paper proposes a feedforward ADRC parameter self-tuning decoupling control strategy based on RBF. Its unique feature is the integration of feedforward control, ADRC, and an RBF neural network to design the decoupling controller, in which feedforward control is used to eliminate the register errors caused by modeled disturbances, ADRC performs the estimation and compensation of unmodeled disturbances, and RBF realizes the online correction of ADRC controller parameters. The simulation of the control performance of the proposed controller with MATLAB shows that the designed controller has a better performance compared with the traditional PID and ADRC control, and the peak register error is reduced by about 32% compared with ADRC, with a better dynamic performance. In the future, we will work on developing engineering applications of the designed controllers in unit flexographic machine control systems (ST code implementation in CODESYS), as well as exploring new control methods (such as linear ADRC).